Section 2 Control Techniques

**70**

*Applied Modern Control*

[1] Willis MJ, Di Massimo C, Montague GA, Tham MT, Morris AJ. Artificial neural networks in process engineering. IEE Proceedings D Control Theory and Applications. IET. 1991;**138**(3):256-266

[2] Dohare RK, Singh K, Kumar R, Upadhyaya SG, Gupta S. Dynamic model of dividing wall column for separation of ternary system. In: Chemca-2011; Australia. 2011

[3] Dohare R, Singh K, Kumar R. Modeling and model predictive control of dividing wall column for separation of BTX. Systems Science & Control Engineering. 2015;**3**(1):142-115

[4] Park S, Han C. A nonlinear soft sensor based on multivariate smoothing procedure for quality estimation in distillation columns. Computers & Chemical Engineering. 2000;**24**:871-877

[5] Delgado A, Kambhampati C, Warwick K. Dynamic recurrent neural network for system identification and control. IEE Proceedings-Control Theory and Applications.

[6] Nogaard M, Ravn OE, Poulsen NK, Hansen LK. Neural Networks for Modelling and Control of Dynamic Systems: A Practitioner's Handbook. London: Springer-Verlag, Inc; 2000

1995;**142**:307-314

**References**

Chapter 5

Espen Oland

Abstract

velocity

73

1. Introduction

from previous works as mentioned above.

Modeling and Attitude Control of

The attitude determination and control system (ADCS) for spacecraft is responsible for determining its orientation using sensor measurements and then applying actuation forces to change the orientation. This chapter details the different components required for a complete attitude determination and control system for satellites moving in elliptical orbits. Specifically, the chapter details the orbital mechanics; perturbations; controller design; actuation methods such as thrusters, reaction wheels, and magnetic torquers; actuation modulation methods such as bang-bang, pulse-width modulation, and pulse-width pulse-frequency; as well as attitude determination using vector measurements combined with mathematical models. In sum, the work describes in a tutorial manner how to put everything

Keywords: ADCS, attitude control, attitude estimation, thrusters, reaction wheels, magnetic torquers, elliptical orbits, PWM, PWPF, bang-bang, Madgwick filter, Sun vector model, magnetic field model, sliding surface control, quaternions, angular

The problem of developing attitude determination and control systems (ADCS) has received much attention in the last century with general books such as [1–4], as well as description of individual ADCS designs for different satellites in works such as [5–8]. While Refs. [1, 2] can be considered excellent foundation books within the topic of spacecraft modeling and control, there is a need for a more concise presentation of the attitude control problem and how this can be modeled in a simple manner, both not only as a tutorial for new researchers but also to give insight into the different components required for ADCS design drawing on ideas and results

This chapter is an extension of [9] and builds on much of the previous work, as well as the research done through the HiNCube project as presented in [10–12]. This work considers the problem of designing a complete ADCS system comprising all the required components. Figure 1 shows the control structure and the required signal paths, giving an overview of the contents in this chapter, as each block is described in detail to put the reader in position to design their own ADCS system. The required sensors for this system are a magnetometer to measure the Earth's magnetic field, a gyro to measure the angular velocity of the satellite, as well as Sun sensors for measuring the direction toward the Sun. Further, the mathematical

Satellites in Elliptical Orbits

together to enable the design of a complete satellite simulator.

#### Chapter 5

## Modeling and Attitude Control of Satellites in Elliptical Orbits

Espen Oland

#### Abstract

The attitude determination and control system (ADCS) for spacecraft is responsible for determining its orientation using sensor measurements and then applying actuation forces to change the orientation. This chapter details the different components required for a complete attitude determination and control system for satellites moving in elliptical orbits. Specifically, the chapter details the orbital mechanics; perturbations; controller design; actuation methods such as thrusters, reaction wheels, and magnetic torquers; actuation modulation methods such as bang-bang, pulse-width modulation, and pulse-width pulse-frequency; as well as attitude determination using vector measurements combined with mathematical models. In sum, the work describes in a tutorial manner how to put everything together to enable the design of a complete satellite simulator.

Keywords: ADCS, attitude control, attitude estimation, thrusters, reaction wheels, magnetic torquers, elliptical orbits, PWM, PWPF, bang-bang, Madgwick filter, Sun vector model, magnetic field model, sliding surface control, quaternions, angular velocity

#### 1. Introduction

The problem of developing attitude determination and control systems (ADCS) has received much attention in the last century with general books such as [1–4], as well as description of individual ADCS designs for different satellites in works such as [5–8]. While Refs. [1, 2] can be considered excellent foundation books within the topic of spacecraft modeling and control, there is a need for a more concise presentation of the attitude control problem and how this can be modeled in a simple manner, both not only as a tutorial for new researchers but also to give insight into the different components required for ADCS design drawing on ideas and results from previous works as mentioned above.

This chapter is an extension of [9] and builds on much of the previous work, as well as the research done through the HiNCube project as presented in [10–12]. This work considers the problem of designing a complete ADCS system comprising all the required components. Figure 1 shows the control structure and the required signal paths, giving an overview of the contents in this chapter, as each block is described in detail to put the reader in position to design their own ADCS system. The required sensors for this system are a magnetometer to measure the Earth's magnetic field, a gyro to measure the angular velocity of the satellite, as well as Sun sensors for measuring the direction toward the Sun. Further, the mathematical

The derivative of the rotation matrix is defined as R\_ <sup>c</sup>

Modeling and Attitude Control of Satellites in Elliptical Orbits

DOI: http://dx.doi.org/10.5772/intechopen.80422

The inverse quaternion is defined as qa,c ¼ ηc,a � ε<sup>Τ</sup>

� �qa,e with

T qc,a

<sup>q</sup>\_c,a <sup>¼</sup> <sup>1</sup> 2 qc,a⊗

.

p. 479). The e<sup>r</sup> axis coincides with the radius vector r<sup>i</sup>

objective (cf. [17]). The desired frame is denoted by F<sup>d</sup> .

<sup>k</sup>, <sup>e</sup><sup>θ</sup> <sup>¼</sup> <sup>e</sup><sup>h</sup> � <sup>e</sup>r, and <sup>e</sup><sup>h</sup> <sup>¼</sup> <sup>h</sup>

quaternions can be found through quaternion multiplication as

be constructed using quaternions as R<sup>c</sup>

qc,e ¼ qc,a⊗qa,e ¼ T qc,a

tial frame is denoted by F<sup>i</sup>

2.2 Orbital mechanics

as <sup>e</sup><sup>r</sup> <sup>¼</sup> <sup>r</sup><sup>i</sup> kri

by F<sup>o</sup> .

75

v1, v2∈R<sup>3</sup>

denotes the cross product operator, which is defined such that for two vectors

Quaternions can be used to parameterize the rotation matrix, where <sup>q</sup>c,a∈S<sup>3</sup> <sup>¼</sup> <sup>q</sup>∈R<sup>4</sup> : <sup>q</sup><sup>Τ</sup><sup>q</sup> <sup>¼</sup> <sup>1</sup> � � denotes the quaternion representing a rotation from frame a to frame c through the angle of rotation ϑc,a around the axis of rotation kc,a.

, S vð Þ<sup>1</sup> v<sup>2</sup> ¼ v<sup>1</sup> � v2, S vð Þ<sup>1</sup> v<sup>2</sup> ¼ �S vð Þ<sup>2</sup> v1, S vð Þ<sup>1</sup> v<sup>1</sup> ¼ 0, and v<sup>Τ</sup>

as q<sup>∗</sup>. A quaternion comprises a scalar and a vector part, where ηc,a denotes the scalar part, while εc,a∈R<sup>3</sup> denotes the vector part. This allows the rotation matrix to

� � <sup>¼</sup> <sup>η</sup>c,a �ε<sup>Τ</sup>

0 ω<sup>a</sup> c,a

" #

tains the unit length property. The quaternion kinematics is defined as

For attitude control, several different frames are needed:

The use of the quaternion product ensures that the resulting quaternion main-

¼ 1 2 T qc,a � � 0

Inertial: The Earth-centered inertial (ECI) has its origin in the center of the Earth, where the x<sup>i</sup> axis points toward the vernal equinox and the z<sup>i</sup> points through the North Pole, while y<sup>i</sup> completes the right-handed orthonormal frame. The iner-

Orbit: The orbit frame has its origin in the center of mass of the satellite (cf. [16],

<sup>k</sup>h<sup>k</sup> where <sup>h</sup> <sup>¼</sup> <sup>r</sup><sup>i</sup> � <sup>r</sup>\_

Body: The body frame has its origin in the center of mass of the satellite, where its axes coincide with the principal axes of inertia. The body frame is denoted by F<sup>b</sup>. Desired: The desired frame can be defined arbitrarily to achieve any given

This section describes how the orbit frame can be related to the inertial frame through the six classical orbital parameters, and for more details, the reader is referred to [1]. Specifically, the objective with this subsection is to find the radius, velocity, and acceleration vector of the orbit, as well as its angular velocity and acceleration. From well-known orbital mechanics, the six classical parameters can be defined as the semimajor axis a, the eccentricity e, the inclination i, the right ascension of the ascending node Ω, the argument of the perigee ω, and the mean anomaly M. The distance to the apogee and perigee from the center of the Earth can

center of the Earth to the center of mass in the satellite. The e<sup>h</sup> axis is parallel to the orbital angular momentum vector, pointing in the normal direction of the orbit. The e<sup>θ</sup> completes the right-handed orthonormal frame where the vectors can be described

<sup>a</sup> <sup>¼</sup> <sup>R</sup><sup>c</sup>

c,a � �<sup>Τ</sup>

<sup>a</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup> <sup>2</sup>ηc,aSð Þþ <sup>ε</sup>c,a <sup>2</sup>S<sup>2</sup>

c,a

ω<sup>a</sup> c,a

∈R<sup>3</sup>

i

" #

εc,a ηc,aI þ Sð Þ εc,a

" #

<sup>a</sup>S ω<sup>a</sup> c,a

� � where <sup>S</sup>ð Þ�

, also sometimes denoted

ð Þ εc,a . Composite

: (1)

: (2)

, which goes from the

. The orbit frame is denoted

<sup>1</sup> S vð Þ<sup>2</sup> v<sup>1</sup> ¼ 0.

#### Figure 1.

This figure shows the different components required for modeling and controlling a satellite in an elliptical orbit and shows the main components required for creating a satellite simulator.

models used together with sensor measurements to determine the attitude of the satellite requires a real-time clock, as the time and date are required to know the direction toward the Sun as well as what the magnetic field looks like at a given day and time. Comparing the sensor measurements with the mathematical models allows for the determination of the attitude of the satellite, something that is done using the Madgwick filter as presented in [13]. With an estimated attitude obtained using the Madgwick filter, the attitude can be controlled to point a sensor onboard the satellite in a desired direction, something that is solved in this chapter using a PD+ attitude controller, calculating the desired torques required in order to make the attitude and angular velocity errors go to zero. In order to create the desired moments, this chapter presents how this can be achieved using a number of different actuators, namely, magnetic torquers, reaction wheels, and thrusters. The orbital mechanics block describes how the satellite moves in its elliptical orbit, while the perturbing forces and moments block describe how the different perturbations affect the satellite. Simulations show the performance of the different methods and should put the reader in a position to simulate and design new attitude determination and control systems for satellites in elliptical orbits.

#### 2. Mathematical modeling

#### 2.1 Notation

This subsection is similar to the author's previous works, e.g., [9, 14]. Let x\_ ¼ dx=dt denote the time derivative of a vector, while the Euclidean length is defined as <sup>k</sup>xk ¼ ffiffiffiffiffiffiffiffi <sup>x</sup><sup>Τ</sup><sup>x</sup> <sup>p</sup> . Superscript denotes frame of reference for a given vector. The rotation matrix is denoted R<sup>c</sup> <sup>a</sup>∈SOð Þ¼ <sup>3</sup> <sup>R</sup>∈R<sup>3</sup>�<sup>3</sup> : <sup>R</sup><sup>Τ</sup><sup>R</sup> <sup>¼</sup> <sup>I</sup>; detð Þ¼ <sup>R</sup> <sup>1</sup> � �, which rotates a vector from frame a to frame c and where I denotes the identity matrix. The inverse rotation is found by taking its transpose, such that R<sup>a</sup> <sup>c</sup> <sup>¼</sup> <sup>R</sup><sup>c</sup> a � �<sup>Τ</sup> . The angular velocity of frame c relative to frame a referenced in frame e is denoted ωe a,c, and angular velocities can be added together as ω<sup>e</sup> a,f <sup>¼</sup> <sup>ω</sup><sup>e</sup> a,c <sup>þ</sup> <sup>ω</sup><sup>e</sup> c,f (cf. [15]).

The derivative of the rotation matrix is defined as R\_ <sup>c</sup> <sup>a</sup> <sup>¼</sup> <sup>R</sup><sup>c</sup> <sup>a</sup>S ω<sup>a</sup> c,a � � where <sup>S</sup>ð Þ� denotes the cross product operator, which is defined such that for two vectors v1, v2∈R<sup>3</sup> , S vð Þ<sup>1</sup> v<sup>2</sup> ¼ v<sup>1</sup> � v2, S vð Þ<sup>1</sup> v<sup>2</sup> ¼ �S vð Þ<sup>2</sup> v1, S vð Þ<sup>1</sup> v<sup>1</sup> ¼ 0, and v<sup>Τ</sup> <sup>1</sup> S vð Þ<sup>2</sup> v<sup>1</sup> ¼ 0.

Quaternions can be used to parameterize the rotation matrix, where <sup>q</sup>c,a∈S<sup>3</sup> <sup>¼</sup> <sup>q</sup>∈R<sup>4</sup> : <sup>q</sup><sup>Τ</sup><sup>q</sup> <sup>¼</sup> <sup>1</sup> � � denotes the quaternion representing a rotation from frame a to frame c through the angle of rotation ϑc,a around the axis of rotation kc,a. The inverse quaternion is defined as qa,c ¼ ηc,a � ε<sup>Τ</sup> c,a � �<sup>Τ</sup> , also sometimes denoted as q<sup>∗</sup>. A quaternion comprises a scalar and a vector part, where ηc,a denotes the scalar part, while εc,a∈R<sup>3</sup> denotes the vector part. This allows the rotation matrix to be constructed using quaternions as R<sup>c</sup> <sup>a</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup> <sup>2</sup>ηc,aSð Þþ <sup>ε</sup>c,a <sup>2</sup>S<sup>2</sup> ð Þ εc,a . Composite quaternions can be found through quaternion multiplication as

$$\mathbf{q}\_{\boldsymbol{\epsilon},\boldsymbol{\epsilon}} = \mathbf{q}\_{\boldsymbol{\epsilon},a} \otimes \mathbf{q}\_{\boldsymbol{\epsilon},\boldsymbol{\epsilon}} = \mathbf{T}\left(\mathbf{q}\_{\boldsymbol{\epsilon},a}\right) \mathbf{q}\_{a,\boldsymbol{\epsilon}}\text{ with}$$

$$\mathbf{T}\left(\mathbf{q}\_{\boldsymbol{\epsilon},a}\right) = \begin{bmatrix} \eta\_{\boldsymbol{\epsilon},a} & -\mathbf{e}\_{\boldsymbol{\epsilon},a}^{\mathrm{T}} \\ \mathbf{e}\_{\boldsymbol{\epsilon},a} & \eta\_{\boldsymbol{\epsilon}} \mathbf{I} + \mathbf{S}(\boldsymbol{\epsilon},a) \end{bmatrix}.\tag{1}$$

The use of the quaternion product ensures that the resulting quaternion maintains the unit length property. The quaternion kinematics is defined as

εc,a ηc,aI þ Sð Þ εc,a

$$\dot{\mathbf{q}}\_{c,a} = \frac{1}{2} \mathbf{q}\_{c,a} \otimes \begin{bmatrix} \mathbf{0} \\ \boldsymbol{\uprho}^a\_{c,a} \end{bmatrix} = \frac{1}{2} \mathbf{T} \begin{pmatrix} \mathbf{q}\_{c,a} \end{pmatrix} \begin{bmatrix} \mathbf{0} \\ \boldsymbol{\uprho}^a\_{c,a} \end{bmatrix}. \tag{2}$$

For attitude control, several different frames are needed:

Inertial: The Earth-centered inertial (ECI) has its origin in the center of the Earth, where the x<sup>i</sup> axis points toward the vernal equinox and the z<sup>i</sup> points through the North Pole, while y<sup>i</sup> completes the right-handed orthonormal frame. The inertial frame is denoted by F<sup>i</sup> .

Orbit: The orbit frame has its origin in the center of mass of the satellite (cf. [16], p. 479). The e<sup>r</sup> axis coincides with the radius vector r<sup>i</sup> ∈R<sup>3</sup> , which goes from the center of the Earth to the center of mass in the satellite. The e<sup>h</sup> axis is parallel to the orbital angular momentum vector, pointing in the normal direction of the orbit. The e<sup>θ</sup> completes the right-handed orthonormal frame where the vectors can be described as <sup>e</sup><sup>r</sup> <sup>¼</sup> <sup>r</sup><sup>i</sup> kri <sup>k</sup>, <sup>e</sup><sup>θ</sup> <sup>¼</sup> <sup>e</sup><sup>h</sup> � <sup>e</sup>r, and <sup>e</sup><sup>h</sup> <sup>¼</sup> <sup>h</sup> <sup>k</sup>h<sup>k</sup> where <sup>h</sup> <sup>¼</sup> <sup>r</sup><sup>i</sup> � <sup>r</sup>\_ i . The orbit frame is denoted by F<sup>o</sup> .

Body: The body frame has its origin in the center of mass of the satellite, where its axes coincide with the principal axes of inertia. The body frame is denoted by F<sup>b</sup>.

Desired: The desired frame can be defined arbitrarily to achieve any given objective (cf. [17]). The desired frame is denoted by F<sup>d</sup> .

#### 2.2 Orbital mechanics

This section describes how the orbit frame can be related to the inertial frame through the six classical orbital parameters, and for more details, the reader is referred to [1]. Specifically, the objective with this subsection is to find the radius, velocity, and acceleration vector of the orbit, as well as its angular velocity and acceleration. From well-known orbital mechanics, the six classical parameters can be defined as the semimajor axis a, the eccentricity e, the inclination i, the right ascension of the ascending node Ω, the argument of the perigee ω, and the mean anomaly M. The distance to the apogee and perigee from the center of the Earth can

models used together with sensor measurements to determine the attitude of the satellite requires a real-time clock, as the time and date are required to know the direction toward the Sun as well as what the magnetic field looks like at a given day and time. Comparing the sensor measurements with the mathematical models allows for the determination of the attitude of the satellite, something that is done using the Madgwick filter as presented in [13]. With an estimated attitude obtained using the Madgwick filter, the attitude can be controlled to point a sensor onboard the satellite in a desired direction, something that is solved in this chapter using a PD+ attitude controller, calculating the desired torques required in order to make the attitude and angular velocity errors go to zero. In order to create the desired moments, this chapter presents how this can be achieved using a number of different actuators, namely, magnetic torquers, reaction wheels, and thrusters. The orbital mechanics block describes how the satellite moves in its elliptical orbit, while the perturbing forces and moments block describe how the different perturbations affect the satellite. Simulations show the performance of the different methods and should put the reader in a position to simulate and design new attitude determina-

This figure shows the different components required for modeling and controlling a satellite in an elliptical orbit

This subsection is similar to the author's previous works, e.g., [9, 14]. Let x\_ ¼ dx=dt denote the time derivative of a vector, while the Euclidean length is

which rotates a vector from frame a to frame c and where I denotes the identity matrix. The inverse rotation is found by taking its transpose, such that R<sup>a</sup>

The angular velocity of frame c relative to frame a referenced in frame e is denoted

<sup>x</sup><sup>Τ</sup><sup>x</sup> <sup>p</sup> . Superscript denotes frame of reference for a given vector.

<sup>a</sup>∈SOð Þ¼ <sup>3</sup> <sup>R</sup>∈R<sup>3</sup>�<sup>3</sup> : <sup>R</sup><sup>Τ</sup><sup>R</sup> <sup>¼</sup> <sup>I</sup>; detð Þ¼ <sup>R</sup> <sup>1</sup> � �,

a,f <sup>¼</sup> <sup>ω</sup><sup>e</sup>

a,c <sup>þ</sup> <sup>ω</sup><sup>e</sup>

<sup>c</sup> <sup>¼</sup> <sup>R</sup><sup>c</sup> a � �<sup>Τ</sup> .

c,f (cf. [15]).

tion and control systems for satellites in elliptical orbits.

and shows the main components required for creating a satellite simulator.

a,c, and angular velocities can be added together as ω<sup>e</sup>

2. Mathematical modeling

The rotation matrix is denoted R<sup>c</sup>

defined as <sup>k</sup>xk ¼ ffiffiffiffiffiffiffiffi

2.1 Notation

Figure 1.

Applied Modern Control

ωe

74

be defined, respectively, as ra and rp, allowing the semimajor axis to be found as <sup>a</sup> <sup>¼</sup> raþrp <sup>2</sup> and the eccentricity of the orbit as <sup>e</sup> <sup>¼</sup> ra�rp raþrp , while the mean motion can be found from <sup>n</sup> <sup>¼</sup> ffiffiffi μ a3 <sup>p</sup> . Here, <sup>μ</sup> <sup>¼</sup> GMEarth, where <sup>G</sup> is the gravitational constant, while MEarth is the mass of the Earth. With knowledge of the mean motion, the mean anomaly can be found as M ¼ n tð Þ¼ � t<sup>0</sup> ψ � esin ð Þ ψ where ψ is the eccentric anomaly, t is the time, and t<sup>0</sup> is the time when passing the perigee. To properly describe where in the orbit the satellite is located, the true anomaly can be found as <sup>θ</sup> <sup>¼</sup> cos �<sup>1</sup> cosð Þ� <sup>ψ</sup> <sup>e</sup> 1�e cosð Þ ψ � �, while its derivative can be found as \_ <sup>θ</sup> <sup>¼</sup> <sup>n</sup>ð Þ <sup>1</sup>þ<sup>e</sup> cosð Þ<sup>θ</sup> <sup>2</sup> <sup>1</sup>�e<sup>2</sup> ð Þ<sup>3</sup> 2 ([18], p. 42). When running a simulation, it is desirable to have a continuously increasing true anomaly, while the direct method will map the angle between 0 and 180° . Instead, by integrating the derivative overtime, a smooth true anomaly can be found that increases continuously. The eccentric anomaly, however, cannot be found analytically, but can be found through an iterative algorithm as described in ([1], p. 26) ψkðÞ¼ t M tðÞþ esin ψ<sup>k</sup>�<sup>1</sup> ð Þ ð Þt , where k is the iteration number. This algorithm is valid as long as 0 < e < 1, which holds for elliptical orbits. From these calculations, the rotation matrix from inertial frame to orbit frame can now be constructed as

$$\mathbf{R}\_{i}^{\theta} = \begin{bmatrix} \cos(\boldsymbol{\omega} + \boldsymbol{\theta})\cos(\boldsymbol{\Omega}) - \cos\left(\boldsymbol{i}\right)\sin\left(\boldsymbol{\omega} + \boldsymbol{\theta}\right)\sin\left(\boldsymbol{\Omega}\right) & \cos\left(\boldsymbol{\omega} + \boldsymbol{\theta}\right)\sin\left(\boldsymbol{\Omega}\right) + \sin\left(\boldsymbol{\omega} + \boldsymbol{\theta}\right)\cos\left(\boldsymbol{i}\right)\cos\left(\boldsymbol{\Omega}\right) & \sin\left(\boldsymbol{\omega} + \boldsymbol{\theta}\right)\sin\left(\boldsymbol{i}\right)\cos\left(\boldsymbol{\Omega}\right) \\ -\sin\left(\boldsymbol{\omega} + \boldsymbol{\theta}\right)\cos\left(\boldsymbol{\Omega}\right) - \cos\left(\boldsymbol{i}\right)\sin\left(\boldsymbol{\Omega}\right)\cos\left(\boldsymbol{\omega} + \boldsymbol{\theta}\right) & -\sin\left(\boldsymbol{\omega} + \boldsymbol{\theta}\right)\sin\left(\boldsymbol{\Omega}\right) + \cos\left(\boldsymbol{i}\right)\cos\left(\boldsymbol{\Omega}\right)\cos\left(\boldsymbol{i}\right)\cos\left(\boldsymbol{\Omega}\right) \\ \sin\left(\boldsymbol{i}\right)\sin\left(\boldsymbol{\Omega}\right) & -\sin\left(\boldsymbol{i}\right)\cos\left(\boldsymbol{\Omega}\right) & \cos\left(\boldsymbol{i}\right) \end{bmatrix} \tag{3}$$

The radius, velocity, and acceleration vector can be defined in the orbit frame, respectively, as ([1], pp. 26–27)

$$\mathbf{r}^{\rho} = \begin{bmatrix} a\cos\left(\psi\right) - a\epsilon & a\sin\left(\psi\right)\sqrt{1 - e^2} & \mathbf{0} \end{bmatrix}^{\mathrm{T}},\tag{4}$$

these values allow for calculating the rotation matrix from the orbit frame to the

Parameter Value Unit <sup>G</sup> <sup>6</sup>:<sup>67408</sup> � <sup>10</sup>�<sup>11</sup> <sup>m</sup><sup>3</sup> kg�<sup>1</sup> <sup>s</sup>�<sup>2</sup> MEarth <sup>5</sup>:<sup>9742</sup> � 1024 kg ra Re þ 1200 km rp Re þ 800 km Re 6378 km i 75 ° Ω 0 ° ω 0 °

all the outputs from this subsystem can easily be found following this procedure and

The attitude dynamics can be derived with the basis in Euler's moment Equation ([1], p. 95). The angular momentum of a rigid body in the body frame is given as

<sup>h</sup><sup>b</sup> <sup>¼</sup> <sup>J</sup>ω<sup>b</sup>

body frame relative to the inertial frame. The angular momentum can be found in

<sup>h</sup><sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>i</sup>

<sup>b</sup>S ω<sup>b</sup> i,b <sup>h</sup><sup>b</sup> <sup>þ</sup> <sup>R</sup><sup>i</sup>

Decomposing the total torque into an actuation component and a perturbing com-

i,b Jω<sup>b</sup>

<sup>p</sup>, allows the rotational dynamics to be written as

. Hence, by differentiating Eq. (9), it is obtained that

<sup>τ</sup><sup>i</sup> <sup>¼</sup> <sup>h</sup>\_ <sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>i</sup>

which can be written in the body frame by using Eq. (8) as

i,b ¼ �<sup>S</sup> <sup>ω</sup><sup>b</sup>

Jω\_ <sup>b</sup>

bhb

The rate of change of angular momentum is equal to the total torque, such that

<sup>o</sup>), the radius vector (r<sup>o</sup>), the velocity vector (v<sup>o</sup>), the acceleration

i, <sup>o</sup>), and angular acceleration vector (ω\_ <sup>i</sup>

i,b, (8)

i,b∈R<sup>3</sup> is the angular velocity of the

: (9)

, (10)

p, (11)

bh\_ b

<sup>a</sup> <sup>þ</sup> <sup>τ</sup><sup>b</sup>

i,b, where the inertia matrix is assumed to be constant.

i,b <sup>þ</sup> <sup>τ</sup><sup>b</sup>

i, <sup>o</sup>). Hence,

inertial frame (R<sup>i</sup>

Table 1.

3.1 Attitude dynamics

the inertial frame as

<sup>τ</sup><sup>i</sup> <sup>¼</sup> <sup>h</sup>\_ <sup>i</sup>

<sup>τ</sup><sup>b</sup> <sup>¼</sup> <sup>S</sup> <sup>ω</sup><sup>b</sup>

77

ponent, <sup>τ</sup><sup>b</sup> <sup>¼</sup> <sup>τ</sup><sup>b</sup>

i,b <sup>J</sup>ω<sup>b</sup>

i,b <sup>þ</sup> <sup>J</sup>ω\_ <sup>b</sup>

<sup>a</sup> <sup>þ</sup> <sup>τ</sup><sup>b</sup>

vector (a<sup>o</sup>), angular velocity (ω<sup>i</sup>

will be used in several other subsystems.

Parameters required for calculation of the orbital dynamics.

Modeling and Attitude Control of Satellites in Elliptical Orbits

DOI: http://dx.doi.org/10.5772/intechopen.80422

3. Attitude dynamics and control

where J∈R<sup>3</sup>x<sup>3</sup> is the inertia matrix, while ω<sup>b</sup>

$$\mathbf{v}^{\rho} = \begin{bmatrix} -\frac{a^2 n}{r} \sin\left(\psi\right) & \frac{a^2 n}{r} \sqrt{1 - e^2} \cos\left(\psi\right) & \mathbf{0} \end{bmatrix}^{\mathrm{T}},\tag{5}$$

$$\mathbf{a}^o = \begin{bmatrix} -\frac{a^3 n^2}{r^2} \cos\left(\psi\right) & -\frac{a^3 n^2}{r^2} \sqrt{\mathbf{1} - e^2} \sin\left(\psi\right) & \mathbf{0} \end{bmatrix}^T,\tag{6}$$

where <sup>r</sup> ¼ kr<sup>i</sup> k is the length of the radius vector. Each of these vectors can be rotated to the inertial frame using Eq. (3), such that <sup>r</sup><sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>o</sup> i ro , <sup>v</sup><sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>i</sup> ovo , and <sup>a</sup><sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>i</sup> <sup>o</sup>a<sup>o</sup>. The angular velocity of the orbit frame relative to the inertial frame can be found as ω<sup>i</sup> i, <sup>o</sup> <sup>¼</sup> <sup>r</sup><sup>i</sup> �v<sup>i</sup> <sup>r</sup><sup>i</sup> ð ÞΤr<sup>i</sup> , while the angular acceleration can be found through differentiation as

$$
\dot{\boldsymbol{\alpha}}\_{i,o}^{i} = \frac{\left(\mathbf{r}^{i} \times \mathbf{a}^{i}\right) \left(\mathbf{r}^{i}\right)^{\mathrm{T}} \mathbf{r}^{i} - 2 \left(\mathbf{r}^{i} \times \mathbf{v}^{i}\right) \left(\mathbf{v}^{i}\right)^{\mathrm{T}} \mathbf{r}^{i}}{\left(\left(\mathbf{r}^{i}\right)^{\mathrm{T}} \mathbf{r}^{i}\right)^{2}}.\tag{7}$$

In order to implement the orbital mechanics in, e.g., a Simulink framework, the required input to the subsystem would be the time (t). Further, the orbital parameters must be defined as given in Table 1 and can be changed depending on the orbit. These constants allow for the calculations of the eccentricity (e), the semimajor axis (a), and mean motion (n). With the mean motion, the mean anomaly (M) can be found and used to approximate the eccentric anomaly (ψ) using the iterative algorithm presented above. The rate of change of the true anomaly ( \_ θ) can also be found and by integration enables the calculation of true anomaly (θ). All


Ω 0 ° ω 0 °

Modeling and Attitude Control of Satellites in Elliptical Orbits

Table 1.

be defined, respectively, as ra and rp, allowing the semimajor axis to be found as

while MEarth is the mass of the Earth. With knowledge of the mean motion, the mean anomaly can be found as M ¼ n tð Þ¼ � t<sup>0</sup> ψ � esin ð Þ ψ where ψ is the eccentric anomaly, t is the time, and t<sup>0</sup> is the time when passing the perigee. To properly describe where in the orbit the satellite is located, the true anomaly can be found as

, while its derivative can be found as \_

Instead, by integrating the derivative overtime, a smooth true anomaly can be found that increases continuously. The eccentric anomaly, however, cannot be found analytically, but can be found through an iterative algorithm as described in ([1], p. 26) ψkðÞ¼ t M tðÞþ esin ψ<sup>k</sup>�<sup>1</sup> ð Þ ð Þt , where k is the iteration number. This algorithm is valid as long as 0 < e < 1, which holds for elliptical orbits. From these calculations, the rotation matrix from inertial frame to orbit frame can now be

p. 42). When running a simulation, it is desirable to have a continuously increasing true anomaly, while the direct method will map the angle between 0 and 180°

cosð Þ ω þ θ cosð Þ� Ω cosð Þi sin ð Þ ω þ θ sin ð Þ Ω cosð Þ ω þ θ sin ð Þþ Ω sin ð Þ ω þ θ cosð Þi cosð Þ Ω sin ð Þ ω þ θ sin ð Þi � sin ð Þ ω þ θ cosð Þ� Ω cosð Þi sin ð Þ Ω cosð Þ� ω þ θ sin ð Þ ω þ θ sin ð Þþ Ω cosð Þ ω þ θ cosð Þi cosð Þ Ω cosð Þ ω þ θ sin ð Þi sin ð Þi sin ð Þ Ω � sin ð Þi cosð Þ Ω cosð Þi

The radius, velocity, and acceleration vector can be defined in the orbit frame,

<sup>1</sup> � <sup>e</sup><sup>2</sup> <sup>p</sup> <sup>0</sup> � �<sup>Τ</sup>

h i<sup>Τ</sup>

r2

h i<sup>Τ</sup>

. The angular velocity of the orbit frame relative to the inertial frame can

<sup>r</sup><sup>i</sup> ð Þ<sup>Τ</sup> ri

In order to implement the orbital mechanics in, e.g., a Simulink framework, the required input to the subsystem would be the time (t). Further, the orbital parameters must be defined as given in Table 1 and can be changed depending on the orbit. These constants allow for the calculations of the eccentricity (e), the

semimajor axis (a), and mean motion (n). With the mean motion, the mean anomaly (M) can be found and used to approximate the eccentric anomaly (ψ) using the iterative algorithm presented above. The rate of change of the true anomaly ( \_

also be found and by integration enables the calculation of true anomaly (θ). All

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>e</sup><sup>2</sup> <sup>p</sup> cosð Þ <sup>ψ</sup> <sup>0</sup>

> ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>e</sup><sup>2</sup> <sup>p</sup> sin ð Þ <sup>ψ</sup> <sup>0</sup>

, while the angular acceleration can be found through dif-

<sup>r</sup><sup>i</sup> � <sup>2</sup> <sup>r</sup><sup>i</sup> � <sup>v</sup><sup>i</sup> � � <sup>v</sup><sup>i</sup> � �<sup>Τ</sup>

i ro

ri

� �<sup>2</sup> : (7)

, <sup>v</sup><sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>i</sup>

k is the length of the radius vector. Each of these vectors can be

r

<sup>r</sup><sup>o</sup> <sup>¼</sup> <sup>a</sup> cosð Þ� <sup>ψ</sup> ae a sin ð Þ <sup>ψ</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>r</sup> sin ð Þ <sup>ψ</sup> <sup>a</sup>2<sup>n</sup>

<sup>r</sup><sup>2</sup> cosðÞ � <sup>ψ</sup> <sup>a</sup>3n<sup>2</sup>

raþrp

<sup>p</sup> . Here, <sup>μ</sup> <sup>¼</sup> GMEarth, where <sup>G</sup> is the gravitational constant,

, while the mean motion can be

<sup>θ</sup> <sup>¼</sup> <sup>n</sup>ð Þ <sup>1</sup>þ<sup>e</sup> cosð Þ<sup>θ</sup> <sup>2</sup> <sup>1</sup>�e<sup>2</sup> ð Þ<sup>3</sup> 2

([18],

.

3 7 5:

(3)

θ) can

, (4)

, (5)

ovo , and

, (6)

<sup>2</sup> and the eccentricity of the orbit as <sup>e</sup> <sup>¼</sup> ra�rp

<sup>a</sup> <sup>¼</sup> raþrp

found from <sup>n</sup> <sup>¼</sup> ffiffiffi

Applied Modern Control

<sup>θ</sup> <sup>¼</sup> cos �<sup>1</sup> cosð Þ� <sup>ψ</sup> <sup>e</sup>

constructed as

Ro <sup>i</sup> ¼ 2 6 4

1�e cosð Þ ψ � �

respectively, as ([1], pp. 26–27)

i, <sup>o</sup> <sup>¼</sup> <sup>r</sup><sup>i</sup> �v<sup>i</sup> <sup>r</sup><sup>i</sup> ð ÞΤr<sup>i</sup>

ω\_ i

where <sup>r</sup> ¼ kr<sup>i</sup>

<sup>a</sup><sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>i</sup> oao

76

be found as ω<sup>i</sup>

ferentiation as

<sup>v</sup><sup>o</sup> <sup>¼</sup> � <sup>a</sup>2<sup>n</sup>

<sup>a</sup><sup>o</sup> <sup>¼</sup> � <sup>a</sup>3n<sup>2</sup>

rotated to the inertial frame using Eq. (3), such that <sup>r</sup><sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>o</sup>

i, <sup>o</sup> <sup>¼</sup> <sup>r</sup><sup>i</sup> � <sup>a</sup><sup>i</sup> � � <sup>r</sup><sup>i</sup> � �<sup>Τ</sup>

μ a3

Parameters required for calculation of the orbital dynamics.

these values allow for calculating the rotation matrix from the orbit frame to the inertial frame (R<sup>i</sup> <sup>o</sup>), the radius vector (r<sup>o</sup>), the velocity vector (v<sup>o</sup>), the acceleration vector (a<sup>o</sup>), angular velocity (ω<sup>i</sup> i, <sup>o</sup>), and angular acceleration vector (ω\_ <sup>i</sup> i, <sup>o</sup>). Hence, all the outputs from this subsystem can easily be found following this procedure and will be used in several other subsystems.

#### 3. Attitude dynamics and control

#### 3.1 Attitude dynamics

The attitude dynamics can be derived with the basis in Euler's moment Equation ([1], p. 95). The angular momentum of a rigid body in the body frame is given as

$$\mathbf{h}^b = \mathbf{J} \mathbf{o}\_{i,b}^b,\tag{8}$$

where J∈R<sup>3</sup>x<sup>3</sup> is the inertia matrix, while ω<sup>b</sup> i,b∈R<sup>3</sup> is the angular velocity of the body frame relative to the inertial frame. The angular momentum can be found in the inertial frame as

$$\mathbf{h}^i = \mathbf{R}\_b^i \mathbf{h}^b. \tag{9}$$

The rate of change of angular momentum is equal to the total torque, such that <sup>τ</sup><sup>i</sup> <sup>¼</sup> <sup>h</sup>\_ <sup>i</sup> . Hence, by differentiating Eq. (9), it is obtained that

$$\boldsymbol{\sigma}^{i} = \dot{\mathbf{h}}^{i} = \mathbf{R}\_{b}^{i} \mathbf{S} (\boldsymbol{\sigma}\_{i,b}^{b}) \mathbf{h}^{b} + \mathbf{R}\_{b}^{i} \dot{\mathbf{h}}^{b},\tag{10}$$

which can be written in the body frame by using Eq. (8) as <sup>τ</sup><sup>b</sup> <sup>¼</sup> <sup>S</sup> <sup>ω</sup><sup>b</sup> i,b <sup>J</sup>ω<sup>b</sup> i,b <sup>þ</sup> <sup>J</sup>ω\_ <sup>b</sup> i,b, where the inertia matrix is assumed to be constant. Decomposing the total torque into an actuation component and a perturbing component, <sup>τ</sup><sup>b</sup> <sup>¼</sup> <sup>τ</sup><sup>b</sup> <sup>a</sup> <sup>þ</sup> <sup>τ</sup><sup>b</sup> <sup>p</sup>, allows the rotational dynamics to be written as

$$\mathbf{J}\dot{\boldsymbol{\alpha}}\_{i,b}^{b} = -\mathbf{S} \left(\boldsymbol{\alpha}\_{i,b}^{b}\right) \mathbf{J}\boldsymbol{\alpha}\_{i,b}^{b} + \boldsymbol{\pi}\_{a}^{b} + \boldsymbol{\pi}\_{p}^{b},\tag{11}$$

where τ<sup>b</sup> <sup>a</sup>∈R<sup>3</sup> denotes the actuation torques (e.g., output from reaction wheels), while τ<sup>b</sup> <sup>p</sup>∈R<sup>3</sup> denotes the perturbing torques (e.g., gravity torque). Further, by using quaternion representation, the update law for the quaternion representing the attitude of the body frame relative to the inertial frame can be written as

$$\dot{\mathbf{q}}\_{i,b} = \frac{1}{2} \mathbf{T} \begin{pmatrix} \mathbf{q}\_{i,b} \end{pmatrix} \begin{bmatrix} \mathbf{0} \\ \boldsymbol{\omega}\_{i,b}^b \end{bmatrix}. \tag{12}$$

3.3 PD+ attitude controller

<sup>2</sup> ω<sup>b</sup> d,b <sup>Τ</sup>

DOI: http://dx.doi.org/10.5772/intechopen.80422

d,bω<sup>b</sup>

o,d � JS <sup>ω</sup><sup>b</sup>

�kpεd,b � kdω<sup>b</sup>

Jω<sup>b</sup>

Modeling and Attitude Control of Satellites in Elliptical Orbits

d,b <sup>þ</sup> <sup>ω</sup><sup>b</sup>

i, <sup>o</sup> <sup>þ</sup> JS <sup>ω</sup><sup>b</sup>

o,b R<sup>b</sup>

where kd is another positive scalar gain and τ<sup>b</sup>

<sup>a</sup> <sup>¼</sup> <sup>τ</sup><sup>b</sup>

 2

d,b 

A PD+ attitude control law can now be chosen as

scalar gain. The derivative is found by using Eqs. (16)–(18) as

d,b <sup>Τ</sup> �<sup>S</sup> <sup>ω</sup><sup>b</sup>

o,b R<sup>b</sup>

dω<sup>d</sup>

[23, 24].

τb <sup>d</sup> <sup>¼</sup> JR<sup>b</sup>

be chosen as <sup>V</sup> <sup>¼</sup> <sup>1</sup>

<sup>V</sup>\_ <sup>¼</sup> kpε<sup>Τ</sup>

<sup>d</sup>ω\_ <sup>d</sup>

actuator dynamics, i.e., τ<sup>b</sup>

obtained that <sup>V</sup>\_ <sup>≤</sup> � kd <sup>ω</sup><sup>b</sup>

4. Perturbing torques

79

the gravity torque is defined as

uniformly asymptotically stable.

�JR<sup>b</sup> <sup>i</sup> ω\_ <sup>i</sup>

Takegaki and Arimoto [19] proposed in 1981 a simple method for position control of robots, something that was extended by [20] to enable trajectory tracking. The so-called PD+ controller has been applied for spacecraft by [21, 22] showing good results. The author has applied this method in previous works such as

In order to design a PD+ attitude controller, let a Lyapunov function candidate

i,b Jω<sup>b</sup>

o,d � JR<sup>b</sup>

<sup>i</sup> ω\_ <sup>i</sup>

required to make the attitude and angular velocity errors go to zero. Assuming no

which are to be defined by the reader, e.g., as part as a guidance block depending on the mission objective. The inertia matrix (J) is assumed to be known, while the

as described above. The other angular velocities are found from the orbital mecha-

or by using the relationship between the quaternions and rotation matrices directly (cf. Section 2A). The error quaternion and angular velocity are found from Eqs. (14) and (15), while the perturbing torques will be described in the following section.

There are different kinds of perturbing torques, such as gravity torque, aerodynamic torque, magnetic field due to the electronics inside the satellite, as well as solar radiation torque. This section only considers the gravity torque. In [16], p. 147,

<sup>r</sup><sup>5</sup> <sup>r</sup><sup>i</sup> � Jr<sup>i</sup>

<sup>2</sup> <sup>þ</sup> kpε<sup>Τ</sup>

i,b <sup>þ</sup> <sup>τ</sup><sup>b</sup>

<sup>d</sup>ω\_ <sup>d</sup>

i, <sup>o</sup> � JS <sup>ω</sup><sup>b</sup>

i,b R<sup>b</sup>

d,b, (20)

i ωi o,i � <sup>τ</sup><sup>b</sup>

<sup>d</sup>, and then by inserting Eq. (20) into Eq. (19), it is

, which is negative semidefinite. By applying the

<sup>a</sup> <sup>þ</sup> <sup>τ</sup><sup>b</sup>

<sup>p</sup> <sup>þ</sup> JS <sup>ω</sup><sup>b</sup>

<sup>d</sup> denotes the desired torque

d,bεd,b where kp is a positive

i,b R<sup>b</sup>

> <sup>p</sup> <sup>þ</sup> <sup>S</sup> <sup>ω</sup><sup>b</sup> i,b Jω<sup>b</sup>

i,b

d,b)=(0, 0) is

o,d, and ω\_ <sup>d</sup>

o,b) can be found

<sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>b</sup> oR<sup>o</sup> i ,

, (21)

o,d,

o,dÞ: (19)

i ωi o,i

d,b þ kp 1 � ηd,b

dω<sup>d</sup>

o,d <sup>þ</sup> JR<sup>b</sup>

Matrosov theorem (cf. [24]), it can be shown that the origin (εd,b, ω<sup>b</sup>

angular velocity vector between the body frame and orbit frame (ω<sup>b</sup>

τb

<sup>g</sup> <sup>¼</sup> <sup>3</sup>GMEarth

The inputs to the control law (Eq. 20) are the desired states qo,d, ω<sup>d</sup>

nics, while the rotation matrices are found as composite rotations, e.g., R<sup>b</sup>

Hence, Eqs. (11) and (12) serve as governing equations describing the attitude and angular velocity of the satellite. The inputs that affect these values are the perturbation and actuation torques, where the latter will be found in the following sections.

#### 3.2 Error dynamics

From Euler's moment equation, the angular acceleration is defined relative to the inertial frame. For attitude control, it is often more interesting controlling the attitude and angular velocity relative to the orbit frame. For that reason, the angular velocity of the body frame relative to the orbit frame can be found as ω<sup>b</sup> o,b <sup>¼</sup> <sup>ω</sup><sup>b</sup> i,b � <sup>R</sup><sup>b</sup> <sup>i</sup> ω<sup>i</sup> i, <sup>o</sup>, which can be differentiated as

$$\mathbf{J}\dot{\boldsymbol{\alpha}}\_{o,b}^{b} = -\mathbf{S}(\boldsymbol{\alpha}\_{i,b}^{b})\mathbf{J}\boldsymbol{\alpha}\_{i,b}^{b} + \boldsymbol{\tau}\_{a}^{b} + \boldsymbol{\tau}\_{p}^{b} + \mathbf{J}\mathbf{S}(\boldsymbol{\alpha}\_{i,b}^{b})\mathbf{R}\_{i}^{b}\boldsymbol{\alpha}\_{o,i}^{i} - \mathbf{J}\mathbf{R}\_{i}^{b}\dot{\boldsymbol{\alpha}}\_{i,o}^{i} \tag{13}$$

giving a description of the attitude dynamics relative to the orbit frame. It is also possible to find the error dynamics, to enable tracking of a desired attitude and angular velocity. Let qo,d, ω<sup>d</sup> o,d, ω\_ <sup>d</sup> o,d∈L<sup>∞</sup> denote a desired quaternion, angular velocity, and acceleration; then, the quaternion and angular velocity error can be found as

$$\mathbf{q}\_{d,b} = \mathbf{q}\_{d,o} \otimes \mathbf{q}\_{o,b},\tag{14}$$

$$
\boldsymbol{\alpha}\_{d,b}^{b} = \boldsymbol{\alpha}\_{o,b}^{b} - \mathbf{R}\_d^{b} \boldsymbol{\alpha}\_{o,d}^{d} \tag{15}
$$

with the kinematics as

$$
\dot{\eta}\_{d,b} = -\frac{1}{2} \mathbf{e}\_{d,b}^{\mathrm{T}} \boldsymbol{\alpha}\_{d,b}^{b}, \tag{16}
$$

$$
\dot{\mathbf{e}}\_{d,b} = \left(\eta\_{d,b}\mathbf{I} + \mathbf{S}(\mathbf{e}\_{d,b})\right) \mathbf{o}\_{d,b}^{b}.\tag{17}
$$

The angular acceleration error can be found by differentiating Eq. (15) as

$$\mathbf{J}\dot{\boldsymbol{\alpha}}\_{d,b}^{b} = -\mathbf{S}(\boldsymbol{\alpha}\_{i,b}^{b})\mathbf{J}\boldsymbol{\alpha}\_{i,b}^{b} + \boldsymbol{\tau}\_{a}^{b} + \boldsymbol{\tau}\_{p}^{b} + \mathbf{J}\mathbf{S}(\boldsymbol{\alpha}\_{i,b}^{b})\mathbf{R}\_{i}^{b}\boldsymbol{\alpha}\_{o,i}^{i} - \mathbf{J}\mathbf{R}\_{i}^{b}\dot{\boldsymbol{\alpha}}\_{i,o}^{i} + \mathbf{J}\mathbf{S}(\boldsymbol{\alpha}\_{o,b}^{b})\mathbf{R}\_{d}^{b}\boldsymbol{\alpha}\_{o,d}^{d} - \mathbf{J}\mathbf{R}\_{d}^{b}\dot{\boldsymbol{\alpha}}\_{o,d}^{d}. \tag{18}$$

Hence, the control objective can be defined as that of making qd,b; ω<sup>b</sup> d,b � � ! ð Þ <sup>0</sup>; <sup>0</sup> , which will make the satellite point in a desired direction and move with a desired angular velocity.

#### 3.3 PD+ attitude controller

where τ<sup>b</sup>

Applied Modern Control

while τ<sup>b</sup>

sections.

ω<sup>b</sup> o,b <sup>¼</sup> <sup>ω</sup><sup>b</sup>

found as

Jω\_ <sup>b</sup>

78

d,b ¼ �<sup>S</sup> <sup>ω</sup><sup>b</sup>

qd,b; ω<sup>b</sup> d,b � �

3.2 Error dynamics

i,b � <sup>R</sup><sup>b</sup> <sup>i</sup> ω<sup>i</sup>

angular velocity. Let qo,d, ω<sup>d</sup>

with the kinematics as

i,b � �Jω<sup>b</sup>

i,b <sup>þ</sup> <sup>τ</sup><sup>b</sup>

move with a desired angular velocity.

<sup>a</sup> <sup>þ</sup> <sup>τ</sup><sup>b</sup>

<sup>p</sup> <sup>þ</sup> JS <sup>ω</sup><sup>b</sup> i,b � �R<sup>b</sup> i ωi

Hence, the control objective can be defined as that of making

o,b ¼ �<sup>S</sup> <sup>ω</sup><sup>b</sup>

i,b � �Jω<sup>b</sup>

Jω\_ <sup>b</sup>

<sup>a</sup>∈R<sup>3</sup> denotes the actuation torques (e.g., output from reaction wheels),

ω<sup>b</sup> i,b

" #

: (12)

<sup>p</sup>∈R<sup>3</sup> denotes the perturbing torques (e.g., gravity torque). Further, by using

quaternion representation, the update law for the quaternion representing the atti-

Hence, Eqs. (11) and (12) serve as governing equations describing the attitude and angular velocity of the satellite. The inputs that affect these values are the perturbation and actuation torques, where the latter will be found in the following

From Euler's moment equation, the angular acceleration is defined relative to the

<sup>p</sup> <sup>þ</sup> JS <sup>ω</sup><sup>b</sup>

giving a description of the attitude dynamics relative to the orbit frame. It is also

o,b � <sup>R</sup><sup>b</sup>

dω<sup>d</sup>

i,b � �R<sup>b</sup>

i ωi

o,d∈L<sup>∞</sup> denote a desired quaternion, angular

qd,b ¼ qd, <sup>o</sup>⊗qo,b, (14)

i, <sup>o</sup> <sup>þ</sup> JS <sup>ω</sup><sup>b</sup>

o,i � JR<sup>b</sup>

<sup>i</sup> ω\_ <sup>i</sup>

o,d, (15)

d,b, (16)

o,b � �R<sup>b</sup>

d,b: (17)

dω<sup>d</sup>

o,d � JR<sup>b</sup>

<sup>d</sup>ω\_ <sup>d</sup> o,d:

(18)

i, <sup>o</sup> (13)

inertial frame. For attitude control, it is often more interesting controlling the attitude and angular velocity relative to the orbit frame. For that reason, the angular

velocity of the body frame relative to the orbit frame can be found as

i, <sup>o</sup>, which can be differentiated as

<sup>a</sup> <sup>þ</sup> <sup>τ</sup><sup>b</sup>

possible to find the error dynamics, to enable tracking of a desired attitude and

velocity, and acceleration; then, the quaternion and angular velocity error can be

i,b <sup>þ</sup> <sup>τ</sup><sup>b</sup>

o,d, ω\_ <sup>d</sup>

ωb d,b <sup>¼</sup> <sup>ω</sup><sup>b</sup>

<sup>η</sup>\_d,b ¼ � <sup>1</sup>

ε\_d,b ¼ ηd,bI þ Sð Þ εd,b

The angular acceleration error can be found by differentiating Eq. (15) as

2 εΤ d,bω<sup>b</sup>

� �ω<sup>b</sup>

o,i � JR<sup>b</sup> i ω\_ i

! ð Þ 0; 0 , which will make the satellite point in a desired direction and

tude of the body frame relative to the inertial frame can be written as

<sup>q</sup>\_ i,b <sup>¼</sup> <sup>1</sup> 2 T qi,b � � 0

Takegaki and Arimoto [19] proposed in 1981 a simple method for position control of robots, something that was extended by [20] to enable trajectory tracking. The so-called PD+ controller has been applied for spacecraft by [21, 22] showing good results. The author has applied this method in previous works such as [23, 24].

In order to design a PD+ attitude controller, let a Lyapunov function candidate be chosen as <sup>V</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> ω<sup>b</sup> d,b <sup>Τ</sup> Jω<sup>b</sup> d,b þ kp 1 � ηd,b <sup>2</sup> <sup>þ</sup> kpε<sup>Τ</sup> d,bεd,b where kp is a positive scalar gain. The derivative is found by using Eqs. (16)–(18) as

$$\dot{V} = k\_p \mathbf{e}\_{d,b}^\mathrm{T} \boldsymbol{\alpha}\_{d,b}^b + \left(\boldsymbol{\alpha}\_{d,b}^b\right)^\mathrm{T} \left(-\mathbf{S} \left(\boldsymbol{\alpha}\_{i,b}^b\right) \mathbf{J} \boldsymbol{\alpha}\_{i,b}^b + \mathbf{r}\_a^b + \mathbf{r}\_p^b + \mathbf{J} \mathbf{S} \left(\boldsymbol{\alpha}\_{i,b}^b\right) \mathbf{R}\_i^b \boldsymbol{\alpha}\_{o,i}^i\right.\tag{19}$$

$$-\mathbf{J} \mathbf{R}\_i^b \boldsymbol{\alpha}\_{i,o}^i + \mathbf{J} \mathbf{S} \left(\boldsymbol{\alpha}\_{o,b}^b\right) \mathbf{R}\_d^b \boldsymbol{\alpha}\_{o,d}^d - \mathbf{J} \mathbf{R}\_d^b \dot{\boldsymbol{\alpha}}\_{o,d}^d \boldsymbol{\alpha}\_{o,d}^d \,. \tag{10}$$

A PD+ attitude control law can now be chosen as

$$\mathbf{r}\_d^b = \mathbf{J} \mathbf{R}\_d^b \dot{\boldsymbol{\alpha}}\_{o,d}^d - \mathbf{J} \mathbf{S} (\boldsymbol{\alpha}\_{o,b}^b) \mathbf{R}\_d^b \boldsymbol{\alpha}\_{o,d}^d + \mathbf{J} \mathbf{R}\_i^b \dot{\boldsymbol{\alpha}}\_{i,o}^i - \mathbf{J} \mathbf{S} (\boldsymbol{\alpha}\_{i,b}^b) \mathbf{R}\_i^b \boldsymbol{\alpha}\_{o,i}^i - \mathbf{r}\_p^b + \mathbf{S} (\boldsymbol{\alpha}\_{i,b}^b) \mathbf{J} \boldsymbol{\alpha}\_{i,b}^b$$
 
$$-k\_p \mathbf{e}\_{d,b} - k\_d \boldsymbol{\alpha}\_{d,b}^b$$

where kd is another positive scalar gain and τ<sup>b</sup> <sup>d</sup> denotes the desired torque required to make the attitude and angular velocity errors go to zero. Assuming no actuator dynamics, i.e., τ<sup>b</sup> <sup>a</sup> <sup>¼</sup> <sup>τ</sup><sup>b</sup> <sup>d</sup>, and then by inserting Eq. (20) into Eq. (19), it is obtained that <sup>V</sup>\_ <sup>≤</sup> � kd <sup>ω</sup><sup>b</sup> d,b 2 , which is negative semidefinite. By applying the Matrosov theorem (cf. [24]), it can be shown that the origin (εd,b, ω<sup>b</sup> d,b)=(0, 0) is uniformly asymptotically stable.

The inputs to the control law (Eq. 20) are the desired states qo,d, ω<sup>d</sup> o,d, and ω\_ <sup>d</sup> o,d, which are to be defined by the reader, e.g., as part as a guidance block depending on the mission objective. The inertia matrix (J) is assumed to be known, while the angular velocity vector between the body frame and orbit frame (ω<sup>b</sup> o,b) can be found as described above. The other angular velocities are found from the orbital mechanics, while the rotation matrices are found as composite rotations, e.g., R<sup>b</sup> <sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>b</sup> oR<sup>o</sup> i , or by using the relationship between the quaternions and rotation matrices directly (cf. Section 2A). The error quaternion and angular velocity are found from Eqs. (14) and (15), while the perturbing torques will be described in the following section.

#### 4. Perturbing torques

There are different kinds of perturbing torques, such as gravity torque, aerodynamic torque, magnetic field due to the electronics inside the satellite, as well as solar radiation torque. This section only considers the gravity torque. In [16], p. 147, the gravity torque is defined as

$$
\sigma\_{\mathcal{g}}^{b} = \frac{\Im G M\_{Earth}}{r^{\mathcal{S}}} \mathbf{r}^{i} \times \mathbf{J} \mathbf{r}^{i},\tag{21}
$$

where the terms have previously been defined. As can be seen from this equation, non-diagonal inertia matrices will induce gravitational torques to align the satellite with the gravity field. This is also sometimes used for passive control, using e.g., a gravity boom to ensure that one side of the satellite is always facing the Earth. For this chapter, the perturbing toque is set equal to the gravity torque such that τb <sup>p</sup> <sup>¼</sup> <sup>τ</sup><sup>b</sup> g .

τb

<sup>Τ</sup> <sup>¼</sup> NA ix iy iz

Modeling and Attitude Control of Satellites in Elliptical Orbits

where <sup>m</sup><sup>b</sup> <sup>¼</sup> mx my mz

DOI: http://dx.doi.org/10.5772/intechopen.80422

Earth's magnetic field is needed.

inverting this equation is given in [25].

enabling the currents to be found as

desired torque τ<sup>b</sup>

torque can be found as

τx,a τy,a τz,a

<sup>m</sup> ¼ τx,m τy,m τz,m <sup>Τ</sup>

<sup>q</sup>o,d <sup>¼</sup> ½ � <sup>1000</sup> <sup>Τ</sup>

where τ<sup>b</sup>

τb

81

ix iy iz <sup>Τ</sup> <sup>¼</sup> <sup>1</sup>

<sup>Τ</sup> <sup>¼</sup> signð Þ <sup>τ</sup>x,d <sup>τ</sup>x,m sign <sup>τ</sup>y,d

.

<sup>a</sup> ¼ τx,a τy,a τz,a <sup>Τ</sup> <sup>m</sup> <sup>¼</sup> S m<sup>b</sup> <sup>b</sup><sup>b</sup>

N is the number of turns of the coils, ið Þ� is the current around a given axis, and A is the area of the coils. The Earth's magnetic field is represented through the vector b<sup>b</sup>

From a control point of view, the physical parameters A and N are defined when the spacecraft is designed, such that the controller needs to dictate which currents that must be sent to the torquers in order to get a desired torque. This means that Eq. (22) must be inverted with regard to m<sup>b</sup>, which is not straightforward due to the cross product, meaning that you obtain rank 2 when inverting the right-hand side,

> S b<sup>b</sup> τ<sup>b</sup> d

> > bb

NA

for the currents and applied resulting in the actuation torque in Eq. (22). Hence, the control law (Eq. 20) can be mapped to a desired magnetic moment (Eq. 23), which then can be used to find the desired current to each of the three coils. Then, the limits

in current will dictate the maximum magnetic moment that can be generated. Consider the HiNCube satellite as shown in Figure 3. The cubesat comprises three orthonormal magnetic torquers with an area <sup>A</sup> <sup>¼</sup> <sup>0</sup>:00757 m2 and with a maximum current of 47.27 mA and N ¼ 100 turns. This gives a maximum magnetic moment of mmax <sup>¼</sup> <sup>0</sup>:03578 mA2. Hence, an implementation of using magnetic torquers for attitude control would encompass mapping the output from the control law to a desired magnetic moment using Eq. (23) and then imposing the maximum magnetic moment on each axis, before finding the resulting actuation torque using Eq. (22). Note that to ensure sign correctness due to the projection, the actuation

, τ<sup>b</sup>

It is here assumed that all three torquers are identical, but depending on satellite configuration, there might be differences in the number of turns and areas. Hence, the

mx my mz <sup>Τ</sup>

<sup>d</sup> can be used to find the magnetic moment m<sup>b</sup> in Eq. (23) and solved

<sup>Τ</sup>

. The gains for the PD+ controller are set kp <sup>¼</sup> <sup>1</sup> � <sup>10</sup>�<sup>5</sup> and

<sup>d</sup> ¼ τx,d τy,d τz,d <sup>Τ</sup>

To show the performance of magnetic torquers, consider again the HiNCube satellite, which had an inertia matrix of <sup>J</sup> <sup>¼</sup> <sup>1</sup>:<sup>67</sup> � <sup>10</sup>�<sup>3</sup><sup>I</sup> kg m2. Consider the problem of making rotating 90° from an initial quaternion <sup>q</sup>o,b <sup>¼</sup> ½ � <sup>0001</sup> <sup>Τ</sup> to

<sup>τ</sup>y,m signð Þ <sup>τ</sup>z,d <sup>τ</sup>z,m

, and

meaning that to use magnetic actuation for attitude control, a good model of the

losing information about one of the axes. To that end, an approximation to

<sup>m</sup><sup>b</sup> <sup>¼</sup>

, (22)

<sup>2</sup> , (23)

: (24)

, (25)

,

<sup>Τ</sup> is the induced magnetic field,

#### 5. Actuators

The control signal must be mapped to an actuator that must generate the desired torque. With limitations in actuation, the saturation must be modeled in order to obtain realistic results when simulating attitude control. This section considers three types of actuators commonly used for spacecraft attitude control: magnetic torquers, reaction wheels, and thrusters.

#### 5.1 Magnetic torquers

Magnetic torquers operate by creating a local magnetic field that interacts with the Earth's magnetic field. In simple terms, magnetic torques can be explained as that of a compass needle. By applying current through a coil, a local magnetic field is created, which will try to align itself with the Earth's magnetic field. This allows the attitude of a spacecraft to be changed and is a very popular approach for small satellites, e.g., cubesats. One of the drawbacks or challenges with magnetic actuation lies in the fact that the Earth's magnetic field goes from the North Pole to the South Pole as shown in Figure 2. <sup>1</sup> As can be seen, when the satellite crosses the North Pole, there will be mainly a downward magnetic field component, reducing the possibility of actuation to only two axes, and similarly along the equator. This subsection is based on [12] and will describe how to model magnetic torquers and how it can be applied for attitude control. A magnetic torquer produces a magnetic torque by applying a current through a coil, which can be expressed as [2].

#### Figure 2.

Magnetic field of the Earth visualized using the IGRF model. The control torque using magnetic torquers is always perpendicular to the magnetic field, such that a the poles, only roll, and pitch control are available, while at the equator, only pitch and yaw control is available [25].

<sup>1</sup> Figure created using the MATLAB script "international geomagnetic reference field (IGRF) model" by Drew Compston.

Modeling and Attitude Control of Satellites in Elliptical Orbits DOI: http://dx.doi.org/10.5772/intechopen.80422

where the terms have previously been defined. As can be seen from this equation, non-diagonal inertia matrices will induce gravitational torques to align the satellite with the gravity field. This is also sometimes used for passive control, using e.g., a gravity boom to ensure that one side of the satellite is always facing the Earth. For this chapter, the perturbing toque is set equal to the gravity torque such that

The control signal must be mapped to an actuator that must generate the desired torque. With limitations in actuation, the saturation must be modeled in order to obtain realistic results when simulating attitude control. This section considers three types of actuators commonly used for spacecraft attitude control: magnetic

Magnetic torquers operate by creating a local magnetic field that interacts with the Earth's magnetic field. In simple terms, magnetic torques can be explained as that of a compass needle. By applying current through a coil, a local magnetic field is created, which will try to align itself with the Earth's magnetic field. This allows the attitude of a spacecraft to be changed and is a very popular approach for small satellites, e.g., cubesats. One of the drawbacks or challenges with magnetic actuation lies in the fact that the Earth's magnetic field goes from the North Pole to the

North Pole, there will be mainly a downward magnetic field component, reducing the possibility of actuation to only two axes, and similarly along the equator. This subsection is based on [12] and will describe how to model magnetic torquers and how it can be applied for attitude control. A magnetic torquer produces a magnetic

torque by applying a current through a coil, which can be expressed as [2].

Magnetic field of the Earth visualized using the IGRF model. The control torque using magnetic torquers is always perpendicular to the magnetic field, such that a the poles, only roll, and pitch control are available,

Figure created using the MATLAB script "international geomagnetic reference field (IGRF) model" by

while at the equator, only pitch and yaw control is available [25].

<sup>1</sup> As can be seen, when the satellite crosses the

τb <sup>p</sup> <sup>¼</sup> <sup>τ</sup><sup>b</sup> g .

5. Actuators

Applied Modern Control

5.1 Magnetic torquers

South Pole as shown in Figure 2.

Figure 2.

Drew Compston.

1

80

torquers, reaction wheels, and thrusters.

$$
\boldsymbol{\sigma}\_{m}^{b} = \mathbf{S}(\mathbf{m}^{b})\mathbf{b}^{b},\tag{22}
$$

where <sup>m</sup><sup>b</sup> <sup>¼</sup> mx my mz <sup>Τ</sup> <sup>¼</sup> NA ix iy iz <sup>Τ</sup> is the induced magnetic field, N is the number of turns of the coils, ið Þ� is the current around a given axis, and A is the area of the coils. The Earth's magnetic field is represented through the vector b<sup>b</sup> , meaning that to use magnetic actuation for attitude control, a good model of the Earth's magnetic field is needed.

From a control point of view, the physical parameters A and N are defined when the spacecraft is designed, such that the controller needs to dictate which currents that must be sent to the torquers in order to get a desired torque. This means that Eq. (22) must be inverted with regard to m<sup>b</sup>, which is not straightforward due to the cross product, meaning that you obtain rank 2 when inverting the right-hand side, losing information about one of the axes. To that end, an approximation to inverting this equation is given in [25].

$$\mathbf{m}^b = \frac{\mathbf{S}\left(\mathbf{b}^b\right)\mathbf{r}\_d^b}{\left\|\mathbf{b}^b\right\|^2},\tag{23}$$

enabling the currents to be found as

$$\begin{bmatrix} \dot{i}\_{\mathbf{x}} & \dot{i}\_{\mathbf{y}} & \dot{i}\_{\mathbf{z}} \end{bmatrix}^{\mathrm{T}} = \frac{1}{\mathrm{NA}} \begin{bmatrix} m\_{\mathbf{x}} & m\_{\mathbf{y}} & m\_{\mathbf{z}} \end{bmatrix}^{\mathrm{T}}.\tag{24}$$

It is here assumed that all three torquers are identical, but depending on satellite configuration, there might be differences in the number of turns and areas. Hence, the desired torque τ<sup>b</sup> <sup>d</sup> can be used to find the magnetic moment m<sup>b</sup> in Eq. (23) and solved for the currents and applied resulting in the actuation torque in Eq. (22). Hence, the control law (Eq. 20) can be mapped to a desired magnetic moment (Eq. 23), which then can be used to find the desired current to each of the three coils. Then, the limits in current will dictate the maximum magnetic moment that can be generated.

Consider the HiNCube satellite as shown in Figure 3. The cubesat comprises three orthonormal magnetic torquers with an area <sup>A</sup> <sup>¼</sup> <sup>0</sup>:00757 m<sup>2</sup> and with a maximum current of 47.27 mA and N ¼ 100 turns. This gives a maximum magnetic moment of mmax <sup>¼</sup> <sup>0</sup>:03578 mA2. Hence, an implementation of using magnetic torquers for attitude control would encompass mapping the output from the control law to a desired magnetic moment using Eq. (23) and then imposing the maximum magnetic moment on each axis, before finding the resulting actuation torque using Eq. (22). Note that to ensure sign correctness due to the projection, the actuation torque can be found as

$$\begin{bmatrix} \tau\_{\mathbf{x},a} & \tau\_{\mathbf{y},a} & \tau\_{\mathbf{z},a} \end{bmatrix}^{\mathrm{T}} = \begin{bmatrix} \operatorname{sign}(\tau\_{\mathbf{x},d})\tau\_{\mathbf{x},m} & \operatorname{sign}(\tau\_{\mathbf{y},d})\tau\_{\mathbf{y},m} & \operatorname{sign}(\tau\_{\mathbf{z},d})\tau\_{\mathbf{z},m} \end{bmatrix}^{\mathrm{T}},\tag{25}$$

where τ<sup>b</sup> <sup>a</sup> ¼ τx,a τy,a τz,a <sup>Τ</sup> , τ<sup>b</sup> <sup>d</sup> ¼ τx,d τy,d τz,d <sup>Τ</sup> , and τb <sup>m</sup> ¼ τx,m τy,m τz,m <sup>Τ</sup> .

To show the performance of magnetic torquers, consider again the HiNCube satellite, which had an inertia matrix of <sup>J</sup> <sup>¼</sup> <sup>1</sup>:<sup>67</sup> � <sup>10</sup>�<sup>3</sup><sup>I</sup> kg m2. Consider the problem of making rotating 90° from an initial quaternion <sup>q</sup>o,b <sup>¼</sup> ½ � <sup>0001</sup> <sup>Τ</sup> to <sup>q</sup>o,d <sup>¼</sup> ½ � <sup>1000</sup> <sup>Τ</sup> . The gains for the PD+ controller are set kp <sup>¼</sup> <sup>1</sup> � <sup>10</sup>�<sup>5</sup> and

kd <sup>¼</sup> <sup>5</sup> � <sup>10</sup>�<sup>3</sup> , and the satellite is assumed to have an orbit of rp ¼ 500 km and ra <sup>¼</sup> 600 km, with inclination of 75° . Figure 4 shows the attitude, angular velocity, and actuation torque. It is evident that magnetic torquers produce very low torque, such that it takes a very long time to change the attitude of the spacecraft (about 1 h). To some extents, this can be improved by being in a lower orbit where the magnetic

field is stronger or by using larger coils with higher currents. Also, note that the actuation signal varies in strength as a function of time, depending on the orbital

Another way of changing the attitude of a satellite is through reaction wheels. Reaction wheels are based on the principle of Newton's third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. This means that by spinning a reaction wheel in one direction, the satellite will rotate in the other direction. Mounting three reaction wheels in an orthogonal configuration enables three-axis attitude control of spacecraft. From Newton's third law, the momentum generated by the reaction wheels will have opposite sign of the momentum of the satellite,

equation similarly as in Section 3, the torque generated by the reaction wheels can

<sup>w</sup> with <sup>h</sup><sup>b</sup>

<sup>w</sup> � <sup>S</sup> <sup>ω</sup><sup>b</sup>

set of three orthonormal reaction wheels, where one produces torques around the xaxis, one around the y-axis, and one around the z-axis of the body frame. Then, the

reaction wheels. To that end, the torque by the reaction wheel can be rewritten as

the angular momentum will be bounded as a function of maximum rotational speed. After imposing the torque and speed constraints, the angular momentum of the

Consider the HiNCube satellite again, where it is possible to use three small reaction wheels as described in [11] where the main idea is to place most of the mass away from the center as shown in Figure 5. The inertia of an individual reaction wheel was found to be Jw <sup>¼</sup> <sup>1</sup>:<sup>46</sup> � <sup>10</sup>�<sup>5</sup> kg m2, and by assuming a maximum rotation speed of 13,700 rpm with maximum torque of τmax ¼ 0:0047 Nm, the maximum

simulation as when using the magnetic torquers, where the gains for the PD+ controller is changed to kp ¼ kd ¼ 2 and the reaction wheels has the limits as defined above. Figure 6 shows the simulation results, where it is obvious that by using reaction wheels, the satellite is able to change its orientation after about 80 s. To some extent, this can be credited to the higher gains, but it lies mainly with the better actuation system that is able to produce higher torque than the reaction wheels. From the figure, the reaction wheels quickly go into saturation of 13, 700 RPM, where the angular velocity also goes into saturation. As the quaternion error goes toward zero, the reaction wheel despin, reducing the angular velocity and the

i,b h<sup>b</sup>

<sup>w</sup> is the torque generated by the reaction wheels. Now, consider a

<sup>w</sup> is the momentum production by the reaction wheels,

<sup>w</sup> where ω<sup>b</sup>

<sup>d</sup>, which shall be achieved by the

<sup>w</sup> allowing the actuation torque to be

<sup>60</sup> <sup>¼</sup> <sup>1</sup>:<sup>52389</sup> � <sup>10</sup>�6. Now, consider the same

<sup>w</sup> must be bounded by the motor torque limit, while

<sup>w</sup> <sup>¼</sup> <sup>J</sup>ww<sup>b</sup>

<sup>w</sup>. By employing Euler's moment

w, (26)

<sup>w</sup> is the angular

position of the satellite.

5.2 Reaction wheels

such that <sup>h</sup>\_ <sup>i</sup> ¼ �h\_ <sup>i</sup>

where τ<sup>b</sup>

<sup>d</sup> � <sup>S</sup> <sup>ω</sup><sup>b</sup>

calculated using Eq. (26).

τb <sup>w</sup> <sup>¼</sup> <sup>τ</sup><sup>b</sup>

83

be found by differentiating h<sup>i</sup>

<sup>w</sup> <sup>¼</sup> <sup>h</sup>\_ <sup>b</sup>

i,b <sup>h</sup><sup>b</sup>

<sup>w</sup> where <sup>h</sup>\_ <sup>i</sup>

PD+ control law dictates a desired torque, τ<sup>b</sup>

reaction wheels is found by integrating h\_ <sup>b</sup>

hmax <sup>¼</sup> Jwω<sup>w</sup> <sup>¼</sup> <sup>1</sup>:<sup>46</sup> � <sup>10</sup>�<sup>5</sup> � <sup>13700</sup> � <sup>2</sup><sup>π</sup>

control objective, is completed.

<sup>w</sup>, where τ<sup>b</sup>

momentum generated by the reaction wheels is found as

h\_ <sup>i</sup> is the momentum acting on the satellite, and τ<sup>b</sup>

Modeling and Attitude Control of Satellites in Elliptical Orbits

DOI: http://dx.doi.org/10.5772/intechopen.80422

<sup>w</sup> <sup>¼</sup> <sup>R</sup><sup>i</sup> bhb

τb <sup>a</sup> ¼ �h\_ <sup>b</sup>

velocity of the reaction wheels and J<sup>w</sup> denotes their inertia. This gives

Figure 3. Magnetic torquers on the HiNCube satellite (shown in brown).

field is stronger or by using larger coils with higher currents. Also, note that the actuation signal varies in strength as a function of time, depending on the orbital position of the satellite.

#### 5.2 Reaction wheels

kd <sup>¼</sup> <sup>5</sup> � <sup>10</sup>�<sup>3</sup>

Applied Modern Control

Figure 3.

Figure 4.

82

ra <sup>¼</sup> 600 km, with inclination of 75°

Magnetic torquers on the HiNCube satellite (shown in brown).

Quaternion error, angular velocity error, and actuation torque using magnetic torquers.

, and the satellite is assumed to have an orbit of rp ¼ 500 km and

and actuation torque. It is evident that magnetic torquers produce very low torque, such that it takes a very long time to change the attitude of the spacecraft (about 1 h). To some extents, this can be improved by being in a lower orbit where the magnetic

. Figure 4 shows the attitude, angular velocity,

Another way of changing the attitude of a satellite is through reaction wheels. Reaction wheels are based on the principle of Newton's third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. This means that by spinning a reaction wheel in one direction, the satellite will rotate in the other direction. Mounting three reaction wheels in an orthogonal configuration enables three-axis attitude control of spacecraft. From Newton's third law, the momentum generated by the reaction wheels will have opposite sign of the momentum of the satellite, such that <sup>h</sup>\_ <sup>i</sup> ¼ �h\_ <sup>i</sup> <sup>w</sup> where <sup>h</sup>\_ <sup>i</sup> <sup>w</sup> is the momentum production by the reaction wheels, h\_ <sup>i</sup> is the momentum acting on the satellite, and τ<sup>b</sup> <sup>w</sup>. By employing Euler's moment equation similarly as in Section 3, the torque generated by the reaction wheels can be found by differentiating h<sup>i</sup> <sup>w</sup> <sup>¼</sup> <sup>R</sup><sup>i</sup> bhb <sup>w</sup> with <sup>h</sup><sup>b</sup> <sup>w</sup> <sup>¼</sup> <sup>J</sup>ww<sup>b</sup> <sup>w</sup> where ω<sup>b</sup> <sup>w</sup> is the angular velocity of the reaction wheels and J<sup>w</sup> denotes their inertia. This gives

$$\mathbf{r}\_a^b = -\dot{\mathbf{h}}\_w^b - \mathbf{S}(a\_{\dot{i},b}^b)\mathbf{h}\_{w^b}^b \tag{26}$$

where τ<sup>b</sup> <sup>w</sup> <sup>¼</sup> <sup>h</sup>\_ <sup>b</sup> <sup>w</sup> is the torque generated by the reaction wheels. Now, consider a set of three orthonormal reaction wheels, where one produces torques around the xaxis, one around the y-axis, and one around the z-axis of the body frame. Then, the PD+ control law dictates a desired torque, τ<sup>b</sup> <sup>d</sup>, which shall be achieved by the reaction wheels. To that end, the torque by the reaction wheel can be rewritten as τb <sup>w</sup> <sup>¼</sup> <sup>τ</sup><sup>b</sup> <sup>d</sup> � <sup>S</sup> <sup>ω</sup><sup>b</sup> i,b <sup>h</sup><sup>b</sup> <sup>w</sup>, where τ<sup>b</sup> <sup>w</sup> must be bounded by the motor torque limit, while the angular momentum will be bounded as a function of maximum rotational speed. After imposing the torque and speed constraints, the angular momentum of the reaction wheels is found by integrating h\_ <sup>b</sup> <sup>w</sup> allowing the actuation torque to be calculated using Eq. (26).

Consider the HiNCube satellite again, where it is possible to use three small reaction wheels as described in [11] where the main idea is to place most of the mass away from the center as shown in Figure 5. The inertia of an individual reaction wheel was found to be Jw <sup>¼</sup> <sup>1</sup>:<sup>46</sup> � <sup>10</sup>�<sup>5</sup> kg m2, and by assuming a maximum rotation speed of 13,700 rpm with maximum torque of τmax ¼ 0:0047 Nm, the maximum momentum generated by the reaction wheels is found as

hmax <sup>¼</sup> Jwω<sup>w</sup> <sup>¼</sup> <sup>1</sup>:<sup>46</sup> � <sup>10</sup>�<sup>5</sup> � <sup>13700</sup> � <sup>2</sup><sup>π</sup> <sup>60</sup> <sup>¼</sup> <sup>1</sup>:<sup>52389</sup> � <sup>10</sup>�6. Now, consider the same simulation as when using the magnetic torquers, where the gains for the PD+ controller is changed to kp ¼ kd ¼ 2 and the reaction wheels has the limits as defined above. Figure 6 shows the simulation results, where it is obvious that by using reaction wheels, the satellite is able to change its orientation after about 80 s. To some extent, this can be credited to the higher gains, but it lies mainly with the better actuation system that is able to produce higher torque than the reaction wheels. From the figure, the reaction wheels quickly go into saturation of 13, 700 RPM, where the angular velocity also goes into saturation. As the quaternion error goes toward zero, the reaction wheel despin, reducing the angular velocity and the control objective, is completed.

rb

<sup>i</sup> ¼ rx ry rz � �<sup>Τ</sup>

u ¼ f <sup>1</sup> f <sup>2</sup> f <sup>3</sup> f <sup>4</sup> � �<sup>Τ</sup>

τb <sup>i</sup> <sup>¼</sup> <sup>r</sup><sup>b</sup> <sup>i</sup> � f b i ¼

tude of each of the thrusters.

above zero, such that

actuator torque to be found as τ<sup>b</sup>

Table 2.

85

Thruster configuration.

, and let them have an azimuth and an elevation angle described

<sup>a</sup> ¼ Bu with

: (28)

<sup>d</sup> where † denotes

(29)

<sup>Τ</sup> denotes the magni-

fi , (27)

ry sin ð Þ χ cosð Þ� γ rz sin ð Þγ rz cosð Þγ cosð Þ� χ rx cosð Þγ sin ð Þ χ rx sin ð Þ� γ ry cosð Þγ cosð Þ χ

> ffiffi 2 p

> ffiffi 2 p

> ffiffi 2 p 4

5 �

4 �

ffiffi 2 p 5

ffiffi 2 p 4

ffiffi 2 p 4

by χ and γ. Then, the torque produced by a given thruster can be found as [1], p. 262

where fi denotes the total thrust from the ith thruster. Given the thruster configuration defined in Table 2, let the vector of thruster signals be denoted

, and then the torque can be found as τ<sup>b</sup>

ffiffi 2 p 5

ffiffi 2 p 4

ffiffi 2 p 4

Given a desired torque from the PD+ control law, it must be mapped to the desired thruster firings, such that the combination of thrusters produces the desired torque. To that end, there are several different modulation methods that can be applied, ranging from a simple bang-bang modulation to more sophisticated pulsewidth pulse-frequency modulation. This section will give an introduction to the different methods and detail how they can be implemented. In general the desired

1. Bang-bang modulation: The easiest approach to thruster firings is bang-bang modulation, where the thruster is fully actuated as long as the ith signal of u<sup>d</sup> is

fi <sup>¼</sup> <sup>f</sup> max if ui>0

where f max denotes the maximum available thrust from the ith thruster. After applying bang-bang modulation, the vector u can be constructed allowing the

Thruster Elevation (γ) Azimuth (χ) rx ry rz f <sup>1</sup> 45 90 �0.5 �0.45 �0.05 f <sup>2</sup> 135 90 �0.5 �0.45 0.05 f <sup>3</sup> �45 90 �0.5 0.45 �0.05 f <sup>4</sup> �135 90 �0.5 0.45 0.05

�

<sup>a</sup> ¼ Bu.

<sup>0</sup> if ui≤<sup>0</sup> ,

Modeling and Attitude Control of Satellites in Elliptical Orbits

DOI: http://dx.doi.org/10.5772/intechopen.80422

B ¼

�

�

torque can be mapped to the desired thruster firings as <sup>u</sup><sup>d</sup> <sup>¼</sup> <sup>B</sup>†τ<sup>b</sup>

the Moore-Penrose pseudoinverse and u<sup>d</sup> ¼ ½ � u<sup>1</sup> u<sup>2</sup> u<sup>3</sup> u<sup>4</sup>

ffiffi 2 p 5

ffiffi 2 p

4 �

4 �

ffiffi 2 p

Figure 5. Example design of a reaction wheel for cubesats (dimensions are in mm) [11].

Figure 6. Quaternion error, angular velocity error, and wheel speeds when using reaction wheels for attitude control.

#### 5.3 Thrusters

The third kind of actuator that will be studied is using reaction control thrusters. This section presents how to map the control signal (Eq. 20) to four thrusters used for attitude control. Let the location of each thruster be denoted by

rb <sup>i</sup> ¼ rx ry rz � �<sup>Τ</sup> , and let them have an azimuth and an elevation angle described by χ and γ. Then, the torque produced by a given thruster can be found as [1], p. 262

$$\mathbf{r}\_i^b = \mathbf{r}\_i^b \times \mathbf{f}\_i^b = \begin{bmatrix} r\_\mathcal{\mathcal{Y}} \sin\left(\boldsymbol{\chi}\right) \cos\left(\boldsymbol{\chi}\right) - r\_\mathcal{x} \sin\left(\boldsymbol{\chi}\right) \\\\ r\_\mathcal{\mathcal{x}} \cos\left(\boldsymbol{\chi}\right) \cos\left(\boldsymbol{\chi}\right) - r\_\mathcal{x} \cos\left(\boldsymbol{\chi}\right) \sin\left(\boldsymbol{\chi}\right) \\\\ r\_\mathcal{\mathcal{x}} \sin\left(\boldsymbol{\chi}\right) - r\_\mathcal{\mathcal{Y}} \cos\left(\boldsymbol{\chi}\right) \cos\left(\boldsymbol{\chi}\right) \end{bmatrix} \mathbf{f}\_i,\tag{27}$$

where fi denotes the total thrust from the ith thruster. Given the thruster configuration defined in Table 2, let the vector of thruster signals be denoted u ¼ f <sup>1</sup> f <sup>2</sup> f <sup>3</sup> f <sup>4</sup> � �<sup>Τ</sup> , and then the torque can be found as τ<sup>b</sup> <sup>a</sup> ¼ Bu with

$$\mathbf{B} = \begin{bmatrix} -\frac{\sqrt{2}}{5} & \frac{\sqrt{2}}{5} & \frac{\sqrt{2}}{5} & -\frac{\sqrt{2}}{5} \\ \frac{\sqrt{2}}{4} & -\frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{4} & -\frac{\sqrt{2}}{4} \\ -\frac{\sqrt{2}}{4} & -\frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{4} \end{bmatrix} . \tag{28}$$

Given a desired torque from the PD+ control law, it must be mapped to the desired thruster firings, such that the combination of thrusters produces the desired torque. To that end, there are several different modulation methods that can be applied, ranging from a simple bang-bang modulation to more sophisticated pulsewidth pulse-frequency modulation. This section will give an introduction to the different methods and detail how they can be implemented. In general the desired torque can be mapped to the desired thruster firings as <sup>u</sup><sup>d</sup> <sup>¼</sup> <sup>B</sup>†τ<sup>b</sup> <sup>d</sup> where † denotes the Moore-Penrose pseudoinverse and u<sup>d</sup> ¼ ½ � u<sup>1</sup> u<sup>2</sup> u<sup>3</sup> u<sup>4</sup> <sup>Τ</sup> denotes the magnitude of each of the thrusters.

1. Bang-bang modulation: The easiest approach to thruster firings is bang-bang modulation, where the thruster is fully actuated as long as the ith signal of u<sup>d</sup> is above zero, such that

$$f\_i = \begin{cases} f\_{\text{max}} & \text{if } \quad u\_i > 0\\ 0 & \text{if } \quad u\_i \le 0 \end{cases} \tag{29}$$

where f max denotes the maximum available thrust from the ith thruster. After applying bang-bang modulation, the vector u can be constructed allowing the actuator torque to be found as τ<sup>b</sup> <sup>a</sup> ¼ Bu.


Table 2. Thruster configuration.

5.3 Thrusters

Figure 6.

84

Figure 5.

Applied Modern Control

The third kind of actuator that will be studied is using reaction control thrusters. This section presents how to map the control signal (Eq. 20) to four thrusters used

Quaternion error, angular velocity error, and wheel speeds when using reaction wheels for attitude control.

for attitude control. Let the location of each thruster be denoted by

Example design of a reaction wheel for cubesats (dimensions are in mm) [11].

2. Bang-bang modulation with dead zone: One of the major drawbacks by using simple bang-bang modulation is when the tracking error has converged to zero, where the thruster firings will continue to maintain the desired attitude. Sensor noise is another source that leads to continuous firings, quickly spending all the propellant. To that end, bang-bang modulation can be augmented with a dead zone, giving

$$f\_i = \begin{cases} f\_{\text{max}} & \text{if } \quad u\_i > D \\ 0 & \text{if } \quad u\_i \le D \end{cases} \tag{30}$$

Figure 8.

Figure 9.

Figure 10.

87

Pulse-width pulse-frequency modulation.

Achieving pulse-width modulated signals for thruster firings.

Modeling and Attitude Control of Satellites in Elliptical Orbits

DOI: http://dx.doi.org/10.5772/intechopen.80422

Attitude control using thrusters with bang-bang modulation.

where D>0 denotes the dead zone. By properly selecting a suitable dead zone enables the thrusters to avoid firing when close to the equilibrium point.


#### Figure 7.

Thruster configuration. The left subfigure shows the definition of azimuth and elevation angles used to dictate the orientation of the thruster, while the right subfigure shows a satellite with thrusters placed and oriented as given in Table 2.

Modeling and Attitude Control of Satellites in Elliptical Orbits DOI: http://dx.doi.org/10.5772/intechopen.80422

2. Bang-bang modulation with dead zone: One of the major drawbacks by using simple bang-bang modulation is when the tracking error has converged to zero, where the thruster firings will continue to maintain the desired attitude.

Sensor noise is another source that leads to continuous firings, quickly spending all the propellant. To that end, bang-bang modulation can be

fi <sup>¼</sup> <sup>f</sup> max if ui><sup>D</sup>

where D>0 denotes the dead zone. By properly selecting a suitable dead zone

3. Pulse-width modulation: Another approach that is often used for thruster firings is by using pulse-width modulation (PWM), where an analogue signal (desired torque) can be mapped to discrete signals using PWM. Instead of changing the thrust level, the duration of the pulses can be changed, leading to a pulse that is proportional to the torque command from the PD+ controller. A simple way of achieving this is by using the intersective technique, which uses a sawtooth signal that is compared to the control signal. When the sawtooth is less than the control signal, the PWM signal is in a high state and otherwise in a low state. This makes it possible to go from continuous control signal to a discrete representation which can be used for thruster firings. Figure 7 shows how to achieve the PWM signal, enabled through a simple comparison of the two

4.Pulse-width pulse-frequency modulation: In addition to controlling the width of the pulse as in PWM, it is also possible to control the frequency of the pulse—something that is done through pulse-width pulse-frequency (PWPF) modulation ([1], p. 265) (Figure 8). The modulation approach comprises a lag filter and a Schmitt trigger as shown in Figure 9. As long as the input to the Schmitt trigger is below Uon, the output is kept at zero and must be larger than

<sup>K</sup> to produce an output, where K is a DC gain, τ is the time constant, Uon and Uoff are the on and off limits for the Schmitt trigger, while Um is the maximum output. Much research has been performed on improving PWPF modulation, and in [26], the authors propose the following settings (cf. Figure 9): 2 <K < 6,

Thruster configuration. The left subfigure shows the definition of azimuth and elevation angles used to dictate the orientation of the thruster, while the right subfigure shows a satellite with thrusters placed and oriented as

0:1 < τ < 0:5, Uon>0:3, Uoff < 0:8Uon, and Um ¼ 1.

enables the thrusters to avoid firing when close to the equilibrium point.

<sup>0</sup> if ui≤<sup>D</sup> ,

(30)

augmented with a dead zone, giving

Applied Modern Control

signals.

Uon

Figure 7.

86

given in Table 2.

Figure 8. Achieving pulse-width modulated signals for thruster firings.

Figure 9. Pulse-width pulse-frequency modulation.

Figure 10. Attitude control using thrusters with bang-bang modulation.

5. Simulations of thruster modulations: Consider a satellite with an inertia matrix as

$$\mathbf{J} = \begin{bmatrix} \mathbf{0.5} & -\mathbf{0.2} & -\mathbf{0.1} \\ -\mathbf{0.2} & \mathbf{0.5} & -\mathbf{0.2} \\ -\mathbf{0.1} & -\mathbf{0.2} & \mathbf{0.5} \end{bmatrix},\tag{31}$$

where the objective is to perform a yaw maneuver of 90° using four thrusters with 0:1 N force, with a specific impulse of 200 s. Figure 10 shows the attitude and angular velocity vectors when using bang-bang modulation, where the satellite is able to make the errors go to zero. However, due to the modulation, the thrusters will continue firing as shown in Figure 11. To that end, consider the bang-bang modulation with dead zone. Let the dead zone be chosen as D ¼ 0:05, and then the satellite obtains an accuracy as shown in Figure 12 where there is a small

Modeling and Attitude Control of Satellites in Elliptical Orbits

DOI: http://dx.doi.org/10.5772/intechopen.80422

Figure 13.

Figure 14.

89

Thruster firings when using bang-bang modulation with dead zone.

Attitude control using thrusters with PWM modulation.

Figure 11. Thruster firings when using bang-bang modulation.

Figure 12. Attitude control using thrusters with bang-bang modulation with dead zone.

5. Simulations of thruster modulations: Consider a satellite with an inertia matrix as

0:5 �0:2 �0:1 �0:2 0:5 �0:2 �0:1 �0:2 0:5

3 7

<sup>5</sup>, (31)

J ¼

Applied Modern Control

Figure 11.

Figure 12.

88

Thruster firings when using bang-bang modulation.

Attitude control using thrusters with bang-bang modulation with dead zone.

2 6 4 where the objective is to perform a yaw maneuver of 90° using four thrusters with 0:1 N force, with a specific impulse of 200 s. Figure 10 shows the attitude and angular velocity vectors when using bang-bang modulation, where the satellite is able to make the errors go to zero. However, due to the modulation, the thrusters will continue firing as shown in Figure 11. To that end, consider the bang-bang modulation with dead zone. Let the dead zone be chosen as D ¼ 0:05, and then the satellite obtains an accuracy as shown in Figure 12 where there is a small

Figure 13. Thruster firings when using bang-bang modulation with dead zone.

Figure 14. Attitude control using thrusters with PWM modulation.

offset from the origin which is proportional to the dead zone. On the other hand, the thruster firings are much less prone to do continuous firings as shown in Figure 13.

The final scenario is using PWPF modulation, where the parameters are chosen as K ¼ 3, τ ¼ 0:2, Uon ¼ 0:35, and Uoff ¼ 0:28. Figure 15 shows the attitude and angular velocity, which go close to zero, while the thruster firings are shown in Figure 16, which is able to constrain the amount of thruster firings, and therefore

For satellite control using thrusters, propellant is a critical resource that must not be wasted. To that end it is desirable to limit the amount of propellant while at the

same time obtain good pointing accuracy (Figure 17). With the basis in

Modeling and Attitude Control of Satellites in Elliptical Orbits

DOI: http://dx.doi.org/10.5772/intechopen.80422

propellant.

Figure 17.

Figure 18.

91

Thruster firings when using PWPF modulation.

Propellant consumption of the different modulation methods.

Now, consider pulse-width modulation. Let the sawtooth signal have an amplitude of 1 and a frequency of 1 Hz. Then, the attitude and angular velocity error is obtained as shown in Figure 14, while the thruster firings are shown in Figure 15. It is possible to tune on sawtooth frequency to improve the performance.

Figure 15. Thruster firings when using PWM modulation.

Figure 16. Attitude control using thrusters with PWPF modulation.

Modeling and Attitude Control of Satellites in Elliptical Orbits DOI: http://dx.doi.org/10.5772/intechopen.80422

offset from the origin which is proportional to the dead zone. On the other hand, the thruster firings are much less prone to do continuous firings as shown in

is possible to tune on sawtooth frequency to improve the performance.

Now, consider pulse-width modulation. Let the sawtooth signal have an amplitude of 1 and a frequency of 1 Hz. Then, the attitude and angular velocity error is obtained as shown in Figure 14, while the thruster firings are shown in Figure 15. It

Figure 13.

Applied Modern Control

Figure 15.

Figure 16.

90

Thruster firings when using PWM modulation.

Attitude control using thrusters with PWPF modulation.

The final scenario is using PWPF modulation, where the parameters are chosen as K ¼ 3, τ ¼ 0:2, Uon ¼ 0:35, and Uoff ¼ 0:28. Figure 15 shows the attitude and angular velocity, which go close to zero, while the thruster firings are shown in Figure 16, which is able to constrain the amount of thruster firings, and therefore propellant.

For satellite control using thrusters, propellant is a critical resource that must not be wasted. To that end it is desirable to limit the amount of propellant while at the same time obtain good pointing accuracy (Figure 17). With the basis in

Figure 17. Thruster firings when using PWPF modulation.

Figure 18. Propellant consumption of the different modulation methods.

Tsiolkovsky's rocket equation, the propellant consumption during thruster firings can be found as mpropellant <sup>¼</sup> <sup>Ð</sup><sup>t</sup> 0 f Ispg dt where f is the force from one of the thrusters and Isp is the specific impulse, while <sup>g</sup> <sup>¼</sup> <sup>9</sup>:81 m/s<sup>2</sup> is the acceleration due to gravity. Figure 18 shows a comparison between the different modulation methods, where it is evident that the PWPF method allows for the least amount of propellant while obtaining close to acceptable performance. The bang-bang modulation will continue spending propellant until running out of fuel but on the other hand obtains the best tracking performance.

<sup>s</sup><sup>o</sup> <sup>¼</sup> <sup>R</sup><sup>o</sup> i

Modeling and Attitude Control of Satellites in Elliptical Orbits

DOI: http://dx.doi.org/10.5772/intechopen.80422

ε, while the Sun vector in orbit frame is denoted by s<sup>o</sup>.

6.2 Magnetic field model

<sup>m</sup><sup>o</sup> <sup>¼</sup> <sup>μ</sup><sup>f</sup>

using Python.<sup>3</sup>

2

3

93

https://github.com/JDeeth/MagDec

https://github.com/scivision/pyigrf12

magnetic field vector in orbit frame as

<sup>μ</sup><sup>f</sup> <sup>¼</sup> <sup>7</sup>:<sup>9</sup> � <sup>10</sup>�<sup>15</sup> Wb-m, while <sup>ω</sup><sup>0</sup> ¼ kω<sup>i</sup>

6.3 Attitude determination using the Madgwick filter

2 6 4

cos λecliptic � � cosð Þ<sup>ε</sup> sin <sup>λ</sup>ecliptic � � sin ð Þ<sup>ε</sup> sin <sup>λ</sup>ecliptic � �

Here, the number of Julian centuries is denoted by TUT1, the mean longitude of the Sun is denoted by λM<sup>⊙</sup> , the mean anomaly for the Sun is denoted by M⊙, the ecliptic longitude is denoted by λecliptic, and the obliquity of the ecliptic is denoted by

Several different geomagnetic models can be applied for attitude determination in conjunction with a magnetometer. The most basic are simple dipole models [27], while more advanced are, e.g., the chaos model or the 12th generation IGRF model [28], which is the most commonly used model for attitude determination. This section presents the simple dipole model by [27], which can be described by the

<sup>a</sup><sup>3</sup> ½ � cosð Þ <sup>ω</sup>0<sup>t</sup> sin ðÞ � <sup>i</sup> cosð Þ<sup>i</sup> 2 sin ð Þ <sup>ω</sup>0<sup>t</sup> sin ð Þ<sup>i</sup> <sup>Τ</sup>

to the geomagnetic equator is denoted by t and the dipole strength is denoted

where the time measured from passing the ascending node of the orbit relative

For a real application, the reader is recommended to apply the IGRF model, which is available in Simulink inside the Aerospace Toolbox, as C++ implementation<sup>2</sup> or

The objective of attitude determination is to find what direction the satellite is

mathematical models that can be compared and used to find the attitude. There are several different kinds of filters applied for attitude estimation, such as the Triad method [29], the Kalman filter [30], or the Mahony filter [31]. The Madgwick filter by [13] has shown good results in attitude estimation based on IMU measurements and is commonly applied in drone applications. The main idea by the filter is to use gradient descent in combination with the complementary filter to fuse sensor data together to produce an estimated quaternion. This section presents an application of the Madgwick filter by using measurements of the Sun vector and the magnetic field vector as well as the acceleration vector (gravity) and shows how to fuse that

pointing. In its core, it mainly requires two vector measurements and two

data together to estimate the attitude of a satellite in an elliptical orbit. Let the quaternion estimate be denoted by q^ ¼ q<sup>1</sup> q<sup>2</sup> q<sup>3</sup> q<sup>4</sup>

the previous estimate is denoted by k � 1. Let the objective function be

sured acceleration, Sun vector, and magnetic field vectors can be defined, respectively, as a<sup>b</sup>, s<sup>b</sup>, and m<sup>b</sup> and can be combined with the mathematical models of the acceleration, Sun vector, and magnetic field vector given in Eqs. (6), (38), and (39) to estimate the attitude. Here, the current estimate is denoted by subscript k, while

3 7

<sup>5</sup>: (38)

, (39)

i, <sup>o</sup>k denotes the angular speed of the orbit.

� �<sup>Τ</sup>

. The mea-

#### 6. Attitude determination

As a preliminary step before trying to estimate the attitude of the satellite, some knowledge of measurement vectors must be known, i.e., what is the direction toward the Sun and how does the magnetic field vector look like at a given position. There are several other quantities that can be measured to obtain the attitude, where star trackers are known to be the most accurate. For the reader to obtain insight into using multiple measurements and combine it to find the attitude, this work presents a Sun vector model and a simplified magnetic field model that can be used for simulation purposes.

#### 6.1 Sun vector model

To find the direction toward the Sun, there are several models that can be applied. The simplest would be to divide a circle into 365 days and have a vector always point toward the Sun. Then, by knowing which day it is, it is straightforward to find the direction toward the Sun. This approach would be coarse, such that more accurate models exist. For example, the Sun vector model in [3], pp. 281–282, has an accuracy of 0:01<sup>∘</sup> and is valid until 2050. First, the time and date must be converted into the Julian date as [3], p. 189.

$$\begin{split} \text{JD} &= \text{367}(yr) - \text{INT} \left( \frac{\Im\left(yr + \text{INT} \left(\frac{mo+9}{12}\right)\right)}{4} \right) + \text{INT} \left(\frac{275mo}{9}\right) + d \\ &+ 1,721,013.5 + \frac{\left(\frac{\epsilon}{60} + \min\right)}{24} + h \end{split} \tag{32}$$

where a real truncation is denoted by INTðÞ and the year, month, day, hour, minute, and second are denoted by yr, mo, d, h, min, s. If the day contains a leap second, 61 s should be used instead of 60<sup>∗</sup> . This gives the Sun vector model as

$$T\_{UT1} = \frac{JD - 2,451,545.0}{36,525},\tag{33}$$

$$
\lambda\_{M\_{\odot}} = 280.460^{\circ} + 36,000.771T\_{UT3} \tag{34}
$$

$$M\_{\odot} = \textbf{357.5277233}^{\*} + \textbf{35,999.05034}T\_{UT1} \tag{35}$$

$$\begin{split} \lambda\_{\text{elliptic}} &= \lambda\_{M\_{\odot}} + 1.914666471^{\circ} \sin \left( M\_{\odot} \right) \\ &+ 0.019994643 \sin \left( 2M \right) \end{split} \tag{36}$$

$$
\varepsilon = 23.439291^\circ - 0.0130042T\_{UT1} \tag{37}
$$

Modeling and Attitude Control of Satellites in Elliptical Orbits DOI: http://dx.doi.org/10.5772/intechopen.80422

$$\mathbf{s}^{\boldsymbol{\sigma}} = \mathbf{R}\_{i}^{\boldsymbol{\sigma}} \begin{bmatrix} \cos \left( \lambda\_{elliptic} \right) \\ \cos \left( \boldsymbol{\varepsilon} \right) \sin \left( \lambda\_{elliptic} \right) \\ \sin \left( \boldsymbol{\varepsilon} \right) \sin \left( \lambda\_{elliptic} \right) \end{bmatrix}. \tag{38}$$

Here, the number of Julian centuries is denoted by TUT1, the mean longitude of the Sun is denoted by λM<sup>⊙</sup> , the mean anomaly for the Sun is denoted by M⊙, the ecliptic longitude is denoted by λecliptic, and the obliquity of the ecliptic is denoted by ε, while the Sun vector in orbit frame is denoted by s<sup>o</sup>.

#### 6.2 Magnetic field model

Tsiolkovsky's rocket equation, the propellant consumption during thruster firings

and Isp is the specific impulse, while <sup>g</sup> <sup>¼</sup> <sup>9</sup>:81 m/s<sup>2</sup> is the acceleration due to gravity. Figure 18 shows a comparison between the different modulation methods, where it is evident that the PWPF method allows for the least amount of propellant while obtaining close to acceptable performance. The bang-bang modulation will continue spending propellant until running out of fuel but on the other hand obtains the best

As a preliminary step before trying to estimate the attitude of the satellite, some

knowledge of measurement vectors must be known, i.e., what is the direction toward the Sun and how does the magnetic field vector look like at a given position. There are several other quantities that can be measured to obtain the attitude, where star trackers are known to be the most accurate. For the reader to obtain insight into using multiple measurements and combine it to find the attitude, this work presents a Sun vector model and a simplified magnetic field model that can be

To find the direction toward the Sun, there are several models that can be applied. The simplest would be to divide a circle into 365 days and have a vector always point toward the Sun. Then, by knowing which day it is, it is straightforward to find the direction toward the Sun. This approach would be coarse, such that more accurate models exist. For example, the Sun vector model in [3], pp. 281–282, has an accuracy of 0:01<sup>∘</sup> and is valid until 2050. First, the time and date must be converted

<sup>7</sup> yr <sup>þ</sup> INT moþ<sup>9</sup>

s <sup>60</sup> ð Þ <sup>∗</sup>þmin <sup>60</sup> þ h <sup>24</sup> ,

� � � � 4 � �

12

where a real truncation is denoted by INTðÞ and the year, month, day, hour, minute, and second are denoted by yr, mo, d, h, min, s. If the day contains a leap

TUT<sup>1</sup> <sup>¼</sup> JD � <sup>2</sup>; <sup>451</sup>; <sup>545</sup>:<sup>0</sup>

<sup>λ</sup>ecliptic <sup>¼</sup> <sup>λ</sup><sup>M</sup><sup>⊙</sup> <sup>þ</sup> <sup>1</sup>:914666471<sup>∘</sup> sin ð Þ <sup>M</sup><sup>⊙</sup>

<sup>þ</sup> INT <sup>275</sup>mo 9 � �

. This gives the Sun vector model as

<sup>36</sup>; <sup>525</sup> , (33)

<sup>λ</sup><sup>M</sup><sup>⊙</sup> <sup>¼</sup> <sup>280</sup>:460<sup>∘</sup> <sup>þ</sup> <sup>36</sup>; <sup>000</sup>:771TUT1, (34) <sup>M</sup><sup>⊙</sup> <sup>¼</sup> <sup>357</sup>:5277233<sup>∘</sup> <sup>þ</sup> <sup>35</sup>; <sup>999</sup>:05034TUT1, (35)

<sup>þ</sup> <sup>0</sup>:019994643 sin 2ð Þ <sup>M</sup> , (36)

<sup>ε</sup> <sup>¼</sup> <sup>23</sup>:439291<sup>∘</sup> � <sup>0</sup>:0130042TUT1, (37)

þ d

(32)

dt where f is the force from one of the thrusters

0 f Ispg

can be found as mpropellant <sup>¼</sup> <sup>Ð</sup><sup>t</sup>

6. Attitude determination

used for simulation purposes.

into the Julian date as [3], p. 189.

JD ¼ 367ð Þ� yr INT

þ 1; 721; 013:5 þ

second, 61 s should be used instead of 60<sup>∗</sup>

92

6.1 Sun vector model

tracking performance.

Applied Modern Control

Several different geomagnetic models can be applied for attitude determination in conjunction with a magnetometer. The most basic are simple dipole models [27], while more advanced are, e.g., the chaos model or the 12th generation IGRF model [28], which is the most commonly used model for attitude determination. This section presents the simple dipole model by [27], which can be described by the magnetic field vector in orbit frame as

$$\mathbf{m}^{\rho} = \frac{\mu\_f}{a^3} \begin{bmatrix} \cos \left( a \nu\_0 t \right) \sin \left( i \right) & -\cos \left( i \right) & 2 \sin \left( a \nu\_0 t \right) \sin \left( i \right) \end{bmatrix}^{\mathrm{T}},\tag{39}$$

where the time measured from passing the ascending node of the orbit relative to the geomagnetic equator is denoted by t and the dipole strength is denoted <sup>μ</sup><sup>f</sup> <sup>¼</sup> <sup>7</sup>:<sup>9</sup> � <sup>10</sup>�<sup>15</sup> Wb-m, while <sup>ω</sup><sup>0</sup> ¼ kω<sup>i</sup> i, <sup>o</sup>k denotes the angular speed of the orbit. For a real application, the reader is recommended to apply the IGRF model, which is available in Simulink inside the Aerospace Toolbox, as C++ implementation<sup>2</sup> or using Python.<sup>3</sup>

#### 6.3 Attitude determination using the Madgwick filter

The objective of attitude determination is to find what direction the satellite is pointing. In its core, it mainly requires two vector measurements and two mathematical models that can be compared and used to find the attitude. There are several different kinds of filters applied for attitude estimation, such as the Triad method [29], the Kalman filter [30], or the Mahony filter [31]. The Madgwick filter by [13] has shown good results in attitude estimation based on IMU measurements and is commonly applied in drone applications. The main idea by the filter is to use gradient descent in combination with the complementary filter to fuse sensor data together to produce an estimated quaternion. This section presents an application of the Madgwick filter by using measurements of the Sun vector and the magnetic field vector as well as the acceleration vector (gravity) and shows how to fuse that data together to estimate the attitude of a satellite in an elliptical orbit.

Let the quaternion estimate be denoted by q^ ¼ q<sup>1</sup> q<sup>2</sup> q<sup>3</sup> q<sup>4</sup> � �<sup>Τ</sup> . The measured acceleration, Sun vector, and magnetic field vectors can be defined, respectively, as a<sup>b</sup>, s<sup>b</sup>, and m<sup>b</sup> and can be combined with the mathematical models of the acceleration, Sun vector, and magnetic field vector given in Eqs. (6), (38), and (39) to estimate the attitude. Here, the current estimate is denoted by subscript k, while the previous estimate is denoted by k � 1. Let the objective function be

<sup>2</sup> https://github.com/JDeeth/MagDec

<sup>3</sup> https://github.com/scivision/pyigrf12

Applied Modern Control

$$\mathcal{G} = \begin{bmatrix} \mathbf{f}(\hat{\mathbf{q}}\_{k-1}, \mathbf{a}^o, \mathbf{a}^b) \\ \mathbf{f}(\hat{\mathbf{q}}\_{k-1}, \mathbf{s}^o, \mathbf{s}^b) \\ \mathbf{f}(\hat{\mathbf{q}}\_{k-1}, \mathbf{m}^o, \mathbf{m}^b) \end{bmatrix}, \tag{40}$$

where the objective is found in an estimated quaternion that minimizes this function, something that can be achieved by using gradient descent. The Jacobian matrix can be found as

$$\mathcal{J} = \begin{bmatrix} \mathbf{J}\_q(\hat{\mathbf{q}}\_{k-1}, \mathbf{a}^o) \\ \mathbf{J}\_q(\hat{\mathbf{q}}\_{k-1}, \mathbf{s}^o) \\ \mathbf{J}\_q(\hat{\mathbf{q}}\_{k-1}, \mathbf{m}^o) \end{bmatrix} \tag{41}$$

<sup>q</sup>^<sup>k</sup> <sup>¼</sup> <sup>q</sup>^<sup>k</sup> kq^kk

velocity based on vector measurements is denoted ω^ <sup>b</sup>

Modeling and Attitude Control of Satellites in Elliptical Orbits

DOI: http://dx.doi.org/10.5772/intechopen.80422

Attitude and angular velocity during the maneuver [9].

bias is denoted by ω<sup>b</sup>

of ω^ <sup>b</sup> <sup>k</sup>, ω<sup>b</sup>

95

Figure 19.

6.4 Simulation

to the orbit frame is denoted ω<sup>b</sup>

<sup>q</sup>o,bð Þ¼ <sup>0</sup> ½ � <sup>0</sup>:5 0:5 0:5 0:<sup>5</sup> <sup>Τ</sup> and <sup>ω</sup><sup>b</sup>

<sup>q</sup>^o,bð Þ¼ <sup>0</sup> ½ � <sup>1000</sup> <sup>Τ</sup> and <sup>ω</sup>^ <sup>b</sup>

chosen as <sup>q</sup>o,d <sup>¼</sup> ½ � <sup>1000</sup> <sup>Τ</sup>

where β and ζ are gains, the time step is denoted by ΔT, the estimated angular

nion is denoted q^<sup>k</sup> describing the body frame relative to the orbit frame. Note that the quaternion must be normalized to ensure unit length and that the first elements

three elements, the projection matrix is defined as <sup>H</sup> <sup>¼</sup> ½ � 0 I <sup>∈</sup>R<sup>3</sup>�4, which has a

Let a satellite have the inertia matrix as given in Eq. (43), which contains nondiagonal terms which therefore will create perturbing moments due to the gravity.

o,b <sup>¼</sup> ½ � <sup>000</sup> <sup>Τ</sup>

sor measurements, the quaternion is converted into Euler angles, where noise is added to the different sensors. Then, creating the rotation matrix from the noisy Euler angles allows the Sun vector, acceleration, and magnetometer models in the orbit frame to be rotated to the body frame, where the measurements now contain noise. The step size

The quaternion and angular velocity error of the satellite are shown in Figure 18. After about 50 s, the objective of making the attitude error and angular velocity

o,d <sup>¼</sup> <sup>ω</sup>\_ <sup>d</sup>

, while ω<sup>d</sup>

column vector of zeros followed by the identity matrix such that ω<sup>b</sup>

Furthermore, let the satellite have the following initial conditions:

of the simulation is 0:01, while the sensors are sampled every 0:1 s.

bias,k∈R<sup>4</sup> are enforced to zero. To map the angular velocity from four to

bias,k∈R4, while the angular velocity of the body frame relative

o,b <sup>¼</sup> ½ � <sup>0</sup>:<sup>1</sup> �0:2 0:<sup>3</sup> <sup>Τ</sup> with

o,b∈R<sup>3</sup> (expected output) and the estimated quater-

, (50)

<sup>k</sup>∈R4, and the estimated gyro

o,b∈R<sup>3</sup> .

. The desired quaternion can be

o,d ¼ 0. To model noise in the sen-

and allows the gradient to be found as <sup>∇</sup> <sup>f</sup> <sup>¼</sup> <sup>J</sup><sup>Τ</sup> <sup>F</sup>. Now, let a vector in the orbit frame obtained from a mathematical model be denoted by <sup>z</sup><sup>o</sup> <sup>¼</sup> ox oy oz � �<sup>Τ</sup> and a vector in the body frame obtained through measurement be denoted by <sup>z</sup><sup>b</sup> <sup>¼</sup> bx by bz � �<sup>Τ</sup> . Then, the functions <sup>f</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup>; <sup>z</sup><sup>o</sup>; ; <sup>z</sup><sup>b</sup> � � and <sup>J</sup><sup>q</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup>; <sup>z</sup><sup>o</sup> � � are given by

<sup>f</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup>; <sup>z</sup><sup>o</sup> ; <sup>z</sup><sup>b</sup> � � <sup>¼</sup> <sup>2</sup>ox <sup>0</sup>:<sup>5</sup> � <sup>q</sup><sup>2</sup> <sup>3</sup> � <sup>q</sup><sup>2</sup> 4 � � <sup>þ</sup> <sup>2</sup>oy <sup>q</sup>1q<sup>4</sup> <sup>þ</sup> <sup>q</sup>2q<sup>3</sup> � � <sup>þ</sup> <sup>2</sup>oz <sup>q</sup>2q<sup>4</sup> � <sup>q</sup>1q<sup>3</sup> � � � bx 2ox q2q<sup>3</sup> � q1q<sup>4</sup> � � <sup>þ</sup> <sup>2</sup>oy <sup>0</sup>:<sup>5</sup> � <sup>q</sup><sup>2</sup> <sup>2</sup> � <sup>q</sup><sup>2</sup> 4 � � <sup>þ</sup> <sup>2</sup>oz <sup>q</sup>1q<sup>2</sup> <sup>þ</sup> <sup>q</sup>3q<sup>4</sup> � � � by 2ox q1q<sup>3</sup> þ q2q<sup>4</sup> � � <sup>þ</sup> <sup>2</sup>oy <sup>q</sup>3q<sup>4</sup> � <sup>q</sup>1q<sup>2</sup> � � <sup>þ</sup> <sup>2</sup>oz <sup>0</sup>:<sup>5</sup> � <sup>q</sup><sup>2</sup> <sup>2</sup> � <sup>q</sup><sup>2</sup> 3 � � � bz 2 6 4 3 7 5, (42) <sup>J</sup><sup>q</sup> <sup>q</sup>^k�1; <sup>z</sup><sup>o</sup> � � <sup>¼</sup> 2oyq<sup>4</sup> � 2ozq<sup>3</sup> 2oyq<sup>3</sup> þ 2ozq<sup>4</sup> �4oxq<sup>3</sup> þ 2oyq<sup>2</sup> � 2ozq<sup>1</sup> �4oxq<sup>4</sup> þ 2oyq<sup>1</sup> þ 2ozq<sup>2</sup> �2oxq<sup>4</sup> þ 2ozq<sup>2</sup> 2oxq<sup>3</sup> � 4oyq<sup>2</sup> þ 2ozq<sup>1</sup> 2oxq<sup>2</sup> þ 2ozq<sup>4</sup> �2oxq<sup>1</sup> � 4oyq<sup>4</sup> þ 2ozq<sup>3</sup> 2oxq<sup>3</sup> � 2oyq<sup>2</sup> 2oxq<sup>4</sup> � 2oyq<sup>1</sup> � 4ozq<sup>2</sup> 2oxq<sup>1</sup> þ 2oyq<sup>4</sup> � 4ozq<sup>3</sup> 2oxq<sup>2</sup> þ 2oyq<sup>3</sup> 2 6 4 3 7 5: (43)

Given the gyro measurement ω<sup>b</sup> gyro (relative to inertial frame), the angular velocity relative to orbit frame can be found as

$$\boldsymbol{\alpha}\_{o,gro}^{b} = \begin{bmatrix} \mathbf{0} \\ \boldsymbol{\alpha}\_{gro}^{b} - \mathbf{R}\_{o}^{b}(\hat{\mathbf{q}}\_{h-1}) \mathbf{R}\_{i}^{o} \boldsymbol{\alpha}\_{i,o}^{i} \end{bmatrix} \in \mathbb{R}^{4},\tag{44}$$

where the rotation matrix from orbit to body frame is constructed using the estimated quaternion and denoted by R<sup>b</sup> <sup>o</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup> � �. The Madgwick filter can now be presented as

$$
\hat{\boldsymbol{\mu}}\_k^b = 2\mathbf{T}(\hat{\mathbf{q}}\_{k-1}^\*) \frac{\nabla \mathbf{f}}{||\nabla \mathbf{f}||},\tag{45}
$$

$$
\boldsymbol{\alpha}\_{bias,k}^{b} = \boldsymbol{\alpha}\_{bias,k-1}^{b} + \zeta \boldsymbol{\hat{o}}\_{k}^{b} \Delta T,\tag{46}
$$

$$\boldsymbol{\alpha}\_{o,b}^{b} = \mathbf{H} \left( \boldsymbol{\alpha}\_{o,gryo}^{b} - \boldsymbol{\alpha}\_{bias,k}^{b} \right), \tag{47}$$

$$\dot{\hat{\mathbf{q}}}\_{k} = \frac{1}{2} \mathbf{T}(\hat{\mathbf{q}}\_{k-1}) \begin{bmatrix} 0 \\ \boldsymbol{\alpha}\_{o,b}^{b} \end{bmatrix} - \beta \frac{\nabla \mathbf{f}}{||\nabla \mathbf{f}||},\tag{48}$$

$$
\hat{\mathbf{q}}\_{k} = \hat{\mathbf{q}}\_{k-1} + \dot{\hat{\mathbf{q}}}\_{k} \Delta T,\tag{49}
$$

Modeling and Attitude Control of Satellites in Elliptical Orbits DOI: http://dx.doi.org/10.5772/intechopen.80422

Figure 19. Attitude and angular velocity during the maneuver [9].

$$
\hat{\mathbf{q}}\_k = \frac{\hat{\mathbf{q}}\_k}{||\hat{\mathbf{q}}\_k||},
\tag{50}
$$

where β and ζ are gains, the time step is denoted by ΔT, the estimated angular velocity based on vector measurements is denoted ω^ <sup>b</sup> <sup>k</sup>∈R4, and the estimated gyro bias is denoted by ω<sup>b</sup> bias,k∈R4, while the angular velocity of the body frame relative to the orbit frame is denoted ω<sup>b</sup> o,b∈R<sup>3</sup> (expected output) and the estimated quaternion is denoted q^<sup>k</sup> describing the body frame relative to the orbit frame. Note that the quaternion must be normalized to ensure unit length and that the first elements of ω^ <sup>b</sup> <sup>k</sup>, ω<sup>b</sup> bias,k∈R<sup>4</sup> are enforced to zero. To map the angular velocity from four to three elements, the projection matrix is defined as <sup>H</sup> <sup>¼</sup> ½ � 0 I <sup>∈</sup>R<sup>3</sup>�4, which has a column vector of zeros followed by the identity matrix such that ω<sup>b</sup> o,b∈R<sup>3</sup> .

#### 6.4 Simulation

F ¼

matrix can be found as

Applied Modern Control

<sup>z</sup><sup>b</sup> <sup>¼</sup> bx by bz � �<sup>Τ</sup>

by

<sup>f</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup>; <sup>z</sup><sup>o</sup> ; <sup>z</sup><sup>b</sup> � � <sup>¼</sup>

<sup>J</sup><sup>q</sup> <sup>q</sup>^k�1; <sup>z</sup><sup>o</sup> � � <sup>¼</sup>

presented as

94

2 6 4 2 6 4

J ¼

frame obtained from a mathematical model be denoted by <sup>z</sup><sup>o</sup> <sup>¼</sup> ox oy oz

vector in the body frame obtained through measurement be denoted by

� � <sup>þ</sup> <sup>2</sup>oy <sup>0</sup>:<sup>5</sup> � <sup>q</sup><sup>2</sup>

� � <sup>þ</sup> <sup>2</sup>oy <sup>q</sup>3q<sup>4</sup> � <sup>q</sup>1q<sup>2</sup>

<sup>3</sup> � <sup>q</sup><sup>2</sup> 4 � � <sup>þ</sup> <sup>2</sup>oy <sup>q</sup>1q<sup>4</sup> <sup>þ</sup> <sup>q</sup>2q<sup>3</sup>

o, gyro <sup>¼</sup> <sup>0</sup> ω<sup>b</sup> gyro � <sup>R</sup><sup>b</sup>

ω^ b

bias,k <sup>¼</sup> <sup>ω</sup><sup>b</sup>

o,b <sup>¼</sup> <sup>H</sup> <sup>ω</sup><sup>b</sup>

ωb

ωb

\_ <sup>q</sup>^<sup>k</sup> <sup>¼</sup> <sup>1</sup> 2 <sup>T</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup> � � <sup>0</sup>

<sup>k</sup> <sup>¼</sup> <sup>2</sup><sup>T</sup> <sup>q</sup>^<sup>∗</sup>

<sup>2</sup>ox <sup>0</sup>:<sup>5</sup> � <sup>q</sup><sup>2</sup>

2 6 4

Given the gyro measurement ω<sup>b</sup>

velocity relative to orbit frame can be found as

ωb

estimated quaternion and denoted by R<sup>b</sup>

2ox q2q<sup>3</sup> � q1q<sup>4</sup>

2ox q1q<sup>3</sup> þ q2q<sup>4</sup>

<sup>f</sup> <sup>q</sup>^k�1; <sup>a</sup><sup>o</sup>; <sup>a</sup><sup>b</sup> � � <sup>f</sup> <sup>q</sup>^k�1; <sup>s</sup><sup>o</sup>; <sup>s</sup><sup>b</sup> � � <sup>f</sup> <sup>q</sup>^k�1; <sup>m</sup><sup>o</sup>; <sup>m</sup><sup>b</sup> � �

where the objective is found in an estimated quaternion that minimizes this function, something that can be achieved by using gradient descent. The Jacobian

> <sup>J</sup><sup>q</sup> <sup>q</sup>^k�1; <sup>a</sup><sup>o</sup> � � <sup>J</sup><sup>q</sup> <sup>q</sup>^k�1; <sup>s</sup><sup>o</sup> � � <sup>J</sup><sup>q</sup> <sup>q</sup>^k�1; <sup>m</sup><sup>o</sup> � �

and allows the gradient to be found as <sup>∇</sup> <sup>f</sup> <sup>¼</sup> <sup>J</sup><sup>Τ</sup> <sup>F</sup>. Now, let a vector in the orbit

3 7

. Then, the functions <sup>f</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup>; <sup>z</sup><sup>o</sup>; ; <sup>z</sup><sup>b</sup> � � and <sup>J</sup><sup>q</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup>; <sup>z</sup><sup>o</sup> � � are given

<sup>2</sup> � <sup>q</sup><sup>2</sup> 4 � � <sup>þ</sup> <sup>2</sup>oz <sup>q</sup>1q<sup>2</sup> <sup>þ</sup> <sup>q</sup>3q<sup>4</sup>

2oyq<sup>4</sup> � 2ozq<sup>3</sup> 2oyq<sup>3</sup> þ 2ozq<sup>4</sup> �4oxq<sup>3</sup> þ 2oyq<sup>2</sup> � 2ozq<sup>1</sup> �4oxq<sup>4</sup> þ 2oyq<sup>1</sup> þ 2ozq<sup>2</sup> �2oxq<sup>4</sup> þ 2ozq<sup>2</sup> 2oxq<sup>3</sup> � 4oyq<sup>2</sup> þ 2ozq<sup>1</sup> 2oxq<sup>2</sup> þ 2ozq<sup>4</sup> �2oxq<sup>1</sup> � 4oyq<sup>4</sup> þ 2ozq<sup>3</sup> 2oxq<sup>3</sup> � 2oyq<sup>2</sup> 2oxq<sup>4</sup> � 2oyq<sup>1</sup> � 4ozq<sup>2</sup> 2oxq<sup>1</sup> þ 2oyq<sup>4</sup> � 4ozq<sup>3</sup> 2oxq<sup>2</sup> þ 2oyq<sup>3</sup>

> <sup>o</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup> � �R<sup>o</sup>

where the rotation matrix from orbit to body frame is constructed using the

<sup>o</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup>

k�1 � � ∇f

bias,k�<sup>1</sup> <sup>þ</sup> <sup>ζ</sup>ω^ <sup>b</sup>

o, gyro � <sup>ω</sup><sup>b</sup>

ω<sup>b</sup> o,b

<sup>q</sup>^<sup>k</sup> <sup>¼</sup> <sup>q</sup>^<sup>k</sup>�<sup>1</sup> <sup>þ</sup> \_

" #

� �

k∇fk

bias,k

� <sup>β</sup> <sup>∇</sup><sup>f</sup> k∇fk

" #

� � <sup>þ</sup> <sup>2</sup>oz <sup>0</sup>:<sup>5</sup> � <sup>q</sup><sup>2</sup>

gyro (relative to inertial frame), the angular

� �. The Madgwick filter can now be

i ωi i, o

� � <sup>þ</sup> <sup>2</sup>oz <sup>q</sup>2q<sup>4</sup> � <sup>q</sup>1q<sup>3</sup>

<sup>5</sup>, (40)

(41)

3 7 5,

> 3 7 5:

(42)

(43)

� �<sup>Τ</sup> and a

� � � bx

� � � by

<sup>2</sup> � <sup>q</sup><sup>2</sup> 3 � � � bz

∈R<sup>4</sup>, (44)

, (45)

<sup>k</sup>ΔT, (46)

q^kΔT, (49)

, (47)

, (48)

Let a satellite have the inertia matrix as given in Eq. (43), which contains nondiagonal terms which therefore will create perturbing moments due to the gravity. Furthermore, let the satellite have the following initial conditions: <sup>q</sup>o,bð Þ¼ <sup>0</sup> ½ � <sup>0</sup>:5 0:5 0:5 0:<sup>5</sup> <sup>Τ</sup> and <sup>ω</sup><sup>b</sup> o,b <sup>¼</sup> ½ � <sup>0</sup>:<sup>1</sup> �0:2 0:<sup>3</sup> <sup>Τ</sup> with <sup>q</sup>^o,bð Þ¼ <sup>0</sup> ½ � <sup>1000</sup> <sup>Τ</sup> and <sup>ω</sup>^ <sup>b</sup> o,b <sup>¼</sup> ½ � <sup>000</sup> <sup>Τ</sup> . The desired quaternion can be chosen as <sup>q</sup>o,d <sup>¼</sup> ½ � <sup>1000</sup> <sup>Τ</sup> , while ω<sup>d</sup> o,d <sup>¼</sup> <sup>ω</sup>\_ <sup>d</sup> o,d ¼ 0. To model noise in the sensor measurements, the quaternion is converted into Euler angles, where noise is added to the different sensors. Then, creating the rotation matrix from the noisy Euler angles allows the Sun vector, acceleration, and magnetometer models in the orbit frame to be rotated to the body frame, where the measurements now contain noise. The step size of the simulation is 0:01, while the sensors are sampled every 0:1 s.

The quaternion and angular velocity error of the satellite are shown in Figure 18. After about 50 s, the objective of making the attitude error and angular velocity

error go to zero is achieved. Since the attitude is not measured directly, the Madgwick filter is used for attitude estimation as shown in Figure 19. Both the quaternion error (estimated truth) and angular velocity error converge close to zero.

7. Conclusion

Author details

Narvik, Norway

Espen Oland

97

new actuation methods and strategies.

DOI: http://dx.doi.org/10.5772/intechopen.80422

Modeling and Attitude Control of Satellites in Elliptical Orbits

This chapter has presented all the components required to create an attitude determination and control system for satellites in elliptical orbits. With this as basis, it is the hope by the author that the work can help in developing new results within attitude determination and control systems, ranging from nonlinear controllers to new understanding in orbital mechanics, attitude determination, new sensors, and

Department of Electrical Engineering, UiT - The Arctic University of Norway,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: espen.oland@uit.no

provided the original work is properly cited.

From the PD+ controller, the desired torque is mapped to the desired thrust firings using bang-bang modulation (Figure 20). Figure 21 shows the thruster firings required to maintain the attitude error close to zero.

Figure 20. Estimation error [9].

Figure 21. Thruster firings to control the attitude [9].

### 7. Conclusion

error go to zero is achieved. Since the attitude is not measured directly, the Madgwick filter is used for attitude estimation as shown in Figure 19. Both the quaternion error (estimated truth) and angular velocity error converge close to

firings required to maintain the attitude error close to zero.

From the PD+ controller, the desired torque is mapped to the desired thrust firings using bang-bang modulation (Figure 20). Figure 21 shows the thruster

zero.

Applied Modern Control

Figure 20. Estimation error [9].

Figure 21.

96

Thruster firings to control the attitude [9].

This chapter has presented all the components required to create an attitude determination and control system for satellites in elliptical orbits. With this as basis, it is the hope by the author that the work can help in developing new results within attitude determination and control systems, ranging from nonlinear controllers to new understanding in orbital mechanics, attitude determination, new sensors, and new actuation methods and strategies.

### Author details

Espen Oland Department of Electrical Engineering, UiT - The Arctic University of Norway, Narvik, Norway

\*Address all correspondence to: espen.oland@uit.no

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[23] Oland E. A command-filtered backstepping approach to autonomous inspections using a quadrotor. In: Proceedings of the 24th Mediterranean Conference on Control and Automation;

[24] Hahn W. Stability of Motion. Berlin/ Heidelberg/New York: Springer-Verlag;

1967. ISBN: 978-3-642-50087-9

[25] Wiśniewski R. Satellite attitude control using only electromagnetic

Ast; 2003. pp. 1-19

2008

1697-1712

2896-2901

[10] Oland E, Houge T, Vedal F. Norwegian student satellite program— HiNCube. In: Proceedings of the 22nd Annual AIAA/USU Conference on Small Satellites; Utah, USA; 2008

[11] Oland E, Schlanbusch R. Reaction wheel design for cubesats. In: Proceedings of the 4th International Conference on Recent Advances in Space Technologies (RAST); Istanbul, Turkey; 2009

[12] Schlanbusch R, Oland E, Nicklasson PJ. Modeling and simulation of a cubesat using nonlinear control in an elliptic orbit. In: Proceedings of the 4th International Conference on Recent Advances in Space Technologies (RAST); Istanbul, Turkey; 2009

[13] Madgwick S, Harrison A, Vaidyanathan R. Estimation of IMU and MARG orientation using a gradient descent algorithm. In: Proceedings of the 2011 IEEE International Conference on Rehabilitation Robotics; Zurich, Switzerland; 2011

[14] Oland E. Autonomous inspection of the International Space Station. In: Proceedings of the 8th International Conference on Mechanical and Aerospace Engineering; Prague, Czech; 2017

[15] Egeland O, Gravdahl JT. Modeling and Simulation for Automatic Control. Trondheim, Norway: Marine Cybernetics; 2002. ISBN: 82-92356-01-0

[16] Schaub H, Junkins JL. Analytical Mechanics of Space Systems. AIAA

Modeling and Attitude Control of Satellites in Elliptical Orbits DOI: http://dx.doi.org/10.5772/intechopen.80422

American Institute of Aeronautics & Ast; 2003. pp. 1-19

References

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Applied Modern Control

Company; 1978

Springer; 2014

593-601

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2013;16(19):283-288

[1] Sidi MJ. Spacecraft Dynamics & Control. New York: Cambridge

[9] Oland E. Attitude determination and control system for satellites in elliptical orbits—A complete solution. In: Proceedings of the 8th International Conference on Recent Advances in Space Technology (RAST); Istanbul,

[10] Oland E, Houge T, Vedal F. Norwegian student satellite program— HiNCube. In: Proceedings of the 22nd Annual AIAA/USU Conference on Small

[11] Oland E, Schlanbusch R. Reaction

[12] Schlanbusch R, Oland E, Nicklasson PJ. Modeling and simulation of a cubesat using nonlinear control in an elliptic orbit. In: Proceedings of the 4th International Conference on Recent Advances in Space Technologies (RAST); Istanbul, Turkey; 2009

Vaidyanathan R. Estimation of IMU and MARG orientation using a gradient descent algorithm. In: Proceedings of the 2011 IEEE International Conference on Rehabilitation Robotics; Zurich,

[14] Oland E. Autonomous inspection of the International Space Station. In: Proceedings of the 8th International Conference on Mechanical and

Aerospace Engineering; Prague, Czech;

[15] Egeland O, Gravdahl JT. Modeling and Simulation for Automatic Control.

Cybernetics; 2002. ISBN: 82-92356-01-0

[16] Schaub H, Junkins JL. Analytical Mechanics of Space Systems. AIAA

Trondheim, Norway: Marine

Satellites; Utah, USA; 2008

wheel design for cubesats. In: Proceedings of the 4th International Conference on Recent Advances in Space Technologies (RAST); Istanbul,

[13] Madgwick S, Harrison A,

Switzerland; 2011

2017

Turkey; 2017

Turkey; 2009

[2] Wertz J, editor. Spacecraft Attitude

Determination and Control. D. Dordrecht, Holland: Reidel Publishing

[3] Vallado DA. Fundamentals of Astrodynamics and Applications. 3rd ed. El Segundo, California: Microcosm Press and Dordrecht, Boston, London: Kluwer Academic Publishers; 2007

[4] Markley FL, Crassidis JL.

Fundamentals of Spacecraft Attitude Determination and Control. El Segundo, California: Microcosm Press and New York, Heidelberg, Dordrecht London:

[5] Alarcon J, Örger N, Kim S, Soon L, Cho M. Aoba VELOX-IV attitude and orbit control system design for a LEO mission applicable to a future lunar mission. In: Proceedings of the 67th International Astronautical Congress; Guadalajara City, Mexico; 2016

[6] Dechao R, Tao S, Lu C, Xiaoqian C, Yong Z. Attitude control system design and on-orbit performance analysis of nano-satellite - tian Tuo 1. Chinese Journal of Aeronautics. 2014;27(3):

[7] Nakajima Y, Murakami N, Ohtani T, Nakamura Y, Hirako K, Inoue K. SDS-4 attitude control system: In-flight results of three axis attitude control for small satellites. IFAC Proceedings Volumes.

[8] Fritz M, Shoer J, Singh L, Henderson T, McGee J, Rose R, Ruf C. Attitude determination and control system design for the CYGNSS microsatellite. In: Proceedings of the IEEE Aerospace Conference; Big Sky, Montana; 2015

[17] Oland E, Andersen TS, Kristiansen R. Subsumption architecture applied to flight control using composite rotations. Automatica. 2016;69:195-200

[18] Kristiansen R. Dynamic synchronization of spacecraft [PhD thesis]. Trondheim, Norway: Norwegian University of Science and Technology; 2008

[19] Takegaki M, Arimoto S. A new feedback method for dynamic control of manipulators. ASME Journal of Dynamic Systems, Measurement, and Control. 1981;103(2):119-125

[20] Paden B, Panja R. Globally asymptotically stable 'PD+' controller for robot manipulators. International Journal of Control. 1988;47(6): 1697-1712

[21] Kristiansen R, Nicklasson PJ, Gravdahl JT. Spacecraft coordination control in 6DOF: Integrator backstepping vs. passivity-based control. Automatica. 2008;44(11): 2896-2901

[22] Schlanbusch R, Loría A, Kristiansen R, Nicklasson PJ. PD+ based output feedback attitude control of rigid bodies. IEEE Transactions on Automatic Control. 2012;57(8):2146-2152

[23] Oland E. A command-filtered backstepping approach to autonomous inspections using a quadrotor. In: Proceedings of the 24th Mediterranean Conference on Control and Automation; Athens, Greece; 2016

[24] Hahn W. Stability of Motion. Berlin/ Heidelberg/New York: Springer-Verlag; 1967. ISBN: 978-3-642-50087-9

[25] Wiśniewski R. Satellite attitude control using only electromagnetic

actuation [PhD thesis]. Aalborg University; 1996

[26] Hu Q, Ma G. Flexible spacecraft vibration suppression using PWPF modulated input component command and sliding mode control. Asian Journal of Control. 2007;9(1):20-29

[27] Psiaki M. Magnetic torquer attitude control via asymptotic periodic linear quadratic regulation. Journal of Guidance, Control, and Dynamics. 2001;24(2):386-394

[28] Thébault E et al. International geomagnetic reference field: The 12th generation. Earth, Planets and Space. 2015;67(79):1-19

[29] Shuster MD, Oh SD. Three-axis attitude determination from vector observations. Journal of Guidance and Control. 1981;4(1):70-77

[30] Farrell JL. Attitude determination by kalman filtering. Automatica. 1970; 6(3):419-430

[31] Mahony R, Hamel T, Pflimlin J. Nonlinear complementary filters on the special orthogonal group. IEEE Transactions on Automatic Control. 2008;54(5):1203-1217

Chapter 6

Abstract

wavelet analysis

1. Introduction

and Alexey Lototsky

which is named associative rules search.

tion methods based on data mining [1].

practice of identification [2, 3].

101

Data Mining-Based Identification

Natalia Bakhtadze, Vladimir Lototsky, Valery Pyatetsky

This chapter presents identification methods using associative search of analogs and wavelet analysis. It investigates the properties of data mining-based identification algorithms which allow to predict: (i) the approach of process variables to critical values and (ii) process transition to chaotic dynamics. The methods proposed are based on the modeling of human operator decision-making. The effectiveness of the methods is illustrated with an example of product quality prediction in oil refining. The development of fuzzy analogs of associative identification models is further discussed. Fuzzy approach expands the application area of associative techniques. Finally, state prediction techniques for manufacturing resources are developed on the basis of binary models and a machine learning procedure,

Keywords: process identification, knowledge base, associative search models,

The reduction of uncertainty in object description in terms of adjustable model has been a key conceptual direction in the identification theory and applications for a long time. In the statistical description of uncertainty, consistent estimates of plant's characteristics can be obtained by analyzing the convergence of the empirical distribution functional with the corresponding "theoretical" values, but this entails appropriate increase of the sample size. The difficulties in implementing this approach, especially for nonlinear and nonstationary objects, along with the increased possibilities of plant history analysis resulted in the advent of identifica-

The use of additional a priori information on the system for its training is considered by some authors today to be one of the key trends in the theory and

One method that implements this approach to identification is the associative search method based on the design of predictive models [2]. They are based on inductive learning, that is, on associative search of analogs by means of intelligent analysis of process history and knowledge base development. The development of a predictive model for a dynamic object by associative search technique (i.e., by building a new model at every time step) is based on the generated and updated

of Nonlinear Systems

#### Chapter 6

## Data Mining-Based Identification of Nonlinear Systems

Natalia Bakhtadze, Vladimir Lototsky, Valery Pyatetsky and Alexey Lototsky

#### Abstract

This chapter presents identification methods using associative search of analogs and wavelet analysis. It investigates the properties of data mining-based identification algorithms which allow to predict: (i) the approach of process variables to critical values and (ii) process transition to chaotic dynamics. The methods proposed are based on the modeling of human operator decision-making. The effectiveness of the methods is illustrated with an example of product quality prediction in oil refining. The development of fuzzy analogs of associative identification models is further discussed. Fuzzy approach expands the application area of associative techniques. Finally, state prediction techniques for manufacturing resources are developed on the basis of binary models and a machine learning procedure, which is named associative rules search.

Keywords: process identification, knowledge base, associative search models, wavelet analysis

#### 1. Introduction

The reduction of uncertainty in object description in terms of adjustable model has been a key conceptual direction in the identification theory and applications for a long time. In the statistical description of uncertainty, consistent estimates of plant's characteristics can be obtained by analyzing the convergence of the empirical distribution functional with the corresponding "theoretical" values, but this entails appropriate increase of the sample size. The difficulties in implementing this approach, especially for nonlinear and nonstationary objects, along with the increased possibilities of plant history analysis resulted in the advent of identification methods based on data mining [1].

The use of additional a priori information on the system for its training is considered by some authors today to be one of the key trends in the theory and practice of identification [2, 3].

One method that implements this approach to identification is the associative search method based on the design of predictive models [2]. They are based on inductive learning, that is, on associative search of analogs by means of intelligent analysis of process history and knowledge base development. The development of a predictive model for a dynamic object by associative search technique (i.e., by building a new model at every time step) is based on the generated and updated

#### Applied Modern Control

knowledge about the system. This approach allows to use any available a priori information about the plant [3].

The stability of a model built using the associative search techniques is investigated in terms of the spectrum analysis of a multi-scale wavelet expansion [4]. Methods based on the wavelet analysis open up a unique possibility to select "frequency-domain windows" as against the well-known windowed Fourier transform.

The development of intelligent identification algorithms for nonlinear and nonstationary objects is important for various applications, in particular, in chemical, oil refining, and power (smart grids) industries; transportation and logistics system; and trading processes (Bakhtadze et al. [1, 2, 4–7]).

#### 2. Control system identification

Consider a traditional problem of dynamic object identification. For input vectors meeting Gauss-Markov assumptions, the least squares parameter estimates are consistent, unbiased, and efficient. However, the development of a closed-loop control system (for identification-based control system synthesis) faces considerable challenges. In a closed loop, the system state depends on control values at earlier time instants, which results in a degeneration problem.

Knowledge in intelligent systems is of two types [10]. The first type of knowledge, that is, declarative knowledge, by means of appropriate ontologies describes different facts, events, and observation. A formal description of skills is called procedural knowledge. Depending on the level of this knowledge, users can be referred to as beginners or experts [11]. These two groups have different structures and ways of thinking. Beginners use so-called inverse reasoning in the procedure for decision-making. They make decisions based on the analysis of the information obtained in the previous step. In contrast to the beginners, experts at an intuitive, subconscious level form the so-called direct reasoning. Thus, cognitive psychology defines knowledge as a collection of symbols stored in the memory of a particular person [12]. The symbols, in turn, can be determined by their structure and the

Knowledge processing in an intelligent system consists in the recovery (associative search) of knowledge by its fragment [14]. The knowledge can be defined as an associative link between images (Figure 1). As an image, we will use "feature sets," that is, components of input vectors or input variables. The set of all associations over the set of images forms the memory of the intelligent system's knowledge base. The associative search process can be either an image reconstruction procedure by a feature set (this set may not be complete; this approach is often used in models of a human associative memory), or the search procedure of other images in the

In Ref. [14], a model of decision-making search by the human operator is proposed, representing the process of associative thinking as a sequence of sets of associations. Association is a pair of images (the image-source and the imageoutput), wherein each image is described by a set of features. This approach is intermediate between neural networks and logical models in the classical theory of

The criterion for the similarity of two images in the general case can be represented as a logical function—a predicate. In the particular case, the features have a numerical expression. The feature sets that form the image are vectors in ndimensional space. In this case, as a criterion of image similarity can be a metric in

Associative search method consists in constructing virtual predictive models. The term "virtual" should be understood as "ad hoc" [2]. The method presumes the construction of a predictive model for a dynamic object as follows. A traditional

archive, similar to the image under study by a certain criterion.

nature of neuron links [13].

Model of a human associative memory.

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

Figure 1.

artificial intelligence.

4. Associative search technique

the space.

103

To develop an informational model of control system's dynamics in a degenerate case, the Moore-Penrose method [8, 9] can be used for getting pseudo-solutions to a linear system by means of least squares techniques.

For a wide class of objects and, in particular, processes, control based on a linear model identification is not satisfactory. At the same time, models constructed by the method of associative search frequently are highly accurate even for nonlinear objects. However, some processes can be characterized by certain "irregularities" in certain time intervals, which affect the accuracy and adequacy of associative models.

Examples of such irregularities (which are often oscillatory in engineering systems) can be:


#### 3. Associative search as intelligent modeling method

The difference between the associative search method based on data mining and traditional identification techniques is as follows. The method does not approximate process dynamics in time; it rather builds a new predictive model of the dynamic object (a "virtual model") at each time step using historical data sets ("associations") generated at the training phase.

As a result, at any time step, process control decision-making by a human individual (process operator, supervisor, plant or enterprise manager, trading operator, etc.) is modeled on the base of his/her knowledge and emerging associations.

Clustering (self-organizing learning) is an effective way to form associations.

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

Figure 1. Model of a human associative memory.

knowledge about the system. This approach allows to use any available a priori

gated in terms of the spectrum analysis of a multi-scale wavelet expansion [4]. Methods based on the wavelet analysis open up a unique possibility to select "frequency-domain windows" as against the well-known windowed Fourier trans-

The development of intelligent identification algorithms for nonlinear and nonstationary objects is important for various applications, in particular, in chemical, oil refining, and power (smart grids) industries; transportation and logistics

Consider a traditional problem of dynamic object identification. For input vectors meeting Gauss-Markov assumptions, the least squares parameter estimates are consistent, unbiased, and efficient. However, the development of a closed-loop control system (for identification-based control system synthesis) faces considerable challenges. In a closed loop, the system state depends on control values at

To develop an informational model of control system's dynamics in a degenerate case, the Moore-Penrose method [8, 9] can be used for getting pseudo-solutions to a

For a wide class of objects and, in particular, processes, control based on a linear model identification is not satisfactory. At the same time, models constructed by the method of associative search frequently are highly accurate even for nonlinear objects. However, some processes can be characterized by certain "irregularities" in certain time intervals, which affect the accuracy and adequacy of associative

Examples of such irregularities (which are often oscillatory in engineering sys-

• seasonal and daily load oscillations in power networks that affect directly the

The difference between the associative search method based on data mining and traditional identification techniques is as follows. The method does not approximate process dynamics in time; it rather builds a new predictive model of the dynamic object (a "virtual model") at each time step using historical data sets ("associa-

As a result, at any time step, process control decision-making by a human individual (process operator, supervisor, plant or enterprise manager, trading operator, etc.) is modeled on the base of his/her knowledge and emerging

Clustering (self-organizing learning) is an effective way to form associations.

• ups and downs of stock market caused by various economic reasons;

system; and trading processes (Bakhtadze et al. [1, 2, 4–7]).

earlier time instants, which results in a degeneration problem.

optimization of power transmission control modes;

• feed source changes in industrial process, and so on.

3. Associative search as intelligent modeling method

tions") generated at the training phase.

linear system by means of least squares techniques.

The stability of a model built using the associative search techniques is investi-

information about the plant [3].

Applied Modern Control

2. Control system identification

form.

models.

tems) can be:

associations.

102

Knowledge in intelligent systems is of two types [10]. The first type of knowledge, that is, declarative knowledge, by means of appropriate ontologies describes different facts, events, and observation. A formal description of skills is called procedural knowledge. Depending on the level of this knowledge, users can be referred to as beginners or experts [11]. These two groups have different structures and ways of thinking. Beginners use so-called inverse reasoning in the procedure for decision-making. They make decisions based on the analysis of the information obtained in the previous step. In contrast to the beginners, experts at an intuitive, subconscious level form the so-called direct reasoning. Thus, cognitive psychology defines knowledge as a collection of symbols stored in the memory of a particular person [12]. The symbols, in turn, can be determined by their structure and the nature of neuron links [13].

Knowledge processing in an intelligent system consists in the recovery (associative search) of knowledge by its fragment [14]. The knowledge can be defined as an associative link between images (Figure 1). As an image, we will use "feature sets," that is, components of input vectors or input variables. The set of all associations over the set of images forms the memory of the intelligent system's knowledge base.

The associative search process can be either an image reconstruction procedure by a feature set (this set may not be complete; this approach is often used in models of a human associative memory), or the search procedure of other images in the archive, similar to the image under study by a certain criterion.

In Ref. [14], a model of decision-making search by the human operator is proposed, representing the process of associative thinking as a sequence of sets of associations. Association is a pair of images (the image-source and the imageoutput), wherein each image is described by a set of features. This approach is intermediate between neural networks and logical models in the classical theory of artificial intelligence.

The criterion for the similarity of two images in the general case can be represented as a logical function—a predicate. In the particular case, the features have a numerical expression. The feature sets that form the image are vectors in ndimensional space. In this case, as a criterion of image similarity can be a metric in the space.

#### 4. Associative search technique

Associative search method consists in constructing virtual predictive models. The term "virtual" should be understood as "ad hoc" [2]. The method presumes the construction of a predictive model for a dynamic object as follows. A traditional

identification algorithm approximates real process in time. As against such algorithms, our method builds a new model at each time step t based on the analysis of the history data set ("associations") formed at the stage of learning and further adaptively corrected in accordance to certain criteria.

Within the present context, linear dynamic model is of the form:

$$\mathbf{y}\_N = \sum\_{i=1}^m a\_i \mathbf{y}\_{N-i} + \sum\_{j=1}^{r\_i} \sum\_{s=1}^S b\_{j,s} \mathbf{x}\_{N-j,s} \,\forall j = \mathbf{1}/N,\tag{1}$$

As a distance (a norm in R<sup>S</sup>

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

dN,N�<sup>j</sup> ¼ ∑

dN,N�<sup>j</sup> ≤ ∑

Let for the current input vector xN:

Figure 3.

105

Approximating hypersurface building.

S s¼1

dN,N�<sup>j</sup> ≤dN þ ∑

DN ≥2dmax

S s¼1

xN,s � xN�j,s

where xN,s are the components of the input vector at the current time instant N.

S s¼1

To derive an approximating hypersurface for the vector xN, we select from the archive of the input data such vectors xN�<sup>j</sup>, j ¼ 1,´N that for a set DN the condition:

> xN�j,s

holds, where DN may be selected, for instance, from the condition (Figure 3):

Under the assumptions that the inputs meet the Gauss-Markov conditions, the

<sup>j</sup> <sup>∑</sup> S s¼1

xN�j,s 

<sup>N</sup> ¼ 2 max

estimates obtained via the LS method are unbiased and statistically effective.

xN�j,s 

By virtue of a property of the norm ("the triangle inequality"), we have:

xN,s j j þ ∑

∑ S s¼1

> S s¼1

inputs, we introduce the value:

) between points of the S-dimensional space of

, ∀j ¼ 1,´N, (2)

, ∀j ¼ 1,´N, (3)

xN,s j j ¼ dN: (4)

≤ DN, ∀j ¼ 1,´N, (5)

: (6)

where yN is the prediction of the object's output at the time instant N, xN is the input vector, m is the memory depth in the output, rs is the memory depth in the input, S is the dimension of the input vectors, and ai and bj,s are tuning coefficients of the model. Model (1) is a regression whose structure is determined by a criterion of similarity of images forming the association.

In general, a new structure is formed for each time instant. The associative model is virtual in the sense that for each time step, it formed a new structure. For each current input vector, the corresponding input vectors and their corresponding outputs are selected from the archive. Further, a system of linear equations with respect to the adjustable coefficients is formed. Its decision in accordance with the least squares method determines the point linear model of a nonlinear object, as well as the output forecast.

Thus, each point of the global nonlinear regression surface is formed as a result of using linear "local" models at each new time step.

The set of values of inputs at each fixed point and the corresponding output replenish the procedural knowledge base.

Unlike classical regression models, for each fixed time instant from the process history, input vectors are selected close to the current input vector in the sense of a certain criterion (rather than the chronological sequence as in regression models). Thus, in Eq. (1), rs is the number of vectors from the archive (from the time instant 1 to the time instant N), selected in accordance to the associative search criterion. A certain set of vectors rs, 1≤rs ≤ N, is selected at each time segment ½ � N � 1; N . The criterion for selecting the input vectors from the archive is described below (Figure 2).

Figure 2. Approximating hypersurface design.

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

identification algorithm approximates real process in time. As against such algorithms, our method builds a new model at each time step t based on the analysis of the history data set ("associations") formed at the stage of learning and further

Within the present context, linear dynamic model is of the form:

rs j¼1 ∑ S s¼1

where yN is the prediction of the object's output at the time instant N, xN is the input vector, m is the memory depth in the output, rs is the memory depth in the input, S is the dimension of the input vectors, and ai and bj,s are tuning coefficients of the model. Model (1) is a regression whose structure is determined by a criterion

In general, a new structure is formed for each time instant. The associative model is virtual in the sense that for each time step, it formed a new structure. For each current input vector, the corresponding input vectors and their corresponding outputs are selected from the archive. Further, a system of linear equations with respect to the adjustable coefficients is formed. Its decision in accordance with the least squares method determines the point linear model of a nonlinear object, as

Thus, each point of the global nonlinear regression surface is formed as a result

Unlike classical regression models, for each fixed time instant from the process history, input vectors are selected close to the current input vector in the sense of a certain criterion (rather than the chronological sequence as in regression models). Thus, in Eq. (1), rs is the number of vectors from the archive (from the time instant 1 to the time instant N), selected in accordance to the associative search criterion. A certain set of vectors rs, 1≤rs ≤ N, is selected at each time segment ½ � N � 1; N . The criterion for selecting the input vectors from the archive is described below

The set of values of inputs at each fixed point and the corresponding output

bj,sxN�j,s, ∀j ¼ 1,´N, (1)

aiyN�<sup>i</sup> <sup>þ</sup> <sup>∑</sup>

adaptively corrected in accordance to certain criteria.

yN ¼ ∑ m i¼1

of similarity of images forming the association.

of using linear "local" models at each new time step.

replenish the procedural knowledge base.

well as the output forecast.

Applied Modern Control

(Figure 2).

Figure 2.

104

Approximating hypersurface design.

As a distance (a norm in R<sup>S</sup> ) between points of the S-dimensional space of inputs, we introduce the value:

$$d\_{N,N-j} = \sum\_{s=1}^{S} \left| \mathbf{x}\_{N,s} - \mathbf{x}\_{N-j,s} \right|, \forall j = \mathbf{1}, \mathbf{'}N,\tag{2}$$

where xN,s are the components of the input vector at the current time instant N. By virtue of a property of the norm ("the triangle inequality"), we have:

$$d\_{N,N-j} \le \sum\_{s=1}^{S} |\boldsymbol{\varkappa}\_{N,s}| + \sum\_{s=1}^{S} |\boldsymbol{\varkappa}\_{N-j,s}|, \forall j = \mathbf{1}, \text{'N'},\tag{3}$$

Let for the current input vector xN:

$$\sum\_{s=1}^{S} |\mathfrak{x}\_{N,s}| = d\_N. \tag{4}$$

To derive an approximating hypersurface for the vector xN, we select from the archive of the input data such vectors xN�<sup>j</sup>, j ¼ 1,´N that for a set DN the condition:

$$d\_{N,N-j} \le d\_N + \sum\_{s=1}^{S} \left| \mathbf{x}\_{N-j,s} \right| \le D\_{N\nu} \,\forall j = \mathbf{1}, 'N,\tag{5}$$

holds, where DN may be selected, for instance, from the condition (Figure 3):

$$D\_N \ge 2d\_N^{\max} = 2 \max\_j \sum\_{s=1}^S \left| \boldsymbol{\omega}\_{N-j,s} \right|. \tag{6}$$

Under the assumptions that the inputs meet the Gauss-Markov conditions, the estimates obtained via the LS method are unbiased and statistically effective.

Figure 3. Approximating hypersurface building.

#### 5. Fuzzy virtual models

Fuzzy models under uncertainty are advisable to apply in decision-making systems in the following cases [3]:

• dynamics of the investigated quality index is described by a complex nonlinear dependence; and

Depending on the features of the object and the purpose of identification, various fuzzy models can be formed. Thus, the Takagi-Sugeno model is most suitable for objects with complex nonlinear dynamics, such as moving objects, in the control

A fuzzy model of the Mamdani type is suitable for problems in the solution of

The singleton-type system may be used in both identification and knowledge-

ð Þ� <sup>x</sup><sup>2</sup> … � <sup>μ</sup>LXð Þ <sup>n</sup>þ<sup>m</sup> <sup>i</sup>

ð Þ� <sup>x</sup><sup>2</sup> … � <sup>μ</sup>LXð Þ <sup>n</sup>þ<sup>m</sup> <sup>i</sup>

ð Þ� <sup>x</sup><sup>2</sup> … � <sup>μ</sup>LXð Þ <sup>n</sup>þ<sup>m</sup> <sup>i</sup>

ð Þ� xnþ<sup>m</sup> ri

(11)

(12)

ð Þ xnþ<sup>m</sup>

ð Þ� xnþ<sup>m</sup> r0<sup>i</sup> þ r1ix<sup>1</sup> þ r2ix<sup>2</sup> þ … þ rð Þ <sup>n</sup>þ<sup>m</sup> ixnþ<sup>m</sup> � �

ð Þ xnþ<sup>m</sup>

<sup>l</sup> x tð Þ � l , θ ¼ 1, … , n, (13)

<sup>0</sup>, … , X<sup>θ</sup>

<sup>1</sup>; … ;c<sup>θ</sup> m � � <sup>¼</sup>

<sup>0</sup>ð Þ x tð Þ ; … ; <sup>X</sup><sup>θ</sup>

<sup>r</sup> are fuzzy sets.

� � are adjustable parameter vectors;

0;c<sup>θ</sup>

<sup>e</sup>u tð Þ, (14)

<sup>r</sup>, X<sup>θ</sup>

Singleton-type fuzzy model performs the mapping <sup>L</sup> : <sup>R</sup>nþ<sup>m</sup> ! <sup>R</sup> where the fuzzy conjunction operator is replaced by a product, and the operator of fuzzy rules aggregation, that is, by summation. The mapping L is defined by the following

ð Þ� x<sup>1</sup> μLX2<sup>i</sup>

ð Þ� <sup>x</sup><sup>2</sup> … � <sup>μ</sup>LXð Þ <sup>n</sup>þ<sup>m</sup> <sup>i</sup>

r,

ky tð Þþ � k ∑

yð Þ¼ t � r ð Þ 1; y tð Þ � 1 ; … ; y tð Þ � r is the state vector; xð Þ¼ t � s ðx tð Þ;

<sup>0</sup>; <sup>b</sup><sup>θ</sup>

<sup>1</sup> ð Þ y tð Þ � <sup>1</sup> ; … ; <sup>Y</sup><sup>θ</sup>

y^ðÞ¼ t c T

<sup>r</sup>, then

ð Þ� x<sup>1</sup> μLX2<sup>i</sup>

is the number of input variables in the model; and <sup>μ</sup>LXij xij � � is the membership

ð Þ� x<sup>1</sup> μLX2<sup>i</sup>

The expression for L mapping in the Takagi-Sugeno model looks as follows:

In Mamdani fuzzy systems, fuzzy logic techniques are used for describing the input vector's x mapping into the output value y, for example, Mamdani approxi-

Let the variables in (1) be fuzzy. In this case, (1) can be represented as a fuzzy

To form the model, product rules with linear finite-difference equations on the right-hand side are defined (for simplicity, we consider one-input case, i.e., P = 1):

> s l¼0 bθ

<sup>1</sup>; … ; <sup>b</sup><sup>θ</sup> s

By re-denoting input variables: ðu0ð Þt ; u1ð Þt ; … ; umð Þt Þ ¼ ð1; y tð Þ � 1 ; … ; y tð Þ � r ;

where m ¼ r ¼ s þ 1, one obtains the analytic form of the fuzzy model, intended

<sup>1</sup>, … , Y<sup>θ</sup>

<sup>r</sup>ð Þ y tð Þ � <sup>r</sup> ;X<sup>θ</sup>

<sup>s</sup>ð Þ x tð Þ � <sup>s</sup> � �,

where <sup>x</sup> <sup>=</sup>½ � <sup>x</sup>1; … ; xnþ<sup>m</sup> <sup>T</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>þ<sup>m</sup>; <sup>q</sup> is the number of rules in a fuzzy model; <sup>n</sup> <sup>þ</sup> <sup>m</sup>

which it is important to form knowledge based on data analysis.

of which the accuracy requirements prevail.

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

L xð Þ¼ <sup>∑</sup><sup>q</sup>

ð Þ� x<sup>1</sup> μLX2<sup>i</sup>

model of Takagi-Sugeno (TS) [15].

yθ

0,…, x tð Þ � <sup>s</sup> is <sup>X</sup><sup>θ</sup>

ðÞ¼ <sup>t</sup> <sup>a</sup><sup>θ</sup>

<sup>0</sup>; a<sup>θ</sup>

<sup>1</sup>; … ; <sup>b</sup><sup>θ</sup> s � �, and membership functions:

<sup>i</sup>¼1μLX1<sup>i</sup>

∑q <sup>i</sup>¼<sup>1</sup>μLX1<sup>i</sup>

> ∑q <sup>i</sup>¼<sup>1</sup>μLX1<sup>i</sup>

mation or a method based on a formal logical proof.

<sup>1</sup> ,…, y tð Þ � <sup>r</sup> is <sup>Y</sup><sup>θ</sup>

<sup>0</sup> þ ∑ r k¼1 aθ

<sup>1</sup>; … ; a<sup>θ</sup> r � �, b<sup>θ</sup> <sup>¼</sup> <sup>b</sup><sup>θ</sup>

x tð Þ � <sup>1</sup> ; … ; x tð ÞÞ � <sup>s</sup> is an input sequence; and <sup>Y</sup><sup>θ</sup>

x tð Þ; … x tð ÞÞ � <sup>s</sup> , finite-difference equation's coefficients: <sup>c</sup><sup>θ</sup>

formation tasks.

expression:

function.

L xð Þ¼ <sup>∑</sup><sup>q</sup>

<sup>i</sup>¼<sup>1</sup>μLX1<sup>i</sup>

If y tð Þ � <sup>1</sup> is <sup>Y</sup><sup>θ</sup>

where: <sup>a</sup><sup>θ</sup> <sup>¼</sup> <sup>a</sup><sup>θ</sup>

<sup>1</sup> ; … ; a<sup>θ</sup>

<sup>1</sup>ð Þ <sup>u</sup>1ð Þ<sup>t</sup> ; … ; <sup>U</sup><sup>θ</sup>

<sup>r</sup>; <sup>b</sup><sup>θ</sup>

<sup>m</sup>ð Þ umð Þ<sup>t</sup> � � <sup>¼</sup> <sup>Y</sup><sup>θ</sup>

for calculating the output ŷð Þt :

x tð Þ is <sup>X</sup><sup>θ</sup>

aθ <sup>0</sup>; a<sup>θ</sup>

Uθ

107

• one or more factors of this dynamics are weakly or not formalized.

In fuzzy systems, the most commonly used technique is the production rule one. The production rule consists of antecedent (or several premises) and consequent. In the general case, the premises are connected by logical operators AND and OR.

Fuzzy systems are based on production-type rules with linguistic variables used as premise and conclusion in the rule.

By renaming the variables, the linear dynamic plant's model can be represented as follows:

$$Y\_N = \sum\_{i=1}^{n+r} a\_i X\_i$$

The fuzzy system based on the production rules has the form:

A fuzzy model with n þ m input variables X ¼ f g X1;X2; … Xnþ<sup>m</sup> defined in space DX ¼ DX<sup>1</sup> � DX<sup>2</sup> � … � DXnþ<sup>m</sup> and with one-dimensional output Y is defined in the domain of reasoning DY.

Clear values of fuzzy variables Xi and Y are denoted by xi and y, respectively. LXi = {LXi,1,…, LXi,li } is the fuzzy domain of definition of the i -th input variable and Xi is the number of linguistic terms on which this fuzzy variable is defined.

LY = {LY1,…, LYly} is the domain of the fuzzy output variable.

l is the number of fuzzy values.

LYj is the name of the output linguistic term.

The rule base in the fuzzy Mamdani system is a set of fuzzy rules such as:

$$R\_j: \text{LX}\_{1,j\_1} \text{AND} \dots \text{AND} \text{LX}\_{n,j\_n} \to \text{LY}\_j. \tag{7}$$

The j-th fuzzy rule in the singleton-type system looks as follows:

$$R\_j: \text{LX}\_{1,j\_1} \text{ AND } \dots \text{AND } \text{LX}\_{n,j\_n} \to r\_j \tag{8}$$

where rj is a real number to estimate the output y. The j-th rule in the Takagi-Sugeno model [15] looks as follows:

$$R\_j: \text{LX}\_{1,j\_1} \text{AND} \dots \text{AND} \text{LX}\_{n+m,j\_{n+m}} \to r\_{0j} + r\_{1j} \text{x}\_1 + r\_{2j} \text{x}\_2 + \dots + r\_{(n+m)j} \text{x}\_{n+m} \tag{9}$$

where the output y is estimated by a linear function.

Thus, the fuzzy system performs the mapping <sup>L</sup> : <sup>R</sup><sup>n</sup>þ<sup>m</sup> ! <sup>R</sup>.

The grade of crisp variable xi membership in the fuzzy notion LXij is determined by membership functions μLXij(xi). The rule base is formed by the criterion of minimum output error which can be defined by the following expressions:

$$\frac{\sum\_{i=1}^{\mathsf{K}} |f(\mathbf{x}\_{i}) - L(\mathbf{x}\_{i})|}{\mathsf{K}}, \frac{\sqrt{\sum\_{i}^{\mathsf{K}} (f(\mathbf{x}\_{i}) - L(\mathbf{x}\_{i}))^{2}}}{\mathsf{K}}, \max\_{i \in \mathsf{K}} |f(\mathbf{x}\_{i}) - L(\mathbf{x}\_{i})| \tag{10}$$

where К is the number of samples.

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

5. Fuzzy virtual models

Applied Modern Control

dependence; and

as follows:

Rj : LX1,j

∑<sup>К</sup>

106

<sup>i</sup>¼<sup>1</sup> f xð Þ�<sup>i</sup> L xð Þ<sup>i</sup> j j <sup>К</sup> ,

where К is the number of samples.

tems in the following cases [3]:

as premise and conclusion in the rule.

defined in the domain of reasoning DY.

l is the number of fuzzy values.

LYj is the name of the output linguistic term.

Rj : LX1,j

Rj : LX1,j

where rj is a real number to estimate the output y.

where the output y is estimated by a linear function.

<sup>1</sup> AND … AND LX<sup>n</sup>þm,j

LXi = {LXi,1,…, LXi,li

Fuzzy models under uncertainty are advisable to apply in decision-making sys-

• dynamics of the investigated quality index is described by a complex nonlinear

In fuzzy systems, the most commonly used technique is the production rule one. The production rule consists of antecedent (or several premises) and consequent. In the general case, the premises are connected by logical operators AND and OR. Fuzzy systems are based on production-type rules with linguistic variables used

By renaming the variables, the linear dynamic plant's model can be represented

YN ¼ ∑ nþт i¼1 aiXi

A fuzzy model with n þ m input variables X ¼ f g X1;X2; … Xnþ<sup>m</sup> defined in space DX ¼ DX<sup>1</sup> � DX<sup>2</sup> � … � DXnþ<sup>m</sup> and with one-dimensional output Y is

Clear values of fuzzy variables Xi and Y are denoted by xi and y, respectively.

and Xi is the number of linguistic terms on which this fuzzy variable is defined.

The rule base in the fuzzy Mamdani system is a set of fuzzy rules such as:

<sup>1</sup> AND … AND LXn,j

<sup>1</sup> AND … AND LXn,j

The grade of crisp variable xi membership in the fuzzy notion LXij is determined

2

<sup>К</sup> , max

by membership functions μLXij(xi). The rule base is formed by the criterion of minimum output error which can be defined by the following expressions:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>i</sup> ð Þ f xð Þ�<sup>i</sup> L xð Þ<sup>i</sup>

} is the fuzzy domain of definition of the i -th input variable

<sup>n</sup>þ<sup>m</sup> ! <sup>r</sup>0<sup>j</sup> <sup>þ</sup> <sup>r</sup>1jx<sup>1</sup> <sup>þ</sup> <sup>r</sup>2jx<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>r</sup>ð Þ <sup>n</sup>þ<sup>m</sup> jxnþ<sup>m</sup> (9)

<sup>n</sup> ! LYj: (7)

<sup>n</sup> ! rj (8)

<sup>i</sup>∈<sup>К</sup> f xð Þ�<sup>i</sup> L xð Þ<sup>i</sup> j j (10)

The fuzzy system based on the production rules has the form:

LY = {LY1,…, LYly} is the domain of the fuzzy output variable.

The j-th fuzzy rule in the singleton-type system looks as follows:

The j-th rule in the Takagi-Sugeno model [15] looks as follows:

Thus, the fuzzy system performs the mapping <sup>L</sup> : <sup>R</sup><sup>n</sup>þ<sup>m</sup> ! <sup>R</sup>.

∑<sup>К</sup>

q

• one or more factors of this dynamics are weakly or not formalized.

Depending on the features of the object and the purpose of identification, various fuzzy models can be formed. Thus, the Takagi-Sugeno model is most suitable for objects with complex nonlinear dynamics, such as moving objects, in the control of which the accuracy requirements prevail.

A fuzzy model of the Mamdani type is suitable for problems in the solution of which it is important to form knowledge based on data analysis.

The singleton-type system may be used in both identification and knowledgeformation tasks.

Singleton-type fuzzy model performs the mapping <sup>L</sup> : <sup>R</sup>nþ<sup>m</sup> ! <sup>R</sup> where the fuzzy conjunction operator is replaced by a product, and the operator of fuzzy rules aggregation, that is, by summation. The mapping L is defined by the following expression:

$$L(\mathbf{x}) = \frac{\sum\_{i=1}^{q} \mu\_{\text{LX}\_{\text{ii}}}(\mathbf{x}\_1) \cdot \mu\_{\text{LX}\_{\text{ii}}}(\mathbf{x}\_2) \cdot \dots \cdot \mu\_{\text{LX}\_{(n+m)i}}(\mathbf{x}\_{n+m}) \cdot r\_i}{\sum\_{i=1}^{q} \mu\_{\text{LX}\_{\text{ii}}}(\mathbf{x}\_1) \cdot \mu\_{\text{LX}\_{\text{ii}}}(\mathbf{x}\_2) \cdot \dots \cdot \mu\_{\text{LX}\_{(n+m)i}}(\mathbf{x}\_{n+m})} \tag{11}$$

where <sup>x</sup> <sup>=</sup>½ � <sup>x</sup>1; … ; xnþ<sup>m</sup> <sup>T</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>þ<sup>m</sup>; <sup>q</sup> is the number of rules in a fuzzy model; <sup>n</sup> <sup>þ</sup> <sup>m</sup> is the number of input variables in the model; and <sup>μ</sup>LXij xij � � is the membership function.

The expression for L mapping in the Takagi-Sugeno model looks as follows:

$$L(\mathbf{x}) = \frac{\sum\_{i=1}^{q} \mu\_{\text{LX}\_{\text{L}}}(\mathbf{x}\_{1}) \cdot \mu\_{\text{LX}\_{\text{L}}}(\mathbf{x}\_{2}) \cdot \dots \cdot \mu\_{\text{LX}\_{(n+m)}}(\mathbf{x}\_{n+m}) \cdot \left(r\_{0i} + r\_{1i}\mathbf{x}\_{1} + r\_{2i}\mathbf{x}\_{2} + \dots + r\_{(n+m)n}\mathbf{x}\_{n+m}\right)}{\sum\_{i=1}^{q} \mu\_{\text{LX}\_{\text{L}}}(\mathbf{x}\_{1}) \cdot \mu\_{\text{LX}\_{\text{L}}}(\mathbf{x}\_{2}) \cdot \dots \cdot \mu\_{\text{LX}\_{(n+m)}}(\mathbf{x}\_{n+m})} \tag{12}$$

In Mamdani fuzzy systems, fuzzy logic techniques are used for describing the input vector's x mapping into the output value y, for example, Mamdani approximation or a method based on a formal logical proof.

Let the variables in (1) be fuzzy. In this case, (1) can be represented as a fuzzy model of Takagi-Sugeno (TS) [15].

To form the model, product rules with linear finite-difference equations on the right-hand side are defined (for simplicity, we consider one-input case, i.e., P = 1):

$$\begin{array}{l} \text{If } \mathfrak{y}(t-1) \text{ is } Y\_1^{\theta}, \dots, \mathfrak{y}(t-r) \text{ is } Y\_r^{\theta}, \\ \mathfrak{x}(t) \text{ is } X\_0^{\theta}, \dots, \mathfrak{x}(t-s) \text{ is } X\_r^{\theta}, \text{ then } \end{array}$$

$$\mathbf{y}^{\theta}(t) = a\_0^{\theta} + \sum\_{k=1}^{r} a\_k^{\theta} \mathbf{y}(t - k) + \sum\_{l=0}^{s} b\_l^{\theta} \mathbf{x}(t - l), \theta = \mathbf{1}, \dots, n,\tag{13}$$

where: <sup>a</sup><sup>θ</sup> <sup>¼</sup> <sup>a</sup><sup>θ</sup> <sup>0</sup>; a<sup>θ</sup> <sup>1</sup>; … ; a<sup>θ</sup> r � �, b<sup>θ</sup> <sup>¼</sup> <sup>b</sup><sup>θ</sup> <sup>0</sup>; <sup>b</sup><sup>θ</sup> <sup>1</sup>; … ; <sup>b</sup><sup>θ</sup> s � � are adjustable parameter vectors; yð Þ¼ t � r ð Þ 1; y tð Þ � 1 ; … ; y tð Þ � r is the state vector; xð Þ¼ t � s ðx tð Þ;

x tð Þ � <sup>1</sup> ; … ; x tð ÞÞ � <sup>s</sup> is an input sequence; and <sup>Y</sup><sup>θ</sup> <sup>1</sup>, … , Y<sup>θ</sup> <sup>r</sup>, X<sup>θ</sup> <sup>0</sup>, … , X<sup>θ</sup> <sup>r</sup> are fuzzy sets. By re-denoting input variables: ðu0ð Þt ; u1ð Þt ; … ; umð Þt Þ ¼ ð1; y tð Þ � 1 ; … ; y tð Þ � r ; x tð Þ; … x tð ÞÞ � <sup>s</sup> , finite-difference equation's coefficients: <sup>c</sup><sup>θ</sup> 0;c<sup>θ</sup> <sup>1</sup>; … ;c<sup>θ</sup> m � � <sup>¼</sup>

aθ <sup>0</sup>; a<sup>θ</sup> <sup>1</sup> ; … ; a<sup>θ</sup> <sup>r</sup>; <sup>b</sup><sup>θ</sup> <sup>1</sup>; … ; <sup>b</sup><sup>θ</sup> s � �, and membership functions:

$$\left(U\_1^{\theta}(u\_1(t)), \dots, U\_m^{\theta}(u\_m(t))\right) = \left(Y\_1^{\theta}(y(t-1)), \dots, Y\_r^{\theta}(y(t-r)), X\_0^{\theta}(\mathbf{x}(t)), \dots, X\_s^{\theta}(\mathbf{x}(t-s))\right),$$

where m ¼ r ¼ s þ 1, one obtains the analytic form of the fuzzy model, intended for calculating the output ŷð Þt :

$$
\hat{\mathbf{y}}(t) = \mathbf{c}^T \widetilde{\boldsymbol{u}}(t), \tag{14}
$$

where <sup>c</sup> <sup>¼</sup> <sup>c</sup><sup>1</sup> <sup>0</sup>; … ;c<sup>n</sup> <sup>0</sup>; … ;c<sup>1</sup> <sup>m</sup>; … ;c<sup>n</sup> m � �<sup>T</sup> is the vector of the adjustable parameters; <sup>e</sup>uT tðÞ¼ <sup>u</sup>0ð Þ<sup>t</sup> <sup>β</sup><sup>1</sup> ð Þ<sup>t</sup> ; … ; <sup>u</sup>0ð Þ<sup>t</sup> <sup>β</sup><sup>θ</sup> ð Þ<sup>t</sup> ; … ; umð Þ<sup>t</sup> <sup>β</sup><sup>1</sup> ð Þ<sup>t</sup> ; … ; umð Þ<sup>t</sup> <sup>β</sup>nð Þ<sup>t</sup> � � is the extended input vector;

$$\boldsymbol{\beta}^{\theta}(t) = \frac{\boldsymbol{U}\_{1}^{\theta}(\boldsymbol{u}\_{1}(t)) \otimes \dots \otimes \boldsymbol{U}\_{m}^{\theta}(\boldsymbol{u}\_{m}(t))}{\sum\_{\theta=1}^{N} \left(\boldsymbol{U}\_{1}^{\theta}(\boldsymbol{u}\_{1}(t)) \otimes \dots \otimes \boldsymbol{U}\_{m}^{\theta}(\boldsymbol{u}\_{m}(t))\right)} \tag{15}$$

Assume the associative search procedure is determined by the predicate

specifications) as a fuzzy conjunction of input variables:

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

for all X1, X2, … Xn from DX ¼ DX1 � DX2 � … � DXn.

Ξ Pa

increases significantly the algorithms'speed.

criterion of minimum distance to the center:

6.1 Virtual clustering ("impostor" method)

satisfying the criterion of associative search.

), which interprets input variables' limits (specified, say, by process

; <sup>R</sup><sup>a</sup> ð Þ¼ fð Þ <sup>X</sup><sup>1</sup> : <sup>x</sup><sup>1</sup> <sup>⊂</sup> <sup>A</sup><sup>1</sup> <sup>∧</sup>ð Þ <sup>X</sup><sup>2</sup> : <sup>x</sup><sup>2</sup> <sup>⊂</sup> <sup>A</sup> … <sup>ð</sup>Xn : xn <sup>⊂</sup> An<sup>g</sup>

) possesses the value FALSE, will be discarded automatically. This reduces

Then, the production rules, where fuzzy variables possess such values that

drastically the number of production rules employed in the fuzzy model and thus

6. Solving the associative search problem by means of clusterization

min <sup>k</sup> <sup>∑</sup> K k¼1

minimum distance between any two of their members can be applied. This approach provides significant savings in computing resources compared to searching through a full search. However, such a combination of clusters does not yet guarantee the solution of the problem. The approach described below looks the

The current input vector at any particular time can be assigned to a specific cluster. This can, for example, be done by the criterion of the minimum distance to

> gk �x´<sup>N</sup> 2

Let ´xN denote the center of the cluster Ar. If additional selection of input vectors from the archive is required (to form a system of a sufficient number of equations to identify the system using the associative search method), clusters with the minimum distance between their centers and x´N are selected for the join. This approach allows not only to discard a significant number of vectors removed from x´N, but also to select from the archive the maximum possible number of vectors

min <sup>k</sup> <sup>∑</sup> K k¼1

The associative search problem is solved by clustering technique (both crisp and

The current vector under investigation is attributed to a certain cluster per the

gk � x´<sup>N</sup> 2 ,

where ´xN ∈X is the current input vector of the control plant under investigation. Within this cluster, the vectors are sought that satisfy the assigned associative criterion. It may turn out that one cannot find within this cluster the number of vectors necessary to solve the problem of forecasting using the method of least squares. In this case, one of the known methods of combining two clusters with the

Ξ(Pa , R<sup>a</sup>

Ξ(Pa , R<sup>a</sup>

techniques

most reasonable.

the center. Let

109

be satisfied for k = r.

fuzzy) in the following way.

is a fuzzy function where ⊗ denotes the minimization operation of fuzzy product.

If for t = 0, the vectorc 0ð Þ¼ 0, the correcting mn � nm matrix Qð Þ 0 (m is the number of input vectors, n is the number of production rules), and the values of u tð Þ, t ¼ 1, … , N are specified, the parameter vector c tð Þ is calculated using the known multi-step LSM:

$$c(t) = c(t-1) + Q(t)\widetilde{u}(t)\left[y(t) - c^T(t-1)\widetilde{u}(t)\right]$$

$$Q(t) = Q(t-1) - \frac{Q(t-1)\widetilde{u}(t)\widetilde{u}^T(t)Q(t-1)}{1 + \widetilde{u}T(t)Q(t-1)\widetilde{u}(t)}\tag{16}$$

Qð Þ¼ 0 γI, γ >>1, where I is the unit matrix.

The above equations show that even in case of one-dimensional input and few production rules, a lot of observations are needed to apply LSM which makes the fuzzy model too unwieldy. Therefore, only a part of the whole set of rules (r < n) should be chosen according to a certain criterion.

The application of the associative search techniques where one or more model parameters are fuzzy is reduced to such determination of the predicate <sup>Ξ</sup> <sup>¼</sup> <sup>Ξ</sup><sup>i</sup> <sup>R</sup><sup>a</sup> <sup>0</sup>, <sup>R</sup><sup>a</sup>; <sup>T</sup><sup>a</sup> � � � � , so that the number of production rules in the TS model is significantly reduced according to some criterion.

For example, the following matrix:

$$\begin{array}{ccccccccc}\beta\_1^{\Theta\_l} & \dots & \beta\_P^{\Theta\_l} & & & & \\ & \dots & \dots & \dots & & & & \\ \beta\_1^{\Theta\_{l-i}} & \dots & \beta\_P^{\Theta\_{l-i}} & & & & \\ \end{array} \tag{17}$$

can be defined for P-dimensional input vectors at time steps t�j, j = 1, …, s. If the rows of this matrix are ranged, say, w.r.t. ∑<sup>P</sup> <sup>p</sup>¼<sup>1</sup> <sup>β</sup><sup>Θ</sup><sup>i</sup> p � � � � � � decrease and a certain number of rows are selected, then such selection combined with condition (4) will determine the predicate Ξ and, respectively, the criterion for selecting the images (sets of input vector) from the history.

Let us range the rows of this matrix, for example, subject to the criterion of descending the values ∑<sup>P</sup> <sup>p</sup>¼<sup>1</sup> <sup>β</sup><sup>Θ</sup><sup>i</sup> p � � � � � �, and select a certain number of rows. Such selection combined with condition (4) defines the predicate <sup>Ξ</sup> <sup>¼</sup> <sup>Ξ</sup><sup>i</sup> <sup>R</sup><sup>a</sup> <sup>0</sup>, <sup>R</sup><sup>a</sup>; <sup>T</sup><sup>a</sup> � � � � , and, respectively, the image selection criterion (sets of input vectors) from the archive.

#### 5.1 Fuzzy associative search

Notwithstanding all benefits delivered by fuzzy techniques, their application significantly reduces the calculation speed that is critical for predicting the dynamics of some plants. This consideration coupled with the principal impossibility of formalizing some factors necessitated the development of algorithms that could combine all advantages of fuzzy approach and associative search algorithms.

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

where <sup>c</sup> <sup>¼</sup> <sup>c</sup><sup>1</sup>

Applied Modern Control

input vector;

product.

<sup>Ξ</sup> <sup>¼</sup> <sup>Ξ</sup><sup>i</sup> <sup>R</sup><sup>a</sup>

<sup>e</sup>uT tðÞ¼ <sup>u</sup>0ð Þ<sup>t</sup> <sup>β</sup><sup>1</sup>

known multi-step LSM:

<sup>0</sup>; … ;c<sup>n</sup>

βθ ðÞ¼ t

<sup>0</sup>; … ;c<sup>1</sup>

� �<sup>T</sup>

ð Þ<sup>t</sup> ; … ; <sup>u</sup>0ð Þ<sup>t</sup> <sup>β</sup><sup>θ</sup>

Qð Þ¼ 0 γI, γ >>1, where I is the unit matrix.

should be chosen according to a certain criterion.

significantly reduced according to some criterion.

For example, the following matrix:

rows of this matrix are ranged, say, w.r.t. ∑<sup>P</sup>

<sup>p</sup>¼<sup>1</sup> <sup>β</sup><sup>Θ</sup><sup>i</sup> p � � � � �

combined with condition (4) defines the predicate <sup>Ξ</sup> <sup>¼</sup> <sup>Ξ</sup><sup>i</sup> <sup>R</sup><sup>a</sup>

input vector) from the history.

5.1 Fuzzy associative search

108

descending the values ∑<sup>P</sup>

<sup>m</sup>; … ;c<sup>n</sup> m

Uθ

c tðÞ¼ c tð Þþ � <sup>1</sup> Q tð Þeu tð Þ y tðÞ� <sup>c</sup>

Q tðÞ¼ Q tð Þ� � <sup>1</sup> Q tð Þ � <sup>1</sup> <sup>e</sup>u tð Þe<sup>u</sup>

parameters are fuzzy is reduced to such determination of the predicate

βΘt

βΘt�<sup>s</sup>

∑<sup>N</sup> <sup>θ</sup>¼<sup>1</sup> <sup>U</sup><sup>θ</sup>

ð Þ<sup>t</sup> ; … ; umð Þ<sup>t</sup> <sup>β</sup><sup>1</sup>

is a fuzzy function where ⊗ denotes the minimization operation of fuzzy

<sup>1</sup> ð Þ <sup>u</sup>1ð Þ<sup>t</sup> <sup>⊗</sup> … <sup>⊗</sup> <sup>U</sup><sup>θ</sup>

If for t = 0, the vectorc 0ð Þ¼ 0, the correcting mn � nm matrix Qð Þ 0 (m is the number of input vectors, n is the number of production rules), and the values of u tð Þ, t ¼ 1, … , N are specified, the parameter vector c tð Þ is calculated using the

The above equations show that even in case of one-dimensional input and few production rules, a lot of observations are needed to apply LSM which makes the fuzzy model too unwieldy. Therefore, only a part of the whole set of rules (r < n)

The application of the associative search techniques where one or more model

<sup>0</sup>, <sup>R</sup><sup>a</sup>; <sup>T</sup><sup>a</sup> � � � � , so that the number of production rules in the TS model is

<sup>1</sup> … <sup>β</sup>Θ<sup>t</sup>

………

<sup>1</sup> … <sup>β</sup>Θt�<sup>s</sup>

of rows are selected, then such selection combined with condition (4) will determine the predicate Ξ and, respectively, the criterion for selecting the images (sets of

Let us range the rows of this matrix, for example, subject to the criterion of

respectively, the image selection criterion (sets of input vectors) from the archive.

Notwithstanding all benefits delivered by fuzzy techniques, their application significantly reduces the calculation speed that is critical for predicting the dynamics of some plants. This consideration coupled with the principal impossibility of formalizing some factors necessitated the development of algorithms that could combine all advantages of fuzzy approach and associative search algorithms.

can be defined for P-dimensional input vectors at time steps t�j, j = 1, …, s. If the

P

P

<sup>p</sup>¼<sup>1</sup> <sup>β</sup><sup>Θ</sup><sup>i</sup> p � � � � �

<sup>1</sup>ð Þ <sup>u</sup>1ð Þ<sup>t</sup> <sup>⊗</sup> … <sup>⊗</sup> <sup>U</sup><sup>θ</sup>

ð Þ<sup>t</sup> ; … ; umð Þ<sup>t</sup> <sup>β</sup>nð Þ<sup>t</sup> � � is the extended

is the vector of the adjustable parameters;

<sup>m</sup>ð Þ umð Þt

<sup>T</sup>ð Þ <sup>t</sup> � <sup>1</sup> <sup>e</sup>u tð Þ� �

<sup>T</sup>ð Þ<sup>t</sup> Q tð Þ � <sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup>uT tð ÞQ tð Þ � <sup>1</sup> <sup>e</sup>u tð Þ (16)

(17)

� decrease and a certain number

<sup>0</sup>, <sup>R</sup><sup>a</sup>; <sup>T</sup><sup>a</sup> � � � � , and,

�, and select a certain number of rows. Such selection

<sup>m</sup>ð Þ umð Þ<sup>t</sup> � � (15)

Assume the associative search procedure is determined by the predicate Ξ(Pa , R<sup>a</sup> ), which interprets input variables' limits (specified, say, by process specifications) as a fuzzy conjunction of input variables:

$$\Xi(P^t, R^t) = \{ (X\_1 : \varkappa\_1 \subset A\_1) \land (X\_2 : \varkappa\_2 \subset A) \dots (X\_n : \varkappa\_n \subset A\_n) \}$$

for all X1, X2, … Xn from DX ¼ DX1 � DX2 � … � DXn.

Then, the production rules, where fuzzy variables possess such values that Ξ(Pa , R<sup>a</sup> ) possesses the value FALSE, will be discarded automatically. This reduces drastically the number of production rules employed in the fuzzy model and thus increases significantly the algorithms'speed.

#### 6. Solving the associative search problem by means of clusterization techniques

The associative search problem is solved by clustering technique (both crisp and fuzzy) in the following way.

The current vector under investigation is attributed to a certain cluster per the criterion of minimum distance to the center:

$$\min\_{\mathbf{k}} \sum\_{k=1}^{K} \left\lVert \mathbf{g}\_k - \mathbf{\acute{x}}\_N \right\rVert^2,$$

where ´xN ∈X is the current input vector of the control plant under investigation.

Within this cluster, the vectors are sought that satisfy the assigned associative criterion. It may turn out that one cannot find within this cluster the number of vectors necessary to solve the problem of forecasting using the method of least squares. In this case, one of the known methods of combining two clusters with the minimum distance between any two of their members can be applied. This approach provides significant savings in computing resources compared to searching through a full search. However, such a combination of clusters does not yet guarantee the solution of the problem. The approach described below looks the most reasonable.

#### 6.1 Virtual clustering ("impostor" method)

The current input vector at any particular time can be assigned to a specific cluster. This can, for example, be done by the criterion of the minimum distance to the center.

Let

$$\min\_{\mathbf{k}} \sum\_{k=1}^{K} \left|| \mathbf{g}\_k - \mathbf{x}\_N \right||^2$$

be satisfied for k = r.

Let ´xN denote the center of the cluster Ar. If additional selection of input vectors from the archive is required (to form a system of a sufficient number of equations to identify the system using the associative search method), clusters with the minimum distance between their centers and x´N are selected for the join. This approach allows not only to discard a significant number of vectors removed from x´N, but also to select from the archive the maximum possible number of vectors satisfying the criterion of associative search.

After the completion of this procedure, assigning x´N as the cluster center A<sup>r</sup> is canceled, and the procedure of the formation of virtual (relevant to the certain time instant) models continues using conventional clustering algorithms.

8. Application of wavelet approach to the analysis of nonstationary

appeared and is developed, as well as its numerous applications.

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

Within the last two decades, applying wavelet transform (WT) to the analysis of nonstationary processes has been widely used. The wavelet transform of signals is a generalization of the spectral analysis, for instance, with regard to the Fourier transform. First papers on the wavelet analysis of time (spatial) series with a pronounced heterogeneity appeared in the end of 1980s [16, 17]. The method was positioned as an alternative to the Fourier transform, localizing the frequencies but not providing the time extension of a process under study. In sequel, the theory of wavelets has

The scope of wavelet analysis today is very wide: it includes the synthesis and processing of nonstationary signals, compression and coding of information, image recognition and image analysis, the study of functions and time-dependent signals and inhomogeneity in space. The approach is effective for tasks where the results of the analysis should contain not only the characteristics of the frequency signal (signal power distribution by frequency components) but also information about local coordinates in which certain groups of frequency components manifest themselves or in which rapid changes in the frequency components of the signal occur. A significant number of practical applications have been created, including in health care, the study of geophysical fields, temporary meteorological series, and predic-

The wavelet analysis method consists in applying a special linear conversion of

dynamics of real objects and processes in depth. For example, it can be processes in manufacturing. The wavelet transform (WT) of a one-dimensional signal is its representation in the form of a generalized Fourier series (or Fourier integral) over a system of basis functions called the "wavelet." A wavelet is characterized by the fact that the function that forms it (a wavelet-formation function or a wavelet matrix) is distinguished by a certain scale (frequency) and localization in time

The time scale is analogous to the oscillation period, that is, it is inverse one with regard to the frequency, and the shift interprets the displacement of the signal over

The wavelet transform performs the projection of a one-dimensional process into a two-dimensional surface in three-dimensional space. The frequency and time

At the same time, it becomes realistic to simultaneously study the properties of the process being studied both in the time domain and in the frequency domain. It becomes possible to investigate the dynamics of the frequency process and its local features. This allows us to identify the coordinates at which certain frequencies

The graphical representation of the wavelet analysis can be displayed in the form of isolines, illustrating the change in the intensities of wavelet transform coefficients at different time scales, and also for revealing local extrema of surfaces.

If a function is used in the Fourier transform that generates an orthonormal basis of space by means of a scale transformation, then the wavelet transform is formed using a basis function localized in a bounded domain, although defined on the

The wavelet transform, as a mathematical tool, serves mainly to analyze data in

Wavelet transformation, as a mathematical tool, provides the ability to analyze data in the time and frequency domains simultaneously. The wavelet transform can

signals. In particular, it becomes possible to study the physical properties or

based on the time shift and the change in the time scale.

are treated as independent variables.

manifest themselves most significantly.

processes

tion of earthquakes [18].

the time axis.

whole numerical axis.

111

the time and frequency domains.

#### 7. Case study: oil refining product quality modeling

Key process equipment of an atmospheric distillation unit comprises of cold and hot crude oil preheat trains, desalter, a flash drum or, instead, a pre-flash column with an overhead reflux drum, atmospheric heaters, and an atmospheric distillation column with a reflux drum and three side stripping columns for middle distillates (typically, kerosene, light diesel and heavy diesel aka atmospheric gas oil). The naphtha streams from both reflux drums are re-combined and further sent to downstream stabilization and rerun facilities. The atmospheric residuum from the bottom of the main atmospheric column is typically streamed to a vacuum distillation section.

To obtain a soft sensor model for the 10% distillation point of a kerosene stream, the lab data for this quality were collected along with process data from the atmospheric column. The predictive model is formed by means of the associative search method. The process data were analyzed, and process variables measured by plant instruments were selected for modeling along with the distillation point sampled at the plant and measured in the refinery's laboratory. Based on the preliminary data analysis, the following linear predictive model was developed:

$$T(t) = \sum\_{i=1}^{4} b\_i F\_i(t-1) + b\_5 F\_5(t-3) + b\_6 F\_6(t-5) + \sum\_{i=7}^{12} b\_i F\_i(t-7),\tag{18}$$

where T tð Þ is the desired estimate; Fið Þ t � j are various process parameters, such as flows, temperatures, and pressures, measured directly at the plant; and b1, … , b<sup>12</sup> are model's coefficients.

The forecast was calculated per linear and associative models for 10,525 time steps (1 step = 10 min). Figure 4 shows simulation results for the steps <sup>t</sup> <sup>¼</sup> <sup>102</sup>´ , 301.

Figure 4. Kerosene 10% distillation point forecast.

After the completion of this procedure, assigning x´N as the cluster center A<sup>r</sup> is canceled, and the procedure of the formation of virtual (relevant to the certain time

Key process equipment of an atmospheric distillation unit comprises of cold and hot crude oil preheat trains, desalter, a flash drum or, instead, a pre-flash column with an overhead reflux drum, atmospheric heaters, and an atmospheric distillation column with a reflux drum and three side stripping columns for middle distillates (typically, kerosene, light diesel and heavy diesel aka atmospheric gas oil). The naphtha streams from both reflux drums are re-combined and further sent to downstream stabilization and rerun facilities. The atmospheric residuum from the bottom of the main atmospheric column is typically streamed to a vacuum distilla-

To obtain a soft sensor model for the 10% distillation point of a kerosene stream, the lab data for this quality were collected along with process data from the atmospheric column. The predictive model is formed by means of the associative search method. The process data were analyzed, and process variables measured by plant instruments were selected for modeling along with the distillation point sampled at the plant and measured in the refinery's laboratory. Based on the preliminary data

where T tð Þ is the desired estimate; Fið Þ t � j are various process parameters, such as flows, temperatures, and pressures, measured directly at the plant; and b1, … , b<sup>12</sup>

The forecast was calculated per linear and associative models for 10,525 time

12 i¼7

biFið Þ t � 7 , (18)

instant) models continues using conventional clustering algorithms.

7. Case study: oil refining product quality modeling

analysis, the following linear predictive model was developed:

biFið Þþ t � 1 b5F5ð Þþ t � 3 b6F6ð Þþ t � 5 ∑

steps (1 step = 10 min). Figure 4 shows simulation results for the steps

tion section.

Applied Modern Control

T tðÞ¼ ∑ 4 i¼1

are model's coefficients.

, 301.

<sup>t</sup> <sup>¼</sup> <sup>102</sup>´

Figure 4.

110

Kerosene 10% distillation point forecast.

#### 8. Application of wavelet approach to the analysis of nonstationary processes

Within the last two decades, applying wavelet transform (WT) to the analysis of nonstationary processes has been widely used. The wavelet transform of signals is a generalization of the spectral analysis, for instance, with regard to the Fourier transform.

First papers on the wavelet analysis of time (spatial) series with a pronounced heterogeneity appeared in the end of 1980s [16, 17]. The method was positioned as an alternative to the Fourier transform, localizing the frequencies but not providing the time extension of a process under study. In sequel, the theory of wavelets has appeared and is developed, as well as its numerous applications.

The scope of wavelet analysis today is very wide: it includes the synthesis and processing of nonstationary signals, compression and coding of information, image recognition and image analysis, the study of functions and time-dependent signals and inhomogeneity in space. The approach is effective for tasks where the results of the analysis should contain not only the characteristics of the frequency signal (signal power distribution by frequency components) but also information about local coordinates in which certain groups of frequency components manifest themselves or in which rapid changes in the frequency components of the signal occur. A significant number of practical applications have been created, including in health care, the study of geophysical fields, temporary meteorological series, and prediction of earthquakes [18].

The wavelet analysis method consists in applying a special linear conversion of signals. In particular, it becomes possible to study the physical properties or dynamics of real objects and processes in depth. For example, it can be processes in manufacturing. The wavelet transform (WT) of a one-dimensional signal is its representation in the form of a generalized Fourier series (or Fourier integral) over a system of basis functions called the "wavelet." A wavelet is characterized by the fact that the function that forms it (a wavelet-formation function or a wavelet matrix) is distinguished by a certain scale (frequency) and localization in time based on the time shift and the change in the time scale.

The time scale is analogous to the oscillation period, that is, it is inverse one with regard to the frequency, and the shift interprets the displacement of the signal over the time axis.

The wavelet transform performs the projection of a one-dimensional process into a two-dimensional surface in three-dimensional space. The frequency and time are treated as independent variables.

At the same time, it becomes realistic to simultaneously study the properties of the process being studied both in the time domain and in the frequency domain. It becomes possible to investigate the dynamics of the frequency process and its local features. This allows us to identify the coordinates at which certain frequencies manifest themselves most significantly.

The graphical representation of the wavelet analysis can be displayed in the form of isolines, illustrating the change in the intensities of wavelet transform coefficients at different time scales, and also for revealing local extrema of surfaces.

If a function is used in the Fourier transform that generates an orthonormal basis of space by means of a scale transformation, then the wavelet transform is formed using a basis function localized in a bounded domain, although defined on the whole numerical axis.

The wavelet transform, as a mathematical tool, serves mainly to analyze data in the time and frequency domains.

Wavelet transformation, as a mathematical tool, provides the ability to analyze data in the time and frequency domains simultaneously. The wavelet transform can provide time-frequency information about a function that in many practical situations is more relevant than information obtained through standard Fourier analysis. In [7], it was shown that a sufficient condition for the stability of plant (1) is as

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup>dxs

<sup>s</sup>¼<sup>1</sup>bs, <sup>2</sup>dxs

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup>dxs

l,kð Þ t � R � 1

l,kð Þ t � R � 2

l,kð Þ t � m þ 1

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup><sup>c</sup> xs L,kð Þ t � 1

<sup>s</sup>¼<sup>1</sup>bs, <sup>2</sup><sup>c</sup> xs L,kð Þ t � 2

L,kð Þ t � R � 1

<sup>s</sup>¼<sup>1</sup>bs,Rc xs L,kð Þ t � R

 

> < 1;

l,kð Þ t � 1

l,kð Þ t � 2

l,kð Þ t � m

 < 1

 

l,kð Þ t � 1

l,kð Þ t � m � 1

l,kð Þ t � R

l,kð Þ t � R þ 1

<sup>s</sup>¼<sup>1</sup>bs,mdxs

l,kð Þ t � m � 2

l,kð Þ t � m � 1

<sup>s</sup>¼<sup>1</sup>bs,Rdxs

 < 1,

 

< 1, … ,

 < 1,

< 1, … ,

< 1, … ,

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup><sup>c</sup> xs L,kð Þ t � 1

L,kð Þ t � R � 2

L,kð Þ t � m

L,kð Þ t � m þ 1

2. if m < R, R ¼ maxrs s¼<sup>1</sup>,´<sup>S</sup> , then the condition for the detailing coefficients:

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup>dxs

<sup>s</sup>¼<sup>1</sup>bs, <sup>2</sup>dxs

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup>dxs

L,kð Þ t � R � 1

l,kð Þ t � R � 1

<sup>s</sup>¼<sup>1</sup>bs,Rdxs

, then the condition for the detailing coefficients:

l,kð Þ t � 1

l,kð Þ t � 2

l,kð Þ t � R

< 1, … ,

l,kð Þ t � 1

 

> < 1

 < 1,

 

> < 1,

 < 1,

 

> < 1,

< 1, … ,

< 1, … ,

(22)

(23)

follows: for ∀k ¼ 1,´N meeting the inequalities is to be provided:

l,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

l,kð Þþ <sup>t</sup> � <sup>2</sup> <sup>∑</sup><sup>S</sup>

�aRþ1d y

l,kð Þþ <sup>t</sup> � <sup>R</sup> <sup>∑</sup><sup>S</sup>

�2amd y l,kð Þ t � m

am�<sup>1</sup>d y

L,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

L,kð Þþ <sup>t</sup> � <sup>2</sup> <sup>∑</sup><sup>S</sup>

�aRþ<sup>1</sup>c y

L,kð Þþ <sup>t</sup> � <sup>R</sup> <sup>∑</sup><sup>S</sup>

�2amc y

am�<sup>1</sup>c y

l,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

l,kð Þþ <sup>t</sup> � <sup>2</sup> <sup>∑</sup><sup>S</sup>

<sup>s</sup>¼<sup>1</sup>bs,mþ<sup>1</sup>dxs

l,kð Þþ <sup>t</sup> � <sup>m</sup> <sup>∑</sup><sup>S</sup>

<sup>s</sup>¼<sup>1</sup>bs,mþ<sup>2</sup>dxs

<sup>s</sup>¼<sup>1</sup>bs,mþ<sup>1</sup>dxs

�2∑<sup>S</sup>

<sup>s</sup>¼<sup>1</sup>bs,R�<sup>1</sup>dxs

∑S

l,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

�∑<sup>S</sup>

�∑<sup>S</sup>

∑S

  2d y lkð Þt

L,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

�aRþ<sup>2</sup>c y

aRþ<sup>1</sup>c y

> 

2c y L,kð Þt

l,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

�aRþ<sup>2</sup>dy

aRþ1d<sup>y</sup>

  2d<sup>y</sup> l,kð Þt

a1dy

 

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

> �a2d y

 

a1d y

> 

for the approximating coefficients:

  a1c y

 

�a2c y

> a1c y

> >

aRc y

> a1d y

 

�a2d y

 

a1d y

 

113

amd y

> 

 

aRd y

> 

1. if <sup>m</sup> <sup>&</sup>gt; R, R <sup>¼</sup> max rs <sup>s</sup>¼1, <sup>S</sup>

There are examples of the use of wavelet analysis in identification problems [5]. In the literature, it is noted that wavelets are used mainly to identify nonlinear systems with a certain structure, where unknown time-varying coefficients can be represented as a linear combination of basis wavelet functions [6, 7]. It was stated that along with the usual ("direct") wavelet analysis, biorthogonal bursts [18], wavelet frames [19], or wavelet networks [20] can be used to identify the system.

There exist many different ways of applying wavelets for linear system identification. In Ref. [21], the identification of systems with a specific input/output structure was studied, in which the parameters are identified via spline-wavelets and their derivatives. In paper [22], an extended use of an orthonormal transformation least squares method is presented in order to reveal useful information from data.

#### 9. Conditions of the associative model stability in the aspect of the analysis of the spectrum of multi-scale wavelet expansion

Let (1) be an associative search model. We represent the multi-scale wavelet decomposition for the current input vector x tð Þ for a fixed level of detail L [7]:

$$\begin{aligned} \varkappa(t) &= \sum\_{k=1}^{N} c\_{L,k}^{x}(t)\varrho\_{L,k}(t) + \sum\_{l=1}^{L} \sum\_{k=1}^{N} d\_{l,k}^{x}(t)\varphi\_{l,k}(t), \\ \varkappa(t) &= \sum\_{k=1}^{N} c\_{L,k}^{y}(t)\varrho\_{L,k}(t) + \sum\_{l=1}^{L} \sum\_{k=1}^{N} d\_{l,k}^{y}(t)\varphi\_{l,k} \Im(t), \end{aligned} \tag{19}$$

where L is the depth of the multi-scale expansion; φL,kð Þt are scaling functions; ψl,kð Þt are the wavelet functions that are obtained from the mother wavelets by tension/combustion and shift

$$
\psi\_{l,k}(t) = \mathcal{2}^{l/2} \psi\_{\text{mother}}\left(\mathcal{2}^l t - k\right),
$$

(as the mother wavelets, in the present case, we consider the Haar wavelets); l is the level of data detailing; cL,k are the scaling coefficients; and dl,k are the detailing coefficients. The coefficients are calculated by use of the Mallat algorithm [17].

Let us expand Eq. (1) over wavelets:

$$\begin{aligned} &\sum\_{k=1}^{N}c\_{Lk}^{\mathcal{V}}(t)\rho\_{Lk}(t)+\sum\_{l=1}^{L}\sum\_{k=1}^{N}d\_{lk}^{\mathcal{V}}(t)\boldsymbol{\mu}\_{lk}(t) = \sum\_{k=1}^{N}\left(\sum\_{i=1}^{m}a\_{i}c\_{Lk}^{\mathcal{V}}(t-i)\boldsymbol{\rho}\_{Lk}(t-i)\right) \\ &+\sum\_{l=1}^{L}\sum\_{k=1}^{N}\left(\sum\_{i=1}^{m}a\_{i}d\_{lk}^{\mathcal{V}}(t-i)\boldsymbol{\mu}\_{lk}(t-i)\right)+\sum\_{k=1}^{N}\left(\sum\_{i=1}^{s}\sum\_{j=1}^{r\_{i}}b\_{jl}c\_{Lk}^{\prime}(t-j)\boldsymbol{\rho}\_{Lk}(t-j)\right) \\ &+\sum\_{l=1}^{L}\sum\_{k=1}^{N}\left(\sum\_{i=1}^{s}\sum\_{j=1}^{r\_{i}}b\_{jl}d\_{lk}^{\prime}(t-j)\boldsymbol{\mu}\_{lk}(t-j)\right) \end{aligned}$$

Let us consider individually the detailing and approximating parts correspondingly:

$$\mu(t)\boldsymbol{\upmu}\_{lk}(t) = \sum\_{i=1}^{m} a\_i d\_{lk}^{\boldsymbol{\upmu}}(t-i)\boldsymbol{\upmu}\_{lk}(t-i) + \sum\_{s=1}^{S} \sum\_{j=1}^{r\_l} b\_{jl} d\_{lk}^{\boldsymbol{\upmu}}(t-j)\boldsymbol{\upmu}\_{lk}(t-j),\tag{20}$$

$$
\sigma\_{Lk}^{\mathcal{I}}(t)\rho\_{Lk}(t) = \sum\_{i=1}^{m} \hat{a}\_i c\_{Lk}^{\mathcal{I}}(t-i)\rho\_{Lk}(t-i) + \sum\_{s=1}^{S} \sum\_{j=1}^{r\_s} \hat{b}\_{jl} c\_{Lk}^{\mathcal{I}}(t-j)\rho\_{Lk}(t-j). \tag{21}
$$

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

provide time-frequency information about a function that in many practical situations is more relevant than information obtained through standard Fourier analysis. There are examples of the use of wavelet analysis in identification problems [5].

In the literature, it is noted that wavelets are used mainly to identify nonlinear systems with a certain structure, where unknown time-varying coefficients can be represented as a linear combination of basis wavelet functions [6, 7]. It was stated that along with the usual ("direct") wavelet analysis, biorthogonal bursts [18], wavelet frames [19], or wavelet networks [20] can be used to identify the system. There exist many different ways of applying wavelets for linear system identification. In Ref. [21], the identification of systems with a specific input/output structure was studied, in which the parameters are identified via spline-wavelets and their derivatives. In paper [22], an extended use of an orthonormal transformation least squares method is presented in order to reveal useful information from data.

9. Conditions of the associative model stability in the aspect of the analysis of the spectrum of multi-scale wavelet expansion

L,kð Þt φL,kðÞþt ∑

L,kð Þt φL,kðÞþt ∑

<sup>ψ</sup>l,kðÞ¼ <sup>t</sup> <sup>2</sup><sup>l</sup>=<sup>2</sup>

x tðÞ¼ ∑ N k¼1 c x

y tðÞ¼ ∑ N k¼1 c y

Let us expand Eq. (1) over wavelets:

L l¼1 ∑ N k¼1 dy

lkð Þ t � i ψlkð Þ t � i � �

!

lkð Þ t � j ψlkð Þ t � j

lkð Þ t � i ψlkð Þþ t � i ∑

Lkð Þ t � i φLkð Þþ t � i ∑

Lkð Þt φLkðÞþt ∑

∑ m i¼1 aid y

∑ S s¼1 ∑ rs j¼1 bsjds

m i¼1 aid<sup>y</sup>

> m i¼1 a^ic y

tension/combustion and shift

Applied Modern Control

∑ N k¼1 c y

þ ∑ L l¼1 ∑ N k¼1

þ ∑ L l¼1 ∑ N k¼1

c y

112

ð Þt ψlkðÞ¼ t ∑

Lkð Þt φLkðÞ¼ t ∑

Let (1) be an associative search model. We represent the multi-scale wavelet decomposition for the current input vector x tð Þ for a fixed level of detail L [7]:

> L l¼1 ∑ N k¼1 dx

L l¼1 ∑ N k¼1 dy

ψmother 2<sup>l</sup>

N k¼1

þ ∑ N k¼1

Let us consider individually the detailing and approximating parts correspondingly:

S s¼1 ∑ rs j¼1 bsjds

> S s¼1 ∑ rs j¼1 ^ bsjc s

(as the mother wavelets, in the present case, we consider the Haar wavelets); l is the level of data detailing; cL,k are the scaling coefficients; and dl,k are the detailing coefficients. The coefficients are calculated by use of the Mallat algorithm [17].

lkð Þt ψlkðÞ¼ t ∑

<sup>t</sup> � <sup>k</sup> � �

∑ m i¼1 aic y

∑ S s¼1 ∑ rs j¼1 bsjc s

Lkð Þ t � i φLkð Þ t � i � �

!

Lkð Þ t � j φLkð Þ t � j

lkð Þ t � j ψlkð Þ t � j , (20)

Lkð Þ t � j φLkð Þ t � j : (21)

where L is the depth of the multi-scale expansion; φL,kð Þt are scaling functions; ψl,kð Þt are the wavelet functions that are obtained from the mother wavelets by

l,kð Þt ψl,kð Þt ,

(19)

l,kð Þt ψl,k7ð Þt ,

In [7], it was shown that a sufficient condition for the stability of plant (1) is as follows: for ∀k ¼ 1,´N meeting the inequalities is to be provided:

1. if <sup>m</sup> <sup>&</sup>gt; R, R <sup>¼</sup> max rs <sup>s</sup>¼1, <sup>S</sup> , then the condition for the detailing coefficients:

$$\begin{cases} \left| a\_l d\_{l,k}^{\mathcal{J}}(t-1) + \sum\_{s=1}^{S} b\_{s,1} d\_{l,k}^{\mathcal{K}}(t-1) \right| < 1, \\ \left| \frac{-a\_l d\_{l,k}^{\mathcal{J}}(t-2) + \sum\_{s=1}^{S} b\_{s,2} d\_{l,k}^{\mathcal{K}}(t-2)}{a\_l d\_{l,k}^{\mathcal{J}}(t-1) + \sum\_{s=1}^{S} b\_{s,1} d\_{l,k}^{\mathcal{K}}(t-1)} \right| < 1, \dots, \\ \left| \frac{-a\_{R+1} d\_{l,k}^{\mathcal{J}}(t-R-1)}{a\_R d\_{l,k}^{\mathcal{J}}(t-R) + \sum\_{s=1}^{S} b\_{s,R} d\_{l,k}^{\mathcal{K}}(t-R)} \right| < 1, \\ \left| \frac{-a\_{R+2} d\_{l,k}^{\mathcal{J}}(t-R-2)}{a\_{R+1} d\_{l,k}^{\mathcal{J}}(t-R-1)} \right| < 1, \dots, \\ \left| \frac{-2a\_{m} d\_{l,k}^{\mathcal{J}}(t-m)}{a\_{m-1} d\_{l,k}^{\mathcal{J}}(t-m+1)} \right| < 1 \end{cases} (22)$$

for the approximating coefficients:

 

$$\begin{array}{c|c} \left| a\_{L\_{L,k}^{\mathcal{I}}}(t-1) + \sum\_{i=1}^{S} b\_{i,1} c\_{L\_{L,k}^{\mathcal{I}}}^{\mathcal{X}}(t-1) \right| < 1, \\ \left| \frac{-a\_{L} c\_{L,k}^{\mathcal{I}}(t-2) + \sum\_{i=1}^{S} b\_{i,2} c\_{L\_{L,k}^{\mathcal{I}}}^{\mathcal{X}}(t-2)}{a\_{1} c\_{L\_{L,k}^{\mathcal{I}}}^{\mathcal{Y}}(t-1) + \sum\_{i=1}^{S} b\_{i,1} c\_{L\_{L,k}^{\mathcal{I}}}^{\mathcal{X}}(t-1)} < 1, \dots, \\ \left| \frac{-a\_{R+1} c\_{L,k}^{\mathcal{I}}(t-R-1)}{a\_{R} c\_{L,k}^{\mathcal{I}}(t-R) + \sum\_{i=1}^{S} b\_{i,R} c\_{L\_{i,k}^{\mathcal{I}}}^{\mathcal{X}}(t-R)} \right| < 1, \\ \left| \frac{-a\_{R+2} c\_{L,k}^{\mathcal{I}}(t-R-2)}{a\_{R+1} c\_{L,k}^{\mathcal{I}}(t-R-1)} \right| < 1, \dots, \\ \left| \frac{-2a\_{m} c\_{L,k}^{\mathcal{I}}(t-m)}{a\_{m-1} c\_{L,k}^{\mathcal{I}}(t-m+1)} \right| < 1, \end{array} \tag{23}$$

2. if m < R, R ¼ maxrs s¼<sup>1</sup>,´<sup>S</sup> , then the condition for the detailing coefficients:

$$\begin{rcmatrix} \left| a\_1 d\_{l,k}^{\mathcal{V}}(t-1) + \sum\_{s=1}^{S} b\_{s,1} d\_{l,k}^{\mathcal{K}\_s}(t-1) \right| & \mathbf{c} \mathbf{1}, \\ \left| \frac{-a\_2 d\_{l,k}^{\mathcal{V}}(t-2) + \sum\_{s=1}^{S} b\_{s,2} d\_{l,k}^{\mathcal{K}\_s}(t-2)}{a\_1 d\_{l,k}^{\mathcal{V}}(t-1) + \sum\_{s=1}^{S} b\_{s,1} d\_{l,k}^{\mathcal{K}\_s}(t-1)} < \mathbf{1}, \dots, \\ \left| \frac{-\sum\_{s=1}^{S} b\_{s,m+1} d\_{l,k}^{\mathcal{K}\_s}(t-m-1)}{a\_m d\_{l,k}^{\mathcal{V}}(t-m) + \sum\_{s=1}^{S} b\_{s,m} d\_{l,k}^{\mathcal{K}\_s}(t-m)} \right| < \mathbf{1}, \\ \left| \frac{-\sum\_{s=1}^{S} b\_{s,m+2} d\_{l,k}^{\mathcal{K}\_s}(t-m-2)}{\sum\_{s=1}^{S} b\_{s,m+1} d\_{l,k}^{\mathcal{K}\_s}(t-m-1)} \right| < \mathbf{1}, \dots, \\ \left| \frac{-2\sum\_{s=1}^{S} b\_{s,m} d\_{l,k}^{\mathcal{K}\_s}(t-R)}{\sum\_{s=1}^{S} b\_{s,R-1} d\_{l,k}^{\mathcal{K}\_s}(t-R+1)} \right| < \mathbf{1} \end{rcmatrix}$$

113

for the approximating coefficients:

$$\begin{array}{c|c} \left| a\_{L\_{L,k}^{y}}^{y}(t-1) + \sum\_{i=1}^{S} b\_{i,1} \mathbf{c}\_{L\_{L}}^{x\_{i}}(t-1) \right| < 1, \\ \left| \frac{-a\_{L} \mathbf{c}\_{L,k}^{y}(t-2) + \sum\_{i=1}^{S} b\_{i,2} \mathbf{c}\_{L,k}^{x\_{i}}(t-2)}{a\_{1} \mathbf{c}\_{L,k}^{y}(t-1) + \sum\_{i=1}^{S} b\_{i,1} \mathbf{c}\_{L,k}^{x\_{i}}(t-1)} < 1, \dots, \\ \left| \frac{-\sum\_{i=1}^{S} b\_{i,m+1} \mathbf{c}\_{L,k}^{x\_{i}}(t-m-1)}{a\_{m} \mathbf{c}\_{L,k}^{y}(t-m) + \sum\_{i=1}^{S} b\_{i,m} \mathbf{c}\_{L,k}^{x\_{i}}(t-m)} \right| < 1, \\ \left| \frac{-\sum\_{i=1}^{S} b\_{i,m+2} \mathbf{c}\_{L,k}^{x\_{i}}(t-m-2)}{\sum\_{i=1}^{S} b\_{i,m+1} \mathbf{c}\_{L,k}^{x\_{i}}(t-m-1)} \right| < 1, \dots, \\ \left| \frac{-2 \sum\_{i=1}^{S} b\_{i,R} \mathbf{c}\_{L,k}^{x\_{i}}(t-R)}{\sum\_{i=1}^{S} b\_{i,R} \mathbf{c}\_{L,k}^{x\_{i}}(t-R+1)} \right| < 1; \end{array} \tag{24}$$

for the approximating coefficients:

a1c y

 

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

10. Prediction of the transfer to chaos

11. Prediction of manufacturing situations

nonlinear NP-hard optimization problems.

cesses under study.

referred to as incidents.

cific production process.

specificity; and

• human resources

115

involved in the j-th operation;

• production equipment.

L,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

c y L,kð Þt

χ<sup>1</sup> j j coincides with the conventional degree of the system stability [23].

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup><sup>c</sup> xs L,kð Þ t � 1

The chaotic system dynamics is characterized by considerable dependence on initial conditions, when as close as needed at the initial time instant trajectories during certain time are diverge by a finite distance. The main characteristics of the chaotic behavior are the speed of divergence of the trajectories defined by the senior Lyapunov exponent. This speed is determined by the Lyapunov exponent whose value represents the degree of instability or degree of sensitivity to the original data. For a linear system with a constant matrix, the senior Lyapunov exponent is χ<sup>1</sup> ¼ max Rλi, where λ<sup>i</sup> are the eigenvalues of the system matrix. In other words,

Thus, (23) and (24) are sufficient conditions for chaotic dynamics prediction, what is a key condition under implementing phase transfers of technological pro-

Optimal routine enterprise resource planning and scheduling are currently based

Present-day industrial sites feature interrelated multi-variable production processes and sophisticated material flow networks; scheduling at such sites poses

It may make sense to develop intelligent predictive models describing the overall current state of resources employed to execute all production operations of a spe-

The state of manufacturing resources should be nevertheless assessed and predicted both to improve control agility and to foresee the situations where schedule execution becomes problematic or impossible. Such situations will be further

The term "production resources" will hereafter mean the following:

and other facilities used for performing the j-th operation;

• input flows characterized by formal properties dependent on production

dij, i ¼ 1, … , N; j ¼ 1, … , M

hij, i ¼ 1, … , H; j ¼ 1, … , M

on detailed mathematical models of production processes [24]. Rescheduling requires model update subject to the current production information.

 < 1:

3. if m ¼ R 6¼ 1, R ¼ maxrs s¼<sup>1</sup>,´<sup>S</sup> , then the condition of the stability for the detailing coefficients:

$$\begin{rcases} \left| \frac{a\_1 d\_{l,k}^{\mathcal{V}}(t-1) + \sum\_{s=1}^{S} b\_{s,1} d\_{l,k}^{\mathcal{K}\_s}(t-1)}{2d\_{l,k}^{\mathcal{V}}(t)} \right| < 1, \\\\ \left| \frac{-a\_2 d\_{l,k}^{\mathcal{V}}(t-2) + \sum\_{s=1}^{S} b\_{s,1} d\_{l,k}^{\mathcal{K}\_s}(t-2)}{a\_1 d\_{l,k}^{\mathcal{V}}(t-1) + \sum\_{s=1}^{S} b\_{s,1} d\_{l,k}^{\mathcal{K}\_s}(t-1)} \right| < 1, \ldots, \\\\ \left| \frac{-2\left[a\_m d\_{l,k}^{\mathcal{V}}(t-m) + \sum\_{s=1}^{S} b\_{s,m} d\_{l,k}^{\mathcal{K}\_s}(t-m) \right]}{a\_{m-1} d\_{l,k}^{\mathcal{V}}(t-m+1) + \sum\_{s=1}^{S} b\_{s,m-1} d\_{l,k}^{\mathcal{K}\_s}(t-m+1)} \right| < 1 \end{rcases} < 1 \end{rcases}$$

for the approximating coefficients:

� � � � � �

$$\begin{rcases} \left| \frac{a\_1 c\_{L,k}^y(t-1) + \sum\_{s=1}^S b\_{s,1} c\_{L,k}^{x\_s}(t-1)}{2c\_{L,k}^y(t)} \right| < 1, \\\\ \left| \frac{-a\_2 c\_{L,k}^y(t-2) + \sum\_{s=1}^S b\_{s,2} c\_{L,k}^{x\_s}(t-2)}{a\_1 c\_{L,k}^y(t-1) + \sum\_{s=1}^S b\_{s,1} c\_{L,k}^{x\_s}(t-1)} \right| < 1, \ldots, \\\\ -2 \left| a\_m c\_{L,k}^y(t-m) + \sum\_{s=1}^S \hat{b}\_{s,m} c\_{L,k}^{x\_s}(t-m) \right| \\\ \frac{a\_{m-1} c\_{L,k}^y(t-m+1) + \sum\_{s=1}^S \hat{b}\_{s,m-1} c\_{L,k}^{x\_s}(t-m+1)}{a\_{m-1} c\_{L,k}^y(t-m+1) + \sum\_{s=1}^S b\_{s,m-1} c\_{L,k}^{x\_s}(t-m+1)} < 1; \end{rcases} < 1; \end{rcases}$$

4. if m ¼ R ¼ 1, R ¼ maxrs s¼<sup>1</sup>,´<sup>S</sup> , then the condition of the stability for the detailing coefficients:

$$\left| \frac{a\_1 d\_{l,k}^{\mathcal{Y}}(t-\mathbf{1}) + \sum\_{s=1}^{\mathcal{S}} b\_{s,1} d\_{l,k}^{\mathcal{X}\_r}(t-\mathbf{1})}{d\_{l,k}^{\mathcal{Y}}(t)} \right| < 1$$

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

for the approximating coefficients:

for the approximating coefficients:

Applied Modern Control

a1c y

� � � � �

�a2c y

� � � � �

a1c y

� � � � �

detailing coefficients:

amc y

> � � � � �

L,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

L,kð Þþ <sup>t</sup> � <sup>2</sup> <sup>∑</sup><sup>S</sup>

<sup>s</sup>¼<sup>1</sup>bs,mþ1<sup>c</sup>

L,kð Þþ <sup>t</sup> � <sup>m</sup> <sup>∑</sup><sup>S</sup>

<sup>s</sup>¼<sup>1</sup>bs,mþ2<sup>c</sup>

<sup>s</sup>¼<sup>1</sup>bs,mþ1<sup>c</sup>

�2∑<sup>S</sup>

<sup>s</sup>¼<sup>1</sup>bs,R�<sup>1</sup><sup>c</sup>

l,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

l,kð Þþ <sup>t</sup> � <sup>2</sup> <sup>∑</sup><sup>S</sup>

l,kð Þþ <sup>t</sup> � <sup>m</sup> <sup>∑</sup><sup>S</sup>

l,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

l,kð Þþ <sup>t</sup> � <sup>m</sup> <sup>þ</sup> <sup>1</sup> <sup>∑</sup><sup>S</sup>

L,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

L,kð Þþ <sup>t</sup> � <sup>2</sup> <sup>∑</sup><sup>S</sup>

L,kð Þþ <sup>t</sup> � <sup>m</sup> <sup>∑</sup><sup>S</sup>

l,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

d y l,kð Þt

L,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

L,kð Þþ <sup>t</sup> � <sup>m</sup> <sup>þ</sup> <sup>1</sup> <sup>∑</sup><sup>S</sup>

2c y L,kð Þt

2d y l,kð Þt

∑S

L,kð Þþ <sup>t</sup> � <sup>1</sup> <sup>∑</sup><sup>S</sup>

�∑<sup>S</sup>

�∑<sup>S</sup>

∑S

� � � � �

a1d y

� � � � �

�a2d y

� � � � �

am�<sup>1</sup>d y

for the approximating coefficients:

� � � � � � a1d y

�<sup>2</sup> amdy

a1c y

� � � � �

�a2c y

� � � � �

am�<sup>1</sup>c y

� � � � � �

detailing coefficients:

114

a1c y

�2 amc y

a1d<sup>y</sup>

� � � � � 2c y L,kð Þt

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup><sup>c</sup> xs L,kð Þ t � 1

<sup>s</sup>¼<sup>1</sup>bs, <sup>2</sup><sup>c</sup> xs L,kð Þ t � 2

L,kð Þ t � m � 1

� � � � �

> � � � � � < 1;

l,kð Þ t � 1

l,kð Þ t � 2

l,kð Þ t � 1

<sup>s</sup>¼<sup>1</sup>bs,mdxs

<sup>s</sup>¼<sup>1</sup>bs,m�<sup>1</sup>dxs

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup><sup>c</sup> xs L,kð Þ t � 1

<sup>s</sup>¼<sup>1</sup>bs, <sup>2</sup><sup>c</sup> xs L,kð Þ t � 2

> s¼1 ^ bs,mc xs L,kð Þ t � m

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup><sup>c</sup> xs L,kð Þ t � 1

h i

s¼1 ^ bs,m�<sup>1</sup>c xs

4. if m ¼ R ¼ 1, R ¼ maxrs s¼<sup>1</sup>,´<sup>S</sup> , then the condition of the stability for the

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup>dxs

l,kð Þ t � 1

� � � � � < 1,

� � � � �

l,kð Þ t � m

l,kð Þ t � m þ 1

� � � � � < 1,

� � � � �

L,kð Þ t � m þ 1

� � � � � < 1

< 1, … ,

� � � � � � < 1;

< 1, … ,

� � � � � � < 1

<sup>s</sup>¼<sup>1</sup>bs,mc xs L,kð Þ t � m

L,kð Þ t � m � 2

L,kð Þ t � R þ 1

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup>dxs

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup>dxs

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup>dxs

h i

L,kð Þ t � m � 1

<sup>s</sup>¼<sup>1</sup>bs, <sup>1</sup><sup>c</sup> xs L,kð Þ t � 1

xs

xs

xs

<sup>s</sup>¼<sup>1</sup>bs,Rc xs L,kð Þ t � R

xs

3. if m ¼ R 6¼ 1, R ¼ maxrs s¼<sup>1</sup>,´<sup>S</sup> , then the condition of the stability for the

� � � � � < 1,

� � � � �

< 1, … ,

(24)

(25)

(26)

� � � � � < 1,

< 1, … ,

$$\left| \frac{a\_1 c\_{L,k}^{\mathcal{Y}}(t-\mathbf{1}) + \sum\_{s=1}^{\mathcal{S}} b\_{s,1} c\_{L,k}^{\chi\_r}(t-\mathbf{1})}{c\_{L,k}^{\mathcal{Y}}(t)} \right| < \mathbf{1}.$$

#### 10. Prediction of the transfer to chaos

The chaotic system dynamics is characterized by considerable dependence on initial conditions, when as close as needed at the initial time instant trajectories during certain time are diverge by a finite distance. The main characteristics of the chaotic behavior are the speed of divergence of the trajectories defined by the senior Lyapunov exponent. This speed is determined by the Lyapunov exponent whose value represents the degree of instability or degree of sensitivity to the original data.

For a linear system with a constant matrix, the senior Lyapunov exponent is χ<sup>1</sup> ¼ max Rλi, where λ<sup>i</sup> are the eigenvalues of the system matrix. In other words, χ<sup>1</sup> j j coincides with the conventional degree of the system stability [23].

Thus, (23) and (24) are sufficient conditions for chaotic dynamics prediction, what is a key condition under implementing phase transfers of technological processes under study.

#### 11. Prediction of manufacturing situations

Optimal routine enterprise resource planning and scheduling are currently based on detailed mathematical models of production processes [24]. Rescheduling requires model update subject to the current production information.

Present-day industrial sites feature interrelated multi-variable production processes and sophisticated material flow networks; scheduling at such sites poses nonlinear NP-hard optimization problems.

The state of manufacturing resources should be nevertheless assessed and predicted both to improve control agility and to foresee the situations where schedule execution becomes problematic or impossible. Such situations will be further referred to as incidents.

It may make sense to develop intelligent predictive models describing the overall current state of resources employed to execute all production operations of a specific production process.

The term "production resources" will hereafter mean the following:


$$d\_{\vec{\eta},} i = \mathbf{1}, \dots, N; j = \mathbf{1}, \dots, M$$

and other facilities used for performing the j-th operation;

• human resources

$$h\_{ij,}i = \mathbf{1}, \dots, H; j = \mathbf{1}, \dots, M$$

involved in the j-th operation;

• other factors

$$f\_{\vec{\eta},}k = \mathbf{1}, \dots, N; j = \mathbf{1}, \dots, M$$

For the resources from the categories 2 and 3, the respective codes will have the same value in all positions (either 1 or 0). <С<sup>3</sup> > is the code of the time before the maintenance end. If a resource is available and operated, the respective code consists of 1s. < С<sup>4</sup> > is the code of the time before the equipment piece fails with the

In the scheduling practice, this time is not less than the operating time. However, resource replacement just during the operation may be sometimes more costeffective. Moreover, the equipment piece may fail unexpectedly. For resource types

Generally, it is hardly possible to formalize all such causes of schedule disruption. Therefore, their consolidation as the "remaining plan execution time" is a way

For the developed binary chain, a forecast may be obtained using data mining techniques. It makes sense to apply the methods named association rules search [25]. A forecast of a state described by a binary chain with an identifier can be obtained by revealing the most probable combination of two binary sets of values at a fixed time instant and at the next instant (a one-step forecast). A more distant

Modern information technologies offer new possibilities for solving identification problems for control and decision-making systems. Data mining methods allow to solve problems that in the general case could not be solved by classical methods,

In this chapter, associative search techniques are presented. The techniques allow the identification of nonlinear systems, without the need to build a bunch of Wiener-Hammerstein models, etc. An alternative is to analyze the current state of the system using the knowledge base and training system. This approach allows the

The algorithms may be successfully applied in the identification of nonlinear nonstationary processes. For these purposes, the multi-scale wavelet expansion is used. By investigating the dynamics of the coefficients of this expansion, one can predict the approach of process parameters to stability limits. Finally, sufficient

The high accuracy of forecasting by associative search technique makes it relevant for studying the dynamics of processes and predicting the transition to chaos. Also, it becomes possible to predict the contingencies of production processes. For

< С<sup>5</sup> > is the time before the scheduled end of the operation. In real-life manufacturing situations, time may be wasted (with the need in schedule update) for the reasons neither stipulated in the production model nor caused by equipment

probability close to 1 (remaining life).

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

prediction horizon is also possible.

or required heuristic approaches.

conditions of stability are derived.

best use of a priori information on the object.

this, the method of searching for associative rules is applied.

12. Conclusion

117

failures.

from categories 1 and 3, < С<sup>4</sup> > has 1s in all positions.

to allow for these hidden factors in the production state model.

affecting the j-th operation such as energy resources and a variety of formal indices and factors related with the production process.

Production resources may be described differently.


Assume a model of a specific manufacturing situation as a dynamic schedule fragment comprising the following components:

$$r\_{\vec{\eta}}(t) = \left\{ <\mathbf{C}\_1 > \mathbf{C}\_2 > \mathbf{C}\_3 > \mathbf{C}\_4 > \mathbf{C}\_5 > \right\}\_{\vec{\eta}t} \tag{27}$$

where

< С<sup>1</sup> > ¼ def < ijt > is a resource identifier including the resource number, the operation number, and the time stamp (the number of characteristics may be increased).

Other components of the resource state vector at the time moment t may be represented by a binary code.

< С<sup>2</sup> > is the code of the numerical value of a state variable; this code is different for each of the above-listed resource types.

< С<sup>3</sup> >, <С<sup>4</sup> > , and < С<sup>5</sup> > will be discussed further.

Consider the resources whose state may be described by some quantitative characteristic, such as inlet flow rate or temperature for chemical processes or an average equipment failure number.

For a specific resource, we assume that the characteristic of its state possesses the values on the half-interval [0; 1) (this half-interval was chosen as an example for simplicity, the results can be easily spread to any other).

This half-interval can be represented as the union

[0; 0.5)∪ [0.5, 1). We will further correspond the symbols {0; 1} to the left and right half-intervals respectively, namely, 0 to the left half-interval, and 1 to the right one.

Each of the two subintervals can be further split in the same way, and, again, the values 0 and 1 can be assigned to the left and the right parts, respectively.

In that way, a finite chain of symbols from {0; 1} has a one-to-one correspondence with a half-interval embedded in [0; 1). For a binary partition, a chain of n symbols corresponds to a half-interval with the length <sup>1</sup> 2n.

This way, for each value of a numerical characteristic at the current time moment, we obtain a code of 0s and 1s. The number of positions, as we show further, will determine the accuracy of prediction.

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

For the resources from the categories 2 and 3, the respective codes will have the same value in all positions (either 1 or 0). <С<sup>3</sup> > is the code of the time before the maintenance end. If a resource is available and operated, the respective code consists of 1s. < С<sup>4</sup> > is the code of the time before the equipment piece fails with the probability close to 1 (remaining life).

In the scheduling practice, this time is not less than the operating time. However, resource replacement just during the operation may be sometimes more costeffective. Moreover, the equipment piece may fail unexpectedly. For resource types from categories 1 and 3, < С<sup>4</sup> > has 1s in all positions.

< С<sup>5</sup> > is the time before the scheduled end of the operation. In real-life manufacturing situations, time may be wasted (with the need in schedule update) for the reasons neither stipulated in the production model nor caused by equipment failures.

Generally, it is hardly possible to formalize all such causes of schedule disruption. Therefore, their consolidation as the "remaining plan execution time" is a way to allow for these hidden factors in the production state model.

For the developed binary chain, a forecast may be obtained using data mining techniques. It makes sense to apply the methods named association rules search [25].

A forecast of a state described by a binary chain with an identifier can be obtained by revealing the most probable combination of two binary sets of values at a fixed time instant and at the next instant (a one-step forecast). A more distant prediction horizon is also possible.

#### 12. Conclusion

• other factors

Applied Modern Control

where < С<sup>1</sup> > ¼

right one.

116

represented by a binary code.

for each of the above-listed resource types.

average equipment failure number.

< С<sup>3</sup> >, <С<sup>4</sup> > , and < С<sup>5</sup> > will be discussed further.

for simplicity, the results can be easily spread to any other). This half-interval can be represented as the union

symbols corresponds to a half-interval with the length <sup>1</sup>

further, will determine the accuracy of prediction.

fij, k ¼ 1, … , N; j ¼ 1, … , M

affecting the j-th operation such as energy resources and a variety of formal

1. Some have qualitative characteristics which take on specific values that may be

2. The state of others such as certain equipment pieces may be exclusively either "working" or "not working." The remaining life time may be known or not for such resources. The process historian may however keep failure statistics for a specific equipment piece; maintenance downtime statistics may be also

maintenance. In case of outage, such resources should be immediately replaced from the backlog. The replacement process is typically fast; therefore, no values other than 1 (OK) and 0 (not OK) should be assigned to such resource.

def < ijt > is a resource identifier including the resource number, the opera-

rijðÞ¼ t f g <С<sup>1</sup> > С<sup>2</sup> >С<sup>3</sup> >С<sup>4</sup> > С<sup>5</sup> > ijt (27)

Assume a model of a specific manufacturing situation as a dynamic schedule

tion number, and the time stamp (the number of characteristics may be increased). Other components of the resource state vector at the time moment t may be

Consider the resources whose state may be described by some quantitative characteristic, such as inlet flow rate or temperature for chemical processes or an

For a specific resource, we assume that the characteristic of its state possesses the values on the half-interval [0; 1) (this half-interval was chosen as an example

[0; 0.5)∪ [0.5, 1). We will further correspond the symbols {0; 1} to the left and right half-intervals respectively, namely, 0 to the left half-interval, and 1 to the

Each of the two subintervals can be further split in the same way, and, again, the

2n.

In that way, a finite chain of symbols from {0; 1} has a one-to-one correspondence with a half-interval embedded in [0; 1). For a binary partition, a chain of n

This way, for each value of a numerical characteristic at the current time moment, we obtain a code of 0s and 1s. The number of positions, as we show

values 0 and 1 can be assigned to the left and the right parts, respectively.

< С<sup>2</sup> > is the code of the numerical value of a state variable; this code is different

available for a specific piece or similar kind of equipment.

3. One more resource type (including human resources) is not subject to

indices and factors related with the production process. Production resources may be described differently.

checked against norms at any moment.

fragment comprising the following components:

Modern information technologies offer new possibilities for solving identification problems for control and decision-making systems. Data mining methods allow to solve problems that in the general case could not be solved by classical methods, or required heuristic approaches.

In this chapter, associative search techniques are presented. The techniques allow the identification of nonlinear systems, without the need to build a bunch of Wiener-Hammerstein models, etc. An alternative is to analyze the current state of the system using the knowledge base and training system. This approach allows the best use of a priori information on the object.

The algorithms may be successfully applied in the identification of nonlinear nonstationary processes. For these purposes, the multi-scale wavelet expansion is used. By investigating the dynamics of the coefficients of this expansion, one can predict the approach of process parameters to stability limits. Finally, sufficient conditions of stability are derived.

The high accuracy of forecasting by associative search technique makes it relevant for studying the dynamics of processes and predicting the transition to chaos. Also, it becomes possible to predict the contingencies of production processes. For this, the method of searching for associative rules is applied.

Applied Modern Control

#### Author details

Natalia Bakhtadze1,2\*, Vladimir Lototsky1 , Valery Pyatetsky<sup>3</sup> and Alexey Lototsky<sup>1</sup> References

[1] Peretzki D, Isaksson A, Carvalho A, Bittencourt C, Forsman K. Data Mining

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

> intelligent soft sensors. IFAC-PapersOnLine. 2017;50:14632-14637. DOI: 10.1016/j.ifacol.2017.08.1742

26:394-395

406-413

[8] Moore E. On the reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society. 1920;

[9] Penrose R. A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society. 1955;51(3):

[10] Larichev OI, Asanov A, Naryzhny Y, Strahov S. Expert system for the diagnostics of acute drug poisonings, applications and innovations in

intelligent systems IX. In: Macintosh A,

International Conference on Knowledge Based Systems and Applied Artificial Intelligence. Cambridge, UK: Springer-

Moulton M, Preece A, editors. Proceedings of the 21 SGES

Verlag; 2001. pp. 159-168

Parc, CA: AAAI Press; 1997

[12] Hunt E. Cognitive science:

[13] Newell A, Simon HA. Human Problem Solving. Englewood Cliffs, NJ:

[14] Gavrilov A. The model of associative memory of intelligent system. In: Proceedings of 6-th Russian-Korean International Symposium on Science and Technology. Novosibirsk.

Prentice-Hall Inc.; 1972

Vol. 1. 2002. pp. 174-177

[15] Takagi T, Sugeno M. Fuzzy identification of systems and its

Definition, status and questions. Annual Review of Psychology. 1989;40:603-629

[11] Patel V, Ramoni M. Cognitive models of directional inference in expert medical reasoning. In: Feltovich P, Ford K, Hofman R, editors. Expertise in Context: Human and Machine. Menlo

of Historic Data for Process Identification. Sweden: Linköping University Electronic Press; 2014.

http://manualzz.com/doc/

[3] Bakhtadze N, Maximov E,

KR-1001.00017

1998-0140

319-23338-3

119

8482583/modeling-and-diagnosisof-friction-and-wear-in-industrial

[2] Bakhtadze N, Kulba V, Lototsky V, Maximov E. Identification-based approach to soft sensors design. IFAC-PapersOnLine. 2007;10:302-307. DOI: 10.3182/20100701-2-PT-4011.00052

Valiakhmetov R. Fuzzy soft sensors for chemical and oil refining processes. IFAC Proceedings Volumes. 2008;41: 4246-4250. DOI: 10.3182/20080706-5-

[4] Bakhtadze N, Lototsky V, Vlasov S, Sakrutina E. Associative search and wavelet analysis techniques in system identification. IFAC Proceedings Volumes. 2012;45:1227-1232. DOI: 10.3182/20120711-3-BE-2027.00242

[5] Bakhtadze N, Sakrutina A. The intelligent identification technique with associative search. International Journal of Mathematical Models and Methods in Applied Sciences. 2015;9:418-431. ISSN:

[6] Bakhtadze N, Lototsky V.

2016. pp. 85-104. ISBN 978-3- 319-23338-3. DOI: 10.1007/978-3-

Knowledge-based models of nonlinear systems based on inductive learning. In: New Frontiers in Information and Production Systems Modelling and Analysis Incentive Mechanisms,

Competence Management, Knowledgebased Production. Heidelberg: Springer;

[7] Bakhtadze N, Sakrutina E, Pyatetsky V. Predicting oil product properties with


\*Address all correspondence to: sung7@yandex.ru

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Data Mining-Based Identification of Nonlinear Systems DOI: http://dx.doi.org/10.5772/intechopen.80968

#### References

[1] Peretzki D, Isaksson A, Carvalho A, Bittencourt C, Forsman K. Data Mining of Historic Data for Process Identification. Sweden: Linköping University Electronic Press; 2014. http://manualzz.com/doc/ 8482583/modeling-and-diagnosisof-friction-and-wear-in-industrial

[2] Bakhtadze N, Kulba V, Lototsky V, Maximov E. Identification-based approach to soft sensors design. IFAC-PapersOnLine. 2007;10:302-307. DOI: 10.3182/20100701-2-PT-4011.00052

[3] Bakhtadze N, Maximov E, Valiakhmetov R. Fuzzy soft sensors for chemical and oil refining processes. IFAC Proceedings Volumes. 2008;41: 4246-4250. DOI: 10.3182/20080706-5- KR-1001.00017

[4] Bakhtadze N, Lototsky V, Vlasov S, Sakrutina E. Associative search and wavelet analysis techniques in system identification. IFAC Proceedings Volumes. 2012;45:1227-1232. DOI: 10.3182/20120711-3-BE-2027.00242

[5] Bakhtadze N, Sakrutina A. The intelligent identification technique with associative search. International Journal of Mathematical Models and Methods in Applied Sciences. 2015;9:418-431. ISSN: 1998-0140

[6] Bakhtadze N, Lototsky V. Knowledge-based models of nonlinear systems based on inductive learning. In: New Frontiers in Information and Production Systems Modelling and Analysis Incentive Mechanisms, Competence Management, Knowledgebased Production. Heidelberg: Springer; 2016. pp. 85-104. ISBN 978-3- 319-23338-3. DOI: 10.1007/978-3- 319-23338-3

[7] Bakhtadze N, Sakrutina E, Pyatetsky V. Predicting oil product properties with intelligent soft sensors. IFAC-PapersOnLine. 2017;50:14632-14637. DOI: 10.1016/j.ifacol.2017.08.1742

[8] Moore E. On the reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society. 1920; 26:394-395

[9] Penrose R. A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society. 1955;51(3): 406-413

[10] Larichev OI, Asanov A, Naryzhny Y, Strahov S. Expert system for the diagnostics of acute drug poisonings, applications and innovations in intelligent systems IX. In: Macintosh A, Moulton M, Preece A, editors. Proceedings of the 21 SGES International Conference on Knowledge Based Systems and Applied Artificial Intelligence. Cambridge, UK: Springer-Verlag; 2001. pp. 159-168

[11] Patel V, Ramoni M. Cognitive models of directional inference in expert medical reasoning. In: Feltovich P, Ford K, Hofman R, editors. Expertise in Context: Human and Machine. Menlo Parc, CA: AAAI Press; 1997

[12] Hunt E. Cognitive science: Definition, status and questions. Annual Review of Psychology. 1989;40:603-629

[13] Newell A, Simon HA. Human Problem Solving. Englewood Cliffs, NJ: Prentice-Hall Inc.; 1972

[14] Gavrilov A. The model of associative memory of intelligent system. In: Proceedings of 6-th Russian-Korean International Symposium on Science and Technology. Novosibirsk. Vol. 1. 2002. pp. 174-177

[15] Takagi T, Sugeno M. Fuzzy identification of systems and its

Author details

Applied Modern Control

118

Natalia Bakhtadze1,2\*, Vladimir Lototsky1

1 V.A. Trapeznikov Institute of Control Sciences, Moscow, Russia

3 National University of Science and Technology "MISIS", Moscow, Russia

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 Bauman Moscow State Technical University, Moscow, Russia

\*Address all correspondence to: sung7@yandex.ru

provided the original work is properly cited.

, Valery Pyatetsky<sup>3</sup> and Alexey Lototsky<sup>1</sup>

#### Applied Modern Control

applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics. 1985;26:116-132

Chapter 7

Abstract

Analysis and Control of Power

Electronic Converters Based on a

System Zero Locations Approach

Jorge-Humberto Urrea-Quintero, Nicolás Muñoz-Galeano

This chapter presents a procedure to design and control power electronic con-

response criterion for dimensioning converter passive elements. For this purpose, a nonideal boost DC-DC converter (converter considering its parasitic losses) is dynamically modeled and analyzed in steady state as an application example. The steady-state model is obtained from the average nonlinear model. The steady-state model allows deducing expressions for equilibrium conversion ratio M Dð Þ and efficiency η of the system. Conditions for the converter conduction modes are analyzed. Simulations are made to see how parasitic losses affect both M Dð Þ and η. Then, inductor current and capacitor voltage ripple analyses are carried out to find lower boundaries for inductor and capacitor values. The values of the boost DC-DC converter passive elements are selected taking into account both steady-state and zero-based analyses. A nonideal boost DC-DC converter and a PI-based current mode control (CMC) structure are designed to validate the proposed procedure. Finally, the boost DC-DC converter is implemented in PSIM and system operating

verters (PECs), which includes a zero-based analysis as a dynamical system

Keywords: power electronic converters, boost DC-DC converter,

and discontinuous conduction mode (DCM) boundaries [3].

zero-based analysis, current mode control, parasitic loss analysis, efficiency

Design procedures of PECs must establish a trade-off between passive elements' values and dynamical performance because of the close dependence between them. Dynamical performance should not be deteriorated and operating requirements must be satisfied [1]. This task generally implies the construction of a nonlinear dynamical model and its implementation in any

Dynamical modeling and steady-state analyses of PECs have received significant attention as tools to model system design [3]. Through dynamical modeling, it is possible to perform an analysis of the system behavior and its relation with passive elements' values [1]. Meanwhile, steady-state analysis provides expressions to determine in PEC: (a) M Dð Þ, (b) η, and (c) continuous conduction mode (CCM)

and Lina-María Gómez-Echavarría

requirements are satisfactorily verified.

1. Introduction

computational tool [2].

121

[16] Daubechies I, Lagarias J. Two-scale difference equations I: Existence and global regularity of solutions. SIAM Journal on Mathematical Analysis. 1991; 22:1388-1410

[17] Mallat S. In: Barlaud M, editor. Wavelet Tour of Signal Processing. San Diego; CA: Academic Press; 1999. 635p

[18] Váňa Z, Preisig H. System identification in frequency domain using wavelets: Conceptual remarks. Systems & Control Letters. 2012;61(10): 1041-1051

[19] Ho K, Blunt S. Adaptive sparse system identification using wavelets. IEEE Trans. Circuits and Systems-II: Analog and Digital Signal Processing. 2003;49(10):656-667

[20] Sureshbabu N, Farrell JA. Waveletbased system identification for nonlinear control. IEEE Transactions on Automatic Control. 1999;44(2):412-417

[21] Preisig HA. Parameter estimation using multi-wavelets. Computer Aided Chemical Engineering. 2010;28:367-372

[22] Carrier J, Stephanopoulos G. Wavelet-based modulation in controlrelevant process identification. AICHE Journal. 1998;44(2):341-360

[23] Fradkov A, Evans R. Control of chaos: Survey—1997–2000. IFAC Proceedings Volumes. 2002;35:131-142

[24] Al-Otabi GA, Stewart MD. Simulation model determines optimal tank farm design. Oil & Gas Journal. 2004;102(7):50-55

[25] Qin JS, Badgwell TA. A survey of industrial model predictive control technology. Control Engineering Practice. 2003;11(7):733-764

#### Chapter 7

applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics. 1985;26:116-132

Applied Modern Control

[16] Daubechies I, Lagarias J. Two-scale difference equations I: Existence and global regularity of solutions. SIAM Journal on Mathematical Analysis. 1991;

[17] Mallat S. In: Barlaud M, editor. Wavelet Tour of Signal Processing. San Diego; CA: Academic Press; 1999. 635p

[19] Ho K, Blunt S. Adaptive sparse system identification using wavelets. IEEE Trans. Circuits and Systems-II: Analog and Digital Signal Processing.

[20] Sureshbabu N, Farrell JA. Wavelet-

nonlinear control. IEEE Transactions on Automatic Control. 1999;44(2):412-417

[21] Preisig HA. Parameter estimation using multi-wavelets. Computer Aided Chemical Engineering. 2010;28:367-372

[22] Carrier J, Stephanopoulos G. Wavelet-based modulation in controlrelevant process identification. AICHE

[23] Fradkov A, Evans R. Control of chaos: Survey—1997–2000. IFAC Proceedings Volumes. 2002;35:131-142

[25] Qin JS, Badgwell TA. A survey of industrial model predictive control technology. Control Engineering Practice. 2003;11(7):733-764

[24] Al-Otabi GA, Stewart MD. Simulation model determines optimal tank farm design. Oil & Gas Journal.

2004;102(7):50-55

120

Journal. 1998;44(2):341-360

based system identification for

[18] Váňa Z, Preisig H. System identification in frequency domain using wavelets: Conceptual remarks. Systems & Control Letters. 2012;61(10):

22:1388-1410

1041-1051

2003;49(10):656-667

## Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

Jorge-Humberto Urrea-Quintero, Nicolás Muñoz-Galeano and Lina-María Gómez-Echavarría

#### Abstract

This chapter presents a procedure to design and control power electronic converters (PECs), which includes a zero-based analysis as a dynamical system response criterion for dimensioning converter passive elements. For this purpose, a nonideal boost DC-DC converter (converter considering its parasitic losses) is dynamically modeled and analyzed in steady state as an application example. The steady-state model is obtained from the average nonlinear model. The steady-state model allows deducing expressions for equilibrium conversion ratio M Dð Þ and efficiency η of the system. Conditions for the converter conduction modes are analyzed. Simulations are made to see how parasitic losses affect both M Dð Þ and η. Then, inductor current and capacitor voltage ripple analyses are carried out to find lower boundaries for inductor and capacitor values. The values of the boost DC-DC converter passive elements are selected taking into account both steady-state and zero-based analyses. A nonideal boost DC-DC converter and a PI-based current mode control (CMC) structure are designed to validate the proposed procedure. Finally, the boost DC-DC converter is implemented in PSIM and system operating requirements are satisfactorily verified.

Keywords: power electronic converters, boost DC-DC converter, zero-based analysis, current mode control, parasitic loss analysis, efficiency

#### 1. Introduction

Design procedures of PECs must establish a trade-off between passive elements' values and dynamical performance because of the close dependence between them. Dynamical performance should not be deteriorated and operating requirements must be satisfied [1]. This task generally implies the construction of a nonlinear dynamical model and its implementation in any computational tool [2].

Dynamical modeling and steady-state analyses of PECs have received significant attention as tools to model system design [3]. Through dynamical modeling, it is possible to perform an analysis of the system behavior and its relation with passive elements' values [1]. Meanwhile, steady-state analysis provides expressions to determine in PEC: (a) M Dð Þ, (b) η, and (c) continuous conduction mode (CCM) and discontinuous conduction mode (DCM) boundaries [3].

#### Applied Modern Control

Multi-resolution PEC models can be constructed where parasitic losses can be taken into account [4]. However, if parasitic losses are not considered, the PEC model is simplified; but models do not adequately represent the PEC behavior in its entire operation range [5]. Moreover, a simplified model cannot predict both M Dð Þ and η nonlinearities and limitations [6].

Parasitic losses are typically modeled as appropriate equivalent series resistances (ESRs) associated with passive elements of PECs [3, 7, 8]. Parasitic losses can be included in the PEC design stage when both dynamical performance and η are taken into account [6]. Several works, [4, 7] to mention some of them, propose different PEC modeling approaches that have included parasitic losses. Nevertheless, in the reviewed literature, a consensus about what is the suitable detailed level of the model does not exist, in which PEC's dynamical behavior can be accurately represented; without the model, deduction becomes a challenge for the designer. However, the trend remains with the so-called average models which describe lowfrequency and neglect high-frequency dynamics (semiconductor switching dynamics) of the system [9].

Average models that take, some or all, parasitic losses into account, have been presented by [1, 10]. Recent works [7, 10–14] show that a practical level of model detail for PECs includes parasitic losses associated with their passive elements and disregards losses due to semiconductor switching. Models with this level of detail are suitable for system design, M Dð Þ derivation, η analysis, and dynamical performance evaluation [12]. Additionally, these models are suitable for control purposes [2, 10].

converter supplies energy to a dominant-current load represented as a Norton equivalent model. Engines and inverters are common dominant-current loads that can be supplied by a boost DC-DC converter. In Figure 1, L is inductor, C is capacitor, and RL and RC are the parasitic losses for L and C, respectively. RL and RC

Circuital scheme of the DC-DC boost converter: configuration (a) h<sup>1</sup> ¼ 1, h<sup>2</sup> ¼ 0. Configuration (b)

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

The boost DC-DC converter operating in CCM can take two configurations according to the switch position as shown in Figure 1. First (Figure 1(a)) and second (Figure 1(b)) configurations correspond to switch H ¼ f g h1; h<sup>2</sup> being turned on h<sup>1</sup> ¼ 1, h<sup>2</sup> ¼ 0 and turned off h<sup>1</sup> ¼ 0, h<sup>2</sup> ¼ 1, respectively. Therefore, the

State variables are inductor current iL and capacitor voltage vC which represent the energy variation in the system. The system inputs are u, DC input voltage source vg , and current source io. Variations of io are useful to represent system current perturbations. The system outputs are output voltage vo and iL. The corresponding

> <sup>R</sup> � <sup>1</sup>

> > ð Þ <sup>1</sup> � <sup>u</sup> iL � vC

<sup>R</sup> � io (2)

detð Þ sI � <sup>A</sup> <sup>C</sup>½ � adjð Þ sI � <sup>A</sup> TB <sup>þ</sup> <sup>D</sup> (3)

, and <sup>y</sup> <sup>¼</sup> ½ � iL; vo <sup>T</sup>. IL and VC, <sup>D</sup> and Io are

ð Þ 1 � u vC þ ϕCð Þ 1 � u io (1)

switching function u can be defined as follows: u ¼ h<sup>1</sup> or u ¼ 1 � h2.

<sup>2</sup> iL <sup>þ</sup> <sup>ϕ</sup><sup>C</sup>

dt <sup>¼</sup> <sup>1</sup>

dynamical model of the system in Figure 1 is given by Eqs. (1) (2), where

1 þ α<sup>C</sup> 

In this chapter, the widely accepted PI-based CMC structure for the boost DC-DC converter is adopted [15, 16]. PI controllers' tuning requires a frequencydomain model. From Eqs. (1) and (2), it is possible to obtain a linear state-space model of the boost DC-DC converter. Next, the frequency-domain model is

The linear state-space model for the boost DC-DC converter is given by Eqs. (4)

states and inputs in their rated values, respectively. d ¼ h i u <sup>o</sup> (average value of u) is

represent all parasitic losses.

DOI: http://dx.doi.org/10.5772/intechopen.80426

Figure 1.

h<sup>1</sup> ¼ 0, h<sup>2</sup> ¼ 1.

α<sup>C</sup> ¼ RC=R and ϕ<sup>C</sup> ¼ RC=1 þ αC.

and (5), where x ¼ ½ � iL; vC

123

dt <sup>¼</sup> vg � RL <sup>þ</sup> <sup>ϕ</sup>Cð Þ <sup>1</sup> � <sup>u</sup>

<sup>C</sup>dvC

obtained by means of the realization given by Eq. (3).

G sðÞ¼ <sup>1</sup>

<sup>T</sup>, <sup>u</sup> <sup>¼</sup> <sup>d</sup>; vg; io <sup>T</sup>

<sup>L</sup> diL

It is clear that based on average models, PECs can be designed to carry out dynamical performance analysis. Notwithstanding, a design procedure is needed that comprises all necessary steps to design and control PECs and fulfills all given operating requirements. This design procedure must be simple and useful.

In the PEC field, few works that take into account dynamical characteristics of the system have been carried out [15–17]. In these works, PEC's design problem is presented as an optimization problem. In consequence, a procedure to easily design and control PECs is still needed. In this chapter, a procedure to easily design and control PECs is introduced. In this procedure, neither an optimization process is carried out nor is the control structure fixed. But, zeros' location impact over the system dynamical responses is analyzed, showing that a careful selection of the PEC passive elements could both avoid electronic device failures due to large overshoots and improve the dynamical system performance.

The structure of the chapter is organized as follows: in Section 2, both time- and frequency-domain models of the boost DC-DC converter are derived. In Section 3, the boost DC-DC converter is studied in steady state. Section 3 is composed of Sections 3.1, 3.2, 3.3, and 3.4. In Sections 3.1 and 3.2, expressions for M Dð Þ and η are derived including some parasitic losses. In Section 3.3, conditions to operate in CCM or DCM are found. In Section 3.4, both inductor current and capacitor voltage ripple analyses are carried out to find lower boundaries for inductor and capacitor values that fulfill ripple requirements. In Section 4, the value of the passive elements is selected such that operating requirements are fulfilled and system dynamical performance is achieved. Mathematical model is contrasted with a PSIM implementation of the boost DC-DC converter. In Section 5, the widely accepted current mode control (CMC) structure for boost DC-DC converters is designed.

#### 2. Nonlinear dynamical modeling

Figure 1 shows a circuital representation of a typical boost DC-DC converter including its parasitic losses associated to the passive elements. The boost DC-DC Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach DOI: http://dx.doi.org/10.5772/intechopen.80426

Figure 1. Circuital scheme of the DC-DC boost converter: configuration (a) h<sup>1</sup> ¼ 1, h<sup>2</sup> ¼ 0. Configuration (b) h<sup>1</sup> ¼ 0, h<sup>2</sup> ¼ 1.

converter supplies energy to a dominant-current load represented as a Norton equivalent model. Engines and inverters are common dominant-current loads that can be supplied by a boost DC-DC converter. In Figure 1, L is inductor, C is capacitor, and RL and RC are the parasitic losses for L and C, respectively. RL and RC represent all parasitic losses.

The boost DC-DC converter operating in CCM can take two configurations according to the switch position as shown in Figure 1. First (Figure 1(a)) and second (Figure 1(b)) configurations correspond to switch H ¼ f g h1; h<sup>2</sup> being turned on h<sup>1</sup> ¼ 1, h<sup>2</sup> ¼ 0 and turned off h<sup>1</sup> ¼ 0, h<sup>2</sup> ¼ 1, respectively. Therefore, the switching function u can be defined as follows: u ¼ h<sup>1</sup> or u ¼ 1 � h2.

State variables are inductor current iL and capacitor voltage vC which represent the energy variation in the system. The system inputs are u, DC input voltage source vg , and current source io. Variations of io are useful to represent system current perturbations. The system outputs are output voltage vo and iL. The corresponding dynamical model of the system in Figure 1 is given by Eqs. (1) (2), where α<sup>C</sup> ¼ RC=R and ϕ<sup>C</sup> ¼ RC=1 þ αC.

$$L\frac{di\_L}{dt} = v\_g - \left(R\_L + \phi\_C(\mathbf{1} - \boldsymbol{u})^2\right)\dot{i}\_L + \left(\frac{\phi\_C}{R} - \mathbf{1}\right)(\mathbf{1} - \boldsymbol{u})v\_C + \phi\_C(\mathbf{1} - \boldsymbol{u})\dot{i}\_o \tag{1}$$

$$\mathbf{C}\frac{dv\_C}{dt} = \left(\frac{1}{1+a\_C}\right)\left((1-u)i\_L - \frac{v\_C}{R} - i\_o\right) \tag{2}$$

In this chapter, the widely accepted PI-based CMC structure for the boost DC-DC converter is adopted [15, 16]. PI controllers' tuning requires a frequencydomain model. From Eqs. (1) and (2), it is possible to obtain a linear state-space model of the boost DC-DC converter. Next, the frequency-domain model is obtained by means of the realization given by Eq. (3).

$$G(s) = \frac{1}{\det(sI - A)} C[\text{adj}(sI - A)]^T B + D \tag{3}$$

The linear state-space model for the boost DC-DC converter is given by Eqs. (4) and (5), where x ¼ ½ � iL; vC <sup>T</sup>, <sup>u</sup> <sup>¼</sup> <sup>d</sup>; vg; io <sup>T</sup> , and <sup>y</sup> <sup>¼</sup> ½ � iL; vo <sup>T</sup>. IL and VC, <sup>D</sup> and Io are states and inputs in their rated values, respectively. d ¼ h i u <sup>o</sup> (average value of u) is

Multi-resolution PEC models can be constructed where parasitic losses can be taken into account [4]. However, if parasitic losses are not considered, the PEC model is simplified; but models do not adequately represent the PEC behavior in its entire operation range [5]. Moreover, a simplified model cannot predict both M Dð Þ

Parasitic losses are typically modeled as appropriate equivalent series resistances (ESRs) associated with passive elements of PECs [3, 7, 8]. Parasitic losses can be included in the PEC design stage when both dynamical performance and η are taken into account [6]. Several works, [4, 7] to mention some of them, propose different PEC modeling approaches that have included parasitic losses. Nevertheless, in the reviewed literature, a consensus about what is the suitable detailed level of the model does not exist, in which PEC's dynamical behavior can be accurately represented; without the model, deduction becomes a challenge for the designer. However, the trend remains with the so-called average models which describe low-

frequency and neglect high-frequency dynamics (semiconductor switching

Average models that take, some or all, parasitic losses into account, have been presented by [1, 10]. Recent works [7, 10–14] show that a practical level of model detail for PECs includes parasitic losses associated with their passive elements and disregards losses due to semiconductor switching. Models with this level of detail are suitable for system design, M Dð Þ derivation, η analysis, and dynamical performance evaluation [12]. Additionally, these models are suitable for control purposes [2, 10]. It is clear that based on average models, PECs can be designed to carry out dynamical performance analysis. Notwithstanding, a design procedure is needed that comprises all necessary steps to design and control PECs and fulfills all given operating requirements. This design procedure must be simple and useful.

In the PEC field, few works that take into account dynamical characteristics of the system have been carried out [15–17]. In these works, PEC's design problem is presented as an optimization problem. In consequence, a procedure to easily design and control PECs is still needed. In this chapter, a procedure to easily design and control PECs is introduced. In this procedure, neither an optimization process is carried out nor is the control structure fixed. But, zeros' location impact over the system dynamical responses is analyzed, showing that a careful selection of the PEC passive elements could both avoid electronic device failures due to large overshoots

The structure of the chapter is organized as follows: in Section 2, both time- and frequency-domain models of the boost DC-DC converter are derived. In Section 3, the boost DC-DC converter is studied in steady state. Section 3 is composed of Sections 3.1, 3.2, 3.3, and 3.4. In Sections 3.1 and 3.2, expressions for M Dð Þ and η are derived including some parasitic losses. In Section 3.3, conditions to operate in CCM or DCM are found. In Section 3.4, both inductor current and capacitor voltage ripple analyses are carried out to find lower boundaries for inductor and capacitor values that fulfill ripple requirements. In Section 4, the value of the passive elements is selected such that operating requirements are fulfilled and system dynamical performance is achieved. Mathematical model is contrasted with a PSIM implementation of the boost DC-DC converter. In Section 5, the widely accepted current mode control (CMC) structure for boost DC-DC converters is designed.

Figure 1 shows a circuital representation of a typical boost DC-DC converter including its parasitic losses associated to the passive elements. The boost DC-DC

and η nonlinearities and limitations [6].

Applied Modern Control

dynamics) of the system [9].

and improve the dynamical system performance.

2. Nonlinear dynamical modeling

122

the duty ratio, a continuous variable, and d∈½ � 0; 1 . In this chapter, d is used as the input control, while D is d in the operation point.

$$
\dot{\mathbf{x}} = A\mathbf{x} + Bu\tag{4}
$$

$$\mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}u \tag{5}$$

Gvovg <sup>¼</sup> <sup>R</sup>ð Þ <sup>ϕ</sup>CCð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>s</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>1</sup> � <sup>D</sup>

Equivalent simplified representation of the boost DC-DC converter.

DOI: http://dx.doi.org/10.5772/intechopen.80426

<sup>C</sup>dvC

1 þ α<sup>C</sup> � � 1 RC � �½ �þ vC

1 1 þ α<sup>C</sup> � � � �

3. Steady-state analysis

dt <sup>¼</sup> <sup>1</sup>

½ �þ vC

Gvo ðÞ¼ s

1 þ α<sup>C</sup>

1 1 þ α<sup>C</sup>

<sup>R</sup> <sup>ϕ</sup>Cð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> � <sup>ϕ</sup>Cð Þ� <sup>1</sup> � <sup>D</sup> RL

Gvoi<sup>0</sup> ¼

Figure 2.

regulates vo.

½ �¼ vo

125

vC: ½ �¼ � <sup>1</sup>

ð Þ RLCð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>s</sup><sup>2</sup> <sup>þ</sup> <sup>L</sup> <sup>þ</sup> RC Rð Þ <sup>L</sup> <sup>þ</sup> <sup>ϕ</sup><sup>C</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> h i<sup>s</sup> <sup>þ</sup> ð Þ RL <sup>þ</sup> <sup>R</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup>

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

� �ð Þ <sup>ϕ</sup>CCð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>s</sup> <sup>þ</sup> <sup>1</sup>

ð Þ RLCð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>s</sup><sup>2</sup> <sup>þ</sup> <sup>L</sup> <sup>þ</sup> RC Rð Þ <sup>L</sup> <sup>þ</sup> <sup>ϕ</sup><sup>C</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> h i<sup>s</sup> <sup>þ</sup> ð Þ RL <sup>þ</sup> <sup>R</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup>

Once the current control loop in the CMC structure is closed, the equivalent simplified representation of the boost DC-DC converter shown in Figure 2 is obtained. Large- and small-signal models of the simplified boost DC-DC converter are given by Eqs. (16) and (17), respectively. The transfer functions of the simplified model are given by Eq. (18). The numerator of Eq. (18) has two components, one for each system input, i.e., iREF and io, respectively. In the CMC structure, Eq. (18) is employed to tune the PI controller in the outer control loop, which

� � iLREF ð Þ� <sup>1</sup> � <sup>D</sup> vC

1 1 þ α<sup>C</sup>

� � ð Þ <sup>1</sup> � <sup>D</sup>

� �RCð Þ� <sup>1</sup> � <sup>D</sup> <sup>1</sup>

½ � ð Þ 1 þ α<sup>C</sup> RCRCs þ R þ RC ð Þ 1 � D

� �

�ð Þ 1 þ α<sup>C</sup> RCRCs þ R þ RC

Once the system model is obtained, the following analysis might be carried out: (1) derivation of the M Dð Þ expression, (2) losses effect and efficiency expression

<sup>R</sup> � io � � (16)

<sup>C</sup> � <sup>1</sup>

ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> ½ � ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> RCs <sup>þ</sup> <sup>1</sup> (18)

1 þ α<sup>C</sup> � �RC

� � iLREF

� � iLREF

1 þ α<sup>C</sup> � � 1

C

io

io

" #

" #

(17)

(14)

(15)

where,

$$A = \begin{bmatrix} -\frac{(R\_L + \phi\_C)(1 - D)^2}{L} & \frac{\left(\frac{\phi\_C}{R} - 1\right)(1 - D)}{L} \\\\ \frac{(1 - D)}{(1 + a\_C)C} & -\frac{1}{RC(1 + a\_C)} \end{bmatrix} \tag{6}$$

$$B = \begin{bmatrix} \left(2\phi\_C I\_L(1 - D) - V\_C\left(\frac{\phi\_C}{R} - 1\right) - \phi\_C I\_o\right) & \frac{1}{L} & \frac{\phi\_C(1 - D)}{L} \\\\ -\frac{I\_L}{(1 + a\_C)C} & 0 & -\frac{1}{(1 + a\_C)C} \end{bmatrix} \tag{7}$$

$$\begin{bmatrix} 1 & 0 & 1 \end{bmatrix}$$

$$\mathbf{C} = \begin{bmatrix} \mathbf{1} & \mathbf{0} \\ \phi\_C(\mathbf{1} - D) & \mathbf{1} - \frac{\phi\_C}{R} \end{bmatrix} \tag{8}$$

$$D = \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} \\ -\phi\_C I\_L & \mathbf{0} & -\phi\_C \end{bmatrix} \tag{9}$$

The transfer functions given by Eqs. (10)–(15) are obtained by applying the realization given by Eq. (3), where GiLd ðÞ¼ s ILð Þs =D sð Þ, GiLvg ðÞ¼ s ILð Þs =Vgð Þs , GiLi<sup>0</sup> ðÞ¼ s ILð Þs =I0ð Þs , GvodðÞ¼ s Voð Þs =D sð Þ, Gvovg ðÞ¼ s Voð Þs =Vg ð Þs , and Gvoi<sup>0</sup> ðÞ¼ s Voð Þs =I0ð Þs .

$$G\_{i\_{L}} = \frac{\left\{ \left( \frac{1}{R} \right) (RC(1 + a\_C)(2R\phi\_C I\_L(1 - D) - V\_C(\phi\_C - R) - R\phi\_C I\_o)s \right\}}{+ (\phi\_C + R)(1 - D)RI\_L - V\_C(\phi\_C - R) - R\phi\_C I\_o)} \tag{10.10}$$

$$G\_{i\_{L}} = \frac{(RLC(1 + a\_C))s^2 + \left[ L + RC(R\_L + \phi\_C)(1 + a\_C)(1 - D)^2 \right]s + (R\_L + R)(1 - D)^2}{(R\_L + R)s + (R\_L + R)(1 - D)^2} \tag{10.11}$$

$$G\_{i1v\_\xi} = \frac{RC(1+a\_C)s+1}{(RLC(1+a\_C))s^2 + \left[L + RC(R\_L + \phi\_C)(1+a\_C)(1-D)^2\right]s + (R\_L + R)(1-D)^2} \tag{11}$$

$$G\_{iLi\_0} = \frac{\phi\_C RC (1 - D)(1 + a\_C)s + R(1 - D)}{(RLC(1 + a\_C))s^2 + \left[L + RC(R\_L + \phi\_C)(1 + a\_C)(1 - D)^2\right]s + (R\_L + R)(1 - D)^2} \tag{12}$$

$$G\_{\rm p,d} = \frac{\left\{ \left[ C\phi\_C (\mathbf{1} - D)(\mathbf{1} + a\_C)(2R\phi\_C I\_L(\mathbf{1} - D) - V\_C(\phi\_C - R) - R\phi\_C I\_o) + L I\_L(\phi\_C - R) \right] \mathbf{s} \right\}}{-L \left( \phi\_C - R \right)(\mathbf{1} - D)^2 \left( 2\phi\_C + R\_L \right) + (\mathbf{1} - D)(2R\phi\_C I\_L(\mathbf{1} - D) - V\_C(\phi\_C - R) - R\phi\_C I\_o)} \right\} \tag{RLC}$$
 
$$\frac{\left( RLC(\mathbf{1} + a\_C) \right) \mathbf{s}^2 + \left[ L + RC(R\_L + \phi\_C)(\mathbf{1} + a\_C)(\mathbf{1} - D)^2 \right] \mathbf{s} + (R\_L + R)(\mathbf{1} - D)^2}{L \left[ (\mathbf{1} - a\_C)(2R\phi\_C I\_o(\mathbf{1} - D) - R\phi\_C I\_o) + (R\_L + R)(\mathbf{1} - D)^2 \right]} \tag{RLC}$$

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach DOI: http://dx.doi.org/10.5772/intechopen.80426

Figure 2. Equivalent simplified representation of the boost DC-DC converter.

the duty ratio, a continuous variable, and d∈½ � 0; 1 . In this chapter, d is used as the

ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>C</sup> � <sup>1</sup>

ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>C</sup> <sup>0</sup> � <sup>1</sup>

<sup>D</sup> <sup>¼</sup> 0 00

The transfer functions given by Eqs. (10)–(15) are obtained by applying the realization given by Eq. (3), where GiLd ðÞ¼ s ILð Þs =D sð Þ, GiLvg ðÞ¼ s ILð Þs =Vgð Þs ,

þð Þ ϕ<sup>C</sup> þ R ð Þ 1 � D RIL � VCð Þ� ϕ<sup>C</sup> � R RϕCIoÞ

½CϕCð Þ 1 � D ð Þ 1 þ α<sup>C</sup> ð2RϕCILð Þ� 1 � D VCð Þ� ϕ<sup>C</sup> � R RϕCIoÞ þ LILð Þ ϕ<sup>C</sup> � R �s

ð Þ RLCð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>s</sup><sup>2</sup> <sup>þ</sup> <sup>L</sup> <sup>þ</sup> RC Rð Þ <sup>L</sup> <sup>þ</sup> <sup>ϕ</sup><sup>C</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> h i

( )

ð Þþ 2ϕ<sup>C</sup> þ RL ð Þ 1 � D ð Þ 2RϕCILð Þ� 1 � D VCð Þ� ϕ<sup>C</sup> � R RϕCIo

GiLi<sup>0</sup> ðÞ¼ s ILð Þs =I0ð Þs , GvodðÞ¼ s Voð Þs =D sð Þ, Gvovg ðÞ¼ s Voð Þs =Vg ð Þs , and

ð Þ RLCð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>s</sup><sup>2</sup> <sup>þ</sup> <sup>L</sup> <sup>þ</sup> RC Rð Þ <sup>L</sup> <sup>þ</sup> <sup>ϕ</sup><sup>C</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> h i

ð Þ RLCð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>s</sup><sup>2</sup> <sup>þ</sup> <sup>L</sup> <sup>þ</sup> RC Rð Þ <sup>L</sup> <sup>þ</sup> <sup>ϕ</sup><sup>C</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> h i

GiLi<sup>0</sup> <sup>¼</sup> <sup>ϕ</sup>CRCð Þ <sup>1</sup> � <sup>D</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>s</sup> <sup>þ</sup> <sup>R</sup>ð Þ <sup>1</sup> � <sup>D</sup> ð Þ RLCð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>s</sup><sup>2</sup> <sup>þ</sup> <sup>L</sup> <sup>þ</sup> RC Rð Þ <sup>L</sup> <sup>þ</sup> <sup>ϕ</sup><sup>C</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> h i

GiLvg <sup>¼</sup> RCð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>s</sup> <sup>þ</sup> <sup>1</sup>

1 0 <sup>ϕ</sup>Cð Þ <sup>1</sup> � <sup>D</sup> <sup>1</sup> � <sup>ϕ</sup><sup>C</sup>

" #

�ϕCIL 0 �ϕ<sup>C</sup> � �

ðRCð Þ 1 þ α<sup>C</sup> ð Þ 2RϕCILð Þ� 1 � D VCð Þ� ϕ<sup>C</sup> � R RϕCIo s

ϕC <sup>R</sup> � <sup>1</sup> � �

x\_ ¼ Ax þ Bu (4) y ¼ Cx þ Du (5)

ð Þ 1 � D

ϕCð Þ 1 � D L

ð Þ 1 þ α<sup>C</sup> C

(6)

(7)

(8)

(9)

9 = ;

(10)

(11)

(12)

(13)

<sup>s</sup> <sup>þ</sup> ð Þ RL <sup>þ</sup> <sup>R</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup>

<sup>s</sup> þ þð Þ RL <sup>þ</sup> <sup>R</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup>

<sup>s</sup> þ þð Þ RL <sup>þ</sup> <sup>R</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup>

<sup>s</sup> <sup>þ</sup> ð Þ RL <sup>þ</sup> <sup>R</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup>

L

RCð Þ 1 þ α<sup>C</sup>

1 L

R

ϕC <sup>R</sup> � <sup>1</sup> � �

� ϕCIo

input control, while D is d in the operation point.

2ϕCILð Þ� 1 � D VC

� IL

<sup>A</sup> <sup>¼</sup> � ð Þ RL <sup>þ</sup> <sup>ϕ</sup><sup>C</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> L

� �

L

C ¼

ð Þ 1 � D

where,

Applied Modern Control

B ¼

Gvoi<sup>0</sup> ðÞ¼ s Voð Þs =I0ð Þs .

1 R � �

�ILð Þ <sup>ϕ</sup><sup>C</sup> � <sup>R</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup>

8 < :

GiLd ¼

Gvod ¼

124

$$G\_{v\_2 i\_2} = \frac{R\left(\phi\_C C(1+a\_C)s + 1\right)(1-D)}{(RLC(1+a\_C))s^2 + \left[L + RC(R\_L + \phi\_C)(1+a\_C)(1-D)^2\right]s + \left(R\_L + R\right)(1-D)^2} \tag{14}$$

$$G\_{v\_2 i\_0} = \frac{R\left(\phi\_C(1-D)^2 - \phi\_C(1-D) - R\_L\right)\left(\phi\_C C(1+a\_C)s + 1\right)}{(RLC(1+a\_C))s^2 + \left[L + RC(R\_L + \phi\_C)(1+a\_C)(1-D)^2\right]s + \left(R\_L + R\right)(1-D)^2} \tag{15}$$

Once the current control loop in the CMC structure is closed, the equivalent simplified representation of the boost DC-DC converter shown in Figure 2 is obtained. Large- and small-signal models of the simplified boost DC-DC converter are given by Eqs. (16) and (17), respectively. The transfer functions of the simplified model are given by Eq. (18). The numerator of Eq. (18) has two components, one for each system input, i.e., iREF and io, respectively. In the CMC structure, Eq. (18) is employed to tune the PI controller in the outer control loop, which regulates vo.

$$C\frac{dv\_C}{dt} = \left(\frac{1}{1+a\_C}\right) \left(i\_{L\_{REF}}(1-D) - \frac{v\_C}{R} - i\_o\right) \tag{16}$$

$$
\begin{bmatrix} \left[ \boldsymbol{v}\_{C} \right] \end{bmatrix} = \left[ -\left( \frac{\mathbf{1}}{\mathbf{1} + a\_{C}} \right) \frac{\mathbf{1}}{RC} \right] \begin{bmatrix} \boldsymbol{v}\_{C} \end{bmatrix} + \left[ \left( \frac{\mathbf{1}}{\mathbf{1} + a\_{C}} \right) \frac{(\mathbf{1} - D)}{C} \quad - \left( \frac{\mathbf{1}}{\mathbf{1} + a\_{C}} \right) \frac{\mathbf{1}}{C} \right] \begin{bmatrix} \boldsymbol{i}\_{L\_{\mathrm{RF}}} \\ \boldsymbol{i}\_{o} \end{bmatrix} \right] \tag{17}
$$
 
$$
\boldsymbol{i}\_{L} \begin{bmatrix} \boldsymbol{v}\_{o} \end{bmatrix} = \left[ \left( \frac{\mathbf{1}}{\mathbf{1} + a\_{C}} \right) \right] \left[ \boldsymbol{v}\_{C} \right] + \left[ \left( \frac{\mathbf{1}}{\mathbf{1} + a\_{C}} \right) \boldsymbol{R}\_{C} (\mathbf{1} - D) \quad - \left( \frac{\mathbf{1}}{\mathbf{1} + a\_{C}} \right) \boldsymbol{R}\_{C} \right] \begin{bmatrix} \boldsymbol{i}\_{L\_{\mathrm{RF}}} \\ \boldsymbol{i}\_{o} \end{bmatrix} \tag{17}
$$

$$\mathbf{G}\_{v\_o}(\mathbf{s}) = \frac{\begin{bmatrix} [(\mathbf{1} + a\_C)R\mathbf{C}\mathbf{R}\_C\mathbf{s} + \mathbf{R} + \mathbf{R}\_C](\mathbf{1} - D) \\ -(\mathbf{1} + a\_C)R\mathbf{C}\mathbf{R}\_C\mathbf{s} + \mathbf{R} + \mathbf{R}\_C \end{bmatrix}}{(\mathbf{1} + a\_C)[(\mathbf{1} + a\_C)R\mathbf{C}\mathbf{s} + \mathbf{1}]} \tag{18}$$

#### 3. Steady-state analysis

Once the system model is obtained, the following analysis might be carried out: (1) derivation of the M Dð Þ expression, (2) losses effect and efficiency expression

derivation, (3) condition analyses of CCM and DCM, and (4) inductor current ΔiL and capacitor voltage ΔvC ripple analysis. The aim of these analyses is to determine suitable passive elements' (L and C) boundaries which satisfy the design requirements.

#### 3.1 First step: derivation of the equilibrium conversion ratio M(D) expression

Steady-state model allows to derive expressions for average rated values for both vC and iL as functions of the system inputs and parameters. The steady-state model is obtained by setting the model given by Eqs. (1) and (2) to zero. Thus, Eqs. (19) and (20) are obtained.

$$I\_L = \frac{V\_C}{R(1 - D)}\tag{19}$$

$$V\_{\mathcal{g}} = \left(\frac{R\_L + R(\mathbf{1} - D)^2}{R(\mathbf{1} - D)}\right) V\_o \tag{20}$$

From Eqs. (1) and (2), it is found that Vo ¼ VC in steady state. The expression of M Dð Þ for the boost DC-DC converter is conveniently written using Eq. (21), where α<sup>L</sup> ¼ RL=R. M Dð Þ indicates the conversion gain factor in voltage in terms of D, R, and RL.

$$M(D) = \frac{V\_o}{V\_\mathcal{g}} = \frac{\left(\mathbf{1} - D\right)}{a\_L + \left(\mathbf{1} - D\right)^2} \tag{21}$$

The equivalent circuit model in Figure 3 allows to compute the converter efficiency η. From Figure 3, it is possible to deduce the efficiency expression given by

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

Simulations of Eqs. (21) and (24) are shown in Figure 4 for several values of α<sup>C</sup>

Figure 4(a) shows how the α<sup>L</sup> ratio affects M Dð Þ: α<sup>L</sup> ¼ 0 is the ideal case for the boost DC-DC converter (without losses) and M Dð Þ in the converter has an increasing trend and eventually tends to infinity. When α<sup>L</sup> increases (real case, converter with losses), M Dð Þ decreases and the curve has a quadratic trend. It can be observed that the higher α<sup>L</sup> value matches to the lower converter conversion ratio M Dð Þ. From Figure 4(b), it is observed that the maximum η value reached by the converter is determined by losses and it is given in D ¼ 0 for every M Dð Þ curve. For the studied case, it is the combination of α<sup>C</sup> and α<sup>L</sup> that determines the maximum value of η, which decreases while D increases, dropping to 0 when D tends to 1.

Figure 4(a) and (b) is shown together to relate M Dð Þ and η. Two different values of D can be selected to reach the same value of M Dð Þ. Nevertheless, higher values of D lead to lower efficiency values. For this reason, it is recommended that

ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>α</sup><sup>L</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> (24)

<sup>η</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup>

and α<sup>L</sup> ratios in order to test how much losses affect both M Dð Þ and η.

the converter operates at low values of D as possible.

(a) Conversion ratio M Dð Þ vs. duty cycle D. (b) Efficiency η vs. duty cycle D.

Eq. (24).

127

Figure 4.

Figure 3.

DC transformer model of the boost DC-DC converter.

DOI: http://dx.doi.org/10.5772/intechopen.80426

Note that M Dð Þ does not depend on RC due to the fact that the capacitor current in the average model is zero, leading to no voltage drop in RC.

It is important to remark that if RL ¼ 0 in Eq. (21), this expression is in agreement with the ideal boost DC-DC M Dð Þ, i.e., M Dð Þ¼ 1=ð Þ 1 � D . However, the converter reaches an efficiency equal to 100% if RL ¼ 0. Additionally, M Dð Þ tends to ∞ when D tends to 1. The above consideration is not true in a real boost DC-DC converter application and, for this reason, an analysis without including parasitic losses is not convenient.

#### 3.2 Second step: losses effect and efficiency expression derivation

This section shows how parasitic losses affect η in the boost DC-DC converter case. Losses effect and efficiency analyses are carried out in order to find suitable values for RL and RC such that the designed PEC fulfills the operating requirements.

The DC transformer correctly represents the relations between DC voltages and currents of the converter. The resulting model can be directly solved to find voltages, currents, losses, and efficiency in the boost DC-DC converter [20].

Eqs. (22) and (23) are obtained from Eqs. (1) and (2). These equations establish that the average value of both iL and vC are equal to zero in steady state. Figure 3 is the representation of Eqs. (22) and (23) as a DC transformer model.

$$\mathbf{0} = V\_{\mathcal{g}} - \left(R\_L + \phi\_C (\mathbf{1} - D)^2\right) I\_L - \left(\frac{1}{1 + a\_C}\right) V\_C (\mathbf{1} - D) \tag{22}$$

$$\mathbf{0} = -\frac{V\_C}{(\mathbf{1} + a\_C)R} + \left(\frac{\mathbf{1}}{\mathbf{1} + a\_C}\right)I\_L(\mathbf{1} - D) \tag{23}$$

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach DOI: http://dx.doi.org/10.5772/intechopen.80426

Figure 3. DC transformer model of the boost DC-DC converter.

derivation, (3) condition analyses of CCM and DCM, and (4) inductor current ΔiL and capacitor voltage ΔvC ripple analysis. The aim of these analyses is to determine suitable passive elements' (L and C) boundaries which satisfy the design require-

3.1 First step: derivation of the equilibrium conversion ratio M(D) expression

IL <sup>¼</sup> VC

Vg <sup>¼</sup> RL <sup>þ</sup> <sup>R</sup>ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> Rð Þ 1 � D !

M Dð Þ¼ Vo

3.2 Second step: losses effect and efficiency expression derivation

ages, currents, losses, and efficiency in the boost DC-DC converter [20].

the representation of Eqs. (22) and (23) as a DC transformer model.

ð Þ <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>C</sup> <sup>R</sup> <sup>þ</sup>

<sup>0</sup> <sup>¼</sup> Vg � RL <sup>þ</sup> <sup>ϕ</sup>Cð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> � �

<sup>0</sup> ¼ � VC

in the average model is zero, leading to no voltage drop in RC.

Vg

From Eqs. (1) and (2), it is found that Vo ¼ VC in steady state. The expression of M Dð Þ for the boost DC-DC converter is conveniently written using Eq. (21), where α<sup>L</sup> ¼ RL=R. M Dð Þ indicates the conversion gain factor in voltage in terms of D, R,

<sup>¼</sup> ð Þ <sup>1</sup> � <sup>D</sup>

Note that M Dð Þ does not depend on RC due to the fact that the capacitor current

It is important to remark that if RL ¼ 0 in Eq. (21), this expression is in agreement with the ideal boost DC-DC M Dð Þ, i.e., M Dð Þ¼ 1=ð Þ 1 � D . However, the converter reaches an efficiency equal to 100% if RL ¼ 0. Additionally, M Dð Þ tends to ∞ when D tends to 1. The above consideration is not true in a real boost DC-DC converter application and, for this reason, an analysis without including parasitic

This section shows how parasitic losses affect η in the boost DC-DC converter case. Losses effect and efficiency analyses are carried out in order to find suitable values for RL and RC such that the designed PEC fulfills the operating requirements. The DC transformer correctly represents the relations between DC voltages and currents of the converter. The resulting model can be directly solved to find volt-

Eqs. (22) and (23) are obtained from Eqs. (1) and (2). These equations establish that the average value of both iL and vC are equal to zero in steady state. Figure 3 is

IL � <sup>1</sup>

1 1 þ α<sup>C</sup> � �

1 þ α<sup>C</sup> � �

ILð Þ 1 � D

VCð Þ 1 � D

Id

Vd

(22)

(23)

Steady-state model allows to derive expressions for average rated values for both vC and iL as functions of the system inputs and parameters. The steady-state model is obtained by setting the model given by Eqs. (1) and (2) to zero. Thus, Eqs. (19)

<sup>R</sup>ð Þ <sup>1</sup> � <sup>D</sup> (19)

<sup>α</sup><sup>L</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> (21)

Vo (20)

ments.

and RL.

126

and (20) are obtained.

Applied Modern Control

losses is not convenient.

Figure 4. (a) Conversion ratio M Dð Þ vs. duty cycle D. (b) Efficiency η vs. duty cycle D.

The equivalent circuit model in Figure 3 allows to compute the converter efficiency η. From Figure 3, it is possible to deduce the efficiency expression given by Eq. (24).

$$\eta = \frac{\left(\mathbf{1} - D\right)^2}{\left(\mathbf{1} + a\_C\right)\left(a\_L + \left(\mathbf{1} - D\right)^2\right)}\tag{24}$$

Simulations of Eqs. (21) and (24) are shown in Figure 4 for several values of α<sup>C</sup> and α<sup>L</sup> ratios in order to test how much losses affect both M Dð Þ and η.

Figure 4(a) and (b) is shown together to relate M Dð Þ and η. Two different values of D can be selected to reach the same value of M Dð Þ. Nevertheless, higher values of D lead to lower efficiency values. For this reason, it is recommended that the converter operates at low values of D as possible.

Figure 4(a) shows how the α<sup>L</sup> ratio affects M Dð Þ: α<sup>L</sup> ¼ 0 is the ideal case for the boost DC-DC converter (without losses) and M Dð Þ in the converter has an increasing trend and eventually tends to infinity. When α<sup>L</sup> increases (real case, converter with losses), M Dð Þ decreases and the curve has a quadratic trend. It can be observed that the higher α<sup>L</sup> value matches to the lower converter conversion ratio M Dð Þ.

From Figure 4(b), it is observed that the maximum η value reached by the converter is determined by losses and it is given in D ¼ 0 for every M Dð Þ curve. For the studied case, it is the combination of α<sup>C</sup> and α<sup>L</sup> that determines the maximum value of η, which decreases while D increases, dropping to 0 when D tends to 1.

Hence, the converter should operate as far as possible with low D values. Additionally, an increase of either α<sup>C</sup> or α<sup>L</sup> causes a decrease in η. Therefore, α<sup>C</sup> and α<sup>L</sup> should tend to zero to guarantee high converter efficiency. Values of α<sup>C</sup> and α<sup>L</sup> were clustered in groups of curves. For α<sup>L</sup> ¼ 0 and α<sup>C</sup> ¼ ½ � 0; 0:05; 0:1 (above lines group), it is noted that while α<sup>C</sup> increases, η slightly decreases. For curves group α<sup>L</sup> ¼ 0:05 and α<sup>C</sup> ¼ ½ � 0; 0:05; 0:1 (middle lines group) and for curves group α<sup>L</sup> ¼ 0:1 and α<sup>C</sup> ¼ ½ � 0; 0:05; 0:1 (below lines group), something similar occurs—η decreases while α<sup>C</sup> increases. However, decreases in η are more notable when α<sup>L</sup> increases than when α<sup>C</sup> increases. The combined effect of high α<sup>L</sup> and α<sup>C</sup> leads to a highly inefficient system with high losses.

#### 3.3 Third step: conditions for the converter conduction model

The CCM is suggested since DCM causes a larger voltage ripple in the boost DC-DC converter case [18, 19]. In consequence, the peak inductor current in DCM is higher than in CCM [22]. By [20], the condition for operating in the CCM is ∣IL∣>∣ΔiL∣ and the condition for operating in the DCM is ∣IL∣<∣ΔiL∣. The DCM condition for the boost DC-DC converter is given by Eq. (25), where Ts ¼ 1=fsw and fsw is the converter switching frequency.

$$D(1-D)^2 \succ \frac{2L}{RT\_s} \tag{25}$$

3.4 Fourth step: inductor current and capacitor voltage ripple analysis

t ¼ Ts and the process repeats.

DOI: http://dx.doi.org/10.5772/intechopen.80426

Figure 6.

Figure 7.

129

approximation.

approximation.

ΔiL and ΔvC analyses are carried out to determine constraint equations for a suitable choice of both L and C values. The carried out analysis in this section is suitable for the boost DC-DC converter operating in CCM. Figure 6 shows both typical inductor voltage vL and inductor current iL linear-ripple approximations. The slope with iL increasing or decreasing is deduced from the analysis of VL at each subinterval of time taken into account. Typical values of current inductor ripple ΔiL lie under 10% of the full-load value of IL [20]. From Figure 6, it is seen that iL begins at the initial value of iLð Þ 0 . After time proceeds, iL increases during the first subinterval (DTs) and decreases during the second subinterval ( 1ð Þ � D Ts), both with a constant slope. Then, the switch changes back to its initial position at time

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

As illustrated in Figure 6(b), both current ripple and inductor magnitudes are related through the slope of iL. The peak inductor current Ipk is equal to IL plus the peak-to-average ripple ΔiL. Ipk flows through inductor and semiconductor devices that comprise the switch. The knowledge of Ipk is necessary when specifying the rating of the device. The ripple magnitude can be calculated through the knowledge of both the slope of iL and the length of the first subinterval (DTs). The iL linearripple approximation is symmetrical to IL; hence during the first time subinterval, iL increases by 2ΔiL (since ΔiL is the peak ripple, the peak-to-peak ripple is 2ΔiL).

In consequence, the inductor value L can be chosen from Eq. (27).

Vg � <sup>α</sup><sup>L</sup> ð Þ 1�D Vo

(a) Typical inductor voltage linear-ripple approximation. (b) Typical current inductor linear-ripple

(a) Typical capacitor current linear-ripple approximation. (b) Typical capacitor voltage linear-ripple

2ΔiL

Eq. (27) is a lower boundary for the L value, where L can be chosen such that a

DTs (27)

L ¼

maximum ΔiL is attained for the boost DC-DC operating condition.

The left side of Eq. (25) is a function that only depends on D. Here, this function is named as K Dð Þ¼ <sup>D</sup>ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup> . The right side of Eq. (25) is a dimensionless function that depends on L, R, and Ts, which is named in this chapter as K ¼ 2L=RTs. If L and R are taken as the converter parameters and fsw is fixed, K is a constant and represents the converter measure to operate in CCM and DCM [20]. Large values of K lead to CCM. Small values of K lead to the DCM for some values of D. K Dð Þ is a function that represents the boundary between DCM and CCM. Then, the minimum value of K must be at least equal to the maximum value of K Dð Þ, i.e., max ð Þ K Dð Þ ≤ min ð Þ K , if it is desired that the converter always operates in CCM, see Figure 5. Therefore, if values for R and Ts are given in the system specifications, a condition for the minimum possible value of L that assures CCM operation is given by Eq. (26), with max ð Þ¼ K Dð Þ 0:148, that is equal to the critical value of K Kð Þ<sup>c</sup> .

$$L > 0.148 \frac{RT\_s}{2} \tag{26}$$

Figure 5. K Dð Þ, K, and conduction mode (CM) conditions.

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach DOI: http://dx.doi.org/10.5772/intechopen.80426

#### 3.4 Fourth step: inductor current and capacitor voltage ripple analysis

ΔiL and ΔvC analyses are carried out to determine constraint equations for a suitable choice of both L and C values. The carried out analysis in this section is suitable for the boost DC-DC converter operating in CCM. Figure 6 shows both typical inductor voltage vL and inductor current iL linear-ripple approximations. The slope with iL increasing or decreasing is deduced from the analysis of VL at each subinterval of time taken into account. Typical values of current inductor ripple ΔiL lie under 10% of the full-load value of IL [20]. From Figure 6, it is seen that iL begins at the initial value of iLð Þ 0 . After time proceeds, iL increases during the first subinterval (DTs) and decreases during the second subinterval ( 1ð Þ � D Ts), both with a constant slope. Then, the switch changes back to its initial position at time t ¼ Ts and the process repeats.

As illustrated in Figure 6(b), both current ripple and inductor magnitudes are related through the slope of iL. The peak inductor current Ipk is equal to IL plus the peak-to-average ripple ΔiL. Ipk flows through inductor and semiconductor devices that comprise the switch. The knowledge of Ipk is necessary when specifying the rating of the device. The ripple magnitude can be calculated through the knowledge of both the slope of iL and the length of the first subinterval (DTs). The iL linearripple approximation is symmetrical to IL; hence during the first time subinterval, iL increases by 2ΔiL (since ΔiL is the peak ripple, the peak-to-peak ripple is 2ΔiL). In consequence, the inductor value L can be chosen from Eq. (27).

$$L = \frac{V\_g - \left(\frac{a\_L}{(1-D)}\right)V\_o}{2\Delta i\_L}DT\_s\tag{27}$$

Eq. (27) is a lower boundary for the L value, where L can be chosen such that a maximum ΔiL is attained for the boost DC-DC operating condition.

Figure 6.

Hence, the converter should operate as far as possible with low D values. Additionally, an increase of either α<sup>C</sup> or α<sup>L</sup> causes a decrease in η. Therefore, α<sup>C</sup> and α<sup>L</sup> should tend to zero to guarantee high converter efficiency. Values of α<sup>C</sup> and α<sup>L</sup> were

The CCM is suggested since DCM causes a larger voltage ripple in the boost DC-DC converter case [18, 19]. In consequence, the peak inductor current in DCM is higher than in CCM [22]. By [20], the condition for operating in the CCM is ∣IL∣>∣ΔiL∣ and the condition for operating in the DCM is ∣IL∣<∣ΔiL∣. The DCM condition for the boost DC-DC converter is given by Eq. (25), where Ts ¼ 1=fsw and fsw is

<sup>D</sup>ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup>

R are taken as the converter parameters and fsw is fixed, K is a constant and

<sup>&</sup>gt; <sup>2</sup><sup>L</sup> RTs

. The right side of Eq. (25) is a dimensionless function

<sup>2</sup> (26)

The left side of Eq. (25) is a function that only depends on D. Here, this function

that depends on L, R, and Ts, which is named in this chapter as K ¼ 2L=RTs. If L and

represents the converter measure to operate in CCM and DCM [20]. Large values of K lead to CCM. Small values of K lead to the DCM for some values of D. K Dð Þ is a function that represents the boundary between DCM and CCM. Then, the minimum value of K must be at least equal to the maximum value of K Dð Þ, i.e.,

max ð Þ K Dð Þ ≤ min ð Þ K , if it is desired that the converter always operates in CCM, see Figure 5. Therefore, if values for R and Ts are given in the system specifications, a condition for the minimum possible value of L that assures CCM operation is given by Eq. (26), with max ð Þ¼ K Dð Þ 0:148, that is equal to the critical value of K Kð Þ<sup>c</sup> .

L>0:148

RTs

(25)

clustered in groups of curves. For α<sup>L</sup> ¼ 0 and α<sup>C</sup> ¼ ½ � 0; 0:05; 0:1 (above lines group), it is noted that while α<sup>C</sup> increases, η slightly decreases. For curves group α<sup>L</sup> ¼ 0:05 and α<sup>C</sup> ¼ ½ � 0; 0:05; 0:1 (middle lines group) and for curves group α<sup>L</sup> ¼ 0:1 and α<sup>C</sup> ¼ ½ � 0; 0:05; 0:1 (below lines group), something similar occurs—η decreases while α<sup>C</sup> increases. However, decreases in η are more notable when α<sup>L</sup> increases than when α<sup>C</sup> increases. The combined effect of high α<sup>L</sup> and α<sup>C</sup> leads to a

3.3 Third step: conditions for the converter conduction model

highly inefficient system with high losses.

Applied Modern Control

the converter switching frequency.

is named as K Dð Þ¼ <sup>D</sup>ð Þ <sup>1</sup> � <sup>D</sup> <sup>2</sup>

Figure 5.

128

K Dð Þ, K, and conduction mode (CM) conditions.

(a) Typical inductor voltage linear-ripple approximation. (b) Typical current inductor linear-ripple approximation.

Figure 7.

(a) Typical capacitor current linear-ripple approximation. (b) Typical capacitor voltage linear-ripple approximation.

Likewise, vC linear-ripple approximation is depicted in Figure 7(b), where a relation between the voltage ripple and the capacitor magnitude is observed. It is seen that vC begins at the initial value of vCð Þ 0 . After time proceeds, vC decreases during the first subinterval (DTs) and increases during the second subinterval ( 1ð Þ � D Ts), both with a constant slope. Then, switch changes back to its initial position at time t ¼ Ts and the process repeats itself. The ripple magnitude can be calculated through the knowledge of both the slope of vC and the length of DTs. The change in vC, �2ΔvC during DTs, is equal to the slope multiplied by DTs. In consequence, Eq. (28) can be used to select the capacitor value of C to obtain a given ΔvC. Eq. (28) is a lower boundary for C value, where C can be chosen such that a maximum ΔvC is attained for the worst boost DC-DC operating condition.

$$C = \frac{V\_o}{2R\Delta v\_C} DT\_s \tag{28}$$

such that constraints like maximum physical admissible currents and voltages, converter efficiency, and converter CM are satisfied by keeping an acceptable

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

Expressions given by Eqs. (27) and (28) were deduced via steady-state analysis for lower inductor L and capacitor C boundaries, respectively. These expressions are suitable to choose L and C values as functions of ΔiL, Δvc, states, and inputs in their steady-state value. Additionally, the expression given by Eq. (26) was deduced from the converter CM analysis, which allows to guarantee the boost DC-DC converter CCM operation in all the operating ranges by choosing suitable L and R

In PECs, it is desired that ΔiL≤ max ð Þ ΔiL and Δvo≤ max ð Þ Δvo should be assured in the entire operation range. Then, based on the worst condition for ΔiL and Δvo, L and C lower boundaries can be deduced such that ripple constraints are satisfied.

2 max ð Þ ΔiL

max ð Þ Vo 2 min ð Þ R max ð Þ Δvo

Eq. (26) also gives a minimum boundary for L value. Then, Eqs. (26) and (29)

Eqs. (29) and (26) depend on RC and RL through α<sup>C</sup> and M Dð Þ, respectively. However, from the steady-state analysis, instead of calculating RC and RL values, it is suitable to establish α<sup>C</sup> and α<sup>L</sup> values. α<sup>C</sup> and α<sup>L</sup> values can be chosen such that the

The maximum boost DC-DC conversion condition corresponds to max ð Þ M Dð Þ .

consequence, loss ratios must be αC<0:05 and αL<0:05 when R ¼ max ð Þ¼ R 100Ω. Figure 8 shows M Dð Þ and η curves for RL ¼ 150mΩ, RC ¼ 70mΩ, and R ¼ 25Ω, i.e., α<sup>L</sup> ¼ 0:006 and α<sup>C</sup> ¼ 0:0034. With these α<sup>L</sup> and α<sup>C</sup> values, it is assured that η≥90% and M Dð Þ ≈ 3:17. Two points are remarked over both M Dð Þ and η curves for the rated converter conversion condition M Dð Þ¼ 2 and M Dð Þ ≈ 3:17. From Figure 8(a),

M Dð Þ ≈ 3:17, the voltage requirement is satisfied. Additionally, from Figure 8(b), it is

max ð Þ <sup>R</sup> ð Þ <sup>1</sup>�<sup>D</sup> min ð Þ Vo 

DTs (29)

DTs (30)

. According to the operating require-

<sup>¼</sup> <sup>30</sup><sup>V</sup> and max ð Þ¼ Vo <sup>95</sup>V, thus the maximum con-

Eqs. (29) and (30) give lower boundaries for L and C, respectively.

� RL

must be evaluated and the maximum L value must be selected as the lower

version condition in the example here presented is max ð Þ M Dð Þ ≈ 3:17. In

it is seen that the converter has sufficient boost capacity to guarantee that for

max Vg

C≥

system efficiency is η≥90% in the entire operating range.

L≥

Then, max ð Þ¼ M Dð Þ max ð Þ Vo = min Vg

ments in Table 1, min Vg

dynamical system performance.

DOI: http://dx.doi.org/10.5772/intechopen.80426

values.

boundary.

Figure 8.

131

(a) Conversion ratio M Dð Þ. (b) Efficiency η.

#### 4. Passive elements' value determination

In this section, the boost DC-DC converter operating requirements are specified. Then, the values of the passive elements are determined such that operating requirements are fulfilled and system dynamical performance is achieved. Finally, the mathematical model is contrasted with a PSIM implementation of the boost DC-DC converter.

#### 4.1 System operating requirements

In the boost DC-DC converter application, typical requirements are: input voltage range, output voltage range, output power range, output current range, operating frequency, output ripple, and efficiency. Unless otherwise noted, the continuous operating mode is assumed. The set of operating requirements for the boost DC-DC converter are specified in Table 1.

#### 4.2 Zeros' location analysis


The values of the passive elements are selected to fulfill the operating requirements. The main interest is to choose suitable values for inductors and capacitors

#### Table 1.

Boost DC-DC converter operating requirements.

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach DOI: http://dx.doi.org/10.5772/intechopen.80426

such that constraints like maximum physical admissible currents and voltages, converter efficiency, and converter CM are satisfied by keeping an acceptable dynamical system performance.

Expressions given by Eqs. (27) and (28) were deduced via steady-state analysis for lower inductor L and capacitor C boundaries, respectively. These expressions are suitable to choose L and C values as functions of ΔiL, Δvc, states, and inputs in their steady-state value. Additionally, the expression given by Eq. (26) was deduced from the converter CM analysis, which allows to guarantee the boost DC-DC converter CCM operation in all the operating ranges by choosing suitable L and R values.

In PECs, it is desired that ΔiL≤ max ð Þ ΔiL and Δvo≤ max ð Þ Δvo should be assured in the entire operation range. Then, based on the worst condition for ΔiL and Δvo, L and C lower boundaries can be deduced such that ripple constraints are satisfied. Eqs. (29) and (30) give lower boundaries for L and C, respectively.

$$L \ge \frac{\max\left(V\_{\mathcal{g}}\right) - \left(\frac{R\_L}{\max\left(R\right)(1-D)} \min\left(V\_{\mathcal{o}}\right)\right)}{2\max\left(\Delta i\_L\right)} DT\_s \tag{29}$$

$$C \geq \frac{\max\left(V\_o\right)}{2\min\left(R\right)\max\left(\Delta v\_o\right)}DT\_s \tag{30}$$

Eq. (26) also gives a minimum boundary for L value. Then, Eqs. (26) and (29) must be evaluated and the maximum L value must be selected as the lower boundary.

Eqs. (29) and (26) depend on RC and RL through α<sup>C</sup> and M Dð Þ, respectively. However, from the steady-state analysis, instead of calculating RC and RL values, it is suitable to establish α<sup>C</sup> and α<sup>L</sup> values. α<sup>C</sup> and α<sup>L</sup> values can be chosen such that the system efficiency is η≥90% in the entire operating range.

The maximum boost DC-DC conversion condition corresponds to max ð Þ M Dð Þ . Then, max ð Þ¼ M Dð Þ max ð Þ Vo = min Vg . According to the operating requirements in Table 1, min Vg <sup>¼</sup> <sup>30</sup><sup>V</sup> and max ð Þ¼ Vo <sup>95</sup>V, thus the maximum conversion condition in the example here presented is max ð Þ M Dð Þ ≈ 3:17. In consequence, loss ratios must be αC<0:05 and αL<0:05 when R ¼ max ð Þ¼ R 100Ω.

Figure 8 shows M Dð Þ and η curves for RL ¼ 150mΩ, RC ¼ 70mΩ, and R ¼ 25Ω, i.e., α<sup>L</sup> ¼ 0:006 and α<sup>C</sup> ¼ 0:0034. With these α<sup>L</sup> and α<sup>C</sup> values, it is assured that η≥90% and M Dð Þ ≈ 3:17. Two points are remarked over both M Dð Þ and η curves for the rated converter conversion condition M Dð Þ¼ 2 and M Dð Þ ≈ 3:17. From Figure 8(a), it is seen that the converter has sufficient boost capacity to guarantee that for M Dð Þ ≈ 3:17, the voltage requirement is satisfied. Additionally, from Figure 8(b), it is

Figure 8. (a) Conversion ratio M Dð Þ. (b) Efficiency η.

Likewise, vC linear-ripple approximation is depicted in Figure 7(b), where a relation between the voltage ripple and the capacitor magnitude is observed. It is seen that vC begins at the initial value of vCð Þ 0 . After time proceeds, vC decreases during the first subinterval (DTs) and increases during the second subinterval ( 1ð Þ � D Ts), both with a constant slope. Then, switch changes back to its initial position at time t ¼ Ts and the process repeats itself. The ripple magnitude can be calculated through the knowledge of both the slope of vC and the length of DTs. The change in vC, �2ΔvC during DTs, is equal to the slope multiplied by DTs. In consequence, Eq. (28) can be used to select the capacitor value of C to obtain a given ΔvC. Eq. (28) is a lower boundary for C value, where C can be chosen such that a maximum ΔvC is attained for the worst boost DC-DC operating condition.

> <sup>C</sup> <sup>¼</sup> Vo 2RΔvC

Then, the values of the passive elements are determined such that operating requirements are fulfilled and system dynamical performance is achieved. Finally, the mathematical model is contrasted with a PSIM implementation of the boost DC-

ing frequency, output ripple, and efficiency. Unless otherwise noted, the

Requirements Values

In this section, the boost DC-DC converter operating requirements are specified.

In the boost DC-DC converter application, typical requirements are: input voltage range, output voltage range, output power range, output current range, operat-

continuous operating mode is assumed. The set of operating requirements for the

The values of the passive elements are selected to fulfill the operating requirements. The main interest is to choose suitable values for inductors and capacitors

Input voltage range 30 V 35 V 40 V Output voltage range 50V 70 V 95 V Output power range 0 W 100 W 300 W Output current range 0A 2A 8A (At 50 V)

Output current ripple 1% 5% 10% Output voltage ripple 0.1% 0.5% 1% Steady-state efficiency 90% 95% 98% Load 25 Ω 50 Ω 100 Ω

Min Typ Max

4. Passive elements' value determination

boost DC-DC converter are specified in Table 1.

Operating frequency 100KHz

Boost DC-DC converter operating requirements.

4.1 System operating requirements

4.2 Zeros' location analysis

DC converter.

Applied Modern Control

Table 1.

130

DTs (28)

seen that for M Dð Þ¼ 2 and M Dð Þ ≈ 3:17, the converter has 97 and 93% of efficiency, respectively.

From Eq. (29), max ð Þ¼ IL max ð Þ VC = min ð Þ R ð Þ 1 � D . Then, on the one hand, if max ð Þ¼ ΔiL 0:1 max ð Þ IL , L≥326:34μH must be selected according to Eq. (29) in order to keep the converter in safe operation [22]. On the other hand, L≥40μH to always operate in CCM by evaluating Eq. (26). The iL ripple-based condition is a less restrictive boundary for L than the CCM-based condition. Therefore, L≥326:34μH is the lower boundary for this element.

If max ð Þ¼ ΔvC 0:01 max ð Þ Vo in order to keep a converter in safe operation [22], C≥14:120μF according to Eq. (30).

Minimum L and C values are selected as system parameters. Next, a simulation of the designed boost DC-DC converter is carried out. Figure 9 shows the step system response for Vg ¼ 35V, Vo ¼ 70V, Io ¼ 0A, L ¼ 326:34μH, C ¼ 14:120μF, R ¼ 50Ω, α<sup>C</sup> ¼ 0:0034, α<sup>L</sup> ¼ 0:006, and fsw ¼ 100kHz.

From Figure 9, it is seen that, with minimum values of L and C, voltage overshoot is O:S:Gvd ¼ 57:4718%, current overshoot is O:S:Gi <sup>L</sup><sup>d</sup> ¼ 187:2323%, and system setting time is ts ¼ 3:03ms. From Figure 9(a), it is seen that the peak current value is 33:1686A, while the steady-state current value is around 11:5477A. From Figure 9(b), it is seen that the peak voltage value is 214:5027 V, while the steadystate voltage value is around 136:2166 V.

to evaluate the effects of large values for both L and C. Figure 10 shows Gvd and

(a) Gvd step system response for varying L and C: overshoot with zeros. (b) GiLd step system response for varying

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

DOI: http://dx.doi.org/10.5772/intechopen.80426

From Figure 10, it is seen that the minimum possible value of C causes maximum overshoot in vo; while a minimum possible value of L causes maximum overshoot in iL. Moreover, minimum C and L values give minimum system setting

In contrast, large values of C cause high overshoot of iL; while large values of L cause high system setting time. In consequence, two additional design requirements are given in order to establish maximum possible values for L and C such that system overshoots and setting time are suitable: (a) maximum duty-ratio-tooutput-voltage overshoot max O:S:Gvd and (b) maximum duty-ratio-to-inductor-

From Figure 10, it is seen that the system dynamical response cannot be modified if the values of both L and C are simultaneously increased. Meanwhile, if either

L ¼ 1mH and C ¼ 15μF are selected by results shown in Figure 10 since with

to 82% and O:S:Gvd is approximately reduced to 4%. Furthermore, ts ¼ 4:32ms, i.e., the system setting time is only increased by 1:3ms. Thus, these L and C values establish a trade-off between system overshoots and performance. It is remarked

Frequency response of both the mathematical model and a PSIM circuital implementation are contrasted in order to validate the dynamical model of the designed boost DC-DC converter via simulation. The boost DC-DC converter was

parameterized with L ¼ 1mH, C ¼ 15μF, Vg ¼ 35V, Vo ¼ 70V, Io ¼ 0A, α<sup>C</sup> ¼ 0:0034, α<sup>L</sup> ¼ 0:006, R ¼ 50Ω, and fsw ¼ 100kHz. In consequence,

<sup>L</sup><sup>d</sup> ≈ 105% and O:S:Gvd ≈ 53%, i.e., O:S:Gi

<sup>L</sup><sup>d</sup> decrease. Nevertheless, larger

<sup>L</sup><sup>d</sup> is approximately reduced

GiLd overshoots and setting time for L∈½ � 326:34μH; 2000μH and

L and C: overshoot with zeros. (c) Step system response varying L and C: setting time.

Ld .

L or C values are increased, both O:S:Gvd and O:S:Gi

values of L have a major impact than larger values of C.

that selected L and C values are commercially available.

IL ¼ 2:8812A and D ¼ 0:5141 in the equilibrium point.

4.3 System frequency response verification

C∈½ � 14:12μF; 100μF .

current overshoot max O:S:Gi

these values O:S:Gi

133

time.

Figure 10.

A designed system with these overshoots needs to oversize its electronic devices such that these devices support both peak voltage and current values without system damage. However, such electronic devices can be expensive and inconvenient. For instance, in this chapter, an analysis of the boost DC-DC converter dynamical characteristics is carried out. This dynamical analysis studies the impact of L and C values over the zeros in transfer functions given by Eqs. (10) and (13), which determine overshoots and system setting time.

In a system as it is known, dynamical response is determined by poles and zeros' location [23]. Zeros are determined by the selected inputs and outputs of the system. Zeros' location is related to some system performance restrictions such as tracking limitations in feedback systems when classical control structures are employed [21, 22]. Moreover, large current or voltage overshoots in converter transient response can cause converter failures. PEC design process could take into account zeros' location due to L and C values such that the right half-plane (RHP) zeros are avoided or their impacts are attenuated. In consequence, the impact of L and C values over the zeros is analyzed to establish a trade-off between their values and the dynamical system response.

In the boost DC-DC converter d is chosen as control input, while vg and io are considered disturbances. Thus, d variations' effect is of primary interest over system output. In consequence, duty ratio-to-voltage-output ð Þ Gvd and duty-ratio-toinductor-current ð Þ GiLd transfer functions are studied. A simulation was carried out

Figure 9. (a) GiLd step system response. (b) Gvd step system response.

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach DOI: http://dx.doi.org/10.5772/intechopen.80426

#### Figure 10.

seen that for M Dð Þ¼ 2 and M Dð Þ ≈ 3:17, the converter has 97 and 93% of efficiency,

If max ð Þ¼ ΔvC 0:01 max ð Þ Vo in order to keep a converter in safe operation

From Figure 9, it is seen that, with minimum values of L and C, voltage

such that these devices support both peak voltage and current values without system damage. However, such electronic devices can be expensive and inconvenient. For instance, in this chapter, an analysis of the boost DC-DC converter dynamical characteristics is carried out. This dynamical analysis studies the impact of L and C values over the zeros in transfer functions given by Eqs. (10) and (13),

location [23]. Zeros are determined by the selected inputs and outputs of the system. Zeros' location is related to some system performance restrictions such as tracking limitations in feedback systems when classical control structures are employed [21, 22]. Moreover, large current or voltage overshoots in converter transient response can cause converter failures. PEC design process could take into account zeros' location due to L and C values such that the right half-plane (RHP) zeros are avoided or their impacts are attenuated. In consequence, the impact of L and C values over the zeros is analyzed to establish a trade-off between their values

system setting time is ts ¼ 3:03ms. From Figure 9(a), it is seen that the peak current value is 33:1686A, while the steady-state current value is around 11:5477A. From Figure 9(b), it is seen that the peak voltage value is 214:5027 V, while the steady-

A designed system with these overshoots needs to oversize its electronic devices

In a system as it is known, dynamical response is determined by poles and zeros'

In the boost DC-DC converter d is chosen as control input, while vg and io are considered disturbances. Thus, d variations' effect is of primary interest over system output. In consequence, duty ratio-to-voltage-output ð Þ Gvd and duty-ratio-toinductor-current ð Þ GiLd transfer functions are studied. A simulation was carried out

Minimum L and C values are selected as system parameters. Next, a simulation of the designed boost DC-DC converter is carried out. Figure 9 shows the step system response for Vg ¼ 35V, Vo ¼ 70V, Io ¼ 0A, L ¼ 326:34μH, C ¼ 14:120μF,

<sup>L</sup><sup>d</sup> ¼ 187:2323%, and

L≥326:34μH is the lower boundary for this element.

R ¼ 50Ω, α<sup>C</sup> ¼ 0:0034, α<sup>L</sup> ¼ 0:006, and fsw ¼ 100kHz.

which determine overshoots and system setting time.

overshoot is O:S:Gvd ¼ 57:4718%, current overshoot is O:S:Gi

[22], C≥14:120μF according to Eq. (30).

state voltage value is around 136:2166 V.

and the dynamical system response.

(a) GiLd step system response. (b) Gvd step system response.

Figure 9.

132

From Eq. (29), max ð Þ¼ IL max ð Þ VC = min ð Þ R ð Þ 1 � D . Then, on the one hand, if max ð Þ¼ ΔiL 0:1 max ð Þ IL , L≥326:34μH must be selected according to Eq. (29) in order to keep the converter in safe operation [22]. On the other hand, L≥40μH to always operate in CCM by evaluating Eq. (26). The iL ripple-based condition is a less restrictive boundary for L than the CCM-based condition. Therefore,

respectively.

Applied Modern Control

(a) Gvd step system response for varying L and C: overshoot with zeros. (b) GiLd step system response for varying L and C: overshoot with zeros. (c) Step system response varying L and C: setting time.

to evaluate the effects of large values for both L and C. Figure 10 shows Gvd and GiLd overshoots and setting time for L∈½ � 326:34μH; 2000μH and C∈½ � 14:12μF; 100μF .

From Figure 10, it is seen that the minimum possible value of C causes maximum overshoot in vo; while a minimum possible value of L causes maximum overshoot in iL. Moreover, minimum C and L values give minimum system setting time.

In contrast, large values of C cause high overshoot of iL; while large values of L cause high system setting time. In consequence, two additional design requirements are given in order to establish maximum possible values for L and C such that system overshoots and setting time are suitable: (a) maximum duty-ratio-tooutput-voltage overshoot max O:S:Gvd and (b) maximum duty-ratio-to-inductorcurrent overshoot max O:S:Gi Ld .

From Figure 10, it is seen that the system dynamical response cannot be modified if the values of both L and C are simultaneously increased. Meanwhile, if either L or C values are increased, both O:S:Gvd and O:S:Gi <sup>L</sup><sup>d</sup> decrease. Nevertheless, larger values of L have a major impact than larger values of C.

L ¼ 1mH and C ¼ 15μF are selected by results shown in Figure 10 since with these values O:S:Gi <sup>L</sup><sup>d</sup> ≈ 105% and O:S:Gvd ≈ 53%, i.e., O:S:Gi <sup>L</sup><sup>d</sup> is approximately reduced to 82% and O:S:Gvd is approximately reduced to 4%. Furthermore, ts ¼ 4:32ms, i.e., the system setting time is only increased by 1:3ms. Thus, these L and C values establish a trade-off between system overshoots and performance. It is remarked that selected L and C values are commercially available.

#### 4.3 System frequency response verification

Frequency response of both the mathematical model and a PSIM circuital implementation are contrasted in order to validate the dynamical model of the designed boost DC-DC converter via simulation. The boost DC-DC converter was parameterized with L ¼ 1mH, C ¼ 15μF, Vg ¼ 35V, Vo ¼ 70V, Io ¼ 0A, α<sup>C</sup> ¼ 0:0034, α<sup>L</sup> ¼ 0:006, R ¼ 50Ω, and fsw ¼ 100kHz. In consequence, IL ¼ 2:8812A and D ¼ 0:5141 in the equilibrium point.

Figure 11. (a) GiLd Bode diagram. (b) Gvd Bode diagram.

Figure 11 presents the boost DC-DC converter Bode diagrams of the PSIM circuital implementation and the mathematical model given by Eqs. (10)–(15). The frequency response of the PSIM circuital implementation matches with the mathematical model. Then, the PSIM circuital implementation is satisfactorily reproduced by the mathematical model.

(i.e., 68:6V–71:4V) in response to random changes (disturbances) in both vg and io. Also, the controller should be able to drive vo within the tolerance for vg variations

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

The inner loop control bandwidth must be 20kHz or less due to the fact that fsw is equal to 100kHz, and the outer loop control bandwidth must be smaller than 1=5 of the inner loop control bandwidth [25], i.e., smaller than 5kHz. Additionally, a robustness index of Ms<2 is desired to establish a trade-off between control perfor-

A PI controller was tuned by acting directly on d to track the inductor current reference iLREF since GiLd exhibits a minimum phase behavior. The inductor current PI controller was tuned by means of the root-locus technique, adopting the following design specifications: damping factor ζ equal to 0:707 and a 20kHz closed-loop

These PI controller design specifications ensure: (a) Zero steady-state error and a satisfactory reference tracking for frequencies below 20kHz; this is observed on transfer function TiLiLREF in Figure 13. (b) Effective disturbance rejection for both input voltage vg and current source io variations, which are observed on transfer functions TiLvg and TiLio in Figure 13, respectively. (c) A closed-loop robustness

A PI controller was tuned to regulate vo since the GvoiLREF ð Þs transfer function given by the Eq. (18) exhibits a minimum phase behavior. This PI controller provides the set-point of the inner control loop. The PI controller of the outer control loop was tuned by means of the root-locus technique considering a damping factor ζ equal to 0:707 and a 5kHz closed-loop bandwidth. The tuned PI controller transfer function GCvo ð Þs is given by Eq. (32). These PI controller design specifications ensure: (a) Zero steady-state error observed on transfer function TvovoREF in Figure 14. (b) Effective disturbance rejection for the current source io variations, which are observed on transfer function Tvoio in Figure 14. (c) A closed-loop

1:27s þ 55218 s

<sup>L</sup> ð Þs is given by Eq. (31).

(31)

bandwidth. The tuned PI controller transfer function GCi

GCi <sup>L</sup> ðÞ¼ s

over a range from 30 to 40V.

Boost DC-DC CMC structure scheme.

DOI: http://dx.doi.org/10.5772/intechopen.80426

mance and robustness [26].

Ms ¼ 1:2.

Figure 12.

robustness Ms ¼ 1:2.

135

#### 5. Control structure design

Nonminimum phase behavior is a well-known result derived from the boost DC-DC converter study [24]. To avoid this system behavior, a CMC structure has been proposed [18, 24]. Nonminimum phase behavior is avoided with this control structure since both GiLd and inductor-current-to-output-voltage GvoiLREF transfer functions have a minimum phase behavior.

The converter control design is focused on imposing a desired low-frequency behavior on the system. Here, a CMC structure for the boost DC-DC converter is designed. The aim is to tune PI controllers such that the control objective is achieved. Figure 12 shows the CMC structure for the DC-DC boost converter. As it is seen in Figure 12, the CMC structure employs two PI controllers: first one for iL control and second one for vo regulation. These PI controllers are arranged in master-slave form; where iL control loop is the inner loop and vo control loop is the outer loop. This master-slave arrangement allows vo regulation while preserving iL within specified safety limits.

In the boost DC-DC converter which operates in a switch-mode power supply and feeds a certain variable load, the d needs adjustments in order to ensure a constant vo for the entire operating range (voltage regulation). Besides, against any system disturbance (vg and io random changes), the d value should be adjusted to drive the system back to the operating point. The PI controller in the outer loop provides the set-point of the inner loop, which acts as the control input of the outer loop. The proportional and integral (PI) controller in the inner loop generates a continuous signal for d, which by means of a pulse width modulation (PWM) is applied to the power switching gate.

#### 5.1 Controller tuning

Controller's tuning task begins with the set of design specifications. The goal of the boost DC-DC converter controller is to maintain vo within 2% of its rated value Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach DOI: http://dx.doi.org/10.5772/intechopen.80426

Figure 12. Boost DC-DC CMC structure scheme.

Figure 11 presents the boost DC-DC converter Bode diagrams of the PSIM circuital implementation and the mathematical model given by Eqs. (10)–(15). The frequency response of the PSIM circuital implementation matches with the mathematical model. Then, the PSIM circuital implementation is satisfactorily reproduced

Nonminimum phase behavior is a well-known result derived from the boost DC-DC converter study [24]. To avoid this system behavior, a CMC structure has been proposed [18, 24]. Nonminimum phase behavior is avoided with this control structure since both GiLd and inductor-current-to-output-voltage GvoiLREF transfer func-

The converter control design is focused on imposing a desired low-frequency behavior on the system. Here, a CMC structure for the boost DC-DC converter is designed. The aim is to tune PI controllers such that the control objective is

achieved. Figure 12 shows the CMC structure for the DC-DC boost converter. As it is seen in Figure 12, the CMC structure employs two PI controllers: first one for iL control and second one for vo regulation. These PI controllers are arranged in master-slave form; where iL control loop is the inner loop and vo control loop is the outer loop. This master-slave arrangement allows vo regulation while preserving iL

In the boost DC-DC converter which operates in a switch-mode power supply and feeds a certain variable load, the d needs adjustments in order to ensure a constant vo for the entire operating range (voltage regulation). Besides, against any system disturbance (vg and io random changes), the d value should be adjusted to drive the system back to the operating point. The PI controller in the outer loop provides the set-point of the inner loop, which acts as the control input of the outer loop. The proportional and integral (PI) controller in the inner loop generates a continuous signal for d, which by means of a pulse width modulation (PWM) is applied to the power switching gate.

Controller's tuning task begins with the set of design specifications. The goal of the boost DC-DC converter controller is to maintain vo within 2% of its rated value

by the mathematical model.

(a) GiLd Bode diagram. (b) Gvd Bode diagram.

Figure 11.

Applied Modern Control

5. Control structure design

within specified safety limits.

5.1 Controller tuning

134

tions have a minimum phase behavior.

(i.e., 68:6V–71:4V) in response to random changes (disturbances) in both vg and io. Also, the controller should be able to drive vo within the tolerance for vg variations over a range from 30 to 40V.

The inner loop control bandwidth must be 20kHz or less due to the fact that fsw is equal to 100kHz, and the outer loop control bandwidth must be smaller than 1=5 of the inner loop control bandwidth [25], i.e., smaller than 5kHz. Additionally, a robustness index of Ms<2 is desired to establish a trade-off between control performance and robustness [26].

A PI controller was tuned by acting directly on d to track the inductor current reference iLREF since GiLd exhibits a minimum phase behavior. The inductor current PI controller was tuned by means of the root-locus technique, adopting the following design specifications: damping factor ζ equal to 0:707 and a 20kHz closed-loop bandwidth. The tuned PI controller transfer function GCi <sup>L</sup> ð Þs is given by Eq. (31). These PI controller design specifications ensure: (a) Zero steady-state error and a satisfactory reference tracking for frequencies below 20kHz; this is observed on transfer function TiLiLREF in Figure 13. (b) Effective disturbance rejection for both input voltage vg and current source io variations, which are observed on transfer functions TiLvg and TiLio in Figure 13, respectively. (c) A closed-loop robustness Ms ¼ 1:2.

$$G\_{C\_L}(s) = \frac{1.27s + 55218}{s} \tag{31}$$

A PI controller was tuned to regulate vo since the GvoiLREF ð Þs transfer function given by the Eq. (18) exhibits a minimum phase behavior. This PI controller provides the set-point of the inner control loop. The PI controller of the outer control loop was tuned by means of the root-locus technique considering a damping factor ζ equal to 0:707 and a 5kHz closed-loop bandwidth. The tuned PI controller transfer function GCvo ð Þs is given by Eq. (32). These PI controller design specifications ensure: (a) Zero steady-state error observed on transfer function TvovoREF in Figure 14. (b) Effective disturbance rejection for the current source io variations, which are observed on transfer function Tvoio in Figure 14. (c) A closed-loop robustness Ms ¼ 1:2.

$$G\_{C\_{\rm v}}(\varsigma) = \frac{0.07994s + 235.1}{\varsigma} \tag{32}$$

(iii) an experiment that simulates a combined change of �35 and �30% around the nominal values of the load and the input voltage, respectively, was carried out. Figure 15 shows the dynamical system response against the perturbations mentioned above. From Figure 15, it is seen that the system stability is not affected by any of the simulated perturbations, which means that the control structure is robust against the system perturbations from both the load and the input voltage up

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

Figure 15(a) shows the closed-loop behavior at unit step changes of io around the operating point corresponding to the full load. Two io unit step changes were applied to evaluate the control structure performance. The first step change was applied at t ¼ 10ms for 10ms, then the current source returns to its rated value io ¼ 0A. The second unit step change was applied at t ¼ 30ms for 10ms, then the current source returns to its rated value io ¼ 0A. In Figure 15(a), a satisfactory tracking of iLREF and regulation of vo to reject load disturbances depicted as changes

Figure 15(b) shows the closed-loop behavior at unit step changes of vg. Two vg

unit step changes were applied to evaluate the control structure capabilities to regulate vo and to evaluate the capabilities of the designed boost DC-DC converter. The first unit step change was applied at t ¼ 10ms for 10ms. This first unit step change was equal to vg ¼ �5V, i.e., the final value of the input voltage was vg ¼ 30V that corresponds with its lower boundary. The second unit step change was applied at t ¼ 30ms for 10ms. This second unit step change was equal to vg ¼ þ5V, i.e., the final value of the input voltage was vg ¼ 40V that corresponds to its upper boundary. In Figure 15(b) a satisfactory reference tracking of iLREF and control regulation of vo to changes in vg is observed. It is important to remark that under the worst condition for vg , the boost DC-DC converter was able to keep vo in its rated value. Finally, Figure 15(c) shows the closed-loop behavior at random unit step changes of both io and vg . These unit step changes were applied such that the designed control structure performance could be evaluated against any random disturbance. From Figure 15(c), it is possible to see that the designed control structure has a satisfactory performance against multiple disturbances within spec-

ified design requirements for the boost DC-DC converter in Table 1.

to 35%.

in io is observed.

DOI: http://dx.doi.org/10.5772/intechopen.80426

Figure 15.

137

Closed-loop behavior at unit steps system disturbances.

#### 5.2 Closed-loop system performance verification

The designed boost DC-DC converter with its control structure was implemented in PSIM to assess the closed-loop system robustness. Three cases were proposed to evaluate the control structure robustness against most common disturbances. (i) An experiment that simulates a change of �35% around the nominal value of the load was carried out. Next, (ii) an experiment that simulates a change of �15% around the nominal value of the input voltage was carried out. Finally,

Figure 13. Inner current control loop transfer functions.

Figure 14. Outer output voltage control loop transfer functions.

#### Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach DOI: http://dx.doi.org/10.5772/intechopen.80426

(iii) an experiment that simulates a combined change of �35 and �30% around the nominal values of the load and the input voltage, respectively, was carried out. Figure 15 shows the dynamical system response against the perturbations mentioned above. From Figure 15, it is seen that the system stability is not affected by any of the simulated perturbations, which means that the control structure is robust against the system perturbations from both the load and the input voltage up to 35%.

Figure 15(a) shows the closed-loop behavior at unit step changes of io around the operating point corresponding to the full load. Two io unit step changes were applied to evaluate the control structure performance. The first step change was applied at t ¼ 10ms for 10ms, then the current source returns to its rated value io ¼ 0A. The second unit step change was applied at t ¼ 30ms for 10ms, then the current source returns to its rated value io ¼ 0A. In Figure 15(a), a satisfactory tracking of iLREF and regulation of vo to reject load disturbances depicted as changes in io is observed.

Figure 15(b) shows the closed-loop behavior at unit step changes of vg. Two vg unit step changes were applied to evaluate the control structure capabilities to regulate vo and to evaluate the capabilities of the designed boost DC-DC converter. The first unit step change was applied at t ¼ 10ms for 10ms. This first unit step change was equal to vg ¼ �5V, i.e., the final value of the input voltage was vg ¼ 30V that corresponds with its lower boundary. The second unit step change was applied at t ¼ 30ms for 10ms. This second unit step change was equal to vg ¼ þ5V, i.e., the final value of the input voltage was vg ¼ 40V that corresponds to its upper boundary. In Figure 15(b) a satisfactory reference tracking of iLREF and control regulation of vo to changes in vg is observed. It is important to remark that under the worst condition for vg , the boost DC-DC converter was able to keep vo in its rated value.

Finally, Figure 15(c) shows the closed-loop behavior at random unit step changes of both io and vg . These unit step changes were applied such that the designed control structure performance could be evaluated against any random disturbance. From Figure 15(c), it is possible to see that the designed control structure has a satisfactory performance against multiple disturbances within specified design requirements for the boost DC-DC converter in Table 1.

Figure 15. Closed-loop behavior at unit steps system disturbances.

GCvo ðÞ¼ s

The designed boost DC-DC converter with its control structure was

implemented in PSIM to assess the closed-loop system robustness. Three cases were proposed to evaluate the control structure robustness against most common disturbances. (i) An experiment that simulates a change of �35% around the nominal value of the load was carried out. Next, (ii) an experiment that simulates a change of �15% around the nominal value of the input voltage was carried out. Finally,

5.2 Closed-loop system performance verification

Applied Modern Control

Figure 13.

Figure 14.

136

Inner current control loop transfer functions.

Outer output voltage control loop transfer functions.

0:07994s þ 235:1 s

(32)

c. The passive elements' value determination based on the system zeros' location. A zero-based analysis allowed to choose the passive elements' values such that a trade-off between operating requirements and system transient response were achieved. This analysis reduced the system outputs' overshoot alleviating

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

d.The proposal of a model-based control structure; particularly, a CMC structure based on PI controllers for automatic converter control was implemented in the boost DC-DC converter, although a control structure does not need to be

e. The procedure was applied to a boost DC-DC converter application taking into account the parasitic losses associated with its passive elements, which allows to investigate the details of its performance, operation, and behavior. It was possible to design a boost DC-DC converter that fulfills all the operating requirements in the entire operating range, even if bounded disturbances appear. Design was based on the nonlinear dynamical model and steady-state analysis. The CMC structure was implemented for the designed boost DC-DC converter. PI controllers were tuned by means of root-locus controller design method. The boost DC-DC converter was implemented in PSIM where system operating requirements, closed-loop performance, and robustness were

This work was partially supported by COLCIENCIAS (Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas) with the doctoral scholarship 727-2015. The authors would also like to thank

Jorge H. Urrea-Quintero carried out most of the work presented here, Nicolás Muñoz-Galeano was the advisor of this work, and Lina M. Gómez had a relevant

contribution with her extensive knowledge about systems theory.

electronic devices'stress and improving the system's performance.

fixed in this procedure.

DOI: http://dx.doi.org/10.5772/intechopen.80426

successfully verified.

"estrategia sostenibilidad UdeA".

The authors declare no conflict of interest.

Acknowledgements

Conflict of interest

Author contributions

139

Figure 16. (a) Instantaneous Power verification. (b) Ripples verification.

In order to carry out system operation requirements verification, case (c) of Figure 15 is taken into account. Figure 16 shows: (a) Pin, Pout and (b) iL and vo, when case (c) of Figure 15 is considered.

From Figure 16(a), it is seen that Pout does not exceed the maximum admissible output power in steady state and is always lower than Pin. ð Þ b . Figure 16(b) shows iL and vo. A zoom was made for the worst simulated system condition. From Figure 16(b), it is seen that even in the worst iL and vo condition, ΔiL and Δvo are below 1%. Accordingly, the designed boost DC-DC converter satisfies both ΔiL and Δvo conditions.

In conclusion, Figure 16 shows that the boost DC-DC converter system operating requirements given in Table 1 are successfully satisfied.

#### 6. Conclusions

In this chapter, a procedure to easily design and control PECs was proposed and zeros' location impact over the system dynamical responses was analyzed, showing that a careful selection of the PEC passive elements could both avoid electronic device failure due to large overshoots and improve the dynamical system performance. Parasitic losses RL and RC were included in order to have a more realistic approach to the system. The presented procedure was composed of:


Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach DOI: http://dx.doi.org/10.5772/intechopen.80426


#### Acknowledgements

In order to carry out system operation requirements verification, case (c) of Figure 15 is taken into account. Figure 16 shows: (a) Pin, Pout and (b) iL and vo,

From Figure 16(a), it is seen that Pout does not exceed the maximum admissible output power in steady state and is always lower than Pin. ð Þ b . Figure 16(b) shows iL and vo. A zoom was made for the worst simulated system condition. From Figure 16(b), it is seen that even in the worst iL and vo condition, ΔiL and Δvo are below 1%. Accordingly, the designed boost DC-DC converter satisfies both ΔiL and

In conclusion, Figure 16 shows that the boost DC-DC converter system operat-

In this chapter, a procedure to easily design and control PECs was proposed and zeros' location impact over the system dynamical responses was analyzed, showing that a careful selection of the PEC passive elements could both avoid electronic device failure due to large overshoots and improve the dynamical system performance. Parasitic losses RL and RC were included in order to have a more realistic

a. The nonlinear dynamical system modeling approach to obtain a mathematical tool and evaluate the system performance. The obtained dynamical model was suitable to describe the dynamical behavior of the system and to derive the

b.The steady-state analysis that allowed to find suitable constraints for passive elements' values. The steady-state analysis was composed of: (a) M Dð Þ expression derivation, (b) losses effect analysis and η expression derivation, (c) conditions for analysis of CCM and DCM, and (d) ΔiL and ΔvC analyses.

when case (c) of Figure 15 is considered.

(a) Instantaneous Power verification. (b) Ripples verification.

ing requirements given in Table 1 are successfully satisfied.

approach to the system. The presented procedure was composed of:

Δvo conditions.

Figure 16.

Applied Modern Control

6. Conclusions

steady-state model.

138

This work was partially supported by COLCIENCIAS (Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas) with the doctoral scholarship 727-2015. The authors would also like to thank "estrategia sostenibilidad UdeA".

#### Conflict of interest

The authors declare no conflict of interest.

#### Author contributions

Jorge H. Urrea-Quintero carried out most of the work presented here, Nicolás Muñoz-Galeano was the advisor of this work, and Lina M. Gómez had a relevant contribution with her extensive knowledge about systems theory.

Applied Modern Control

#### Author details

Jorge-Humberto Urrea-Quintero<sup>1</sup> \*, Nicolás Muñoz-Galeano<sup>1</sup> and Lina-María Gómez-Echavarría<sup>2</sup>

1 Faculty of Engineering, Universidad de Antioquia, Medellín, Colombia

References

[1] Liu J, Hu J, Xu L. Dynamic modeling and analysis of z source converter— Derivation of AC small signal model and

DOI: http://dx.doi.org/10.5772/intechopen.80426

affine modelling. Application to a stepdown converter. IET Power Electronics. 2014;7(6):1482-1498

25(4):2655-2669

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach

[9] Chiniforoosh S et al. Definitions and applications of dynamic average models for analysis of power systems. IEEE Transactions on Power Delivery. 2010;

[10] Trejos A, Gonzalez D, Ramos-Paja CA. Modeling of step-up grid-connected

[11] Kapat S, Krein PT. Formulation of PID control for DC-DC converters based on capacitor current: A geometric context. IEEE Transactions on Power Electronics. 2012;27(3):1424-1432

[12] Kabalo M, Paire D, Blunier B, Bouquain D, Godoy Simões M, Miraoui A. Experimental evaluation of fourphase floating interleaved boost

2013;6(2):215-226

1106-1113

converter design and control for fuel cell applications. IET Power Electronics.

[13] Mapurunga Caracas JV, De Carvalho Farias G, Moreira Teixeira LF, De Souza Ribeiro LA. Implementation of a highefficiency, high-lifetime, and low-cost

converter for an autonomous photovoltaic water pumping system. IEEE Transactions on Industry Applications. 2014;50(1):631-641

[14] Evzelman M, Ben-Yaakov S. Simulation of hybrid converters by average models. IEEE Transactions on Industry Applications. 2014;50(2):

[15] Gezgin C, Heck BS, Bass RM. Integrated design of power stage and controller for switching power supplies. In: IEEE Workshop on Computers in Power Electronics; 1996; pp. 36-44

photovoltaic systems for control purposes. Energies. 2012;5(6):1900

[2] Arango E, Ramos-Paja CA, Calvente

[3] Liang TJ, Tseng KC. Analysis of integrated boost-flyback step-up converter. IEE Proceedings—Electric Power Applications. 2005;152(2):

[4] Davoudi A, Jatskevich J, Chapman PL, Bidram A. Multi-resolution modeling of power electronics circuits

[5] Beldjajev V, Roasto I. Efficiency and

bi-directional current doubler rectifier. Przeglad Elektrotechniczny. 2012;88(8):

[6] Galigekere VP, Kazimierczuk MK. Analysis of PWM Z-source DC-DC converter in CCM for steady state. IEEE Transactions on Circuits and Systems I.

[7] Geyer T, Papafotiou G, Frasca R, Morari M. Constrained optimal control of the step-down DC-DC converter.

[8] Vlad C, Rodriguez-Ayerbe P, Godoy E, Lefranc P. Advanced control laws of DC-DC converters based on piecewise

IEEE Transactions on Power Electronics. 2008;23(5):2454-2464

using model-order reduction techniques. IEEE Transactions on Circuits and Systems I. 2013;60(3):

voltage characteristics of the

design-oriented analysis. IEEE Transactions on Power Electronics.

2007;22(5):1786-1796

2013;6(10):5570

217-225

80-823

124-129

141

2012;59(4):854-863

J, Giral R, Serna-Garces SI. Asymmetrical interleaved DC/DC switching converters for photovoltaic and fuel cell applications—Part 2: Control-oriented models. Energies.

2 Department of Process and Energy, Universidad Nacional de Colombia, Sede Medellín, Colombia

\*Address all correspondence to: humberto.urrea@udea.edu.co

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Analysis and Control of Power Electronic Converters Based on a System Zero Locations Approach DOI: http://dx.doi.org/10.5772/intechopen.80426

#### References

[1] Liu J, Hu J, Xu L. Dynamic modeling and analysis of z source converter— Derivation of AC small signal model and design-oriented analysis. IEEE Transactions on Power Electronics. 2007;22(5):1786-1796

[2] Arango E, Ramos-Paja CA, Calvente J, Giral R, Serna-Garces SI. Asymmetrical interleaved DC/DC switching converters for photovoltaic and fuel cell applications—Part 2: Control-oriented models. Energies. 2013;6(10):5570

[3] Liang TJ, Tseng KC. Analysis of integrated boost-flyback step-up converter. IEE Proceedings—Electric Power Applications. 2005;152(2): 217-225

[4] Davoudi A, Jatskevich J, Chapman PL, Bidram A. Multi-resolution modeling of power electronics circuits using model-order reduction techniques. IEEE Transactions on Circuits and Systems I. 2013;60(3): 80-823

[5] Beldjajev V, Roasto I. Efficiency and voltage characteristics of the bi-directional current doubler rectifier. Przeglad Elektrotechniczny. 2012;88(8): 124-129

[6] Galigekere VP, Kazimierczuk MK. Analysis of PWM Z-source DC-DC converter in CCM for steady state. IEEE Transactions on Circuits and Systems I. 2012;59(4):854-863

[7] Geyer T, Papafotiou G, Frasca R, Morari M. Constrained optimal control of the step-down DC-DC converter. IEEE Transactions on Power Electronics. 2008;23(5):2454-2464

[8] Vlad C, Rodriguez-Ayerbe P, Godoy E, Lefranc P. Advanced control laws of DC-DC converters based on piecewise

affine modelling. Application to a stepdown converter. IET Power Electronics. 2014;7(6):1482-1498

[9] Chiniforoosh S et al. Definitions and applications of dynamic average models for analysis of power systems. IEEE Transactions on Power Delivery. 2010; 25(4):2655-2669

[10] Trejos A, Gonzalez D, Ramos-Paja CA. Modeling of step-up grid-connected photovoltaic systems for control purposes. Energies. 2012;5(6):1900

[11] Kapat S, Krein PT. Formulation of PID control for DC-DC converters based on capacitor current: A geometric context. IEEE Transactions on Power Electronics. 2012;27(3):1424-1432

[12] Kabalo M, Paire D, Blunier B, Bouquain D, Godoy Simões M, Miraoui A. Experimental evaluation of fourphase floating interleaved boost converter design and control for fuel cell applications. IET Power Electronics. 2013;6(2):215-226

[13] Mapurunga Caracas JV, De Carvalho Farias G, Moreira Teixeira LF, De Souza Ribeiro LA. Implementation of a highefficiency, high-lifetime, and low-cost converter for an autonomous photovoltaic water pumping system. IEEE Transactions on Industry Applications. 2014;50(1):631-641

[14] Evzelman M, Ben-Yaakov S. Simulation of hybrid converters by average models. IEEE Transactions on Industry Applications. 2014;50(2): 1106-1113

[15] Gezgin C, Heck BS, Bass RM. Integrated design of power stage and controller for switching power supplies. In: IEEE Workshop on Computers in Power Electronics; 1996; pp. 36-44

Author details

Applied Modern Control

Jorge-Humberto Urrea-Quintero<sup>1</sup>

provided the original work is properly cited.

Lina-María Gómez-Echavarría<sup>2</sup>

Sede Medellín, Colombia

140

\*, Nicolás Muñoz-Galeano<sup>1</sup> and

1 Faculty of Engineering, Universidad de Antioquia, Medellín, Colombia

2 Department of Process and Energy, Universidad Nacional de Colombia,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: humberto.urrea@udea.edu.co

[16] Pomar M, Gutierrez G, de Prada Moraga C, and Normey J E. Rico. Integrated design and control applied to a buck boost converter. In: European Control Conference; 2007, pp. 948-954

[17] Mariethoz S, Almer S, Morari M. Integrated control and circuit design; an optimization approach applied to the buck converter. In: American Control Conference (ACC); 2010, pp. 3305-3310

[18] Alvarez-Ramirez J, Espinosa-Pérez G, Noriega-Pineda D. Current-mode control of DC-DC power converters: A backstepping approach. International Journal of Robust and Nonlinear Control. 2003;13(5):421-442

[19] Qiu Y, Liu H, Chen X. Digital average current-mode control of pwm dc-dc converters without current sensors. IEEE Transactions on Industrial Electronics. 2010;57(5):1670-1677

[20] Erickson RW, Maksimovic D. Fundamentals of Power Electronics. Springer Science & Business Media; 2007

[21] Ping L, Xin M, Bo Z, Zhao-ji L. Analysis of the stability and ripple of PSM converter in DCM by EB model. In: International Conference on Communications, Circuits and Systems; 2007. pp. 1240-1243

[22] Liu S-L, Liu J, Mao H, Zhang Y. Analysis of operating modes and output voltageripple of boost DC-DC convertersand its design considerations. IEEE Transactions on Power Electronics. 2008;23(4):1813-1821

[23] Hoagg JB, Bernstein DS. Nonminimum-phase zeros: Much to do about nothing—classical control revisited. Part II. IEEE Control Systems Society. 2007;27(3):45-57

[24] Chen Z, Gao W, Hu J, Ye X. Closedloop analysis and cascade control of a nonminimum phase boost converter.

IEEE Transactions on Power Electronics. 2011;26(4):1237-1252

[25] Louganski KP, Lai J-S. Current phase lead compensation in single-phase pfc boost converters with a reduced switching frequency to line frequency ratio. IEEE Transactions on Power Electronics. 2007;22(1):113-119

Chapter 8

Olga Tolochko

simulation results.

1. Introduction

143

chronous motor (PMSM).

stator voltage constraint, simulation

Abstract

Energy Efficient Speed Control of

In this chapter, methods for the structural realization of a speed control system for the interior permanent magnet synchronous motor (IPMSM) using the "maximum torque per ampere" (MTA) and "maximum torque per volt" (MTV) optimal control strategies are considered. In the system in constant torque region, is a technique for adapting the speed controller to the presence of the reactive motor torque component, which improves the quality of the transient processes, is proposed. It is also recommended to approximate the dependence of the flux-forming current component on the motor torque by the "dead zone" nonlinearity, which will simplify the optimal control algorithm and avoid solving the fourth-degree algebraic equation in real time. For the speed control with field weakening technique, a novel system is recommended. In this system, the control algorithms are switched by the variable of the direct stator current component constraint generated in accordance with the MTA law: the upper limit is calculated in accordance with the "field weakening control" (FWC) strategy, and the lower limit in accordance with the MTV strategy. The steady-state stator voltage constraint is implemented through the variable quadrature stator current component limitation. The effectiveness of the proposed solutions is confirmed by the

Keywords: interior permanent magnet synchronous motor, optimal control, maximum torque per ampere, maximum torque per volt, field weakening control,

Currently, more attention is being paid to improving the energy efficiency of managing electromechanical plants. The solution for this problem is of particular importance for electric drive systems with autonomous power sources, in particular for electric vehicles, allowing them to increase their mileage between recharges. One of the motors widely used in electric vehicles is the permanent magnet syn-

Depending on the magnets' location in the rotor, PMSMs are divided into motors with surface-mounted magnets (SPMSM—surface permanent magnet synchronous machine) and interior-mounted magnets (IPMSM—interior PMSM). The surface allocation of the magnets prevents the engine from operating at high speed. With

Interior Permanent Magnet

Synchronous Motor

[26] Vilanova R, Alfaro VM. Control PID robusto: Una visión panorámica. Revista Iberoamericana de Automatica e Informatica Industrial (RIAI). 2011; 8(3):141-158

#### Chapter 8

[16] Pomar M, Gutierrez G, de Prada Moraga C, and Normey J E. Rico. Integrated design and control applied to a buck boost converter. In: European Control Conference; 2007, pp. 948-954

Applied Modern Control

IEEE Transactions on Power Electronics. 2011;26(4):1237-1252

[25] Louganski KP, Lai J-S. Current phase lead compensation in single-phase pfc boost converters with a reduced switching frequency to line frequency ratio. IEEE Transactions on Power Electronics. 2007;22(1):113-119

[26] Vilanova R, Alfaro VM. Control PID robusto: Una visión panorámica. Revista

Iberoamericana de Automatica e Informatica Industrial (RIAI). 2011;

8(3):141-158

[17] Mariethoz S, Almer S, Morari M. Integrated control and circuit design; an optimization approach applied to the buck converter. In: American Control Conference (ACC); 2010, pp. 3305-3310

[18] Alvarez-Ramirez J, Espinosa-Pérez G, Noriega-Pineda D. Current-mode control of DC-DC power converters: A backstepping approach. International Journal of Robust and Nonlinear Control. 2003;13(5):421-442

[19] Qiu Y, Liu H, Chen X. Digital average current-mode control of pwm dc-dc converters without current sensors. IEEE Transactions on Industrial Electronics. 2010;57(5):1670-1677

[20] Erickson RW, Maksimovic D. Fundamentals of Power Electronics. Springer Science & Business Media;

[21] Ping L, Xin M, Bo Z, Zhao-ji L. Analysis of the stability and ripple of PSM converter in DCM by EB model. In: International Conference on

2007. pp. 1240-1243

Communications, Circuits and Systems;

convertersand its design considerations.

Nonminimum-phase zeros: Much to do about nothing—classical control revisited. Part II. IEEE Control Systems

[24] Chen Z, Gao W, Hu J, Ye X. Closedloop analysis and cascade control of a nonminimum phase boost converter.

[22] Liu S-L, Liu J, Mao H, Zhang Y. Analysis of operating modes and output

voltageripple of boost DC-DC

IEEE Transactions on Power Electronics. 2008;23(4):1813-1821

[23] Hoagg JB, Bernstein DS.

Society. 2007;27(3):45-57

142

2007

## Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

Olga Tolochko

#### Abstract

In this chapter, methods for the structural realization of a speed control system for the interior permanent magnet synchronous motor (IPMSM) using the "maximum torque per ampere" (MTA) and "maximum torque per volt" (MTV) optimal control strategies are considered. In the system in constant torque region, is a technique for adapting the speed controller to the presence of the reactive motor torque component, which improves the quality of the transient processes, is proposed. It is also recommended to approximate the dependence of the flux-forming current component on the motor torque by the "dead zone" nonlinearity, which will simplify the optimal control algorithm and avoid solving the fourth-degree algebraic equation in real time. For the speed control with field weakening technique, a novel system is recommended. In this system, the control algorithms are switched by the variable of the direct stator current component constraint generated in accordance with the MTA law: the upper limit is calculated in accordance with the "field weakening control" (FWC) strategy, and the lower limit in accordance with the MTV strategy. The steady-state stator voltage constraint is implemented through the variable quadrature stator current component limitation. The effectiveness of the proposed solutions is confirmed by the simulation results.

Keywords: interior permanent magnet synchronous motor, optimal control, maximum torque per ampere, maximum torque per volt, field weakening control, stator voltage constraint, simulation

#### 1. Introduction

Currently, more attention is being paid to improving the energy efficiency of managing electromechanical plants. The solution for this problem is of particular importance for electric drive systems with autonomous power sources, in particular for electric vehicles, allowing them to increase their mileage between recharges. One of the motors widely used in electric vehicles is the permanent magnet synchronous motor (PMSM).

Depending on the magnets' location in the rotor, PMSMs are divided into motors with surface-mounted magnets (SPMSM—surface permanent magnet synchronous machine) and interior-mounted magnets (IPMSM—interior PMSM). The surface allocation of the magnets prevents the engine from operating at high speed. With

the internal allocation of the magnets, the mechanical strength of the rotor increases, and this defect is eliminated. SPMSM has a symmetrical magnetic system, since the magnetic permeability of air and permanent magnets is practically the same. The electromagnetic system in IPMSM is asymmetric. Hence, IPMSM is a salient pole motor. This leads to the occurrence, along with the active component of the torque, of an additional reactive component, through which it is possible to obtain larger power/weight, torque/current, and torque/voltage ratios.

set the reference torque of the motor at the output of the speed controller and use

In [18], a comparison of the MTA strategy with such additional control methods of energy efficiency increasing as "constant torque angle" (CTA), "unity power factor" (UPF), "constant mutual flux linkage" (CMFL) and "angle control of air gap flux current phasor" (ACAGF). It is shown that UPF control yields a comparatively low voltage requirement but very low torque/current ratio. On comparing UPF with CMFL control, it should be noted that the voltage requirement for CMFLC is next to UPFC but can produce much higher torque/current ratio, which is quite a bit

In [19], an equation of the fourth-order polynomial about the direct component of the stator current is derived, the coefficients of which depend on the torque, velocity, and quadrature current. This equation minimizes the total loss as the sum of copper, iron, and stray losses. The loss minimizing solutions are obtained by a

In [20], the MTA control strategy of IPMSM and its flux-weakening control strategy are described. In this chapter, electromagnetic torque and the relationship of the direct and quadrature axis currents can be derived with the curve fitting method directly. The approximation is performed by a second-order polynomial using the method of least squares. In this case, the approximating curve, unlike the approximated one, does not fall into a point with coordinates [0, 0], which reduces the accuracy of regulation in the initial section. Flux-weakening control (FWC) algorithm in this paper is phase shifting. In this case, the system is implemented with two control channels, speed and torque, which complicate the configuration of

In [21], a flux-weakening scheme for the IPMSM is proposed. This is done by an additional external voltage regulator of the pulse width modulated (PWM) inverter,

In [22], a novel field weakening technology of IPMM is described. Here, closed-

In [23], field weakening control of fast dynamics and variable DC-link voltage are achieved by suitable combination of look-up table and voltage feedback con-

In [24], a voltage-constraint tracking (VCT) field weakening control scheme for

In [26], a concept for optimal torque control of IPMSM has been presented. The schema is based on look-up tables, where saturation effects can be considered. A consideration of the permanent magnets' demagnetization effect during flux weak-

As the review performed shows, many authors suggest optimizing control systems using pre-calculated tables. The disadvantages of this solution are the reduction of the real-time performance and reliability of the system, increased demands on the amount of processor memory, and the impossibility of adapting the control system to changing parameters and signal disturbances. The use of analytical expressions makes it possible to adapt by on-line identification of the current values of the main parameters. A lot of work is devoted to this problem, for example, [25, 27–31].

loop control using the output voltage and feed-forward control with the pre-

IPMSM drives is proposed. The control algorithm is presented in the form of a complicated block diagram with numerous branching and computations. In [25] an approach to minimize the electrical losses of the interior permanent magnet synchronous motors is presented. Two control strategies based on unsaturated and saturated motor model are analyzed. To overcome the problem of parameters unavailable, a procedure is proposed to estimate the parameters of the loss minimi-

another LUT to calculate the reference peak stator current.

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

smaller than using the MTA method.

DOI: http://dx.doi.org/10.5772/intechopen.80424

simple numerical approach or using a LUT.

the system and worsen its dynamic properties.

calculated tables are combined.

troller.

145

zation conditions.

which controls the beginning of the flux-weakening and its level.

ening showed the restrictions between the system parameters.

For IPMSM, optimal control strategies have been developed [1–5]. They increase the energy efficiency of electric drives in steady-state conditions by forming the relationship between the stator current orthogonal components, corresponding to the chosen optimality criterion.

With IPMSM speed control, three ranges are possible, each with different control algorithms. The greatest difficulty in the practical implementation of such systems is the organization of control switching from one algorithm to another while maintaining stator current and voltage constraints.

There are different approaches to improving the energy efficiency of the studied electric drive systems. First of all, they differ in the kind of losses that are subject to minimization.

Losses in IPMSM frequency systems consist of losses in windings (copper losses), losses in magnetic conductor (iron losses), losses from higher harmonics in windings and grooves (stray losses), switching losses in the frequency converter, and mechanical losses [6, 7]. In turn, the losses in steel consist of losses for reversal of magnetization (hysteresis losses) and losses from eddy currents, and mechanical losses due to friction losses and losses from air or liquid resistance. The commutation, mechanical, and hysteresis losses are determined by approximate empirical formulas. Therefore, to their analysis, many papers [6–11] are devoted. There are also methods for determining them experimentally [12, 13].

Optimization methods have also different input and output signals in minimizing expressions. For systems of torque and speed control, the most logical is the use of the dependences of the direct and quadrature stator current components on the electromagnetic torque. However, these dependencies are most complex. For example, many of them have the form of equations of the fourth degree, which either have to be solved by iterative methods in real time, or approximated by simpler equations by the least squares method, or represented in the form of precomputed look-up tables (LUTs), the search in which it is performed by interpolation methods. Therefore, optimization equations often are found in the form of dependencies between the components of the stator currents or in the form of the amplitude-phase trajectory of the stator current. In both cases, the quality of transient processes deteriorates.

Minimization of copper losses is provided by the "maximum torque per ampere" (MTA) management strategy, and the "maximum torque per volt" (MTV) strategy minimizes steel losses from eddy currents. These strategies use variable speed ranges in different ranges: MTA for under the rated speed and MTV for over the rated speed.

In [14–16], the static characteristics of the drive using the MTA strategy were analyzed without and with taking into account the saturation of the steel and the demagnetization of the permanent magnets. Based on the analysis performed, a control system is synthesized, in which the speed controller generates a reference to the amplitude of the stator current, and the phase angle is determined from the previously calculated LUT.

It was shown in [17] that in the flux-weakening constant power region, the stator current sometimes becomes uncontrollable in transients because of the current regulator saturation. To eliminate this disadvantage, the authors proposed to

#### Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor DOI: http://dx.doi.org/10.5772/intechopen.80424

set the reference torque of the motor at the output of the speed controller and use another LUT to calculate the reference peak stator current.

In [18], a comparison of the MTA strategy with such additional control methods of energy efficiency increasing as "constant torque angle" (CTA), "unity power factor" (UPF), "constant mutual flux linkage" (CMFL) and "angle control of air gap flux current phasor" (ACAGF). It is shown that UPF control yields a comparatively low voltage requirement but very low torque/current ratio. On comparing UPF with CMFL control, it should be noted that the voltage requirement for CMFLC is next to UPFC but can produce much higher torque/current ratio, which is quite a bit smaller than using the MTA method.

In [19], an equation of the fourth-order polynomial about the direct component of the stator current is derived, the coefficients of which depend on the torque, velocity, and quadrature current. This equation minimizes the total loss as the sum of copper, iron, and stray losses. The loss minimizing solutions are obtained by a simple numerical approach or using a LUT.

In [20], the MTA control strategy of IPMSM and its flux-weakening control strategy are described. In this chapter, electromagnetic torque and the relationship of the direct and quadrature axis currents can be derived with the curve fitting method directly. The approximation is performed by a second-order polynomial using the method of least squares. In this case, the approximating curve, unlike the approximated one, does not fall into a point with coordinates [0, 0], which reduces the accuracy of regulation in the initial section. Flux-weakening control (FWC) algorithm in this paper is phase shifting. In this case, the system is implemented with two control channels, speed and torque, which complicate the configuration of the system and worsen its dynamic properties.

In [21], a flux-weakening scheme for the IPMSM is proposed. This is done by an additional external voltage regulator of the pulse width modulated (PWM) inverter, which controls the beginning of the flux-weakening and its level.

In [22], a novel field weakening technology of IPMM is described. Here, closedloop control using the output voltage and feed-forward control with the precalculated tables are combined.

In [23], field weakening control of fast dynamics and variable DC-link voltage are achieved by suitable combination of look-up table and voltage feedback controller.

In [24], a voltage-constraint tracking (VCT) field weakening control scheme for IPMSM drives is proposed. The control algorithm is presented in the form of a complicated block diagram with numerous branching and computations. In [25] an approach to minimize the electrical losses of the interior permanent magnet synchronous motors is presented. Two control strategies based on unsaturated and saturated motor model are analyzed. To overcome the problem of parameters unavailable, a procedure is proposed to estimate the parameters of the loss minimization conditions.

In [26], a concept for optimal torque control of IPMSM has been presented. The schema is based on look-up tables, where saturation effects can be considered. A consideration of the permanent magnets' demagnetization effect during flux weakening showed the restrictions between the system parameters.

As the review performed shows, many authors suggest optimizing control systems using pre-calculated tables. The disadvantages of this solution are the reduction of the real-time performance and reliability of the system, increased demands on the amount of processor memory, and the impossibility of adapting the control system to changing parameters and signal disturbances. The use of analytical expressions makes it possible to adapt by on-line identification of the current values of the main parameters. A lot of work is devoted to this problem, for example, [25, 27–31].

the internal allocation of the magnets, the mechanical strength of the rotor

obtain larger power/weight, torque/current, and torque/voltage ratios.

while maintaining stator current and voltage constraints.

also methods for determining them experimentally [12, 13].

the chosen optimality criterion.

Applied Modern Control

sient processes deteriorates.

previously calculated LUT.

rated speed.

144

minimization.

increases, and this defect is eliminated. SPMSM has a symmetrical magnetic system, since the magnetic permeability of air and permanent magnets is practically the same. The electromagnetic system in IPMSM is asymmetric. Hence, IPMSM is a salient pole motor. This leads to the occurrence, along with the active component of the torque, of an additional reactive component, through which it is possible to

For IPMSM, optimal control strategies have been developed [1–5]. They increase the energy efficiency of electric drives in steady-state conditions by forming the relationship between the stator current orthogonal components, corresponding to

With IPMSM speed control, three ranges are possible, each with different control algorithms. The greatest difficulty in the practical implementation of such systems is the organization of control switching from one algorithm to another

There are different approaches to improving the energy efficiency of the studied electric drive systems. First of all, they differ in the kind of losses that are subject to

Optimization methods have also different input and output signals in minimizing expressions. For systems of torque and speed control, the most logical is the use of the dependences of the direct and quadrature stator current components on the electromagnetic torque. However, these dependencies are most complex. For example, many of them have the form of equations of the fourth degree, which either have to be solved by iterative methods in real time, or approximated by simpler equations by the least squares method, or represented in the form of precomputed look-up tables (LUTs), the search in which it is performed by interpolation methods. Therefore, optimization equations often are found in the form of dependencies between the components of the stator currents or in the form of the amplitude-phase trajectory of the stator current. In both cases, the quality of tran-

Minimization of copper losses is provided by the "maximum torque per ampere" (MTA) management strategy, and the "maximum torque per volt" (MTV) strategy minimizes steel losses from eddy currents. These strategies use variable speed ranges in different ranges: MTA for under the rated speed and MTV for over the

In [14–16], the static characteristics of the drive using the MTA strategy were analyzed without and with taking into account the saturation of the steel and the demagnetization of the permanent magnets. Based on the analysis performed, a control system is synthesized, in which the speed controller generates a reference to the amplitude of the stator current, and the phase angle is determined from the

It was shown in [17] that in the flux-weakening constant power region, the stator current sometimes becomes uncontrollable in transients because of the current regulator saturation. To eliminate this disadvantage, the authors proposed to

Losses in IPMSM frequency systems consist of losses in windings (copper losses), losses in magnetic conductor (iron losses), losses from higher harmonics in windings and grooves (stray losses), switching losses in the frequency converter, and mechanical losses [6, 7]. In turn, the losses in steel consist of losses for reversal of magnetization (hysteresis losses) and losses from eddy currents, and mechanical losses due to friction losses and losses from air or liquid resistance. The commutation, mechanical, and hysteresis losses are determined by approximate empirical formulas. Therefore, to their analysis, many papers [6–11] are devoted. There are

Methods based on the use of certain analytical expressions or pre-calculated tables belong to the group "loss model-based control" (LMBC). An alternative for them are the search algorithms, which are most often used in minimizing total losses. Their feature is the measurement of losses or input and output power and the use of iterative methods to find the optimal solution in real time. The search algorithms do not require the knowledge of the motor model and parameters, but in the search process, they have a negative effect on the transients and steady-state values of the controlled coordinates.

This chapter accesses simple speed control field oriented control (FOC) vector systems of IPMSM for constant torque and field weakening operation ranges without using pre-calculated tables, search algorithms, on-line solving of algebraic equations using iterative methods and without the presence of additional control loops. The quality of proposed control systems is investigated via simulation.

The chapter content is structured as follows. In Section 2, the problem is formulated. In Section 3, the IPMSM speed control system in the constant torque operating range is analyzed. In Section 4, speed control system with new field weakening technique is analyzed and designed. Section 5 contains conclusions on the chapter.

3. Speed control of IPMSM in constant torque region

nent of the electromagnetic torque denoted by the bold line.

ktψpmτω

PCu <sup>¼</sup> <sup>1</sup>:5Ri<sup>2</sup>

strategy is called maximal torque per ampere (MTA).

idMTA ¼ � <sup>ψ</sup>pm

In conventional IPMSM FOC system, a reference signal i

2 Ld � Lq � � �

are calculated by the formulas:

current loops.

Rated speed Rated torque

Table 1.

Moment of inertia Number of pole pairs Stator resistance d-axis inductance q-axis inductance

Specifications of IPMSM.

Permanent magnet flux linkage

DOI: http://dx.doi.org/10.5772/intechopen.80424

ksc <sup>¼</sup> <sup>i</sup> ∗ <sup>q</sup>ð Þs <sup>Δ</sup>ωð Þ<sup>s</sup> <sup>¼</sup> <sup>J</sup>

stator current, are determined as:

and has a known solution:

147

The block diagram of the IPMSM, designed according to Eqs. (1), is shown in Figure 1. The synthesis of the vector control system is traditionally performed according to the block diagram shown in Figure 1, neglecting the rotation EMF feedback and the crosslinks denoted by dashed lines, as well as the reactive compo-

Parameters Designation Values 1 Values 2

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

nr, rpm Tr, Nm ψpm, Wb J, kg m<sup>2</sup> p R, Ώ Ld, mH Lq, mH

4000 1.8 0.0844 0.45 � <sup>10</sup>�<sup>3</sup> 3 2.21 9.77 8.72

2000 1.67 0.0785 0.5 � <sup>10</sup>�<sup>3</sup> 2 0.87 14.94 22.78

When using PI current controllers (PI-CCs), synthesized by the series correction method, the dotted links are usually compensated by adding corresponding links with opposite signs to the output signals of the PI-CCs. If the asymmetry of the motors magnetic system is not taken into consideration, the proportional gain (P-) and the transfer function of the proportional-integral (PI-) speed controller (SC)

, WscðÞ¼ s

where τω ¼ 2τ<sup>i</sup> and τω<sup>i</sup> ¼ 2τω; τi—integral time-constants of open speed and

In the first speed range, the copper losses, proportional to the square of the

<sup>s</sup> , i<sup>2</sup> <sup>s</sup> ¼ i 2 <sup>q</sup> þ i 2

Then the control problem can be formulated as follows: to find such a relation between the orthogonal components of the stator current, at a given torque, at which the amplitude of the current (2) would be smallest possible. This control

This optimal control problem is a classical variation task for the conditional extremum, which requires minimizing the expression (3) ensuring the additional torque equation from (1). Such a problem is solved by the Euler-Lagrange method

input of the d-axis stator current (CCd). When implementing the MTA-strategy, the easiest way to take the reference for the q-axis current is from the speed

i ∗ <sup>q</sup>ð Þs <sup>Δ</sup>ωð Þ<sup>s</sup> <sup>¼</sup> ksc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψ2 pm 4 Ld � Lq � �<sup>2</sup> <sup>þ</sup> <sup>i</sup>

2 q vuut : (4)

<sup>d</sup><sup>0</sup> ¼ 0 is applied to the

∗

τωis þ 1 τωis

<sup>d</sup>: (3)

(2)

#### 2. Problem formulation

For the mathematical description of IPMSM, we use the following symbols: ud, uq, id, iq, ψd, ψq— d- and q-axis voltage, current and flux linkage; Ld, Lq—direct and quadrature stator inductance (Ld<Lq); ΔL ¼ Ld�Lq; R—stator resistance; τ<sup>d</sup> ¼ Ld=R, τ<sup>q</sup> ¼ Lq=R; ω, ωe—mechanical and electrical rotor speed; p—number of pole pairs; ψpm—permanent magnet flux linkage; J—moment of inertia; T—motor torque, TL—load torque.

The differential equations of IPMSM in the d-q rotating reference frame used in the FOC vector systems synthesis are written as follows:

$$\begin{cases} u\_d = i\_d R + L\_d \frac{di\_d}{dt} - \alpha\_\epsilon \wp\_q, \quad \alpha\_\epsilon = p\alpha, \qquad \wp\_q = L\_q i\_q, \\ u\_q = i\_q R + L\_q \frac{di\_q}{dt} + \alpha\_\epsilon \wp\_d, \qquad \wp\_d = L\_d i\_d + \wp\_{pm}, \\ T = k\_t \left[ \wp\_{pm} i\_q + \left( L\_d - L\_q \right) i\_d i\_q \right], \quad k\_t = 1.5p, \\ J \frac{d\alpha}{dt} = T - T\_L. \end{cases} \tag{1}$$

Table 1 presents the parameters of two motors with different degrees of magnetic asymmetry, for which the research in this chapter is performed.

The task of energy efficient optimal control is to minimize total losses or one of their kinds. Mechanical and switching losses can be most effectively reduced at the design stage of the frequency converter, the motor and the mechanism. Designing a control system based on loss models, usually allows minimizing the copper losses, iron losses or their sum. This chapter discusses methods based on models of copper losses and iron losses from eddy currents.

During analysis and synthesis, the saturation of steel is not taken into account, the methods for parameters identifying and perturbations estimation are not considered.

The graphs in this chapter are presented in p.u. units: y ¼ y=yb. As base values, we used: Tb <sup>¼</sup> Tr, ib <sup>¼</sup> Tr<sup>=</sup> ktψpm � �, <sup>ω</sup><sup>b</sup> <sup>¼</sup> <sup>ω</sup><sup>r</sup> <sup>¼</sup> <sup>π</sup>nr=30, and ub <sup>¼</sup> er <sup>¼</sup> <sup>p</sup>ωrψpm.


Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor DOI: http://dx.doi.org/10.5772/intechopen.80424

Table 1. Specifications of IPMSM.

Methods based on the use of certain analytical expressions or pre-calculated tables belong to the group "loss model-based control" (LMBC). An alternative for them are the search algorithms, which are most often used in minimizing total losses. Their feature is the measurement of losses or input and output power and the use of iterative methods to find the optimal solution in real time. The search algorithms do not require the knowledge of the motor model and parameters, but in the search process, they have a negative effect on the transients and steady-state

This chapter accesses simple speed control field oriented control (FOC) vector systems of IPMSM for constant torque and field weakening operation ranges without using pre-calculated tables, search algorithms, on-line solving of algebraic equations using iterative methods and without the presence of additional control loops. The quality of proposed control systems is investigated via simulation.

The chapter content is structured as follows. In Section 2, the problem is formulated. In Section 3, the IPMSM speed control system in the constant torque operating range is analyzed. In Section 4, speed control system with new field weakening technique is analyzed and designed. Section 5 contains conclusions on the chapter.

For the mathematical description of IPMSM, we use the following symbols: ud, uq, id, iq, ψd, ψq— d- and q-axis voltage, current and flux linkage; Ld, Lq—direct and quadrature stator inductance (Ld<Lq); ΔL ¼ Ld�Lq; R—stator resistance; τ<sup>d</sup> ¼ Ld=R, τ<sup>q</sup> ¼ Lq=R; ω, ωe—mechanical and electrical rotor speed; p—number of pole pairs; ψpm—permanent magnet flux linkage; J—moment of inertia; T—motor

The differential equations of IPMSM in the d-q rotating reference frame used in

� �idiq h i

Table 1 presents the parameters of two motors with different degrees of mag-

The task of energy efficient optimal control is to minimize total losses or one of their kinds. Mechanical and switching losses can be most effectively reduced at the design stage of the frequency converter, the motor and the mechanism. Designing a control system based on loss models, usually allows minimizing the copper losses, iron losses or their sum. This chapter discusses methods based on models of copper

During analysis and synthesis, the saturation of steel is not taken into account, the methods for parameters identifying and perturbations estimation are not con-

The graphs in this chapter are presented in p.u. units: y ¼ y=yb. As base values,

dt � <sup>ω</sup>eψq, <sup>ω</sup><sup>e</sup> <sup>¼</sup> <sup>p</sup>ω, <sup>ψ</sup><sup>q</sup> <sup>¼</sup> Lq iq,

, kt ¼ 1:5p,

, ω<sup>b</sup> ¼ ω<sup>r</sup> ¼ πnr=30, and ub ¼ er ¼ pωrψpm.

(1)

dt <sup>þ</sup> <sup>ω</sup>eψd, <sup>ψ</sup><sup>d</sup> <sup>¼</sup> Ld id <sup>þ</sup> <sup>ψ</sup>pm,

the FOC vector systems synthesis are written as follows:

did

diq

netic asymmetry, for which the research in this chapter is performed.

� �

T ¼ kt ψpmiq þ Ld � Lq

ud ¼ idR þ Ld

uq ¼ iqR þ Lq

dt <sup>¼</sup> <sup>T</sup> � TL:

J dω

losses and iron losses from eddy currents.

we used: Tb ¼ Tr, ib ¼ Tr= ktψpm

sidered.

146

values of the controlled coordinates.

Applied Modern Control

2. Problem formulation

torque, TL—load torque.

8

>>>>>>>>>><

>>>>>>>>>>:

#### 3. Speed control of IPMSM in constant torque region

The block diagram of the IPMSM, designed according to Eqs. (1), is shown in Figure 1. The synthesis of the vector control system is traditionally performed according to the block diagram shown in Figure 1, neglecting the rotation EMF feedback and the crosslinks denoted by dashed lines, as well as the reactive component of the electromagnetic torque denoted by the bold line.

When using PI current controllers (PI-CCs), synthesized by the series correction method, the dotted links are usually compensated by adding corresponding links with opposite signs to the output signals of the PI-CCs. If the asymmetry of the motors magnetic system is not taken into consideration, the proportional gain (P-) and the transfer function of the proportional-integral (PI-) speed controller (SC) are calculated by the formulas:

$$k\_{\rm sc} = \frac{i\_q^\*(\mathfrak{s})}{\Delta \alpha(\mathfrak{s})} = \frac{J}{k\_l \mathfrak{y}\_{pm} \mathfrak{r}\_{\alpha}}, \quad \mathcal{W}\_{\rm sc}(\mathfrak{s}) = \frac{i\_q^\*(\mathfrak{s})}{\Delta \alpha(\mathfrak{s})} = k\_{\rm sc} \frac{\mathfrak{r}\_{\rm oc} \mathfrak{s} + 1}{\mathfrak{r}\_{\rm av} \mathfrak{s}} \tag{2}$$

where τω ¼ 2τ<sup>i</sup> and τω<sup>i</sup> ¼ 2τω; τi—integral time-constants of open speed and current loops.

In the first speed range, the copper losses, proportional to the square of the stator current, are determined as:

$$P\_{\rm Cu} = \mathbf{1.5Ri}\_{\rm s}^2, \quad \dot{\mathbf{r}}\_{\rm s}^2 = \dot{\mathbf{r}}\_{\rm q}^2 + \dot{\mathbf{r}}\_{\rm d}^2. \tag{3}$$

Then the control problem can be formulated as follows: to find such a relation between the orthogonal components of the stator current, at a given torque, at which the amplitude of the current (2) would be smallest possible. This control strategy is called maximal torque per ampere (MTA).

This optimal control problem is a classical variation task for the conditional extremum, which requires minimizing the expression (3) ensuring the additional torque equation from (1). Such a problem is solved by the Euler-Lagrange method and has a known solution:

$$i\_{dMTA} = -\frac{\nu\_{pm}}{2\left(L\_d - L\_q\right)} - \sqrt{\frac{\nu\_{pm}^2}{4\left(L\_d - L\_q\right)^2} + i\_q^2}.\tag{4}$$

In conventional IPMSM FOC system, a reference signal i ∗ <sup>d</sup><sup>0</sup> ¼ 0 is applied to the input of the d-axis stator current (CCd). When implementing the MTA-strategy, the easiest way to take the reference for the q-axis current is from the speed

controller and to obtain a reference to the d-axis current from it is through nonlinear functional transformation (4). A fragment of the block diagram for the comparison of the "zero direct current strategy" and the simplest implementation of the MTA strategy is shown in Figure 2, where.

$$\mathcal{W}\_{CCd}(\mathfrak{s}) = \frac{(\mathfrak{r}\_d \mathfrak{s} + \mathfrak{1})R}{\mathfrak{r}\_i \mathfrak{s}}, \quad \mathcal{W}\_{CCq}(\mathfrak{s}) = \frac{(\mathfrak{r}\_q \mathfrak{s} + \mathfrak{1})R}{\mathfrak{r}\_i \mathfrak{s}}.$$

The effectiveness of the MTA strategy is explained in Figure 3, which shows in p.u. units a MTA parabola, calculated using expression (4), equal currents circles (Eq. (3)), and equal torques hyperbolae, calculated according to the equation

$$i\_q(i\_d, T) = \frac{T}{k\_t \left[\wp\_{pm} + (L\_d - L\_q)i\_d\right]}.\tag{5}$$

of the stator current for a given torque. Therefore, the equal torques hyperbolae lie outside the equal currents circles, that is, any ratio of the orthogonal current components that is different from the optimal one leads to an increase in the modulus of

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

DOI: http://dx.doi.org/10.5772/intechopen.80424

To compare currents when using the control strategies under study, attention should be drawn to the cross points of constant currents circles with the q-axis that defines the current modules for the MTA strategy, and the cross points of constant torques hyperbolae with the same axis that defines the peak current for the strategy

Analysis of the location of these points shows that the advantage of the MTAstrategy increases with increasing electromagnetic torque. This radically differentiates the IPMSM optimal control from the optimal control of induction motor, in which the efficiency of the MTA strategy increases with decreasing electromagnetic

We will perform a study of the static and dynamic properties of IPMSM speed control systems with compared control strategies for the IMPSM with parameters presented in column "Values1" from Table 1 with time constants τ<sup>i</sup> ¼ 2τμ = 0.4 ms,

Transient processes during acceleration with the desired electromagnetic torque without load up to the nominal speed and stepwise load torque response for P-SC are shown in Figure 4, and for the PI-SC – in Figure 5: (a) with "zero d-current

From the comparison of the transients, it can be seen that in the conventional control system (a) the torque and current curves practically coincide and have the current and torque overshoot values about 5% during the acceleration of and 10%

the current vector.

Current locus diagram.

"id ¼ 0."

Figure 3.

torque.

149

τω ¼ 2τ<sup>i</sup> = 0.8 ms.

strategy"; (b) with the MTA strategy (Eq. (4)).

during step disturbance change in the system with P-SC.

The intersection points Pi of these hyperbolae with the MTA trajectory determine the optimal distribution of stator current components for given torque values. Circles with centers at the coordinate system origin, drawn through the Pi points with dashed lines, have radii equal to the total stator currents isMTA with an optimal distribution of their components.

It can be seen from Figure 3 that a parabolic MTA trajectory intersects equal torques hyperbolae at an angle of 90°, which provides the minimum possible values

Figure 1. IPMSM block diagram in rotating rotor reference frame.

#### Figure 2.

Block diagram for the comparison of the "zero direct current strategy" and the simplest implementation of MTA strategy.

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor DOI: http://dx.doi.org/10.5772/intechopen.80424

Current locus diagram.

controller and to obtain a reference to the d-axis current from it is through nonlinear functional transformation (4). A fragment of the block diagram for the comparison of the "zero direct current strategy" and the simplest implementation

, WCCqðÞ¼ <sup>s</sup> <sup>τ</sup>qs <sup>þ</sup> <sup>1</sup> � �<sup>R</sup>

� �id

<sup>τ</sup>is :

h i : (5)

τis

iqð Þ¼ id; <sup>T</sup> <sup>T</sup>

The effectiveness of the MTA strategy is explained in Figure 3, which shows in p.u. units a MTA parabola, calculated using expression (4), equal currents circles (Eq. (3)), and equal torques hyperbolae, calculated according to the equation

kt ψpm þ Ld � Lq

The intersection points Pi of these hyperbolae with the MTA trajectory determine the optimal distribution of stator current components for given torque values. Circles with centers at the coordinate system origin, drawn through the Pi points with dashed lines, have radii equal to the total stator currents isMTA with an optimal

It can be seen from Figure 3 that a parabolic MTA trajectory intersects equal torques hyperbolae at an angle of 90°, which provides the minimum possible values

Block diagram for the comparison of the "zero direct current strategy" and the simplest implementation of MTA

of the MTA strategy is shown in Figure 2, where.

Applied Modern Control

distribution of their components.

IPMSM block diagram in rotating rotor reference frame.

Figure 2.

Figure 1.

strategy.

148

WCCdðÞ¼ <sup>s</sup> ð Þ <sup>τ</sup>ds <sup>þ</sup> <sup>1</sup> <sup>R</sup>

of the stator current for a given torque. Therefore, the equal torques hyperbolae lie outside the equal currents circles, that is, any ratio of the orthogonal current components that is different from the optimal one leads to an increase in the modulus of the current vector.

To compare currents when using the control strategies under study, attention should be drawn to the cross points of constant currents circles with the q-axis that defines the current modules for the MTA strategy, and the cross points of constant torques hyperbolae with the same axis that defines the peak current for the strategy "id ¼ 0."

Analysis of the location of these points shows that the advantage of the MTAstrategy increases with increasing electromagnetic torque. This radically differentiates the IPMSM optimal control from the optimal control of induction motor, in which the efficiency of the MTA strategy increases with decreasing electromagnetic torque.

We will perform a study of the static and dynamic properties of IPMSM speed control systems with compared control strategies for the IMPSM with parameters presented in column "Values1" from Table 1 with time constants τ<sup>i</sup> ¼ 2τμ = 0.4 ms, τω ¼ 2τ<sup>i</sup> = 0.8 ms.

Transient processes during acceleration with the desired electromagnetic torque without load up to the nominal speed and stepwise load torque response for P-SC are shown in Figure 4, and for the PI-SC – in Figure 5: (a) with "zero d-current strategy"; (b) with the MTA strategy (Eq. (4)).

From the comparison of the transients, it can be seen that in the conventional control system (a) the torque and current curves practically coincide and have the current and torque overshoot values about 5% during the acceleration of and 10% during step disturbance change in the system with P-SC.

Figure 4. Transients in IPMSM speed control system with P-SC: (a) used strategy i ∗ <sup>d</sup> ¼ 0; (b) used MTA strategy with constant P-SС gain; and (c) used MTA strategy with variable P-SС gain.

In the PI-SC system, the corresponding values are 52 and 65%. The dynamic deviations of the d-current from 0 does not exceed 12 and 25% of the rated value. When applying the MTA strategy without changing the setting of SC adjustments (b), the ratio T=is increases in steady-state region, but current and torque overshoots increase up to 25% in during the acceleration and to 35% during step disturbance change in the system from P-SC and to 80 and 90% in a system with PI-SC. In this case, the transient oscillation increases significantly.

into account of the reactive torque, the P-SC must be designed according to the

� � h i <sup>¼</sup> <sup>J</sup>

∗

Transient processes in the investigated system after the proposed speed controller adaptation to reactive torque effect are shown in the Figure 4c and Figure 5c. They testify the efficiency of the developed technique, since it significantly reduces

τωktψdð Þ<sup>t</sup> : (6)

<sup>d</sup> ¼ 0; (b) used MTA-strategy

τωkt ψpm þ idð Þt Ld � Lq

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

DOI: http://dx.doi.org/10.5772/intechopen.80424

ksc<sup>1</sup> <sup>¼</sup> <sup>J</sup>

Transients in IPMSM speed control system with PI-SC: (a) used strategy i

without SС adaptation; and (c) used MTA strategy with SС adaptation.

equation

151

Figure 5.

From the analysis of the block diagram in Figure 1, it follows that this shortcoming is due to incomplete plant compensation by the speed controller. Taking

Figure 5.

In the PI-SC system, the corresponding values are 52 and 65%. The dynamic deviations of the d-current from 0 does not exceed 12 and 25% of the rated value. When applying the MTA strategy without changing the setting of SC adjustments (b), the ratio T=is increases in steady-state region, but current and torque overshoots increase up to 25% in during the acceleration and to 35% during step disturbance change in the system from P-SC and to 80 and 90% in a system with PI-SC. In

∗

<sup>d</sup> ¼ 0; (b) used MTA strategy with

From the analysis of the block diagram in Figure 1, it follows that this shortcoming is due to incomplete plant compensation by the speed controller. Taking

this case, the transient oscillation increases significantly.

Transients in IPMSM speed control system with P-SC: (a) used strategy i

constant P-SС gain; and (c) used MTA strategy with variable P-SС gain.

Figure 4.

Applied Modern Control

150

Transients in IPMSM speed control system with PI-SC: (a) used strategy i ∗ <sup>d</sup> ¼ 0; (b) used MTA-strategy without SС adaptation; and (c) used MTA strategy with SС adaptation.

into account of the reactive torque, the P-SC must be designed according to the equation

$$k\_{\varkappa1} = \frac{J}{\tau\_o k\_t \left[\varphi\_{pm} + i\_d(t) \left(L\_d - L\_q\right)\right]} = \frac{J}{\tau\_o k\_t \varphi\_d(t)}.\tag{6}$$

Transient processes in the investigated system after the proposed speed controller adaptation to reactive torque effect are shown in the Figure 4c and Figure 5c. They testify the efficiency of the developed technique, since it significantly reduces the torque and currents overshoots, as well as overvoltage and transient oscillations without quality loss of the steady-state energy parameters. Correction of the SC (Eq. 6) improves the dynamics of the system but requires putting the division block after the SC.

Let us consider another approach to the MTA algorithm implementation, in which the SC forms the electromagnetic torque reference. In this case, the P-SC gain and the PI-SC transfer function are calculated by the equations:

$$k\_{\kappa2} = \frac{J}{\tau\_w}, \; w\_{\kappa2}(p) = \frac{i\_q^\*(s)}{\Delta o(s)} = k\_{\kappa2} \frac{\tau\_{o\vartheta}s + \mathbf{1}}{\tau\_{o\vartheta}s} \tag{7}$$

the MTA–and "id ¼ 0"–strategies for the two IPMSMs from Table 1, presented in Figure 8. In the latter case, the steady-state value of the peak current coincides with

iq0ð Þ¼ <sup>T</sup> is0ð Þ¼ <sup>T</sup> <sup>T</sup>

the increase of the motor torque and its magnetic asymmetry j j ΔL ¼ Ld � Lq

8 ><

>:

Figure 8 shows that, firstly, the effectiveness of the MTA strategy increases with

secondly, the trajectories idMTAð Þ T can be linearized quite easily. For the first motor (Figure 8a), the difference between the compared strategies with T≤0:5Tr is almost nonexistent, and where this difference becomes significant, the plot idMTAð Þ T becomes practically linear. Therefore, the dependence calculated by the Eq. (11) can be replaced by the "dead zone" nonlinearity idMTAkð Þ T type, which practically does

For the second motor, the nonlinear characteristic can generally be approximated by a straight line drawn through the original curve end points. Finding the parameters of the approximating straight by the least-squares method in this case

Block diagram fragment using MTA criteria optimization with expressions (4), (5) and delay block.

kmψpm

0 if Tj j≤Tz, klð Þ T � Tz if T>Tz, klð Þ T þ Tz if T< � Tz,

: (10)

� � �

� and,

(11)

its q-component and is determined from the torque equation:

DOI: http://dx.doi.org/10.5772/intechopen.80424

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

idMTAkð Þ¼ T

where Tz—dead band limit and kl—linear gain.

not change the T=is ratio:

Figure 7.

Figure 8.

153

Optimal and quasi-optimal curves i T

� �.

If we express d-axis current id, through torque T and q-axis current iq from the 4-th equation of the system (1), and substitute the resulting expression into the lefthand side of Eq. (3), then after transformations, we can write an incomplete fourth degree algebraic equation, that reflects the relationship between the electromagnetic torque and the q-axis stator current in an implicit form:

$$i\_{q\rm MTA}{}^4 + \frac{T\nu\_{pm}}{k\_t \left(L\_d - L\_q\right)^2} i\_{q\rm MTA} - \left(\frac{T\nu\_{pm}}{k\_t \left(L\_d - L\_q\right)}\right)^2 = 0.\tag{8}$$

Eq. (8) can be solved only by numerical methods, which increase the requirements for the microprocessor control devices.

When the system of Eqs. (7), (8) and (4) is used for synthesis, its block diagram acquires the form of Figure 6.

To avoid solution of Eq. (8) in real time, the q-axis stator current reference can be calculated using Eq. (5) with iq ¼ i ∗ <sup>q</sup>, id ¼ i ∗ <sup>d</sup>, and <sup>T</sup> <sup>¼</sup> <sup>T</sup><sup>∗</sup>. In this case, taking into account that, in turn, i ∗ <sup>d</sup> depends on i ∗ <sup>q</sup>, an algebraic loop is formed in a block diagram. This leads to the need to include one sample time delay in the algorithm, as shown in Figure 7.

In order to use Eq. (5) in the control algorithm without i ∗ dMTA signal delay, it is necessary to generate this signal first as a function of the electromagnetic torque. However, the idMTAð Þ T dependence is expressed by an even more complicated equation than iqMTAð Þ T :

$$\left(\dot{\mathbf{i}}\_{\mathrm{dMTA}}\,^{2} + \frac{\left(\boldsymbol{\nu}\_{\mathrm{pm}}\right)}{\left(\boldsymbol{L}\_{d} - \boldsymbol{L}\_{q}\right)}\dot{\mathbf{i}}\_{\mathrm{do}}\right)^{2} + \frac{T\boldsymbol{\nu}\_{\mathrm{pm}}}{k\_{T}\left(\boldsymbol{L}\_{d} - \boldsymbol{L}\_{q}\right)^{2}}\sqrt{\dot{\mathbf{i}}\_{\mathrm{dMTA}}\,^{2} + \frac{\left(\boldsymbol{\nu}\_{\mathrm{pm}}\right)}{\left(\boldsymbol{L}\_{d} - \boldsymbol{L}\_{q}\right)}\dot{\mathbf{i}}\_{\mathrm{dMTA}} - \left(\frac{T\boldsymbol{\nu}\_{\mathrm{pm}}}{k\_{T}\left(\boldsymbol{L}\_{d} - \boldsymbol{L}\_{q}\right)}\right)^{2}}\tag{9}$$

Therefore, we approximate the curve obtained by the numerical solution of Eq. (9). To do this, we analyze the dependence curves of the orthogonal stator current components and peak values of the stator current from the motor torque for

Figure 6. Fragment of the MTA-optimal block diagram, using Eqs. (8), (4) and SC with a gain (7).

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor DOI: http://dx.doi.org/10.5772/intechopen.80424

the MTA–and "id ¼ 0"–strategies for the two IPMSMs from Table 1, presented in Figure 8. In the latter case, the steady-state value of the peak current coincides with its q-component and is determined from the torque equation:

$$i\_{q0}(T) = i\_{s0}(T) = \frac{T}{k\_m \nu\_{pm}}.\tag{10}$$

Figure 8 shows that, firstly, the effectiveness of the MTA strategy increases with the increase of the motor torque and its magnetic asymmetry j j ΔL ¼ Ld � Lq � � � � and, secondly, the trajectories idMTAð Þ T can be linearized quite easily. For the first motor (Figure 8a), the difference between the compared strategies with T≤0:5Tr is almost nonexistent, and where this difference becomes significant, the plot idMTAð Þ T becomes practically linear. Therefore, the dependence calculated by the Eq. (11) can be replaced by the "dead zone" nonlinearity idMTAkð Þ T type, which practically does not change the T=is ratio:

$$\dot{\mathbf{u}}\_{dMTAk}(T) = \begin{cases} \mathbf{0} & \text{if} \quad |T| \le T\_x, \\ k\_l(T - T\_x) & \text{if} \quad T > T\_x, \\ k\_l(T + T\_x) & \text{if} \quad T < -T\_x, \end{cases} \tag{11}$$

where Tz—dead band limit and kl—linear gain.

For the second motor, the nonlinear characteristic can generally be approximated by a straight line drawn through the original curve end points. Finding the parameters of the approximating straight by the least-squares method in this case

Figure 7.

the torque and currents overshoots, as well as overvoltage and transient oscillations without quality loss of the steady-state energy parameters. Correction of the SC (Eq. 6) improves the dynamics of the system but requires putting the division block

Let us consider another approach to the MTA algorithm implementation, in which the SC forms the electromagnetic torque reference. In this case, the P-SC gain

> i ∗ <sup>q</sup>ð Þs <sup>Δ</sup>ωð Þ<sup>s</sup> <sup>¼</sup> ks<sup>с</sup><sup>2</sup>

If we express d-axis current id, through torque T and q-axis current iq from the 4-th equation of the system (1), and substitute the resulting expression into the lefthand side of Eq. (3), then after transformations, we can write an incomplete fourth degree algebraic equation, that reflects the relationship between the electromag-

� �<sup>2</sup> iqMTA � <sup>T</sup>ψpm

Eq. (8) can be solved only by numerical methods, which increase the require-

When the system of Eqs. (7), (8) and (4) is used for synthesis, its block diagram

To avoid solution of Eq. (8) in real time, the q-axis stator current reference can

diagram. This leads to the need to include one sample time delay in the algorithm, as

necessary to generate this signal first as a function of the electromagnetic torque. However, the idMTAð Þ T dependence is expressed by an even more complicated

s

idMTA<sup>2</sup> <sup>þ</sup>

Therefore, we approximate the curve obtained by the numerical solution of Eq. (9). To do this, we analyze the dependence curves of the orthogonal stator current components and peak values of the stator current from the motor torque for

∗ <sup>q</sup>, id ¼ i ∗

∗

τωis þ 1 τωis

kt Ld � Lq � � !<sup>2</sup>

<sup>q</sup>, an algebraic loop is formed in a block

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ψpm Ld � Lq � � idMTA

∗

(7)

¼ 0: (8)

dMTA signal delay, it is

¼ 0:

(9)

� <sup>T</sup>ψpm kT Ld � Lq � � !<sup>2</sup>

<sup>d</sup>, and <sup>T</sup> <sup>¼</sup> <sup>T</sup><sup>∗</sup>. In this case, taking into

and the PI-SC transfer function are calculated by the equations:

netic torque and the q-axis stator current in an implicit form:

Tψpm kt Ld � Lq

, ws<sup>с</sup>2ð Þ¼ p

ksc<sup>2</sup> <sup>¼</sup> <sup>J</sup> τω

iqMTA 4 þ

be calculated using Eq. (5) with iq ¼ i

acquires the form of Figure 6.

account that, in turn, i

equation than iqMTAð Þ T :

ψpm Ld � Lq � � ido !<sup>2</sup>

shown in Figure 7.

idMTA<sup>2</sup> <sup>þ</sup>

Figure 6.

152

ments for the microprocessor control devices.

∗

þ

<sup>d</sup> depends on i

In order to use Eq. (5) in the control algorithm without i

Tψpm kT Ld � Lq � �<sup>2</sup>

Fragment of the MTA-optimal block diagram, using Eqs. (8), (4) and SC with a gain (7).

after the SC.

Applied Modern Control

Block diagram fragment using MTA criteria optimization with expressions (4), (5) and delay block.

Figure 8. Optimal and quasi-optimal curves i T � �.

makes no sense, since the goal of the approximation is the minimum deviation from the optimal trajectories of not d- or q-components of the stator current, but its amplitude. The approximations obtained for the nonlinear dependence determined by Eq. (11) can be called quasi-optimal. Their use makes it possible to greatly simplify the system with the implementation of the MTA-strategy, as shown in Figure 9.

It is known that when using the MTA strategy, the desired torque value is provided not only with a smaller amplitude of the stator current but also with a lower voltage amplitude of the voltage, which further increases the energy efficiency of the control method in question. The plots of the steady-state voltages are shown in Figure 10.

To compare the systems under investigation and to confirm the correctness of the plots in Figures 8 and 10, we perform simulation using the example of the second motor from Table 1.

Transients are shown in Figure 11 (id ¼ 0-strategy) and Figure 12 (quasioptimal MTA strategy (block diagram Figure 9)).

Comparison of the transients shows that in the steady state, the values of the currents, voltages, and their components coincide with the values obtained from the static characteristics in Figure 8b and Figure 10b. Improvement of energy indicators (reduction of current and voltage amplitudes at the same values of torque and speed) occurs without deteriorating the quality of transient processes. The orthogonal components and voltage amplitude decrease not only for steady-state values of the electromagnetic torque but also during its change.

4. Three-range speed control system of IPMSM

Transients in the system used quasi-optimal MTA-strategy.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 <sup>q</sup> <sup>þ</sup> <sup>u</sup><sup>2</sup> d

≈ω<sup>2</sup> e

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

Lqiq

r

q

u2 <sup>s</sup> ¼

from where follows:

value (first range);

three ways:

155

Figure 11.

Figure 12.

Transients in the system used id ¼ 0 strategy.

DOI: http://dx.doi.org/10.5772/intechopen.80424

The idea of three-range speed regulation follows from the approximated equation for the peak steady-state stator voltage, which can be obtained from the first two equations of system (1), excluding voltage losses on resistance and inductance:

Lqiq

<sup>ω</sup><sup>e</sup> <sup>≈</sup> us ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �<sup>2</sup> <sup>þ</sup> <sup>ψ</sup>pm <sup>þ</sup> Ldid

It follows from expression (13) that in the IPMSM, the speed can be increased in

1. due to a change in the amplitude of the stator voltage from 0 to the nominal

2. due to the pseudo-weakening of the permanent magnets field by increasing the

d-axis stator current in the negative direction (second range); and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �<sup>2</sup>

� �<sup>2</sup> <sup>r</sup> : (13)

, (12)

� �<sup>2</sup> <sup>þ</sup> <sup>ψ</sup>pm <sup>þ</sup> Ldid

Figure 9. Block diagram fragment of the MTA-quasi-optimal system.

Figure 10. Optimal and quasi-optimal curves u T .

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor DOI: http://dx.doi.org/10.5772/intechopen.80424

Figure 11. Transients in the system used id ¼ 0 strategy.

makes no sense, since the goal of the approximation is the minimum deviation from the optimal trajectories of not d- or q-components of the stator current, but its amplitude. The approximations obtained for the nonlinear dependence determined by Eq. (11) can be called quasi-optimal. Their use makes it possible to greatly simplify the system with the implementation of the MTA-strategy, as shown in

It is known that when using the MTA strategy, the desired torque value is provided not only with a smaller amplitude of the stator current but also with a lower voltage amplitude of the voltage, which further increases the energy efficiency of the control method in question. The plots of the steady-state voltages are

To compare the systems under investigation and to confirm the correctness of the plots in Figures 8 and 10, we perform simulation using the example of the

Transients are shown in Figure 11 (id ¼ 0-strategy) and Figure 12 (quasi-

Comparison of the transients shows that in the steady state, the values of the currents, voltages, and their components coincide with the values obtained from the static characteristics in Figure 8b and Figure 10b. Improvement of energy indicators (reduction of current and voltage amplitudes at the same values of torque and speed) occurs without deteriorating the quality of transient processes. The orthogonal components and voltage amplitude decrease not only for steady-state

Figure 9.

Figure 9.

Figure 10.

154

Optimal and quasi-optimal curves u T

.

shown in Figure 10.

Applied Modern Control

second motor from Table 1.

optimal MTA strategy (block diagram Figure 9)).

Block diagram fragment of the MTA-quasi-optimal system.

values of the electromagnetic torque but also during its change.

Figure 12. Transients in the system used quasi-optimal MTA-strategy.

#### 4. Three-range speed control system of IPMSM

The idea of three-range speed regulation follows from the approximated equation for the peak steady-state stator voltage, which can be obtained from the first two equations of system (1), excluding voltage losses on resistance and inductance:

$$\boldsymbol{u}\_{s}^{2} = \sqrt{\boldsymbol{u}\_{q}^{2} + \boldsymbol{u}\_{d}^{2}} \approx \boldsymbol{o}\_{e}^{2}\sqrt{\left(\boldsymbol{L}\_{q}\dot{\boldsymbol{i}}\_{q}\right)^{2} + \left(\boldsymbol{\psi}\_{pm} + \boldsymbol{L}\_{d}\dot{\boldsymbol{i}}\_{d}\right)^{2}},\tag{12}$$

from where follows:

$$
\omega\_t \approx \frac{\mathfrak{u}\_t}{\sqrt{\left(L\_q \dot{\mathfrak{u}}\_q\right)^2 + \left(\mathfrak{w}\_{pm} + L\_d \dot{\mathfrak{u}}\_d\right)^2}} \,\tag{13}
$$

It follows from expression (13) that in the IPMSM, the speed can be increased in three ways:


3. due to weakening of the stator field by decreasing the modulus of the q-axis stator current (third range).

The speed control in the first range using the MMA strategy is discussed in the previous section.

The transition to the second range occurs when the stator voltage reaches the nominal value, which is the maximum permissible steady-state voltage value. The relationship between the stator current components in this mode, which is called the field weakening control (FWC), is determined from Eq. (13) with substitutions us ¼ usmax and ω<sup>e</sup> ¼ ωep ¼ max ω<sup>e</sup> ð Þ , pω<sup>r</sup> :

$$i\_{dFWC}(i\_q, u\_{\varepsilon \max}, \alpha\_{\varepsilon p}) = \frac{-\nu\_{pm} + \sqrt{u\_{\varepsilon \max}^2 / \alpha\_{\varepsilon p}^2 - L\_q^2 i\_q^2}}{L\_d}. \tag{14}$$

Substituting (14) into the torque equation, we obtain implicitly the dependence of the q-axis stator current on the torque, voltage, and speed of the motor [1]:

$$i\_{qFWC}^4 + p\_2 i\_{qFWC}^2 + p\_1 i\_{qFWC} + p\_0 = 0,\tag{15}$$

idMTV <sup>¼</sup> <sup>ψ</sup>dMTV � <sup>ψ</sup>pm

strategy in the following form:

idMTV iq

current at the level:

decreases.

Figure 13.

157

Steady-state dq-trajectories of IPMSM.

Ld

DOI: http://dx.doi.org/10.5772/intechopen.80424

� � ¼ � <sup>ψ</sup>pm

2 Ld � Lq

iqmax<sup>i</sup> � � �

� � � <sup>2</sup> � Lq

<sup>ψ</sup>dMTV <sup>¼</sup> �Lqψpm <sup>þ</sup>

, iqMTV ¼

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

ψ<sup>2</sup> pmL<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi usmax=ωep � �<sup>2</sup> � <sup>ψ</sup><sup>2</sup> dMTV <sup>q</sup> Lq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 <sup>s</sup> max � i 2 d

� � :

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψ2 pm 4 Ld � Lq � �<sup>2</sup> <sup>þ</sup> <sup>i</sup>

2 q vuut : (20)

: (21)

<sup>q</sup> þ 8 Ld � Lq � �<sup>2</sup> usmax=ωep � �<sup>2</sup> <sup>q</sup>

4 Ld � Lq

� Lq Ld

q

Using the Euler–Lagrange equations to find the minimum of the stator voltage (Eq. (12)), taking into account the torque equation as an additional condition, we obtain the dependence between the components of the stator current for the MTV

> Ld � �

� ¼ iq ið Þ¼ <sup>s</sup>max; id

In many papers, the current constraint is achieved by limiting the q-axis stator

Once again, we emphasize that the MTV strategy is applied in the third range, when the reserves for increasing the speed due to the weakening of the field of permanent magnets are exhausted, and regulation is carried out by decreasing the q-component of the stator current, so that the amplitude of the current also

The p.u. MTA and MTV trajectories calculated using Eqs. (4) and (20) are shown in Figure 13. The same graph shows two constant currents circles of

,

(19)

where

$$p\_2 = \frac{\wp\_{pm}^2 L\_q^2 - \Delta L^2 u\_{s\max}^2 / o\_{cp}^2}{L\_q^2 \Delta L^2}, \quad p\_1 = \frac{4T L\_d L\_q \wp\_{pm}}{3\text{z}\_p L\_q^2 \Delta L^2}, \quad p\_0 = \frac{4T^2 L\_d^2}{9\text{z}\_p^2 L\_q^2 \Delta L^2}.\tag{16}$$

When adjusting the motor speed in the second range in the current constraint mode, Eq. (16) is modified [1, 2]:

$$i\_{dFWC}(i\_{i\max}, u\_{\max}, \alpha\_{ep}) = \frac{-\boldsymbol{\nu}\_{pw}\boldsymbol{L}\_{q} + \sqrt{\boldsymbol{\nu}\_{pw}^{2}\boldsymbol{L}\_{q}^{2} - \left(\boldsymbol{L}\_{d}^{2} - \boldsymbol{L}\_{q}^{2}\right)\left(\boldsymbol{L}\_{q}^{2}i\_{i\max}^{2} + \boldsymbol{\nu}\_{pw}^{2} - \boldsymbol{u}\_{\max}^{2}/\alpha\_{ep}^{2}\right)}{\boldsymbol{L}\_{d}^{2} - \boldsymbol{L}\_{q}^{2}}. \tag{17}$$

As the speed increases, iron losses due to eddy currents become more and more significant:

$$P\_{\rm Fe} \approx k\_{\rm et} \mu\_{\rm s}^2 \alpha\_{\rm e}^2 \approx k\_{\rm et} u\_{\rm s}^2 = \mathbf{1.5} \cdot u\_{\rm s}^2 / R\_{\rm Fe} \tag{18}$$

where kec is the eddy current gain, RFe is the fictitious resistance of the steel, which is inserted into the motor equivalent circuit to simulate this kind of losses.

This makes it advisable to indirectly limit the iron losses by applying the "maximal torque per volt" (MTV), or the "minimal flux per torque" (MFT) control strategies in the third range. Expressing currents through flux linkages

$$i\_d = \frac{\left\|\nu\_d - \left\|\nu\_{pm}\right\|\_q}{L\_d}, \quad i\_q = \frac{\sqrt{\left\|\nu\_s^2 - \left\|\nu\_d^2\right\|}}{L\_q}.$$

and substituting them into the torque equation, we obtain

$$T = \frac{k\_t}{L\_d L\_q} \sqrt{\mathcal{W}\_s^2 - \mathcal{W}\_d^2} \cdot \left[ L\_q \mathcal{W}\_{pm} - \left( L\_d - L\_q \right) \mathcal{W}\_d \right].$$

Analyzing the last expression for the extremum, we obtain the following equations that ensure the maximum of the torque:

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor DOI: http://dx.doi.org/10.5772/intechopen.80424

$$i\_{dMTV} = \frac{\left(\nu\_{dMTV} - \nu\_{pm}\right)}{L\_d}, \quad i\_{qMTV} = \frac{\sqrt{\left(u\_{s\text{max}}/\omega\_{cp}\right)^2 - \nu\_{dMTV}^2}}{L\_q},$$

$$\nu\_{dMTV} = \frac{-L\_q \nu\_{pm} + \sqrt{\nu\_{pm}^2 L\_q^2 + 8\left(L\_d - L\_q\right)^2 \left(u\_{s\text{max}}/\omega\_{cp}\right)^2}}{4\left(L\_d - L\_q\right)}.\tag{19}$$

Using the Euler–Lagrange equations to find the minimum of the stator voltage (Eq. (12)), taking into account the torque equation as an additional condition, we obtain the dependence between the components of the stator current for the MTV strategy in the following form:

$$i\_{dMTV}(i\_q) = -\frac{\wp\_{pm}}{2(L\_d - L\_q)} \cdot \left(2 - \frac{L\_q}{L\_d}\right) - \frac{L\_q}{L\_d} \sqrt{\frac{\wp\_{pm}^2}{4\left(L\_d - L\_q\right)^2} + i\_q^2}.\tag{20}$$

In many papers, the current constraint is achieved by limiting the q-axis stator current at the level:

$$\left|i\_{q\max}\right| = i\_q(i\_{\text{max}}, i\_d) = \sqrt{i\_{\text{smax}}^2 - i\_d^2}.\tag{21}$$

Once again, we emphasize that the MTV strategy is applied in the third range, when the reserves for increasing the speed due to the weakening of the field of permanent magnets are exhausted, and regulation is carried out by decreasing the q-component of the stator current, so that the amplitude of the current also decreases.

The p.u. MTA and MTV trajectories calculated using Eqs. (4) and (20) are shown in Figure 13. The same graph shows two constant currents circles of

Figure 13. Steady-state dq-trajectories of IPMSM.

3. due to weakening of the stator field by decreasing the modulus of the q-axis

The speed control in the first range using the MMA strategy is discussed in the

The transition to the second range occurs when the stator voltage reaches the nominal value, which is the maximum permissible steady-state voltage value. The relationship between the stator current components in this mode, which is called the field weakening control (FWC), is determined from Eq. (13) with substitutions

�ψpm þ

Substituting (14) into the torque equation, we obtain implicitly the dependence

3zpL<sup>2</sup>

<sup>d</sup> � <sup>L</sup><sup>2</sup> q � �

> L2 <sup>d</sup> � <sup>L</sup><sup>2</sup> q

<sup>s</sup> <sup>¼</sup> <sup>1</sup>:<sup>5</sup> � <sup>u</sup><sup>2</sup>

q

� Lqψpm � Ld � Lq

As the speed increases, iron losses due to eddy currents become more and more

<sup>e</sup> ≈kecu<sup>2</sup>

imal torque per volt" (MTV), or the "minimal flux per torque" (MFT) control

where kec is the eddy current gain, RFe is the fictitious resistance of the steel, which is inserted into the motor equivalent circuit to simulate this kind of losses. This makes it advisable to indirectly limit the iron losses by applying the "max-

, iq ¼

Analyzing the last expression for the extremum, we obtain the following equa-

When adjusting the motor speed in the second range in the current constraint

of the q-axis stator current on the torque, voltage, and speed of the motor [1]:

2

qΔL<sup>2</sup> , p<sup>1</sup> <sup>¼</sup> <sup>4</sup>TLdLqψpm

ψ<sup>2</sup> pmL<sup>2</sup> <sup>q</sup> � <sup>L</sup><sup>2</sup>

sω<sup>2</sup>

strategies in the third range. Expressing currents through flux linkages

id <sup>¼</sup> <sup>ψ</sup><sup>d</sup> � <sup>ψ</sup>pm Ld

and substituting them into the torque equation, we obtain

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψ2 <sup>s</sup> � <sup>ψ</sup><sup>2</sup> d

<sup>T</sup> <sup>¼</sup> kt LdLq

tions that ensure the maximum of the torque:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qFWC þ p1iqFWC þ p<sup>0</sup> ¼ 0, (15)

qΔL<sup>2</sup> , p<sup>0</sup> <sup>¼</sup> <sup>4</sup>T<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψ2 <sup>s</sup> � ψ<sup>2</sup> d

� �ψ<sup>d</sup> h i

:

Lq

L2 qi 2 <sup>s</sup> max þ ψ<sup>2</sup> L2 d

qΔL<sup>2</sup> : (16)

: (17)

9z<sup>2</sup> pL2

pm � u<sup>2</sup>

<sup>s</sup> =RFe, (18)

smax=ω<sup>2</sup> ep

ep � <sup>L</sup><sup>2</sup> qi 2 q

: (14)

u2 smax=ω<sup>2</sup>

Ld

q

stator current (third range).

us ¼ usmax and ω<sup>e</sup> ¼ ωep ¼ max ω<sup>e</sup> ð Þ , pω<sup>r</sup> :

idFWC iq; us max;ωep � � <sup>¼</sup>

> i 4

> > u2 <sup>s</sup> max=ω<sup>2</sup> ep

�ψpmLq þ

PFe ≈ kecψ<sup>2</sup>

qFWC þ p2i

previous section.

Applied Modern Control

where

<sup>p</sup><sup>2</sup> <sup>¼</sup> <sup>ψ</sup><sup>2</sup>

idFWC is max; usmax;ωep � � <sup>¼</sup>

significant:

156

pmL<sup>2</sup>

mode, Eq. (16) is modified [1, 2]:

<sup>q</sup> � <sup>Δ</sup>L<sup>2</sup>

L2

corresponding to Eq. (21), three constant torques hyperbolae, and three constant velocities ellipses, calculated from Eq. (13) with us ¼ usmax.

Acceleration begins from point A via almost instantaneous transition to point B or to point B1 on the trajectory of MTA. With a simultaneous increase in the amplitude and frequency of the stator voltage, the change in speed is not accompanied by a transition to another ellipse. After reaching the nominal voltage amplitude value (point B or B1), which determines the upper limit of this value in the steady state us≤usr ¼ usmax, the execution of MTA strategy becomes impossible.

Transition from the MTA parabola to MTV parabola occurs either along the maximum allowable current circumference (arch B1-D) or a constant torque hyperbola (hyperbolic line B-D1), or first along the constant torque hyperbola with increase of the current, and then along the overcurrent circumference (path BCD). Movement along the hyperbola is carried out with an underloaded motor with constant torque and voltage. The power increases due to the increase in speed and current. Movement along the circumference occurs at a constant power. These trajectories control the speed in the second range in accordance with the FWC strategy (Eq. 14–17).

Further increase in the speed to the desired value occurs in the third range along the MTV trajectory (Eq. (19)). In this mode (segment D-E), the orthogonal components of the stator current are reduced, resulting in the motor coming out of the current-limiting mode and operating at the reduced values of both stator current amplitude and power.

After the speed has reached its set value, the transition from the acceleration mode to the steady state occurs. In this case, the motion of the working point occurs along an ellipse of constant velocity until it intersects with the load torque hyperbola (segment E-GL or E-G0).

by current limit block CL. The dynamic saturation block Sat2 for the stator voltage in no-load mode is switched in series with the block Sat1, and the constraint signal is formed by the steady-state voltage limit block SSVL

Block diagram of the three-range IPMSM speed control system considering voltage and current constrains.

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

DOI: http://dx.doi.org/10.5772/intechopen.80424

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi usmax=ωep � �<sup>2</sup> � Ldid <sup>þ</sup> <sup>ψ</sup>pm � �<sup>2</sup>

obtained from Eq. (12) at us ¼ usmax. Thus, at the output of the block Sat2, we

�; iq max<sup>u</sup> � � �

us <sup>≤</sup>ulim <sup>¼</sup> udc<sup>=</sup> ffiffiffi

The d-axis current reference signal in the first region is calculated from the Eq. (4) as a function of the signal (22), in the second range from the Eq. (14) as the function of signal (21) and in the third region from the Eq. (20) with the input signal (22). The switching of the control algorithm from MTA to FWC and from FWC to MTV occurs by dynamically limiting the signal of the MTA block at the top

Transients in the system shown in Figure 14, obtained via simulation, are shown

The coordinates of the characteristic points of the transients in Figure 15 coincide with the corresponding points of the diagrams in Figure 13. The stator current does not exceed its maximum permissible value, and the stator voltage in no-load

3

To ensure that the motor voltage in dynamic modes does not exceed the inverter

=Lq, (22)

<sup>p</sup> (23)

according to the equation

� � �

obtain a signal iqu

Figure 14.

in Figure 15.

159

iq max <sup>u</sup> ¼

� ¼ min iq � � �

level idFWC<sup>∗</sup> and at the bottom level idMTV<sup>∗</sup>.

r

�; iq max<sup>i</sup> � � �

with the blocks of dynamic voltage limit (DVL) and Sat4.

DC link voltage, the output of the regulator CCq is limited at the level

� � �.

Concentric ellipses of constant velocities have a center at the point H with coordinates id<sup>0</sup> ¼ ψpm=Ld, iq<sup>0</sup> ¼ 0, in which the MTV line ends. At this point, a complete demagnetization of the motor takes place, which theoretically makes it possible to achieve an arbitrarily large steady-state velocity (ω ! ∞) at zero-load torque (TL ¼ 0). In practice, these points are unattainable, and the real range of speed control in the third range is limited by the magnitude of the load torque, the mechanical strength of the rotor, and the constraint of the d-axis stator current in the steady state at about id0=2 to prevent the permanent magnets from being irreversibly demagnetized.

As can be seen from the above Eqs. (4, 5, 8, 9, 14–17, 19–21), a general algorithm for controlling the speed of IPMSM in the three range requires significant computational resources associated with the need for a numerical solution of algebraic equations in real time and complex logic branching for the purpose of organizing controlled switching, etc. Replacing a single algorithm with multidimensional lookup tables of data is associated with the preliminary calculation of a large number of curves and the organization of the search in these tables.

Meanwhile, the analysis of the control trajectories in Figure 13 suggests the possibility of a structural implementation of the control algorithm, which is presented in Figure 14.

In it, the speed controller SC forms the torque reference. To avoid solving fourth-degree algebraic equations in real time (Eqs. 8, 9, 15) and computations according to formulas (19), the q-axis stator current reference is determined by the Eq. (5) with <sup>T</sup> <sup>¼</sup> <sup>T</sup><sup>∗</sup>, iq <sup>=</sup> iq <sup>∗</sup>. To prevent the formation of an algebraic loop, we use in this equation not the reference id <sup>∗</sup>, but the feedback signal id.

The limitation of the stator current amplitude is achieved by the dynamic saturation block Sat1 for the q-axis current at the level (21), which is calculated Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor DOI: http://dx.doi.org/10.5772/intechopen.80424

Figure 14.

corresponding to Eq. (21), three constant torques hyperbolae, and three constant

Transition from the MTA parabola to MTV parabola occurs either along the maximum allowable current circumference (arch B1-D) or a constant torque hyperbola (hyperbolic line B-D1), or first along the constant torque hyperbola with increase of the current, and then along the overcurrent circumference (path BCD). Movement along the hyperbola is carried out with an underloaded motor with constant torque and voltage. The power increases due to the increase in speed and current. Movement along the circumference occurs at a constant power. These trajectories control the speed in the second range in accordance with the FWC

Further increase in the speed to the desired value occurs in the third range along the MTV trajectory (Eq. (19)). In this mode (segment D-E), the orthogonal components of the stator current are reduced, resulting in the motor coming out of the current-limiting mode and operating at the reduced values of both stator current

After the speed has reached its set value, the transition from the acceleration mode to the steady state occurs. In this case, the motion of the working point occurs along an ellipse of constant velocity until it intersects with the load torque hyper-

Concentric ellipses of constant velocities have a center at the point H with coordinates id<sup>0</sup> ¼ ψpm=Ld, iq<sup>0</sup> ¼ 0, in which the MTV line ends. At this point, a complete demagnetization of the motor takes place, which theoretically makes it possible to achieve an arbitrarily large steady-state velocity (ω ! ∞) at zero-load torque (TL ¼ 0). In practice, these points are unattainable, and the real range of speed control in the third range is limited by the magnitude of the load torque, the mechanical strength of the rotor, and the constraint of the d-axis stator current in the steady state at about id0=2 to prevent the permanent magnets from being

As can be seen from the above Eqs. (4, 5, 8, 9, 14–17, 19–21), a general algorithm for controlling the speed of IPMSM in the three range requires significant computational resources associated with the need for a numerical solution of algebraic equations in real time and complex logic branching for the purpose of organizing controlled switching, etc. Replacing a single algorithm with multidimensional lookup tables of data is associated with the preliminary calculation of a large number of

Meanwhile, the analysis of the control trajectories in Figure 13 suggests the possibility of a structural implementation of the control algorithm, which is

In it, the speed controller SC forms the torque reference. To avoid solving fourth-degree algebraic equations in real time (Eqs. 8, 9, 15) and computations according to formulas (19), the q-axis stator current reference is determined by the

The limitation of the stator current amplitude is achieved by the dynamic saturation block Sat1 for the q-axis current at the level (21), which is calculated

<sup>∗</sup>. To prevent the formation of an algebraic loop, we use

<sup>∗</sup>, but the feedback signal id.

curves and the organization of the search in these tables.

or to point B1 on the trajectory of MTA. With a simultaneous increase in the amplitude and frequency of the stator voltage, the change in speed is not accompanied by a transition to another ellipse. After reaching the nominal voltage amplitude value (point B or B1), which determines the upper limit of this value in the steady

state us≤usr ¼ usmax, the execution of MTA strategy becomes impossible.

Acceleration begins from point A via almost instantaneous transition to point B

velocities ellipses, calculated from Eq. (13) with us ¼ usmax.

strategy (Eq. 14–17).

Applied Modern Control

amplitude and power.

bola (segment E-GL or E-G0).

irreversibly demagnetized.

presented in Figure 14.

Eq. (5) with <sup>T</sup> <sup>¼</sup> <sup>T</sup><sup>∗</sup>, iq <sup>=</sup> iq

158

in this equation not the reference id

Block diagram of the three-range IPMSM speed control system considering voltage and current constrains.

by current limit block CL. The dynamic saturation block Sat2 for the stator voltage in no-load mode is switched in series with the block Sat1, and the constraint signal is formed by the steady-state voltage limit block SSVL according to the equation

$$\dot{a}\_{q\max u} = \sqrt{\left(u\_{\text{smax}}/a\_{ep}\right)^2 - \left(L\_d i\_d + \mu\_{pm}\right)^2}/L\_q,\tag{22}$$

obtained from Eq. (12) at us ¼ usmax. Thus, at the output of the block Sat2, we obtain a signal iqu � � � � ¼ min iq � � � �; iq max<sup>i</sup> � � � �; iq max<sup>u</sup> � � � � � �.

To ensure that the motor voltage in dynamic modes does not exceed the inverter DC link voltage, the output of the regulator CCq is limited at the level

$$
u\_s \le 
u\_{\rm lim} = 
u\_{\rm dc} / \sqrt{3} \tag{23}$$

with the blocks of dynamic voltage limit (DVL) and Sat4.

The d-axis current reference signal in the first region is calculated from the Eq. (4) as a function of the signal (22), in the second range from the Eq. (14) as the function of signal (21) and in the third region from the Eq. (20) with the input signal (22). The switching of the control algorithm from MTA to FWC and from FWC to MTV occurs by dynamically limiting the signal of the MTA block at the top level idFWC<sup>∗</sup> and at the bottom level idMTV<sup>∗</sup>.

Transients in the system shown in Figure 14, obtained via simulation, are shown in Figure 15.

The coordinates of the characteristic points of the transients in Figure 15 coincide with the corresponding points of the diagrams in Figure 13. The stator current does not exceed its maximum permissible value, and the stator voltage in no-load

mode is equal to the nominal stator voltage, and in dynamic modes, it is limited by

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

In this chapter, we presented the IPMSM speed control systems. Two options for improving the quality of the transient processes for the speed regulation by changing the stator voltage using the MTA strategy are proposed. In a system in which the speed controller generates q-axis stator current reference, a method is proposed for adapting the speed controller to the presence of the reactive component of the electromagnetic torque. In a system in which the speed controller forms the electromagnetic torque reference, it is suggested to approximate the dependence i

using the "dead zone" nonlinearity and thus avoid solving the fourth-order equation in real time. At the same time, it was possible to ensure high energy parameters without the use of pre-calculated tables and organization of search in them without

For the system with a three-range speed regulation, a system is proposed using MTA, FWC, and MTV strategies with automatic switching between them and stator current and voltage constraints without using additional control loops is proposed.

The analytical researches are confirmed by the simulation results.

National Technical University of Ukraine, Igor Sikorsky Polytechnic Institute,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: tolochko.ola@gmail.com

provided the original work is properly cited.

∗ <sup>d</sup> <sup>T</sup><sup>∗</sup> ð Þ

the output voltage of the inverter DC link.

DOI: http://dx.doi.org/10.5772/intechopen.80424

degrading the quality of transient processes.

5. Conclusions

Author details

Olga Tolochko

Kyiv, Ukraine

161

Figure 15. Transients in the 3-range speed control system of IPMSM.

mode is equal to the nominal stator voltage, and in dynamic modes, it is limited by the output voltage of the inverter DC link.

### 5. Conclusions

In this chapter, we presented the IPMSM speed control systems. Two options for improving the quality of the transient processes for the speed regulation by changing the stator voltage using the MTA strategy are proposed. In a system in which the speed controller generates q-axis stator current reference, a method is proposed for adapting the speed controller to the presence of the reactive component of the electromagnetic torque. In a system in which the speed controller forms the electromagnetic torque reference, it is suggested to approximate the dependence i ∗ <sup>d</sup> <sup>T</sup><sup>∗</sup> ð Þ using the "dead zone" nonlinearity and thus avoid solving the fourth-order equation in real time. At the same time, it was possible to ensure high energy parameters without the use of pre-calculated tables and organization of search in them without degrading the quality of transient processes.

For the system with a three-range speed regulation, a system is proposed using MTA, FWC, and MTV strategies with automatic switching between them and stator current and voltage constraints without using additional control loops is proposed.

The analytical researches are confirmed by the simulation results.

### Author details

Olga Tolochko National Technical University of Ukraine, Igor Sikorsky Polytechnic Institute, Kyiv, Ukraine

\*Address all correspondence to: tolochko.ola@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Figure 15.

Applied Modern Control

160

Transients in the 3-range speed control system of IPMSM.

### References

[1] Schröder D. Elektrische Antriebe – Regelung von Antriebssystemen. 3. bearbeitete Auflage. Berlin, Heidelberg: Springer; 2009. p. 1336. DOI: 10.1007/ 978-3-540-89613-5

[2] Krishnan R. Permanent magnet synchronous and brushless DC motor drives. Virginia Tech, Blacksburg, USA: CRC press; 2010. p. 611. ISBN 978-0- 8247-5384-9

[3] Bose BK. Modern power electronics and AC drives. New Jersey: Prentice Hall PTR; 2002. p. 711. ISBN 0130167436, 9780130167439

[4] Sul S-K. Control of electric machine drive systems. Willey-IEEE Press; 2011. p. 424. ISBN: 978-0-470-59079-9

[5] Doncker RD, Pulle DWJ, Veltman A. Advanced electrical drives. Analysis, modeling, control. Berlin: Springer; 2011. p. 455. DOI: 10.1007/978-94- 007-0181-6

[6] Agrawal J, Bodkhe S. Steady-state analysis and comparison of control strategies for PMSM. Modelling and Simulation in Engineering; 2015. p. 11. DOI: 10.1155/2015/306787

[7] Morimoto S, Takeda Y, Hatanaka K, Tong Y, Hirasa T. Design and control system of permanent magnet synchronous motor for high torque and high efficiency operation. In: Industry Applications Conference Record of the 1991 IEEE Industry; 1991; 1. pp. 176-181. DOI: 10.1109/IAS.1991.178151

[8] Morimoto S, Hatanaka K, Tong Y, Takeda Y, Hirasa T. High performance servo drive system of salient pole permanent magnet synchronous motor. In: Industry Applications Society Annual Meeting of the IEEE Conference Record Proceedings; 1; 28 September–4

October 1991; pp. 463-468. DOI: 10.1109/IAS.1991.178196

[9] Morimoto S, Sanada M, Takeda Y. Wide speed operation of interior permanent magnet synchronous motors with high performance current regulator. IEEE Transactions on Industry Applications. 1994;IA-30(4): 920-926. DOI: 10.1109/28.297908

magnet synchronous motor (PMSM) in AC servo system. In: Electrical Machines and Systems. ICEMS; 17–20 October 2008; Wuhan, China. pp. 602-607. INSPEC Accession Number: 10458526

DOI: http://dx.doi.org/10.5772/intechopen.80424

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor

[21] Kim J.-M., Sul S.-K. Speed control of interior permanent magnet synchronous motor drive for the flux weakening operation. In: IEEE Transactions on Industry Applications, Vol. 33, No. 1, January/February 1997. p. 43–48. DOI:

[22] Bae B-H, Patel N, Schulz S, Sul S-K. New field weakening technique for high saliency interior permanent magnet motor. In: 38th IAS Annual Meeting on Industry Applications Conference, 2003; 12–16 October 2003; Salt Lake City, USA. pp. 898-905. DOI: 10.1109/

[23] Lee JH, Lee JH, Park JH, Won CY. Field-weakening strategy in condition of DC-link voltage variation using on electric vehicle of IPMSM. In: Electrical Machines and Systems (ICEMS). 20–23 August 2011; Beijing, China; 2011. 27 Jan/Feb. pp. 38-44. DOI: 10.1109/

10.1109/28.567075

IAS.2003.1257641

ICEMS.2011.6073676

TIE.2007.909087

jamdsm.4.1157

[24] Sue S-M, Pan C-T. Voltageconstraint-tracking-based field-

weakening control of IPM synchronous motor drives. In: IEEE Transactions on Industrial Electronics; 55(1); January 2008. рp. 340-347. DOI: 10.1109/

[25] Hao S, Shi J, Hao M, Mizugaki Y. Closed-loop parameter identification of permanent magnet synchronous motor considering nonlinear influence factors. Journal of Advanced Mechanical Design, Systems and Manufacturing. 2010;4(6):1157-1165. DOI: 10.1299/

[26] Meyer M, Bocker J. Optimum control for interior permanent magnet synchronous motors (IPMSM) in constant torque and flux weakening range. In: Power Electronics and Motion Control Conference, 2006. EPE-PEMC

2006. 12th International Power Electronics and Motion Control

[15] Navrapescu V, Kisck DO, Popescu M, Kisck M, Andronescu G. Modeling of iron losses in salient pole permanent magnet synchronous motors. In: 7th International Conference on Power Electronics (ICPE '07); 22–26 October 2007; Daegu, South Korea. pp. 352-357. DOI: 10.1109/ICPE.2007.4692408

[16] Urasaki N, Senjyu T, Uezato K. A novel calculation method for iron loss resistance suitable in modeling permanent-magnet synchronous motors. IEEE Transactions on Energy Conversion. 2003;18(1):41-47. DOI:

10.1109/TEC.2002.808329

PEDS.1997.627416

[17] Wijenayake AH, Schmidt PB. Modeling and analysis of permanent magnet synchronous motor by taking saturation and core loss into account. In: 1997 International Conference on Power Electronics and Drive Systems; 26–29 May 1997; Singapore. DOI: 10.1109/

[18] Yan Y, Zhu J, Guo Y. A permanent magnet synchronous motor model with core loss. Journal of the Japan Society of

Applied Electromagnetic and Mechanics. 2007;15:147-150

[19] Mansouri A, Trabelsi H. On the performances investigation and iron losses computation of an inset surface mounted permanent magnet motor. Systems, Signals and Devices. 2012. P. 1-5

[20] Mi C, Slemo GR, Bonert R. Minimization of iron losses of permanent magnet synchronous

10.1109/TEC.2004.832091

163

machines. IEEE Transactions on Energy Conversion. 2005;20:121-127. DOI:

[10] Morimoto S, Hatanaka K, Tong Y, Takeda Y, Hirasa T. Servo drive system and control characteristics of salient pole permanent magnet synchronous motor. IEEE Transactions on Industry Applications. 1993;29(2):338-343. DOI: 10.1109/28.216541

[11] Lee J-G, Nam K-H, Lee S-H, Choi S-H, Kwon S-W. A lookup table based loss minimizing control for FCEV permanent magnet synchronous motors. In: 2007 IEEE Vehicle Power and Propulsion Conference Journal of Electrical Engineering Technology; 9–12 September 2007; Arlington, USA; 4(2). pp. 201-210. DOI: 10.1109/VPPC. 2007.4544120

[12] Huang S, Chen Z, Huang K, Gao J. Maximum torque per ampere and fluxweakening control for PMSM based on curve fitting. In: Vehicle Power and Propulsion Conference (VPPC), 2010 IEEE; 1–3 September 2010; Lille, France. DOI: 10.1109/VPPC. 2010.5729024

[13] Ma L, Sanada M, Morimoto S, Takeda Y. Prediction of iron loss in rotating machines with rotational loss included magnetics. In: IEEE Transactions on Magnetics; 2003. 39(4). pp. 2036-2041. DOI: 10.1109/TMAG. 2003.812706

[14] Li L, Huang X, Baoquan Kao B, Yan B. Research of core loss of permanent

Energy Efficient Speed Control of Interior Permanent Magnet Synchronous Motor DOI: http://dx.doi.org/10.5772/intechopen.80424

magnet synchronous motor (PMSM) in AC servo system. In: Electrical Machines and Systems. ICEMS; 17–20 October 2008; Wuhan, China. pp. 602-607. INSPEC Accession Number: 10458526

References

Applied Modern Control

978-3-540-89613-5

8247-5384-9

9780130167439

007-0181-6

162

[1] Schröder D. Elektrische Antriebe – Regelung von Antriebssystemen. 3. bearbeitete Auflage. Berlin, Heidelberg: Springer; 2009. p. 1336. DOI: 10.1007/

October 1991; pp. 463-468. DOI:

with high performance current regulator. IEEE Transactions on Industry Applications. 1994;IA-30(4): 920-926. DOI: 10.1109/28.297908

[9] Morimoto S, Sanada M, Takeda Y. Wide speed operation of interior permanent magnet synchronous motors

[10] Morimoto S, Hatanaka K, Tong Y, Takeda Y, Hirasa T. Servo drive system and control characteristics of salient pole permanent magnet synchronous motor. IEEE Transactions on Industry Applications. 1993;29(2):338-343. DOI:

[11] Lee J-G, Nam K-H, Lee S-H, Choi S-H, Kwon S-W. A lookup table based loss minimizing control for FCEV permanent magnet synchronous motors. In: 2007 IEEE Vehicle Power and Propulsion Conference Journal of Electrical Engineering Technology; 9–12 September 2007; Arlington, USA; 4(2). pp. 201-210. DOI: 10.1109/VPPC.

[12] Huang S, Chen Z, Huang K, Gao J. Maximum torque per ampere and fluxweakening control for PMSM based on curve fitting. In: Vehicle Power and Propulsion Conference (VPPC), 2010 IEEE; 1–3 September 2010; Lille, France. DOI: 10.1109/VPPC.

[13] Ma L, Sanada M, Morimoto S, Takeda Y. Prediction of iron loss in rotating machines with rotational loss

Transactions on Magnetics; 2003. 39(4). pp. 2036-2041. DOI: 10.1109/TMAG.

[14] Li L, Huang X, Baoquan Kao B, Yan B. Research of core loss of permanent

included magnetics. In: IEEE

10.1109/IAS.1991.178196

10.1109/28.216541

2007.4544120

2010.5729024

2003.812706

[2] Krishnan R. Permanent magnet synchronous and brushless DC motor drives. Virginia Tech, Blacksburg, USA: CRC press; 2010. p. 611. ISBN 978-0-

[3] Bose BK. Modern power electronics and AC drives. New Jersey: Prentice Hall PTR; 2002. p. 711. ISBN 0130167436,

[4] Sul S-K. Control of electric machine drive systems. Willey-IEEE Press; 2011. p. 424. ISBN: 978-0-470-59079-9

[5] Doncker RD, Pulle DWJ, Veltman A. Advanced electrical drives. Analysis, modeling, control. Berlin: Springer; 2011. p. 455. DOI: 10.1007/978-94-

[6] Agrawal J, Bodkhe S. Steady-state analysis and comparison of control strategies for PMSM. Modelling and Simulation in Engineering; 2015. p. 11.

[7] Morimoto S, Takeda Y, Hatanaka K, Tong Y, Hirasa T. Design and control

synchronous motor for high torque and high efficiency operation. In: Industry Applications Conference Record of the 1991 IEEE Industry; 1991; 1. pp. 176-181.

[8] Morimoto S, Hatanaka K, Tong Y, Takeda Y, Hirasa T. High performance servo drive system of salient pole permanent magnet synchronous motor. In: Industry Applications Society

Annual Meeting of the IEEE Conference Record Proceedings; 1; 28 September–4

DOI: 10.1155/2015/306787

system of permanent magnet

DOI: 10.1109/IAS.1991.178151

[15] Navrapescu V, Kisck DO, Popescu M, Kisck M, Andronescu G. Modeling of iron losses in salient pole permanent magnet synchronous motors. In: 7th International Conference on Power Electronics (ICPE '07); 22–26 October 2007; Daegu, South Korea. pp. 352-357. DOI: 10.1109/ICPE.2007.4692408

[16] Urasaki N, Senjyu T, Uezato K. A novel calculation method for iron loss resistance suitable in modeling permanent-magnet synchronous motors. IEEE Transactions on Energy Conversion. 2003;18(1):41-47. DOI: 10.1109/TEC.2002.808329

[17] Wijenayake AH, Schmidt PB. Modeling and analysis of permanent magnet synchronous motor by taking saturation and core loss into account. In: 1997 International Conference on Power Electronics and Drive Systems; 26–29 May 1997; Singapore. DOI: 10.1109/ PEDS.1997.627416

[18] Yan Y, Zhu J, Guo Y. A permanent magnet synchronous motor model with core loss. Journal of the Japan Society of Applied Electromagnetic and Mechanics. 2007;15:147-150

[19] Mansouri A, Trabelsi H. On the performances investigation and iron losses computation of an inset surface mounted permanent magnet motor. Systems, Signals and Devices. 2012. P. 1-5

[20] Mi C, Slemo GR, Bonert R. Minimization of iron losses of permanent magnet synchronous machines. IEEE Transactions on Energy Conversion. 2005;20:121-127. DOI: 10.1109/TEC.2004.832091

[21] Kim J.-M., Sul S.-K. Speed control of interior permanent magnet synchronous motor drive for the flux weakening operation. In: IEEE Transactions on Industry Applications, Vol. 33, No. 1, January/February 1997. p. 43–48. DOI: 10.1109/28.567075

[22] Bae B-H, Patel N, Schulz S, Sul S-K. New field weakening technique for high saliency interior permanent magnet motor. In: 38th IAS Annual Meeting on Industry Applications Conference, 2003; 12–16 October 2003; Salt Lake City, USA. pp. 898-905. DOI: 10.1109/ IAS.2003.1257641

[23] Lee JH, Lee JH, Park JH, Won CY. Field-weakening strategy in condition of DC-link voltage variation using on electric vehicle of IPMSM. In: Electrical Machines and Systems (ICEMS). 20–23 August 2011; Beijing, China; 2011. 27 Jan/Feb. pp. 38-44. DOI: 10.1109/ ICEMS.2011.6073676

[24] Sue S-M, Pan C-T. Voltageconstraint-tracking-based fieldweakening control of IPM synchronous motor drives. In: IEEE Transactions on Industrial Electronics; 55(1); January 2008. рp. 340-347. DOI: 10.1109/ TIE.2007.909087

[25] Hao S, Shi J, Hao M, Mizugaki Y. Closed-loop parameter identification of permanent magnet synchronous motor considering nonlinear influence factors. Journal of Advanced Mechanical Design, Systems and Manufacturing. 2010;4(6):1157-1165. DOI: 10.1299/ jamdsm.4.1157

[26] Meyer M, Bocker J. Optimum control for interior permanent magnet synchronous motors (IPMSM) in constant torque and flux weakening range. In: Power Electronics and Motion Control Conference, 2006. EPE-PEMC 2006. 12th International Power Electronics and Motion Control

Conference; 10 February 2006; Portoroz, Slovenia. pp. 282-286. DOI: 10.1109/EPEPEMC.2006.4778413

Chapter 9

Dynamics

Navin Khaneja

Abstract

time optimality.

1. Introduction

two coupled spin <sup>1</sup>

tion for unitary evolution

165

Convexity, Majorization and Time

In this chapter, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. In a two qubit system, the ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of unitary evolution, g, are decomposed into fast generators k (local Hamiltonians) and slow generators p (couplings) as a Cartan decomposition

g ¼ p ⊕ k. Using this decomposition, we exploit some convexity ideas to completely

A rich class of model control problems arise when one considers dynamics of

field of quantum information processing and computing [1] and is fundamental in multidimensional NMR spectroscopy [2, 3]. Numerous experiments in NMR spectroscopy, involve synthesizing unitary transformations [4–6] that require interaction between the spins (evolution of the coupling Hamiltonian). These experiments involve transferring, coherence and polarization from one spin to another and involve evolution of interaction Hamiltonians [2]. Similarly, many protocols in quantum communication and information processing involve synthesizing entangled states starting from the separable states [1, 7, 8]. This again

A typical feature of many of these problems is that evolution of interaction Hamiltonians takes significantly longer than the time required to generate local unitary transformations (unitary transformations that effect individual spins only). In NMR spectroscopy [2, 3], local unitary transformations on spins are obtained by application of rf-pulses, whose strength may be orders of magnitude larger than the couplings between the spins. Given the Schróedinger equa-

requires evolution of interaction Hamiltonians between the qubits.

. The dynamics of two coupled spins, forms the basis for the

characterize the reachable set and time optimal control for these problems. The main contribution of the chapter is, we carry out a global analysis of

Keywords: Kostant convexity, spin dynamics, Cartan decomposition, Cartan

subalgebra, Weyl group, time optimal control

2

Optimal Control of Coupled Spin

[27] Inoue Y, Yamada K, Morimoto S, Sanada M. Effectiveness of voltage error compensation and parameter identification for model-based sensorless control of IPMSM. IEEE Transactions on Industry Applications. 2009;45(1):213-221. DOI: 10.1109/ TIA.2008.2009617

[28] Mink F, Kubasiak N, Ritter B, Binder A. Parametric model and identification of PMSM considering the influence of magnetic saturation. In: 13th International Conference on Optimization of Electrical and Electronic Equipment. 24–26 May 2012; Brasov, Romania; 2012. pp. 444-452. DOI: 10.1109/OPTIM.2012.6231768

[29] Senjyu T, Kinjo K, Urasaki N, Uezato K. Parameter measurement for PMSM using adaptive identification. In: Industrial Electronics, 2002. ISIE 2002. Proceedings of the 2002 IEEE International Symposium; 3. pp. 711-716

[30] Tolochko ОІ, Bozhko VV. Inductances identification of IPMSM by the recurrent method of least squares. Scientific papers of Donetsk National Technical University. Series: Electrical engineering and power engineering; 2012;12(2):234-238

[31] Trandafilov VN, Bozhko VV, Tolochko ОІ. Inertia identification of electric drive by an unnormalized gradient method. Scientific papers of Donetsk National Technical University. Series: Electrical engineering and power engineering. 2012;10(180):194-199

#### Chapter 9

Conference; 10 February 2006; Portoroz, Slovenia. pp. 282-286. DOI: 10.1109/EPEPEMC.2006.4778413

Applied Modern Control

compensation and parameter identification for model-based sensorless control of IPMSM. IEEE Transactions on Industry Applications. 2009;45(1):213-221. DOI: 10.1109/

[28] Mink F, Kubasiak N, Ritter B, Binder A. Parametric model and identification of PMSM considering the influence of magnetic saturation. In: 13th International Conference on Optimization of Electrical and

Electronic Equipment. 24–26 May 2012; Brasov, Romania; 2012. pp. 444-452. DOI: 10.1109/OPTIM.2012.6231768

International Symposium; 3. pp. 711-716

Inductances identification of IPMSM by the recurrent method of least squares. Scientific papers of Donetsk National Technical University. Series: Electrical engineering and power engineering;

[29] Senjyu T, Kinjo K, Urasaki N, Uezato K. Parameter measurement for PMSM using adaptive identification. In: Industrial Electronics, 2002. ISIE 2002.

Proceedings of the 2002 IEEE

[30] Tolochko ОІ, Bozhko VV.

[31] Trandafilov VN, Bozhko VV, Tolochko ОІ. Inertia identification of electric drive by an unnormalized gradient method. Scientific papers of Donetsk National Technical University. Series: Electrical engineering and power engineering. 2012;10(180):194-199

2012;12(2):234-238

164

TIA.2008.2009617

[27] Inoue Y, Yamada K, Morimoto S, Sanada M. Effectiveness of voltage error

## Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

Navin Khaneja

#### Abstract

In this chapter, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. In a two qubit system, the ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of unitary evolution, g, are decomposed into fast generators k (local Hamiltonians) and slow generators p (couplings) as a Cartan decomposition g ¼ p ⊕ k. Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems. The main contribution of the chapter is, we carry out a global analysis of time optimality.

Keywords: Kostant convexity, spin dynamics, Cartan decomposition, Cartan subalgebra, Weyl group, time optimal control

#### 1. Introduction

A rich class of model control problems arise when one considers dynamics of two coupled spin <sup>1</sup> 2 . The dynamics of two coupled spins, forms the basis for the field of quantum information processing and computing [1] and is fundamental in multidimensional NMR spectroscopy [2, 3]. Numerous experiments in NMR spectroscopy, involve synthesizing unitary transformations [4–6] that require interaction between the spins (evolution of the coupling Hamiltonian). These experiments involve transferring, coherence and polarization from one spin to another and involve evolution of interaction Hamiltonians [2]. Similarly, many protocols in quantum communication and information processing involve synthesizing entangled states starting from the separable states [1, 7, 8]. This again requires evolution of interaction Hamiltonians between the qubits.

A typical feature of many of these problems is that evolution of interaction Hamiltonians takes significantly longer than the time required to generate local unitary transformations (unitary transformations that effect individual spins only). In NMR spectroscopy [2, 3], local unitary transformations on spins are obtained by application of rf-pulses, whose strength may be orders of magnitude larger than the couplings between the spins. Given the Schróedinger equation for unitary evolution

$$\dot{U} = -i \left[ H\_c + \sum\_{j=1}^{n} u\_j H\_j \right] U, \quad U(0) = I,\tag{1}$$

obtained by evolution of the local Hamiltonians are called local unitary trans-

Hc ¼ ∑JαβIαSβ: (9)

5: (10)

5: (11)

� �: (12)

U, Uð Þ¼ 0 1 , (14)

10 0 0 01 0 0 0 0 �1 0 00 0 �1

10 00 0 �100 0 0 �1 0 00 01

The coupling Hamiltonian can be written as

DOI: http://dx.doi.org/10.5772/intechopen.80567

Written explicitly, some of these matrices take the form

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

Iz <sup>¼</sup> <sup>σ</sup>z⊗<sup>1</sup> <sup>¼</sup> <sup>1</sup>

IzSz <sup>¼</sup> <sup>σ</sup>z⊗σ<sup>z</sup> <sup>¼</sup> <sup>1</sup>

2

4

�i Iα; Sβ;IαS<sup>β</sup> � �,

The Lie algebra g ¼ suð Þ 4 has a direct sum decomposition g ¼ p ⊕ k, where

� �, <sup>p</sup> ¼ �i IαS<sup>β</sup>

Here k is a subalgebra of g made from local Hamiltonians and p nonlocal Hamil-

This decomposition of a real semi-simple Lie algebra g ¼ p ⊕ k satisfying (13) is

This special structure of Cartan decomposition arising in dynamics of two coupled spins in Eq. (1), motivates study of a broader class of time optimal control

ujð Þt Xj

where U ∈SU nð Þ, the special Unitary group (determinant 1, n � n matrices U

½ � k;k ⊂k, ½ � k; p ⊂p, ½ � p; p ⊂p: (13)

is conjugate transpose). Where Xj ∈ k ¼ so nð Þ, skew symmetric

less skew-Hermitian matrices. For the coupled two spins, the generators �iHc, � iHj ∈suð Þ 4 and the evolution operator U tð Þ in Eq. (1) is an element of

tonians. In Eq. (1), we have �iHj ∈k and �iHc ∈p, It is easy to verify that

Consider the following canonical problems. Given the evolution

!

j

k ¼ �i Iα; S<sup>β</sup>

called the Cartan decomposition of the Lie algebra g [24].

<sup>U</sup>\_ <sup>¼</sup> Xd <sup>þ</sup> <sup>∑</sup>

SUð Þ 4 , the 4 � 4, unitary matrices of determinant 1.

for α, β ∈ f g x; y; z , form the basis for the Lie algebra g ¼ suð Þ 4 , the 4 � 4, trace-

formations.

and

problems.

such that UU′ ¼ 1,

matrices and

167

0

The 15 operators,

where Hc represents a coupling Hamiltonian, and uj are controls that can be switched on and off. What is the minimum time required to synthesize any unitary transformation in the coupled spin system, when the control generators Hj are local Hamiltonians and are much stronger than the coupling between the spins (uj can be made large). Design of time optimal rf-pulse sequences is an important research subject in NMR spectroscopy and quantum information processing [4, 9–21], as minimizing the time to execute quantum operations can reduce relaxation losses, which are always present in an open quantum system [22, 23]. This problem has a special mathematical structure that helps to characterize all the time optimal trajectories [4]. The special mathematical structure manifested in the coupled two spin system, motivates a broader study of control systems with the same properties.

The Hamiltonian of a spin <sup>1</sup> <sup>2</sup> can be written in terms of the generators of rotations on a two dimensional space and these are the Pauli matrices �iσx, � iσy, � iσz, where,

$$
\sigma\_x = \frac{1}{2} \begin{bmatrix} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} \end{bmatrix}; \quad \sigma\_y = \frac{1}{2} \begin{bmatrix} \mathbf{0} & -i \\ i & \mathbf{0} \end{bmatrix}; \quad \sigma\_x = \frac{1}{2} \begin{bmatrix} \mathbf{0} & \mathbf{1} \\ \mathbf{1} & \mathbf{0} \end{bmatrix}. \tag{2}
$$

Note

$$\left[\sigma\_{\mathbf{x}}, \sigma\_{\mathbf{y}}\right] = i\sigma\_{\mathbf{z}}, \quad \left[\sigma\_{\mathbf{y}}, \sigma\_{\mathbf{z}}\right] = i\sigma\_{\mathbf{x}}, \quad \left[\sigma\_{\mathbf{z}}, \sigma\_{\mathbf{x}}\right] = i\sigma\_{\mathbf{y}}.\tag{3}$$

where ½ �¼ A; B AB � BA is the matrix commutator and

$$
\sigma\_x^2 = \sigma\_y^2 = \sigma\_x^2 = \frac{1}{4},
\tag{4}
$$

The Hamiltonian for a system of two coupled spins takes the general form

$$H\_0 = \sum a\_a \sigma\_a \otimes \mathbf{1}\_a + \sum b\_\beta \mathbf{1}\_\beta \otimes \sigma\_\beta + \sum f\_{a\beta} \ \sigma\_a \otimes \sigma\_\beta,\tag{5}$$

where α, β ∈f g x; y; z . The Hamiltonians σα⊗1 and 1 ⊗σβ are termed local Hamiltonians and operate on one of the spins. The Hamiltonian

$$H\_{\mathfrak{c}} = \sum f\_{a\beta} \cdot \sigma\_a \otimes \sigma\_\beta,\tag{6}$$

is the coupling or interaction Hamiltonian and operates on both the spins. The following notation is therefore common place in the NMR literature.

$$I\_a = \sigma\_a \otimes \mathbf{1}; \quad \mathbb{S}\_\beta = \mathbf{1} \otimes \sigma\_\beta. \tag{7}$$

The operators I<sup>α</sup> and S<sup>β</sup> commute and therefore exp �i∑αaαI<sup>α</sup> þ ∑βbβS<sup>β</sup> � � <sup>¼</sup>

$$\exp\left(-i\sum\_{a}a\_{a}I\_{a}\right)\exp\left(-i\sum\_{\beta}b\_{\beta}\mathbb{S}\_{\beta}\right) = \left(\exp\left(-i\sum\_{a}a\_{a}\sigma\_{a}\right)\otimes\mathbf{1}\right)\left(\mathbf{1}\otimes\exp\left(-i\sum\_{\beta}b\_{\beta}\sigma\_{\beta}\right). \tag{8}$$

The unitary transformations of the kind

$$\exp\left(-i\sum\_{a}a\_{a}\sigma\_{a}\right)\otimes\exp\left(-i\sum\_{\beta}b\_{\beta}\sigma\_{\beta}\right),$$

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics DOI: http://dx.doi.org/10.5772/intechopen.80567

obtained by evolution of the local Hamiltonians are called local unitary transformations.

The coupling Hamiltonian can be written as

$$H\_{\mathfrak{c}} = \sum I\_{a\beta} I\_{a} \mathbb{S}\_{\beta} \,. \tag{9}$$

Written explicitly, some of these matrices take the form

$$I\_x = \sigma\_x \otimes \mathbf{1} \quad = \frac{1}{2} \begin{bmatrix} \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & -\mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & -\mathbf{1} \end{bmatrix} . \tag{10}$$

and

<sup>U</sup>\_ ¼ �i Hc <sup>þ</sup> <sup>∑</sup>

The Hamiltonian of a spin <sup>1</sup>

Applied Modern Control

<sup>σ</sup><sup>z</sup> <sup>¼</sup> <sup>1</sup> 2

σx; σ<sup>y</sup>

exp �i ∑ β bβS<sup>β</sup> !

The unitary transformations of the kind

exp �i ∑ α aασα � �

Note

exp �i ∑ α aαI<sup>α</sup> � �

166

1 0 0 �1 � �

� � <sup>¼</sup> <sup>i</sup>σz, <sup>σ</sup>y; <sup>σ</sup><sup>z</sup>

where ½ �¼ A; B AB � BA is the matrix commutator and

σ2 <sup>x</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup>

Hamiltonians and operate on one of the spins. The Hamiltonian

n j¼1 ujHj

where Hc represents a coupling Hamiltonian, and uj are controls that can be switched on and off. What is the minimum time required to synthesize any unitary transformation in the coupled spin system, when the control generators Hj are local Hamiltonians and are much stronger than the coupling between the spins (uj can be made large). Design of time optimal rf-pulse sequences is an important research subject in NMR spectroscopy and quantum information processing [4, 9–21], as minimizing the time to execute quantum operations can reduce relaxation losses, which are always present in an open quantum system [22, 23]. This problem has a special mathematical structure that helps to characterize all the time optimal trajectories [4]. The special mathematical structure manifested in the coupled two spin system, motivates a broader study of control systems with the same properties.

on a two dimensional space and these are the Pauli matrices �iσx, � iσy, � iσz, where,

<sup>y</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup>

The Hamiltonian for a system of two coupled spins takes the general form

where α, β ∈f g x; y; z . The Hamiltonians σα⊗1 and 1 ⊗σβ are termed local

is the coupling or interaction Hamiltonian and operates on both the spins. The following notation is therefore common place in the NMR literature.

The operators I<sup>α</sup> and S<sup>β</sup> commute and therefore exp �i∑αaαI<sup>α</sup> þ ∑βbβS<sup>β</sup>

¼ exp �i ∑

α aασα � �

⊗ exp �i ∑

β bβσβ !

� �

<sup>z</sup> <sup>¼</sup> <sup>1</sup>

H<sup>0</sup> ¼ ∑ aασα⊗1 þ ∑ b<sup>β</sup> 1 ⊗σβ þ ∑ Jαβ σα⊗σβ, (5)

0 �i i 0 � �

; <sup>σ</sup><sup>y</sup> <sup>¼</sup> <sup>1</sup> 2 U, Uð Þ¼ 0 I, (1)

<sup>2</sup> can be written in terms of the generators of rotations

� � <sup>¼</sup> <sup>i</sup>σx, ½ �¼ <sup>σ</sup>z; <sup>σ</sup><sup>x</sup> <sup>i</sup>σy, (3)

Hc ¼ ∑ Jαβ σα⊗σβ, (6)

I<sup>α</sup> ¼ σα⊗1; S<sup>β</sup> ¼ 1 ⊗σβ: (7)

⊗1

,

0 1 1 0 � �

<sup>4</sup> , (4)

� �

1 ⊗ exp �i ∑

¼

(8)

,

β bβσβ !

: (2)

; <sup>σ</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup> 2

" #

$$I\_x \mathbb{S}\_x = \sigma\_x \mathbb{S} \mathbb{G} \sigma\_x = \frac{1}{4} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} . \tag{11}$$

The 15 operators,

$$-i\left\{I\_a, \mathbf{S}\_{\beta}, I\_a \mathbf{S}\_{\beta}\right\},$$

for α, β ∈ f g x; y; z , form the basis for the Lie algebra g ¼ suð Þ 4 , the 4 � 4, traceless skew-Hermitian matrices. For the coupled two spins, the generators �iHc, � iHj ∈suð Þ 4 and the evolution operator U tð Þ in Eq. (1) is an element of SUð Þ 4 , the 4 � 4, unitary matrices of determinant 1.

The Lie algebra g ¼ suð Þ 4 has a direct sum decomposition g ¼ p ⊕ k, where

$$\mathfrak{k} = -i \{ I\_a, \mathbb{S}\_{\beta} \}, \quad \mathfrak{p} = -i \{ I\_a \mathbb{S}\_{\beta} \}. \tag{12}$$

Here k is a subalgebra of g made from local Hamiltonians and p nonlocal Hamiltonians. In Eq. (1), we have �iHj ∈k and �iHc ∈p, It is easy to verify that

$$[\mathfrak{k}, \mathfrak{k}] \subset \mathfrak{k}, \qquad [\mathfrak{k}, \mathfrak{p}] \subset \mathfrak{p}, \quad [\mathfrak{p}, \mathfrak{p}] \subset \mathfrak{p}. \tag{13}$$

This decomposition of a real semi-simple Lie algebra g ¼ p ⊕ k satisfying (13) is called the Cartan decomposition of the Lie algebra g [24].

This special structure of Cartan decomposition arising in dynamics of two coupled spins in Eq. (1), motivates study of a broader class of time optimal control problems.

Consider the following canonical problems. Given the evolution

$$\dot{U} = \left(X\_d + \sum\_j u\_j(t)X\_j\right)U, \quad U(0) = \mathbf{1} \quad , \tag{14}$$

where U ∈SU nð Þ, the special Unitary group (determinant 1, n � n matrices U such that UU′ ¼ 1, 0 is conjugate transpose). Where Xj ∈ k ¼ so nð Þ, skew symmetric matrices and

$$X\_d = -i \begin{bmatrix} \lambda\_1 & \mathbf{0} & \dots & \mathbf{0} \\ \mathbf{0} & \lambda\_2 & \dots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \dots & \lambda\_n \end{bmatrix}, \quad \sum \lambda\_i = \mathbf{0}.$$

We assume Xj � � LA, the Lie algebra (Xj and its matrix commutators) generated by generators Xj is all of so nð Þ. We want to find the minimum time to steer this system between points of interest, assuming no bounds on our controls ujð Þt . Here again we have a Cartan decomposition on generators. Given g ¼ su nð Þ, traceless skew-Hermitian matrices, generators of SU nð Þ, we have g ¼ p ⊕ k, where p ¼ �iA, where A is traceless symmetric and k ¼ so nð Þ. As before, Xd ∈p and Xj ∈k. We want to find time optimal ways to steer this system. We call this SU nð Þ SO nð Þ problem. For <sup>n</sup> <sup>¼</sup> 4, this system models the dynamics of two coupled nuclear spins in NMR spectroscopy.

In general, U is in a compact Lie group G (such as SU nð Þ), with Xd, Xj in its real semisimple (no abelian ideals) Lie algebra g and

$$\dot{U} = \left(X\_d + \sum\_j u\_j(t)X\_j\right) U, \quad U(0) = \mathbf{1} \; . \tag{15}$$

exp ð Þ a<sup>1</sup> is synthesized by evolution of Hamiltonian Xd. Time optimal strategy

and time-optimality is characterized by synthesis of commuting Hamiltonians

The chapter is organized as follows. In Section 2, we study the SU nð Þ

Given Lie algebra g, we use killing form h i x; y ¼ tr adxady

on g. When g ¼ su nð Þ, we also use the inner product h i x; y ¼ tr x′

2. Time optimal control for SU nð Þ=SO nð Þ problem

λ<sup>1</sup> 0 … 0 0 λ<sup>2</sup> … 0 ⋮ ⋮⋱⋮ 0 0 … λ<sup>n</sup>

permutations. This is called Schur convexity.

3 7

<sup>5</sup>, where <sup>∑</sup><sup>i</sup>

U ∈ SU nð Þ, U ¼ Θ<sup>1</sup> exp ð Þ Ω Θ<sup>2</sup> where Θ1, Θ<sup>2</sup> ∈ SO nð Þ and

Let U tð Þ∈SU nð Þ be a solution to the differential Eq. (14)

tonians is derived using convexity ideas [4, 28]. The remaining chapter develops

where K1, K2, W<sup>k</sup> take negligible time to synthesize using unbounded controls ui

Remark 1. Birkhoff's convexity states, a real n � n matrix A is doubly stochastic (∑<sup>i</sup> Aij ¼ ∑<sup>j</sup> Aij ¼ 1, for Aij ≥0) if it can be written as convex hull of permutation matrices Pi (only one 1 and everything else zero in every row and column). Given

diagð Þ <sup>X</sup> is a column vector containing diagonal entries of <sup>X</sup> and Bij <sup>¼</sup> <sup>Θ</sup>ij � �<sup>2</sup> and hence ∑<sup>i</sup> Bij ¼ ∑<sup>j</sup> Bij ¼ 1, making B a doubly stochastic matrix, which can be

diagonal of a symmetric matrix ΘXΘ<sup>T</sup>, lies in convex hull of its eigenvalues and its

Remark 2. G ¼ SU nð Þ has a closed subgroup K ¼ SO nð Þ and a Cartan decomposition of its Lie algerbra g ¼ su nð Þ as g ¼ p ⊕ k, for k ¼ so nð Þ and p ¼ �iA where A is

> λ<sup>1</sup> … 0 0 ⋱ 0 0 0 λ<sup>n</sup>

3 7 5,

written as convex sum of permutations. Therefore B diagð Þ¼ X ∑<sup>i</sup>

traceless symmetric and a is maximal abelian subalgebra of p, such that

Ω ¼ �i

2 6 4

Remark 3. We now give a proof of the reachable set (16), for the SU nð Þ

<sup>k</sup> . This characterization of time optimality, involving commuting Hamil-

k � � <sup>K</sup><sup>2</sup> <sup>¼</sup> <sup>K</sup><sup>1</sup>

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

<sup>k</sup> where WkXdW�<sup>1</sup>

<sup>K</sup> problem. The main contribution of the chapter is,

, we have diag <sup>Θ</sup>XΘ<sup>T</sup> � � <sup>¼</sup> B diag Xð Þ where

λ<sup>i</sup> ¼ 0. KAK decomposition in Eq. (17) states for

<sup>W</sup><sup>k</sup> exp ð Þ tkXd <sup>W</sup>�<sup>1</sup>

Y k

<sup>k</sup> all commute.

SO nð Þ problem. In

αiPi diagð Þ X , i.e.

SO nð Þ problem.

� � as an inner product

y � �. We call this

<sup>k</sup> K2:

suggests evolving Xd and its conjugates WkXdW�<sup>1</sup>

exp tkWkXdW�<sup>1</sup>

Written as evolution

Y k

DOI: http://dx.doi.org/10.5772/intechopen.80567

Section 3, we study the general <sup>G</sup>

standard inner product.

Θ ∈SO nð Þ and X ¼

λ<sup>1</sup> … 0 0 ⋱ 0 0 0 λ<sup>n</sup>

λ<sup>i</sup> ¼ 0.

a ¼ �i

2 6 4

where ∑<sup>i</sup>

169

we carry out a global analysis of time optimality.

G ¼ K<sup>1</sup>

WkXdW�<sup>1</sup>

these notions.

Given the Cartan decomposition g ¼ p ⊕ k, where Xd ∈ p, Xj � � LA ¼ k and K ¼ exp ð Þk (product of exponentials of k) a closed subgroup of G, We want to find the minimum time to steer this system between points of interest, assuming no bounds on our controls ujð Þt . Since Xj � � LA ¼ k, any rotation (evolution) in subgroup K can be synthesized with evolution of Xj [25, 26]. Since there are no bounds on ujð Þ<sup>t</sup> , this can be done in arbitrarily small time [4]. We call this <sup>G</sup> <sup>K</sup> problem.

The special structure of this problem helps in complete description of the reachable set [27]. The elements of the reachable set at time T, takes the form U Tð Þ∈

$$\mathcal{S} = K\_1 \exp\left(T \sum\_k a\_k \,\,\mathcal{W}\_k \mathcal{X}\_d \mathcal{W}\_k^{-1}\right) \mathcal{K}\_2 \tag{16}$$

where <sup>K</sup>1, K2, <sup>W</sup><sup>k</sup> <sup>∈</sup> exp ð Þ<sup>k</sup> , and <sup>W</sup>kXdW�<sup>1</sup> <sup>k</sup> all commute, and αk>0, ∑α<sup>k</sup> ¼ 1. This reachable set is formed from evolution of K1, K<sup>2</sup> and commuting Hamiltonians WkXdW�<sup>1</sup> <sup>k</sup> . Unbounded control suggests that K1, K2, W<sup>k</sup> can be synthesized in negligible time.

This reachable set can be understood as follows. The Cartan decomposition of the Lie algebra g, in Eq. (13) leads to a decomposition of the Lie group G [24]. Inside p is contained the largest abelian subalgebra, denoted as a. Any X ∈p is AdK conjugate to an element of <sup>a</sup>, i.e. <sup>X</sup> <sup>¼</sup> Ka1K�<sup>1</sup> for some <sup>a</sup><sup>1</sup> <sup>∈</sup>a.

Then, any arbitrary element of the group G can be written as

$$G = K\_0 \exp\left(X\right) = K\_0 \exp\left(Ad\_K(a\_1)\right) = K\_1 \exp\left(a\_1\right) K\_2 \tag{17}$$

for some X ∈ p where Ki ∈ K and a<sup>1</sup> ∈a. The first equation is a fact about geodesics in G=K space [24], where K ¼ exp ð Þk is a closed subgroup of G. Eq. (17) is called the KAK decomposition [24].

The results in this chapter suggest that K<sup>1</sup> and K<sup>2</sup> can be synthesized by unbounded controls Xi in negligible time. The time consuming part of the evolution Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics DOI: http://dx.doi.org/10.5772/intechopen.80567

exp ð Þ a<sup>1</sup> is synthesized by evolution of Hamiltonian Xd. Time optimal strategy suggests evolving Xd and its conjugates WkXdW�<sup>1</sup> <sup>k</sup> where WkXdW�<sup>1</sup> <sup>k</sup> all commute.

Written as evolution

Xd ¼ �i

We assume Xj

Applied Modern Control

copy.

U Tð Þ∈

WkXdW�<sup>1</sup>

168

negligible time.

� �

to find time optimal ways to steer this system. We call this SU nð Þ

<sup>U</sup>\_ <sup>¼</sup> Xd <sup>þ</sup> <sup>∑</sup>

j

� �

The special structure of this problem helps in complete description of the reachable set [27]. The elements of the reachable set at time T, takes the form

k

This reachable set is formed from evolution of K1, K<sup>2</sup> and commuting Hamiltonians

<sup>k</sup> . Unbounded control suggests that K1, K2, W<sup>k</sup> can be synthesized in

This reachable set can be understood as follows. The Cartan decomposition of the Lie algebra g, in Eq. (13) leads to a decomposition of the Lie group G [24]. Inside

for some X ∈ p where Ki ∈ K and a<sup>1</sup> ∈a. The first equation is a fact about geodesics in G=K space [24], where K ¼ exp ð Þk is a closed subgroup of G. Eq. (17) is

The results in this chapter suggest that K<sup>1</sup> and K<sup>2</sup> can be synthesized by unbounded controls Xi in negligible time. The time consuming part of the evolution

p is contained the largest abelian subalgebra, denoted as a. Any X ∈p is AdK

α<sup>k</sup> WkXdW�<sup>1</sup>

G ¼ K<sup>0</sup> exp ð Þ¼ X K<sup>0</sup> exp ð Þ¼ AdKð Þ a<sup>1</sup> K<sup>1</sup> exp ð Þ a<sup>1</sup> K2, (17)

� �

k

Given the Cartan decomposition g ¼ p ⊕ k, where Xd ∈ p, Xj

on ujð Þ<sup>t</sup> , this can be done in arbitrarily small time [4]. We call this <sup>G</sup>

S ¼ K<sup>1</sup> exp T ∑

conjugate to an element of <sup>a</sup>, i.e. <sup>X</sup> <sup>¼</sup> Ka1K�<sup>1</sup> for some <sup>a</sup><sup>1</sup> <sup>∈</sup>a. Then, any arbitrary element of the group G can be written as

where <sup>K</sup>1, K2, <sup>W</sup><sup>k</sup> <sup>∈</sup> exp ð Þ<sup>k</sup> , and <sup>W</sup>kXdW�<sup>1</sup>

called the KAK decomposition [24].

!

semisimple (no abelian ideals) Lie algebra g and

bounds on our controls ujð Þt . Since Xj

λ<sup>1</sup> 0 … 0 0 λ<sup>2</sup> … 0 ⋮ ⋮⋱⋮ 0 0 … λ<sup>n</sup>

by generators Xj is all of so nð Þ. We want to find the minimum time to steer this system between points of interest, assuming no bounds on our controls ujð Þt . Here again we have a Cartan decomposition on generators. Given g ¼ su nð Þ, traceless skew-Hermitian matrices, generators of SU nð Þ, we have g ¼ p ⊕ k, where p ¼ �iA, where A is traceless symmetric and k ¼ so nð Þ. As before, Xd ∈p and Xj ∈k. We want

this system models the dynamics of two coupled nuclear spins in NMR spectros-

ujð Þt Xj

K ¼ exp ð Þk (product of exponentials of k) a closed subgroup of G, We want to find the minimum time to steer this system between points of interest, assuming no

group K can be synthesized with evolution of Xj [25, 26]. Since there are no bounds

In general, U is in a compact Lie group G (such as SU nð Þ), with Xd, Xj in its real

, ∑λ<sup>i</sup> ¼ 0:

SO nð Þ problem. For <sup>n</sup> <sup>¼</sup> 4,

LA ¼ k and

<sup>K</sup> problem.

K2, (16)

<sup>k</sup> all commute, and αk>0, ∑α<sup>k</sup> ¼ 1.

U, Uð Þ¼ 0 1 : (15)

� �

LA ¼ k, any rotation (evolution) in sub-

LA, the Lie algebra (Xj and its matrix commutators) generated

$$G = K\_1 \prod\_k \exp\left(t\_k \mathcal{W}\_k X\_d \mathcal{W}\_k^{-1}\right) \\ \ K\_2 = K\_1 \prod\_k \mathcal{W}\_k \exp\left(t\_k X\_d\right) \mathcal{W}\_k^{-1} \ K\_2.$$

where K1, K2, W<sup>k</sup> take negligible time to synthesize using unbounded controls ui and time-optimality is characterized by synthesis of commuting Hamiltonians WkXdW�<sup>1</sup> <sup>k</sup> . This characterization of time optimality, involving commuting Hamiltonians is derived using convexity ideas [4, 28]. The remaining chapter develops these notions.

The chapter is organized as follows. In Section 2, we study the SU nð Þ SO nð Þ problem. In Section 3, we study the general <sup>G</sup> <sup>K</sup> problem. The main contribution of the chapter is, we carry out a global analysis of time optimality.

Given Lie algebra g, we use killing form h i x; y ¼ tr adxady � � as an inner product on g. When g ¼ su nð Þ, we also use the inner product h i x; y ¼ tr x′ y � �. We call this standard inner product.

### 2. Time optimal control for SU nð Þ=SO nð Þ problem

Remark 1. Birkhoff's convexity states, a real n � n matrix A is doubly stochastic (∑<sup>i</sup> Aij ¼ ∑<sup>j</sup> Aij ¼ 1, for Aij ≥0) if it can be written as convex hull of permutation matrices Pi (only one 1 and everything else zero in every row and column). Given

$$\Theta \in \text{SO}(n) \text{ and } X = \begin{bmatrix} \lambda\_1 & 0 & \dots & 0 \\ 0 & \lambda\_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \lambda\_n \end{bmatrix} \text{, we have } \operatorname{diag} \left( \Theta X \Theta^T \right) = B \text{ } \operatorname{diag}(X) \text{ where } \operatorname{diag}(X) = \operatorname{diag}(X) \text{ and } \operatorname{diag}(X) = \operatorname{diag}(X) \text{ for some } n \in \mathbb{N}$$

diagð Þ <sup>X</sup> is a column vector containing diagonal entries of <sup>X</sup> and Bij <sup>¼</sup> <sup>Θ</sup>ij � �<sup>2</sup> and hence ∑<sup>i</sup> Bij ¼ ∑<sup>j</sup> Bij ¼ 1, making B a doubly stochastic matrix, which can be written as convex sum of permutations. Therefore B diagð Þ¼ X ∑<sup>i</sup> αiPi diagð Þ X , i.e. diagonal of a symmetric matrix ΘXΘ<sup>T</sup>, lies in convex hull of its eigenvalues and its permutations. This is called Schur convexity.

Remark 2. G ¼ SU nð Þ has a closed subgroup K ¼ SO nð Þ and a Cartan decomposition of its Lie algerbra g ¼ su nð Þ as g ¼ p ⊕ k, for k ¼ so nð Þ and p ¼ �iA where A is traceless symmetric and a is maximal abelian subalgebra of p, such that

a ¼ �i λ<sup>1</sup> … 0 0 ⋱ 0 0 0 λ<sup>n</sup> 2 6 4 3 7 <sup>5</sup>, where <sup>∑</sup><sup>i</sup> λ<sup>i</sup> ¼ 0. KAK decomposition in Eq. (17) states for U ∈ SU nð Þ, U ¼ Θ<sup>1</sup> exp ð Þ Ω Θ<sup>2</sup> where Θ1, Θ<sup>2</sup> ∈ SO nð Þ and

$$
\boldsymbol{\Omega} = -i \begin{bmatrix} \lambda\_1 & \dots & \mathbf{0} \\ \mathbf{0} & \ddots & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \lambda\_n \end{bmatrix},
$$

where ∑<sup>i</sup> λ<sup>i</sup> ¼ 0.

Remark 3. We now give a proof of the reachable set (16), for the SU nð Þ SO nð Þ problem. Let U tð Þ∈SU nð Þ be a solution to the differential Eq. (14)

$$\dot{U} = \left(X\_d + \sum\_i u\_i X\_i\right) U, \quad U(\mathbf{0}) = I.$$

To understand the reachable set of this system we make a change of coordinates P tðÞ¼ K′ ð Þ<sup>t</sup> U tð Þ, where, <sup>K</sup>\_ <sup>¼</sup> <sup>∑</sup><sup>i</sup> uiXi � �K. Then

$$
\dot{P}(t) = Ad\_{K^\circ(t)}(X\_d)P(t), \quad Ad\_K(X\_d) = K X K^{-1}.
$$

If we understand reachable set of P tð Þ, then the reachable set in Eq. (14) is easily derived.

Theorem 1. Let P tð Þ∈SU nð Þ be a solution to the differential equation

$$
\dot{P} = \mathcal{A}d\_{K(t)}(\mathbf{X}\_d)P,
$$

and K tð Þ∈ SO nð Þ and Xd ¼ �i λ<sup>1</sup> 0 … 0 0 λ<sup>2</sup> … 0 ⋮ ⋮⋱⋮ 0 0 λ<sup>n</sup> 2 6 6 6 4 3 7 7 7 5 . The elements of the reachable

set at time T, take the form K<sup>1</sup> exp ð Þ �iμT K2, where K1, K<sup>2</sup> ∈SO nð Þ and μ≺λ (μ lies in convex hull of λ and its permutations), where λ ¼ ð Þ λ1; …; λ<sup>n</sup> ′ .

Proof. As a first step, discretize the evolution of P tð Þ, as piecewise constant evolution, over steps of size τ. The total evolution is then

$$P\_n = \prod\_i \exp\left(Ad\_{k\_i}(X\_d)\tau\right),\tag{18}$$

AdKð Þ¼ Xd Ω<sup>1</sup> þ K1aK′

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

AdKð Þ¼ Xd <sup>Ω</sup>′

ΩK.

<sup>1</sup> ¼ K′

Note iR⊥a and onto a<sup>⊥</sup>, by appropriate choice of Ω2. Given AdKð Þ Xd <sup>∈</sup>p, we decompose it as

AdKð Þ¼ Xd P AdKð Þ Xd

with P denoting the projection onto a (a ¼ �i

Consider the case, when A is degenerate. Let,

A ¼

AdKð Þ¼ Xd P AdKð Þ Xd

w.r.t to standard inner product and AdKð Þ Xd

gives <sup>Ω</sup>2. Let <sup>a</sup> <sup>¼</sup> P AdKð Þ Xd

order in Δ and

WLOG, we arrange

Consider the decomposition

orthogonal complement.

171

, for Ω<sup>2</sup> ∈so nð Þ.

kl ¼ exp i ϕ<sup>k</sup> � ϕ<sup>l</sup> f g ð Þ ð Þ Ω<sup>2</sup> kl ¼ cos ϕ<sup>k</sup> � ϕ<sup>l</sup> ð Þð Þ Ω<sup>2</sup> kl

<sup>1</sup>ð Þ� K<sup>1</sup> gives

Multiplying both sides with K′

DOI: http://dx.doi.org/10.5772/intechopen.80567

<sup>1</sup>K and Ω′

where, K ¼ K′

AΩ2A† � �

We evaluate AΩ2A†

<sup>1</sup> þ K1AΩ2A′

<sup>1</sup> þ a þ AΩ2A′


<sup>⊥</sup> <sup>¼</sup> <sup>Ω</sup>′

2 6 4

<sup>1</sup> <sup>¼</sup> AdKð Þ Xd

0 �Is�<sup>s</sup>

� � <sup>þ</sup> AdKð Þ Xd

λ<sup>1</sup> … 0 0 ⋱ 0 0 0 λ<sup>n</sup>

such that S is skew symmetric and R is traceless symmetric matrix with iR∈ p.

� � <sup>þ</sup> AdKð Þ Xd

Eq. (24), <sup>ϕ</sup><sup>k</sup> � <sup>ϕ</sup><sup>l</sup> 6¼ <sup>0</sup>, <sup>π</sup>, we can solve for ð Þ <sup>Ω</sup><sup>2</sup> kl such that iR <sup>¼</sup> AdKð Þ Xd

With this choice of Ω1, Ω<sup>2</sup> and a, P tð Þ þ Δ and Q tð Þ þ Δ are matched to first

P tð Þ� <sup>þ</sup> <sup>Δ</sup> Q tð Þ¼ <sup>þ</sup> <sup>Δ</sup> <sup>o</sup> <sup>Δ</sup><sup>2</sup> � �:

A<sup>1</sup> 0 … 0 0 A<sup>2</sup> … 0 ⋮ ⋮⋱⋮ 0 0 An

where Ak is nk fold degenerate (modulo sign) described by nk � nk block.

Ak <sup>¼</sup> exp <sup>i</sup>ϕ<sup>k</sup> ð Þ Ir�<sup>r</sup> <sup>0</sup>

where P denotes projection onto nk � nk blocks in Eq. (25) and AdKð Þ Xd

� � and choose <sup>Ω</sup>′

K′

<sup>1</sup>: (22)

: (23)

: (24)

λ<sup>i</sup> ¼ 0.)

<sup>⊥</sup>. This

<sup>⊥</sup>, the

þisin ϕ<sup>k</sup> � ϕ<sup>l</sup> ð Þð Þ Ω<sup>2</sup> kl |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Rkl

<sup>1</sup> þ a þ AΩ2A′

3 7

<sup>⊥</sup> to the orthogonal component. In

,

<sup>5</sup>, where <sup>∑</sup><sup>i</sup>

<sup>⊥</sup> � <sup>A</sup>Ω2A† ¼ �S∈k.

5, (25)

� �: (26)

⊥,

For t ∈½ � ð Þ n � 1 τ; nτ , choose small step Δ, such that t þ Δ<nτ, then P tð Þ¼ þ Δ exp ð Þ AdKð Þ Xd Δ P tð Þ.

$$\text{By KAK}, P(t) = K\_1 \underbrace{\begin{bmatrix} \exp\left(i\phi\_1\right) & 0 & 0 & 0\\ 0 & \exp\left(i\phi\_2\right) & 0 & 0\\ 0 & 0 & \ddots & 0\\ \hline 0 & 0 & 0 & \exp\left(i\phi\_n\right) \end{bmatrix} K\_2,$$

$$\overbrace{A}$$

where K1, K<sup>2</sup> ∈SO nð Þ. To begin with, assume eigenvalues ϕ<sup>j</sup> � ϕ<sup>k</sup> 6¼ nπ, where n is an integer. When we take a small step of size Δ, P tð Þ changes to P tð Þ þ Δ as K1, K2, A change to

$$K\_1(t+\Delta) = \exp\left(\Omega\_1 \Delta\right) K\_1, \quad K\_2(t+\Delta) = \exp\left(\Omega\_2 \Delta\right) K\_2, \quad A(t+\Delta) = \exp\left(a\Delta\right) A,$$

where, Ω1, Ω<sup>2</sup> ∈ k and a∈a. Let Q tð Þ¼ þ Δ K1ð Þ t þ Δ A tð Þ þ Δ K2ð Þ t þ Δ , which can be written as

$$Q(t+\Delta) = \exp\left(\Omega\_1 \Delta\right) K\_1 \exp\left(a\Delta\right) A \exp\left(\Omega\_2 \Delta\right) K\_2. \tag{19}$$

$$Q(t+\Delta) = \exp\left(\Omega\_1 \Delta\right) \exp\left(K\_1 a K\_1^\prime \Delta\right) \exp\left(K\_1 A \Omega\_2 A^\prime K\_1^\prime \Delta\right) P(t). \tag{20}$$

Observe

$$P(t+\Delta) = \exp\left(Ad\_K(X\_d)\Delta\right)P(t). \tag{21}$$

We equate P tð Þ þ Δ and Q tð Þ þ Δ to first order in Δ. This gives,

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics DOI: http://dx.doi.org/10.5772/intechopen.80567

$$Ad\_K(\mathbf{X}\_d) = \mathbf{Q}\_1 + K\_1 a K\_1^\prime + K\_1 A \mathbf{Q}\_2 A^\prime K\_1^\prime . \tag{22}$$

Multiplying both sides with K′ <sup>1</sup>ð Þ� K<sup>1</sup> gives

$$Ad\_{\overline{K}}(X\_d) = \Omega\_1^{'} + a + A\Omega\_2 A^{'}.\tag{23}$$

where, K ¼ K′ <sup>1</sup>K and Ω′ <sup>1</sup> ¼ K′ ΩK. We evaluate AΩ2A† , for Ω<sup>2</sup> ∈so nð Þ.

<sup>U</sup>\_ <sup>¼</sup> Xd <sup>þ</sup> <sup>∑</sup>

ð Þ<sup>t</sup> U tð Þ, where, <sup>K</sup>\_ <sup>¼</sup> <sup>∑</sup><sup>i</sup>

P t \_ðÞ¼ AdK′

and K tð Þ∈ SO nð Þ and Xd ¼ �i

P tð Þ¼ þ Δ exp ð Þ AdKð Þ Xd Δ P tð Þ.

Q tð Þ¼ þ Δ exp ð Þ Ω1Δ exp K1aK′

By KAK, P tðÞ¼ K<sup>1</sup>

change to

can be written as

Observe

170

P tðÞ¼ K′

Applied Modern Control

derived.

i uiXi � �

Theorem 1. Let P tð Þ∈SU nð Þ be a solution to the differential equation

in convex hull of λ and its permutations), where λ ¼ ð Þ λ1; …; λ<sup>n</sup> ′

Pn <sup>¼</sup> <sup>Y</sup> i

For t ∈½ � ð Þ n � 1 τ; nτ , choose small step Δ, such that t þ Δ<nτ, then

evolution, over steps of size τ. The total evolution is then

uiXi � �K. Then

To understand the reachable set of this system we make a change of coordinates

If we understand reachable set of P tð Þ, then the reachable set in Eq. (14) is easily

<sup>P</sup>\_ <sup>¼</sup> AdK tð Þð Þ Xd P,

λ<sup>1</sup> 0 … 0 0 λ<sup>2</sup> … 0 ⋮ ⋮⋱⋮ 0 0 λ<sup>n</sup>

set at time T, take the form K<sup>1</sup> exp ð Þ �iμT K2, where K1, K<sup>2</sup> ∈SO nð Þ and μ≺λ (μ lies

Proof. As a first step, discretize the evolution of P tð Þ, as piecewise constant

exp iϕ<sup>1</sup> ð Þ 000 0 exp iϕ<sup>2</sup> ð Þ 0 0 0 0 ⋱ 0 0 0 0 exp iϕ<sup>n</sup> ð Þ


where K1, K<sup>2</sup> ∈SO nð Þ. To begin with, assume eigenvalues ϕ<sup>j</sup> � ϕ<sup>k</sup> 6¼ nπ, where n is an integer. When we take a small step of size Δ, P tð Þ changes to P tð Þ þ Δ as K1, K2, A

K1ð Þ¼ t þ Δ exp ð Þ Ω1Δ K1, K2ð Þ¼ t þ Δ exp ð Þ Ω2Δ K2, Atð Þ¼ þ Δ exp ð Þ aΔ A,

We equate P tð Þ þ Δ and Q tð Þ þ Δ to first order in Δ. This gives,

where, Ω1, Ω<sup>2</sup> ∈ k and a∈a. Let Q tð Þ¼ þ Δ K1ð Þ t þ Δ A tð Þ þ Δ K2ð Þ t þ Δ , which

Q tð Þ¼ þ Δ exp ð Þ Ω1Δ K<sup>1</sup> exp ð Þ aΔ A exp ð Þ Ω2Δ K2: (19)

P tð Þ¼ þ Δ exp ð Þ AdKð Þ Xd Δ P tð Þ: (21)

K′

<sup>1</sup><sup>Δ</sup> � �P tð Þ: (20)

<sup>1</sup><sup>Δ</sup> � � exp <sup>K</sup>1AΩ2A′

exp ð Þ Adki

ð Þ<sup>t</sup> ð Þ Xd P tð Þ, AdKð Þ¼ Xd KXK�<sup>1</sup>

U, Uð Þ¼ 0 I:

:

. The elements of the reachable

.

ð Þ Xd τ , (18)

K2,

$$\left\{{A\Omega\_2 A^\dagger}^\dagger\right\}\_{kl} = \exp\left\{i(\phi\_k - \phi\_l)\right\} (\Omega\_2)\_{kl} = \underbrace{\cos\left(\phi\_k - \phi\_l\right)(\Omega\_2)\_{kl}}\_{S\_{kl}} + \underbrace{i\sin\left(\phi\_k - \phi\_l\right)(\Omega\_2)\_{kl}}\_{R\_{kl}}.\tag{24}$$

such that S is skew symmetric and R is traceless symmetric matrix with iR∈ p. Note iR⊥a and onto a<sup>⊥</sup>, by appropriate choice of Ω2.

Given AdKð Þ Xd <sup>∈</sup>p, we decompose it as

$$Ad\_{\overline{K}}(\mathbf{X}\_d) = P\left(Ad\_{\overline{K}}(\mathbf{X}\_d)\right) + Ad\_{\overline{K}}(\mathbf{X}\_d)^\perp = \boldsymbol{\Omega}\_1^\prime + \boldsymbol{a} + A\boldsymbol{\Omega}\_2\boldsymbol{A}^\prime,$$

with P denoting the projection onto a (a ¼ �i λ<sup>1</sup> … 0 0 ⋱ 0 0 0 λ<sup>n</sup> 2 6 4 3 7 <sup>5</sup>, where <sup>∑</sup><sup>i</sup> λ<sup>i</sup> ¼ 0.)

w.r.t to standard inner product and AdKð Þ Xd <sup>⊥</sup> to the orthogonal component. In Eq. (24), <sup>ϕ</sup><sup>k</sup> � <sup>ϕ</sup><sup>l</sup> 6¼ <sup>0</sup>, <sup>π</sup>, we can solve for ð Þ <sup>Ω</sup><sup>2</sup> kl such that iR <sup>¼</sup> AdKð Þ Xd <sup>⊥</sup>. This gives <sup>Ω</sup>2. Let <sup>a</sup> <sup>¼</sup> P AdKð Þ Xd � � and choose <sup>Ω</sup>′ <sup>1</sup> <sup>¼</sup> AdKð Þ Xd <sup>⊥</sup> � <sup>A</sup>Ω2A† ¼ �S∈k.

With this choice of Ω1, Ω<sup>2</sup> and a, P tð Þ þ Δ and Q tð Þ þ Δ are matched to first order in Δ and

$$P(t+\Delta) - Q(t+\Delta) = o\left(\Delta^2\right).$$

Consider the case, when A is degenerate. Let,

$$A = \begin{bmatrix} A\_1 & \mathbf{0} & \dots & \mathbf{0} \\ \mathbf{0} & A\_2 & \dots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & A\_n \end{bmatrix},\tag{25}$$

where Ak is nk fold degenerate (modulo sign) described by nk � nk block. WLOG, we arrange

$$A\_k = \exp\left(i\phi\_k\right) \begin{bmatrix} I\_{r \times r} & \mathbf{0} \\ \mathbf{0} & -I\_{s \times s} \end{bmatrix}.\tag{26}$$

Consider the decomposition

$$\operatorname{Ad}\_{\overline{K}}(\mathcal{X}\_d) = P\Big(\operatorname{Ad}\_{\overline{K}}(\mathcal{X}\_d)\Big) + \operatorname{Ad}\_{\overline{K}}(\mathcal{X}\_d)^\perp,$$

where P denotes projection onto nk � nk blocks in Eq. (25) and AdKð Þ Xd <sup>⊥</sup>, the orthogonal complement.

$$P\left(\begin{bmatrix}X\_{11} & X\_{12} & \dots & X\_{1n} \\ X\_{21} & X\_{22} & \dots & X\_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ X\_{n1} & X\_{n2} & \dots & X\_{nn} \end{bmatrix}\right) = \begin{bmatrix}X\_{11} & 0 & \dots & 0 \\ 0 & X\_{22} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & X\_{nn} \end{bmatrix},\tag{27}$$

as H1a1H′

fore diagonal of H′

Now using H1AH†

<sup>1</sup> has block diagonal form which is perpendicular to Adkð Þ Xd

� � � � <sup>H</sup>1AH†

1K′ <sup>1</sup><sup>Δ</sup> � � exp <sup>K</sup>1AΩ2A′

<sup>¼</sup> P tð Þ <sup>þ</sup> <sup>Δ</sup> <sup>P</sup><sup>T</sup>ð Þ <sup>t</sup> <sup>þ</sup> <sup>Δ</sup> <sup>I</sup> <sup>þ</sup> <sup>o</sup> <sup>Δ</sup><sup>2</sup> � � � � :

exp i2ϕ<sup>1</sup> ð Þ 0 … 0 0 exp i2ϕ<sup>2</sup> ð Þ … 0 ⋮ ⋮⋱⋮ 0 0 … exp i2ϕ<sup>n</sup> ð Þ

K′ 1

<sup>1</sup> þ K1AΩ2A′

� � <sup>¼</sup> AdKð Þ Xd :

<sup>1</sup> <sup>þ</sup> <sup>A</sup>Ω2A′ � � <sup>¼</sup> AdKð Þ Xd : Q tð Þ� <sup>þ</sup> <sup>Δ</sup> P tð Þ¼ <sup>þ</sup> <sup>Δ</sup> <sup>o</sup> <sup>Δ</sup><sup>2</sup> � �P tð Þ: Q tð Þ¼ <sup>þ</sup> <sup>Δ</sup> <sup>I</sup> <sup>þ</sup> <sup>o</sup> <sup>Δ</sup><sup>2</sup> � � � � P tð Þ <sup>þ</sup> <sup>Δ</sup> : Q tð Þ <sup>þ</sup> <sup>Δ</sup> Q tð Þ <sup>þ</sup> <sup>Δ</sup> <sup>T</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup> <sup>o</sup> <sup>Δ</sup><sup>2</sup> � � � � P tð Þ <sup>þ</sup> <sup>Δ</sup> <sup>P</sup><sup>T</sup>ð Þ <sup>t</sup> <sup>þ</sup> <sup>Δ</sup> <sup>I</sup> <sup>þ</sup> <sup>o</sup> <sup>Δ</sup><sup>2</sup> � � � �

Let <sup>F</sup> <sup>¼</sup> P tð Þ <sup>þ</sup> <sup>Δ</sup> <sup>P</sup><sup>T</sup>ð Þ <sup>t</sup> <sup>þ</sup> <sup>Δ</sup> and <sup>G</sup> <sup>¼</sup> Q tð Þ <sup>þ</sup> <sup>Δ</sup> <sup>Q</sup><sup>T</sup>ð Þ <sup>T</sup> <sup>þ</sup> <sup>Δ</sup> we relate the eigenvalues, of F and G. Given F, G, as above, with ∣F � G∣ # ε, and a ordered set of

<sup>1</sup> and G ¼ U2Dð Þ μ U<sup>2</sup>′, where Dð Þλ is diagonal with diagonal as λ, let

i α<sup>i</sup> λ′

>j j λ � μ 2 .

<sup>2</sup> � tr Dð Þ<sup>λ</sup> ′

Dð Þλ

<sup>λ</sup> QQ<sup>T</sup>ð Þ <sup>t</sup> <sup>þ</sup> <sup>Δ</sup> � � <sup>¼</sup> <sup>λ</sup> PP<sup>T</sup>ð Þ <sup>t</sup> <sup>þ</sup> <sup>Δ</sup> � � <sup>þ</sup> <sup>o</sup> <sup>Δ</sup><sup>2</sup> � �:

exp i2ϕ<sup>1</sup> ð Þ exp i2ϕ<sup>2</sup> ð Þ ⋮ exp i2ϕ<sup>n</sup> ð Þ

<sup>1</sup>P Ad ð Þ <sup>k</sup>ð Þ Xd H<sup>1</sup> is same as diagonal of H′

<sup>2</sup> ¼ A, from 28, we have

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

Q tð Þ¼ <sup>þ</sup> <sup>Δ</sup> exp ð Þ <sup>Ω</sup>1<sup>Δ</sup> <sup>K</sup>1H<sup>1</sup> exp ð Þ <sup>a</sup><sup>Δ</sup> AH†

1K′

<sup>1</sup> þ H1aH′

Choose an ordering of λð Þ G call μ that minimizes ∣λð Þ� F λð Þ G ∣.

<sup>¼</sup> j j <sup>λ</sup> <sup>2</sup> <sup>þ</sup> j j <sup>μ</sup>

spondence) of eigenvalues of G, such that ∣λð Þ� F λð Þ G ∣<ε.

� 2

UDð Þ <sup>μ</sup> <sup>U</sup>′ <sup>þ</sup> UDð Þ <sup>μ</sup> <sup>U</sup>′ � �′

where Pi are permutations. Therefore j j <sup>F</sup> � <sup>G</sup> <sup>2</sup>

� � <sup>¼</sup> <sup>∑</sup>

Q tð Þ¼ <sup>þ</sup> <sup>Δ</sup> exp ð Þ <sup>Ω</sup>1<sup>Δ</sup> <sup>K</sup><sup>1</sup> exp P AdKð Þ Xd <sup>Δ</sup>

DOI: http://dx.doi.org/10.5772/intechopen.80567

where the above expression can be written as

Q tð Þ¼ þ Δ exp ð Þ Ω1Δ exp K1H1aH′

Ω<sup>1</sup> þ K1H1aH′

Ω′

P tð Þ <sup>þ</sup> <sup>Δ</sup> <sup>P</sup><sup>T</sup>ð Þ¼ <sup>t</sup> <sup>þ</sup> <sup>Δ</sup> <sup>K</sup><sup>1</sup>

eigenvalues of F, denote λð Þ¼ F

F ¼ U1Dð Þλ U′

j j <sup>F</sup> � <sup>G</sup> <sup>2</sup> <sup>¼</sup> <sup>D</sup>ð Þ� <sup>λ</sup> UDð Þ <sup>μ</sup> <sup>U</sup>′ �

By Schur convexity,

tr Dð Þλ ′

� �

<sup>1</sup>U2,

Therefore,

173

U ¼ U′

where Ω1, H1, a, Ω2, are chosen such that

<sup>⊥</sup>. There-

<sup>1</sup>AdKð Þ Xd H1.

<sup>2</sup> exp ð Þ Ω2Δ K2: (32)

, there exists an ordering (corre-

UDð Þ μ U′ þ ð Þ UDð Þ μ U ′

Pið Þþ μ Pið Þ μ ′

� �,

λ

� �,

Dð Þλ

<sup>2</sup> exp ð Þ Ω2Δ K2: (33)

K′ <sup>1</sup><sup>Δ</sup> � �P tð Þ:

where Xij are blocks.

Then we write

$$Q(t+\Delta) = \exp\left(\Omega\_1 \Delta\right) \mathbf{K}\_1 \exp\left(P\left(Ad\_{\overline{K}}(X\_d)\Delta\right)\right) \mathbf{A} \exp\left(\Omega\_2 \Delta\right) \mathbf{K}\_2. \tag{28}$$

where in Eq. (24) we can solve for ð Þ <sup>Ω</sup><sup>2</sup> kl such that iR <sup>¼</sup> AdKð Þ Xd <sup>⊥</sup>. This gives <sup>Ω</sup>2. Choose, AdKð Þ Xd <sup>⊥</sup> � <sup>A</sup>Ω2A† <sup>¼</sup> <sup>Ω</sup>′ <sup>1</sup> ∈ k, this gives Ω<sup>1</sup> ¼ K1Ω′ 1K′ 1. Again P tð Þ� <sup>þ</sup> <sup>Δ</sup> Q tð Þ¼ <sup>þ</sup> <sup>Δ</sup> <sup>o</sup> <sup>Δ</sup><sup>2</sup> � �: We write Eq. (28) slightly differently. Let H<sup>1</sup> be a rotation formed from block diagonal matrix

$$H\_1 = \begin{bmatrix} \Theta\_1 & 0 & \dots & 0 \\ 0 & \Theta\_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \Theta\_n \end{bmatrix},\tag{29}$$

where Θ<sup>k</sup> is nk � nk sub-block in SO nð Þ<sup>k</sup> . H<sup>1</sup> ¼ exp ð Þ h<sup>1</sup> is chosen such that

$$H\_1^\top P \Big( A d\_{\overline{K}}(X\_d) \Big) H\_1 = a$$

is a diagonal matrix. Let <sup>H</sup><sup>2</sup> <sup>¼</sup> exp <sup>ð</sup>A�<sup>1</sup> h1A |fflfflfflffl{zfflfflfflffl} h2 Þ, where h<sup>2</sup> is skew symmetric, such

that

$$h\_1 = \begin{bmatrix} \theta\_1 & 0 & \dots & 0 \\ 0 & \theta\_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \theta\_n \end{bmatrix}, h\_2 = \begin{bmatrix} \hat{\theta}\_1 & 0 & \dots & 0 \\ 0 & \hat{\theta}\_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \hat{\theta}\_n \end{bmatrix},\tag{30}$$

where

θk, θ ^<sup>k</sup> is nk � nk sub-block in so nð Þ<sup>k</sup> , related by (see 26)

$$\hat{\theta}\_{k} = A\_{\dot{k}} \cdot \theta\_{k} A\_{k} \quad \theta\_{k} = \begin{bmatrix} \stackrel{r \times r}{\theta\_{11}} & \theta\_{12} \\ -\stackrel{\theta\_{12}}{\theta\_{12}} & \stackrel{\theta\_{22}}{\theta\_{22}} \end{bmatrix}, \hat{\theta}\_{k} = \begin{bmatrix} \theta\_{11} & -\theta\_{12} \\ \theta\_{12}^{\dagger} & \theta\_{22} \end{bmatrix} \tag{31}$$

Note H′ <sup>1</sup>P Ad ð Þ <sup>k</sup>ð Þ Xd H<sup>1</sup> ¼ a lies in convex hull of eigenvalues of Xd. This is true if we look at the diagonal of H′ <sup>1</sup>AdKð Þ Xd H1, it follows from Schur Convexity. The diagonal of H′ <sup>1</sup>Adkð Þ Xd <sup>⊥</sup>H<sup>1</sup> is zero as its inner product

$$\operatorname{tr}\left(a\_1 H\_1^\prime A d\_k(X\_d)^\perp H\_1\right) = \operatorname{tr}\left(H\_1 a\_1 H\_1^\prime A d\_k(X\_d)^\perp\right) = \mathbf{0} \dots$$

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics DOI: http://dx.doi.org/10.5772/intechopen.80567

as H1a1H′ <sup>1</sup> has block diagonal form which is perpendicular to Adkð Þ Xd <sup>⊥</sup>. Therefore diagonal of H′ <sup>1</sup>P Ad ð Þ <sup>k</sup>ð Þ Xd H<sup>1</sup> is same as diagonal of H′ <sup>1</sup>AdKð Þ Xd H1.

Now using H1AH† <sup>2</sup> ¼ A, from 28, we have

P

Applied Modern Control

where Xij are blocks. Then we write

<sup>Ω</sup>2. Choose, AdKð Þ Xd

that

where θk, θ

Note H′

diagonal of H′

172

we look at the diagonal of H′

<sup>1</sup>Adkð Þ Xd

tr a1H′

0

BBB@

X<sup>11</sup> X<sup>12</sup> … X1<sup>n</sup> X<sup>21</sup> X<sup>22</sup> … X2<sup>n</sup> ⋮ ⋮⋱⋮ Xn<sup>1</sup> Xn<sup>2</sup> … Xnn

Q tð Þ¼ <sup>þ</sup> <sup>Δ</sup> exp ð Þ <sup>Ω</sup>1<sup>Δ</sup> <sup>K</sup><sup>1</sup> exp P AdKð Þ Xd <sup>Δ</sup>

<sup>⊥</sup> � <sup>A</sup>Ω2A† <sup>¼</sup> <sup>Ω</sup>′

H<sup>1</sup> ¼

H′

θ<sup>1</sup> 0 … 0 0 θ<sup>2</sup> … 0 ⋮ ⋮⋱⋮ 0 0 … θ<sup>n</sup>

^<sup>k</sup> is nk � nk sub-block in so nð Þ<sup>k</sup> , related by (see 26)

^θ<sup>k</sup> <sup>¼</sup> Ak′θkAk, <sup>θ</sup><sup>k</sup> <sup>¼</sup> <sup>θ</sup><sup>11</sup>

<sup>1</sup>Adkð Þ Xd

� �

is a diagonal matrix. Let <sup>H</sup><sup>2</sup> <sup>¼</sup> exp <sup>ð</sup>A�<sup>1</sup>

h<sup>1</sup> ¼

where in Eq. (24) we can solve for ð Þ <sup>Ω</sup><sup>2</sup> kl such that iR <sup>¼</sup> AdKð Þ Xd

P tð Þ� <sup>þ</sup> <sup>Δ</sup> Q tð Þ¼ <sup>þ</sup> <sup>Δ</sup> <sup>o</sup> <sup>Δ</sup><sup>2</sup> � �: We write Eq. (28) slightly differently. Let H<sup>1</sup> be a rotation formed from block diagonal matrix

1

� � � �

Θ<sup>1</sup> 0 … 0 0 Θ<sup>2</sup> … 0 ⋮ ⋮⋱⋮ 0 0 … Θ<sup>n</sup>

where Θ<sup>k</sup> is nk � nk sub-block in SO nð Þ<sup>k</sup> . H<sup>1</sup> ¼ exp ð Þ h<sup>1</sup> is chosen such that

<sup>1</sup>P AdKð Þ Xd � �

z}|{ r�r

�θ†

<sup>⊥</sup>H<sup>1</sup> is zero as its inner product

<sup>⊥</sup>H<sup>1</sup>

θ<sup>12</sup>

<sup>1</sup>P Ad ð Þ <sup>k</sup>ð Þ Xd H<sup>1</sup> ¼ a lies in convex hull of eigenvalues of Xd. This is true if

¼ tr H1a1H′

<sup>1</sup>AdKð Þ Xd H1, it follows from Schur Convexity. The

<sup>1</sup>Adkð Þ Xd <sup>⊥</sup> � �

¼ 0:

3 7 7 5, θ

<sup>12</sup> θ<sup>22</sup> |{z} s�s

<sup>1</sup> ∈ k, this gives Ω<sup>1</sup> ¼ K1Ω′

H<sup>1</sup> ¼ a

θ

0 θ

^<sup>1</sup> 0 … 0

^<sup>2</sup> … 0 ⋮ ⋮⋱⋮ 0 0 … θ

> ^<sup>k</sup> <sup>¼</sup> <sup>θ</sup><sup>11</sup> �θ<sup>12</sup> θ† <sup>12</sup> θ<sup>22</sup> � �

^n

h1A |fflfflfflffl{zfflfflfflffl} h2

CCCA ¼

X<sup>11</sup> 0 … 0 0 X<sup>22</sup> … 0 ⋮ ⋮⋱⋮ 0 0 … Xnn

A exp ð Þ Ω2Δ K2: (28)

1. Again

5, (29)

Þ, where h<sup>2</sup> is skew symmetric, such

5, (30)

(31)

1K′

, (27)

<sup>⊥</sup>. This gives

$$Q(t+\Delta) = \exp\left(\Omega\_1 \Delta\right) K\_1 \exp\left(P\left(Ad\overline{K}(X\_d)\Delta\right)\right) H\_1 A H\_2^\dagger \exp\left(\Omega\_2 \Delta\right) K\_2. \tag{32}$$

$$Q(t+\Delta) = \exp\left(\Omega\_1 \Delta\right) K\_1 H\_1 \exp\left(a\Delta\right) A H\_2^\dagger \exp\left(\Omega\_2 \Delta\right) K\_2. \tag{33}$$

where the above expression can be written as

$$Q(t+\Delta) = \exp\left(\Omega\_1 \Delta\right) \exp\left(K\_1 H\_1 a H\_1^\prime K\_1^\prime \Delta\right) \exp\left(K\_1 A \Omega\_2 A^\prime K\_1^\prime \Delta\right) P(t).$$

where Ω1, H1, a, Ω2, are chosen such that

$$\left(\Delta\_1 + K\_1 H\_1 a H\_1' K\_1 + K\_1 A \Delta\_2 A' K\_1\right) = A d\_K(X\_d).$$

$$\left(\Delta\_1' + H\_1 a H\_1' + A \Delta\_2 A'\right) = A d\_K(X\_d).$$

$$Q(t + \Delta) - P(t + \Delta) = o\left(\Delta^2\right) P(t).$$

$$Q(t + \Delta) = \left(I + o\left(\Delta^2\right)\right) P(t + \Delta).$$

$$Q(t + \Delta)Q(t + \Delta)^T = \left(I + o\left(\Delta^2\right)\right) P(t + \Delta)P^T(t + \Delta)\left(I + o\left(\Delta^2\right)\right)$$

$$= P(t + \Delta)P^T(t + \Delta)\left[I + o\left(\Delta^2\right)\right].$$

$$P(t + \Delta)P^T(t + \Delta) = K\_1 \begin{bmatrix} \exp\left(i2\phi\_1\right) & 0 & \dots & 0\\ 0 & \exp\left(i2\phi\_2\right) & \dots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \dots & \exp\left(i2\phi\_n\right) \end{bmatrix} K\_1^T.$$

Let <sup>F</sup> <sup>¼</sup> P tð Þ <sup>þ</sup> <sup>Δ</sup> <sup>P</sup><sup>T</sup>ð Þ <sup>t</sup> <sup>þ</sup> <sup>Δ</sup> and <sup>G</sup> <sup>¼</sup> Q tð Þ <sup>þ</sup> <sup>Δ</sup> <sup>Q</sup><sup>T</sup>ð Þ <sup>T</sup> <sup>þ</sup> <sup>Δ</sup> we relate the eigenvalues, of F and G. Given F, G, as above, with ∣F � G∣ # ε, and a ordered set of 2 3

$$\text{eigenvalues of F, denote } \lambda(F) = \begin{bmatrix} \exp\left(i2\phi\_1\right) \\ \exp\left(i2\phi\_2\right) \\ \vdots \\ \exp\left(i2\phi\_n\right) \end{bmatrix}, \text{ there exists an ordering (correct)}$$

spondence) of eigenvalues of G, such that ∣λð Þ� F λð Þ G ∣<ε.

Choose an ordering of λð Þ G call μ that minimizes ∣λð Þ� F λð Þ G ∣.

F ¼ U1Dð Þλ U′ <sup>1</sup> and G ¼ U2Dð Þ μ U<sup>2</sup>′, where Dð Þλ is diagonal with diagonal as λ, let U ¼ U′ <sup>1</sup>U2,

$$\left| \left| F - \mathbf{G} \right|^2 = \left| D(\boldsymbol{\lambda}) - \mathbf{U} \mathbf{D}(\boldsymbol{\mu}) \mathbf{U}' \right|^2 = \left| \boldsymbol{\lambda} \right|^2 + \left| \boldsymbol{\mu} \right|^2 - \text{tr} \left( \mathbf{D}(\boldsymbol{\lambda})^\prime \mathbf{U} \mathbf{D}(\boldsymbol{\mu}) \mathbf{U}' + (\mathbf{U} \mathbf{D}(\boldsymbol{\mu}) \mathbf{U})^\prime \mathbf{D}(\boldsymbol{\lambda}) \right),$$

By Schur convexity,

$$\operatorname{tr}\Big(\operatorname{D}(\boldsymbol{\lambda})^{\stackrel{\
\operatorname{'}}}\operatorname{UD}(\boldsymbol{\mu})\boldsymbol{U}^{\stackrel{\
\operatorname{'}}}+\Big(\operatorname{UD}(\boldsymbol{\mu})\boldsymbol{U}^{\stackrel{\
\operatorname{'}}}\Big)\boldsymbol{D}(\boldsymbol{\lambda})\Big)=\sum\_{i}a\_{i}\Big(\boldsymbol{\lambda}^{\stackrel{\
\operatorname{'}}}\boldsymbol{P}\_{i}(\boldsymbol{\mu})+\boldsymbol{P}\_{i}(\boldsymbol{\mu})^{\stackrel{\
\operatorname{'}}}\Big),$$

where Pi are permutations. Therefore j j <sup>F</sup> � <sup>G</sup> <sup>2</sup> >j j λ � μ 2 . Therefore,

$$
\lambda \left( Q Q^T (t + \Delta) \right) = \lambda \left( P P^T (t + \Delta) \right) + o \left( \Delta^2 \right).
$$

The difference

$$\begin{split} \sigma \left( \Delta^2 \right) &= \underbrace{\exp \left( \left( \Omega\_1 + K\_1 H\_1 a H\_1' K\_1' + K\_1 A \Omega\_2 A' K\_1' \right) \Delta \right)}\_{\exp \left( A d\_K (X\_d) \Delta \right)} \\ &- \exp \left( \Omega\_1 \Delta \right) \exp \left( K\_1 H\_1 a H\_1' K\_1' \Delta \right) \exp \left( K\_1 A \Omega\_2 A' K\_1' \Delta \right), \end{split}$$

is regulated by size of Ω2, which is bounded by ∣Ω2∣ # <sup>∥</sup>Xd<sup>∥</sup> sin ð Þ <sup>ϕ</sup>i�ϕ<sup>j</sup> , where sin ϕ<sup>i</sup> � ϕ<sup>j</sup> � � is smallest non-zero difference. <sup>Δ</sup> is chosen small enough such that

∣o Δ<sup>2</sup> � �∣<εΔ.

For each point t∈½ � 0; T , we choose an open nghd N tðÞ¼ t � Nt ð Þ ; t þ Nt , such that ot <sup>Δ</sup><sup>2</sup> � �<ε<sup>Δ</sup> for <sup>Δ</sup> <sup>∈</sup> N tð Þ. N tð Þ forms a cover of 0½ � ; <sup>T</sup> . We can choose a finite subcover centered at t1, …, tn (see Figure 1A). Consider trajectory at points P tð Þ<sup>1</sup> , …, …P tð Þ<sup>n</sup> . Let ti,iþ<sup>1</sup> be the point in intersection of N tð Þ<sup>i</sup> and N tð Þ <sup>i</sup>þ<sup>1</sup> . Let Δþ <sup>i</sup> ¼ ti,iþ<sup>1</sup> � ti and Δ� <sup>i</sup>þ<sup>1</sup> ¼ tiþ<sup>1</sup> � ti,iþ1. We consider points P tð Þ<sup>i</sup> ,P tð Þ <sup>i</sup>þ<sup>1</sup> ,P tð Þ i,iþ<sup>1</sup> ,Q ti þ Δ<sup>þ</sup> i � � ,Q tiþ<sup>1</sup> � Δ� iþ1 � � as shown in Figure 1B.


Then we get the following recursive relations.

$$
\lambda \left( Q\_{i+} Q\_{i+}^T \right) = \exp \left( 2a\_i^+ \Delta\_i^+ \right) \ \lambda \left( P\_i P\_i^T \right) \tag{34}
$$

making Δ ! 0 and hence ε ! 0, we can make this arbitrarily small. In Eq. (18), Pn ! P Tð Þ as τ ! 0. Hence P Tð Þ belongs to compact set K<sup>1</sup> exp ð Þ μT K2. q.e.d.

Corollary 1. Let U tð Þ∈SU nð Þ be a solution to the differential equation

i uiXi � �U,

U Tð Þ∈K<sup>1</sup> exp ð Þ �iμT K2, where K1, K<sup>2</sup> ∈SO nð Þ and μ≺λ, where λ ¼ ð Þ λ1; …; λ<sup>n</sup> ′ and

. The elements of reachable set at time T, takes the form

uiXi � �K. Then

j

exp �itjXd

αjPjð Þλ T !

> π 2 Iz � �

K2

� �Kj, <sup>∑</sup>tj <sup>¼</sup> <sup>T</sup>:

ð Þ<sup>t</sup> ð Þ Xd V tð Þ:

<sup>U</sup>\_ <sup>¼</sup> Xd <sup>þ</sup> <sup>∑</sup>

where f g Xi LA, the Lie algebra generated by Xi, is so nð Þ and

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

the set S ¼ K<sup>1</sup> exp ð Þ �iμT K<sup>2</sup> belongs to the closure of reachable set.

ð Þ<sup>t</sup> U tð Þ, where, <sup>K</sup>\_ <sup>¼</sup> <sup>∑</sup><sup>i</sup>

V t \_ ðÞ¼ AdK′

¼ K<sup>1</sup> Y j

W ¼ exp �iπIySy

We can synthesize Kj in negligible time, therefore ∣U Tð Þ� U∣<ε, for any desired

� � exp �<sup>i</sup>

Remark 4. We now show how Remark 2 and Theorem 1 can be mapped to results on decomposition and reachable set for coupled spins/qubits. Consider the

A. Collection of overlapping neighborhoods forming the finite subcover. B. Depiction of Pi, Piþ<sup>1</sup>, Qiþ, Qi�,

From Theorem 1, we have V Tð Þ∈K<sup>1</sup> exp ð Þ �iμT K2. Therefore

U ¼ K<sup>1</sup> exp ð Þ �iμT K<sup>2</sup> ¼ K<sup>1</sup> exp �i ∑

ε. Hence U is in closure of reachable set. q.e.d.

Xd ¼ �i

λ<sup>1</sup> 0 … 0 0 λ<sup>2</sup> … 0 ⋮ ⋮⋱⋮ 0 0 … λ<sup>n</sup>

DOI: http://dx.doi.org/10.5772/intechopen.80567

Proof. Let V tðÞ¼ K′

U Tð Þ∈K<sup>1</sup> exp ð Þ �iμT K2. Given

transformation

Figure 1.

175

Pi,iþ<sup>1</sup> as in proof of Theorem 1.

$$
\lambda \left( P\_{i,i+1} P\_{i,i+1}^T \right) = \lambda \left( Q\_{i+} Q\_{i+}^T \right) + o \left( \left( \Delta\_i^+ \right)^2 \right) \tag{35}
$$

$$
\lambda \left( \mathbf{Q}\_{(i+1)-} \mathbf{Q}\_{(i+1)-}^T \right) = \lambda \left( \mathbf{P}\_{i,i+1} \mathbf{P}\_{i,i+1}^T \right) + o\left( \left( \boldsymbol{\Delta}\_{i+1}^- \right)^2 \right) \tag{36}
$$

$$\exp\left(-2a\_{i+1}^{-}\Delta\_{i+1}^{-}\right)\ \lambda\left(P\_{i+1}P\_{i+1}^{T}\right)=\lambda\left(Q\_{(i+1)-}Q\_{(i+1)-}^{T}\right)\tag{37}$$

where a<sup>þ</sup> <sup>i</sup> and a� <sup>i</sup>þ<sup>1</sup> correspond to a in Eq. (33) and lie in the convex hull of the eigenvalues Xd.

Adding the above equations,

$$\lambda\left(P\_{i+1}P\_{i+1}^T\right) = \exp\left(o\left(\Delta^2\right)\right)\exp\left(2\left(a\_i^+\Delta\_i^+ + a\_{i+1}^-\Delta\_{i+1}^-\right)\right.\tag{38}$$

$$\lambda(P\_n P\_n^T) = \exp\left(\underbrace{\sum o\left(\Delta^2\right)}\_{\le \epsilon T}\right) \exp\left(2\sum\_i a\_i^+ \Delta\_i^+ + a\_{i+1}^- \Delta\_{i+1}^-\right) \ \lambda\left(P\_1 P\_1^T\right). \tag{39}$$

where o Δ<sup>2</sup> � � in Eq. (38) is diagonal.

$$\lambda(P\_n P\_n^T) = \exp\left(\underbrace{\sum o(\Delta^2)}\_{\le \epsilon T}\right) \cdot \exp\left(2T \sum\_k a\_k P\_k(\lambda)\right) \\ \lambda(P\_1 P\_1^T) = \exp\left(\underbrace{\sum o(\Delta^2)}\_{\le \epsilon T}\right) \cdot \exp\left(2\mu T\right) \\ \lambda(P\_1 P\_1^T), \tag{40}$$

where μ≺λ and P<sup>1</sup> ¼ I.

$$P\_n = K\_1 \exp\left(\frac{1}{2} \underbrace{\sum \rho(\Delta^2)}\_{\leq \epsilon T} \right) \quad \exp\left(\mu T\right) \ K\_2. \tag{41}$$

Note, ∣Pn � K<sup>1</sup> exp ð Þ μT K2∣ ¼ oð Þε . This implies that Pn belongs to the compact set K<sup>1</sup> exp ð Þ μT K2, else it has minimum distance from this compact set and by

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics DOI: http://dx.doi.org/10.5772/intechopen.80567

making Δ ! 0 and hence ε ! 0, we can make this arbitrarily small. In Eq. (18), Pn ! P Tð Þ as τ ! 0. Hence P Tð Þ belongs to compact set K<sup>1</sup> exp ð Þ μT K2. q.e.d.

Corollary 1. Let U tð Þ∈SU nð Þ be a solution to the differential equation

$$
\dot{U} = \left(X\_d + \sum\_i u\_i X\_i\right) U,
$$

where f g Xi LA, the Lie algebra generated by Xi, is so nð Þ and Xd ¼ �i λ<sup>1</sup> 0 … 0 0 λ<sup>2</sup> … 0 ⋮ ⋮⋱⋮ 0 0 … λ<sup>n</sup> 2 6 6 6 4 3 7 7 7 5 . The elements of reachable set at time T, takes the form

U Tð Þ∈K<sup>1</sup> exp ð Þ �iμT K2, where K1, K<sup>2</sup> ∈SO nð Þ and μ≺λ, where λ ¼ ð Þ λ1; …; λ<sup>n</sup> ′ and the set S ¼ K<sup>1</sup> exp ð Þ �iμT K<sup>2</sup> belongs to the closure of reachable set.

Proof. Let V tðÞ¼ K′ ð Þ<sup>t</sup> U tð Þ, where, <sup>K</sup>\_ <sup>¼</sup> <sup>∑</sup><sup>i</sup> uiXi � �K. Then

$$
\dot{V}(t) = A d\_{K\_\cdot(t)}(X\_d) V(t) .
$$

From Theorem 1, we have V Tð Þ∈K<sup>1</sup> exp ð Þ �iμT K2. Therefore U Tð Þ∈K<sup>1</sup> exp ð Þ �iμT K2. Given

$$\begin{aligned} U = K\_1 \exp\left(-i\mu T\right) K\_2 &= K\_1 \exp\left(-i\sum\_j a\_j P\_j(\lambda) T\right) K\_2 \\ &= K\_1 \prod\_j \exp\left(-i t\_j X\_d\right) K\_j, \quad \Sigma t\_j = T. \end{aligned}$$

We can synthesize Kj in negligible time, therefore ∣U Tð Þ� U∣<ε, for any desired ε. Hence U is in closure of reachable set. q.e.d.

Remark 4. We now show how Remark 2 and Theorem 1 can be mapped to results on decomposition and reachable set for coupled spins/qubits. Consider the transformation

Figure 1.

A. Collection of overlapping neighborhoods forming the finite subcover. B. Depiction of Pi, Piþ<sup>1</sup>, Qiþ, Qi�, Pi,iþ<sup>1</sup> as in proof of Theorem 1.

The difference

Applied Modern Control

sin ϕ<sup>i</sup> � ϕ<sup>j</sup> � �

∣o Δ<sup>2</sup> � �∣<εΔ.

<sup>i</sup> ¼ ti,iþ<sup>1</sup> � ti and Δ�

where a<sup>þ</sup>

eigenvalues Xd.

P tð Þ<sup>i</sup> ,P tð Þ <sup>i</sup>þ<sup>1</sup> ,P tð Þ i,iþ<sup>1</sup> ,Q ti þ Δ<sup>þ</sup>

<sup>i</sup> and a�

<sup>λ</sup> Piþ<sup>1</sup>PT iþ1

λ PnPT n

� � <sup>¼</sup> exp <sup>ð</sup>∑<sup>o</sup> <sup>Δ</sup><sup>2</sup> � �

where μ≺λ and P<sup>1</sup> ¼ I.

λ PnPT n

174

Adding the above equations,

� � <sup>¼</sup> exp <sup>ð</sup>∑<sup>o</sup> <sup>Δ</sup><sup>2</sup> � �


where o Δ<sup>2</sup> � � in Eq. (38) is diagonal.

Δþ

<sup>o</sup> <sup>Δ</sup><sup>2</sup> � � <sup>¼</sup> exp <sup>Ω</sup><sup>1</sup> <sup>þ</sup> <sup>K</sup>1H1aH′

� exp ð Þ Ω1Δ exp K1H1aH′

i � � |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Qi<sup>þ</sup>

Then we get the following recursive relations.

<sup>λ</sup> QiþQ<sup>T</sup> iþ � � <sup>¼</sup> exp 2a<sup>þ</sup>

i,iþ1 � � <sup>¼</sup> <sup>λ</sup> QiþQ<sup>T</sup>

ð Þ� iþ1

<sup>λ</sup> Pi,iþ<sup>1</sup>PT

� �

� � <sup>¼</sup> exp <sup>o</sup> <sup>Δ</sup><sup>2</sup> � � � � exp 2 <sup>a</sup><sup>þ</sup>


Þ exp 2T ∑

Pn ¼ K<sup>1</sup> exp

k

Þ exp 2 ∑ i a<sup>þ</sup> <sup>i</sup> Δ<sup>þ</sup> <sup>i</sup> þ a�

αkPkð Þλ � �

> 1 2

0

B@

∑o Δ<sup>2</sup> � � |fflfflfflffl{zfflfflfflffl} # <sup>ε</sup><sup>T</sup>

Note, ∣Pn � K<sup>1</sup> exp ð Þ μT K2∣ ¼ oð Þε . This implies that Pn belongs to the compact set K<sup>1</sup> exp ð Þ μT K2, else it has minimum distance from this compact set and by

<sup>i</sup>þ<sup>1</sup>Δ� iþ1 � � <sup>λ</sup> Piþ<sup>1</sup>P<sup>T</sup>

<sup>λ</sup> <sup>Q</sup>ð Þ� <sup>i</sup>þ<sup>1</sup> <sup>Q</sup><sup>T</sup>

exp �2a�

is regulated by size of Ω2, which is bounded by ∣Ω2∣ # <sup>∥</sup>Xd<sup>∥</sup>

1K′

For each point t∈½ � 0; T , we choose an open nghd N tðÞ¼ t � Nt ð Þ ; t þ Nt , such that ot <sup>Δ</sup><sup>2</sup> � �<ε<sup>Δ</sup> for <sup>Δ</sup> <sup>∈</sup> N tð Þ. N tð Þ forms a cover of 0½ � ; <sup>T</sup> . We can choose a finite subcover centered at t1, …, tn (see Figure 1A). Consider trajectory at points P tð Þ<sup>1</sup> , …, …P tð Þ<sup>n</sup> . Let ti,iþ<sup>1</sup> be the point in intersection of N tð Þ<sup>i</sup> and N tð Þ <sup>i</sup>þ<sup>1</sup> . Let

<sup>i</sup>þ<sup>1</sup> ¼ tiþ<sup>1</sup> � ti,iþ1. We consider points

,Q tiþ<sup>1</sup> � Δ�

� �Δ � � |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} exp ð Þ AdKð Þ Xd Δ

> 1K′ <sup>1</sup><sup>Δ</sup> � � exp <sup>K</sup>1AΩ2A′

is smallest non-zero difference. Δ is chosen small enough such that

iþ1 � � |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} Qð Þ� <sup>i</sup>þ<sup>1</sup>

> <sup>i</sup> Δ<sup>þ</sup> i � � λ PiPT

iþ � � <sup>þ</sup> <sup>o</sup> <sup>Δ</sup><sup>þ</sup>

i,iþ1 � � <sup>þ</sup> <sup>o</sup> <sup>Δ</sup>�

� � <sup>¼</sup> <sup>λ</sup> <sup>Q</sup>ð Þ� <sup>i</sup>þ<sup>1</sup> <sup>Q</sup><sup>T</sup>

<sup>i</sup>þ<sup>1</sup> correspond to a in Eq. (33) and lie in the convex hull of the

<sup>i</sup>þ<sup>1</sup>Δ� iþ1 � � λ PiP†

> <sup>i</sup>þ<sup>1</sup>Δ� iþ1

� � <sup>¼</sup> exp <sup>ð</sup>∑<sup>o</sup> <sup>Δ</sup><sup>2</sup> � �

<sup>¼</sup> <sup>λ</sup> Pi,iþ<sup>1</sup>PT

iþ1

<sup>i</sup> Δ<sup>þ</sup> <sup>i</sup> þ a�

λ P1PT 1

1

� �

<sup>1</sup> þ K1AΩ2A′

K′ 1

K′ <sup>1</sup>Δ � �,

, where

sin ð Þ <sup>ϕ</sup>i�ϕ<sup>j</sup>

as shown in Figure 1B.

iþ1 � �<sup>2</sup> � �

� �

� �: � (38)


CA exp ð Þ <sup>μ</sup><sup>T</sup> <sup>K</sup>2: (41)

ð Þ� iþ1

i

<sup>Þ</sup> exp 2ð Þ <sup>μ</sup><sup>T</sup> <sup>λ</sup> <sup>P</sup>1PT

1 � �,

(40)

λ P1P<sup>T</sup> 1 � �: (39)

� � (34)

(35)

(36)

(37)

i

i � �<sup>2</sup> � �

The transformation maps the algebra k ¼ suð Þ� 2 suð Þ¼ 2 f g Iα; S<sup>α</sup> to k<sup>1</sup> ¼ soð Þ 4 , four dimensional skew symmetric matrices, i.e., AdWð Þ¼ k k1. The transformation maps p ¼ IαS<sup>β</sup> � � to <sup>p</sup><sup>1</sup> ¼ �iA, where <sup>A</sup> is traceless symmetric and maps a ¼ �i IxSx;IySy;IzSz � � to <sup>a</sup><sup>1</sup> ¼ �<sup>i</sup> � Sz <sup>2</sup> ; Iz <sup>2</sup> ;IzSz � �, space of diagonal matrices in <sup>p</sup>1, such that axIxSx þ ayIySy þ azIzSz gets mapped to the four vector (the diagonal) ð Þ¼ λ1; λ2; λ3; λ<sup>4</sup> ay þ az � ax; ax þ ay � az; � ax þ ay þ az � �; ax <sup>þ</sup> az � ay � �.

V t \_ ðÞ¼ AdK′

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

<sup>V</sup> <sup>¼</sup> <sup>Y</sup> i

i

i

V ¼ K<sup>1</sup> exp �i αIxSx þ βIySy þ γIzSz

WVW′ <sup>¼</sup> <sup>Y</sup>

WVW′ ¼ J<sup>1</sup> exp ð Þ �iμ J<sup>2</sup> ¼ J<sup>1</sup> exp �i ∑

V ¼ K<sup>1</sup> exp T ∑

Furthermore U ¼ KV. Hence the proof. q.e.d.

Lie algebra g and a∈p be its Cartan subalgebra. Let a∈a. ad<sup>2</sup>

eigenvectors with nonzero (negative) eigenvalues �λ<sup>2</sup>

in basis orthonormal wrt to the killing form. We can diagonalize ad<sup>2</sup>

<sup>a</sup>, i.e., ½ �¼ a;X ∑<sup>i</sup>

This is a contradiction. Yi are orthogonal, implies Xi are orthogonal,

3. Time optimal control for G=K problem

Multiplying both sides with W′

Consider the product

DOI: http://dx.doi.org/10.5772/intechopen.80567

from Theorem 1, we have

which we can write as

pose it in eigenvectors of ad<sup>2</sup>

eigenvectors of ad<sup>2</sup>

177

where using μ≺λT, we get,

Then,

ð Þ<sup>t</sup> ð Þ �iXd V tð Þ:

exp ð Þ AdKi ð Þ �iXd Δt

exp AdWKiW′ �iWXdW′ � �Δ<sup>t</sup> � �

> αjPjð Þλ !

� � � � K2, α≥β≥∣γ∣,

J2, J1, J<sup>2</sup> ∈SOð Þ 4 , μ≺λT

α<sup>i</sup> ¼ 1:

i

� �T (42) α # axT (43)

� �T: (44)

<sup>i</sup> . Let Xi <sup>¼</sup> <sup>a</sup>;Yi ½ �

<sup>a</sup> : p ! p is symmetric

<sup>λ</sup><sup>i</sup> , λi>0.

βjYj, where ai are zero

<sup>a</sup>. Let Yi be

where Ki ∈SUð Þ2 ⊗SUð Þ2 and Xd ¼ axIxSx þ ayIySy þ azIzSz, where ax≥ay≥∣az∣.

Observe WKiW′ ∈SOð Þ 4 and WXdW′ ¼ diagð Þ λ1; λ2; …; λ<sup>4</sup> . Then using results

j

ð Þ� W, we get

α<sup>i</sup> ai; bi ð Þ ;ci � �K2, <sup>α</sup>i>0 <sup>∑</sup>

α þ β � γ # ax þ ay � az

α þ β þ γ # ax þ ay þ az

Remark 5. Stabilizer: Let g ¼ p ⊕ k be Cartan decomposition of real semisimple

adað Þ¼ Yi λiXi, adað Þ¼� Xi λiYi:

Xi are independent, as ∑αiXi ¼ 0 implies �∑αiλiYi ¼ 0. Since Yi are independent, Xi are independent. Given X⊥Xi, then ½ �¼ a;X 0, otherwise we can decom-

αiai þ ∑<sup>j</sup>

<sup>a</sup>. Since 0 <sup>¼</sup> Xaa <sup>½</sup> ½ � ;<sup>X</sup> <sup>i</sup> ¼ �∥½ � <sup>a</sup>;<sup>X</sup> <sup>∥</sup><sup>2</sup> � , which means ½ �¼ <sup>a</sup>;<sup>X</sup> 0.

Corollary 2. Canonical decomposition. Given the decomposition of SU(4) from Remark 2, we can write

$$U = \exp\left(\Omega\_1\right) \exp\left(-i\begin{bmatrix}\lambda\_1 & \dots & 0\\0 & \ddots & 0\\0 & 0 & \lambda\_4\end{bmatrix}\right) \exp\left(\Omega\_2\right),$$

where Ω1, Ω<sup>2</sup> ∈soð Þ 4 . We write above as

$$U = \exp\left(\Omega\_1\right) \exp\left(-i\left(-\frac{a\_x}{2}\mathcal{S}\_x + \frac{a\_y}{2}I\_x + a\_z I\_x \mathcal{S}\_x\right)\right) \exp\left(\Omega\_2\right),$$

Multiplying both sides with W′ ð Þ: W gives

$$W^\prime UW = K\_1 \exp\left(-i a\_x I\_x \mathcal{S}\_x + a\_\mathcal{Y} I\_\mathcal{Y} \mathcal{S}\_\mathcal{Y} + a\_x I\_x \mathcal{S}\_x\right) K\_2 J$$

where K1, K<sup>2</sup> ∈SUð Þ� 2 SUð Þ2 local unitaries and we can rotate to ax≥ay≥∣az∣.

Corollary 3. Digonalization. Given �iHc ¼ �i∑αβJαβIαSβ, there exists a local unitary K such that

$$K(-iH\_c)K' = -i\left(a\_x I\_x \mathbb{S}\_x + a\_\jmath I\_\jmath \mathbb{S}\_\jmath + a\_x I\_x \mathbb{S}\_x\right), a\_x \ge a\_\jmath \ge |a\_x|.$$

Note Wð Þ �iHc W′ ∈p1. Then choose Θ ∈SO nð Þ such that ΘWð Þ �iHc W′ <sup>Θ</sup>′ ¼ �<sup>i</sup> � ax <sup>2</sup> Sz <sup>þ</sup> ay <sup>2</sup> Iz þ azIzSz � � and hence

$$\left(\boldsymbol{W}^{\prime}\exp\left(\boldsymbol{\Omega}\right)\boldsymbol{W}\right)\left(-i\boldsymbol{H}\_{\boldsymbol{\varepsilon}}\right)\left(\boldsymbol{W}\exp\left(\boldsymbol{\Omega}\right)\boldsymbol{W}^{\prime}\right)^{\prime} = -i\left(\boldsymbol{a}\_{\boldsymbol{x}}\boldsymbol{I}\_{\boldsymbol{x}}\boldsymbol{S}\_{\boldsymbol{x}} + \boldsymbol{a}\_{\boldsymbol{\gamma}}\boldsymbol{I}\_{\boldsymbol{\gamma}}\boldsymbol{S}\_{\boldsymbol{\gamma}} + \boldsymbol{a}\_{\boldsymbol{x}}\boldsymbol{I}\_{\boldsymbol{x}}\boldsymbol{S}\_{\boldsymbol{x}}\right).$$

where K ¼ W′ exp ð Þ Ω W is a local unitary. We can rotate to ensure ax≥ay≥∣az∣.

Corollary 4. Given the evolution of coupled qubits <sup>U</sup>\_ ¼ �i Hc <sup>þ</sup> <sup>∑</sup><sup>j</sup> ujHj � �U, we can diagonalize Hc ¼ ∑αβJαβIαS<sup>β</sup> by local unitary Xd ¼ K′ HcK ¼ axIxSx þ ayIySyþ azIzSz, ax≥ay≥∣az∣, which we write as triple ax; ay; az � �. From this, there are 24 triples obtained by permuting and changing sign of any two by local unitary. Then U Tð Þ∈S where

$$S = K\_1 \exp\left(T \sum\_i a\_i(a\_i, b\_i, c\_i)\right) K\_2, \quad a\_i > 0 \quad \sum\_i a\_i = 1.$$

Furthermore S belongs to the closure of the reachable set. Alternate description of S is

$$U = K\_1 \exp\left(-i\left(aI\_x\mathbb{S}\_x + \beta I\_\mathcal{Y}\mathbb{S}\_\mathcal{Y} + \gamma I\_x\mathbb{S}\_x\right)\right)K\_2, \quad a \ge \beta \ge |\gamma|.$$

α # axT and α þ β � γ # ax þ ay � az � �T. Proof. Let V tðÞ¼ K′ ð Þ<sup>t</sup> U tð Þ, where <sup>K</sup>\_ ¼ �i∑<sup>j</sup> ujXj � �K. Then Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics DOI: http://dx.doi.org/10.5772/intechopen.80567

$$
\dot{V}(t) = \mathcal{A}d\_{K^\vee(t)}(-i\mathcal{X}\_d)V(t) \,.
$$

Consider the product

The transformation maps the algebra k ¼ suð Þ� 2 suð Þ¼ 2 f g Iα; S<sup>α</sup> to k<sup>1</sup> ¼ soð Þ 4 , four dimensional skew symmetric matrices, i.e., AdWð Þ¼ k k1. The transformation

� �.

λ<sup>1</sup> … 0 0 ⋱ 0 0 0 λ<sup>4</sup>

> ay 2

� � � �

3 7 5

Iz þ azIzSz

� �K2,

� �, ax≥ay≥∣az∣:

1

CA exp ð Þ <sup>Ω</sup><sup>2</sup> ,

¼ �i axIxSx þ ayIySy þ azIzSz � �:

� �. From this, there are 24

α<sup>i</sup> ¼ 1:

i

K. Then

K2, αi>0 ∑

ujXj � �

ujHj � �

HcK ¼ axIxSx þ ayIySyþ

U, we

exp ð Þ Ω<sup>2</sup> ,

� �, space of diagonal matrices in <sup>p</sup>1,

� �; ax <sup>þ</sup> az � ay

� � to <sup>p</sup><sup>1</sup> ¼ �iA, where <sup>A</sup> is traceless symmetric and maps

<sup>2</sup> ; Iz <sup>2</sup> ;IzSz

such that axIxSx þ ayIySy þ azIzSz gets mapped to the four vector (the diagonal)

Corollary 2. Canonical decomposition. Given the decomposition of SU(4)

2 6 4

<sup>2</sup> Sz <sup>þ</sup>

ð Þ: W gives

UW ¼ K<sup>1</sup> exp �iaxIxSx þ ayIySy þ azIzSz

where K1, K<sup>2</sup> ∈SUð Þ� 2 SUð Þ2 local unitaries and we can rotate to ax≥ay≥∣az∣.

Corollary 3. Digonalization. Given �iHc ¼ �i∑αβJαβIαSβ, there exists a local

where K ¼ W′ exp ð Þ Ω W is a local unitary. We can rotate to ensure ax≥ay≥∣az∣.

Kð Þ �iHc K′ ¼ �i axIxSx þ ayIySy þ azIzSz

<sup>2</sup> Iz þ azIzSz � � and hence

Corollary 4. Given the evolution of coupled qubits <sup>U</sup>\_ ¼ �i Hc <sup>þ</sup> <sup>∑</sup><sup>j</sup>

triples obtained by permuting and changing sign of any two by local unitary.

α<sup>i</sup> ai; bi ð Þ ;ci � �

Furthermore S belongs to the closure of the reachable set. Alternate description

� � � � K2, α≥β≥∣γ∣,

Note Wð Þ �iHc W′ ∈p1. Then choose Θ ∈SO nð Þ such that

<sup>2</sup> Sz <sup>þ</sup> ay

can diagonalize Hc ¼ ∑αβJαβIαS<sup>β</sup> by local unitary Xd ¼ K′

i

U ¼ K<sup>1</sup> exp �i αIxSx þ βIySy þ γIzSz

� �T.

ð Þ<sup>t</sup> U tð Þ, where <sup>K</sup>\_ ¼ �i∑<sup>j</sup>

azIzSz, ax≥ay≥∣az∣, which we write as triple ax; ay; az

S ¼ K<sup>1</sup> exp T ∑

α # axT and α þ β � γ # ax þ ay � az

<sup>W</sup>′ exp ð Þ <sup>Ω</sup> <sup>W</sup> � �ð Þ �iHc <sup>W</sup> exp ð Þ <sup>Ω</sup> <sup>W</sup>′ � �′

0

B@

maps p ¼ IαS<sup>β</sup>

a ¼ �i IxSx;IySy;IzSz

Applied Modern Control

from Remark 2, we can write

� � to <sup>a</sup><sup>1</sup> ¼ �<sup>i</sup> � Sz

ð Þ¼ λ1; λ2; λ3; λ<sup>4</sup> ay þ az � ax; ax þ ay � az; � ax þ ay þ az

U ¼ exp ð Þ Ω<sup>1</sup> exp �i

where Ω1, Ω<sup>2</sup> ∈soð Þ 4 . We write above as

Multiplying both sides with W′

W′

<sup>Θ</sup>′ ¼ �<sup>i</sup> � ax

unitary K such that

Then U Tð Þ∈S where

Proof. Let V tðÞ¼ K′

of S is

176

ΘWð Þ �iHc W′

<sup>U</sup> <sup>¼</sup> exp ð Þ <sup>Ω</sup><sup>1</sup> exp �<sup>i</sup> � ax

$$V = \prod\_i \exp\left(Ad\_{K\_i}(-iX\_d)\Delta t\right)$$

where Ki ∈SUð Þ2 ⊗SUð Þ2 and Xd ¼ axIxSx þ ayIySy þ azIzSz, where ax≥ay≥∣az∣. Then,

$$WWW' = \prod\_i \exp\left(Ad\_{W\mathbb{K}\_iW'}\left(-i\mathsf{WX}\_dW'\right)\Delta t\right).$$

Observe WKiW′ ∈SOð Þ 4 and WXdW′ ¼ diagð Þ λ1; λ2; …; λ<sup>4</sup> . Then using results from Theorem 1, we have

$$
\hat{\mu}\_1 \mathbf{W} \mathbf{V} \mathbf{W}^\dagger = I\_1 \exp \left( -i\mu \right) I\_2 = I\_1 \exp \left( -i \sum\_j a\_j \mathbf{P}\_j(\lambda) \right) I\_2, \quad I\_1, I\_2 \in \text{SO}(4), \quad \mu \preccurlyeq \mathbf{Z} \mathbf{T},
$$

Multiplying both sides with W′ ð Þ� W, we get

$$V = K\_1 \exp\left(T \sum\_i a\_i(a\_i, b\_i, c\_i)\right) K\_2, \quad a\_i > 0 \quad \sum\_i a\_i = 1.$$

which we can write as

$$V = K\_1 \exp\left(-i\left(aI\_x\mathbb{S}\_x + \beta I\_\mathcal{Y}\mathbb{S}\_y + \gamma I\_x\mathbb{S}\_x\right)\right)K\_2, \quad a \ge \beta \ge |\gamma|J$$

where using μ≺λT, we get,

$$a + \beta - \gamma \le (a\_x + a\_y - a\_x)T \tag{42}$$

$$a \uplus a\_{\mathfrak{x}} T \tag{43}$$

$$a + \beta + \gamma \le (a\_{\infty} + a\_{\gamma} + a\_{x})T. \tag{44}$$

Furthermore U ¼ KV. Hence the proof. q.e.d.

#### 3. Time optimal control for G=K problem

Remark 5. Stabilizer: Let g ¼ p ⊕ k be Cartan decomposition of real semisimple Lie algebra g and a∈p be its Cartan subalgebra. Let a∈a. ad<sup>2</sup> <sup>a</sup> : p ! p is symmetric in basis orthonormal wrt to the killing form. We can diagonalize ad<sup>2</sup> <sup>a</sup>. Let Yi be eigenvectors with nonzero (negative) eigenvalues �λ<sup>2</sup> <sup>i</sup> . Let Xi <sup>¼</sup> <sup>a</sup>;Yi ½ � <sup>λ</sup><sup>i</sup> , λi>0.

$$ad\_a(Y\_i) = \lambda\_i X\_i, \quad ad\_a(X\_i) = -\lambda\_i Y\_i.$$

Xi are independent, as ∑αiXi ¼ 0 implies �∑αiλiYi ¼ 0. Since Yi are independent, Xi are independent. Given X⊥Xi, then ½ �¼ a;X 0, otherwise we can decompose it in eigenvectors of ad<sup>2</sup> <sup>a</sup>, i.e., ½ �¼ a;X ∑<sup>i</sup> αiai þ ∑<sup>j</sup> βjYj, where ai are zero eigenvectors of ad<sup>2</sup> <sup>a</sup>. Since 0 <sup>¼</sup> Xaa <sup>½</sup> ½ � ;<sup>X</sup> <sup>i</sup> ¼ �∥½ � <sup>a</sup>;<sup>X</sup> <sup>∥</sup><sup>2</sup> � , which means ½ �¼ <sup>a</sup>;<sup>X</sup> 0. This is a contradiction. Yi are orthogonal, implies Xi are orthogonal,

a; Yi ½ � a; Yj � � � � <sup>¼</sup> a, a; Yi ½ �Yj � � <sup>¼</sup> <sup>λ</sup><sup>2</sup> <sup>i</sup> YiYj � � <sup>¼</sup> <sup>0</sup> � . Let <sup>k</sup><sup>0</sup> <sup>∈</sup> <sup>k</sup> satisfy ½ �¼ <sup>a</sup>;k<sup>0</sup> 0. Then k<sup>0</sup> ¼ f g Xi ⊥.

<sup>Y</sup>~<sup>i</sup> denote eigenvectors that have <sup>λ</sup><sup>i</sup> as non-zero integral multiples of <sup>π</sup>. <sup>X</sup>~<sup>i</sup> are ada related to <sup>Y</sup>~<sup>i</sup> . We now reserve Yi for non-zero eigenvectors that are not integral multiples of π.

Let

$$\mathfrak{f} = \{a\_i\} \oplus \tilde{Y}\_{\mathfrak{p}}, \qquad \mathfrak{h} = \mathfrak{k}\_0 \oplus \tilde{X}\_{\mathfrak{p}}.$$

X~i , Xl, kj where kj forms a basis of k0, forms a basis of k. Let A ¼ exp ð Þ a . AkA�<sup>1</sup> <sup>¼</sup> <sup>A</sup> <sup>∑</sup><sup>i</sup> αiXi þ ∑<sup>l</sup> <sup>α</sup>lX~<sup>l</sup> <sup>þ</sup> <sup>∑</sup><sup>j</sup> αjkj � �A�, where <sup>k</sup><sup>∈</sup> <sup>k</sup> AkA�<sup>1</sup> <sup>¼</sup> <sup>∑</sup><sup>i</sup> <sup>α</sup><sup>i</sup> cosð Þ <sup>λ</sup><sup>i</sup> Xi � sin ð Þ <sup>λ</sup><sup>i</sup> Yi <sup>½</sup> � þ <sup>∑</sup><sup>l</sup> � <sup>α</sup>lX~<sup>l</sup> <sup>þ</sup> <sup>∑</sup><sup>j</sup> αjkj

The range of <sup>A</sup>ð Þ� <sup>A</sup>�<sup>1</sup> in <sup>p</sup>, is perpendicular to <sup>f</sup>. Given <sup>Y</sup> <sup>∈</sup><sup>p</sup> such that <sup>Y</sup> <sup>∈</sup> <sup>f</sup> <sup>⊥</sup>. The norm ∥X∥ of X ∈ k, such that p part of AXA�<sup>1</sup> � � <sup>p</sup> ¼ Y satisfies

$$\|X\| \le \frac{\|Y\|}{\sin \lambda\_s}.\tag{45}$$

if exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> 6¼ a, then ar ½ � ; exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> ∈k. The bracket ar ½ � ; exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> is AdA<sup>2</sup> invariant and, hence, belongs to h. We can choose h<sup>1</sup> so that gradient is not zero. Hence exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> ∈a. For z∈p such that

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

h i a; exp ð Þ h<sup>0</sup> z exp ð Þ �h<sup>0</sup> ¼ h i exp ð Þ �h<sup>0</sup> a exp ð Þ h<sup>0</sup> ; z ¼ 0,

Remark 6. Kostant's convexity: [28] Given the decomposition g ¼ p ⊕ k, let a⊂p and X ∈a,. Let W<sup>i</sup> ∈ exp ð Þk such that WiXW<sup>i</sup> ∈a are distinct, Weyl points. Then projection (w.r.t killing form) of AdKð Þ X on a lies in convex hull of these Weyl points. The C be the convex hull and let projection P Ad ð Þ <sup>K</sup>ð Þ X lie outside this Hull. Then there is a separating hyperplane a, such that h i AdKð Þ X ; a <h i C; a . W.L.O.G we can take a to be a regular element. We minimize h i AdKð Þ X ; a , with choice of K and find that minimum happens when ½ �¼ AdKð Þ X ; a 0, i.e. AdKð Þ X is a Weyl point.

<sup>i</sup> , for αi>0 and ∑<sup>i</sup>

projection w.r.t inner product that satisfies h i x; ½ � y; z ¼ h½ � x; y ; z�i, like standard inner

<sup>X</sup>\_ <sup>¼</sup> AdK tð Þð Þ Xd X, Pð Þ¼ <sup>0</sup> <sup>1</sup>

where Xd ∈a, the Cartan subalgebra a ∈p and K tð Þ∈ exp k, a closed subgroup of

i

P tð Þ¼ þ Δ exp ð Þ AdKð Þ Xd Δ P tðÞ¼ exp ð Þ AdKð Þ Xd Δ K<sup>1</sup> exp ð Þ a K<sup>2</sup>

exp ð Þ AdKð Þ Xd <sup>Δ</sup> <sup>K</sup>1AK<sup>2</sup> <sup>¼</sup> Ka exp <sup>a</sup>0<sup>Δ</sup> <sup>þ</sup> <sup>C</sup>Δ<sup>2</sup> � �AKb <sup>¼</sup> Ka exp <sup>a</sup> <sup>þ</sup> <sup>a</sup>0<sup>Δ</sup> <sup>þ</sup> <sup>C</sup>Δ<sup>2</sup> � �Kb, (46)

� �


where P is projection w.r.t killing form and a<sup>0</sup> ∈ f, the centralizer in p as defined in Remark 5, CΔ<sup>2</sup> ∈f is a second order term that can be made small by choosing Δ.

αiWið Þ Xd � �K2,

<sup>þ</sup>AdKð Þ Xd

<sup>1</sup>′, K<sup>2</sup>′′ ∈K such that

⊥:

Theorem 2 Given a compact Lie group G and Lie algebra g. Consider the Cartan decomposition of a real semisimple Lie algebra g ¼ p ⊕ k. Given the

P Tð Þ¼ K<sup>1</sup> exp T ∑

AdKð Þ¼ Xd P AdKð Þ Xd

Proof. As in proof of Theorem 1, we define

<sup>1</sup> K,

To show Eq. (46), we show there exists K′

where K1, K<sup>2</sup> ∈ exp ð Þk and Wið Þ Xd ∈a are Weyl points, αi>0 and ∑<sup>i</sup>

α<sup>i</sup> ¼ 1. The result is true with a

α<sup>i</sup> ¼ 1.

as exp ð Þ �h<sup>0</sup> a exp ð Þ h<sup>0</sup> is AdA<sup>2</sup> invariant, hence exp ð Þ �h<sup>0</sup> a exp ð Þ h<sup>0</sup> ∈f. In above, we worked with killing form. For g ¼ su nð Þ, we may use standard inner

<sup>⊥</sup>, we have exp ð Þ <sup>h</sup><sup>0</sup> <sup>z</sup> exp ð Þ �h<sup>0</sup> <sup>∈</sup>a<sup>⊥</sup>.

DOI: http://dx.doi.org/10.5772/intechopen.80567

αiWiXW�<sup>1</sup>

z∈f

product.

Hence P Ad ð Þ <sup>K</sup>ð Þ X ∈ ∑<sup>i</sup>

product on g ¼ su nð Þ.

control system

G. The end point

and show that

Ka, Kb ∈ exp ð Þk .

179

where for <sup>K</sup> <sup>¼</sup> <sup>K</sup>�<sup>1</sup>

where λ<sup>2</sup> <sup>s</sup> is the smallest nonzero eigenvalue of �ad<sup>2</sup> <sup>a</sup> such that λ<sup>s</sup> is not an integral multiple of π.

A2 kA�<sup>2</sup> stabilizes h∈k and f∈p. If k∈k, is stabilized by A<sup>2</sup> ð Þ� <sup>A</sup>�<sup>2</sup> , λ<sup>i</sup> ¼ nπ, i.e., k∈h. This means h is an subalgebra, as the Lie bracket of ½ � y; z ∈ k for y, z∈ h is stabilized by A<sup>2</sup> ð Þ� <sup>A</sup>�<sup>2</sup> .

Let <sup>H</sup> <sup>¼</sup> exp ð Þ <sup>h</sup> , be an integral manifold of <sup>h</sup>. Let <sup>H</sup><sup>~</sup> <sup>∈</sup> <sup>K</sup> be the solution to A2 HA<sup>~</sup> �<sup>2</sup> <sup>¼</sup> <sup>H</sup><sup>~</sup> or <sup>A</sup><sup>2</sup> <sup>H</sup><sup>~</sup> � HA<sup>~</sup> �<sup>2</sup> <sup>¼</sup> 0. <sup>H</sup><sup>~</sup> is closed, <sup>H</sup> <sup>∈</sup> <sup>H</sup><sup>~</sup> . We show that <sup>H</sup><sup>~</sup> is a manifold. Given element H<sup>0</sup> ∈ H~ ∈ K, where K is closed, we have a exp B<sup>k</sup> δ � � nghd of H0, in exp ð Þ <sup>B</sup><sup>δ</sup> ball nghd of <sup>H</sup>0, which is one to one. For <sup>x</sup>∈B<sup>k</sup> δ, <sup>A</sup><sup>2</sup> exp ð Þ <sup>x</sup> <sup>A</sup>�<sup>2</sup> <sup>¼</sup> exp ð Þ <sup>x</sup> , implies,

$$\begin{aligned} A^2 \exp\left(\sum\_i a\_i \mathbf{X}\_i + \sum\_l \beta\_l \breve{\mathbf{X}}\_l + \sum\_j \gamma\_j k\_j \right) H\_0 A^{-2} &= \exp\left(\sum\_i a\_i \cos\left(2\lambda\_i \mathbf{X}\_i - \sin\left(2\lambda\_i \right) \mathbf{Y}\_i\right)\right) \\ + \sum\_l \beta\_l \breve{\mathbf{X}}\_l + \sum\_j \gamma\_j k\_j \right) H\_0 &= \exp\left(\sum\_i a\_i \mathbf{X}\_i + \sum\_l \beta\_l \breve{\mathbf{X}}\_l + \sum\_j \gamma\_j k\_j \right) H\_0 A^{-2} \end{aligned}$$

then by one to one property of exp ð Þ B<sup>δ</sup> , we get α<sup>i</sup> ¼ 0 and x∈h. Therefore exp B<sup>h</sup> δ � �H<sup>0</sup> is a nghd of <sup>H</sup>0.

Given a sequence Hi ∈ exp ð Þ h converging to H0, for n large enough Hn ∈ exp B<sup>h</sup> δ � �H0. Then <sup>H</sup><sup>0</sup> is in invariant manifold exp ð Þ <sup>h</sup> . Hence exp ð Þ <sup>h</sup> is closed and hence compact.

Let y∈f, then there exists a h<sup>0</sup> ∈ h such that exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> ∈a. We maximize the function ar h i ; exp ð Þ h y exp ð Þ h , over the compact group exp ð Þ h , for regular element ar ∈a and h i :; : is the killing form. At the maxima, we have at t ¼ 0, d dt ar h i ; exp ð Þ h1t ð Þ exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> exp ð Þ �h1t ¼ 0.

$$\langle a\_r, [h\_1 \exp\left(h\_0\right) y \exp\left(-h\_0\right)] \rangle = -\langle h\_1, [a\_r \exp\left(h\_0\right) y \exp\left(-h\_0\right)] \rangle.$$

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics DOI: http://dx.doi.org/10.5772/intechopen.80567

if exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> 6¼ a, then ar ½ � ; exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> ∈k. The bracket ar ½ � ; exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> is AdA<sup>2</sup> invariant and, hence, belongs to h. We can choose h<sup>1</sup> so that gradient is not zero. Hence exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> ∈a. For z∈p such that z∈f <sup>⊥</sup>, we have exp ð Þ <sup>h</sup><sup>0</sup> <sup>z</sup> exp ð Þ �h<sup>0</sup> <sup>∈</sup>a<sup>⊥</sup>.

$$\langle \mathfrak{a}, \exp \left( h\_0 \right) z \exp \left( -h\_0 \right) \rangle = \langle \exp \left( -h\_0 \right) \mathfrak{a} \exp \left( h\_0 \right), z \rangle = \mathfrak{0},$$

as exp ð Þ �h<sup>0</sup> a exp ð Þ h<sup>0</sup> is AdA<sup>2</sup> invariant, hence exp ð Þ �h<sup>0</sup> a exp ð Þ h<sup>0</sup> ∈f. In above, we worked with killing form. For g ¼ su nð Þ, we may use standard inner product.

Remark 6. Kostant's convexity: [28] Given the decomposition g ¼ p ⊕ k, let a⊂p and X ∈a,. Let W<sup>i</sup> ∈ exp ð Þk such that WiXW<sup>i</sup> ∈a are distinct, Weyl points. Then projection (w.r.t killing form) of AdKð Þ X on a lies in convex hull of these Weyl points. The C be the convex hull and let projection P Ad ð Þ <sup>K</sup>ð Þ X lie outside this Hull. Then there is a separating hyperplane a, such that h i AdKð Þ X ; a <h i C; a . W.L.O.G we can take a to be a regular element. We minimize h i AdKð Þ X ; a , with choice of K and find that minimum happens when ½ �¼ AdKð Þ X ; a 0, i.e. AdKð Þ X is a Weyl point. Hence P Ad ð Þ <sup>K</sup>ð Þ X ∈ ∑<sup>i</sup> αiWiXW�<sup>1</sup> <sup>i</sup> , for αi>0 and ∑<sup>i</sup> α<sup>i</sup> ¼ 1. The result is true with a projection w.r.t inner product that satisfies h i x; ½ � y; z ¼ h½ � x; y ; z�i, like standard inner product on g ¼ su nð Þ.

Theorem 2 Given a compact Lie group G and Lie algebra g. Consider the Cartan decomposition of a real semisimple Lie algebra g ¼ p ⊕ k. Given the control system

$$\dot{X} = Ad\_{K(t)}(X\_d)X, \ \ P(\mathbf{0}) = \mathbf{1}$$

where Xd ∈a, the Cartan subalgebra a ∈p and K tð Þ∈ exp k, a closed subgroup of G. The end point

$$P(T) = K\_1 \exp\left(T \sum\_i a\_i \mathcal{W}\_i(X\_d)\right) K\_2.$$

where K1, K<sup>2</sup> ∈ exp ð Þk and Wið Þ Xd ∈a are Weyl points, αi>0 and ∑<sup>i</sup> α<sup>i</sup> ¼ 1. Proof. As in proof of Theorem 1, we define

$$P(t + \Delta) = \exp\left(A d\_K(X\_d)\Delta\right) \\ P(t) = \exp\left(A d\_K(X\_d)\Delta\right) K\_1 \exp\left(a\right) K\_2$$

and show that

a; Yi ½ � a; Yj

related to <sup>Y</sup>~<sup>i</sup>

multiples of π. Let

⊥.

Applied Modern Control

AkA�<sup>1</sup> <sup>¼</sup> <sup>A</sup> <sup>∑</sup><sup>i</sup>

where λ<sup>2</sup>

stabilized by A<sup>2</sup>

A<sup>2</sup> exp ∑

exp B<sup>h</sup> δ � �

d

178

Hn ∈ exp B<sup>h</sup>

HA<sup>~</sup> �<sup>2</sup> <sup>¼</sup> <sup>H</sup><sup>~</sup> or <sup>A</sup><sup>2</sup>

i

δ � �

and hence compact.

A2

multiple of π. A2

k<sup>0</sup> ¼ f g Xi

X~i

� � � � <sup>¼</sup> a, a; Yi ½ �Yj

� � <sup>¼</sup> <sup>λ</sup><sup>2</sup>

αiXi þ ∑<sup>l</sup>

norm ∥X∥ of X ∈ k, such that p part of AXA�<sup>1</sup>

AkA�<sup>1</sup> <sup>¼</sup> <sup>∑</sup><sup>i</sup>

ð Þ� <sup>A</sup>�<sup>2</sup> .

<sup>A</sup><sup>2</sup> exp ð Þ <sup>x</sup> <sup>A</sup>�<sup>2</sup> <sup>¼</sup> exp ð Þ <sup>x</sup> , implies,

αiXi þ ∑ l

<sup>i</sup> YiYj

<sup>f</sup> <sup>¼</sup> f g ai <sup>⊕</sup> <sup>Y</sup>~<sup>i</sup>

<sup>α</sup>lX~<sup>l</sup> <sup>þ</sup> <sup>∑</sup><sup>j</sup>

<sup>s</sup> is the smallest nonzero eigenvalue of �ad<sup>2</sup>

kA�<sup>2</sup> stabilizes h∈k and f∈p. If k∈k, is stabilized by A<sup>2</sup>

fold. Given element H<sup>0</sup> ∈ H~ ∈ K, where K is closed, we have a exp B<sup>k</sup>

in exp ð Þ <sup>B</sup><sup>δ</sup> ball nghd of <sup>H</sup>0, which is one to one. For <sup>x</sup>∈B<sup>k</sup>

<sup>β</sup>lX<sup>~</sup> <sup>l</sup> <sup>þ</sup> <sup>∑</sup> j γj kj

dt ar h i ; exp ð Þ h1t ð Þ exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> exp ð Þ �h1t ¼ 0.

<sup>β</sup>lX<sup>~</sup> <sup>l</sup> <sup>þ</sup> <sup>∑</sup> j γjkj !

!

þ ∑ l

H<sup>0</sup> is a nghd of H0.

� �

� � <sup>¼</sup> <sup>0</sup> � . Let <sup>k</sup><sup>0</sup> <sup>∈</sup> <sup>k</sup> satisfy ½ �¼ <sup>a</sup>;k<sup>0</sup> 0. Then

<sup>Y</sup>~<sup>i</sup> denote eigenvectors that have <sup>λ</sup><sup>i</sup> as non-zero integral multiples of <sup>π</sup>. <sup>X</sup>~<sup>i</sup> are ada

, Xl, kj where kj forms a basis of k0, forms a basis of k. Let A ¼ exp ð Þ a .

αjkj

The range of <sup>A</sup>ð Þ� <sup>A</sup>�<sup>1</sup> in <sup>p</sup>, is perpendicular to <sup>f</sup>. Given <sup>Y</sup> <sup>∈</sup><sup>p</sup> such that <sup>Y</sup> <sup>∈</sup> <sup>f</sup>

<sup>∥</sup>X<sup>∥</sup> # <sup>∥</sup>Y<sup>∥</sup>

k∈h. This means h is an subalgebra, as the Lie bracket of ½ � y; z ∈ k for y, z∈ h is

Let <sup>H</sup> <sup>¼</sup> exp ð Þ <sup>h</sup> , be an integral manifold of <sup>h</sup>. Let <sup>H</sup><sup>~</sup> <sup>∈</sup> <sup>K</sup> be the solution to

. We now reserve Yi for non-zero eigenvectors that are not integral

, <sup>h</sup> <sup>¼</sup> <sup>k</sup><sup>0</sup> <sup>⊕</sup> <sup>X</sup>~<sup>i</sup>

<sup>α</sup><sup>i</sup> cosð Þ <sup>λ</sup><sup>i</sup> Xi � sin ð Þ <sup>λ</sup><sup>i</sup> Yi <sup>½</sup> � þ <sup>∑</sup><sup>l</sup> � <sup>α</sup>lX~<sup>l</sup> <sup>þ</sup> <sup>∑</sup><sup>j</sup>

� �

sin λ<sup>s</sup>

A�, where k∈ k

<sup>p</sup> ¼ Y satisfies

<sup>H</sup><sup>~</sup> � HA<sup>~</sup> �<sup>2</sup> <sup>¼</sup> 0. <sup>H</sup><sup>~</sup> is closed, <sup>H</sup> <sup>∈</sup> <sup>H</sup><sup>~</sup> . We show that <sup>H</sup><sup>~</sup> is a mani-

<sup>H</sup>0A�<sup>2</sup> <sup>¼</sup> exp <sup>∑</sup>

H<sup>0</sup> ¼ exp ∑

H0. Then H<sup>0</sup> is in invariant manifold exp ð Þ h . Hence exp ð Þ h is closed

then by one to one property of exp ð Þ B<sup>δ</sup> , we get α<sup>i</sup> ¼ 0 and x∈h. Therefore

Let y∈f, then there exists a h<sup>0</sup> ∈ h such that exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> ∈a. We maximize the function ar h i ; exp ð Þ h y exp ð Þ h , over the compact group exp ð Þ h , for regular element ar ∈a and h i :; : is the killing form. At the maxima, we have at t ¼ 0,

ar h i ; ½ � h<sup>1</sup> exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> ¼ �h i h1; ½ � ar exp ð Þ h<sup>0</sup> y exp ð Þ �h<sup>0</sup> ,

Given a sequence Hi ∈ exp ð Þ h converging to H0, for n large enough

δ,

αiXi þ ∑ l

i

�

i

,

αjkj

<sup>a</sup> such that λ<sup>s</sup> is not an integral

δ

α<sup>i</sup> cos 2ð Þ λ<sup>i</sup> Xi � sin 2ð Þ λ<sup>i</sup> Yi

<sup>β</sup>lX<sup>~</sup> <sup>l</sup> <sup>þ</sup> <sup>∑</sup> j γjkj

!

, λ<sup>i</sup> ¼ nπ, i.e.,

� � nghd of H0,

H0,

: (45)

ð Þ� <sup>A</sup>�<sup>2</sup>

<sup>⊥</sup>. The

$$\exp\left(A d\_K(\mathcal{X}\_d) \Delta\right) \mathcal{K}\_1 A \mathcal{K}\_2 = \mathcal{K}\_a \exp\left(a\_0 \Delta + C \Delta^2\right) A \mathcal{K}\_b = \mathcal{K}\_a \exp\left(a + a\_0 \Delta + C \Delta^2\right) \mathcal{K}\_b,\tag{46}$$

where for <sup>K</sup> <sup>¼</sup> <sup>K</sup>�<sup>1</sup> <sup>1</sup> K,

$$Ad\overline{K}(X\_d) = \underbrace{P\Big(Ad\overline{K}(X\_d)\Big)}\_{a\_0} + Ad\overline{K}(X\_d)^\perp.$$

where P is projection w.r.t killing form and a<sup>0</sup> ∈ f, the centralizer in p as defined in Remark 5, CΔ<sup>2</sup> ∈f is a second order term that can be made small by choosing Δ. Ka, Kb ∈ exp ð Þk .

To show Eq. (46), we show there exists K′ <sup>1</sup>′, K<sup>2</sup>′′ ∈K such that

$$\underbrace{\exp\left(k\mathfrak{f}\right)}\_{\text{Kf}}\exp\left(Ad\overline{\underset{K}{\mathbf{x}}(X\_{d})}\Delta\right)\underbrace{\exp\left(Ak\_{2}^{\star}\mathbf{A}^{-1}\right)}\_{\text{Kf}}=\exp\left(a\_{0}\Delta+C\Delta^{2}\right),\tag{47}$$

Where, using bound in <sup>c</sup>0′ # Mc <sup>~</sup> 0, which gives <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>b</sup><sup>0</sup> <sup>þ</sup> <sup>c</sup>0′<sup>Δ</sup> # <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>b</sup><sup>0</sup> <sup>þ</sup> <sup>c</sup>0. Using the bound again, we obtain, <sup>b</sup>0′ # Mb <sup>~</sup> 0. We can decompose, ð Þ <sup>b</sup>0′ <sup>þ</sup> <sup>c</sup>0′ <sup>Δ</sup>, into

This gives, <sup>a</sup>0″# M bð Þ <sup>0</sup>′ <sup>þ</sup> <sup>c</sup>0′ <sup>Δ</sup>, <sup>b</sup><sup>1</sup> # M bð Þ <sup>0</sup>′ <sup>þ</sup> <sup>c</sup>0′ <sup>Δ</sup> and <sup>c</sup><sup>1</sup> # M bð Þ <sup>0</sup>′ <sup>þ</sup> <sup>c</sup>0′ <sup>Δ</sup>. This gives <sup>a</sup><sup>1</sup> # <sup>a</sup><sup>0</sup> <sup>þ</sup> MM b <sup>~</sup> ð Þ <sup>0</sup> <sup>þ</sup> <sup>c</sup><sup>0</sup> <sup>Δ</sup> <sup>b</sup><sup>1</sup> # MM b <sup>~</sup> ð Þ <sup>0</sup> <sup>þ</sup> <sup>c</sup><sup>0</sup> <sup>Δ</sup> <sup>c</sup><sup>1</sup> # MM b <sup>~</sup> ð Þ <sup>0</sup> <sup>þ</sup> <sup>c</sup><sup>0</sup> <sup>Δ</sup>

ð Þ b<sup>0</sup> þ c<sup>0</sup> .

#

ð Þ b<sup>0</sup> þ c<sup>0</sup>

<sup>þ</sup>AdKð Þ Xd


<sup>2</sup>MM<sup>~</sup> <sup>Δ</sup>ð Þ <sup>b</sup><sup>0</sup> <sup>þ</sup> <sup>c</sup><sup>0</sup> <sup>1</sup> � <sup>2</sup>MM<sup>~</sup> <sup>Δ</sup>

⊥

,
