Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft

*Yuichi Ikeda*

## **Abstract**

Recent space programs require agile and large-angle attitude maneuvers for applications in various fields such as observational astronomy. To achieve agility and large-angle attitude maneuvers, it will be required to design an attitude control system that takes into account nonlinear motion because agile and large-angle rotational motion of a spacecraft in such missions represents a nonlinear system. Considerable research has been done about the nonlinear attitude tracking control of spacecraft, and these methods involve a continuous-time control framework. However, since a computer, which is a digital device, is employed as a spacecraft controller, the control method should have discrete-time control or sampled-data control framework. This chapter considers discrete-time nonlinear attitude tracking control problem of spacecraft. To this end, a Euler approximation system with respect to tracking error is first derived. Then, we design a discrete-time nonlinear attitude tracking controller so that the closed-loop system consisting of the Euler approximation system becomes input-to-state stable (ISS). Furthermore, the exact discrete-time system with a derived controller is indicated semiglobal practical asymptotic (SPA) stable. Finally, the effectiveness of proposed control method is verified by numerical simulations.

**Keywords:** spacecraft, attitude tracking control, discrete-time nonlinear control

#### **1. Introduction**

Recent space programs require agile and large-angle attitude maneuvers for applications in various fields such as observational astronomy [1–3]. To achieve agility and large-angle attitude maneuvers, it will be required to design an attitude control system that takes into account nonlinear motion because agile and largeangle rotational motion of a spacecraft in such missions represents a nonlinear system.

Considerable research has been done about the nonlinear attitude tracking control of spacecraft [4–12], and these methods involve a continuous-time control framework. However, since a computer, which is a digital device, is employed as a spacecraft controller, the control method should have discrete-time control or sampled-data control framework.

Although a sampled-data control method for nonlinear system did not advance because it is difficult to discretize a nonlinear system, a control method based on the Euler approximate model has been proposed in recent years [13, 14] and is applied to ship control [15]. Although our research group has proposed a sampled-data

control method using backstepping [16] and a discrete-time control method based on sliding mode control [17] for spacecraft control problem, these methods are disadvantageous because control input amplitude depends on the sampling period *T* as the control law is of the form *u* ¼ *a x*ð Þþ ð Þ *b x*ð Þ*=T* .

*ωe*ðÞ¼ *t ω*ðÞ�*t C t*ð Þ*ωd*ð Þ*t ,* (4)

<sup>1</sup> <sup>þ</sup> k k *<sup>σ</sup>e*ð Þ*<sup>t</sup>* <sup>2</sup> � �<sup>2</sup> *:* (5)

*G*ð Þ *σe*ð Þ*s ωe*ð Þ*s ds,* (8)

*,* ∀*wk* ∈ R3

*,*

(9)

*σ*\_*e*ðÞ¼ *t G*ð Þ *σe*ð Þ*t ωe*ð Þ*t ,* (6)

where *C t*ð Þ∈R3�<sup>3</sup> is the direction cosine matrix from f g*<sup>b</sup>* to f g*<sup>d</sup>* that expresses

Substituting Eqs. (3) and (4) into Eqs. (1) and (2) using the identity

*C t* \_ðÞ¼�*ωe*ð Þ*<sup>t</sup>* �*C t*ð Þ yields the following relative motion equations:

*ω*\_ *<sup>d</sup>*ð Þ*t* , and the disturbance *w t*ð Þ, the following assumption is made.

*σe,k*þ<sup>1</sup> ¼ *σe,k* þ

�*J Ckω*\_ *d,k* � *ωe,k*

**3. Discrete-time nonlinear attitude tracking control**

satisfies the following equation:

k k *xk*þ<sup>1</sup> <sup>≤</sup>*ρ*ð Þþ k k *<sup>x</sup>*<sup>0</sup> *; <sup>k</sup> <sup>γ</sup>*ð Þ k k *wk ,* <sup>∀</sup>*xk* <sup>∈</sup> R3

ðð Þ *<sup>k</sup>*þ<sup>1</sup> *<sup>T</sup> kT*

<sup>8</sup> *<sup>σ</sup>e*ð Þ*<sup>t</sup>* � ð Þ<sup>2</sup> � 4 1 � k k *<sup>σ</sup>e*ð Þ*<sup>t</sup>* <sup>2</sup> � �*σe*ð Þ*<sup>t</sup>* �

½�f g *ωe*ðÞþ*t C t*ð Þ*ωd*ð Þ*t* �*J*f g *ωe*ðÞþ*t C t*ð Þ*ωd*ð Þ*t*

Hereafter, we assume that the variables of spacecraft *σ*ð Þ*t* and *ω*ð Þ*t* are directly measurable and *J* is known. In addition, regarding the desired states *σd*ð Þ*t* , *ωd*ð Þ*t* ,

**Assumption 1**: the desired states *σd*ð Þ*t* , *ωd*ð Þ*t* , and *ω*\_ *<sup>d</sup>*ð Þ*t* are uniformly continuous and bounded ∀*t*∈ ½ Þ 0*;* ∞ . The disturbance *w t*ð Þ is uniformly bounded ∀*t* ∈½ Þ 0*;* ∞ . From Eqs. (A4) and (A5) in Appendix, the exact discrete-time model of relative

> ðð Þ *<sup>k</sup>*þ<sup>1</sup> *<sup>T</sup> kT*

�*J Cs*ð Þ*ω*\_ *<sup>d</sup>*ðÞ�*s ωe*ð Þ*s* � f g *C s*ð Þ*ωd*ð Þ*s* þ *uk* þ *wk*�*ds*

and the Euler approximate model of relative motion equations are obtained as

*<sup>ω</sup>e,k*þ<sup>1</sup> <sup>¼</sup> *<sup>ω</sup>e,k* � *TJ*�<sup>1</sup> �f g *<sup>ω</sup>e,k* <sup>þ</sup> *Ckωd,k* � <sup>½</sup> *<sup>J</sup>*f g *<sup>ω</sup>e,k* <sup>þ</sup> *Ckωd,k*

We derive a controller based on the backstepping approach that makes the closed-loop system consisting of the Euler approximate modes (10) and (11) become input-to-state stable (ISS), i.e., the state variable of closed-loop system

where *ρ*ð Þ� is the class KL function and *γ*ð Þ� is the class K function. To this end, assume that *ωe,k* is the virtual input to subsystem (10), and derive the stabilizing function *α<sup>k</sup>* that *σe,k* is asymptotic convergence to zero. Then, derive the control

�*J Ct*ð Þ*ω*\_ *<sup>d</sup>*ðÞ�*<sup>t</sup> <sup>ω</sup>e*ð Þ*<sup>t</sup>* � f g *C t*ð Þ*ωd*ð Þ*<sup>t</sup>* <sup>þ</sup> *u t*ð Þþ *w t*ð Þ� (7)

�f g *ωe*ðÞþ*s C s*ð Þ*ωd*ð Þ*s* � ½ *J*f g *ωe*ðÞþ*s C s*ð Þ*ωd*ð Þ*s*

*σe,k*þ<sup>1</sup> ¼ *σe,k* þ *TG* ð Þ *σe,k ωe,k,* (10)

� f g *Ckωd,k* <sup>þ</sup> *uk* <sup>þ</sup> *wk*�*:* (11)

the following Eq. [7]:

*C t*ðÞ¼ *I*<sup>3</sup> þ

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

�1

*ω*\_ *<sup>e</sup>*ðÞ¼ *t J*

motion equations is obtained as

*ωe,k*þ<sup>1</sup> ¼ *ωe,k* þ

*xk* <sup>¼</sup> *<sup>σ</sup><sup>T</sup>*

**63**

*e,kω<sup>T</sup> e,k* h i*<sup>T</sup>*

For these facts, about the spacecraft attitude control problem that requires agile and large-angle attitude maneuvers, this chapter proposed a discrete-time nonlinear attitude tracking control in which the control input amplitude is independent of the sampling period *T*. The effectiveness of proposed control method is verified by numerical simulations.

The following notations are used throughout the chapter. Let R and N denote the real and the integer numbers. Rn and R*n*�*<sup>m</sup>* are the sets of real vectors and matrices. For real vector *<sup>a</sup>*∈R*n*, *<sup>a</sup><sup>T</sup>* is the vector transpose, k k*<sup>a</sup>* denotes the Euclidean norm, and *a*� ∈R3�<sup>3</sup> is the skew symmetric matrix

$$a^\times = \begin{bmatrix} \mathbf{0} & -a\_3 & a\_2 \\ a\_3 & \mathbf{0} & -a\_1 \\ -a\_2 & a\_1 & \mathbf{0} \end{bmatrix}$$

derived from vector *a*∈R<sup>3</sup> . For real symmetric matrix *A*, *A* . 0 means the positive definite matrix. The identity matrix of size 3 � 3 is denoted by *<sup>I</sup>*3. *<sup>λ</sup>*max *<sup>A</sup>* ∈R and *λ*min *<sup>A</sup>* ∈R are the maximal and the minimal eigenvalues of a matrix *A*, respectively.

#### **2. Relative equation of motion and discrete-time model for spacecraft**

In this chapter, as the kinematics represents the attitude of the spacecraft with respect to the inertia frame f g*i* , the modified Rodrigues parameters (MRPs) [5] are used. The rotational motion equations of the spacecraft's body-fixed frame f g*b* are given by the following equations:

$$\dot{\sigma}(t) = G(\sigma(t))o(t), \tag{1}$$

$$G(\sigma(t)) = \frac{1}{2} \left\{ \frac{1 - \left\| \sigma(t) \right\|^2}{2} I\_3 + \sigma(t)\sigma(t)^T + \sigma(t)^\times \right\},$$

$$\dot{o}(t) = f^{-1} \{-o(t)^\times J o(t) + u(t) + w(t)\}, \tag{2}$$

where Eq. (1) is the kinematics that represents the attitude of f g*b* with respect to the f g*<sup>i</sup>* , Eq. (2) is the rotation dynamics, *<sup>σ</sup>*ð Þ*<sup>t</sup>* <sup>∈</sup>R<sup>3</sup> [�] is the MRPs, *<sup>ω</sup>*ð Þ*<sup>t</sup>* <sup>∈</sup> R3 [rad/s] is the angular velocity, *u t*ð Þ<sup>∈</sup> R3 [Nm] is the control torque (input), *w t*ð Þ∈R<sup>3</sup> [Nm] is the disturbance input, and *J* ∈ R3�<sup>3</sup> [kg m<sup>2</sup> ] is the moment of inertia.

We consider a control problem in which a spacecraft tracks a desired attitude (MRPs) *<sup>σ</sup>d*ð Þ*<sup>t</sup>* <sup>∈</sup> R3 and angular velocity *<sup>ω</sup>d*ð Þ*<sup>t</sup>* <sup>∈</sup>R<sup>3</sup> in fixed frame f g*<sup>d</sup>* . The MRPs of the relative attitude *<sup>σ</sup>e*ð Þ*<sup>t</sup>* <sup>∈</sup> R3 and the relative angular velocity *<sup>ω</sup>e*ð Þ*<sup>t</sup>* <sup>∈</sup> R3 in the frame f g*b* are given by

$$\sigma\_{\varepsilon}(t) = \frac{N\_{\varepsilon}(t)}{1 + \left\|\sigma(t)\right\|^2 \left\|\sigma\_d(t)\right\|^2 + 2\sigma\_d(t)^T \sigma(t)},\tag{3}$$

$$N\_{\varepsilon}(t) = \left(1 - \left\|\sigma\_d(t)\right\|^2\right)\sigma(t) - \left(1 - \left\|\sigma(t)\right\|^2\right)\sigma\_d(t) + 2\sigma(t)^\times \sigma\_d(t),$$

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft DOI: http://dx.doi.org/10.5772/intechopen.87191*

control method using backstepping [16] and a discrete-time control method based on sliding mode control [17] for spacecraft control problem, these methods are disadvantageous because control input amplitude depends on the sampling period *T*

For these facts, about the spacecraft attitude control problem that requires agile and large-angle attitude maneuvers, this chapter proposed a discrete-time nonlinear attitude tracking control in which the control input amplitude is independent of the sampling period *T*. The effectiveness of proposed control method is verified by

The following notations are used throughout the chapter. Let R and N denote the real and the integer numbers. Rn and R*n*�*<sup>m</sup>* are the sets of real vectors and matrices. For real vector *<sup>a</sup>*∈R*n*, *<sup>a</sup><sup>T</sup>* is the vector transpose, k k*<sup>a</sup>* denotes the Euclidean norm,

> 0 �*a*<sup>3</sup> *a*<sup>2</sup> *a*<sup>3</sup> 0 �*a*<sup>1</sup> �*a*<sup>2</sup> *a*<sup>1</sup> 0

3 7 5

. For real symmetric matrix *A*, *A* . 0 means the

*σ*\_ðÞ¼ *t G*ð Þ *σ*ð Þ*t ω*ð Þ*t ,* (1)

�<sup>1</sup> �*ω*ð Þ*<sup>t</sup>* � f g *<sup>J</sup>ω*ðÞþ*<sup>t</sup> u t*ðÞþ *w t*ð Þ *,* (2)

] is the moment of inertia.

k k *<sup>σ</sup>d*ð Þ*<sup>t</sup>* <sup>2</sup> <sup>þ</sup> <sup>2</sup>*σd*ð Þ*<sup>t</sup> <sup>T</sup>σ*ð Þ*<sup>t</sup> ,* (3)

*σd*ðÞþ*t* 2*σ*ð Þ*t* �*σd*ð Þ*t ,*

*,*

*<sup>I</sup>*<sup>3</sup> <sup>þ</sup> *<sup>σ</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*ð Þ*<sup>t</sup> <sup>T</sup>* <sup>þ</sup> *<sup>σ</sup>*ð Þ*<sup>t</sup>* �

( )

where Eq. (1) is the kinematics that represents the attitude of f g*b* with respect to the f g*<sup>i</sup>* , Eq. (2) is the rotation dynamics, *<sup>σ</sup>*ð Þ*<sup>t</sup>* <sup>∈</sup>R<sup>3</sup> [�] is the MRPs, *<sup>ω</sup>*ð Þ*<sup>t</sup>* <sup>∈</sup> R3 [rad/s] is the angular velocity, *u t*ð Þ<sup>∈</sup> R3 [Nm] is the control torque (input), *w t*ð Þ∈R<sup>3</sup> [Nm]

We consider a control problem in which a spacecraft tracks a desired attitude (MRPs) *<sup>σ</sup>d*ð Þ*<sup>t</sup>* <sup>∈</sup> R3 and angular velocity *<sup>ω</sup>d*ð Þ*<sup>t</sup>* <sup>∈</sup>R<sup>3</sup> in fixed frame f g*<sup>d</sup>* . The MRPs of the relative attitude *<sup>σ</sup>e*ð Þ*<sup>t</sup>* <sup>∈</sup> R3 and the relative angular velocity *<sup>ω</sup>e*ð Þ*<sup>t</sup>* <sup>∈</sup> R3 in the

*Ne*ð Þ*t*

*<sup>σ</sup>*ðÞ�*<sup>t</sup>* <sup>1</sup> � k k *<sup>σ</sup>*ð Þ*<sup>t</sup>* <sup>2</sup> � �

*<sup>A</sup>* ∈R

as the control law is of the form *u* ¼ *a x*ð Þþ ð Þ *b x*ð Þ*=T* .

and *a*� ∈R3�<sup>3</sup> is the skew symmetric matrix

derived from vector *a*∈R<sup>3</sup>

given by the following equations:

*G*ð Þ¼ *σ*ð Þ*t*

is the disturbance input, and *J* ∈ R3�<sup>3</sup> [kg m<sup>2</sup>

*σe*ðÞ¼ *t*

*Ne*ðÞ¼ *<sup>t</sup>* <sup>1</sup> � k k *<sup>σ</sup>d*ð Þ*<sup>t</sup>* <sup>2</sup> � �

<sup>1</sup> <sup>þ</sup> k k *<sup>σ</sup>*ð Þ*<sup>t</sup>* <sup>2</sup>

frame f g*b* are given by

**62**

1 2

*ω*\_ðÞ¼ *t J*

<sup>1</sup> � k k *<sup>σ</sup>*ð Þ*<sup>t</sup>* <sup>2</sup> 2

and *λ*min

respectively.

*a*� ¼

2 6 4

positive definite matrix. The identity matrix of size 3 � 3 is denoted by *<sup>I</sup>*3. *<sup>λ</sup>*max

*<sup>A</sup>* ∈R are the maximal and the minimal eigenvalues of a matrix *A*,

**2. Relative equation of motion and discrete-time model for spacecraft**

In this chapter, as the kinematics represents the attitude of the spacecraft with respect to the inertia frame f g*i* , the modified Rodrigues parameters (MRPs) [5] are used. The rotational motion equations of the spacecraft's body-fixed frame f g*b* are

numerical simulations.

*Gyroscopes - Principles and Applications*

$$
\rho\_{\mathbf{t}}(\mathbf{t}) = \boldsymbol{\alpha}(\mathbf{t}) - \mathbf{C}(\mathbf{t})\boldsymbol{\alpha}\_d(\mathbf{t}), \tag{4}
$$

where *C t*ð Þ∈R3�<sup>3</sup> is the direction cosine matrix from f g*<sup>b</sup>* to f g*<sup>d</sup>* that expresses the following Eq. [7]:

$$C(t) = I\_3 + \frac{8\left(\sigma\_\epsilon(t)^\times\right)^2 - 4\left(1 - \left\|\sigma\_\epsilon(t)\right\|^2\right)\sigma\_\epsilon(t)^\times}{\left(1 + \left\|\sigma\_\epsilon(t)\right\|^2\right)^2}. \tag{5}$$

Substituting Eqs. (3) and (4) into Eqs. (1) and (2) using the identity *C t* \_ðÞ¼�*ωe*ð Þ*<sup>t</sup>* �*C t*ð Þ yields the following relative motion equations:

$$
\dot{\sigma}\_{\varepsilon}(t) = G(\sigma\_{\varepsilon}(t)) a\_{\varepsilon}(t), \tag{6}
$$

$$\dot{\boldsymbol{\alpha}}\_{\varepsilon}(t) = \boldsymbol{f}^{-1}[-\{\boldsymbol{\alpha}\_{\varepsilon}(t) + \mathbf{C}(t)\boldsymbol{\alpha}\_{d}(t)\}^{\boldsymbol{\upbeta}}] \boldsymbol{f}\{\boldsymbol{\alpha}\_{\varepsilon}(t) + \mathbf{C}(t)\boldsymbol{\alpha}\_{d}(t)\} \tag{7}$$

$$-J\{\mathbf{C}(t)\dot{o}\_d(t) - a\_\epsilon(t)^\times \mathbf{C}(t)a\_d(t)\} + \omega(t) + w(t)\tag{7}$$

Hereafter, we assume that the variables of spacecraft *σ*ð Þ*t* and *ω*ð Þ*t* are directly measurable and *J* is known. In addition, regarding the desired states *σd*ð Þ*t* , *ωd*ð Þ*t* , *ω*\_ *<sup>d</sup>*ð Þ*t* , and the disturbance *w t*ð Þ, the following assumption is made.

**Assumption 1**: the desired states *σd*ð Þ*t* , *ωd*ð Þ*t* , and *ω*\_ *<sup>d</sup>*ð Þ*t* are uniformly continuous and bounded ∀*t*∈ ½ Þ 0*;* ∞ . The disturbance *w t*ð Þ is uniformly bounded ∀*t* ∈½ Þ 0*;* ∞ .

From Eqs. (A4) and (A5) in Appendix, the exact discrete-time model of relative motion equations is obtained as

$$
\sigma\_{\epsilon,k+1} = \sigma\_{\epsilon,k} + \int\_{kT}^{(k+1)T} G(\sigma\_{\epsilon}(s)) \alpha\_{\epsilon}(s) \, ds,\tag{8}
$$

$$\begin{aligned} \boldsymbol{\dot{\alpha}\_{\epsilon,k+1}} &= \boldsymbol{\alpha\_{\epsilon,k}} + \int\_{kT}^{(k+1)T} [-\{\boldsymbol{\alpha}\_{\epsilon}(\boldsymbol{\varsigma}) + \mathbf{C}(\boldsymbol{\varsigma})\boldsymbol{\alpha\_{d}}(\boldsymbol{\varsigma})\}^{\times}] \boldsymbol{f} \{\boldsymbol{\alpha}\_{\epsilon}(\boldsymbol{\varsigma}) + \mathbf{C}(\boldsymbol{\varsigma})\boldsymbol{\alpha\_{d}}(\boldsymbol{\varsigma})\} \\ &- \boldsymbol{f} \{\mathbf{C}(\boldsymbol{\varsigma})\boldsymbol{\dot{\alpha}\_{d}}(\boldsymbol{\varsigma}) - \boldsymbol{\alpha\_{\epsilon}}(\boldsymbol{\varsigma})^{\times}\mathbf{C}(\boldsymbol{\varsigma})\boldsymbol{\alpha\_{d}}(\boldsymbol{\varsigma})\} + \boldsymbol{u\_{k}} + \boldsymbol{w\_{k}}] d\boldsymbol{s} \end{aligned} \tag{9}$$

and the Euler approximate model of relative motion equations are obtained as

$$
\sigma\_{\varepsilon,k+1} = \sigma\_{\varepsilon,k} + T\mathcal{G}\left(\sigma\_{\varepsilon,k}\right)a\_{\varepsilon,k} \tag{10}
$$

$$\begin{split} \boldsymbol{\alpha}\_{\boldsymbol{\epsilon},k+1} &= \boldsymbol{\alpha}\_{\boldsymbol{\epsilon},k} - \boldsymbol{T} \boldsymbol{\beta}^{-1} [-\{\boldsymbol{\alpha}\_{\boldsymbol{\epsilon},k} + \mathbf{C}\_{k} \boldsymbol{\alpha}\_{d,k}\}^{\times}]^{\times} \boldsymbol{f} \{\boldsymbol{\alpha}\_{\boldsymbol{\epsilon},k} + \mathbf{C}\_{k} \boldsymbol{\alpha}\_{d,k}\} \\ &- \boldsymbol{J} \{\mathbf{C}\_{k} \boldsymbol{\dot{\alpha}}\_{\boldsymbol{d},k} - \boldsymbol{\alpha}\_{\boldsymbol{\epsilon},k} \times \mathbf{C}\_{k} \boldsymbol{\alpha}\_{d,k}\} + \boldsymbol{u}\_{k} + \boldsymbol{w}\_{k} \big]. \end{split} \tag{11}$$

#### **3. Discrete-time nonlinear attitude tracking control**

We derive a controller based on the backstepping approach that makes the closed-loop system consisting of the Euler approximate modes (10) and (11) become input-to-state stable (ISS), i.e., the state variable of closed-loop system *xk* <sup>¼</sup> *<sup>σ</sup><sup>T</sup> e,kω<sup>T</sup> e,k* h i*<sup>T</sup>* satisfies the following equation:

$$||\varkappa\_{k+1}|| \le \rho(||\varkappa\_0||,k) + \gamma(||\varkappa\_k||), \quad \forall \varkappa\_k \in \mathbb{R}^3, \quad \forall w\_k \in \mathbb{R}^3,$$

where *ρ*ð Þ� is the class KL function and *γ*ð Þ� is the class K function. To this end, assume that *ωe,k* is the virtual input to subsystem (10), and derive the stabilizing function *α<sup>k</sup>* that *σe,k* is asymptotic convergence to zero. Then, derive the control

input *uk* that closed-loop system becomes ISS. Here, regarding the variable *σe,k*, the following assumption is made.

*Tf* <sup>1</sup>*bk*

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

0 , *f* <sup>1</sup> ,

0 , *f* <sup>1</sup> ,

Therefore, if *f* <sup>1</sup> satisfies Eq. (17) and *ωe,k* ! *αk*ð Þ *k* ! ∞ , then *σe,k* ! 0.

The error variable between the state *ωe,k* and *α<sup>k</sup>* is defined as

2 *Tbk*

4

In addition, since 2≤ð Þ 1*=bk* ≤4 under Assumption 2, the range of *f* <sup>1</sup> that holds

The control input *uk* that makes the closed-loop system becomes ISS is derived.

*α<sup>k</sup>* � *α<sup>k</sup>*þ<sup>1</sup> ¼ *Tf* <sup>1</sup> *G*ð Þ *σe,k zk* � *f* <sup>1</sup>*bkσe,k*

� *J C*f g *<sup>k</sup>ω*\_ *d,k* � ð Þ� *zk* þ *α,k Ckωd,k* þ *uk* þ *wk*�*:*

*uk* ¼ f g *zk* þ *α,k* þ *Ckωd,k* � *J z*f g *<sup>k</sup>* þ *α,k* þ *Ckωd,k*

� *<sup>f</sup>* <sup>2</sup>*Jzk,*

*zk* <sup>þ</sup> *TJ*�<sup>1</sup>

<sup>2</sup> <sup>¼</sup> k k *Xk*

2

*, Xk* <sup>¼</sup> *<sup>σ</sup><sup>T</sup>*

*e,kz<sup>T</sup> k <sup>T</sup>*

where *f* <sup>2</sup> ∈R is the feedback gain. The candidate Lyapunov function for

þ *J C*f g *<sup>k</sup>ω*\_ *d,k* � ð Þ� *zk* þ *α,k Ckωd,k*

� *f* <sup>1</sup>*J G*ð Þ *σe,k zk* � *f* <sup>1</sup>*bkσe,k*

*zk*þ<sup>1</sup> ¼ 1 � *Tf* <sup>2</sup>

*V*2ð Þ¼ *k V*1ð Þþ *k* k k *zk*

is obtained as

Eq. (15) is obtained as

**3.2 Derivation of control input** *uk*

From Eq. (18), subsystem (10) becomes

From Eqs. (18) and (19) and the following equation

the discrete-time equation with respect to *zk* is

*zk*þ<sup>1</sup> ¼ *zk* þ *Tf* <sup>1</sup> *G*ð Þ *σe,k zk* � *f* <sup>1</sup>*bkσe,k*

<sup>þ</sup> *TJ*�<sup>1</sup>

Now, by setting *uk* to

Eq. (20) becomes

**65**

Eqs. (19) and (21) is defined as

<sup>2</sup> � <sup>2</sup>*Tf* <sup>1</sup>*bk* , <sup>0</sup> (15)

*:* (16)

*<sup>T</sup> :* (17)

(20)

*wk,* (21)

*:* (22)

*zk* ≔ *ωe,k* � *αk:* (18)

*σe,k*þ<sup>1</sup> ¼ *σe,k* þ *TG* ð Þ *σe,k* ð Þ *zk* þ *α<sup>k</sup> :* (19)

*,*

½�f g *zk* þ *α,k* þ *Ckωd,k* � *J z*f g *<sup>k</sup>* þ *α,k* þ *Ckωd,k*

**Assumption 2:** *σe,k* lies in the region that satisfies the following equation:

$$\mathbf{0} \le \|\sigma\_{e,k}\| \le \mathbf{1}, \quad \forall k \dots$$

**Remark 1:** from the relational expression

$$
\sigma\_{e,k} = \frac{e\_{e,k}}{1 + \eta\_{e,k}},
$$

where *εe,k* ∈R<sup>3</sup> and *ηe,k* ∈R are the quaternion *ε<sup>T</sup> e,kηe,k* h i*<sup>T</sup>* � � � � � � � � <sup>¼</sup> �

1*,* k k *εe,k* ≤1*, ηe,k* � � � �≤1*,* ∀*k*Þ. Assumption 2 is equivalent to *ηe,k* ∈½ � 0*;* 1 .

In addition, Lemmas when using the derivation of the control law are shown below.

**Lemma 1:** for all *σ* ∈R<sup>3</sup> , the following equations hold [5]:

$$b\sigma^T G(\sigma) = b\sigma^T, \quad G(\sigma)^T G(\sigma) = b^2 I\_3, \quad \left(b = \frac{\mathbf{1} + \left\|\sigma\right\|^2}{4} > \mathbf{0}\right).$$

**Lemma 2:** when the quadratic equation

$$a\mathfrak{x}^2 + b\mathfrak{x} + c = \mathbf{0} \\ (a, b, c \in \mathbb{R}).$$

has two distinct real roots *x* ¼ *α, β α*ð Þ , *β* , if *a* . 0, then the solution of the quadratic inequality

$$a\mathbf{x}^2 + b\mathbf{x} + c \le \mathbf{0}$$

is *α* , *x* , *β*.

### **3.1 Derivation of virtual input** *α<sup>k</sup>*

Assume that *ωe,k* is the virtual input to subsystem (10), and define the stabilizing function such that

$$a\_{\varepsilon,k} = a\_k = -f\_{\,1} \sigma\_{\varepsilon,k},\tag{12}$$

where *f* <sup>1</sup> ∈ R is the feedback gain. The candidate Lyapunov function for (10) is defined as

$$\left|V\_1(k) = \left|\sigma\_{\epsilon,k}\right|\right|^2. \tag{13}$$

From Lemma 1, the difference of Eq. (13) along the trajectories of the closedloop system is given by

$$
\Delta V\_1(k) = V\_1(k+1) - V\_1(k) = \left\{ \left( T f\_1 b\_k \right)^2 - 2T f\_1 b\_k \right\} ||\sigma\_{\varepsilon,k}||^2. \tag{14}
$$

From Lemma 2, *ΔV*1ð Þ*k* becomes negative, i.e., the range of *f* <sup>1</sup> that holds the following equation

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft DOI: http://dx.doi.org/10.5772/intechopen.87191*

$$\left(\left(Tf\_1b\_k\right)^2 - 2Tf\_1b\_k < 0\right) \tag{15}$$

is obtained as

input *uk* that closed-loop system becomes ISS. Here, regarding the variable *σe,k*, the

0≤k k *σe,k* ≤1*,* ∀*k:*

**Assumption 2:** *σe,k* lies in the region that satisfies the following equation:

*<sup>σ</sup>e,k* <sup>¼</sup> *<sup>ε</sup>e,k*

�≤1*,* ∀*k*Þ. Assumption 2 is equivalent to *ηe,k* ∈½ � 0*;* 1 . In addition, Lemmas when using the derivation of the control law are shown

, the following equations hold [5]:

*ax*<sup>2</sup> <sup>þ</sup> *bx* <sup>þ</sup> *<sup>c</sup>* <sup>¼</sup> <sup>0</sup>ð Þ *<sup>a</sup>; <sup>b</sup>;c*∈<sup>R</sup>

has two distinct real roots *x* ¼ *α, β α*ð Þ , *β* , if *a* . 0, then the solution of the

*ax*<sup>2</sup> <sup>þ</sup> *bx* <sup>þ</sup> *<sup>c</sup>* , <sup>0</sup>

Assume that *ωe,k* is the virtual input to subsystem (10), and define the

where *f* <sup>1</sup> ∈ R is the feedback gain. The candidate Lyapunov function for (10) is

2

� �<sup>2</sup> � <sup>2</sup>*Tf* <sup>1</sup>*bk* n o

*V*1ð Þ¼ *k* k k *σe,k*

From Lemma 1, the difference of Eq. (13) along the trajectories of the closed-

From Lemma 2, *ΔV*1ð Þ*k* becomes negative, i.e., the range of *f* <sup>1</sup> that holds the

*ΔV*1ð Þ¼ *k V*1ð Þ� *k* þ 1 *V*1ð Þ¼ *k Tf* <sup>1</sup>*bk*

1 þ *ηe,k*

*,*

� � �

*<sup>I</sup>*3*, b* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> k k*<sup>σ</sup>* <sup>2</sup>

*ωe,k* ¼ *α<sup>k</sup>* ¼ �*f* <sup>1</sup>*σe,k,* (12)

*:* (13)

k k *σe,k* 2

*:* (14)

4

!

. 0

*:*

�

*e,kηe,k* h i*<sup>T</sup>* �

� � � � <sup>¼</sup>

following assumption is made.

*Gyroscopes - Principles and Applications*

1*,* k k *εe,k* ≤1*, ηe,k* � � �

quadratic inequality

is *α* , *x* , *β*.

defined as

**3.1 Derivation of virtual input** *α<sup>k</sup>*

stabilizing function such that

loop system is given by

following equation

**64**

**Lemma 1:** for all *σ* ∈R<sup>3</sup>

below.

**Remark 1:** from the relational expression

where *εe,k* ∈R<sup>3</sup> and *ηe,k* ∈R are the quaternion *ε<sup>T</sup>*

*<sup>σ</sup>TG*ð Þ¼ *<sup>σ</sup> <sup>b</sup>σT, G*ð Þ *<sup>σ</sup> TG*ð Þ¼ *<sup>σ</sup> <sup>b</sup>*<sup>2</sup>

**Lemma 2:** when the quadratic equation

$$0 < f\_1 < \frac{2}{Tb\_k}.\tag{16}$$

In addition, since 2≤ð Þ 1*=bk* ≤4 under Assumption 2, the range of *f* <sup>1</sup> that holds Eq. (15) is obtained as

$$0 < f\_1 < \frac{4}{T}.\tag{17}$$

Therefore, if *f* <sup>1</sup> satisfies Eq. (17) and *ωe,k* ! *αk*ð Þ *k* ! ∞ , then *σe,k* ! 0.

#### **3.2 Derivation of control input** *uk*

The error variable between the state *ωe,k* and *α<sup>k</sup>* is defined as

$$z\_k \coloneqq \alpha\_{\varepsilon,k} - \alpha\_k. \tag{18}$$

The control input *uk* that makes the closed-loop system becomes ISS is derived. From Eq. (18), subsystem (10) becomes

$$
\sigma\_{\varepsilon,k+1} = \sigma\_{\varepsilon,k} + T\mathcal{G}\left(\sigma\_{\varepsilon,k}\right)(z\_k + a\_k). \tag{19}
$$

From Eqs. (18) and (19) and the following equation

$$a\_k - a\_{k+1} = T f\_1 \{ G(\sigma\_{e,k}) z\_k - f\_1 b\_k \sigma\_{e,k} \},$$

the discrete-time equation with respect to *zk* is

$$\begin{split} z\_{k+1} &= z\_k + T f\_1 \left\{ G(\sigma\_{\epsilon,k}) z\_k - f\_1 b\_k \sigma\_{\epsilon,k} \right\} \\ &+ T f^{-1} [-\{ z\_k + a\_{,k} + C\_k a\_{d,k} \} \times f \{ z\_k + a\_{,k} + C\_k a\_{d,k} \} \\ &- f \{ C\_k \dot{\alpha}\_{d,k} - (z\_k + a\_{,k}) \times C\_k a\_{d,k} \} + u\_k + w\_k ]. \end{split} \tag{20}$$

Now, by setting *uk* to

$$\begin{aligned} u\_k &= \{z\_k + a\_{,k} + \mathcal{C}\_k a\_{d,k}\} \times f\{z\_k + a\_{,k} + \mathcal{C}\_k a\_{d,k}\} \\ &+ f\{\mathcal{C}\_k \dot{\alpha}\_{d,k} - (z\_k + a\_{,k}) \times \mathcal{C}\_k a\_{d,k}\} \\ &- f\, f\{\mathcal{G}(\sigma\_{e,k}) z\_k - f\, \_1 b\_k \sigma\_{e,k}\} - f\, \_2 Iz\_k, \end{aligned}$$

Eq. (20) becomes

$$z\_{k+1} = (1 - Tf\_2)z\_k + Tf^{-1}w\_k.\tag{21}$$

where *f* <sup>2</sup> ∈R is the feedback gain. The candidate Lyapunov function for Eqs. (19) and (21) is defined as

$$V\_2(k) = V\_1(k) + \|\mathbf{z}\_k\|^2 = \|X\_k\|^2,\\ X\_k = \left[\sigma\_{e,k}^T \mathbf{z}\_k^T\right]^T. \tag{22}$$

As Eq. (14) is given by

$$\Delta V\_1(k) = \left(Tb\_k\right)^2 \left\|z\_k\right\|^2 + \left\{ \left(Tf\_1b\_k\right)^2 - 2Tf\_1b\_k \right\} \left\|\sigma\_{\epsilon,k}\right\|^2 + 2Tb\_k\left(1 - Tf\_1b\_k\right)z\_k^T\sigma\_{\epsilon,k}^T$$

must hold true in order to obtain a real number. As the denominator of Eq. (27)

in order to hold Eq. (27). From Lemma 2, the range of *f* <sup>1</sup> that holds for Eq. (28) is

1 þ

<sup>2</sup> . <sup>0</sup> ) <sup>0</sup> , *<sup>T</sup>* ,

must hold in order to have the real number. As 2≤ ð Þ 1*=bk* ≤4 under Assumption

2

2

<sup>2</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>4</sup> � *<sup>T</sup>*<sup>2</sup> <sup>p</sup>

Therefore, if *f* <sup>1</sup> satisfies Eq. (32) under Assumption 2, Eqs. (27) and (28) hold.

*T* �

*T* þ

*Tf* <sup>2</sup>

2*T*<sup>2</sup>

Therefore, if *f* <sup>1</sup> and *f* <sup>2</sup> satisfy Eqs. (32) and (33) under Assumption 2, then

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ *Tbk*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ *Tbk*

q

*Tbk*

<sup>¼</sup> <sup>2</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>4</sup> � *<sup>T</sup>*<sup>2</sup> <sup>p</sup> *<sup>T</sup> ,*

<sup>¼</sup> <sup>2</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>4</sup> � *<sup>T</sup>*<sup>2</sup> <sup>p</sup> *T*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>f</sup>* <sup>1</sup> *Tf* <sup>1</sup> � <sup>4</sup> � � <sup>s</sup>

<sup>1</sup> � 4*f* <sup>1</sup> þ *T*

*<sup>f</sup>* <sup>1</sup> *Tf* <sup>1</sup> � <sup>4</sup> � � <sup>s</sup>

<sup>1</sup> � 4*f* <sup>1</sup> þ *T*

*Tf* <sup>2</sup>

2*T*<sup>2</sup>

*Tf* <sup>2</sup>

2*T*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>f</sup>* <sup>1</sup> *Tf* <sup>1</sup> � <sup>4</sup> � � <sup>s</sup>

<sup>1</sup> � 4*f* <sup>1</sup> þ *T*

, *f* <sup>1</sup> ,

<sup>1</sup> � 2*f* <sup>1</sup> þ *Tbk* , 0 (28)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ *Tbk*

> 1 *bk*

0 , *T* , 2*:* (31)

*<sup>T</sup>* ð Þ <sup>0</sup> , *<sup>T</sup>* , <sup>2</sup> *:* (32)

*,*

*,*

ð Þ 0 , *T* , 2 *:* (33)

2

*,* (29)

(30)

is the same as Eq. (24), the following equation must hold

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

1 �

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

and the following Eq.

2, *T* must satisfy the condition

In addition, since

Furthermore, since

1 *T* �

*Qk* , 0.

**67**

q

max*bk*

min*bk*

<sup>2</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>4</sup> � *<sup>T</sup>*<sup>2</sup> <sup>p</sup> *T*

max *bk*

min *bk*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>f</sup>* <sup>1</sup> *Tf* <sup>1</sup> � <sup>4</sup> � � <sup>s</sup>

<sup>1</sup> � 4*f* <sup>1</sup> þ *T*

*Tf* <sup>2</sup>

2*T*<sup>2</sup>

1 �

1 þ

under Assumption 2, the condition (29) is given by

<sup>2</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ð Þ � *ck*

<sup>2</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ð Þ � *ck*

p

p

under Assumption 2, the condition (26) is given by.

, *f* <sup>2</sup> ,

q

q

*Tbk*

*Tbk*

, *f* <sup>1</sup> ,

<sup>2</sup>*<sup>T</sup>* <sup>¼</sup> <sup>1</sup>

<sup>2</sup>*<sup>T</sup>* <sup>¼</sup> <sup>1</sup>

1 *T* þ

Summarizing the above, the following theorem can be obtained.

obtained as

*Tbkf* 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ *Tbk*

*Tbk*

1 � ð Þ *Tbk*

2

from Eq. (18), by using completing square, the difference of Eq. (22) along the trajectories of the closed-loop system is given by

$$\begin{split} \Delta V\_{2}(k) &= \left(T^{2}f\_{2}^{2} - 2Tf\_{2} + T^{2}b\_{k}^{2}\right) \|x\_{k}\|^{2} + \left\{\left(Tf\_{1}b\_{k}\right)^{2} - 2Tf\_{1}^{\prime}b\_{k}\right\} \|\sigma\_{\epsilon,k}\|^{2} \\ &\quad + 2Tb\_{k}\left(1 - Tf\_{1}b\_{k}\right)z\_{k}^{T}\sigma\_{\epsilon,k}^{T} + T^{2}w\_{k}^{T}f^{-2}w\_{k} + 2T(1 - Tf\_{2})w\_{k}^{T})^{-1}x\_{k} \\ &\leq \left(2T^{2}f\_{2}^{2} - 4Tf\_{2} + T^{2}b\_{k}^{2} + 1\right) \|x\_{k}\|^{2} + \left\{\left(Tf\_{1}b\_{k}\right)^{2} - 2Tf\_{1}b\_{k}\right\} \|\sigma\_{\epsilon,k}\|^{2} \\ &\quad + 2Tb\_{k}\left(1 - Tf\_{1}b\_{k}\right)z\_{k}^{T}\sigma\_{\epsilon,k}^{T} + 2\left(\frac{T}{\lambda\_{l}}\right)^{2} \|w\_{k}\|^{2} \\ &= X\_{k}^{T}Q\_{k}X\_{k} + 2\left(\frac{T}{\lambda\_{l}}\right)^{2} \|w\_{k}\|^{2}, \end{split} \tag{23}$$

where

$$\lambda\_l = ||f||,\\\ Q\_k = \begin{bmatrix} Q\_{11,k} & Q\_{12,k} \\ Q\_{12,k}^T & Q\_{22,k} \end{bmatrix},\\\ Q\_{11,k} = \left\{ \left( T f\_1 b\_k \right)^2 - 2 T f\_1 b\_k \right\} I\_{34},$$

$$\mathcal{Q}\_{12,k} = T b\_k \left( 1 - T f\_1 b\_k \right) I\_{34},\\\ Q\_{22,k} = \left( 2 T^2 f\_2^2 - 4 T f\_2 + T^2 b\_k^2 + 1 \right) I\_{34}.$$

In Eq. (23), if *Qk* , 0, then

$$
\Delta V\_2(k) \le -\left|\lambda\_{Q\_k}^{\min}\right| \|X\_k\|^2 + 2\left(\frac{T}{\lambda\_I}\right)^2 \|w\_k\|^2,
$$

where *λ*min *Qk* , 0∈R is the minimum eigenvalue of *Qk* and the condition of ISS holds [18]. Hereafter, conditions of *f* <sup>1</sup> and *f* <sup>2</sup> which the matrix *Qk* holds *Qk* , 0 are derived under Assumption 2.

From Schur complement, condition *Qk* , 0 is equivalent to the following equations:

$$\left(\left(Tf\_1 b\_k\right)^2 - 2Tf\_1 b\_k \le 0,\tag{24}$$

$$-2T^2f\_2^2 - 4Tf\_2 + c\_k < 0 \quad \left(c\_k = \frac{Tb\_kf\_1^2 - 2f\_1 - Tb\_k}{Tb\_kf\_1^2 - 2f\_1}\right). \tag{25}$$

Condition (24) is the same as Eq. (15), and assume that Eq. (24) holds. From Lemma 2, the range of *f* <sup>2</sup> that holds for Eq. (25) is obtained as

$$\frac{2 - \sqrt{2(2 - c\_k)}}{2T} < f\_2 < \frac{2 + \sqrt{2(2 - c\_k)}}{2T},\tag{26}$$

and the following Eq.

$$2 - c\_k > 0 \quad \Rightarrow \quad \frac{Tb\_k f\_1^2 - 2f\_1 + Tb\_k}{Tb\_k f\_1^2 - 2f\_1} > 0 \tag{27}$$

must hold true in order to obtain a real number. As the denominator of Eq. (27) is the same as Eq. (24), the following equation must hold

$$\left\| Tb\_{b}f\_{1}^{2} - \mathfrak{H}\_{1} + Tb\_{k} \le 0 \right\| \tag{28}$$

in order to hold Eq. (27). From Lemma 2, the range of *f* <sup>1</sup> that holds for Eq. (28) is obtained as

$$\frac{\mathbf{1} - \sqrt{\mathbf{1} - \left(Tb\_k\right)^2}}{Tb\_k} < f\_1 < \frac{\mathbf{1} + \sqrt{\mathbf{1} - \left(Tb\_k\right)^2}}{Tb\_k},\tag{29}$$

and the following Eq.

As Eq. (14) is given by

*f* 2

≤ 2*T*<sup>2</sup> *f* 2

<sup>¼</sup> *<sup>X</sup><sup>T</sup>*

where

where *λ*min

equations:

**66**

2 k k *zk*

*Gyroscopes - Principles and Applications*

<sup>2</sup> <sup>þ</sup> *Tf* <sup>1</sup>*bk*

*b*2 *k*

> *<sup>k</sup> σ<sup>T</sup> e, <sup>k</sup>* <sup>þ</sup> *<sup>T</sup>*<sup>2</sup>

> > *b*2

*<sup>k</sup> σ<sup>T</sup> e, <sup>k</sup>* þ 2

k k *wk* 2 *,*

<sup>12</sup>*,k Q*22*,k*

*<sup>Δ</sup>V*2ð Þ*<sup>k</sup>* <sup>≤</sup> � *<sup>λ</sup>*min

� �*I*3*, Q*22*,k* <sup>¼</sup> <sup>2</sup>*T*<sup>2</sup>

*Qk* � � �

*Tf* <sup>1</sup>*bk*

Lemma 2, the range of *f* <sup>2</sup> that holds for Eq. (25) is obtained as

<sup>2</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ð Þ � *ck*

2 � *ck* . 0 )

p 2*T*

<sup>2</sup> � <sup>4</sup>*Tf* <sup>2</sup> <sup>þ</sup> *ck* , <sup>0</sup> *ck* <sup>¼</sup> *Tbkf*

� � �k k *Xk*

" #

trajectories of the closed-loop system is given by

� �k k *zk*

� �*z<sup>T</sup>*

<sup>2</sup> � <sup>4</sup>*Tf* <sup>2</sup> <sup>þ</sup> *<sup>T</sup>*<sup>2</sup>

� �*z<sup>T</sup>*

*λJ* � �<sup>2</sup>

*<sup>λ</sup><sup>J</sup>* <sup>¼</sup> k k*<sup>J</sup> , Qk* <sup>¼</sup> *<sup>Q</sup>*11*,k <sup>Q</sup>*12*,k Q<sup>T</sup>*

*Q*12*,k* ¼ *Tbk* 1 � *Tf* <sup>1</sup>*bk*

In Eq. (23), if *Qk* , 0, then

derived under Assumption 2.

and the following Eq.

2*T*<sup>2</sup> *f* 2

*<sup>k</sup>* <sup>þ</sup> <sup>1</sup> � �k k *zk*

<sup>2</sup> � <sup>2</sup>*Tf* <sup>2</sup> <sup>þ</sup> *<sup>T</sup>*<sup>2</sup>

þ 2*Tbk* 1 � *Tf* <sup>1</sup>*bk*

þ 2*Tbk* 1 � *Tf* <sup>1</sup>*bk*

*<sup>k</sup> QkXk* <sup>þ</sup> <sup>2</sup> *<sup>T</sup>*

� �<sup>2</sup> � <sup>2</sup>*Tf* <sup>1</sup>*bk* n o

from Eq. (18), by using completing square, the difference of Eq. (22) along the

<sup>2</sup> <sup>þ</sup> *Tf* <sup>1</sup>*bk*

*w<sup>T</sup> k J* �2

*T λJ* � �<sup>2</sup> k k *σe,k*

� �<sup>2</sup> � <sup>2</sup>*Tf* <sup>1</sup>*bk* n o

<sup>2</sup> <sup>þ</sup> *Tf* <sup>1</sup>*bk*

k k *wk* 2

*, Q*11*,k* ¼ *Tf* <sup>1</sup>*bk*

*f* 2

<sup>2</sup> <sup>þ</sup> <sup>2</sup> *<sup>T</sup> λJ* � �<sup>2</sup>

*Qk* , 0∈R is the minimum eigenvalue of *Qk* and the condition of ISS

holds [18]. Hereafter, conditions of *f* <sup>1</sup> and *f* <sup>2</sup> which the matrix *Qk* holds *Qk* , 0 are

Condition (24) is the same as Eq. (15), and assume that Eq. (24) holds. From

, *f* <sup>2</sup> ,

*Tbkf* 2

> *Tbkf* 2 <sup>1</sup> � 2*f* <sup>1</sup>

From Schur complement, condition *Qk* , 0 is equivalent to the following

*wk* þ 2*T* 1 � *Tf* <sup>2</sup>

� �<sup>2</sup> � <sup>2</sup>*Tf* <sup>1</sup>*bk* n o

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*Tbk* <sup>1</sup> � *Tf* <sup>1</sup>*bk*

k k *σe,k* 2

> *k J* �1 *zk*

> > k k *σe,k* 2

> > > *I*3*,*

*:* (25)

<sup>2</sup>*<sup>T</sup> ,* (26)

. 0 (27)

� �*w<sup>T</sup>*

� �<sup>2</sup> � <sup>2</sup>*Tf* <sup>1</sup>*bk* n o

*b*2

<sup>2</sup> � <sup>4</sup>*Tf* <sup>2</sup> <sup>þ</sup> *<sup>T</sup>*<sup>2</sup>

k k *wk* 2 *,*

� �<sup>2</sup> � <sup>2</sup>*Tf* <sup>1</sup>*bk* , <sup>0</sup>*,* (24)

<sup>1</sup> � 2*f* <sup>1</sup> � *Tbk*

2

!

*Tbkf* 2 <sup>1</sup> � 2*f* <sup>1</sup>

<sup>2</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ð Þ � *ck*

p

<sup>1</sup> � 2*f* <sup>1</sup> þ *Tbk*

*<sup>k</sup>* <sup>þ</sup> <sup>1</sup> � �*I*3*:*

� �*zT*

*<sup>k</sup> σ<sup>T</sup> e,k*

(23)

*ΔV*1ð Þ¼ *k* ð Þ *Tbk*

*<sup>Δ</sup>V*2ð Þ¼ *<sup>k</sup> <sup>T</sup>*<sup>2</sup>

$$\mathbf{1} - (Tb\_k)^2 > \mathbf{0} \quad \Rightarrow \quad \mathbf{0} \le T \le \frac{1}{b\_k} \tag{30}$$

must hold in order to have the real number. As 2≤ ð Þ 1*=bk* ≤4 under Assumption 2, *T* must satisfy the condition

$$0 \le T \le 2.\tag{31}$$

In addition, since

$$\max\_{b\_k} \frac{1 - \sqrt{1 - \left(Tb\_k\right)^2}}{Tb\_k} = \frac{2 - \sqrt{4 - T^2}}{T},$$

$$\min\_{b\_k} \frac{1 + \sqrt{1 - \left(Tb\_k\right)^2}}{Tb\_k} = \frac{2 + \sqrt{4 - T^2}}{T}$$

under Assumption 2, the condition (29) is given by

$$\frac{2 - \sqrt{4 - T^2}}{T} < f\_1 < \frac{2 + \sqrt{4 - T^2}}{T} (0 < T < 2). \tag{32}$$

Therefore, if *f* <sup>1</sup> satisfies Eq. (32) under Assumption 2, Eqs. (27) and (28) hold. Furthermore, since

$$\max\_{b\_k} \frac{2 - \sqrt{2(2 - c\_k)}}{2T} = \frac{1}{T} - \sqrt{\frac{T f\_1^2 - 4f\_1 + T}{2T^2 f\_1 (T f\_1 - 4)}},$$

$$\min\_{b\_k} \frac{2 + \sqrt{2(2 - c\_k)}}{2T} = \frac{1}{T} + \sqrt{\frac{T f\_1^2 - 4f\_1 + T}{2T^2 f\_1 (T f\_1 - 4)}},$$

under Assumption 2, the condition (26) is given by.

$$\frac{1}{T} - \sqrt{\frac{T f\_1^2 - 4f\_1 + T}{2T^2 f\_1 (T f\_1 - 4)}} < f\_2 < \frac{1}{T} + \sqrt{\frac{T f\_1^2 - 4f\_1 + T}{2T^2 f\_1 (T f\_1 - 4)}} (0 < T < 2). \tag{33}$$

Therefore, if *f* <sup>1</sup> and *f* <sup>2</sup> satisfy Eqs. (32) and (33) under Assumption 2, then *Qk* , 0.

Summarizing the above, the following theorem can be obtained.

**Theorem 1:** if sampling period *T* and feedback gains *f* <sup>1</sup> and *f* <sup>2</sup> satisfy Eqs. (31), (32), and (33) under Assumption 2, then the closed-loop systems (10) and (11) with the following control law

$$u\_k = \{z\_k + a\_{,k} + \mathcal{C}\_k \boldsymbol{o}\_{d,k}\}^\times \{z\_k + a\_{,k} + \mathcal{C}\_k \boldsymbol{o}\_{d,k}\}$$

$$+ f\{\mathcal{C}\_k \dot{\boldsymbol{o}}\_{d,k} - (\mathbf{z}\_k + a\_{,k})^\times \mathcal{C}\_k \boldsymbol{o}\_{d,k}\}$$

$$- f\_1 f\{\mathcal{G}(\sigma\_{\epsilon,k}) \mathbf{z}\_k - f\_1 \mathbf{b}\_k \sigma\_{\epsilon,k}\} - f\_2 f\_2 \mathbf{z}\_k \tag{34}$$

$$= \boldsymbol{o}\_k \, ^\times f \boldsymbol{o}\_k + f\{\mathcal{C}\_k \dot{\boldsymbol{o}}\_{d,k} - (\mathbf{z}\_k + a\_{,k})^\times \mathcal{C}\_k \boldsymbol{o}\_{d,k}\}$$

$$- f\_1 f\_2 f \sigma\_{\epsilon,k} - f\{f\_1 \mathcal{G}(\sigma\_{\epsilon,k}) + f\_2 I\_3\} \boldsymbol{o}\_{\epsilon,k}$$

Euler approximate systems (10) and (11). Then, the following theorem can be

**Theorem 2:** control input (34) is SPA stabilizing for exact discrete-time systems

The properties of the proposed method are discussed in the numerical study. For

*, σ*ð Þ¼ 0

*, f* <sup>1</sup> ¼ 0*:*6*, f* <sup>2</sup> ¼ 0*:*8*:*

0 0 0 3 7 <sup>5</sup>*, <sup>ω</sup>*ð Þ¼ <sup>0</sup>

0 0 0

2 6 4

2 6 4

3 7 5kgm<sup>2</sup>

The moment of inertia *J* is from [1]. The initial values *σ*ð Þ 0 correspond to Euler angles of 1–2-3 system of *<sup>θ</sup>*ð Þ¼ <sup>0</sup> ½ � *<sup>θ</sup>*1ð Þ <sup>0</sup> *<sup>θ</sup>*2ð Þ <sup>0</sup> *<sup>θ</sup>*3ð Þ <sup>0</sup> *<sup>T</sup>* <sup>¼</sup> ½ � <sup>000</sup> *<sup>T</sup>* ½ � deg . The feedback gains *f* <sup>1</sup> and *f* <sup>2</sup> satisfy Eqs. (25) and (28) for all cases of *T*. The desired states *σd*ð Þ*t* , *ωd*ð Þ*t* , and *ω*\_ *<sup>d</sup>*ð Þ*t* in this simulation are the switching maneuver as shown

1*:*0 : Case 1 0*:*5 : Case 2 0*:*1 : Case 3

obtained by Theorem A.1 in Appendix.

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

this purpose, parameter setting of simulation is as follows:

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

8 ><

>:

7050*:*0 �0*:*536 43*:*9 �0*:*536 2390 1640*:*0 43*:*9 1640*:*0 6130*:*0

*T* ¼

**4. Numerical simulation**

(8) and (9).

*J* ¼

in **Figure 1**.

**Figure 1.**

**69**

*Switching maneuver.*

2 6 4

becomes ISS.

Then, we show that the pair ð Þ *uk;V*2ð Þ*k* is semiglobal practical asymptotic (SPA) stabilizing pair for the Euler approximate systems (10) and (11). Hereafter, suppose that sampling period *T* and feedback gains *f* <sup>1</sup> and *f* <sup>2</sup> satisfy Eqs. (31), (32), and (33) under Assumption 2. By using the following coordinate transformation

$$X\_k = \begin{bmatrix} \mathbf{1} & \mathbf{0} \\ f\_1 & \mathbf{1} \end{bmatrix} \begin{bmatrix} \sigma\_{\epsilon,k} \\ a\_{\epsilon,k} \end{bmatrix} = Z \overline{X}\_k.$$

Lyapunov function *V*2ð Þ*k* and its difference *ΔV*2ð Þ*k* can be rewritten as

$$V\_2(k) = \overline{X}\_k^T Z^T Z \overline{X}\_k = \overline{X}\_k^T R \overline{X}\_k,$$

$$\Delta V\_2(k) = \overline{X}\_k^T Z^T Q\_k Z \overline{X}\_k + 2\left(\frac{T}{\lambda\_l}\right)^2 ||w\_k||^2 = \overline{X}\_k^T \overline{Q\_k} \overline{X}\_k + 2\left(\frac{T}{\lambda\_l}\right)^2 ||w\_k||^2.$$

Since *R* . 0 and *Qk* , 0, *V*2ð Þ*k* and *ΔV*2ð Þ*k* satisfy following equations:

$$\lambda\_R^{\min} \left|| \overline{X}\_k \right||^2 \le V\_2(k) \le \lambda\_R^{\max} \left|| \overline{X}\_k \right||^2,\tag{35}$$

$$
\Delta V\_2(k) \le -\left| \lambda\_{\overline{Q}\_k}^{\min} \right| \left|| \overline{X}\_k \right||^2 + 2 \left( \frac{T}{\lambda\_I} \right)^2 \left\| w\_k \right\|^2. \tag{36}
$$

In addition, *Xk* is bounded, and *V*2ð Þ*k* is radially unbounded from Eqs. (35) and (36). Hence, the control input (34) satisfies the following equation under Assumption 1:

$$\|\|u\_k\|\| \le M,\tag{37}$$

where *M* is a positive constant. Furthermore, *V*2ð Þ*k* also satisfies the following equation for all *x, z*<sup>∈</sup> R6 with maxf g k k*<sup>x</sup> ;* k k*<sup>z</sup>* <sup>≤</sup> <sup>Δ</sup>:

$$\begin{aligned} \left| V\_2(\mathbf{x}) - V\_2(\mathbf{z}) \right| &= \left| \mathbf{x}^T \mathbf{R} \mathbf{x} - \mathbf{z}^T \mathbf{R} \mathbf{z} \right| = \left| (\mathbf{x} + \mathbf{z})^T \mathbf{R} (\mathbf{x} - \mathbf{z}) \right| \\ &= \lambda\_\mathcal{R}^{\max} ||\mathbf{x} + \mathbf{z}|| ||\mathbf{x} - \mathbf{z}|| \le 2 \Delta \lambda\_\mathcal{R}^{\max} ||\mathbf{x} - \mathbf{z}||, \end{aligned} \tag{38}$$

where Δ is a positive constant. Therefore, from Eqs. (35) to (38), Lyapunov function *V*2ð Þ*k* and control input *uk* satisfied Eqs. (A8)–(A11) in Definition 2 under Assumptions 1 and 2, and the pair ð Þ *uk;V*2ð Þ*k* becomes SPA stabilizing pair for the

Euler approximate systems (10) and (11). Then, the following theorem can be obtained by Theorem A.1 in Appendix.

**Theorem 2:** control input (34) is SPA stabilizing for exact discrete-time systems (8) and (9).

### **4. Numerical simulation**

**Theorem 1:** if sampling period *T* and feedback gains *f* <sup>1</sup> and *f* <sup>2</sup> satisfy Eqs. (31), (32), and (33) under Assumption 2, then the closed-loop systems (10) and (11) with

*uk* ¼ f g *zk* þ *α,k* þ *Ckωd,k* �*J z*f g *<sup>k</sup>* þ *α,k* þ *Ckωd,k*

� *<sup>f</sup>* <sup>2</sup>*Jzk*

Then, we show that the pair ð Þ *uk;V*2ð Þ*k* is semiglobal practical asymptotic (SPA) stabilizing pair for the Euler approximate systems (10) and (11). Hereafter, suppose that sampling period *T* and feedback gains *f* <sup>1</sup> and *f* <sup>2</sup> satisfy Eqs. (31), (32), and (33)

> *ωe,k*

*<sup>k</sup> <sup>Z</sup>TZXk* <sup>¼</sup> *<sup>X</sup><sup>T</sup>*

k k *wk*

<sup>≤</sup>*V*2ð Þ*<sup>k</sup>* <sup>≤</sup> *<sup>λ</sup>*max

In addition, *Xk* is bounded, and *V*2ð Þ*k* is radially unbounded from Eqs. (35) and (36). Hence, the control input (34) satisfies the following equation under Assump-

where *M* is a positive constant. Furthermore, *V*2ð Þ*k* also satisfies the following

where Δ is a positive constant. Therefore, from Eqs. (35) to (38), Lyapunov function *V*2ð Þ*k* and control input *uk* satisfied Eqs. (A8)–(A11) in Definition 2 under Assumptions 1 and 2, and the pair ð Þ *uk;V*2ð Þ*k* becomes SPA stabilizing pair for the

¼ *ZXk,*

*<sup>k</sup> RXk,*

*<sup>k</sup> QkXk* <sup>þ</sup> <sup>2</sup> *<sup>T</sup>*

k k *wk* 2

k k *uk* ≤ *M,* (37)

*TR x*ð Þ � *<sup>z</sup>*

*<sup>R</sup>* k k *x* � *z ,*

 

(38)

*λJ* <sup>2</sup>

k k *wk* 2 *:*

*,* (35)

*:* (36)

<sup>2</sup> <sup>¼</sup> *<sup>X</sup><sup>T</sup>*

*<sup>R</sup> Xk* 2

¼ ð Þ *x* þ *z*

 

*<sup>R</sup>* k k *<sup>x</sup>* <sup>þ</sup> *<sup>z</sup>* k k *<sup>x</sup>* � *<sup>z</sup>* <sup>≤</sup>2Δ*λ*max

�*Jω<sup>k</sup>* þ *J Ckω*\_ *d,k* � ð Þ *zk* þ *α,k* � f g *Ckωd,k*

*ωe,k*

(34)

þ *J Ckω*\_ *d,k* � ð Þ *zk* þ *α,k* � f g *Ckωd,k*

� *f* <sup>1</sup> *f* <sup>2</sup>*Jσe,k* � *J f* <sup>1</sup>*G*ð Þþ *σe,k f* <sup>2</sup>*I*<sup>3</sup>

� *f* <sup>1</sup> *J G*ð Þ *σe,k zk* � *f* <sup>1</sup>*bkσe,k*

under Assumption 2. By using the following coordinate transformation

Lyapunov function *V*2ð Þ*k* and its difference *ΔV*2ð Þ*k* can be rewritten as

*λJ* <sup>2</sup>

Since *R* . 0 and *Qk* , 0, *V*2ð Þ*k* and *ΔV*2ð Þ*k* satisfy following equations:

 *Xk* 2 <sup>þ</sup> <sup>2</sup> *<sup>T</sup> λJ* <sup>2</sup>

*Qk* 

*Xk* <sup>¼</sup> 1 0 *f* <sup>1</sup> 1 *σe,k*

*<sup>V</sup>*2ð Þ¼ *<sup>k</sup> <sup>X</sup><sup>T</sup>*

*<sup>k</sup> <sup>Z</sup>TQkZXk* <sup>þ</sup> <sup>2</sup> *<sup>T</sup>*

*λ*min *<sup>R</sup> Xk* 2

equation for all *x, z*<sup>∈</sup> R6 with maxf g k k*<sup>x</sup> ;* k k*<sup>z</sup>* <sup>≤</sup> <sup>Δ</sup>:

*<sup>Δ</sup>V*2ð Þ*<sup>k</sup>* <sup>≤</sup> � *<sup>λ</sup>*min

j j *<sup>V</sup>*2ð Þ� *<sup>x</sup> <sup>V</sup>*2ð Þ*<sup>z</sup>* <sup>¼</sup> *<sup>x</sup>TRx* � *<sup>z</sup>TRz*

<sup>¼</sup> *<sup>λ</sup>*max

¼ *ω<sup>k</sup>*

the following control law

*Gyroscopes - Principles and Applications*

becomes ISS.

*<sup>Δ</sup>V*2ð Þ¼ *<sup>k</sup> <sup>X</sup><sup>T</sup>*

tion 1:

**68**

The properties of the proposed method are discussed in the numerical study. For this purpose, parameter setting of simulation is as follows:

$$J = \begin{bmatrix} 7050.0 & -0.536 & 43.9 \\ -0.536 & 2390 & 1640.0 \\ 43.9 & 1640.0 & 6130.0 \end{bmatrix} \text{kg} \text{m}^2, \sigma(\mathbf{0}) = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \rho(\mathbf{0}) = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \text{rad/s}$$

$$T = \begin{cases} 1.0 : \text{Case 1} \\ 0.5 : \text{Case 2}, f\_1 = 0.6, f\_2 = 0.8. \\ 0.1 : \text{Case 3} \end{cases}$$

The moment of inertia *J* is from [1]. The initial values *σ*ð Þ 0 correspond to Euler angles of 1–2-3 system of *<sup>θ</sup>*ð Þ¼ <sup>0</sup> ½ � *<sup>θ</sup>*1ð Þ <sup>0</sup> *<sup>θ</sup>*2ð Þ <sup>0</sup> *<sup>θ</sup>*3ð Þ <sup>0</sup> *<sup>T</sup>* <sup>¼</sup> ½ � <sup>000</sup> *<sup>T</sup>* ½ � deg . The feedback gains *f* <sup>1</sup> and *f* <sup>2</sup> satisfy Eqs. (25) and (28) for all cases of *T*. The desired states *σd*ð Þ*t* , *ωd*ð Þ*t* , and *ω*\_ *<sup>d</sup>*ð Þ*t* in this simulation are the switching maneuver as shown in **Figure 1**.

**Figure 1.** *Switching maneuver.*

#### **Figure 2.**

*Time histories of MRPs σ*ð Þ*t and σe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, σd*ð Þ*t ).*

The results of the numerical simulation are shown in **Figures 2**–**5**. The relative attitude *σe*ð Þ*t* and relative angular velocity *ωe*ð Þ*t* converge to the neighborhood of ð Þ¼ *σe*ð Þ*t ;ωe*ð Þ*t* ð Þ 0*;* 0 , and the control input amplitude *u t*ð Þ does not depend on the sampling period *T* although there is a slight difference in the

*Time histories of control input u t*ð Þ *(solid line, case 1; dashed-dotted line, case 2; and dashed line, case 3).*

This chapter considers the spacecraft attitude tracking control problem that requires agile and large-angle attitude maneuvers and proposed a discrete-time nonlinear attitude tracking control that the amplitude of the control input does not depend on the sampling period *T*. The effectiveness of proposed control method is verified by numerical simulations. Extension to the guarantee of stability as

This section shows preliminary results for nonlinear sampled-data control

where *x t*ð Þ∈R*<sup>n</sup>* is the state variable and *x t*ð Þ∈R*<sup>m</sup>* is the control input. The function *f xt* ð Þ ð Þ*; u t*ð Þ in Eq. (A1) is assumed to be such that, for each initial condition and each constant control input, there exists a unique solution defined on some

and zero-order hold (D/A converter), and the control signal is assumed to be

The nonlinear system (A1) is assumed to be between a sampler (A/D converter)

*u t*ðÞ¼ *u kT* ð Þ ≕ *u k*ð Þ*,* ∀*t*∈½ � *kT;*ð Þ *k* þ 1 *T , k*∈f g0 ∪ N*,* (A2)

*x t* \_ðÞ¼ *f xt* ð Þ ð Þ*; u t*ð Þ *, x*ð Þ¼ 0 *x*0*, f*ð Þ¼ 0*;* 0 0*,* (A1)

sampled-data control system will be subject to future work.

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

**Appendix: sampled-data control of nonlinear system**

Let us consider the following nonlinear system:

maximal value of *u t*ð Þ.

**5. Conclusion**

**Figure 5.**

[13, 14, 19].

intervals of *x*½0*; τ*Þ.

**71**

piecewise constant, that is,

#### **Figure 3.**

*Time histories of attitude angles θ*ð Þ*t and θe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, θd*ð Þ*t ).*

#### **Figure 4.**

*Time histories of angular velocities ω*ð Þ*t and ωe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, ωd*ð Þ*t ).*

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft DOI: http://dx.doi.org/10.5772/intechopen.87191*

#### **Figure 5.**

**Figure 2.**

**Figure 3.**

**Figure 4.**

**70**

*and case 3; dotted line, ωd*ð Þ*t ).*

*case 3; dotted line, θd*ð Þ*t ).*

*dotted line, σd*ð Þ*t ).*

*Gyroscopes - Principles and Applications*

*Time histories of MRPs σ*ð Þ*t and σe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3;*

*Time histories of attitude angles θ*ð Þ*t and θe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line, and*

*Time histories of angular velocities ω*ð Þ*t and ωe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line,*

*Time histories of control input u t*ð Þ *(solid line, case 1; dashed-dotted line, case 2; and dashed line, case 3).*

The results of the numerical simulation are shown in **Figures 2**–**5**. The relative attitude *σe*ð Þ*t* and relative angular velocity *ωe*ð Þ*t* converge to the neighborhood of ð Þ¼ *σe*ð Þ*t ;ωe*ð Þ*t* ð Þ 0*;* 0 , and the control input amplitude *u t*ð Þ does not depend on the sampling period *T* although there is a slight difference in the maximal value of *u t*ð Þ.

### **5. Conclusion**

This chapter considers the spacecraft attitude tracking control problem that requires agile and large-angle attitude maneuvers and proposed a discrete-time nonlinear attitude tracking control that the amplitude of the control input does not depend on the sampling period *T*. The effectiveness of proposed control method is verified by numerical simulations. Extension to the guarantee of stability as sampled-data control system will be subject to future work.

#### **Appendix: sampled-data control of nonlinear system**

This section shows preliminary results for nonlinear sampled-data control [13, 14, 19].

Let us consider the following nonlinear system:

$$\dot{\boldsymbol{x}}(t) = f(\boldsymbol{x}(t), \boldsymbol{u}(t)), \boldsymbol{x}(0) = \boldsymbol{x}\_0, f(0, 0) = 0,\tag{A1}$$

where *x t*ð Þ∈R*<sup>n</sup>* is the state variable and *x t*ð Þ∈R*<sup>m</sup>* is the control input. The function *f xt* ð Þ ð Þ*; u t*ð Þ in Eq. (A1) is assumed to be such that, for each initial condition and each constant control input, there exists a unique solution defined on some intervals of *x*½0*; τ*Þ.

The nonlinear system (A1) is assumed to be between a sampler (A/D converter) and zero-order hold (D/A converter), and the control signal is assumed to be piecewise constant, that is,

$$u(t) = u(kT) =: u(k), \forall t \in [kT, (k+1)T], k \in \{0\} \cup \mathbb{N},\tag{A2}$$

where *T* . 0 is a sampling period. In addition, assume that the state variable

$$\mathfrak{x}(k) := \mathfrak{x}(kT) \tag{A3}$$

is measurable at each sampling instance. The exact discrete-time model and Euler approximate model of the nonlinear sampled-data systems (A1)–(A3) are expressed as follows, respectively:

$$\mathfrak{x}\_{k+1} = \mathfrak{x}\_k + \int\_{kT}^{(k+1)T} f(\mathfrak{x}(s), u\_k) \, ds =: F\_T^\epsilon(\mathfrak{x}\_k, u\_k), \tag{A4}$$

$$\varkappa\_{k+1} = \varkappa\_k + T f(\varkappa\_k, \mu\_k) =: F\_T^{Euler}(\varkappa\_k, \mu\_k), \tag{A5}$$

where we abbreviate *x k*ð Þ and *u k*ð Þ to *xk* and *uk*. For the stability of the exact discrete-time model (A4) (*F<sup>e</sup> T*) and Euler approximate model (A5) (*FEuler <sup>T</sup>* ), the following definitions are used [13, 14, 19].

**Definition 1:** consider the following discrete-time nonlinear system:

$$\mathbf{x}\_{k+1} = F\_T(\mathbf{x}\_k, \boldsymbol{\mu}\_T(\mathbf{x}\_k)),\tag{A6}$$

where *xk* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is the state variable and *uT*ð Þ *xk* <sup>∈</sup> <sup>R</sup>*<sup>m</sup>* is a control input. The family of controllers *uT*ð Þ *xk* SPA stabilizes the system (A6) if there exists a class KL function *β*ð Þ� such that for any strictly positive real numbers ð Þ *D; ν* , there exists *<sup>T</sup>*<sup>∗</sup> . 0, and such that for all *<sup>T</sup>* <sup>∈</sup> <sup>0</sup>*; <sup>T</sup>*<sup>∗</sup> ð Þ and all initial state *<sup>x</sup>*<sup>0</sup> with k k *<sup>x</sup>*<sup>0</sup> <sup>≤</sup> *<sup>D</sup>*, the solution of the system satisfies

$$\|\mathbf{x}\_k\| \le \beta(\|\mathbf{x}\_0\|, kT) + \nu, \forall k \in \{\mathbf{0}\} \cup \mathbf{N}.\tag{A7}$$

**Definition 2:** let *T*^ . 0 be given, and for each *T* ∈ 0*; T*^ � �, let functions *VT* : <sup>R</sup>*<sup>n</sup>* ! R and *uT* : <sup>R</sup>*<sup>n</sup>* ! <sup>R</sup>*<sup>m</sup>* be defined. The pair of families ð Þ *uT;VT* is a SPA stabilizing pair for the system (A7) if there exist a class K<sup>∞</sup> functions *α*1, *α*2, and *α*<sup>3</sup> such that for any pair of strictly positive real numbers ð Þ Δ*; δ* , there exists a triple of strictly positive real numbers *<sup>T</sup>*<sup>∗</sup> ð Þ *; <sup>L</sup>; <sup>M</sup> <sup>T</sup>*<sup>∗</sup> <sup>≤</sup>*T*^ � � such that for all *x, z*<sup>∈</sup> <sup>R</sup>*<sup>n</sup>* with maxf g k k*<sup>x</sup> ;* k k*<sup>z</sup>* <sup>≤</sup> <sup>Δ</sup>, and *<sup>T</sup>* <sup>∈</sup> <sup>0</sup>*; <sup>T</sup>* <sup>∗</sup> ð Þ:

$$a\_1(||\mathfrak{x}||) \le V\_T(\mathfrak{x}) \le a\_2(||\mathfrak{x}||),\tag{A8}$$

$$V\_T(F\_T(\mathfrak{x}, \mathfrak{u}\_T(\mathfrak{x}))) - V\_T(\mathfrak{x}) \le -a\_3(\|\mathfrak{x}\|) + T\delta,\tag{A9}$$

$$|V\_T(\mathbf{x}) - V\_T(\mathbf{z})| \le L||\mathbf{x} - \mathbf{z}||,\tag{A10}$$

$$\|\|\mu\_T(\mathbf{x})\|\| \le M. \tag{A11}$$

**Author details**

Shonan Institute of Technology, Fujisawa, Japan

provided the original work is properly cited.

\*Address all correspondence to: ikeda@mech.shonan-it.ac.jp

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

© 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,

Yuichi Ikeda

**73**

In addition, if there exists *<sup>T</sup>*∗ ∗ . 0 such that Eqs. (A8)–(A11) with *<sup>δ</sup>* <sup>¼</sup> 0 hold for all *<sup>x</sup>*∈R*<sup>n</sup>* and *<sup>T</sup>* <sup>∈</sup> <sup>0</sup>*; <sup>T</sup>* ∗ ∗ ð Þ, then the pair ð Þ *uT;VT* is globally asymptotic (GA) stabilizing pair for the system (A6).

Using the above definitions, the following theorem is obtained by literatures [13, 14, 19].

**Theorem A.1:** if the pair ð Þ *uT;VT* is SPA stabilizing for *<sup>F</sup>Euler <sup>T</sup>* , then *uT* is SPA stabilizing for *F<sup>e</sup> T* .

Hence, if we can find a family of pairs of ð Þ *uT;VT* that is a GA or SPA stabilizing pair for *FEuler <sup>T</sup>* , then the controller *uT* will stabilize the exact model *F<sup>e</sup> T*.

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft DOI: http://dx.doi.org/10.5772/intechopen.87191*

where *T* . 0 is a sampling period. In addition, assume that the state variable

is measurable at each sampling instance. The exact discrete-time model and Euler approximate model of the nonlinear sampled-data systems (A1)–(A3) are

*f xs* ð Þ ð Þ*; uk ds* <sup>≕</sup> *<sup>F</sup><sup>e</sup>*

*T*) and Euler approximate model (A5) (*FEuler*

*xk*þ<sup>1</sup> ¼ *FT*ð Þ *xk; uT*ð Þ *xk ,* (A6)

k k *xk* ≤*β*ð Þþ k k *x*<sup>0</sup> *; kT ν,* ∀*k*∈f g0 ∪ N*:* (A7)

*α*1ð Þ k k*x* ≤*VT*ð Þ *x* ≤*α*2ð Þ k k*x ,* (A8)

j j *VT*ð Þ� *x VT*ð Þ*z* ≤ *L x*k k � *z ,* (A10)

k k *uT*ð Þ *x* ≤ *M:* (A11)

*<sup>T</sup>* , then *uT* is SPA

*T*.

*VT*ð*FT*ð Þ *x; uT*ð Þ *x* Þ � *VT*ð Þ *x* ≤ � *α*3ð Þþ k k*x Tδ,* (A9)

ðð Þ *<sup>k</sup>*þ<sup>1</sup> *<sup>T</sup> kT*

*xk*þ<sup>1</sup> <sup>¼</sup> *xk* <sup>þ</sup> *Tf x*ð Þ *<sup>k</sup>; uk* <sup>≕</sup> *<sup>F</sup>Euler*

**Definition 1:** consider the following discrete-time nonlinear system:

of controllers *uT*ð Þ *xk* SPA stabilizes the system (A6) if there exists a class KL function *β*ð Þ� such that for any strictly positive real numbers ð Þ *D; ν* , there exists *<sup>T</sup>*<sup>∗</sup> . 0, and such that for all *<sup>T</sup>* <sup>∈</sup> <sup>0</sup>*; <sup>T</sup>*<sup>∗</sup> ð Þ and all initial state *<sup>x</sup>*<sup>0</sup> with k k *<sup>x</sup>*<sup>0</sup> <sup>≤</sup> *<sup>D</sup>*, the

**Definition 2:** let *T*^ . 0 be given, and for each *T* ∈ 0*; T*^ � �, let functions *VT* : <sup>R</sup>*<sup>n</sup>* ! R and *uT* : <sup>R</sup>*<sup>n</sup>* ! <sup>R</sup>*<sup>m</sup>* be defined. The pair of families ð Þ *uT;VT* is a SPA stabilizing pair for the system (A7) if there exist a class K<sup>∞</sup> functions *α*1, *α*2, and *α*<sup>3</sup> such that for any pair of strictly positive real numbers ð Þ Δ*; δ* , there exists a triple of

strictly positive real numbers *<sup>T</sup>*<sup>∗</sup> ð Þ *; <sup>L</sup>; <sup>M</sup> <sup>T</sup>*<sup>∗</sup> <sup>≤</sup>*T*^ � � such that for all *x, z*<sup>∈</sup> <sup>R</sup>*<sup>n</sup>*

In addition, if there exists *<sup>T</sup>*∗ ∗ . 0 such that Eqs. (A8)–(A11) with *<sup>δ</sup>* <sup>¼</sup> 0 hold for all *<sup>x</sup>*∈R*<sup>n</sup>* and *<sup>T</sup>* <sup>∈</sup> <sup>0</sup>*; <sup>T</sup>* ∗ ∗ ð Þ, then the pair ð Þ *uT;VT* is globally asymptotic (GA)

Using the above definitions, the following theorem is obtained by literatures

Hence, if we can find a family of pairs of ð Þ *uT;VT* that is a GA or SPA stabilizing

*<sup>T</sup>* , then the controller *uT* will stabilize the exact model *F<sup>e</sup>*

**Theorem A.1:** if the pair ð Þ *uT;VT* is SPA stabilizing for *<sup>F</sup>Euler*

where we abbreviate *x k*ð Þ and *u k*ð Þ to *xk* and *uk*. For the stability of the exact

where *xk* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is the state variable and *uT*ð Þ *xk* <sup>∈</sup> <sup>R</sup>*<sup>m</sup>* is a control input. The family

expressed as follows, respectively:

*Gyroscopes - Principles and Applications*

discrete-time model (A4) (*F<sup>e</sup>*

solution of the system satisfies

with maxf g k k*<sup>x</sup> ;* k k*<sup>z</sup>* <sup>≤</sup> <sup>Δ</sup>, and *<sup>T</sup>* <sup>∈</sup> <sup>0</sup>*; <sup>T</sup>* <sup>∗</sup> ð Þ:

stabilizing pair for the system (A6).

*T* .

[13, 14, 19].

stabilizing for *F<sup>e</sup>*

pair for *FEuler*

**72**

following definitions are used [13, 14, 19].

*xk*þ<sup>1</sup> ¼ *xk* þ

*x k*ð Þ ≔ *x kT* ð Þ (A3)

*<sup>T</sup>*ð Þ *xk; uk ,* (A4)

*<sup>T</sup>* ), the

*<sup>T</sup>* ð Þ *xk; uk ,* (A5)
