2. Preliminaries

engineering and application value, and has been employed in Apollo entry guidance [6] and shuttle entry guidance [7]. Nevertheless, in reference trajectory reentry guidance, since HSV owns particular characteristics such as strong nonlinear, large flight envelope, complex entry environment, and precise terminal guidance accuracy requirement, it is difficult for HSV to track the nominal profile properly. Many scholars have done continuous research on it [8–12]. For following nominal profile problem in reentry reference trajectory guidance, namely trajectory tracking law, traditional PID control law of shuttle was given in [7]. Roenneke et al. [13] derived a linear control law tracking the drag reference in drag state space to achieve guidance command. In [14], a feedback linearization method for shuttle entry guidance trajectory tracking law to extend its application range was presented. In [15, 16], a feedback tracking law is designed by taking advantage of the linear structure of system dynamics in the energy space to achieve bounded tracking of the flat outputs. These foregoing approaches improved performances of trajectory tracking laws for aerial vehicles. However, they are not applied to HSV which owns particular characteristics. Dukeman [17] proposed a linear trajectory tracking law based on linear quadratic regulator (LQR) by constructing weighting matrices with Bryson principle [18], and the tracking law was very robust with respect to varying initial conditions and worked satisfactorily even for entries from widely different orbits than that of the reference profile. The capacity of the approach in [17] against other reentry process interferences such as aerodynamic parameter error, nevertheless, was relatively poor, and simulations demonstrated that performances for HSV tracking nominal profile under various disturbances depended directly on weighting matrices in LQR. In this study, one focuses on constructing

LQR weighting matrices to strengthen robustness of HSV trajectory tracking law.

under different disturbances.

134 Advances in Some Hypersonic Vehicles Technologies

The weighting matrices Q and R are the most important parameters in LQR optimization and determine the output performances of systems [19]. Trial-and-error method has been employed to construct these matrices, which is simple, but primarily depends on people's experience and intuitive adjustment. In trial-and-error method, elements of weighting matrices must be repeatedly experimented to get a proper value, and is not feasible for application in large scale system. In [18, 20], certain general guidelines were followed to construct weighting matrices simply and normally, but might not lead to satisfactory responses. Connecting closedloop poles to feedback gains for LQR were presented in [21–24] using pole-assignment approach, which resulted in more accurate responses. However, it was difficult to balance state and control variables and to account for control effectiveness using the approaches in [21–24]. A trade-off between penalties on the state and control inputs for optimization of the cost function was considered in [25], where specified closed-loop eigenvalues were obtained, but the computation normally needed more iterations. Genetic algorithm (GA) can be applied to find a global optimal solution [26–28], and the differential evolution algorithms inspired from GA are efficient evolution strategies for fast optimization technique [29–31]. However, the approaches in [26–31] have little improvement for HSV profile-following performances

In this study, a novel method to construct weighting matrices with time-varying parameters on the basis of Bryson principle is proposed. This idea employs current flight states to provide flexible and accurate feedback gains in HSV trajectory tracking law under various interferences

In this section, the concepts and basic results on reentry dynamics, LQR, Bryson principle, and their applications in hypersonic vehicle trajectory tracking law are introduced, which are the research foundation of the following sections.

#### 2.1. Reentry dynamics

For a lifting reentry vehicle, the common control variables are the bank angle σ, and the angle of attack α. The state variables include the radial distance from the Earth center to the vehicle r, the longitude θ, the latitude ϕ, the Earth-relative velocity v, the flight path angle γ, and the heading angle ψ. The three-degree-of-freedom point-mass dynamics for the vehicle over a sphere rotating Earth are expressed as [32]:

$$
\dot{r} = \upsilon \sin \upsilon,\tag{1}
$$

$$\dot{\theta} = \frac{v \cos \varphi \sin \psi}{r \cos \phi},\tag{2}$$

$$\dot{\phi} = \frac{v \cos \gamma \cos \psi}{r} \,\tag{3}$$

$$\psi = -D - g\sin\chi + \omega^2 r \cos\phi (\sin\chi\cos\phi - \cos\chi\sin\phi\cos\psi). \tag{4}$$

$$\gamma = \frac{1}{v} \left[ L\cos\sigma - g\cos\chi + \frac{v^2\cos\chi}{r} + 2\omega v\cos\phi\sin\psi + \omega^2 r\cos\phi \left(\cos\chi\cos\phi + \sin\chi\sin\phi\cos\psi\right) \right], \tag{5}$$

$$\dot{\psi} = \frac{1}{v} \left[ \frac{v^2 \cos \gamma \sin \psi \tan \phi}{r} - 2\omega v \{ \tan \gamma \cos \phi \cos \psi - \sin \phi \} \right. \\ \quad + \frac{\omega^2 r}{\cos \gamma} \sin \phi \cos \phi \sin \psi + \frac{L \sin \sigma}{\cos \gamma} \Big], \tag{6}$$

where ω is the Earth's self-rotation rate, and g is the gravitational acceleration. L and D are the aerodynamic lift and drag accelerations defined by

$$L = \frac{1}{2m}\rho v^2 \mathcal{C}\_L \mathcal{S}, \ D = \frac{1}{2m}\rho v^2 \mathcal{C}\_D \mathcal{S},\tag{7}$$

where m is the mass of the vehicle, S is the reference area, CL is the lift coefficient, and CD is the drag coefficient. ρ is the atmospheric density expressed as an exponential model [33].

$$
\rho = \rho\_0 e^{-\beta \hbar},
\tag{8}
$$

In order to obtain a proper quadratic performance index, elements of weighting matrices must

The basic principle of Bryson principle is to normalize the contributions, and then the states and the control terms may behave effectively within the definition of the quadratic cost function. The normalization is accomplished by using the anticipated maximum values of the

Q tðÞ¼ diag <sup>q</sup>1ð Þ<sup>t</sup> ; …; qnð Þ<sup>t</sup> � � , RtðÞ¼ diag½ � <sup>r</sup>1ð Þ<sup>t</sup> ;…;rmð Þ<sup>t</sup> : (14)

Hypersonic Vehicles Profile-Following Based on LQR Design Using Time-Varying Weighting Matrices

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> rmð Þ<sup>τ</sup> umð Þ<sup>τ</sup>

http://dx.doi.org/10.5772/intechopen.70659

<sup>2</sup> <sup>¼</sup> … <sup>¼</sup> rmumð Þ max

(16)

137

(18)

<sup>2</sup> <sup>¼</sup> <sup>1</sup>: (17)

<sup>2</sup> <sup>þ</sup> <sup>r</sup>1ð Þ<sup>τ</sup> <sup>u</sup>1ð Þ<sup>τ</sup>

xið Þ max , i ¼ 1, 2, …, n, ujð Þ max , j ¼ 1, 2, …, m:

<sup>2</sup> <sup>¼</sup> <sup>r</sup>1u1ð Þ max

<sup>2</sup> , i ¼ 1, 2, …n ,

<sup>2</sup> , j ¼ 1, 2, …m:

Then, the elements to construct the weighting matrices can be obtained as time-invariant param-

Through simple calculation, Bryson principle can generate better results in a short time, which minimizes the quadratic index value in a proper scope. Because of that, Bryson principle is

In reentry reference trajectory guidance, the chief challenge of following the nominal profile lies in generating a proper compensatory signal, and LQR can solve this problem effectively. After generating a feasible reference entry profile containing altitude z, velocity v, flight path angle γ as reference parameters, and bank angle σref as guidance reference signal, one can download this profile into the onboard computer. After the vehicle enters the atmosphere, the deviations between nominal profile and the actual real-time data can be obtained by

<sup>2</sup> � �dτ: (15)

be chosen properly, and Bryson principle can solve this problem effectively.

Then, develop the quadratic index in the following expression.

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> qnð Þ<sup>τ</sup> xnð Þ<sup>τ</sup>

Normalize all the contributions to 1 with the help of all the maximum values.

Determine each maximum value of all the states and control terms.

�

<sup>2</sup> <sup>¼</sup> … <sup>¼</sup> qnxnð Þ max

8 >>><

>>>:

widely applied to the selection of weighting matrices in LQR.

2.3. Reentry reference trajectory guidance based on LQR

qi <sup>¼</sup> <sup>1</sup> xið Þ max

rj <sup>¼</sup> <sup>1</sup> ujð Þ max

q1ð Þτ x1ð Þτ

q1x1ð Þ max

J ¼ ðtf t

eters.

individual states and control quantities. The method can be explained as follows. First, define the weighting matrices Q(t) and R(t) as diagonal matrices, namely:

where ρ<sup>0</sup> is the atmospheric density at sea level, h is the altitude of the vehicle, and β is a constant.

To guide the vehicle from the initial point to the terminal interface with multiple constraints, a reference trajectory is usually optimized offline, and a profile-following law is utilized to track the reference trajectory onboard. In the longitudinal profile-following, the linear quadratic regulator (LQR) law is a good choice [10].

#### 2.2. Linear quadratic regulator

This subsection introduces LQR and Bryson principle. For a linear system, the dynamics can be described by

$$\begin{cases} \dot{\mathbf{x}}(t) = A(t)\mathbf{x}(t) + B(t)\boldsymbol{\mu}(t),\\ \mathbf{y}(t) = \mathbf{C}(t)\mathbf{x}(t), \end{cases} \tag{9}$$

where A(t), B(t) and C(t) are system matrices, x(t)=[x1(t), x2(t), ……, xn(t)]<sup>T</sup> is the state, and u(t)=[u1(t), u2(t), ……, un(t)]<sup>T</sup> is the control input.

The quadratic performance index required to be minimized can be written as

$$J(t, t\_f) = \int\_{t}^{t\_f} \left[ \mathbf{x}^T(\tau) \mathbf{Q}(\tau) \mathbf{x}(\tau) + \mathbf{u}^T(\tau) \mathbf{R}(\tau) \mathbf{u}(\tau) \right] d\tau,\tag{10}$$

where the weighting matrix Q(t) is symmetrical positive semi-definite and weighting matrix R(t) is symmetrical positive definite. The specific procedure of LQR minimizing quadratic performance index is as follows.

The Riccati equation is given as

$$P(t)A(t) - P(t)B(t)R(t)^{-1}B(t)^TP(t) + Q(t) + A(t)^TP(t) = 0. \tag{11}$$

After getting the solution P(t) corresponding to each time instant t by solving Eq. (11), the feedback gain matrix can be obtained as

$$K(t) = R(t)^{-1} B^T(t) P(t). \tag{12}$$

Based on Eq. (12), the control input can be designed as

$$
\mu(t) = -K(t)\mathbf{x}(t). \tag{13}
$$

In order to obtain a proper quadratic performance index, elements of weighting matrices must be chosen properly, and Bryson principle can solve this problem effectively.

The basic principle of Bryson principle is to normalize the contributions, and then the states and the control terms may behave effectively within the definition of the quadratic cost function. The normalization is accomplished by using the anticipated maximum values of the individual states and control quantities. The method can be explained as follows.

First, define the weighting matrices Q(t) and R(t) as diagonal matrices, namely:

$$Q(t) = \text{diag}\left[q\_1(t), \dots, q\_n(t)\right] \quad , \quad R(t) = \text{diag}[r\_1(t), \dots, r\_m(t)]. \tag{14}$$

Then, develop the quadratic index in the following expression.

where m is the mass of the vehicle, S is the reference area, CL is the lift coefficient, and CD is the

�βh

, (8)

(9)

drag coefficient. ρ is the atmospheric density expressed as an exponential model [33].

constant.

described by

regulator (LQR) law is a good choice [10].

136 Advances in Some Hypersonic Vehicles Technologies

u(t)=[u1(t), u2(t), ……, un(t)]<sup>T</sup> is the control input.

J t; tf � � <sup>¼</sup>

2.2. Linear quadratic regulator

performance index is as follows. The Riccati equation is given as

feedback gain matrix can be obtained as

ρ ¼ ρ0e

where ρ<sup>0</sup> is the atmospheric density at sea level, h is the altitude of the vehicle, and β is a

To guide the vehicle from the initial point to the terminal interface with multiple constraints, a reference trajectory is usually optimized offline, and a profile-following law is utilized to track the reference trajectory onboard. In the longitudinal profile-following, the linear quadratic

This subsection introduces LQR and Bryson principle. For a linear system, the dynamics can be

x t \_ðÞ¼ A tð Þx tð Þþ B tð Þu tð Þ,

where A(t), B(t) and C(t) are system matrices, x(t)=[x1(t), x2(t), ……, xn(t)]<sup>T</sup> is the state, and

where the weighting matrix Q(t) is symmetrical positive semi-definite and weighting matrix R(t) is symmetrical positive definite. The specific procedure of LQR minimizing quadratic

After getting the solution P(t) corresponding to each time instant t by solving Eq. (11), the

K tðÞ¼ R tð Þ�<sup>1</sup>

xTð Þ<sup>τ</sup> <sup>Q</sup>ð Þ<sup>τ</sup> <sup>x</sup>ð Þþ <sup>τ</sup> <sup>u</sup><sup>T</sup>ð Þ<sup>τ</sup> <sup>R</sup>ð Þ<sup>τ</sup> <sup>u</sup>ð Þ<sup>τ</sup> � �dτ, (10)

B tð ÞTP tð Þþ Q tð Þþ A tð ÞTP tðÞ¼ <sup>0</sup>: (11)

<sup>B</sup><sup>T</sup>ð Þ<sup>t</sup> P tð Þ: (12)

u tðÞ¼�K tð Þx tð Þ: (13)

y tðÞ¼ C tð Þx tð Þ,

The quadratic performance index required to be minimized can be written as

�

ðtf t

P tð ÞA tð Þ� P tð ÞB tð ÞR tð Þ�<sup>1</sup>

Based on Eq. (12), the control input can be designed as

$$J = \int\_{t}^{t\_f} \left( q\_1(\tau) \mathbf{x}\_1(\tau)^2 + \dots + q\_n(\tau) \mathbf{x}\_n(\tau)^2 + r\_1(\tau) u\_1(\tau)^2 + \dots + r\_m(\tau) u\_m(\tau)^2 \right) d\tau. \tag{15}$$

Determine each maximum value of all the states and control terms.

$$\begin{cases} \mathbf{x}\_i(\text{max}) \; , \; i = 1, 2, \dots, n \\ \mathbf{u}\_j(\text{max}) \; , \; j = 1, 2, \dots, m. \end{cases} \tag{16}$$

Normalize all the contributions to 1 with the help of all the maximum values.

$$
\eta\_1 \mathbf{x}\_1 (\text{max})^2 = \dots = \eta\_n \mathbf{x}\_n (\text{max})^2 = r\_1 \mu\_1 (\text{max})^2 = \dots = r\_m \mu\_m (\text{max})^2 = 1. \tag{17}
$$

Then, the elements to construct the weighting matrices can be obtained as time-invariant parameters.

$$\begin{cases} q\_i = \frac{1}{\chi\_i \left(\max\right)^2} \quad , \quad i = 1, 2, \dots \text{n} \\\\ r\_j = \frac{1}{\mu\_j \left(\max\right)^2} \quad , \quad j = 1, 2, \dots \text{m} \end{cases} \tag{18}$$

Through simple calculation, Bryson principle can generate better results in a short time, which minimizes the quadratic index value in a proper scope. Because of that, Bryson principle is widely applied to the selection of weighting matrices in LQR.

#### 2.3. Reentry reference trajectory guidance based on LQR

In reentry reference trajectory guidance, the chief challenge of following the nominal profile lies in generating a proper compensatory signal, and LQR can solve this problem effectively. After generating a feasible reference entry profile containing altitude z, velocity v, flight path angle γ as reference parameters, and bank angle σref as guidance reference signal, one can download this profile into the onboard computer. After the vehicle enters the atmosphere, the deviations between nominal profile and the actual real-time data can be obtained by navigation facilities. The deviations contain altitude error zδ, velocity error v<sup>δ</sup> and flight path angle error γδ.

Denote the compensatory bank angle by σδ. In order to minimize the deviations and keep the aerial vehicle tracking nominal profile properly, the optimal feedback gain of guidance compensatory signal σδ can be calculated by LQR in the following algorithm.

Algorithm 1. The actual guidance signal in trajectory tracking law based on LQR can be determined in the following procedure.

Step 1. Based on perturbation theory, one establishes the linear equations of motion by taking the deviations as state parameters.

$$\begin{cases} \delta \mathbf{x}'(t) = A(t)\delta \mathbf{x}(t) + B(t)\delta u(t), \\ y(t) = \mathbf{C}(t)\delta \mathbf{x}(t). \end{cases} \tag{19}$$

It has been verified in [17, 34] that using Algorithm 1, the aerial vehicle performs well in

Hypersonic Vehicles Profile-Following Based on LQR Design Using Time-Varying Weighting Matrices

http://dx.doi.org/10.5772/intechopen.70659

In this section, the problem of HSV profile-following using trajectory tracking law based on

The traditional reference guidance is not suitable for hypersonic vehicles because of its particular characteristics, including strong nonlinear, large flight envelope and complex entry environment. In the process of entry flight, it is difficult to constrain the deviation between real profile and nominal profile into a proper scope. Furthermore, strict terminal accuracy requirement demands that hypersonic vehicles track nominal profile precisely, that is, deviations in

Consequently, in the reference profile-following of HSV based on LQR, new problems occur in the selection of weighting matrices. In the initial flight stage, it is assumed that the deviations

> zδ<sup>0</sup> ¼ 3 km, vδ<sup>0</sup> ¼ 200m=s:

Let zδ<sup>1</sup> and vδ<sup>1</sup> be the anticipated maximum deviations accuracy in the terminal stage,

zδ<sup>1</sup> ¼ 0:5 km, vδ<sup>1</sup> ¼ 20m=s:

Substituting zδ0, vδ0, zδ<sup>1</sup> and vδ<sup>1</sup> into Eq. (21), one can get the weighting matrix Q<sup>0</sup> and Q<sup>1</sup> as

3 7

<sup>5</sup>: (27)

2 6 4

3 7 <sup>5</sup>, Q<sup>1</sup> <sup>¼</sup>

From Eqs. (25) and (26), one sees that zδ<sup>0</sup> and vδ<sup>0</sup> are bigger than zδ<sup>1</sup> and vδ1, respectively. The weighting matrix Q0, which is determined by zδ<sup>0</sup> and vδ0, can effectively eliminate the large initial stage deviations between the real and nominal profiles. Nevertheless, the capacity of Q<sup>0</sup> for resisting disturbance in the process of flight is not strong enough to satisfy the terminal accuracy requirement. On the contrary, the weighting matrix Q<sup>1</sup> constructed by zδ<sup>1</sup> and vδ<sup>1</sup> can eliminate the disturbance in the process of flight effectively. However, facing the existence of large deviations in the initial entry flight, it is difficult to keep HSV tracking the nominal

Therefore, compared with Q0, the weighting matrix Q<sup>1</sup> is not applicable to initial deviations, and has good robustness to deal with disturbance in the process of entry flight. The

(25)

139

(26)

of altitude and velocity are zδ<sup>0</sup> and vδ0, respectively, which are chosen to be

�

�

2 6 4

profile properly, which will further influence the terminal accuracy.

Q<sup>0</sup> ¼

tracking reference trajectory.

3. Problem statement

LQR in Section 2 is presented.

the terminal stage must be smaller.

expressed as

where δx(t)=[zδ(t), vδ(t), γδ(t)]<sup>T</sup> and δu(t) = σδ(t).

Step 2. Construct the quadratic performance index as follows:

$$J(t, t\_f) = \int\_{t}^{t\_f} \left[ \delta \mathbf{x}^T(\tau) Q \delta \mathbf{x}(\tau) + \delta u(\tau) R \delta u(\tau) \right] d\tau. \tag{20}$$

Step 3. The weighting matrices Q and R in Eq. (20) are determined by Bryson principle. Since altitude z and velocity v are main factors in profile-following, q3, which is the weighting element of path angle γ, can be ignored. Using Eq. (18), the other elements of weighting matrices can be obtained as

$$q\_1 = \frac{1}{z\_{\delta\_{\text{max}}}} \quad \text{ } \quad q\_2 = \frac{1}{\upsilon\_{\delta\_{\text{max}}}} \text{ } \text{ } \quad r\_1 = \frac{1}{\sigma\_{\delta\_{\text{max}}}} \text{ } \tag{21}$$

where zδmax and vδmax are anticipated maximum deviations between the actual profile and the nominal profile, and σδmax is the maximum allowable modification of guidance signal σ. Based on Eq. (21), one can get the weighting matrices:

$$\mathcal{Q} = \text{diag}\left[q\_1, q\_2\right] \; , \; \mathcal{R} = r\_1. \tag{22}$$

Step 4. In order to minimize the index J in Eq. (20), one calculates the Riccati Eqs. (11) and (12) to obtain the optimal feedback gain K(t). Then, the compensatory signal can be obtained as

$$
\delta u(t) = -K(t)\delta x(t). \tag{23}
$$

Step 5. The actual guidance signal σ(t) which consists of guidance reference signal u(t) and guidance compensatory signal δu(t) can be obtained. It can be shown that

$$
\sigma(t) = u(t) + \delta u(t) = \sigma\_{ref} - \mathbf{K}(t)\delta \mathbf{x}(t). \tag{24}
$$

It has been verified in [17, 34] that using Algorithm 1, the aerial vehicle performs well in tracking reference trajectory.
