Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft

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.

1. Introduction

Recent space programs require agile and large-angle attitude maneuvers for applications in various fields such as observational astronomy [1, 2, 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 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 [4, 5, 6, 7, 8, 9, 10, 11, 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 $u=a\left(x\right)+\left(b\left(x\right)/T\right)$.

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 $\mathrm{R}$ and $\mathrm{N}$ denote the real and the integer numbers. Rn and ${\mathrm{R}}^{n\times m}$ are the sets of real vectors and matrices. For real vector $a\in {\mathrm{R}}^n$, $a^T$ is the vector transpose, $‖a‖$ denotes the Euclidean norm, and $a^{\times }\in {\mathrm{R}}^{3\times 3}$ is the skew symmetric matrix

$$a^{\times }=\left[\begin{array}{ccc}0& -a_3& a_2\\ a_3& 0& -a_1\\ -a_2& a_1& 0\end{array}\right]$$

derived from vector $a\in {\mathrm{R}}^3$. For real symmetric matrix $A$, $A>0$ means the positive definite matrix. The identity matrix of size $3\times 3$ is denoted by $I_3$. ${\lambda }_A^{\max }\in \mathrm{R}$ and ${\lambda }_A^{\min }\in \mathrm{R}$ 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 $\left\{i\right\}$, the modified Rodrigues parameters (MRPs) [5] are used. The rotational motion equations of the spacecraft’s body-fixed frame $\left\{b\right\}$ are given by the following equations:

$$\dot{\sigma }\left(t\right)=G\left(\sigma \left(t\right)\right)\omega \left(t\right),$$E1

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

$$\dot{\omega }\left(t\right)=J^{-1}\left\{-\omega {\left(t\right)}^{\times }\mathit{J\omega }\left(t\right)+u\left(t\right)+w\left(t\right)\right\},$$E2

where Eq. (1) is the kinematics that represents the attitude of $\left\{b\right\}$ with respect to the $\left\{i\right\}$, Eq. (2) is the rotation dynamics, $\sigma \left(t\right)\in {\mathrm{R}}^3$ [−] is the MRPs, $\omega \left(t\right)\in {\mathrm{R}}^3$ [rad/s] is the angular velocity, $u\left(t\right)\in {\mathrm{R}}^3$ [Nm] is the control torque (input), $w\left(t\right)\in {\mathrm{R}}^3$ [Nm] is the disturbance input, and $J\in {\mathrm{R}}^{3\times 3}$ [kg m²] is the moment of inertia.

We consider a control problem in which a spacecraft tracks a desired attitude (MRPs) ${\sigma }_d\left(t\right)\in {\mathrm{R}}^3$ and angular velocity ${\omega }_d\left(t\right)\in {\mathrm{R}}^3$ in fixed frame $\left\{d\right\}$. The MRPs of the relative attitude ${\sigma }_e\left(t\right)\in {\mathrm{R}}^3$ and the relative angular velocity ${\omega }_e\left(t\right)\in {\mathrm{R}}^3$ in the frame $\left\{b\right\}$ are given by

$${\sigma }_e\left(t\right)=\frac{N_e\left(t\right)}{1+{‖\sigma \left(t\right)‖}^2{‖{\sigma }_d\left(t\right)‖}^2+2{\sigma }_d{\left(t\right)}^T\sigma \left(t\right)},$$E3

$$N_e\left(t\right)=\left(1-{‖{\sigma }_d\left(t\right)‖}^2\right)\sigma \left(t\right)-\left(1-{‖\sigma \left(t\right)‖}^2\right){\sigma }_d\left(t\right)+2\sigma {\left(t\right)}^{\times }{\sigma }_d\left(t\right),$$

$${\omega }_e\left(t\right)=\omega \left(t\right)-C\left(t\right){\omega }_d\left(t\right),$$E4

where $C\left(t\right)\in {\mathrm{R}}^{3\times 3}$ is the direction cosine matrix from $\left\{b\right\}$ to $\left\{d\right\}$ that expresses the following Eq. [7]:

$$C\left(t\right)=I_3+\frac{8{\left({\sigma }_e{\left(t\right)}^{\times }\right)}^2-4\left(1-{‖{\sigma }_e\left(t\right)‖}^2\right){\sigma }_e{\left(t\right)}^{\times }}{{\left(1+{‖{\sigma }_e\left(t\right)‖}^2\right)}^2}.$$E5

Substituting Eqs. (3) and (4) into Eqs. (1) and (2) using the identity $\dot{C}\left(t\right)=-{\omega }_e{\left(t\right)}^{\times }C\left(t\right)$ yields the following relative motion equations:

$${\dot{\sigma }}_e\left(t\right)=G\left({\sigma }_e\left(t\right)\right){\omega }_e\left(t\right),$$E6

$$\begin{array}{cc}{{\displaystyle \dot{\omega }}}_e\left(t\right)=& J^{-1}[-{\left\{{\omega }_e\left(t\right)+C\left(t\right){\omega }_d\left(t\right)\right\}}^{\times }J\left\{{\omega }_e\left(t\right)+C\left(t\right){\omega }_d\left(t\right)\right\}\\ & \left.-J\left\{C\left(t\right){{\displaystyle \dot{\omega }}}_d\left(t\right)-{\omega }_e{\left(t\right)}^{\times }C\left(t\right){\omega }_d\left(t\right)\right\}+u\left(t\right)+w\left(t\right)\right]\end{array}$$E7

Hereafter, we assume that the variables of spacecraft $\sigma \left(t\right)$ and $\omega \left(t\right)$ are directly measurable and $J$ is known. In addition, regarding the desired states ${\sigma }_d\left(t\right)$, ${\omega }_d\left(t\right)$, ${\dot{\omega }}_d\left(t\right)$, and the disturbance $w\left(t\right)$, the following assumption is made.

Assumption 1: the desired states ${\sigma }_d\left(t\right)$, ${\omega }_d\left(t\right)$, and ${\dot{\omega }}_d\left(t\right)$ are uniformly continuous and bounded $\forall t\in \left[0,\infty \right)$. The disturbance $w\left(t\right)$ is uniformly bounded $\forall t\in \left[0,\infty \right)$.

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

$${\sigma }_{e,k+1}={\sigma }_{e,k}+{\int }_{\mathit{kT}}^{\left(k+1\right)T}G\left({\sigma }_e\left(s\right)\right){\omega }_e\left(s\right)\mathit{ds},$$E8

$$\begin{array}{cc}{\omega }_{e,k+1}=& {\omega }_{e,k}+{\int }_{\mathit{kT}}^{\left(k+1\right)T}\left[-{\left\{{\omega }_e\left(s\right)+C\left(s\right){\omega }_d\left(s\right)\right\}}^{\times }J\left\{{\omega }_e\left(s\right)+C\left(s\right){\omega }_d\left(s\right)\right\}\right)\\ & \left.-J\left\{C\left(s\right){\dot{\omega }}_d\left(s\right)-{\omega }_e{\left(s\right)}^{\times }C\left(s\right){\omega }_d\left(s\right)\right\}+u_k+w_k\right]\mathit{ds}\end{array}$$E9

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

$${\sigma }_{e,k+1}={\sigma }_{e,k}+\mathit{TG}\left({\sigma }_{e,k}\right){\omega }_{e,k},$$E10

$$\begin{array}{c}{\omega }_{e,k+1}={\omega }_{e,k}-T{J}^{-1}\left[-{\left\{{\omega }_{e,k}+C_k{\omega }_{d,k}\right\}}^{\times }J\left\{{\omega }_{e,k}+C_k{\omega }_{d,k}\right\}\right)\\ \left.-J\left\{C_k{\dot{\omega }}_{d,k}-{{\omega }_{e,k}}^{\times }{C}_k{\omega }_{d,k}\right\}+u_k+w_k\right].\end{array}$$E11

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 $x_k={\left[{\sigma }_{e,k}^T{\omega }_{e,k}^T\right]}^T$ satisfies the following equation:

$$‖x_{k+1}‖\le \rho \left(‖x_0‖,k\right)+\gamma \left(‖w_k‖\right),\forall x_k\in {\mathrm{R}}^3,\forall w_k\in {\mathrm{R}}^3,$$

where $\rho \left(·\right)$ is the class KL function and $\gamma \left(·\right)$ is the class K function. To this end, assume that ${\omega }_{e,k}$ is the virtual input to subsystem (10), and derive the stabilizing function ${\alpha }_k$ that ${\sigma }_{e,k}$ is asymptotic convergence to zero. Then, derive the control input $u_k$ that closed-loop system becomes ISS. Here, regarding the variable ${\sigma }_{e,k}$, the following assumption is made.

Assumption 2:${\sigma }_{e,k}$ lies in the region that satisfies the following equation:

$$0\le ‖{\sigma }_{e,k}‖\le 1,\forall k.$$

Remark 1: from the relational expression

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

where ${\epsilon }_{e,k}\in {\mathrm{R}}^3$ and ${\eta }_{e,k}\in \mathrm{R}$ are the quaternion $(‖{\left[{\epsilon }_{e,k}^T{\eta }_{e,k}\right]}^T‖=1,‖{\epsilon }_{e,k}‖\le 1,\left|{\eta }_{e,k}\right|\le 1,\forall k)$. Assumption 2 is equivalent to ${\eta }_{e,k}\in \left[0,1\right]$.

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

Lemma 1: for all $\sigma \in {\mathrm{R}}^3$, the following equations hold [5]:

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

Lemma 2: when the quadratic equation

$${\mathit{ax}}^2+\mathit{bx}+c=0\left(a,b,c\in \mathrm{R}\right)$$

has two distinct real roots $x=\alpha ,\beta \left(\alpha <\beta \right)$, if $a>0$, then the solution of the quadratic inequality

$${\mathit{ax}}^2+\mathit{bx}+c<0$$

is $\alpha <x<\beta$.

3.1 Derivation of virtual input ${\mathit{\alpha }}_{\mathit{k}}$

Assume that ${\omega }_{e,k}$ is the virtual input to subsystem (10), and define the stabilizing function such that

$${\omega }_{e,k}={\alpha }_k=-f_1{\sigma }_{e,k},$$E12

where $f_1\in \mathrm{R}$ is the feedback gain. The candidate Lyapunov function for (10) is defined as

$$V_1\left(k\right)={‖{\sigma }_{e,k}‖}^2.$$E13

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

$$\Delta V_1\left(k\right)=V_1\left(k+1\right)-V_1\left(k\right)=\left\{{\left(T{f}_1{b}_k\right)}^2-2T{f}_1{b}_k\right\}{‖{\sigma }_{e,k}‖}^2.$$E14

From Lemma 2, $\Delta V_1\left(k\right)$ becomes negative, i.e., the range of $f_1$ that holds the following equation

$${\left(T{f}_1{b}_k\right)}^2-2T{f}_1{b}_k<0$$E15

is obtained as

$$0<f_1<\frac{2}{T{b}_k}.$$E16

In addition, since $2\le \left(1/b_k\right)\le 4$ under Assumption 2, the range of $f_1$ that holds Eq. (15) is obtained as

$$0<f_1<\frac{4}{T}.$$E17

Therefore, if $f_1$ satisfies Eq. (17) and ${\omega }_{e,k}\to {\alpha }_k\left(k\to \infty \right)$, then ${\sigma }_{e,k}\to 0$.

3.2 Derivation of control input ${\mathit{u}}_{\mathit{k}}$

The error variable between the state ${\omega }_{e,k}$ and ${\alpha }_k$ is defined as

$$z_k≔{\omega }_{e,k}-{\alpha }_k.$$E18

The control input $u_k$ that makes the closed-loop system becomes ISS is derived.

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

$${\sigma }_{e,k+1}={\sigma }_{e,k}+\mathit{TG}\left({\sigma }_{e,k}\right)\left(z_k+{\alpha }_k\right).$$E19

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

$${\alpha }_k-{\alpha }_{k+1}=T{f}_1\left\{G\left({\sigma }_{e,k}\right)z_k-f_1{b}_k{\sigma }_{e,k}\right\},$$

the discrete-time equation with respect to $z_k$ is

$$\begin{array}{cc}{z}_{k+1}& =z_k+T{f}_1\left\{G\left({\sigma }_{e,k}\right)z_k-f_1{b}_k{\sigma }_{e,k}\right\}\\ & + T{J}^{-1}\left[-\left\{z_k+{\alpha }_{,k}+C_k{\omega }_{d,k}\right\}\times J\left\{z_k+{\alpha }_{,k}+C_k{\omega }_{d,k}\right\}\right)\\ & \left.- J\left\{C_k{\dot{\omega }}_{d,k}-\left(z_k+{\alpha }_{,k}\right)\times C_k{\omega }_{d,k}\right\}+u_k+w_k\right].\end{array}$$E20

Now, by setting $u_k$ to

$$\begin{array}{cc}{u}_k=& \left\{z_k+{\alpha }_{,k}+C_k{\omega }_{d,k}\right\}\times J\left\{z_k+{\alpha }_{,k}+C_k{\omega }_{d,k}\right\}\\ & + J\left\{C_k{\dot{\omega }}_{d,k}-\left(z_k+{\alpha }_{,k}\right)\times C_k{\omega }_{d,k}\right\}\\ & - f_{1}J\left\{G\left({\sigma }_{e,k}\right)z_k-f_1{b}_k{\sigma }_{e,k}\right\}-f_{2}J{z}_k,\end{array}$$

Eq. (20) becomes

$$z_{k+1}=\left(1-T{f}_2\right)z_k+T{J}^{-1}{w}_k,$$E21

where $f_2\in \mathrm{R}$ is the feedback gain. The candidate Lyapunov function for Eqs. (19) and (21) is defined as

$$V_2\left(k\right)=V_1\left(k\right)+{‖z_k‖}^2={‖X_k‖}^2,X_k={\left[{\sigma }_{e,k}^T{z}_k^T\right]}^T.$$E22

As Eq. (14) is given by

$$\Delta V_1\left(k\right)={\left(T{b}_k\right)}^2{‖z_k‖}^2+\left\{{\left(T{f}_1{b}_k\right)}^2-2T{f}_1{b}_k\right\}{‖{\sigma }_{e,k}‖}^2+2T{b}_k\left(1-T{f}_1{b}_k\right)z_k^T{\sigma }_{e,k}^T$$

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

$$\begin{array}{cc}\Delta V_2\left(k\right)=& \left(T^2{f}_2^2-2T{f}_2+T^2{b}_k^2\right){‖z_k‖}^2+\left\{{\left(T{f}_1{b}_k\right)}^2-2T{f}_1{b}_k\right\}{‖{\sigma }_{e,k}‖}^2\\ & +2T{b}_k\left(1-T{f}_1{b}_k\right)z_k^T{\sigma }_{e,k}^T+T^2{w}_k^T{J}^{-2}{w}_k+2T\left(1-T{f}_2\right)w_k^T{J}^{-1}{z}_k\\ & \le \left(2T^2{f}_2^2-4T{f}_2+T^2{b}_k^2+1\right){‖z_k‖}^2+\left\{{\left(T{f}_1{b}_k\right)}^2-2T{f}_1{b}_k\right\}{‖{\sigma }_{e,k}‖}^2\\ & +2T{b}_k\left(1-T{f}_1{b}_k\right)z_k^T{\sigma }_{e,k}^T+2{\left(\frac{T}{{\lambda }_J}\right)}^2{‖w_k‖}^2\\ =& X_k^T{Q}_k{X}_k+2{\left(\frac{T}{{\lambda }_J}\right)}^2{‖w_k‖}^2,\end{array}$$E23

where

$${\lambda }_J=‖J‖,Q_k=\left[\begin{array}{cc}{Q}_{11,k}& Q_{12,k}\\ Q_{12,k}^T& Q_{22,k}\end{array}\right],Q_{11,k}=\left\{{\left(T{f}_1{b}_k\right)}^2-2T{f}_1{b}_k\right\}{I}_3,$$

$$Q_{12,k}=T{b}_k\left(1-T{f}_1{b}_k\right)I_3,Q_{22,k}=\left(2T^2{f}_2^2-4T{f}_2+T^2{b}_k^2+1\right)I_3.$$

In Eq. (23), if $Q_k<0$, then

$$\Delta V_2\left(k\right)\le -\left|{\lambda }_{Q_k}^{\min }\right|{‖X_k‖}^2+2{\left(\frac{T}{{\lambda }_J}\right)}^2{‖w_k‖}^2,$$

where ${\lambda }_{Q_k}^{\min }<0\in \mathrm{R}$ is the minimum eigenvalue of $Q_k$ and the condition of ISS holds [18]. Hereafter, conditions of $f_1$ and $f_2$ which the matrix $Q_k$ holds $Q_k<0$ are derived under Assumption 2.

From Schur complement, condition $Q_k<0$ is equivalent to the following equations:

$${\left(T{f}_1{b}_k\right)}^2-2T{f}_1{b}_k<0,$$E24

$$2T^2{f}_2^2-4T{f}_2+c_k<0\left(c_k=\frac{T{b}_k{f}_1^2-2f_1-T{b}_k}{T{b}_k{f}_1^2-2f_1}\right).$$E25

Condition (24) is the same as Eq. (15), and assume that Eq. (24) holds. From Lemma 2, the range of $f_2$ that holds for Eq. (25) is obtained as

$$\frac{2-\sqrt{2\left(2-c_k\right)}}{2T}<f_2<\frac{2+\sqrt{2\left(2-c_k\right)}}{2T},$$E26

and the following Eq.

$$2-c_k>0\Rightarrow \frac{T{b}_k{f}_1^2-2f_1+T{b}_k}{T{b}_k{f}_1^2-2f_1}>0$$E27

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

$$T{b}_k{f}_1^2-2f_1+T{b}_k<0$$E28

in order to hold Eq. (27). From Lemma 2, the range of $f_1$ that holds for Eq. (28) is obtained as

$$\frac{1-\sqrt{1-{\left(T{b}_k\right)}^2}}{T{b}_k}<f_1<\frac{1+\sqrt{1-{\left(T{b}_k\right)}^2}}{T{b}_k},$$E29

and the following Eq.

$$1-{\left(T{b}_k\right)}^2>0\Rightarrow 0<T<\frac{1}{b_k}$$E30

must hold in order to have the real number. As $2\le \left(1/b_k\right)\le 4$ under Assumption 2, $T$ must satisfy the condition

$$0<T<2.$$E31

In addition, since

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

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

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

$$\frac{2-\sqrt{4-T^2}}{T}<f_1<\frac{2+\sqrt{4-T^2}}{T}\left(0<T<2\right).$$E32

Therefore, if $f_1$ satisfies Eq. (32) under Assumption 2, Eqs. (27) and (28) hold. Furthermore, since

$$\underset{b_k}{max}\frac{2-\sqrt{2\left(2-c_k\right)}}{2T}=\frac{1}{T}-\sqrt{\frac{T{f}_1^2-4f_1+T}{2T^2{f}_1\left(T{f}_1-4\right)}},$$

$$\underset{b_k}{min}\frac{2+\sqrt{2\left(2-c_k\right)}}{2T}=\frac{1}{T}+\sqrt{\frac{T{f}_1^2-4f_1+T}{2T^2{f}_1\left(T{f}_1-4\right)}},$$

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

$$\frac{1}{T}-\sqrt{\frac{T{f}_1^2-4f_1+T}{2T^2{f}_1\left(T{f}_1-4\right)}}<f_2<\frac{1}{T}+\sqrt{\frac{T{f}_1^2-4f_1+T}{2T^2{f}_1\left(T{f}_1-4\right)}}\left(0<T<2\right).$$E33

Therefore, if $f_1$ and $f_2$ satisfy Eqs. (32) and (33) under Assumption 2, then $Q_k<0$.

Summarizing the above, the following theorem can be obtained.

Theorem 1: if sampling period $T$ and feedback gains $f_1$ and $f_2$ satisfy Eqs. (31), (32), and (33) under Assumption 2, then the closed-loop systems (10) and (11) with the following control law

$$\begin{array}{cc}{u}_k=& {\left\{z_k+{\alpha }_{,k}+C_k{\omega }_{d,k}\right\}}^{\times }J\left\{z_k+{\alpha }_{,k}+C_k{\omega }_{d,k}\right\}\\ & +J\left\{C_k{\dot{\omega }}_{d,k}-{\left(z_k+{\alpha }_{,k}\right)}^{\times }{C}_k{\omega }_{d,k}\right\}\\ & -f_{1}J\left\{G\left({\sigma }_{e,k}\right)z_k-f_1{b}_k{\sigma }_{e,k}\right\}-f_{2}J{z}_k\\ =& {{\omega }_k}^{\times }J{\omega }_k+J\left\{C_k{\dot{\omega }}_{d,k}-{\left(z_k+{\alpha }_{,k}\right)}^{\times }{C}_k{\omega }_{d,k}\right\}\\ & -f_1{f}_{2}J{\sigma }_{e,k}-J\left\{f_{1}G\left({\sigma }_{e,k}\right)+f_2{I}_3\right\}{\omega }_{e,k}\end{array}$$E34

becomes ISS.

Then, we show that the pair $\left(u_k,V_2\left(k\right)\right)$ is semiglobal practical asymptotic (SPA) stabilizing pair for the Euler approximate systems (10) and (11). Hereafter, suppose that sampling period $T$ and feedback gains $f_1$ and $f_2$ satisfy Eqs. (31), (32), and (33) under Assumption 2. By using the following coordinate transformation

$$X_k=\left[\begin{array}{cc}1& 0\\ f_1& 1\end{array}\right]\left[\begin{array}{c}{\sigma }_{e,k}\\ {\omega }_{e,k}\end{array}\right]=Z{\overline{X}}_k,$$

Lyapunov function $V_2\left(k\right)$ and its difference $\Delta V_2\left(k\right)$ can be rewritten as

$$V_2\left(k\right)={\overline{X}}_k^T{Z}^{T}Z{\overline{X}}_k={\overline{X}}_k^{T}R{\overline{X}}_k,$$

$$\Delta V_2\left(k\right)={\overline{X}}_k^T{Z}^T{Q}_{k}Z{\overline{X}}_k+2{\left(\frac{T}{{\lambda }_J}\right)}^2{‖w_k‖}^2={\overline{X}}_k^T{\overline{Q}}_k{\overline{X}}_k+2{\left(\frac{T}{{\lambda }_J}\right)}^2{‖w_k‖}^2.$$

Since $R>0$ and ${\overline{Q}}_k<0$, $V_2\left(k\right)$ and $\Delta V_2\left(k\right)$ satisfy following equations:

$${\lambda }_R^{\min }{‖{\overline{X}}_k‖}^2\le V_2\left(k\right)\le {\lambda }_R^{\max }{‖{\overline{X}}_k‖}^2,$$E35

$$\Delta V_2\left(k\right)\le -\left|{\lambda }_{{\overline{Q}}_k}^{\min }\right|{‖{\overline{X}}_k‖}^2+2{\left(\frac{T}{{\lambda }_J}\right)}^2{‖w_k‖}^2.$$E36

In addition, ${\overline{X}}_k$ is bounded, and $V_2\left(k\right)$ is radially unbounded from Eqs. (35) and (36). Hence, the control input (34) satisfies the following equation under Assumption 1:

$$‖u_k‖\le M,$$E37

where $M$ is a positive constant. Furthermore, $V_2\left(k\right)$ also satisfies the following equation for all $x,z\in {\mathrm{R}}^6$ with $max\left\{‖x‖,‖z‖\right\}\le \mathrm{\Delta }$:

$$\begin{array}{cc}\left|V_2\left(x\right)-V_2\left(z\right)\right|& =\left|x^T\mathit{Rx}-z^T\mathit{Rz}\right|=\left|{\left(x+z\right)}^{T}R\left(x-z\right)\right|\\ & ={\lambda }_R^{\max }‖x+z‖‖x-z‖\le 2\mathrm{\Delta }{\lambda }_R^{\max }‖x-z‖,\end{array}$$E38

where $\mathrm{\Delta }$ is a positive constant. Therefore, from Eqs. (35) to (38), Lyapunov function $V_2\left(k\right)$ and control input $u_k$ satisfied Eqs. (A8)–(A11) in Definition 2 under Assumptions 1 and 2, and the pair $\left(u_k,V_2\left(k\right)\right)$ 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

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

$$J=\left[\begin{array}{ccc}7050.0& -0.536& 43.9\\ -0.536& 2390& 1640.0\\ 43.9& 1640.0& 6130.0\end{array}\right]\mathrm{kg}{\mathrm{m}}^2,\sigma \left(0\right)=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],\omega \left(0\right)=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\mathrm{rad}/\mathrm{s}$$

$$T=\left\{\begin{array}{c}1.0:\text{Case}1\\ 0.5:\text{Case}2\\ 0.1:\text{Case}3\end{array}\right.,f_1=0.6,f_2=0.8.$$

The moment of inertia $J$ is from [1]. The initial values $\sigma \left(0\right)$ correspond to Euler angles of 1–2-3 system of $\theta \left(0\right)={\left[{\theta }_1\left(0\right){\theta }_2\left(0\right){\theta }_3\left(0\right)\right]}^T={\left[000\right]}^T\left[deg\right]$. The feedback gains $f_1$ and $f_2$ satisfy Eqs. (25) and (28) for all cases of $T$. The desired states ${\sigma }_d\left(t\right)$, ${\omega }_d\left(t\right)$, and ${\dot{\omega }}_d\left(t\right)$ in this simulation are the switching maneuver as shown in Figure 1.

Figure 1.

Switching maneuver.

The results of the numerical simulation are shown in Figures 2–5. The relative attitude ${\sigma }_e\left(t\right)$ and relative angular velocity ${\omega }_e\left(t\right)$ converge to the neighborhood of $\left({\sigma }_e\left(t\right),{\omega }_e\left(t\right)\right)=\left(0,0\right)$, and the control input amplitude $u\left(t\right)$ does not depend on the sampling period $T$ although there is a slight difference in the maximal value of $u\left(t\right)$.

Figure 2.

Time histories of MRPs $\sigma \left(t\right)$ and ${\sigma }_e\left(t\right)$ (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, ${\sigma }_d\left(t\right)$).

Figure 3.

Time histories of attitude angles $\theta \left(t\right)$ and ${\theta }_e\left(t\right)$ (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, ${\theta }_d\left(t\right)$).

Figure 4.

Time histories of angular velocities $\omega \left(t\right)$ and ${\omega }_e\left(t\right)$ (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, ${\omega }_d\left(t\right)$).

Figure 5.

Time histories of control input $u\left(t\right)$ (solid line, case 1; dashed-dotted line, case 2; and dashed line, case 3).

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.

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

Let us consider the following nonlinear system:

$$\dot{x}\left(t\right)=f\left(x\left(t\right),u\left(t\right)\right),x\left(0\right)=x_0,f\left(0,0\right)=0,$$EA1

where $x\left(t\right)\in {\mathrm{R}}^n$ is the state variable and $x\left(t\right)\in {\mathrm{R}}^m$ is the control input. The function $f\left(x\left(t\right),u\left(t\right)\right)$ 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 $x\left[\left.0,\tau \right)\right.$.

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,

$$u\left(t\right)=u\left(\mathit{kT}\right)≕u\left(k\right),\forall t\in \left[\mathit{kT},\left(k+1\right)T\right],k\in \left\{0\right\}\cup \mathrm{N},$$EA2

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

$$x\left(k\right)≔x\left(\mathit{kT}\right)$$EA3

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:

$$x_{k+1}=x_k+{\int }_{\mathit{kT}}^{\left(k+1\right)T}f\left(x\left(s\right),u_k\right)\mathit{ds}≕F_T^e\left(x_k,u_k\right),$$EA4

$$x_{k+1}=x_k+\mathit{Tf}\left(x_k,u_k\right)≕F_T^{\mathit{\text{Euler}}}\left(x_k,u_k\right),$$EA5

where we abbreviate $x\left(k\right)$ and $u\left(k\right)$ to $x_k$ and $u_k$. For the stability of the exact discrete-time model (A4) ($F_T^e$) and Euler approximate model (A5) ($F_T^{\mathit{\text{Euler}}}$), the following definitions are used [13, 14, 19].

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

$$x_{k+1}=F_T\left(x_k,u_T\left(x_k\right)\right),$$EA6

where $x_k\in {\mathrm{R}}^n$ is the state variable and $u_T\left(x_k\right)\in {\mathrm{R}}^m$ is a control input. The family of controllers $u_T\left(x_k\right)$ SPA stabilizes the system (A6) if there exists a class KL function $\beta \left(·\right)$ such that for any strictly positive real numbers $\left(D,\nu \right)$, there exists $T^{\ast }>0$, and such that for all $T\in \left(0,T^{\ast }\right)$ and all initial state $x_0$ with $‖x_0‖\le D$, the solution of the system satisfies

$$‖x_k‖\le \beta \left(‖x_0‖,\mathit{kT}\right)+\nu ,\forall k\in \left\{0\right\}\cup \mathrm{N}.$$EA7

Definition 2: let $\widehat{T}>0$ be given, and for each $T\in \left(0,\widehat{T}\right)$, let functions $V_T:{\mathrm{R}}^n\to \mathrm{R}$ and $u_T:{\mathrm{R}}^n\to {\mathrm{R}}^m$ be defined. The pair of families $\left(u_T,V_T\right)$ is a SPA stabilizing pair for the system (A7) if there exist a class ${\mathrm{K}}_{\infty }$ functions ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$ such that for any pair of strictly positive real numbers $\left(\mathrm{\Delta },\delta \right)$, there exists a triple of strictly positive real numbers $\left(T^{\ast },L,M\right)\left(T^{\ast }\le \widehat{T}\right)$ such that for all $x,z\in {\mathrm{R}}^n$ with$max\left\{‖x‖,‖z‖\right\}\le \mathrm{\Delta }$, and $T\in \left(0,T^{\ast }\right)$:

$${\alpha }_1\left(‖x‖\right)\le V_T\left(x\right)\le {\alpha }_2\left(‖x‖\right),$$EA8

$$V_T\left(F_T\left(x,u_T\left(x\right)\right)\right)-V_T\left(x\right)\le -{\alpha }_3\left(‖x‖\right)+\mathit{T\delta },$$EA9

$$\left|V_T\left(x\right)-V_T\left(z\right)\right|\le L‖x-z‖,$$EA10

$$‖u_T\left(x\right)‖\le M.$$EA11

In addition, if there exists $T^{\ast \ast }>0$ such that Eqs. (A8)–(A11) with $\delta =0$ hold for all $x\in {\mathrm{R}}^n$ and $T\in \left(0,T^{\ast \ast }\right)$, then the pair $\left(u_T,V_T\right)$ 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 $\left(u_T,V_T\right)$ is SPA stabilizing for $F_T^{\mathit{\text{Euler}}}$, then $u_T$ is SPA stabilizing for $F_T^e$ .

Hence, if we can find a family of pairs of $\left(u_T,V_T\right)$ that is a GA or SPA stabilizing pair for $F_T^{\mathit{\text{Euler}}}$, then the controller $u_T$ will stabilize the exact model $F_T^e$.