Fault-tolerant sliding mode attitude tracking control for flexible spacecraft with disturbance and modeling uncertainty

Abstract

This article investigates the attitude tracking control problem for a class of flexible spacecraft with a redundant four reaction wheels’ setting. In this study, inertia uncertainties, external disturbance, wheel torque saturation, and configuration misalignment are taken into account simultaneously. Two types of sliding mode controllers are presented to solve this design problem. First, supposing the norm upper bound of external disturbances, bounds of inertia, and input uncertainties are available, a terminal sliding mode attitude tracking control strategy is developed, where the controller and steering laws are synthesized together. Second, supposing these norm upper bounds are unavailable, an adaptive control law is designed to cope with attitude tracking problem. Under both control schemes, the system trajectory can be ensured to arrive on the specified sliding surface in finite time. Finally, an illustrative example is given to verify the effectiveness of the proposed attitude tracking control methodologies.

Introduction

Satellite control systems refer to one of the most significant mechatronics systems, which have received extensive research attention in practical spacecraft engineering. With the development of modern micro-satellite technology, the fast integrated techniques provide feasible way to build low-cost, fast, and light-weight spacecrafts and have been thus widely employed to various space missions including surveillance, Earth imaging, and telecommunication.^1–3 Among these fast integrated satellites, plug-and-play satellite is one of the most significant application examples which has attracted considerable research attention. ⁴ However, in practical control systems, model uncertainty and nonlinearity, inertial uncertainty, external disturbance, control signal delay, and other constraints always exist inevitably,^5–9 and actuator alignment errors also occur frequently due to finite-manufacturing tolerance or warping of spacecraft structure. ¹⁰ These phenomena may degrade the performances of the spacecraft control systems, and thus, a variety of designed approaches have been developed to investigate robust control design problems.^11–16

Among the existing methods for spacecraft attitude control system design, sliding mode control is an effective robust control scheme due to its insensitiveness to matched external disturbances or nonlinearity, which has been thus applied broadly to cope with attitude and/or orbit maneuver and regulation problems.^17–19 To mention a few, a quaternion-based attitude tracking control law is proposed for rigid satellite in the presence of unknown inertia matrix. ²⁰ A robust control strategy is developed for rigid body satellite to achieve high-precision attitude tracking, where time-varying mass and unknown external disturbance are considered together. ²¹ An adaptive sliding mode attitude tracking control scheme is proposed in the presence of actuator misalignment, uncertain inertia, and external disturbance. ¹⁰

Although there have been a few published results concerned with attitude tracking control problem of spacecraft with or without consideration of actuator misalignment, inertia uncertainty, and unknown external disturbance, most of the obtained results are only focused on rigid body, while flexible spacecraft has received little research attention.^22,23 Moreover, most of the achieved methods are based on the assumption that full knowledge of inertia, faults, or/and disturbance is available to the controller, while these assumptions do not hold in practice due to physical constraints. Moreover, the existing attitude control results mostly only consider the asymptotic stability, that is, the system trajectories converge to zero in infinite settling time. In fact, finite-time stabilization can provide faster convergence to the origin and more appropriate disturbance rejection, which is desirable to be considered in spacecraft control systems design. However, to the best of the authors’ knowledge, the finite-time attitude tracking problem has not been fully investigated for flexible spacecraft with actuator misalignment, unknown inertia, and external disturbance. It is, therefore, the objective of this article to solve this problem by developing two new types of attitude control schemes.

In this article, we investigate the asymptotic and finite-time attitude stabilization problem with simultaneous consideration of the flexible appendage, actuator misalignment, and external disturbance. First, based on the assumption that the bound knowledge of external disturbances and inertia and input uncertainties are known, a finite-time sliding mode control scheme is designed to solve the attitude tracking problem. Second, in the case that these norm upper bounds are unknown, an adaptive control law is designed to revisit attitude tracking problem. In light of both of these control methods, the reachability of the specified sliding surface can be guaranteed in finite time. Finally, a practical example is presented to demonstrate the validness of the designed attitude tracking control strategies.

The remaining part of this article is organized as follows. Section “Problem formulation” formulates the design problem and objective of this article. Section “Terminal finite-time sliding mode control strategy” presents the detailed controller design process. Section “Adaptive sliding mode control scheme” builds illustrative simulation results to verify the effectiveness of the proposed methods. Finally, the conclusion is discussed in section “Numerical simulation.”

Problem formulation

The attitude kinematics

With the representation of nonsingular unit quaternion, the satellite kinematics equation can be described as follows

$$\stackrel{.}{q}=\left[\begin{array}{c}-\frac{1}{2}{q}_v^T\\ \frac{1}{2}\left(q_0{I}_3+q_v^{\times }\right)\end{array}\right]\omega$$

(1)

where $\omega$ represents the satellite angular velocity, $q=(q_0,q_v)\in \mathbb{R}\times {\mathbb{R}}^3$ with

$$q_v^{\times }=\left[\begin{array}{ccc}0& -q_3& q_2\\ q_3& 0& -q_1\\ -q_2& q_1& 0\end{array}\right]$$

and $q_v^T{q}_v+q_0^2=1$ .

Flexible spacecraft dynamics

The total angular momentum of a flexible spacecraft can be represented as

$$H=J\omega +\mathrm{Dh}+{\delta }^T\stackrel{·}{\eta }$$

where J represents the inertia matrix, $D\in {\mathbb{R}}^{3\times 4}$ is the reaction wheel orientation matrix, h is the angular momentum of the satellite, $\delta$ is the coupling matrix, and $\eta$ represents the vibration mode. If misalignment of the wheels is considered, we have

$$H=J\omega +(D+\mathrm{\Delta }D)h+{\delta }^T\stackrel{·}{\eta }$$

where $\mathrm{\Delta }D$ denotes the unknown misalignment matrix. Thus, the flexible dynamics can be written as

$$\{\begin{array}{c}J\stackrel{·}{\omega }+{\delta }^T\stackrel{··}{\eta }=-\omega \times (J\omega +{\delta }^T\stackrel{·}{\eta }+(D+\mathrm{\Delta }D)h)-(D+\mathrm{\Delta }D)\stackrel{·}{h}+d(t)\hfill \\ \multicolumn{1}{c}{\stackrel{··}{\eta }+C\stackrel{·}{\eta }+K\eta +\delta \stackrel{·}{\omega }=0}\end{array}$$

where

$$C=\left[\begin{array}{ccc}2{\zeta }_1{\Lambda }_1^{\frac{1}{2}}\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{0}& \cdots \hfill & 2{\zeta }_N{\Lambda }_N^{\frac{1}{2}}\hfill \end{array}\right],K=\left[\begin{array}{ccc}{\Lambda }_1\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{0}& \cdots \hfill & {\Lambda }_N\hfill \end{array}\right]$$

denotes the damping and stillness matrices, respectively; ${\zeta }_i$ and ${\omega }_i$ are the damping ratio and natural frequency of the ith-order mode, respectively; and ${\Lambda }_i={\omega }_i^2$ ; and $d(t)$ denotes the external disturbance.

For the four-wheel setting

$$u(t)={\left[\begin{array}{cccc}{\tau }_1(t)\hfill & {\tau }_2(t)\hfill & {\tau }_3(t)\hfill & {\tau }_4(t)\hfill \end{array}\right]}^T=-\stackrel{·}{h}$$

be the output torque of the reaction wheels, and $u_r(t)=-(D+\mathrm{\Delta }D)\stackrel{·}{h}=(D+\mathrm{\Delta }D)u(t)$ be the system control torque. Then, we have

$$\{\begin{array}{c}J\stackrel{·}{\omega }+{\delta }^T\stackrel{··}{\eta }=-\omega \times (J\omega +{\delta }^T\stackrel{·}{\eta }+(D+\mathrm{\Delta }D)h)+u_r\left(t\right)+d\left(t\right)\hfill \\ \multicolumn{1}{c}{\stackrel{··}{\eta }+C\stackrel{·}{\eta }+K\eta +\delta \stackrel{·}{\omega }=0}\end{array}$$

(2)

In system (2), define an auxiliary state $\xi =\delta \omega +\stackrel{·}{\eta }$ , and we have

$$\stackrel{·}{\xi }=\delta \stackrel{·}{\omega }+\stackrel{··}{\eta }=-C\xi +C\delta \omega -K\eta$$

(3)

Substituting equation (3) into equation (2) yields

$$(J-{\delta }^T\delta )\stackrel{·}{\omega }=-{\omega }^{\times }J\omega +u_r(t)+\overline{d}(t)$$

(4)

with the generalized disturbance vector $\overline{d}(t)$ denoted as

$$\begin{array}{l}\overline{d}(t)={\delta }^T\left[\begin{array}{cc}K& C\end{array}\right]\left[\begin{array}{c}\eta \\ \xi \end{array}\right]-{\delta }^{T}C\delta \omega -{\omega }^{\times }{\delta }^T(\xi -\delta \omega )\\ -{\omega }^{\times }\left(D+\mathrm{\Delta }D\right)h+d(t)\end{array}$$

Recall the relationship of the total control torque with the individual wheel

$$u_r(t)=(D+\mathrm{\Delta }D)u(t)$$

(5)

As a result, system (4) becomes

$$(J-{\delta }^T\delta )\stackrel{·}{\omega }=-{\omega }^{\times }J\omega +(D+\mathrm{\Delta }D)u(t)+\overline{d}(t)$$

(6)

Attitude-error quaternion–based kinematics and dynamics

Now, considering the kinematics in equation (1), define $q_d=(q_{d0},q_{\mathrm{dv}})\in {\mathbb{R}}^4$ as the desired attitude, where $q_{d0}\in {\mathbb{R}}^1$ , $q_{\mathrm{dv}}\in {\mathbb{R}}^3$ , and ${\omega }_d$ is the desired angular velocity. Then, $q_d$ should satisfy the following kinematics

$${\stackrel{·}{q}}_d=\left[\begin{array}{c}-\frac{1}{2}{q}_{\mathrm{dv}}^T\hfill \\ \multicolumn{1}{c}{\frac{1}{2}(q_{d0}{I}_3+q_{\mathrm{dv}}^{\times })}\end{array}\right]{\omega }_d$$

(7)

Define attitude tracking error

$$q_e={\left[\begin{array}{cc}{e}_0\hfill & e\hfill \end{array}\right]}^T,e_0\in R,e\in R^3$$

(8)

and define the relative velocity ${\omega }_e$ as

$${\omega }_e=\omega -\tilde{R}{\omega }_d$$

(9)

where $\tilde{R}\in R^{3\times 3}$ denotes the rotation matrix

$$\tilde{R}=(e_0^2-e^{T}e)I_3+2e{e}^T-2e_0{e}^{\times }$$

(10)

From equation (9), we have (note that $\stackrel{·}{\stackrel{~}{R}}=-{\omega }_e^{\times }\tilde{R}$ )

$${\stackrel{·}{\omega }}_e=\stackrel{·}{\omega }-\stackrel{·}{\stackrel{~}{R}}{\omega }_d-\tilde{R}{\stackrel{·}{\omega }}_d=\stackrel{·}{\omega }+{\omega }_e^{\times }\tilde{R}{\omega }_d-\tilde{R}{\stackrel{·}{\omega }}_d$$

(11)

Substituting equation (11) into equation (6) yields

$$\begin{array}{l}(J-{\delta }^T\delta )({\stackrel{.}{\omega }}_e-{\omega }_e^{\times }\tilde{R}{\omega }_d+\tilde{R}{\stackrel{.}{\omega }}_d)\\ =-{({\omega }_e+\tilde{R}{\omega }_d)}^{\times }J({\omega }_e+\tilde{R}{\omega }_d)+(D+\mathrm{\Delta }D)u(t)+\overline{d}(t)\end{array}$$

(12)

Defining $J_0=J-{\delta }^T\delta$ and recalling $\omega ={\omega }_e+\tilde{R}{\omega }_d$ , equation (12) can be rewritten as

$$\begin{array}{l}{J}_0{\stackrel{.}{\omega }}_e=-{\omega }^{\times }J\omega +J_0({\omega }_e^{\times }\tilde{R}{\omega }_d-\tilde{R}{\stackrel{.}{\omega }}_d)\\ +(D+\mathrm{\Delta }D)u(t)+\overline{d}(t)\end{array}$$

(13)

Moreover, the attitude-error quaternion satisfies

$$\{\begin{array}{c}{\stackrel{·}{e}}_0=-\frac{1}{2}{e}^T{\omega }_e\hfill \\ \stackrel{·}{e}=\frac{1}{2}(e^{\times }+e_0{I}_3){\omega }_e\hfill \end{array}$$

(14)

To achieve the attitude tracking objective, we make the following assumptions for systems (13)–(14).

• Assumption 1. The external disturbance $\overline{d}(t)$ satisfies

$$||\overline{d}(t)||\le d_m$$

where $d_m>0$ is a known constant.

• Assumption 2. The uncertain matrix $\mathrm{\Delta }D$ satisfies

$$||D^T(D{D}^T{)}^{-1}\mathrm{\Delta }D||\le \alpha ,||\mathrm{\Delta }D{D}^T(D{D}^T{)}^{-1}||\le \alpha$$

where $\alpha >0$ is a known constant.

• Assumption 3. The inertia matrix is supposed to satisfy

$$||J||\le \overline{J}$$

where $\overline{J}>0$ is the upper bound.

• Then, the design objective is to make the attitude error and velocity error converge to zero

$$\underset{t\to \infty }{lim}e=0,\underset{t\to \infty }{lim}{\omega }_e=0$$

(15)

Terminal finite-time sliding mode control strategy

We define the following sliding mode surface

$$s(t)={\omega }_e(t)+{\delta }_{1}e(t)+{\delta }_2\mathrm{sgn}(e{)}^{\beta }$$

(16)

where $s(t)=[\begin{array}{ccc}{s}_1^T\hfill & s_2^T\hfill & s_3^T\hfill \end{array}{]}^T\in R^3,0<\beta <1$ , and $\mathrm{sgn}(e{)}^{\beta }=[|e_1{|}^{\beta }\mathrm{sgn}(e_1),|e_2{|}^{\beta }\mathrm{sgn}(e_2),|e_3{|}^{\beta }\mathrm{sgn}(e_3){]}^T$ .

Lemma 1

Consider kinematics system (14) and sliding mode surface (equation (16)) with $s(t)=0$ under the following controller

$$u(t)=u_1(t)+u_2(t)+u_3(t)$$

with

$$\begin{array}{c}{u}_1(t)=-(1-\alpha {)}^{-1}\rho ·D^T(D{D}^T{)}^{-1}s(t)\hfill \\ u_2(t)=-(1-\alpha {)}^{-1}\gamma ·D^T(D{D}^T{)}^{-1}\mathrm{sig}(s{)}^{\beta }\hfill \\ u_3(t)=-(1-\alpha {)}^{-1}·D^T(D{D}^T{)}^{-1}f(t)·\frac{s(t)}{||s(t)||}\hfill \end{array}$$

(17)

where $\rho =\mathrm{diag}\{{\rho }_1,{\rho }_2,{\rho }_3\},\gamma =\mathrm{diag}\{{\gamma }_1,{\gamma }_2,{\gamma }_3\},{\rho }_i,{\gamma }_i>0(i=1,2,3)$ , and

$$\begin{array}{l}f(t)=\Vert {\delta }_1{e}^{\times }-{\delta }_2{(sgn{(e)}^{\beta })}^{\times }+{(\tilde{R}{\omega }_d)}^{\times }\Vert ·\Vert J_{0}w\Vert \\ +\Vert J_0({\omega }_e^{\times }(t)\tilde{R}{\omega }_d-\tilde{R}{\stackrel{.}{w}}_d)\Vert \\ +\Vert \frac{1}{2}{J}_0{\delta }_1(e^{\times }+e_0{I}_3){\omega }_e(t)\Vert \\ +\Vert \frac{1}{2}{J}_0{\delta }_2|e{|}^{\beta -1}\beta (e^{\times }+e_0{I}_3){\omega }_e(t)\Vert +d_m\end{array}$$

(18)

Then, ${\xi }_1(t)=0$ and ${\xi }_2(t)=0$ can be achieved in finite time.

Proof 1

For dynamic system (13), we define the following Lyapunov function

$$V_1(t)=\frac{1}{2}{s}^T{J}_{0}s$$

(19)

We now consider the sliding mode surface (equation (16)) with two cases of equations (1) and (2).

1. When $e_1>0$ , we have

   $$\mathrm{sgn}(e_1{)}^{\beta }=(e_1{)}^{\beta },(\mathrm{sgn}(e_1{)}^{\beta })\prime =\beta e_1^{\beta -1}{\stackrel{·}{e}}_1=|e_1{|}^{\beta -1}\beta {\stackrel{·}{e}}_1$$

2. When $e_1<0$ , we have

   $$\mathrm{sgn}(e_1{)}^{\beta }=(-e_1{)}^{\beta }(-1)=(-1{)}^{\beta +1}{e}_1^{\beta }$$

   and

   $$(\mathrm{sgn}(e_1{)}^{\beta })\prime =(-1{)}^{\beta +1}\beta e_1^{\beta -1}{\stackrel{·}{e}}_1=|e_1{|}^{\beta -1}\beta {\stackrel{·}{e}}_1$$

As a result, we have

$$\frac{d(\mathrm{sgn}{(e)}^{\beta })}{\mathrm{dt}}=|e{|}^{\beta -1}\beta \stackrel{·}{e}$$

(20)

Hence, from equation (16), we have $\stackrel{.}{s}(t)={\stackrel{.}{w}}_e(t)+s_1\stackrel{.}{e}(t)+{\delta }_2|e{|}^{\beta -1}\beta \stackrel{.}{e}$ , and we now recall the Lyapunov function (equation (19)) and it can be calculated as

$${\stackrel{·}{V}}_1(t)=s^T{J}_0\stackrel{·}{s}(t)=s^T{J}_0({\stackrel{·}{w}}_e(t)+{\delta }_1\stackrel{·}{e}(t)+{\delta }_2|e{|}^{\beta -1}\beta {\stackrel{·}{e}}_1)$$

(21)

recalling that

$$\begin{array}{c}{\stackrel{·}{w}}_e(t)=J_0^{-1}\{-w^{\times }\mathrm{Jw}+J_0({\omega }_e^{\times }\tilde{R}{\omega }_d-\tilde{R}{\stackrel{·}{w}}_d)\hfill \\ +(D+\mathrm{\Delta }D)u(t)+\overline{d}(t)\}\hfill \\ \stackrel{·}{e}(t)=\frac{1}{2}(e^{\times }+e_0{I}_3){\omega }_e(t)\hfill \end{array}$$

Then, equation (21) is derived as

$${\stackrel{·}{V}}_1(t)=s^T\left(\begin{array}{c}-w^{\times }{J}_{0}w+J_0({\omega }_e^{\times }\tilde{R}{\omega }_d-\tilde{R}{\stackrel{·}{w}}_d)+(D+\mathrm{\Delta }D)u(t)+\overline{d}(t)+J_0{\delta }_1\frac{1}{2}(e^{\times }+e_0{I}_3){\omega }_e(t)\hfill \\ \multicolumn{1}{c}{+J_0{\delta }_2|e{|}^{\beta -1}\beta \frac{1}{2}(e^{\times }+e_0{I}_3){\omega }_e(t)}\end{array}\right)$$

(22)

Note that

$${\omega }_e(t)=s(t)-{\delta }_{1}e(t)-{\delta }_2\mathrm{sgn}(e{)}^{\beta }$$

which implies that

$w(t)={\omega }_e(t)+\tilde{R}{\omega }_d=s(t)-{\delta }_{1}e(t)-{\delta }_2\mathrm{sgn}(e{)}^{\beta }+\tilde{R}{\omega }_d$ , and also notice that $s^T{s}^{\times }=[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array}]$ , and equation (22) becomes

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1(t)=s^T({\delta }_1{e}^{\times }(t)-{\delta }_2(\mathrm{sgn}(e{)}^{\beta }{)}^{\times }+(\tilde{R}{\omega }_d{)}^{\times })J_0\omega }\\ +J_0({\omega }_e^{\times }\tilde{R}{\omega }_d-\tilde{R}{\stackrel{·}{w}}_d)+J_0{\delta }_1\frac{1}{2}(e^{\times }+e_0{I}_3){\omega }_e(t)\hfill \\ +\frac{1}{2}{J}_0{\delta }_2|e{|}^{\beta -1}\beta (e^{\times }+e_0{I}_3){\omega }_e(t)\hfill \\ +s^T\mathrm{\Delta }\mathrm{Du}(t)+s^T\mathrm{Du}(t)+s^T\overline{d}(t)\hfill \end{array}$$

(23)

Note that $||J_0||=||J-{\delta }^T\delta ||\le ||J||+||{\delta }^T\delta ||$ , we have

$$||J_0||\le ||\overline{J}||+||{\delta }^T\delta ||\stackrel{\mathrm{\Delta }}{=}\theta$$

Hence, equation (23) implies

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1(t)\le ||s||·||{\delta }_1{e}^{\times }(t)-{\delta }_2(\mathrm{sgn}(e{)}^{\beta }{)}^{\times }+(\tilde{R}{\omega }_d{)}^{\times })||}\\ \times ||J_0||·||w||+||s||·||J_0({\omega }_e^{\times }(t)\tilde{R}{\omega }_d-\tilde{R}{\stackrel{·}{w}}_d)||\hfill \\ +||s||·\Vert \frac{1}{2}{J}_0{\delta }_1(e^{\times }+e_0{I}_3){\omega }_e(t)\Vert \hfill \\ +||s||·\Vert \frac{1}{2}{J}_0{\delta }_2|e{|}^{\beta -1}\beta (e^{\times }+e_0{I}_3){\omega }_e(t)\Vert \hfill \\ +s^T\mathrm{Du}(t)+s^T\mathrm{\Delta }\mathrm{Du}(t)+||s||d_{\max }\hfill \end{array}$$

(24)

In equation (24), consider the term

$$s^T\mathrm{Du}(t)+s^T\mathrm{\Delta }\mathrm{Du}(t)=s^{T}D·(1+D^{+}\mathrm{\Delta }D)u(t)$$

(25)

where $D^{+}=D^T(D{D}^T{)}^{-1}$ denotes the pseudo-inverse matrix. Based on Assumption A2, equation (25) becomes

$$\begin{array}{l}{s}^{T}Du(t)+s^T\mathrm{\Delta }Du(t)\\ =(1-\alpha )s^{T}Du(t)+\alpha s^{T}Du(t)+s^T\mathrm{\Delta }Du(t)\end{array}$$

(26)

Substituting equations (17)–(18) into equation (24) and considering equation (26) yield

$${\stackrel{·}{V}}_1(t)\le ||s||f(t)+(1-\alpha )s^T\mathrm{Du}(t)+\alpha s^T\mathrm{Du}(t)+s^T\mathrm{\Delta }\mathrm{Du}(t)$$

(27)

Note that

$$\begin{array}{c}\alpha s^{T}D(u_1(t)+u_2(t)+u_3(t))\hfill \\ =-\alpha (1-\alpha {)}^{-1}{s}^T(\rho s(t)+\gamma \mathrm{sig}{(s)}^{\beta }+f(t)\frac{s(t)}{||s(t)||})\hfill \\ =-\frac{\alpha }{1-\alpha }(\rho ||s(t)|{|}^2+\gamma s^T\mathrm{sig}(s{)}^{\beta }+f(t)||s(t)||)\hfill \end{array}$$

(28)

In equation (28), recalling the definition of term $\mathrm{sgn}(s{)}^{\beta }$ , $s^T(t)\mathrm{sig}(s{)}^{\beta }=|s{|}^{\beta +1}$ and equation (28) becomes

$$\begin{array}{l}\alpha s^{T}D(u_1(t)+u_2(t)+u_3(t))\\ =-\alpha {(1-\alpha )}^{-1}(\rho ||s(t)|{|}^2+\gamma |s{|}^{\beta +1}+f(t)||s(t)||\end{array}$$

(29)

However, consider the term in equation (27)

$$\begin{array}{l}{s}^T\mathrm{\Delta }D(u_1(t)+u_2(t)+u_3(t))\le \alpha {(1-\alpha )}^{-1}(\rho ||s(t)|{|}^2\\ +\gamma |s{|}^{\beta +1}+||f(t)||·||s(t)||)\end{array}$$

(30)

Combining equations (29) and (30) yields

$$\alpha s^T\mathrm{Du}(t)+s^T\mathrm{\Delta }\mathrm{Du}(t)\le 0$$

Thus, equation (27) becomes

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1(t)\le ||s||f(t)+(1-\alpha )s^T\mathrm{Du}(t)=||s||f(t)}\\ -s^T(\rho s(t)+\gamma \mathrm{sig}{(s)}^{\beta }+f(t)\frac{s(t)}{||s(t)||})\hfill \\ =-\rho ||s(t)|{|}^2-\gamma \sum _{i=1}^3|s_i{|}^{\beta +1}\hfill \\ \le -{\rho }_{\min }||s(t)|{|}^2-{\gamma }_{\min }\sum _{i=1}^3||s_i|{|}^{\beta +1}\le -{\rho }_{\min }||s(t)|{|}^2\hfill \end{array}$$

$\mathrm{for}\forall$ , $s(t)\ne 0$ holds, which means that the state trajectory of the satellite control systems will achieve on the predefined sliding surface in finite time. We thus complete the proof.

Adaptive sliding mode control scheme

In this section, we will revisit the attitude tracking problem with an adaptive sliding mode control scheme. An adaptive strategy is adopted to estimate the upper bound of the generalized disturbance, and an adaptive control law is integrated with a specific steering law.

We define the following sliding mode surface

$$s(t)={\omega }_e(t)+K_{1}e(t)$$

(31)

where $s(t)=[{\hat{s}}_1,{\hat{s}}_2,{\hat{s}}_3]\in {\mathbb{R}}^3$ and $K_1=\mathrm{diag}[k_1,k_2,k_3],k_1>0,k_2>0,\mathrm{and}{k}_3>0$ are the parameter matrix to be designed.

Lemma 2

Considering the sliding mode surface (equation (31)), the spacecraft dynamics (equation (13)) and kinematics (equation (14)) can be stabilized by the following controller

$$\begin{array}{l}u(t)=-{(1-\alpha )}^{-1}\rho D^T{(D{D}^T)}^{-1}\frac{s(t)}{||s(t)||}\\ -{(1-\alpha )}^{-1}({\psi }_1(t)+{\psi }_2(t)||{\omega }_e||\\ +d_m(t))D^T{(D{D}^T)}^{-1}sign(s)\end{array}$$

(32)

with an adaptive law

$$\{\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{\psi }}_1(t)}& =C_1||s||\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\psi }}_2(t)}& =C_2||s||||{\omega }_e||\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{d}}_m(t)}& =C_3||s||\hfill \end{array}$$

(33)

Proof

We define the following Lyapunov candidate

$$V_2(t)=\frac{1}{2}{s}^T(t)J_{0}s(t)$$

(34)

Recalling equations (22)–(23), it is derived that

$$\begin{array}{l}{\stackrel{.}{V}}_2(t)=s^T(-K_1{e}^{\times }(t)+{(\tilde{R}{\omega }_d)}^{\times })J\omega \\ +J_0({\omega }_e^{\times }\tilde{R}{\omega }_d-\tilde{R}{\stackrel{.}{\omega }}_d)\\ +J_0{K}_1\frac{1}{2}(e^{\times }+e_0{I}_3){\omega }_e(t)+s^T\mathrm{\Delta }Du(t)\\ +s^{T}Du(t)+s^T\overline{d}(t)\end{array}$$

(35)

Assume that

$$\Vert (-{(K_{1}e(t))}^{\times }+{(\tilde{R}{\omega }_d)}^{\times })J\omega +J_0({\omega }_e^{\times }(t)\tilde{R}{\omega }_d-\tilde{R}{\stackrel{·}{\omega }}_d)+J_0{K}_1\frac{1}{2}(e^{\times }+e_0{I}_3)||\le {\psi }_1+{\psi }_2||{\omega }_e\Vert$$

(36)

where ${\psi }_1>0$ and ${\psi }_2>0$ are unknown constants. Then, equation (35) yields

$$\begin{array}{l}{\stackrel{.}{V}}_2(t)\le ||s||({\psi }_1+{\psi }_2||{\omega }_e||)\\ +||s||d_m+||s||||\mathrm{\Delta }Du(t)||+s^{T}u(t)\end{array}$$

(37)

Define a new Lyapunov function

$$V_3(t)=V_2(t)+\frac{1}{2}·{\frac{1}{C}}_1{\tilde{\psi }}_1^2(t)+\frac{1}{2}·{\frac{1}{C}}_2{\tilde{\psi }}_2^2(t)+\frac{1}{2}·{\frac{1}{C}}_3{\tilde{d}}_m^2(t)$$

(38)

where ${\tilde{\psi }}_1(t)={\psi }_1-{\psi }_1(t)$ , ${\tilde{\psi }}_2(t)={\psi }_2-{\psi }_2(t)$ , ${\tilde{d}}_m(t)=d_m-d_m(t)$ , and $C_1,C_2,C_3>0$ . Then

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3(t)={\stackrel{·}{V}}_2(t)+{\frac{1}{C}}_1{\tilde{\psi }}_1(t){\tilde{\psi }}_1(t)}\\ +{\frac{1}{C}}_2{\tilde{\psi }}_2(t){\tilde{\psi }}_2(t)+{\frac{1}{C}}_3{\tilde{d}}_m(t){\tilde{d}}_m(t)\hfill \\ \le ||s||({\psi }_1+{\psi }_2||{\omega }_e||+d_m)+||s||||\mathrm{\Delta }\mathrm{Du}(t)||\hfill \\ +s^{T}u(t)+{\frac{1}{C}}_1{\tilde{\psi }}_1(t){\tilde{\psi }}_1(t)+{\frac{1}{C}}_2{\tilde{\psi }}_2(t){\tilde{\psi }}_2(t)\hfill \\ +{\frac{1}{C}}_3{\tilde{d}}_m(t){\tilde{d}}_m(t)\hfill \end{array}$$

(39)

Note that ${\psi }_1={\psi }_1(t)+{\tilde{\psi }}_1(t)$ , ${\psi }_2={\psi }_2(t)+{\tilde{\psi }}_2(t)$ , and $d_m=d_m(t)+{\tilde{d}}_m(t)$ . Thus, equation (38) becomes

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3(t)\le ||s||({\psi }_1(t)+{\psi }_2(t)||{\omega }_e||+d_m(t))}\\ +||s||({\tilde{\psi }}_1(t)+{\tilde{\psi }}_2(t)||{\omega }_e||+{\tilde{\psi }}_m(t))\hfill \\ +{\frac{1}{C}}_1{\tilde{\psi }}_1(t){\tilde{\psi }}_1(t)+{\frac{1}{C}}_2{\tilde{\psi }}_2(t){\tilde{\psi }}_2(t)+{\frac{1}{C}}_3{\tilde{d}}_m(t){\tilde{d}}_m(t)\hfill \\ +||s||||\mathrm{\Delta }\mathrm{Du}(t)||+s^T\mathrm{Du}(t)\hfill \end{array}$$

(40)

Substituting the adaptive sliding mode controller (equation (32)) into equation (40) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3(t)\le ||s||({\psi }_1(t)+\psi (t{)}_2||{\omega }_e||+d_m(t))+||s||({\tilde{\psi }}_1(t)}\\ +{\tilde{\psi }}_2(t)||{\omega }_e||+{\tilde{d}}_m(t))\hfill \\ -{\tilde{\psi }}_1(t)||s||-{\tilde{\psi }}_2(t)||s||||{\omega }_e||-{\tilde{d}}_m(t)||s||\hfill \\ +||s||||\mathrm{\Delta }\mathrm{Du}(t)||+s^T\mathrm{Du}(t)\hfill \\ =||s||({\psi }_1(t)+{\psi }_2(t)||{\omega }_e||+d_m(t))\hfill \\ +||s||||\mathrm{\Delta }\mathrm{Du}(t)||+s^T\mathrm{Du}(t)\hfill \\ =||s||({\psi }_1(t)+\psi (t{)}_2||{\omega }_e||+d_m(t))\hfill \\ +||s||||\mathrm{\Delta }\mathrm{Du}(t)||-\frac{1}{1-\alpha }(\rho ||s(t)||\hfill \\ +({\psi }_1(t)+\psi (t{)}_2||{\omega }_e||+d_m(t))|s(t)|)\hfill \end{array}$$

(41)

Note that

$$\begin{array}{l}||s||||\mathrm{\Delta }Du(t)||\le ||s(t)||\frac{\alpha }{1-\alpha }\rho +\frac{\alpha }{1-\alpha }({\psi }_1(t)\\ +{\psi }_2(t)||{\omega }_e||+d_m(t))|s(t)|\end{array}$$

(42)

Substituting equation (42) into equation (41) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3(t)\le ||s(t)||({\psi }_1(t)+{\psi }_2(t)||{\omega }_e||+d_m(t))}\\ +\frac{\alpha }{1-\alpha }\{||s(t)||\rho +({\psi }_1(t)+\psi {(t)}_2||{\omega }_e||+d_m(t))|s(t)|\}\hfill \\ -\frac{\alpha }{1-\alpha }(||s(t)||\rho +({\psi }_1(t)+\psi {(t)}_2||{\omega }_e||+d_m(t))|s(t)|)\hfill \\ -(\rho ||s||+({\psi }_1(t)+\psi {(t)}_2||{\omega }_e||+d_m(t))|s(t)|)\hfill \\ \le -\rho ||s(t)||\hfill \\ <0\hfill \end{array}$$

(43)

$\mathrm{for}\forall$ , $s(t)\ne 0$ . We complete the proof.

To avoid chattering, the saturation function and boundary layer method are considered in controller (equation (32)), and we modify equation (32) as the following form

$$\begin{array}{l}u(t)=-{(1-\alpha )}^{-1}\rho D^T{(D{D}^T)}^{-1}\frac{s}{\Vert s\Vert +\Vert \epsilon \Vert }\\ -{(1-\alpha )}^{-1}({\psi }_1(t)+{\psi }_2(t)||{\omega }_e||+d_m(t))D^T{(D{D}^T)}^{-1}sat(s,\kappa )\end{array}$$

(44)

where $\mathrm{sat}(s,\kappa )=\{\begin{array}{cc}\kappa \hfill & s\ge \kappa \hfill \\ \multicolumn{1}{c}{s}& \Vert s\Vert <\kappa \hfill \\ \multicolumn{1}{c}{-\kappa }& s<-\kappa \hfill \end{array}$ .

Numerical simulation

In this section, we present a practical example to illustrate the effectiveness of the proposed control approach in this article. The inertia matrix is selected as

$$J_0=\left[\begin{array}{ccc}350\hfill & 3\hfill & 4\hfill \\ \multicolumn{1}{c}{3}& 280\hfill & 10\hfill \\ \multicolumn{1}{c}{4}& 10\hfill & 190\hfill \end{array}\right]$$

and the installation matrix of the reaction wheels is set as

$$D=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & \frac{1}{\sqrt{3}}\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & \frac{1}{\sqrt{3}}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & \frac{1}{\sqrt{3}}\hfill \end{array}\right]$$

Moreover, the coupling matrix is chosen as

$$\delta =\left[\begin{array}{ccc}6.45637\hfill & 1.27814\hfill & 2.15629\hfill \\ \multicolumn{1}{c}{-1.25619}& 0.91756\hfill & -1.67264\hfill \\ \multicolumn{1}{c}{1.11687}& 2.48901\hfill & -0.83674\hfill \\ \multicolumn{1}{c}{1.23637}& -2.6581\hfill & -1.12503\hfill \end{array}\right]\left(\mathrm{kg}{\mathrm{m}}^2\right)$$

and the external disturbance is set as $d(t)={10}^{-4}\times d\mathrm{N}\mathrm{m}$

$$d=\left[\begin{array}{c}-3+4\cos (0.2\pi t)-\cos (0.4\pi t)+2{\omega }_1\sin (0.11t)\hfill \\ \multicolumn{1}{c}{4+3\sin (0.2\pi t)-2\cos (0.4\pi t)+{\omega }_2\cos (0.11t)}\\ \multicolumn{1}{c}{-3+4\sin (0.2\pi t)-3\sin (0.4\pi t)-2{\omega }_3\cos (0.11t)}\end{array}\right]$$

The misalignment matrix is set as $\mathrm{\Delta }D=0.2\%D$ ; the damping ratios of the vibration suppression are chosen as ${\zeta }_1=0.005607$ , ${\zeta }_2=0.008620$ , ${\zeta }_3=0.01283$ , and ${\zeta }_4=0.02516$ ; the natural frequencies of the vibration suppression are ${\omega }_{n1}=0.7681\mathrm{rad}/\mathrm{s}$ , ${\omega }_{n2}=1.1038\mathrm{rad}/\mathrm{s}$ , ${\omega }_{n3}=1.8733\mathrm{rad}/\mathrm{s}$ , and ${\omega }_{n1}=2.5496\mathrm{rad}/\mathrm{s}$ ; ${\Lambda }_i={\omega }_{\mathrm{ni}}^2$ ; and the desired angular velocity is

$${\omega }_d=0.02{\left[\begin{array}{ccc}\sin \left(\frac{\pi }{100}t\right)\hfill & \sin \left(\frac{2\pi }{100}t\right)\hfill & \sin \left(\frac{3\pi }{100}t\right)\hfill \end{array}\right]}^T(\mathrm{rad}/\mathrm{s})$$

The initial attitude is $q=[\begin{array}{cccc}0.9950\hfill & 0.1\hfill & 0\hfill & 0\hfill \end{array}{]}^T$ ; the initial velocities are ${\omega }_x=0.05\circ /s$ , ${\omega }_y=-0.01\circ /s$ , and ${\omega }_z=0.{01}^{\circ }/s$ ; the initial condition of the adaptive parameters is ${\psi }_1(0)=0,{\psi }_2(0)=0,d_m(0)=0$ ; the maximum output torque is ${\tau }_{\max }=0.5\mathrm{N}\mathrm{m}$ ; the maximum angular momentum is $h_{\max }=10\mathrm{N}\mathrm{m}\mathrm{s}$ ; and we set $C_1=0.1,C_2=0.1,C_3=1$ and $\alpha =0.005,\rho =2,\kappa =0.1,\epsilon =0.001$ .

Figure 1 shows the trajectory of the attitude error, and it can be seen that the attitude tracking can be achieved to a certain degree with initial attitude error and velocity error. Figure 2 presents the velocity error history. It is obvious that the velocity error converges to $0.01\circ /s$ under the misalignment of the actuators through adaptive sliding mode controller (equation (44)). Figure 3 illustrates the reaction wheel’s output torque which is bounded by the physical constraint. Figure 4 demonstrates the angular momentum of each reaction wheel, which avoids the saturation behavior. Figure 5 shows the estimation of unknown parameters. Finally, Figure 6 shows the curves of vibration suppression; all the vibration modes are bounded.

Figure 1.

The curve of attitude error.

Figure 2.

The curve of angular velocity error.

Figure 3.

The curve of reaction wheel’s output torque.

Figure 4.

The curve of reaction wheel’s angular momentum.

Figure 5.

The curve of estimation parameter.

Figure 6.

The curve of vibration suppression.

Conclusion

In this article, the attitude tracking problem for flexible spacecraft with inertia uncertainty, external disturbance, actuator misalignment, and input saturation has been investigated. Two types of sliding mode control laws are developed to solve the attitude tracking control problem of flexible satellite. The first one is based on terminal sliding mode approach, and the second one employs adaptive control scheme. The effectiveness of the proposed control laws is verified through practical simulation examples.