Adaptive output feedback integral sliding mode attitude tracking control of spacecraft without unwinding

Abstract

This article studies an output feedback attitude tracking control problem for rigid spacecraft in the presence of parameter uncertainties and external disturbances. First, an anti-unwinding attitude control law is designed using the integral sliding mode control technique to achieve accurate tracking responses and robustness against inertia uncertainties and external disturbances. Next, the derived control law is combined with a suitable tuning law to relax the knowledge about the bounds of uncertainties and disturbances. The stability results are rigorously proved using the Lyapunov stability theory. In addition, a new finite-time sliding mode observer is developed to estimate the first time derivative of attitude. A new adaptive output feedback attitude controller is designed based on the estimated results, and angular velocity measurements are not required in the design process. A Lyapunov-based analysis is provided to demonstrate the uniformly ultimately bounded stability of the observer errors. Numerical simulations are given to illustrate the effectiveness of the proposed control method.

Introduction

Attitude control of rigid spacecraft has become one of the most interesting problems among researchers during the past decades. Spacecraft control problems, for example, satellite maneuvering, satellite surveillance, spacecraft formation flying, and spacecraft on-orbit service ¹ are challenging control problems. Since dynamic and kinematic equations of spacecraft are coupled and highly nonlinear, an attitude controller design is usually difficult. In practical situations, the model parameters of a spacecraft may not be acquired exactly and the spacecraft always faces model uncertainties and external disturbances. Thus, the attitude control of spacecraft requires solution of very challenging and interesting mathematical problems of great practical importance.

A large variety of nonlinear control schemes have been employed to improve the closed-loop performance of the attitude tracking control system such as adaptive attitude control^2,3 sliding mode control (SMC),^4,5 output feedback control,^6,7 linear matrix inequality-based control, ⁸ passivity-based control,^9,10 and fuzzy control. ¹¹ Among these methods, SMC has been successfully employed^12,13 to solve attitude control problem especially in the presence of model uncertainties and external disturbances satisfying the matched uncertainty condition. ¹⁴ SMC can provide a lot of attractive properties, such as insensitivity to model uncertainty, disturbance rejection, and fast dynamic response. However, the control techniques mentioned above only guarantee asymptotic stability and convergence in infinite time. The ability of control methods to offer fast maneuver performance is highly desirable in many space missions.

To obtain fast maneuver performance, finite-time control (FTC) methods are usually considered. It is well known that FTC methods can drive the system states to the equilibrium in finite time. This leads to rapid convergence and strong disturbance rejection properties. The Lyapunov-based approach and homogeneous theory are two main methods to design a finite-time controller. Based on homogeneous theory, finite-time controllers have been designed in previous studies.^15–18 Du et al. ¹⁷ have developed finite-time attitude tracking and attitude synchronization using a power integrator. The finite-time stability of the closed-loop system is achieved but the robustness is not strong. In Du and Li, ¹⁸ local and global continuous saturated finite-time controllers have been developed based on the homogeneous method but inertia uncertainties are not considered in the design. The Lyapunov-based approach often depends on terminal sliding mode control (TSMC). Recently, TSMC has been proposed in which a nonlinear sliding surface is synthesized to obtain convergence of system states in finite time.^19,20 TSMC has been employed by previous studies^21–23 to design a robust FTC for spacecraft attitude tracking. A major drawback of TSMC is the singularity problem. To overcome this problem, the nonsingular TSM (NTSM),^24–27 time-varying sliding mode, ²⁸ and integral sliding mode control (ISMC)^29,30 were constructed for developing attitude control laws. To compensate for mismatched uncertainties, continuous dynamics SMC and continuous NTSM control have been developed in Yang et al.^31,32 In Yang et al., ³³ a disturbance observer-based SMC approach has been designed to handle mismatched uncertainties.

Another enhanced version of SMC is higher-order sliding mode control (HOSMC).^34–36 HOSMC maintains the disturbance attenuation ability of SMC and also yields improved accuracy and performance. A practical implementation of HOSMC has been successfully applied to many real-life applications^37,38 and spacecraft attitude tracking maneuver. ³⁹ Recently, various FTC laws have been developed using second-order sliding mode control (SOSMC) concepts. A smooth SOSMC law has been applied by Shtessel et al. ⁴⁰ to a missile guidance system. In this article, the homogeneity approach^15,16 has been used to prove the finite-time convergence of the closed-loop system. Two SOSMC schemes have been designed by Pukdeboon ⁴¹ to solve the attitude tracking control problem. In this article, a strong Lyapunov function ⁴² was used for proving the finite-time stability of the closed-loop system. However, adaptive output feedback integral SMC of spacecraft attitude tracking has been rarely studied in practical implementation.

Most of the above-mentioned control methods are designed based on full state feedback and require the measurement of angular velocities. However, in practice, the use of angular velocity sensors may be restricted due to cost/weight constraints. Therefore, a partial-state feedback control law that does not require measurements of angular velocities is practically useful and highly desirable. To the best of the authors’ knowledge, there are no application of the adaptive integral sliding mode control (AISMC) technique to the solution for attitude tracking of spacecraft without unwinding when measurements of angular velocities are not available. In this article, an AISMC scheme is designed for attitude tracking control of spacecraft with inertia uncertainties and external disturbances. Based on NTSM and adaptive parameter-tuning strategy, the finite-time convergence to a small neighborhood around the sliding surface is realized. A sliding mode observer is designed to estimate unknown variables, so angular velocity measurements are not required. The stability and robustness of the proposed method are verified using Lyapunov stability theory.

The main contributions of this article are as follows:

1. A new adaptive integral sliding mode controller is developed to force the attitude of a rigid spacecraft to track the desired attitude in finite time and achieve high tracking precision performance in the presence of inertia uncertainties and external disturbances. Moreover, this controller can eliminate the unwinding phenomenon.

2. A new finite-time sliding mode observer is developed to estimate the first time derivative of quaternion. The finite-time convergence of error dynamics is proven. Then, the estimate results are used to develop a new adaptive output feedback controller. This avoids the measurements of the angular velocities.

This article is organized as follows. Section “Nonlinear model of spacecraft and problem formulation” describes spacecraft attitude dynamics and kinematics.^43,44 The problem formulation is also given. The proposed AISMC algorithms for a rigid spacecraft are discussed in section “Finite-time attitude control via ISMC.” In section “Output feedback via sliding mode observer,” a new sliding mode observer is designed such that the observer error dynamics converge to a bounded region containing the origin. In section “Simulations,” simulation results are presented to show the performance of the proposed controller. In section “Conclusion,” we present conclusions.

Nonlinear model of spacecraft and problem formulation

Spacecraft attitude dynamics and kinematics

The unit quaternion is employed to represent the attitude of the spacecraft for global representation without singularities. It is widely used to represent the attitude kinematics of rigid spacecraft owing to its non-trigonometric expression and nonsingular computations. ⁴⁵ The unit quaternion Q is defined by

$$Q=\left[\begin{array}{c}\hat{e}\sin (\varphi /2)\hfill \\ \multicolumn{1}{c}{\cos (\varphi /2)}\end{array}\right]=\left[\begin{array}{c}q\hfill \\ \multicolumn{1}{c}{q_4}\end{array}\right]$$

(1)

where $\hat{e}=[{\hat{e}}_1{\hat{e}}_2{\hat{e}}_3{]}^T\in {\mathcal{R}}^3$ is the Euler rotation axis; $\varphi \in \mathcal{R}$ is the Euler rotation angle; and $q\in {\mathcal{R}}^3$ and $q_4\in \mathcal{R}$ are the vector components and the scalar of the unit quaternion Q, respectively. They are subject to the constraint $q^{T}q+q_4^2=1$ . Consider the time derivative of Q. The kinematic equations are given by^43,44

$$\stackrel{·}{q}=\frac{1}{2}(q_4{I}_3+q^{\times })\omega$$

$${\stackrel{·}{q}}_4=-\frac{1}{2}{q}^T\omega$$

(2)

where $I_3$ is a $3\times 3$ identity matrix, and for any vector $a\in {\mathcal{R}}^3$ , a skew-symmetric matrix $a^{\times }$ is defined by

$$a^{\times }=\left[\begin{array}{ccc}0\hfill & -a_3\hfill & a_2\hfill \\ \multicolumn{1}{c}{a_3}& 0\hfill & -a_1\hfill \\ \multicolumn{1}{c}{-a_2}& a_1\hfill & 0\hfill \end{array}\right]$$

Kinematics of attitude errors

To define the attitude error using quaternions, we let the desired quaternion be $Q_d=[q_d^T{q}_{4d}{]}^T\in {\mathcal{R}}^3\times \mathcal{R}$ with $q_d=[q_{1d}{q}_{2d}{q}_{3d}{]}^T$ . Also the attitude error $Q_e=[q_e^T{q}_{4e}{]}^T\in {\mathcal{R}}^3\times \mathcal{R}$ with $q_e=[q_{1e}{q}_{2e}{q}_{3e}{]}^T$ is denoted as the relative attitude error from the body-fixed reference frame to the desired reference frame. Using the quaternion multiplication law, the error quaternion ⁴⁵

$$Q_e=\left[\begin{array}{c}{q}_{4d}q-q_4{q}_d-q_d^{\times }q\hfill \\ \multicolumn{1}{c}{q_4{q}_{4d}+q^T{q}_d}\end{array}\right]$$

(3)

satisfying the constraint

$$Q_e^T{Q}_e=(q^{T}q+q_4^2)(q_d^T{q}_d+q_{4d}^2)=1$$

(4)

The kinematics for the attitude error of a rigid spacecraft ⁴³ can be expressed as

$$\begin{array}{c}{\stackrel{·}{q}}_e=\frac{1}{2}T(Q_e){\omega }_e\hfill \\ {\stackrel{·}{q}}_{4e}=-\frac{1}{2}{q}_e^T{\omega }_e\hfill \end{array}$$

(5)

where $T(Q_e)=q_e^{\times }+q_{4e}{I}_3$ .

Dynamic equations of the error rate

The dynamic equation for a rigid spacecraft rotating under the influence of body-fixed devices is given by Edwards et al. ³⁵ as

$$J\stackrel{·}{\omega }=-{\omega }^{\times }J\omega +u+d$$

(6)

where $\omega =[{\omega }_1{\omega }_2{\omega }_3{]}^T$ is the angular rate of the spacecraft, $u=[u_1{u}_2{u}_3{]}^T$ represents the control vector, $d=[d_1{d}_2{d}_3{]}^T$ are bounded external disturbances, and J is the inertia matrix.

Next, let us denote ${\omega }_d=[{\omega }_{1d}{\omega }_{2d}{\omega }_{3d}{]}^T$ as the desired angular velocity vector. The corresponding rotation matrix is provided by $C=(q_{4e}^2-2q_e^T{q}_e)I_3+2q_e{q}_e^T-2q_{4e}{q}_e^{\times }$ . It should be noted that $||C||=1$ and $\stackrel{·}{C}=-{\omega }_e^{\times }C$ . Now, the angular velocity error can be obtained as ${\omega }_e=\omega -{\omega }_r$ , where ${\omega }_r=C{\omega }_d$ . Then, the dynamic equation of the error rate can be written in the form ¹³ as

$$\begin{array}{c}\multicolumn{1}{c}{J{\stackrel{·}{\omega }}_e=-({\omega }_e+{\omega }_r{)}^{\times }J({\omega }_e+{\omega }_r)+u}\\ +d+J({\omega }_e^{\times }C{\omega }_d-C{\stackrel{·}{\omega }}_d)\hfill \end{array}$$

(7)

Assumption 1

We assume that the inertia matrix in equation (7) is in the form $J=J_0+\Delta J$ where $J_0$ is the known nonsingular constant matrix and $\Delta J$ denotes the uncertainties.

Now, the spacecraft attitude dynamics become

$$J_0{\stackrel{·}{\omega }}_e=-{\omega }^{\times }{J}_0\omega -J_0{\stackrel{·}{\omega }}_r+u+\tilde{d}$$

(8)

where ${\stackrel{·}{\omega }}_r=C{\stackrel{·}{\omega }}_d-{\omega }_e^{\times }C{\omega }_d$ and $\tilde{d}=d-\Delta J\stackrel{·}{\omega }-{\omega }^{\times }\Delta J\omega$ .

Assumption 2

The total uncertainty vector $\tilde{d}$ is assumed to be bounded, but the upper bound of $\tilde{d}$ is unknown in advance, that is, $||\tilde{d}||\le \lambda$ where $\lambda$ is an unknown positive constant.

Remark 1

For Assumption 2, it is reasonable because the external unknown disturbances including environmental disturbance, solar radiation, and magnetic effects are all bounded in practice.

Lemmas

We now give some lemmas that will be used in later sections.

Lemma 1

If $p\in (0,1)$ , then the following inequality holds ⁴⁶

$$\sum _{i=1}^3|x_i{|}^{1+p}\ge {(\sum _{i=1}^3|x_i{|}^2)}^{\frac{1+p}{2}}$$

(9)

Lemma 2

Suppose $V(x)$ is a smooth positive definite function (defined on $U\subset R^n$ ) and $\stackrel{·}{V}(x)+\varsigma V^{\iota }(x)$ is negative semi-definite on $U\subset R^n$ for $\iota \in (0,1)$ and $\varsigma \in R^{+}$ , then there exists an area $U_0\subset R^n$ such that any $V(x)$ which starts from $U_0\subset R^n$ can reach $V(x)\equiv 0$ , in finite time. ¹⁵ Moreover, if $T_r$ is the time needed to reach $V(x)\equiv 0$ , then

$$T_r\le \frac{V^{1-\iota }(x_0)}{\varsigma (1-\iota )}$$

(10)

where $V(x_0)$ is the initial value of $V(x)$ .

Lemma 3

For any numbers ${\lambda }_1>0$ , ${\lambda }_2>0$ , $0<\varpi <1$ , an extended Lyapunov condition of finite-time stability can be given in the form of fast terminal sliding mode as ⁴⁷

$$\stackrel{·}{V}(x)+{\lambda }_{1}V(x)+{\lambda }_2{V}^{\varpi }(x)\le 0$$

where the settling time can be estimated by

$$T_r\le \frac{1}{{\lambda }_1(1-\varpi )}\ln \left(\frac{{\lambda }_1{V}^{1-\varpi }(x_0)+{\lambda }_2}{{\lambda }_2}\right)$$

(11)

Lemma 4

Considering the system $\stackrel{·}{x}=f(x,u)$ . ²³ Suppose there exists a Lyapunov function V(x) defined on a neighborhood $U\subset R^n$ of the origin, scalars $\vartheta >0$ , $0<\varrho <1$ , and $0<\theta <\infty$ , such that

$$\stackrel{·}{V}(x)+\vartheta V(x{)}^{\varrho }\le \theta ,x\in U\backslash \{0\}$$

(12)

Then, the trajectory of this system is practical finite-time stable.

Problem statement

In this article, $q_d,{\omega }_d$ , and ${\stackrel{·}{\omega }}_d$ are assumed to be bounded and measurements of attitude are always available. In this article, the objective is to design a control law which forces the states of closed-loop systems (2) and (6) to track the desired attitude motion $q_d$ and desired angular velocity ${\omega }_d$ in finite time. Moreover, that objective is to be guaranteed under the conditions of (1) no angular velocity measurements and (2) uncertain inertia parameters and external disturbances. In other words, the control objective is to design a control law which forces the states of the closed-loop systems (5) and (8) to reach a desired region containing the origin. This can be expressed as

$$\underset{t\to T}{lim}(q_e(t),{\omega }_e(t))\in {\Omega }_c$$

(13)

where T is a finite time and ${\Omega }_c$ denotes an open set around the origin.

Finite-time attitude control via ISMC

In this section, a novel controller is developed to achieve high-precision attitude tracking control. This control law is designed such that attitude tracking errors converge to a desired region in finite time. A proof of finite-time convergence of the closed-loop system is given based on Lyapunov stability theory.

For the spacecraft systems (5) and (8), we define the sliding surface as

$$s={\omega }_e+k\mathrm{sign}(q_{4e})q_e+\underset{0}{\overset{t}{\int }}\varphi d\tau$$

(14)

where k is a positive constant. By the concepts of ISMC, we choose

$$\varphi =c_1\mathrm{sign}(q_{4e})q_e+c_2{\omega }_e-k\mathrm{sign}(q_{4e}){\stackrel{·}{q}}_e$$

(15)

where $c_1$ and $c_2$ are positive constants. Based on ISMC, $\varphi (0)$ is set as $\varphi (0)=-({\omega }_e(0)+k\mathrm{sign}(q_{4e}(0))q_e(0))$ to guarantee $s(0)=0$ .

We now consider the spacecraft systems (5) and (8) in the presence of the disturbance $\tilde{d}(t)$ . The proposed control law is designed as

$$u(t)=u_{\mathrm{eq}}(t)+u_s(t)$$

(16)

where

$$u_{\mathrm{eq}}={\omega }^{\times }{J}_0\omega +J_0{\stackrel{·}{\omega }}_r-c_1{J}_0\mathrm{sign}(q_{4e})q_e-c_2{J}_0{\omega }_e$$

(17)

with $\gamma \in (0,1)$ , $\kappa$ and $\gamma$ being positive constants

$$u_s(t)=-\kappa {\mathrm{sign}}^{\gamma }(s(t))-\lambda \mathrm{sign}(s(t))$$

(18)

is the switching control that is used to compensate for the disturbance. In equation (18), the function $\mathrm{sign}(s)$ and ${\mathrm{sign}}^{\gamma }(s)$ are defined as

$$\mathrm{sign}(s)=\left[\begin{array}{c}\mathrm{sign}(s_1)\hfill \\ \multicolumn{1}{c}{\mathrm{sign}(s_2)}\\ \multicolumn{1}{c}{\mathrm{sign}(s_3)}\end{array}\right]\mathrm{and}{\mathrm{sign}}^{\gamma }(s)=\left[\begin{array}{c}|s_1{|}^{\gamma }\mathrm{sign}(s_1)\hfill \\ \multicolumn{1}{c}{|s_2{|}^{\gamma }\mathrm{sign}(s_2)}\\ \multicolumn{1}{c}{|s_3{|}^{\gamma }\mathrm{sign}(s_3)}\end{array}\right]$$

Next, the convergence of the system state errors to the origin is analyzed in the following theorem.

Theorem 1

Consider the spacecraft system in the presence of the disturbance $\tilde{d}(t)$ . If the proposed controller is defined by equation (16), then the sliding surface $s(t)=0$ is reached in finite time.

Proof

Premultiplying (14) by $J_0$ and differentiating the result with respect to time, one can obtain

$$\begin{array}{c}\multicolumn{1}{c}{J_0\stackrel{·}{s}(t)=-{\omega }^{\times }{J}_0\omega -J_0{\stackrel{·}{\omega }}_r+u+\tilde{d}}\\ +\varphi +k\mathrm{sign}(q_{4e})T(Q_e){\omega }_e\hfill \\ \multicolumn{1}{c}{=-{\omega }^{\times }{J}_0\omega -J_0{\stackrel{·}{\omega }}_r+u+\tilde{d}}\\ +c_1{J}_0\mathrm{sign}(q_{4e})q_e+c_2{J}_0{\omega }_e\hfill \end{array}$$

(19)

Substituting equation (15) into equation (19), $J_0\stackrel{·}{s}$ can be rearranged as

$$J_0\stackrel{·}{s}=-\kappa {\mathrm{sign}}^{\gamma }(s)-\lambda \mathrm{sign}(s)+\tilde{d}$$

(20)

Consider the following candidate positive definite function

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

(21)

which satisfies the following

$${\sigma }_{\min }(J_0)||s|{|}^2\le V_1\le {\sigma }_{\max }(J_0)||s|{|}^2$$

(22)

and ${\sigma }_{\min }(J_0)$ and ${\sigma }_{\max }(J_0)$ are the minimum and maximum singular value of the matrix $J_0$ .

The first time derivative of $V_1(t)$ is obtained as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1(t)=s^T{J}_0(-\kappa {\mathrm{sign}}^{\gamma }(s)-\lambda \mathrm{sign}(s)+\tilde{d})}\\ =-\kappa (|s_1{|}^{\gamma +1}+|s_2{|}^{\gamma +1}+|s_3{|}^{\gamma +1})\hfill \\ -\lambda (|s_1|+|s_2|+|s_3|)+s^T\tilde{d}\hfill \end{array}$$

(23)

By Lemma 1 and Assumption 1, equation (23) becomes

$$\begin{array}{c}\multicolumn{1}{c}{V_1(t)\le -\kappa ||s|{|}^{\gamma +1}-(\lambda -||\tilde{d}||)||s||}\\ \le -\kappa ||s|{|}^{\gamma +1}\hfill \end{array}$$

(24)

which can be written as

$$V_1(t)\le -\overline{\kappa }{V}_1^{\frac{\gamma +1}{2}}$$

(25)

where $\overline{\kappa }=\kappa (2/({\sigma }_{\max }(J_0){)}^{(\gamma +1)/2}$ . By Lemma 2, the sliding surface $s(t)$ converges to zero in finite time and the tracking variables $q_e$ and ${\stackrel{·}{q}}_e$ approach to zero in finite time. The proof is completed. □

In practice, the upper bound of the disturbance $\tilde{d}(t)$ is often unknown and therefore it is difficult to determine the upper bound of $\tilde{d}(t)$ . In the following theorem, an adaptive law is applied to estimate the unknown upper bound of the disturbance. The proposed adaptive control law is defined as

$$u(t)=u_{\mathrm{eq}}(t)+u_{\mathrm{sa}}(t)$$

(26)

where $u_{\mathrm{sa}}(t)=-\kappa {\mathrm{sign}}^{\gamma }(s(t))-\hat{\lambda }\mathrm{sign}(s(t))$ is the switching control that is used to compensate for the disturbance.

In equation (26), an adaptive law $\hat{\lambda }$ is updated by

$$\stackrel{·}{\hat{\lambda }}=\eta (||s||-\delta \hat{\lambda })$$

(27)

with $\eta$ and $\delta$ being positive constants.

Remark 2

The adaptive updating provided in equation (26) dynamically adjusts the compensation for the system uncertainties, rather than offering the precise estimate of $\lambda$ such that that attitude tracking is attained. This form of updating law has been widely used in adaptive attitude controller designs.^30,48,49 Similar to an adaptive law without the term $(-\delta \hat{\lambda })$ , the uniformly ultimate boundedness of s is guaranteed by the Lyapunov method. With the updated law (27), when s converges to zero, the trajectory of $\hat{\lambda }$ goes to zero. If the term $(-\delta \hat{\lambda })$ is not included, it can only ensure uniformly ultimate boundedness of $\hat{\lambda }$ . When this term is included, the trajectory of $\hat{\lambda }$ is reduced significantly and then converges to a value near zero.

Next, the convergence of the system state errors to the origin is analyzed in the following theorem.

Theorem 2

Consider the spacecraft system in the presence of the disturbance $\tilde{d}(t)$ . If the proposed adaptive controller is defined by equation (25), then the sliding surface $s(t)$ is stabilized to a neighborhood of zero in finite time.

Proof

Substituting equation (26) into equation (19), $\stackrel{·}{s}(t)$ becomes

$$\stackrel{·}{s}(t)=-\kappa {\mathrm{sign}}^{\gamma }(s(t))-\hat{\lambda }\mathrm{sign}(s)+\tilde{d}$$

(28)

Consider the following candidate positive definite function

$$V_2(t)=\frac{1}{2}{s}^T{J}_{0}s+\frac{1}{2\eta }{\tilde{\lambda }}^2$$

(29)

where $\tilde{\lambda }=\lambda -\hat{\lambda }$ , with $\hat{\lambda }$ being the estimated value for $\lambda$ .

Differentiating equation (29), one has

$${\stackrel{·}{V}}_2(t)=s^T{J}_0\stackrel{·}{s}+\frac{1}{\eta }\tilde{\lambda }\stackrel{·}{\stackrel{~}{\lambda }}$$

(30)

Substituting equations (27) and (28) into equation (30), we obtain

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2(t)=s(t)(-\kappa {\mathrm{sign}}^{\gamma }(s(t))-\hat{\lambda }\mathrm{sign}(s)+\tilde{d})}\\ -\frac{1}{\eta }\tilde{\lambda }(\eta |s(t)|-\delta \hat{\lambda })\hfill \\ \le -\kappa ||s|{|}^{\gamma }-\hat{\lambda }||s||+||\tilde{d}||||s||\hfill \\ -\tilde{\lambda }||s||+\delta \tilde{\lambda }\hat{\lambda }\hfill \\ =-\kappa ||s|{|}^{\gamma }-(\lambda -||\tilde{d}||)||s||+\delta \tilde{\lambda }\hat{\lambda }\hfill \end{array}$$

(31)

Using $\lambda >||\tilde{d}||$ , then ${\stackrel{·}{V}}_2(t)$ can be written as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2(t)\le -\kappa ||s|{|}^{\gamma }+\delta \tilde{\lambda }}\\ \le -\kappa {\left(\frac{2}{{\sigma }_{\max }(J_0)}\right)}^{\frac{\gamma +1}{2}}{\left(\frac{1}{2}{s}^T{J}_{0}s\right)}^{\frac{\gamma +1}{2}}+\delta \tilde{\lambda }\hat{\lambda }\hfill \end{array}$$

(32)

Note that

$$\delta \tilde{\lambda }\hat{\lambda }=\delta (\lambda \tilde{\lambda }-{\tilde{\lambda }}^2)$$

Applying Young’s inequality with $\epsilon >0$ , one obtains

$$\begin{array}{cc}\multicolumn{1}{c}{\delta \tilde{\lambda }\hat{\lambda }}& \le \delta (\frac{1}{2\epsilon }{\tilde{\lambda }}^2+\frac{\epsilon }{2}{\lambda }^2-{\tilde{\lambda }}^2)\hfill \\ =-\delta \overline{\epsilon }{\tilde{\lambda }}^2+\delta \frac{\epsilon }{2}{\lambda }^2\hfill \end{array}$$

(33)

where $\overline{\epsilon }=(2\epsilon -1)/2\epsilon$ . Now, ${\stackrel{·}{V}}_2$ becomes

$${\stackrel{·}{V}}_2\le -\kappa {\left(\frac{2}{{\sigma }_{\max }(J_0)}\right)}^{\frac{\gamma +1}{2}}{\left(\frac{1}{2}{s}^T{J}_{0}s\right)}^{\frac{\gamma +1}{2}}-\delta \overline{\epsilon }{\tilde{\lambda }}^2+\delta \frac{\epsilon }{2}{\lambda }^2$$

(34)

If $\delta \overline{\epsilon }{\tilde{\lambda }}^2>1$ , it yields

$$(\delta \overline{\epsilon }{\tilde{\lambda }}^2{)}^{\frac{\gamma +1}{2}}\le \delta \overline{\epsilon }{\tilde{\lambda }}^2$$

(35)

Thus, equation (34) becomes

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2(t)\le -\kappa {\left(\frac{2}{{\sigma }_{\max }(J_0)}\right)}^{\frac{\gamma +1}{2}}{\left(\frac{1}{2}{s}^T{J}_{0}s\right)}^{\frac{\gamma +1}{2}}-(\delta \overline{\epsilon }{\tilde{\lambda }}^2{)}^{\frac{\gamma +1}{2}}+\delta \frac{\epsilon }{2}{\lambda }^2}\\ =-\beta [{\left(\frac{1}{2}{s}^T{J}_{0}s\right)}^{\frac{\gamma +1}{2}}+{\left(\frac{1}{2\eta }{\tilde{\lambda }}^2\right)}^{\frac{\gamma +1}{2}}]+{\Theta }_1\hfill \end{array}$$

(36)

where $\beta =\kappa (2/({\sigma }_{\max }(J_0){)}^{(\gamma +1)/2}$ , $\eta =({\beta }^{\frac{2}{\gamma +1}})/2\delta \overline{\epsilon }$ , and ${\Theta }_1=((\delta \epsilon )/2){\lambda }^2$

If $\delta \overline{\epsilon }{\tilde{\lambda }}^2\le 1$ , it yields

$$(\delta \overline{\epsilon }{\tilde{\lambda }}^2{)}^{\frac{\gamma +1}{2}}\le \delta \overline{\epsilon }{\tilde{\lambda }}^2+\nu$$

(37)

where $0<\nu <1$ . Thus, equation (34) becomes

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2(t)\le -\kappa {\left(\frac{2}{{\sigma }_{\max }(J_0)}\right)}^{\frac{\gamma +1}{2}}{\left(\frac{1}{2}{s}^T{J}_{0}s\right)}^{\frac{\gamma +1}{2}}}\\ -(\delta \overline{\epsilon }{\tilde{\lambda }}^2{)}^{\frac{\gamma +1}{2}}+\delta \frac{\epsilon }{2}{\lambda }^2+\nu \hfill \\ =-\beta [{\left(\frac{1}{2}{s}^T{J}_{0}s\right)}^{\frac{\gamma +1}{2}}+{\left(\frac{1}{2\eta }{\tilde{\lambda }}^2\right)}^{\frac{\gamma +1}{2}}]+{\Theta }_1+\nu \hfill \end{array}$$

(38)

Combining equations (36) and (38), one obtains

$${\stackrel{·}{V}}_2(t)\le -\beta [{\left(\frac{1}{2}{s}^T{J}_{0}s\right)}^{\frac{\gamma +1}{2}}+{\left(\frac{1}{2\eta }{\tilde{\lambda }}^2\right)}^{\frac{\gamma +1}{2}}]+{\Theta }_2$$

(39)

where ${\Theta }_2={\Theta }_1+\nu$ . Using Lemma 1, we can obtain

$${\stackrel{·}{V}}_2(t)\le -\beta {[\left(\frac{1}{2}{s}^T{J}_{0}s\right)+\left(\frac{1}{2\eta }{\tilde{\lambda }}^2\right)]}^{\frac{\gamma +1}{2}}+{\Theta }_2$$

(40)

which can be further written as

$${\stackrel{·}{V}}_2(t)\le -\beta V_2^{\frac{\gamma +1}{2}}+{\Theta }_2$$

(41)

It follows from equation (41) that there must exist $0<\varpi <1$ such that

$${\stackrel{·}{V}}_2(t)\le -\varpi \beta V_2^{\frac{\gamma +1}{2}}-(1-\varpi )\beta V_2^{\frac{\gamma +1}{2}}+{\Theta }_2$$

(42)

According to Lemma 4, $V_2$ is decreased and the trajectories converge to

$$V_2^{\frac{\gamma +1}{2}}(t)\le \frac{{\Theta }_2}{(1-\varpi )\beta }$$

(43)

in finite time which implies that the trajectories of the closed-loop system are bounded in finite time $T_2$ in the sense that

$$\underset{t\to T_2}{lim}s(t)\in (\Vert s(t)\Vert \le \sqrt{\frac{2}{{\sigma }_{\min }(J_0)}}{\left(\frac{{\Theta }_2}{(1-\varpi )\beta }\right)}^{\frac{1}{\gamma +1}}=\Delta )$$

(44)

where $\Delta$ is a small region containing the origin of the closed-loop system. □

Output feedback via sliding mode observer

In this section, a sliding mode observer is designed based on sliding mode technique.

Observer design

The proposed finite time observer is given by

$$\begin{array}{c}{\stackrel{·}{x}}_1=-{\rho }_1(x_1-v(t))+x_2\hfill \\ {\stackrel{·}{x}}_2=-{\rho }_2{\mathrm{sign}}^{(2\beta -1)}(x_1-v(t))-{\rho }_3(x_1-v(t))\hfill \\ -{\rho }_4\mathrm{sign}(x_1-v(t))\hfill \end{array}$$

(45)

where the second-order differentiator signal $v(t)$ is bounded, and $||\stackrel{··}{v}(t)||\le L$ , L is a positive constant. ${\rho }_1=\mathrm{diag}({\rho }_{11},{\rho }_{12},{\rho }_{13})$ , ${\rho }_2=\mathrm{diag}({\rho }_{21},{\rho }_{22},{\rho }_{23})$ , ${\rho }_3=\mathrm{diag}({\rho }_{31},{\rho }_{32},{\rho }_{33})$ , ${\rho }_4=\mathrm{diag}({\rho }_{41},{\rho }_{42},{\rho }_{43})$ , and $\beta \in (0.5,1)$ . We next show that there exists a finite time $t_s>0$ such that

$$x_1=v(t),x_2=\stackrel{·}{v}(t)$$

(46)

for $t\ge t_s$ . Let $z_1=x_1-v(t),z_2=x_2-\stackrel{·}{v}(t)$ , and $\chi =\stackrel{··}{v}(t)$ . The error system is

$$\begin{array}{c}{\stackrel{·}{z}}_{1i}=-{\rho }_{1i}{z}_{1i}+z_{2i}\hfill \\ {\stackrel{·}{z}}_{2i}=-{\rho }_{2i}|z_{1i}{|}^{(2\beta -1)}\mathrm{sign}(z_{1i})-{\rho }_{3i}{z}_{1i}-{\rho }_{4i}\mathrm{sign}(z_{1i})-{\chi }_i\hfill \end{array}$$

(47)

Remark 3

The system (47) is non-homogeneous and the homogeneity method cannot be applied to prove the finite-time stability. However, a rigorous Lyapunov analysis ⁴² can be used to ensure the uniformly ultimately bounded (UUB) stability in finite time of the proposed observer.

Next, to ensure the finite-time convergence of the proposed observer (45), we prove that the observer error states in (47) converge to the region containing the origin in finite time.

Theorem 3

Consider the observer error system (47), and the bounded differentiable signal $v(t)\in {\mathbf{C}}^2$ , there exist constants ${\rho }_{1i}$ , ${\rho }_{2i}$ , ${\rho }_{3i}$ , and ${\rho }_{4i}(i=1,2,3)$ and $\beta \in (0.5,1)$ such that the error dynamics $z_1$ and $z_2$ converge to the neighborhood of origin in finite time.

Proof

Consider the following candidate strong Lyapunov function

$$\begin{array}{c}\multicolumn{1}{c}{V_3(z)=\frac{{\rho }_{2i}}{\beta }|z_{1i}{|}^{2\beta }+{\rho }_{3i}{z}_{1i}^2+\frac{1}{2}{z}_{2i}^2+2{\rho }_{4i}|z_{1i}|}\\ +\frac{1}{2}{(z_{2i}-{\rho }_{1i}{z}_{1i})}^2\hfill \end{array}$$

(48)

which can be written as

$$V_3(z)={\xi }^T\Pi \xi$$

(49)

where $\xi =[|z_{1i}{|}^{\beta }\mathrm{sign}(z_{1i})|z_{1i}{|}^{1/2}{z}_{1i}{z}_{2i}{]}^T$ and

$${\Omega }_1=\frac{1}{2}\left[\begin{array}{cccc}\frac{2{\rho }_{2i}}{\beta }\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 4{\rho }_{4i}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 2{\rho }_{3i}+{\rho }_{1i}^2\hfill & -{\rho }_{1i}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -{\rho }_{1i}\hfill & 2\hfill \end{array}\right]$$

It satisfies

$${\sigma }_{\min }({\Omega }_1)\Vert \xi {\Vert }^2\le V_3\le {\sigma }_{\max }({\Omega }_1)\Vert \xi {\Vert }^2$$

(50)

where $||\xi |{|}^2=|z_{1i}{|}^{2\beta }+|z_{1i}|+z_{1i}^2+z_{2i}^2$ and ${\sigma }_{\min }({\Omega }_1)$ and ${\sigma }_{\max }({\Omega }_1)$ are the minimum and maximum singular value of the matrix $\Pi$ .

Taking time derivative to (48), one obtains

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3(z)=2{\rho }_{2i}|z_{1i}{|}^{2\beta -1}\mathrm{sign}(z_{1i}){\stackrel{·}{z}}_{1i}+(2{\rho }_{3i}+{\rho }_{1i}^2)z_{1i}{\stackrel{·}{z}}_{1i}}\\ +2z_{2i}{\stackrel{·}{z}}_{2i}-{\rho }_{1i}{z}_{1i}{\stackrel{·}{z}}_{2i}\hfill \\ -{\rho }_{1i}{z}_{2i}{\stackrel{·}{z}}_{1i}-2{\rho }_{4i}\mathrm{sign}(z_{1i}){\stackrel{·}{z}}_{1i}\hfill \end{array}$$

(51)

Substituting equation (47) into equation (51), one has

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3(z)=2{\rho }_{2i}|z_{1i}{|}^{2\beta -1}\mathrm{sign}(z_{1i})(-{\rho }_{1i}{z}_{1i}+z_{2i})}\\ +(2{\rho }_{3i}+{\rho }_{1i}^2)z_{1i}(-{\rho }_{1i}{z}_{1i}+z_{2i})\hfill \\ +2z_{2i}(-{\rho }_{2i}|z_{1i}{|}^{\frac{\beta -1}{2}}\mathrm{sign}(z_{1i})-{\rho }_{3i}{z}_{1i}-{\rho }_{4i}\mathrm{sign}(z_{1i})-{\chi }_i)\hfill \\ -{\rho }_{1i}{z}_{1i}(-{\rho }_{2i}|z_{1i}{|}^{\frac{\beta -1}{2}}\mathrm{sign}(z_{1i})-{\rho }_{3i}{z}_{1i}-{\rho }_{4i}\mathrm{sign}(z_{1i})-{\chi }_i)\hfill \\ -{\rho }_{1i}{z}_{2i}(-{\rho }_{1i}{z}_{1i}+z_{2i})-2{\rho }_{4i}\mathrm{sign}(z_{1i})(-{\rho }_{1i}{z}_{1i}+z_{2i})\hfill \end{array}$$

(52)

Multiplying out brackets and using algebraic manipulation, one obtains

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3(z)=-{\rho }_{1i}{\rho }_{2i}|z_{1i}{|}^{2\beta }-{\rho }_{1i}{\rho }_{4i}|z_{1i}|-{\rho }_{1i}({\rho }_{3i}+{\rho }_{1i}^2)z_{1i}^2}\\ -{\rho }_{1i}{z}_{2i}^2+2{\rho }_{1i}^2{z}_{1i}{z}_{2i}+{\rho }_{1i}{z}_{2i}\chi -2z_{2i}\chi \hfill \end{array}$$

(53)

Thus, the derivative of $V_3$ can be written as

$${\stackrel{·}{V}}_3=-{\xi }^T{\Omega }_2\xi -\chi \Gamma \xi$$

(54)

where

$${\Omega }_2={\rho }_{1i}\left[\begin{array}{cccc}{\rho }_{2i}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\rho }_{4i}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & {\rho }_{3i}+{\rho }_{1i}^2\hfill & -{\rho }_{1i}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -{\rho }_{1i}\hfill & 1\hfill \end{array}\right]$$

and

$$\Gamma =[\begin{array}{cccc}0\hfill & 0\hfill & {\rho }_{1i}\hfill & -2\hfill \end{array}].$$

With positive values of ${\rho }_{1i}$ , ${\rho }_{2i}$ , ${\rho }_{3i}$ , and ${\rho }_{4i}$ , it can be ensured that ${\Omega }_2$ is positive definite.

Therefore, we have

$${\stackrel{·}{V}}_3\le -{\sigma }_{\min }({\Omega }_2)||\xi |{|}^2+L||\Gamma ||||\xi ||$$

(55)

We can change equation (48) into the following form

$${\stackrel{·}{V}}_3\le -({\sigma }_{\min }({\Omega }_2)||\xi ||-L||\Gamma ||)||\xi ||$$

(56)

Using equation (50), it can be further obtained from (56) that

$${\stackrel{·}{V}}_3\le -({\sigma }_{\min }({\Omega }_2)||\xi ||-L||\Gamma ||)\frac{V_3^{1/2}}{{\sigma }_{\max }({\Omega }_1)}$$

(57)

If the observer gains are selected such that ${\sigma }_{\min }({\Omega }_2)||\xi ||>L||\Gamma ||$ , then the error system (47) can converge to the small region $||\xi ||\le \frac{L||\Gamma ||}{{\sigma }_{\min }({\Omega }_2)}\stackrel{\Delta }{=}\Upsilon$ in finite time. This implies that ${\stackrel{·}{V}}_3<0$ when $\xi$ enters the compact set $D_1\stackrel{\Delta }{=}\{\xi |||\xi ||\le \Upsilon \}$ , and also $V_3$ decrease monotonically. Thus, the observer error $\xi$ is eventually forced into the set $D_1$ and then it cannot go out of $D_1$ . With the given initial estimates, $||\xi (0)||=0$ is guaranteed. According to the Lyapunov stability theory and LaSalle’s invariance principle, ⁵⁰ all state of differentiation errors are confined into $D_1$ in finite time $T_0$ , which also indicates the UUB stability of $\xi$ . This completes the proof. □

Controller design

Output feedback attitude tracking controllers can be designed using the observer outputs $x_{2i}(i=1,2,3)$ to estimate the angular velocity errors. In this case, the estimated angular velocity errors can be obtained as

$${\hat{\omega }}_e=2T^{-1}(Q_e)x_2$$

Note that from the problem statement, the measurements of quaternion are available. Thus, $Q_e$ is known and can be used in the following sliding surface and controller designs.

For the output feedback controller law, the chosen sliding surface for the controller is

$$s={\hat{\omega }}_e+k\mathrm{sign}(q_{4e})q_e+\underset{0}{\overset{t}{\int }}\hat{\varphi }d\tau$$

(58)

where $\hat{\varphi }=c_1\mathrm{sign}(q_{4e})q_e+c_2{\hat{\omega }}_e-k\mathrm{sign}(q_{4e}){\stackrel{·}{\hat{q}}}_e$ . Also, the proposed output feedback law is defined as

$$\begin{array}{c}\multicolumn{1}{c}{u={\hat{\omega }}^{\times }{J}_0\hat{\omega }+J_0{\stackrel{·}{\omega }}_r-c_1{J}_0\mathrm{sign}(q_{4e})q_e}\\ -c_2{J}_0{\hat{\omega }}_e-\kappa {\mathrm{sign}}^{\gamma }(s(t))-\hat{\lambda }\mathrm{sign}(s)\hfill \end{array}$$

(59)

where $\hat{\omega }={\hat{\omega }}_e+{\omega }_d$ .

We next design a controller with observer under inertia uncertainties, external disturbances, and control input saturations. Consider the rigid spacecraft system with actuator constraints

$$J_0{\stackrel{·}{\omega }}_e=-{\omega }^{\times }{J}_0\omega -J_0{\stackrel{·}{\omega }}_r+\mathrm{sat}(u)+\tilde{d}$$

(60)

where $\mathrm{sat}(u)=[\mathrm{sat}(u_1)\mathrm{sat}(u_2)\mathrm{sat}(u_3){]}^T$ is the vector of actuator torques and forces generated by the actuators (or thrusters), in which $\mathrm{sat}(u_i),i=1,2,3$ denotes the nonlinear saturation characteristic of the actuators and is of the form

$$\mathrm{sat}(u_i)=\{\begin{array}{cc}{u}_{\mathrm{mi}},\hfill & u_i>u_{\mathrm{mi}}\hfill \\ \multicolumn{1}{c}{u_i,}& |u_i|\ge u_{\mathrm{mi}}\hfill \\ \multicolumn{1}{c}{-u_{\mathrm{mi}},}& u_i<-u_{\mathrm{mi}}\hfill \end{array}$$

(61)

where $u_{\mathrm{mi}}>0,i=1,2,3$ are maximum torques generated by actuators. We can define an auxiliary variable

$${\vartheta }_i=\{\begin{array}{cc}{u}_{\mathrm{mi}}-u_i,\hfill & u_i>u_{\mathrm{mi}}\hfill \\ \multicolumn{1}{c}{0,}& |u_i|\ge u_{\mathrm{mi}}\hfill \\ \multicolumn{1}{c}{-u_{\mathrm{mi}}-u_i,}& u_i<-u_{\mathrm{mi}}\hfill \end{array}$$

(62)

Using equations (61) and (62), one has

$$\mathrm{sat}(u)=u+\vartheta$$

(63)

Substituting equation (63) into equation (60), one obtains

$$J_0{\stackrel{·}{\omega }}_e=-{\omega }^{\times }{J}_0\omega -J_0{\stackrel{·}{\omega }}_r+u+\vartheta +\tilde{d}$$

(64)

According to Theorem 2, one can see that if $\lambda$ in equation (31) is chosen such that $\lambda >||\vartheta ||+||\tilde{d}||$ , then the same form of ${\stackrel{·}{V}}_2$ is obtained. Thus, the stability of the closed-loop system can be analyzed by following the same steps presented in Theorem 2.

Remark 4

With the use of the sliding surface (14) or (58), the unwinding phenomena can be avoided. When the sliding surface $s=0$ is achieved, the term $k\mathrm{sign}(q_{4e})q_e$ can guarantee that $q_e$ reaches zero and $q_{4e}$ reaches the equilibrium point without changing the sign of $q_{4e}$ . When $q_{4e}(0)>0$ , one has $q_{4e}(0)\to 1$ . On the other hand, when $q_{4e}(0)<0$ , one obtains $q_{4e}(0)\to -1$ . Thus, the sliding surface (14) or (58) can prevent unwinding.

Remark 5

For a given practical system, the input difference $\vartheta$ cannot be too large. If this input difference is too big, it means that the actuator cannot give high enough control torques to make system stable. ⁵¹ From the controllability of a practical system, it is reasonable that $||\vartheta ||$ is always bounded by a constant.

Simulations

An example of a rigid-body satellite given by Lu et al. ¹³ is presented with numerical simulations to demonstrate the comparison of the developed adaptive output feedback law (58) and the adaptive sliding mode control (ASMC) method in Lu et al. ¹³ The spacecraft is assumed to have the nominal inertia matrix

$$J=\left[\begin{array}{ccc}20\hfill & 1.2\hfill & 0.9\hfill \\ \multicolumn{1}{c}{1.2}& 17\hfill & 1.4\hfill \\ \multicolumn{1}{c}{0.9}& 1.4\hfill & 15\hfill \end{array}\right]\mathrm{kg}{\mathrm{m}}^2$$

(65)

and the parameter uncertainties

$$\Delta J=\mathrm{diag}\left[\begin{array}{ccc}\sin (0.1t)\hfill & 2\sin (0.2t)\hfill & 3\sin (0.2t)\hfill \end{array}\right]\mathrm{kg}{\mathrm{m}}^2$$

(66)

The attitude control problem is considered in the presence of external disturbance $d(t)$ . The external disturbances are given by

$$d(t)=0.1\left[\begin{array}{c}\sin (0.1t)\hfill \\ \multicolumn{1}{c}{\sin (0.2t)}\\ \multicolumn{1}{c}{\sin (0.3t)}\end{array}\right]\mathrm{N\; m}$$

(67)

In this numerical simulation, we assume that the desired angular velocity is given by

$${\omega }_d(t)=0.05\left[\begin{array}{c}\sin \left(\frac{\pi t}{100}\right)\hfill \\ \multicolumn{1}{c}{\sin \left(\frac{2\pi t}{100}\right)}\\ \multicolumn{1}{c}{\sin \left(\frac{3\pi t}{100}\right)}\end{array}\right]\mathrm{rad}/\mathrm{s}$$

(68)

Numerical simulations are given as follows.

Case I

We check whether both controller (59) and ASMC method in Lu et al. ¹³ can prevent the unwinding phenomenon or not. The initial and desired conditions of quaternion are set as $Q(0)=[-0.46180.19150.7999-0.3320{]}^T$ and $Q_d(0)=[000-1{]}^T$ , respectively. The initial value of the angular velocity is assumed to be $\omega (0)=[-0.04-0.010.14{]}^T\mathrm{rad}/\mathrm{s}$ . The control torques are limited by $|u_i|\le 3.0\mathrm{N}\mathrm{m},i=1,2,3$ .

For the ASMC method in Lu et al., ¹³ simulations are carried out with the parameters given as $\tau =2.5I_3$ , $\sigma =0.01I_3$ , $\rho =I_3$ , $k=I_3$ , and $r=0.9$ . The parameters for the adaptive law are chosen as $p_i=0.5,i=1,2,3$ and the initial conditions ${\hat{c}}_i=1.0,i=1,2,3$ . The control parameters for the output feedback law (59) are selected to be $c_1=0.02,c_2=0.1$ , $\kappa =2.5$ , and $\gamma =0.9$ . The parameters for the adaptive law (27) are set as $\eta =10,\delta =0.01$ with the initial condition $\hat{\lambda }(0)=0.1,i=1,2,3$ . For the proposed observer (45), the parameters are given as $\beta =2/3$ , ${\rho }_{1i}=10.5$ , ${\rho }_{2i}=2.5$ , ${\rho }_{3i}=1.0$ , and ${\rho }_{4i}=0.2(i=1,2,3)$ .

Figures 1 and 2 show that the quaternion errors obtained by the controller (59) are smoother than the ASMC method in Lu et al. ¹³ From Figure 3, one can see that the unwinding phenomenon occurs when the ASMC method is used. Since the initial condition of $q_{4e}$ is $-0.3320$ , it should converge to the equilibrium point $q_{4e}=-1$ instead of $q_{4e}=1$ . On the other hand, as shown in Figure 4 the controller (59) can force $q_{4e}$ to the equilibrium point $q_{4e}=-1$ . This controller can prevent the unwinding phenomenon. Figures 5 and 6 depict responses of angular velocity errors made by ASMC and estimated angular velocity errors obtained by the controller (59). These responses converge to zero after 20 s. The effect of external disturbances and inertia uncertainties is not shown in both quaternion and angular velocity tracking error responses. From Figures 7 and 8, it can be seen that for the ASMC method in Lu et al., ¹³ the control torque signals slowly converge to the steady-state level. As shown in Figures 9 and 10, responses of estimated parameters for the ASMC method in Lu et al. ¹³ converge to constants, whereas the estimated parameter for the controller (59) is reduced significantly and converges to a small value near zero. Figures 11–13 depict responses of the estimated angular velocity errors which are obtained by applying results from the proposed observer (45). From these figures, estimated angular velocities converge to the true values in finite time.

Figure 1.

Attitude tracking errors under ASMC—Case I.

Figure 2.

Attitude tracking errors under the controller (59)—Case I.

Figure 3.

Scalar quaternion errors under ASMC—Case I.

Figure 4.

Scalar quaternion errors under the controller (59)—Case I.

Figure 5.

Angular velocity errors under ASMC—Case I.

Figure 6.

Estimated angular velocity errors under the controller (59)—Case I.

Figure 7.

Torque input under ASMC—Case I.

Figure 8.

Torque input under the controller (59)—Case I.

Figure 9.

Estimated parameters under ASMC—Case I.

Figure 10.

Estimated parameters under the controller (59)—Case I.

Figure 11.

First component of the estimated angular velocity errors—Observer (45).

Figure 12.

Second component of the estimated angular velocity errors—Observer (45).

Case II

The controller (59) and the ASMC in Lu et al. ¹³ are compared based on responses for disturbance rejection in simulation. The initial and desired conditions of quaternion are set as $Q(0)=[-0.20.40.70.55678{]}^T$ and $Q_d(0)=[0001{]}^T$ , respectively. The initial value of the angular velocity is assumed to be $\omega (0)=[0.060.04-0.05{]}^T\mathrm{rad}/\mathrm{s}$ . The control torques are limited by $|u_i|\le 3.0\mathrm{N}\mathrm{m},i=1,2,3$ .

As shown in Figures 14 and 15, both controllers have good disturbance rejection abilities. The effect of uncertainties and disturbances is not shown. The controller (59) gives smoother responses of attitude tracking errors when compared to those of the ASMC method in Lu et al. ¹³ Similarly, Figures 16 and 17 depict good responses of angular and estimated angular velocity errors obtained by the controller (59) and the ASMC method, respectively. From Figures 18 and 19, one can see the control torques obtained by the controller (59) fastly converge to the steady-state levels. As shown in Figures 20 and 21, the estimated parameters for the ASMC method are finally bounded by constant values, while the estimated parameter for the controller (59) converges to a small neighborhood of zero.

Figure 13.

Third component of the estimated angular velocity errors—Observer (45).

Figure 14.

Attitude tracking errors under ASMC—Case II.

Figure 15.

Attitude tracking errors under the controller (59)—Case II.

Figure 16.

Angular velocity errors under ASMC—Case II.

Figure 17.

Estimated angular velocity errors under the controller (59)—Case II.

Figure 18.

Torque input under ASMC—Case II.

Figure 19.

Torque input under the controller (59)—Case II.

Figure 20.

Estimated parameters under ASMC—Case II.

Case III

We use another set of initial conditions for quaternion and angular velocities. The performance of the controller (59) and ASMC in Lu et al. ¹³ are compared. The initial and desired conditions of quaternion are set as $Q(0)=[0.4-0.10.150.8986{]}^T$ and $Q_d(0)=[0001{]}^T$ , respectively. The initial value of the angular velocity is assumed to be $\omega (0)=[0.1-0.1-0.05{]}^T\mathrm{rad}/\mathrm{s}$ . The restrictions on torque magnitude with $|u_i|\le 2.0\mathrm{N}\mathrm{m},i=1,2,3$ are considered.

As shown in Figures 22 and 23, the controller (59) achieves smoother attitude tracking errors than those of ASMC in Lu et al. ¹³ Figure 24 depicts finite-time convergence to zero of the angular velocity errors. Similarly, from Figure 24, we can see that controller (59) guarantees finite-time convergence to zero of the estimated angular velocity errors. As shown in Figures 26 and 27, the control torques obtained by the ASMC method in Lu et al. ¹³ show higher variation during the first 10 s. Thus, good performance of the controller (59) is evident.

Figure 21.

Estimated parameters under the controller (59)—Case II.

Figure 22.

Attitude tracking errors under ASMC—Case III.

Figure 23.

Attitude tracking errors under the controller (59)—Case III.

Figure 24.

Angular velocity errors under ASMC—Case III.

Figure 25.

Estimated angular velocity errors under the controller (59)—Case III.

Figure 26.

Torque input under ASMC—Case III.

The simulation results obtained from the controller (59) and the ASMC method in Lu et al. ¹³ have been compared. Several initial conditions of quaternion and angular velocities are considered. The controller (59) gives smoother responses of quaternion tracking errors and smaller magnitude of torque is required for the first 10 s. Moreover, this controller eliminates the unwinding phenomenon. From simulation results, the controller (59) seems to be a more suitable scheme to deal with practical high-precision attitude tracking control of a rigid spacecraft because it relaxes the requirements of the upper bound of uncertainties and disturbances and provides good tracking outputs.

Figure 27.

Torque input under the controller (59)—Case III.

Conclusion

In this article, an adaptive output feedback ISMC law has been developed for the attitude tracking control problem of a rigid spacecraft. In the presence of external disturbances and inertia uncertainties, the proposed attitude control strategy achieves the control objective and eliminates the unwinding phenomenon. Using Lyapunov stability theory, we have proved that the error dynamics converge to a desired region containing the origin in finite time. Besides, a new sliding mode observer has been developed to estimate the time derivative of attitude errors. Then, with the estimated results, we have derived a new adaptive output feedback integral sliding mode attitude controller without the need of angular velocity measurements. Numerical simulations on attitude control of a spacecraft model are also provided to demonstrate the performance of the proposed controller.