Adaptive finite-time attitude control for spacecraft with input quantization and measurement uncertainties

Abstract

The issue of adaptive finite-time attitude control is explored for rigid spacecraft with input quantization and measurement uncertainties, in which a hysteretic quantizer is introduced. The measurement uncertainties and external disturbances are compensated via adaptive control technique. By combining adding a power integrator technique with a new coordinate transformation, an adaptive finite-time controller is proposed to derive system states into a neighborhood of the origin in finite time. Finally, the control performance of the developed control scheme is demonstrated via simulation results.

Introduction

In the past decades, the attitude control issue for spacecraft system has been investigated by many scholars. A large amount of control schemes have been reported to enhance the control performance for the attitude control system, for example, geometric approach, ¹ Lyapunov control,^2–4 passivity-based control, ⁵ adaptive control,^6,7 sliding mode control,^8–10 and finite-time control (FTC).^11–15 Among them, FTC techniques can gain higher control precision, faster convergence speed, and better disturbance attenuation and rejection properties. It is worth noticing that the adaptive control scheme provides a good method to attenuate the unknown inertia matrix and external disturbance influences, and some adaptive FTC approaches have been proposed for spacecraft systems with matched uncertainties.^16–18 However, it is not focused on measurement uncertainties (MUs) in the aforesaid references. In practice, the attitude and angular velocity of spacecraft system are measured with MUs from many sensors. If the measurement error is not explicitly considered, the static precision and the control performance of the spacecraft system may be destroyed.^19–21 Some control schemes are proposed for spacecraft systems with MUs,^19–21 but they only guarantee that system states converge to zero or a neighborhood of the origin in infinite time. Few results are paid attention to investigating the finite-time attitude stabilization issue for spacecraft systems with MUs.

The above survey shows that a wealth of research work has been achieved to obtain good performance for attitude control systems. However, the quantization error of the attitude control torque was not taken into account in the aforementioned results. The quantization error can be rationally left out if the quantization accuracy is high. However, when the actuator module and attitude control module are connected via a low-price wireless network, the quantization accuracy is coarse. So it is necessary to put a big spotlight on the quantization error in the analysis and design of attitude controller. The attitude controllers with input quantization have been designed for spacecraft system via different control methods.^22–25 However, in previous studies,^22–25 the additive MUs were not explicitly concerned. To the best of authors’ knowledge, there are few results to study the problem of finite-time attitude stabilization for spacecraft system with input quantization, MUs, and external disturbances.

From what has been discussed above, the issue of input quantization and MUs in the spacecraft attitude control system is explored carefully. A novel adaptive finite-time attitude controller is constructed for the spacecraft attitude stabilization system in the presence of MUs, input quantization, and external disturbances. Due to the mismatched parameterized unknown functions stemmed from the additive MUs, a virtual control law with an adaptive parameter term is designed. Then, a novel adaptive finite-time controller is obtained by combining a new coordinate transformation with a sliding mode differentiator to surmount the problem of explosion of terms. Furthermore, the stability of the closed-loop system is analyzed via Lyapunov function method. Finally, simulation results are provided to verify the effectiveness of the obtained controller.

Rigid spacecraft systems and preliminaries

The following attitude kinematics and dynamics equations are given based on modified Rodrigues parameters (MRPs)

$$J\stackrel{·}{\mathcal{W}}=\mathcal{W}\times J\mathcal{W}+q\left(u\left(t\right)\right)+d\left(t\right)$$

(1)

$$\stackrel{·}{\sigma }=G\left(\sigma \right)\mathcal{W}$$

(2)

where $\mathcal{W}=[{\mathcal{W}}_1,{\mathcal{W}}_2,{\mathcal{W}}_3{]}^T$ and $\sigma =[{\sigma }_1,{\sigma }_2,{\sigma }_3{]}^T$ mean the angular velocity and the attitude of spacecraft, respectively; $J=\mathrm{diag}\{J_1,J_2,J_3\}$ is the inertia matrix; and $d(t)=[d_1(t),d_2(t),d_3(t){]}^T$ is the external disturbances, $G(\sigma )=\frac{1}{2}[\frac{1-{\sigma }^T\sigma }{2}{I}_3-s(\sigma )+\sigma {\sigma }^T]$ with $I_3$ being the identity matrix and $s(\sigma )=[0{\sigma }_3-{\sigma }_2;-{\sigma }_{3}0{\sigma }_1;{\sigma }_2-{\sigma }_{1}0]$ satisfies the following properties ²⁶

$$\begin{array}{c}{\sigma }^{T}G\left(\sigma \right)\mathcal{W}=\frac{1+{\sigma }^T\sigma }{4}{\sigma }^T\mathcal{W},G\left(\sigma \right)G^T\left(\sigma \right)={\left(\frac{1+{\sigma }^T\sigma }{4}\right)}^2{I}_3,\hfill \\ ||G\left(\sigma \right)||\le \frac{1}{4}+\frac{1}{2}\sigma +\frac{3}{4}||\sigma |{|}^2,||s\left(\sigma \right)||=||\sigma ||\hfill \end{array}$$

(3)

$q(u(t))=[q(u_1(t)),q(u_2(t)),q(u_3(t){]}^T$ denotes a quantized input, which is defined as^27,28

$$q\left(u_i\right)=\{\begin{array}{cc}{\varsigma }_{i,j},\hfill & \mathrm{if}\frac{{\varsigma }_{i,j}}{1+{\delta }_i}<u_i\le {\varsigma }_{i,j},q^{-}\ge {\varsigma }_{i,j}\hfill \\ \multicolumn{1}{c}{}& \mathrm{or}{\varsigma }_{i,j}\le u_i<\frac{{\varsigma }_{i,j}}{1-{\delta }_i},q^{-}\le {\varsigma }_{i,j}\hfill \\ \multicolumn{1}{c}{(1+{\delta }_i){\varsigma }_{i,j},}& \mathrm{if}{\varsigma }_{i,j}<u_i\le \frac{{\varsigma }_{i,j}}{1-{\delta }_i},q^{-}\ge (1+{\delta }_i){\varsigma }_{i,j}\hfill \\ \multicolumn{1}{c}{}& \mathrm{or}\frac{{\varsigma }_{i,j}}{1-{\delta }_i}\le u_i<{\varsigma }_{i,j+1},q^{-}\le (1+{\delta }_i){\varsigma }_{i,j}\hfill \\ \multicolumn{1}{c}{0,}& \mathrm{if}0\le u_i\le \frac{{\varsigma }_{i,j}}{1+{\delta }_i}\hfill \\ \multicolumn{1}{c}{}& \mathrm{or}\frac{{\varsigma }_{i,1}}{1+{\delta }_i}<u_i<{\varsigma }_{i,1},q^{-}=0\hfill \\ \multicolumn{1}{c}{-q(-u_i),}& \mathrm{if}{u}_i<0\hfill \end{array}$$

(4)

In equation (4), ${\delta }_i=(1-{ϵ}_i)/(1+{ϵ}_i)$ and ${\varsigma }_{i,j}={\alpha }_i{ϵ}_i^{1-j}$ , where $0<{ϵ}_i<1$ , ${\alpha }_i>0$ , and $j=1,2,3,\dots$ The parameters ${\alpha }_i$ and ${ϵ}_i$ mean the size of the dead zone for $q(u_i)$ and a measure of quantization density, respectively. $q^{-}(t)$ denotes the latest value of $Q$ before the time instant $t$ and $q^{-}(0)=0$ . Mathematically, $q^{-}(t)=0$ for $t\in [0,T_{i,i}]$ and $q^{-}(t)=q(u_i(T_{i,h}))$ for $t\in (T_{i,h},T_{i,h+1}]$ , where $T_{i,h}(h=1,2,3,\dots )$ with $0\le T_{i,1}<T_{i,2}<T_{i,3}<\cdots \le +\infty$ being the time instants when $q(u_i)$ makes transitions.

Due to measure the attitude and angular velocity via many kinds of sensors, and the sensors are effected by complicated space environment, so that the measurement quantities are generally uncertain. Then the attitude and angular velocity can be defined as

$$\hat{\sigma }=\sigma +{\varpi }_1,\hat{\mathcal{W}}=\mathcal{W}+{\varpi }_2$$

(5)

where ${\varpi }_1$ and ${\varpi }_2$ mean the measurement errors. Calculating the first derivative of equation (5) along with system equations (1) and (2) yields

$$\stackrel{·}{\hat{\sigma }}=G\left(\hat{\sigma }\right)\hat{\mathcal{W}}+{\zeta }_1$$

(6)

$$J\stackrel{·}{\hat{\mathcal{W}}}=s\left(\hat{\mathcal{W}}\right)J\hat{\mathcal{W}}+q\left(u\right)+{\zeta }_2$$

(7)

where $G(\hat{\sigma })=\frac{1}{2}[\frac{1-{\hat{\sigma }}^T\hat{\sigma }}{2}{I}_3-s(\hat{\sigma })+\hat{\sigma }{\hat{\sigma }}^T]$ , ${\zeta }_1={\stackrel{·}{\varpi }}_1+G(\sigma ){\varpi }_2+\frac{1}{4}((2{\hat{\sigma }}^T{\varpi }_1-{\varpi }_1^T{\varpi }_1)I_3-2s({\varpi }_1)-2\hat{\sigma }{\varpi }_1^T+2{\varpi }_1^T{\varpi }_1)\hat{\mathcal{W}}$ , and ${\zeta }_2=d(t)+J{\stackrel{·}{\varpi }}_2+s({\varpi }_2)J\hat{\mathcal{W}}+s(\hat{\mathcal{W}})J{\varpi }_2-s({\varpi }_2)J{\varpi }_2$ .

Some assumptions and lemmas are presented as follows.

Assumption 1

The external disturbances $d(t)$ and the inertia matrix $J$ satisfy $||d(t)||\le \kappa$ and $||J||\le \overline{J}$ , where $\kappa >0$ and $\overline{J}>0$ .

Remark 1

It is reasonable to assume that the external disturbances $d(t)$ and the inertia matrix $J$ are bounded, since all the environmental disturbances due to aerodynamic drag, solar radiation pressure, gravitation, or magnetic forces are bounded and the inertia matrix is the constant matrix with some uncertainties.

Assumption 2

The MUs ${\varpi }_i$ $(i=1,2)$ and their derivative ${\stackrel{·}{\varpi }}_i$ are bounded such that $||{\varpi }_i||\le {\overline{\varpi }}_i$ and $||{\stackrel{·}{\varpi }}_i||\le {\overline{\stackrel{·}{\varpi }}}_i$ with unknown constants ${\overline{\varpi }}_i$ and ${\overline{\stackrel{·}{\varpi }}}_i$ .^24,25

Remark 2

Because the properly sensors can effectively measure data within a limit of allowable error, it is a reasonable condition.

Based on the definitions of ${\zeta }_i(i=1,2)$ and Assumptions 1 and 2 plus properties (equation (3)), we have the following assumption.

Assumption 3

${\zeta }_1$ and ${\zeta }_2$ satisfy the following inequalities ²⁵

$$||{\zeta }_1||\le {\mathbf{b}}_1^T{\varphi }_1,||{\zeta }_2||\le {\mathbf{b}}_2^T{\varphi }_2$$

(8)

where ${\mathbf{b}}_1=[b_{11},b_{12},b_{13},b_{14},b_{15}{]}^T$ , ${\varphi }_1=[1,||\hat{\sigma }||,||\hat{\sigma }||||\hat{\mathcal{W}}||,||\hat{\mathcal{W}}||,||\hat{\sigma }|{|}^2{]}^T$ , ${\mathbf{b}}_2=[b_{21},b_{22}{]}^T$ , ${\varphi }_2=[1,||\hat{\mathcal{W}}||{]}^T$ , $b_{11}={\overline{\stackrel{·}{\varpi }}}_1+\frac{1}{4}{\overline{\varpi }}_2+{\overline{\varpi }}_1+\frac{1}{4}{\overline{\varpi }}_1^2{\overline{\varpi }}_2$ , $b_{12}=\frac{1}{2}{\overline{\varpi }}_2+\frac{3}{2}{\overline{\varpi }}_1{\overline{\varpi }}_2+\frac{1}{2}{\overline{\varpi }}_1$ , $b_{13}={\overline{\varpi }}_1$ , $b_{14}=\frac{3}{4}{\overline{\varpi }}_1^2$ , $b_{15}=\frac{3}{4}{\overline{\varpi }}_2$ , $b_{21}=\kappa +\overline{J}{\overline{\stackrel{·}{\varpi }}}_2+\overline{J}{\overline{\varpi }}_2^2$ , and $b_{22}=2{\overline{\varpi }}_2\overline{J}$ .

Lemma 1

For any $x_i\in R$ , $i=1,2,\dots ,n$ , the inequality ²⁹

$${(\sum _{i=1}^n|x_i|)}^p\le \sum _{i=1}^n|x_i{|}^p$$

holds, where $p\in (0,1]$ .

According to Zhou et al., ²⁸ the hysteretic quantizer $q(u_i)$ can be decomposed as

$$q\left(u_i\right)=u_i+{\varrho }_i$$

(9)

where ${\varrho }_i=q(u_i)-u_i\in R$ .

Lemma 2

The following inequalities satisfy ²⁸

$${\varrho }_i^2\le {\delta }_i^2{u}_i^2,\forall |u_i|\ge {\alpha }_i$$

(10)

$${\varrho }_i^2\le {\alpha }_i^2,\forall |u_i|\le {\alpha }_i$$

(11)

Lemma 3

Consider the following system^30,31

$$\stackrel{·}{\hat{\beta }}=-\gamma \hat{\beta }+o\rho \left(t\right)$$

(12)

where $\gamma >0$ and $o>0$ , and $\rho (t)$ is a positive function; then $\hat{\beta }\ge 0$ for $\forall t\ge t_0$ for any given bounded initial condition $\hat{\beta }(t_0)\ge 0$ .

Controller design and stability analysis

In this section, by using a power integrator technique, an adaptive FTC strategy will be presented. Meanwhile, the stability analysis of the closed-loop system is established via Lyapunov function theory.

Controller design

• Step 1: The following equation is considered

$$\stackrel{·}{\hat{\sigma }}=G\left(\hat{\sigma }\right)\hat{\mathcal{W}}+{\zeta }_1$$

(13)

Based on Assumption 3 and Young’s inequality, we have

$$\begin{array}{c}\multicolumn{1}{c}{{\hat{\sigma }}^T{\zeta }_1\le ||\hat{\sigma }||||{\zeta }_1||\le ||\hat{\sigma }||{\mathbf{b}}_1^T{\varphi }_1}\\ \multicolumn{1}{c}{\le \frac{||\hat{\sigma }{||}^2||{\mathbf{b}}_1{||}^2||{\varphi }_1{||}^2}{4a_1}+a_1}\\ \multicolumn{1}{c}{\le \frac{{\theta }_1||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4a_1}+a_1}\end{array}$$

(14)

where ${\theta }_1=||{\mathbf{b}}_1|{|}^2$ and $a_1>0$ . Choose a Lyapunov function as

$$V_0=\frac{1}{2}{\hat{\sigma }}^T\hat{\sigma }+\frac{{\gamma }_1}{2}{\tilde{\theta }}_1^2$$

(15)

where ${\gamma }_1>0$ and ${\tilde{\theta }}_1={\theta }_1-{\hat{\theta }}_1$ , and ${\hat{\theta }}_1$ is the estimation of ${\theta }_1$ .

The time derivative of equation (15) leads to

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_0={\hat{\sigma }}^T(G\left(\hat{\sigma }\right)\hat{\mathcal{W}}+{\zeta }_1)-{\gamma }_1{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1}\\ \multicolumn{1}{c}{=\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{\hat{\sigma }}^T\hat{\mathcal{W}}+{\hat{\sigma }}^T{\zeta }_1-{\gamma }_1{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1}\end{array}$$

(16)

Substituting equation (14) into equation (16) leads to

$$\begin{array}{c}{\stackrel{·}{V}}_0\le \frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{\hat{\sigma }}^T\hat{\mathcal{W}}+\frac{{\theta }_1||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4a_1}+a_1-{\gamma }_1{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1\hfill \\ =\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{\hat{\sigma }}^T{\hat{\mathcal{W}}}^{*}+\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{\hat{\sigma }}^T(\hat{\mathcal{W}}-{\hat{\mathcal{W}}}^{*})+\frac{{\theta }_1||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4a_1}+a_1-{\gamma }_1{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1\hfill \end{array}$$

(17)

where ${\hat{\mathcal{W}}}^{*}$ is the virtual control law, which can be designed as

$${\hat{\mathcal{W}}}^{*}=-k_1{\hat{\sigma }}^{1/p}-\frac{{\hat{\theta }}_1\hat{\sigma }||{\varphi }_1{||}^2}{(1+{\hat{\sigma }}^T\hat{\sigma })a_1}$$

(18)

where $p>1$ is ratio of two odds and $k_1>0$ . Combining equations (17) with (18) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_0\le -\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{\hat{\sigma }}^T{k}_1{\hat{\sigma }}^{1/p}-{\gamma }_1{\tilde{\theta }}_1({\stackrel{·}{\hat{\theta }}}_1-\frac{||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1})}\\ \multicolumn{1}{c}{+\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{\hat{\sigma }}^T(\hat{\mathcal{W}}-{\hat{\mathcal{W}}}^{*})+a_1}\\ \multicolumn{1}{c}{=-\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{k}_1\sum _{i=1}^3{\hat{\sigma }}_i^d-{\gamma }_1{\tilde{\theta }}_1({\stackrel{·}{\hat{\theta }}}_1-\frac{||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1})}\\ \multicolumn{1}{c}{+\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}\sum _{i=1}^3{\hat{\sigma }}_i({\hat{\mathcal{W}}}_i-{\hat{\mathcal{W}}}_i^{*})+a_1}\end{array}$$

(19)

where $d=1+(1/p)$ .

• Step 2: The Lyapunov function is constructed as

$$V=V_0+\sum _{i=1}^3{V}_i+\frac{{\gamma }_2}{2}{\tilde{\theta }}_2^2$$

(20)

where

$$\begin{array}{c}{V}_i=\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}\underset{{\overline{\hat{\mathcal{W}}}}_i^{*}}{\overset{{\overline{\hat{\mathcal{W}}}}_i}{\int }}{(s^p-{\overline{\hat{\mathcal{W}}}}_i^{*p})}^{2-\frac{1}{p}}\mathrm{ds}\hfill \\ \overline{\hat{\mathcal{W}}}={[{\overline{\hat{\mathcal{W}}}}_1,{\overline{\hat{\mathcal{W}}}}_2,{\overline{\hat{\mathcal{W}}}}_3]}^T=\hat{\mathcal{W}}+\frac{{\hat{\theta }}_1\hat{\sigma }||{\varphi }_1{||}^2}{(1+{\hat{\sigma }}^T\hat{\sigma })a_1}\hfill \\ {\overline{\hat{\mathcal{W}}}}^{*}={[{\overline{\hat{\mathcal{W}}}}_1^{*},{\overline{\hat{\mathcal{W}}}}_2^{*},{\overline{\hat{\mathcal{W}}}}_3^{*}]}^T={\hat{\mathcal{W}}}^{*}+\frac{{\hat{\theta }}_1\hat{\sigma }||{\varphi }_1{||}^2}{(1+{\hat{\sigma }}^T\hat{\sigma })a_1}\hfill \end{array}$$

(21)

${\gamma }_2>0,{\tilde{\theta }}_2={\theta }_2-{\hat{\theta }}_2$ , where ${\hat{\theta }}_2$ is the estimation of ${\theta }_2$ ; ${\theta }_2$ will be defined later.

Remark 3

In equation (21), if we define $V_i=\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}{\int }_{{\hat{\mathcal{W}}}_i^{*}}^{{\hat{\mathcal{W}}}_i}{(s^p-{\hat{\mathcal{W}}}_i^{*p})}^{2-\frac{1}{p}}\mathrm{ds}$ , it may lead to singularity when computing the derivative of ${\hat{\mathcal{W}}}_i^{*p}$ . To solve this problem, a new coordinate transformation is given and then the Lyapunov function (equation (21)) is constructed. Taking the derivative of equation (20) yields

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{k}_1\sum _{i=1}^3{\hat{\sigma }}_i^d-{\gamma }_1{\tilde{\theta }}_1({\stackrel{·}{\hat{\theta }}}_1-\frac{||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1})-{\gamma }_2{\tilde{\theta }}_2{\stackrel{·}{\hat{\theta }}}_2}\\ \multicolumn{1}{c}{+\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}\sum _{i=1}^3{\hat{\sigma }}_i({\hat{\mathcal{W}}}_i-{\hat{\mathcal{W}}}_i^{*})+a_1}\\ \multicolumn{1}{c}{-\sum _{i=1}^3\frac{1}{2^{1-\frac{1}{p}}}\frac{\partial {\overline{\hat{\mathcal{W}}}}_i^{*p}}{\partial t}\underset{{\overline{\hat{\mathcal{W}}}}_i^{*}}{\overset{{\overline{\hat{\mathcal{W}}}}_i}{\int }}{(s^p-{\overline{\hat{\mathcal{W}}}}_i^{*p})}^{1-\frac{1}{p}}\mathrm{ds}+\sum _{i=1}^3\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}{\xi }_i^{2-\frac{1}{p}}{\stackrel{·}{\overline{\hat{\mathcal{W}}}}}_i}\end{array}$$

(22)

where ${\xi }_i={\overline{\hat{\mathcal{W}}}}_i^p-{\overline{\hat{\mathcal{W}}}}_i^{*p}$ . By employing Lemmas A.1-2 in Qian and Lin, ³² we have

$${\hat{\sigma }}_i({\overline{\hat{\mathcal{W}}}}_i-{\overline{\hat{\mathcal{W}}}}_i^{*})\le 2^{1-\frac{1}{p}}|{\hat{\sigma }}_i||{\xi }_i{|}^{\frac{1}{p}}\le 2^{1-\frac{1}{p}}(\frac{p}{1+p}|{\hat{\sigma }}_i{|}^d+\frac{1}{1+p}|{\xi }_i{|}^d)$$

(23)

Combining equation (13) with equation (18) generates

$${\left[\frac{\partial {\overline{\hat{\mathcal{W}}}}_1^{*p}}{\partial t}\frac{\partial {\overline{\hat{\mathcal{W}}}}_2^{*p}}{\partial t}\frac{\partial {\overline{\hat{\mathcal{W}}}}_3^{*p}}{\partial t}\right]}^T=-\frac{\partial \left(k_1^p\hat{\sigma }\right)}{\partial \hat{\sigma }}\frac{\partial \hat{\sigma }}{\partial t}=-k_1^p(G\left(\hat{\sigma }\right)\hat{\mathcal{W}}+{\zeta }_1)$$

(24)

Similar to Du et al., ³³ we obtain

$$|\frac{\partial {\overline{\hat{\mathcal{W}}}}_i^{*p}}{\partial t}|\le \frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{k}_1^p(|{\hat{\mathcal{W}}}_1|+|{\hat{\mathcal{W}}}_2|+|{\hat{\mathcal{W}}}_3|)+k_1^p|{\zeta }_{1i}|,i=1,2,3$$

(25)

Substituting equations (23) and (25) into equation (22) leads to

$$\begin{array}{c}\stackrel{·}{V}\le -\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}(k_1-2^{1-\frac{1}{p}}\frac{p}{1+p})\sum _{i=1}^3{\hat{\sigma }}_i^d+2^{1-\frac{1}{p}}\frac{1}{1+p}\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}\sum _{i=1}^3{\xi }_i^d\hfill \\ +\sum _{i=1}^3\frac{1}{2^{1-\frac{1}{p}}}(\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{k}_1^p(|{\hat{\mathcal{W}}}_1|+|{\hat{\mathcal{W}}}_2|+|{\hat{\mathcal{W}}}_3|)+k_1^p|{\zeta }_{1i}|)|{\overline{\hat{\mathcal{W}}}}_i-{\overline{\hat{\mathcal{W}}}}_i^{*}|{\xi }_i^{1-\frac{1}{p}}\hfill \\ +\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}\left[{\xi }_1^{2-\frac{1}{p}}{\xi }_2^{2-\frac{1}{p}}{\xi }_3^{2-\frac{1}{p}}\right](J^{-1}s\left(\hat{\mathcal{W}}\right)J\hat{\mathcal{W}}+J^{-1}q\left(u\right)+J^{-1}{\zeta }_2+{\stackrel{·}{\alpha }}_1)\hfill \\ -{\gamma }_1{\tilde{\theta }}_1({\stackrel{·}{\hat{\theta }}}_1-\frac{||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1})-{\gamma }_2{\tilde{\theta }}_2{\stackrel{·}{\hat{\theta }}}_2+a_1\hfill \end{array}$$

(26)

where ${\alpha }_1=\frac{{\hat{\theta }}_1\hat{\sigma }||{\varphi }_1{||}^2}{(1+{\hat{\sigma }}^T\hat{\sigma })a_1}$ . For any $i,m=1,2,3$ , using Lemmas A.1-2 in Qian and Lin ³² produces

$$\begin{array}{l}\left|{\widehat{W}}_m\right||{\overline{\widehat{W}}}_i-{\overline{\widehat{W}}}_i^{*}||{\xi }_i{|}^{1-\frac{1}{p}}\le 2^{1-\frac{1}{p}}|{\widehat{W}}_m\left|\right|{\xi }_i{|}^{\frac{1}{p}}|{\xi }_i{|}^{1-\frac{1}{p}}\\ \le 2^{1-\frac{1}{p}}|\left(\left|{\xi }_i\right||{\widehat{W}}_m-{\widehat{W}}_m^{*}|+\left|{\xi }_i\right|\left|k_1{\widehat{\sigma }}_m^{1/p}\right|+\left|{\xi }_i\right|\left|\frac{{\widehat{\theta }}_1{\widehat{\sigma }}_m\left|\right|{\varphi }_1|{|}^2}{\left(1+{\widehat{\sigma }}^T\widehat{\sigma }\right)a_1}\right|\right)\\ \le 2^{1-\frac{1}{p}}\left(2^{1-\frac{1}{p}}\left|{\xi }_i\right||{\xi }_m{|}^{\frac{1}{p}}+k_1|{\xi }_i\left|\right|{\widehat{\sigma }}_m{|}^{\frac{1}{p}}+\left|{\xi }_i\right||{\widehat{\sigma }}_m{|}^{\frac{1}{p}}{\rho }_m\right)\\ \le 2^{1-\frac{1}{p}}\left(\frac{\left(2^{1-\frac{1}{p}}+k_1+{\rho }_m^d\right)p}{1+p}|{\xi }_i{|}^d+\frac{2^{1-\frac{1}{p}}}{1+p}|{\xi }_m{|}^d+\frac{1+k_1}{1+p}|{\widehat{\sigma }}_m{|}^d\right)\\ \left|{\zeta }_{1i}\right||{\overline{\widehat{W}}}_i-{\overline{\widehat{W}}}_i^{*}|{\xi }_i^{1-\frac{1}{p}}\le 2^{1-\frac{1}{p}}{\mathbf{b}}_1^T{\varphi }_1\left|{\xi }_i\right|\le \frac{{\left(2^{1-\frac{1}{p}}\right)}^2{\theta }_1\left|\right|{\varphi }_1\left|{|}^2\right|{\xi }_i{|}^2}{4a_1}+a_1\end{array}$$

(27)

where ${\rho }_m=\frac{|{\hat{\theta }}_1||{\hat{\sigma }}_m{|}^{1-\frac{1}{p}}||{\varphi }_1{||}^2}{(1+{\hat{\sigma }}^T\hat{\sigma })a_1}$ .

The last term in equation (26) needs to calculate the first derivative. In order to reduce the complexity of count, a first-order sliding mode differentiator is employed to estimate its derivative. We have

$$\begin{array}{c}{\stackrel{·}{\rho }}_{20}={\nu }_{20}={\rho }_{21}-{\mu }_{20}|{\rho }_{20}-{\alpha }_1{|}^{\frac{1}{2}}\mathrm{sign}({\rho }_{20}-{\alpha }_1)\hfill \\ {\stackrel{·}{\rho }}_{21}=-{\mu }_{21}\mathrm{sign}({\rho }_{21}-{\nu }_{20})\hfill \end{array}$$

(28)

where ${\mu }_{20}>0$ , ${\mu }_{21}>0$ , and ${\rho }_{20}$ , ${\rho }_{21}$ , and ${\nu }_{20}$ are system states.

On the basis of equation (28) and Levant’s study, ³⁴ we obtain

$${\stackrel{·}{\alpha }}_1={\nu }_{20}+o_2$$

(29)

where $o_2$ is the estimation error of sliding mode differentiator. According to Levant, ³⁴ we have $|o_2|\le {\varrho }_2$ with ${\varrho }_2>0$ . Combining equations (26) to (29) and considering equation (9) plus using Lemma 2 lead to

$$\begin{array}{c}\stackrel{·}{V}\le -\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}(k_1-\frac{2^{1-\frac{1}{p}}p+3(1+k_1)k_1^p}{1+p})\sum _{i=1}^3{\hat{\sigma }}_i^d+2^{1-\frac{1}{p}}\frac{1}{1+p}\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}\sum _{i=1}^3{\xi }_i^d\hfill \\ +\sum _{i=1}^3(\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{k}_1^p(\frac{(2^{1-\frac{1}{p}}+k_1+{\rho }_1^d+{\rho }_2^d+{\rho }_3^d)p+3\times 2^{1-\frac{1}{p}}}{1+p}|{\xi }_i{|}^d)+\frac{2^{1-\frac{1}{p}}{\theta }_1||{\varphi }_1{||}^2|{\xi }_i{|}^2}{4a_1})\hfill \\ +\sum _{i=1}^3\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}{\xi }_i^{2-\frac{1}{p}}{\tau }_i+\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}\left[{\xi }_1^{2-\frac{1}{p}}{\xi }_2^{2-\frac{1}{p}}{\xi }_3^{2-\frac{1}{p}}\right](J^{-1}(u+\varrho )+J^{-1}{\zeta }_2+o_2)\hfill \\ -{\gamma }_1{\tilde{\theta }}_1({\stackrel{·}{\hat{\theta }}}_1-\frac{||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1})-{\gamma }_2{\tilde{\theta }}_2{\stackrel{·}{\hat{\theta }}}_2+a_1+\frac{3a_1}{2^{1-\frac{1}{p}}}\hfill \\ \le -\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}(k_1-\frac{2^{1-\frac{1}{p}}p+3(1+k_1)k_1^p}{1+p})\sum _{i=1}^3{\hat{\sigma }}_i^d+2^{1-\frac{1}{p}}\frac{1}{1+p}\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}\sum _{i=1}^3{\xi }_i^d\hfill \\ +\sum _{i=1}^3(\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}{k}_1^p(\frac{(2^{1-\frac{1}{p}}+k_1+{\rho }_1^d+{\rho }_2^d+{\rho }_3^d)p+3\times 2^{1-\frac{1}{p}}}{1+p}|{\xi }_i{|}^d)+\frac{2^{1-\frac{1}{p}}{\theta }_1||{\varphi }_1{||}^2|{\xi }_i{|}^2}{4a_1})\hfill \\ +\sum _{i=1}^3\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}{\xi }_i^{2-\frac{1}{p}}{\tau }_i+\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}\left[{\xi }_1^{2-\frac{1}{p}}{\xi }_2^{2-\frac{1}{p}}{\xi }_3^{2-\frac{1}{p}}\right](J^{-1}(u+\varrho )+J^{-1}{\zeta }_2+o_2)\hfill \\ -{\gamma }_1{\tilde{\theta }}_1({\stackrel{·}{\hat{\theta }}}_1-\frac{||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1})-{\gamma }_2{\tilde{\theta }}_2{\stackrel{·}{\hat{\theta }}}_2+a_1+\frac{3a_1}{2^{1-\frac{1}{p}}}\hfill \\ \le -\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}(k_1-\frac{2^{1-\frac{1}{p}}p+3(1+k_1)k_1^p}{1+p})\sum _{i=1}^3{\hat{\sigma }}_i^d+2^{1-\frac{1}{p}}\frac{1}{1+p}\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}\sum _{i=1}^3{\xi }_i^d\hfill \\ +\sum _{i=1}^3(\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}\frac{(2^{1-\frac{1}{p}}+k_1+{\rho }_1^d+{\rho }_2^d+{\rho }_3^d)p+3\times 2^{1-\frac{1}{p}}}{1+p}{k}_1^p|{\xi }_i{|}^d+\frac{2^{1-\frac{1}{p}}{\theta }_1||{\varphi }_1{||}^2|{\xi }_i{|}^2}{4a_1})\hfill \\ +\sum _{i=1}^3\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}{\xi }_i^{2-\frac{1}{p}}{\tau }_i+\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}\left[{\xi }_1^{2-\frac{1}{p}}{\xi }_2^{2-\frac{1}{p}}{\xi }_3^{2-\frac{1}{p}}\right](J^{-1}u+J^{-1}{\zeta }_2+o_2)\hfill \\ +\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}\sum _{i=1}^3{J}_i^{-1}|{\xi }_i{|}^{2-\frac{1}{p}}({\delta }_i|u_i|+{\alpha }_i)\hfill \\ -{\gamma }_1{\tilde{\theta }}_1({\stackrel{·}{\hat{\theta }}}_1-\frac{{\Vert \hat{\sigma }\Vert }^2{\Vert {\varphi }_1\Vert }^2}{4{\gamma }_1{a}_1})-{\gamma }_2{\tilde{\theta }}_2{\stackrel{·}{\hat{\theta }}}_2+a_1+\frac{3a_1}{2^{1-\frac{1}{p}}}\hfill \end{array}$$

(30)

where $\tau =[{\tau }_1,{\tau }_2,{\tau }_3{]}^T=J^{-1}s(\hat{\mathcal{W}})J\hat{\mathcal{W}}+{\nu }_{20}$ .

Note that

$$\begin{array}{c}{J}_i^{-1}|{\xi }_i{|}^{2-\frac{1}{p}}|({\alpha }_i+{\zeta }_{2i}+J_i{o}_{2i})|\le J_i^{-1}|{\xi }_i{|}^{2-\frac{1}{p}}{{\overline{\mathbf{b}}}_2}^T{\varphi }_2\hfill \\ \le \frac{J_i^{-2}{(|{\xi }_i{|}^{2-\frac{1}{p}}|)}^2||{\overline{\mathbf{b}}}_2{||}^2||{\varphi }_2{||}^2}{4a_1}+a_1=\frac{{\theta }_2{J}_i^{-2}{(|{\xi }_i{|}^{2-\frac{1}{p}}|)}^2||{\varphi }_2{||}^2}{4a_1}+a_1\hfill \end{array}$$

(31)

where ${\theta }_2=||{\overline{\mathbf{b}}}_2|{|}^2$ , ${\overline{\mathbf{b}}}_2=[{\overline{b}}_{21},b_{22}]$ , and ${\overline{b}}_{21}=b_{21}+{\alpha }_i+\overline{J}{\varrho }_2$ .

The controller and adaptive laws are designed as

$$u=-{(I-\delta )}^{-1}J{[{\overline{u}}_1,{\overline{u}}_2,{\overline{u}}_3]}^T$$

(32)

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{u}}_i=(2-\frac{1}{p})2^{1-\frac{1}{p}}(k_2+\chi ){\xi }_i^{\frac{2}{p}-1}+\frac{2^{1-\frac{1}{p}}{\hat{\theta }}_1||{\varphi }_1{||}^2{\xi }_i^{\frac{1}{p}}}{4a_1}}\\ \multicolumn{1}{c}{+\frac{{\hat{\theta }}_2{J}_i^{-2}{\xi }_i^{2-\frac{1}{p}}||{\varphi }_2{||}^2}{4a_1}+\frac{{\xi }_i^{2-\frac{1}{p}}|{\tau }_i{|}^2}{|{\xi }_i{|}^{2-\frac{1}{p}}|{\tau }_i|+{\eta }_i}}\end{array}$$

(33)

$${\stackrel{·}{\hat{\theta }}}_1=\frac{||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1}+\sum _{i=1}^3\frac{2^{1-\frac{1}{p}}{\xi }_i^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1}-\frac{{\iota }_1}{{\gamma }_1}{\hat{\theta }}_1$$

(34)

$${\stackrel{·}{\hat{\theta }}}_2=\sum _{i=1}^3\frac{J_i^{-2}{\left({\xi }_i^{2-\frac{1}{p}}\right)}^2||{\varphi }_2{||}^2}{4a_1{\gamma }_2}-\frac{{\iota }_2}{{\gamma }_2}{\hat{\theta }}_2$$

(35)

where $\delta =\mathrm{diag}\{{\delta }_1,{\delta }_2,{\delta }_3\}$ , $k_2\ge \frac{p{2}^{1-\frac{1}{p}}+3}{1+p}+k_3$ , $k_3>0$ , $1<p<2$ , ${\eta }_i>0$ , $i=1,2,3$ , $\chi =2^{1-\frac{1}{p}}\frac{1}{1+p}\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}+\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}\frac{(2^{1-\frac{1}{p}}+k_1+{\rho }_1^d+{\rho }_2^d+{\rho }_3^d)p+3\times 2^{1-\frac{1}{p}}}{1+p}{k}_1^p$ , ${\iota }_j>0$ , and $j=1,2$ .

The design block diagram procedure of the proposed control scheme is shown in Figure 1.

Figure 1.

Block diagram of control system.

Remark 4

The adaptive finite-time controller is designed in equation (32) with adaptive laws (equations (34) and (35)). In equation (33), it contains four parts: the first part is used to guarantee the closed-loop system that has a good control performance, the second and third parts are employed to reject the unknown parameters, and the fourth part is designed to cope with the term ${\tau }_i$ due to the existence of quantizer.

Remark 5

The input constraint problem is an interesting topic. Based on the present work, we can introduce the auxiliary system to remove the influence of the input constraints.^35,36

Stability analysis

Now, the stability analysis of the closed-loop system is illustrated as follows.

Theorem 1

Consider the rigid spacecraft (equations (6) and (7)) with Assumptions 1 to 3 and the controller (equation (32)). Then all the states of the closed-loop system are bounded and can converge to a neighborhood of origin in finite time.

Proof. By using Lemma 3, we have ${\hat{\theta }}_1\ge 0$ and ${\hat{\theta }}_2>0$ . Then we obtain $[{\xi }_1^{2-\frac{1}{p}}{\xi }_2^{2-\frac{1}{p}}{\xi }_3^{2-\frac{1}{p}}]J^{-1}u\le 0$ . Furthermore, one has

$$\begin{array}{c}\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}\sum _{i=1}^3{J}_i^{-1}|{\xi }_i{|}^{2-\frac{1}{p}}{\delta }_i|u_i|\hfill \\ \le -\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}[{\xi }_1^{2-\frac{1}{p}}{\xi }_2^{2-\frac{1}{p}}{\xi }_3^{2-\frac{1}{p}}]J^{-1}\delta u\hfill \end{array}$$

(36)

Combining equations (30) and (33) with equation (36) leads to

$$\begin{array}{c}\stackrel{·}{V}\le -\frac{1+{\hat{\sigma }}^T\hat{\sigma }}{4}(k_1-\frac{2^{1-\frac{1}{p}}p+3(1+k_1)k_1^p}{1+p})\sum _{i=1}^3{\hat{\sigma }}_i^d-k_2\sum _{i=1}^3{\xi }_i^d\hfill \\ +\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}(\sum _{i=1}^3|{\xi }_i{|}^{2-\frac{1}{p}}|{\tau }_i|-\sum _{i=1}^3\frac{{\left({\xi }_i^{2-\frac{1}{p}}\right)}^2|{\tau }_i{|}^2}{|{\xi }_i{|}^{2-\frac{1}{p}}|{\tau }_i|+{\eta }_i})\hfill \\ -{\gamma }_1{\tilde{\theta }}_1({\stackrel{·}{\hat{\theta }}}_1-\frac{||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1}-\frac{||\xi {||}^2||{\varphi }_2{||}^2}{2{\gamma }_1{a}_1})-{\gamma }_2{\tilde{\theta }}_2({\stackrel{·}{\hat{\theta }}}_2-\sum _{i=1}^3\frac{J_i^{-2}{\left({\xi }_2^{2-\frac{1}{p}}\right)}^2||{\varphi }_2{||}^2}{4a_1{\gamma }_2})+2a_1+\frac{3a_1}{2^{1-\frac{1}{p}}}\hfill \\ \le -\frac{k_3}{4}\sum _{i=1}^3{\hat{\sigma }}_i^d-k_2\sum _{i=1}^3{\xi }_i^d+\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}\sum _{i=1}^3\frac{|{\xi }_i{|}^{2-\frac{1}{p}}|{\tau }_i|{\eta }_i}{|{\xi }_i{|}^{2-\frac{1}{p}}|{\tau }_i|+{\eta }_i}\hfill \\ -{\gamma }_1{\tilde{\theta }}_1({\stackrel{·}{\hat{\theta }}}_1-\frac{||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1}-\frac{||\xi {||}^2||{\varphi }_2{||}^2}{2{\gamma }_1{a}_1})-{\gamma }_2{\tilde{\theta }}_2({\stackrel{·}{\hat{\theta }}}_2-\sum _{i=1}^3\frac{J_i^{-2}{\left({\xi }_2^{2-\frac{1}{p}}\right)}^2||{\varphi }_2{||}^2}{4a_1{\gamma }_2})+2a_1+\frac{3a_1}{2^{1-\frac{1}{p}}}\hfill \end{array}$$

(37)

In fact

$$0\le \frac{\mathcal{X}\mathcal{Y}}{\mathcal{X}+\mathcal{Y}}\le \mathcal{Y},\forall \mathcal{X}\ge 0,\mathcal{Y}>0$$

Then one gets

$$\begin{array}{c}\stackrel{·}{V}\le -\frac{k_3}{4}\sum _{i=1}^3{\hat{\sigma }}_i^d-k_2\sum _{i=1}^3{\xi }_i^d-{\gamma }_1{\tilde{\theta }}_1({\stackrel{·}{\hat{\theta }}}_1-\frac{||\hat{\sigma }{||}^2||{\varphi }_1{||}^2}{4{\gamma }_1{a}_1}-\frac{||\xi {||}^2||{\varphi }_2{||}^2}{2{\gamma }_1{a}_1})\hfill \\ -{\gamma }_2{\tilde{\theta }}_2({\stackrel{·}{\hat{\theta }}}_2-\sum _{i=1}^3\frac{J_i^{-2}{\left({\xi }_2^{2-\frac{1}{p}}\right)}^2||{\varphi }_2{||}^2}{4a_1{\gamma }_2})+2a_1+\frac{3a_1}{2^{1-\frac{1}{p}}}+\sum _{i=1}^3\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}{\eta }_i\hfill \end{array}$$

(38)

Substituting equations (34) and (35) into equation (38) yields

$$\stackrel{·}{V}\le -\frac{k_3}{4}\sum _{i=1}^3{\hat{\sigma }}_i^d-k_2\sum _{i=1}^3{\xi }_i^d+{\iota }_1{\tilde{\theta }}_1{\hat{\theta }}_1+{\iota }_2{\tilde{\theta }}_2{\hat{\theta }}_2+2a_1+\frac{3a_1}{2^{1-\frac{1}{p}}}+\sum _{i=1}^3\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}{\eta }_i$$

(39)

Note that

$$\begin{array}{c}{\iota }_i{\tilde{\theta }}_i{\hat{\theta }}_i={\iota }_i{\tilde{\theta }}_i({\theta }_i-{\tilde{\theta }}_i)=-{\iota }_i{\tilde{\theta }}_i^2+{\iota }_i{\tilde{\theta }}_i{\theta }_i\hfill \\ \le -{\iota }_i{\tilde{\theta }}_i^2+\frac{1}{4}{\iota }_i{\tilde{\theta }}_i^2+{\iota }_i{\theta }_i^2=-\frac{3}{4}{\iota }_i{\tilde{\theta }}_i^2+{\iota }_i{\theta }_i^2\hfill \end{array}$$

(40)

Next, by using Lemma 2.4 in Qian and Lin, ³⁷ the following inequality holds

$${\left(\frac{3}{4}{\iota }_i{\tilde{\theta }}_i^2\right)}^{\frac{d}{2}}\le \frac{3}{4}{\iota }_i{\tilde{\theta }}_i^2+(1-\frac{d}{2})\kappa$$

(41)

where $\kappa =(d/2{)}^{\frac{d/2}{1-d/2}}$ .

Combining equations (39) and (40) with equation (41) gives rise to

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -\frac{k_3}{4}\sum _{i=1}^3{\hat{\sigma }}_i^d-k_2\sum _{i=1}^3{\xi }_i^d-{\left(\frac{3}{4}{\iota }_1{\tilde{\theta }}_1^2\right)}^{\frac{d}{2}}-{\left(\frac{3}{4}{\iota }_2{\tilde{\theta }}_2^2\right)}^{\frac{d}{2}}}\\ \multicolumn{1}{c}{+{\iota }_1{\theta }_1^2+{\iota }_2{\theta }_2^2+2(1-\frac{d}{2})\kappa +2a_1+\frac{3a_1}{2^{1-\frac{1}{p}}}+\sum _{i=1}^3\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}{\eta }_i}\\ \multicolumn{1}{c}{\le -k(\sum _{i=1}^3{\hat{\sigma }}_i^d+\sum _{i=1}^3{\xi }_i^d+{\tilde{\theta }}_1^d+{\tilde{\theta }}_2^d)+\mathbb{N}}\end{array}$$

(42)

where $k=\max \{\frac{k_3}{4},k_2,{\left(\frac{3}{4}{\iota }_1\right)}^{\frac{d}{2}},{\left(\frac{3}{4}{\iota }_2\right)}^{\frac{d}{2}}\}$ and $\mathbb{N}={\iota }_1{\theta }_1^2+{\iota }_2{\theta }_2^2+2(1-\frac{d}{2})\kappa +2a_1+\frac{3a_1}{2^{1-\frac{1}{p}}}+\sum _{i=1}^3\frac{1}{(2-\frac{1}{p})2^{1-\frac{1}{p}}}{\eta }_i$ .

From equation (20), by employing Lemma A.1 in Qian and Lin ³² and some calculations, we have

$$V\le \lambda (\sum _{i=1}^3({\hat{\sigma }}_i^2+{\xi }_i^2)+{\tilde{\theta }}_1^2+{\tilde{\theta }}_2^2)$$

(43)

where $\lambda =\max \{\frac{1}{2},\frac{1}{2-1/p},\frac{{\gamma }_1}{2},\frac{{\gamma }_2}{2}\}$ . Employing Lemma 2 produces

$$V^{\frac{d}{2}}\le {\lambda }^{\frac{d}{2}}(\sum _{i=1}^3({\hat{\sigma }}_i^d+{\xi }_i^d)+{\tilde{\theta }}_1^d+{\tilde{\theta }}_2^d)$$

(44)

Let $c_1=(k/2){\lambda }^{-\frac{d}{2}}$ . Combining equation (38) with equation (44) leads to

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}+c_1{V}^{\frac{d}{2}}\le -k(\sum _{i=1}^3({\hat{\sigma }}_i^2+{\xi }_i^2)+{\tilde{\theta }}_1^2+{\tilde{\theta }}_2^2)+\mathbb{N}}\\ \multicolumn{1}{c}{+c_1{\lambda }^{\frac{d}{2}}(\sum _{i=1}^3({\hat{\sigma }}_i^2+{\xi }_i^2)+{\tilde{\theta }}_1^2+{\tilde{\theta }}_2^2)}\\ \multicolumn{1}{c}{\le -\frac{k}{2}(\sum _{i=1}^3({\hat{\sigma }}_i^2+{\xi }_i^2)+{\tilde{\theta }}_1^2+{\tilde{\theta }}_2^2)+\mathbb{N}}\end{array}$$

(45)

Then we have

$$\stackrel{·}{V}\le -c_1{V}^{\frac{d}{2}}+\mathbb{N}$$

(46)

Based on Lemma 3.6 in Zhu et al., ³⁸ the system (equations (6) and (7)) is practically finite-time stability. Moreover, the compact set $\mathcal{D}$ is

$$\mathcal{D}=\{\underset{t\to T}{lim}V\le \min \left\{{\left[\frac{\mathbb{N}}{c_1(1-\varpi )}\right]}^{\frac{2}{d}}\right\}\},0<\varpi <1$$

(47)

Hence, the proof is completed.

Remark 6

From equation (47), in order to obtain a good control performance, we can add the value of $c_1$ and $d$ and reduce the value of $\mathbb{N}$ with suitable selection of the design parameters in $c_1,$ $d$ , and $\mathbb{N}$ . Thus, specific control performance and control action are achieved via regulation parameters in practice.

Simulation results

In this section, some simulation and comparison results are given to demonstrate the effectiveness of the obtained controller. The parameter of the spacecraft system is presented as

$$J=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0.63\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0.85\hfill \end{array}\right]\mathrm{kg}.{\mathrm{m}}^2$$

The initial states are selected as $\sigma (0)=(0.50.40.2{)}^T$ and $\mathcal{W}(0)=(0.20.30.4{)}^T\mathrm{rad}/\mathrm{s}$ . Choose $d(t)$ as

$$d\left(t\right)={[0.5\sin \left(t\right)0.4\cos \left(t\right)0.2\sin \left(t\right)]}^T$$

To demonstrate the superiority of the proposed scheme (adaptive finite-time controller), the control method (fixed-time controller plus disturbance observer) in Sun et al. ²⁴ is used for comparison.

The quantization parameters ${\delta }_i$ and ${\alpha }_i$ are chosen as ${\delta }_1=0.3,{\delta }_2=0.4,{\delta }_3=0.3,{\alpha }_1={\alpha }_2={\alpha }_3=0.5$ . Meanwhile, the measurement errors are assumed as ${\varpi }_1=0.1\mathrm{rand}(3,1)$ and ${\varpi }_2=0.1\mathrm{rand}(3,1)$ .

The controller parameters are selected as

$$k_1=4,k_2=1.2,p=\frac{13}{11},a_1=100,{\iota }_1={\iota }_2=0.0001$$

$${\eta }_1={\eta }_2={\eta }_3=0.01,{\gamma }_1={\gamma }_2=0.001,{\mu }_{20}=2,{\mu }_{21}=4$$

The controller parameters in Sun et al. ²⁴ are chosen as

$$k_{11}=5,k_{12}=5,k_{21}=2,k_{22}=1,{\eta }_1={\eta }_2={\eta }_3=0.01$$

$$p=\frac{7}{5},q=\frac{5}{3},{\lambda }_1=5,{\lambda }_2=5,\alpha =100,\gamma =\frac{7}{5}$$

In Figures 2 to 4, the simulation results are presented. Response curves of system states under two controllers are presented in Figures 2 and 3. According to Figures 2 and 3, we find that the proposed control method can obtain better controller performance than the controller in Sun et al. ²⁴ Figure 4 shows the quantized signal $q(u)$ . In Figure 5, the adaptive parameters ${\hat{\theta }}_1$ and ${\hat{\theta }}_2$ are depicted.

Figure 2.

Response curves of attitude $\sigma$ under two controllers: (a) ${\sigma }_1$ , (b) ${\sigma }_2$ , and (c) ${\sigma }_3$ .

Figure 3.

Response curves of attitude velocity $\mathcal{W}$ under two controllers: (a) ${\mathcal{W}}_1$ , (b) ${\mathcal{W}}_2$ , and (c) ${\mathcal{W}}_3$ .

Figure 4.

Curves of quantized input under two controllers: (a) $q(u_1)$ , (b) $q(u_2)$ , and (c) $q(u_3)$ .

Figure 5.

Curves of adaptive parameters: (a) ${\hat{\theta }}_1$ and (b) ${\hat{\theta }}_2$ .

To demonstrate the effectiveness in a practical environment, the band-limit white noises are imposed on the system, response curves of the angular velocity and the attitude are presented in Figures 6 and 7. Comparing Figures 2 and 3 shows that the tracking performance becomes poor. Curves of control inputs are given in Figure 8.

Figure 6.

Response curves of attitude $\sigma$ under two controllers and parameter uncertainties: (a) ${\sigma }_1$ , (b) ${\sigma }_2$ , and (c) ${\sigma }_3$ .

Figure 7.

Response curves of attitude velocity $\mathcal{W}$ under two controllers and parameter uncertainties: (a) ${\mathcal{W}}_1$ , (b) ${\mathcal{W}}_2$ , and (c) ${\mathcal{W}}_3$ .

Figure 8.

Curves of quantized input under two controllers and parameter uncertainties: (a) $q(u_1)$ , (b) $q(u_2)$ , and (c) $q(u_3)$ .

Conclusion and further work

The issue of adaptive quantized FTC for rigid spacecraft with MUs has been studied in this article. By using power technique, an adaptive finite-time controller has been presented, which can ensure that all the signals in the closed-loop systems can converge to a neighborhood of region in finite time. Finally, the effectiveness of the proposed scheme has been demonstrated via simulation results. Further work will focus on the output feedback attitude control for spacecraft system with input and output quantization and MUs.