Robust fixed-time attitude stabilization control of flexible spacecraft with actuator uncertainty

Abstract

A robust fixed-time control framework is presented to stabilize flexible spacecraft’s attitude system with external disturbance, uncertain parameters of inertia, and actuator uncertainty. As a stepping stone, a nonlinear system having faster fixed-time convergence property is preliminarily proposed by introducing a time-varying gain into the conventional fixed-time stability method. This gain improves the convergence rate. Then, a fixed-time observer is proposed to estimate the uncertain torque induced by disturbance, uncertain parameters of inertia, and actuator uncertainty. Fixed-time stability is ensured for the estimation error. Using this estimated knowledge and the full-states’ measurements, a nonsingular terminal sliding controller is finally synthesized. This is achieved via a nonsingular and faster terminal sliding surface with faster convergence rate. The closed-loop attitude stabilization system is proved to be fixed-time stable with the convergence time independent of initial states. The attitude stabilization performance is robust to disturbance and uncertainties in inertia and actuators. Simulation results are also shown to validate the attitude stabilization performance of this control approach.

Appendices

Appendix A (Proof of Lemma 1)

Defining a new variable $W = y^{1 - pk}$, it can be obtained from (9) that

$$\begin{aligned} \begin{aligned} \dot{W}&= - \left( 1 - pk\right) y^{-pk}\left( \xi \left( y\right) ^{\frac{1}{k}}\alpha y^p + \xi \left( y\right) ^{\frac{1}{k}}\beta y^\lambda \right) ^k\\&= - \left( 1 - pk\right) \left( \xi \left( y\right) ^{\frac{1}{k}}\alpha + \xi \left( y\right) ^{\frac{1}{k}}\beta W^\eta \right) ^k \end{aligned} \end{aligned}$$

(28)

where $\eta = \frac{\lambda - p}{1 - pk}$.

Since $1-pk>0$ and $\xi \left( y\right) >1$, it follows from (28) that

$$\begin{aligned} \begin{aligned} \dot{W} \le - \left( 1 - pk\right) \left( \alpha + \beta W^\eta \right) ^k \end{aligned} \end{aligned}$$

(29)

Applying the result in [39] and the comparison principle [44], it can be proved from (29) that W is fixed-time stable. Moreover, solving (28), one can get the settling time as

$$\begin{aligned} \begin{aligned} T_1&= \frac{1}{\left( 1 - pk\right) }\int _0^{W_0} \frac{1}{\left( \xi \left( y\right) ^{\frac{1}{k}}\alpha + \xi \left( y\right) ^{\frac{1}{k}}\beta W^\eta \right) ^k}dW\\&= \frac{1}{\left( 1 - pk\right) }\left( \int _1^{W_0} \frac{1}{\xi \left( y\right) \left( \alpha + \beta W^{\bar{\eta }}\right) ^k}dW\right. \\&\quad +\, \, \left. \int _0^1 \frac{1}{\xi \left( y\right) \left( \alpha + \beta W^{\frac{1}{k}}\right) ^k}dW\right) \end{aligned} \end{aligned}$$

(30)

where $\bar{\eta } = \frac{q - p}{1 - pk}$ and $W_0=(y(0))^{1 - pk}$.

If $\xi (\varvec{y}) = 1$, then one has

$$\begin{aligned} \begin{aligned} T_1^\prime&= \frac{1}{\left( 1 - pk\right) }\left( \int _1^{W_0} \frac{1}{\left( \alpha + \beta W^{\bar{\eta }}\right) ^k}dW\right. \\&\quad +\, \, \left. \int _0^1 \frac{1}{\left( \alpha + \beta W^{1/k}\right) ^k}dW\right) \end{aligned} \end{aligned}$$

(31)

Since $1 \le \xi (y) \le a$, then $1/a \le 1/\xi (y) \le 1$. Hence, for all $W_0$, it is concluded that

$$\begin{aligned} T_1 < T_1^\prime \end{aligned}$$

(32)

On other hand, $T_1^\prime$ is also the settling time of the fixed-time system given in [39]. To this end, one can prove that the settling time provided by the proposed system (9) is less than [39]. The convergence rate of the system (9) is faster than [39].

From (30), it be proved that $T_1^\prime$ is bounded as

$$\begin{aligned} \begin{aligned} T_1^\prime&\le \frac{1}{\left( 1 - pk\right) }\left( \int _1^{W_0} \frac{1}{\beta ^kW^{\bar{\eta }k}}dW + \int _0^1 \frac{1}{\alpha ^k + \beta ^kW}dW\right) \\&\le \frac{1}{\left( 1 - pk\right) }\left( \frac{1 - W_0^{1 - \bar{\eta }k}}{\beta ^k\left( \bar{\eta }k - 1\right) } + \frac{1}{\beta ^k}\ln \left( 1 + \left( \beta /\alpha \right) ^k\right) \right) \end{aligned} \end{aligned}$$

(33)

Since $\bar{\eta }k > 1$ and $W_0 > 0$, one has

$$\begin{aligned} T_1^\prime \le \frac{1}{\beta ^k\left( qk - 1\right) } + \frac{1}{\beta ^k\left( 1 - pk\right) }\ln \left( 1 + \left( \beta /\alpha \right) ^k\right) \end{aligned}$$

(34)

which does not depend on the initial condition.

Appendix B (Proof of Theorem 1)

It is obtained from (13) and (14) that the estimation error of the observer satisfies

$$\begin{aligned} \begin{aligned} \dot{\varvec{e}}&= \dot{\varvec{z}} - \dot{\hat{\varvec{z}}} = \dot{\varvec{z}} + l_2l_3\hat{\varvec{z}} - \frac{1}{l_2}\dot{\varvec{y}} - l_3\varvec{y}\\&\quad -\, \, \left( \xi (\varvec{e})^{1/k_1}\alpha _1\mathrm{{sig}}(\varvec{e})^{\frac{2p_1k_1 - 1}{k_1}}\right. \\&\quad +\, \, \left. \xi (\varvec{e})^{1/k_1}\beta _1\mathrm{{sig}}(\varvec{e})^{\frac{2\lambda _1k_1 - 1}{k_1}}\right) ^{k_1}\\&= - l_2l_3\varvec{e} - \left( \xi (\varvec{e})^{1/k_1}\alpha _1\mathrm{{sig}}(\varvec{e})^{\frac{2p_1k_1 - 1}{k_1}}\right. \\&\quad +\, \, \left. \xi (\varvec{e})^{1/k_1}\beta _1\mathrm{{sig}}(\varvec{e})^{\frac{2\lambda _1k_1 - 1}{k_1}}\right) ^{k_1} \end{aligned} \end{aligned}$$

(35)

Define a Lyapunov candidate function as $V_1 = 0.5\varvec{e}^\mathrm{{T}}\varvec{e}$, it leaves its time derivative as

$$\begin{aligned} \begin{aligned} \dot{V}_1&= \varvec{e}^\mathrm{{T}}\dot{\varvec{e}} \le - \varvec{e}^\mathrm{{T}}\left( \xi (\varvec{e})^{1/k_1}\alpha _1\mathrm{{sig}}(\varvec{e})^{\frac{2p_1k_1 - 1}{k_1}}\right. \\&\quad +\, \, \left. \xi (\varvec{e})^{1/k_1}\beta _1\mathrm{{sig}}(\varvec{e})^{\frac{2\lambda _1k_1 - 1}{k_1}}\right) ^{k_1}\\&\le - \sum \limits _{i = 1}^3 \left( \xi (\varvec{e})^{1/k_1}\alpha _1\left| e_i \right| ^{\frac{2p_1k_1 - 1}{k_1} + \frac{1}{k_1}}\right. \\&\quad +\, \, \left. \xi (\varvec{e})^{1/k_1}\beta _1\left| e_i \right| ^{\frac{2\lambda _1k_1 - 1}{k_1} + \frac{1}{k_1}} \right) ^{k_1}\\&\le - \left( \xi ^{1/k_1}\mu _1V_1^{p_1} + \xi ^{1/k_1}\mu _2V_1^{\lambda _1}\right) ^{k_1} \end{aligned} \end{aligned}$$

(36)

Applying Lemma 1 and the comparison principle [44], it is concluded that $V(\varvec{e}) \equiv 0$ is met for $t \ge T_e$, where the settling time $T_e$ satisfies (15).

Appendix C (Proof of Theorem 2)

From (16) and (17), it follows that

$$\begin{aligned} \begin{aligned} \varvec{d}_e&= \varvec{d}_{l} - l_1\varvec{x}_2 + \varvec{M}^{-1}(\varvec{x}_1)\left( \varvec{C}_1(\varvec{x}_1 \text{, } \varvec{x}_2)\varvec{x}_2\right. \\&\quad +\, \, \left. \varvec{C}_2(\varvec{x}_1 \text{, } \varvec{x}_2)\right) - \hat{d}_{l} + l_1\varvec{x}_2\\&\quad -\, \, \varvec{M}^{-1}(\varvec{x}_1)\left( \varvec{C}_1(\varvec{x}_1 \text{, } \varvec{x}_2)\varvec{x}_2 + \varvec{C}_2(\varvec{x}_1 \text{, } \varvec{x}_2)\right) \\&= \varvec{d}_{l} - \hat{\varvec{d}}_{l} \end{aligned} \end{aligned}$$

(37)

Substituting (17) in (37) gives

$$\begin{aligned} \begin{aligned} \varvec{d}_e&= \varvec{d}_{l} - \frac{l_1l_2\hat{\varvec{z}} + \dot{\varvec{y}}}{l_2}\\&= \varvec{d}_{l} - \frac{l_1l_2\hat{\varvec{z}} - l_1l_2\varvec{z} + l_2\varvec{d}_{l}}{l_2} = l_1\varvec{e} \end{aligned} \end{aligned}$$

(38)

Because $\varvec{e}(t) \equiv \varvec{0}$ is achieved in Theorem 1 for $t \ge T_e$, $\varvec{d}_e(t) = \varvec{0}$ is achieved for $t \ge T_e$. It is inferred that $\varvec{d}$ is estimated utilizing $\varvec{d}_\mathrm{est}$ after $T_e$.

Appendix D (Proof of Theorem 3)

When $\varvec{S} = \varvec{0}$ is reached, from (19), one has

$$\begin{aligned} \begin{aligned} \dot{x}_{1i}&= - \left( h(x_{1i})\right) ^{1/\gamma }\mathrm{{sig}}^{1/\gamma }\left( x_{1i}\right) \\&= - \left( \xi (\varvec{x}_1)^{1/k_2}\alpha _1\left| x_{1i} \right| ^{p_2 - 1/k_2\gamma }\right. \\&\quad \,\, -\, \, \left. \xi (\varvec{x}_1)^{1/k_2}\beta _2\left| x_{1i} \right| ^{\lambda _2 - 1/k_2\gamma }\right) ^{k_2}\mathrm{{sig}}^{1/\gamma }\left( x_{1i}\right) \\&= - \left( \xi (\varvec{x}_1)^{1/k_2}\alpha _2\left| x_{1i} \right| ^{p_2}\right. \\&\quad \,\, -\, \, \left. \xi (\varvec{x}_1)^{1/k_2}\beta _2\left| x_{1i} \right| ^{\lambda _2}\right) ^{k_2}\mathrm{{sgn}}\left( x_{1i}\right) \end{aligned} \end{aligned}$$

(39)

Defining a new variable $\Xi _i = \left| x_{1i} \right| ^{1 - p_2k_2}$, (39) is expressed as

$$\begin{aligned} \begin{aligned} \dot{\Xi }_i&= - \left( 1 - p_2k_2\right) \dot{x}_{1i}\left| x_{1i} \right| ^{-p_2k_2}\mathrm{{sgn}}(x_{1i})\\&= - \left( 1 - p_2k_2\right) \left| x_{1i} \right| ^{-p_2k_2}\left( \xi (\varvec{x}_1)^{1/k_2}\alpha _2\left| x_{1i} \right| ^{p_2}\right. \\&\quad \, +\, \, \left. \xi (\varvec{x}_1)^{1/k_2}\beta _2\left| x_{1i} \right| ^{\lambda _2}\right) ^{k_2}\\&= - \left( 1 - p_2k_2\right) \left( \xi (\varvec{x}_1)^{\frac{1}{k_2}}{\alpha _2} + \xi (\varvec{x}_1)^{\frac{1}{k_2}}\beta _2\Xi _i^{\eta _2}\right) ^{k_2} \end{aligned} \end{aligned}$$

(40)

where $\eta _2 = \frac{\lambda _2 - p_2}{1 - p_2k_2}$. Similar to Lemma 1, the system state converges to zero after a fixed time given by (21).

Appendix E (Proof of Theorem 4)

Select another Lyapunov candidate function $V_s = \varvec{S}^\mathrm{{T}}\varvec{S}$. Applying (8), one can calculate the time derivative of $V_s$ as

$$\begin{aligned} \begin{aligned} \dot{V}_s&= 2\varvec{S}^\mathrm{{T}}\left( \dot{\varvec{H}}(\varvec{x}_1)\varvec{x}_1 + \varvec{H}(\varvec{x}_1)\dot{\varvec{x}}_1 \right. \\&\quad \quad +\, \gamma \mathrm{{diag}}\left( \left| x_{2i} \right| ^{\gamma - 1}\right) \\&\quad \times \, \left. \left( - \varvec{M}^{-1}(\varvec{x}_1)\left( \varvec{C}_1(\varvec{x}_1 \text{, } \varvec{x}_2)\varvec{x}_2 \right. \right. \right. \\&\quad \quad \left. \left. \left. +\, \varvec{C}_2(\varvec{x}_1 \text{, } \varvec{x}_2)\right) \right. \right. \\&\quad +\, \, \left. \left. \varvec{u}(\varvec{x}_1) + \varvec{d}(\varvec{x}_1,\varvec{\chi },\varvec{\omega })\right) \right) \end{aligned} \end{aligned}$$

(41)

Substituting the controller (23) into (41) yields

$$\begin{aligned} \begin{aligned} \dot{V}_s&= \frac{2}{\rho _0}\varvec{S}^\mathrm{{T}}\mathrm{{diag}}\left( \mu _\sigma \left( \left| x_{2i}\right| ^{\gamma - 1}\right) \right) \\&\quad \times \, \left( \xi ^{1/k_3}\alpha _3\mathrm{{sig}}(\varvec{S})^{(2p_3k_3 - 1)/k_3}\right. \\&\quad +\, \, \left. \xi ^{1/k_3}\beta _3\mathrm{{sig}}(\varvec{S})^{(2\lambda _3k_3 - 1)/k_3}\right) ^{k_3}\\&\quad +\, \, \gamma \varvec{S}^\mathrm{{T}}\mathrm{{diag}}\left( \left| x_{2i} \right| ^{\gamma - 1}\right) \left( \varvec{d} - \varvec{d}_\mathrm{est}\right) \end{aligned} \end{aligned}$$

(42)

Since $\varvec{d}_e = \varvec{d} - \varvec{d}_\mathrm{est} = \varvec{0}$ for $t > T_e$, (42) can be simplified as

$$\begin{aligned} \begin{aligned} \dot{V}_s&= \frac{2}{\rho _0}\varvec{S}^\mathrm{{T}}\mathrm{{diag}}\left( \mu _\sigma \left( \left| x_{2i} \right| ^{\gamma - 1}\right) \right) \\&\quad \times \, \left( \xi ^{\frac{1}{k_3}}\alpha _3\mathrm{{sig}}(\varvec{S})^{(2p_3k_3 - 1)/k_3}\right. \\&\quad +\, \, \left. \xi ^{\frac{1}{k_3}}\beta _3\mathrm{{sig}}(\varvec{S})^{(2\lambda _3k_3 - 1)/k_3}\right) ^{k_3}\\&{\le } {-} \sum \limits _{i = 1}^3 \left( \xi ^{\frac{1}{k_3}}\alpha _3\rho _0^{-\frac{1}{k_3}}\mu _\sigma ^{\frac{1}{k_3}}\left( \left| x_{2i} \right| ^{\gamma {-} 1}\right) \left| S_i \right| ^{\frac{2p_3k_3 {-} 1}{k_3} {+} \frac{1}{k_3}}\right. \\&+\, \, \left. \xi ^{\frac{1}{k_3}}\beta _3\rho _0^{-\frac{1}{k_3}}\mu _\sigma ^{\frac{1}{k_3}}\left( \left| x_{2i}\right| ^{\gamma - 1}\right) \left| S_i \right| ^{\frac{2\lambda _3k_3 - 1}{k_3} + \frac{1}{k_3}}\right) ^{k_3}\\&\le - \left( \xi ^{\frac{1}{k_3}}\mu _3V^{p_3} + \xi ^{1/k_3}\mu _4V^{\lambda _3}\right) ^{k_3} \end{aligned} \end{aligned}$$

(43)

where $\mu _3 = \alpha _3\rho _0^{-1/k_3}\mu _\sigma ^{1/k_3}\left( \left| x_{2i} \right| ^{\gamma - 1}\right)$ and $\mu _4 = \beta _3\rho _0^{-1/k_3}$$\mu _\sigma ^{1/k_3}\left( \left| x_{2i} \right| ^{\gamma - 1}\right)$. Applying the comparison principle [44] and the result in Lemma 1, it is ready to conclude that $V_s \equiv 0$ after the settling time $T_1$ satisfying (26).

After reaching the sliding surface $\varvec{S}=\varvec{0}$, it can be obtained from Theorem 2 that the states will be zero after the settling time $T_s$. Then, one can prove that the attitude Euler angles and the rotation velocity are fixed-time stable with the settling time $T_c$ satisfying $T_{c}<T_{s}+T_{1}$ regardless any initial states.