Adaptive model-free fault-tolerant control for autonomous underwater vehicles subject to actuator failure

Abstract

This study aims to devise an adaptive model-free fault-tolerant control system based on sliding mode theory for the trajectory tracking of an autonomous underwater vehicle (AUV) equipped with four rotatable thrusters. The proposed control system addresses challenges such as actuator faults, dynamic uncertainty, and time-varying exogenous disturbances. Ensuring the boundedness of the switching gain and achieving uniformly ultimately bounded performance necessitates a priori boundedness of the uncertainty. To address these requirements, a novel control framework is developed by integrating sliding mode and adaptive control techniques for AUV trajectory tracking missions. The effectiveness of the controller is assessed in the presence of partial loss of effectiveness, bias faults, and complete failure of a rotatable actuator. To manage the absence of one actuator, the elimination of column method is introduced into the actuator distribution. The proposed model-free fault-tolerant structure significantly enhances the system reliability. Numerical results validate the efficacy of the proposed methodology in successfully handling the total failure of a single actuator.

Appendix

According to property 1, $\Vert M\Vert ,\Vert C\Vert ,\Vert D\Vert$ and $\Vert G\Vert$ satisfy property 2. Therefore, Eq. (15) is further introduced as:

$$\begin{aligned} & E = \left[ {\begin{array}{*{20}c} e & {\dot{e}} \\ \end{array} } \right]^{T} ,\quad \left\| e \right\| \le \left\| E \right\|\& \left\| {\dot{e}} \right\| \le \left\| E \right\| \\ & \left\| \Xi \right\| \le \overline{c}\left\| {\dot{\eta }} \right\|^{2} + \overline{f}\left\| {\dot{\eta }} \right\| + \overline{g} + \overline{\tau }_{d} + \overline{m}\left\| {\ddot{\eta }_{d} } \right\| + \overline{m}\left\| {k_{1} } \right\|.\left\| {\dot{e}} \right\| \\ & \quad \quad + \overline{m}\left\| \lambda \right\|.\left[ {\left\| {\dot{e}} \right\| + \left\| {k_{1} } \right\|.\left\| e \right\|} \right] + \overline{c}\left[ {\left\| {\dot{e}} \right\| + \left\| {k_{1} } \right\|.\left\| {\dot{e}} \right\| + \left\| {k_{2} } \right\|.\left\| {\int {\left( {\dot{e} + k_{1} e} \right)} } \right\|} \right] \\ & \quad \Rightarrow \left\| \Xi \right\| \le \overline{c}\left\| {\dot{e} + \dot{\eta }_{d} } \right\|^{2} + \overline{g} + \overline{f}\left\| {\dot{e} + \dot{\eta }_{d} } \right\| + \overline{m}\left\| {\ddot{\eta }_{d} } \right\| + \overline{\tau }_{d} + \overline{m}\left\| {k_{1} } \right\|.\left\| {\dot{e}} \right\| + \overline{m}\left\| {k_{2} } \right\|.\left\| {\dot{e}} \right\| \\ & \quad \quad + \overline{m}\left\| {k_{2} } \right\|\left\| {k_{1} } \right\|.\left\| e \right\| + \overline{c}\left\| {\dot{e}} \right\| + \overline{c}\left\| {k_{1} } \right\|.\left\| e \right\| + \overline{c}\left\| {k_{2} } \right\|.\left\| {\int {\left( {\dot{e} + k_{1} e} \right)} } \right\| \\ & \quad \Rightarrow \left\| \Xi \right\| \le \overline{c}\left\| {\dot{e}} \right\|^{2} + 2\overline{c}\left\| {\dot{e}} \right\|.\left\| {\dot{\eta }_{d} } \right\| + \overline{c}\left\| {\dot{\eta }_{d} } \right\|^{2} + \overline{g} + \overline{f}\left\| {\dot{e}} \right\| + \overline{f}\left\| {\dot{\eta }_{d} } \right\| + \overline{m}\left\| {\ddot{\eta }_{d} } \right\| + \overline{\tau }_{d} \\ & \quad \quad + \overline{m}\left\| {k_{1} } \right\|.\left\| {\dot{e}} \right\| + \overline{m}\left\| {k_{2} } \right\|.\left\| {\dot{e}} \right\| + \overline{m}\left\| {k_{2} } \right\|\left\| {k_{1} } \right\|.\left\| e \right\| + \overline{c}\left\| {\dot{e}} \right\| + \overline{c}\left\| {k_{1} } \right\|.\left\| e \right\| + \overline{c}\left\| {k_{2} } \right\|.N \\ \end{aligned}$$

(34)

By defining ${\rm E}={\left[\begin{array}{cc}e& \dot{e}\end{array}\right]}^{T},\Vert e\Vert \le \Vert E\Vert \&\Vert \dot{e}\Vert \le \Vert E\Vert$, Eq. (34) can be rewritten as follows:

$$\begin{aligned} & \left\| \Xi \right\| \le \Omega_{1} \left\| E \right\|^{0} + \Omega_{2} \left\| E \right\|^{1} + \Omega_{3} \left\| E \right\|^{2} \\ & \Omega_{1} = \overline{c}\left\| {\dot{\eta }_{d} } \right\|^{2} + \overline{g} + \overline{f}\left\| {\dot{\eta }_{d} } \right\| + \overline{\tau }_{d} + \overline{m}\left\| {\ddot{\eta }_{d} } \right\| \\ & \Omega_{2} = 2\overline{c}\left\| {\dot{\eta }_{d} } \right\| + \overline{f} + \overline{m}\left\| {k_{2} } \right\| + \overline{m}\left\| {k_{1} } \right\| + \overline{m}\left\| {k_{2} } \right\|\left\| {k_{1} } \right\| + \overline{c} + \overline{c}\left\| {k_{1} } \right\| + \overline{c}\left\| {k_{2} } \right\|.N \\ & \Omega_{3} = \overline{c} \\ \end{aligned}$$

(35)

In Eq. (21), ${\Omega }_{i}$ represents unknown terms, and it is important to note that Assumption 3 does not impose a priori bounds on the lumped term.