Event-triggered sliding mode control for a class of nonlinear systems

ABSTRACT

Event-triggering strategy is one of the real-time control implementation techniques which aims at achieving minimum resource utilisation while ensuring the satisfactory performance of the closed-loop system. In this paper, we address the problem of robust stabilisation for a class of nonlinear systems subject to external disturbances using sliding mode control (SMC) by event-triggering scheme. An event-triggering scheme is developed for SMC to ensure the sliding trajectory remains confined in the vicinity of sliding manifold. The event-triggered SMC brings the sliding mode in the system and thus the steady-state trajectories of the system also remain bounded within a predesigned region in the presence of disturbances. The design of event parameters is also given considering the practical constraints on control execution. We show that the next triggering instant is larger than its immediate past triggering instant by a given positive constant. The analysis is also presented with taking delay into account in the control updates. An upper bound for delay is calculated to ensure stability of the system. It is shown that with delay steady-state bound of the system is increased than that of the case without delay. However, the system trajectories remain bounded in the case of delay, so stability is ensured. The performance of this event-triggered SMC is demonstrated through a numerical simulation.

Keywords:

• Nonlinear systems
• event-triggered control
• sliding mode control
• robust stabilisation

Acknowledgments

The authors would like to thank the anonymous reviewers for their comments and suggestions which helped a lot in improving this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Based on the Assumptions 2.1 and 2.2, it follows immediately that f(x) is also Lipschitz with Lipschitz constant L = L₁ + 2L₂ + ‖B₁‖ for any $\mathcal{D}\ni \xi =(z,y)$ and is shown as $$\begin{array}{cc}\hfill \parallel f\left({\xi }_1\right)-f\left({\xi }_2\right)\parallel & =\parallel f(z_1,y_1)-f(z_2,y_2)\parallel \hfill \\ & \le \parallel f_1\left(z_1\right)-f_1\left(z_2\right)\parallel +\parallel B_1(y_1-y_2)\parallel \hfill \\ & +|f_2(z_1,y_1)-f_2(z_2,y_2)|\hfill \\ & \le L_1\parallel z_1-z_2\parallel +\parallel B_1\parallel |y_1-y_2|\hfill \\ & +L_2\parallel z_1-z_2\parallel +L_2|y_1-y_2|\hfill \\ & =(L_1+L_2)\parallel z_1-z_2\parallel \hfill \\ & +(\parallel B_1\parallel +L_2\left)\right|y_1-y_2|\hfill \\ & \le (L_1+L_2)\parallel {\xi }_1-{\xi }_2\parallel \hfill \\ & +(\parallel B_1\parallel +L_2)\parallel {\xi }_1-{\xi }_2\parallel \hfill \\ & =(L_1+2L_2+\parallel B_1\parallel )\parallel {\xi }_1-{\xi }_2\parallel \hfill \\ & =L\parallel {\xi }_1-{\xi }_2\parallel .\hfill \end{array}$$