Robust Adaptive Sliding Mode Control of Markovian Jump Systems with Uncertain Mode-dependent Time-varying Delays and Partly Unknown Transition Probabilities

TL;DR

This paper introduces a robust adaptive sliding mode control approach for Markovian jump systems with mode-dependent delays and partly unknown transition probabilities, ensuring stability and finite-time convergence.

Contribution

It develops a unified LMI-based design method for sliding mode control applicable to systems with fully or partially unknown transition probabilities and unknown delays.

Findings

Proposed delay-dependent stability conditions via LMIs.

Designed a sliding mode controller ensuring finite-time convergence.

Validated effectiveness through numerical examples.

Abstract

This paper deals with the problems of stochastic stability and sliding mode control for a class of continuous-time Markovian jump systems with mode-dependent time-varying delays and partly unknown transition probabilities. The design method is general enough to cover a wide spectrum of systems from those with completely known transition probability rates to those with completely unknown transition probability rates. Based on some mode-dependent Lyapunov-Krasovski functionals and making use of the free-connection weighting matrices, new delay-dependent conditions guaranteeing the existence of linear switching surfaces and the stochastic stability of sliding mode dynamics are derived in terms of linear matrix inequalities (LMIs). Then, a sliding mode controller is designed such that the resulted closed-loop system's trajectories converge to predefined sliding surfaces in a finite time and remain there for all subsequent times. This paper also proposes an adaptive sliding mode controller design method which applies to cases in which mode-dependent time-varying delays are unknown. All the conditions obtained in this paper are in terms of LMI feasibility problems. Numerical examples are given to illustrate the effectiveness of the proposed methods.