Estimation and Synthesis of Reachable Set for Singular Markovian Jump Systems

Yucai Ding,Hui Liu

doi:10.1155/2018/5139840

Abstract

The problems of reachable set estimation and state-feedback controller design are investigated for singular Markovian jump systems with bounded input disturbances. Based on the Lyapunov approach, several new sufficient conditions on state reachable set and output reachable set are derived to ensure the existence of ellipsoids that bound the system states and output, respectively. Moreover, a state-feedback controller is also designed based on the estimated reachable set. The derived sufficient conditions are expressed in terms of linear matrix inequalities. The effectiveness of the proposed results is illustrated by numerical examples.

Highlights

The research on singular systems has attracted significant attention in the past years due to the fact that singular systems can better describe a larger class of physical systems such as robotic systems, electric circuits, and mechanical systems
When singular systems experience abrupt changes in their structures, it is natural to model them as singular Markovian jump systems [1, 2]
The exact shape of reachable sets of a dynamic system is very complex and hard to obtain; for this reason a number of researchers began to turn their attention to the reachable set estimation problem

Summary

Introduction

The research on singular systems has attracted significant attention in the past years due to the fact that singular systems can better describe a larger class of physical systems such as robotic systems, electric circuits, and mechanical systems. In [14] sufficient conditions for the existence of bounding ellipsoids containing the reachable set of continuous-time linear systems with time-varying delays were derived by using the Lyapunov-Razumikhin function. By constructing suitable LyapunovKrasovskii functional, LMI-based sufficient conditions for the existence of controller guaranteeing the ellipsoid bounds as small as possible have been derived for continuous-time delay systems [34] and discrete-time periodic systems [36]. It should be pointed out that the reachable set estimation and synthesis problems of singular Markovian jump systems are much more difficult and challenging than that for nonsingular Markovian jump systems since the ellipsoid containing the reachable set is not directly related to quadratic Lyapunov functions. Throughout this paper, Rn denotes the ndimensional Euclidean space; AT represents the transpose of A; Sym(M) stands for M + MT; X > 0 (

Problem Formulation

Main Results

Numerical Examples

Conclusions