Exponential Estimates and Stabilization of Discrete‐Time Singular Time‐Delay Systems Subject to Actuator Saturation

Jinxing Lin

doi:10.1155/2012/414373

Abstract

This paper is concerned with exponential estimates and stabilization of a class of discrete‐time singular systems with time‐varying state delays and saturating actuators. By constructing a decay‐rate‐dependent Lyapunov‐Krasovskii function and utilizing the slow‐fast decomposition technique, an exponential admissibility condition, which not only guarantees the regularity, causality, and exponential stability of the unforced system but also gives the corresponding estimates of decay rate and decay coefficient, is derived in terms of linear matrix inequalities (LMIs). Under the proposed condition, the exponential stabilization problem of discrete‐time singular time‐delay systems subject actuator saturation is solved by designing a stabilizing state feedback controller and determining an associated set of safe initial conditions, for which the local exponential stability of the saturated closed‐loop system is guaranteed. Two numerical examples are provided to illustrate the effectiveness of the proposed results.

Highlights

Singular time-delay systems STDSs arise naturally in many engineering fields such as electric networks, chemical processes, lossless transmission lines, and so forth 1
A STDS is a mixture of delay differential equations and delay difference equations; such a complex nature of STDS leads to abundant dynamics, for example, non-strictly proper transcendental equations, irregularity, impulses or non-causality
In 16, 17, an exponential estimates approach for SSTDs was presented by employing the graph theory to establish an explicit expression of the state variables of fast subsystem in terms of those of slow subsystem and the initial conditions, which allows to prove the exponential stability of the fast subsystem

Summary

Introduction

Singular time-delay systems STDSs arise naturally in many engineering fields such as electric networks, chemical processes, lossless transmission lines, and so forth 1. In 10, , the STDS was decomposed into slow differential and fast algebraic subsystems and the exponential stability of the slow subsystem was proved by using the Lyapunov method. The solutions of the fast subsystem was bounded by an exponential term using a function inequality. This approach cannot give an estimate of the convergence rate of the system. To overcome this difficulty, Shu and Lam and Lin et al adopted the Lyapunov-Krasovskii function method 14, 15 and some improvements have been obtained. The first aim is to develop effective approach to give the exponential estimates of discrete-time STDSs

Objectives

Results

Conclusion