Robust Observer-Based Feedback Control for Lipschitz Singularly Perturbed Systems

Yanyan Wang,Wei Liu

doi:10.1155/2015/585301

Abstract

The observer-based feedback control for singularly perturbed systems (SPSs) with Lipschitz constraint is addressed. A sufficient condition, independent of the perturbation parameter, for a full-order observer is presented in terms of linear matrix inequality (LMI) such that observation error is exponentially stable for all sufficiently small perturbation parameters. Then, for observer-based feedback control, a proper controller is constructed to guarantee the input-to-state stability of the system with regard to the observation error. Considering the convergence of observation error, the stability of the system can be obtained based on the input-to-state stability property. It is shown that the proposed method is simple and easy to operate. Moreover, the upper bound of the small perturbation parameter for stability of systems can be explicitly estimated with a workable computation way. Finally, two numerical examples show the effectiveness of the proposed method.

Highlights

Perturbed systems or two-time-scale systems are usually described by state-space models in which a small parameter multiplies the time derivatives of some of the system states
For observer-based feedback control, we find the feasible solutions of linear matrix inequality (LMI) (24) as follows: 0.3724 0
This paper has presented the observer design and observerbased control for continuous singularly perturbed system with Lipschitz constraint

Summary

Introduction

Perturbed systems or two-time-scale systems are usually described by state-space models in which a small parameter multiplies the time derivatives of some of the system states. Reference [19] considers the robust stability of linear shift-invariant discrete-time singularly perturbed systems; a composite observer-based controller is given such that the closed-loop system is stable. Motivated by the above works, we, in this paper, discuss robust observer-based feedback control for continuous singularly perturbed systems with Lipschitz constraint Such a system has been studied without considering observer in [4]. Compared with the previous results, the newly developed method has the following advantages: (1) is the existence of the observer established, where observer gain matrix can be obtained by solving a simple LMI, and an estimate for the stability bound of observation error is given; (2) a LMI based sufficient condition for input-to-state stability of system is provided, under which the control gain matrix can be solved efficiently and much more complex equation is not involved.

Problem Formulation

Observer Design

Observer-Based Feedback Control

Numerical Examples

Conclusion