RobustH∞Control for a Class of Nonlinear Distributed Parameter Systems via Proportional-Spatial Derivative Control Approach

Cheng-Dong Yang,Jun-Wei Wang,Jianlong Qiu

doi:10.1155/2014/631071

Abstract

This paper addresses the problem of robustH∞control design via the proportional-spatial derivative (P-sD) control approach for a class of nonlinear distributed parameter systems modeled by semilinear parabolic partial differential equations (PDEs). By using the Lyapunov direct method and the technique of integration by parts, a simple linear matrix inequality (LMI) based design method of the robustH∞P-sD controller is developed such that the closed-loop PDE system is exponentially stable with a given decay rate and a prescribedH∞performance of disturbance attenuation. Moreover, a suboptimalH∞controller is proposed to minimize the attenuation level for a given decay rate. The proposed method is successfully employed to address the control problem of the FitzHugh-Nagumo (FHN) equation, and the achieved simulation results show its effectiveness.

Highlights

A significant research area that has received a lot of attention over the past few decades is the control design for distributed parameter systems (DPSs) modeled by parabolic partial differential equations (PDEs)
Different from the spatial differential linear matrix inequalities (SDLMIs)-based control designs in [18, 19], the main result of this study is presented in terms of standard linear matrix inequalities (LMIs), which can be directly verified via the existing convex optimization techniques [20, 21]
We have addressed the problem of robust H∞ proportionalspatial derivative (P-sD) state-feedback controller design for a class of semilinear parabolic PDE systems with external disturbances

Summary

Introduction

A significant research area that has received a lot of attention over the past few decades is the control design for distributed parameter systems (DPSs) modeled by parabolic partial differential equations (PDEs). Fridman and Orlov [12] have presented exponential stabilization with H∞ performance in terms of linear matrix inequalities (LMIs) for uncertain semilinear parabolic and hyperbolic systems via a robust collocated static output feedback boundary controller These results [9,10,11,12] are only applicable for boundary control PDE systems. Notice that the results reported in [15,16,17,18,19] are presented in terms of spatial differential linear matrix inequalities (SDLMIs), which can be only approximately solved on the basis of standard finite difference method and the existing convex optimization techniques [20, 21] Despite these efforts, to the best of the authors’ knowledge, there are still few results on the robust H∞ control design via the original PDE model of semi-linear parabolic PDE systems with external disturbances, which motivates this study.

Preliminaries and Problem Formulation

Simulation Study on the FHN Equation

Findings

Conclusions