In-Domain Control of a Heat Equation: An Approach Combining Zero-Dynamics Inverse and Differential Flatness

Jun Zheng,Guchuan Zhu

doi:10.1155/2015/187284

Abstract

This paper addresses the set-point control problem of a one-dimensional heat equation with in-domain actuation. The proposed scheme is based on the framework of zero-dynamics inverse combined with flat system control. Moreover, the set-point control is cast into a motion planning problem of a multiple-input, multiple-output system, which is solved by a Green’s function-based reference trajectory decomposition. The validity of the proposed method is assessed through the analysis of the invertibility of the map generated by Green’s function and the convergence of the regulation error. The performance of the developed control scheme and the viability of the proposed approach are confirmed by numerical simulation of a representative system.

Highlights

Control of parabolic partial differential equations (PDEs) is a long-standing problem in PDE control theory and practice
The control scheme developed in this paper is based-on the framework of zero-dynamics inverse (ZDI), which was introduced by Byrnes and Gilliam in [17] and has been exploited and developed in a series of works
The implementation of such control schemes requires resolving the corresponding zero-dynamics, which may be very difficult for generic regulation problems, such as set-point control considered in the present work

Summary

Introduction

Control of parabolic partial differential equations (PDEs) is a long-standing problem in PDE control theory and practice. The control scheme developed in this paper is based-on the framework of zero-dynamics inverse (ZDI), which was introduced by Byrnes and Gilliam in [17] and has been exploited and developed in a series of works (see, e.g., [18] and the references therein) It is pointed out in [19] that “for certain boundary control systems it is very easy to model the system’s zero dynamics, which, in turn, provides a simple systematic methodology for solving certain problems of output regulation.”. The implementation of such control schemes requires resolving the corresponding zero-dynamics, which may be very difficult for generic regulation problems, such as set-point control considered in the present work To overcome this difficulty, we resort to the theory of flat systems [13, 21].

Problem Setting

Control Design and Implementation

Motion Planning

Simulation Study

Conclusion