A unified Lyapunov-based design for a dynamic compensator of linear parabolic MIMO PDEs

Abstract

This paper addresses the problem of dynamic compensator design for exponential stabilisation of linear parabolic partial differential equations (PDEs) with multiple actuation control inputs and multiple non-collocated observation outputs. Both in-domain control and boundary control are considered. A new observer-based dynamic compensator is constructed by the non-collocated observation outputs such that the resulting closed-loop coupled PDEs are exponentially stable. By constructing a Lyapunov function candidate and using Poincaré–Wirtinger inequality's variants, a sufficient condition for the existence of such dynamic compensator is presented in terms of standard linear matrix inequalities (LMIs). The closed-loop well-posedness analysis result is also established by the method of $C_0$-semigroup and its perturbations by bounded/unbounded linear operators. Finally, numerical simulation results are presented to support the proposed design method.

Keywords:

• Distributed parameter systems
• non-collocated control and observation
• Lyapunov direct method
• observer-based feedback control

Acknowledgments

The author gratefully acknowledges the helpful comments and suggestions from the Editor and Anonymous Reviewers, which have improved the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported in part by the Beijing Natural Science Foundation under Grant 4192037, and in part by the National Natural Science Foundation of China (NSFC) under Grant 61403026.