Spatial domain decomposition approach to dynamic compensator design for linear space-varying parabolic MIMO PDEs

• Published in: IET control theory & applications Vol. 14; no. 1; pp. 39 - 51
• Main Author: Wang, Jun-Wei
• Format: Journal Article
• Language: English
• Published: The Institution of Engineering and Technology 01.01.2020
• Subjects: actuators; algebraic linear matrix inequalities; asymptotic stability; closed loop systems; closed‐loop coupled PDEs; compensation; control system synthesis; distributed parameter systems; dynamic compensator design; exponential stabilisation; feedback; linear matrix inequalities; linear space‐varying parabolic MIMO; LMI‐based conditions; Lyapunov methods; multiple‐input–multiple‐output partial differential equations; nonlinear control systems; observers; observer‐based dynamic feedback compensator; observer‐based feedback control technique; partial areas; partial differential equations; periodic boundary conditions; Research Article; spatial domain decomposition approach; subdomain conditions; sufficient conditions; LMI-based conditions; observer-based feedback control technique; partial areas; asymptotic stability; feedback; sufficient conditions; actuators; linear space-varying parabolic MIMO; linear matrix inequalities; distributed parameter systems; observer-based dynamic feedback compensator; closed-loop coupled PDEs; periodic boundary conditions; multiple-input–multiple-output partial differential equations; control system synthesis; exponential stabilisation; closed loop systems; subdomain conditions; spatial domain decomposition approach; algebraic linear matrix inequalities; nonlinear control systems; dynamic compensator design; compensation; partial differential equations; observers; Lyapunov methods
• ISSN: 1751-8644; 1751-8652; 1751-8652
• DOI: 10.1049/iet-cta.2019.0404

This study addresses the problem of dynamic compensator design for exponential stabilisation of linear space-varying parabolic multiple-input–multiple-output (MIMO) partial differential equations (PDEs) subject to periodic boundary conditions. With the aid of the observer-based feedback control technique, an observer-based dynamic feedback compensator, whose implementation requires only a few actuators and sensors active over partial areas of the spatial domain, is constructed such that the resulting closed-loop coupled PDEs is exponentially stable. The spatial domain is divided into multiple subdomains according to the minimum of the actuators' number and the sensors' one. By Lyapunov direct method and two general variants of Poincaré-Wirtinger inequality at each subdomain, sufficient conditions for the existence of such feedback compensator are developed and presented in terms of algebraic linear matrix inequalities (LMIs) in space. Based on the extreme value theorem, LMI-based sufficient and necessary conditions are presented for the feasibility of algebraic LMIs in space. Finally, numerical simulation results are presented to support the proposed design method.