Observer-based piecewise control of reaction–diffusion systems with the non-collocated output feedback

• Published in: Journal of applied mathematics & computing Vol. 69; no. 6; pp. 4187 - 4211
• Main Authors: Zhong, Jiaqi, Feng, Yan, Chen, Xiaolei, Zeng, Cheng
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2023; Springer Nature B.V
• Subjects: Actuators; Computational Mathematics and Numerical Analysis; Controllers; Differential calculus; Liapunov direct method; Liapunov functions; Linear matrix inequalities; Lipschitz condition; Mathematical analysis; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Nonlinear control; Nonlinear differential equations; Original Research; Output feedback; Partial differential equations; Theory of Computation; 93D05; Nonlinear reaction–diffusion systems; Piecewise stabilization; 35K57; 93C41; Non-collocated output feedback; Observer-based control
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-023-01921-4

This paper mainly deals with an observer-based control problem for the nonlinear reaction–diffusion systems subject to non-collocated output feedback. Different with the existing non-collocated controllers, the proposed methodology constructs a more general Lyapunov function with segmented spatial regions to guarantee the piecewise stabilization of nonlinear partial differential equations (PDEs) with multi-spatiotemporal states. The sufficient linear matrix inequalities (LMIs) condition is first derived by combining the improved Lyapunov direct method, Lipschitz condition, mean value theorem of integrals and a variant of Wirtinger’s inequality to obtain the local gain matrices of collocated control. Further, a nonlinear PDEs observer-based piecewise controller is proposed to overcome the non-collocated distribution between finite piecewise actuators and sensors. Finally, a comparison simulation for the FitzHugh–Nagumo (FHN) system is presented to demonstrate the effectiveness and superiority of proposed non-collocated piecewise controller.