Observability and observer design for a class of hyperbolic PDEs with van de Pol type boundary conditions

• Published in: Communications in nonlinear science & numerical simulation Vol. 127; p. 107537
• Main Authors: Xiang, Qiaomin, Wu, Ze-Hao, Deng, Feiqi, Wu, Chufen
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2023
• Subjects: Distributed parameter systems; Nonlinear van de Pol type boundary conditions; Observability; Observer design; Nonlinear van de Pol type boundary conditions; Distributed parameter systems; Observability; Observer design
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2023.107537

This paper focuses on observability and observer design for nonlinear complex dynamical systems described by a class of hyperbolic partial differential equations (PDEs) with nonlinear van de Pol type boundary conditions. The systems exhibit complex dynamics due to its imbalance of energy flows. Both the exact observability and approximate observability of the systems with different boundary output measurements are shown by using methods of characteristics and boundary nonlinear reflections. Motivated by the approximate observability of the systems, a PDE state observer by using the boundary displacement measurement only is designed, and a sufficient condition to guarantee the estimation error systems to be exponentially stable is given. Theoretical results are proved rigorously, with some numerical simulations performed to validate the effect of the proposed observer. •Observability and observer design for hyperbolic PDEs with nonlinear van de Pol type boundary conditions are addressed.•Only boundary displacement measurement is required in the design of the PDE state observer.•The methods of characteristics and boundary reflection to analyze convergence of the PDE state observer are developed.•The exponential convergence (not just the available asymptotic one) of the PDE state observer is obtained.