Late‐lumping fuzzy boundary geometric control of nonlinear partial differential systems

• Published in: International journal of robust and nonlinear control Vol. 30; no. 16; pp. 6473 - 6501
• Main Authors: Raab, Sadia, Habbi, Hacene, Maidi, Ahmed
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 10.11.2020; Wiley Subscription Services, Inc
• Subjects: Boundary control; Computer simulation; Control methods; Control stability; Control theory; Controllers; Differential geometry; distributed parameter system; Distributed parameter systems; Fuzzy control; fuzzy modeling; geometric control; late‐lumping; Liapunov direct method; Lumping; Mathematical models; Nonlinear control; Nonlinear systems; Numerical methods; Operators (mathematics); partial differential equation; Partial differential equations; Performance indices; stabilization
• ISSN: 1049-8923; 1099-1239
• DOI: 10.1002/rnc.5108

Summary In this article, a fuzzy boundary geometric controller that stabilizes a class of nonlinear distributed parameter systems (DPSs) is proposed. The design procedure relies on the use of Takagi‐Sugeno (T‐S) type fuzzy partial differential equation (PDE) model, which approximates the dynamical behavior of the nonlinear DPS. The T‐S fuzzy PDE model is constructed through “fuzzy blending” of local linear PDE models of infinite characteristic indexes. This is a challenging task in the design procedure of fuzzy PDE model‐based boundary controller in the framework of the well‐established geometric control theory. To overcome this constraint, it is proposed in this article to resort to the concept of extended operator in order to transform the T‐S fuzzy PDE model with boundary control to an equivalent fuzzy PDE model with punctual control and finite characteristic index. Based on the developed fuzzy model, a fuzzy boundary geometric controller is derived and sufficient conditions of exponential stability of the resulting closed‐loop system are established by employing the Lyapunov direct method. The stabilizing performance of the proposed fuzzy PDE model‐based boundary geometric controller is evaluated on benchmark control problems and compared with other existing control methods via numerical simulations.