Adaptive Fuzzy Boundary Observer Design for Uncertain Linear Coupled Hyperbolic Partial Differential Equation Systems

Joint uncertainties and state estimation of a class of linearly coupled hyperbolic partial differential equation systems in the presence of unstructured and structured uncertainties are studied in this article. For unstructured uncertainties that are completely unknown, by employing Takagi–Sugeno fuzzy logic systems to approximate the unstructured uncertainties, a novel adaptive fuzzy boundary observer is developed to estimate both unknown system states as well as unknown weights in the fuzzy logic system, and the estimation errors are ultimately bounded. Therein, in the design of the proposed observer, a set of swapping filters and an infinite dimensional backstepping technique are combined. On the other hand, for structured uncertainties that can be described in a concrete parameterized form, the proposed method can easily achieve the exact estimation of weights and states to their true values. The rigorous proof is provided to show that the ultimately bounded estimation errors for the case of unstructured uncertainties and the exponential convergent estimation errors for the case of structured uncertainties can be realized. Finally, three illustrative simulations are carried out to show the feasibility and effectiveness of the developed methods in this article.