Fuzzy Observer Based Control for Nonlinear Coupled Hyperbolic PDE-ODE Systems

In this work, the fuzzy observer-based control problem is investigated for a class of nonlinear coupled systems, which consists of a hyperbolic partial differential equation (PDE) containing nonlinearities and a nonlinear ordinary differential equation (ODE). The nonlinear coupled system is represented as a Takagi-Sugeno (T-S) fuzzy coupled hyperbolic PDE-ODE model. Based on the T-S fuzzy model, a novel Lyapunov functional approach is proposed to design a fuzzy observer based control strategy. More specifically, a fuzzy observer is presented to estimate the state variables of the fuzzy coupled PDE-ODE system with the measurements of the PDE, and the exponential convergence of the observer error is proved. Then, a fuzzy controller is given utilizing the estimated states as feedback variables, and it is proved that the evolution profiles of the PDE and the trajectory of the ODE in the closed-loop fuzzy system converge exponentially to the desired values, respectively. The sufficient existence conditions of the fuzzy observer based controller are formulated in terms of a set of space differential linear matrix inequalities (SDLMIs). A recursive algorithm based on the finite-difference approximation and the linear matrix inequality techniques are provided to solve the SDLMIs. Finally, the results are applied to case study of a predator-prey system, and the simulations are performed to illustrate the effectiveness of the proposed observer based control law.