Robust adaptive fuzzy control for a class of nonlinear coupled ODE-beam systems with boundary uncertainty

This paper investigates a robust adaptive fuzzy control problem for a class of nonlinear ordinary differential equations (ODEs) coupled with a beam equation subject to boundary uncertainty. A fuzzy control scheme based on the Takagi-Sugeno (T-S) fuzzy model is employed for the nonlinear ODE subsystem with ODE state feedback. For the beam with boundary uncertainty, a robust adaptive fuzzy boundary control scheme is adopted, where a linear controller via boundary measurements is used to stabilize the beam, and a robust adaptive fuzzy compensator is utilized to counteract the boundary uncertainty. The asymptotic stabilization condition is derived for the nonlinear coupled ODE-beam system by means of the Lyapunov's direct method, which is provided in terms of a set of bilinear matrix inequalities (BMIs). Furthermore, a two-step procedure is presented to solve the BMI feasibility problem by the existing linear matrix inequality (LMI) optimization techniques. Finally, a simulation study is conducted on a flexible spacecraft to demonstrate the effectiveness of the proposed method.