Hybrid Robust Boundary and Fuzzy Control for Disturbance Attenuation of Nonlinear Coupled ODE-Beam Systems With Application to a Flexible Spacecraft

This paper introduces a hybrid robust boundary and fuzzy control design for disturbance attenuation of a class of coupled systems described by nonlinear ordinary differential equations (ODEs) and two nonlinear beam equations. Initially, a Takagi–Sugeno (T–S) model is employed to exactly represent the nonlinear ODE subsystem. Then, a fuzzy controller is designed for the ODE subsystem based on the T–S fuzzy model, and a robust boundary controller via beam boundary measurements is proposed for the nonlinear beam subsystem. Such a hybrid robust boundary and fuzzy controller is developed in terms of a set of space-dependent bilinear matrix inequalities (BMIs) by Lyapunov's direct method, which can exponentially stabilize the coupled system in the absence of disturbances and achieve an prescribed
$H_{\infty }$
performance of disturbance attenuation in the presence of disturbances. Furthermore, in order to make the level of disturbance attenuation as small as possible, a suboptimal
$H_{\infty }$
control problem is formulated as a BMI optimization problem. A two-step procedure is subsequently presented to solve this BMI optimization problem by the existing linear matrix inequality optimization techniques. Finally, the proposed control method is applied to the control of a flexible spacecraft to illustrate its effectiveness.