Exponentially stabilizing fuzzy controller design for a nonlinear ODE-beam cascaded system and its application to flexible air-breathing hypersonic vehicle

Abstract

This paper addresses fuzzy control design for a class of nonlinear systems which are described by nonlinear ordinary differential equations (ODEs) cascaded with an Euler-Bernoulli beam (EBB) equation. Two design difficulties are involved in the control design addressed in this paper. The first one is caused by the EBB equation whose spatiotemporal dynamics is affected by the output of the nonlinear ODE subsystem through its differential equation rather than boundary conditions. A state differential transformation is introduced for the EBB equation to bring the output of the ODE subsystem in the differential equation to its boundary conditions. The second one comes from the nonlinear ODE subsystem. To overcome this difficulty, it is assumed that an exact Takagi-Sugeno (T-S) fuzzy ODE model is utilized by the sector nonlinearity approach to describe the dynamics of nonlinear ODE subsystem. Based on the T-S fuzzy model, a composite Lyapunov function is constructed to develop a fuzzy controller via the ODE state feedback and boundary feedback of transformed EBB equation to exponentially stabilize the nonlinear ODE-EBB cascaded system. The design procedure is presented in terms of bilinear matrix inequalities (BMIs). Moreover, a simple design method is also provided in terms of linear matrix inequalities (LMIs) from the obtained design procedure. Finally, numerical simulations on flight control with vibration suppression of a flexible air-breathing hypersonic vehicle are given to illustrate the effectiveness of the proposed design method.