H∞ finite-time control with a PDE state constraint for a class of nonlinear coupled ODE-PDE systems

This paper addresses the H_∞ finite-time control problem subject to a state constraint of the partial differential equation (PDE) for a class of coupled systems described by nonlinear ordinary differential equations (ODEs) and a linear parabolic PDE. Initially, the Karhunen-Loève decomposition (KLD) and the singular perturbation technique are applied to the PDE system to derive a finite dimensional ODE model which accurately describes the dominant dynamics of the PDE system. By combining the original ODE system with the slow model of the PDE system, a nonlinear coupled ODE system is obtained, which is subsequently represented by the Takagi-Sugeno (T-S) fuzzy model. Meanwhile, the PDE state constraint is converted into a state constraint exerted on the coupled ODE system. Then, an H_∞ fuzzy control design is developed to stabilize the original ODE system in a finite time with a terminal time as small as possible, and achieve an optimized H_∞ performance of disturbance attenuation, while the PDE state constraint is respected. Finally, the proposed design method is applied to the control of a hypersonic rocket car to illustrate its effectiveness.