Finite time stabilization with a PDE state constraint for a class of nonlinear ODE-PDE systems

This paper addresses the finite time stabilization problem subject to a constraint of partial differential equation (PDE) state for a class of coupled systems described by nonlinear ordinary differential equations (ODEs) and a linear parabolic PDE. Initially, the modal decomposition and singular perturbation techniques are applied to the PDE system to derive the finite dimensional ODE model which accurately describes the dynamics of the dominant (slow) modes of the PDE system. By augmenting the original ODE system by the slow system of the PDE system, a coupled ODE system can be obtained, which is subsequently represented by the Takagi-Sugeno (T-S) fuzzy model. Meanwhile, the PDE state constraint is also converted into a state constraint exerted on the coupled ODE system. Then, based on the T-S fuzzy model, a fuzzy control design is developed in terms of linear matrix inequalities (LMIs), such that the original ODE system is finite time quasi-contractively stable with a terminal time as small as possible, 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.