Finite-time composite control for a class of singularly perturbed nonlinear systems via successive Galerkin approximation

This paper presents a new algorithm of the finite-time closed-loop composite control for a class of singularly perturbed nonlinear systems with respect to performance criteria, using the successive Galerkin approximation (SGA). The singularly perturbed nonlinear system is decomposed into two subsystems of a slow-time scale and a fast-time scale via singular perturbation theory, and then two optimal control laws are obtained to each subsystem using the SGA method. Composite control theory guarantees near-optimal closed-loop performance but the resulting problem is difficult to solve for nonlinear systems. To overcome the difficulties inherent in the optimal control problem, the suitable optimal feedback control law can be constructed in term of the approximated solution to a Hamilton-Jacobi-Bellman equation using SGA. The finite-time composite control law for singularly perturbed nonlinear systems is designed by a linear combination of the slow and fast variables.