(A) study on nonlinear singularly perturbed systems using gain scheduling

Abstract

In this thesis, we studied on a two-time-scale system which have the slow state and the fast state. These system is called a singularly perturbed system. In these system, the slow state plays a role of the slowly varying input. But this variable isn``t the external variable but the internal variable. The interaction between the slow and fast state exists. Thus, we needs another interconnection conditions. Many physical systems have dynamical phenomena occurring in different time scales. There are circuits, robots, car suspension, and so on. There exist the slow manifold and the fast state rapidly converges into the slow manifold. Therefore the dynamics is degenerated into the reduced model and controlled by the slow state and when the fast state is on the slow manifold. The behavior of the system is constrained in O(ε) bound of the slow manifold. The controller type is divided into the slow controller and the fast controller which control the slow and the fast dynamics respectively. First, the slow controller is designed to stabilize the slow dynamics where the fast state is substituted with the slow manifold, the function of the slow state and the input. If the slow controller is designed, the slow manifold appears. Then, the fast controller is designed to control the fast dynamics for a fixed slow variable so that it is inactive on the slow manifold. Finally, the overall controller is the simple addition of the slow and the fast controller. But the slow manifold isn``t the invariant manifold because of the existence of the small constant. The development of the integral manifold approach solved the approximation problem. In this approach, the fast state is more rapidly attracted on the manifold so that the behavior of the system is more rapidly converged into the reduced system. The manifold condition is given by the partial differential equation, depended on the manifold itself. For a particular nonlinear system, the affine dynamic systems, it was shown that...