Stabilization of nonlinear singularly perturbed system

Abstract

The singularly perturbed system has two-time scale properties, and we can divide the whole system into the boundary-layer and the reduced model. Analysis and control of the nonlinear system is getting more difficult as the dimension grows. Thus, the merit of the singularly perturbed system is maximized in the nonlinear system. The stability of the singularly perturbed system can be checked through the boundary-layer model and the reduced model. If the boundary-layer model is uniformly exponentially stable and the reduced model is exponentially stable, then the whole system is exponentially stable. Therefore, without finding the Lyapunov function for the whole system, we can check the stability of the whole system. To stabilize the singularly perturbed system, the composite feedback control method is used. In this method, we design a controller to stabilize the reduced model and to stabilize the boundary layer model separately. By combining these two controller, we can stabilize the whole system. Though the singularly perturbed system has some benefits, it also has some problems. First, uncertainties in the singularly perturbed system make the analysis difficult and in some cases make it impossible. Especially, in the nonlinear system, the uncertainties change the manifold entirely different and possibly make the manifold nonexistent. In this paper, we find conditions such that the manifold exists when uncertainties are added to the nominal singularly perturbed system. We also measure the change of the manifold due to the uncertainties. Using this measured change, we find the conditions that the boundary layer model is uniformly exponentially stable and the reduced model is exponentially stable. Through this procedure, we can find the condition that the whole singularly perturbed system with uncertainties is exponentially stable. Based on this, we propose a robust composite controller design method. Another problem of the singularly perturbed system is r...