Stability analysis and control of non-standard nonlinear singularly perturbed system with non-separation of slow subsystem and fast subsystem

Abstract

In many practical systems, the presence of some parameters such as small time constants, masses, moments of inertia, and resistances give rise to singularly perturbed systems that has two-time-scale properties. These singularly perturbed systems have been studied in several decades in terms of the stability analysis and the controller design. Using the singular perturbation technique, the singularly perturbed system is decomposed into two lower-order subsystems, that is, the slow subsystem (reduced system) and the fast subsystem (boundary-layer system). From the slow subsystem and the fast subsystem, the stability of a high dimensional system (singularly perturbed system) can be analyzed based on the lower-order subsystem. If the slow subsystem is exponentially stable and the fast subsystem is uniformly exponentially stable, then the whole system (singularly perturbed system) is exponentially stable. From these property, the controller is also designed to stabilize the singularly perturbed system based on the slow subsystem and the fast subsystem. It is called the composite controller, that consists of the slow controller and the fast controller. From the singularly perturbed system, there are some studies about the non-standard singularly perturbed systems. The non-standard singularly perturbed systems are defined by their own purposes such as the relaxation of the stability condition, multiple perturbation parameter, the existence of time-delay terms. We also define the non-standard nonlinear singularly perturbed system whose the state equations are characterized by the nominal terms with the additive perturbation terms that contain the perturbation parameter. Specifically, the non-standard nonlinear singularly perturbed system is divided into two standard singularly perturbed systems. Based on the well-known standard singularly perturbed system`s properties, we discuss about the exponential stability of the non-standard nonlinear singularly perturbed system...