Stability analysis and controller design of nonlinear singularly perturbed systems with uncertainties

Abstract

In these days, systems are getting complex more and more. As a result, there exist a pronounced burden to analyze that complicated system. At this point of view, singular perturbation theory has been attracting attention from researchers in many different fields. The singulary perturbed system has two-time scale properties, and we can separate the whole system into the boundary-later and the reduced model. From this separation, we are able to reduce its complexity when analyzing and controlling the system. The stability of the singularly perturbed system can be checked through the boundary-layer model and the reduced model. If the boundary-model is uniformly exponentially stable and the reduced model is exponentially stable, then the whole system is exponentially stable. Therefore, it is possible to check out the whole system`s stability without a lot of difficulties to find out Lyapunov function for the whole system.

Even though the singular perturbed system has strong advantages, it remains some problems. If uncertainties are added into the singularly perturbed system, it is difficult to analyze the system and in some cases it makes analysis impossible. The uncertainties would change the manifold entirely different and possibly make the manifold nonexistent in the system, especially in nonlinear system. In this paper, we change the system structure so that we can deal with nonlinear term directly. The existing system structure can cover linear term only. To achieve this objective, we suggest novel way to change variable in the boundary-layer system. Furthermore, we also extend the system structures multiplied by input in both the slow and fast dynamic. As a result, the system structure multiplied by input can include not only an argument x, but also an argument z. Then, based on these changed structures, we find conditions for the existence of manifold even if certain bound uncertainties are added to the nominal singularly perturbed system. Also, we fig...