Some Problems Of Singularly Perturbed Control Systems

The feature of singularly perturbed control is principle of two-time-scale design. Namely, first the original system is decomposed into two subsystems -a slow reduced system and a fast boundary-layer system, state-feedback then proceed for each lower-order subsystems, and the results combined to yield a composite state-feedback control for the original systems. When the system is linear, the stability of the whole systems is determined by the stability of its two slow and fast subsystems. But for the nonlinear systems, which stability is determined by some additional condition distinguished as cascade condition besides stability of two subsystems. Body of the work of the dissertation that deals with two problems as follows:1, first, the global robust stabilization problem for a class of nonlinear uncertain two-times scale systems is presented. The robust composite feedback control for the whole systems is obtained by applying the two-time scale design approach and the upper stability bound of ε is also given. Finally, some examples show the application of the result.2, second, the cascade condition of nonlinear singularly perturbed systems is considered. We present a small gain growth condition for a class of nonlinear singularly perturbed systems, under which the asymptotic stability of nonlinear singularly perturbed systems is guaranteed by the asymptotic stability of its fast and slow subsystems. Moreover, we provide an explicit and closed form of stability bound on perturbation parameter and some example- les in application.The contributions of the work: 1. the problem of robust stability of two-time-scale systems with nonlinear uncertainties has already solved, but corresponding problem of robust stabilization still remain unsettled for the difficulty of cascade condition. In this work the global robust stabilization problem for a class of nonlinear uncertain two-times scale systems are presented without any cascade condition. 2. We present a cascade condition for a class of nonlinear singularly perturbed systems, extend the quadratic-type asymptotic stability cascade condition, and furthermore show the characteristic of a small gain growth condition for general asymptotic stability cascade condition.