Algebraic approach to time scale analysis of singularly perturbed linear systems

Abstract

In this paper we develop an algebraic approach to the multiple time scale analysis of perturbed linear systems based on the examination of the Smith form of the system matrix viewed as a matrix over a ring of functions in the perturbation parameter. This perspective allows us to obtain a strengthened version of the results of Coderch et al. (1983) and to provide a bridge between these complex but general results and previous explicit, conceptually simple, but somewhat restrictive results such as those described by Kokotovic (1981) and Chow (1982). In addition, our algebraic framework allows us to investigate a variety of other problems. In this paper we study the problem of developing valid time scale decompositions in cases in which weak damping terms discarded in the approaches of Kokotovic (1981), Chow (1982) and Coderch et al. (1983) must be retained. Also, our approach exposes the role of the invariant factors of the system matrix in determining its time scales. This leads naturally to the problem of...