Decomposition and stability for multiparameter singular perturbation problems

Time-scale separation and stability of linear time-varying and time-invariant multiparameter singular perturbation problems are analyzed. The first problem considered in the paper is that of deriving upper bounds on the small "parasitic" parameters ensuring the existence of an invertible, bounded transformation exactly separating fast and slow dynamics. This problem is most interesting for the time-varying case. The analysis of this problem in the time-varying case requires the two-time-scale setting introduced by H. K. Khalil and P. V. Kokotovic [16]. This entails that the mutual ratios of the small parameters are bounded by known positive constants. The second problem considered is to derive parameter bounds ensuring that the system in question is uniformly asymptotically stable. The results on decomposition are used to facilitate the derivation of these latter bounds. Fortunately, the analysis of decomposition and stability questions for time-invariant multiparameter singular perturbation problems requires no restriction on the relative magnitudes of the small parameters. A concept of "strong
D
-stability" is introduced and shown to greatly simplify the stability analysis of time-invariant multiparameter problems.