A unified framework for stability in systems with two time scales

Prof. Andrew Teel, Department of Electrical and Computer Engineering, University of California, Santa Barbara

If singular perturbation theory, averaging theory and slowly varying systems theory all appear in the same nonlinear systems textbook, they typically appear in separate chapters and, other than having a common two time scale property, appear to be unrelated. For systems fitting these theories, stability results are typically stated for equilibria and don't allow exogenous disturbances.

Despite this tradition, it is possible to develop a framework that covers all of these stability problems simultaneously, extends the classical studies to systems with disturbances, and addresses input-to-state stability of general attracting sets. The key step in doing this is to allow the ``boundary layer'' of a singularly perturbed system to have complex asymptotic behavior. Convergence of the boundary layer to an asymptotically stable equilibrium manifold, which is the classical assumption, is a special case. (This generalized view of singular perturbations has recently been pioneered by Z. Artstein.) In turn, the notion of stability of sets plays a fundamental role in enabling the results.

In this talk, a unified framework for stability in systems with two time scales will be developed and connections to many classical results will be made.