A Lyapunov characterization of robust stabilization

TLDR: In this article, the existence of smooth Lyapunov functions is shown to imply the presence of feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances.

Abstract: One of the fundamental facts in control theory (Artstein’s theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper establishes that the existence of smooth Lyapunov functions implies the existence of, in general discontinuous, feedback stabilizers which are insensitive (or robust) to small errors in state measurements. Conversely, it is shown that the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. Moreover, it is established that, for general nonlinear control systems under persistently acting disturbances, the existence of smooth Lyapunov functions is equivalent to the existence of (in general, discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances. ∗Supported in part by Russian Fund for Fundamental Research Grant 96-01-00219 and by the Rutgers Center for Systems and Control (SYCON). Work done while visiting Rutgers University, Mathematics Department. On leave from Steklov Institute of Mathematics, Moscow 117966, Russia †Supported in part by US Air Force Grant AFOSR-94-0293