Topological Aspects of Universal Adaptive Stabilization

Abstract

In this paper we consider two problems in “non–identifier-based,” universal adaptive control within the framework of Mårtensson [Adaptive Stabilization, Ph.D. thesis, Lund Institute of Technology, 1986]. In this framework, any linear system stabilizable by constant linear output feedback is adaptively stabilized by an adaptive piecewise-linear output feedback control law. The essential feature we exploit is that of a piecewise-linear output feedback which switches through a set of feedback matrices, with switching controlled by an output-driven differential equation. For each initial condition the state of the system converges to zero and the time-varying gain matrix converges to a “limit gain.” In this setting we consider two related problems. The first concerns the sensitivity of closed-loop solutions under small perturbations of the initial data. The second concerns generic properties, with respect to the set of initial conditions, of stabilization by the limit gain. We adopt a topological approach, based on a decomposition of the dynamics of the resultant nonlinear, closed-loop system into a sequence of homeo/diffeomorphisms derived from the switching nature of the dynamics. Using this decomposition we show that the set of initial conditions for which solutions are stable under small perturbations and the limiting gain is stabilizing has full Lebesgue measure and dense interior. This latter result has been conjectured in the literature. The results are illustrated by examples of planar control systems where the sets of initial conditions yielding nonstabilizing limit gains are computed.