TOPOLOGICAL ASPECTS OF UNIVERSAL ADAPTIVE STABILIZATION

In this paper we consider two problems in ''non-identifier-based,'' un iversal adaptive control within the framework of Martensson [Adaptive Stabilization, Ph.D. thesis, Lund Institute of Technology, 1986]. In t his framework, any linear system stabilizable by constant linear outpu t feedback is adaptively stabilized by an adaptive piecewise-linear ou tput feedback control law. The essential feature we exploit is that of a piecewise-linear output feedback which switches through a set of fe edback matrices, with switching controlled by an output-driven differe ntial equation. For each initial condition the state of the system con verges to zero and the time-varying gain matrix converges to a ''limit gain.'' In this setting we consider two related problems. The first c oncerns the sensitivity of closed-loop solutions under small perturbat ions of the initial data. The second concerns generic properties, with respect to the set of initial conditions, of stabilization by the lim it gain. We adopt a topological approach, based on a decomposition of the dynamics of the resultant nonlinear, closed-loop system into a seq uence of homeo/diffeomorphisms derived from the switching nature of th e dynamics. Using this decomposition we show that the set of initial c onditions 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. T he results are illustrated by examples of planar control systems where the sets of initial conditions yielding nonstabilizing limit gains ar e computed.