Invariant manifolds and asymptotic properties of adaptive nonlinear stabilizers

A classical question in adaptive control is that of convergence of the parameter estimates to constant values in the absence of persistent excitation. The author provides an affirmative answer for a class of adaptive stabilizers for nonlinear systems. Then the author studies their asymptotic behavior by considering the problem of whether the parameter estimates converge to stabilizing values-the values which would guarantee stabilization if used in a nonadaptive controller. The author approaches this problem by studying invariant manifolds and shows that except for a set of initial conditions of Lebesgue measure zero, the parameter estimates do converge to stabilizing values. Finally, the author determines a (sufficiently large) time instant after which the adaptation can be disconnected at any time without destroying the closed-loop system stability.