GEOMETRIC ASYMPTOTIC PROPERTIES OF ADAPTIVE NONLINEAR-SYSTEMS WITH PARTIAL EXCITATION/

In this paper we continue the study of geometric/asymptotic properties of adaptive nonlinear systems. The long-standing question of whether the parameter estimates converge to stabilizing values-stabilizing if used in a nonadaptive controller-is addressed in the general set-point regulation case. The key quantifier of excitation in an adaptive syst em is the rank r of the regressor matrix at the resulting equilibrium. Our earlier paper showed that when either r = 0 or r = p (where p is the number of uncertain parameters), the set of initial conditions lea ding to destabilizing estimates is of measure zero. Intuition suggests the same for the intermediate case 0 < r < p studied in this paper. W e present a surprising result: the set of initial conditions leading t o destabilizing estimates can have positive measure. We present result s for the backstepping design with tuning functions; the same results can be established for other Lyapunov-based adaptive designs.