On parameter convergence of nonlinearly parameterized adaptive systems: Analysis via contraction and first Lyapunov´s methods

Adaptive systems have been traditionally analyzed using classical second Lyapunov´s method and/or passivity theory. These powerful tools permit the establishment of conditions for convergence and stability (asymptotic and ℒ₂) of adaptive estimators and controllers. A natural, alternative framework to analyze adaptive systems invokes the notion of (flow) contraction, which imposes that some distance between any pair of solutions is monotonically decreasing with time. In this paper we investigate the contraction properties of adaptive systems designed following the principles of immersion and invariance. In particular, we use contraction theory and a recent extension of first Lyapunov´s method to give conditions under which adaptive estimators and controllers-designed for nonlinearly parameterized, nonlinear systems-ensure exponential parameter convergence.