Adaptive Regulation with Global KL Guarantees

In the absence of persistency of excitation (PE), referring to adaptive control systems as "asymptotically stable" typically indicates insufficient understanding of stability concepts. While the state is indeed regulated to zero and the parameter estimate has some limit, namely, the overall state converges to some equilibrium, the equilibrium reached is not unique (and not even necessarily stable) but is dependent on the initial condition. The equilibrium set in the absence of PE is not uniformly attractive (from an open set containing the equilibrium set); hence, asymptotic stability does not hold and KL estimates are unavailable for the full state. In this paper we pursue adaptive control design with KL guarantees on the regulated state, something that is possible but previously unachieved with smooth, time-invariant and non-hybrid adaptive controllers. This property is referred to as Uniform Global Asymptotic Output Stability, where the regulated state is thought of as a system output. We provide designs for (i) systems with a matched uncertainty and (ii) systems in the parametric strict feedback form. To guarantee KL estimates in the absence of PE, our designs employ time-invariant nonlinear damping terms, which depend both on the state and the parameter estimate. With an example, we illustrate the theory.