Adaptation in Nonlinear Dynamical Systems

We propose a solution to the problem of adaptive control and parameter estimation in systems with unstable target dynamics. Models of uncertainties are allowed to be nonlinearly parameterized, and required to be smooth and monotonic functions of linear functionals of the parameters. The mere assumption of existence of nonlinear operator gains for the target dynamics is sufficient to guarantee that system solutions are bounded, reach a neighborhood of the target set, and the mismatches between the modeled uncertainties and their compensator converge to zero. With respect to parameter convergence, a standard persistent excitation condition suffices to ensure that it is exponential. When a weaker, nonlinear version of persistent excitation is satisfied, asymptotic convergence is guaranteed. The spectrum of possible applications ranges from tyre-road slip control to asynchronous message transmission in spiking neural oscillators.