Stabilization of Nonlinear Systems Nonlinearly Depending on Fast Time-Varying Parameters: An Immersion and Invariance Approach

The problem of stabilization of nonlinear systems, which depend nonlinearly on fast time-varying parameters, is considered in the technical note. It is assumed that, if the plant parameters were known, a static state-feedback controller that achieves the stabilization objective with a Lyapunov-like function that is independent of the parameters is known. A constructive procedure to update the unknown parameters of the controller, based on the immersion and invariance approach, is proposed. The main contribution of the paper is to show that the proposed controller guarantees global convergence to zero of the systems state for arbitrary time variations of the plant parameters provided the controller parameters are bounded. To ensure the latter condition, an assumption, that in the single parameter case is strictly weaker than the monotonicity condition invoked in previous studies, is imposed. Stabilization is achieved via a, rather unique, combination of gradient-like parameter estimation and the construction of a monotonic signal that counters the deleterious effect of the parameter variations. Several simulation examples illustrate the applicability of the suggested method.