Parameter convergence in nonlinearly parameterized systems

A large class of problems in parameter estimation concerns nonlinearly parametrized systems. Over the past few years, a stability framework for estimation and control of such systems has been established. We address the issue of parameter convergence in such systems in this paper. Systems with both convex/concave and general parameterizations are considered. In the former case, sufficient conditions are derived under which parameter estimates converge to their true values using a min-max algorithm. In the latter case, to achieve parameter convergence a hierarchical min-max algorithm is proposed where the lower level consists of a min-max algorithm and the higher level component updates the bounds on the parameter region within which the unknown parameter is known to lie. Using this hierarchical algorithm, a necessary and sufficient condition is established for global parameter convergence in systems with a general nonlinear parameterization. In both cases, the conditions needed are shown to be stronger than linear persistent excitation conditions that guarantee parameter convergence in linearly parametrized systems. Explanations and examples of these conditions and simulation results are included to illustrate the nature of these conditions. A general definition of nonlinear persistent excitation that leads to parameter convergence is proposed at the end of this paper.