Persistence of excitation conditions in passive learning control

Abstract

This paper analyzes the stability of both the system state and parameter estimates in passive learning control applications. The analysis is valid for any linear in parameter approximator. This class of approximators includes many of those commonly used: radial basis functions, splines, wavelets, certain fuzzy systems, and Cerebellar Model Articulation Controller networks. Stability results are presented for both parametric (known model structure with unknown parameters) and nonparametric (unknown model structure resulting in ε-approximation error) adaptive control applications. The main contribution of the article is an analysis of the persistence of excitation conditions required for parameter convergence. In addition, to a general analysis, the article presents a specific analysis pertinent to approximators that are composed of basis elements with local support. In particular, the analysis shows that, as long as a reduced dimension subvector of the regressor vector is persistently exciting, then a specialized form of exponential convergence will be achieved. This condition is critical, since the general persistence of excitation conditions are not practical in most control applications.