On the Persistence of Excitation in the Adaptive Estimation of Distributed Parameter Systems

Persistence of excitation is a sufficient condition for parameter convergence in a class of adaptive, or on-line, identification schemes for dynamical systems. In the case of abstract linear parabolic and hyperbolic distributed parameter systems, this condition requires that a family of bounded linear functionals be norm bounded away from zero. In general, this condition on the plant is difficult to verify. The level of persistence of excitation of the plant and its implications are considered for a simple parabolic and hyperbolic system for the purpose of gaining insight into its effect in the more general case. Its effect on the qualitative and quantitative behavior of the estimators is investigated. Numerical studies illustrating the results of the analysis are discussed.