A result on exponential tracking error convergence and persistent excitation

Conditions are investigated for exponential convergence of the tracking error in feedforward adaptive systems having insufficient excitation. Particular attention is paid to the adaptive gradient algorithm with periodic excitation in the overparameterized case. A main result is that for a bounded periodic regressor, the tracking error converges exponentially without regard to parameter convergence or to the degree of overparameterization. These results weaken the persistent excitation (PE) conditions and parameter convergence renditions generally considered necessary to ensure exponential tracking convergence in this class of systems.