Convergence results for Widrow’s adaptive controller

Widrow and colleagues have proposed an adaptive control scheme, for the task of following a prescribed reference trajectory, which relies on adaptively identifying a finite impulse response model of the inverse of the unknown plant. The structure of the proposed controller is unlike any other proposed, and no theoretical convergence results have so far been demonstrated, despite excellent simulation experience. In this paper, a convergence result is established, which basically demands a persistency of excitation condition on the reference trajectory. This condition in fact guarantees exponentially fast decay of errors to zero. Without it however, it is possible for convergence not to occur (in contrast to some other adaptive control algorithms). Key tools in establishing the conclusion are provided by new results on the propagation of the persistent excitation property through time-invariant and slowly time-varying systems.