Strong Consistency and Optimal Tracking of the {\AA}str\"om-Wittenmark Self-Tuning Regulator with Unknown Input Gain

Abstract

We study strong consistency and optimal tracking of the {\AA}str\"om-Wittenmark self-tuning regulator with unknown input gain. Existing results establish stability, optimal tracking and parameter consistency for the unmodified recursion under growth conditions on reference information. Other approaches obtain performance guarantees by adjusting the estimates used in feedback or adding decaying probing signals. For a class of minimum-phase linear systems with martingale difference noise, we establish joint guarantees for ordinary least squares with certainty-equivalent control without reference excitation, gain adjustment or added probing. For each bounded reference, average input and output energy are almost surely bounded, average squared tracking error converges almost surely to the noise variance, and all parameter estimates are strongly consistent whenever at least two parameters are estimated. For the consistency result, the key is a logarithmic lower bound on the cumulative squared difference between an auxiliary least-squares prediction of the noise and the reference. If the gain error persisted, this noise information and the actual least-squares recursion would give incompatible lower and upper bounds on the same weighted squared sum. This contradiction establishes gain convergence before stability and full parameter consistency.