Strong Consistency, Optimal Tracking and Sharp Rates for the Åström-Wittenmark Self-Tuning Regulator with Unknown Input Gain

Abstract

We study strong consistency and optimal tracking of the Astrom-Wittenmark self-tuning regulator with unknown input gain. Existing results establish consistency of all parameter estimates for the unmodified recursion under additional growth conditions on reference information. Other approaches obtain stability and optimal tracking by adjusting the estimates used in feedback or adding decaying probing signals. For a class of minimum-phase linear systems with martingale difference noise, ordinary least squares (LS) with certainty-equivalent control achieves stability and optimal tracking almost surely for each bounded reference, provided the recursion is well defined. No reference excitation, adjustment of feedback estimates or added probing is required. When at least two parameters are estimated, all parameter estimates are strongly consistent, with squared norm of the estimation error $O(\log \log \log n/\log n)$ almost surely. A law of the iterated logarithm under zero reference shows that this rate cannot in general be improved. We also derive a logarithm law for cumulative squared tracking residuals, with an explicit dependence of the leading term on reference energy. A chi-square limit yields asymptotically exact parameter confidence ellipsoids for each bounded reference. The proof combines a logarithmic lower bound on the cumulative squared difference between an auxiliary LS prediction of the noise and the reference with estimates from the LS recursion. If the gain estimate were inconsistent, these bounds would yield incompatible estimates for the same weighted sum of squares, establishing gain consistency without assuming stability.

arXiv comment32 pages