Recursive Identification of Binary-Valued Systems under Uniform Persistent Excitations

TL;DR

This paper introduces a new stochastic approximation algorithm for identifying binary-valued systems with persistent excitation, achieving almost sure convergence and optimal mean square rates without requiring projections or periodic inputs.

Contribution

The paper presents the first SA-based recursive identification algorithm for set-valued systems under uniform persistent excitation, overcoming previous limitations and providing strong convergence guarantees.

Findings

Almost sure convergence rate of $O(rac{\

Mean square convergence rate of $O(1/k)$.

Numerical example confirms effectiveness and theoretical results.

Abstract

This paper studies the control-oriented identification problem of set-valued moving average systems with uniform persistent excitations and observation noises. A stochastic approximation-based (SA-based) algorithm without projections or truncations is proposed. The algorithm overcomes the limitations of the existing empirical measurement method and the recursive projection method, where the former requires periodic inputs, and the latter requires projections to restrict the search region in a compact set.To analyze the convergence property of the algorithm, the distribution tail of the estimation error is proved to be exponentially convergent through an auxiliary stochastic process. Based on this key technique, the SA-based algorithm appears to be the first to reach the almost sure convergence rate of $ O(\sqrt{\ln\ln k/k}) $ theoretically in the non-periodic input case. Meanwhile, the mean square convergence is proved to have a rate of $ O(1/k) $, which is the best one even under accurate observations. A numerical example is given to demonstrate the effectiveness of the proposed algorithm and theoretical results.