Recursive Identification of Binary-Valued Systems under Uniform Persistent Excitations

TL;DR

This work addresses online identification of binary-valued moving-average systems under uniform persistent excitation by proposing a projection-free stochastic-approximation algorithm. A novel auxiliary process, SPAO, links the average of binary observations to the algorithm's convergence, enabling explicit tail bounds and convergence rates. The main results show almost-sure convergence at the rate $O\left(\sqrt{\ln\ln k / k}\right)$ and mean-square convergence at $O\left(1/k\right)$ when the step-size is properly chosen, which are optimal among online binary-valued identification methods under non-periodic inputs. The approach advances online quantized-system identification and provides a constructive framework (SPAO) that can extend to a broader class of recursive algorithms and robustness analyses with fixed finite-level quantizers.

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.

Figures (5)

• Figure 1: Convergence of Algorithm \ref{['algo']}.
• Figure 2: The trajectory of $k\lVert \tilde{\theta}_k\rVert^2/ \ln \ln k$.
• Figure 3: The trajectory of $k\lVert \tilde{\theta}_k\rVert^2$ in 200 repeated experiments.
• Figure 4: Empirical mean square convergence rates under different $\beta$.
• Figure 5: The trajectory of $\lVert \tilde{\theta}_k\rVert^2$ for the wrong variance case.

Theorems & Definitions (56)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8
• Lemma 1
• Lemma 2