A Unified Recursive Identification Algorithm with Quantized Observations Based on Weighted Least-Squares Type Criteria

TL;DR

This paper introduces a novel recursive system identification algorithm that effectively estimates parameters from quantized observations, achieving convergence and efficiency in both noisy and noise-free Gaussian systems, with extensions to output-error models.

Contribution

It proposes a weighted least-squares based recursive identification method that handles quantized data and proves its convergence with optimal rates, including variance estimation and system extensions.

Findings

Convergence in almost sure and $L^{p}$ senses with specified rates.

Accurate variance estimation from quantized observations.

Validation through numerical examples confirming theoretical results.

Abstract

This paper investigates system identification problems with Gaussian inputs and quantized observations under fixed thresholds. By reinterpreting the nonlinear effects induced by quantization as the product of the unknown parameter and an unknown nonlinear coefficient, this work establishes a novel weighted least-squares criterion that enables linear estimation of unknown parameters under quantized observations. Subsequently, a two-step recursive identification algorithm is constructed by estimating two unknown terms, which is capable of handling both Gaussian noisy and noise-free linear systems. Convergence analysis of this identification algorithm is conducted, demonstrating convergence in both almost sure and $L^{p}$ senses under mild conditions, with respective rates of $O(\sqrt{ \log \log k/k})$ and $O(1/k^{p/2})$, where $k$ denotes the time step. In particular, this algorithm offers an asymptotically efficient estimation of the variance of Gaussian variables using quantized observations. Furthermore, extensions to output-error systems are discussed, enhancing the applicability and relevance of the proposed methods. Two numerical examples are provided to validate these theoretical advancements.