A Unified Recursive Identification Algorithm with Quantized Observations Based on Weighted Least-Squares Type Criteria
Xingrui Liu, Ying Wang, Yanlong Zhao
TL;DR
This work addresses parameter identification for linear systems with Gaussian inputs when outputs are observed only through fixed-threshold quantizers. It introduces a novel weighted least-squares type criterion that rewrites quantization nonlinearities as a product of the unknown parameter and an unknown coefficient ρ(δ_y), enabling a two-step recursive algorithm: (1) ML-type estimation of the system output variance δ_y, and (2) a WLS-type estimation of γ = ρ(δ_y) θ, yielding θ = γ/ρ(δ_y). The approach delivers almost-sure convergence at rate $O(\,\sqrt{\log\log k / k}\,)$ and $L^p$ convergence at rate $O(1/k^{p/2})$, with the first step achieving asymptotic efficiency for δ_y, and extends to dynamic output-error (OE) systems via Durbin’s method, using stationary mixing auxiliary processes to handle non-iid data. Numerical examples validate the theory and show improved performance over existing stochastic-approximation methods, demonstrating practical applicability to quantized data scenarios. The framework reduces reliance on threshold design and prior model specifics, providing a robust path for quantized-system identification under Gaussian excitation.
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.