A unified recursive identification algorithm with quantized observations based on weighted least-squares type criteria

Abstract: This paper investigates system identification problems with Gaussian inputs and quantized observations under fixed thresholds. A new formulation for the predictor of quantized observations is introduced, establishing a linear correlation with the parameter estimations through a probabilistic relationship among quantized observations, Gaussian inputs, and system parameters. Subsequently, a novel weighted least-squares criterion is proposed, and a two-step recursive identification algorithm is constructed, which is capable of addressing both 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. Additionally, asymptotic normality is established, and an expression for the asymptotic variance is provided when the weight coefficients are properly selected. 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.