Asymptotically efficient adaptive identification under saturated output observation

TL;DR

This paper develops a new adaptive algorithm for identifying stochastic systems with saturated output observations, achieving asymptotic efficiency without restrictive assumptions, and demonstrating optimal performance theoretically and numerically.

Contribution

It introduces a novel adaptive Newton-type algorithm that attains asymptotic efficiency for stochastic systems with saturated outputs under general conditions.

Findings

Parameter estimates are strongly consistent and asymptotically normal.

The mean square error reaches the Cramer-Rao bound asymptotically.

Numerical results show the proposed method outperforms existing algorithms.

Abstract

As saturated output observations are ubiquitous in practice, identifying stochastic systems with such nonlinear observations is a fundamental problem across various fields. This paper investigates the asymptotically efficient identification problem for stochastic dynamical systems with saturated output observations. In contrast to most of the existing results, our results do not need the commonly used but stringent conditions such as periodic or independent assumptions on the system signals, and thus do not exclude applications to stochastic feedback systems. To be specific, we introduce a new adaptive Newton-type algorithm on the negative log-likelihood of the partially observed samples using a two-step design technique. Under some general excitation data conditions, we show that the parameter estimate is strongly consistent and asymptotically normal by employing the stochastic Lyapunov function method and limit theories for martingales. Furthermore, we show that the mean square error of the estimates can achieve the Cramer-Rao bound asymptotically without resorting to i.i.d data assumptions. This indicates that the performance of the proposed algorithm is the best possible that one can expect in general. A numerical example is provided to illustrate the superiority of our new adaptive algorithm over the existing related ones in the literature.