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Asymptotically efficient adaptive identification under saturated output observation

Lantian Zhang, Lei Guo

TL;DR

The paper addresses the problem of identifying stochastic systems under saturated output observations without relying on i.i.d. data or strong excitation. It introduces a two-step online adaptive Newton algorithm that achieves strong consistency, asymptotic normality, and asymptotic attainment of the Cramér-Rao bound by tailoring adaptation gains to approach the Fisher information inverse. Theoretical results are supported by martingale-based convergence proofs and a stochastic Lyapunov framework, with numerical simulations showing superior performance over existing methods. This work broadens applicability to stochastic feedback settings where data are non-i.i.d. and saturations are present, delivering near-optimal identification performance in general settings.

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.

Asymptotically efficient adaptive identification under saturated output observation

TL;DR

The paper addresses the problem of identifying stochastic systems under saturated output observations without relying on i.i.d. data or strong excitation. It introduces a two-step online adaptive Newton algorithm that achieves strong consistency, asymptotic normality, and asymptotic attainment of the Cramér-Rao bound by tailoring adaptation gains to approach the Fisher information inverse. Theoretical results are supported by martingale-based convergence proofs and a stochastic Lyapunov framework, with numerical simulations showing superior performance over existing methods. This work broadens applicability to stochastic feedback settings where data are non-i.i.d. and saturations are present, delivering near-optimal identification performance in general settings.

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.
Paper Structure (13 sections, 14 theorems, 153 equations, 2 figures, 1 algorithm)

This paper contains 13 sections, 14 theorems, 153 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions
  4. Problem Setup
  5. Main results
  6. Fisher information
  7. Adaptive asymptotically efficient algorithm
  8. Convergence analysis
  9. Efficiency analysis
  10. Proofs of the main results
  11. Numerical simulation
  12. Concluding remarks
  13. Lemmas and proofs

Key Result

Lemma 3.1

\newlabellem230 For the nonlinear system $(eq1)$-$(eq2)$, if Assumptions assum2-assum4 hold, then for each $n\geq 1$, the Fisher information matrix is given by where

Figures (2)

  • Figure 1: Mean square error
  • Figure 2: Covariance and CR lower bound

Theorems & Definitions (24)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5
  • Remark 3.6
  • Theorem 3.7
  • Remark 3.8
  • ...and 14 more