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Non-Asymptotic Bounds for Closed-Loop Identification of Unstable Nonlinear Stochastic Systems

Seth Siriya, Jingge Zhu, Dragan Nešić, Ye Pu

TL;DR

The paper addresses finite-time identification of unstable nonlinear stochastic systems in a closed-loop setting using regularised least squares with linear parameter uncertainty. It introduces regional excitation to certify informative regressor data and derives non-asymptotic probabilistic bounds on the parameter estimation error over the exciting region; strengthening to global excitation yields infinite-horizon bounds with convergence to the true parameters. The contributions include the regional-excitation framework, data-dependent and global bounds, and concrete examples (piecewise affine systems and a controlled double integrator) illustrating when regional excitation suffices and how global excitation yields stronger guarantees. The results bridge non-asymptotic system identification with unstable systems, enabling finite-time, high-probability performance guarantees for closed-loop identification under realistic control policies. The work advances practical identification by relaxing stability assumptions and providing verifiable, region-based data informativeness criteria that align with nonlinear dynamics and noisy closed-loop operation.

Abstract

We consider the problem of least squares parameter estimation from single-trajectory data for discrete-time, unstable, closed-loop nonlinear stochastic systems, with linearly parameterised uncertainty. Assuming a region of the state space produces informative data, and the system is sub-exponentially unstable, we establish non-asymptotic guarantees on the estimation error at times where the state trajectory evolves in this region. If the whole state space is informative, high probability guarantees on the error hold for all times. Examples are provided where our results are useful for analysis, but existing results are not.

Non-Asymptotic Bounds for Closed-Loop Identification of Unstable Nonlinear Stochastic Systems

TL;DR

The paper addresses finite-time identification of unstable nonlinear stochastic systems in a closed-loop setting using regularised least squares with linear parameter uncertainty. It introduces regional excitation to certify informative regressor data and derives non-asymptotic probabilistic bounds on the parameter estimation error over the exciting region; strengthening to global excitation yields infinite-horizon bounds with convergence to the true parameters. The contributions include the regional-excitation framework, data-dependent and global bounds, and concrete examples (piecewise affine systems and a controlled double integrator) illustrating when regional excitation suffices and how global excitation yields stronger guarantees. The results bridge non-asymptotic system identification with unstable systems, enabling finite-time, high-probability performance guarantees for closed-loop identification under realistic control policies. The work advances practical identification by relaxing stability assumptions and providing verifiable, region-based data informativeness criteria that align with nonlinear dynamics and noisy closed-loop operation.

Abstract

We consider the problem of least squares parameter estimation from single-trajectory data for discrete-time, unstable, closed-loop nonlinear stochastic systems, with linearly parameterised uncertainty. Assuming a region of the state space produces informative data, and the system is sub-exponentially unstable, we establish non-asymptotic guarantees on the estimation error at times where the state trajectory evolves in this region. If the whole state space is informative, high probability guarantees on the error hold for all times. Examples are provided where our results are useful for analysis, but existing results are not.

Paper Structure

This paper contains 26 sections, 22 theorems, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Discrete-Time Stochastic System and Least-Squares Estimator
  4. Standing Assumptions
  5. Main Results
  6. Error Bounds With Regional Excitation
  7. Understanding Thm. \ref{['sysid:theorem:rls-error-exciting']}
  8. Error Bounds With Global Excitation
  9. Examples
  10. Example 1: Piecewise Affine System with Random Controls
  11. The Benefit of Regional Excitation
  12. Simulation of the PWA System
  13. Example 2: Controlled Double Integrator
  14. Proofs
  15. Proofs for Error Bounds With Regional Excitation
  16. ...and 11 more sections

Key Result

Lemma 1

Consider feature map $\psi$ and controller family $\alpha$ satisfying Ass. sysid:assump:measurable, and noise distributions $\mu_w,\mu_s$ satisfying Ass. sysid:assump:process-noise. Suppose there exists a subset $\mathcal{X} \subseteq \mathbb{X}$ and constants $c_{\textnormal{PE1}}>0$ and $c_{\textn

Figures (2)

  • Figure 1: Log scale plot of estimation error averaged over 100 sample paths
  • Figure 2: Dependency graph for theoretical results. Everything inside the dashed-blue box depends on Lem. \ref{['sysid:lemma:random-variables']}.

Theorems & Definitions (35)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Definition 2
  • Lemma 1
  • ...and 25 more