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A Framework for Adaptive Stabilisation of Nonlinear Stochastic Systems

Seth Siriya, Jingge Zhu, Dragan Nešić, Ye Pu

TL;DR

This work develops a modular certainty-equivalence adaptive-control framework for discrete-time nonlinear stochastic systems with linearly parameterised uncertainty, combining a parameterised stabilising policy with regularised least squares (RLS) parameter estimation. It derives non-asymptotic, high-probability guarantees: (i) under regional excitation and a robustly positive invariant (RPI) set, a time-varying estimation error bound $e(t,\delta,x_0)$ and almost-sure invariance of the state within the RPI region; (ii) under global excitation and a global stochastic Lyapunov function, high-probability stability bounds hold for any initial condition; and (iii) the results are illustrated on a piecewise-affine system and an input-constrained Gaussian-disturbed linear system. The analysis avoids requiring a priori bounded parameter sets and emphasizes informative data through excitation regions, enabling stability guarantees even when global stabilisability is unavailable. The framework thereby provides a practical, verifiable approach to learning-based adaptive stabilization for a broad class of stochastic nonlinear systems with linear-in-parameters structure, with clear conditions for regional and global stability guarantees. The results have potential impact on reliable autonomous control in uncertain, noisy environments where online learning and adaptation are essential.

Abstract

We consider the adaptive control problem for discrete-time, nonlinear stochastic systems with linearly parameterised uncertainty. Assuming access to a parameterised family of controllers that can stabilise the system in a bounded set within an informative region of the state space when the parameter is well-chosen, we propose a certainty equivalence learning-based adaptive control strategy, and subsequently derive stability bounds on the closed-loop system that hold for some probabilities. We then show that if the entire state space is informative, and the family of controllers is globally stabilising with appropriately chosen parameters, high probability stability guarantees can be derived.

A Framework for Adaptive Stabilisation of Nonlinear Stochastic Systems

TL;DR

This work develops a modular certainty-equivalence adaptive-control framework for discrete-time nonlinear stochastic systems with linearly parameterised uncertainty, combining a parameterised stabilising policy with regularised least squares (RLS) parameter estimation. It derives non-asymptotic, high-probability guarantees: (i) under regional excitation and a robustly positive invariant (RPI) set, a time-varying estimation error bound and almost-sure invariance of the state within the RPI region; (ii) under global excitation and a global stochastic Lyapunov function, high-probability stability bounds hold for any initial condition; and (iii) the results are illustrated on a piecewise-affine system and an input-constrained Gaussian-disturbed linear system. The analysis avoids requiring a priori bounded parameter sets and emphasizes informative data through excitation regions, enabling stability guarantees even when global stabilisability is unavailable. The framework thereby provides a practical, verifiable approach to learning-based adaptive stabilization for a broad class of stochastic nonlinear systems with linear-in-parameters structure, with clear conditions for regional and global stability guarantees. The results have potential impact on reliable autonomous control in uncertain, noisy environments where online learning and adaptation are essential.

Abstract

We consider the adaptive control problem for discrete-time, nonlinear stochastic systems with linearly parameterised uncertainty. Assuming access to a parameterised family of controllers that can stabilise the system in a bounded set within an informative region of the state space when the parameter is well-chosen, we propose a certainty equivalence learning-based adaptive control strategy, and subsequently derive stability bounds on the closed-loop system that hold for some probabilities. We then show that if the entire state space is informative, and the family of controllers is globally stabilising with appropriately chosen parameters, high probability stability guarantees can be derived.

Paper Structure

This paper contains 14 sections, 11 theorems, 90 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup and Framework
  3. Discrete-Time Stochastic System and Adaptive Control Framework
  4. Standing Assumptions
  5. Main Results
  6. Stability and Estimation with Regional Control and Excitation
  7. Stability with Global Control and Excitation
  8. Examples
  9. Example 1: Piecewise-Affine System
  10. Example 2: Input-Constrained Stochastic Linear System
  11. Proofs
  12. Estimation Error Bounds and Invariance Guarantee
  13. Stability Guarantees
  14. Conclusion

Key Result

Lemma 1

Consider feature map $\psi$ and controller family $\alpha$ satisfying Assumption ac:assump:measurable, and noise distributions $\mu_w,\mu_s$ satisfying Assumption ac:assump:process-noise. Suppose there exists a subset $\mathcal{X} \subseteq \mathbb{X}$ and constants $c_{\textnormal{PE1}}>0$ and $c_{ Then, $(\psi,\alpha,\mu_s,\mu_w)$ is $(\mathcal{X},c_{\textnormal{PE}},p_{\textnormal{PE}})$-region

Figures (1)

  • Figure 1: Median and $90$th percentile of $\vert X(t) \vert$ over 100 trials.

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Remark 1
  • Theorem 1
  • Remark 2
  • Definition 4
  • Theorem 2
  • Remark 3
  • ...and 9 more