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Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimation

Mohamed Serry, Maxwell Fitzsimmons, Jun Liu

TL;DR

A novel framework for the accurate estimation of safe (state-constrained) and robust DOAs for discrete-time nonlinear uncertain systems with continuous dynamics, open safe sets, compact disturbance sets, and uniformly locally $\ell_p$-stable compact RISs is proposed.

Abstract

Analyzing nonlinear systems with attracting robust invariant sets (RISs) requires estimating their domains of attraction (DOAs). Despite extensive research, accurately characterizing DOAs for general nonlinear systems remains challenging due to both theoretical and computational limitations, particularly in the presence of uncertainties and state constraints. In this paper, we propose a novel framework for the accurate estimation of safe (state-constrained) and robust DOAs for discrete-time nonlinear uncertain systems with continuous dynamics, open safe sets, compact disturbance sets, and uniformly locally $\ell_p$-stable compact RISs. The notion of uniform $\ell_p$ stability is quite general and encompasses, as special cases, uniform exponential and polynomial stability. The DOAs are characterized via newly introduced value functions defined on metric spaces of compact sets. We establish their fundamental mathematical properties and derive the associated Bellman-type (Zubov-type) functional equations. Building on this characterization, we develop a physics-informed neural network (NN) framework to learn the corresponding value functions by embedding the derived Bellman-type equations directly into the training process. To obtain certifiable estimates of the safe robust DOAs from the learned neural approximations, we further introduce a verification procedure that leverages existing formal verification tools. The effectiveness and applicability of the proposed methodology are demonstrated through four numerical examples involving nonlinear uncertain systems subject to state constraints, and its performance is compared with existing methods from the literature.

Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimation

TL;DR

A novel framework for the accurate estimation of safe (state-constrained) and robust DOAs for discrete-time nonlinear uncertain systems with continuous dynamics, open safe sets, compact disturbance sets, and uniformly locally -stable compact RISs is proposed.

Abstract

Analyzing nonlinear systems with attracting robust invariant sets (RISs) requires estimating their domains of attraction (DOAs). Despite extensive research, accurately characterizing DOAs for general nonlinear systems remains challenging due to both theoretical and computational limitations, particularly in the presence of uncertainties and state constraints. In this paper, we propose a novel framework for the accurate estimation of safe (state-constrained) and robust DOAs for discrete-time nonlinear uncertain systems with continuous dynamics, open safe sets, compact disturbance sets, and uniformly locally -stable compact RISs. The notion of uniform stability is quite general and encompasses, as special cases, uniform exponential and polynomial stability. The DOAs are characterized via newly introduced value functions defined on metric spaces of compact sets. We establish their fundamental mathematical properties and derive the associated Bellman-type (Zubov-type) functional equations. Building on this characterization, we develop a physics-informed neural network (NN) framework to learn the corresponding value functions by embedding the derived Bellman-type equations directly into the training process. To obtain certifiable estimates of the safe robust DOAs from the learned neural approximations, we further introduce a verification procedure that leverages existing formal verification tools. The effectiveness and applicability of the proposed methodology are demonstrated through four numerical examples involving nonlinear uncertain systems subject to state constraints, and its performance is compared with existing methods from the literature.
Paper Structure (32 sections, 28 theorems, 126 equations, 4 figures, 1 table)

This paper contains 32 sections, 28 theorems, 126 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Problem formulation
  4. Properties of DOA
  5. Value functions
  6. Properties of the value functions
  7. Characterizing the value functions: Bellman-type equations
  8. DOA estimation
  9. Finite-horizon approximation of the value functions
  10. Physics-informed and set-based neural network learning
  11. Verified safe robust ROAs from neural network approximation
  12. Initial ROA
  13. Enlarging the initial ROA through verification
  14. NN ROAs via verification
  15. Computation of candidate values for $c_{2}$, $\omega_{1}$, and $\omega_{2}$
  16. ...and 17 more sections

Key Result

Lemma 1

Let $X\subseteq \mathbb{R}^{n}$ be nonempty and open and $\Omega \in \mathcal{K}(X)$. Then there exists $\delta>0$ such that $\Omega+\delta \mathbb{B}_{n}\subseteq X.$

Figures (4)

  • Figure 1: Certified ROAs for Example \ref{['sec:TwoMachine']} under Scenario 1 (uncertainty neglected, top) and Scenario 2 (full disturbance considered, bottom). Red circles indicate the state constraints. The nominally trained neural network fails to yield a certifiable ROA under Scenario 2.
  • Figure 2: Certified ROAs for Example \ref{['sec:PolynomiallyStable']} under Scenario 1 (uncertainty neglected, top) and Scenario 2 (full disturbance considered, bottom). Red lines indicate the state constraints.
  • Figure 3: Certified ROAs for Example \ref{['sec:Pendulum']} under Scenario 1 (uncertainty neglected, top) and Scenario 2 (full disturbance considered, bottom). The nominally trained neural network fails to yield a certifiable ROA under Scenario 2.
  • Figure 4: Certified ROAs for Example \ref{['sec:Rational']}. The ROA based on parameter-dependent Lyapunov functions is digitized from coutinho2013local.

Theorems & Definitions (60)

  • Lemma 1: See the proof in Appendix \ref{['Proof_lem:enlargement_of_compact']}
  • Lemma 2: See the proof in Appendix \ref{['Proof_lem:CompactSetsMetric']}
  • Lemma 3: See the proof in Appendix \ref{['Proof_lem:FCts']}
  • Lemma 4: See the proof in Appendix \ref{['Proof_lem:SemiGroup']}
  • Lemma 5: Follows from Lemmas \ref{['lem:FCts']} and \ref{['lem:SemiGroup']}
  • Remark 1
  • Remark 2: Generality of $\ell_p$ stability
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 50 more