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Data-driven certificates of constraint enforcement and stability for unmodeled, discrete dynamical systems using tree data structures

Amy K. Strong, Ali Kashani, Claus Danielson, Leila J. Bridgeman

TL;DR

The paper presents a data-driven, model-free framework for certificates of constraint enforcement and stability in unmodeled, Lipschitz discrete-time systems. It combines a tree-based subdivision of the state constraint set with Lipschitz-based over-approximations to synthesize a true invariant set from deterministic samples, obviating the need for an initial invariant. A discontinuous piecewise affine Lyapunov function is then synthesized over the invariant set to certify asymptotic convergence to an invariant approximation of the minimal pi set, enabling uniform ultimate boundedness guarantees. The approach provides finite-time invariance guarantees and explicit data requirements, demonstrated on linear and nonlinear examples, with invariant-set sizes competitive to existing methods and clear applicability to safety-critical scenarios with limited system knowledge.

Abstract

This paper addresses the critical challenge of developing data-driven certificates for the stability and safety of unmodeled dynamical systems by leveraging a tree data structure and an upper bound of the system's Lipschitz constant. Previously, an invariant set was synthesized by iteratively expanding an initial invariant set. In contrast, this work iteratively prunes the constraint set to synthesize an invariant set -- eliminating the need for a known, initial invariant set. Furthermore, we provide stability assurances by characterizing the asymptotic stability of the system relative to an invariant approximation of the minimal positive invariant set through synthesis of a discontinuous piecewise affine Lyapunov function over the computed invariant set. The proposed method takes inspiration from subdivision techniques and requires no prior system knowledge beyond Lipschitz continuity.

Data-driven certificates of constraint enforcement and stability for unmodeled, discrete dynamical systems using tree data structures

TL;DR

The paper presents a data-driven, model-free framework for certificates of constraint enforcement and stability in unmodeled, Lipschitz discrete-time systems. It combines a tree-based subdivision of the state constraint set with Lipschitz-based over-approximations to synthesize a true invariant set from deterministic samples, obviating the need for an initial invariant. A discontinuous piecewise affine Lyapunov function is then synthesized over the invariant set to certify asymptotic convergence to an invariant approximation of the minimal pi set, enabling uniform ultimate boundedness guarantees. The approach provides finite-time invariance guarantees and explicit data requirements, demonstrated on linear and nonlinear examples, with invariant-set sizes competitive to existing methods and clear applicability to safety-critical scenarios with limited system knowledge.

Abstract

This paper addresses the critical challenge of developing data-driven certificates for the stability and safety of unmodeled dynamical systems by leveraging a tree data structure and an upper bound of the system's Lipschitz constant. Previously, an invariant set was synthesized by iteratively expanding an initial invariant set. In contrast, this work iteratively prunes the constraint set to synthesize an invariant set -- eliminating the need for a known, initial invariant set. Furthermore, we provide stability assurances by characterizing the asymptotic stability of the system relative to an invariant approximation of the minimal positive invariant set through synthesis of a discontinuous piecewise affine Lyapunov function over the computed invariant set. The proposed method takes inspiration from subdivision techniques and requires no prior system knowledge beyond Lipschitz continuity.

Paper Structure

This paper contains 12 sections, 9 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Geometric Conditions of Invariance
  4. Lyapunov Functions
  5. Data Driven Invariant Set Synthesis
  6. Data Driven Lyapunov Function Synthesis
  7. Convergence to a set
  8. Decrease Condition
  9. Lyapunov Function Conditions
  10. Numerical Examples
  11. Linear Dynamical System
  12. Nonlinear Dynamical System

Figures (5)

  • Figure 1: A representation of different stages in Algorithm \ref{['alg:findSet']} for a single node in the tree data structure. Each blue square is a member of the candidate invariant set, $\hat{\mathcal{S}}.$
  • Figure 2: Area and number of partitions of the candidate invariant set over each iteration of Algorithm \ref{['alg:findSet']}. The area of $\hat{\mathcal{S}}$ is compared to the maximal invariant set area calculated from MPT3 (MPT3).
  • Figure 3: A discontinuous Lyapunov function shows asymptotic convergence of \ref{['eq:exLinSys']} to the small invariant set, $\mathcal{B}_\mathcal{E} = [-0.25,0.375]\times [-0.375,0.25]$.
  • Figure 4: Area and number of partitions of the candidate invariant set over each iteration of Algorithm \ref{['alg:findSet']}.
  • Figure 5: A discontinuous Lyapunov function shows asymptotic convergence of \ref{['eq:exNonLinSys']} to the set, $\mathcal{B}_\mathcal{E} = [-0.21875,0.21875]\times [-0.21875, 0.21875]$.