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Data driven synthesis of provable invariant sets via stochastically sampled data

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

TL;DR

This work tackles safety-critical control by constructing provable PI sets for possibly unmodeled discrete-time systems using only pre-collected data and a known Lipschitz bound $L$. The authors develop a data-driven, geometric approach that over-approximates one-step evolution with balls around sampled states and leverages $\epsilon$-net set coverage to certify invariance, implemented via a tree-based partitioning algorithm. They prove that the method yields a PI set that approximates the maximal PI set and provide deterministic and probabilistic bounds linking the number of samples $M$ to the volume of the resulting invariant set. The approach requires no full system model, only data $\mathcal{D}=\{\mathbf{x}_j, \mathbf{x}_j^+\}$ and Lipschitz information, and is validated on unmodeled linear and nonlinear 2D systems, showing PI-set volumes grow with more data and correlate with model-based or barrier-function benchmarks. This provides a practical, provable pathway to data-driven safety guarantees in environments with unmodeled dynamics.

Abstract

Positive invariant (PI) sets are essential for ensuring safety, i.e. constraint adherence, of dynamical systems. With the increasing availability of sampled data from complex (and often unmodeled) systems, it is advantageous to leverage these data sets for PI set synthesis. This paper uses data driven geometric conditions of invariance to synthesize PI sets from data. Where previous data driven, set-based approaches to PI set synthesis used deterministic sampling schemes, this work instead synthesizes PI sets from any pre-collected data sets. Beyond a data set and Lipschitz continuity, no additional information about the system is needed. A tree data structure is used to partition the space and select samples used to construct the PI set, while Lipschitz continuity is used to provide deterministic guarantees of invariance. Finally, probabilistic bounds are given on the number of samples needed for the algorithm to determine of a certain volume.

Data driven synthesis of provable invariant sets via stochastically sampled data

TL;DR

This work tackles safety-critical control by constructing provable PI sets for possibly unmodeled discrete-time systems using only pre-collected data and a known Lipschitz bound . The authors develop a data-driven, geometric approach that over-approximates one-step evolution with balls around sampled states and leverages -net set coverage to certify invariance, implemented via a tree-based partitioning algorithm. They prove that the method yields a PI set that approximates the maximal PI set and provide deterministic and probabilistic bounds linking the number of samples to the volume of the resulting invariant set. The approach requires no full system model, only data and Lipschitz information, and is validated on unmodeled linear and nonlinear 2D systems, showing PI-set volumes grow with more data and correlate with model-based or barrier-function benchmarks. This provides a practical, provable pathway to data-driven safety guarantees in environments with unmodeled dynamics.

Abstract

Positive invariant (PI) sets are essential for ensuring safety, i.e. constraint adherence, of dynamical systems. With the increasing availability of sampled data from complex (and often unmodeled) systems, it is advantageous to leverage these data sets for PI set synthesis. This paper uses data driven geometric conditions of invariance to synthesize PI sets from data. Where previous data driven, set-based approaches to PI set synthesis used deterministic sampling schemes, this work instead synthesizes PI sets from any pre-collected data sets. Beyond a data set and Lipschitz continuity, no additional information about the system is needed. A tree data structure is used to partition the space and select samples used to construct the PI set, while Lipschitz continuity is used to provide deterministic guarantees of invariance. Finally, probabilistic bounds are given on the number of samples needed for the algorithm to determine of a certain volume.

Paper Structure

This paper contains 13 sections, 12 theorems, 7 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Geometric conditions of positive invariance
  4. Data driven geometric conditions of invariance
  5. Set Coverage
  6. Main Results
  7. Data Driven PI Sets and Set Coverage
  8. Algorithm for Data Driven PI Set Synthesis
  9. Sample Bounds
  10. Numerical examples
  11. Unmodeled linear system
  12. Unmodeled nonlinear system
  13. Conclusions

Key Result

Lemma 1

The set $\mathcal{S} \subseteq \Omega$ is pi for mapping $T: \mathbb{R}^n {\rightarrow} \mathbb{R}^n,$ if and only if $\text{Suc}(\mathcal{S})\subseteq \mathcal{S}$ or $\mathcal{S} \subseteq \text{Pre}(\mathcal{S})$.

Figures (4)

  • Figure 1: The volume of the pi set for \ref{['eq:exLinSys']} within $\mathcal{X}$ found using \ref{['alg:findSet']} is compared with the number samples used within the data set. The dotted black line shows the volume of the maximal pi set found using the model-based geometric method kerrigan2000robust via MPT3.
  • Figure 2: The pi set found by \ref{['alg:findSet']} for \ref{['eq:exLinSys']} is shown for a data set of (a) $M=100$ uniformly sampled points and (b) $M=10,000$ uniformly sampled points. Each blue square is a leaf node of the tree data structure. In both figures, the solid black line denotes the boundary of the maximal pi set of \ref{['eq:exLinSys']} found by the seminal, model-based geometric method kerrigan2000robust via MPT3.
  • Figure 3: The volume of the pi set for \ref{['eq:exNonlinSys']} within $\mathcal{X}$ found using \ref{['alg:findSet']} is compared with the number samples used within the data set. The dotted black line shows the volume of the pi set found using a barrier function.
  • Figure 4: The pi set found by \ref{['alg:findSet']} for \ref{['eq:exNonlinSys']} is shown for a data set of (a) $M=3,000$ uniformly sampled points and (b) $M=10,000$ uniformly sampled points. Each blue square is a leaf node of the tree data structure. In comparison, the red dashed curve denotes the boundary of the pi set induced via the barrier function, $V(\mathbf{x}) = x_1^2 + x_2^2$, which is determined using the system model.

Theorems & Definitions (27)

  • Definition II.1: Invariant Setalberto2007invariance
  • Definition II.2: $\lambda-$contractive Set blanchini2015set
  • Definition II.3: Precursor Set alberto2007invariance
  • Definition II.4: Successor Set borrelli2017predictive
  • Lemma 1: Invariant Setalberto2007invariancedorea1999b
  • Lemma 2: strong2025data
  • Lemma 3: strong2025data
  • Definition II.5: Covering number wainwright2019high
  • Lemma 4: Proposition 4.2.12 from vershynin2018high
  • Lemma 5: Lemma 3 korda2020computing
  • ...and 17 more