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Data-driven Invariance for Reference Governors

Ali Kashani, Claus Danielson

TL;DR

The paper addresses constraint enforcement for unmodeled nonlinear systems by learning a constraint-admissible invariant set directly from data and using it to synthesize a data-driven Reference Governor (RG). A Lyapunov-like function $V(x;\\bar{r})=\\varphi(x;\\bar{r})^T P \\varphi(x;\\bar{r})$ with kernel-based lifting defines a PI set $\\mathcal{O}({\\bar{r}})$, tightened for robustness, and a data-driven SDP (reduced to an LP) computes $P$ from finite data. By inner-approximating CI sets as unions of PI sets over a reference subset $\\bar{\\mathcal{R}}$, the RG reduces to a tractable one-step policy selection problem that preserves recursive feasibility. The approach is validated on a linear illustrative system and a nonlinear autonomous lane-keeping problem, showing effective constraint enforcement and practical computational scalability via LP relaxations. These results enable safe constraint handling for complex CPS with limited models, using offline data and scalable optimization.

Abstract

This paper presents a novel approach to synthesizing positive invariant sets for unmodeled nonlinear systems using direct data-driven techniques. The data-driven invariant sets are used to design a data-driven reference governor that selects a reference for the closed-loop system to enforce constraints. Using kernel-basis functions, we solve a semi-definite program to learn a sum-of-squares Lyapunov-like function whose unity level-set is a constraint admissible positive invariant set, which determines the constraint admissible states as well as reference inputs. Leveraging Lipschitz properties of the system, we prove that tightening the model-based design ensures robustness of the data-driven invariant set to the inherent plant uncertainty in a data-driven framework. To mitigate the curse-of-dimensionality, we repose the semi-definite program into a linear program. We validate our approach through two examples: First, we present an illustrative example where we can analytically compute the maximum positive invariant set and compare with the presented data-driven invariant set. Second, we present a practical autonomous driving scenario to demonstrate the utility of the presented method for nonlinear systems.

Data-driven Invariance for Reference Governors

TL;DR

The paper addresses constraint enforcement for unmodeled nonlinear systems by learning a constraint-admissible invariant set directly from data and using it to synthesize a data-driven Reference Governor (RG). A Lyapunov-like function with kernel-based lifting defines a PI set , tightened for robustness, and a data-driven SDP (reduced to an LP) computes from finite data. By inner-approximating CI sets as unions of PI sets over a reference subset , the RG reduces to a tractable one-step policy selection problem that preserves recursive feasibility. The approach is validated on a linear illustrative system and a nonlinear autonomous lane-keeping problem, showing effective constraint enforcement and practical computational scalability via LP relaxations. These results enable safe constraint handling for complex CPS with limited models, using offline data and scalable optimization.

Abstract

This paper presents a novel approach to synthesizing positive invariant sets for unmodeled nonlinear systems using direct data-driven techniques. The data-driven invariant sets are used to design a data-driven reference governor that selects a reference for the closed-loop system to enforce constraints. Using kernel-basis functions, we solve a semi-definite program to learn a sum-of-squares Lyapunov-like function whose unity level-set is a constraint admissible positive invariant set, which determines the constraint admissible states as well as reference inputs. Leveraging Lipschitz properties of the system, we prove that tightening the model-based design ensures robustness of the data-driven invariant set to the inherent plant uncertainty in a data-driven framework. To mitigate the curse-of-dimensionality, we repose the semi-definite program into a linear program. We validate our approach through two examples: First, we present an illustrative example where we can analytically compute the maximum positive invariant set and compare with the presented data-driven invariant set. Second, we present a practical autonomous driving scenario to demonstrate the utility of the presented method for nonlinear systems.
Paper Structure (12 sections, 6 theorems, 38 equations, 6 figures, 2 algorithms)

This paper contains 12 sections, 6 theorems, 38 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Reference Governor Synthesis Problem
  3. Reference Governors
  4. Data-driven Reference Governor
  5. PI Inner-Approximations of Maximal CI
  6. Model-Driven Approach
  7. Data-Driven Approach
  8. Computational Tractability
  9. Numerical Examples
  10. Illustrative Example
  11. Autonomous Lane-Keeping
  12. Conclusion

Key Result

Lemma 1

rg eq:rg is recursively feasible for all the states $x\in\textnormal{Pre}(\mathcal{C})$ in the one-step predecessor $\textnormal{Pre}(\mathcal{C})$ of $\mathcal{C}$ if and only if $\mathcal{C}$ is ci.

Figures (6)

  • Figure 1: Reference governor (rg) design for a system equipped with an inner-controller.
  • Figure 2: The blue polyhedron is the maximal admissible set $\Lambda_\infty$ of the system \ref{['eq:linear']} and the orange balloon like set is our data driven admissible set $\bar{\Lambda}$, produced by Algorithm \ref{['alg:mci']}. Figures are the projection of the corresponding sets onto (a) $x_1, x_2$ (b) $x_1, r$, and (c) $x_2, r$ planes, where $x_1$, $x_2$, and $r$ are respectively position, velocity and reference.
  • Figure 3: Reference tracking and constraint enforcement for the linear system \ref{['eq:linear']} by our data-driven rg (Algorithm \ref{['alg:rg']}).
  • Figure 4: Admissible set for the nonlinear kinetic bicycle model \ref{['eq:bicycle']} generated by Algorithm \ref{['alg:mci']}. Each slice is a pi set $\mathcal{O}(\bar{r})$ for the constant $\bar{r}$.
  • Figure 5: Constraint enforcement in tracking of infeasible references for the nonlinear kinetic bicycle model \ref{['eq:bicycle']} at nominal speed $v=45 mph$.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Proposition 1
  • Corollary 1
  • Theorem 1
  • Theorem 2
  • Proposition 2