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A Hamilton-Jacobi-Bellman Approach to Ellipsoidal Approximations of Reachable Sets for Linear Time-Varying Systems

Vincent Liu, Chris Manzie, Peter M. Dower

TL;DR

The paper tackles the challenge of computing reachable sets for high-dimensional systems by leveraging viscosity supersolutions and subsolutions of Hamilton-Jacobi-Bellman equations to obtain under- and over-approximations of backwards reachable sets. It builds ellipsoidal approximations by evolving unions (under-approximations) and intersections (over-approximations) of ellipsoids for time-varying linear systems with ellipsoidal input and terminal sets, using matrix differential equations to achieve polynomial-time computation. A key contribution is showing that these ellipsoidal bounds can be made tight along system trajectories and that they inherit recursive feasibility/infeasibility properties, providing rigorous guarantees for control design and safety verification. The numerical example corroborates substantial computational advantages over grid-based methods while maintaining tight bounds, highlighting practical relevance for high-dimensional autonomous systems.

Abstract

Reachable sets for a dynamical system describe collections of system states that can be reached in finite time, subject to system dynamics. They can be used to guarantee goal satisfaction in controller design or to verify that unsafe regions will be avoided. However, general-purpose methods for computing these sets suffer from the curse of dimensionality, which typically prohibits their use for systems with more than a small number of states, even if they are linear. In this paper, we demonstrate that viscosity supersolutions and subsolutions of a Hamilton-Jacobi-Bellman equation can be used to generate, respectively, under-approximating and over-approximating reachable sets for time-varying nonlinear systems. Based on this observation, we derive dynamics for a union and intersection of ellipsoidal sets that, respectively, under-approximate and over-approximate the reachable set for linear time-varying systems subject to an ellipsoidal input constraint and an ellipsoidal terminal (or initial) set. We demonstrate that the dynamics for these ellipsoids can be selected to ensure that their boundaries coincide with the boundary of the exact reachable set along a solution of the system. The ellipsoidal sets can be generated with polynomial computational complexity in the number of states, making our approximation scheme computationally tractable for continuous-time linear time-varying systems of relatively high dimension.

A Hamilton-Jacobi-Bellman Approach to Ellipsoidal Approximations of Reachable Sets for Linear Time-Varying Systems

TL;DR

The paper tackles the challenge of computing reachable sets for high-dimensional systems by leveraging viscosity supersolutions and subsolutions of Hamilton-Jacobi-Bellman equations to obtain under- and over-approximations of backwards reachable sets. It builds ellipsoidal approximations by evolving unions (under-approximations) and intersections (over-approximations) of ellipsoids for time-varying linear systems with ellipsoidal input and terminal sets, using matrix differential equations to achieve polynomial-time computation. A key contribution is showing that these ellipsoidal bounds can be made tight along system trajectories and that they inherit recursive feasibility/infeasibility properties, providing rigorous guarantees for control design and safety verification. The numerical example corroborates substantial computational advantages over grid-based methods while maintaining tight bounds, highlighting practical relevance for high-dimensional autonomous systems.

Abstract

Reachable sets for a dynamical system describe collections of system states that can be reached in finite time, subject to system dynamics. They can be used to guarantee goal satisfaction in controller design or to verify that unsafe regions will be avoided. However, general-purpose methods for computing these sets suffer from the curse of dimensionality, which typically prohibits their use for systems with more than a small number of states, even if they are linear. In this paper, we demonstrate that viscosity supersolutions and subsolutions of a Hamilton-Jacobi-Bellman equation can be used to generate, respectively, under-approximating and over-approximating reachable sets for time-varying nonlinear systems. Based on this observation, we derive dynamics for a union and intersection of ellipsoidal sets that, respectively, under-approximate and over-approximate the reachable set for linear time-varying systems subject to an ellipsoidal input constraint and an ellipsoidal terminal (or initial) set. We demonstrate that the dynamics for these ellipsoids can be selected to ensure that their boundaries coincide with the boundary of the exact reachable set along a solution of the system. The ellipsoidal sets can be generated with polynomial computational complexity in the number of states, making our approximation scheme computationally tractable for continuous-time linear time-varying systems of relatively high dimension.
Paper Structure (14 sections, 20 theorems, 111 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 14 sections, 20 theorems, 111 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background Theory
  3. Hamilton-Jacobi-Bellman Reachability Analysis
  4. Viscosity Solutions of HJB Equations
  5. Approximating Reachable Sets
  6. Under-approximating Backwards Reachable Sets
  7. Over-approximating Backwards Reachable Sets
  8. Generating Viscosity Supersolutions and Subsolutions
  9. Approximate Reachability for LTV Systems
  10. Ellipsoidal Under-approximation of Backwards Reachable Sets
  11. Ellipsoidal Over-approximation of Backwards Reachable Sets
  12. Numerical Implementation
  13. Numerical Example
  14. Conclusion

Key Result

Theorem 1

Let $f:[0,T]\times{\mathbb{R}}^n\times{\mathbb{U}}\rightarrow{\mathbb{R}}$ satisfy Assumptions assumption: background: flow field conditions and assumption: background: convex flow field. Consider the HJB PDE given by where $g\in{\mathscr{C}}\left({\mathbb{R}}^n;\,{\mathbb{R}}\right)$ defines the terminal set $\mathcal{X}$ in eq: background: closed terminal set, and the Hamiltonian $H$ is defined

Figures (4)

  • Figure 1: Diagram depicting the backwards reachable set $\mathcal{G}(t)$ and the forwards reachable set $\mathcal{F}(t)$, which enclose all admissible trajectories heading to or away from the set $\mathcal{X}$.
  • Figure 2: Comparison of the union of ellipsoids $\underline{\mathcal{G}}(t)$ and the backwards reachable set $\mathcal{G}(t)$ for \ref{['eq: numerical ex: forced oscillator system description']} at time $t = 0$. Grey dotted lines give the outline of each individual ellipsoid in $\underline{\mathcal{G}}(t)$. Animation: https://youtu.be/XNGjqlYKxnU.
  • Figure 3: Comparison of the intersection of ellipsoids $\overline{\mathcal{G}}(t)$ and the backwards reachable set $\mathcal{G}(t)$ for \ref{['eq: numerical ex: forced oscillator system description']} at time $t = 0$. Grey dotted lines give the outline of each individual ellipsoid in $\overline{\mathcal{G}}(t)$. Animation: https://youtu.be/-0sMzyfY2GY.
  • Figure 4: Comparison of computation times for the sets $\underline{\mathcal{G}}(t)$, $\overline{\mathcal{G}}(t)$, and $\mathcal{G}(t)$ as the number of states $n$ increases.

Theorems & Definitions (47)

  • Remark 1
  • Definition 1
  • Theorem 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Definition 4
  • Lemma 2
  • Theorem 2
  • Definition 5
  • ...and 37 more