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Projection-free computation of robust controllable sets with constrained zonotopes

Abraham P. Vinod, Avishai Weiss, Stefano Di Cairano

TL;DR

The paper tackles robust controllable set computation for discrete-time linear systems with additive uncertainty by introducing a projection-free, least-squares framework based on constrained zonotopes. It develops both inner and outer approximations of Pontryagin differences where the minuend is a CZ and the subtrahend is symmetric, convex, and compact, with closed-form results for several common subtrahends (ellipsoids, zonotopes, and their unions). These approximations feed into efficient RC-set recursions, yielding scalable inner RC sets for high-dimensional problems and enabling exactness under invertible CZ representations. The approach is validated through case studies including a 100D RC set and abort-safe spacecraft rendezvous, demonstrating orders-of-magnitude speedups over traditional polytope-based methods while maintaining meaningful accuracy. Overall, the work provides a practical, scalable solution for robust planning and verification in high-dimensional systems with structured uncertainties.

Abstract

We study the problem of computing robust controllable sets for discrete-time linear systems with additive uncertainty. We propose a tractable and scalable approach to inner- and outer-approximate robust controllable sets using constrained zonotopes, when the additive uncertainty set is a symmetric, convex, and compact set. Our least-squares-based approach uses novel closed-form approximations of the Pontryagin difference between a constrained zonotopic minuend and a symmetric, convex, and compact subtrahend. Unlike existing approaches, our approach does not rely on convex optimization solvers, and is projection-free for ellipsoidal and zonotopic uncertainty sets. We also propose a least-squares-based approach to compute a convex, polyhedral outer-approximation to constrained zonotopes, and characterize sufficient conditions under which all these approximations are exact. We demonstrate the computational efficiency and scalability of our approach in several case studies, including the design of abort-safe rendezvous trajectories for a spacecraft in near-rectilinear halo orbit under uncertainty. Our approach can inner-approximate a 20-step robust controllable set for a 100-dimensional linear system in under 15 seconds on a standard computer.

Projection-free computation of robust controllable sets with constrained zonotopes

TL;DR

The paper tackles robust controllable set computation for discrete-time linear systems with additive uncertainty by introducing a projection-free, least-squares framework based on constrained zonotopes. It develops both inner and outer approximations of Pontryagin differences where the minuend is a CZ and the subtrahend is symmetric, convex, and compact, with closed-form results for several common subtrahends (ellipsoids, zonotopes, and their unions). These approximations feed into efficient RC-set recursions, yielding scalable inner RC sets for high-dimensional problems and enabling exactness under invertible CZ representations. The approach is validated through case studies including a 100D RC set and abort-safe spacecraft rendezvous, demonstrating orders-of-magnitude speedups over traditional polytope-based methods while maintaining meaningful accuracy. Overall, the work provides a practical, scalable solution for robust planning and verification in high-dimensional systems with structured uncertainties.

Abstract

We study the problem of computing robust controllable sets for discrete-time linear systems with additive uncertainty. We propose a tractable and scalable approach to inner- and outer-approximate robust controllable sets using constrained zonotopes, when the additive uncertainty set is a symmetric, convex, and compact set. Our least-squares-based approach uses novel closed-form approximations of the Pontryagin difference between a constrained zonotopic minuend and a symmetric, convex, and compact subtrahend. Unlike existing approaches, our approach does not rely on convex optimization solvers, and is projection-free for ellipsoidal and zonotopic uncertainty sets. We also propose a least-squares-based approach to compute a convex, polyhedral outer-approximation to constrained zonotopes, and characterize sufficient conditions under which all these approximations are exact. We demonstrate the computational efficiency and scalability of our approach in several case studies, including the design of abort-safe rendezvous trajectories for a spacecraft in near-rectilinear halo orbit under uncertainty. Our approach can inner-approximate a 20-step robust controllable set for a 100-dimensional linear system in under 15 seconds on a standard computer.
Paper Structure (29 sections, 9 theorems, 56 equations, 7 figures, 4 tables, 5 algorithms)

This paper contains 29 sections, 9 theorems, 56 equations, 7 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Set representations
  4. Set operations
  5. Robust controllable set
  6. Inner-approximation of Pontryagin difference
  7. Full-dimensional constrained zonotopes
  8. Inner-approximation of the Pontryagin difference
  9. Relation of Algo. \ref{['algo:inner_approx']} with existing two-stage approach for zonotopic subtrahend
  10. Outer-approximation of Pontryagin difference and sufficient conditions for exactness
  11. Outer-approximating convex polyhedron for a given constrained zonotope
  12. Outer-approximation of the Pontryagin difference
  13. Sufficient conditions for exactness
  14. Implementation considerations
  15. Inner-approximation of robust controllable sets
  16. ...and 14 more sections

Key Result

Proposition 1

The following statements hold: 1) Every representation $(G_C,c_C,A_C,b_C)$ of a full-dimensional polytope $\mathcal{C}$ satisfies $\operatorname{rank}(G_C)=n\leq {N_{C}}$. 2) A polytope is full-dimensional if and only if the polytope is non-empty and it has a MinRow representation. 3) Algo. algo:min

Figures (7)

  • Figure 1: Relationship between various representations discussed in the paper. For any pair of representations $R_1, R_2$, a dashed arrow from $R_1$ to $R_2$ shows that $R_1$ is also $R_2$.
  • Figure 2: Robust controllable sets computed using the recursion in Sec. \ref{['sub:set_recursion_no_inv_A']} for Sec. \ref{['sub:ex1']} with a ball-shaped $\mathcal{W}$ (left) and an ellipsoidal $\mathcal{W}$ (right). We compare the sets obtained with the proposed approximations to the sets from existing approaches (exact MPT3 and the two-stage approach yang_efficient_2022). The insets in the left figure show that the proposed approach with ellipsoidal $\mathcal{W}$ (cyan) provides sufficiently accurate inner-approximations of the exact RC set (white).
  • Figure 3: Snapshots of the $100$-step robust controllable sets computed using Sec. \ref{['sub:set_recursion_inv_A']} for Sec. \ref{['sub:ex2']} at recursion steps $t\in\{0, 40, 80\}$. Our approach computes inner-approximations of the RC set that are similar to those obtained using the exact approach MPT3 and the two-stage approach yang_efficient_2022, with significantly lower computational effort (see Table \ref{['tab:comparison_yang']}).
  • Figure 4: Chain of ${N_M}$ mass-spring-damper systems.
  • Figure 5: Time taken by various methods to compute the RC sets for varying system dimension $n$. The proposed method takes $12.52$ seconds to inner-approximate $20$-step RC sets for a $100$-dimensional system. In contrast, existing methods (the exact computation using MPT3 MPT3 and the two-stage approach in yang_efficient_2022) require longer computation time to tackle lower dimensional systems. We also report the time taken by the proposed method to inner-approximate $40$-step RC sets.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Remark 1
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • ...and 12 more