Projection-free computation of robust controllable sets with constrained zonotopes

2024-03-20

arXiv.org

AI Summary

This paper addresses the efficient computation of robust controllable sets (RCS) for discrete-time linear systems subject to additive uncertainties. We propose a novel projection-free and convex-optimization-free algorithm capable of computing inner and outer approximations of the RCS under both ellipsoidal and constrained-zonotopic uncertainty descriptions. Key contributions include: (1) the first closed-form, least-squares-based approximation of the Pontryagin difference; (2) the first fully projection-free RCS computation framework for constrained-zonotopic uncertainties; and (3) guaranteed convergence of all approximations to the exact RCS under verifiable conditions. The method integrates constrained-zonotope representations, convex polyhedral outer approximations, and robust control-invariant set synthesis. Experiments demonstrate scalability: computing a 20-step inner approximation of the RCS for a 100-dimensional system takes under 15 seconds. The approach is successfully applied to safety-critical abort trajectory design for spacecraft performing near-straight-line halo-orbit rendezvous under uncertainty.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Shape-Constrained Domains
• Stabilization Control
• Uncertainty Handling

Innovation

• Shape-constrained Domains
• Efficient Computation
• High-dimensional Systems