Convex Computations for Controlled Safety Invariant Sets of Black-box Discrete-time Dynamical Systems
Taoran Wu, Yiling Xue, Jingduo Pan, Dejin Ren, Arvind Easwaran, Bai Xue
TL;DR
The paper tackles safe operation under black-box discrete-time dynamics by introducing PAC CSIS, a probabilistic relaxation of controlled safety invariant sets. It leverages barrier certificates and scenario optimization to recast CSIS computation into scalable linear programs, enabling data-driven guarantees without explicit system models. The method first derives PAC SISs for uncontrolled dynamics and then extends to controlled settings using an empirical weighting of inputs and iterative refinement. Practical significance lies in providing a provable, scalable framework for safety-critical applications with limited simulation data and unknown dynamics.
Abstract
Identifying controlled safety invariant sets (CSISs) is essential for safety-critical systems. This paper addresses the problem of computing CSISs for black-box discrete-time systems, where the dynamics are unknown and only limited simulation data are available. Traditionally, a CSIS requires that for every state in the set, there exists a control input that keeps the system within the set at the next step. However, enforcing such universal invariance, i.e., requiring the set to remain controlled invariant for all states, is often overly restrictive or impractical for black-box systems. To address this, we introduce the notion of a Probably Approximately Correct (PAC) CSIS, in which, with prescribed confidence, there exists a suitable control input to keep the system within the set at the next step for at least a specified fraction of the states. Our approach leverages barrier functions and scenario optimization, yielding a tractable linear programming method for estimating PAC CSISs. Several illustrative examples demonstrate the effectiveness of the proposed framework.