Finding Control Invariant Sets via Lipschitz Constants of Linear Programs

TL;DR

This paper introduces a novel method for expanding control invariant sets using Lipschitz constants of LPs, enabling safe set enlargement and verification through finite LPs, with applications in safety filters for control systems.

Contribution

It proposes a new approach to expand control invariant sets safely by leveraging Lipschitz constants and differentiable optimization, transforming continuous verification into finite LPs.

Findings

The method guarantees control invariance during set expansion.

Lipschitz constants enable finite verification of continuous invariance.

Application demonstrated in safety filters within CBF frameworks.

Abstract

Control invariant sets play an important role in safety-critical control and find broad application in numerous fields such as obstacle avoidance for mobile robots. However, finding valid control invariant sets of dynamical systems under input limitations is notoriously difficult. We present an approach to safely expand an initial set while always guaranteeing that the set is control invariant. Specifically, we define an expansion law for the boundary of a set and check for control invariance using Linear Programs (LPs). To verify control invariance on a continuous domain, we leverage recently proposed Lipschitz constants of LPs to transform the problem of continuous verification into a finite number of LPs. Using concepts from differentiable optimization, we derive the safe expansion law of the control invariant set and show how it can be interpreted as a second invariance problem in the space of possible boundaries. Finally, we show how the obtained set can be used to obtain a minimally invasive safety filter in a Control Barrier Function (CBF) framework. Our work is supported by theoretical results as well as numerical examples.