Verifying Contracts for Perturbed Control Systems using Linear Programming

Verifying specifications for large-scale control systems is of utmost importance, but can be hard in practice as most formal verification methods can not handle high-dimensional dynamics. Contract theory has been proposed as a modular alternative to formal verification in which specifications are defined by assumptions on the inputs to a component and guarantees on its outputs. In this paper, we present linear-programming-based tools for verifying contracts for control systems. We first consider the problem of verifying contracts defined by time-invariant inequalities for unperturbed systems. We use $k$-induction to show that contract verification can be achieved by considering a collection of implications between inequalities, which are then recast as linear programs. We then move our attention to perturbed systems. We present a comparison-based framework, verifying that a perturbed system satisfies a contract by checking that the corresponding unperturbed system satisfies a robustified (and $\epsilon$-approximated) contract. In both cases, we present explicit algorithms for contract verification, proving their correctness and analyzing their complexity. We also demonstrate the verification process for two case studies, one considering a two-vehicle autonomous driving scenario, and one considering formation control of a multi-agent system.