Formal Abstraction of Linear Systems via Polyhedral Lyapunov Functions

TL;DR

This paper introduces a method to create finite abstractions of linear systems using polyhedral Lyapunov functions, enabling system verification against complex temporal logic specifications.

Contribution

It presents a novel algorithm for constructing bisimulation quotients of linear systems that preserve observations over polytopic regions, facilitating formal verification.

Findings

Successfully generates finite bisimulation quotients for linear systems.

Guarantees bisimulation within each Lyapunov sublevel set.

Enables verification of linear systems against Linear Temporal Logic formulas.

Abstract

In this paper we present an abstraction algorithm that produces a finite bisimulation quotient for an autonomous discrete-time linear system. We assume that the bisimulation quotient is required to preserve the observations over an arbitrary, finite number of polytopic subsets of the system state space. We generate the bisimulation quotient with the aid of a sequence of contractive polytopic sublevel sets obtained via a polyhedral Lyapunov function. The proposed algorithm guarantees that at iteration $i$, the bisimulation of the system within the $i$-th sublevel set of the Lyapunov function is completed. We then show how to use the obtained bisimulation quotient to verify the system with respect to arbitrary Linear Temporal Logic formulas over the observed regions.