Temporal Logic Resilience for Dynamical Systems

TL;DR

This paper introduces a formal framework using finite temporal logic to quantify and compute the resilience of cyber-physical systems against disturbances, with exact and approximate solutions for linear and nonlinear systems, demonstrated on practical examples.

Contribution

It develops a novel resilience metric based on temporal logic specifications and provides computational methods for both linear and nonlinear systems, including new classes of temporal specifications.

Findings

Exact solutions for linear systems and reachability problems.

Approximate solutions for finite-horizon reachability.

Successful demonstrations on building temperature, cruise control, and motor systems.

Abstract

We consider the notion of resilience for cyber-physical systems, that is, the ability of the system to withstand adverse events while maintaining acceptable functionality. We use finite temporal logic to express the requirements on the acceptable functionality and define the resilience metric as the maximum disturbance under which the system satisfies the temporal requirements. We fix a parameterized template for the set of disturbances and form a robust optimization problem under the system dynamics and the temporal specifications to find the maximum value of the parameter. Additionally, we introduce two novel classes of specifications: closed and convex finite temporal logics specifications, offering a comprehensive analysis of the resilience metric within these specific frameworks. From a computational standpoint, we present an exact solution for linear systems and exact-time reachability and finite-horizon safety, complemented by an approximate solution for finite-horizon reachability. Extending our findings to nonlinear systems, we leverage linear approximations and SMT-based approaches to offer viable computational methodologies. The theoretical results are demonstrated on the temperature regulation of buildings, adaptive cruise control and DC motors.