Quantitative Resilience of Linear Systems

TL;DR

This paper introduces a method to quantify the resilience of linear systems with actuator malfunctions by deriving bounds on reach times using Lyapunov theory, demonstrated on a temperature control system.

Contribution

It provides a novel approach to quantify system resilience under actuator failures by establishing analytical bounds without requiring exact control solutions.

Findings

Derived bounds on reach times for linear systems with actuator malfunctions.

Quantitative resilience can be effectively bounded using Lyapunov-based methods.

Application demonstrated on a temperature control system.

Abstract

Actuator malfunctions may have disastrous consequences for systems not designed to mitigate them. We focus on the loss of control authority over actuators, where some actuators are uncontrolled but remain fully capable. To counteract the undesirable outputs of these malfunctioning actuators, we use real-time measurements and redundant actuators. In this setting, a system that can still reach its target is deemed resilient. To quantify the resilience of a system, we compare the shortest time for the undamaged system to reach the target with the worst-case shortest time for the malfunctioning system to reach the same target, i.e., when the malfunction makes that time the longest. Contrary to prior work on driftless linear systems, the absence of analytical expression for time-optimal controls of general linear systems prevents an exact calculation of quantitative resilience. Instead, relying on Lyapunov theory we derive analytical bounds on the nominal and malfunctioning reach times in order to bound quantitative resilience. We illustrate our work on a temperature control system.