Stochastic Analysis of Control Systems Subject to Communication and Computation Faults

Abstract

Control theory allows one to design controllers that are robust to external disturbances, model simplification, and modelling inaccuracy. Researchers have investigated whether the robustness carries on to the controller’s digital implementation, mostly looking at how the controller reacts to either communication or computational problems. Communication problems are typically modelled using random variables (i.e., estimating the probability that a fault will occur during a transmission), while computational problems are modelled using deterministic guarantees on the number of deadlines that the control task has to meet. These fault models allow the engineer to both design robust controllers and assess the controllers’ behaviour in the presence of isolated faults. Despite being very relevant for the real-world implementations of control system, the question of what happens when these faults occur simultaneously does not yet have a proper answer. In this paper, we answer this question in the stochastic setting, using the theory of Markov Jump Linear Systems to provide stability contracts with almost sure guarantees of convergence. For linear time-invariant Markov jump linear systems, mean square stability implies almost sure convergence – a property that is central to our investigation. Our research primarily emphasises the validation of this property for closed-loop systems that are subject to packet losses and computational overruns, potentially occurring simultaneously. We apply our method to two case studies from the recent literature and show their robustness to a comprehensive set of faults. We employ closed-loop system simulations to empirically derive performance metrics that elucidate the quality of the controller implementation, such as the system settling time and the integral absolute error.