Throughput and Stabilisability Analysis of Mode-Constrained Stochastic Switching Max-Plus Linear Systems

Abstract

Switching max-plus linear (SMPL) systems written in max-plus algebra form a robust framework to model discrete-event systems governed by synchronisation whose behaviour may switch over time. Their evolution is described by a max-plus linear state-space representation that may change by switching modes. In their typical form, switching may depend on the system’s previous state, previous mode, a discrete control signal, and exogenous stochastic signals.

In this work, we investigate modelling options and performance analyses for SMPL systems and the application of control to such systems. We propose to model stochastic systems whose mode sequences are constrained in some form as discrete hybrid stochastic automata. For such systems, we offer definitions with which to predict system performance measured in throughput in the form of finite-horizon approximations of a class of asymptotic performance metrics. We validate these approximations of a system’s growth rate using a Monte Carlo method and corresponding statistical analyses. We use these analyses to form stabilisability guarantees for SMPL systems as a function of the growth rate of their reference signal. Lastly, we propose a model predictive control framework for stabilising stochastic mode-constrained SMPL systems. We validate these contributions by considering three control cases regarding stabilising mode-constrained deterministic and stochastic SMPL systems under discrete and hybrid control.