Analytic Expressions in Stochastic Max-Plus-Linear Algebra and their Application in Model Predictive Control

The class of max-plus-linear systems can model discrete event systems with synchronization but no choice. Model mismatch and/or disturbances can be characterized as stochastic uncertainties. In stochastic max-plus-linear systems one often needs to compute the expectation of a max-plus-scaling (MPS) function or the chance constraint of a MPS function. The algorithms available in literature are either computationally too expensive or only give an approximation. In this article, we derive an analytic expression for both the expectation and the chance constraint of a MPS function. Both can be written in the form of a piecewise polynomial function in the components of the control variables. The analytic function can be derived offline and can be evaluated online in a quick and efficient way. We also show how the expressions can be used in a model predictive control setting and show the efficiency of the proposed approach with a worked example.