T.J.J. van den Boom and B. De Schutter, "Model predictive control of manufacturing systems with max-plus algebra," Chapter 12 in Formal Methods in Manufacturing (J. Campos, C. Seatzu, and X. Xie, eds.), Industrial Information Technology, CRC Press, ISBN 978-1466561557, pp. 343-378, Feb. 2014.
Manufacturing systems can often be modeled as max-plus-linear (MPL) systems. MPL systems are discrete-event systems with synchronization but no choice and they are linear in the so-called max-plus algebra, which has addition maximization as its basic operations. In this chapter we present an in-depth account of the model predictive control (MPC) framework for MPL systems. MPC is an on-line model based controller design method that is very popular in the process industry and that can also be extended to MPL systems. A key advantage of MPC is that it can accommodate constraints on the inputs and outputs of the controlled system. In MPC the optimal control signal is obtained by an optimization over all possible future control sequences. In general, the resulting MPL-MPC optimization problem is nonlinear and nonconvex. However, we show that if the control objective is piecewise affine, the constraints are linear, and if the control objective and the constraints depend monotonically on the outputs of the system, which is a frequently occurring situation for manufacturing systems, the MPL-MPC optimization can be recast into a linear programming problem, which can be solved very efficiently. Subsequently we focus on implementation and timing aspects, closed-loop behavior, and tuning rules for MPL-MPC. We derive sufficient conditions for stability and formulate a closed-loop expression for the unconstrained MPL-MPC controller. In the case of perturbed operation due to modeling errors and/or noise we need a robust MPL-MPC controller. We show that under quite general conditions the resulting optimization problems can be solved very efficiently. For the bounded error case we also derive an MPL-MPC controller by optimizing over feedback policies, rather than open-loop input sequences. In general, this results in increased feasibility and a better performance. Finally we discuss robust MPC for MPL systems with stochastic uncertainty.