A stabilizing model predictive controller for uncertain max-plus-linear systems and uncertain switching max-plus-linear systems ⋆ Ton van den Boom ∗ Bart De Schutter ∗ ∗ Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands a.j.j.vandenboom@tudelft.nl, b.deschutter@tudelft.nl, http://dcsc.tudelft.nl Abstract: We first discuss conditions for stability for uncertain max-plus-linear systems and for uncertain switching max-plus-linear systems, where in the uncertainty description the system matrices live in a union of polyhedra. Based on the newly derived stability conditions, a stabilizing model predictive controller is derived for both uncertain max-plus-linear systems and uncertain switching max-plus-linear systems. Keywords: Discrete event and hybrid systems, max-plus-linear systems, model uncertainty, model predictive control, robust control 1. INTRODUCTION The class of discrete event systems (DES) essentially con- sists of man-made systems that contain a finite number of resources that are shared by several users all of which contribute to the achievement of some common goal (Bac- celli et al., 1992). In general, models that describe the behavior of a DES are nonlinear in conventional algebra. However, there is a class of DES that can be described by a model that is linear in the max-plus algebra. This class of max-plus-linear (MPL) systems can only characterize synchronization and no concurrency or choice. In switching MPL systems the system can switch between different modes of operation, in which the mode switching depends on a switching mechanism. In this paper we will consider the control of both uncertain MPL systems and uncertain switching MPL systems. In contrast to conventional linear systems, where noise and disturbances are usually modeled by including an extra term in the system equations (i.e., the noise is considered to be additive), the influence of noise and disturbances in (switching) MPL systems is not max-plus-additive, but max-plus-multiplicative. This means that the system ma- trices will be perturbed and as a consequence the system properties will change. Ignoring the noise can lead to a bad tracking behavior or even to an unstable closed loop. A second important feature is modeling errors. Uncertainty in the modeling or identification phase leads to errors in the system matrices. It is clear that both modeling errors, and noise and disturbances perturb the system by introducing uncertainty in the system matrices. Sometimes ⋆ Research partially funded by the Dutch Technology Founda- tion STW project “Model-predictive railway traffic management” (11025), and by the European 7th Framework Network of Excellence project “Highly-complex and networked systems (HYCON2)”. it is difficult to distinguish the two from one another, but usually fast changes in the system matrices will be con- sidered as noise and disturbances, whereas slow changes or permanent errors are considered as model mismatch. In this paper we extend the results of van den Boom and De Schutter (2002b) and consider MPL systems and switch- ing MPL systems in a single framework. We distinguish between two characterizations: static uncertainty, in which the system is uncertain but the system matrices are con- stant, and dynamic uncertainty, in which the uncertainty in the system is affected by bounded noise. We will also show that for both characterizations we can derive stability conditions under quite general assumptions. Model predictive control (MPC) (Camacho and Bordons, 1995; Maciejowski, 2002) is a model-based predictive con- trol approach that has its origins in the process industry and that has mainly been developed for linear or non- linear time-driven systems. Its main ingredients are: a prediction model, a performance criterion to be optimized over a given horizon, constraints on inputs and outputs, and a receding horizon approach. We show that for MPC of (switching) MPL systems with the given uncertainty description we can use the results of van den Boom and De Schutter (2002b). The uncertainty description in this paper is related to the interval uncertainty given by Lhommeau et al. (2004); Corronc et al. (2010). For other classes of DES uncertainty results can be found in Cardoso et al. (1999); Liu (1993); Park and Lim (1999); Young and Garg (1995), and the references therein. This paper is organized as follows. In Section 2 we give a concise introduction to the max-plus algebra and (S)MPL systems with uncertainty. In Section 3 we derive stability conditions for (S)MPL systems with uncertainty, and in Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 Copyright by the International Federation of Automatic Control (IFAC) 8663