The objective of this paper is to provide a concise introduction to the max-plus algebra and to max-plus linear discrete-event systems. We present the basic concepts of the max-plus algebra and explain how it can be used to model a specific class of discrete-event systems with synchronization but no concurrency. Such systems are called max-plus linear discrete-event systems because they can be described by a model that is “linear” in the max-plus algebra. We discuss some key properties of the max-plus algebra and indicate how these properties can be used to analyze the behavior of max-plus linear discrete-event systems. Next, some control approaches for max-plus linear discrete-event systems, including residuation-based control and model predictive control, are presented briefly. Finally, we discuss some extensions of the max-plus algebra and of max-plus linear systems.

The current focus on the massive CO2 reduction highlights the need for the rapid development of technology for the production, storage, transportation and distribution of renewable energy. In addition to electricity, we need other forms of energy carriers that are more suitable for energy storage and transportation. Hydrogen is one of the main candidates for this purpose, since it can be produced from solar or wind energy and then stored; once needed, it can be converted back to electricity using fuel cells. Another important aspect of future energy systems is sector coupling, where different sectors, e.g. mobility and energy, work together to provide better services. In such an integrated system, electric vehicles – both battery and hydrogen-based fuel cell – can provide, when parked, electricity services, such as backup power and balancing; when driving they produce no emissions. In this paper we present the concept design and energy management of such an integrated energy and mobility system in a real-life environment at the Shell Technology Centre in Amsterdam. Our results show that storage using hydrogen and salt caverns is much cheaper than using large battery storage systems. We also show that the integration of electric vehicles into the electricity network is technically and economically feasible and that they can provide a flexible energy buffer. Ultimately, the results of this study show that using both electricity and hydrogen as energy carriers can create a more flexible, reliable and cheaper energy system at an office building.

We present shrinking horizon model predictive control for discrete-time linear systems under stochastic disturbances with constraints encoded as signal temporal logic (STL) specification. The control objective is to satisfy a given STL specification with high probability against stochastic uncertainties while maximizing the robust satisfaction of an STL specification with minimum control effort. We formulate a general solution, which does not require precise knowledge of probability distributions of (possibly dependent) stochastic disturbances; only the bounded support of the density functions and moment intervals are used. For the specific case of disturbances that are normally distributed, we optimize the controllers by utilizing knowledge of the probability distribution of the disturbance. We show that in both cases, the control law can be obtained by solving optimization problems with linear constraints at each step. We experimentally demonstrate effectiveness of this approach by synthesizing a controller for a heating, ventilation, and air conditioning system.