This thesis is devoted to the class of Max-Plus Linear Parameter Varying (MP-LPV) systems. Recently, this class is introduced as an extension of the class of Max-Plus-Linear (MPL) systems in the field of max-plus algebra. The MPL framework is useful when modeling Discrete Event Systems (DES). Describing DES in conventional algebra results in nonlinear system descriptions, but when described as a max-plus linear system, the model becomes ’linear’ in max-plus algebra. The extension class of MP-LPV systems is introduced as a tool for parametric modeling, providing the possibility to model uncertain and nonlinear dynamics in a parameterized linear system structure. MP-LPV systems are the max-plus algebraic analogue to the conventional class of Linear Parameter Varying systems. The system matrices of MP-LPV systems can depend on the varying parameter in different manners. Problems arise when the varying parameter depends on the state vector itself. The resulting system description is then implicit. Due to properties of max-plus algebra, it cannot always be guaranteed that such implicit MP-LPV systems have a solution. This leads to the solvability problem. In this thesis, we first define different levels of implicitness in MP-LPV systems, and present frameworks for each level to solve these solvability problems. The results will thereafter be illustrated with a case study that describes an urban railway system. Then, we will present a first model predictive controller for a MP-LPV system with the urban railway system as application. Research about MP-LPV systems has so far been about modeling and system analysis, and little research has been done in control approaches. This model predictive controller can therefore be considered as a first step in controller design for MP-LPV systems. Lastly, the foundation is laid for analyzing closed-loop stability of MP-LPV systems subject to model predictive control.

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.

In this research, we introduce and discuss an extension of max-plus linear systems, called max-plus linear parameter varying systems. In this extension, uncertainty is modelled as a varying parameter, making it possible to leverage non-linearities and non-max-plus terms within the model. Issues arise when the varying parameter is dependent on previous and current states, so-called implicit max-plus linear parameter varying systems. Implicit maxplus linear parameter varying systems are not guaranteed to have a solution. In this work, we introduce some analytical and graph-theoretical methods to analyze the solvability of implicit max-plus linear parameter varying systems. Using these methods, we show that it is not possible to conclude structural solvability for general implicit max-plus linear parameter varying systems. Although, we show that, under mild assumptions, there exist conditions for which the implicit system is solvable. These conditions are explained and applied on an implicit max-plus linear parameter varying model with a state-dependent varying parameter. We illustrate our results with an urban railway network with passenger-dependent dwell time.

A switching max-plus linear model is a framework to describe the discrete dynamics of the timing of events. To influence these systems one can choose the routes of jobs and the orderings of operations as input for the system. In this thesis the techniques of model predictive control are used to find good input values. The problem of finding the optimal input is however NP-hard, which means there is no guarantee to find the optimal in a reasonable amount of time. This is an issue for model predictive control on real applications of the switching max-plus model. In applications, on-line performance is used where there is limited time to compute the input values for control.\\ This thesis takes a look into methods to reduce the computational complexity of the MPC-SMPL problem. Alternative formulations such as reparameterization, model based-partitioning and the cutting plane method are developed and tested for the MPC-SMPL problem. To solve the MPC-SMPL problem 3 heuristics are designed and implemented for simulation. The heuristics are partition-based optimization, tabu search and simulated annealing. The goal is to find a strategy that obtains the best solution to the problem in a limited amount of time.