Max-Plus Linear Parameter Varying Systems

Abstract

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.