Modeling and Control of Switching Max-Plus-Linear Systems

Abstract

The operation of many systems can be described by the timing of events. When the system behavior can be described by equations that are "linear'' in the max-plus algebra, which has maximization and addition as its basic operations, the system is called a max-plus-linear system. In many of these systems the order of the events may need to be changed due to changes in the conditions, or the requirements. Such systems that can change the order of events are called switching max-plus-linear (SMPL) systems. In this thesis we consider two application of SMPL systems. The first application of SMPL systems models the railway traffic networks and is used for on-line rescheduling of railway traffic in the case of delays. In this thesis a macroscopic model for the railway traffic network is presented that can model the effects on the railway traffic of several control actions. For every set of control actions the new event order and times are determined. In order to solve the on-line rescheduling problem for a railway traffic network a global model predictive control (MPC) approach and four distributed model predictive control (DMPC) approaches are proposed. The second type of SMPL system models legged locomotion for different gaits. In this thesis the steady state cyclic behavior of the max-plus-linear systems describing the gaits, and the transition to the steady state cyclic behavior, are analyzed. It is shown that the steady state behavior can be uniquely defined for all gaits. With the steady state behavior uniquely defined for all gaits we were able to determine optimal gait switches.