Stability for Discrete Event Max-Min-Plus (MMP) and Max-Min-Plus-Scaling (MMPS) Systems

Max-Plus Lyapunov Functions for Stability Analysis and Control

Master thesis (2023)

Authors

E.J.E. Peijnenburg Mechanical Engineering

Contributors

A.J.J. van den Boom Team Ton van den Boom - Mechanical, Maritime and Materials Engineering (supervisor 1)
S. Markkassery Team Ton van den Boom - Mechanical, Maritime and Materials Engineering (supervisor 2)
Faculty
Mechanical Engineering
Copyright: © 2023 Emiel Peijnenburg
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Published Date

12-10-2023

Language

English

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Faculty

Mechanical Engineering

Abstract

This research presents a framework for analysing the stability and control of discrete-event systems, specifically emphasising max-min-plus (MMP) and max-min-min-plus-scaling (MMPS) systems. These systems are valuable modelling tools for various applications, including production systems and urban railway traffic management, respectively. However, a critical challenge in discrete-event systems is the lack of a generalised approach to assessing the stability of time signals, particularly in the context of MMPS systems. To address this challenge,
this research will use max-plus Lyapunov functions already used to study the buffer stability in discrete-event switching-max-plus-linear (SMPL) systems.

This thesis provides a framework to use max-plus Lyapunov functions to determine buffer stability of MMP and MMPS systems, focusing on their time signals. The max-plus Lyapunov function uses a buffer for each pair of states. The system is considered stable if the difference converges to the buffer levels for every pair of states. Given the structure of MMP and MMPS systems, the difference between the states after one state update will often be bounded. To determine this boundedness of the buffer levels, a novel concept of "fully correlated" MMP and MMPS systems is introduced. Using the properties of fully correlated systems, an algorithm is proposed to determine the buffer levels for both MMP and MMPS systems. We also derive analytical methods using Markov properties to assess the additive eigenvalue of fully correlated time-invariant monotonic MMPS systems. Using the property of fully correlatedness, it is also derived that fully correlated time-invariant non-monotonic MMPS systems will always have a bounded buffer and growth rate and can have multiple additive eigenvalues. The findings show that fully correlated time-invariant systems consistently exhibit bounded growth rates.

In addition to providing theoretical insights, this study demonstrates the practical use of max-plus Lyapunov functions as a control Lyapunov function (CLF) in model predictive control (MPC). A novel control technique is proposed to stabilise naturally unstable discrete event systems. This approach has been effectively applied to stabilise inherently unstable discrete-event max-plus-linear (MPL) and MMP systems, indicating the practical significance of the proposed framework.