This thesis dives deep into the concepts of solvability and control of implicit Max-Min-Plus-Scaling (MMPS) systems. An advanced mathematical framework used to model discrete-event systems combining max-plus, min-plus, and conventional algebraic operations. These systems have a broad spectrum of applications in fields such as scheduling, transportation, and performance evaluation of networks. An initial overview of MMPS systems, and necessary background is provided through the mathematical preliminaries, including max-plus and min-plus algebra, spectral theory, and their graph-theoretical interpretations. This thesis recognizes the distinction between explicit and implicit MMPS systems, where the latter involves current state dependencies, leading to challenges in analysis and solvability. The focus of the thesis will solely lie in researching implicit MMPS systems, and is split into two main parts.
The first part providing novel theoretical concepts regarding control and solvability of implicit MMPS systems.The main contribution of the first part lies in extending the existing solvability theory. This thesis shows that previously proposed solvability conditions are merely sufficient, but not necessary. A graph-theoretic interpretation of solvability is introduced by analyzing the structure matrix $S$, and conditions are developed to identify circuit subsystems, which pinpoint implicit dependencies within the system. The thesis further proposes a classification of solvability into uniquely solvable-, parametrically solvable-, parametrically unsolvable-, and strictly unsolvable modes and derives a necessary and sufficient condition for solvability using rank tests on linear algebraic subsystems. Furthermore, the control of implicit MMPS systems is explored by proposing open-loop and closed-loop control strategies. The effects of these control strategies on system properties such as time-invariance and solvability are analytically derived.
In the second part, the theoretical results are supported by application to an urban railway system (URS), which is augmented in order to accommodate complex passenger flows, and controlled using the developed implicit MMPS control framework. Results of the simulation demonstrate the system's stability and effectiveness of the control strategies under various disturbances.
Overall, this thesis provides significant theoretical advancements in implicit MMPS system analysis, and offers practical methodologies and illustrative examples regarding modeling and controlling complex discrete-event systems.

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.