This Master's thesis employs the Max-Min-Plus-Scaling (MMPS) mathematical framework to model, analyse, and control an Urban Railways System (URS) featuring a bidirectional line and origin-destination passenger flows. MMPS systems utilise max-plus, min-plus, and conventional algebraic operations, and provide a powerful modelling language for Discrete Event Systems (DES) such as transportation networks, computer networks, and manufacturing plants. A comprehensive overview of MMPS systems is provided, covering the mathematical foundations, existing analysis techniques, and control strategies.

The thesis then advances the existing theory on the periodicity of MMPS systems by introducing the Interconnected Max-Min-Plus-Scaling (I-MMPS) framework, which incorporates dependencies on states x(k-d), with d>1, extending beyond the single-step dependency assumed in standard MMPS systems. The ABCD canonical form is derived for I-MMPS systems, and the existing analysis and control frameworks are extended accordingly, establishing conditions for time-invariance, bounded-buffer stability, and solvability, as well as a fixed controller design methodology.

These theoretical contributions are then applied to improve the existing unidirectional URS model. A return trip is incorporated to achieve bidirectional operation and origin-destination passenger flows are introduced without appending states to the state vector, keeping the computational complexity of eigenvalue computation to a minimum. The resulting Bidirectional Origin-Destination Urban Railway System (BiOD-URS) is analysed through a case study, examining its eigenvalue structure, dynamic behaviour, and response to various disturbances. Finally, state feedback controllers are designed and applied to the BiOD-URS to improve nominal performance and disturbance rejection, including a growth rate assignment controller that drives the system to a uniform steady state.

Overall, this thesis advances MMPS system theory through the development of the I-MMPS framework, and demonstrates its practical applicability through a comprehensive modelling, analysis, and control study of a complex urban railway system. ... Read more

While reinforcement learning (RL) and supervised learning provide powerful approaches for finding optimal controllers for complex systems, ensuring safety remains a critical challenge. In control problems, safety is typically defined as maintaining state and input constraint satisfaction throughout the system’s evolution. The key issue lies in balancing constraint satisfaction with computational efficiency in the presence of inevitable learning errors. This PhD thesis addresses this challenge across linear, piecewise affine (PWA), and nonlinear systems with various constraint structures. ... Read more

This thesis explores the analysis, periodicity, and scalable modelling of Max-Min-Plus-Scaling systems, a versatile approach to modelling Discrete Event systems. Unlike traditional continuous-time or discrete-time systems that evolve through differential or difference equations, DE systems progress through discrete events. MMPS systems rely only on maximisation, minimisation, addition, and scaling, making them highly suitable for modelling processes with synchronisation and/or competition such as energy delivery, transportation, and manufacturing.
The work is divided into three main segments. First, a new Mixed-Integer Linear Program ming based method is developed for analysing growth rates and fixed points of general implicit MMPS systems. This extends an existing MILP formulation for homogeneous and non-expansive explicit MMPS systems, introducing adaptations for general implicit cases.
A dedicated preprocessing step and search strategy are introduced, resulting in an analysis method that significantly reduces computational requirements. Secondly, the dynamical and stability behaviour of periodic MMPS systems with periods greater than one is examined.
A new canonical form is proposed, enabling the use of existing analysis tools on periodic systems, along with a method for determining the stability of periodic orbits. Thirdly, a modelling framework for transportation systems is introduced, featuring a connectable, node based toolbox and an algorithm that transforms high-level system descriptions into sets of equations.
All developed methods, theories, and tools are demonstrated on a real-world 4-node transportation system. The results confirm the efficiency of the new MILP approach, reveal periodic behaviour and stable periodic orbits, and highlight fixed points, all within the proposed transportation network framework. ... Read more

This thesis examines the use of a potential field model for simulating pedestrian dynamics in complex environments. The study first reviewed the different types of pedestrian dynamics models, highlighting their strengths and weaknesses. From this study, a research gap emerged regarding hybrid pedestrian movement models. As a base for such a model, a microscopic pedestrian dynamics model has been developed combining potential fields and gradient descent optimization as the drivers for trajectory selection. In this formulation, agents follow trajectories along the gradient of the potential field, naturally balancing goal seeking behavior with obstacle and inter-agent avoidance. The potential field approach was also discussed as a foundation for hybrid models, in which microscopic and macroscopic modeling strategies are combined to exploit the advantages of both. To evaluate and calibrate the model, real-world trajectory data from a bidirectional corridor experiment and a bottleneck experiment were used. A surrogate model was constructed to accelerate the optimization process, given the high computational cost of the original simulation model. The surrogate model enabled systematic parameter calibration and sensitivity analysis, focusing on three key parameters: the goal potential function weight (KG), the wall potential function weight (KW), and the obstacle potential function weight (KO). The results demonstrated that the optimized model is capable of reproducing key crowd phenomena observed in the empirical datasets. A sensitivity analysis further showed the relative importance of the potential function weights across different key performance indicators. Moreover, predictive uncertainty analysis confirmed that the model exhibited relatively high confidence around the optimum and avoided regions of overfitting. Despite these contributions, the research was constrained by the computational cost of the simulation model. The reliance on a surrogate model limited the optimization to a small subset of parameters, assuming that other model parameters were already sufficiently calibrated. This assumption likely introduced some biases, such as underestimation of obstacle repulsion, leading to overly frequent close inter-agent encounters. In conclusion, this thesis has demonstrated that potential field models, when combined with real-world data and surrogate-based optimization, provide a valid and powerful framework for simulating pedestrian dynamics in complex environments. Their ability to model pedestrian trajectories through potential functions and gradient descent makes them conceptually simple yet effective, while their extensibility offers a pathway toward hybrid models. Nevertheless, computational burden and limited parameter coverage remain key challenges, highlighting the need for more efficient implementations and broader parameter optimization in future research. ... Read more

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.
... Read more

Model Predictive Control (MPC) is an effective reference tracking strategy for automated vehicle control, particularly useful during emergency evasive maneuvers such as double lane changes. This control method often requires a high-fidelity vehicle model to accurately capture nonlinearities and uncertainties, significantly increasing computational demand. Hybrid systems modeling frameworks have been developed to approximate these nonlinearities, thereby reducing computational complexities while maintaining satisfactory tracking performance. However, existing benchmarks that evaluate the impact of these hybrid approximations on tracking performance and computational demands are lacking. Establishing such comparative benchmarks is crucial for understanding how different levels of model complexity affect the overall efficiency and effectiveness of model predictive approaches in automated driving scenarios.

This research addresses this gap by presenting a comparative analysis of various levels of hybrid model complexity. It assesses their tracking performance and computational demand using both MPC formulation and Model Predictive Contouring Control (MPCC) formalism in different emergency maneuver scenarios.
Four hybrid approximations of the nonlinear single-track vehicle model with varying complexity levels are considered and employed as prediction models in both MPC and MPCC optimization problems. The closed-loop behavior of these control frameworks is simulated in double-lane change maneuver scenarios, evaluating tracking performance scenarios with varying levels of curvature. Additionally, variations in friction and velocity are evaluated in different scenarios to assess controller robustness to model uncertainty.

Results indicate that reducing the complexity of hybrid approximations can decrease computational demand, albeit at the expense of tracking performance. Moreover, MPC formalism offers a more robust approach to tracking performance and provides a feasible solution in a broader range of scenarios than the MPCC framework. By shedding light on the impact of different complexity levels for the hybrid approximation of the nonlinear model and the control optimization problem formulation, this work offers comprehensive guidelines for hybrid MPC applications for automated driving in emergency scenarios. ... Read more

This thesis offers a detailed exploration of the integration of input signals and control mechanisms within max-min-plus-scaling (MMPS) systems, a subclass of discrete event (DE) systems. Unlike traditional control systems, which rely on continuous evolution modeled by
differential equations, DE systems progress through the occurrence of discrete events. MMPS
systems enhance this adaptability by encompassing maximization, minimization, scaling, and
addition, creating a framework for modeling and managing various processes, including logistics networks and urban railway systems.

The primary objective of this thesis is to introduce input signals into MMPS systems and
systematically investigate control strategies. This involves establishing a structure accommodating these input signals while preserving essential properties such as time invariance. The study examines both open-loop and closed-loop control strategies, focusing on the latter to implement optimization-based control to optimize system performance through effective
feedback control.

This thesis is organized, beginning with the mathematical foundation of MMPS systems and
progressing to the development of control methods. The implementation of these methods is
validated through practical applications such as manufacturing systems and the urban railway system, demonstrating their effectiveness.

By advancing our understanding of control in MMPS systems, this research provides a systematic methodology that integrates control and illustrates how optimization-based techniques
can enhance overall performance. The insights gained from this work lay a groundwork for
future research, potentially extending beyond transportation systems to other discrete eventdriven industries. Engaging with this research has the potential to control numerous fields, promising innovative solutions and improved efficiencies across sectors. ... Read more

Inland waterway transport is a low CO2 emission alternative to road transport. A shift towards more inland waterway transport could also help reduce road congestion and noise pollution. Infrastructure bottlenecking, particularly at locks, is part of the reasons preventing this shift. Congestion is leading to delays. Locks can be physically improved, or the passage of the vessels through the locks can be optimized through scheduling. Recent work introduced a novel switching-max-plus-linear system approach to scheduling vessels passing through networks of waterways and locks, also introducing a novel routing component to the scheduling problem. Switching max-plus-linear systems are a convenient way to model scheduling systems using max-plus-linear algebra. The switching-max-plus-linear model only considered locks with a single chamber that can only process one vessel at a time. Additionally, it only considered four specific waterway network configurations, rather than any arbitrary network configuration. Real locks can have multiple chambers, and they can process multiple vessels at the same time if they are placed according to regulations in the two-dimensional space of the chamber. The scope of this report was then to build upon this switching max-plus-linear model by adding support for arbitrary network configurations, and multi-chamber, multi-vessel locks with proper twodimensional ship placement, to answer the main research question: How can multi-vessel, multichamber locks with ship placement be integrated into the SMPL IWT scheduling model? Three mathematical scheduling models formulated as SMPL systems were introduced, each subsequent model building on the previous one. The first introduced support for arbitrary network configurations. The second introduced support for multi-chamber locks and allowed vessels to pass through lock chambers at the same time, provided that their assigned one-dimensional sizes fit into the assigned one-dimensional capacity of the chamber. The third introduced support for twodimensional ship placement through a Tetris-like placement sequence also modeled as a switching max-plus-linear system. The models were translated to mixed integer linear programming models, and arrival time and arrival time offset objectives were added, so they could be used as the rules by which a scheduler would build and solve offline scheduling optimization models on a case-by-case basis. Auxiliary objectives to promote vessels slowing down, rather than waiting stationary in the waiting areas, were also added. The models were all found to be working as intended and implemented correctly through the use of a number of verification cases and tests. Complexity tests showed that the solution times for all three models already became larger than a practical limit of 15-20 minutes for simple scenarios with 10-15 vessels. A heuristic model that mimics how vessels are assigned to lockages in practice was built. Comparisons to the scheduling models on the verification cases showed negligible differences for the models in single-lock cases, but it indicated that the multi-vessel, multi-chamber models may outperform practice in multi-lock cases. Future research is recommended to focus on online optimization to account for disturbances, distributed optimization to reduce calculation times, and validation of the model’s performance with real data. ... Read more

The transition towards a more sustainable future has become increasingly necessary due to rising greenhouse gas emissions. With the growing size of modern cities, transport has become a major contributor, with road transport accounting for up to 70% of total emissions [3]. As a result, Electric Vehicles (EVs) are increasingly adopted as a replacement for Internal Combustion Engine Vehicles (ICEVs), which are the main source of emissions in road transport.

However, EVs still face two main bottlenecks: significantly slower charging times compared to refuelling ICEVs, and Li-ion battery degradation over time, which affects lifespan. Therefore, a charging strategy is required that mitigates these effects. This leads to the research question: finding a model-based real-time control charging strategy that reduces charging time and degradation.

To develop such a strategy, accurate battery and degradation models are required to capture internal states such as State of Charge (SOC) and degradation mechanisms such as Solid Electrolyte Interphase (SEI) growth and lithium plating. Three electrochemical Li-ion battery models are considered: the Pseudo 2-Dimensional (P2D) model, the Electrolyte Enhanced Single Particle Model (SPMe), and the Extended Single Particle Model (ESPM). The P2D model is a full-order model with high accuracy but high computational cost. The SPMe is a simplified version with reduced accuracy but lower computational cost. Both models are unsuitable for real-time control. The ESPM is developed as a further simplification to achieve real-time feasibility while maintaining accuracy.

The P2D and SPMe models are implemented using PyBaMM, while the ESPM is implemented in MATLAB. Results show that the ESPM maintains over 90% similarity with reference models for currents up to 2C, making it suitable for control applications.

A degradation model including SEI growth and lithium plating is also implemented in MATLAB, calibrated to match capacity fade of the LGM50 battery. These degradation mechanisms are incorporated into the control framework to be minimised during charging.

Finally, a Nonlinear Model Predictive Control (NMPC) strategy is developed to optimise the trade-off between charging time, SEI growth, lithium plating, and current constraints. The resulting health-conscious fast charging strategy achieves a charging time of 34 minutes with an estimated battery lifespan of approximately 800 cycles. This charging time is comparable to DC fast charging standards while reducing degradation effects, saving around 100 cycles over the battery lifetime. ... Read more

In this thesis a novel method for the realisation of Max-Min-Plus-Scaling (MMPS) functions is presented. It has previously been shown that continuous piecewise affine (PWA) functions, conjunctive MMPS functions (also lattices- or min-max functions) and kripfganz MMPS functions (difference between two convex functions) can each describe the same function. Each form has its own specific use cases and benefits and thus it may be desired to rewrite a specific function described in one form, into another. Current techniques are input dependent and not available for each combination, while some techniques blow the number of parameters. The technique presented in this thesis however is input and output independent and rigid in its construction. It fills up the gaps where some realisation were not possible yet, it generates a rigid and predictable output. The necessary and sufficient conditions for the existence of each form are proposed. Additionally, it is explained how redundant terms may be removed in order to ensure a minimal representation. An algorithm is given for the decomposition and for the construction of each canonical form and supported with some worked examples. The decomposition provides the tools to efficiently map between different descriptions. ... Read more

This thesis extensively examines the influential factors affecting the performance of approximations of Model Predictive Control (MPC) control laws using neural networks. MPC is a control strategy that solves an optimization problem at each timestep. This problem can be computationally complex and could be too slow to compute for online control. Sometimes an explicit solution for MPC exists, but this can become very large in memory and is not always available. That is why approximations with neural networks might offer a benefit. Under certain conditions, the explicit solution yields a piecewise affine (PWA) control law. A PWA model class is equivalent to the so-called Max-Min-Plus-Scaling (MMPS) model class, which is a generalization of max-plus and min-plus algebra. Neural networks are made up of neurons, which make use of activation functions. A feed-forward neural network with some specific activation function can yield an MMPS function. This inspires us to research the use of different activation functions in approximating MPC control laws. Additionally, we investigate different sampling strategies and the use of max-plus and min-plus layers in neural networks.

We do this by setting up different PWA and non-PWA control laws for two inverted pendulum systems and training several neural networks to approximate these control laws. We first observe a significantly better performance in approximating the PWA control laws compared to the non-PWA control laws. When varying the activation functions of the neural networks we find that for PWA control laws a MMPS activation function can offer a better performance, but it is not guaranteed for all MMPS functions. We also find that networks with custom max-plus layers can offer a similar performance on approximating control laws compared to networks with traditional layers. When investigating what sampling strategy is most beneficial we find comparable performance with a stratified sampling strategy and a uniform sampling strategy. Depending on what areas of the control law you want to capture with the most detail, you can choose the most viable sampling strategy. With this, we have researched various factors that influence the performance of approximations of MPC control laws. The thesis ends with a recommendation to research even more factors that might offer even better approximations. ... Read more

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. ... Read more

Sorting systems form an example of event driven systems. These types of systems are referred to as discrete event systems (DES), and they consist of jobs that need to be performed at available resources. In an autonomous sorting system, jobs consist of robots receiving and delivering parcels at the correct locations. With scheduling, optimal allocation of the jobs to those resources over time is computed, where the decisions that need to be made are routing, ordering and synchronization. The behaviour of DES is often described by non-linear models, but max-plus linear (MPL) systems are a class of DES that can be described by a model that is linear in the max-plus algebra. This algebra uses two operators maximization and addition. Allowing different routes and switching between orders of jobs extends an MPL system to a switching max-plus linear (SMPL) system. Robots in a sorting system often have many routes to choose from, and need to make order choices with respect to other robots in the system.

In this thesis, a general SMPL model is made for the autonomous sorting system at software company Prime Vision, which can be applied to any sorting area design. The solution to the scheduling problem for the model results in a time schedule for the active robots at the correct locations in the sorting area, as well as the optimal decisions on routing, ordering and synchronization. The optimization problem is solved with a model predictive scheduling (MPS) approach and recast as a mixed integer linear programming (MILP) problem. The model is created in Python and the optimization problem is solved with Gurobi. The resulting schedule is visualized with a simulation, in which the decisions of the robots are clearly shown. An idea for implementation of the optimization into the sorting system is given as well. ... Read more

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. ... Read more

The goal in this thesis is to make a prediction on the total processing time for logistical systems to complete their tasks. For simple systems that can be described using a regular max plus state space model this is done by calculating the systems eigenvalue and multiplying that by the number of iterations required. But, for more complex systems that can only be described using a switching max-plus state space model such as a production line that can change during production, or a rail network where some rail segments become unavailable at times. If these changes to the system are fully within control of the operator they are easily accounted for when predicting the total processing time. But, if the system switches stochastically then one would need to simulate all possible permutations of operation to know the average processing time of the system. Alternatively, one might approximate this average processing time using some simplified model. We found three main methods to predict the total processing time. Firstly, we can fit a generalized extreme value distribution to a histogram of the systems performance and then extrapolate this distribution function. Secondly, we can fit a marginal cost model to a limited simulation of the system and then extrapolate based on that model. Lastly, we can rewrite the expectation of how long the system might take to go through N iterations to an inequality by adding an arbitrary diagonal matrix S, which if minimized can be used to approximate total processing time. All three prediction methods can, when fitted properly predict at least within 15% accuracy with the fitted extreme value distribution generally performing best. We also found that it is possible to save total processing time using a controller based on the marginal cost model if it is possible to exert some limited control over the mode switching process. ... Read more

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. ... Read more

Delay Management is proven to reduce the average passenger delay. However, the current method of Delay Management either uses fast basic or slow accurate passenger rerouting. The former has a downfall in that it is inaccurate since it assumes that passengers delay is the periodicity of the train of the original transfer, while in reality, faster alternatives are often available. The latter reroutes passengers simultaneously with the delay management resulting in computation times larger than the time window of a Train Dispatcher to make a decision. This thesis proposes an alternative method of Delay Management which calculates the alternative routes before the Delay Management optimization using a modified Dijkstra algorithm. This method outperforms the fast basic approach to reducing passenger delay with a similar computation time, thereby outperforming the slow accurate approach. ... Read more

In this thesis, research is done on the influence and benefits of an iterative interaction between a scheduler and its subsystems for an updated scheduler which minimizes to a certain cost. This is done by providing a case study of a beer brewery. The scheduler is obtained by using a switching max-plus linear approach. The subsystems will be simulated, estimated and predicted using the system dynamics of the beer brewing case study. The processes discussed in the beer brewing process are mashing, brewing and fermentation. The simulation is done by filling in the system dynamics with addition of noise. The estimation is done by using an extended Kalman filter, and the predictions are done by filling in the system dynamics with the estimated states and no addition of noise. The updating of the scheduler is done by receiving the estimations and predictions of the subsystems and thereafter using model predictive control on the switching max-plus scheduler, also called model predictive scheduling. The results for the case study are shown as a substantiation of the conclusions drawn.
Ultimately, discussions and further research are given for the reliability of the conclusions and extensions on the above summarized research. ... Read more

As the demand for the railways is expected to raise in the future, researchers are looking for ways to improve the railway capacity to increase the transport ability. Virtual coupling is a new solution based on the moving block technology, which further shortens train separation from absolute braking distance to relative braking distance. This will also affect the train service schedules for optimal operation under virtual coupling. In this study, a mixed integer quadratic programming method has been proposed to schedule the train services under virtual coupling on a network. Train operation as well as the headway and capacity benefit have been analyzed. In comparison to moving block, virtual coupling will change train operation on shared routes, and the capacity will be improved by different percentages for different timetable patterns and different service braking rates with or without speed limit restrictions.
... Read more

Inland waterways form a natural network infrastructure with the capacity for waterborne transport of people and goods for moving freight from seaports to the hinterland. Recently, Inland Waterway Transport (IWT) has been promoted more extensively by the European Union and various governments as it plays a crucial role in reducing road congestion and CO2 emissions from transport. However, the advantages of IWT are not fully exploited due to inefficiencies in the logistics system, such as long waiting times at locks and sub-optimal navigation on waterways. Currently, no scheduling at infrastructures or routing optimisation of the overall waterway network is happening. The scheduling of vessels through a lock is usually performed on a First In First Out basis, providing an opportunity for improvement. Hence, this thesis aims to design a scheduling strategy for generating an optimal plan for sending inland vessels through a waterway network with minimal delays, yielding a significant positive impact on the modal shift towards IWT. 
A promising approach to scheduling problems is by using Switching Max-Plus-Linear (SMPL) systems. SMPL systems have proven to be effective in various Discrete-Event Systems and transportation networks. Using SMPL models is convenient since non-linear scheduling problems can be described linearly using Max-Plus operators without compromising on the system dynamics. Moreover, as the SMPL systems can be transformed into Mixed-Integer-Linear-Programming (MILP) problems, it is also possible to use fast optimisers for solving the scheduling problems.
This thesis will show how one can describe IWT systems, consisting of; waterways, vessels and locks, as SMPL systems. The optimal schedule for the inland vessels is determined based on multiple input parameters, including waterway network lay-out, the sailing speeds of vessels and arrival deadlines of the vessels. The scheduler will return the individual vessel routing and overall vessel order in the waterway network. This routing and order selection is defined using binary control variables, turning the IWT scheduling problem into a MILP problem, which will allow finding the solution to large scale IWT scheduling problems in a reasonable computation time. Furthermore, this thesis will show how the goal of minimising the cumulative arrival times of all vessels in a network can be achieved. This is done for different types of waterway network cases, for which the results are shown and analysed. ... Read more