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S. Markkassery
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Max-Min-Plus-Scaling Approach to Urban Railway Systems
Modelling, Analysis, and Control of Bidirectional Urban Railway Systems with Origin–Destination Passenger Flows in the MMPS Framework
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. ...
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. ...
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.
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.
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. ...
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. ...
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.
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.
Solving Solvability of Implicit Max-Min-Plus-Scaling Systems
A deep dive into solvability and control of implicit Max-Min-Plus-Scaling systems
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.
...
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 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.
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.
Stability for Discrete Event Max-Min-Plus (MMP) and Max-Min-Plus-Scaling (MMPS) Systems
Max-Plus Lyapunov Functions for Stability Analysis and Control
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. ...
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. ...
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.
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.