ref_max_plus.bib

@book{baccelliSynchronizationLinearityAlgebra2001,
  title = {Synchronization and Linearity: An Algebra for Discrete Event Systems},
  shorttitle = {Synchronization and Linearity},
  author = {Baccelli, Fran{\c c}ois and Cohen, Guy and Olsder, Geert Jan and Quadrat, Jean-Pierre},
  year = {2001},
  edition = {Web edition},
  publisher = {Wiley},
  address = {Chichester},
  url = {https://www.rocq.inria.fr/metalau/cohen/documents/BCOQ-book.pdf},
  isbn = {978-0-471-93609-1},
  langid = {english}
}

@book{butkovicMaxlinearSystemsTheory2010,
  title = {Max-Linear {{Systems}}: {{Theory}} and {{Algorithms}}: {{Theory}} and {{Algorithms}}},
  shorttitle = {Max-Linear {{Systems}}},
  author = {Butkovi{\v c}, Peter},
  year = {2010},
  series = {Springer {{Monographs}} in {{Mathematics}}},
  publisher = {Springer},
  address = {London},
  url = {https://doi.org/10.1007/978-1-84996-299-5},
  abstract = {This book provides a first detailed and self-contained account of linear-algebraic aspects of max-algebra for general matrices. Among the main features of the proposed book is the presentation of the fundamental max-algebraic theory, in one place in a comprehensive and unified form, made with all proofs and in full generality (that is for both irreducible and reducible matrices). Advanced material within the book also provides the reader with a wealth of information that has never been published before.},
  isbn = {978-1-84996-298-8}
}

@article{cohenAlgebraicToolsPerformance1989,
  title = {Algebraic Tools for the Performance Evaluation of Discrete Event Systems},
  author = {Cohen, G. and Moller, P. and Quadrat, J.-P. and Viot, M.},
  year = {1989},
  month = jan,
  journal = {Proceedings of the IEEE},
  volume = {77},
  number = {1},
  pages = {39--85},
  issn = {1558-2256},
  doi = {10.1109/5.21069},
  abstract = {It is shown that a certain class of Petri nets called event graphs can be represented as linear time-invariant finite-dimensional systems using some particular algebras. This sets the ground on which a theory of these systems can be developed in a manner which is very analogous to that of conventional linear system theory. Some preliminary basic developments in that direction are shown. Several ways in which one can consider event graphs as linear systems are described. These correspond to approaches in the time domain, in the event domain, and in a two-dimensional domain. In each of these approaches, a different algebra has to be used for models to remain linear, but the common feature of these algebras is that they all fall into the axiomatic definition of 'dioids'. A unified presentation of basic algebraic results on dioids is provided.}
}

@article{cohenLinearsystemtheoreticViewDiscreteevent1985,
  title = {A Linear-System-Theoretic View of Discrete-Event Processes and Its Use for Performance Evaluation in Manufacturing},
  author = {Cohen, G. and Dubois, D. and Quadrat, J. and Viot, M.},
  year = {1985},
  month = mar,
  journal = {IEEE Transactions on Automatic Control},
  volume = {30},
  number = {3},
  pages = {210--220},
  issn = {1558-2523},
  doi = {10.1109/TAC.1985.1103925},
  abstract = {A discrete-event system is a system whose behavior can be described by means of a set of time-consuming activities, performed according to a prescribed ordering. Events correspond to starting or ending some activity. An analogy between linear systems and a class of discrete-event systems is developed. Following this analogy, such discrete-event systems can be viewed as linear, in the sense of an appropriate algebra. The periodical behavior of closed discrete-event systems, i.e., involving a set of repeatedly performed activities, can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra. This problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph. The potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized; the case of a flowshop-like production process is analyzed in detail.}
}

@article{cottenceauModelReferenceControl2001,
  title = {Model Reference Control for Timed Event Graphs in Dioids},
  author = {Cottenceau, Bertrand and Hardouin, Laurent and Boimond, Jean-Louis and Ferrier, Jean-Louis},
  year = {2001},
  month = sep,
  journal = {Automatica},
  volume = {37},
  number = {9},
  pages = {1451--1458},
  issn = {0005-1098},
  doi = {10.1016/S0005-1098(01)00073-5},
  url = {https://www.sciencedirect.com/science/article/pii/S0005109801000735},
  urldate = {2022-10-06},
  abstract = {This paper deals with feedback controller synthesis for timed event graphs in dioids. We discuss here the existence and the computation of a controller which leads to a closed-loop system whose behavior is as close as possible to the one of a given reference model and which delays as much as possible the input of tokens inside the (controlled) system. The synthesis presented here is mainly based on residuation theory results and some Kleene star properties.},
  langid = {english}
}

@article{deschutterAnalysisControlMaxplus2020,
  title = {Analysis and Control of Max-plus Linear Discrete-Event Systems: {{An}} Introduction},
  shorttitle = {Analysis and Control of Max-plus Linear Discrete-Event Systems},
  author = {De Schutter, Bart and {van den Boom}, Ton and Xu, Jia and Farahani, Samira S.},
  year = {2020},
  month = mar,
  journal = {Discrete Event Dynamic Systems},
  volume = {30},
  number = {1},
  pages = {25--54},
  issn = {1573-7594},
  doi = {10.1007/s10626-019-00294-w},
  url = {https://doi.org/10.1007/s10626-019-00294-w},
  urldate = {2022-09-10},
  abstract = {The objective of this paper is to provide a concise introduction to the max-plus algebra and to max-plus linear discrete-event systems. We present the basic concepts of the max-plus algebra and explain how it can be used to model a specific class of discrete-event systems with synchronization but no concurrency. Such systems are called max-plus linear discrete-event systems because they can be described by a model that is ``linear'' in the max-plus algebra. We discuss some key properties of the max-plus algebra and indicate how these properties can be used to analyze the behavior of max-plus linear discrete-event systems. Next, some control approaches for max-plus linear discrete-event systems, including residuation-based control and model predictive control, are presented briefly. Finally, we discuss some extensions of the max-plus algebra and of max-plus linear systems.},
  langid = {english}
}

@inproceedings{deschutterMaxplusAlgebraMaxplus2008,
  title = {Max-plus Algebra and Max-plus Linear Discrete Event Systems: {{An}} Introduction},
  shorttitle = {Max-plus Algebra and Max-plus Linear Discrete Event Systems},
  booktitle = {2008 9th {{International Workshop}} on {{Discrete Event Systems}}},
  author = {De Schutter, Bart and {van den Boom}, Ton},
  year = {2008},
  month = may,
  pages = {36--42},
  doi = {10.1109/WODES.2008.4605919},
  abstract = {We provide an introduction to the max-plus algebra and explain how it can be used to model a specific class of discrete event systems with synchronization but no concurrency. Such systems are called max-plus linear discrete event systems because they can be described by a model that is ldquolinearrdquo in the max-plus algebra. We discuss some key properties of the max-plus algebra and indicate how these properties can be used to analyze the behavior of max-plus linear discrete event systems. We also briefly present some control approaches for max-plus linear discrete event systems, including model predictive control. Finally, we discuss some extensions of the max-plus algebra and of max-plus linear systems.}
}

@article{deschutterMethodFindAll1996,
  title = {A Method to Find All Solutions of a System of Multivariate Polynomial Equalities and Inequalities in the Max Algebra},
  author = {De Schutter, Bart and De Moor, Bart},
  year = {1996},
  month = mar,
  journal = {Discrete Event Dynamic Systems},
  volume = {6},
  number = {2},
  pages = {115--138},
  issn = {1573-7594},
  doi = {10.1007/BF01797235},
  url = {https://doi.org/10.1007/BF01797235},
  urldate = {2022-11-24},
  abstract = {In this paper we show that finding solutions of a system of multivariate polynomial equalities and inequalities in the max algebra is equivalent to solving an Extended Linear Complementarity Problem. This allows us to find all solutions of such a system of multivariate polynomial equalities and inequalities and provides a geometrical insight in the structure of the solution set. We also demonstrate that this enables us to solve many important problems in the max algebra and the max-min-plus algebra such as matrix decompositions, construction of matrices with a given characteristic polynomial, state space transformations and the (minimal) state space realization problem.},
  langid = {english}
}

@techreport{deschutterModelPredictiveControl2000,
  type = {Technical Report},
  title = {On Model Predictive Control for Max-Min-plus-Scaling Discrete Event Systems},
  author = {De Schutter, Bart and {van den Boom}, T.},
  year = {2000},
  month = jun,
  number = {bds:00-04},
  pages = {21},
  address = {Delft, NL},
  institution = {{Control Systems Engineering, Faculty of Information Technology and Systems, Delft University of Technology}},
  url = {https://pub.deschutter.info/abs/00_04.html},
  abstract = {We extend the model predictive control framework, which is very popular in the process industry due to its ability to handle constraints on inputs and outputs, to a class of discrete event systems that can be modeled using the operations maximization, minimization, addition and scalar multiplication, and that we call max-min-plus-scaling systems. We show that this class encompasses several other classes of discrete event systems such as maxplus-linear systems, bilinear max-plus systems, polynomial max-plus systems, separated max-min-plus systems and regular max-min-plus systems. In general the model predictive control problem for max-min-plus-scaling systems leads to a nonlinear non-convex optimization problem, that can also be solved using extended linear complementarity problems. We show that under certain conditions the optimization problem reduces to a convex programming problem, which can be solved very efficiently.},
  langid = {english}
}

@techreport{deschutterModelPredictiveControl2000,
  type = {Technical Report},
  title = {Model Predictive Control for Max-plus-Linear Discrete-Event Systems: {{Extended}} Report \& {{Addendum}}},
  author = {De Schutter, Bart and {van den Boom}, Ton},
  year = {2000},
  month = nov,
  number = {bds:99-10a},
  address = {Delft, The Netherlands},
  institution = {Delft University of Technology},
  url = {https://pub.deschutter.info/abs/99_10a.html},
  abstract = {Model predictive control (MPC) is a very popular controller design method in the process industry. A key advantage of MPC is that it can accommodate constraints on the inputs and outputs. Usually MPC uses linear discrete-time models. In this report we extend MPC to a class of discrete-event systems that can be described by models that are "linear" in the max-plus algebra, which has maximization and addition as basic operations. In general the resulting optimization problem are nonlinear and non-convex. However, if the control objective and the constraints depend monotonically on the outputs of the system, the model predictive control problem can be recast as problem with a convex feasible set. If in addition the objective function is convex, this leads to a convex optimization problem, which can be solved very efficiently.},
  langid = {english}
}

@article{deschutterModelPredictiveControl2001,
  title = {Model Predictive Control for Max-plus-Linear Discrete Event Systems},
  author = {De Schutter, Bart and {van den Boom}, Ton},
  year = {2001},
  month = jul,
  journal = {Automatica},
  volume = {37},
  number = {7},
  pages = {1049--1056},
  issn = {0005-1098},
  doi = {10.1016/S0005-1098(01)00054-1},
  url = {https://www.sciencedirect.com/science/article/pii/S0005109801000541},
  urldate = {2022-09-10},
  abstract = {Model predictive control (MPC) is a very popular controller design method in the process industry. A key advantage of MPC is that it can accommodate constraints on the inputs and outputs. Usually MPC uses linear discrete-time models. In this paper we extend MPC to a class of discrete-event systems that can be described by models that are ``linear'' in the max-plus algebra, which has maximization and addition as basic operations. In general, the resulting optimization problem are nonlinear and nonconvex. However, if the control objective and the constraints depend monotonically on the outputs of the system, the model predictive control problem can be recast as problem with a convex feasible set. If in addition the objective function is convex, this leads to a convex optimization problem, which can be solved very efficiently.},
  langid = {english}
}

@inproceedings{deschutterModelPredictiveControl2001a,
  title = {Model Predictive Control for Max-Min-plus-Scaling Systems},
  booktitle = {Proceedings of the 2001 {{American Control Conference}}. ({{Cat}}. {{No}}.{{01CH37148}})},
  author = {De Schutter, B. and {van den Boom}, T.J.J.},
  year = {2001},
  month = jun,
  volume = {1},
  pages = {319-324 vol.1},
  issn = {0743-1619},
  doi = {10.1109/ACC.2001.945564},
  abstract = {We further extend the model predictive control framework, which is very popular in the process industry due to its ability to handle constraints on inputs and outputs, to a class of discrete event systems that can be modeled using the operations maximization, minimization, addition and scalar multiplication. This class encompasses max-plus-linear systems, min-max-plus systems, bilinear max-plus systems and polynomial max-plus systems. In general the model predictive control problem for max-min-plus-scaling systems leads to a nonlinear non-convex optimization problem, that can also be reformulated as an optimization problem over the solution set of an extended linear complementarity problem. We also show that under certain conditions the optimization problem reduces to a convex programming problem, which can be solved very efficiently.}
}

@inproceedings{deschutterModelPredictiveControl2002,
  title = {Model Predictive Control for Max-Min-plus-Scaling Systems - Efficient Implementation},
  booktitle = {Sixth {{International Workshop}} on {{Discrete Event Systems}}, 2002. {{Proceedings}}.},
  author = {De Schutter, B. and {van den Boom}, T.J.J.},
  year = {2002},
  month = oct,
  pages = {343--348},
  doi = {10.1109/WODES.2002.1167709},
  abstract = {In previous work we have introduced model predictive control (MPC) for max-plus-linear and max-min-plus(-scaling) discrete-event systems. For max-plus-linear systems there are efficient algorithms to solve the corresponding MPC optimization problems. However, previously, for max-min-plus(-scaling) systems the only approach was to consider a limited subclass of decoupled max-min-plus systems or to use nonlinear nonconvex optimization algorithms, which are not efficient if the size of the system or the MPC optimization problem is large. In this paper we present a more efficient approach that is based on canonical forms for max-min-plus-scaling functions and in which the MPC optimization problem is reduced to a set of linear programming problems.}
}

@article{farahaniOptimizationStochasticMax2017,
  title = {On Optimization of Stochastic Max--Min-plus-Scaling Systems---{{An}} Approximation Approach},
  author = {Farahani, Samira S. and {van den Boom}, Ton and De Schutter, Bart},
  year = {2017},
  month = sep,
  journal = {Automatica},
  volume = {83},
  pages = {20--27},
  issn = {0005-1098},
  doi = {10.1016/j.automatica.2017.05.001},
  url = {https://www.sciencedirect.com/science/article/pii/S0005109817302637},
  urldate = {2022-10-10},
  abstract = {A large class of discrete-event and hybrid systems can be described by a max--min-plus-scaling (MMPS) model, i.e., a model in which the main operations are maximization, minimization, addition, and scalar multiplication. Accordingly, optimization of MMPS systems appears in different problems defined for discrete-event and hybrid systems. For a stochastic MMPS system, this optimization problem is computationally highly demanding as often numerical integration has to be used to compute the objective function. The aim of this paper is to decrease such computational complexity by applying an approximation method that is based on the moments of a random variable and that can be computed analytically.},
  langid = {english}
}

@book{heidergottMaxWorkModeling2005,
  title = {Max {{Plus}} at {{Work}}: {{Modeling}} and {{Analysis}} of {{Synchronized Systems}}: {{A Course}} on {{Max-Plus Algebra}} and {{Its Applications}}},
  shorttitle = {Max {{Plus}} at {{Work}}},
  author = {Heidergott, Bernd and Olsder, Geert Jan and van der Woude, Jacob},
  year = {7 listopadu 2005},
  series = {Princeton {{Applied Mathematics}}},
  publisher = {Princeton University Press},
  address = {Princeton, NJ},
  url = {https://press.princeton.edu/books/hardcover/9780691117638/max-plus-at-work},
  abstract = {Trains pull into a railroad station and must wait for each other before leaving again in order to let passengers change trains. How do mathematicians then calculate a railroad timetable that accurately reflects their comings and goings? One approach is to use max-plus algebra, a framework used to model Discrete Event Systems, which are well suited to describe the ordering and timing of events. This is the first textbook on max-plus algebra, providing a concise and self-contained introduction to the topic. Applications of max-plus algebra abound in the world around us. Traffic systems, computer communication systems, production lines, and flows in networks are all based on discrete even systems, and thus can be conveniently described and analyzed by means of max-plus algebra. The book consists of an introduction and thirteen chapters in three parts. Part One explores the introduction of max-plus algebra and of system descriptions based upon it. Part Two deals with a real application, namely the design of timetables for railway networks. Part Three examines various extensions, such as stochastic systems and min-max-plus systems. The text is suitable for last-year undergraduates in mathematics, and each chapter provides exercises, notes, and a reference section.},
  isbn = {978-0-691-11763-8}
}

@article{katzWHATTropicalGeometry2017,
  title = {{{WHAT IS}}...{{Tropical Geometry}}?},
  author = {Katz, Eric},
  year = {2017},
  month = apr,
  journal = {Notices of the American Mathematical Society},
  volume = {64},
  number = {04},
  pages = {380--382},
  issn = {0002-9920, 1088-9477},
  doi = {10.1090/noti1507},
  url = {http://www.ams.org/notices/201704/rnoti-p380.pdf},
  urldate = {2023-10-05},
  langid = {english}
}

@article{komendaMaxplusAlgebraHistory2018,
  title = {Max-plus Algebra in the History of Discrete Event Systems},
  author = {Komenda, J. and Lahaye, S. and Boimond, J. -L. and {van den Boom}, T.},
  year = {2018},
  month = jan,
  journal = {Annual Reviews in Control},
  volume = {45},
  pages = {240--249},
  issn = {1367-5788},
  doi = {10.1016/j.arcontrol.2018.04.004},
  url = {https://www.sciencedirect.com/science/article/pii/S1367578818300129},
  urldate = {2022-09-20},
  abstract = {This paper is a survey of the history of max-plus algebra and its role in the field of discrete event systems during the last three decades. It is based on the perspective of the authors but it covers a large variety of topics, where max-plus algebra plays a key role.},
  langid = {english}
}

@book{maclaganIntroductionTropicalGeometry2015,
  title = {Introduction to {{Tropical Geometry}}},
  author = {Maclagan, Diane and Sturmfels, Bernd},
  year = {2015},
  month = apr,
  series = {Graduate {{Studies}} in {{Mathematics}}},
  volume = {161},
  publisher = {American Mathematical Society},
  issn = {1065-7339},
  doi = {10.1090/gsm/161},
  url = {https://www.ams.org/gsm/161},
  urldate = {2023-10-05},
  abstract = {Advancing research. Creating connections.},
  isbn = {978-1-4704-2221-9 978-1-4704-2235-6 978-1-4704-3731-2 978-0-8218-5198-2},
  langid = {english}
}

@article{maiaOptimalClosedloopControl2003,
  title = {Optimal Closed-Loop Control of Timed {{EventGraphs}} in Dioids},
  author = {Maia, C.A. and Hardouin, L. and {Santos-Mendes}, R. and Cottenceau, B.},
  year = {2003},
  month = dec,
  journal = {IEEE Transactions on Automatic Control},
  volume = {48},
  number = {12},
  pages = {2284--2287},
  issn = {1558-2523},
  doi = {10.1109/TAC.2003.820666},
  abstract = {This note deals with the model-reference control of timed event graphs using the dioid algebra and the residuation theory. It proposes a control structure based on a precompensator and a feedback controller to improve the controlled system performance. It is shown that this approach always leads to an optimal behavior of the closed-loop system. An example is given to illustrate the proposed approach.}
}

@unpublished{MainPdf,
  type = {Draft of a Book},
  title = {Tropical {{Geometry}}},
  author = {Mikhalkin, Grigory and Rau, Johannes},
  year = {2018},
  month = nov,
  url = {https://www.math.uni-tuebingen.de/user/jora/downloads/main.pdf},
  urldate = {2023-10-05}
}

@article{menguyJustintimeControlTimed2000,
  title = {Just-in-Time Control of Timed Event Graphs: Update of Reference Input, Presence of Uncontrollable Input},
  shorttitle = {Just-in-Time Control of Timed Event Graphs},
  author = {Menguy, E. and Boimond, J.-L. and Hardouin, L. and Ferrier, J.-L.},
  year = {2000},
  month = nov,
  journal = {IEEE Transactions on Automatic Control},
  volume = {45},
  number = {11},
  pages = {2155--2159},
  issn = {1558-2523},
  doi = {10.1109/9.887652},
  abstract = {A linear system theory has been developed for the class of discrete-event systems subject to synchronization. This paper presents the just-in-time control of such systems when reference input is updated and/or in the presence of uncontrollable input(s), the proposed controls are the solutions to an optimization problem under equality constraint.}
}

@article{necoaraModelPredictiveControl2008,
  title = {Model Predictive Control for Uncertain Max--Min-plus-Scaling Systems},
  author = {Necoara, I. and De Schutter, B. and Van Den Boom, T. and Hellendoorn, H.},
  year = {2008},
  month = may,
  journal = {International Journal of Control},
  volume = {81},
  number = {5},
  pages = {701--713},
  publisher = {Taylor \& Francis},
  issn = {0020-7179},
  doi = {10.1080/00207170601094404},
  url = {https://www.dcsc.tudelft.nl/},
  urldate = {2022-10-10},
  abstract = {In this paper we extend the classical min--max model predictive control framework to a class of uncertain discrete event systems that can be modelled using the operations maximization, minimization, addition and scalar multiplication, and that we call max--min-plus-scaling (MMPS) systems. Provided that the stage cost is an MMPS expression and considering only linear input constraints then the open-loop min--max model predictive control problem for MMPS systems can be transformed into a sequence of linear programming problems. Hence, the min--max model predictive control problem for MMPS systems can be solved efficiently, despite the fact that the system is non-linear. A min--max feedback model predictive control approach using disturbance feedback policies is also presented, which leads to improved performance compared to the open-loop approach.}
}

@misc{quadratJuliaMaxMin2022,
  title = {Julia's (Max,+) and (Min,+) {{Algebra Toolbox}}},
  author = {Quadrat, Quentin},
  year = {2022},
  month = sep,
  url = {https://github.com/Lecrapouille/MaxPlus.jl},
  urldate = {2022-09-16},
  copyright = {Unlicense}
}

@unpublished{rauFirstExpeditionTropical2017,
  title = {A {{First Expedition}} to {{Tropical Geometry}}},
  author = {Rau, Johannes},
  year = {2017},
  month = apr,
  url = {https://www.math.uni-tuebingen.de/user/jora/downloads/FirstExpedition.pdf},
  langid = {english}
}

@article{speyerTropicalMathematics2009,
  title = {Tropical {{Mathematics}}},
  author = {Speyer, David and Sturmfels, Bernd},
  year = {2009},
  month = jun,
  journal = {Mathematics Magazine},
  volume = {82},
  number = {3},
  pages = {163--173},
  url = {https://math.berkeley.edu/~bernd/mathmag.pdf},
  urldate = {2023-10-05}
}

@misc{stanczykMaxPlusAlgebraToolbox2016,
  title = {Max-{{Plus Algebra Toolbox}} for {{Matlab}}},
  author = {Sta{\'n}czyk, Jaros{\l}aw},
  year = {2016},
  month = jun,
  url = {http://www.stanczyk.pro/mpa/}
}

@article{tebaniMinPlusRealizableControl2021,
  title = {Min-{{Plus}} Realizable Control Design for Partially Observable Timed Event Graphs under Marking Constraints},
  author = {Tebani, Karima and Amari, Said},
  year = {2021},
  month = jan,
  journal = {European Journal of Control},
  volume = {57},
  pages = {33--40},
  issn = {0947-3580},
  doi = {10.1016/j.ejcon.2020.12.002},
  url = {https://www.sciencedirect.com/science/article/pii/S0947358020305434},
  urldate = {2022-10-06},
  abstract = {This paper deals with a control problem of discrete event systems subject to capacity constraints. Combined use of timed event graphs and Min-Plus algebra is a well-known approach and efficient for handling timed behavior and mathematical modelling of discrete event systems. However, in current literature, many of control approaches assume that system states are fully observable, which is not the case in our study. Hence, we propose in this paper a feedback control method to guarantee the respect of marking constraints imposed for some paths of partially observable timed event graphs. We demonstrate that if each loop of the considered TEG contains at last one observable transition, we can derive a realisable control law satisfying a set of constraints.},
  langid = {english}
}

@inproceedings{vandenboomModelingFrameworkModel2013,
  title = {A Modeling Framework for Model Predictive Scheduling Using Switching Max-plus Linear Models},
  booktitle = {52nd {{IEEE Conference}} on {{Decision}} and {{Control}}},
  author = {{van den Boom}, Ton J.J. and Lopes, Gabriel Delgado and De Schutter, Bart},
  year = {2013},
  month = dec,
  pages = {5456--5461},
  issn = {0191-2216},
  doi = {10.1109/CDC.2013.6760748},
  abstract = {In this paper we discuss a modeling framework for model predictive scheduling of a class of semi-cyclic discrete event systems that can be described by switching max-plus linear models. We study the structure of the system matrices and derive how routing, ordering, and synchronization can be manipulated by a set of control variables. In addition, we show that this leads to a system matrix that is linear in the control variables. We define the model predictive scheduling design problem to optimize the schedule, and we show that the problem can be recast as a mixed integer linear programming (MILP) problem.}
}

@article{vandenboomModellingControlDiscrete2006,
  title = {Modelling and Control of Discrete Event Systems Using Switching Max-plus-Linear Systems},
  author = {{van den Boom}, T. J. J. and De Schutter, B.},
  year = {2006},
  month = oct,
  journal = {Control Engineering Practice},
  series = {The {{Seventh Workshop On Discrete Event Systems}} ({{WODES2004}})},
  volume = {14},
  number = {10},
  pages = {1199--1211},
  issn = {0967-0661},
  doi = {10.1016/j.conengprac.2006.02.006},
  url = {https://www.sciencedirect.com/science/article/pii/S0967066106000232},
  urldate = {2022-09-10},
  abstract = {In this paper we consider the modelling and control of discrete event systems using switching max-plus-linear systems. In switching max-plus-linear systems we can switch between different modes of operation. In each mode the discrete event system is described by a max-plus-linear state space model with different system matrices for each mode. The switching allows us to change the structure of the system, to break synchronization and to change the order of events. We will give some examples of this type of systems. We define the model predictive control design problem for this type of discrete event system, and we show that solving this problem in general leads to a mixed integer optimization problem.},
  langid = {english}
}

@incollection{vandenboomModelPredictiveControl2014,
  title = {Model {{Predictive Control}} of {{Manufacturing Systems}} with {{Max-Plus Algebra}}},
  booktitle = {Formal {{Methods}} in {{Manufacturing}}},
  author = {{van den Boom}, Ton J. J. and De Schutter, Bart},
  editor = {Campos, Javier and Seatzu, Carla and Xie, Xiaolan},
  year = {2014},
  pages = {343--378},
  publisher = {CRC Press},
  address = {Boca Raton},
  url = {https://www.dcsc.tudelft.nl/~bdeschutter/pub/rep/14_002.pdf},
  abstract = {This chapter considers the problem of designing an model predictive controller (MPC) for the class of max-plus linear (MPL) discrete event systems and gives and extensive overview of the results. It considers the case where the input, output and state sequence must satisfy a set of linear inequality constraints. The chapter discusses the background in MPC for time-driven systems and gives some basic results in max-plus algebra and MPL systems. It gives an analytic expression for the controller and provides sufficient conditions for stability. The chapter analyses the performance of the MPC controller. It treats robust MPC in the case of perturbed operation due to modelling errors and/or noise. The chapter also considers both the bounded perturbation case and the stochastic perturbation case. It shows that under quite general conditions, the resulting optimization problems can be solved very efficiently.},
  isbn = {978-1-315-21614-0}
}

@article{xuOptimisticOptimizationModel2016,
  title = {Optimistic Optimization for Model Predictive Control of Max-plus Linear Systems},
  author = {Xu, Jia and {van den Boom}, Ton and De Schutter, Bart},
  year = {2016},
  month = dec,
  journal = {Automatica},
  volume = {74},
  pages = {16--22},
  issn = {0005-1098},
  doi = {10.1016/j.automatica.2016.07.002},
  url = {https://www.sciencedirect.com/science/article/pii/S0005109816302709},
  urldate = {2022-10-06},
  abstract = {Model predictive control for max-plus linear discrete-event systems usually leads to a nonsmooth nonconvex optimization problem with real valued variables, which may be hard to solve efficiently. An alternative approach is to transform the given problem into a mixed integer linear programming problem. However, the computational complexity of current mixed integer linear programming algorithms increases in the worst case exponentially as a function of the prediction horizon. The focus of this paper is on making optimistic optimization suited to solve the given problem. Optimistic optimization is a class of algorithms that can find an approximation of the global optimum for general nonlinear optimization. A key advantage of optimistic optimization is that one can specify the computational budget in advance and guarantee bounds on the suboptimality with respect to the global optimum. We prove that optimistic optimization can be applied for the given problem by developing a dedicated semi-metric and by proving it satisfies the necessary requirements for optimistic optimization. Moreover, we show that the complexity of optimistic optimization is exponential in the control horizon instead of the prediction horizon. Hence, using optimistic optimization is more efficient when the control horizon is small and the prediction horizon is large.},
  langid = {english}
}