Fixed zeros in the model matching problem for systems over a semiring

Computer-integrated engineering systems are event-driven and often asynchronous, exhibiting concurrent, sequential, competitive and coordinated activities among their components. They are often complex and large in scale. These systems include integrated manufacturing systems, concurrent distributed systems, computer operating systems, communication networks, and intelligent machines. They fall into a class of systems called discrete event dynamic systems (DEDSs). The relations of events, their evolution over time and their order of appearance are of main interest. Compared with systems modeled as differential or difference equations, e.g. a manufacturing process, one has to deal with qualitative or abrupt changes characterized as events, instead of with quantitative changes in conventional continuous or discrete time systems. An investigation into the properties of and design methods for a DEDS has to start with an appropriate mathematical representation.

Discrete event dynamical systems are used to model a number of engineering applications ranging from communication networks, distributed computing systems to manufacturing systems. A new analytical model is proposed for the analysis of asynchronous, optimistic distributed simulation of discrete event dynamical systems. The performance of traditional timewarp algorithms and that of a new rollback algorithm, Wolf, are examined in this framework and a comparison is attempted. A few preliminary implementations of Wolf on a distributed computing system confirm analytical results predicting a considerable reduction in error propagation, a decrease in the number of cancellations and an enhancement in the forward computation.

The forecast of delay time is of great assistance to decision making in train operation adjustment when the schedule is disturbed either by infrastructure fault or natural hazard. This paper presents a railway delay propagation model to forecast the delay time, described by discrete event dynamic system (DEDS) and formulated by max-plus algebra theory. On the basis of the train operation regulations and headway constraints, a system matrix of max-plus algebra is acquired to illustrate the mechanism of delay propagation. And then a function to predict the delay time is proposed to solve the model, with two advantages: Firstly, the specific delay time is able to be calculated; secondly, the result of the prediction is comparatively precise due to the highly match of the model to the actual operation. Finally, by analysis of the prediction, this paper offers the decision support in train adjustment, from which the dispatcher can proactively conduct countermeasures to alleviate the propagation and even stop it.

Impetuous multiplication of mobility and road traffic proliferation lead to concerns for increasing the attractiveness of urban public transport. Compared to private car use, urban public transport attractiveness is conditioned, in particular, by travel times and certainty of respecting the transport schedules, meaning planned traffic stability. Traffic schedules are considered to be more stable as the primary delays from the announced schedule have low probabilities and values and their propagation as repeated delays is least noticed in time and space. Solutions for assuring traffic stability must take into consideration contradictory aspects, because introducing time reserves in the schedules means time travel extensions. In order to assure the stability of planned traffic, present paper develops studies of various models and methods that aim to reduce inherent primary delays. Thereby, for studying repeated delays on a complex network, a mathematical model adequate to a Discrete Event Dynamic System (DEDS), that in MAX-PLUS algebra becomes a linear system, was used. The paper concludes with a case study on an integrated network resulted from the superposition of Bucharest’s existing suburban rail network with the underground network designed for 2030, being identified measures for improving the stability indicators. Traffic stability is assessed on the basis of two indicators: instability coefficient and delay elimination rate. Main measure for improving stability indicators is the growth of time reserves taking into consideration the quality requirements resulting from the condition of maintaining a reduced planned travel time.

The traditional models of distributed computing systems employ the mathematical formalisms of discrete event dynamical systems along with Petri Nets. However, such models are complex to analyze and computationally expensive. Interestingly, the evolving distributed computing systems closely resemble the discrete dynamical systems. This paper constructs and analyzes the model of distributed computing systems by applying mathematical formalisms of the discrete dynamical systems. The proposed model embeds the partial ordering of states under happened-before relation in the set of globally observable states. The stability of the proposed model is analyzed using the first order linear approximation.

This paper extends a rule-based controller formulation described in the references to include the maxplus formulation in the case of general Petri nets, not only event graphs. This potentially allows design of conflict resolution inputs (i.e. dispatching rules) for the re-entrant flow line and the job shop problem in the maxplus algebra, as well as in a new algebra being developed for discrete event dynamical systems in the references. >

ABSTRACTIn this paper, the static output feedback stabilisation of discrete event dynamic systems (DEDSs) is investigated via the semi-tensor product (STP) of matrices. Firstly, the dynamics of DEDSs modelled by deterministic Moore-type finite automata are converted into an matrix expression in the STP frame. Secondly, necessary and sufficient conditions for the existence of a static output feedback control pattern, stabilising the controlled discrete event dynamic systems to some equilibrium point, are given, and constructive algorithms to seek the static output feedback control pattern including an effective specific solution algorithm and an analytic solution algorithm are proposed. Thirdly, the equilibrium-based static output feedback stabilisation of DEDSs is extended to the set-based static output feedback stabilisation and necessary and sufficient conditions and constructive algorithms to seek the corresponding control pattern are provided. Finally, some examples are presented to illustrate the effectiveness of the proposed approach.

Consider a discrete dynamical system given by a square matrix $M \in \mathbb{Q}^{d \times d}$ and a starting point $s \in \mathbb{Q}^d$. The orbit of such a system is the infinite trajectory $\langle s, Ms, M^2s, \ldots\rangle$. Given a collection $T_1, T_2, \ldots, T_m \subseteq \mathbb{R}^d$ of semialgebraic sets, we can associate with each $T_i$ an atomic proposition $P_i$ which evaluates to true at time $n$ if, and only if, $M^ns \in T_i$. This gives rise to the LTL Model-Checking Problem for discrete linear dynamical systems: given such a system $(M,s)$ and an LTL formula over such atomic propositions, determine whether the orbit satisfies the formula. The main contribution of the present paper is to show that the LTL Model-Checking Problem for discrete linear dynamical systems is decidable in dimension 3 or less.

Algebraic Characteristics and Structure Decomposition for “Linear” Discrete Event Dynamic Systems

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

The running time of high-speed train is generally not late, and it can run normally in most cases. However, when severe weather conditions, train components and equipment, accidents or emergencies occur, it will lead to train operation delay and traffic congestion. Therefore, when an accident occurs, we need to adjust the train time or route timely and accurately. As an important algebraic system, max-plus algebra is widely used in the field of industrial production control. In industrial production, the most production mode is the discrete system , but the characteristics and the ability of discrete systems depends on the periodic of system and the number of workpiece produced by the system in unit time, and the characteristics of the system are closely related to the properties of the matrix, especially, the eigenvalues and eigenvectors of the matrix in the sense of max-plus algebra. Therefore, this paper studies the max-plus algebra theory and the solution of eigenvalues and eigenvectors of matrices in the sense of max-plus algebra, establish the operation time matrix to optimize the train operation adjustment model of high-speed railway, and analyze the failure propagation model.

Discrete event dynamic systems (DEDS), are systems, which change state over time, based on current state and state transition rules. Each state is separated from its neighbour by a step rather than a continuum of infinitesimal states. This continuum in DEDS needs a representation between 0 and 1. The application of fuzzy logic in defining and modelling the DEDS is suggested as a possible solution because the nature of DEDS may be asynchronous, distributed, parallel, non-deterministic and/or stochastic. This results in the generalisation of crisp finite automata into fuzzy automata for a DEDS. In this paper, a DEDS in the form of hopper feed system is considered. The system is reformulated using state vectors and event transition matrices and is extended to fuzzy vectors and matrices. An algorithm is proposed for the final control action to be implemented using a programmable logic controller (PLC).

The paper describes a novel approach to model any deterministic manufacturing system. A function block diagram type of model is proposed. A block can be a single machine or a factory. The model is expressed as a system of linear event timing equations using max-plus algebra. One of the advantages of block diagram representation is the simplicity of algebraic relations between the sub-system transfer functions [1]. It is expected that the proposed model will make it possible to develop and use techniques similar to those found in classical transfer function block diagrams.

This paper is an updated and extended version of the paper "The QR Decomposition and the Singular Value Decomposition in the Symmetrized Max-Plus Algebra" (B. De Schutter and B. De Moor, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 378--406). The max-plus algebra, which has maximization and addition as its basic operations, can be used to describe and analyze certain classes of discrete-event systems, such as flexible manufacturing systems, railway networks, and parallel processor systems. In contrast to conventional algebra and conventional (linear) system theory, the max-plus algebra and the max-plus-algebraic system theory for discrete-event systems are at present far from fully developed, and many fundamental problems still have to be solved. Currently, much research is going on to deal with these problems and to further extend the max-plus algebra and to develop a complete max-plus-algebraic system theory for discrete-event systems. In this paper we address one of the remaining gaps in the max-plus algebra by considering matrix decompositions in the symmetrized max-plus algebra. The symmetrized max-plus algebra is an extension of the max-plus algebra obtained by introducing a max-plus-algebraic analogue of the $-$-operator. We show that we can use well-known linear algebra algorithms to prove the existence of max-plus-algebraic analogues of basic matrix decompositions from linear algebra such as the QR decomposition, the singular value decomposition, the Hessenberg decomposition, the LU decomposition, and so on. These max-plus-algebraic matrix decompositions could play an important role in the max-plus-algebraic system theory for discrete-event systems.

ALGEBRAIC CHARACTERISTICS AND STRUCTURE DECOMPOSITION FOR “LINEAR” DISCRETE EVENT DYNAMIC SYSTEMS