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24
Algebraic Tools for the Performance Evaluation of Discrete Event Systems
 IEEE Proceedings: Special issue on Discrete Event Systems
, 1989
"... In this paper, it is shown that a certain class of Petri nets called event graphs can be represented as linear "timeinvariant" finitedimensional systems using some particular algebras. This sets the ground on which a theory of these systems can be developped in a manner which is very ana ..."
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Cited by 96 (6 self)
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In this paper, it is shown that a certain class of Petri nets called event graphs can be represented as linear "timeinvariant" finitedimensional systems using some particular algebras. This sets the ground on which a theory of these systems can be developped in a manner which is very analogous to that of conventional linear system theory. Part 2 of the paper is devoted to showing some preliminary basic developments in that direction. Indeed, there are several ways in which one can consider event graphs as linear systems: these ways correspond to approaches in the time domain, in the event domain and in a twodimensional domain. In each of these approaches, a di#erent algebra has to be used for models to remain linear. However, the common feature of these algebras is that they all fall into the axiomatic definition of "dioids". Therefore, Part 1 of the paper is devoted to a unified presentation of basic algebraic results on dioids. 1 Introduction Definitions and examples of Discrete ...
Ergodic Theorems for Stochastic Operators and Discrete Event Networks
, 1995
"... We present a survey of the main ergodic theory techniques which are used in the study of iterates of monotone and homogeneous stochastic operators. It is shown that ergodic theorems on discrete event networks (queueing networks and/or Petri nets) are a generalization of these stochastic operator the ..."
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Cited by 14 (2 self)
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We present a survey of the main ergodic theory techniques which are used in the study of iterates of monotone and homogeneous stochastic operators. It is shown that ergodic theorems on discrete event networks (queueing networks and/or Petri nets) are a generalization of these stochastic operator theorems. Kingman's subadditive ergodic Theorem is the key tool for deriving what we call rst order ergodic results. We also show how to use backward constructions (also called Loynes schemes in network theory) in order to obtain second order ergodic results. We will propose a review of systems entering the framework insisting on two models, precedence constraints networks and Jackson type networks.
Complexity reduction in MPC for stochastic maxpluslinear discrete event systems by variability expansion.
, 2007
"... Abstract Model predictive control (MPC) is a popular controller design technique in the process industry. Recently, MPC has been extended to a class of discrete event systems that can be described by a model that is "linear" in the maxplus algebra. In this context both the perturbations ..."
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Cited by 5 (3 self)
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Abstract Model predictive control (MPC) is a popular controller design technique in the process industry. Recently, MPC has been extended to a class of discrete event systems that can be described by a model that is "linear" in the maxplus algebra. In this context both the perturbationsfree case and for the case with noise and/or modeling errors in a bounded or stochastic setting have been considered. In each of these cases an optimization problem has to be solved online at each event step in order to determine the MPC input. This paper considers a method to reduce the computational complexity of this optimization problem, based on variability expansion. In particular, it is shown that the computational load is reduced if one decreases the level of "randomness" in the system.
Series Expansions of Lyapunov Exponents and Forgetful Monoids
, 2000
"... We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for ..."
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Cited by 5 (0 self)
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We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for Tetrislike heaps of pieces models, we give a series expansion formula for the Lyapunov exponent, as a function of the probability law. In the case of rational probability laws, we show that the Lyapunov exponent is an analytic function of the parameters of the law, in a domain that contains the absolute convergence domain of a partition function associated to a special "forgetful" monoid, defined by generators and relations.
Taylor Expansions for Poisson Driven (max; +)Linear Systems
, 1995
"... We give a Taylor expansion for the mean value of the canonical stationary state variables fWng = fXn \Gamma Tng of open (max; +)linear stochastic systems with Poisson input process, that is systems with (transient) state variables fXng satisfying the vectorial recursion Xn+1 = AnXn \Phi Bn+1Tn+1 ..."
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Cited by 4 (1 self)
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We give a Taylor expansion for the mean value of the canonical stationary state variables fWng = fXn \Gamma Tng of open (max; +)linear stochastic systems with Poisson input process, that is systems with (transient) state variables fXng satisfying the vectorial recursion Xn+1 = AnXn \Phi Bn+1Tn+1 in this algebra, where fTng is a Poisson point process, and fAng and fBng are sequences of random matrices satisfying certain independence properties. Probabilistic expressions are derived for coefficients of all orders, under certain integrability conditions: the kth coefficient in the Taylor expansion of the ith component IEW i n of IEWn is the expectation of a polynomial p k+1 (D i 0 ; : : : ; D i k ), known in explicit form, of the random variables D i 0 ; : : : ; D i k , where D i n = (A \Gamma1 : : : A \Gamman B \Gamman ) i . The polynomials fp k g are of independent combinatorial interest: their monomials belong to a subset of those obtained in the multinomial expansion...
B.: An approximation approach for model predictive control of stochastic maxplus linear systems
 In: Proc. 10th Int. Workshop Discrete Event Systems
, 2010
"... Abstract: Model Predictive Control (MPC) is a modelbased control method based on a receding horizon approach and online optimization. In previous work we have extended MPC to a class of discreteevent systems, namely the maxplus linear systems, i.e., models that are "linear" in the maxp ..."
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Cited by 3 (2 self)
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Abstract: Model Predictive Control (MPC) is a modelbased control method based on a receding horizon approach and online optimization. In previous work we have extended MPC to a class of discreteevent systems, namely the maxplus linear systems, i.e., models that are "linear" in the maxplus algebra. Lately, the application of MPC for stochastic maxpluslinear systems has attracted a lot of attention. At each event step, an optimization problem then has to be solved that is, in general, a highly complex and computationally hard problem. Therefore, the focus of this paper is on decreasing the computational complexity of the optimization problem. To this end, we use an approximation approach that is based on the pth raw moments of a random variable. This method results in a much lower computational complexity and computation time while still guaranteeing a good performance.
Products of irreducible random matrices . . .
, 1995
"... We consider the recursive equation “x(n + 1) = A(n) ⊗ x(n) ” where x(n + 1) and x(n) are R kvalued vectors and A(n) is an irreducible random matrix of size k × k. The matrixvector multiplication in the (max,+) algebra is defined by (A(n) ⊗ x(n))i = maxj(Aij(n) + xj(n)). This type of equation ..."
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Cited by 3 (0 self)
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We consider the recursive equation “x(n + 1) = A(n) ⊗ x(n) ” where x(n + 1) and x(n) are R kvalued vectors and A(n) is an irreducible random matrix of size k × k. The matrixvector multiplication in the (max,+) algebra is defined by (A(n) ⊗ x(n))i = maxj(Aij(n) + xj(n)). This type of equation can be used to represent the evolution of Stochastic Event Graphs which include cyclic Jackson Networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence {A(n), n ∈ N} is i.i.d or more generally stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices C such that
Calculating the lyapunov exponent for generalized linear systems with exponentially distributed elements of the transition matrix
 Vestnik St. Petersburg Univ. Math
, 2009
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Methodologies for discrete event dynamic systems: a survey
 Publisher: Koninkliijke Vlaamse Ingenieursvereniging
, 1995
"... In recent years both industry and the academic world are becoming more and more interested in techniques to model, analyse and control complex systems such as flexible production systems, parallel processing systems, railway traffic networks and so on. This kind of systems are typical examples of di ..."
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Cited by 2 (0 self)
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In recent years both industry and the academic world are becoming more and more interested in techniques to model, analyse and control complex systems such as flexible production systems, parallel processing systems, railway traffic networks and so on. This kind of systems are typical examples of discrete event dynamic systems, the subject of an emerging discipline in systems and control theory. In this paper we present an overview of some of the important methodologies for discrete event dynamic systems. All these methods have been studied intensively by the authors in the framework of projects. 1