G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot, "A linear-systemtheoretic view of discrete-event processes and its use for performance evaluation in manufacturing". IEEE Trans. Automat. Control, 30, pp. 210-220, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....methodology for maxplus linear systems. In Section 3 we discuss the closedloop properties of max plus algebraic MPC. Next we discuss the tuning of the parameters in MPC for maxplus linear systems. We conclude with some illustrative examples. 2 The MPC problem for max plus linear systems In [1, 2, 3] it has been shown that discrete event systems with only synchronization and no concurrency can be modeled by a max plus algebraic model of the following form: x(k 1) B u(k) 1) y(k) x(k) 2) with A 2 R , B 2 R n m l n where m is the number of inputs and l the number of ....

....parameters (N p , N c , and to select appropriate due dates r(k) Again we assume that we are dealing with a SISO system (so l = m = 1) Furthermore, we will assume irreducibility of the system . In many applications, for example in manufacturing systems, this assumption is not restrictive [2]. The selection of appropriate parameters has to lead to a stabilizing and e ective control law. The MPC algorithm computes the vector of controls using optimization of the cost criterion (3) with additional conditions (4) 6) and (7) For now we will not consider constraints of the form (5) ....

[Article contains additional citation context not shown here]

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot, \A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing," IEEE Transactions on Automatic Control, vol. 30, no. 3, pp. 210-220, Mar. 1985.

....a. For i belonging to the critical graph, i is an eigenvector of A. b. i and :j are different iff i and j belong to two different s.c.s. of the critical graph. c. Every eigenvector of A is a linear combination of the critical columns :i . Theorem 3 (Cohen Dubois Quadrat Viot [5, 6]) Let A be irreducible, ae(A) its eigenvalue, c its cyclicity. 9N; 8n N; A n c ae(A) Omega c Omega On Fig. 3, we have represented the set of eigenvectors (gray area) of matrix A and also the effect of matrix A on a ball (for a more complicated example, involving a cat, see the ....

....: where v is a finite pattern satisfying the ratio constraint. Problem 1: How to evaluate (v) the throughput associated with a given pattern. Problem 2: How to choose the best schedule among patterns of a given length, i.e. maxf(v) j jvj = ng Event Graph Algorithm (Cohen Dubois Quadrat Viot [6], Hillion Proth [18] d c p41 p42 p32 Figure 21: Timed Event Graph associated with the schedule (abcd) For a given periodic schedule, one is able to build a timed Event Graph representing the system. This is done by replacing some of the choice places by circuits, forcing the periodic ....

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot. A linear system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automatic Control, AC-30:210--220, 1985.

.... algorithms, shared memory systems, PERT graphs, see [43] or [23] ffl Queueing theory : G=G=1=1 queue (see #5) queues in series, queues in series with blocking, fork join networks [3] ffl Operations research and manufacturing : Job shop models, event graphs (a subclass of Petri nets) see [17] [28] and [3] ffl Economy or control theory : dynamic optimization, see [46] ffl Physics of crystal structures : Frenkel Kontorova model, see [24] Among the very large and complete literature on the theoretical aspects of deterministic (max, systems, let us quote only [3] 37] and the ....

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot. A linear system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automatic Control, AC-30:210220, 1985.

....to model DEDS. Petri Nets is one of the most common formalism. More precisely, a sub class of Petri Nets, Event Graphs, appears to be very efficient in describing models with synchronization, blocking and or fork join properties. We can mention Job Shop models (see Cohen, Dubois, Quadrat Viot [16] or [4] cyclic Jackson Networks (see Section 2.2) or asymmetric exclusion models as examples. On the other hand, Event Graphs cannot be used to model systems with routings. We can describe the evolution of an Event Graph by the daters associated with the transitions (nodes) of the graph. It is ....

....if A is aperiodic (Def. 3.4) 4 Deterministic Spectral Theory We recall some results of the deterministic spectral theory in the Rmax algebra. Theorem 4.2 is due to Cuninghame Green [18] Versions of Theorem 4. 4 were proved in [34] 19] and [24] Under the form proposed here, the result is from [16]. Theorem 4.5 is due to Cohen, Dubois, Quadrat and Viot [15] and [16] A complete treatment of the spectral theory can be found in [4] We want to find non trivial solutions to the equation: x = x ; where A 2 R is an irreducible matrix, x is a column vector (the eigenvector ) and is a ....

[Article contains additional citation context not shown here]

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot. A linear system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automatic Control, AC-30:210--220, 1985.

....be represented by dater functions which provide the occurrence time of all the possible events in the system. The most satisfactory results are relative to the subclass of timed Event Graphs, which can be modeled by finite dimensional recurrent linear systems over the (max, semiring, see [2] [9]. There exists a striking connection between the two cases: both (max, linear systems and conventional automata are specializations of (max, automata, i.e. automata with multiplicities [14] over the (max, semiring. This constatation leads to the natural question: What is the modeling power ....

.... The assignment of the jobs on the machines is fixed but not the order on which the jobs are processed by the machines (the schedule) The classical modeling associates with each chosen schedule an event graph (i.e. a (max, linear system) whose size grows with the period of the schedule, see [9], 18] 2] On the other hand, the representation by heap model is independent of the (even non periodic) schedule which is considered. This is particularly interesting for the successive evaluation of a large number of schedules. For a periodic schedule, we propose a new heapbased algorithm to ....

[Article contains additional citation context not shown here]

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot. A linear system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automatic Control, AC-30:210--220, 1985.

....average time separation of events (TSE) a generic problem in analyzing timed concurrent and distributed systems. In this paper, we focus on bounding average TSE s of such systems that can be modeled using stochastic timed marked graphs (e.g. 1] also referred to as timed event graphs (e.g. [2, 3]) or decision free Petri nets (e.g. 4] Although marked graphs are a restrict class of Petri nets [5] they constitute an adequate model for many real world systems, including many asynchronous circuits and embedded systems. In different application domains, the TSE s are typically ....

....yield estimation. For instance, the average throughput of an asynchronous pipeline is the inverse of the average time separation of consecutive output requests. Over the last two decades, the problem of computing the average TSE s in basic classes of Petri nets has been studied extensively (e.g. [11, 4, 3, 2, 12, 13, 14, 15, 9, 16, 10]) although the majority of the work was on average system throughput. The pioneer work by Karp [11] solved this problem for mean cycle time for marked graphs with deterministic delays. In the stochastic cases where delays are not fixed, the problem has been addressed using Markovian analysis ....

G. Cohen, D. Dubois, J.-P. Quadrat, and M. Viot. A linear-system-theoretic view of discreteevent processes and its use for performance evaluation in manufacturing. IEEE Transactions on Automatic Control, 30(3):210--220, March 1985.

....of speaking of structural fixed points, Gunawardena [8] speaks of a balanced mapping if it has such a fixed point. Theorem 1 A max plus system z(k 1) A Omega z(k) has a structural fixed point if and only if the precedence graph of A is strongly connected. This follows directly from [5] or [1] The essence is that the precedence graph is invariant with respect to finite changes in the parameter values of A. Theorem 2 A nondegenerated separated system with both the max part and the min part strongly connected (see [12] does not have a structural fixed point. This follows ....

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot (1985), `A linear system-theoretic view of discrete event processes and its use for performance evaluation in manufacturing', IEEE Transactions on Automatic Control, AC-30 210--220.

....x, then we get (a special case of) an ELCP. 4 The ELCP and Max Linear Time Invariant DESs In general the description of DESs is nonlinear. However, there exists a class of DESs the so called max linear DESs for which the description becomes linear when we express it in the max plus algebra [1, 2]. Loosely speaking we could say that this subclass corresponds to the class of deterministic timeinvariant DESs in which only synchronization and no concurrency occurs. The basic operations of the max plus algebra are maximization (represented by ) and addition (represented by ) There exists a ....

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot, \A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing, " IEEE Trans. on Aut. Control, vol. 30, no. 3, pp. 210-220, Mar. 1985.

.... Event Systems (DES) a convenient abstraction for many man made systems such as communication networks, digital circuits, or manufacturing systems, can usually be modelled by topical maps and IFS; the extrema of the asymptotic height then correspond to the best and the worst throughput of the DES [Ba, Co, BV, GM1, GM2]. Among topical maps, a special role is played by max plus maps which appear in the modelling of event graphs, 1 bounded Petri nets and Tetris like heap models [Ga2] Topical IFS also appear in other contexts, for example, in various problems of automata and formal language theory [Pin, Sim] ....

G. Cohen, D. Dubois, J. P. Quadrat and M. Viot, A linear system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing, IEEE Trans. Aut. Cont. 30 (1985), pp. 210--220

....in the last forty years. It appears in Operations Research for optimization problems (see [9, 14] it is a useful tool to study some decision problems in formal language theory (see [16, 17, 21, 22] and it has an important role in the modelling and analysis of Discrete Event Systems (see [1, 6, 13]) In all of these applications, linear functions over the (max; semiring play a preeminent role. To give just one example, the dates of occurrence of events in a Timed Event Graph, a class of Discrete Event Systems, are given by the iterates of a linear function over the (max; semiring, see ....

....In all of these applications, linear functions over the (max; semiring play a preeminent role. To give just one example, the dates of occurrence of events in a Timed Event Graph, a class of Discrete Event Systems, are given by the iterates of a linear function over the (max; semiring, see [1, 6]. It is natural to study the direct generalization of linear functions: bilinear functions over the (max; semiring. There is another possible way to introduce and motivate our study. Trees are one of the most important structure in computer science. They constitute a basic data structure; they ....

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot. A linear system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automatic Control, 30:210--220, 1985.

....are: the completion of a part on a machine, a machine breakdown, or a bu#er becoming empty. In general, models that describe the behavior of a DES are nonlinear, but there exists a class of DESs the max linear DESs for which the model becomes linear when formulated in the max plus algebra [1, 3, 4]. One of the open problems in the max plus algebraic system theory for DESs is the minimal realization problem, which can be stated as follows: given the impulse response of a max linear DES, determine a model of smallest possible size the impulse response of which coincides with the given impulse ....

....eigenvalue of A and that v is a corresponding max plus algebraic eigenvector of A. It can be shown that every matrix A#R nn has at least 1 and at most n max plus algebraic eigenvalues (see, e.g. 1] In particular, irreducible matrices have only one max plus algebraic eigenvalue (see, e.g. [3]) For algorithms to determine max plus algebraic eigenvalues and eigenvectors the interested reader is referred to [1, 3, 15] and the references cited therein. Theorem 2.4. If A#R nn is irreducible, then #k 0 #N such that #kk 0 : A# k c = ## c# A# k (3) where # is the (unique) ....

[Article contains additional citation context not shown here]

G. Cohen, D. Dubois, J.P. Quadrat, M. Viot, A linearsystem -theoretic view of discrete-event processes and its use for performance evaluation in manufacturing, IEEE Trans. Automat. Control 30 (3) (1985) 210 -- 220.

....any unifying notation nor any unifying theory for DEDS. A considerable amount of work has been done in different areas to describe and analyze DEDS, and to develop controllers for DEDS. Models for DEDS have been developed based on temporal logic [23, 17] queueing theory [9] and minimax algebra [5]. A DEDS is easily described in automata theory, and work in this area has been done by Ramadge and Wonham [20] Inan and Variaya [12] Caines, Greiner and Wang [4] and others. Another way of describing DEDS is by using Communicating Sequential Processes developed by Hoare [10] Benveniste and ....

G. Cohen, D.Dubois, J. P. Quadrat, and M. Viot. A linear-system-theoretic view of discreteevent processes and its use for performance evaluation in manufacturing. IEEE Transactions on Automatic Control, AC-30(3):210--220, 1985.

....networks, parallel processing systems and railroad trac networks. There exists a wide range of frameworks to model and to analyze DES: Petri nets, formal languages, computer simulation, perturbation analysis and so on. We concentrate on a subclass of DES that can be described with the max algebra [1, 2]. Although the description of these systems is nonlinear in linear algebra, the model becomes linear when we formulate it in the max algebra. In this paper we only consider systems that can be described by max linear time invariant state space models. One of the main advantages of an analytic ....

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot, \A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing," IEEE Transactions on Automatic Control, vol. AC-30, no. 3, pp. 210-220, Mar. 1985.

.... can also be seen as a vectorial generalization of (9) x j (p) n i=1 inf qp fx i (q) A i;j (p Gamma q)g m l=1 inf qp fu l (q) B l;j (p Gamma q)g; 0 p 2 N; j = 1; n: 12) This class of equations was first introduced and studied by Cohen, Dubois, Quadrat and Viot (1983) [9] in the particular case when the entries of the matrices A( and B( are all integer valued and have finite support, namely all entries of A(r) and B(r) are equal to 1, for all r K. The interpretation is then as follows: if A i;j (r) h 1, there is an arc from transition i to transition j ....

....of 1. the fork join network model considered in [17] 2] to non additive service processes; 2. the models considered in [1] and [7] to time varying service processes. The focus in [1] and [8] was on worst case performance bounds; 3. the counter equations for timed event graphs considered in [9] and [4] to the time varying, infinite support and continuous time case. 7 2.3 Dater Equations In this section, we describe another class of models. The relationship with the above class is discussed in the last subsection. 2.3.1 Isolated Queue Both the isolated queue and network equations given ....

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot. A linear system-theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automat. Control, 30:210--220, 1985.

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G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot, "A linear-systemtheoretic view of discrete-event processes and its use for performance evaluation in manufacturing". IEEE Trans. Automat. Control, 30, pp. 210-220, 1985.

No context found.

Cohen G.D., Dubois, J.P. Quadrat, and M. Viot (1985). A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automat. Control, 30, 210-220.

No context found.

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot. A linear system-theoretic view of discreteevent processes and its use for performance evaluation in manufacturing. IEEE Trans. Automatic Control, AC-30:210220, 1985.

No context found.

G. Cohen, D. Dubois, J. Quadrat, and M. Viot. A linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Trans. on Automatic Control, AC{30:210-220, 1985.

No context found.

G. Cohen, D. Dubois, J.-P. Quadrat, and M. Viot. A linear system-theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Transactions on Automatic Control, AC-30:210--220, 1985.

No context found.

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot. A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Transactions on Automatic Control, 30:210--220, 1985.

No context found.

G. Cohen, D. Dubois, J.-P. Quadrat, and M. Viot. A linear system-theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Transactions on Automatic Control, AC-30:210--220, 1985.

No context found.

G. Cohen, D. Dubois, J.-P. Quadrat, and M. Viot, "A linear system-theoretic view of discrete event processes and its use for performance evaluation in manufacturing," IEEE Transactions on Automatic Control, vol. AC-30, pp. 210--220, 1985.

No context found.

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot, \A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing," IEEE Transactions on Automatic Control, vol. 30, no. 3, pp. 210-220, Mar. 1985.

No context found.

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot, \A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing," IEEE Transactions on Automatic Control, vol. 30, no. 3, pp. 210-220, Mar. 1985.

No context found.

G. Cohen, D. Dubois, J.P. Quadrat, and M. Viot. A linear system-theoretic view of discreteevent processes and its use for performance evaluation in manufacturing. IEEE Trans. Automatic Control, AC-30:210--220, 1985.

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