G. Cohen

A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing

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 in finding the shortest paths from one node to all other nodes in a graph. The potentiality of this approach for the performance evaluation of repetitive production processes is illustrated on an example.

Algebraic tools for the performance evaluation of discrete event systems

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. >

Duality and separation theorems in idempotent semimodules

Abstract We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert’s projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring.

Max-Plus Algebra and System Theory: Where We Are and Where to Go Now

Abstract More than sixteen years after the beginning of a linear theory for certain discrete event systems in which max-plus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison with classical linear system theory, there are areas which are practically untouched, mostly because the corresponding mathematical tools are yet to be fabricated. This is the case of the geometric approach of systems which is known, in the classical theory. to provide another important insight to system-theoretic and control-synthesis problems. beside the algebraic machinery. A preliminary discussion of geometric aspects in the max-plus algebra and their use for system theory is proposed in the last part of the paper.

Linear system theory for discrete event systems

In this paper, we pursue the Analogy between classical linear System Theory and a new linear Theory for Discrete-Event Dynamic Systems which has been introduced for the first time in [3] and which is based on the algebra of dioids. We first define the particular class of timed Petri nets that can be described by linear recurrent equations and then develop concepts such as transfer matrix representations of such systems, stability, observability, controllability, feed-back stabilization, realization of transfer matrices etc...

Linear systems in (max, +) algebra

The general system of linear equations in the (max, +) algebra is studied. A symmetrization of this algebra and a new notion called balance which generalizes classical equations are introduced. This construction results in the linear closure of the (max, +) algebra in the sense that every non-degenerate system of linear balances has a unique solution given by Cramers rule.<<ETX>>

Timed-event graphs with multipliers and homogeneous min-plus systems

The authors study fluid analogues of a subclass of Petri nets, called fluid timed event graphs with multipliers, which are a timed extension of weighted T systems studied in the Petri net literature. These event graphs can be studied naturally, with a new algebra, analogous to the min-plus algebra, but defined on piecewise linear concave increasing functions, endowed with the pointwise minimum as addition and the composition of functions as multiplication. A subclass of dynamical systems in this algebra, which have a property of homogeneity, can be reduced to standard min-plus linear systems after a change of counting units. The authors give a necessary and sufficient condition under which a fluid timed-event graph with multipliers can be reduced to a fluid timed event graph without multipliers. In the fluid case, this class corresponds to the so-called expansible timed-event graphs with multipliers of Munier (1993), or to conservative weighted T-systems. The change of variable is called here a potential. Its restriction to the transition nodes of the event graph is a T-semiflow.

A linear-system-theoretic view of discrete-event processes

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 in finding the shortest paths from one node to all other nodes in a graph. The potentiality of this approach for the performance evaluation of repetitive production processes is illustrated on an example.

From First to Second-Order Theory of Linear Discrete Event Systems

Abstract For timed event graphs, linear models were obtained using dioid algebra. After describing backward equations which solve an optimal tracking problem and which introduce co-state variables, this paper presents preliminary results concerning the matrix of ratios (i.e. conventional differences) of co-states over states: this matrix sounds like a Riccati matrix, although a neat analogue to a Riccati equation has not been found yet.

Asymptotic throughput of continuous timed Petri nets

We set up a connection between continuous timed Petri nets (the fluid version of usual timed Petri nets) and Markov decision processes. We characterize the subclass of continuous timed Petri nets corresponding to undiscounted average cost structure. This subclass satisfies conservation laws and shows a linear growth: one obtains as mere application of existing results for dynamic programming the existence of an asymptotic throughput. This rate can be computed using Howard type algorithms, or by an extension of the well known cycle time formula for timed event graphs. We present an illustrating example and briefly sketch the relation with the discrete case.