S. Zerhouni

Max-plus algebra for discrete event systems-some links to structural controllability and structural observability

One can consider a system of production to be a dynamic system. Moreover, it is interesting to describe the system by equations in which state vector, input and output vectors appear. This can help when carrying out synthesis of the system. The Petri net representation of a system allows one to define three vectors mentioned above. If the corresponding Petri net is an event graph (i.e. each place has exactly one input and one output transition), it is possible to apply a mathematical model for this system. The model is known in literature as max algebra or max-plus algebra R/sub max/=(R/spl cup.

On a new method of Markov chain reduction

The practical usefulness of Markov models and Markovian decision process has been severely limited due to their extremely large dimension. Thus, a reduced model without sacrificing significant accuracy can be very interesting. The homogeneous finite Markov chains long-run behaviour is given by the persistent states, obtained after the decomposition in classes of connected states. In this paper we expound a new reduction method for ergodic classes formed by such persistent states. An ergodic class has a steady-state independent of the initial distribution. This class constitutes an irreducible finite ergodic Markov chain, which evolves independently after the capture of the event. The reduction is made according to the significance of steady-state probabilities. For being treatable by this method, the ergodic chain must have the Two-Time-Scale property. The presented reduction method is an approximate method. We begin with an arrangement of irreducible Markov chain states, in decreasing order of their ste...

Simplification of Stochastic Petri Nets Using a Decoupling Method

Abstract This paper concerns the simplification of stochastic Petri net model, possessing the property of double scale of time. This simplification uses the singular perturbation method. The stochastic Petri net is transformed in a Markov chain in discrete time and discrete state space. Then, by application of singular perturbations, our model is decomposed in two subsystems. The part that has the greatest influence on the system is only preserved. Then the reverse process determines the simplified graph that corresponds to the most influential part of the system. An example stemming from an industrial repair cycle validates the step used.

Modelling and Control of an Electroplating Line in the Max Algebra

Abstract A modelling of the Hoist Scheduling Problem for electroplating lines is developed. For the modelling, a class of min-max P-time event graphs is introduced, and the max algebra theory is applied. Concurrence appears for resource utilization. This is why a simple rule is introduced to solve it. Interpretation of the rule provides a control of process sequencing, and a solution of the problem.

Control of nonautonomous discrete event systems using dioid algebra

In this article, finite nonautonomous discrete event systems are studied in the dioid of formal series. If such a system can be described by a timed event Petri net, a max algebra model of the system can be written. Furthermore, the input-output relation of the nonautonomous system appears in the dioid of formal series. The greatest subsolution of a system of linear equations in the dioid of formal series is found. This subsolution can be used to calculate the latest admissible input of the nonautonomous discrete event system, such that the output of the system is achieved. The application of the theory to an assembly line is given.

Minimal recurrence equation formulation of discrete event systems in max algebra

In this paper, discrete event systems modelled by timed-event Petri nets are studied in the algebraic structure R/sub max/ that is an idempotent semifield. It is called max algebra or max-plus algebra in the literature. In this algebra, the equations describing evolution of the system have linear form and this allows the system to be analysed like systems in the conventional algebra. The application of the idea of an annulation polynomial or the idea of free vectors in the max algebra allow the authors to obtain a minimal recurrence equation of the system.

Stochastic Petri nets simplification with singular perturbations

In this paper, we introduce a new simplification of stochastic Petri net models. This simplification uses the singular perturbation method for discrete event systems in continuous time. We adapt this method for stochastic Petri net models. The model studied should have the double time scale property in order to apply this method of simplification. The decoupling method gives us two sub-systems, a fast and a slow evolution. These evolutions are the probabilities to be in a certain marking of the stochastic Petri net. For these two sub-systems, we only preserve the slow evolution of the marking probabilities, which yields the most precision given by the singular perturbation in continuous time. The main advantage of this method is to reduce the number of places and/or transitions of the stochastic Petri net. The calculation of the performance rates is then simplified. For complex stochastic Petri net models, this method allows one to draw the sub-system with a slow evolution.

Contribution to the modelling and control methodology of complex systems

The paper presents the use of hierarchical coloured Petri nets (HCPN) in the area of modelling and control methodology of reactive systems. The goal is to give some facilities to a system designers by introducing a generic models elaborated by the HCPN that allows verification, analysis and validation of the results in the preliminary stage of design. The models are destined to the modelling of sequential systems.<<ETX>>

Use of an homographic transformation jointly to the singular perturbation for the resolution of Markov chains. Application to the operational safety study

Our work concerns the adaptation of the singular perturbation method jointly to the homographic transformation to the category of ergodic Markov chains which presents the two-time-scale property. For Markov chains, the two-time-scale property becomes a property of two-weighting-scale of the states in the system evolution. This lead us to call the slow and fast parts of a decomposed system strong and respectively weak. The limit resolution methodology of the Markov chains by the method of singular perturbation assumes firstly the detection of the irreducible classes of the chain, and secondly, the decomposition of each final ergodic classes presenting the two-weighting-scale property. In the resolution at the limit of the decomposed system, we struck the problem of the stochasticity of the subsystems obtained using directly the singular perturbation method. Indeed, the strong and weak submatrix are not stochastic matrix. In our method, we use the homographic transformation in order to make stochastic the strong part matrix.<<ETX>>

Contribution for human body behaviour research at the time of a jump reception

The paper studies the initial phase associated to human body flexion, at the time of a jump reception, beginning with standing up position. It views the modelling of vertical efforts induced at the hand and feet interface levels of a vehicle drive post, according to a series of trials. The trial rig contains a driving post, connected with a float where the assayer is sitting down. The float moves vertically and is provided with an adjustable height, and at full stroke the float movement is absorbed by a pneumatic suspension with adjustable design features. The instrumentation of interfaces with the pilot gives readings of the components of efforts and absolute acceleration.<<ETX>>