The purpose of the paper is to consider some special types of Petri nets, introduced by Lien (1976), and to propose a complete and unified approach for the study of their structural properties by using techniques of linear algebra of matrices. We distinguish four subclasses: forward-conflict-free, backward-conflict-free, forward-concurrent-free, and backward-concurrent-free Petri nets. A modification of the classical incidence matrix results in a square matrix, called a modified incidence matrix, with nonpositive (nonnegative) off-diagonal elements when backward-(forward-) conflict-free or concurrent-free Petri nets are considered. The modified incidence matrix eigenvalues are computed and theorems on matrices of this type are used to prove several sufficient and/or necessary conditions for structural boundedness, liveness, repetitiveness, conservativeness, and consistency of these four subclasses of Petri nets.

In this paper, a method for the calculation of the transfer function of timed event graphs is developed. The transfer function has the form of a formal series in one or two formal variables. The knowledge of this formal series plays a crucial role in the max-plus system theory. It is very important in the investigation of the problem of controllability of a system, because it provides us with the input-output relation as an equation in the dioid of formal series. The direct approach to this calculation requires the computation of a star operation of a matrix with entries in this dioid,which seems to be difficult for matrices of large dimensions. For this reason, we propose another approach using only the max-plus modeling of a system, that is, the dater's description. We have considered the multivariable case with several input and output transitions. We can obtain the transfer function as the output corresponding to a special control. This control follows from the dater's description of the zero and identity elements of this dioid. To illustrate this approach we give two examples. The purpose of the first one is to compare different approaches to this problem; the second one may serve as an application. We use our approach to determine the transfer function of a manufacturing system.

In this paper, a method for the calculation of transfer function of timed event graphs is developed. The transfer function has form of a formal series in one or two formal variables. The knowledge of this formal series plays a crucial role in the max-plus system theory. Namely, it is very important in the investigation of the problem of controllability of a system, because it provides the input-output relation as an equation in the dioid of formal series. The direct approach to this calculation requires the computation of a star operation of a matrix with entries in this dioid which seems to be difficult for matrices of large dimensions. This is why the authors propose another approach using only the max-plus modelling of a system. The authors have considered the multivariable case with several input and output transitions. The authors can obtain the transfer function as the output corresponding to a special control. To illustrate this approach the authors give two examples. The purpose of the first one is to compare different approaches to this problem, whereas the second one may serve as an application. The authors use their approach to determine the transfer function of a manufacturing system.

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.

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.>