Philippe Declerck

State Estimation of Timed Labeled Petri Nets With Unobservable Transitions

The aim of this paper is to reconstruct the least/greatest sequence of unobservable transitions in timed Petri nets based on the online observation of firing occurrences of some transitions on a sliding horizon. The Petri net, which can be unbounded and can contain self-loops and circuits, is described under an algebraic form composed of A.x ≤ b which expresses the possible time sequence x and the fundamental marking relation. Under the assumption of Backward/Forward Conflict Freeness of the unobservable-induced subnet, we show the existence of a finite least/greatest sequence with respect to the data known on a given horizon. A technique of computation using linear programming is given.

Optimal Control Synthesis of Timed Event Graphs With Interval Model Specifications

The purpose of this technical note is the optimal control synthesis of a timed event graph when the state and control trajectories should follow the specifications defined by an interval model. The problem is reformulated in the fixed point form and the spectral theory gives the conditions of existence of a solution.

From Extremal Trajectories to Token Deaths in P-time Event Graphs

In this technical note, we consider the (max,+) model of P-time Event Graphs whose behaviors are defined by lower and upper bound constraints. The extremal trajectories of the system starting from an initial interval are characterized with a particular series of matrices for a given finite horizon. Two dual polynomial algorithms are proposed to check the existence of feasible trajectories. The series of matrices are used in the determination of the maximal horizon of consistency and the calculation of the date of the first token deaths.

State estimation and detection of changes in time interval models

This paper presents an estimation approach for Time Event Graphs such as P-Time Event Graphs and Time Stream Event Graphs. It is assumed that the nominal behavior is known and that transitions are partitioned as observable and unobservable transitions. The technique is applied to the detection of changes which are (possibly small) finite variations of dynamic models compared to this nominal behavior. The detected changes provide indications that can be used in future maintenance operations. Using the algebra of dioids, the approach uses a receding-horizon estimation of the greatest state and analyzes the consistency of the data.

Extremal trajectories in P-time Event Graphs: application to control synthesis with specifications

This paper presents a modelling and an analysis of P-time Event Graphs in the field of topical algebra. A particular serie of matrices is introduced whose evolution determines the system behavior and the existence of a trajectory without token deaths. The extremal trajectories obeying to an interval of desired output are deduced. If every event is controllable, the Just-In-Time control of Timed Event Graph is solved when additional specifications are given by a P-time Event Graph.

Predictive control of Timed Event Graphs with specifications defined by P-time Event Graphs

The aim of this paper is the predictive control of Timed Event Graphs with specifications defined by P-time Event Graphs. We propose a fixed-point approach which leads to a pseudo-polynomial algorithm. As the performance of the algorithm is crucial in on-line control, we highlight an important case where the resolution of this first algorithm is efficient. The second technique is a space controller on a horizon leading to a strongly polynomial algorithm.

Causality Phenomenon and Compromise Technique for Predictive Control of Timed Event Graphs with Specifications Defined by P-Time Event Graphs

Abstract An imperative condition of operation in Predictive Control is that a control must be applied after the end of its calculation. In this paper, we analyze and formalize this causality phenomenon which depends on both the computer time and the control problem. Two techniques are proposed. When the causality forbids the complete convergence of the algorithm, we propose a compromise technique which needs the analysis of the partial satisfaction of the specifications at each iteration of the algorithm. The plant is described by a Timed Event Graph while the specifications are defined by a P-time Event Graph.

Estimation, Prediction and Control in (max,+) Systems

Abstract Discrete event systems undergo perturbations such as failures that disrupt the control system and reduce the anticipation capacities of the future evolution of the process. Using the (max,+) algebra, processes modelled by a timed event graph may be represented by a linear model. The knowledge of the model and of the initial conditions enable to characterise the state vector with a state equation iteration but perturbations may generate a misappreciation of the state vector. The aim of this paper is to estimate the unknown state, to predict the output trajectory and to calculate the control when the state vector is unknown.

Trajectory Tracking Control of a Timed Event Graph with Specifications Defined by a P-time Event Graph

The aim of this paper is a trajectory tracking control of Timed Event Graphs with specifications defined by a P-time Event Graph. Two problems are solved on a fixed horizon knowing the current state: The optimal control for favorable past evolution; The prediction of the earliest future evolution of the process. These two parts make up an on-line control which is used on a sliding horizon. Completely defined in (max, +) algebra, the proposed approach is a Model Predictive Control using the componentwise order relation.

DETECTION OF CHANGES BY OBSERVER IN TIMED EVENT GRAPHS AND TIME STREAM EVENT GRAPHS

Abstract A state-based approach for detection of changes in systems modelled as Timed Event Graph and Time Stream Event Graph is presented. We assume that the net in its nominal behavior is known and transitions are partitioned as observable and unobservable transitions. Considered faults are (possibly small) variations of dynamical models by respect to this nominal behavior. Using the algebra of dioids, the approach follows the same principle as the observers used in continuous systems.