Sébastien Lahaye

Just in Time Control of Constrained (max,+)-Linear Systems

This paper deals with just in time control of (max,+)-linear systems. The output tracking problem, considered in previous studies, is generalized by considering additional constraints in the control objective. The problem is formulated as an extremal fixed point computation. This control is applied to timetables computation for urban bus networks.

Supervisory Control of (max,+) Automata: A Behavioral Approach

A behavioral framework for control of (max,+) automata is proposed. It is based on behaviors (formal power series) and a generalized version of the Hadamard product, which is the behavior of a generalized tensor product of the plant and controller (max,+) automata in their linear representations. In the tensor product and the Hadamard product, the uncontrollable events that can neither be disabled nor delayed are distinguished. Supervisory control of (max,+) automata is then studied using residuation theory applied to our generalization of the Hadamard product of formal power series. This yields a notion of controllability of formal power series as well as (max,+)-counterparts of supremal controllable languages. Finally, rationality as an equivalent condition to realizability of the resulting controller series is discussed together with hints on future use of this approach.

Linear Periodic Systems Over Dioids

A specification of the linear system theory over dioids is proposed for periodic systems. Using the conventional periodic system theory as a guideline, we study periodic systems for which the underlying algebraic structure is a dioid. The focus is on representations (impulse response and state model) associated with such systems, the properties of these representations as well as the state space realization.

Control of (max,+)-linear systems minimizing delays

In this paper, we develop a new control technique for discrete event dynamic systems subject to synchronization phenomena. We propose a feedback controller for (max, + )-linear systems which delays input events as little as possible while constraints on internal or output events are satisfied. The synthesis is mainly based on new results about fixed points of antitone (i.e., order reversing) mappings.

Optimal control of (min,+) linear time-varying systems

The class of discrete event dynamic systems involving only synchronization phenomena can be seen as linear time-invariant systems in a particular algebraic structure called (min,+) algebra. In the same framework, this paper deals with linear time-varying systems, that is, systems whose parameters may change as functions of time. For example, in a manufacturing system the number of working machines, or the number of trains running in a closed network of railway connections, can vary as functions of time. For such systems, the output tracking problem is optimally solved under just-in-time criterion.

Modeling and Control of Hybrid Timed Event Graphs with Multipliers Using (Min, +) Algebra

We study a subclass of Petri nets, called hybrid timed event graphs with multipliers, or equivalently, hybrid timed weighted marked graphs, composed of continuous and discrete graphs interconnected among themselves. Such graphs can be modeled by using a particular algebra, called dioid, defined on a set of operators and endowed with the pointwise minimum operation as addition and the composition operation as multiplication. A just in time control method of these graphs based on residuation theory is proposed.

Compositions of (max, +) automata

This paper presents a compositional modeling approach by means of (max, +) automata. The motivation is to be able to model a complex discrete event system by composing sub-models representing its elementary parts. A direct modeling of safe timed Petri nets using (max, +) automata is first introduced. Based on this result, two types of synchronous product of (max, +) automata are proposed to model safe timed Petri nets obtained by merging places and/or transitions in subnets. An asynchronous product is finally proposed to represent particular bounded timed Petri nets.

On the linearizability of discrete timed event graphs with multipliers using (min,+) algebra

Abstract We deal with the reduction of timed event graphs with multipliers to timed event graphs without multipliers which allows (min, +) linear representations of graphs. The proposed linearization method uses an algebra defined on a set of operators and endowed with the pointwise minimum as addition and the composition as multiplication. The method considers a change of variables which does not increase the number of transitions of graph, but the graph is supposed consistent and conservative and its (unique) elementary T-semiflow has at least one component equal to one.

Control of (max,+) automata: Logical and timing aspects

A new framework for control of (max,+) automata is introduced. The tensor product of their linear representations used in this paper is an extension of parallel composition from Boolean to (max,+) automata and can be nicely applied to both logical and timing aspects of supervisory control. Case of uncontrollable events that can neither be disabled nor delayed is studied within a behavioral framework. Optimal (least restrictive) control of (max,+) automata is studied using residuation theory applied to Hadamard product of (multivariable) formal power series.

Just-in-time control of time-varying discrete event dynamic systems in (max,+) algebra

We deal with timed event graphs whose holding times associated with places are variable. Defining a first-in-first-out functioning rule, we show that such graphs can be linearly described in (max,+) algebra. Moreover, this linear representation allows extending the just-in-time control synthesis existing for timed event graphs with constant holding times. An example is proposed in order to illustrate how the approach can be applied as a just-in-time strategy for production lines.