A Characterization of Fields in the Calculus of Variations by Morse M., Littauer S. B.

By Morse M., Littauer S. B.

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1 The Choice of a Formalism Hybrid systems have been studied for more than a decade. In this time interval, a large number of hybrid system modelling formalisms have been proposed. A detailed review of all these formalisms is not presented. The references at the end of the book can be used for this purpose (Antsaklis and Koutsoukos, 2003), (Gueguen and Lefebvre, 2001), (Champagnat, 1998). Briefly, hybrid system modelling formalisms can be grouped into three classes. The first one comprises the extensions of continuous formalisms by the introduction of discrete variables, such as ordinary differential equations with Boolean variables.

The other is a transition tb_z from the used interface of class Cz. Definition 8: The interface provided by a class Ci is composed of: x x A set of public variables Xpb_i (Definition 5). A set of transitions Tp_i, where Tp_i  Ti. Definition 9: The interface used by a class Ci is composed of: x x A set of image variables Xim_i (Definitions 5 and 6). A set of transitions Tu_i, where Tu_i  Ti and Tu_i ˆ Tp_i = ‡. z of class Cz calls the method ta_i of the class Ci, it must know which object of Ci will perform the method.

Aa associates an action to each transition. The action defines the value of the Ax variables of the transition output arcs using the Ax variables of the transition input arcs. M0 is the initial marking of the net. Each token is a vector of variables similar to the arc vectors. The initial marking defines the value of the token variables. 2 Modelling of Hybrid Systems 19 In a predicate transition net, a transition is enabled (or not) for a specific set of tokens in its input places. If the arc variables are replaced by the values of the token variables and the enabling condition is true, then the transition is enabled.

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