By Arjan J. van der Schaft, Hans Schumacher

This e-book is ready dynamical structures which are "hybrid" within the feel that they comprise either non-stop and discrete country variables. lately there was elevated examine curiosity within the learn of the interplay among discrete and non-stop dynamics. the current quantity presents a primary test in publication shape to collect suggestions and strategies facing hybrid platforms from quite a few components, and to examine those from a unified perspective.
The authors have selected a method of exposition that's mostly in response to illustrative examples instead of at the summary theorem-proof structure as the systematic research of hybrid platforms remains to be in its infancy. The examples are taken from many alternative program parts, starting from energy converters to communique protocols and from chaos to mathematical finance.
Subjects lined comprise the subsequent: definition of hybrid platforms; description codecs; life and area of expertise of suggestions; distinct subclasses (variable-structure platforms, complementarity systems); reachability and verification; balance and stabilizability; keep watch over layout equipment. The publication may be of curiosity to scientists from a variety of disciplines together with: laptop technology, regulate thought, dynamical procedure idea, structures modeling and simulation, and operations research.

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Below we shall apply this rule in a situation where multiple impact occurs (as it was done in [63]); of course one may debate whether such an extension is "correct". 14) V2 ~--- V3 ~ 0. How do we model this case with the above impact rule? The problem is t h a t not only the left mass is colliding with the middle mass, but that also the middle mass is in contact with the right mass; we thus encounter a multiple collision. 2. 13) for single collisions, and these two ways lead to different answers!

Chapter 1. Modeling of hybrid systems 24 Various other function spaces might be defined. Since any a t t e m p t at completeness would be futile, we do not list any further examples, but we shall feel free to use variants below. At this point we can discuss the notion of solution for EFFs. T h e notion of solution is based on the evaluation of the elementary clauses in an E F F at specific time points. To be precise, a clause of the form r x, w, P ) = 0 will be said to evaluate to TRUE for an element of Z/oc/C1/L~oc at a time t E ~ \ $ T if r x(t), w(t), P(t)) = O.

Basically there are three different approaches to the simulation of hybrid systems, which each may be worked out in many different ways. The three approaches may be briefly described as follows (cf. [114]). 1. The smoothing method. In this method, one tries to replace the hybrid model by a smooth model which is in some sense close to it. For instance, diodes in an electrical network may be described as ideal diodes (possibly plus some other elements), which will give rise to regime-switching dynamics, or as strongly nonlinear resistors, which gives rise to smooth dynamics.

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