By Arjan J. van der Schaft, Hans Schumacher

ISBN-10: 1852332336

ISBN-13: 9781852332334

This publication is set dynamical structures which are "hybrid" within the experience that they include either non-stop and discrete kingdom variables. lately there was elevated examine curiosity within the learn of the interplay among discrete and non-stop dynamics. the current quantity offers a primary try in booklet shape to compile suggestions and techniques facing hybrid structures from a number of parts, and to examine those from a unified perspective.

The authors have selected a style of exposition that's principally in response to illustrative examples instead of at the summary theorem-proof layout as the systematic research of hybrid structures remains to be in its infancy. The examples are taken from many various program components, starting from strength converters to conversation protocols and from chaos to mathematical finance.

Subjects coated comprise the subsequent: definition of hybrid platforms; description codecs; lifestyles and area of expertise of ideas; distinctive subclasses (variable-structure platforms, complementarity systems); reachability and verification; balance and stabilizability; regulate layout tools. The publication may be of curiosity to scientists from a variety of disciplines together with: computing device technological know-how, regulate thought, dynamical process conception, structures modeling and simulation, and operations research.

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**Additional info for An introduction to hybrid dynamical systems**

**Sample text**

We do not necessarily have w(t-) = w(t +) for teST. 2. Towards a definition of hybrid systems 23 The space we have just introduced is very general; however, we shall often want to work with smaller and more manageable spaces. An example of such a space is the following. 17. The (rE, x, w, P, S), where - - space NZ/1/CI/~ ~ consists of tuples TE is a time evolution specified by a pair (ET, m) where ET is a set of isolated points in l~, and re(t) = 1 for all t E ET x is a continuous function from TE to X which is continuously differentiable on R \ ET and which satisfies x(t +) = x(t-) for all t E ET - w is a continuous function from I~ \ ET to W - P is a continuous function from TE to L S is a function from E to A.

For some examples of hybrid systems having more than one location, where the existence of an extension after the accumulation point is more problematic, we refer to [82]. 4 Temperature control system In a simple model to be used in t e m p e r a t u r e control of a room (cf. [1]), we have one continuous variable (room temperature, denoted by O(t), and taking values in ~) and one discrete variable (status of the heater, denoted by H i t ) and taking values in {on, o f f } ) . 2. Examples 39 where f(-, ,) is a (sufficiently smooth) function of the continuous state x, the discrete state H, and the continuous external variable w, which may for example contain the outside temperature.

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.

### An introduction to hybrid dynamical systems by Arjan J. van der Schaft, Hans Schumacher

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