By Arjan J. van der Schaft, Hans Schumacher

ISBN-10: 1852332336

ISBN-13: 9781852332334

This booklet is ready dynamical structures which are "hybrid" within the experience that they include either non-stop and discrete country variables. lately there was elevated examine curiosity within the examine of the interplay among discrete and non-stop dynamics. the current quantity presents a primary test in publication shape to collect strategies and strategies facing hybrid structures from quite a few components, and to examine those from a unified perspective.

The authors have selected a method of exposition that's mostly according to illustrative examples instead of at the summary theorem-proof layout as the systematic learn of hybrid structures continues to be in its infancy. The examples are taken from many alternative software components, starting from energy converters to communique protocols and from chaos to mathematical finance.

Subjects coated comprise the next: definition of hybrid platforms; description codecs; life and specialty of ideas; targeted subclasses (variable-structure platforms, complementarity systems); reachability and verification; balance and stabilizability; regulate layout equipment. The booklet might be of curiosity to scientists from a variety of disciplines together with: laptop technology, keep an eye on conception, dynamical method concept, platforms modeling and simulation, and operations research.

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

**Sample text**

The combined vector field f(x) defined by f(x) := fl(x) for h(x) > 0 and f(x) := f2(x) for h(x) < 0 is in general discontinuous on the switching surface h(x) -- O. 2. 5: Collision to an elastic wall is easy to come up with examples exhibiting multiple solutions for the continuous state x. 16) for all t. ) to be differentiable. 16) in the sense of Carathdodory. 17) also obviates the need for specifying the value of f on the surface {x I h(x) = 0}, at least for cases in which solutions arrive at this surface from one side and leave it immediately on the other side.

Furthermore, the hybrid nature of such systems is usually in first instance described by "if-then" or "either-or'statements, in the sense that in one location of the hybrid system a particular subset of the total set of differential and algebraic equations has to be satisfied, while in another location a different subset of equations should hold. Thus, while in the (generalized) hybrid automaton model the dynamics associated with every location are in principle completely independent, in most "physical" examples (as we will see in Chapter 2) the set of equations describing the various activities or modes (continuous dynamics associated to the locations) will remain almost the same, replacing one or more equations by some others.

We list a few choices t h a t m a y turn out to be useful. In each case, we use a continuous state space X , a continuous communication space W, a discrete state space L, and a discrete communication space A. It is convenient to use some topological concepts. Recall t h a t a "time evolution" is a set of the form = ( m \ E T ) UE with E ---- UteET{(t ~ where ST is a subset of 1~ and m is a function from ET- to N U {oo}. We say t h a t the time evolution TE and the set of time events $ are specified by the pair (ET, m).

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

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