By Arjan J. van der Schaft, Hans Schumacher

ISBN-10: 1852332336

ISBN-13: 9781852332334

This e-book is set dynamical platforms which are "hybrid" within the experience that they comprise either non-stop and discrete kingdom variables. lately there was elevated learn curiosity within the research of the interplay among discrete and non-stop dynamics. the current quantity offers a primary test in booklet shape to compile techniques and techniques facing hybrid platforms from a number of components, and to examine those from a unified perspective.

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

Subjects lined comprise the next: definition of hybrid platforms; description codecs; life and strong point of recommendations; particular subclasses (variable-structure platforms, complementarity systems); reachability and verification; balance and stabilizability; keep watch over layout equipment. The ebook might be of curiosity to scientists from a variety of disciplines together with: computing device technological know-how, regulate concept, dynamical process conception, structures modeling and simulation, and operations research.

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

**Example text**

13) specializes for e -- 1 to the event-clause = V~ ~ Vi. 14) an event with multiplicity 2, representing the transfer of the velocity Vl -- 1 of the first mass to the second and then to the third mass, with the velocities of the first and second mass being equal to zero after the collision. This behavior is definitely different from the behavior for e = 1 derived above, but seems to be reasonably close, at least for small time, to what one observes experimentally for "Newton's cradle". 14) as above, this gives rise to an event with multiplicity equal to c~.

The letter Z (for Zeno) refers to the fact that in the definition the set of event times 87- is allowed to have accumulation points; for instance ET- = {~1 I n e Z \ {0}} U {0}, or even the Cantor set, could be sets of event times. Nevertheless, since E~- is assumed to be closed, these accumulation points are necessarily elements of ET-. g. ) The symbol co indicates that event times can be of arbitrarily high or even infinite multiplicity. T h e symbols C 1 and L~oc indicate the degree of smoothness that is required for the trajectories of the continuous state variables and the continuous communication variables on the open set I~ \ ET-.

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. Similarly, in a mechanical system with unilateral constraints one might describe collisions as instantaneous, and then one must allow jumps in velocities; or one might describe them in terms of a compression and a decompression phase, and in that case jumps in velocities may be avoided.

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

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