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

By Arjan J. van der Schaft, Hans Schumacher

This e-book is set dynamical structures which are "hybrid" within the feel that they comprise either non-stop and discrete kingdom variables. lately there was elevated learn curiosity within the learn of the interplay among discrete and non-stop dynamics. the current quantity offers a primary test in publication shape to collect thoughts and strategies facing hybrid platforms from a number of parts, 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 research of hybrid platforms continues to be in its infancy. The examples are taken from many various software components, starting from energy converters to communique protocols and from chaos to mathematical finance.
Subjects coated comprise the subsequent: definition of hybrid platforms; description codecs; lifestyles and strong point of strategies; distinct subclasses (variable-structure platforms, complementarity systems); reachability and verification; balance and stabilizability; keep watch over layout equipment. The booklet could be of curiosity to scientists from quite a lot of disciplines together with: desktop technological know-how, keep an eye on concept, dynamical approach conception, structures modeling and simulation, and operations research.

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As an example consider the following model of an elastic collision, taken from [80]. 5. [xl] [0] + mode 1: x2 -k -d x2 u 1 Chapter 2. Examples of hybrid dynamical systems 46 Note that for d ~ 0 the overall dynamics is not continuous on the surface y -xl = 0, so that the standard theory of existence and uniqueness of solutions of differential equations does not apply. Nevertheless, it can be readily checked that the system has unique solutions for every initial condition (as we expect).

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~. In fact, we obtain the following distribution of velocities at the subsequents stages t ~ t ~, t ~ , t ~ , ... of the time event at time t: Vl V2 V3 t~ 1 0 0 ~ i 1 0 t~ i 1 i ~ ~ Chapter 2. 4 1 1 1 ~ Hence the outcome of this event with multiplicity co is the same as the outcome derived by the first method, with an event of multiplicity 1.

724]). Reasoning in this way, a point could be made that it is almost always possible to provide a smooth model t h a t is "closer to reality" than a competing hybrid model. Nevertheless, one can easily come up with examples that would be relatively awkward to describe on the basis of smoothing. If one looks at a bouncing ping-pong ball on a flat table, then the nonsmooth model comes to mind immediately since the time during which the ball is in contact with the table Chapter 1. Modeling of hybrid systems 26 is very short in comparison with the flight phases.

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