By Arjan J. van der Schaft, Hans Schumacher

This ebook is ready dynamical platforms which are "hybrid" within the experience that they include 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 presents a primary try out in e-book shape to assemble thoughts and strategies facing hybrid platforms from quite a few components, and to examine those from a unified perspective.

The authors have selected a style of exposition that's mostly in accordance with 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 alternative software components, starting from energy converters to communique protocols and from chaos to mathematical finance.

Subjects coated contain the next: definition of hybrid structures; description codecs; life and strong point of options; particular subclasses (variable-structure platforms, complementarity systems); reachability and verification; balance and stabilizability; keep an eye on layout equipment. The ebook can be of curiosity to scientists from quite a lot of disciplines together with: desktop technological know-how, regulate thought, dynamical approach idea, platforms modeling and simulation, and operations research.

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

**Example text**

Theoretically, the state after the j u m p should satisfy certain constraints exactly; finite word length effects however will cause small deviations in the order of the machine precision. Such deviations m a y cause Chapter 1. Modeling of hybrid systems 28 an interaction with the mode selection module; in particular it may appear that a certain constraint is violated so that a new event is detected. In this way it may happen that a cycling between different modes occurs ("livelock"), and the simulator does not return to a situation in which motion according to some continuous dynamics is generated, so that effectively the simulation stops.

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. Interesting variations of both modeling approaches may be obtained by considering e. g. )3 ---- V4 : 0. Another interesting issue which may be studied in the context of this example concerns the continuous dependence of solutions on initial conditions. 2 below. 7 Variable-structure system Consider a control system described by equations of the form x(t) = f(x(t), u(t)), where u(t) is the scalar control input.

Are event times. Moreover, if m ( t i ) is finite, then S(ti+l) S(t'~(tD~). This defines the behavior of a discrete-event system. Note, however, t h a t in general the set of "discrete time instants" . . , ~247176 § ' t~~, . ,t'~(tl)U, ~1 . §176, ~§ , t~2~, . '~2 § . ' v§3. ' ~§3 , " of this resulting discrete-event system may have a very complicated structure. Analogously, the projection of the hybrid behavior on the behavior of the continuous variables defines a continuous-time behavior defined on the time axis ~ minus the event times and their accumulation points.