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

By Arjan J. van der Schaft, Hans Schumacher

This booklet is ready dynamical platforms which are "hybrid" within the feel that they include either non-stop and discrete nation 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 booklet shape to assemble options and techniques facing hybrid platforms from numerous components, and to examine those from a unified perspective.
The authors have selected a style of exposition that's mostly in keeping with illustrative examples instead of at the summary theorem-proof layout as the systematic research of hybrid platforms remains to be in its infancy. The examples are taken from many alternative program components, starting from energy converters to verbal exchange protocols and from chaos to mathematical finance.
Subjects lined contain the next: definition of hybrid structures; description codecs; lifestyles and specialty of recommendations; designated subclasses (variable-structure platforms, complementarity systems); reachability and verification; balance and stabilizability; regulate layout equipment. The booklet might be of curiosity to scientists from quite a lot of disciplines together with: computing device technological know-how, keep watch over conception, dynamical procedure idea, platforms modeling and simulation, and operations research.

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When the water level rises to 10 units, the monitor sends a switch-off signal, which after a delay of two seconds results in the pump turning off. When the water level falls to 5 units, the monitor sends a switch-on signal, which after a delay of again two seconds causes the pump to switch on. There are several ways in which one may write down equations to describe the system, which may be related to various ways in which the controller may be implemented. For instance the monitor should send a switch-off signal when the water level reaches 10 units and is rising, but not when the level reaches 10 units on its way down.

13b) expresses conservation of m o m e n t u m for simple impacts. Below we shall apply this rule in a situation where multiple impact occurs (as it was done in [63]); of course one may debate whether such an extension is "correct". 14) V2 ~--- V3 ~ 0. How do we model this case with the above impact rule? The problem is t h a t not only the left mass is colliding with the middle mass, but that also the middle mass is in contact with the right mass; we thus encounter a multiple collision. 2. 13) for single collisions, and these two ways lead to different answers!

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).

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