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

By Arjan J. van der Schaft, Hans Schumacher

This e-book is set dynamical platforms which are "hybrid" within the experience that they include either non-stop and discrete country variables. lately there was elevated learn curiosity within the examine of the interplay among discrete and non-stop dynamics. the current quantity offers a primary try out in e-book shape to compile strategies and techniques 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 in line with 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 various software parts, starting from energy converters to verbal exchange protocols and from chaos to mathematical finance.
Subjects lined comprise the subsequent: definition of hybrid structures; description codecs; life and forte of suggestions; designated subclasses (variable-structure platforms, complementarity systems); reachability and verification; balance and stabilizability; keep watch over layout tools. The booklet may be of curiosity to scientists from quite a lot of disciplines together with: desktop technological know-how, regulate conception, dynamical method conception, platforms modeling and simulation, and operations research.

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Ar will be denoted by A. As a simple mnemonic device, the discrete variables are denoted by capital letters P and S, while the continuous variables x and w are in lowercase. As noted before, the notation also serves to emphasize the nature of a "discrete variable" taking values in the discrete spaces L and A, rather than being an element of these discrete spaces. The communication variables may be used to link several parts of a system description to each other. One may also consider "open" systems in which the behavior of the communication variables is not completely determined by the system description itself; in such cases the communication variables may be thought of as providing a link to the (unmodeled) outside world.

Prescribing data at 0 - rather than at 0 allows 0 to be an event time. In this example, event times must actually have accumulation points; for instance if we set q(0-) = 0 and 4 ( 0 - ) = 1, then it is easily verified t h a t bounces take place at times 2, 2 + 2e, 2 + 2e + 2 e 2 , . . , so t h a t we have an accumulation point at 12_--7. Nevertheless we are still able to define a solution: gT = {2 ~j=0 k-1 e1 I k e N} U { ~ 2 } q(t) = ekt - ~1 ( t - 2 ~ j =k -01 eJ) 2 kml . for t e (2 Ej=o e~, 2 ~-]~=o e J), k = 0,1,2,...

31] and the references quoted therein. We will only consider the following seemingly simple case, which is sometimes known as Newton's cradle (with three balls). Consider three point masses with unit mass, moving on a straight line. The positions of the three masses will be denoted by ql, q2, q3, with ql < q2 < q3. , gravitational forces). Collisions (events) take place whenever the positions qi and the velocities v~ satisfy relations of the form qi = qj vi > vj for some i E {1, 2, 3} with j = i + 1.

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