By Arjan J. van der Schaft, Hans Schumacher
This e-book is set dynamical structures which are "hybrid" within the experience that they comprise either non-stop and discrete nation variables. lately there was elevated examine curiosity within the examine of the interplay among discrete and non-stop dynamics. the current quantity presents a primary test in booklet shape to compile options and strategies facing hybrid platforms from quite a few parts, and to examine those from a unified perspective.
The authors have selected a method of exposition that's principally in response to illustrative examples instead of at the summary theorem-proof structure as the systematic examine of hybrid structures remains to be in its infancy. The examples are taken from many alternative software components, starting from strength converters to communique protocols and from chaos to mathematical finance.
Subjects lined contain the next: definition of hybrid structures; description codecs; lifestyles and forte of recommendations; exact subclasses (variable-structure structures, complementarity systems); reachability and verification; balance and stabilizability; keep watch over layout equipment. The booklet might be of curiosity to scientists from quite a lot of disciplines together with: machine technological know-how, keep watch over thought, dynamical procedure conception, platforms modeling and simulation, and operations research.
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Extra info for An introduction to hybrid dynamical systems
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. This may be implemented by the sign of the derivative of y, by looking at the status of the pump, or by looking at the signal last sent by the monitor. Under the assumptions of the model these methods are all equivalent in the sense that they produce the same behavior; however there can be differences in robustness with respect to unmodeled effects.
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.
The three approaches may be briefly described as follows (cf. ). 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