en>GirasoleDE at 21:14, 6 May 2013

2013-05-06T21:14:40Z

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{{refimprove|date=July 2011}}
A variable structure system, or '''VSS''', is a [[Classification of discontinuities|discontinuous]] [[nonlinear system]] of the form
:<math>\dot{\mathbf{x}} = \varphi( \mathbf{x}, t )</math>
where <math>\mathbf{x} \triangleq [x_1, x_2, \ldots, x_n]^{\operatorname{T}} \in \mathbb{R}^n</math> is the state vector, <math>t \in \mathbb{R}</math> is the time variable, and <math>\varphi(\mathbf{x},t) \triangleq [ \varphi_1(\mathbf{x},t), \varphi_2(\mathbf{x},t), \ldots, \varphi_n(\mathbf{x},t) ]^{\operatorname{T}} : \mathbb{R}^{n+1} \mapsto \mathbb{R}^n</math> is a ''[[piecewise continuous]]'' function.<ref name="EFF06">{{Cite book|title=Advances in Variable Structure and Sliding Mode Control|editor1-last=Edwards|editor1-first=Cristopher|editor2-last=Fossas Colet|editor2-first=Enric|editor3-last=Fridman|editor3-first=Leonid|publisher=Springer-Verlag|location=Berlin|date=2006|series=Lecture Notes in Control and Information Sciences|volume=vol 334|isbn=978-3-540-32800-1|postscript=}}</ref> Due to the ''piecewise'' continuity of these systems, they behave like different continuous nonlinear systems in different regions of their [[state space (controls)|state space]]. At the boundaries of these regions, their [[dynamic system|dynamics]] switch abruptly. Hence, their ''structure'' ''varies'' over different parts of their state space.

The development of [[variable structure control]] depends upon methods of analyzing variable structure systems, which are special cases of [[hybrid system|hybrid dynamical system]]s.

== See also ==

*[[Variable structure control]]
*[[Sliding mode control]]
*[[Hybrid system]]
*[[Nonlinear control]]
*[[Robust control]]
*[[Optimal control]]
*[[H-bridge]] – A topology that combines four switches forming the four legs of an "H". Can be used to drive a motor (or other electrical device) forward or backward when only a single supply is available. Often used in actuator sliding-mode control systems.
*[[Switching amplifier]] – Uses switching-mode control to drive continuous outputs
*[[Delta-sigma modulation]] – Another (feedback) method of encoding a continuous range of values in a signal that rapidly switches between two states (i.e., a kind of specialized sliding-mode control)
*[[Pulse density modulation]] – A generalized form of delta-sigma modulation.
*[[Pulse-width modulation]] – Another modulation scheme that produces continuous motion through discontinuous switching.

== Further reading ==
{{refbegin}}
*{{cite book
| last = Khalil
| first = H.K.
| authorlink = Hassan K. Khalil
| year = 2002
| edition = 3rd
| url = http://www.egr.msu.edu/~khalil/NonlinearSystems/
| isbn = 0-13-067389-7
| title = Nonlinear Systems
| publisher = [[Prentice Hall]]
| location = Upper Saddle River, NJ}}
*{{cite book
| author = [http://www.ece.osu.edu/~utkin/ Utkin, V.I.]
| title = Sliding Modes in Control and Optimization
| publisher = Springer-Verlag
| date = 1992
| pages =
| isbn = 978-0-387-53516-6
}}
*{{cite book
| editor-last = Zinober
| editor-first = A.S.I.
| title = Deterministic control of uncertain systems
| publisher = Peter Peregrinus Press
| place = London
| date = 1990
| isbn = 978-0-86341-170-0
| editor-link = Alan S.I. Zinober
}}
*{{cite book
| editor-last = Zinober
| editor-first = Alan S.I.
| title = Variable Structure and Lyapunov Control
| publisher = Springer-Verlag
| place = London
| date = 1994
| isbn = 978-3-540-19869-7
| doi = 10.1007/BFb0033675
}}
{{refend}}

== References ==
{{reflist|colwidth=30em}}

{{Systems}}

{{tech-stub}}

[[Category:Nonlinear systems]]
[[Category:Dynamical systems]]
[[Category:Concepts in physics]]
[[Category:Nonlinear control]]

Fractional quantum mechanics - Revision history

en>GirasoleDE at 21:14, 6 May 2013