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| A '''hybrid system''' is a [[dynamic system]] that exhibits both continuous and discrete dynamic behavior – a system that can both ''flow'' (described by a [[differential equation]]) and ''jump'' (described by a [[difference equation]] or [[Graph (mathematics)|control graph]]). Often, the term "hybrid dynamic system" is used, to distinguish over hybrid systems such as those that combine [[neural net]]s and [[fuzzy logic]], or electrical and mechanical drivelines. A hybrid system has the benefit of encompassing a larger class of systems within its structure, allowing for more flexibility in modelling dynamic phenomena.
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| In general, the ''state'' of a hybrid system is defined by the values of the ''continuous variables'' and a discrete ''control mode''. The state changes either continuously, according to a ''[[flow conditioning|flow condition]]'', or discretely according to a ''control graph''. Continuous flow is permitted as long as so-called ''invariants'' hold, while discrete transitions can occur as soon as given ''jump conditions'' are satisfied. Discrete transition may be associated with ''events''.
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| ==Examples==
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| Hybrid systems have been used to model several systems, including [[physical system]]s with ''impact'', logic-dynamic [[Controller (control theory)|controllers]], and even [[Internet]] congestion.
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| ===Bouncing ball===
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| A canonical example of a hybrid system is '''the bouncing ball''', a physical system with impact. Here, the ball (thought of as a point-mass) is dropped from an initial height and bounces off the ground, dissipating its energy with each bounce. The ball exhibits continuous dynamics between each bounce; however, as the ball impacts the ground, its velocity undergoes a discrete change modeled after an [[inelastic collision]]. A mathematical description of the bouncing ball follows. Let <math>x_1</math> be the height of the ball and <math>x_2</math> be the velocity of the ball. A hybrid system describing the ball is as follows:
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| When <math>x \in C = \{x_1 > 0\}</math>, flow is governed by
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| <math>
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| \dot{x_1} = x_2,
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| \dot{x_2} = -g
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| </math>,
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| where <math>g</math> is the acceleration due to gravity. These equations state that when the ball is above ground, it is being drawn to the ground by gravity.
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| When <math>x \in D = \{x_1 = 0\}</math>, jumps are governed by
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| <math>
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| x_1^+ = x_1,
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| x_2^+ = -\gamma x_2
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| </math>,
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| where <math>0 < \gamma < 1</math> is a dissipation factor. This is saying that when the height of the ball is zero (it has impacted the ground), its velocity is reversed and decreased by a factor of <math>\gamma</math>. Effectively, this describes the nature of the inelastic collision.
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| The bouncing ball is an especially interesting hybrid system, as it exhibits [[Zeno of Elea|Zeno]] behavior. Zeno behavior has a strict mathematical definition, but can be described informally as the system making an ''infinite'' number of jumps in a ''finite'' amount of time. In this example, each time the ball bounces it loses energy, making the subsequent jumps (impacts with the ground) closer and closer together in time.
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| It is noteworthy that the dynamical model is complete if and only if one adds the contact force between the ground and the ball. Indeed, without forces, one cannot properly define the bouncing ball and the model is, from a mechanical point of view, meaningless. The simplest contact model that represents the interactions between the ball and the ground, is the complementarity relation between the force and the distance (the gap) between the ball and the ground. This is written as
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| <math>
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| 0 \leq \lambda \perp x_1 \geq 0
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| </math>
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| Such a contact model does not incorporate magnetic forces, nor gluing effects. When the complementarity relations are in, one can continue to integrate the system after the impacts have accumulated and vanished: the equilibrium of the system is well-defined as the static equilibrium of the ball on the ground, under the action of gravity compensated by the contact force <math>\lambda</math>. One also notices from basic convex analysis that the complementarity relation can equivalently be rewritten as the inclusion into a normal cone, so that the bouncing ball dynamics is a differential inclusion into a normal cone to a convex set. See Chapters 1, 2 and 3 in Acary-Brogliato's book cited below (Springer LNACM 35, 2008). See also the other references on non-smooth mechanics.
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| == Hybrid Systems [[Formal verification|Verification]] ==
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| There are approaches to automatically proving properties of hybrid systems (e.g., some of the tools mentioned below). Most verification tasks are undecidable,<ref>Thomas A. Henzinger, Peter W. Kopke, Anuj Puri, and Pravin Varaiya: What's Decidable about Hybrid Automata, Journal of Computer and System Sciences, 1998</ref> so tools cannot succeed always with verification. Instead they usually try to be able to verify interesting benchmark problems. A possible theoretical characterization of this is algorithms that succeed with hybrid systems verification in all robust cases <ref>Martin Fränzle: Analysis of Hybrid Systems: An ounce of realism can save an infinity of states, Springer LNCS 1683</ref> implying that many problem for hybrid systems, while being undecidable, are at least quasi-decidable <ref>Stefan Ratschan: Safety verification of non-linear hybrid systems is quasi-decidable, Formal Methods in System Design, volume 44, pp. 71-90, 2014</ref>
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| ==Other modeling approaches==
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| Two basic hybrid system modeling approaches can be classified, an implicit and an explicit one. The explicit approach is often represented by a [[hybrid automaton]], a [http://symbolaris.com/info/KeYmaera-guide.html#HP hybrid program] or a hybrid [[Petri net]]. The implicit approach is often represented by guarded equations to result in systems of [[differential algebraic equation]]s (DAEs) where the active equations may change, for example by means of a [[hybrid bond graph]].
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| As a unified simulation approach for hybrid system analysis, there is a method based on [[DEVS]] formalism in which integrators for differential equations are quantized into atomic [[DEVS]] models. These methods generate traces of system behaviors in discrete event system manner which are different from discrete time systems. Detailed of this approach can be found in references [Kofman2004] [CF2006] [Nutaro2010] and the software tool [[PowerDEVS]].
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| ==Tools==
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| * [http://embedded.eecs.berkeley.edu/research/hytech// HyTech]: A Model Checker for Hybrid Systems
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| * [http://hsolver.sourceforge.net/ HSolver]: Verification of Hybrid Systems
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| * [http://www-verimag.imag.fr/~frehse/phaver_web/ PHAVer]: Polyhedral Hybrid Automaton Verifyer
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| * [http://spaceex.imag.fr/ SpaceEx]: State-Space Explorer
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| * [http://symbolaris.com/info/KeYmaera.html KeYmaera]: A Hybrid Theorem Prover for Hybrid Systems
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| * [[PowerDEVS]]: A general-purpose software tool for DEVS modeling and simulation oriented to the simulation of hybrid systems
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| * [http://www.mathworks.com/videos/hyeq-a-toolbox-for-simulation-of-hybrid-dynamical-systems-81992.html HyEQ]: A Hybrid System Solver for Matlab
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| ==See also==
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| * [[Sliding mode control]]
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| * [[Variable structure system]]
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| * [[Variable structure control]]
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| * [[Joint spectral radius]]
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| <!--
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| *[[Hybrid Systems Visual Modeler]]
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| -->
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| ==Further reading==
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| * {{citation
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| |last1=Henzinger
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| |first1=Thomas A.
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| |contribution=The Theory of Hybrid Automata
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| |title=11th Annual Symposium on Logic in Computer Science (LICS)
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| |series=IEEE Computer Society Press
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| |year=1996
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| |pages=278–292
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| |url=http://www.eecs.berkeley.edu/~tah/Publications/the_theory_of_hybrid_automata.html}}
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| * {{citation
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| |doi=10.1016/0304-3975(94)00202-T
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| |first1=Rajeev
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| |last1=Alur
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| |first2=Costas
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| |last2=Courcoubetis
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| |first3=Nicolas
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| |last3=Halbwachs
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| |first4=Thomas A.
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| |last4=Henzinger
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| |first5=Pei-Hsin
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| |last5=Ho
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| |first6=Xavier
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| |last6=Nicollin
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| |first7=Alfredo
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| |last7=Olivero
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| |first8=Joseph
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| |last8=Sifakis
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| |first9=Sergio
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| |last9=Yovine
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| |title=The algorithmic analysis of hybrid systems.
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| |journal=Theoretical Computer Science
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| |volume=138
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| |issue=1
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| |pages=3–34
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| |year=1995
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| |url=http://www.eecs.berkeley.edu/~tah/Publications/the_algorithmic_analysis_of_hybrid_systems.html}}
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| * {{citation
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| |last1=Goebel
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| |first1=Rafal
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| |last2=Sanfelice
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| |first2=Ricardo G.
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| |last3=Teel
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| |first3=Andrew R.
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| |title=Hybrid dynamical systems
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| |journal=IEEE Control Systems Magazine
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| |year=2009
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| |volume=29
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| |issue=2
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| |pages=28–93
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| |doi=10.1109/MCS.2008.931718}}
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| * {{citation
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| |last1=Acary
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| |first1=Vincent
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| |last2=Brogliato
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| |first2=Bernard
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| |title=Numerical Methods for Nonsmooth Dynamical Systems
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| |journal=Lecture Notes in Applied and Computational Mechanics
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| |year=2008
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| |volume=35 }}
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| * [Kofman2004] {{citation
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| |doi=10.1137/S1064827502418379
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| |last1=Kofman
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| |first1=E
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| |title=Discrete Event Simulation of Hybrid Systems
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| |journal=SIAM Journal on Scientific Computing
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| |year=2004
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| |volume=25
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| |issue=5
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| |pages=1771–1797
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| }}
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| * [CF2006] {{Citation|author = Francois E. Cellier and Ernesto Kofman| year = 2006| title = Continuous System Simulation| publisher = Springer| isbn = 978-0-387-26102-7 |edition=first}}
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| * [Nutaro2010] {{Citation|author = James Nutaro| year = 2010| title = Building Software for Simulation: Theory, Algorithms, and Applications in C++| publisher = Wiley|edition=first}}
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| ==External links==
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| *[http://www.dii.unisi.it/hybrid/ieee/ IEEE CSS Committee on Hybrid Systems]
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| == References ==
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| {{Reflist}}
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| [[Category:Systems theory]]
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| [[Category:Differential equations]]
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| [[Category:Dynamical systems]]
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| [[Category:Control theory]]
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