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| In [[automata theory]], a '''hybrid automaton''' (plural: ''hybrid automata'' or ''hybrid automatons'') is a mathematical model for precisely describing systems in which digital computational processes interact with analog physical processes. A hybrid automaton is a [[finite state machine]] with a finite set of continuous variables whose values are described by a set of [[ordinary differential equations]]. This combined specification of discrete and continuous behaviors enables dynamic systems that comprise both digital and analog components to be modeled and analyzed.
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| ==Examples==
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| A simple example is a room-thermostat-heater system where the temperature of the room evolves according to laws of [[thermodynamics]] and the state of the heater (on/off); the thermostat senses the temperature, performs certain computations and turns the heater on and off. In general, hybrid automata have been used to model and analyze a variety of [[embedded systems]] including [[vehicle control system]]s, [[air traffic control]] systems, [[mobile robots]], and processes from [[systems biology]].
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| ==Formal Definition==
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| An '''Alur-Henzinger hybrid automaton''' <math>H</math> comprises the following components:<ref>Henzinger, T.A. "The Theory of Hybrid Automata". Proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS), pages 278-292, 1996.</ref>
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| * A finite set <math>X = \{x_1, ..., x_n\}</math> of real-numbered variables. The number <math>n</math> is called the ''dimension'' of <math>H</math>. Let <math>\dot{X}</math> be the set <math>\{\dot{x}_1, . . . , \dot{x}_n\}</math> of dotted variables that represent first derivatives during continuous change, and let <math>X'</math> be the set <math>\{x'_1, ..., x'_n\}</math> of primed variables that represent values at the conclusion of discrete change.
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| * A finite [[multidigraph]] <math>(V, E)</math>. The vertices in <math>V</math> are called ''control modes''. The edges in <math>E</math> are called ''control switches''.
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| * Three vertex labeling functions ''init'', ''inv'', and ''flow'' that assign to each control mode <math>v\in V</math> three predicates. Each initial condition ''init''<math>(v)</math> is a predicate whose free variables are from <math>X</math>. Each invariant condition ''inv''<math>(v)</math> is a predicate whose free variables are from <math>X</math>. Each flow condition ''flow''<math>(v)</math> is a predicate whose free variables are from <math>X\cup \dot{X}</math>.
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| So this is a [[labeled multidigraph]].
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| * An edge labeling function ''jump'' that assigns to each control switch <math>e\in E</math> a predicate. Each jump condition ''jump''<math>(e)</math> is a predicate whose free variables are from <math>X\cup X'</math>.
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| * A finite set <math>\Sigma</math> of events, and an edge labeling function ''event'': <math>E \rightarrow \Sigma</math> that assigns to each control switch an event.
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| ==Related models==
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| Hybrid automata come in several flavors: The '''Alur-Henzinger hybrid automaton''' is a popular model; it was developed primarily for algorithmic analysis of hybrid systems [[model checking]]. The [http://www-cad.eecs.berkeley.edu/~tah/HyTech/ HyTech] model checking tool is based on this model. The '''Hybrid Input/Output Automaton''' model has been developed more recently. This model enables compositional modeling and analysis of hybrid systems. Another formalism which is useful to model implementations of hybrid automaton is the [[lazy linear hybrid automaton]]. A sub-class of hybrid automata are [[timed automaton|timed automata]] in which all continuous variables have derivative 1. State reachability is decidable for this sub-class, which is why it is an interesting formalism for formal verification.
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| ==References==
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| <references />
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| ==Further reading==
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| *[[Rajeev Alur]], Costas Courcoubetis, Nicolas Halbwachs, Thomas A. Henzinger, Pei-Hsin Ho, Xavier Nicollin, Alfredo Olivero, Joseph Sifakis, and Sergio Yovine ''The algorithmic analysis of hybrid systems''. Theoretical Computer Science, volume 138(1), pages 3–34, 1995.
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| *[[Nancy Lynch]], Roberto Segala, Frits Vaandrager, ''Hybrid I/O Automata''. Information and Computation, volume 185(1), pages 103–157, 2003.
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| [[Category:Automata theory]]
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| [[Category:Differential equations]]
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