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| {{Refimprove|date=December 2009}}
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| A '''state variable''' is one of the set of [[variable (mathematics)|variables]] that are used to describe the mathematical "state" of a [[dynamical system]]. Intuitively, the state of a system describes enough about the system to determine its future behaviour. Models that consist of coupled first-order differential equations are said to be in state-variable form.<ref>
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| {{cite book
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| |author=William J. Palm III
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| |year=2010
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| |title=System Dynamics
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| |edition=2nd
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| |publisher=
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| |page=225
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| |isbn=
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| }}</ref>
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| ==Examples==
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| *In [[mechanics|mechanical systems]], the position coordinates and [[Velocity|velocities]] of mechanical parts are typical state variables; knowing these, it is possible to determine the future state of the objects in the system.
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| *In a [[Thermodynamics|thermodynamic system]], properties such as [[thermodynamic temperature|temperature]], [[pressure]], [[volume]], [[internal energy]], [[enthalpy]], and [[entropy]] are state variables.
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| *In [[electronic circuit]]s, the [[voltage]]s of the nodes and the [[Electric current|current]]s through components in the circuit are usually the state variables.
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| *In [[ecosystem model]]s, population sizes (or concentrations) of plants, animals and resources (nutrients, organic material) are typical state variables.
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| ==Control systems engineering==
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| In [[control engineering]] and other areas of science and engineering, state variables are used to represent the states of a general system. The state variables can be used to describe the [[state space (controls)|state space]] of the system. The equations relating the current state and output of a system to its current input and past states are called the state equations. The state equations for a [[linear time invariant]] system can be expressed using coefficient [[Matrix (mathematics)|matrices]]:
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| :<math>A\in</math> '''R'''<sup>''N*N''</sup>, <math>\quad B\in</math> '''R'''<sup>''N*L''</sup>, <math>\quad C\in</math> '''R'''<sup>''M*N''</sup>, <math>\quad D\in</math> '''R'''<sup>''M*L''</sup>,
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| where ''N'', ''L'' and ''M'' are the dimensions of the vectors describing the state, input and output, respectively.
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| ===Discrete-time systems===
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| The state variable representing the current state of a discrete-time system (i.e. digital system) is <math>x[n]\,</math>, where n is the [[Discrete time|discrete]] point at which the system is being evaluated. The discrete-time state equations are
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| : <math> x[n+1] = Ax[n] + Bu[n]\,\!</math> , which describes the next state of the system (x[n+1]) with respect to current state and inputs u[n] of the system.
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| : <math> y[n] = Cx[n] + Du[n]\,\!</math> , which describes the output y[n] with respect to current states and inputs u[n] to the system.
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| ===Continuous time systems===
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| The state variable representing the current state of a continuous-time system (i.e. analog system) is <math>x(t)\,</math>, and the continuous time state equations are
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| : <math> \frac{dx(t)}{dt} \ = Ax(t) + Bu(t)\,\!</math> , which describes the next state of the system <math> \frac{dx(t)}{dt} \,\!</math> with respect to current state x(t) and inputs u(t) of the system. | |
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| : <math> y(t) = Cx(t) + Du(t)\,\!</math> , which describes the output y(t) with respect to current states x(t) and inputs u(t) to the system.
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| ==See also==
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| *[[State space (controls)]]
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| *[[Control theory]]
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| *[[Equation of state]]
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| *[[State (computer science)]]
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| *[[Dynamical systems]]
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| *[[State (functional analysis)]]
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| *[[State diagram]]
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| *[[State variable filter]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:State Variable}}
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| [[Category:Control theory]]
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