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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">State variable</span></span>

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<p>A <b>state variable</b> is one of the set of <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a> that are used to describe the mathematical "state" of a <a href="Dynamical_system" title="Dynamical system">dynamical system</a>. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of any external forces affecting the system. Models that consist of coupled first-order <a href="Differential_equation" title="Differential equation">differential equations</a> are said to be in state-variable form.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>In <a href="Thermodynamics" title="Thermodynamics">thermodynamics</a>, state variables are defined as large-scale characteristics or aggregate properties of a system which provide a <a href="Macroscopic_scale" title="Macroscopic scale">macroscopic description</a> of it.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In general, state variables have the following properties in common:

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<ul><li>They don't involve any special assumptions concerning the structure of matter, fields or radiation.</li>

<li>They are few in number needed to describe the system.</li>

<li>They are fundamental, as suggested by our sensory perceptions.</li>

<li>They can be, in general, directly measured. <sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>In <a href="Mechanics" title="Mechanics">mechanical systems</a>, the position coordinates and <a href="Velocity" title="Velocity">velocities</a> of mechanical parts are typical state variables; knowing these, it is possible to determine the future state of the objects in the system.</li>

<li>In <a href="Thermodynamics" title="Thermodynamics">thermodynamics</a>, a state variable is an independent variable of a <a href="State_function" title="State function">state function</a>. Examples include <a href="Internal_energy" title="Internal energy">internal energy</a>, <a href="Enthalpy" title="Enthalpy">enthalpy</a>, <a href="Thermodynamic_temperature" title="Thermodynamic temperature">temperature</a>, <a href="Pressure" title="Pressure">pressure</a>, <a href="Volume" title="Volume">volume</a> and <a href="Entropy" title="Entropy">entropy</a>. <a href="Heat" title="Heat">Heat</a> and <a href="Work_(Thermodynamics)" class="mw-redirect" title="Work (Thermodynamics)">work</a> are not state functions, but <a href="Process_function" title="Process function">process functions</a>.</li>

<li>In <a href="Electronics" title="Electronics">electronic</a>/<a href="Electrical_circuit" class="mw-redirect" title="Electrical circuit">electrical circuits</a>, the <a href="Voltage" title="Voltage">voltages</a> of the nodes and the <a href="Electric_current" title="Electric current">currents</a> through components in the circuit are usually the state variables. In any electrical circuit, the number of state variables are equal to the number of (independent) storage elements, which are inductors and capacitors. The state variable for an inductor is the current through the inductor, while that for a capacitor is the voltage across the capacitor.</li>

<li>In <a href="Ecosystem_model" title="Ecosystem model">ecosystem models</a>, population sizes (or concentrations) of plants, animals and resources (nutrients, organic material) are typical state variables.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Control_systems_engineering">Control systems engineering</h2></div>

<p>In <a href="Control_engineering" title="Control engineering">control engineering</a> and other areas of science and engineering, state variables are used to represent the states of a general system. The set of possible combinations of state variable values is called the <a href="State_space_(controls)" class="mw-redirect" title="State space (controls)">state space</a> of the system. The equations relating the current state of a system to its most recent input and past states are called the state equations, and the equations expressing the values of the output variables in terms of the state variables and inputs are called the output equations. As shown below, the state equations and output equations for a <a href="Linear_time_invariant" class="mw-redirect" title="Linear time invariant">linear time invariant</a> system can be expressed using coefficient <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>: <i>A</i>, <i>B</i>, <i>C</i>, and <i>D</i>

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<annotation encoding="application/x-tex">{\displaystyle A\in \mathbb {R} ^{N\times N},\quad B\in \mathbb {R} ^{N\times L},\quad C\in \mathbb {R} ^{M\times N},\quad D\in \mathbb {R} ^{M\times L},}</annotation>

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</math></span><img src="./bf156581e45b828b83231879db346118be0e375c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:53.561ex; height:3.009ex;" alt="{\displaystyle A\in \mathbb {R} ^{N\times N},\quad B\in \mathbb {R} ^{N\times L},\quad C\in \mathbb {R} ^{M\times N},\quad D\in \mathbb {R} ^{M\times L},}" loading="lazy"></span></dd></dl>

<p>where <i>N</i>, <i>L</i> and <i>M</i> are the dimensions of the vectors describing the state, input and output, respectively.

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<div class="mw-heading mw-heading3"><h3 id="Discrete-time_systems">Discrete-time systems</h3></div>

<p>The state vector (vector of state variables) representing the current state of a <a href="Discrete-time" class="mw-redirect" title="Discrete-time">discrete-time</a> system (i.e. digital system) is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]}">

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</math></span><img src="./864cbbefbdcb55af4d9390911de1bf70167c4a3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle x[n]}" loading="lazy"></span>, where <i>n</i> is the discrete point in time at which the system is being evaluated. The discrete-time state equations are

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<annotation encoding="application/x-tex">{\displaystyle x[n+1]=Ax[n]+Bu[n],}</annotation>

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</math></span><img src="./1b3b41ed33294fd2c2d99283bf6e5e132ee92a38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.15ex; height:2.843ex;" alt="{\displaystyle x[n+1]=Ax[n]+Bu[n],}" loading="lazy"></span></dd></dl>

<p>which describes the next state of the system (<i>x</i>[<i>n</i>+1]) with respect to current state and inputs <i>u</i>[<i>n</i>] of the system. The output equations are

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<annotation encoding="application/x-tex">{\displaystyle y[n]=Cx[n]+Du[n],}</annotation>

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</math></span><img src="./dc61ab54f3baacfb7cb73fb75910c1ae248641a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.156ex; height:2.843ex;" alt="{\displaystyle y[n]=Cx[n]+Du[n],}" loading="lazy"></span></dd></dl>

<p>which describes the output <i>y</i>[<i>n</i>] with respect to current states and inputs <i>u</i>[<i>n</i>] to the system.

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<div class="mw-heading mw-heading3"><h3 id="Continuous_time_systems">Continuous time systems</h3></div>

<p>The state vector representing the current state of a <a href="Continuous-time" class="mw-redirect" title="Continuous-time">continuous-time</a> system (i.e. analog system) is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dx(t)}{dt}}=Ax(t)+Bu(t),}</annotation>

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<p>which describes the continuous rate of change <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {dx(t)}{dt}}}">

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<annotation encoding="application/x-tex">{\displaystyle y(t)=Cx(t)+Du(t),}</annotation>

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<p>which describes the output <i>y</i>(<i>t</i>) with respect to current states <i>x</i>(<i>t</i>) and inputs <i>u</i>(<i>t</i>) to the system.

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<ul><li><a href="State_space_(controls)" class="mw-redirect" title="State space (controls)">State space (controls)</a></li>

<li><a href="Control_(optimal_control_theory)" title="Control (optimal control theory)">Control (optimal control theory)</a></li>

<li><a href="Control_theory" title="Control theory">Control theory</a></li>

<li><a href="Equation_of_state" title="Equation of state">Equation of state</a></li>

<li><a href="State_(computer_science)" title="State (computer science)">State (computer science)</a></li>

<li><a href="Dynamical_systems" class="mw-redirect" title="Dynamical systems">Dynamical systems</a></li>

<li><a href="State_(functional_analysis)" title="State (functional analysis)">State (functional analysis)</a></li>

<li><a href="State_diagram" title="State diagram">State diagram</a></li>

<li><a href="State_variable_filter" title="State variable filter">State variable filter</a></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFPalm,_III_William_J.2009" class="citation book cs1">Palm, III William J. (2009). <i>System Dynamics</i> (2nd ed.). McGraw-Hill Medical Publishing. p. 420. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-126779-3</bdi>.</cite></span>

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<li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFZemanskyDittman1997" class="citation book cs1">Zemansky, Mark Waldo; Dittman, Richard (1997). <i>Heat and thermodynamics: an intermediate textbook</i> (7th ed.). New York: McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-017059-9</bdi>.</cite></span>

</li>

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