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| '''Realization''', in the [[system theory]] context refers to a [[State space (controls)|state space]] model implementing a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of ([[time-variant system|time-varying]]) [[Matrix (mathematics)|matrices]] <math>[A(t),B(t),C(t),D(t)]</math> such that
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| : <math>\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t)</math>
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| : <math>\mathbf{y}(t) = C(t) \mathbf{x}(t) + D(t) \mathbf{u}(t)</math>
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| with <math>(u(t),y(t))</math> describing the input and output of the system at time <math>t</math>.
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| ==LTI System==
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| For a [[linear time-invariant system]] specified by a [[transfer function|transfer matrix]], <math> H(s) </math>, a realization is any quadruple of matrices <math> (A,B,C,D) </math> such that <math> H(s) = C(sI-A)^{-1}B+D</math>.
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| === Canonical realizations ===
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| Any given transfer function which is [[strictly proper]] can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system)):
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| Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:
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| :<math> H(s) = \frac{n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{s^{4} + d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}</math>.
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| The coefficients can now be inserted directly into the state-space model by the following approach:
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| :<math>\dot{\textbf{x}}(t) = \begin{bmatrix}
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| -d_{1}& -d_{2}& -d_{3}& -d_{4}\\
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| 1& 0& 0& 0\\
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| 0& 1& 0& 0\\
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| 0& 0& 1& 0
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| \end{bmatrix}\textbf{x}(t) +
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| \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix}\textbf{u}(t)</math>
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| :<math> \textbf{y}(t) = \begin{bmatrix} n_{1}& n_{2}& n_{3}& n_{4} \end{bmatrix}\textbf{x}(t)</math>.
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| This state-space realization is called '''controllable canonical form''' (also known as phase variable canonical form) because the resulting model is guaranteed to be [[Controllability|controllable]] (i.e., because the control enters a chain of integrators, it has the ability to move every state).
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| The transfer function coefficients can also be used to construct another type of canonical form
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| :<math>\dot{\textbf{x}}(t) = \begin{bmatrix}
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| -d_{1}& 1& 0& 0\\
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| -d_{2}& 0& 1& 0\\
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| -d_{3}& 0& 0& 1\\
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| -d_{4}& 0& 0& 0
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| \end{bmatrix}\textbf{x}(t) +
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| \begin{bmatrix} n_{1}\\ n_{2}\\ n_{3}\\ n_{4} \end{bmatrix}\textbf{u}(t)</math>
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| :<math> \textbf{y}(t) = \begin{bmatrix} 1& 0& 0& 0 \end{bmatrix}\textbf{x}(t)</math>.
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| This state-space realization is called '''observable canonical form''' because the resulting model is guaranteed to be [[Observability|observable]] (i.e., because the output exits from a chain of integrators, every state has an effect on the output).
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| ==General System==
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| ===<math>D = 0</math>===
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| If we have an input <math>u(t)</math>, an output <math>y(t)</math>, and a [[weighting pattern]] <math>T(t,\sigma)</math> then a realization is any triple of matrices <math>[A(t),B(t),C(t)]</math> such that <math>T(t,\sigma) = C(t) \phi(t,\sigma) B(\sigma)</math> where <math>\phi</math> is the [[state-transition matrix]] associated with the realization.<ref>{{cite book|first=Roger W.|last=Brockett|title=Finite Dimensional Linear Systems|publisher=John Wiley & Sons|year=1970|isbn=978-0-471-10585-5}}</ref>
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| ==System identification==
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| {{main|System identification}}
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| System identification techniques take the experimental data from a system and output a realization. Such techniques can utilize both input and output data (e.g. [[eigensystem realization algorithm]]) or can only include the output data (e.g. [[frequency domain decomposition]]). Typically an input-output technique would be more accurate, but the input data is not always available.
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| ==References==
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| {{Reflist}}
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| [[Category:Models of computation]]
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| [[Category:Systems theory]]
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