# Realization (systems) - LTI System - Canonical Realizations

Canonical Realizations

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)):

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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The coefficients can now be inserted directly into the state-space model by the following approach: $dot{textbf{x}}(t) = begin{bmatrix} -d_{1}& -d_{2}& -d_{3}& -d_{4}\ 1& 0& 0& 0\ 0& 1& 0& 0\ 0& 0& 1& 0 end{bmatrix}textbf{x}(t) + begin{bmatrix} 1\ 0\ 0\ 0\ end{bmatrix}textbf{u}(t)$
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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 controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).

The transfer function coefficients can also be used to construct another type of canonical form $dot{textbf{x}}(t) = begin{bmatrix} -d_{1}& 1& 0& 0\ -d_{2}& 0& 1& 0\ -d_{3}& 0& 0& 1\ -d_{4}& 0& 0& 0 end{bmatrix}textbf{x}(t) + begin{bmatrix} n_{1}\ n_{2}\ n_{3}\ n_{4} end{bmatrix}textbf{u}(t)$
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This state-space realization is called observable canonical form because the resulting model is guaranteed to be observable (i.e., because the output exits from a chain of integrators, every state has an effect on the output).