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:

${\displaystyle H(s)={\frac {n_{3}s^{3}+n_{2}s^{2}+n_{1}s+n_{0}}{s^{4}+d_{3}s^{3}+d_{2}s^{2}+d_{1}s+d_{0}}}}$.

The coefficients can now be inserted directly into the state-space model by the following approach:

${\displaystyle {\dot {\textbf {x}}}(t)={\begin{bmatrix}-d_{3}&-d_{2}&-d_{1}&-d_{0}\\1&0&0&0\\0&1&0&0\\0&0&1&0\end{bmatrix}}{\textbf {x}}(t)+{\begin{bmatrix}1\\0\\0\\0\\\end{bmatrix}}{\textbf {u}}(t)}$

${\displaystyle {\textbf {y}}(t)={\begin{bmatrix}n_{3}&n_{2}&n_{1}&n_{0}\end{bmatrix}}{\textbf {x}}(t)}$.

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

${\displaystyle {\dot {\textbf {x}}}(t)={\begin{bmatrix}-d_{3}&1&0&0\\-d_{2}&0&1&0\\-d_{1}&0&0&1\\-d_{0}&0&0&0\end{bmatrix}}{\textbf {x}}(t)+{\begin{bmatrix}n_{3}\\n_{2}\\n_{1}\\n_{0}\end{bmatrix}}{\textbf {u}}(t)}$

${\displaystyle {\textbf {y}}(t)={\begin{bmatrix}1&0&0&0\end{bmatrix}}{\textbf {x}}(t)}$.

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).

D = 0

If we have an input ${\displaystyle u(t)}$, an output ${\displaystyle y(t)}$, and a weighting pattern ${\displaystyle T(t,\sigma )}$ then a realization is any triple of matrices ${\displaystyle [A(t),B(t),C(t)]}$ such that ${\displaystyle T(t,\sigma )=C(t)\phi (t,\sigma )B(\sigma )}$ where ${\displaystyle \phi }$ is the state-transition matrix associated with the realization.^[1]