Editing State-space representation (section)

=== Canonical realizations ===
{{main|Realization (systems)}}

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:
<math display="block"> \mathbf{G}(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>

The coefficients can now be inserted directly into the state-space model by the following approach:
<math display="block">\dot{\mathbf{x}}(t) = \begin{bmatrix}
    0&     1&     0&    0\\
    0&     0&     1&    0\\
    0&     0&     0&    1\\
-d_4 & -d_3 & -d_2 & -d_1
\end{bmatrix}\mathbf{x}(t) + 
\begin{bmatrix} 0\\ 0\\ 0\\ 1 \end{bmatrix}\mathbf{u}(t)</math>

<math display="block"> \mathbf{y}(t) = \begin{bmatrix} n_4 & n_3 & n_2 & n_1 \end{bmatrix} \mathbf{x}(t). </math>

This state-space realization is called '''controllable 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
<math display="block">\dot{\mathbf{x}}(t) = \begin{bmatrix}
                               0&   0&  0&  -d_{4}\\
                               1&   0&  0& -d_{3}\\
                               0&   1&  0& -d_{2}\\
                               0&   0&  1&  -d_{1}
                             \end{bmatrix}\mathbf{x}(t) + 
                             \begin{bmatrix} n_{4}\\ n_{3}\\ n_{2}\\ n_{1} \end{bmatrix}\mathbf{u}(t)</math><math display="block"> \mathbf{y}(t) = \begin{bmatrix} 0& 0& 0& 1 \end{bmatrix}\mathbf{x}(t). </math>

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).