Editing State-space representation (section)

=== Proper transfer functions ===
Transfer functions which are only [[proper transfer function|proper]] (and not [[strictly proper]]) can also be realised quite easily. The trick here is to separate the transfer function into two parts: a strictly proper part and a constant. 
<math display="block"> \mathbf{G}(s) = \mathbf{G}_\mathrm{SP}(s) + \mathbf{G}(\infty). </math>

The strictly proper transfer function can then be transformed into a canonical state-space realization using techniques shown above. The state-space realization of the constant is trivially <math>\mathbf{y}(t) = \mathbf{G}(\infty)\mathbf{u}(t)</math>. Together we then get a state-space realization with matrices ''A'', ''B'' and ''C'' determined by the strictly proper part, and matrix ''D'' determined by the constant.

Here is an example to clear things up a bit:
<math display="block"> \mathbf{G}(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}
                      = \frac{s + 2}{s^2 + 2s + 1} + 1</math>
which yields the following controllable realization
<math display="block">\dot{\mathbf{x}}(t) = \begin{bmatrix}
                               -2& -1\\
                                1&      0\\
                             \end{bmatrix}\mathbf{x}(t) + 
                             \begin{bmatrix} 1\\ 0\end{bmatrix}\mathbf{u}(t)</math><math display="block"> \mathbf{y}(t) = \begin{bmatrix} 1& 2\end{bmatrix}\mathbf{x}(t) + \begin{bmatrix} 1\end{bmatrix}\mathbf{u}(t)</math>
Notice how the output also depends directly on the input. This is due to the <math>\mathbf{G}(\infty)</math> constant in the transfer function.