A Short-Cut to Controller Canonical Form

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When converting a transfer function to state-space form by hand, the step of pulling out the direct path, like we did in going from Eq. $\,$ (G.13) to Eq. $\,$ (G.14), can be bypassed [28, p. 87].

**Figure:** Direct-form-II realization of Eq. $\,$ (G.7). Note that this form has a delay-free path from input to output.
$\begin{figure}\input fig/ssexdf2.pstex_t \end{figure}$

Figure G.2 gives the standard direct-form-II structure for a second-order IIR filter. Unlike Fig.G.1, it includes a direct path from the input to the output. The filter coefficients are all given directly by the transfer function, Eq. $\,$ (G.13).

This form can be converted directly to state-space form by carefully observing all paths from the input and state variables to the output. For example, reaches the output through gain 2 on the right, but also via gain $-1/2\cdot 1$ on the left and above. Therefore, its contribution to the output is , as obtained in the DF-II realization with direct-path pulled out shown in Fig.G.1. The state variable reaches the output with gain $3 - 1/3\cdot 1 = 8/3$ , again as we obtained before. Finally, it must also be observed that the gain of the direct path from input to output is .