A Short-Cut to Controller Canonical Form

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. (E.13) to Eq. (E.14), can be bypassed [29, p. 87].

Figure E.2: Direct-form-II realization of Eq. (E.7). Note that this form has a delay-free path from input to output.

Figure E.2 gives the standard direct-form-II structure for a second-order IIR filter. Unlike Fig.E.1, it includes a direct path from the input to the output. The filter coefficients are all given directly by the transfer function, Eq. (E.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, $x_1(n)$ 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 $(2 - 1/2)x_1(n) = (3/2) x_1(n)$, as obtained in the DF-II realization with direct-path pulled out shown in Fig.E.1. The state variable $x_2(n)$ 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 $1$.