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.(G.13) to Eq.(G.14), can be bypassed [28, p. 87].

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 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 , again as we obtained before. Finally, it must also be observed that the gain of the direct path from input to output is .

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.