Converting to state variable form

Converting a digital filter to state-space form is easy because there are various ``canonical forms'' for state-space models which can be written by inspection given the strictly proper transfer-function coefficients.

The canonical forms useful for transfer-function to state-space conversion are controller canonical form (also called control or controllable canonical form ) and observer canonical form (or observable canonical form ) [ 28, p. 80], [ 37]. These names come from the field of control theory [ 28] which is concerned with designing feedback laws to control the dynamics of real-world physical systems. State-space models are used extensively in the control field to model physical systems.

The name ``controller canonical form'' reflects the fact that the input signal can ``drive'' all modes (poles) of the system. In the language of control theory, we may say that all of the system poles are controllable from the input . In observer canonical form, all modes are guaranteed to be observable . Controllability and observability of a state-space model are discussed further in §G.7.3 below.

1. Determine the filter transfer function .
2. If is not strictly proper ( ), ``pull out'' the delay-free path to obtain a feed-through gain in parallel with a strictly proper transfer function.
3. Write down the state-space representation by inspection using controller canonical form for the strictly proper transfer function. (Or use the matlab function tf2ss .)

One might worry that choosing controller canonical form may result in unobservable modes. However, this will not happen if and have no common factors. In other words, if there are no pole-zero cancellations in the transfer function , then either controller or observer canonical form will yield a controllable and observable state-space model.

The transfer function can be written, by inspection, as