Realization (systems)


In systems theory, a realization of a state space model is an implementation of a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of matrices such that
with describing the input and output of the system at time.

LTI System

For a linear time-invariant system specified by a transfer matrix,, a realization is any quadruple of matrices such that.

Canonical realizations

Any given transfer function which is strictly proper can easily be transferred into state-space by the following approach ):
Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:
The coefficients can now be inserted directly into the state-space model by the following approach:
This state-space realization is called controllable canonical form because the resulting model is guaranteed to be controllable.
The transfer function coefficients can also be used to construct another type of canonical form
This state-space realization is called observable canonical form because the resulting model is guaranteed to be observable.

General System

''D'' = 0

If we have an input, an output, and a weighting pattern then a realization is any triple of matrices such that where is the state-transition matrix associated with the realization.

System identification

System identification techniques take the experimental data from a system and output a realization. Such techniques can utilize both input and output data or can only include the output data. Typically an input-output technique would be more accurate, but the input data is not always available.