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 (time-varying) matrices
describing the input and output of the system at time
For a linear time-invariant system specified by a transfer matrix,
, a realization is any quadruple of matrices
Any given transfer function which is strictly proper can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system)): 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 (also known as phase variable canonical form) because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).
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 (i.e., because the output exits from a chain of integrators, every state has an effect on the output).
then a realization is any triple of matrices
[1] System identification techniques take the experimental data from a system and output a realization.
Such techniques can utilize both input and output data (e.g. eigensystem realization algorithm) or can only include the output data (e.g. frequency domain decomposition).
Typically an input-output technique would be more accurate, but the input data is not always available.