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 (time-varying) matrices formula_1 such that
with formula_4 describing the input and output of the system at time formula_5.
LTI System.
For a linear time-invariant system specified by a transfer matrix, formula_6, a realization is any quadruple of matrices formula_7 such that formula_8.
Canonical realizations.
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).
General System.
"D" = 0.
If we have an input formula_14, an output formula_15, and a weighting pattern formula_16 then a realization is any triple of matrices formula_17 such that formula_18 where formula_19 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 (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.