# Realization (systems)

**Realization**, in the system theory context refers to a state space model implementing a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of (time-varying) 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 (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

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.^{[1]}

## System identification

{{#invoke:main|main}} 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.

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}