# Realization (systems)

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In [systems theory](/source/Systems_theory), a **realization** of a [state space](/source/State_space_(controls)) 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](/source/Time-variant_system)) [matrices](/source/Matrix_(mathematics)) [A(t),B(t),C(t),D(t)] such that

- \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t)
- \mathbf{y}(t) = C(t) \mathbf{x}(t) + D(t) \mathbf{u}(t)

with (u(t),y(t)) describing the input and output of the system at time t.

## LTI System

For a [linear time-invariant system](/source/Linear_time-invariant_system) specified by a [transfer matrix](/source/Transfer_function), H(s), a realization is any quadruple of matrices (A,B,C,D) such that H(s) = C(sI-A)^{-1}B+D.

### Canonical realizations

Any given transfer function which is [strictly proper](/source/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:

- H(s) = \frac{n_{3}s^{3} + n_{2}s^{2} + n_{1}s + n_{0}}{s^{4} + d_{3}s^{3} + d_{2}s^{2} + d_{1}s + d_{0}}.

The coefficients can now be inserted directly into the state-space model by the following approach:

- \dot{\textbf{x}}(t) = \begin{bmatrix} -d_{3}& -d_{2}& -d_{1}& -d_{0}\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0 \end{bmatrix}\textbf{x}(t) + \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix}\textbf{u}(t)

- \textbf{y}(t) = \begin{bmatrix} n_{3}& n_{2}& n_{1}& n_{0} \end{bmatrix}\textbf{x}(t).

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](/source/Controllability) (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

- \dot{\textbf{x}}(t) = \begin{bmatrix} -d_{3}& 1& 0& 0\\ -d_{2}& 0& 1& 0\\ -d_{1}& 0& 0& 1\\ -d_{0}& 0& 0& 0 \end{bmatrix}\textbf{x}(t) + \begin{bmatrix} n_{3}\\ n_{2}\\ n_{1}\\ n_{0} \end{bmatrix}\textbf{u}(t)

- \textbf{y}(t) = \begin{bmatrix} 1& 0& 0& 0 \end{bmatrix}\textbf{x}(t).

This state-space realization is called **observable canonical form** because the resulting model is guaranteed to be [observable](/source/Observability) (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 u(t), an output y(t), and a [weighting pattern](/source/Weighting_pattern) T(t,\sigma) then a realization is any triple of matrices [A(t),B(t),C(t)] such that T(t,\sigma) = C(t) \phi(t,\sigma) B(\sigma) where \phi is the [state-transition matrix](/source/State-transition_matrix) associated with the realization.[1]

## System identification

Main article: [System identification](/source/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](/source/Eigensystem_realization_algorithm)) or can only include the output data (e.g. [frequency domain decomposition](/source/Frequency_domain_decomposition)). Typically an input-output technique would be more accurate, but the input data is not always available.

## See also

- [Grey box model](/source/Grey_box_model)
- [Statistical Model](/source/Statistical_Model)
- [System identification](/source/System_identification)

## References

1. Brockett, Roger W. (1970). *Finite Dimensional Linear Systems*. John Wiley & Sons. ISBN 978-0-471-10585-5.

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