System Realization

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 (t),B(t),C(t),D(t)/math> such that
: $\dot(t) = A(t) \mathbf(t) + B(t) \mathbf(t)$
: $\mathbf(t) = C(t) \mathbf(t) + D(t) \mathbf(t)$
with $(u(t),y(t))$ describing the input and output of the system at time $t$.

LTI System

For a linear time-invariant system specified by a transfer matrix, $H(s)$, a realization is any quadruple of matrices $(A,B,C,D)$ such that $H(s) = C(sI-A)^B+D$.

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:
:$H(s) = \frac$.

The coefficients can now be inserted directly into the state-space model by the following approach:
:$\dot(t) = \begin -d_& -d_& -d_& -d_\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0 \end\textbf(t) + \begin 1\\ 0\\ 0\\ 0\\ \end\textbf(t)$

:$\textbf(t) = \begin n_& n_& n_& n_ \end\textbf(t)$.

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 (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
:$\dot(t) = \begin -d_& 1& 0& 0\\ -d_& 0& 1& 0\\ -d_& 0& 0& 1\\ -d_& 0& 0& 0 \end\textbf(t) + \begin n_\\ n_\\ n_\\ n_ \end\textbf(t)$

:$\textbf(t) = \begin 1& 0& 0& 0 \end\textbf(t)$.

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 $u(t)$, an output $y(t)$, and a weighting pattern $T(t,\sigma)$ then a realization is any triple of matrices (t),B(t),C(t)/math> such that $T(t,\sigma) = C(t) \phi(t,\sigma) B(\sigma)$ where $\phi$ 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.

See also

* Grey box model
* Statistical Model
* System identification

References

{{Reflist

Models of computation
Systems theory