Realization (systems)

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)^{-1}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{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 (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 (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 [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 associated with the realization.

System identification

Main article: 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.

References

References

1. Brockett, Roger W.. (1970). "Finite Dimensional Linear Systems". John Wiley & Sons.