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 $[A(t),B(t),C(t),D(t)]$ 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 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.^[1]

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

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

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

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