Invariance mechanics

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 $[A(t),B(t),C(t),D(t)]$ 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:

$H(s)=\frac{n_1{s}^3+n_2{s}^2+n_{3}s+n_4}{s^4+d_1{s}^3+d_2{s}^2+d_{3}s+d_4}$.

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

$\dot{\mathbf{\text{x}}}(t)=\left[\begin{array}{cccc}-d_1& -d_2& -d_3& -d_4\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right]\mathbf{\text{x}}(t)+\left[\begin{array}{c}1\\ 0\\ 0\\ 0\\ \end{array}\right]\mathbf{\text{u}}(t)$

$\mathbf{\text{y}}(t)=\left[\begin{array}{cccc}{n}_1& n_2& n_3& n_4\end{array}\right]\mathbf{\text{x}}(t)$.

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

$\dot{\mathbf{\text{x}}}(t)=\left[\begin{array}{cccc}-d_1& 1& 0& 0\\ -d_2& 0& 1& 0\\ -d_3& 0& 0& 1\\ -d_4& 0& 0& 0\end{array}\right]\mathbf{\text{x}}(t)+\left[\begin{array}{c}{n}_1\\ n_2\\ n_3\\ n_4\end{array}\right]\mathbf{\text{u}}(t)$

$\mathbf{\text{y}}(t)=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]\mathbf{\text{x}}(t)$.

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 $T(t,\sigma )=C(t)\varphi (t,\sigma )B(\sigma )$ where $\varphi$ is the state-transition matrix associated with the realization.

System identification

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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

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