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

$\dot{𝐱}(t)=A(t)𝐱(t)+B(t)𝐮(t)$

$𝐲(t)=C(t)𝐱(t)+D(t)𝐮(t)$

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:

$\dot{\mathbf{\text{𝐱}}}(t)=\left[\begin{array}{cccc}-d_3& -d_2& -d_1& -d_0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right]\mathbf{\text{𝐱}}(t)+\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right]\mathbf{\text{𝐮}}(t)$

$\mathbf{\text{𝐲}}(t)=\left[\begin{array}{cccc}{n}_3& n_2& n_1& n_0\end{array}\right]\mathbf{\text{𝐱}}(t)$.

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

$\dot{\mathbf{\text{𝐱}}}(t)=\left[\begin{array}{cccc}-d_3& 1& 0& 0\\ -d_2& 0& 1& 0\\ -d_1& 0& 0& 1\\ -d_0& 0& 0& 0\end{array}\right]\mathbf{\text{𝐱}}(t)+\left[\begin{array}{c}{n}_3\\ n_2\\ n_1\\ n_0\end{array}\right]\mathbf{\text{𝐮}}(t)$

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

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

Template:Main 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