[A(t),B(t),C(t),D(t)]
| x |
(t)=A(t)x(t)+B(t)u(t)
y(t)=C(t)x(t)+D(t)u(t)
(u(t),y(t))
t
For a linear time-invariant system specified by a transfer matrix,
H(s)
(A,B,C,D)
H(s)=C(sI-A)-1B+D
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)=
| n3s3+n2s2+n1s+n0 | |
| s4+d3s3+d2s2+d1s+d0 |
The coefficients can now be inserted directly into the state-space model by the following approach:
| bf{x |
bf{y}(t)=\begin{bmatrix}n3&n2&n1&n0\end{bmatrix}bf{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
| bf{x |
bf{y}(t)=\begin{bmatrix}1&0&0&0\end{bmatrix}bf{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).
If we have an input
u(t)
y(t)
T(t,\sigma)
[A(t),B(t),C(t)]
T(t,\sigma)=C(t)\phi(t,\sigma)B(\sigma)
\phi
See 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.