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 [math]\displaystyle{ [A(t),B(t),C(t),D(t)] }[/math] such that

[math]\displaystyle{ \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t) }[/math]

[math]\displaystyle{ \mathbf{y}(t) = C(t) \mathbf{x}(t) + D(t) \mathbf{u}(t) }[/math]

with [math]\displaystyle{ (u(t),y(t)) }[/math] describing the input and output of the system at time [math]\displaystyle{ t }[/math].

LTI System

For a linear time-invariant system specified by a transfer matrix, [math]\displaystyle{ H(s) }[/math], a realization is any quadruple of matrices [math]\displaystyle{ (A,B,C,D) }[/math] such that [math]\displaystyle{ H(s) = C(sI-A)^{-1}B+D }[/math].

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:

[math]\displaystyle{ 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}} }[/math].

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

[math]\displaystyle{ \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) }[/math]

[math]\displaystyle{ \textbf{y}(t) = \begin{bmatrix} n_{3}& n_{2}& n_{1}& n_{0} \end{bmatrix}\textbf{x}(t) }[/math].

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

[math]\displaystyle{ \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) }[/math]

[math]\displaystyle{ \textbf{y}(t) = \begin{bmatrix} 1& 0& 0& 0 \end{bmatrix}\textbf{x}(t) }[/math].

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 [math]\displaystyle{ u(t) }[/math], an output [math]\displaystyle{ y(t) }[/math], and a weighting pattern [math]\displaystyle{ T(t,\sigma) }[/math] then a realization is any triple of matrices [math]\displaystyle{ [A(t),B(t),C(t)] }[/math] such that [math]\displaystyle{ T(t,\sigma) = C(t) \phi(t,\sigma) B(\sigma) }[/math] where [math]\displaystyle{ \phi }[/math] is the state-transition matrix associated with the realization.

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.

See also

• Grey box model
• Statistical Model
• System identification