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).

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.

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.