Realization (systems) - LTI System - Canonical Realizations

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:


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

dot{textbf{x}}(t) = begin{bmatrix} -d_{1}& -d_{2}& -d_{3}& -d_{4}\ 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)

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_{1}& 1& 0& 0\ -d_{2}& 0& 1& 0\ -d_{3}& 0& 0& 1\ -d_{4}& 0& 0& 0 end{bmatrix}textbf{x}(t) + begin{bmatrix} n_{1}\ n_{2}\ n_{3}\ n_{4} end{bmatrix}textbf{u}(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).

Read more about this topic:  Realization (systems), LTI System

Famous quotes containing the words realizations and/or canonical:

    These marbles, the works of the dreamers and idealists of old, live on, leading and pointing to good. They are the works of visionaries and dreamers, but they are realizations of soul, the representations of the ideal. They are grand, beautiful, and true, and they speak with a voice that echoes through the ages. Governments have changed; empires have fallen; nations have passed away; but these mute marbles remain—the oracles of time, the perfection of art.
    Herman Melville (1819–1891)

    If God bestowed immortality on every man then when he made him, and he made many to whom he never purposed to give his saving grace, what did his Lordship think that God gave any man immortality with purpose only to make him capable of immortal torments? It is a hard saying, and I think cannot piously be believed. I am sure it can never be proved by the canonical Scripture.
    Thomas Hobbes (1579–1688)