|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| '''Realization''', in the [[system theory]] context refers to a [[State space (controls)|state space]] model implementing a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of ([[time-variant system|time-varying]]) [[Matrix (mathematics)|matrices]] <math>[A(t),B(t),C(t),D(t)]</math> such that | | The writer is known as Araceli Gulledge. Her friends say it's not great for her but what she enjoys doing is flower arranging and she is attempting to make it a profession. I am a manufacturing and distribution officer. Delaware is our beginning place.<br><br>My website :: extended car warranty ([http://Glskating.com/groups/great-ideas-about-auto-repair-that-you-can-use/ get redirected here]) |
| : <math>\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t)</math> | |
| : <math>\mathbf{y}(t) = C(t) \mathbf{x}(t) + D(t) \mathbf{u}(t)</math> | |
| with <math>(u(t),y(t))</math> describing the input and output of the system at time <math>t</math>.
| |
| | |
| ==LTI System==
| |
| For a [[linear time-invariant system]] specified by a [[transfer function|transfer matrix]], <math> H(s) </math>, a realization is any quadruple of matrices <math> (A,B,C,D) </math> such that <math> 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> H(s) = \frac{n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{s^{4} + d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}</math>.
| |
| | |
| The coefficients can now be inserted directly into the state-space model by the following approach:
| |
| :<math>\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)</math>
| |
| | |
| :<math> \textbf{y}(t) = \begin{bmatrix} n_{1}& n_{2}& n_{3}& n_{4} \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 [[Controllability|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>\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)</math>
| |
| | |
| :<math> \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 [[Observability|observable]] (i.e., because the output exits from a chain of integrators, every state has an effect on the output).
| |
| | |
| ==General System==
| |
| ===<math>D = 0</math>===
| |
| If we have an input <math>u(t)</math>, an output <math>y(t)</math>, and a [[weighting pattern]] <math>T(t,\sigma)</math> then a realization is any triple of matrices <math>[A(t),B(t),C(t)]</math> such that <math>T(t,\sigma) = C(t) \phi(t,\sigma) B(\sigma)</math> where <math>\phi</math> is the [[state-transition matrix]] associated with the realization.<ref>{{cite book|first=Roger W.|last=Brockett|title=Finite Dimensional Linear Systems|publisher=John Wiley & Sons|year=1970|isbn=978-0-471-10585-5}}</ref>
| |
| | |
| ==System identification==
| |
| {{main|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.
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| [[Category:Models of computation]]
| |
| [[Category:Systems theory]]
| |
The writer is known as Araceli Gulledge. Her friends say it's not great for her but what she enjoys doing is flower arranging and she is attempting to make it a profession. I am a manufacturing and distribution officer. Delaware is our beginning place.
My website :: extended car warranty (get redirected here)