Re: Butter Filters

[A S Hodel]

State space realizations are not unique.  (Similarity transformations 
allow for a change
of the "invisible" state variables in a state space realization.  See 
Brogan's book, Modern
Control Theory, for details, or look at my lecture notes

        ftp://ftp.eng.auburn.edu/pub/hodel/7500/ControlsNotes.pdf

If the original system was created with tf2sys, Octave's state space 
realizations are
in controllable canonical form.  If made with zp2sys, then the state 
space realization
is in a in a quasi-modal form (not to be confused with the Victor Hugo 
novel character
of a similar name).

Example: tf form:
octave:6> sys = tf2sys([1 2 3],[4 5 6 7 8]); sysout(sys,"ss");
Input(s)
      1: u_1

Output(s):
     1: y_1

state-space form:
4 continuous states, 0 discrete states
State(s):
    1: x_1
    2: x_2
    3: x_3
    4: x_4

A matrix: 4 x 4
   0.00000   1.00000   0.00000   0.00000
   0.00000   0.00000   1.00000   0.00000
   0.00000   0.00000   0.00000   1.00000
  -2.00000  -1.75000  -1.50000  -1.25000
B matrix: 4 x 1
  0
  0
  0
  1
C matrix: 1 x 4
  0.75000  0.50000  0.25000  0.00000
D matrix: 1 x 1
0


Same example, but in zp form.  Here the A-matrix is in an upper 
triangular  form with 2x2 blocks corresponding to complex conjugate 
pairs:
octave:8> [zer,pol,k] = sys2zp(sys);
octave:9> sysout(zp2sys(zer,pol,k),"ss");
Input(s)
      1: u_1

Output(s):
     1: y_1

state-space form:
4 continuous states, 0 discrete states
State(s):
    1: x_1
    2: x_2
    3: x_3
    4: x_4

A matrix: 4 x 4
   0.00000   1.00000   0.00000   0.00000
  -1.42101  -1.93614   1.00000   0.00000
   0.00000   0.00000   0.00000   1.00000
   0.00000   0.00000  -1.40745   0.68614
B matrix: 4 x 1
  0
  0
  0
  1
C matrix: 1 x 4
  0.39475  0.01597  0.25000  0.00000
D matrix: 1 x 1
0


In both cases, you should get C * inv( s I - A ) * B to be identical.  
That's your transfer function.

On Monday, February 17, 2003, at 06:58 AM, Doug Stewart wrote:

> I have made the changes to butter.m, so that it now does the same thing
> as Matlab's butter.m
> I am in the testing phase, and I found that the state space model from
> Matlab is different than the SS model from Octave. I am a little weak 
> on
> SS theory but if I remember corectly SS models can be displayed in
> different ways.
> Is this true??  Would some one please clue me in so I can finish
> testing this, and donate it to the Octave group.
>
> I will do  cheby1, cheby2 and ellip after I get this tested.
> Doug Stewart
>
>
> Doug Stewart wrote:
>  _
>  _ Paul Kienzle wrote:
>  _
>  __ Doug Stewart wrote:
>  __
>  ___ Is there a M file to make laplace space filters rather than Z 
> space
>  ___ filters?
>  __
>  __
>  __
>  __ You just need to skip the bilinear command in butter.m.  How
>  __ about checking for a file arg of 's' on the command line and using
>  __ that to decide if you are going to call bilinear.  Patches for 
> butter,
>  __ cheby1, cheby2 and ellip will be much appreciated.
>  __
>  __ Paul Kienzle
>  __ address@hidden
>  _
>  _
>  _ I will gladly make these changes. Just thought that they might by 
> done
>  _ allready by someone else. If not I will do it.
>  _ Doug Stewart
>  _
>
>
>
>
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> -------------------------------------------------------------
A. S. Hodel, Assoc. Prof, Dept. Elect & Comp Eng, Auburn University, AL 
36849-5201
(334) 844-1854 200 Broun Hall address@hidden 
http://www.eng.auburn.edu/~scotte



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