RE: zeros

At 12:23 AM 11/12/2001 -0300, you wrote:
>Dear Dr. Skogestad:
>Thanks for your quick response.
>
>Regarding item 1) I simulated d(x)/d(t)=Ax for non-zero initial conditions
>with:
>
>ode45('model',[0 10], [1 1])
>
>where
>
> function [x]=model(t,x)
> x(1)=-1*x(1)+5*x(2);
> x(2)=0*x(1)-2*x(2);
>
>which resembles inverse response behaviour.

This is sort of an inverse response, but it is NOT related to a RHP-zero in
the transfer function, but rather to the
initial condition (or equivalently to a particular combination of
disturbances).

Specifically, f we let d1=x0 denote the disturbance then the transfer
function inv(sI-A)*B (where B=A) does not have
any RHP-zero in any of its elements.

However, if we create a "combined" disturbance d2=[1,1]*d (where d is a scalar)
then this transfer function has a RHP-zero at s=0.5 in the 1,1-element. To
show this use
the commands:

>> s = sym('s')

s =

s

>> A = [-1 5; 0 -2]

A =

     -1 5
      0 -2

>> Bd=A*[1; 1]

Bd =

      4
     -2

>> inv(s*I-A)*Bd

ans =

[ 4/(s+1)-10/(s+1)/(s+2)]
[ -2/(s+2)]

>> simplify(ans)

ans =

[ 2*(2*s-1)/(s+1)/(s+2)]
[ -2/(s+2)]

>>

>Regarding item 2) I used the following matlab command to evaluate the
>generalized eigenvalue problem:
>
>EIG Eigenvalues and eigenvectors.
> E = EIG(X) is a vector containing the eigenvalues of a square
> matrix X.
>
> [V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and a
> full matrix V whose columns are the corresponding eigenvectors so
> that X*V = V*D.
>
> [V,D] = EIG(X,'nobalance') performs the computation with balancing
> disabled, which sometimes gives more accurate results for certain
> problems with unusual scaling.
>
> E = EIG(A,B) is a vector containing the generalized eigenvalues
> of square matrices A and B.
>
> [V,D] = EIG(A,B) produces a diagonal matrix D of generalized
> eigenvalues and a full matrix V whose columns are the
> corresponding eigenvectors so that A*V = B*V*D.
>
> See also CONDEIG, EIGS.
>For eqn. (4.61), A=M and B=Ig.
>In order to solve exercise 4.7 (page 137) I applied EIG(M, Ig) and tried
>with different values of c1 until a change of sign in the only not
>undetermined generalized eigenvalue was observed for c1=1, which is the
>actual solution according to the the document of odd exercises solutions
>available in your website.

In this case there is only one solution with xz (the two first elements in
each column of V) nonzero. Thus the system has only one zero.

For example, let c1=10
>> M = [A B; C D]

M =

     10 0 1 0
      0 -1 0 1
     10 1 0 0
     10 0 0 1

>> Ig

Ig =

      1 0 0 0
      0 1 0 0
      0 0 0 0
      0 0 0 0

>> [V,E]=eig(M,Ig)

V =

          0 -0.1000 0.0000 0.0000
          0 1.0000 0 0
     1.0000 1.0000 1.0000 1.0000
          0 1.0000 -0.0000 0.0000

E =

   1.0e-015 *

        Inf 0 0 0
          0 0.1574 0 0
          0 0 Inf 0
          0 0 0 Inf

 From V we see that only the second column of V has a nonzero xz. Thus,
from E the system has a zero a s=o (=

>
>With kind regards.
>
>Aníbal M. BLANCO
>PLAPIQUI (UNS-CONICET)
><http://www.plapiqui.edu.ar/>http://www.plapiqui.edu.ar/
>>----- Original Message -----
>>From: <mailto:skoge@chembio.ntnu.no>Sigurd Skogestad
>>To: <mailto:ablanco@plapiqui.edu.ar>Anibal Blanco
>>Sent: Sunday, November 11, 2001 12:49 PM
>>Subject: Re: zeros
>>
>>Dear Anibal,
>>
>>Thank you for your email.
>>
>>I need some more information in order to help you.
>>Could you please supply me with the specific matlab commands you have
>>been using.
>>
>>Best regards,
>>Sigurd Skogestad
>>
>>At 10:50 AM 11/8/2001 -0300, you wrote:
>>>Dear Dr. Skogestad:
>>>I'm a PhD student interested in chemical processes dynamics and control.
>>>I've been reading your excellent book MULTIVARIABLE FEEDBACK CONTROL
>>>Analysis and Design; which covers far more material than traditional
>>>linear control text books.
>>>
>>>I'm particularly interested at the moment in zeros dynamics and I have a
>>>couple of doubts that you probably could clarify.
>>>
>>>1) The solution of the (2*2) system d(x)/d(t)=Ax for A=[-1 5 | 0 -2]
>>>grows for a while before decaying for a non zero initial condition (one
>>>of the states), resembling inverse response behaviour. Is this issue
>>>related to multivariable RHP zeros? If so, of what state space model?.
>>>
>>>2) In section 4.5.1 of your book it is established that zeros may be
>>>evaluated by solving (4.61) generalized eigenvalue problem. For the
>>>system of exercise 4.7, for example, generalized eigenvalue solution
>>>(matlab eig) drives to three undetermined and one real generalized
>>>eigenvalues. Are the determined eigenvalues the actual zeros of the system?
>>>
>>>Thanks a lot.
>>>Best regards.
>>>
>>>
>>>Aníbal M. BLANCO
>>>PLAPIQUI (UNS-CONICET)
>>><http://www.plapiqui.edu.ar/>http://www.plapiqui.edu.ar/
Received on Wed Nov 21 23:15:17 2001