From: "Vinay kariwala" <kariwala@ualberta.ca> 
To: "Sigurd Skogestad" <skoge@chemeng.ntnu.no> 
Subject: RE: Poster for AIChE 2003 
Date: Thu, 6 Nov 2003 12:13:03 -0700 


Dear Dr. Skogestad,


The ACC paper (2004) is attached.


SISO unstable plant with time delay:
By slightly rearranging the expression for a SISO system with
a single unstable pole, it can be shown that the system cannot be stabilized
without input saturation if T > (1/p) log_e (1/2p),

To derive this expression for admissible time delay for an unstable SISO
system, consider
G(s) = exp(-Ts)/(s-p), where the system has been scaled such that maximum
allowable value of input movement is from -1 to +1.

Then using Proposition 2 in this paper, ||KS||_inf = 2p exp(pT)

Since ||KS||_L1 >= ||KS||_inf, the system cannot be stabilized without input
saturation if ||KS||_inf > 1 or 2p exp(pT) > 1. Then, taking the natural
logarithm and rearranging this expression, we get

T > (1/p) log_e (1/2p)

which gives the maximum allowable value for time delay. Please let me know,
if you have any questions about this.


>
>Another fine point, which I realized while working on this poster (but have
>not put there due to space constraints) is that:
>No Branch and Bound method can be constructed to find the optimal
>combination of variables based on input usage minimization (infinity norm).
>If you look at the table of alternatives for the TE process in the poster,
>use of y22 is optimal when only single output is to be used to stabilize
all
>poles, but this output does not appear when use of more outputs is allowed.


Sounds reasonable that this can happen.


>My first guess is that this happens due to the angles between pole vectors,
>but I need to think more about it.
>
>Vinay Kariwala
>280D Dept of Chemical Engineering
>University of Alberta, Edmonton, Canada T6G2G6
>1-(780)-492-6268 (Phone) 1-(780)-492-2881 (Fax)
>Web: http://www.ualberta.ca/~kariwala
>