Page 352 - Design of Simple and Robust Process Plants

338 Chapter 8 Instrumentation, Automation of Operation and Control
6. Evaluate losses of the selected controlled variables, which are kept constant,
due to disturbances and implementation errors versus the optimal case.
7. Analyze and select the sets of CVs and MVs with acceptable losses as candi-
dates for self-optimizing control. These selected CVs and MVs will be subject
for further evaluation after interaction analysis.

8.4.8
Evaluate Static Interaction of the Selected Pairing Options
Steady-state analysis is the first step to evaluate the pre-selected pairings of CVs and
MVs on interaction of loops. This is performed by perturbations of the input of static
simulations. The functions to be determined are the process gains and the relative
gain analysis (RGA). The process or open loop static gain is the change of Dy i in
output y i relative to a change in Du i in input u j where all other inputs remain
constant

y i
u j
u m6ˆjˆ0
The other outputs may or may not change.
Another open loop gain is defined as the change Dy i in output y i relative to a
change in Du j in input u j where all other outputs remain constant Also, in this
case the other input may or may not change now

y i
u j
y 16ˆiˆ0
The ratio between these open loop gains defines the relative gain k ij between y i
and input u j
,

k ij = y i y i
u j u j
u m6ˆjˆ0 y 16ˆiˆ0
In words it is phrased as:
Process gain as seen by a given controller with all other loops open
Process gain as seen by a given controller with all other loops closed
When k ij = 0 then y i does not respond to u j and should not be used to control y i .
If k ij = 1 then y i does not respond to any other u m6ˆj and the loop is not affected by
other loops this is the preferred case If 0 < k ij <1or k ij > 1 then interaction is pres-
ent. The more that k ij deviates from 1, the larger the interaction.
Now, the level of interaction can be determined by putting the relative gains in an
array, called RGA which is a square matrix. For multiple inputs and outputs
u 1 u 2 ¼.u n
2 3
y 1
k 11 k 21 ::::
1n
y
K = 6 k 21 k 22 ::::k 2n 7 2
7
6
4 : 5
k nl :::::::::k nl y n