Rga
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Interpreting the RGA
What is the RGA?
• Relative Gain Array is an analytical tool used to determine the optimal input-output variable pairings for a multi-input-multi-output (MIMO) system.
Example of RGA Analysis: 2 x 2 system• Steady-state process model,
The RGA is defined as:
where the relative gain, ij, relates the ith controlled variable and the jth manipulated variable
2221212
2121111
UKUKY
UKUKY
2221
1211RGA
gain loop-closed
gain loop-openij
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Scaling Properties:
i) ij is dimensionless
ii)
For 2 x 2 system,
Recommended Controller Pairing
Corresponds to the ij which has the largest positive value.
0.1j
iji
ij
211112
2211
211211 1,
KKKK
1
1
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Significance of the RGA
• Below is a general relative gain array
Significance of the RGA
• Below is a general relative gain array• Different columns represent the different
manipulated (input) variables
Significance of the RGA
• Below is a general relative gain array• Different columns represent the different
manipulated (input) variables• Different rows represent the different controlled
(output) variables
Significance of the RGA
• The values in the array describe the relationship between the input and output variables
• Negative values indicate an unstable relationship
• A value of 0 indicates no relationship• A value of 1 indicates that specific input variable
is the only influence on that output variable
Example 1
• Assume a mixing tank with constant mass and two inputs as shown below:
• wA and wB are manipulated flowrates entering the tank
• w is the flowrate leaving the tank and xA is the concentration of A in the tank
Example from http://eweb.chemeng.ed.ac.uk/courses/control/restricted/course/advanced/casestudy/exercise2.html
Example 1 (cont.)
• This process can be modeled by the following equations:
w = wA + wB
xA = wA/(wA + wB)
• The RGA can be solved for this system:
Relative G ain A rrayx 1 x
1 x x
Example 1
Solution• What does the RGA tell us?
– If a concentration of xA=0.5 is desired, either wA or wB can be used
– If a concentration of xA>0.5 is desired, then the concentration loop should be paired with wA
– If a concentration of xA<0.5 is desired, then the concentration loop should be paired with wB
Example 2
Run R(kg/min) S(kg/min) xD xB
1 125 22 0.97 0.04
2 150 22 0.93 0.06
3 150 20 0.91 0.08
Adapted from http://eweb.chemeng.ed.ac.uk/courses/control/restricted/course/advanced/casestudy/exercise2.html
2211
211211
1
1
KK
KK
Relative G ain A rrayx 1 x
1 x x
Example 2
Solution• RGA Matrix
• Pairing of variables: RGA matrix value should be 1)positive, then 2)close to 1.
• Because each combination has only one positive value, that is the combination to be paired (R,xD and P,xB)
2 -1
-1 2
Example 3
• Suppose you calculate the following RGA matrix. How should pairing of the variables occur?
-.25 0 1.25
0.75 0.8 -0.55
0.5 0.2 0.3
Example 3
Solution• In the first row,
only x3 gives a positive result, and then we go with the closest values to 1 for the others.
• y1,x3
y2,x2
y3,x1
-.25 0 1.25
0.75 0.8 -0.55
0.5 0.2 0.3
Decoupling Control Systems
Basic Idea: Use additional controllers to compensate for processinteractions and thus reduce control loop interactions
Ideally, decoupling control allows setpoint changes to affect onlythe desired controlled variables.
Typically, decoupling controllers are designed using a simple process model (e.g. steady state model or transfer function model)
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Design Equations:
We want cross-controller, GC12, to cancel out the effect of U2 on Y1.Thus, we would like,
Since U2 0 (in general), then
Similarly, we want G21 to cancel the effect of M1 on C2. Thus, werequire that...
cf. with design equations for FF control based on block diagramanalysis
12 11 2 12 2 0P PT G U G U
1212
11
P
P
GT
G
21 22 1 21 1
2121
22
0P P
P
P
T G U G U
GT
G
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Process Interaction
Corrective Action (via “cross-controller” or “decoupler”).Ideal Decouplers:
1212
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2121
22
( )( )
( )
( )( )
( )
P
P
P
P
G sT s
G s
G sT s
G s
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