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

11

2121

22

( )( )

( )

( )( )

( )

P

P

P

P

G sT s

G s

G sT s

G s

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