relative gain measure of interaction · relative gain measure of interaction we have seen that...

RELATIVE GAIN MEASURE OF INTERACTION

We have seen that interaction is important. It affectswhether feedback control is possible, and if possible, itsperformance.

Do we have a quantitative measure of interaction?

The answer is yes, we have several! Here, we will learnabout the RELATIVE GAIN ARRAY.

Our main challenge is to understand the correct interpretations of the RGA.


OUTLINE OF THE PRESENTATION

1. DEFINITION OF THE RGA

2. EVALUATION OF THE RGA

3. INTERPRETATION OF THE RGA

4. PRELIMINARY CONTROL DESIGNIMPLICATIONS OF RGA

Let’s start here to build understanding

RELATIVE GAIN MEASURE OF INTERACTION+ +

+ +

G11(s)

G21(s)

G12(s)

G22(s)

Gd2(s)

Gd1(s)D(s)

CV1(s)

CV2(s)MV2(s)

MV1(s)The relative gainbetween MVj andCVi is λλλλij . It givenin the followingequation.

Explain in words.

closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ Assumesthatloopshave anintegralmode







Now, how do we determine

the value?


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ

[ ] [ ][ ]

[ ] [ ] [ ]

∂∂==

∂∂==

−

ji

ji

CVMVCVKMV

MVCVMVKCV

ij

ij

kI

k

1

The relative gain array is the element-by-elementproduct of K with K-1

( ) ( )( )jiijij kIkKK ==Λ −⊗ λ T

1

1. The RGA can becalculated from open-loop values.


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ

1. The RGA can becalculated from open-loop values.

The relative gain array for a 2x2 system is given in thefollowing equation.

2211

211211

1

1

KKKK−

=λ

What is true for the RGA to have 1’s on diagonal?


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ

2. The RGA elementsare scale independent.

=

=

2

1

2

1

2

1

2

1

109101

1091010

MVMV

CVCV

MVMV

CVCV *

. *

Original units Modified units

10910910

10

22

1

21

1

−−

/or

or

CVCVCV

MVMV

MVChanging the units ofthe CV or the capacityof the valve does notchange λλλλij .


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ

3. The rows andcolumns of the RGAsum to 1.0.

109910

2

1

21

−−

CVCV

MVMV

For a 2x2 system, how many elements are independent?


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ

4. In some cases, theRGA is very sensitiveto small errors in thegains, Kij.

2211

211211

1

1

KKKK−

=λ

When is this equation very sensitive to errors in theindividual gains?


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ

5. We can evaluate theRGA of a system withintegrating processes,such as levels.

Redefine the output as the derivative of the level; then,calculate as normal.

outFmmAdtdLA −+== 21ε

m2m1m

L

A

ρ = density

D = density







How do weuse values to

evaluate behavior?


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλMVj →→→→ CVi

λλλλij < 0 In this case, the steady-state gains have differentsigns depending on the status (auto/manual) ofother loops

A

A

CA0

CA

A →→→→ BSolvent

A

Discussinteraction inthis system.


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k


λλλλij < 0 In this case, the steady-state gains have differentsigns depending on the status (auto/manual) ofother loops

We can achieve stable multiloop feedback by using thesign of the controller gain that stabilizes the multiloopsystem.

Discuss what happens when the other interacting loop isplaced in manual!


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k


λλλλij < 0 the steady-state gains have different signs

For λλλλij < 0 , one of three BAD situations occurs

1. Multiloop is unstable with all in automatic.

2. Single-loop ij is unstable when others are in manual.

3. Multiloop is unstable when loop ij is manual and otherloops are in automatic

0 100 200 300 4000.975

0.98

0.985

0.99

0.995IAE = 0.3338 IS E = 0.0012881

XD

, Dis

tilla

te L

t Key

0 100 200 300 4000.005

0.01

0.015

0.02

0.025

0.03IAE = 0.58326 IS E = 0.0041497

XB

, Bot

tom

s Lt

Key

0 100 200 300 400 50013.3

13.4

13.5

13.6

13.7

13.8

Time

Reb

oile

d V

apor

0 100 200 300 400 5008.5

8.6

8.7

8.8

8.9

9

Time

Ref

lux

Flow

FR →→→→ XB

FRB →→→→ XD

Example of pairing on a negative RGA (-5.09). XBcontroller has a Kc with opposite sign from single-loopcontrol! The system goes unstable when a constraint isencountered.


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k


λλλλij = 0 In this case, the steady-state gain is zero when allother loops are open, in manual.

T

L

Could this controlsystem work?

What would happen ifone controller were inmanual?


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k


0<λλλλij<1 In this case, the steady-state (ML) gain is largerthan the SL gain.

What would be the effect on tuning of opening/closing theother loop?

Discuss the case of a 2x2 system paired on λλλλij = 0.1


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k


λλλλij= 1 In this case, the steady-state gains are identical inboth the ML and the SL conditions.

+ +

+ +

G11(s)

G21(s)

G12(s)

G22(s)

Gd2(s)

Gd1(s)D(s)

CV1(s)

CV2(s)MV2(s)

MV1(s)

What is generallytrue when λλλλij= 1 ?

Does λλλλij= 1 indicateno interaction?


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ

λλλλij= 1 In this case, thesteady-state gains areidentical in both theML and the SLconditions.

AC

TC

Zero heatof reaction

Solvent

Reactant

FS >> FR

Calculate therelative gain.

Discussinteraction inthis system.


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ

λλλλij= 1 In this case, thesteady-state gains areidentical in both theML and the SLconditions.

=

....

..k

k

K22

11

=

........k....

....kk

k

K

n1

2221

11

0

0

0

IRGA =

=

11

11

1 0

0Lowerdiagonal gainmatrix

Diagonal gainmatrixDiagonal gainmatrix

Both give an RGAthat is diagonal!


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k


1<λλλλij In this case, the steady-state (ML) gain is largerthan the SL gain.

What would be the effect on tuning of opening/closingthe other loop?

Discuss a the case of a 2x2 system paired on λλλλij = 10.


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k


λλλλij= ∞∞∞∞ In this case, the gain in the ML situation is zero.We conclude that ML control is not possible.

How can we improve the situation?







Let’s evaluatesome design

guidelines basedon RGA


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ

Proposed Guideline #1

Select pairings that donot have any λλλλij<0

• Review the interpretation, i.e., the effect on behavior.

• What would be the effect if the rule were violated?

• Do you agree with the Proposed Guideline?


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ


Select pairings that donot have any λλλλij=0





• We conclude that the RGA provides excellent insightinto the INTEGRITY of a multiloop control system.

• INTEGRITY: A multiloop control system has goodintegrity when after one loop is turned off, theremainder of the control system remains stable.

• “Turning off” can occur when (1) a loop is placed inmanual, (2) a valve saturates, or (3) a lower levelcascade controller no lower changes the valve (inmanual or reached set point limit).

• Pairings with negative or zero RGA’s have poorintegrity

RGA and INTEGRITY


closed loopsother

open loopsother

constant

constant

=

=

=

=

j

i

j

i

CVj

i

MVj

i

ij

MVCV

MVCV

MVCV

MVCV

k

kλ


Select a pairing that hasRGA elements as close aspossible to λλλλij=1




FR →→→→ XD

FRB →→→→ XB

FD →→→→ XD

FRB →→→→ XB

0 50 100 150 2000.98

0.982

0.984

0.986

0.988IAE = 0.26687 IS E = 0.00052456

XD

, lig

ht k

ey

0 50 100 150 2000.02

0.021

0.022

0.023

0.024IAE = 0.25454 IS E = 0.0004554

XB

, lig

ht k

ey

0 50 100 150 2008.5

8.6

8.7

8.8

8.9

9S AM = 0.31512 S S M = 0.011905

Time

Ref

lux

flow

0 50 100 150 20013.5

13.6

13.7

13.8

13.9

14S AM = 0.28826 S S M = 0.00064734

Time

Reb

oile

d va

por

0 50 100 150 2000.98

0.982

0.984

0.986

0.988IAE = 0.059056 IS E = 0.00017124

XD

, lig

ht k

ey

0 50 100 150 2000.019

0.02

0.021

0.022

0.023IAE = 0.045707 IS E = 8.4564e-005

XB

, lig

ht k

ey

0 50 100 150 2008.46

8.48

8.5

8.52

8.54S AM = 0.10303 S S M = 0.0093095

Time

Ref

lux

flow

0 50 100 150 20013.5

13.6

13.7

13.8

13.9

14S AM = 0.55128 S S M = 0.017408

Time

Reb

oile

d va

por

RGA = 6.09 RGA = 0.39

For set point response, RGA closer to 1.0 is better

FR →→→→ XD

FRB →→→→ XBFD →→→→ XD

FRB →→→→ XB

0 50 100 150 200

0.975

0.98

IAE = 0.14463 IS E = 0.00051677

XD

, lig

ht k

ey

0 50 100 150 2000

0.005

0.01

0.015

0.02

0.025IAE = 0.32334 IS E = 0.0038309

XB

, lig

ht k

ey

0 50 100 150 2008.5

8.55

8.6

8.65

8.7S AM = 0.21116 SS M = 0.0020517

Time

Ref

lux

flow

0 50 100 150 20013.1

13.2

13.3

13.4

13.5

13.6SAM = 0.38988 S S M = 0.0085339

Time

Reb

oile

d va

por

RGA = 6.09 RGA = 0.39

0 50 100 150 2000.95

0.96

0.97

0.98

0.99IAE = 0.45265 ISE = 0.0070806

XD

, lig

ht k

ey

0 50 100 150 2000

0.005

0.01

0.015

0.02

0.025

0.03IAE = 0.31352 ISE = 0.0027774

XB

, lig

ht k

ey

0 50 100 150 2008

8.1

8.2

8.3

8.4

8.5

8.6S AM = 0.51504 SS M = 0.011985

Time

Ref

lux

flow

0 50 100 150 20011

11.5

12

12.5

13

13.5

14S AM = 4.0285 S SM = 0.6871

TimeR

eboi

led

vapo

r

For set point response, RGA farther from 1.0 is better


• Tells us about the integrity of multiloopsystems and something about thedifferences in tuning as well.

• Uses only gains from feedback process!

• Does not use following information- Control objectives- Dynamics- Disturbances

• Lower diagonal gain matrix can havestrong interaction but gives RGAs = 1

Can we designcontrols withoutthis information?

The RGA gives useful conclusions from S-S information

Powerful resultsfrom limitedinformation!

“Interaction?”

relative gain measure of interaction · relative gain measure of interaction we have seen that...

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