Post on 19-May-2019
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.
RELATIVE GAIN MEASURE OF INTERACTION
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
RELATIVE GAIN MEASURE OF INTERACTION
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
Now, how do we determine
the value?
RELATIVE GAIN MEASURE OF INTERACTION
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.
RELATIVE GAIN MEASURE OF INTERACTION
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?
RELATIVE GAIN MEASURE OF INTERACTION
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 .
RELATIVE GAIN MEASURE OF INTERACTION
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?
RELATIVE GAIN MEASURE OF INTERACTION
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?
RELATIVE GAIN MEASURE OF INTERACTION
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
RELATIVE GAIN MEASURE OF INTERACTION
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
How do weuse values to
evaluate behavior?
RELATIVE GAIN MEASURE OF INTERACTION
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.
RELATIVE GAIN MEASURE OF INTERACTION
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
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!
RELATIVE GAIN MEASURE OF INTERACTION
closed loopsother
open loopsother
constant
constant
=
=
=
=
j
i
j
i
CVj
i
MVj
i
ij
MVCV
MVCV
MVCV
MVCV
k
kλMVj →→→→ CVi
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.
RELATIVE GAIN MEASURE OF INTERACTION
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 gain is zero when allother loops are open, in manual.
T
L
Could this controlsystem work?
What would happen ifone controller were inmanual?
RELATIVE GAIN MEASURE OF INTERACTION
closed loopsother
open loopsother
constant
constant
=
=
=
=
j
i
j
i
CVj
i
MVj
i
ij
MVCV
MVCV
MVCV
MVCV
k
kλMVj →→→→ CVi
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
RELATIVE GAIN MEASURE OF INTERACTION
closed loopsother
open loopsother
constant
constant
=
=
=
=
j
i
j
i
CVj
i
MVj
i
ij
MVCV
MVCV
MVCV
MVCV
k
kλMVj →→→→ CVi
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?
RELATIVE GAIN MEASURE OF 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.
RELATIVE GAIN MEASURE OF 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.
=
....
..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!
RELATIVE GAIN MEASURE OF INTERACTION
closed loopsother
open loopsother
constant
constant
=
=
=
=
j
i
j
i
CVj
i
MVj
i
ij
MVCV
MVCV
MVCV
MVCV
k
kλMVj →→→→ CVi
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.
RELATIVE GAIN MEASURE OF INTERACTION
closed loopsother
open loopsother
constant
constant
=
=
=
=
j
i
j
i
CVj
i
MVj
i
ij
MVCV
MVCV
MVCV
MVCV
k
kλMVj →→→→ CVi
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?
RELATIVE GAIN MEASURE OF INTERACTION
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 evaluatesome design
guidelines basedon RGA
RELATIVE GAIN MEASURE OF INTERACTION
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?
RELATIVE GAIN MEASURE OF INTERACTION
closed loopsother
open loopsother
constant
constant
=
=
=
=
j
i
j
i
CVj
i
MVj
i
ij
MVCV
MVCV
MVCV
MVCV
k
kλ
Proposed Guideline #2
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?
RELATIVE GAIN MEASURE OF INTERACTION
• 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
RELATIVE GAIN MEASURE OF INTERACTION
closed loopsother
open loopsother
constant
constant
=
=
=
=
j
i
j
i
CVj
i
MVj
i
ij
MVCV
MVCV
MVCV
MVCV
k
kλ
Proposed Guideline #3
Select a pairing that hasRGA elements as close aspossible to λλλλij=1
• 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?
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
RELATIVE GAIN MEASURE OF INTERACTION
• 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?”