Transcript No Slide Title

RELATIVE GAIN MEASURE OF INTERACTION
We have seen that interaction is important. It affects
whether feedback control is possible, and if possible, its
performance.
Do we have a quantitative measure of interaction?
The answer is yes, we have several! Here, we will learn
about the RELATIVE GAIN ARRAY.
Our main challenge is to understand the correct
interpretations of the RGA.
RELATIVE GAIN
We are here, and making progress all the time!
•
Defining control objectives
•
Robustness
•
Controllability & Observability
•
Integrity
•
Interaction & Operating window
•
Control for profit
•
The Relative Gain
•
Optimization-based design methods
•
Multiloop Tuning
•
•
Performance and the RDG
•
SVD and Process directionality
Process design
- Series and self-regulation
- Zeros (good/bad/ugly)
- Recycle systems
- Staged systems
RELATIVE GAIN MEASURE OF INTERACTION
OUTLINE OF THE PRESENTATION
Let’s start
here to build
understanding
1.
DEFINITION OF THE RGA
2.
EVALUATION OF THE RGA
3.
INTERPRETATION OF THE RGA
4.
EXTENSIONS OF RGA
5.
PRELIMINARY CONTROL DESIGN IMPLICATIONS
OF RGA
RELATIVE GAIN MEASURE OF INTERACTION
The relative gain
between MVj and
CVi is ij . It is
defined in the
following equation.
Explain in words.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant
MV1(s)
G11(s)
+
G21(s)
+
CV1(s)
Gd1(s)
D(s)
G12(s)
MV2(s)

G22(s)
Gd2(s)
+
  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
+
loops open
loops closed
CV2(s)
What have
we assumed
about the
other
controllers?
RELATIVE GAIN MEASURE OF INTERACTION
OUTLINE OF THE PRESENTATION
1.
DEFINITION OF THE RGA
Now, how do
we determine
the value?
2.
EVALUATION OF THE RGA
3.
INTERPRETATION OF THE RGA
4.
EXTENSIONS OF RGA
5.
PRELIMINARY CONTROL DESIGN IMPLICATIONS
OF RGA
RELATIVE GAIN MEASURE OF INTERACTION
1. The RGA can be
calculated from openloop gains (only).
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
CV   K MV 
  CV i

k ij  

 MV j

 MV i
Open-loop
MV   K 1 CV 
  MV i

kI ij  

 CV j

 CV j
Closed-loop
The relative gain array is the element-by-element product of K
with K-1. ( = product of ij elements, not normal matrix
multiplication)
  K

K 
1
T
 ij  k ij kI
ji

RELATIVE GAIN MEASURE OF INTERACTION
1. The RGA can be
calculated from openloop values.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
The relative gain array for a 2x2 system is given in the
following equation.
1
 11 
1
K 12 K 21
K 11 K 22
What is true for the RGA to have 1’s on diagonal?
RELATIVE GAIN MEASURE OF INTERACTION
2. The RGA elements
are scale independent.
What is the effect of changing the units
of the CV, expressing CV as % of
instrument range, or changing the
capacity of the final element on ij ?
Original units
 CV 1  10


 CV 2   9
Modified units
10   MV 1 


10   MV 2 
Can we prove that this
is general?
 CV 1   1


* 
 CV 2   . 09
MV or1
CV 1
10 MV 1
10
CV 2 or CV 2 / 10
9
MV 2
9
10
10   MV * 1 


1   MV 2 
RELATIVE GAIN MEASURE OF INTERACTION
3. The rows and
columns of the RGA
sum to 1.0.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant
MV 1
MV 2
CV 1
10
9
CV 2
9
10

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
For a 2x2 system, how many elements are independent?
RELATIVE GAIN MEASURE OF INTERACTION
3. The rows and
columns of the RGA
sum to 1.0.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
Class exercise: prove this statement.
Hint: a matrix and its inverse commute, i.e.,
K K-1 = K-1 K = I
loops open
loops closed
RELATIVE GAIN MEASURE OF INTERACTION
3. The rows and
columns of the RGA
sum to 1.0.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
K K-1 = I = K-1 K
From the left hand equation, the elements of I are equal to
n
k
n
ik kI kj

k 1
0 if i  j and 1 if i  j sum of col i 
k
n
im kI mi

m 1

ij
j1
From the right hand equation, the elements of I are equal to
n
 kI
k 1
n
ik k kj

0 if i  j and 1 if i  j sum of row j 
k
m 1
n
jm kI mj


i 1
ij
RELATIVE GAIN MEASURE OF INTERACTION
4. In some cases, the
RGA is very sensitive
to small errors in the
gains, Kij.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
1
 11 
1
K 12 K 21
K 11 K 22
When is this equation very sensitive to errors in the
individual gains?
RELATIVE GAIN MEASURE OF INTERACTION
4. In some cases, the
RGA is very sensitive
to small errors in the
gains, Kij.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant
We must perform a thorough study to ensure that
numerical derivatives are sufficiently accurate!
C hange in F D us ed in
finite diffe re nce for
der iv ative
 11 for a positive
c hange in F D
 11 for a neg ative
c hange in F D
2%
0.5%
0.2%
0.05%
.796
.673
.629
.605
.301
.508
.562
.588
A ver age  1 1 for
positive and neg ative
c hanges in F D
.548
.590
.596
.597
From McAvpy, 1983
The x must be sufficiently small (be careful about roundoff).
RELATIVE GAIN MEASURE OF INTERACTION
4. In some cases, the
RGA is very sensitive
to small errors in the
gains, Kij.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant
We must perform a through study to ensure that
numerical derivatives are sufficiently accurate!
C onv erge nc e
tole ranc e of
eq uatio ns (so m e o f
all e rrors sq uar ed )
-4
10
-6
10
-8
10
-1 0
10
-1 6
10
 11 for a positive
c hange in F D
 11 for a neg ative
c hange in F D
A ver age  1 1 for
positive and neg ative
c hanges in F D
-4.605
-.096
.556
.622
.629
8.080
1.068
.615
.568
.562
-.887
.503
.586
.595
.596
Average gains
from +/-
From McAvpy, 1983
The convergence tolerance must be sufficiently small.
RELATIVE GAIN MEASURE OF INTERACTION
  CV
i

  MV
j




 MV k  constant
5. The relative gain
 ij 
elements are
  CV 
i 

independent of the
  MV 
j 

CV
control design for the
“ij” inputs and outputs
being considered.
Solvent
k
 constant
Reactant
FS >> FR
AC
TC

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
RELATIVE GAIN MEASURE OF INTERACTION
6. A permutation in the
gain matrix (changing CVs
and MVs) results in the
same permutation in the RG
Array.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant
Process gain
 CV 1  10

 
 CV 2   9
10   MV 1 


10   MV 2 
 CV 2   9

 
 CV 1  10
10   MV 1 


10   MV 2 

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
RGA
MV 1
MV 2
CV 1
10
9
CV 2
9
10
??
loops open
loops closed
RELATIVE GAIN MEASURE OF INTERACTION
OUTLINE OF THE PRESENTATION
1.
DEFINITION OF THE RGA
How do we
use values to
evaluate
behavior?
2.
EVALUATION OF THE RGA
3.
INTERPRETATION OF THE RGA
4.
EXTENSIONS OF RGA
5.
PRELIMINARY CONTROL DESIGN IMPLICATIONS
OF RGA
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
ij < 0 In this case, the steady-state gains have different
signs depending on the status (auto/manual) of
the other loops.
A
Solvent
CA0
CSTR with
A B
A
CA
A
Discuss
interaction
and RGA in
this system.
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
ij < 0 In this case, the steady-state gains have different
signs depending on the status (auto/manual) of
other loops
We can achieve stable multiloop feedback by using the
sign of the controller gain that stabilizes the multiloop
system.
Discuss what happens when the other interacting loop is
placed in manual!
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
ij < 0 the steady-state gains have different signs
For ij < 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 other
loops are in automatic
Example of pairing on a negative RGA (-5.09). XB
controller has a Kc with opposite sign from single-loop
control! The system goes unstable when a constraint is
encountered. But, we can achieve stable control with
pairing on negative RGA!
FR  XB
IAE = 0.3338 ISE = 0.0012881
IAE = 0.58326 ISE = 0.0041497
0.98
Reboiled Vapor
FV
0.985
0.975
0
XD
FR
0.99
100
200
300
400
0.02
0.015
0.01
0.005
0
13.8
9
13.7
8.9
13.6
13.5
13.4
13.3
0
XB
0.025
Reflux Flow
XD, Distillate Lt Key
FV  XD
0.03
XB, Bottoms Lt Key
0.995
100
200
300
400
8.8
8.7
8.6
100
200
300
Time
400
500
8.5
0
100
200
300
Time
400
500
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
ij < 0 the steady-state gains have different signs
For ij < 0 , one of three situations occurs
1. The process gij(s) has a RHP zero
2. The overall plant has a RHP zero
3. The system with gij(s) removed has a RHP zero
See Skogestad and Postlethwaite, 1996
loops open
loops closed
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
ij = 0 In this case, the steady-state gain is zero when all
other loops are open, in manual.
Heating tank
without boiling
T
L
Could this control
system work?
What would happen if
one controller were in
manual?
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
0<ij<1 In this case, the multiloop (ML) steady-state
gain is larger than the single-loop (SL) gain.
What would be the effect on tuning of opening/closing the
other loop?
Discuss the case of a 2x2 system paired on ij = 0.1
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
ij= 1 In this case, the steady-state gains are identical in
both the ML and the SL conditions.
MV1(s)
What is generally
true when ij= 1 ?
G11(s)
+
G21(s)
+
CV1(s)
Gd1(s)
D(s)
Does ij= 1 indicate
no interaction?
G12(s)
MV2(s)
G22(s)
Gd2(s)
+
+
CV2(s)
RELATIVE GAIN MEASURE OF INTERACTION
ij= 1 In this case, the
steady-state gains are
identical in both the ML
and the SL conditions.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
Solvent
CSTR with
zero heat
of reaction
Reactant
FS >> FR
AC
TC
Determine the
relative gain.
Discuss
interaction in
this system.
RELATIVE GAIN MEASURE OF INTERACTION
ij= 1 In this case, the
steady-state gains are
identical in both the ML
and the SL conditions.
Diagonal gain
matrix
Lower
diagonal gain
matrix
 k 11


K  




0
k 22
..
0
 k 11

k
 21
K   ..

 ..
k
 n1
..
0
k 22
..
..
..
..
..






.. 






.. 
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
Both give an RGA
that is diagonal!
1


RGA  




0
1
1
0
1



 I


1 
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
1<ij In this case, the steady-state multiloop (ML) gain is
smaller than the single-loop (SL) gain.
What would be the effect on tuning of opening/closing
the other loop?
Discuss a the case of a 2x2 system paired on ij = 10.
RELATIVE GAIN MEASURE OF INTERACTION
FR  XD
FRB  XB
FD  XD
Rel. vol = 1.2, R = 1.2 Rmin
XD, XB
.998,.02
.998,.02
.998,.02
.98, .02
.98, .02
.98, .02
.98, .002
.98, .002
.98, .002
F eed
C om p.
.25
.50
.75
.25
.50
.75
.25
.50
.75
RGA
RGA
46.4
45.4
66.5
36.5
30.8
37.8
66.1
46
48.8
.07
.113
.233
.344
.5
.65
.787
.887
.939
FRB  XB
From McAvpy, 1983
1.
2.
Do level loops affect the composition RGA’s?
Does the process operation affect RGA’s?
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
ij=  In this case, the gain in the ML situation is zero.
We conclude that ML control is not possible.
Have we seen this result before?
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
Let’s extend the
concept
3.
INTERPRETATION OF THE RGA
4.
EXTENSIONS OF RGA
5.
PRELIMINARY CONTROL DESIGN IMPLICATIONS
OF RGA
RELATIVE GAIN MEASURE OF INTERACTION
The relative gain between
MVj and CVi is ij .
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant
The basic definition involves steady-state gain information.
•
Some plants are unstable
•
Control performance is influenced by dynamics
•
Many plants have an unequal number of MVs and CVs
•
Control design involves structures other than single-loop
•
Disturbances are not considered!
RELATIVE GAIN MEASURE OF INTERACTION
We can evaluate the
RGA of a system with
integrating processes,
such as levels.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
Redefine the output as the derivative of the level; then, calculate as
normal. (Note that L is unstable, but dL/dt is stable.)
m1
m2
 = density
L
A
D = density
A
dL
dt
D 
 A   m 1  m 2  F out
m1 
m1  m 2
 de nsity of slurry
RELATIVE GAIN MEASURE OF INTERACTION
We can evaluate the
RGA of a system with
integrating processes,
such as levels.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
Redefine the output as the derivative of the level; then,
calculate as normal.
m1
m2
 = density
m1
m2
L

D/
(1  D /  )
A
D
(1  D /  )
D/
D = density
RELATIVE GAIN MEASURE OF INTERACTION
We can evaluate the
RGA of dynamics
processes
FR  XD
A frequency-dependent RGA can be
calculated using the transfer
functions in place of the steady-state
gains.
FRB  XB
MV1(s)
G11(s)
+
G21(s)
+
CV1(s)
Gd1(s)
D(s)
G12(s)
MV2(s)
G22(s)
Gd2(s)
+
+
CV2(s)
Bode plots of the individual transfer functions for a
distillation tower
0
amplitude, XD(jw)/FV(jw)
amplitude, XD(jw)/FR(jw)
0
10
-5
10
-4
10
-2
0
10
10
frequency, radians/min
10
-4
10
0
-2
10
0
10
2
10
0
amplitude, XB(jw)/FV(jw)
amplitude, XB(jw)/FR(jw)
-5
10
2
10
-2
10
-4
10
-4
10
10
-2
10
0
10
2
10
10
-2
10
-4
10
-4
10
-2
10
0
10
2
10
Bode plot of the RGA 11 element. What frequency range is
most important for feedback control?
frequency dependent RGA for distillation tower
1
amplitude ratio
10
0
10
-1
10
-3
10
-2
10
-1
0
10
10
frequency, rad/time
1
10
2
10
RELATIVE GAIN MEASURE OF INTERACTION
The basic definition involves steady-state gain information.
•
Some plants are unstable
•
Control performance is influenced by dynamics
•
Many plants have an unequal number of MVs and CVs
•
Control design involves structures other than single-loop
•
Disturbances are not considered!
Apparently, there is a lot more to learn.
We better plan to address these issues in the
remainder of the course
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.
EXTENSIONS OF RGA
5.
PRELIMINARY CONTROL DESIGN IMPLICATIONS
OF RGA
Let’s evaluate
some design
guidelines based
on RGA
RELATIVE GAIN MEASURE OF INTERACTION
Proposed Guideline #1
Select pairings that do
not have any ij<0
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
• 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
Proposed Guideline #2
Select pairings that do
not have any ij=0
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
• 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
RGA and INTEGRITY
•
We conclude that the RGA provides excellent insight
into the INTEGRITY of a multiloop control system.
•
INTEGRITY: A multiloop control system has good
integrity when after one loop is turned off, the
remainder of the control system remains stable.
•
“Turning off” can occur when (1) a loop is placed in
manual, (2) a valve saturates, or (3) a lower level
cascade controller no lower changes the valve (in
manual or reached set point limit).
•
Pairings with negative or zero RGA’s have poor
integrity
RELATIVE GAIN MEASURE OF INTERACTION
Proposed Guideline #3
Select a pairing that has
RGA elements as close as
possible to ij=1
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
• 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?
For set point response,
RGA closer to 1.0 is better
FR  XD
FD  XD
FRB  XB
FRB  XB
RGA = 6.09
RGA = 0.39
IAE = 0.25454 ISE = 0.0004554
IAE = 0.045707 ISE = 8.4564e-005
IAE = 0.059056 ISE = 0.00017124
0.986
0.023
0.986
0.022
0.982
0.98
0.022
0.98
0.02
0
50
100
150
200
0
SAM = 0.31512 SSM = 0.011905
50
100
150
200
SAM = 0.28826 SSM = 0.00064734
8.6
50
100
150
200
13.8
13.7
50
100
Time
150
200
8.5
8.46
0
50
100
Time
150
200
100
150
200
14
8.48
13.5
50
13.9
13.8
13.7
13.6
13.6
0
0
SAM = 0.55128 SSM = 0.017408
8.52
Reflux flow
Reboiled vapor
8.7
0.019
0
8.54
13.9
8.8
0.02
SAM = 0.10303 SSM = 0.0093095
14
8.9
0.021
0.982
0.021
9
8.5
0.984
Reboiled vapor
0.984
XB, light key
0.023
XD, light key
0.988
XB, light key
0.024
Reflux flow
XD, light key
IAE = 0.26687 ISE = 0.00052456
0.988
0
50
100
Time
150
200
13.5
0
50
100
Time
150
200
For feed composition disturbance response,
RGA farther from 1.0 is better
FR  XD
FD  XD
FRB  XB
FRB  XB
RGA = 6.09
RGA = 0.39
IAE = 0.14463 ISE = 0.00051677
IAE = 0.45265 ISE = 0.0070806
IAE = 0.32334 ISE = 0.0038309
0.025
0.015
0.01
100
150
0
200
0
100
150
200
8.55
50
100
Time
150
0
200
50
100
150
0
200
14
13.5
8.5
13.5
13.4
13.3
8.4
8.3
8.2
8.1
8
0
50
100
Time
150
200
50
100
150
200
SAM = 4.0285 SSM = 0.6871
8.6
13.1
0
SAM = 0.51504 SSM = 0.011985
13.6
13.2
0
0.95
Reflux flow
Reboiled vapor
Reflux flow
8.6
8.5
50
SAM = 0.38988 SSM = 0.0085339
8.65
0.01
0.005
SAM = 0.21116 SSM = 0.0020517
8.7
0.02
0.015
Reboiled vapor
50
0.97
0.96
0.005
0
XB, light key
0.98
XD, light key
XB, light key
XD, light key
0.975
0.03
0.025
0.02
0.98
IAE = 0.31352 ISE = 0.0027774
0.99
13
12.5
12
11.5
0
50
100
Time
150
200
11
0
50
100
Time
150
200
RELATIVE GAIN MEASURE OF INTERACTION
Using guidelines #1 and #2, the control possibilities for this
example process were reduced from 36 to 4.
RELATIVE GAIN MEASURE OF INTERACTION
The RGA gives useful conclusions from S-S information, but
not enough to design process control
• Tells us about the integrity of multiloop
systems and something about the
differences in tuning as well.
• Uses only gains from feedback process!
• Does not use following information
- Control objectives
- Dynamics
- Disturbances
• Lower diagonal gain matrix can have
strong interaction but gives RGAs = 1
Powerful results
from limited
information!
Can we design
controls without
this information?
“Interaction?”
INTERACTION IN FEEDBACK SYSTEMS
Workshop on Relative Gain Array
Workshop on Relative Gain Array: Problem 1
The RGA has been evaluated, but the regulatory control
system (below the loops being analyzed using RGA) has been
modified. Instead of adjusting a valve directly, one of the
loops being evaluated will adjust a flow controller set point
(which adjusts the same valve). How would you evaluate the
new RGA?
Workshop on Relative Gain Array: Problem 2
You have decided to pair on a loop that has a negative RGA
element. Discuss the tuning that is appropriate for this loop.
A
Solvent
CA0
CSTR with
A B
A
CA
A
Workshop on Relative Gain Array: Problem 3
Discuss what information can be obtained from the RGA and
some information that cannot.
 ij 
  CV
i

  MV
j




 MV k  constant
  CV
i

  MV
j




 CV k  constant

  CV
i

  MV
j




 other
  CV
i

  MV
j




 other
loops open
loops closed
Workshop on Relative Gain Array: Problem 4
You would like to evaluate the
steady-state RGA for a
process, but the feedback
controllers must remain in
automatic status. How can
you obtain the needed data?
T o fla r e
PA H
PC -1
PV-3
P3
TA L
T5
LC -1
15
F7
14
dP-1
13
T6
A C -1
T1 0
•
Explain a procedure that
might yield the needed
information.
3
TC -7
2
dP-2
F4
1
LA H
LA L
LC -3
•
Discuss the practicality of the
approach.
F9
A C -2
F8
Workshop on Relative Gain Array: Problem 5
Determine the relative gain array for the TC and VC controllers.
Discuss the behavior of this design and generalize the
conclusions.
To the SP of the feed flow
FC
MV from the VC controller
LC
TC
SP = 95% open
VC
MV from the TC controller
CV to the VC controller
RGA WORKSHOP PROBLEM 6
Three CSTR's with the configuration in the figure and with the design parameters below are
considered in this example; the common data is given below, and the case-specific data and
steady-states are given in the table.
F=1 M3 , V=1 M3 , CA0=2.0 kg-moles/M3, Cp=1 cal/(g C), =106 g/M3 , ko = 1.0x1010
min-1 , E/R = 8330.1 K-1 , (Fc)s=15 M3/min , Cpc=1 cal/(g K) , c=106 g/M3 , b=0.5
Case
I (Example 3.10)
II
III
-?Hrxn 106 cal/(kg-mole)
130
13
-30
a (cal/min)/? K
1.678x106
1.678x106
0.7746x106
T0 ? K
323
370
370
Tcin ? K
365
365
420 (heating)
Ts ? K
394
368.3
392.7
CAs kg-mole/M3
0.265
0.80
0.28
Gain matrices for the three Cases, about the appropriate steady-state
Input
variable
Case I
CA
Case II
T
CA
Case III
T
CA
T
CA0
- 0.161
23.8
0.3615
1.309
0.2214
-6.267
Fc
0.0158
-1.28
.0034
-0.1144
-0.0085
0.657
T0
-.0026
0.211
-.0049
0.1678
-.0032
0.243
F
-.0948
26.33
0.4251
1.864
.4527
-16.30
RGA WORKSHOP. PROBLEM 6 (continued)
A. Calculate the relative gain arrays for the three cases shown in the table below.
CASE I
CA0
FC
CASE II
CASE III
CA0
CA0
FC
FC
CA
T
Notes:
1. CA0 is controlled by adjusting the reactant valve
2. FC is achieved by adjusting the valve on the pipe to the heat exchanger coils.
3. For Cases I and II, the coils provide coling, and for Case III, the coils provide heating.
4. The feed total flow and temperature controllers are in operation for Cases I-III
B. Determine conclusions for control design for each case. Explain each result on physical
grounds.
C. Discuss relationships among the cases.