12thrhegwghntjrjkrjrhehrrnfhrrbegge720937.ppt

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1. Process Control: Designing Process and Control
Systems for Dynamic Performance
Copyright © Thomas Marlin 2016
The copyright holder provides a royalty-free license for use of this material at non-profit
educational institutions
Part IV:Relative Gain Array
Chapter 20. Multiloop Control

2. Chapter 20. MultiloopControl,
Part IV: Relative Gain Array
Instructor for this lesson
Dr. Tom Marlin
Course architect and instructor
McMaster University

3. CHAPTER 20: Multi-loop Control: Relative Gain Array
AC-1
AC-2
R
FV
Controller “off”
R
FV
AC-1
AC-2
Controller “on”

4. CHAPTER 20: Multi-loop Control: Relative Gain Array
Goals
•Use Relative Gain Array to quantify key aspects of
interaction
•Understand how strong interaction can lead to changes
in process gain
•Use the Relative Gain Array to determine key
performance characteristic –Integrity
•Use Relative Gain Array as one factor in control design
RGA = Relative Gain Array

5. •Definition of the RGA
•Key RGA mathematical properties
•Evaluation of RGA for process systems
•Interpretation of relative gain values
•Control design guidelines based on RGA
•Workshop
CHAPTER 20: Multi-loop Control: Relative Gain Array
OUTLINE

6. CHAPTER 20: Multi-loop Control: Relative Gain Array
Let’s see how interaction can “flip the sign” of a process gain
+
+ +
-
+
G
c2(s)
G
11(s)
G
21(s)
G
12(s)
G
22(s)
CV
1(s)
CV
2(s)
MV
2(s)
MV
1(s)
SP
2(s)
“direct path”
“interaction path”
CV
1(s)= CV
D(s)+ CV
I(S)
CV
D(s)
CV
I(S)
When|CV
D(s)| < |CV
I(S)|and CV
D(s)/CV
I(S) < 0 , the sign of the process
gain changes

7. CHAPTER 20: Multi-loop Control: Relative Gain Array
What condition can “flip the sign” of a process gain?
+
+ +
-
+
G
c2(s)
G
11(s)
G
21(s)
G
12(s)
G
22(s)
CV
2(s)
MV
2(s)
MV
1(s)
SP
2(s)
“direct path”
“interaction path”
CV
1(s) = CV
D(s)+ CV
I(S)
CV
D(s)
CV
I(S)
We will limit our analysis to the steady-state behavior.
•K
P1= CV
1/ MV
1
•K
P1* K
C1> 0 for negative (stabilizing) feedback control
•When the following conditions hold,
|CV
D(s)| < |CV
I(S)|and CV
D(s)/CV
I(S) < 0
The sign of K
P1changes depending on whether the other control loop
is “on”. The sign of K
C1is unchanged, so when the G
C1control is on,
the system becomes unstable!

8. CHAPTER 20: Multi-loop Control: Relative Gain Array
Integrity and interaction –a key property.
+
+ +
-
+
G
c2(s)
G
11(s)
G
21(s)
G
12(s)
G
22(s)
CV
2(s)
MV
2(s)
MV
1(s)
SP
2(s)
“direct path”
“interaction path”
CV
1(s) = CV
D(s)+ CV
I(S)
CV
D(s)
CV
I(S)
When the following conditions hold,
|CV
D(s)| < |CV
I(S)|and CV
D(s)/CV
I(S) < 0
The system does not have integrity. (This is BAD.)
INTEGRITY: A control system has integrity when after one or more
loops is placed “off”, the remaining control system can be stable without
changing the signs of controllers that remain “on”.
How can we determine whether a system will have integrity?
The Relative Gain Array --RGA

9. The relative gain between MV
jand
CV
iis designated by 
ijand defined
below. The Arrayis the matrix
with elements 
ij.ps closedother loo
j
i
ps openother loo
j
i
constantCV
j
i
constantMV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
λ
k
k












































CHAPTER 20: Multi-loop Control: Relative Gain Array
Definition of Relative Gain.
=
∆????????????
??????
∆????????????
1
∆????????????
??????+∆????????????
??????
∆????????????
1
Other loops have integral modes, so they keep their CV’s
constant (at steady state).

10. •Definition of the RGA
•Key RGA mathematical properties
•Evaluation of RGA for process systems
•Interpretation of relative gain values
•Control design guidelines based on RGA
•Workshop
CHAPTER 20: Multi-loop Control: Relative Gain Array
OUTLINE

11. 

j
i
CV
j
i
MV
j
i
CV
MV
CVKMV
MV
CV
MVKCV



















ij
1
ij
kI
k The relative gain array is the element-by-element product of K with
K
-1
. (= product of ijelements, not normal matrix multiplication) 
jiijij kIkKK 

 
T
1
1. The RGA can be
calculated from open-
loop gains (only).
Open-loop: MV
independentclosed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

Closed-loop: CV
independent
Relative
Gain Array
CHAPTER 20: Multi-loop Control: Relative Gain Array

12. 1. The RGA can be
calculated from open-
loop values.
The relative gain array for a 2x2 system is given in the
following equation.2211
2112
11
1
1
KK
KK


What is true for the RGA to have 1’s on diagonal?closed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

CHAPTER 20: Multi-loop Control: Relative Gain Array

13. 2. The RGA elements are scale independent.





































2
1
2
1
2
1
2
1
109
101
109
1010
MV
MV
CV
CV
MV
MV
CV
CV *
.

*
Original units Modified units
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?109*
910
*
22
1
2
1
1


CVCV
CV
MV
MV
MV
or
or
CV
*
2= CV
2/10
MV
*
1= 10MV
1
CHAPTER 20: Multi-loop Control: Relative Gain Array

14. 3. The rows and columns of the RGA sum to 1.0.109
910
2
1
21


CV
CV
MVMV
For a 2x2 system, how many elements are independent?closed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

CHAPTER 20: Multi-loop Control: Relative Gain Array

15. 4. In some cases, the RGA is very sensitive to small errors
in the gains, K
ij.2211
2112
11
1
1
KK
KK


When is this equation very sensitive to errors in the
individual gains?closed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

CHAPTER 20: Multi-loop Control: Relative Gain Array2211
2112
KK
KK
When the term approaches 1.0

16. 5. A permutation in the gain matrix
(changing CVs and MVs) results in the
same permutation in the RG Array.closed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

109
1010
2
1
2
1



















MV
MV
CV
CV 109
910
2
1
21




CV
CV
MVMV
Process Gain Matrix RGA
1010
109
2
1
1
2



















MV
MV
CV
CV 910
109
1
2
21




CV
CV
MVMV
CHAPTER 20: Multi-loop Control: Relative Gain Array

17. •Definition of the RGA
•Key RGA mathematical properties
•Evaluation of RGA for process systems
•Interpretation of relative gain values
•Control design guidelines based on RGA
•Workshop
CHAPTER 20: Multi-loop Control: Relative Gain Array
OUTLINE

18. CHAPTER 20: Multi-loop Control: Relative Gain Array
The steady-state gains used to calculate the RGA can be
determined by several methods.
Analytical models Experimentation Numerical
derivatives from
flowsheetAMMASSAA
MSA
xFxvkxvk
Fvkvk


21
21 '
22
12
1'
12
12
2
21
)(
*
)(
*
'
'''
v
vkvk
kvk
v
vkvk
kvk
x
vkvkF
ssAs
sA
ssAs
AS
AM
SAM


















??????=
??????&#3627408460;
??????&#3627408485;
≈
∆&#3627408460;
∆&#3627408485;
Most models are
too complex for this
method
Experimental errors
likely give
unreliable answers
Check linearization
using +/-x ,
reduce x until
accuracy obtained.

19. Specific approach used to evaluate the RGA of a system
with integrating processes, such as levels.
Redefine the dependent variable as the derivative of the level; then,
calculate RGA as normal. (Note that L is unstable, but dL/dtis stable.)slurry of density



21
1
21
mm
m
D
FmmA
dt
dL
A
out


m
2
m
1m
L
A
= density
D = density
CHAPTER 20: Multi-loop Control: Relative Gain Array
Model

20. 

/)/(
)/(/
DDD
DD
mm


1
1
21 Specific approach used to evaluate the RGA of a system
with integrating processes, such as levels.
Redefine the dependent variable as the derivative of the level; then,
calculate RGA as normal. (Note that L is unstable, but dL/dtis stable.)
CHAPTER 20: Multi-loop Control: Relative Gain Array
m
2
m
1m
L
A
= density
D = density
Relative Gain Array

21. XD, XB Feed
Comp.
11 11
.998,.02 .25 46.4 .07
.998,.02 .50 45.4 .113
.998,.02 .75 66.5 .233
.98, .02 .25 36.5 .344
.98, .02 .50 30.8 .5
.98, .02 .75 37.8 .65
.98, .002 .25 66.1 .787
.98, .002 .50 46 .887
.98, .002 .75 48.8 .939
1. Other control loops (not in RGA) affect RGA!
2.Process operating conditions affects the RGA!
Rel. vol = 1.2, R = 1.2 R
min
Data from McAvoy, 1983
CHAPTER 20: Multi-loop Control: Relative Gain Array
&#3627408459;
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&#3627408459;
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&#3627408479;&#3627408466;&#3627408463;??????????????????
&#3627408459;
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=??????
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22. •Definition of the RGA
•Key RGA mathematical properties
•Evaluation of RGA for process systems
•Interpretation of relative gain values
•Control design guidelines based on RGA
•Workshop
CHAPTER 20: Multi-loop Control: Relative Gain Array
OUTLINE

23. CHAPTER 20: Multi-loop Control: Relative Gain Array

ij> 0 In this case, the multiloopsteady-state gain and the
single-loop gain have the same sign!
MV
jCV
iclosed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

K
ij
•The sign of K
ijdoes not depend on the “on/off” status of other
control loops in the plant.
•The sign of the controller gain, K
ci, for negative feedback does not
depend on the “on/off” status of other control loops in the plant.
•Because the magnitude of K
ijcan change significantly between
single and multiloop, there is no guarantee of stability or good
performance (without changing K
cimagnitude).

24. 0<
ij<1 In this case, the multiloop(ML) process steady-
state gain is larger than the single-loop (SL) gain.
•What would be the effect on dynamic performance of
closing the other loops?
The SL tuning requires a larger controller gain, K
ci. If
the controller is tuned well for SL, closing the other
loops (without changing K
ci) will result in aggressive
feedback control, perhaps even instability.
CHAPTER 20: Multi-loop Control: Relative Gain Array
MV
jCV
iclosed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

K
ij

25. 
ij= 1 In this case, the steady-state gains are identical in
both the ML and the SL conditions.
+ +
+ +
G
11(s)
G
21(s)
G
12(s)
G
22(s)
G
d2(s)
G
d1(s)
D(s)
CV
1(s)
CV
2(s)MV
2(s)
MV
1(s)
•What is generally true when 
ij= 1 ?
•Does 
ij= 1 indicate nointeraction?
CHAPTER 20: Multi-loop Control: Relative Gain Arrayclosed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k


26. AC
TC
CSTR with
zero heat of
reaction
Solvent
Reactant
F
S>> F
R
CHAPTER 20: Multi-loop Control: Relative Gain Array

ij= 1 In this case, the steady-state gains are identical in
both the ML and the SL conditions.
•Cooling flow affect the reaction rate
•Small reactant flow does not affect
temperature
??????=
&#3627408485;&#3627408486;
0&#3627408487;
Λ=
10
01

27. 
















..
..
..
k
k
K
22
11 
















........k
....
....
kk
k
K
n1
2221
11 0
0
0I

















1
1
1
1
1
0
0
Lower
diagonal gain
matrix
Diagonal gain
matrix
Diagonal gain
matrix
Both give an RGA
that is identity matrix!
CHAPTER 20: Multi-loop Control: Relative Gain Array

ij= 1 In this case, the steady-state gains are identical in
both the ML and the SL conditions.
An identity RGA does not mean no interaction, it signifies
that no “two-way” process interaction exists

28. 1<
ijIn this case, the steady-state multiloop (ML) gain is
smaller than the single-loop (SL) gain.
CHAPTER 20: Multi-loop Control: Relative Gain Array
MV
jCV
iclosed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

K
ij
•What would be the effect on dynamic performance of
closing the other loops?
The SL tuning requires a smaller controller gain, K
ci. If
the controller is tuned well for SL, closing the other
loops (without changing K
ci) will result very slow
feedback.
•We cannot increase the tuning for multiloopbecause of
stability issues. (See multilooptuning exercise.)

29. 
ij= 0 In this case, the steady-state gain is zero when all
other loops are open, in manual.
TC
LC
•Could this control system work? Yes
•What would happen if TC were “off”? There would be
no causal relationship between heating flow and level.
Heating tank
without boiling
CHAPTER 20: Multi-loop Control: Relative Gain Array
Sometimes called
“nested loops”

30. 
ij< 0 In this case, the steady-state gain has a different sign
depending on the status (auto/manual) of the other loops.
A2C
A1C
C
A0
C
A
CSTR with
A BSolvent
Pure A
•With A2C “off”, increasing pure A flow increases C
A0,
which increases C
A.
CHAPTER 20: Multi-loop Control: Relative Gain Array
F
S>> F
A
•With A2C “on”, increasing pure A flow does not affect
C
A0, which is controlled. The result is the total flow
rate increases, which decreases space time and C
A
K
p> 0
K
p< 0

31. 
ij= In this case, the gain in the ML situation is zero.
We conclude that ML control is not possible.
CHAPTER 20: Multi-loop Control: Relative Gain Array
Constant
speed
motor
Centrifugal
pump
F1
v
1
v
2
P
1 P
2
We have encountered this condition when we learned
about controllability. 
ij= is equivalent to an
uncontrollable system.
The level and temperature cannot be controlled by
adjusting v1 and v2.

32. •Definition of the RGA
•Key RGA mathematical properties
•Evaluation of RGA for process systems
•Interpretation of relative gain values
•Control design guidelines based on RGA
•Workshop
CHAPTER 20: Multi-loop Control: Relative Gain Array
OUTLINE

33. •“Turning off” can occur when (1) a loop
is placed in manual, (2) a valve saturates,
or (3) a lower level cascade controller no
longer changes the valve (in manual or
reached set point limit).
RGA and INTEGRITY
•We conclude that the RGAprovides important insight
into the INTEGRITY of a multiloopcontrol system.
•INTEGRITY: A stable multiloopcontrol system has
good integrity when after one loop is turned off, the
remainder of the control system remains stable.
Not
unusual
CHAPTER 20: Multi-loop Control: Relative Gain Array

34. Guideline #1: Select pairings that do not have any 
ij< 0closed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

CHAPTER 20: Multi-loop Control: Relative Gain Array
Integrity is an important property for a control system.
CV-MV pairings related to negative relative gain elements
have poor integrity.
Avoid negative relative gains

35. Guideline #2: Select pairings that do not have any 
ij=0closed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

CHAPTER 20: Multi-loop Control: Relative Gain Array
Turning one loop “off” should not affect the ability of the
remaining loops to perform correctly. A loop should not
lose controllability due to another loop being “off”.
Avoid zero relative gains
Caution: Exceptions shown in Chapter 22

36. Guideline #3: Select a pairing that has RGA elements as
close as possible to 
ij=1
With RGA elements near 1.0, the tuning of single-loop
and multiloopare nearly the same. Therefore, complex
real-time retuning depending on loop status is not
required.
Pairing one elements near 1.0 reduces “interaction”,
which sometimes (not always) contributes to good control
performance.
Cautions:
1. Dynamics is also important
2. Important exceptions shown in Chapter 21 based on disturbances
CHAPTER 20: Multi-loop Control: Relative Gain Array

37. Lot’s of improvement! But refer to the next
slide for a caution.
How did we do?
CHAPTER 20: Multi-loop Control, Relative Gain Array
•Use Relative Gain Array to quantify key aspects of
interaction
•Understand how strong interaction can lead to changes in
process gain
•Use the Relative Gain Array to determine key performance
characteristic –Integrity
•Use Relative Gain Array as one factor in control design

38. •Limited to (a) selected CV/MV (b) operating conditions, (c) design
on other loops implemented and external to RGA analysis
•RGA interpretation changes for unstable plants
•RGA based on steady-state information, while control
performance is influenced by dynamics
•Many plants have an unequal number of MVs and CVs
•Control design involves structures other than multiple single-loop
•Control performance is influenced by disturbances, which are not
considered in RGA!
Apparently, there is a lot moreto learn!
CHAPTER 20: Multi-loop Control: Relative Gain Array
Caution: There is more learning in multiloopsystems

39. •PC-EDUCATION WEB SITE
-Process Control Learning Support –Chapter 20
+ Quiz (with solutions)
+ Tutorials (with solutions)
•TEXTBOOK
-Chapter 20
LEARNING RESOURCES
CHAPTER 20: Multi-loop Control, MultiloopTuning
Bristol, E., "On a New Measure of Interaction for Multivariable Process Control,"
IEEE Trans. Auto. Control, AC-I1, 133-134 (1966).
McAvoy, T., Interaction Analysis, Instrument Society of America, Research
Triangle Park, NC, 19836.
Edgar Bristol invented the RGA

40. CITATIONS
None!
CHAPTER 20: Multi-loop Control, MultiloopTuning

41. •Definition of the RGA
•Key RGA mathematical properties
•Evaluation of RGA for process systems
•Interpretation of relative gain values
•Control design guidelines based on RGA
•Workshop
CHAPTER 20: Multi-loop Control: Relative Gain Array
OUTLINE

42. Workshops on Relative Gain Array
CHAPTER 20: Multi-loop Control: Relative Gain Array

43. TC
1
FC
1
A
1
T
2 F
QT
Q
T C
The effluent temperature and
composition for the two packed
bed reactors are controlled by
adjusting the interstagequench
flow rate and temperature. The
dynamics are given below.
Determine the relative gain
array for this process.































)(
)(
1870.0
654.0
1917.0
841.1
1092.0
746.0
1786.0
265.2
)(
)(
786.0445.0
538.2326.1
sT
sF
s
e
s
e
s
e
s
e
sC
sT
Q
Q
ss
ss
Workshop 1: Packed bed reactor
CHAPTER 20: Multi-loop Control: Relative Gain Array

44. 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?
CHAPTER 20: Multi-loop Control: Relative Gain Array
Workshop 2. Effect of cascade on RGA

45. Discuss what information can be obtained from the RGA
and some important information for control design that
cannot be obtained.closed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

CHAPTER 20: Multi-loop Control: Relative Gain Array
Workshop 3. Information from RGA

46. 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?
•Explain a procedure that
might yield the needed
information.
•Discuss the practicality of
the approach.1
2
3
13
14
15
LC-1
LC-3
dP-1
dP-2
To flare
T5
T6
TC-7
AC-1
LAH
LAL
PAH
PC-1
P3
F4
F7
F8
F9
PV-3
TAL
T10
AC-2
CHAPTER 20: Multi-loop Control: Relative Gain Array
Workshop 4. RGA from closed-loop data

47. Youhavedecidedtopaironaloopthathasanegative
RGAelement.Discussthetuningthatisappropriatefor
theseloops.
A
A
C
A0
C
A
CSTR with
A B
Solvent
A
CHAPTER 20: Multi-loop Control: Relative Gain Array
Workshop 5. Pair loop on negative RGA element

48. 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 M
3
, V=1 M
3
, CA0=2.0 kg-moles/M
3
, Cp=1 cal/(g C), =10
6
g/M
3
, ko = 1.0x10
10
min
-1
,
E/R = 8330.1 K
-1
, (Fc)s=15 M
3
/min , Cpc=1 cal/(g K) , c=10
6
g/M
3
, b=0.5
Case I (Example 3.10) II III
- rxn 10
6
cal/(kg-mole) 130 13 -30
a (cal/min)/K 1.678x10
6
1.678x10
6
0.7746x10
6

T0 K 323 370 370
Tcin K 365 365 420 (heating)
Ts K 394 368.3 392.7
CAs kg-mole/M
3
0.265 0.80 0.28
CHAPTER 20: Multi-loop Control: Relative Gain Array
Workshop 6. Effect of operating conditions on RGA

49. CHAPTER 20: Multi-loop Control: Relative Gain Array
Workshop 6. continuedA. Calculate the relative gain arrays for the three cases shown in the table below.


CASE I CASE II CASE III

CA0 FC CA0 FC CA0 FC


CA

T



Notes:
1. CA0 is controlled by adjusting the reactant valve in a cascade
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 cooling, 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 based on principles
of the process.


C. Discuss relationships among the cases.

50. Prove the statement that the rows and columns of the
RGA sum to 1.0.closed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

Hint: a matrix and its inverse commute, i.e.,
K K
-1
= K
-1
K = I
CHAPTER 20: Multi-loop Control: Relative Gain Array
Workshop 7. RGA row-column sum

51. 





n
1i
n
1m
n
1j
n
1m
j row of sum ji if 1 and ji if 0
i col of sum ji if 1 and ji if 0
ijmjjm
n
k
kjik
ijmiim
n
k
kjik
kIkkkI
kIkkIk


1
1 The rows and columns
of the RGA sum to 1.0.closed loopsother
open loopsother
constant
constant












































j
i
j
i
CV
j
i
MV
j
i
ij
MV
CV
MV
CV
MV
CV
MV
CV
k
k

K K
-1
= I = K
-1
K
From the left hand equation, the elements of I are equal to
From the right hand equation, the elements of I are equal to
CHAPTER 20: Multi-loop Control: Relative Gain Array
Workshop 7. RGA row-column sum

52. TC
LC
FC
VC
SP = 95% open
To the SP of the feed flow
CV to the VC controller
MV from the VC controller
MV from the TC controller
Determine the relative gain array for the TC and VC
controllers. Discuss the behavior of this design and
generalize the conclusions.
CHAPTER 20: Multi-loop Control: Relative Gain Array
Workshop 8. Pair on zero RGA element