NAMP for North American Mobility In Higher Education
Program
PIECE
NAMP
Module 5
Controllability
Analysis
Introducing
Process integration
for Environmental Control in Engineering Curricula
Module
5 – Controllability
Analysis
PIECE
1
PIECE
NAMP integration for Environmental Control in Engineering Curricula
Process
Paprican
PIECE
École
Polytechnique
de Montréal
Universidad
Autónoma de
San Luis Potosí
University of
Ottawa
Universidad de
Guanajuato
North Carolina
State University
Instituto
Mexicano del
Petróleo
Program
North American
Mobility in Higher Education
Module
5 –for
Controllability
Analysis
Texas A&M
University
NAMP
2
NAMP
PIECE
Module 5
This module was created by:
Stacey Woodruff
Universidad de
Guanajuato
University of
Ottawa
From
Host University
Universidad de
Guanajuato
Carlos Carreón
Module 5 – Controllability Analysis
University of
Ottawa
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Project Summary
Objectives
Create web-based modules to assist universities to address
the introduction to Process Integration into Engineering
curricula
Make these modules widely available in each of the
participating countries
Participating institutions
Six universities in three countries (Canada, Mexico and the
USA)
Two research institutes in different industry sectors:
petroleum (Mexico) and pulp and paper (Canada)
Each of the six universities has sponsored 7 exchange
students during the period of the grant subsidised in part by
each of the three countries’ governments
Module 5 – Controllability Analysis
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Structure of Module 5
What is the structure of this module?
All modules are divided into 3 tiers, each with a specific goal:
Tier I: Background Information
Tier II: Case Study Applications
Tier III: Open-Ended Design Problem
These tiers are intended to be completed in that particular
order. In the first tier, students are quizzed at various points
to measure their degree of understanding, before proceeding
to the next two tiers.
Module 5 – Controllability Analysis
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Purpose of Module 5
What is the purpose of this module?
It is the objective of this module to cover the
basic aspects of Controllability Analysis. It is
targeted to be an integral part of a
fundamental/and or advanced Control course.
This module is intended for students with some
basic understanding of the fundamental
concepts of control.
Module 5 – Controllability Analysis
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Tier I
Background Information
Module 5 – Controllability Analysis
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• Statement of Intent
– Define Stability
– Demonstrate simple methods for stability analysis,
mostly for Single-Input Single-Output (SISO)
systems
– Understand interaction between control loops in
Multiple-Input Multiple-Output (MIMO) systems
– Demonstrate the Relative Gain Array
– Investigate controllability analysis for continuous and
discrete systems
– Comprehend singular value decomposition (SVD)
Module 5 – Controllability Analysis
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Stability
A dynamic system is stable if the system output
response is bounded for all bounded inputs. A
stable system will tend to return to its
equilibrium point following a disturbance.
Conversely, an unstable system will have the
tendency to move away from its equilibrium
point following a disturbance.
Module 5 – Controllability Analysis
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• Why is the stability of a system important??
When a system becomes unstable it can be
A DISASTER!!!!!
Module 5 – Controllability Analysis
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• Example
The concept of stability is illustrated in the following
figure. The sphere in (a) is stable as it will return to its
original equilibrium after a small disturbance whereas
the sphere in (b) is unstable as it moves away from its
equilibrium point and never comes back. The sphere in
(c) is said to be marginally stable.
(a)
Module 5 – Controllability Analysis
(b)
(c)
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Quiz #1
• Why is it important that a system is
stable?
• List two examples of systems that have
become unstable.
Module 5 – Controllability Analysis
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There are many ways of determining if a
system is stable such as :




Roots of Characteristic Equation
Bode Diagrams
Nyquist Plots
Simulation
Module 5 – Controllability Analysis
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• Roots of Characteristic Equation
One can determine if a system is stable based on the nature
of the roots of its characteristic equations. Consider the
following system:
D(s)
Y*(s) + (s)
-
GC (s)
U(s)
Ym(s)
Module 5 – Controllability Analysis
G1(s)
M(s)
G3 (s)
G2 (s)
+
Y(s)
+
G4 (s)
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From the previous diagram, we can see that the
output Y is influenced in the following manner.
Gc G1G2
G3
Y(s) =
Y*(s) +
D(s)
1 + GOL
1 + GOL
Where
GOL = Gc G1G2G4
GOL is the open loop transfer function.
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For the moment, let’s consider that there is only a change in set point,
therefore, the previous equation reduces to the closed loop transfer
function,
Y(s) =
Gc G1G2
Gc G1G2
Gc G1G2 (s)
1
Y*(s) =
=
1 + Gc G1G2G4
s 1 + Gc G1G2G4 s(s - r1 )(s - r2 )(s - r3 )...(s - rn )
The roots r1, r2, r3… rn are those of the characteristic equation
1+GcG1G2G4 =0
and (s) is a function that arises from the rearrangement. The roots of the
characteristic equation (denominator) are the poles of the transfer function
whereas the roots of the numerator are the zeros.
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• The nature of the roots of the characteristic equation can dictate if a system
is stable or not due to the fact that if there is one (or more) root on the
right half of the complex plane, the response will contain a term that grows
exponentially, leading to an unstable system.
Imaginary
Part
Imaginary
Part
Imaginary
Part
Real Part
Real Part
φ
φ
time
Negative real root
Stable
Region
Imaginary
Part
Stable
Region
Real Part
time
Unstable
Region Real
Part
Positive real root
Imaginary
Part
Real Part
φ
time
Complex Roots (Negative real parts)
Module 5 – Controllability Analysis
φ
time
Complex Roots (Positive real parts)
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• Routh Test
The Routh test (Routh stability criterion) is a very useful tool in
determining whether or not a closed-loop system is stable provided
the characteristic equation is available. The Routh stability criterion
is based on a characteristic equation that is in the form
ansn + an-1sn-1 + ... + a1s + a0 = 0
A necessary (but not sufficient) condition of stability is that all of
the coefficients (a0, a1, a2, …etc.) must be positive.
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Routh Array
When all coefficients are positive, a Routh Array must be
constructed as follows:
Row
1
2
3
an
an -1
b1
an -2
an -3
b2
an -4
an -5
b3
...
...
...
4
c1
c2
c3
...
n+1
}
The first two rows are
filled in using the
coefficients of the
characteristic
equation. Subsequent
rows are calculated as
shown in the next
page.
The system is stable if ALL the elements in the first column
are positive!
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Routh Array
After the coefficients of the characteristic equation are input in
the array, the coefficients, b1, b2 … bn and subsequently c1…cn
should be calculated as follows and input into the array.
an -1an -2 - anan -3
b1 =
an -1
b2 =
an -1an -4 - anan -5
...
an -1
c1 =
b1an -3 - an -1b2
b1
b a - an -1b3
c 2 = 1 n -5
...
b1
Module 5 – Controllability Analysis
Row
1
2
3
an
an -1
b1
an -2
an -3
b2
an -4
an -5
b3
...
...
...
4
c1
c2
c3
...
n+1
Pivot to calculate all bi
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Routh Test Theorems
Theorem 1- The necessary and sufficient condition for stability (i.e. All
roots with negative real parts) is that all elements of the first
column of the Routh Array must be positive and non zero.
Routh Test Example 1- Consider the following characteristic equation:
Row
1(s3)
s3 + 4.583s2 + 6.38s + 15.625 = 0
1
6.38
2(s2)
4.583
15.625
3(s1)
2.97
0
4(s0)
15.625
0
Module 5 – Controllability Analysis
All of the elements in the first column
of this Routh Array are positive,
therefore the system is stable.
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Routh Test Example 2- It is possible to determine for which values of Kc the
system remains stable
1+K c
3
2
s + 4.583s + 6.38s +
=0
0.384
Row
1(s3)
2(s2)
3(s1)
1
4.583
6.38
(1+Kc)/0.384
29.24 - (1+K c )/0.384
4.583
0
29.24-(1-Kc)/0.384>0 → Kc <10.23
1+Kc >0
4(s0)
(1+Kc)/0.384
Module 5 – Controllability Analysis
→ Kc>-1
(Kc is positive)
0
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Theorem 2- If some of the elements of the first column are negative, the
number of roots on the right hand side of the imaginary axis is equal to the
number of sign changes in the first column.
Routh Test Example 3 – If the characteristic equation of a system is given by
the following equation, is the system stable?
Row
s4 + 6s3 + 11s2 + 36s + 120 = 0
1(s4)
1
11
120
2(s3)
6
36
0
3(s2)
5
120
4(s1)
-108
0
5(s0)
120
Module 5 – Controllability Analysis
There are 2 sign changes.
Therefore, the system has two
roots in the right-hand plane, and
the system is unstable.
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Theorem 3- If one pair of roots is on the imaginary axis,
equidistant from the origin, and all the other roots are in the
left-hand plane, all the elements of the nth row will vanish.
The location of the pair of imaginary roots can be found by
solving the auxiliary equation:
Cs2+D=0
where the coefficients C and D are the elements of the array
in the (n-1)th row. These roots are also the roots of the
characteristic equation.
Module 5 – Controllability Analysis
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Routh Test Example 4 – Determine the stability of the system having the
following characteristic equation:
s4 + 3s3 + 6s2 + 12s + 8 = 0
Row
1(s4)
1
6
2(s3)
3
12
3(s2)
2
8
4(s1)
0
4(s1)
4
5(s0)
8
8
d
(2 s 2  8)  4 s
ds
Module 5 – Controllability Analysis
The derivative taken indicates
that a 4 should be placed in
the s row (Row 4). The
procedure is carried out.
There are no sign changes in
the first column, indicating
that there are no roots located
on the right-hand side of the
plane.
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Quiz #2
• In what cases can the Routh test be used to determine
stability?
• Is the system having the following characteristic equation
stable?
s4 + 7s3 + 6s2 + 1 = 0
• If a system has two negative real roots, is the system
stable?
• If a system has one negative real root and one positive real
root is the system stable?
Module 5 – Controllability Analysis
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Frequency Response
• One very useful method of determining system stability, even
when transportation lags exist, is Frequency Response.
• Frequency response is a method concerning the response of a
process or system to a sustained sinusoidal plot.
• Frequency Response Stability Criteria
Two principal criteria:
1. Bode Stability Criterion
2. Nyquist Stability Criterion
Module 5 – Controllability Analysis
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Bode stability criterion
A closed-loop system is unstable if the Frequency Response of the
open-loop Transfer Function, GOL=GCG1G2G4, has an amplitude ratio
greater than one at the critical frequency, ωc. Otherwise the closed-loop
system is stable.
Note: ωc is the value of ω where the open-loop phase angle is -1800.
Thus,
The Bode Stability criterion provides information on the closedloop stability from open-loop frequency response information.
Module 5 – Controllability Analysis
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Bode Stability Criterion- Example 1
A process has the following transfer function: G2 (s) =
2
(0.5s + 1)3
With a value of G1=0.1 and G4=10. If proportional control is used, determine
closed-loop stability for 3 values of Kc: 1, 4, and 20. GOL=GCG1G2G4
Solution:
2K c
2
GOL = Gc G1G2G4 = (K c )(0.1)
(10) =
3
(0.5s+1)
(0.5s+1)3
Kc
AROL for Kc
Stable?
1
0.25
Yes
4
1
Marginally
20
5
No
Module 5 – Controllability Analysis
You will find the Bode
plots on the next slide
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Bode plots for GOL = 2Kc/(0.5s + 1)3
Module 5 – Controllability Analysis
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• Nyquist Stability Criterion
The Nyquist stability criterion is the most powerful stability test that is
available for linear systems described by transfer function models.
Consider an open-loop transfer function, GOL(s) that is proper and has no
unstable pole-zero cancellations. Let N be the number of times that the
Nyquist plot of GOL(s) encircles the (-1, 0) point in a clockwise direction.
Also, let P denote the number of poles of GOL(s) that lie to the right of the
imaginary axis. Then, Z=N+P, where Z is the number of roots (or zeros)
of the characteristic equation that lie to the right of the imaginary axis.
The closed-loop system is stable, if and only if Z=0.
Module 5 – Controllability Analysis
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 Example 9.2 – Find the amplitude ratio and the phase lag of the
following process for  = 0.1 and 0.4.
U(s)
X(s)
1
5s + 1
e
First system:
1
AR =
=
2 2
 ω +1
Second system:
AR = 1
Z(s)
-0.3s
1
(5)2 ( )2 + 1
;  = -  = -0.3
1
=
25( )2 + 1
1.2
s3 + 2.3s2 + 1.7s + 0.4
Y(s)
;  = tan-1 (- ) = tan-1 (-5 )
180

Third system:
G(j ) =
AR =
1.2
1.2
=
(j )3 + 2.3(j )2 + 1.7(j ) + 0.4
0.4 - 2.3 2 + 1.7 -  3 j

1.2
 0.4 - 2.3 
2
2

+ 1.7 - 
Module 5 – Controllability Analysis
3

2
;  = tan
 
-1

 - (1.7 -  3 ) 

2 
 0.4 - 2.3 
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 Example 9.2 – Find AR and  (from known equations)
G(jω) = G1(jω) G2 (jω) ... Gn (jω)



1

 1 
G(jω) = 
 25( )2 + 1 

 

1.2

0.4 - 2.3 2

2

+ 1.7 -  3

2





 G(jω) =  G1(jω) +  G2 (jω) + ... +  Gn (jω)
 - (1.7 -  3 ) 
 G(jω) = tan (-5 ) - 0.3 + tan 
2 
 0.4 - 2.3 
If (0.4 – 2.33) < 0 then  –  or  – 180o
-1
-1
 = 0.1  AR = 2.60 ;  = - 0.915 s-1 or - 52.4o
 = 0.4  AR = 0.87 ;  = - 2.75 s-1 or - 157.3o
Module 5 – Controllability Analysis
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 Example 9.2 – Find AR and  … Nyquist plot
90
Im
Re
180
=0.4
0
1
2
0
3
=0.1

Module 5 – Controllability Analysis
270
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Quiz #3
• Name two methods of determining stability using
frequency response.
• What does an amplitude ratio (AR) of 1 signify? An
amplitude ratio of less than 1?
• What does a value of Z=0 signify?
Module 5 – Controllability Analysis
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• Multiple Input Multiple Output (MIMO) Systems
Co oling unit
Reflux Rec eiver
Nap tha
Light g as o il
Hea vy g as oil
High b oiling Re sid ue
FEED PUMPS
Air FuelGas
CRUDE OIL FEED
STORAGE TANKS
PIPESTILL
FRACTIONATOR
FURNAC E
Module 5 – Controllability Analysis
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When dealing with Multiple Input Multiple
Output systems, we have to ask ourselves two
main questions.
1. How to pair the input and output variables
2. How to design the individual single-loop
controllers
Module 5 – Controllability Analysis
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Let’s consider the following system:
Loop 1
+
Gc1
m1
G11
y1
+
+
G12
G21
+
Gc2
-
m2
G22
+
+
y2
Loop 2
y1(s) = G11(s)m1(s) + G12(s)m2(s)
y2(s) = G21(s)m1(s) + G22(s)m2(s)
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We will perform 2 small “experiments” to demonstrate MIMO
system interactions.
Let´s consider m1 as a candidate to pair with y1.
Experiment #1
When a unit step change is made to the input variable m1, with all
loops open, the output y1 will change, and so will y2, but for now,
we are primarily concerned with the effect on y1. After steadystate is reached, let’s consider the change in y1 as a result of the
change in m1, y1m ; this will represent the main effect of m1 on
y1.
Δy1m = K11
Keep in mind that no other input variables have been changed,
and that all loops are open, so no feedback control is required.
Module 5 – Controllability Analysis
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Experiment #2-Unit step change in m1 with Loop 2 closed.
These things will happen as a result of the unit step change in m1.
1- y1 changes because of G11, but because of interactions via the
element G21, y2 changes as well.
2- Under feedback control, Loop 2 wards off this interaction effect
on y2 by manipulating m2 until y2 is returned to its initial state
before the disturbance.
3-The changes in m2 will now affect y1 via the G12 transfer
element.
The changes in y1 are from two different sources.
(1) the DIRECT INFLUENCE of m1 on y1 (Δy1m)
(2) the Indirect Influence, from the retaliatory action from Loop 2
in warding off the interaction effect of m1 on y2 (Δy1r)
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After dynamic transients die away and steady-state is reached,
the net change observed in y1 is given by:
Δy1*= Δy1m+ Δy1r
This net change is the sum of the main effect of m1 on y1 and the
interactive effect provoked by m1 interacting with the other loop.
K12 K 21
  K11 *
y*  K11  1 
K11K 22
A good measure of how well a system can be controlled (λ) if m1
is used to control y1 is:
y1m
y1m
11 

y * y1m  y1r
Module 5 – Controllability Analysis
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Loop Pairing on the Basis of Interaction Analysis
Case 1 : λ11=1
This case is only possible if y1r is equal to zero. In physical
terms, this means that the main effect of m1 on y1, when all the
loops are opened, and the total effect, measured when the
other loop is closed, are identical.
This will be the case if:
• m1 does not affect y2, and thus, there is no retaliatory control
action from m2, or
• m1 does affect y2, but the retaliatory control action from m2
does not cause any change in y1 because m2 does not affect y1.
Under these circumstances, m1 is the perfect input
variable to control y1 because there will be NO
interaction problems.
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Case 2 : λ11=0
This condition indicates that m1 has no effect on y1,
therefore  y1m will be zero in response to a change in m1.
Note that under these circumstances, m2 is the perfect
input variable for controlling y2, NOT y1. Since m1 does not
affect y1, y1 can be controlled with m2 without any
interaction with y1.
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Case 3 : 0 < λ11< 1
This condition indicates that the direction of the
interaction effect is in the same direction as that of
the main effect. In this case the total effect is greater
than the main effect. For λ11>0.5, the main effect
contributes MORE to the total effect than the
interaction effect, and as the contribution of the main
effect increases, the closer to a value of 1 λ11
becomes. For λ11<0.5, the contribution from the
interaction effect dominates, as this contribution
increases, λ11 moves closer to zero. For λ11=0.5, the
contributions of the main effect and the interaction
effect are equal.
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Case 4 : λ11>1
This is the condition where y1r is the opposite sign of y1m, but it is
smaller in absolute value. In this case y1* (y1r +y1m) is less than
the main effect y1m, and therefore a larger controller action m1 is
needed to achieve a given change in y1 in the closed loop than in the
open loop. For a very large and positive λ11 the interaction effect almost
cancels out the main effect and closed-loop control of y1 using m1 will
be very difficult to achieve.
Case 5 : λ11< 0
This is the case when  y1r is not only opposite in sign, but also
larger in absolute value to  y1m. The pairing of m1 with y1 in this
case is not very desirable because the direction of the effect of m1 on y1
in the open loop is opposite to the direction in the closed loop. The
consequences of using such a pairing could be catastrophic.
Module 5 – Controllability Analysis
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Quiz#4
• What is a MIMO system?
• What does λ11=1 signify? If this is the case, is m1 a
good input variable to control y1?
• If λ11 is very large and positive, is m1 a good input
variable to control y1?
Module 5 – Controllability Analysis
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Relative Gain Array (RGA)
The quantity λ11 is defined as the Relative Gain between input m1
and output y1.
λij is defined as the relative gain between output yi and input mj,
as the ratio of two steady-state gains:
ij 
 y i

 m j


loops
all
open
 y i

 m j


loops closed
all
except for
 open-loopgain 
ij  
 for loop i under
closed-loopgain

 the control of m
j
the m j loop
Module 5 – Controllability Analysis
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When the relative gain is calculated for all of the
input/output combinations of a multivariable system,
the results are placed into a matrix as follows and this
array produces
 11 12


21
22


 

n1 n 2
 1n 
 2 n 
 

 nn 
THE RELATIVE GAIN ARRAY
Module 5 – Controllability Analysis
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PROPERTIES OF THE
RELATIVE GAIN ARRAY
•
Properties of the Relative Gain Array
1. The elements of the RGA across any row, or down
any column sum up to 1. i.e.:
n

i 1
ij
n
  ij  1
j 1
2. λij is dimensionless; therefore, neither the units, nor
the absolute value actually taken by the variables mj,
or yi affect it.
Module 5 – Controllability Analysis
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PROPERTIES OF THE
RELATIVE GAIN ARRAY
3. The value λij is a measure of the steadystate interaction expected in the ith loop of
the multivariable system if its output (yi) is
paired with input (mj); in particular, λij =1
indicates that mj affects yi without interacting
with the other loops. Conversely, if λij=0 this
indicates that mj has no effect on yi.
Module 5 – Controllability Analysis
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NAMP
PIECE
PROPERTIES OF THE
RELATIVE GAIN ARRAY
4. Let Kij* represent the loop i steady-state gain when all
loops (other than loop i) are closed, whereas, Kij
represents the normal open loop gain.
Kij * 
1
ij
Kij
This equation has the very important implication: that
1/λij tells us by what factor the open loop gain
between output yi and input mj will be changed when
the loop are closed.
Module 5 – Controllability Analysis
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NAMP
PIECE
PROPERTIES OF THE
RELATIVE GAIN ARRAY
5. When λij is negative, it indicates a situation in
which loop i, with all loops open, will produce
a change in yi in response to a change in mj
in totally the opposite direction to that when
all the other loops are closed. Such
input/output pairings are potentially unstable
and should be avoided.
Module 5 – Controllability Analysis
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NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
•
Calculating the Relative Gain Array
There are two ways of calculating the Relative Gain
Array
1. The “First Principles” Method
2. The Matrix Method
Module 5 – Controllability Analysis
53
NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
•First Principles Method
Let’s consider a 2x2 system as we encountered before. First, we
must observe that the Relative Gain Array deals with steady-state
systems, and therefore , must only be concerned with the steady
state form of this model which is:
y1=K11m1 +K12m2
(Eq. 1a)
y 2 =K 21m1 +K 22m2
(Eq. 1b)
In order to calculate the λ11 we defined earlier, we need to
evaluate the partial derivatives as was explained on slide 47.
 y 


Recall:

m


i
ij 
Module 5 – Controllability Analysis
j
 y i

 m j
all loops
open


loops closed
all
except for
the m j loop
54
NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
Due to the fact that the equations found on the previous slide
represent steady-state, open-loop conditions, the differentiation
for the numerator portion of the relative gain is:
 y1 


 K11
 m1 all loops open
The second partial derivative (the denominator) requires Loop 2
to be closed, so that in response to changes in m1 , the second
control variable m2 can be used to restore y2 to its initial value
of 0. To obtain the second partial derivative, we first find from
Eq. 1b the value of the m2 must be to maintain y2=0 in the face
of changes in m1, what effect this will have on y1 is deduced by
substituting this value of m2 into Equation 1a.
Module 5 – Controllability Analysis
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NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
The computation of the denominator of λ11
Set y2=0 and solve m2 in Eq. 1b.
K 21
m2  
K 22
m1
Substituting this value of m2 into Eq. 1a. gives:
K12 K 21
y1  K11m1 
m1
K 22
Having eliminated m2 from the equation, we now may
differentiate with respect to m1.
 y1 
 K 12 K 21 



 K 11 1 
 m1  loop 2 closed
 K 11 K 22 
Module 5 – Controllability Analysis
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NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
We then substitute the numerator and denominator into the
definition of λ11 which yields:
11 
K 11
 K 12K 21 
K11 1
 K 11K 22 
This equation simplifies to the form:
1
11 
1
Module 5 – Controllability Analysis
where
K12K 21
=
K11K 22
57
NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
This exercise should be repeated for all λij’s so
that the RGA can be constructed.
For Practice, repeat this exercise and verify the
following.

12  21 
1
Module 5 – Controllability Analysis
and
1
22  11 
1
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NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
• Thus the RGA for this 2x2 system is given by:
 
 1
1  
1 




1



1 
1  

Note, that if we define
1
  11 
1
The RGA can be rewritten as follows
Module 5 – Controllability Analysis
1 
 


 
1  
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NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
• The Matrix Method for Calculating RGA
Let K be the matrix of steady-state gains of the transfer
function matrix G(s) i.e.:
lim G ( s )  K
s 0
Whose elements are Kij, further, let R be the transpose
of the inverse of this steady state matrix (K)
 
R K
Module 5 – Controllability Analysis
1 T
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NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
With elements rij it is possible to show that the
elements or the RGA can be obtained from the
elements of these two matrices as:
ij  K ij rij
It is important to note that the equation above indicates
an element-by-element multiplication of the
corresponding elements of the two matrices, K and R,
DO NOT TAKE THE PRODUCT OF THESE MATRICES!
Module 5 – Controllability Analysis
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NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
•Example- Matrix Method of Calculating RGA.
Find the RGA for the 2x2 system represented by Equations 1a and
1b and compare them with the results obtained using the First
Principles Method.
Solution:
For this system, the steady-state gain matrix (K) is the following.
 K11
K 
 K 21
Module 5 – Controllability Analysis
K12 

K 22 
62
NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
From the definition of the inverse matrix we know that
K
1
1

K
 K 22  K12 
 K

K
21
11 

Where the determinant of K, |K| is:
K  K11K 22  K12 K 21
Therefore, by taking the transpose of the K-1 matrix, we obtain the
R matrix
 
R K
Module 5 – Controllability Analysis
1 T
1

K
 K 22
 K
 12
 K 21 

K11 
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NAMP
PIECE
COMPUTING THE RELATIVE
GAIN ARRAY
Since we now have the R and K matrices, we can perform an
element by element multiplication to obtain the elements (λij)
of the RGA (Λ)
K 11K 22
11 =
K
K 11K 22
OR 11 = K K - K K
11 22
12 21
here is the first element of the matrix. Try on your own to
compute the other 3 elements of the RGA.
 K 11K 22
 K

 -K 21K 12

Module 5 – Controllability Analysis
 K
-K 12K 21 
K 

K 22K 11 

K 
64
NAMP
PIECE
• Example of RGA for the Wood and Berry Distillation,
using the Matrix Method
Find the RGA for Wood and Berry Distillation column whose
transfer function matrix is
 12.8e  s
 18.9e 3 s 

G ( s )  16.7 s7 s1
 6.6e
10.9 s  1

21.0s  1 
 19.4e 3 s 
14.4 s  1 
Solution: For this system, the steady-state gain matrix is easily
extracted from the transfer function matrix by setting s=0.
12.8  18.9
K  G(0)  

6
.
6

19
.
4


Module 5 – Controllability Analysis
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NAMP
PIECE
The next step is to determine the inverse of the matrix K:
K
1
0.157  0.153


0
.
053

0
.
104


Once the inverse is calculated, the transpose of this matrix must
be calculated to yield the matrix R.
0.053 
 0.157
R  (K )  


0
.
153

0
.
104


1 T
After these two matrices are computed, it is time to calculate the
RGA by multiplying the matrices element by element.
 2  1


 1 2 
Module 5 – Controllability Analysis
Note that all of the rows
and columns sum to one.
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NAMP
PIECE
LOOP PAIRING USING THE
RELATIVE GAIN ARRAY
• Loop Pairing using the RGA
Now that we know how to compute the RGA, we will now
consider how it can be used to guide the pairing of input and
output variables in order to obtain the control configuration
with minimal loop interaction.
On the following slides, we will investigate how to interpret the
elements of the RGA (λij). We will use the five scenarios
presented early to interpret the implications of the values of λij
Module 5 – Controllability Analysis
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NAMP
PIECE
LOOP PAIRING USING THE
RELATIVE GAIN ARRAY
Case 1: λij=1, the open loop gain is the equal to
the closed loop gain.
Loop interactions implications : This situation indicates that
loop i will not be subject to retaliatory effects from other loops
when they are closed, therefore mj can control yi without
interference from other control loops. If any of the other
elements in the transfer function matrix are nonzero, the ith
loop will experience some disturbances from other control
loops, but these are NOT provoked from actions in the ith loop.
Recommendation for pairing: In this case, the pairing if mj
with yi would be ideal.
Module 5 – Controllability Analysis
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NAMP
PIECE
LOOP PAIRING USING THE
RELATIVE GAIN ARRAY
Case 2: λij=0, the open loop gain between mj and
yi is zero.
Loop interactions implications : mj has no direct influence on
yi (keep in mind that mj may still have an effect on other
control loops)
Recommendation for pairing: Do NOT pair yi with mj, it
would be more advantageous to pair mj with another output
variable, since we are led to believe that yi will not be
influenced by the loop containing mj.
Module 5 – Controllability Analysis
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NAMP
PIECE
LOOP PAIRING USING THE
RELATIVE GAIN ARRAY
Case 3: 0<λij<1, the open loop gain between yi
and mj is smaller than the closed loop gain.
Loop interactions implications : The closed loop gain is
the sum of the open loop gain and the retaliatory effect, from
the other loops,
a) The loops are interacting, but
b) They interact in such a way that the retaliatory effect from the
other loops is in the same direction as the main effect of mj on
yi.
Module 5 – Controllability Analysis
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PIECE
Loop interactions implications :
The loop interactions “assist” mj on controlling yi, The extent of
this assistance is dependent on how close λij is to 0.5
When:
λij =0.5: the main effect of mj on yi is exactly the same as the
retaliatory effect.
0.5<λij <1, the retaliatory effects are less than the main effect
0<λij< 0.5, the retaliatory effect is larger than the main effect.
Recommendation for pairing: If possible, avoid pairing yi with
mj if λij<0.5
Module 5 – Controllability Analysis
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PIECE
LOOP PAIRING USING THE
RELATIVE GAIN ARRAY
Case 4: λij>1, the open loop gain between yi and mj is
larger than the closed loop gain.
Loop interactions implications : The loops interact, and the
retaliatory effect from the other loops acts in opposition to the
main effect of mj on yi, (which means that the loop gain will be
reduced when the other loops are closed), but the main effect
is still dominant, otherwise λij would be negative. For large
values of λij, the controller gain for loop i will have to be chosen
much larger than when all loops are open. This would cause
loop i to be stable when the other loops are open.
Recommendation for pairing: The higher the value of λij , the
greater the opposition mj experiences from the other loops in
trying to control yi. Therefore try not to pair yi with mj with if
the5 –value
of λij Analysis
is large.
Module
Controllability
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PIECE
LOOP PAIRING USING THE
RELATIVE GAIN ARRAY
Case 5: λij<0, the open loop and closed loop
gains between yj and mi have opposite signs.
Loop interactions implications : The loops interact, and
the retaliatory effect from the other loops is not only in
opposition, but it is greater in absolute value to the main effect
of mj on yi. This is potentially dangerous because if the other
loops are opened, loop i could become very unstable.
Recommendation for pairing: Avoid pairing mj with yi
because of the retaliatory effect that mj provokes from the other
loops acts in opposition to, and dominates the main effect on
yi.
Module 5 – Controllability Analysis
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NAMP
PIECE
Quiz#5
• What advantages does the Matrix Method have over
the First Principles Method?
• What does λ with a value of 1 signify, and should mj
and yi be paired together?
• What does λ with a value less than zero of signify, and
should mj and yi be paired together?
Module 5 – Controllability Analysis
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NAMP
PIECE
• Basic Loop Pairing Rules
From what we have learned about loop pairing, it is natural that the
ideal RGA would take the form
1 0 0  0 
0 1 0  0 


  0 0 1  0 


0






0 0 0  1 
This is known as the identity matrix, in which each row and column
only contains one non-zero element whose value is unity (1). This
ideal RGA is produced when the transfer matrix G(s) has one of
two forms, only a diagonal element, or is in lower triangular from.
The first situation indicates that there is no interaction between the
loops. The second case indicates that there is a one-way
interaction (which is explained on the next slide).
Module 5 – Controllability Analysis
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If the G(s) indicates that there is a one-way interaction( the transfer
function matrix is in triangular form), it will yield an RGA of the identity
matrix, but it can not be treated as if there are no interactions or
influences. Please consider the following example.
 1

G( s)   s  1
3

 3s  1

0 
4 

4 s  1
yields an RGA   

1 0

0
1


Note that since the element g12(s) is zero, the input m2 does not have
an effect on the output y1, however, the input m1 does influence the
output y2 as can be seen due to the fact that the g21 element is
nonzero. Upsets in Loop 1 requiring action by m1 would have to also be
handled by the controller of Loop 2. So, even though the RGA is ideal,
Loop 2 would be at a disadvantage. Thus, in deciding on loop pairing,
one should distinguish between ideal RGAs produced from diagonal or
triangular transfer function matrices.
Module 5 – Controllability Analysis
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PIECE
• RULE #1
Pair input and output variables that have positive RGA
elements closest to 1.0.
Consider the following examples to demonstrate this rule.
For a 2x2 system with output variables y1 and y2, to be paired
with m1 and m2
If the RGA is…
0.8 0.2


0
.
2
0
.
8


Then it is recommended to pair m1 with y1 and m2 with y2, which
is quite often referred to a the 1-1/2-2 pairing.
Module 5 – Controllability Analysis
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Now, consider the 2x2 system whose transfer matrix is:
 1.5  0.5


 0.5 1.5 
In this case, a 1-1/2-2 pairing is preferred as to avoid pairing on a
negative RGA element. Usually, we will try to avoid pairing on RGA
elements greater than 1, but pairing on negative RGA elements is
worse.
Recall the Wood and Berry distillation column example we saw on
Slide 65, it’s RGA is:
In this case, it is
 2  1
desirable for a 1

1/2-2 pairing

1
2


Module 5 – Controllability Analysis
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On the other hand, for the 2x2 systems whose RGA is
0.3 0.7


0.7 0.3
y1 should be paired with m2 and y2 should be paired with m1,
this is referred to as 1-2/2-1 pairing. (as the elements 1-2,2-1
are closer to a value of 1 and all elements in the RGA are
positive.)
Module 5 – Controllability Analysis
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PIECE
Let’s consider the following 3x3 matrix:
 1.95
   0.66
 0.29
 0.65
1.88
 0.23
 0.3 
 0.22
1.52 
The same general guidelines, we applied to the 2x2 systems can
also be applied here. It can be seen that although the diagonal
elements are all greater than 1, the other elements are all
negative, suggesting that a 1-1/2-2/3-3 pairing would be
preferable.
Module 5 – Controllability Analysis
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PIECE
NIEDERLINSKI INDEX
Niederlinski Index
Pairing Rule #1 is usually sufficient in most
cases, it is often necessary to use this rule in
conjunction with the theorem found on the
next slide developed by Niederlinski and later
modified by Grosdidier et al. This theorem is
especially useful if the system is 3x3 or larger.
Module 5 – Controllability Analysis
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PIECE
NIEDERLINSKI INDEX
Consider the n x n multivariable system whose input-output
variables have been paired y1-u1, y2-u2…..yn-un, resulting in a
transfer function model of the form:
.
y(s)=G(s) u(s)
Let each element of G(s), gij(s) be,
1.Rational, and
2.Open-loop stable
Module 5 – Controllability Analysis
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PIECE
Let n individual feedback controllers (which have integral action)
be designed for each loop so that each one of the resulting n
feedback loops is stable when all of the other n-1 loops are
open.
Under closed-loop conditions in all n loops, the multivariable will
be unstable for all possible values of controller parameters if
the Niederlinski Index N defined below is negative.
On the following slides
there are important
points to help us use
this result properly.
Module 5 – Controllability Analysis
N
G ( 0)
n
0
(Eq. N)
 g ii (0)
i 1
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NAMP
PIECE
NIEDERLINSKI INDEX
Important Points for us to consider:
1.The result is both necessary and sufficient for 2x2 systems; for
higher dimensional systems, it only provides sufficient
conditions (if Equation N holds it is definitely unstable, but if
Eq. N does not hold, the system may or may not be unstable:
the stability will be dictated by the values taken by the
controller parameters).
2.For 2x2 systems the Niederlinski index becomes
N  1
where ζ defined as follows as
seen on Slide 57
Module 5 – Controllability Analysis
K 12K 21
 
K 11K 22
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NAMP
PIECE
NIEDERLINSKI INDEX
2. For a 2x2 system with a negative relative gain, ζ >1, the Niederlinski
index is always negative; hence 2x2 systems paired with
negative relative gains are ALWAYS structurally unstable.
3. This theorem is designed for systems with rational transfer function
elements, therefore, this technically excludes systems containing
time-delays. However, since Eq.N depends on Steady State gains
(s=0, therefore, the gains are independent of time-delays). Due to
this fact, the results of this theorem also provide important
information about time-delay systems as well, but is not very
rigorous. USE CAUTION WHEN APPLYING Eq.N TO SYSTEMS
WITH TIME DELAYS.
Module 5 – Controllability Analysis
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• RULE #2
Any loop pairing is unacceptable
if it leads to a control system
configuration for which the
Niederlinski Index is negative.
Module 5 – Controllability Analysis
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Summary of using RGA for Loop Pairing
1. Given the transfer matrix G(s), obtain the steady-state gain
matrix K=G(0), and from this obtain the RGA, Λ, also calculate
the determinant of the K and the product of the elements on
the main diagonal
2. Use Rule #1 to obtain tentative loop pairing suggestions from
the RGA by pairing the positive elements which are closest
to one.
3. Use the Niederlinski condition (Eq. N) to verify the stability
status of the of the control configuration obtained using Step
2, if the selected pairing is unacceptable, choose another.
Module 5 – Controllability Analysis
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•Applying Loop Pairing Rules
Loop Pairing Example 1: Calculate the RGA for the system whose
steady-state gain matrix is given below and investigate the loop
pairing suggested upon applying Rule #1.
K = G(0)
Module 5 – Controllability Analysis
5
3

= 1


1


1
1
3
1

1

1


1
3

88
NAMP
PIECE
First, we need to take the inverse of this matrix, then take the
transpose of this matrix to obtain R, being:
 10  4.5  4.5
   4.5
1
4.5 
 4.5 4.5
1 
The next step is to determine the RGA by multiplying the
elements of the K and R matrices.
 4.5  4.5
 6


R   4.5
3
4.5 
 4.5 4.5
3 
Module 5 – Controllability Analysis
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Rule #1 would suggest a 1-1,2-2,3-3 pairing
To calculate the Niederlinski Index we need to find :
• The determinant of the K matrix which is :|K|=-0.148
• The product of the main diagonal which is :
 5  1  1  5
K ii      

 3  3  3  27
i 1
n
It is clear that when the determinant is divided by the product of
the elements of the main diagonal it will yield a negative
number which leads to a…
NEGATIVE NIEDERLINSKI INDEX which violates Rule
#2.
Module 5 – Controllability Analysis
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PIECE
This example provides a situation where the pairing suggested by
Rule #1 is disqualified by Rule #2. Due to this fact, we need
to investigate another loop pairing. Let’s try the possible
pairing of 1-1,2-3,3-2, which would give a RGA of:
 10  4.5  4.5


   4.5 4.5
1 
 4.5
1
4.5 
Module 5 – Controllability Analysis
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The new K is:
5
3

K  G (0)   1

1


1
1
1
3

1
1

3
1


It is clear that the element in 2-2 has been interchanged
with the element 2-3 and the element 3-3 has been
interchanged with the old element 2-2.
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We need to calculate the determinant and product of the
elements of the main diagonal of the new matrix K:
|K|=0.1481 while the product of the elements is equal to 5/3.
Therefore, the Niederlinski Index is
N
K
0.148

0
5/3
n
K
ii
i 1
Clearly, this Niederlinski Index is positive, so we come to
the conclusion that this system is no longer structurally
unstable.
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Loop Pairing Example 2: Consider the system with the steady
state gain matrix as seen below
1  0.1
1
K  G(0)   0.1 2
 1 
 2  3
1 
• The determinant of this matrix is 0.53.
The RGA is :
  1.89 3.59  0.7 
   0.13 3.02  1.89
 3.02  5.61 3.59 
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From the RGA seen, there is only one feasible pairing, because all
of the other pairings violate Rule 2. The only feasible pairing is
a 1-1,2-2,3-3 pairing, but you will notice that this pairing
violates Rule 1, as the RGA element 1-1 is negative, but
according to the Niederlinski Theorem this system would NOT
be structurally unstable.
If the first loop is opened (the m1, y1 elements dropped from the
process model) the new steady-state gain matrix relating the 2
remaining input variables with the 2 remaining output variables
is:
 2  1
K 


3
1


~
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It is easy to see that if the first loop is open, the Niederlinski
Index of the remaining two loops would be negative, indicating
that the system would be structurally unstable. As a
consequence, this system will only be stable if all loops are
CLOSED, such a system is said to have a low degree of
integrity.
There are some examples of higher order systems
where it is possible to pair on negative RGA values
and still have a structurally stable system (this is NOT
possible for 2x2 systems).
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• Summary of Loop Pairing using RGA
Always pair on positive RGA elements that are the
closest to 1 in value. Thereafter, use the Niederlinski
Index to check if the resulting configuration is
structurally stable. Whenever possible, try to avoid
pairing on negative RGA elements; for 2x2 systems
such pairings always lead to unstable
configurations, while for systems of higher
dimension, they can lead to a condition which, at
best has a low degree of integrity.
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Quiz #6
• What does a positive Niederlinski Index indicate?
• According to Rule 1, should elements be paired on
positive or negative elements?
• In what case should a favourable pairing from
Rule 1 be discarded?
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•Loop Pairing for Non-linear systems.
LOOP PAIRING FOR NONLINEAR SYSTEMS
Example 1- RGA and Loop pairing of non-linear systems. The
process shown is a blending process, the objective is to control
both the total product flow rate (F) and the product composition
(x) as calculated in terms of the mole fraction of A in the blend.
Obtain the RGA for the system and suggest which input variable
to pair with each output.
FC
x
Analyzer
FA
FB
GC
Blending
Module 5 – Controllability Analysis
F
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Total Mass Balance:
PIECE
FA  FB  F
Mass Balance on Component A
FA
x
FA  FB
Solution: Notice that for this system, the two output variables are F and
x, and the input variable are FA and FB, from now on, we will refer to
the input variables as m1 and m2 for the input feeds of A and B
respectively.
Therefore, our Overall Mass Balance becomes
F  m1  m2
(Eq 1)
(which is linear)
(Eq 2)
(which is NON-linear)
And the Component A Mass Balance becomes
Module 5 – Controllability Analysis
m1
x
m1  m 2
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Since this is a 2x2 system, we only need to obtain the (1,1)
element of the RGA given by:
Recall:
 F 


 m1  both loopsopen

 F 


 m1  second loop closed
To calculate the numerator, take the derivative of the first
equation with both loops open with respect to m1 , yielding
 F 


 1
 m1  both loopsopen
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In order to calculate the denominator, loop 2 must be closed, and we
will have to determine the value of m2 so that when a change occurs in
m1, x will return to its steady state value (x*).
To determine the value of m2 in this case, we must set x=x* in Equation
2 and solve for m2 in terms of m1 and x*, the result is:
m1
m2 =
-m1
x*
When loop 2 is closed, the mole fraction of the the component A in the
output at x*, m2 will respond to changes in m1, to determine the
relationship, we have to substitute the value of m2 above into the
Overall Mass Balance (Equation 1) yielding:
m1
F=m1 +
-m1
x*
Module 5 – Controllability Analysis
or
m1
F=
x*
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The next step is to differentiate the expression of F obtained in the last
step with respect to m1 yielding:
 F 
1




m
x*

1 second loop
closed
If the numerator and denominator are substituted into the statement for
the relative gain (λ), we get:
1

 x*
1/ x *
For a 2x2 matrix recall that the RGA is given by…
Therefore the RGA of this system is:
 x * 1  x *


1  x * x * 
Module 5 – Controllability Analysis
1 
 


1





Where x* is the
desired mole
fraction of A in
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Some things to consider about these results:
1. The RGA is dependent on the steady-state value of x* desired
for the composition of the blend; it is NOT constant as it was
in the linear systems we dealt with before.
2. It is implied that the recommended loop-pairing will depend
on the steady-state operating point.
3. Due to the fact that x* is a mole fraction, it is bounded
between 0 and 1 (0 < x*< 1) and therefore, none of the
elements in the RGA will be negative. The implication of this
fact, is that in the worst possible scenario is that there will be
large interactions between the input variables if the input and
output variables are paired improperly, but the system will not
become unstable.
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A loop pairing strategy for this system is as follows:
1. If x* is close to 1, the first implication is that m1 is larger than m2 . If
we look at the RGA, the following pairing would be recommended, Fm1, x-m2.(ie. The larger flow rate is used to control the total flow rate
out and the smaller flow rate is used to control the composition.)
2. This is the most reasonable pairing because: when the product
composition is close to one (x* close to 1), we have almost pure A
coming out of the system, and so we can modify the flow rate out
quite easily by changing the flow rate of A into the blending without
changing the composition of the blend significantly. Similarly if we
alter the composition, the additional small amounts of material B will
not have a significant impact on the flow rate of the blend out of the
system. Thus, the flow controller will not interact strongly with the
composition controller if the pairing : F-m1 and x-m2 is used, but if
the opposite pairing was used, the interaction would be severe.
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3. When the steady-state product composition is closer to 0, the
RGA suggests that the loop pairing stated in point 2 should be
switched, i.e. m2 (FB) should be paired with the outgoing flow
rate (F-m2) and m1(FA) should be paired with the composition
(x-m1). If you analyze the effects that each variable has as
done in point 2, you will see that the physics of this system
dictates such a pairing.
4. An interesting situation arises when the composition (x*) is
equal to 0.5 (x*=0.5). In this case it does not matter which
input variable is used to control which output variable. The
observed interactions will be equal and significant in either
case.
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LOOP PAIRING FOR PURE
INTEGRATOR MODES
Loop Pairing for Systems with Pure Integrator
Modes:
Since we have seen that interaction analysis using the RGA is
carried out using steady-state information, an interesting
situation occurs when dealing with systems that contain pure
integrator elements (i.e. if s was set to zero, an element would
become undefined), since pure integrator elements show no
steady-state. Several suggestions are available to deal with this
problem, but we will use the industrial application of the a deethanizer to demonstrate one method to recommend a loop
pairing strategy.
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Pure Integrator System Example 1 - The transfer function for a
2x2 subsystem extracted from a larger system for an industrial
de-ethanizer is given below. Obtain the RGA and use it to
recommend loop pairings.

1.318e 2.5 s

20 s  1

G(s)  

0.0385(182 s  1)

 ( 27 s  1)(10 s  1)( 6.5s  1)
 e 4 s 

3s 

0.36 

s 
Solution- Our usual course of action to determine the RGA is to
normally calculate the K matrix which is G(s) when s=0.
Unfortunately, we can see that elements (1,2) and (2,2) contain
pure integrator elements represented by 1/s, which if we set s=0
would yield an undefined number.
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Let’s make the substitution,
PIECE
I
1
s
If I is substituted into G(s), K becomes:
I 

1.318


K  lim  lim
3
s 0
I  

0.038 0.36I 
The relative gain parameter (λ)
Module 5 – Controllability Analysis


1
  lim 
I 
 1  0.038 x 0.333I

1.138 x 0.36I







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We can see that in the λ term the Is cancel out, so we obtain
λ=0.97
Therefore the resulting RGA is
0.97 0.03


0.03 0.97 
It is quite obvious that it is desirable to pain in a 1-1,2-2
fashion.
If you encounter a system in which there the Is do not cancel
out, you will have to consult another reference.
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LOOP PAIRING FOR NONSQUARE SYSTEMS
• Loop Pairing for Non-Square Systems
In the previous slides, we have discussed how obtain RGAs and
how to use them for input/output pairings when the process
has an equal number of input and output variables (square
systems).
There are some cases, where multivariable systems do not have
the same number of input and output variables, these are
referred to as non-square systems.
The most obvious problem with non-square systems is that after
the input/output pairing, there will always be either an input or
an output that is not paired (a residual ).
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With non-square systems, we are faced with
two questions.
1) Which input/output variables should be
paired together?
2) Which variables are redundant and which
take an active role in control?
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Classifying Non-Square Systems
We have 2 types of non-square systems,
1) Underdefined- there are fewer input variables than output
variables.
2) Overdefined- there are more input variables than output
variables.
Thus, a multivariable system with n output and m input
variables, whose transfer function matrix will
therefore be n x m in dimension is:
UNDERDEFINED if m<n and OVERDEFINED if m>n
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B
Underdefined Systems
n outputs
m inputs
As seen in the system above, there are less inputs m than there
are outputs n, thus is defined as an underdefined system.
m=the number of inputs = 2
m<n
n=the number of outputs = 4
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Underdefined Systems
The main issue with underdefined systems is that not all
outputs can be controlled, since we do not have
enough input variables.
The loop pairing is easier if we make the following
consideration
By economic considerations, or other such means,
decide which m of the n output variables are the most
important, these m output variables should be paired
with the m input variables; the less important (n-m)
output variables will not be under any control.
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Overdefined Systems
B4
n outputs
m inputs
As seen in the system above, there are less inputs m than there
are outputs n, thus is defined as an underdefined system.
m=the number of inputs = 3
n=the number of outputs = 2
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m>n
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Overdefined Systems
Deciding the loop pairing of overdefined systems presents a real
challenge. In this case, there is an excess of input variables,
therefore we can achieve arbitrary control of the fewer output
variables in more than one way.
The situation we are faced with is as follows: since there are m
input variables to control n output variable (m>n), there are many
more input variables to choose from in pairing the inputs and the
outputs, and therefore, there will be several different square
subsystems from which the pairing is possible. There are  m 
n 
possible square subsystems.
Module 5 – Controllability Analysis
 m  m!
Recall that:  n  = n! (m-n)!
 
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The Variable Pairing Strategy for Overdefined Systems
is:
1. Determine all of the
m
 
n 
subsystems from a given model.
2.Obtain the RGAs for each of the square subsystems.
3.Examine the RGAs and chose the best subsystem on the basis
of the overall character of its RGA (in terms of how close it is to
the ideal RGA).
4. After determining the best subsystem, use its RGA to decide
which input variable within its subsystem to pair with each output
variable.
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LOOP PAIRING IN THE ABSENCE
OF PROCESS MODELS
Loop Pairing in the Absence of Process Models
Sometimes, situations arise where a process model is not
available, but it is still possible to determine their RGAs from
experimental data. There are 2 approaches as follows:
Approach 1- Experimentally determine the steady-state gain
matrix K, by implementing a step change in the process input
variables, one at a time, and observing the ultimate change in
each output variable.
Let y1j be the observed change in the value of the output
variable 1 in response to a change of  mj in the jth input variable
mj ; then , by definition of the steady-state gain:
k1 j 
Module 5 – Controllability Analysis
y1 j
m j
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In general, the steady-state gain between the
ith variable and the jth variable will be given by
k ij 
yij
m j
Thus, the elements of the K matrix can be
calculated, and once the K matrix is known, it is
easy to calculate the RGA.
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Approach 2- It is possible to determine each element of the RGA
directly from experimentation.
As you may recall, each RGA element (λij) can be obtained by
performing two experiments. The first experiment determines the openloop steady-state gain by measuring the response of yi to input mj ,
when all other loops are open. In the second experiment, all other loops
are closed – using PI controllers to ensure that there will be no steadystate offsets – and the response of yi to input mj is redetermined. By
definition, the ratio of these two gains is the desired relative gain
element ( λij ).
The second approach is more time consuming, and involves too many
upsets to the process; for these reasons it is not desirable in practice.
Therefore, the first approach is preferred.
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Final Comments on the RGA
1.The RGA requires only steady-state process information, it is
therefore easy to calculate and easy to use.
2. The main criticism of the RGA is that the RGA only provides
information about the steady-state interactions within a
process systems, and therefore, dynamic factors are not taken
into account by the RGA analysis.
3. The RGA only suggests input/output pairing such that the
interaction effects are minimized; it provides no guidance
about other factors which may influence the pairing.
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Other Factors Influencing the Choice of Loop Pairing
1.Constraints on the input variable: It is possible that the
best pairing obtained from the RGA will result in a choice of
input variable for yi that is severely limited by some constraint
(ex. maximum feed concentration) in a way that it can not
carry out the assigned control task.
2.The presence of a time-delay, inverse-response, or other
slow dynamics in the best RGA pairing: Since the RGA is
based on steady-state information, sometimes, the best RGA
pairing results can result in very slow closed-loop response if
there are long time delays, significant inverse response or
large time constants. If this is the case, it would be more
suitable to pair on more unfavourable RGA elements if the
slow elements could be omitted to improve system
performance.
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Other Factors Influencing the Choice of Loop Pairing
3. Timescale Decoupling of Loop Dynamics: Often timescale
issues arise that can influence the choice of loop pairing. For
example, in a 2x2 system, it may be that for a given pairing, the
RGA indicates a serious loop interaction. However, if at the same
time, one of the loops responds a great deal faster than the
other, there can be a timescale decoupling of the loops. This can
occur if the fast loop responds so fast that the effect on the slow
loop seems to be a constant disturbance, in opposition, the slow
loop does not respond at all to the high-frequency disturbances
coming from the fast loop. This indicates that loops with large
differences in closed-loop response times can be paired even
when the RGA indicates that the pairing is unfavourable.
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Quiz#7
• What system information is needed to construct the
RGA?
• What is the difference between a underdefined and
overdefined system?
• What is a difficulty in overdefined systems?
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Controller Design Procedure-Multiloop Controller Design
There are 2 stages in the design of multiple single-loop controllers
for multivariable systems:
•Judicious choice of loop pairing
•Controller tuning for each individual loop
We have discussed this first point a great deal in the past slides,
this should signify importance of the choice of loop pairing in
controller design.
Now, we must address the issue of tuning the individual
controllers.
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It should be obvious that when the RGA for a process is
close to ideal (ie. λij is very close to 1) that the
multiloop controllers are very likely to function very well
if they are designed properly.
However, when the RGA indicates strong interactions for
the chosen loop pairing (ie. λij is very large or negative)
the controller is not likely to perform well even if it is
tuned well.
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•Controller Tuning for Multiloop Systems
The main challenge in controller tuning is the interactions between the
different control loops of a multi-loop system. Due to this fact, it can be
risky to adopt the obvious strategy of tuning each controller individually
without considering the other controllers and hoping that when all the
loops are closed that the overall system performance will be adequate.
The procedure that is normally followed in practice is the following:
1.With the other loops on manual control, tune each control loop
independently until satisfactory closed-loop performance is obtained.
2.Restore all the controllers to joint operation under automatic control
and readjust the tuning parameters until the overall closed-loop
performance is satisfactory in all the loops.
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When the interactions between the control loops are not too
significant, the procedure mentioned before can be quite useful.
However, for systems with significant interactions, the
readjustment of the tuning in Step 2 can be difficult and tedious.
One can cut down on the amount of guesswork that goes into
such a procedure by noting that in almost all cases, the controllers
will need to be made more conservative (ie. the controller gains
will have to be reduced and the integral times increased) when all
the loops are closed in comparison to when all of the individual
controllers are operating individually, with all of the other loops
open. The process of this changing of the control parameters is
referred to as “detuning”.
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One method of “detuning” for a 2x2 system is as follows:
1.Use any of the single-loop tuning rules (Ziegler-Nichols, Cohen
and Coon, etc) to obtain starting values for the individual
controllers; let the controller gains be Kci*.
2. These gains should be reduced using the following expressions
that depend on the relative gain parameter λ:
(   2   ) K *   1.0
ci

K ci  
2
      K ci *   1.0
It may still be necessary to “retune” these controllers after they
have been put in operation; however, this will not require as
much effort as if one were starting from scratch.
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Design of Multivariable Controllers
PIECE
DESIGN OF MULTIVARIABLE
CONTROLLERS-Introduction
In the next section, we will discuss the design of true
multivariable controllers that utilize all of the available process
output information jointly to determine what the complete input
vector u should be. Thus each control command from the
multivariable controller will be based on all of the output
variables, not just based on one. In principle, it will be possible to
eliminate all of the interactions between the process variables.
The objective of the next section is to present some of the
principles and techniques used for designing multivariable
controllers, as designing multivariable controllers is one of the
more challenging problems faced in industrial process control. We
will start by addressing loop decoupling, the most widely used
multivariable controller technique. We will then address Singular
Value Decomposition (SVD) which is a means of determining
when it is structurally unstable to apply decoupling to a system.
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yd
PIECE
-
+
ε1
1
v1
gc1
u1
+
g11
y1
+
+
+
gI1
g12
Please consider the following system:
g21
gI2
yd
+
ε2
2
gc1
v2
+
+
u2
g22
+
+
y2
-
Module 5 – Controllability Analysis
Figure 1-D
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Let’s assume that the input/output variable pairing has been
determined to be: y1-u1, y2-u2 … yn-un pairings.
Under the multiple, independent, single-loop control strategy,
each controller gci operates according to:
The controller transfer
function multiplied by
the difference in the set
point of yi(ydi) and the
actual yi output
ui=gci(ydi-yi)
OR
ui=gciεi
The difference
between the
desired yi and
the actual yi
output.
The output error
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However, a true multivariable controller must decide on ui, not
using only εi, but using the entire set of ε1, ε2 … εn.
Thus, the controller actions are obtained by:
u1=f1 (ε1, ε2 , … εn)
u2=f2 (ε1, ε2 , … εn)
u3=f3 (ε1, ε2 , … εn)
…
un=fn(ε1, ε2 , … εn)
The design problem is to find the f1(.),f2(.)…fn(.) so that each of
the output variable errors is driven to zero.
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DECOUPLING INTRODUCTION
Decoupling:
In Decoupling, as seen in the Figure on Slide 132, additional
transfer function blocks are introduced between the single-loop
controllers and the process, functioning as links between the
otherwise independent controllers. The actual control action
experienced by the process will now contain information from all
of the controllers. For example, a 2x2 system, whose individual
controller outputs are gc1ε1 and gc2ε2 if the decoupling blocks for
each loop have transfer functions of gI1 and gI2 respectively, then
the control equations will be given by:
u1=gc1ε1+gI1 (gc2ε2)
u2=gc2ε2+gI2 (gc1ε1)
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Decoupling Introduction
We know from our discussion of input/output pairing that the
pairing of y1-u1, y2-u2,…yn-un couplings are desirable; it is however
the yi-uj cross-couplings, by which yi is influenced by uj (for all i
and all j with i≠j), that are undesirable: they are responsible for
the control loop interactions.
It is clear that any technique that eliminates the undesired crosscoupling will improve the performance of control systems. It is
however NOT possible to ELIMINATE the cross-couplings; that is
a physical impossibility since it will require altering the physical
nature of the system. Consider an example of this on the
following slide.
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Cold flow rate
Hot flow rate
Module 5 – Controllability Analysis
PIECE
It is not possible to stop the
hot stream from affecting
the temperature of the
stirred tank, even though
the main objective of this
stream is to maintain the
tank level. It is also true
that we can not prevent the
cold stream from affecting
the tank level even though
controlling the temperature
is its main responsibility.
137
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yd
PIECE
+ ε
-
Gc
Single Loop
Controller
v
GI
u
G
y
Interaction
Compensation
The main objective in decoupling is to compensate for the effect of
interactions as a result of cross-coupling of the process variables. As
shown in the figure above, this can be achieved by introducing an
additional transfer function “block”( the Interaction Compensator)
between the Single Loop Controllers and the process. This Interaction
Compensator, together with the Single Loop Controllers now form the
multivariable decoupling controller. In the ideal case, the decoupler
causes the control loops to act as if they are totally independent of each
other, reducing the tuning task so that it will be possible to use SISO
design
techniques.
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The design problem is to find the element GI (the
compensator) to satisfy one of the following objectives.
•Dynamic Decoupling- To eliminate interactions from all
control loops, at every instant in time
•Steady-State Decoupling- To only eliminate steady-state
interactions from all loops; in this case dynamic interactions are
tolerated. Although this type of decoupling is less rigorous than
this dynamic decoupling, it leads to much simpler decoupler
designs.
•Partial Decoupling- To eliminate dynamic or steady-state
interactions in a subset of the control loops. This focuses only on
the critical loops with the strongest interactions, leaving those
with weak interactions to act without decoupling.
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SIMPLIFIED DECOUPLING
Design of Ideal Decouplers - Simplified Decoupling
First we will consider some important aspects of the block
diagram in Figure 1-D (found on slide 132)
1. There are two compensator blocks gI1 gI2 , one for each loop.
2. There is a new notation: the controller outputs are now v1
and v2, while the actual control action implemented on the
process remains as u1 and u2. This distinction is necessary
because the output of the controllers and the control action to
be implemented on the process no longer have to be the
same.
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SIMPLIFIED DECOUPLING
3. Without the compensator, u1=v1 and u2=v2 and the process
model remains
y1=g11u1+g12u2
y2=g12u1+g22u2
The interactions persist, as u2 is still cross-coupled with and
affecting y1 through the g12 element, and u1 affects y2 by crosscoupling through g21.
4. With the interaction compensator, Loop 2 is “informed” of
changes in v1 through gI2, so that u2 “what the process actually
feels” is adjusted accordingly. The same process is preformed
by Loop 1 by gI1 which adjusts u1 from information about v2.
Module 5 – Controllability Analysis
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The design question is now posed:
What should gI1 and gI2 be if the effects of loop
interactions are to be completely neutralized?
To answer this:
Let’s consider Loop 1 in Figure 1-D where the process model is :
y1=g11u1+g12u2
y2=g12u1+g22u2
Because of the compensators, the equations governing the control
action are:
u1=v1+gI1v2
u2=v2+gI1v1
Module 5 – Controllability Analysis
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If we substitute the expressions for u1 and u2 into the expressions
for y1 and y2 seen on the previous slide the system is defined as:
y1= g11(v1+gI1v2) + g12(v2+gI1v1 )
y2=g12(v1+gI1v2) +g22(v2+gI1v1 )
Which Yields
y1=(g11+g12gI2)v1+(g11gI1+g12)v2 (Eq.1-D)
y2=(g21+g22gI2)v1+(g22+g12gI1)v2 (Eq.2-D)
Module 5 – Controllability Analysis
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In order to only have v1 affect y1 and to eliminate the effect of v2
on y1, we must choose a value of gI1 so that the coefficient of v2
in Eq.1-D will disappear i.e.:
g11gI1+g12=0
Then solving for gI1
g12
gI1 = g11
A similar procedure can be done for Loop 2, which eliminates any
influences of v1 on y2, with the manipulation of Eq 2-D we obtain
a value of:
g21
gI2 = g22
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The transfer functions seen on the previous slide are the
decouplers needed to exactly compensate for the effect of loop
interactions in the 2x2 system shown in Figure 1-D.
If we now substitute our expressions for gI1 and gI2 into Equations
1-D and 2-D respectively we will yield:

g12g21 
y1=  g11  v1
g22 

Module 5 – Controllability Analysis

g12g21 
y 2 =  g22  v2
g11 

145
NAMP
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Now the system is completely decoupled with only v1 affecting y1,
and v2 affecting y2.
We can see in the figure below the equivalent block diagram
where the loops appear to act independently and therefore, can
be individually tuned.
yd1 +
yd2 +
-
gc1
gc2
v1
g12g21
g11 g22
v2
g g
g22 - 12 21
g11
Module 5 – Controllability Analysis
y1
y2
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NAMP
PIECE
Let’s consider that the closed loop system is under steady state. If
the steady state gain for an element gij =Kij, observe how the
system is expressed at steady-state.

K K 
y1=  K 11 - 12 21  v1
K 22 


K 12K 21 
y 2 =  K 22  v2
K 11 

Recall the definition of λ for a 2x2 system:
Then the system simplifies to:
K 
y1 =  11  v1
  
Module 5 – Controllability Analysis

1
K 12K 21
1
K 11K 22
K 
y 2 =  22  v 2
  
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When we examine the simplified decoupling , the effective closedloop steady-state gain in each loop is the ratio of the open-loop
gain and the relative gain parameter (λ).
Note that when λ is very large, the effective closed-loop gains
become very small, and control system performance may be
jeopardized.
It is important to note that when dealing with systems with
dimensions larger than 2x2, the simplified decoupling method can
become very tedious. For an N x N system there are (N2-1)
compensators. The same principles as used for a 2 x 2 system are
applicable, but the work becomes very cumbersome.
On the next slide we will see an example of a 3 x 3 system, which
has 6 compensator blocks, it is clear that using simplified
decoupling in this situation would be very tedious.
Module 5 – Controllability Analysis
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yd1
+
PIECE
-
gc1
v1
u1
+
gI12
+
yd2
+
v2
gc2
gI23
+
-
gc3
v3
gI32
Module 5 – Controllability Analysis
+
+
g13
u3 u 2 u1
+
+
u2
g21
g22
g23
+
gI31
yd3
y1
+
g12
+
gI13
gI21
g11
+
+
y2
+
g31
+
+
+
u3
g32
g33
+
+
y3
+
149
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GENERALIZED DECOUPLING
Generalized Decoupling
Please refer to Figure 1-D which we will use this figure to outline
a more generalized procedure for decoupler design.
yd
-
+
ε1
1
gc1
v1
u1
+
gI1
+
ε2
+
2
-
gc1
v2
Module 5 – Controllability Analysis
+
g12
gI2
yd
g11
+
g21
+
+
u2
Figure 1-D
g22
+
+
y2
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PIECE
GENERALIZED DECOUPLING
1. We can observe from Figure 1-D that:
y=Gu
u=GIv
So that:
y=GGIv
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GENERALIZED DECOUPLING
2. In order to eliminate all interactions, y must be related to v
through a diagonal matrix, let us call it GR(s), now we must chose
GI such that
GGI=GR(s)
And the compensated input/output relation becomes:
y=GR(s)v
Where GR represents the equivalent diagonal process that the
diagonal controllers GC are required to control.
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3. Therefore, the compensator (GI) must be given by:
GI=G-1 GR
4. The compensator obtained depends on what GR is selected.
The elements of GR should be chosen to provide the desired
decoupled behaviour with the simplest possible decoupler. A
common choice for GR is:
GR=Diag[G(s)]
Ie. The diagonal elements of G(s) are retained as the elements of
the diagonal matrix GR, however, other choices have been used.
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The Relationship between Generalized and Simplified
Decoupling
“Generalized” decoupling may be related to simplified decoupling, by
noting that for simplified decoupling applied to a 2x2 system, the
compensator transfer function matrix is given by:
1
GI = 
gI2
gI1 

1 
While for a 3x3 system, the compensatory matrix GI takes the form:
 1

GI = gI21

g

I31

Module 5 – Controllability Analysis
gI12
1
gI32
gI13 

gI23 

1 
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NAMP
PIECE
Quiz #8
• What is the main objective of decoupling?
• What is a downfall of simple decoupling?
• Is it often easy to achieve perfect decoupling?
Module 5 – Controllability Analysis
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LIMITATIONS OF
DECOUPLING
Some Limitations of the Application of Decoupling
There are some limitations to the application of decoupling, and
we must keep these in mind in order to maintain a proper
perspective when designing decouplers.
Perfect decoupling is only possible if the process model is perfect,
which is hardly ever the case, so perfect decoupling in practice
is impossible.
Perfect dynamic decouplers are based on model inverses. As such,
they can only be implemented if such inverses are both causal
and stable.
Module 5 – Controllability Analysis
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PIECE
LIMITATIONS OF
DECOUPLING
To illustrate the idea of stable and casual, please consider the 2x2
compensators we saw in Figure 1-D whose transfer functions are
GI1 and GI2 must be casual (no e+αs terms) and stable.
To satisfy causality for the 2x2 system, any time delays in g11
must be smaller than the time delays in g12 and a similar condition
must hold for g22 and g21.
To satisfy stability, a second condition that g11 and g22 must not
have any right hand plane zeros and also g12 and g21 must not
have any right hand plane poles. This leads to the following
general conditions that must be satisfied in order to implement
simplified dynamic decoupling for N x N systems.
Module 5 – Controllability Analysis
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LIMITATIONS OF
DECOUPLING
1.Causality: In order to ensure causality in the compensator
transfer functions the time-delay structure in G(s) must be such
that the smallest time-delay in each row occurs on the diagonal.
For simplified decoupling, this is an absolute requirement, but it is
possible to add delays to the inputs u1,u2…un, to satisfy the
requirement if the original process G does not comply. This is
equivalent to defining a modified process as Gm:
Gm=GD
Where D is a
diagonal matrix of
time delays
Module 5 – Controllability Analysis
e d11s

D(s)= 


 0
e d22s
0 



dnn s 
e 
158
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PIECE
LIMITATIONS OF
DECOUPLING
The simplified decoupler is then designed by using the elements
of Gm rather than G, and the matrix D must be inserted into the
control loop as shown below:
modified process Gm
+
yd
ε
-
Gc
v
Single Loop
Controllers
GI
Decoupler
u
D
Delays
G
Process
y
In the case of generalized decoupling, one may use the modified
process Gm as above, or alternatively, the time delays in the
diagonal matrix GR can be adjusted, in order that the elements of
GI=(GD)-1GR are casual. This is equivalent to requiring that GR-1GD
have the smallest delay in each row on the diagonal.
Module 5 – Controllability Analysis
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PIECE
LIMITATIONS OF
DECOUPLING
2. Stability- In order to ensure the stability of the
compensator transfer functions, the causality condition
must be satisfied and there are no Right Hand Plane
zeros of the process G(s). This is an absolute
requirement for simplified decoupling and reduces to
the condition that there are no Right Hand Plane zeros
in the diagonal elements of G and that the off-diagonal
elements of G are stable. For generalized decoupling,
this may be performed by adjusting the dynamics of GR
in order that the elements of GI=G-1GR be stable.
Module 5 – Controllability Analysis
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PIECE
PARTIAL DECOUPLING
Partial Decoupling
If some loop interactions are weak or if some of the loops do not
need to achieve high performance, the partial decoupling is a
method one should consider. If this is the case, only a subset of
the control loops where the interactions are important and high
performance is important are focused on.
Typically partial decoupling is considered for 3x3 or higher
dimension systems. The main advantage is the reduction of
dimensionality. Partial decoupling is also applicable to 2x2
systems, in this case, one of the compensator blocks is set to zero
for the loop that is to be excluded from decoupling.
Module 5 – Controllability Analysis
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PIECE
STEADY-STATE
DECOUPLING
Steady-State Decoupling
The difference between dynamic decoupling and steady-state
decoupling is that dynamic decoupling uses the complete,
dynamic version of each transfer function element to obtain the
decoupler, and steady-state decoupling only uses the steady-state
gain portion of each of the transfer elements.
Therefore, if each transfer function element gij(s), has a steadystate gain term Kij, and if the gain matrix is defined as K, the
steady-state decoupling results in the same way as it did for a 2x2
system that we discussed earlier.
Module 5 – Controllability Analysis
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STEADY-STATE DECOUPLING
FOR A 2X2 SYSTEM
Simplified steady-state decoupling for a 2x2 system
Here:
K12
gI1 = K11
and
K 21
gI2 = K 22
These expressions to describe the transfer function of the
compensator block are simple, constant, numerical values so
they will always be realizable and can be implemented.
Module 5 – Controllability Analysis
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PIECE
STEADY-STATE DECOUPLING
FOR A 2X2 SYSTEM
Simplified steady-state decoupling for a 2x2 system
In this case, the decoupler matrix is given by:
-1
GI =K K R
Where KR is the steady-state version of GR(s). The inversion
indicated is a matrix of numbers, and therefore, the
inversion will always be realizable and easily implemented.
The main advantages of steady-state decoupling are that the
design involves simple numerical computations and that
the resulting decouplers are always realizable.
Module 5 – Controllability Analysis
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Quiz #9
• What 2 conditions must a system satisfy to achieve
perfect dynamic decoupling?
• What is the main advantage of partial-decoupling?
• Why is steady-state decoupling a favorable method if
applicable?
Module 5 – Controllability Analysis
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PIECE
SINGULAR VALUE DECOMPOSITION
Singular Value Decompostion
Any real n x m matrix K, it is possible to find orthogonal (unitary)
matrices W and V such that
WTAV=∑
Here ∑ is the m x n matrix described below:
s 0
  0 0 


where
 1 0
0 
2
s


0 0
0
0
0
0
0 


r 
Where, for p=min(m,n), the diagonal elements of S:
σ1> σ2> … > σr> 0,(r > p), together with σr+1=0, σp=0 are called the
singular values of A; these are the positive square roots of the
eigenvalues of ATA; r is the rank of A .
Module 5 – Controllability Analysis
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SINGULAR VALUE DECOMPOSITION
W is the m x m matrix
W=w1 w2
wm 
Whose columns wi, i=1,2,…,m are called the left singular vectors
of A; these are normalized (orthonormal) eigenvectors of AAT.
V is the n x n matrix:
V=v1 v2
vn 
Whose n columns vi, i=1,2,…,n are called the right singular
vectors of A; these are normalized (orthonormal) eigenvectors of
ATA.
Module 5 – Controllability Analysis
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PIECE
SINGULAR VALUE DECOMPOSITION
Because they are composed of orthonormal vectors, the matrices
W and V are orthogonal (or unitary) matrices i.e.
WTW=I=WWT
So that
Also
So that
Module 5 – Controllability Analysis
W-1=WT
VTV=I=V VT
V-1=VT
168
NAMP
PIECE
SINGULAR VALUE DECOMPOSITION
By applying these properties of unitary matrices, we can obtain
the relationship:
A=W ∑ VT
Analogously to the eigenvalue/eigenvector expression for square
matrices, we have the more general pair of expressions
Avi= σiwi
ATiwi= σivi
Module 5 – Controllability Analysis
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SINGULAR VALUE DECOMPOSITION
The ratio of the largest to the smallest singular value is
designated the condition member of A:
ie.
1
 (A) 
p
This gives the most reliable indication of how close A is
to being singular. Note that for a singular matrix,
κ(A)=∞, thus nearness to singularity is indicated by
excessively large (but finite) values for κ(A)
Module 5 – Controllability Analysis
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PIECE
SINGULAR VALUE DECOMPOSITION
EXAMPLE
Example - Singular Value Decomposition of a 3x2 matrix
1 2


A   2 1
 2 1 
Therefore,
 9 2 
A A= 

 2 6 
T
Module 5 – Controllability Analysis
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NAMP
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SINGULAR VALUE DECOMPOSITION
EXAMPLE
The eigenvalues are obtained as 10 and 5 , thus the
singular values of A are :
σ1, σ2=√10 and √5
Ordered so that σ1>σ2 as required for SVD analysis, the
next step is to determine the 3x2 matrix ∑.
Module 5 – Controllability Analysis
 10

 0

 0
0 

5

0 

172
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SINGULAR VALUE DECOMPOSITION
EXAMPLE
Right Singular Values
The first eigenvector or ATA corresponding to λ1 is obtained from
adj(ATA- λ1 I)
-4 2 
T
adj(A A- 1 I) = 

2

1


A possible choice for the eigenvector is the second column.
Normalizing this with √22+12= √5, the norm of the vector, we
obtain the first right singular vector v1 corresponding to σ1= √10
 2 
 
5
v1 =  
 1 
 
5

Module 5 – Controllability Analysis
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SINGULAR VALUE DECOMPOSITION
EXAMPLE
In the same way, the second normalized eigenvalue corresponding
to λ2=5 is:
 1 
 
5

v2 =
 2 
 
 5
Therefore:
 2

5
v =
 1

 5
Module 5 – Controllability Analysis
1 

5
2 

5
174
NAMP
PIECE
SINGULAR VALUE DECOMPOSITION
EXAMPLE
From V we can determine VT to be:
VT
 2 1 


5
5

=
2 
 1


5
5


You can verify that V is a unitary matrix by evaluating
VTV and confirming that the product is I.
Module 5 – Controllability Analysis
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SINGULAR VALUE DECOMPOSITION
EXAMPLE
Left Singular Values
For the given matrix:
5 0 0 


T
AA  0 5 5 
0 5 5 
The Eigenvalues of this 3x3 matrix are obtained from the
characteristic equation which in this case is:
(5-λ) [(5- λ)2--25]=0
Ie. λ1,λ2, λ3= 10,5,0
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Note that the non-zero eigenvalues of AAT
eigenvalues of ATA.
 5 0
For λ1=10
(AA T - 1 I) =  0 5
 0 5
SINGULAR VALUE DECOMPOSITION
EXAMPLE
are identical to the
0
5 
5 
To find the adjoint of this matrix, we first find the cofactors and
take the transpose of the matrix of cofactors. In this case,
0 
0 0
adj(AA T - 1 I) = 0 25 25 
0 25 25 
Module 5 – Controllability Analysis
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SINGULAR VALUE DECOMPOSITION
EXAMPLE
And by normalizing any of the non-zero columns, we obtain the
first left singular value of A, and by a similar procedure the second
and third eigenvectors can be determined using values of λ2, =5
and λ3=0


 0 


 1 
w1 = 

2


 1 


 2
Module 5 – Controllability Analysis
1 
w 2 = 0 
0 




w3 = 





0 

1 

2
1 

2
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SINGULAR VALUE DECOMPOSITION
EXAMPLE
When the 3 eigenvalues are combined:

 0

 1
W =
 2
 1

 2
1
0
0

0 

1 

2
1 

2
Now we have all of the elements desired to decompose the
matrix. You can verify that A=W ∑ VT by multiplying the elements
we have determined.
Module 5 – Controllability Analysis
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STEADY-STATE DECOUPLING BY
SINGULAR VALUE DECOMPOSITION
Steady-State Decoupling by Singular Value
Decomposition
The Singular Value Decomposition (SVD) of the steady-state gain
matrix of a process is another approach to steady-state
decoupling.
The SVD of a process gain matrix K can be written as:
K=W ∑ VT
then applying the SVD of K, the steady state model becomes:
y= W ∑ VT u
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STEADY-STATE DECOUPLING BY
SINGULAR VALUE DECOMPOSITION
We will multiply both sides by WT (recall the orthogonality
properties of W), our expression becomes:
WTy= ∑ VT u
Recall that when the matrix K is a square matrix ∑ is a diagonal
matrix of singular values. This allows us to define a new output
variables η and new input variables μ where:
η= WTy
And
μ =∑ VT u
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STEADY-STATE DECOUPLING BY
SINGULAR VALUE DECOMPOSITION
Now, the process model becomes
η=∑μ
Because ∑ is diagonal, this indicates that the system is completely
decoupled at steady state.
yd
WT
ηd +
Gc∑
-
μ
V
u
G
y
η
WT
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STEADY-STATE DECOUPLING BY
SINGULAR VALUE DECOMPOSITION
The implication of this is the following: instead of
controlling y with u, the transformed variables will
convert the original system (with cross-coupling
among the process variables) to a system that has no
cross-coupling. The open-loop gain of each loop of the
transformed system is indicated clearly by the singular
values and conditioning is automatically accessed from
the condition number.
A controller can now be designed for the equivalent
(steady-state) system which controls η by using μ. If
this controller is designated Gc∑ then the scheme
would be implemented as seen in the previous slide.
Module 5 – Controllability Analysis
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REFERENCES
References:
• Ogunnaike,B.,Ray,W. Process Dynamics, Modeling, and
Control. Oxford University Press, New York (1994)
• Seborg, D., et al. Process Dynamics and Control.
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