Program for North American Mobility In Higher Education

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NAMP
Program for North American Mobility In Higher
Education
Module 5
Controllability Analysis
PIECE
Introducing Process integration for Environmental
Control in Engineering Curricula
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PIECE
Process integration for Environmental Control in
Engineering Curricula
NAMP
Program for North American Mobility in Higher
Education
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Module 5
This module was created by
Stacey Woodruff
From
Host University
Carlos Carreón
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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

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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.

6
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.

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Tier IBackground Information
8

• 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)

9

• 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.

10

• Why is the stability of a system important??
• When a system becomes unstable it can
  be
• A DISASTER!!!!!

11

• 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.

12

• Quiz 1
• Why is it important that a system is stable?
• List two examples of systems that have become
  unstable.

13

• There are many ways of determining if a system is
  stable such as
• Roots of Characteristic Equation
• Bode Diagrams
• Nyquist Plots
• Simulation

14

• 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

15

• From the previous diagram, we can see that the
  output Y is influenced in the following manner.
• GOL is the open loop transfer function.

16

• For the moment, lets consider that there is only
  a change in set point, therefore, the previous
  equation reduces to the closed loop transfer
  function,
• The roots r1, r2, r3 rn are those of the
  characteristic equation
• 1GcG1G2G4 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.

17

• 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
f
f
time
time
Negative real root
Positive real root
Stable Region
Unstable Region
Real Part
Imaginary Part
Imaginary Part
Stable Region
Real Part
Real Part
f
f
time
time
Complex Roots (Negative real parts)
Complex Roots (Positive real parts)
18

• 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
• A necessary (but not sufficient) condition of
  stability is that all of the coefficients (a0,
  a1, a2, etc.) must be positive.

19
Routh Array

• When all coefficients are positive, a Routh Array
  must be constructed as follows
• The system is stable if ALL the elements in the
  first column are positive!

The first two rows are filled in using the
coefficients of the characteristic equation.
Subsequent rows are calculated as shown in the
next page.

20
Routh Array

• After the coefficients of the characteristic
  equation are input in the array, the
  coefficients, b1, b2 bn and subsequently c1cn
  should be calculated as follows and input into
  the array.

21

• 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
• 
  
• All of the elements in the first
  column of this
  Routh Array are positive,
• therefore the system is stable.

22

• Routh Test Example 2- It is possible to
  determine for which values of Kc the system
  remains stable

29.24-(1-Kc)/0.384gt0 ? Kc lt10.23 1Kc gt0 ? Kcgt-1
(Kc is positive)
23

• 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?

There are 2 sign changes. Therefore, the system
has two roots in the right-hand plane, and the
system is unstable.
24

• 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
• 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.

Cs2D0
25

• Routh Test Example 4 Determine the stability of
  the system having the following characteristic
  equation

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.
26

• Quiz 2
• In what cases can the Routh test be used to
  determine stability?
• Is the system having the following characteristic
  equation stable?
• 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?

27

• 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

28

• Bode stability criterion
• A closed-loop system is unstable if the Frequency
  Response of the
• open-loop Transfer Function, GOLGCG1G2G4, has an
  amplitude ratio
• greater than one at the critical frequency, ?c.
  Otherwise the closed-loop system is stable.
• Note ?c is the value of ? where the open-loop
  phase angle is -1800.
• Thus,
• The Bode Stability criterion provides information
  on the closed-loop stability from open-loop
  frequency response information.

29

• Bode Stability Criterion- Example 1
• A process has the following transfer function
• With a value of G10.1 and G410. If proportional
  control is used, determine closed-loop stability
  for 3 values of Kc 1, 4, and 20. GOLGCG1G2G4
• Solution

Kc AROL for Kc Stable?
1 0.25 Yes
4 1 Marginally
20 5 No
You will find the Bode plots on the next slide
30
Bode plots for GOL 2Kc/(0.5s 1)3
31

• 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, ZNP,
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 Z0.
32

• Example 9.2 Find the amplitude ratio and the
  phase lag of the following process for ? 0.1
  and 0.4.

33

• Example 9.2 Find AR and ? (from known equations)

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• Example 9.2 Find AR and ? Nyquist plot

Im
Re
35

• 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 Z0 signify?

36

• Multiple Input Multiple Output (MIMO) Systems

37

• 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

38

• Lets consider the following system

Loop 1
-
y1
m1
Gc1
G11



G12
G21


y2

m2
G22
Gc2
-
Loop 2
y1(s) G11(s)m1(s) G12(s)m2(s) y2(s)
G21(s)m1(s) G22(s)m2(s)
39

• We will perform 2 small experiments to
  demonstrate MIMO system interactions.
• Lets 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
  steady-state is reached, lets consider the
  change in y1 as a result of the change in m1,
  ?y1m this will represent the main effect of m1
  on y1.
• ?y1m K11
• Keep in mind that no other input variables have
  been changed, and that all loops are open, so no
  feedback control is required.

40

• 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 (?y1m)
• (2) the Indirect Influence, from the retaliatory
  action from Loop 2 in warding off the interaction
  effect of m1 on y2 (?y1r)

41

• After dynamic transients die away and
  steady-state is reached, the net change observed
  in y1 is given by
• ?y1 ?y1m ?y1r
• 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.
• A good measure of how well a system can be
  controlled (?) if m1 is used to control y1 is

42

• Loop Pairing on the Basis of Interaction Analysis
• Case 1 ?111
• 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.

43

• Case 2 ?110
• 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.

44

• Case 3 0 lt ?11lt 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
  ?11gt0.5, 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 ?11 becomes. For
  ?11lt0.5, the contribution from the interaction
  effect dominates, as this contribution increases,
  ?11 moves closer to zero. For ?110.5, the
  contributions of the main effect and the
  interaction effect are equal.

45

• Case 4 ?11gt1
• 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 ?11 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 ?11lt 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.

46

• Quiz4
• What is a MIMO system?
• What does ?111 signify? If this is the case, is
  m1 a good input variable to control y1?
• If ?11 is very large and positive, is m1 a good
  input variable to control y1?

47

• Relative Gain Array (RGA)
• The quantity ?11 is defined as the Relative Gain
  between input m1 and output y1.
• ?ij is defined as the relative gain between
  output yi and input mj, as the ratio of two
  steady-state gains

48

• 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
• THE RELATIVE GAIN ARRAY

49
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.
• 2. ?ij is dimensionless therefore, neither the
  units, nor the absolute value actually taken by
  the variables mj, or yi affect it.

50
PROPERTIES OF THE RELATIVE GAIN ARRAY

• 3. The value ?ij is a measure of the
  steady-state interaction expected in the ith loop
  of the multivariable system if its output (yi) is
  paired with input (mj) in particular, ?ij 1
  indicates that mj affects yi without interacting
  with the other loops. Conversely, if ?ij0 this
  indicates that mj has no effect on yi.

51
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.
• This equation has the very important implication
  that 1/?ij tells us by what factor the open loop
  gain between output yi and input mj will be
  changed when the loop are closed.

52
PROPERTIES OF THE RELATIVE GAIN ARRAY

• 5. When ?ij 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.

53
COMPUTING THE RELATIVE GAIN ARRAY

• Calculating the Relative Gain Array
• There are two ways of calculating the Relative
  Gain Array
• The First Principles Method
• The Matrix Method

54
COMPUTING THE RELATIVE GAIN ARRAY

• First Principles Method
• Lets 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
• In order to calculate the ?11 we defined earlier,
  we need to evaluate the partial derivatives as
  was explained on slide 47.
• Recall

(Eq. 1a)
(Eq. 1b)
55
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
• 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 y20 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.

56
COMPUTING THE RELATIVE GAIN ARRAY

• The computation of the denominator of ?11
• Set y20 and solve m2 in Eq. 1b.
• Substituting this value of m2 into Eq. 1a. gives
• Having eliminated m2 from the equation, we now
  may differentiate with respect to m1.

57
COMPUTING THE RELATIVE GAIN ARRAY

• We then substitute the numerator and denominator
  into the definition of ?11 which yields
• This equation simplifies to the form
• where

58
COMPUTING THE RELATIVE GAIN ARRAY

• This exercise should be repeated for all ?ijs so
  that the RGA can be constructed.
• For Practice, repeat this exercise and verify the
  following.
• and

59
COMPUTING THE RELATIVE GAIN ARRAY

• Thus the RGA for this 2x2 system is given by
• Note, that if we define
• The RGA can be rewritten as follows

60
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.
• Whose elements are Kij, further, let R be the
  transpose of the inverse of this steady state
  matrix (K)

61
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
• 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!

62
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.

63
COMPUTING THE RELATIVE GAIN ARRAY

• From the definition of the inverse matrix we know
  that
• Where the determinant of K, K is
• Therefore, by taking the transpose of the K-1
  matrix, we obtain the R matrix

64
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 (?ij) of the RGA (?)
• OR
• here is the first element of the matrix. Try on
  your own to compute the other 3 elements of the
  RGA.

65

• 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
• Solution For this system, the steady-state gain
  matrix is easily extracted from the transfer
  function matrix by setting s0.

66

• The next step is to determine the inverse of the
  matrix K
• Once the inverse is calculated, the transpose of
  this matrix must be calculated to yield the
  matrix R.
• After these two matrices are computed, it is time
  to calculate the RGA by multiplying the matrices
  element by element.
• 
  
• 
  

Note that all of the rows and columns sum to one.
67
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 (?ij). We
  will use the five scenarios presented early to
  interpret the implications of the values of ?ij

68
LOOP PAIRING USING THE RELATIVE GAIN ARRAY

• Case 1 ?ij1, 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.

69
LOOP PAIRING USING THE RELATIVE GAIN ARRAY

• Case 2 ?ij0, 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.

70
LOOP PAIRING USING THE RELATIVE GAIN ARRAY

• Case 3 0lt?ijlt1, 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.

71

• Loop interactions implications
• The loop interactions assist mj on controlling
  yi, The extent of this assistance is dependent on
  how close ?ij is to 0.5
• When
• ?ij 0.5 the main effect of mj on yi is exactly
  the same as the retaliatory effect.
• 0.5lt?ij lt1, the retaliatory effects are less than
  the main effect
• 0lt?ijlt 0.5, the retaliatory effect is larger than
  the main effect.
• Recommendation for pairing If possible, avoid
  pairing yi with mj if ?ijlt0.5

72
LOOP PAIRING USING THE RELATIVE GAIN ARRAY

• Case 4 ?ijgt1, 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 ?ij
  would be negative. For large values of ?ij, 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 ?ij , 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 the value of ?ij is large.

73
LOOP PAIRING USING THE RELATIVE GAIN ARRAY

• Case 5 ?ijlt0, 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.

74

• Quiz5
• What advantages does the Matrix Method have over
  the First Principles Method?
• What does ? with a value of 1 signify, and
  should mj and yi be paired together?
• What does ? with a value less than zero of
  signify, and should mj and yi be paired together?

75

• Basic Loop Pairing Rules
• From what we have learned about loop pairing, it
  is natural that the ideal RGA would take the form
• 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).

76

• 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.
• yields an RGA
• 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.

77

• 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
• 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.

78

• Now, consider the 2x2 system whose transfer
  matrix is
• 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, its RGA is

In this case, it is desirable for a 1-1/2-2
pairing
79

• On the other hand, for the 2x2 systems whose RGA
  is
• 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.)

80

• Lets consider the following 3x3 matrix
• 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.

81
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.

82
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,
• Rational, and
• Open-loop stable

83

• 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.
(Eq. N)
84
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
• where ? defined as follows as
• seen on Slide 57

85
NIEDERLINSKI INDEX

• 2. For a 2x2 system with a negative relative
  gain, ? gt1, 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 (s0, 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.

• This leads us to.

86

• RULE 2
• Any loop pairing is unacceptable if it leads to a
  control system configuration for which the
  Niederlinski Index is negative.

87

• Summary of using RGA for Loop Pairing
• Given the transfer matrix G(s), obtain the
  steady-state gain matrix KG(0), and from this
  obtain the RGA, ?, also calculate the determinant
  of the K and the product of the elements on the
  main diagonal
• Use Rule 1 to obtain tentative loop pairing
  suggestions from the RGA by pairing the positive
  elements which are closest to one.
• 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.

88

• 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)
89

• First, we need to take the inverse of this
  matrix, then take the transpose of this matrix to
  obtain R, being
• The next step is to determine the RGA by
  multiplying the elements of the K and R matrices.

90

• 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
• 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.

91

• 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. Lets try the possible
  pairing of 1-1,2-3,3-2, which would give a RGA
  of

92

• The new K is
• 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.

93

• We need to calculate the determinant and product
  of the elements of the main diagonal of the new
  matrix K
• K0.1481 while the product of the elements is
  equal to 5/3.
• Therefore, the Niederlinski Index is
• Clearly, this Niederlinski Index is positive, so
  we come to the conclusion that this system is no
  longer structurally unstable.

94

• Loop Pairing Example 2 Consider the system with
  the steady state gain matrix as seen below
• The determinant of this matrix is 0.53.
• The RGA is

95

• 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

96

• 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).

97

• 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.

98

• 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?

99
LOOP PAIRING FOR NON-LINEAR SYSTEMS

• Loop Pairing for Non-linear 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.

GC
FC
x
Analyzer
FA
F
Blending
FB
100

• Total Mass Balance
• Mass Balance on Component A
• 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
• (Eq 1) (which is linear)
• And the Component A Mass Balance becomes
• 
  (Eq 2)
  (which is NON-linear)

101

• Since this is a 2x2 system, we only need to
  obtain the (1,1) element of the RGA given by
• Recall
• To calculate the numerator, take the derivative
  of the first equation with both loops open with
  respect to m1 , yielding

102

• 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 xx in Equation 2 and solve for m2 in
  terms of m1 and x, the result is
• 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

103

• The next step is to differentiate the expression
  of F obtained in the last step with respect to m1
  yielding
• If the numerator and denominator are substituted
  into the statement for the relative gain (?), we
  get
• For a 2x2 matrix recall that the RGA is given by
• Therefore the RGA of this system is

Where x is the desired mole fraction of A in the
product.
104

• Some things to consider about these results
• 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.
• It is implied that the recommended loop-pairing
  will depend on the steady-state operating point.
• Due to the fact that x is a mole fraction, it is
  bounded between 0 and 1 (0 lt xlt 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.

105

• 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,
  F-m1, 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.

106

• 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 (x0.5). 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.

107
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 de-ethanizer to demonstrate
  one method to recommend a loop pairing strategy.

108

• 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.
• Solution- Our usual course of action to determine
  the RGA is to normally calculate the K matrix
  which is G(s) when s0. Unfortunately, we can see
  that elements (1,2) and (2,2) contain pure
  integrator elements represented by 1/s, which if
  we set s0 would yield an undefined number.

109

• Lets make the substitution,
• If I is substituted into G(s), K becomes
• The relative gain parameter (?)

110

• We can see that in the ? term the Is cancel out,
  so we obtain
• ?0.97
• Therefore the resulting RGA is
• 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.

111
LOOP PAIRING FOR NON-SQUARE 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 ).

112

• 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?

113

• Classifying Non-Square Systems
• We have 2 types of non-square systems,
• Underdefined- there are fewer input variables
  than output variables.
• 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 mltn and OVERDEFINED if mgtn

114
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. mthe number
of inputs 2 nthe number of outputs 4
mltn
115

• 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.

116
Overdefined 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. mthe number
of inputs 3 nthe number of outputs 2
mgtn
117

• 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 (mgtn), 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
  possible square subsystems.

Recall that
118

• The Variable Pairing Strategy for Overdefined
  Systems is
• 1. Determine all of the 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.

119
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

120

• In general, the steady-state gain between the ith
  variable and the jth variable will be given by
• Thus, the elements of the K matrix can be
• calculated, and once the K matrix is known, it is
  easy to calculate the RGA.

121

• Approach 2- It is possible to determine each
  element of the RGA directly from experimentation.
• As you may recall, each RGA element (?ij) can be
  obtained by performing two experiments. The first
  experiment determines the open-loop 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
  steady-state 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 ( ?ij ).
• 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.

122

• 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.

123

• 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.

124

• 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.

125

• Quiz7
• 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?

126

• 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.

127

• It should be obvious that when the RGA for a
  process is close to ideal (ie. ?ij 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. ?ij
  is very large or negative) the controller is not
  likely to perform well even if it is tuned well.

128

• 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.

129

• 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.

130

• 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 ?
• 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.

131
DESIGN OF MULTIVARIABLE CONTROLLERS-Introduction

• Design of Multivariable Controllers
• 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.

132
y1
-

e1
yd1
v1
u1


gc1
g11


gI1
g12
Please consider the following system
g21
gI2


y2
v2
u2

e2

yd2

gc1
g22
-
Figure 1-D
133

• Lets 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
• uigci(ydi-yi)
• OR
• uigciei

The difference between the desired yi and the
actual yi output.
The controller transfer function multiplied by
the difference in the set point of yi(ydi) and
the actual yi output
The output error
134

• However, a true multivariable controller must
  decide on ui, not using only ei, but using the
  entire set of e1, e2 en.
• Thus, the controller actions are obtained by

u1f1 (e1, e2 , en) u2f2 (e1, e2 , en) u3f3
(e1, e2 , en) unfn(e1, e2 , en)
The design problem is to find the
f1(.),f2(.)fn(.) so that each of the output
variable errors is driven to zero.
135
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 gc1e1 and gc2e2
  if the decoupling blocks for each loop have
  transfer functions of gI1 and gI2 respectively,
  then the control equations will be given by
• u1gc1e1gI1 (gc2e2)
• u2gc2e2gI2 (gc1e1)

136

• 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 whic