PART II. INTERACTION ANALYSIS - Kalyana Veluvoluncbs.knu.ac.kr/Teaching/MCS_Files/Part2.pdf · 2015-03-05 · Part II Interaction Analysis 2 feedback control involved. Experiment

Part II Interaction Analysis 1

PART II. INTERACTION ANALYSIS

2.1 PRELIMINARY CONSIDERATIONS 2.1.1 A MEASURE OF CONTROL LOOP INTERACTIONS Consider a 2x2 system, y1, y2, and m1, m2.

Fig. 2.1.1 Closed-loop TITO Control System

When both loops are open, m1 and m2 can be manipulated independently and the effect of inputs on outputs is:

1 11 1 12 2

2 21 1 22 2

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

y s g s m s g s m s y s g s m s g s m s

= += +

where each transfer function element consists of an unspecified dynamic portion, and a steady-state gain term Kij.

Consider m1 as a candidate input variable to pair with y1, and perform two “experiments” on the system.

Experiment 1: Unit step change in m1 with all loops open, as shown in Fig. 2.1.2

Fig. 2.1.2 TITO System with Both Loops are Open

In this case, the output variable y1 will change, and so will y2. After steady state has been achieved, let the change in y1 as a result of m1 be ∆y1m; this is the main effect of m1 on y1. and

1 11my K∆ =

As no other input variable has been changed, and all the control loops are opened, there is no

KNU/EECS/ELEC835001 Dr. Kalyana Veluvolu


feedback control involved.

Experiment 2: Unit step change in m1 with Loop 2 closed, as shown in Fig. 2.1.3

Fig. 2.1.3. TITO System with Loop 2 Closed and Loop 1 Open

In this case, the controller gc2 is charged with the responsibility of using the other input variable m2 to ward off any upset in y2 occurring as a result of this step change in m1. m1 has both a direct influence on y1 and an indirect influence with the path given by the dotted line. The followings happen to the process due to the change in m1:

1. y1 changes because of g11, but also because of interactions via the g21 element, so does y2.

2. Under feedback control, Loop 2 wards off this interaction effect on y2 by manipulating m2 until y2 is restored to its initial state before the occurrence of the “disturbance.”

3. The changes in m2 return to affect y1 via the g12 transfer function element.

Thus, the changes observed in y1 are from two different sources:

1. The direct influence of m1 on y1 ; ∆y1m and

2. The retaliatory action from the Loop 2 controller in warding off the interaction effect of m1

on y2 say, ∆y1r.

At steady state, the net change in y1, ∆y1* given by:

1 1 1* m ry y y∆ = ∆ + ∆ (2.1.1)

The quantity ∆y1* will be given by:

*11

2211

211211

*1 1 K

KKKK

Ky =⎟⎟⎠

⎞⎜⎜⎝

⎛−=∆ (2.1.2)

A measure of the process is controlled if m1 is used for y1 is:

*1

111 y

y m

∆∆

=λ (2.1.3a)

or

rm

m

yyy

11

111 ∆+∆

∆=λ (2.1.3b)

Thus the quantity, λ11, provides a measure of the extent of interaction in using m1 to control y1 while using m2 to control y2.



A similar set of “experiments” that using m2 as the input for y1 can be performed. By properly interpreting the measures, we can quantify the degree of steady-state interaction involved with each control configuration, and thus determine which configuration minimizes steady-state interaction.

2.1.2 LOOP PAIRING ON THE BASIS OF INTERACTION ANALYSIS

The quantity λ11 is a measure of loop interaction.

1. λ11 =1

Only if 1ry∆ is zero which means that the main effect of m1 on y1 when all the loops are opened, and the total effect, when the other loop is closed are identical: m1 does not affect y2, or m1 affect y2, but control action from m2 does not cause any changes in y1. In this case, m1 is the perfect input variable to use for y1.

2. λ11 =0

This indicates m1 has no effect on y1, 1my∆ be zero to a change in m1. m1 is the perfect input variable for controlling not y1 but y2:

3. 0<λ11<1

For λ11 > 0.5, the main effect contributes more to the total effect. For λ11 <0.5, the contribution from interaction effect dominates. For λ11 = 0.5, the contributions from the two are exactly equal.

4. λ11 > 1

When ∆y1r, is opposite in sign to 1my∆ , but smaller in value, the total effect, *1ry∆ , is less than the

main effect, ∆y1m, a larger m1 is required to achieve a given change in y1 in the closed loop than in the open loop. For λ11 very large and positive, the interactive effect almost cancels the main effect and closed-loop control of y1 by m1 will be very difficult to achieve.

5. λ11 <0

When ∆y1r is not only opposite in sign to 1my∆ , but is larger in absolute value, the pairing of m1

with y1 is not desirable because the direction of the effect of m1 on y1 in the open loop is opposite to the direction in the closed 1oop.

2.2 THE RELATIVE GAIN ARRAY (RGA) Define λij, the relative gain between output variable yi and input variable mj, as the ratio of two steady-state gains:

loopmtheforexceptclosedloopsforgainloopclosed

gainloopopen

loopjpt mlosed exceall loop cmy

penall loop omy

j

j

i

j

i

ij

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=λ (2.2.1)

When the relative gain is calculated for all the input/output combinations of a multivariable



system, it results in an array of the form:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=Λ

nnnn

n

n

λλλ

λλλλλλ

...............

...

...

21

22221

11211

(2.2.2)

the result is the relative gain array (RGA) and has become the most widely used measure of interaction.

2.2.1 PROPERTIES OF THE RGA Important properties of the RGA:

1. The elements of the RGA across any row, or down any column, sum up to 1, i.e.:

111

== ∑∑==

n

jij

n

iij λλ (2.2.3)

2. λij is dimensionless, neither the units, nor the absolute values taken by the variables mj or yi affect it.

3. The value of λij is a measure of the steady-state interaction expected in the ith loop if its output yi is paired with mj.

4. Let Kij* represent the loop i steady-state gain when all the other loops are closed, whereas Kij represents the normal, open-loop gain. By the definition of λij, that:

ijij

ij KKλ1* = (2.2.4)

1/λij tells by what factor the open-loop gain between yi and mj will be altered when the other loops are closed.

5. When λij is negative, it is indicative of a situation in which loop i, with all loops open, will produce a change in yi in response to a change in mj totally opposite in direction to that when the other loops are closed. Such input/output pairings are potentially unstable and should be avoided.

2.2.2 CALCULATING RGA’S FROM FIRST PRINCIPLES

We illustrate the procedure by a 2 x 2 system. For the steady-state, the model is:

2221212

2121111

m K m KymKm Ky

+=+= (2.2.5)

To obtain λ11 from the definition, we first need to evaluate the partial derivatives for open-loop conditions, the numerator partial derivative is obtained:

11 1

1 Kmy

openloopsall

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ (2.2.6)

The second partial derivative is to close Loop 2, to obtain the second partial derivative, the value m2 must take to keep y2 = 0 in changes in m1; Setting y2 =0 and solving for m2 gives:



122

212 m

KKm −= (2.2.7)

and substituting this into y1 gives:

122

21121111 m

KKKmKy −= (2.2.8)

Having eliminated m2 we may then differentiate, so that:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

2211

211211

21

1 1KKKKK

my

closedloop

(2.2.9)

and we obtain:

ζλ

−=

11

11 (2.2.10)

where

2211

2112

KKKK

=ζ

As an exercise, to illustrate the procedure and confirm that the other relative gains, 222112 and ,, λλλ are given by:

ζζλλ

−−

==12112

ζλλ

−==

11

1122

Thus the RGA for the 2 x 2 system is given by:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−

−−

−=Λ

ζζζ

ζζ

ζ

11

1

111

(2.2.11a)

It is useful to observe that if we define:

ζλλ

−==

11

11

for this 2 x 2 system, the RGA may be written:

⎥⎦

⎤⎢⎣

⎡−

−=Λ

λλλλ

11

(2.2.11b)

The single element λ is sufficient to determine the RGA for a 2 x 2 system, it is called the relative gain parameter.

Using first-principles method of obtaining the RGA, it will be too tedious to apply in cases where the system is of higher dimension than 2 x 2.

2.2.3 THE MATRIX METHOD FOR CALCULATING RGA’S Let K be the matrix of steady-state gains of G(s), i.e.:



KsGs

=→

)(lim0

(2.2.12)

whose elements are Kij; let R be the transpose of the inverse of K, i.e.: -1( )TR K= (2.2.13)

with elements rij. Then, the elements of the RGA can be obtained from the elements of these two matrices according to:

ij ij ijK rλ = ⊗ (2.2.14)

The ⊗ in equation (2.2.14) indicates an element-by-element multiplication of the corresponding elements of K and R,.

Example: RGA for a 2 x2 system using matrix method

For the 2 x 2 system, the steady-state gain matrix is given by:

⎥⎦⎤

⎢⎣⎡=

2221

1211KKKKK

and from the definition of the inverse of a matrix, we have:

⎥⎦

⎤⎢⎣

⎡−

−=−

1121

12221 1KKKK

KK

where |K|, the determinant of K, is given by:

|K| = K11 K22 - K12 K21

and taking the transpose of the matrix given:

( ) 22 211

12 11

1T K KR K

K KK− −⎡ ⎤

= = ⎢ ⎥−⎣ ⎦

Carry out a term-by-term multiplication of the elements of the matrices K and R, we obtain:

21122211

221111 KKKK

KK−

=λ

which, for 11 22 0 K K ≠ simplifies to:

ζλ

−=

11

11

with

2211

2112

KKKK

=ζ

as we earlier obtained. By considering each element of the K and R matrices, we can generate an expression for each of the other λij.

#

Example: RGA for the wood and binary distillation column



Find the RGA for the Wood and Berry binary distillation column whose transfer function matrix is given as:

14.144.19

19.106.6

10.219.18

17.168.12

)( 37

3

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

+

+−

+= −−

−−

se

se

se

se

sG ss

ss

For this system, the steady-state gain matrix is obtained from the transfer function matrix by setting s=0, giving:

⎥⎦

⎤⎢⎣

⎡−−

==4.196.69.188.12

)0(GK

its inverse and transpose of inverse are obtained as:

1 0.157 0.1530.053 0.104

K − −⎡ ⎤= ⎢ ⎥−⎣ ⎦

1 0.157 0.053( )

0.153 0.104TR K − ⎡ ⎤

= = ⎢ ⎥− −⎣ ⎦

term-by-term multiplication of Kij and rij gives the RGA as:

⎥⎦

⎤⎢⎣

⎡−

−=Λ

0.20.10.10.2

#

2.3 LOOP PAIRING USING RGA 2.3.1 INTERPRETING THE RGA ELEMENTS

1. λij = 1, the open-loop gain and closed-loop gain between yi and mj are identical:

Implication for loop interactions: Loop i will not be subject to retaliatory action from other loops when they are closed, mj can control yi without interference from the other control loops. If any of the gik elements in the transfer function matrix are nonzero, then the ith loop will experience some disturbances from control actions taken in the other loops, but they are not provoked by control actions in the ith loop.

Pairing recommendation: pairing yi with mj.

2. λij = 0, open-loop gain between yi and mj is zero:

Implication for loop interactions: mj has no direct influence on yi, even though it may affect other output variables.

Pairing recommendation: do not pair yi with mj.

3. 0< λij <1, indicating that the open-loop gain between yi and mj is smaller than the closed-loop gain:

Implication for loop interactions: As the closed-loop gain is the sum of the open-loop gain and the retaliatory effect from the other loops: The loops are definitely interacting, but the retaliatory effect from the other loops is in the same direction as the main effect of mj on yi. Thus the loop interactions “assist” mj in controlling yi. The extent is indicated by how close λij is to 0.5:

λij = 0.5, the main effect of mj on yi is exactly identical to the retaliatory effect from other loops



0.5 < λij < 1 the retaliatory effect from other loops is lower than the main effect of mj on yi;

0 < λij <0.5, the retaliatory effect is more substantial than the main effect.

Pairing recommendation: Avoid pairing mj with yi if λij ≤ 0.5.

4. λij > 1, indicating that the open-loop gain between yi and mj is larger than the closed-loop gain:

Implication for loop interactions: the loops interact, and the retaliatory effect from other loops acts in opposition to the main effect of mj on yi; main effect is still dominant. For large λij values, the controller gain for loop i will have to be much larger than when other loops are open. This could cause loop i to become unstable when other loops are open.

Pairing recommendation: where possible, do not pair mj with yi if λij takes a very high value.

5. λij < 0; indicating that the open-loop and closed-loop gains between yi and mj have opposite signs:

Implication for loop interactions: the loops interact, and the retaliatory effect from the other loops is not only in opposition to the main effect of mj on yi, it is also the more dominant. This is a dangerous situation because opening the other loops will likely cause 1oop i to become unstable.

Pairing recommendation: avoid pairing mj with yi.

2.3.2 BASIC LOOP PAIRING RULES

The ideal situation occurs when the RGA takes the form:

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=Λ

1...000...............0...1000...0100...001

It is obtained when the process transfer function matrix is diagonal, or triangular; the first indicates no interaction among the loops, while the latter indicates one-way interaction.

For one-way interaction case, consider the 2 x 2 system:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++

+=

144

133

01

1

)(

ss

ssG

Calculation of the RGA yields:

⎥⎦

⎤⎢⎣

⎡=Λ

1001

Because element g12(s) is zero, the input m2 has no effect on output y1, any control action in Loop 2 has no effect on Loop 1. However, the input m1 does influence the output y2 through the nonzero g21(s) element. Upsets in Loop 1 requiring control action by m1 would appear as a disturbance, which have to be handled by the feedback controller of Loop 2. Hence, even the RGA is ideal, Loop 2 would be at a disadvantage

In the non-ideal cases, the following pairing rule should be used:



Rule 1: Pair input and output variables that have positive RGA elements that are closest to 1.0.

min 1.0 kij kλ − ∀∑ (2.3.1)

where kijλ denotes the paired RGA elements corresponding to the kth alternative.

Thus, for a 2 x 2 system, if the RGA is:

⎥⎦

⎤⎢⎣

⎡=Λ

8.02.02.08.0

then it is recommended pair to pair y1 with m1, and y2 with m2. On the other hand, for the 2 x 2 system whose RGA is:

⎥⎦

⎤⎢⎣

⎡=Λ

3.07.07.03.0

the recommended pairing is 1-2/2-1 pairing

Fro the following RGA:

⎥⎦

⎤⎢⎣

⎡−

−=Λ

5.15.05.05.1

it recommends the 1-1/2-2 pairing to avoid pairing on a negative RGA element. Despite pairing on RGA values greater than 1 is not very desirable, to pair on a negative RGA value is worse still.

For the 3 x 3 distillation column with the RGA:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−

=Λ52.123.029.022.088.166.03.065.095.1

1-1/2-2/3-3 pairing is recommended.

2.4 ADDITIONAL RULES 2.4.1 NIEDERLINSKI THEOREM

Theorem: Consider an n x n multivariable system whose input and output have been paired as follows: y1 — u1 , y2 — u2 , ... , yn — un, resulting in a transfer function model of the form:

( ) ( ) ( )y s G s u s= (2.4.1)

with ( )ijg s be Rational, and Open-loop stable, and let n individual feedback controllers (with integral action) be designed for each loop so that each of the resulting n feedback control loops is stable when all the other n-1 loops are open.

Under closed-loop conditions in all n loops, the system will be unstable for all possible values of controller parameters if the Niederlinski index NI defined below is negative, i.e.:

0)0(

)0(

1

<≡

∏=

n

iiig

GNI (2.4.2)

Note:



1. It necessary and sufficient only for 2 x 2 systems; for higher dimensional systems, it is only sufficient conditions: if it holds then the system is definitely unstable; if it does not hold, the system may, or may not be unstable: the stability depend on the values taken by the controller parameters.

2. For 2 x 2 systems the Niederlinski index becomes:

11 22 12 21 12 21

11 22 11 22

1 1K K K K K KNIK K K K

ζ ζ−= = = − = − (2.4.3)

If ζ > 1, the Niederlinski index is always negative; hence 2 x 2 systems paired with negative relative gains are always structurally unstable.

3. The theorem is for systems with rational transfer function elements, thereby excluding time-delay systems. However, since it depends only on steady-state gains, the results of this theorem provide useful information about time-delay systems also, but the analysis is no longer rigorous. It should be applied with caution when time delays are involved.

2.4.2 NIEDERLINSKI PARING RULE Any loop pairing is unacceptable if it leads to a control system configuration for which the Niederlinski index is negative.

The strategy for using the RGA and Niederlinski index for loop pairing may be summarized as follows:

1. Given G(s), obtain steady-state gain matrix K = G(0), obtain the RGA, Λ; obtain the determinant of K, and the product of the elements on its main diagonal.

2. Use Rule #1 to obtain tentative loop pairing suggestions from the RGA, by pairing on positive elements which are closest to 1.0.

3. Use Niederlinski’s condition to verify the stability status of the control configuration resulting from 2; if the pairing is unacceptable, select another.

4. Variables should be paired in such a way that the resulting pairing corresponds to an NI closest to 1.0. The NI interaction rule is based on empirical observations of the definition of the NI and is justified largely from the relationship between the size of the RGA and that of the NI. The NI interaction rule has been found to be capable of avoiding ambiguities in using the RGA interaction rule.

2.4.3 APPLICATIONS OF THE LOOP PAIRING RULES

Example: Loop pairing for a 3 x 3 system

Calculate RGA for system with steady-state gain matrix:

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

==

3111

1311

1135

)0(GK

By taking the inverse, and calculating the RGA, we obtain:



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−=Λ

15.45.45.415.45.45.410

from which, by Rule #1, the 1-1/2-2/3-3 pairing is recommended.

According to this pairing, the steady-state gain matrix for the system has determinant given by:

| K | = -0.148

and the product of the diagonal elements of K is:

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=∏

= 275

31

31

353

1iiiK

Thus, for this system, this pairing will lead to a negative Niederlinski index and an unstable configuration.

This is the case in which Rule #2 disqualifies a loop pairing suggested by Rule #1;

#

Example: an alternative loop pairing is considered for a possible pairing of 1-1/2-3/3-2 which has relative gain array:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−=Λ

5.415.415.45.4

5.45.410

and new steady-state gain matrix:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

==13/113/111

1135

)0(GK

with determinant 274

=K and Niederlinski index:

454

3/527/4

==N

with this pairing the system is no longer structurally unstable. #

2.5 SPECIAL TOPICS 2.5.1 LOOP PAIRING FOR NONLINEAR SYSTEMS Assuming that a process model is available, two approaches can be used to obtain RGA’s for nonlinear systems:

1. By first principles, using steady-state version of nonlinear model, it is possible to obtain analytical expressions.

2. By linearizing the nonlinear model around a specific steady state and using the approximate K matrix to obtain the RGA.



The second approach is usually less tedious than the first one, and is more popular.

As nonlinear systems are depend on specific steady-state process conditions, it’s parameters are different at different steady-state operating conditions. Thus, the RGA’s for nonlinear systems are functions of process steady-state operating conditions and will change as these operating conditions change.

Example: RGA and loop pairing for nonlinear systems

The process is used to blend two process streams containing pure material A and pure material B, with respective flowrates, FA and FB. The objective is to control both the total product flowrate F, and the product composition, represented as x, and calculated in terms of mole fraction of material A in the blend.

The mathematical model, assuming negligible mixing dynamics, is obtained from total and component mass balances:

Total Mass Balance:

F = FA+FB

Component A Mass Balance:

BA

A

FFFx+

=

The process two output variables are F and x, and input variables are FA and FB, refer as m1 and m2 respectively.

F= m1+m2 which is linear, and:

21

1

mmmx+

=

which is nonlinear.

To obtain RGA for the process, we only need to obtain

closedloopond

openloopsboth

mFmF

sec1

1

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=λ

For both loops open:



1 1

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

openloopsbothmF

Upon closing the second loop, setting x = x* and solving for m2 in terms of m1 and x*, in order that for any change in m1, x is restored to its desired steady-state value x*:

11

2 *m

xmm −=

When the second loop is closed, m2 will respond to any changes in m1, we will have:

11

1 *m

xmmF −+=

or

*1

xmF =

It represents the changes to expect in F as a result of changes in m1, when the second loop is closed. Differentiate m1 gives:

*1 losed econd

1 xmF

cloops =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

with the final result that:

**/1

1 xx

==λ

and therefore, the RGA for this blending system is:

⎥⎦

⎤⎢⎣

⎡−

−**1

*1*xx

xx

where x* is the desired mole fraction of species A in the blend.

The loop pairing strategy can be developed as follows:

1. If x* is close to 1, the first implication is that m1 is larger than m2, the recommended pairing is F- m1 and x-m2.

2.When the steady-state product composition is closer to 0, the RGA suggests that we switch loop pairings, and use m2 to control F, and m1 to control x.

3. When x* = 0.5; in this case, it matters not which input variable is used to control which output variable; the observed interactions will be significant in either case. #

The technique of utilizing an approximate linear model (obtained by linearization around an operating steady state) is not only more convenient in such cases; it is also consistent with other aspects of nonlinear control system analysis and design we have presented so far.

Example: RGA and Loop-pairing for the Stirred Mixing Tank

The stirred mixing tank is a multivariable system in which the cold and hot stream flowrates are to be used in controlling the liquid level and tank temperature.



The nonlinear modeling equations, and an approximate transfer function, obtained by linearizing around some nominal steady-state liquid level hs and temperature Ts was given as

1 1

( ) ( )2 2

( )( ) ( )

( ) ( )

c cc s c s

H s C s

c s c sc s c s

K KA s A sA h A h

G sT T T T

K KA h s A h sA h A h

⎡ ⎤⎢ ⎥

+ +⎢ ⎥⎢ ⎥= ⎢ ⎥− −⎢ ⎥⎢ ⎥+ +⎢ ⎥⎣ ⎦

Obtain the RGA for this system and use it to recommend which of the input variables should be used to control the liquid level and the tank temperature.

From the approximate transfer function matrix, we obtain the steady-state gain matrix for the process as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−==

s

sC

s

sH

ss

hTT

hTT

hh

KGK )()(

221)0(

its output variables are represented as follows:

y1= liquid level;

y2 = tank temperature

and the input variables are represented as:

m1 = hot stream flowrate;

m2 = cold stream flowrate

The RGA for this system is obtained as:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−−

−−−

−−

=Λ

HC

sC

HC

sH

HC

sH

HC

sC

TTTT

TTTT

TTTT

TTTT

)(

)(

again that the RGA depends on the steady-state operating temperature Ts and it is expected that different loop pairing suggestions for different steady-state operating conditions.

Introducing numerical values for the parameters of this process as follows:

Hot stream temperature, TH = 650C

Cold stream temperature, TC = 150C

Investigate loop pairing for this process under four different steady-state operating conditions:

Condition 1: Ts > 400C (in particular Ts = 550C)

Under these conditions, the operating steady-state temperature is closer to the hot stream temperature and the RGA is:

⎥⎦

⎤⎢⎣

⎡=Λ

8.02.02.08.0



and the RGA recommends a 1-1 /2-2 pairing, i.e.:

Hot stream (m1) to control liquid level, (y1),and

Cold stream (m2) to control temperature, (y2).

As the cold stream temperature (150C) is much farther away from the steady-state operating tank temperature (550C) than the hot stream temperature (650C), small changes are required in the cold stream flowrate to produce noticeable changes in the tank temperature, the temperature controller can do the job effectively without causing much upset in the liquid level.

Condition 2: Ts <400C (In particular Ts = 250C)

The operating steady-state temperature is closer to the cold stream temperature. The RGA in this case is:

⎥⎦

⎤⎢⎣

⎡=Λ

2.08.08.02.0

and the RGA recommends a pairing the 1-2/2-1 pairing, in which:

Liquid level (y1) is controlled by the cold stream (m2 ).and

Temperature (y2) is controlled by the hot stream (m1).

This is the only configuration that minimizes the steady-state interaction effects among the process variables.

Condition 3: Ts = 400C

In this case, the operating temperature is exactly equidistant from the cold and hot stream temperatures. The RGA in this case is:

⎥⎦

⎤⎢⎣

⎡=Λ

5.05.05.05.0

and, as we might also expect from the physics of this situation, either pairing is equally bad.

Condition 4: Ts = TH

In this case, the tank is to operate at the same temperature as the hot stream. Observe that the RGA in this case is:

⎥⎦

⎤⎢⎣

⎡=Λ

1001

and it suggests that we can achieve perfect control of the level without interacting with the temperature if we use the hot stream for this purpose. It also indicates that variations in the hot stream flowrate are not able to affect the tank temperature. #

2.5.2 LOOP PAIRING FOR SYSTEMS WITH INTEGRATOR Since interaction analysis via the RGA involves steady-state information, for processes that contain pure integrator elements that has no steady-state, following example to suggest a possible (if exist) alternative way of dealing with such systems.

Example: Obtain the RGA for following system



⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++++

−+=

−−

sssss

se

se

sG

ss

36.0)15.6)(110)(127(

)1182(038.03120

318.1

)(

45.2

To obtain the steady-state gain matrix K, let’s consider

sI 1

=

We then have:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −==

∞>−>− I

IsGK

Is 36.0038.03

318.1lim)(lim0

thus the relative gain parameter for this system may be obtained:

1lim0.038 0.33311.138 0.36

I II

λ−>∞

⎛ ⎞⎜ ⎟

= ⎜ ⎟×⎜ ⎟+⎜ ⎟×⎝ ⎠

Since the variables I cancel, we obtain λ = 0.97 and the RGA is:

⎥⎦

⎤⎢⎣

⎡=Λ

97.003.003.097.0

Obviously 1-1/2-2 pairing is recommended.

2.5.3 LOOP PAIRING FOR NONSQUARE SYSTEMS

Classifying Nonsquare Systems Some multivariable systems are nonsquare systems, which have an unequal number of input and output variables, and their transfer function matrices are not square.

Two types: systems with fewer inputs than output variables, and those with more input than output variables.

Systems that have fewer inputs than output variables are referred to as underdefined systems, as it has insufficient input variables to control the system output.

Systems with more input than output variables are referred to as overdefined. the system has an excess of input variables.

Underdefined Systems For such systems, not all the outputs can be controlled. The loop pairing decision is, however, much simpler:

By economic considerations, or other means, decide which m of the n output variables are the most important; these will be paired with the m available input variables; the less important n - m output variables will not be under any form of control.

Example: Loop paring for a Binary Distillation Column

Consider the binary distillation column has its sidestream draw-off rate set at a fixed amount, the



remaining manipulated variables m1 and m2 are the overhead reflux rate and the reboiler steam pressure; three output variables: the overhead mole fraction of ethanol, the sidestream mole fraction of ethanol, and the Tray #19 temperature are to be controlled. Since we have lost one control variable, the process model now becomes:

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+++

+−

+−

+

+−

+

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−

−−

2

1

2.9

2.15.6

6.2

3

2

1

)18.18)(189.3()161.11(87.0

115.868.33

109.7012.0

125.311.1

106.90049.0

17.666.0

mm

sses

se

se

se

se

se

yyy

ss

ss

ss

It is impossible to control all three output variables with only two input variables; however, the sidestream mole fraction of ethanol is deemed the least important of the output variables, leaving two output variables to be controlled with the two input variables.

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+++

+−

+−

+=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−

2

1

2.9

6.2

3

1

)18.18)(189.3()161.11(87.0

115.868.33

106.90049.0

17.666.0

mm

sses

se

se

se

y

y

ss

ss

along with the additional relation

22.1

1

5.6

109.7012.0

125.311.1 m

sem

sey s

s

+−

++

= −−

The modified (square) subsystem’s steady-state gain matrix is

⎥⎦

⎤⎢⎣

⎡−

−=

87.068.330049.066.0~

K

and the RGA is:

⎥⎦

⎤⎢⎣

⎡−

−=

4.14.04.04.1

λ

The recommended input/output pairing scheme is:

y1 - m1; overhead reflux to control overhead composition 3/3

y3 – m2; reboiler steam pressure to control Tray #19 temperature #

Overdefined Systems In this case, we have an excess of input variables, and we can actually achieve arbitrary control of the fewer output variables in more than one way. Since there are m input variables to n output variables (and m > n), there are many more input variables to choose from in pairing the inputs and the outputs, and there will be several different square subsystems from which the pairing

possibilities are to be determined; there are v = ⎟⎟⎠

⎞⎜⎜⎝

⎛nm possible square subsystems,

! !( - )!

mvn m n

= .

The strategy is:



1. Determine all the v possible square subsystems.

2. Obtain the RGA’s for each of these square subsystems.

3. Examine these RGA’s and pick the best subsystem on the basis of the overall character of its RGA.

4. Use its RGA to determine which input variable within this subsystem to pair with which output variable.

Example: RGA and Loop paring for an Overdefined System

A multivariable system has two outputs y1 and y2 that can be controlled by any of three available inputs m1, m2, and m3 with

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

+−

+

+++=⎥⎦

⎤⎢⎣

⎡−−−

−−−

3

2

1

4.02.05.0

03.03.02.0

2

1

16.1001.0

1003.0

15.1004.0

18.204.0

15.207.0

135.0

mmm

se

se

se

se

se

se

yy

sss

sss

Two of the three input variables will be used for control, while the third input will set at a fixed value, and will be redundant.

Solution: All possible 2x2 subsystems and RGA are listed as:

Subsystem 1 (Utilizing m1 and m2 for control):

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

+

++=⎥⎦

⎤⎢⎣

⎡−−

−−

2

12.05.0

3.02.0

2

1

1003.0

15.1004.0

15.207.0

135.0

mm

se

se

se

se

yy

ss

ss

⎥⎦

⎤⎢⎣

⎡=Λ

843.0157.0157.0843.0

1


⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

+

++=⎥⎦

⎤⎢⎣

⎡−−

−−

3

14.05.0

03.02.0

2

1

16.1001.0

15.1004.0

18.204.0

135.0

mm

se

se

se

se

yy

ss

ss

⎥⎦

⎤⎢⎣

⎡=Λ

758.0242.0242.0758.0

2


⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

+−

++=⎥⎦

⎤⎢⎣

⎡−−

−−

3

24.02.0

03.03.0

2

1

16.1001.0

1003.0

18.204.0

15.207.0

mm

se

se

se

se

yy

ss

ss

⎥⎦

⎤⎢⎣

⎡−

−=Λ

4.14.24.24.1

3



Based on inspection, Subsystem 1 offers the best possible control, as its RGA is the closest to the ideal situation. The input variables m1 and m2 are to be used for control and m3 is to be redundant. #

2.5.4 LOOP PAIRING WITHOUT PROCESS MODELS When process models are not available, one of the following two approaches can be used to obtain RGA:

1. Experimentally determine the steady-state gain matrix K, by implementing step changes in the process input variables, one at a time, and observing the change in each of the output variables. Let ∆y1j be the observed change in the value of output variable 1 in response to a change of ∆mj in the jth input variable mj then, by definition of the steady-state gain:

j

jj m

yk

∆

∆= 1

1

and, in general, the steady-state gain between the ith output variable and the jth input variable will be given by:

ijij

j

yk

m∆

=∆

The elements of the K matrix are thus obtained. Once this gain matrix is known, we can easily generate the RGA.

2. As each RGA element λij can be determined upon performing two experiments; the first determines the open-loop steady-state gain by measuring the response of yi to input mj, when all the other loops are opened; in the second experiment, all the 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 gains gives the desired relative gain element.

It should be clear that, of these two experimental procedures, the second one is more time consuming, and involves too many upsets to the process; for these reasons it is not likely to be popular in practice. The first procedure is to be preferred.

2.5.5 TIMESCALE DECOUPLING

This issue can influence the choice of loop pairing.

Example: in a 2 x 2 system with a certain pairing, the RGA indicates serious loop interactions. However, if one of the loops responds very much faster than the other, then there can be timescale decoupling of the loops.

Timescale decoupling occurs when the fast loop responds so fast that the effect of the slow loop appears as a constant disturbance; conversely the slow loop does not respond at all to the high-frequency disturbances coming from the fast loop. This means that we can safely pair loops with large differences in closed-loop response times even when the RGA is not favorable.

#

Example: Loop Pairing with Timescale Decoupling

Consider the in-line blending problem, when we wish a blend which contains 20% species A, the RGA is.



⎥⎦

⎤⎢⎣

⎡=Λ

2.08.08.02.0

Based on RGA analysis, pairing the total flowrate, y1, to the flowrate of A, m1, and the blend composition, y2, to the flowrate of B,m2, would not be a recommended pairing. However, if the flow measurement and valve dynamics were very rapid, then the closed-loop response of the y1 — m1 loop would be extremely fast (on the order of seconds). By contrast, if the composition analyser had a significant time constant (on the order of minutes), then the y2 — m2 loop would be 10— 100 times slower than the y1 — m1 loop. In this case, timescale decoupling mean the control loops could be considered virtually independent of each other, and no serious interactions would occur. #

Example: Another form of Timescale Decoupling

Consider the 2 x 2 system with the transfer function:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−

+

++=

1104

11

12

1102

)(

ss

sssG

with RGA given by:

⎥⎦

⎤⎢⎣

⎡=Λ

8.02.02.08.0

the y1 — m1 / y2 — m2 pairing would be recommended.

The closed-loop response for a unit set-point change in y1 using this pairing and a diagonal PI controller

)3.0,4,5.0,4( 2211 =−=== ICIC KK ττ

is shown below.

The performance is not too bad considering that the open-loop time constants on the diagonal are 10 minutes.

However, note that in spite of an adverse RGA parameter, λ= 0.2, the reverse y1 — m2 / y2 — m1 pairing could have great advantages because with this configuration, the open-loop time constants on the diagonal are only 1 minute. In fact with this reverse pairing, the closed-loop response for the same set-point change using a diagonal PI controller



)3.0,20,3.0,10( 2211 ==== ICIC KK ττ

is as follows. The performance in this case is dramatically better than with the RGA recommended pairing. The reason is that the control loops are able to respond so rapidly that the interactions that appear more slowly are easily dealt with.

The lesson is that the RGA provides only a guide to steady-state interactions and must be used together with all other engineering considerations in choosing the loop pairing.

#

2.6 OTHER LOOP PAIRING METHODS 2.6.1 RIA BASED LOOP PARING

This is a RGA-like approach. The relative interaction for loop i jy u− is defined as the ratio of two elements: the increment of the process gain after all other n-1 control loops are closed and the apparent gain in the same loop when all other control loops are open, i.e.,

( / ) ( / )

( / )i ji j all other loops close except for loop y u i j all loops open

iji j all loops open

y u y u

y uϕ −

∂ ∂ − ∂ ∂=

∂ ∂. (2.6.1)

The loop interaction can also be exposed by investigating the response of an output to its input in the presence of interaction from other loops through the hidden loops. This situation can be best depicted by the Figure below. Focusing on an arbitrary control loop with ju paired with iy , output in response to any change in the input consists of two parallel paths: the direct forward path via the interaction free process and the parallel path via the hidden loops due to off-diagonal elements of the process.

(a) absolute interaction



(b) relative interaction

Figure 2.6.1. Definition of dynamic interaction in ju - iy loop

Here, ( )ija s is the absolute interaction, which can be obtained by the signal flow technique, the relative interaction can be then represented as the ratio of the absolute interaction and the independent interaction free process as

( )( )( )ij

ij

ijs

a sg s

ϕ = . (2.6.2)

At steady-state, assuming 1) the closed-loop system possesses integrity against any single-loop failure and 2) each controller in each control loop contains integral action, we obtain

1

1(0) (0)(0)ij ij

ji

a gG−= − .

Hence, at steady-state we obtain the relative interaction

1

1 1(0) (0)ij

ij jig Gϕ −= − .

All the individual relative interaction terms constitute a relative interaction array (RIA) as

[ ]ij ijRIAϕ = .

Compared with the definition of the RGA, it is easy to give

1 1ijij

ϕλ

= − . (2.6.3)

Clearly, the RIA provides an equivalent interaction measure to the RGA. Loop interaction and implications can be briefly characterized by the RIA as follows:

(1) 0ijϕ = ⇒ no interaction;

(2) 0ijϕ > ⇒ interaction acts in the same direction as interaction-free process; 1.0ijϕ > ⇒ interaction dominates over interaction-free process gain;

(3) 0ijϕ < ⇒ interaction acts in the reverse direction as interaction-free process gain;

(4) 1.0ijϕ < − ⇒ reverse interaction dominates over interaction-free process gain.

Under RIA based pairing criterion, variables should be paired in such a way so that:

(1) All the RIA elements are closest to 0;



(2) NI is positive;

(3) All the RIA elements are greater than -1;

(4) RIA elements close to -1 are avoided. It is important to notice that both very large positive and very large negative RGA elements drive loop interaction toward the inverse direction of the original interaction-free process gain and that both very small positive and very small negative RGA elements are also, actually even more, harmful to process control. Apparently, in contrast to the RIA, the RGA exhibits discontinuous behavior when used to characterize loop interaction. Although the above interaction rule seems to be equivalent to the RGA interaction rule, the closeness to a value (0 in RIA rule, 1.0 in RGA rule) has different implications. Specifically, the distance of the RGA elements from 1.0 may not realistically reflect the amount of interaction as noted previously, whereas the distance of the RIA does. In particular, the RGA interaction rule mainly targets at dismissing large RGA elements, which makes the robustness rule redundant, whereas the RIA interaction rule essentially aims at eliminating small RGA elements (large interaction), which complements the robustness rule.

However, the RIA does not encode any more information on the process than the RGA, although they provide valuable alternative viewpoints and more smooth interpretation. The common feature of those indices is that they only use the system model at zero frequency.

2.6.2. DYNAMIC RGA (DRGA) BASED LOOP PARING DRGA is another approach to evaluate the interactions, which may overcome some limitations from the RGA based approaches. Since the DRGA is most valuable for screening alternate control system designs, the requirement of an extensive controller design tends to defeat the utility of these methods. A better approach to define a useful DRGA should involve relatively little user interaction in the controller design aspect of the analysis. Assuming that a dynamic state space model of the process is available, the proportional output optimal feedback controller can be used to define the DRGA. The scaled linear state space dynamic model is given by

x Ax Buy Cx

= +⎧⎨ =⎩

. (2.6.4)

The objective function to be minimized is defined as

0

1 ( )2

T TJ y Qy u Ru dt∞

= +∫ . (2.6.5)

The optimal control problem to be solved is a linear quadratic regulator with output feedback. Here, Q and R are typically taken as identity matrices, since u and y have been scaled by their operating ranges. Such a choice for Q and R treats all measurements and manipulated variables equally. An output feedback matrix gives the admissible controls as

u Kx= − (2.6.6)

where K is an optimal feedback controller.

By solving the expected value of J for the case where the initial state forcing lies on a unit sphere around the origin, the DRGA which is based on the optimal proportional controller K can be defined for the ( , )thi j element of the DRGA as



0,

0,

||

k

k

i j u k iDij

i j u k i

u yu y

λ ≠ ≠

= ≠

∂ ∂=

∂ ∂, (2.6.7)

where both terms give the gain of iu to jy during a transient in which the process is controlled using the optimal output proportional gain matrix, K . Further more, from u Kx= − , the partial derivatives can be calculated to give the following result

ˆˆ1ij

Dij ij jiji

KK K

Kλ

−= =

−, (2.6.8)

where K̂ is equal to 1K − . The interpretation of Dijλ and paring rule are given as:

• 1.0Dijλ ≈ , the manipulated variable under consideration does not interact with the other manipulated variables so far as jy is concerned. Thus, pairing jy with iu would be one candidate for SISO control.

• If a manipulated variable has a negative value of Dijλ , then its behavior switches sign depending on how aggressively the other manipulated variables are moving. Such a pairing should be avoided since interaction is likely to be high.

• Large values of Dijλ also indicate large changes in the behavior of a manipulated variable, which in turn indicate that interaction is high.

While in all examples the new DRGA indicates the best pairings to use and it accurately assesses the extent of interaction present, the disadvantages are also obvious:

(1) This pairing rule is still a heuristic;

(2) Complex computing: a conversion from a transfer function model to a state space model and a linear quadratic optimal control solution are required;

(3) The value of and Dijλ is sensitive to the weigh matrices, Q and R .

2.6.3. GRDG BASED LOOP PARING Another interesting dynamic interaction measure is the generalized relative dynamic gain (GRDG). The GRDG also takes into account the dynamics of the closed-loops. The element

11( )sλ of GRDG for a TITO process is defined as:

211 22

211

211 22 12 21

2

( )( ) ( )( )( ) ( )( ) ( ) ( ) ( )

( )

r sg s g sy ss r sg s g s g s g s

y s

λ =−

(2.6.9)

where the variables 2 ( )r s and 2 ( )y s are the set-point and the process output of the other loop,

2

2

( )( )

r sy s

is the desired dynamics of the second loop, respectively. The transfer functions 11( )g s ,

12 ( )g s , 21( )g s and 22 ( )g s are the elements of the process transfer matrix ( )G s .

The loop paring rule is as below:

(1) Eliminate the infeasible pairings by normal RGA procedure;



(2) Calculate interaction potentials to obtain the most feasible pairings.

For easier control and tuning, the specifications on the closed-loop set-point responses have to be chosen in a frequency band where interaction is reduced so the system behaves more like SISO systems. In order to do so, the closed-loop response specifications are chosen in frequency band where the GRDG is close to one since, as for RGA, it means that the interaction is low.

Because the zeros in a transfer function affect the process dynamic, the GRDG, which takes into account the dynamic part of the transfer function, is preferred to the RGA, which only considers the steady-state. However, as can be seen from 11( )sλ , the representation and computation of the RDGA will be dramatically complex if the system dimension is higher than 2 2× .

2.6.4. SSV BASED LOOP PARING A shortcoming of many currently available interaction measures is their limited theoretical basis. Using the notion of structural singular value (SSV), a dynamic interaction measure for multivariable systems under feedback with diagonal or block diagonal controllers can be represented.

Let ( )G s be the process, ( )G s be a matrix consisted of the block diagonal elements of ( )G s . A block diagonal controller

1 2( ) ( ( ), ( ), ..., ( ))c c c cnG s diag G s G s G s=

is to be designed for the system

11 22( ) ( ( ), ( ), ..., ( ))nnG s diag G s G s G s=

such that block diagonal closed-loop system with the transfer matrix 1( ) ( ) ( )( ( ) ( ))c cH s G s G s I G s G s −= +

is stable. An interaction measure expresses the constraints imposed on the choice of the closed-loop transfer matrix ( )H s for the block diagonal system, which guarantee that the full closed-loop system

1( ) ( ) ( )( ( ) ( ))c cH s G s G s I G s G s −= +

is stable.

Using relative error, 1( ) ( ( ) ( )) ( )E s G s G s G s−= − ,

which is arised from the “approximation” of the full system ( )G s by the block diagonal system ( )G s , the general interaction measure can be stated as:

Assume that ( )G s and ( )G s have the same RHF poles and that ( )H s is stable. Then the closed-loop system ( )H s is stable if

1max max( ( )) ( ( ))H j E jσ ω σ ω ω−< ∀ ,

This equation implies that for stability the magnitude of the diagonal blocks ( )iH s has to be constrained by the reciprocal of the max ( ( ))E jσ ω of the relative error matrix. The value of

max ( ( ))E jσ ω depends on the structure assumed for ( )H s .



Although all popular interaction measures are either explicitly or implicitly based on this definition and thus it has a rigorous theoretical basis, this definition has its limitations and therefore the results should be interpreted with caution. The reason is that interaction measure is based on the block diagonal ( )H s which might not be indicative of the actual full closed-loop transfer function matrix ( )H s . Though the definition of the interaction measure guarantees ( )H s to be stable it can be very badly behaved. The interaction measure might indicate “small” interactions but the performance could be arbitrarily poor. Furthermore, the following disadvantages also need to notice as they are obstacles for it can be understood and accepted by control engineers:

(1) When applied to loop paring, all pairing possibilities need to screen and calculate to find the best one;

(2) The computation of max ( ( ))E jσ ω needs complicated digital optimization;

(3) The max ( ( ))E jσ ω gives equal preference to all the loops. In some cases this may impose an early and perhaps unnecessary roll-off for some of the ( )iH s . The introduction of a weight matrix with block diagonal structure equal to that of ( )H s can circumvent this problem. Nevertheless, this makes the application more difficult.


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PART II. INTERACTION ANALYSIS - Kalyana Veluvoluncbs.knu.ac.kr/Teaching/MCS_Files/Part2.pdf · 2015-03-05 · Part II Interaction Analysis 2 feedback control involved. Experiment