L19 MV Controllability RGA

7/31/2019 L19 MV Controllability RGA

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EEB5213/EAB4233 Plant ProcessControl Systems

Controllability &

Relative Gain Array


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Objectives

At the end of this lecture, students should be able to:

Describe interaction and its effects on amultivariable system

Determine the controllability of a multivariable

system given the model transfer functions

Apply the Relative Gain Array (RGA) method toquantitatively indicate favorable interaction in a

given multivariable process


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Multivariable Control A necessary but not sufficient requirement for a

multivariable control system is the degrees of

freedom (DoF).

OK, but this does not ensurethat we can control the CVs

of interest !

Degrees ofFreedom

Number of valves Number of controlled variables


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Controllability

A system is controllable if its CVs can be maintained

at their set points, in the steady-state, in spite of

disturbances entering the system.

Steady-statemodel of 2x2

system

A system is controllable when the matrix of process gains,

Kcan be inverted, i.e. when the determinant ofK 0.


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Controllability

For the toy autos in the figure :

Are they independently controllable ?

Does interaction exist ?

Lets do the toy

autos first; then,do some

processes


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Controllability

For the auto toys in this figure :




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Controllability

For the auto toys in this figure:




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Example #1 : BlendingProcess

Are the CVs independently controllable ? Does interaction exist ?

Yes, this system is controllable !


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Example #2 : Distillation Tower

Are the CVs independently

controllable ?



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Example #2 : Distillation Tower

Det (K) = 1.54 x 10-3 0

Small but not zero (each gain is small)

The system is controllable !


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Multiloop Control

When any of the loops in a multivariable process is

in closed mode, we must consider both direct andinteraction dynamic paths in the system.


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Relative Gain Array (RGA)

A quantitative measure of interaction developed by

Bristol (1966). ij is the relative gain betweenMVj and CVi.

closedloopsother

openloopsother

constant

constant

j

i

j

i

j

i

j

i

i j

MV

CV

MV

CV

MV

CV

MV

CV

k

k

CV

MV

Assumes that loops

have integral mode


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The RGA can be calculated from open-loop gain

values. The relative gain array is the element-by-element

product ofKwithK-1.

))((1

1

jiijij

j

i

j

i

kIkKK

CV

MV

CVKMV

MVCVMVKCV

T

ij

ij

kI

k


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The relative gain array for a 2x2 system is given in

the following equation :

The rows and columns of the RGA sum to 1.0

What is true for theRGA to have 1s on

diagonal ?


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Relative Gain Array (RGA) The RGA elements are scale independent.

Changing the units of the CV or the capacity of the

valve does not change ij.

Relative Gain Array


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In some cases, the RGA is very sensitive to small errors

in the gains, Kij.

When is this equation verysensitive to individual gain

errors?


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We can achieve stable multiloop feedback control byusing the sign of the controller gain.

What will happen when the other interacting loops

are placed in manual mode?

How Do We InterpretRGA?

closedloopsother

openloopsother

j

i

j

i

ij

MV

CV

MVCV

ij < 0 In this case, the steady-state gains have

different signs depending on the status(auto/manual) of other loops.


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How Do We Interpret RGA?

Forij< 0, one of three BAD situations can occur :

1. Multiloop system is unstable when all loops in auto.

2. Single-loop ij is unstable when others are in manual.

3. Multiloop is unstable when loop ij is manual and other

loops are in automatic.

ij < 0 In this case, the steady-state gains have different

signs depending on the status (auto/manual) ofother loops.


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ij = 0 In this case, the steady-state gain is zero whenall other loops are open, in manual.

Could this control system work?

What would happen if one controller were in manual?

closedloopsother

openloopsother

j

i

j

i

ij

MV

CV

MV

CV


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What would be the effect

of opening/closing the

other loop on tuning ?

Discuss the case of 2x2system paired on ij = 0.1.

0 < ij < 1 In this case, the steady-state gain is larger

when all other loops are closed (in auto).

closedloopsother

openloopsother

j

i

j

i

ij

MVCV

MV

CV


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ij = 1 In this case, the steady-state gains are identicalin both situations; whether other loops are

open or closed.

What is generally true when ij = 1?

Does ij = 1 indicate no interaction?

closedloopsother

openloopsother

j

i

j

i

ij

MV

CV

MV

CV


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What would be the effect of

opening/closing the other

loop on tuning?

Discuss the case of 2x2system paired on ij = 10.

ij > 1 In this case, the steady-state gain is larger when

all other loops are open, in manual.

closedloopsother

openloopsother

j

i

j

i

ij

MV

CV

MV

CV


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ij = In this case, the gain when all other loops are

closed is zero. We conclude that multiloop

control is not possible.

closedloopsother

openloopsother

j

i

j

i

ij

MV

CV

MV

CV


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Relative Gain Array

Integrity: A multiloop control

system has good integrity

when after a loop is turned

off, the remainder of the

control system remains

stable.

Proposed Guideline #1

Select pairings that do

not haveij < 0.

Turning off can occur when (i) a loop is placed in

manual, (ii) a valve saturates or (iii) the secondary

controller in a cascade system can no longer change

the valve (it has reached its limit or it is switched off).


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Relative Gain Array

Pairings with negative or

zero RGAs have poor

integrity

Proposed Guideline #2

Select pairings that donot haveij = 0.

Pairing with RGA closest to

1 has the least amount of

interaction.

Do you agree with the

proposed guidelines?

Proposed Guideline #3Select pairings that

have RGA elements as

close as possible to 1.


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RGA Example

Determine the relative gain array for the binary

distillation column.

09.6

)1253.0(0747.0

)1173.0)(0667.0(0.1

1

0.1

1

2211

2112

11

KK

KK


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RGA Example

Using 11, determine the whole RGA and eliminate

unfavourable ij.

FR FV

xD

6.09 - 5.09

xB - 5.09 6.09

Negative ij, unfavourable

interaction, to be eliminated


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Decoupling

We can reduce the effect of unfavorable interaction

using decoupling.

Decoupling retains the single-loop control

algorithms and reduces/eliminates the effects of

interaction.

Three possible approaches :

1. Implicit decoupling Modified CVs

2. Implicit decoupling Modified MVs

3. Explicit decoupling Controller compensation


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Decoupling

Reducing the effects of unfavourable interaction using

explicit decoupling.

E li i D li


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Explicit Decoupling

Design approach:

Perfect decoupling compensates for interactions.

)(

)(

)( sG

sG

sD ii

ij

ij

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L19 MV Controllability RGA