Chapter 15

PID Controllers Applied to MIMO Processes

2×2 Example of a MIMO Process

G' 11 (s)

++

y 1

y 2

G' 21 (s)

G' 12 (s)

G' 22 (s)++

c 2

c 1

Process

Example of a 2×2 MIMO Process

AT

LC

LC

AT

DL

B

V

• Two inputs: Setpoints for flow controller on steam and reflux.

• Two outputs: Composition of products B and D

Configuration Selection (Choosing the u/y Pairings)

• That is, which manipulated variable is to be used to control which controlled variable.

• Choosing an inferior configuration can dramatically reduce control performance.

• For many processes, configuration selection is a difficult and challenging process (e.g., dual composition control for distillation).

Single Loop Controllers Applied to a 2×2 MIMO Process

y 1

y 2c 2

c 1G C1

+

-G C2

y 1,sp

y 2,sp

G' 11 (s)++

G' 21 (s)

G' 12 (s)

G' 22 (s)

++

Control Loop 1

Control Loop 2

+ -

Example of Single Loop PID Controllers Applied to 2×2 Process

• L is adjusted by PID controller to maintain composition of D at its setpoint.

• Steam flow is adjusted by PID controller to maintain composition of B at its setpoint.

AT

LC

LC

AT

Dy

L

Bx

VFz

PT

AC

AC

Coupling Effect of Loop 2 on y1

y 1

y 2c 2

c 1G C1

+

-G C2

y 1,sp

y 2,sp

G' 11 (s)

++

G' 21 (s)

G' 12 (s)

G' 22 (s)

++

Control Loop 1

Control Loop 2

+ -

Example of Coupling

• L is adjusted to maintain the composition of D which causes changes in the composition of B.

• The bottom loop changes the flow rate of steam to correct for the effect of the reflux changes which causes changes in the composition of D.

AT

LC

LC

AT

Dy

L

Bx

VFz

PT

AC

AC

The Three Factors that Affect Configuration Selection

• Coupling

• Dynamic response

• Sensitivity to Disturbances

Steady-State Coupling

2

2

1

1

1

1

11

222112112211

2221

1211

:evaluationrequireselementoneonlyTherefore,

11

y

c

cy

cy

RGA

Relative Gain Array

• When 11 is equal to unity, no coupling is present.

• When 11 is greater than unity, coupling works in the opposite direction as the primary effect.

• When 11 is less than unity, coupling works in the same direction as the primary effect.

Numerator of 11

y 2c 2

c

+

-G C2

y 2,sp

G' 11 (s)++

G' 21 (s)

G' 12 (s)

G' 22 (s)

++Control Loop 2

y 1

21

1

cc

y

Denominator of 11

y 1

y 2c 2

c 1

+

-G C2

y 2,sp

G' 11 (s)

++

G' 21 (s)

G' 12 (s)

G' 22 (s)

++Control Loop 2

21

1

yc

y

RGA Example

1111

1

111

1

2221

1211

oneffectcoupling

0.1

0.205.0

1.00.1

2

2

yKc

y

Kc

y

KK

KK

y

c

c1 = 1.0y2 = K21 c1 = 0.05

y 1

y 2c 2

c 1

+

-G C2

y 2,sp

G' 11 (s)

++

G' 21 (s)

G' 12 (s)

G' 22 (s)

++Control Loop 2

c2 = -y2/K22 = -0.05/2 =-0.025

y 1

y 2c 2

c 1

+

-G C2

y 2,sp

G' 11 (s)++

G' 21 (s)

G' 12 (s)

G' 22 (s)

++Control Loop 2

(y1)coup = c2 K12 = -0.025(0.1) =-.0025

y 1

y 2c 2

c 1

+

-G C2

y 2,sp

G' 11 (s)++

G' 21 (s)

G' 12 (s)

G' 22 (s)

++Control Loop 2

Calculation of RGA

decoupledhighlyissystem

thethatindicatesresultThis

0025.19975.0

0.1

9975.01

0025.01

11

1

1

2

yc

y

RGA Calculation for 2×2 System

2211

2112

22

211211

1111

1

1

KKKK

KKK

K

K

RGA Analysis

• RGA is a good measure of the coupling effect of a configuration if all the input/output relationships have the same general dynamic behavior.

• Otherwise, it can be misleading.

Example Showing Dynamic Factors

94.0

)2(1)3.0(4.0

1

1StateSteady

1100

0.2)(

110

4.0)(

110

3.0)(

1100

0.1)(

2221

1211

RGA

ssG

ssG

ssG

ssG

Dynamic Example

• Note that the off-diagonal terms possess dynamics that are 10 times faster than the diagonal terms.

• As a result, adjustments in c1 to correct y1 result in changes in y2 long before y1 can be corrected. Then the other control loop makes adjustments in c2 to correct y2, but y1 changes long before y2. Thus adjustments in c1 cause changes in y1 from the coupling long before the direct effect.

Direct Pairing (Thin Line) and Reverse Pairing (Thick Line)

0 100 200 300 400 500Time

y1

y2

Dynamic RGA

1100)110(7.16

1

1

:examplethisFor

1)(

:processorderfirstaFor

)()(

)()(1

1)(

22

2211

22

2211

211211

p

pKiG

iGiG

iGiG

Dynamic RGA for Direct (a) and Reverse (b) Pairings

• Consider the frequency, , corresponding to desired closed loop response which indicates b better than a

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10

11

a

b

Overall Dynamic Considerations

• Pairings of manipulate and controlled variables should be done so that each controlled variable responds as quickly as possible to changes in its manipulated variable.

Sensitivity to Disturbances• In general, each configuration has a different sensitivity to a

disturbance. Note that thick and thin line represent the results for different configurations

Time

Pro

duct

Im

puri

ty Bottom Product

Overhead Product

Configuration Selection

• It is the combined effect of coupling, dynamic response, and sensitivity to disturbances that determines the control performance for a particular control configuration for a MIMO process.

Configuration Selection for a C3 Splitter

06.0),(

70.1)/,/(

3.25),(

94.0),(

)11(RGAionConfigurat

VD

BVDL

VL

BL

(L,V) Configuration Applied to the C3 Splitter

AT

LC

LC

AT

Dy

L

Bx

VFz

PT

AC

AC

Reflux Ratio Applied to the Overhead of the C3 Splitter

LC

AT

×

AC

FT

L

D

L/D

Configuration Selection Example

• L, L/D, and V are the least sensitive to feed composition disturbances.

• L and V have the most immediate effect on the product compositions followed by L/D and V/B with D and B yielding the slowest response.

Control Performance

91.1098.0),(

00.2095.0)/,/(

3.13250.0),(

49.1067.0),(

BottomsforIAEOverheadforIAEionConfigurat

VD

BVDL

VL

BL

Analysis of Configuration Selection Example

• Note that (L,V) is the worst configuration in spite of the fact that it is the least susceptible to disturbances and the fastest acting configuration, but it is the most coupled.

• Even though (D,V) had an RGA of 0.06, it had decent control performance.

• (L,B) is best since it has good decoupling and the overhead product is most important.

Tuning Decentralized Controllers

• When a particular loop is 3 times or more faster than the rest of the loops, tune it first.

• When tuning two or more loop with similar dynamics, use ATV identification with online tuning

TZNIIT

ZNcc

TZNIIT

ZNcc

FFKK

FFKK

/:loopSecond

/:loopFirst

One-Way Decoupler

y 1

y 2

c 2

c 1G' 11 ++

G' 21

G' 12

G' 22

++

D 1(s)

++G C1

y 1,sp + -

+

-G C2

y 2,sp

)(

)()(

11

121 sG

sGsD

Overview

• The combined effect of coupling, sensitivity to disturbances, and dynamic response determine the performance of a configuration

• Implement tuning of fast loops first and use a single tuning factor when several loops are tuned together.

• One-way decoupling can be effective when the most important controlled variable suffers from significant coupling.