Coupling Analysis of Multivariable Systems( 多变量系统的关联分析 )

Lei XIE

Zhejiang University, Hangzhou, P. R. China

Control Problem Discussion

For the two-input-two-input controlled system, design your control schemes.

Suppose that

;,21

221121 FF

FCFCCFFF

;12

,14

1,

1

5.0 5

s

e

C

A

sC

C

sF

F smm

%.40%,60,25,75 210

20

1 CCFF

Initial states:

F1, C1 F2, C2

FT03

FSP

C F

FC03

ASP

AT11

AC11

u1 u2

Examples of Multivariable Control Systems: Blending

Tank

For the two-input-two-input controlled system, the simplest control schemes?

#1: F1 F, F2 C;

#2: F1 C, F2 F.

F1, C1 F2, C2

FT03

FSP

C F

FC03

ASP

AT11

AC11

u1 u2

Which one is better ?

Multi-Loop Control Scheme #1

for a Blending Process

F1, C1 F2, C2

FT03

C F

FC03

ASPAT11

u1 u2F1SP

FC01

FT01

F2SP

FC02

FT02

FSP

AC11

1 1 1

2 2 2

,y u FF

y u FC

Controlled Process

y1u1

y1sp

PID1

y2u2

y2sp

PID2

Multi-Loop Control Scheme #2

for a Blending Process

F1, C1 F2, C2

FT03

C F

FC03

ASPAT11

u1 u2F1SP

FC01

FT01

F2SP

FC02

FT02

FSP

AC11

1 1 1

2 2 2

,y u FF

y u FC

Controlled Process

y1u2

y1sp

PID1

y2u1

y2sp

PID2

Simulation Results of Multiloop Scheme (u1-y1, u2-

y2)

0 50 100 150 20085

90

95

100

105

110

T/h

r

F, SingleLoop

0 50 100 150 20085

90

95

100

105

110

T/h

r

F, MultiLoop

0 50 100 150 20042

44

46

48

50

52

Time, min

%

C, SingleLoop

0 50 100 150 20042

44

46

48

50

52C, MultiLoop

%

Time, min

Analyze response differences

Simulation Results of Multiloop Scheme (u1-y2, u2-

y1)

0 50 100 150 20085

90

95

100

105

110

T/h

r

F, SingleLoop

0 50 100 150 20085

90

95

100

105

110

T/h

r

F, MultiLoop

0 50 100 150 20042

44

46

48

50

52

Time, min

%

C, SingleLoop

0 50 100 150 20042

44

46

48

50

52

Time, min

%C, MultiLoop

Analyze response differences

Simulation Result Analysis

G11(s)

G21(s)

G12(s)

G22(s)

u1

u2

y1m

y2m

y1sp

PID1+

_

MV1

“ A”

“ M”

y2spPID2

+_

MV2

“ A”

“ M”

Different Control Paths for PID2

Analysis on the Control Path for Controller PID2

)()()(

)(222

2

2 sGsGsu

syP

m

Case 1: PID1 is in “M” model

)(

)()()(

)()()(1

)()()()(

)(

)(

11

122122

12111

121222

2

2

sG

sGsGsG

sGsGsG

sGsGsGsG

su

sy

C

CP

m

Case 2: PID1 is in “A” model

Analysis on the Control Path for Controller PID2 (cont.)

)()()(

)(222

2

2 sGsGsu

syP

m

Case 1: PID1 is in “M” model

)()()(1

)()()()(

)(

)(12

111

121222

2

2 sGsGsG

sGsGsGsG

su

sy

C

CP

m

Case 2: PID1 is in “A” model

22222 )0( KGKP

11

122122

12111

12122

'22 )0(

)0()0(1

)0()0()0(

K

KKK

GGG

GGGK

C

C

Concept of Relative Gain

Definition of Relative Gain Calculation of Relative Gain Meaning of Relative Gain Matrix Calculation of Relative Gain Matrix

2

1

2221

1211

2

1

)()(

)()(

u

u

sGsG

sGsG

y

y Relative gain for control path u2y2 ?

Concept of Relative Gain for a 2*2 Multivariable

Process

Open-loop Gain

K22

Closed-loop Gain

K’22

Relative Gain

Kinds of Process Gain u1(s)

u2(s)

y1(s)

y2(s)

u1(s)

u2(s)

y1(s)

y2(s)

PID1r1(s)+

_

1

1

2 2 0 2222

2 2 220

u

y

y u K

y u K

Relative gain for other paths ?

Example of Relative Gain Calculation

1

1

2 2 0 2222

2 2 220

;u

y

y u K

y u K

1 1

2 2

3.0 0.4

2.0 0.2

y u

y u

Steady-state model in incremental mode:

122 2 2 0

0.2,u

K y u

1 1 2

1 2

0 3 0.4 0

4 / 30*

y u u

u u

1

22 2 2 2 2 20

72.0 (4 / 30 ) 0.2 0.2

3yK y u u u u

22

3;

7 Other relative gains ?

Computation of Relative Gain for 2*2 Control System

u1(s)

u2(s)

y1(s)

y2(s)2221212

2121111

uKuKy

uKuKy

11 11 2211

12 21 11 22 12 2111

22

K K KK K K K K KK

K

Steady-state equation:

2 21 1 22 2 2 21 22 1

12 211 11 1 12 2 11 1

22

0 0y K u K u u K K u

K Ky K u K u K u

K

μ11 ?

Definition of Relative Gain Matrix

0

0

e

e

i j u ijij

iji j y

y u K

Ky u

gain when all other loops are open

gain when all other loops are closed

1 2

1 11 12 1

2 21 22 2

1 2

n

n

n

n n n nn

u u u

y

y

y

The relative gain describes the “effect” on this loop of the other loops.

1ij ij

ij

K K

Calculation Example of Relative Gain Matrix

2

2

1 1 0 1111

1 1 110

3 3;

3 0.4 ( 10) 7u

y

y u K

y u K

1 1

2 2

3.0 0.4

2.0 0.2

y u

y u

Steady-state model in incremental mode:

12

4;

7

1 2

1 11 12

2 21 22

u u

y

y

21

4;

7 22

3;

7

1 2

1

2

3 47 7

347 7

u u

y

y

Properties of Relative Gain Matrix ?

Property of Relative Gain Matrix

1 2

1 11 12 1

2 21 22 2

1 2

n

n

n

n n n nn

u u u

y

y

y

1 1

1n n

ij iji j

Summation of all the terms in each row and in each column must equal 1.

1 2

1 11 11

2 11 11

1

1

u u

y

y

2×2 systems:

1 2 3

1 11 12

2 21 22

3

?

?

? ? ?

u u u

y

y

y

3×3 systems:

Computation of Relative Gain for n×n Systems

uy 1, yu

0e

iij

j u

yK

u

0 0

1e e

j iji

ji y y

u yH

uy

Tij

Note: “●” means the multiplication of matrix elements

0

0

,e

e

i j uij ij ji

i j y

y uK H

y u

Application of Relative Gain

Meaning of Relative Gain CVs and MVs Pairing Application Examples

1 2

1

2

0.2 0.8

0.8 0.2

u u

y

y

Example 1: 1 2

1

2

2 1

1 2

u u

y

y

Example 2:

Problem: which are the best pairs and why?

Significance of Relative Gain

0

0

e

e

i j u ijij

iji j y

y u K

Ky u

1ij ij

ij

K K

1ij : no interaction between the particular loop and all other loops, or possible offsetting interaction.

0ij : the open-loop gain is very small, or the closed-loop gain is very large.

ij : the open-loop gain is very large, or the closed-loop gain is very small.

0ij : the signs of the closed-loop gain and the open-loop gain are different

Pairing Rule of Multiloop Systems

Bristol (1966) : To minimize the interaction between loops, always pair on RGM elements that are closest to 1.0. Avoid negative pairings.

1 2

1

2

0.2 0.8

0.8 0.2

u u

y

y

Example 1: 1 2

1

2

2 1

1 2

u u

y

y

Example 2:

Problem: which are the best pairs and why?

Pairing of a Blending Process

1 1 1

2 2 2

,y u FF

y u FC

21

22112

211

uu

uCuCy

uuy

FC

AC

F1, C1

F2, C2

F, C

Blending Tank

FC

FC

Steady-state model:

Problem: it is a nonlinear model, how can you analyze the coupling between two loop ?

Suppose C1 > C2

Pairing of a Blending Process (cont.)

1. Obtaining steady-state process gain:

2 1

1 2 2 2 1 12 221 222 2

1 21 2 1 2

,u u

C C u C C uy yK K

u uu u u u

21

22112

211

uu

uCuCy

uuy

1 1

2 21 22 2

1 1y u

y K K u

Pairing of a Blending Process (cont.)

2. Obtaining relative gain matrix:

1111

12 21 2111

22 22

1

2 1 2

1

1

1

1;

1

KK K K

KK K

uu u uu

1 1

2 21 22 2

1 1y u

y K K u

1 2

1 21

1 2 1 2

2 12

1 2 1 2

u u

u uy

u u u u

u uy

u u u u

Pairing of a Blending Process (cont.)

3. Pairing CVs and MVs using RGM

1 2

1 2

1 2 1 2

2 1

1 2 1 2

F F

F FF

F F F F

F FC

F F F F

If F1>F2, the correct pairing is F － F1, C － F2;

If F2>F1, the correct pairing is F － F2, C － F1.

If F2=F1, which is the correct pairing ?

Exercise 8.1It is assumed that the open-loop gain matrix of a 3×3 multivariable controlled process is

1 1

2 2

3 3

0.58 0.36 0.36

0 0.61 0 .

1 1 1

y u

y u

y u

Please calculate the relative gain matrix of the process, select the best pairing of CVs and MVs, and analyze if a decoupler is needed.

Next Topic: Decoupling Control of Multivariable Systems

When Necessary to Design Decoupler Linear Decoupler Design from Block

Diagrams Nonlinear Decoupler Design from Basic

Principles Application Examples

Problem Discussion for Next Topic

For the controlled system, design your decoupling control systems and simulate your solution with SimuLink. If F1, C1 F2, C2

FT03

C F

FC03

ASP

AT11

F1SP

FC01

FT01

F2SP

FC02

FT02

FSP

AC11

Am Fm

u2u1

F1m F2m

;,21

221121 FF

FCFCCFFF

;12

,14

1,

1

5.0 5

s

e

C

A

sC

C

sF

F smm

%.40%,60,25,75 210

20

1 CCFF

Initial states: