coupling analysis of multivariable systems ( 多变量系统的关联分析 ) lei xie zhejiang...
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Coupling Analysis of Multivariable Systems( 多变量系统的关联分析 )
Lei XIE
Zhejiang University, Hangzhou, P. R. China
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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
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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 ?
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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
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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
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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
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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
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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
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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
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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
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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 ?
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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 ?
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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 ?
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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 ?
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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
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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 ?
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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:
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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
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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?
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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
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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?
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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
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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
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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
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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 ?
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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.
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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
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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: