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Multivariable Control Systems ECSE 6460. Fall 2009 Lecture 15: 23 October 2009. State Space Plant Model. Given a 2 input - 2 output plant:. Coupled Plant. Control Problem. We want the outputs y1 and y2 to track two reference signals r1 and r2 simultaneously. - PowerPoint PPT Presentation
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Multivariable Control SystemsECSE 6460
Fall 2009
Lecture 15: 23 October 2009
State Space Plant Model
Given a 2 input - 2 output plant:
Coupled Plant
0 2 4 60
2
4
6
8
10
12
To:
Out
(2)
From: In(2)
0 2 4 6
Coupled Plant Step Responses
Time (sec)
Am
plitu
de -6
-4
-2
0
2From: In(1)
To:
Out
(1)
Control Problem
We want the outputs y1 and y2 to track two reference signals r1 and r2 simultaneously.
Performance criteria for step reference response:Overshoot less than 5%Three scenarios for minimum rise time =
[30, 3, 0.03] sec
Geometric I/O Decoupling
Geometric I/O Decoupling
Geometric I/O Decoupling
0
500
1000
1500From: In(1)
To:
Out
(1)
0 500 1000 15000
500
1000
1500
To:
Out
(2)
From: In(2)
0 500 1000 1500
Decoupled Plant Step Responses
Time (sec)
Am
plitu
de
Controller Design
C
F
GK(s)+ +
+-
Performance (wc = 0.1)
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
To:
Out
(2)
From: In(2)
0 20 40 60 80 100
System: TI/O: In(2) to Out(2)Time (sec): 23.1Amplitude: 0.9
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de 0
0.2
0.4
0.6
0.8
1
System: TI/O: In(1) to Out(1)Time (sec): 23Amplitude: 0.9
From: In(1)T
o: O
ut(1
)
Performance (wc = 1 )
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
To:
Out
(2)
From: In(2)
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de
0 2 4 6 8 10
System: TI/O: In(2) to Out(2)Time (sec): 2.3Amplitude: 0.9
0
0.2
0.4
0.6
0.8
1
System: TI/O: In(1) to Out(1)Time (sec): 2.31Amplitude: 0.901
From: In(1)T
o: O
ut(1
)
Performance (wc = 100)
0 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
1
To:
Out
(2)
From: In(2)
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de 0
0.2
0.4
0.6
0.8
1
System: TI/O: In(1) to Out(1)Time (sec): 0.023Amplitude: 0.9
From: In(1)
To:
Out
(1)
0 0.02 0.04 0.06 0.08 0.1
System: TI/O: In(2) to Out(2)Time (sec): 0.023Amplitude: 0.9
RGA computation
Diagonal vs off-diagonal pairing
10-5
100
105
1010
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Diagonal (blue) and Off-diagonal RGA (red)
RGA Number
10-5
100
105
1010
0
0.5
1
1.5
2
2.5
3
3.5
4RGA number
Independent Design
Idea: Design a diagonal controller while disregarding the coupling between the two I/O pairs.
CK(s)+
-
Independent Design
Performance (wc = 0.1)
-150
-100
-50
0
50From: In(1)
To:
Out
(1)
100
105
-200
-150
-100
-50
0
50
To:
Out
(2)
From: In(2)
100
105
Sensitivity Functions
Frequency (rad/sec)
Mag
nitu
de (
dB)
Performance (wc = 0.1)
0 20 40 60 80 100
-0.5
0
0.5
1
To:
Out
(2)
From: In(2)
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de 0
0.2
0.4
0.6
0.8
1
System: TI/O: In(1) to Out(1)Time (sec): 23.2Amplitude: 0.9
From: In(1)T
o: O
ut(1
)
0 20 40 60 80 100
System: TI/O: In(2) to Out(2)Time (sec): 23.4Amplitude: 0.902
Performance (wc = 0.1)
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de
0 5 10 15 20 25 30 35 40 45 50-0.5
0
0.5
1
System: TOutput: Out(2)Time (sec): 47.4Amplitude: 0.9
To:
Out
(2)
0
0.2
0.4
0.6
0.8
1
System: TOutput: Out(1)Time (sec): 23.6Amplitude: 0.9
To:
Out
(1)
Increase wc to 0.2
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de 0
0.2
0.4
0.6
0.8
1
System: TOutput: Out(1)Time (sec): 12Amplitude: 0.9
To:
Out
(1)
0 5 10 15 20 25-0.5
0
0.5
1
System: TOutput: Out(2)Time (sec): 24.2Amplitude: 0.9
To:
Out
(2)
Performance (wc = 1)
-150
-100
-50
0
50From: In(1)
To:
Out
(1)
100
105
-150
-100
-50
0
50
To:
Out
(2)
From: In(2)
100
105
Sensitivity Functions
Frequency (rad/sec)
Mag
nitu
de (
dB)
Performance (wc = 1)
0
0.2
0.4
0.6
0.8
1From: In(1)
To:
Out
(1)
0 2 4 6 8 10
-0.5
0
0.5
1
To:
Out
(2)
From: In(2)
0 2 4 6 8 10
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de
Performance (wc = 1)
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de 0
0.2
0.4
0.6
0.8
1
System: TOutput: Out(1)Time (sec): 2.87Amplitude: 0.9
To:
Out
(1)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.5
0
0.5
1
System: TOutput: Out(2)Time (sec): 4.88Amplitude: 0.854
To:
Out
(2)
Increase wc to 2.5
0
0.2
0.4
0.6
0.8
1
System: TOutput: Out(1)Time (sec): 1.42Amplitude: 0.9
To:
Out
(1)
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
System: TOutput: Out(2)Time (sec): 2.72Amplitude: 0.9
To:
Out
(2)
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de
Performance (wc = 100)
-150
-100
-50
0
50From: In(1)
To:
Out
(1)
100
105
-100
-80
-60
-40
-20
0
20
To:
Out
(2)
From: In(2)
100
105
Sensitivity Functions
Frequency (rad/sec)
Mag
nitu
de (
dB)
Performance (wc = 100)
-3
-2
-1
0
1
2
3x 10
4 From: In(1)T
o: O
ut(1
)
0 0.02 0.04 0.06 0.08 0.1-2000
-1000
0
1000
2000
To:
Out
(2)
From: In(2)
0 0.02 0.04 0.06 0.08 0.1
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de
Performance (wc = 100)
-6
-4
-2
0
2
4
6x 10
5T
o: O
ut(1
)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-4
-2
0
2
4x 10
4
To:
Out
(2)
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de
Independent Design
Not proper!
Performance (wc = 0.2)
-4
-2
0
2
4
6From: In(1)
To:
Out
(1)
0 10 20 30 40 50-10
-5
0
5
10
To:
Out
(2)
From: In(2)
0 10 20 30 40 50
Close Loop Plant Step Responses
Time (sec)
Am
plitu
de
Sinusoidal tracking (wr = 1000)
-1
-0.5
0
0.5
1T
o: O
ut(1
)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-1.5
-1
-0.5
0
0.5
1
1.5
To:
Out
(2)
Linear Simulation Results
Time (sec)
Am
plitu
de
Sinusoidal tracking (wr = 1000)
-1
-0.5
0
0.5
1T
o: O
ut(1
)
0 0.01 0.02 0.03 0.04 0.05 0.06-1.5
-1
-0.5
0
0.5
1
1.5
To:
Out
(2)
Linear Simulation Results
Time (sec)
Am
plitu
de
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