Fractional Order Relay Feedback Experiments for

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Zhuo Li
PhD Student, EECS, UC Merced
Member of the MESA Lab
zli32@ucmerced.edu
6/12/2013
Outlines
• Background
• Identification
– The relay feedback technique
– relay meets fractional calculus
– relay meets fractional order systems
• Decoupling
– The experiment platform
– When decoupling meet fractional order systems
• Some random thinking
2
Background
3
MEMS
Micro-electro-mechanical systems
Inside an accelerometer
http://memsblog.wordpress.com/2011/01/05/chipworks-2/
4
Nano fabrication, wafer processing
Demand:
High precision
High yield
Repeatability
Efficiency
Massive production
Fabrication of SiC nano-pillars by inductively coupled SF6/O2 plasma etching
J H Choi1,2, L Latu-Romain2, E Bano1, F Dhalluin2, T Chevolleau2 and T Baron2
2012 J. Phys. D: Appl. Phys. 45 235204
Challenges:
Difficult to sense
High nonlinearity
Multi variable
Synchronization
5
Mission for control engineers
•
•
•
•
•
Temperature
Pressure
Gas flow
RF power
etc ……
• Advanced modeling techniques
• Advanced control technologies
6
The relay feedback technique
7
The time line
Z-N Critical
Oscillation
P feedback
1942
Astrom &
Hugglund
Relay feedback
tuning
1984
Ramirez, R. W
Use FFT for
relay
1985
Luyben
W Li
Using relay for Relay with
identification time delay
1987
1991
CC Yu
Biased
relay
A Leva
1992
1993
K.K Tan
Waller
Modified Relay Two channel
Relay
1996
1997
J Lee et. al
Relay with FO
integrator behind
2011
Astrom, 1984, Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins
Luyben, 1987, Derivation of Transfer Functions for Highly Nonlinear Distillation Columns
Li, 1991, An improved auto tune identi๏ฌcation method
……
8
Varieties of relay feedbacks
Relay type
Process phase
Phase pre-know
Describing function
Phase shift range
Ideal
−๐›‘
Yes
One point
With hysteresis, ๐›†
−๐›‘ + ๐›Ÿ,
๐›†
๐ฌ๐ข๐ง๐›Ÿ =
๐€
No
๐Ÿ’๐‡
๐›‘๐€
๐Ÿ’๐‡ −๐ฃ๐›Ÿ
๐ž
๐›‘๐€
3rd and 4th
quadrant
With time delay
๐ž−๐‹๐ฌ
−๐›‘ + ๐‹
Yes
๐Ÿ’๐‡ −๐ฃ๐‹
๐ž
๐›‘๐€
3rd and 4th
quadrant
delay ๐ž−๐‹๐ฌ , behind
Same as above
-
-
-
Yes
๐Ÿ’๐‡
๐ฃ
๐›‘๐€
One point
With an integrator,
๐Ÿ
๐ฌ
๐Ÿ
๐ฌ
−
๐›‘
๐Ÿ
integrator, , behind
-
-
-
-
TC relay
−๐›‘ + ๐›Ÿ
๐‡
tan๐›Ÿ = ๐ข
Yes
๐Ÿ’๐‡๐ฉ ๐Ÿ’๐‡๐ข
−
๐ฃ
๐›‘๐€
๐›‘๐€
3rd quadrant
Biased ideal relay
−๐›‘
Yes
๐Ÿ’๐‡ ๐œ๐ŸŽ๐ฎ
+
๐›‘๐€ ๐œ๐ŸŽ๐ฏ
One point
Biased with
hysteresis
−๐›‘ + ๐›Ÿ,
๐›†
๐ฌ๐ข๐ง๐›Ÿ =
๐€
No
-
3rd and 4th
quadrant
๐‡๐
9
The frequency response
Im
-180
Re
Ideal Relay
Relay with
hysteresis
2 channel relay
Relay plus time
delay
-90
Relay plus an
integrator
10
When relay feedback meets with
fractional order integrator
11
1
๐‘ ๐›ผ
1
๐‘ ๐›ผ
12
Block diagrams
−
๐ด๐‘ ๐‘–๐‘›(๐œ”๐‘ก)
๐ด
๐ด
๐‘๐‘œ๐‘  ๐œ”๐‘ก
๐œ”
1.5
๐ป
1
๐œ”
0.5
0
−๐ด ๐œ”
Relay with integer order integrator
๐‘’
+ -
๐ด
− ๐‘—
2
1
๐‘ 
−
๐ด๐‘ ๐‘–๐‘›(๐œ”๐‘ก)
๐ด
๐œ‹
๐‘ ๐‘–๐‘› ๐œ”๐‘ก − ๐›ผ
๐œ”๐›ผ
2
1.5
๐ด
2๐‘—
1
๐‘ ๐›ผ
1
0.5
0
0
-0.5
-0.5
-1
-1
0
1
๐‘ฃ
2
3
4
5
6
7
8
9
10
๐ด
๐œ‹
๐‘ ๐‘–๐‘› ๐œ”๐‘ก − ๐›ผ
๐œ”๐›ผ
2
1 −๐œ‹๐›ผ๐‘—
๐‘’ 2
๐œ”๐›ผ
4๐ป๐œ”๐›ผ
๐œ‹๐ด
-1.5
-1
๐ด
2๐œ”
0
1
๐‘ข
−
4๐ป๐œ”
๐œ‹๐ด
2
3
4
5
6
7
8
9
10
2๐ป
๐œ‹
−
4๐ป
๐‘—
๐œ‹๐ด
1.5
1
0.5
-1.5
-1
๐‘’
-H
-1
๐‘ฃ
−
1
๐‘—๐œ”
-0.5
-1.5
-1
0
1
−
2
3
4
5
6
7
8
9
10
๐‘ข
2๐ป ๐œ‹(1−๐›ผ)๐‘—
๐‘’2
๐œ‹
Relay with fractional order integrator
4๐ป −๐œ‹๐›ผ๐‘—
๐‘’ 2
๐œ‹๐ด
13
Varieties of relay feedbacks
Relay type
Process phase
Phase pre-know
Describing function
Phase shift range
Ideal
−๐›‘
Yes
One point
With hysteresis, ๐›†
−๐›‘ + ๐›Ÿ,
๐›†
๐ฌ๐ข๐ง๐›Ÿ =
๐€
No
๐Ÿ’๐‡
๐›‘๐€
๐Ÿ’๐‡ −๐ฃ๐›Ÿ
๐ž
๐›‘๐€
3rd and 4th
quadrant
With time delay
๐ž−๐‹๐ฌ
−๐›‘ + ๐‹
Yes
๐Ÿ’๐‡ −๐ฃ๐‹
๐ž
๐›‘๐€
3rd and 4th
quadrant
delay ๐ž−๐‹๐ฌ , behind
Same as above
-
-
-
Yes
๐Ÿ’๐‡
๐ฃ
๐›‘๐€
One point
With an integrator,
๐Ÿ
๐ฌ
๐Ÿ
๐ฌ
−
๐›‘
๐Ÿ
integrator, , behind
-
-
-
-
TC relay
−๐›‘ + ๐›Ÿ
๐‡
tan๐›Ÿ = ๐ข
Yes
๐Ÿ’๐‡๐ฉ ๐Ÿ’๐‡๐ข
−
๐ฃ
๐›‘๐€
๐›‘๐€
3rd quadrant
๐›‘
๐›‚
๐Ÿ
Yes
๐Ÿ’๐‡ ๐Ÿ
๐›‘๐€ (๐ฃ)๐›‚
3rd and 4th
quadrant
๐‡๐
With FO integrator,
๐Ÿ
๐ฌ๐›‚
−๐›‘ +
Biased ideal relay
−๐›‘
Yes
๐Ÿ’๐‡ ๐œ๐ŸŽ๐ฎ
+
๐›‘๐€ ๐œ๐ŸŽ๐ฏ
One point
Biased with
hysteresis
−๐›‘ + ๐›Ÿ,
๐›†
๐ฌ๐ข๐ง๐›Ÿ =
๐€
No
-
3rd and 4th
quadrant
14
The frequency response
Im
-180
Re
Ideal Relay
Relay with
hysteresis
2 channel relay
Relay plus time
delay
Relay plus an
FO integrator
-90
Relay plus an
integrator
15
The frequency response
Im
-180
Re
Ideal Relay
Relay with
hysteresis
2 channel relay
Relay plus time
delay
Relay plus an
FO integrator
-90
Relay plus an
integrator
16
17
Simulation Eg.1
2
๐บ ๐‘  =
๐‘’ −0.1๐‘ 
2๐‘  + 1
With Hysteresis, eps= ๏‚ฑ0.3
Ideal relay, H=1
A
To
T
Error (%)
L
1
1
0
0
-1
-1
0
Ideal
Hysteresis
Delay
0.097
0.383
0.846
0.39
1.552
3.612
1.623
1.624
1.632
18.86
18.79
18.4
0.099
TC
FO int
0.7
0.195
0.133
2.853
0.743
0.536
1.589
2.183
1.627
20.53
9.16
18.65
10
0
With time delay, L=1
5
10
With integrator
1
1
0
0
0.201
0.135
-1
-1
0
Integrator
5
0.126
0.099
0.102
5
10
0
5
10
With FO integrator, ๏ก =0.1
TC relay, Hi=Hp=1
2
1
0
0
-2
-1
0
5
Time [sec]
10
0
5
Time [sec]
10
12.8
๐‘’ −๐‘ 
16.7๐‘  + 1
๐บ ๐‘  =
Simulation Eg.2
R. K. WOOD and M. W. BERRY Model
“Terminal composition control of a binary distillation column”
๐›ผ = 0.1:0.1:1.8
๐Ÿ
๐ฌ๐›‚
๏ก
๏ก
1/s , ๏ก = 0.1
1/s , ๏ก = 0.2
1
1
0
0
-1
0
20
40
60
80
100
120
140
160
180
200
-1
0
20
40
60
80
๏ก
100
120
140
160
180
1/s , ๏ก = 0.4
2
2
0
0
A
To
T
Error (%)
L
0.1
0.853
4.42
13.4
19.73
1.03
0.3
1.09
5.7
13.53
19.01
1.06
0.5
1.5
7.86
13.54
18.91
1.1
0.7
2.29
12.1
13.55
18.87
1.18
0.9
3.81
20.5
13.57
18.72
1.29
1.1
7.17
42
13.64
18.31
2
1.3
10.8
81.8
14.8
11.4
3.25
1.4
11.9
111
16.47
1.35
1.05
1.5
12.5
148
19.63
17.56
2.1
1.6
12.7
196
24.98
49.56
1.43
12.8
243
30.51
12.8
235 20 29.44
๏ก 1.7
1/s , ๏ก = 1.3
1.8
200
๏ก
1/s , ๏ก = 0.3
๐›ผ
20
0
-2
0
20
40
60
80
100
120
140
160
180
200
-2
0
20
40
60
80
๏ก
0
0
20
40
60
80
100
120
140
160
180
200
-2
0
60
80
100
20
40
60
80
120
140
160
180
200
-5
0
20
40
60
-20
120
140
160
180
200
100
120
140
160
180
200
๏ก
1/s , ๏ก = 1
5
0
0
-10
20
40
60
80
100
120
140
160
180
200
0
20
40
60
80
0
0
60
80
100
600
800
-20
0
200
400
600
800
๏ก
1/s , ๏ก = 1.6
20
0
0
-20
0
200
400
600
800
-20
0
100
120
140
160
180
200
400
600
800
๏ก
1/s , ๏ก = 1.8
20
20
0
0
200
๏ก
1/s , ๏ก = 1.2
10
40
400
1/s , ๏ก = 1.7
๏ก
20
200
๏ก
1/s , ๏ก = 1.1
10
0
0
20
10
-5
-10
100
80
๏ก
1/s , ๏ก = 0.9
0
0
200
๏ก
0
40
13.2
1/s , ๏ก = 0.8
5
20
180
1/s , ๏ก = 1.5
0
๏ก
0
160
๏ก
1/s , ๏ก = 0.7
5
-5
140
1/s , ๏ก = 0.6
2
0
120
7.62
๏ก
1/s , ๏ก = 0.5
2
-2
100
๏ก82.68
1/s , ๏ก = 1.4
76.31
120
140
160
180
200
-10
-20
0
20
40
60
80
100
120
140
160
180
200
0
200
400
600
800
-20
0
200
400
600
800
Advantages
• Wider phase range
• Phase can be predetermined
๐œ™ = arcsin
๐œ€
๐ด
๐œ‹
2
, ๐œ‘๐‘ = −๐œ‹ + ๐›ผ
• Non-zero initial part (efficient)
Save a quarter cycle time !
Think about some slow processes
e.g. distillation column
Relay with time delay
Relay with FO integrator
20
When relay feedback meets with
fractional order system
21
Equations for relay identification
For integer order systems
๐พ
๐บ ๐‘  =
๐‘’ −๐ฟ๐‘ 
๐‘‡๐‘  + 1
๐พ
๐บ ๐‘—๐œ”
๐‘‡=
๐ฟ=
2
−1
For fractional order system (Proposed method)
๐พ
๐บ ๐‘  = ๐›ผ
๐‘’ −๐ฟ๐‘  , 0 < ๐›ผ < 2
๐‘‡๐‘  + 1
๐œ‹
๐œ‹
๐พ
− cos 2 ๐›ผ + cos2 2 ๐›ผ + (
)2 − 1
๐บ(๐‘—๐œ”)
๐‘‡=
๐œ”๐›ผ
๐œ”
1
(∠๐บ − arctan(๐‘‡๐œ”))
๐œ”
๐›ผ sin ๐œ‹ ๐›ผ
๐‘‡๐œ”
1
2
๐ฟ=
∠๐บ − arctan
๐œ‹
๐œ”
๐‘‡๐œ” ๐›ผ cos 2 ๐›ผ + 1
Equations for IO are special cases of those for FO
22
Simulation
Relay with integrator
2
• ๐บ ๐‘  =
12.8
๐‘’ −๐‘ 
0.5
16.7๐‘  +1
0
-2
0
5
10
15
20
25
30
35
40
45
50
30
35
40
45
50
30
35
40
45
50
35
40
45
50
Relay with time delay
2
0
-2
0
5
10
15
20
25
Ideal relay
1
0
-1
0
5
10
15
20
25
Relay with hysteresis
1
0
-1
0
5
10
15
Ideal
relay
๐ด๐‘œ
0.7049
๐‘‡๐‘œ
2.4520
๐‘‡
13.9949
Error
L
16.2%
0.9314
20
25
30
Time [sec]
With
delay
With
integrator
1.2369
1.0962
8.1960
6.2240
14.2198
14.0771
14.85%
15.71%
1.0945
0.8251
With
hyst
0.9002
4.1150
14.0679
15.76%
1.1830
23
Experimental implementation
Raw Data from
Platform on slide 27
50
30
40
28
300
26
250
25
200
24
150
30
26
23
100
22
20
24
50
10
22
0
0
20
350
21
20
-50
0
50
100
150
200
250
300
0
50
19
150
100
Order: ๏ก = 0.8
29
Identified by relay feedback
28
0.16
๐บ ๐‘  =
๐‘’ −1.08๐‘ 
0.8
15.31๐‘  + 1
Identified by curve fitting
Using Dr.Podlubny’s mlf
๐บ ๐‘  =
24
relay with hyst
300
0.1584
๐‘’ −0.96๐‘ 
15.79๐‘  0.8 + 1
250
23
Raw data
Model response
22
21
0
20
40
60
0.18
80
Time [sec]
100
120
140
Order
scanning
0.16
0.14
0.12
160
25
200
24
23
150
22
100
21
20
50
19
0
Temperature [๏‚ฐC]
25
Relay signal: PWM duty cycle /255
26
Fitting error: least mean squares
Temperature ๏‚ฐC
27
18
-50
0.1
0
0.08
0.06
0.5
0.55
0.6
0.65
0.7
0.75
0.8
Fractioanl order ๏ก
0.85
0.9
0.95
1
10
20
30
40
50
60
70
80
90
100
24
Future work
• Other model structures
• Using relay transient
๐บ ๐‘  =
๐พ
−๐ฟ๐‘  … …
๐‘’
(๐‘‡๐‘  + 1)๐‘›
25
The experiment platform
26
The development highlights
•
•
•
•
•
•
Thermoelectric modules Power
H-bridge, heating/cooling
IR thermo meters
Two inputs four outputs
MOSFET
Real-time control
Product of multiple failures
Arduino
PC
(Matlab)
Side product
27
The hardware configuration
Heat
sink
Peltier
Load
28
A video demo
29
The four modes
Power on cooling – heat pumping
Power off cooling – annealing/natural
dissipation
Power on heating – electrical heating
Power off heating – thermo cycle
30
Performance testing
•
•
PID control with anti-windup
Testing with actuator only having cooling capability
Set point
Control signal
Temperature
The non-minimum phase temperature data
Input/255 [Duty cycle]
Fitting using second order model
Fitting using fractional order model
100
Commemorate order
50
0
0
50
100
150
200
250
300
0
50
100
150
Time [sec]
200
250
300
Temperature ๏‚ฐC
30
25
20
15
๐‘‡4๐‘  0.5 + 1
๐บ ๐‘  =
๐‘’ −๐ฟ๐‘ 
1.5
0.5
๐‘‡1๐‘  + ๐‘‡2๐‘  + ๐‘‡3๐‘  + 1
4
4
Temperature data
Model response
2
1
๐บ ๐‘ 
0
๐พ(๐‘‡3 ๐‘  + 1) −๐ฟ๐‘ 
=
๐‘’
๐‘‡1 ๐‘  2 + ๐‘‡2 ๐‘  + 1
-1
-2
2
1
0
-1
-2
0
20
40
60
80
100
120
Time [sec]
140
160
180
Temperature data
Model response
3
Temperature ๏‚ฐC
Temperature ๏‚ฐC
3
200
[K T1 T2 T3] = [1.7048 198.8152 53.7816 39.3604]
0
20
40
60
80
100
120
Time [sec]
140
160
180
200
[K T1 T2 T3] = [2716 -877 349.3 -6.1]
Decoupling
33
34
The conventional techniques
35
Conventional Decoupling
• Ideal decoupling
• Simple decoupling
• Inverted decoupling
36
Example – simplified decoupling
• System
• ๐บ ๐‘  =
0.5 −1.5๐‘ 
๐‘’
๐‘ +1
2
๐‘’ −๐‘ 
0.5๐‘ +1
1
๐‘’ −0.5๐‘ 
2๐‘ +1
1
๐‘ +1
Step Response
From: In(1)
From: In(2)
1
• Decoupler
๐‘”
− 21
๐‘”22
1
To: Out(1)
0
Original response
After decoupling
-0.5
-1
-1.5
2
1
To: Out(2)
1
๐‘”12
−
๐‘”11
Amplitude
• D ๐‘  =
0.5
0
-1
-2
-3
0
2
4
6
8
10
12
14 0
Time (seconds)
2
4
6
8
10
12
14
Example – modified simplified
• System
• ๐บ ๐‘  =
0.5 −1.5๐‘ 
๐‘’
๐‘ +1
2
๐‘’ −๐‘ 
0.5๐‘ +1
1
๐‘’ −0.5๐‘ 
2๐‘ +1
1
๐‘ +1
Step Response
From: In(1)
From: In(2)
1
• Decoupler
−๐‘ฃ
๐‘”21
๐‘”22
๐‘ฃ
To: Out(1)
0
Original response
After decoupling
-0.5
-1
-1.5
2
1
To: Out(2)
๐‘ฃ
๐‘”12
−๐‘ฃ
๐‘”11
Amplitude
• D ๐‘  =
0.5
0
-1
-2
-3
0
2
4
6
8
10
12
14 0
Time (seconds)
2
4
6
8
10
12
14
What if the process is fractional order
39
Fractional order decoupler
40
Random thinkings
41
Another example
Credit: Dr.Richard Migan
Zhuo Li
42
Some diffusion data
43
Temperature in a sealed room –
bounded diffusion
•
•
Half order plus delay
Using NILT/Mittag leffler
•
•
Half order plus delay
Using NILT/Mittag leffler
•
๐บ ๐‘  =
K
•
๐บ ๐‘  =
•
•
[K T L] = 2.1232 22.8021 9.7312
Fitting error (least mean squares): 0.0700
๐‘’ −๐ฟ๐‘ 
T๐‘ 0.5 +1
[K T L] = 6.0031 5.2222 14.7917
•
•
Fitting error (least mean squares): 0.2214
5
−๐ฟ๐‘ 
4
3
Temperature ๏‚ฐC
Raw data
Model response
2
1
0
Raw data
Model response
3
2
1
0
0
20
40
60
80
100
Time [sec]
120
140
160
-1
180
0
20
25
40
60
80
100
Time [sec]
120
140
160
180
20
fitting error
Temperature ๏‚ฐC
T๐‘ +1 ๐‘’
5
4
-1
K
15
10
5
0
0.2
0.4
0.6
0.8
1
order: ๏ก
1.2
1.4
1.6
44
Thank you
45
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