16 part2.activevibrationcontrolpiezo fuzzy
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Instructor: Dr. SongDept. of Mechanical Engineering
116. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
Acknowledgement: The help from Haichang Gu is greatly appreciated.
Topic 16. Active Vibration ControlUsing Piezoelectric Materials
Part 2. Fuzzy Control Methods
Dr. G. Song, Associate Professor
University of Houston
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Instructor: Dr. SongDept. of Mechanical Engineering
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Outline
1. Introduction
2. Positive position feedback control with gain tuning
2.1 PPF control
2.2 Traditional fuzzy gain tuning2.3 Batch least squares fuzzy gain tuning
3. Experimental set-up
4. Experimental result
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1.Introduction
For the structural vibration control, positive position feedback (PPF)
control was first proposed by Goh and Caughey in 1985. To ensure
vibration is quickly suppressed, a large scalar gain is often used in a
PPF controller. However, PPF control with a large gain will cause
initial overshoot, which is undesirable in many situations.
A fuzzy gain tuner is proposed to tune the scalar gain in the positive
position feedback control. The fuzzy system is trained by the desired
input-output data by batch least squares algorithm so that the fuzzy
system can behave as we desired. The experiments are applied in an11-foot-long I-beam with piezoceramic patch sensors and actuators.
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416. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
2. Positive position feedback(PPF)control with gain tuning
2.1 PPF control
In PPF control, structural position information is fed to a compensator.
The output of the compensator, magnified by a scalar gain, is fed
directly back to the structure.
Figure 1 The block diagram of the PPF (positive position feedback) controller
02 2 =++
02 2ccc =++
2
c2
G
+
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Positive position feedback(PPF)control with gain tuning
The equations describing PPF operation are given as
Structure: (1a)
Compensator: (1b)
: modal coordinate describing displacement of the structure
: damping ratio
: natural frequency of the structure
G : the feedback gain: the compensator coordinate
: compensator damping ratio
: the frequency of the compensator.
=++ 22 G2
=++ 2c2
ccc2
c
c
6
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616. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
2.2 Traditional fuzzy gain tuning
A tradition fuzzy system is designed to tune the PPF scalar gain G during the
vibration control process. Sensor signal is input and the scalar gain is theoutput.
Gaussion membership function is defined as the membership function of inputvariable and output variable.
(2)
We divide the universe of discourse of the input fuzzy variable (sensor signal)into 4 overlapping fuzzy sets {s1, s2, s3, s4}. We also divide the universe ofdiscourse of the output fuzzy variable (PPF scalar gain) into 4 overlappingfuzzy sets {m1, m2, m3, m4}.
2
2)(
)(
ax
ex
=
7
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716. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
The number indicates the degree to which the data x belongs to a
fuzzy set. The centers of the 4 membership functions for the input are
defined as 0.2,0.3,0.4,0.5, the width of each membership function is0.0541.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60
0.2
0.4
0.6
0.8
1
The membership functions for the input
s1 s2 s3 s4
)x(
8
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816. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
The centers of the output membership functions are 0.4, 1.166, 1.8333,
2.5.
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
The membership functions for the output
m1 m2 m3 m4
9
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916. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
The fuzzy inference rules:
If the vibration amplitude is s1, then the PPF gain is m4;If the vibration amplitude is s2, then the PPF gain is m3;
If the vibration amplitude is s3, then the PPF gain is m2;
If the vibration amplitude is s4, then the PPF gain is m1
Defuzzification:
(3)
where is the input membership function, x is the input variable , is thecenter of the output membership function
=
=
=
4
1l l
4
1l
l
l
*
)x(
)x(y
y
)x(ily
1016 A i i i C i i i i C
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1016. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
The input(error)-output(gain) map of the fuzzy gain tuner
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.5
1
1.5
2
2.5
The input-output map of the fuzzy gain tuner
absolute value of error
gainvalue
1116 A ti Vib ti C t l U i Pi l t i M t i l F C t l M th d
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1116. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
(1) Get the desired data
5 10 15 20 25 30-10
-5
0
5
10
timesensorvolta
geforvibrationamplitud
5 10 15 20 25 300.5
1
1.5
2
2.5
3
t ime
desiredgainvalue
2.3 Batch least squares fuzzy gain tuning
1216 A ti Vib ti C t l U i Pi l t i M t i l F C t l M th d
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1216. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
We use this input (sensor voltage of the vibration amplitude)-output (PPF
scalar gain) data map sets to train the fuzzy system.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.5
1
1.5
2
2.5
3The desired input-ouput map
Vibration amplitude
gain
1316 A ti Vib ti C t l U i Pi l t i M t i l F C t l M th d
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1316. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
We consider a fuzzy system described in equation (4)
(4)
bi is the center of the output membership function for the ith rule.
(5)
R is the number of rules.
If we define
(6)
=
=
==
R
1i i
R
1i ii
)x(
)x(b)|x(fy
===
= ++==R
i
i
RR
R
i
i
R
i
i
R
i
ii
x
xb
x
xb
x
xb
y
11
11
1
1
)(
)(
)(
)(
)(
)(
"
T
R
i
i
R
R
i
i x
x
x
xX ]
)(
)(,
)(
)([
11
1
==
=
"
T
Rbbb ],,[ 21 "=
Batch least squares algorithm
1416 A ti Vib ti C t l U i Pi l t i M t i l F C t l M th d
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1416. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
Then (7)
(M is the number of the training data)
(8)
We choose to be a measure of how good the approximation is for all the data for
a given .
Xy T=
T
M
yyyY ],,[21
"=
[ ]
[ ]
[ ]
=
TM
T
T
X
X
X
#
2
1
= YE
EE2
1)(V T=
Batch least squares algorithm
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1516. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
So (9)
We assume that is invertible and letting
(10)
(11)
This is so called Batch Least Squares algorithm.
In our experiment, the estimation of the centers of the output membership functions
+== TTTTTTT YYYYEEV2
Y)(YY)(YYYYYV2 T1TTT1TTTTTTTT ++=
)Y)(()Y)((Y))(I(Y T1TTTT1TT1TT +=
Y)( T1T =
TT
R21 ]6627.0,0968.1,9233.0,4640.2[]b,b,b[ == "
Batch least squares algorithm
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1616. Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
3.Experiment set-up
The experiment is implemented in a pultruded fiber-reinforced polymer (FRP)
composites thin-walled I-beam using smart sensors and actuators. The 3.35-
meter long beam is cantilevered at one end and PZT patches are bonded on it.
Base to Cantilever the 11-Foot Composite I-Beam
Peizo Patches as Sensors and Actuators
PC with Real-time Control System
Power Amplifier for Peizo Actuators
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I-beam properties
bw
bh
bt
b3m
Quality Description Units Value
L Beam Length m 3.35
Beam width mm 100
Beam height mm 102
Beam thickness mm 6
Beam density Kg/ 1850
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Two patches(model No.PZT QP-40W) are bonded on each of the top and
bottom flange surfaces.There are also two PZT patches (model No.10W)
bonded on the beam that act as sensors for the feedback of the signals in
the active control algorithms.
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One sensor is on the top flange to measure the beam vibration in the
strong direction and the other is bonded to the web of the I-beam to
measure its vibration in the weak direction. In the research, we only activecontrol vibrations along the strong direction of the beam.
2016 Active Vibration Control Using Piezoelectric Materials Fuzzy Control Methods
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twL 08.081.316.10 038.081.308.5
31d 1210350 1210350
T
3K
P
PE 10109.6 10109.6
Quality Description Units
QP40WQP10W
Dimensions cm
Lateral strain coefficient C/N
Dielectric constant 1800 1800
PZT density Kg/m3 7700 7700
Youngs Modulus N/m2
Properties of PZT patches used on the beam
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The Simulink Model
Vibration suppression algorithm is designed in the MATLAB/Simulink
and then downloaded to the dSPACE digital signal processor for
implementation.
Define w before running simulation
w1=7.4
1
sensor2
limit
Terminator2
Terminator1
Switch1
Switch
Sum2
Sum1
In1Out1
SubSystem
Sine Wave
Saturation
function3
S-Function for
Fuzzy system
RTI Data
Product
w1*w1
Pre_PPF_Gain
-w1*w1
Post_PPF_Gain
1
s +2*w1*2*pi*0.5s+(2*w1*pi)^22
PPF compensator
1
PPF Gain
Ground3
Ground2
Ground1
du/dt
Derivative
DAC #1
DAC #2
DAC #3
DAC #4
DS1102DAC
ADC #1
ADC #2
ADC #3
ADC #4
DS1102ADC
0
Constant
0
Clock
0
Bias
Band-Limited
White Noise
|u|
Abs
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Graphical User Interface
The dSPACE system comes with an analog to digital converter and a digit toanalog converter. The dSPACE ControlDesk module is used to develop a
Graphical User Interface(GUI) for online parameter adjustment and real time
data acquisition. This module allows to save the data in object oriented *.mat
format, which is then utilized in the matlab for further processing.
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Matlab codes were written to analyze the data for modal analysis andenergy drops at various modes. The input signals to the PZT actuators in
this experiment are amplified using voltage amplifiers, which have an
amplification factor of 20. These amplified signals drive the PZT actuatorsand are used to excite the I-beam. The sensor signals from both the weak
and strong directions of the beam are captured and only the strong
direction signal is used for feedback control.
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g y
4. Experimental resultFour experiments are implemented on the I-beam.
a)free vibration
b)standard PPF control
c)tradition fuzzy gain tuning PPF control
d) Batch least squares(BLS) fuzzy gain tuning PPF control
In each experiment, the beam is excited by the sinusoidal signals of
its first mode frequency with a combination of white noise for the
initial 5 seconds. The active vibration control is implemented after the
initial 5 seconds.
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g y
4.1 Free vibration
5 10 15 20 25-8
-6
-4
-2
0
2
4
6
8Time response of the free vibration
Time(Seconds)
Voltagefromt
hesensor
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g y
In the PPF control, the vibration was successfully suppressed in 5 seconds,
however, there is initial overshoot at the beginning stage, which may cause
damage to the instrument. So we need to use the fuzzy gain tuning method to
depress the initial overshoot of PPF control.
5 10 15 20 25-10
-8
-6
-4
-2
0
2
4
6
8
10The comparison between the free vibration and the PPF control
Time(s)
Sensor
voltageforvibrationamp
litude
Free vibration
PPF control
4.2 The standard positive position feedback control
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g y
The traditional fuzzy method is used to tune the scalar gain in PPFcontrol. The initial overshoot was suppressed, and the vibration wassuccessfully depressed in 5 seconds. However, the vibrationsuppression during 5 to 7 seconds is not satisfying.
5 10 15 20 25-10
-8
-6
-4
-2
0
2
4
6
8
10The comparison between the free vibration and the traditional fuzzy gain tuning PPF control
Time(s)
S
ensorvoltageforvibrationa
mplitude
Free vibration
Traditonal fuzzy gain tuning PPF control
4.3 Traditional fuzzy gain tuning PPF
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g y
4.4 Batch least squares fuzzy gain tuning PPF control
PPF control with batch least squares fuzzy gain tuner behaves muchbetter than the other two in terms of successfully reducing the initial
overshoot and quickly suppressing vibration.
5 10 15 20 25 30-10
-8
-6
-4
-2
0
2
4
6
8
10
time
se
nsorvoltageforvibrationam
plitude
The comparison between free vibratin and the batch least square gain tuning
free vibration
Batch least squares fuzzy gain tuning
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6 7 8 9 10 11-10
-8
-6
-4
-2
0
2
4
6
8
time
senso
rvoltageforvibrationamplitude
The comparison between the training data and the batch least square gain tuning
The training data
Batch leaast square fuzzy gain tuning
The comparison between the training data and the batch least squaresfuzzy gain tuning
In the comparison of the training data and the experimental data, theexperimental data match the desired data. This means the fuzzy gain tuner is
trained to behave in the way we desired. We can also train the fuzzy system for
other different requirements of vibration control, provided that the desired
input-output map data sets could be gotten.
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In order to analyze the vibration control effect, PSD plot is used to
show the energy distribution on frequency domain.
(a) Spectral Analysis
The goal of spectral estimation is to describe the
distribution (over frequency) of the power contained in a
signal, based on a finite set of data. Estimation of power
spectra is useful in a variety of applications, including
the detection of signals buried in wide-band noise.
4.5 The power spectrum density comparison of the experimental results
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Spectral analysis
The power spectrum of a stationary random process Xn is
mathematically related to the correlation sequence by thediscrete-time Fourier transform. In terms of normalized
frequency, this is given by
(12)
This can be written as a function of physical frequency f (e.g.,
in hertz) by using the relation , where is the samplingfrequency.
(13)
=
=m
mj
xxxx e)m(R)(S
=
=m
f/jfm2
xxxxse)m(R)f(S
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The correlation sequence can be derived from the power
spectrum by use of the inverse discrete-time Fourier
transform.
(14)
The average power of the sequence Xn over the entire
Nyquist interval is represented by(15)
dff
e)f(Sd2
e)(S)m(R
2/f
2/f s
f/jfm2
xx
mj
xxxx
x
x
s
=
=
dff
)f(Sd
2
)(S)0(R
2/f
2/f s
xxxxxx
x
x
=
=
Spectral analysis
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Power spectral density
The quantities
(16) and (17)
from the above expression are defined as thepower spectral density (PSD)
of the stationary random signal Xn.The average power of a signal over a particular frequency band[ ] ,
, can be found by integrating the PSD over that band.
You can see from the above expression that Pxx
( ) represents the power
content of a signal in an infinitesimal frequency band, which is why we
call it the power spectral density. The units of the PSD are power (e.g.,
watts) per unit of frequency.
=
2
)(S)(P xxxx
s
xxxx
f
)f(S)f(P =
21,
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(b) The PSD comparison of the 4 experiments
We have implemented the experiments of free vibration,
standard PPF control, traditional fuzzy gain tuning PPF
control, BLS fuzzy gain tuning PPF control. In each
experiment, the beam is excited by the sinusoidal signals at
its first mode frequency with a combination of white noisefor the initial 5 seconds. The active vibration control is
implemented after the initial 5 seconds.
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The power spectrum density comparison of these 4 experiments between
5-6 seconds
20 40 60 80 100 120
-80
-70
-60
-50
-40
-30
-20
-10
frequency (Hz)
energylevelinDecibels
The PSD comparisons between 5-6 seconds
free vibrationstandard PPF control
tradition fuzzy gain tuning
batch least squares fuzzy gain tuning
5 6 7 8 9 10 11 12 13 14 15-10
-8
-6
-4
-2
0
2
4
6
8
10The comparison between the free vibration and the PPF control
Time(s)
Sensorvoltageforvib
rationamplitude
Free vibration
PPF control
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The enlarged power spectrum density comparison of the experiments
between 5-6 seconds
The more energy drop at first mode frequency ,
the better result the vibration control is
4 6 8 10 12 14 16 18
-30
-25
-20
-15
-10
-5
frequency (Hz)
energylevelinDecibels
The PSD comparisons between 5-6 seconds
free vibration
standard PPF control
tradition fuzzy gain tuning
batch least squares fuzzy gain tuning
The method
The energy level
for 1st modal
frequency(dB)
The energy dropped
For the 1st modal
frequency(dB)(5-6S)
Free
vibration-1.08
_
Standard
PPF control-2.55 1.47
Traditional
fuzzy gain
tuning
-1.18 0.1
BLS fuzzy
gain tuning -13.88 12.8
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The power spectrum density comparison of these 4 experiments between
6-8 seconds
20 40 60 80 100 120-100
-80
-60
-40
-20
0
frequency (Hz)
energyl
evelinDecibels
The PSD comparisons between 6-8 seconds
free vibration
standard PPF control
tradition fuzzy gain tuningbatch least squares fuzzy gain tuning
5 6 7 8 9 10 11 12 13 14 15-10
-8
-6
-4
-2
0
2
4
6
8
10The comparison between the free vibration and the PPF control
Time(s)
Sensorvoltage
forvibrationamplitude
Free vibration
PPF control
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5 10 15 20 25 30 35
-20
-15
-10
-5
0
5
10
15
frequency (Hz)
energylevelinDecibels
The PSD comparisons between 6-8 seconds
free vibration
standard PPF controltradition fuzzy gain tuning
batch least squares fuzzy gain tuning
The
method
The energy
level
for 1st modal
frequency(dB)
The energy dropped
For the 1st modal
frequency (dB)(6-8S)
Free
vibration
14 _
Standard
PPF
control
-2.2 16.2
Traditional
fuzzy gain
tuning
5.9 8.1
BLS fuzzy
gain tuning
-7.4 21.4
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The power spectrum density comparison of the experiments
between 8-10 seconds
20 40 60 80 100 120
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
frequency (Hz)
energy
levelinDecibels
The PSD comparisons between 8-10 seconds
free vibration
standard PPF controltradition fuzzy gain tuning
batch least squares fuzzy gain tuning
5 6 7 8 9 10 11 12 13 14 15-10
-8
-6
-4
-2
0
2
4
6
8
10The comparison between the free vibration and the PPF control
Time(s)
Sensorvoltageforvibrationamplitude
Free vibration
PPF control
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4 6 8 10 12 14 16 18 20
-50
-40
-30
-20
-10
0
10
frequency (Hz)
energyleve
linDecibels
The PSD comparisons between 8-10 seconds
free vibrationstandard PPF control
tradition fuzzy gain tuning
batch least squares fuzzy gain tuning
The method
The energy level
for 1st modal
frequency(dB)
The energy dropped
For the 1st modal frequency
(dB)(8-10S)
Free
vibration11
_
Standard
PPF control-21.7
32.7
Traditional
fuzzy gain
tuning
-13.3 24.3
BLS fuzzy
gain tuning
-26.68 37.68
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Instructor: Dr. SongDept. of Mechanical Engineering
The power spectrum density comparison of the experiments
between 5-10 seconds
20 40 60 80 100 120-100
-80
-60
-40
-20
0
20
frequency (Hz)
energyle
velinDecibels
The PSD comparisons between 5-10 seconds
free vibration
standard PPF controltradition fuzzy gain tuning
batch least squares fuzzy gain tuning
5 6 7 8 9 10 11 12 13 14 15-10
-8
-6
-4
-2
0
2
4
6
8
10The comparison between the free vibration and the PPF control
Time(s)
Sensorvoltagefo
rvibrationamplitude
Free vibration
PPF control
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Instructor: Dr. SongDept. of Mechanical Engineering
0 5 10 15 20
-20
-10
0
10
20
30
40
frequency (Hz)
energylevelinDecibels
The PSD comparisons between 5-10 seconds
free vibration
standard PPF control
tradition fuzzy gain tuning
batch least squares fuzzy gain tuning
24.728.15BLS fuzzy
gain tuning
12.8720Traditional
fuzzy gain
tuning
19.5213.35Standard
PPF control
-32.87Freevibration
The energydropped
For the 1st
modal
frequency
(dB)(5-10S)
The energylevel
for 1st modal
frequency(d
B)
The method
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Instructor: Dr. SongDept. of Mechanical Engineering
The power spectrum density comparison of the experiments between 5-30
seconds
20 40 60 80 100 120
-100
-80
-60
-40
-20
0
20
frequency (Hz)
energylevelinDecibels
The PSD comparisons between 5-30 seconds
free vibration
standard PPF control
tradition fuzzy gain tuning
batch least squares fuzzy gain tuning
5 10 15 20 25 30
-10
-8
-6
-4
-2
0
2
4
6
8
10
time
sensorvoltageforvibrationamplitude
The comparison between free vibratin and the batch least square gain tuning
free vibration
Batch least squares fuzzy gain tuning
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Instructor: Dr. SongDept. of Mechanical Engineering
10 15 20 25 30
-80
-60
-40
-20
0
20
frequency (Hz)
energylevelinDecibels
The PSD comparisons between 5-30 seconds
free vibration
standard PPF control
tradition fuzzy gain tuningbatch least squares fuzzy gain tuning
The
method
The energy
level
for 1st
modalfrequency(dB)
The energy dropped
For the 1st modal
frequency (dB)(5-30S)
Free
vibration
34 -
StandardPPF
control
13.35 20.65
Traditional
fuzzy gain
tuning
20.3 13.7
BLS fuzzy
gain
tuning
8.59 25.41
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From the four experiments, the following observations and conclusion are
obtained:
1.The standard PPF control can effectively suppress the vibration, however, it is
accompanied with initial overshoot if the scalar gain is large.2.The traditional fuzzy system can be applied in tuning the scalar gain of PPF
control, however, due to the trial and error method in the choice of the fuzzy
system parameters, it cannot guarantee a satisfied overall result. In our
experiment, it suppresses the initial overshoot of PPF control, but the vibrationsuppression around the beginning 2 to 3 seconds is not effective.
3.The batch least squares fuzzy system trains the fuzzy system with desired data
by batch least squares algorithm so that the fuzzy gain tuner could behave in
the way we wish. The experimental data clearly reveal that the batch leastsquares fuzzy gain tuning has suppressed the initial overshoot and the result is
better than the standard PPF control and the traditional fuzzy gain tuning
method.
Conclusions:
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