16 part2.activevibrationcontrolpiezo fuzzy

7/28/2019 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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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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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


6/45Instructor: Dr. SongDept. of Mechanical Engineering


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




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




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




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




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




(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





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





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





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=


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[ == "






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





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





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.



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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.



30/45


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.



35/45


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


Time(s)

Sensorvoltageforvib

rationamplitude

Free vibration

PPF control



36/45


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)



free vibration

standard PPF control



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



37/45


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


free vibration


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


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)



free vibration

standard PPF controltradition 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


free vibration



5 6 7 8 9 10 11 12 13 14 15-10

-8

-6

-4

-2

0

2

4

6

8


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


free vibrationstandard PPF control



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



41/45


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


free vibration



5 6 7 8 9 10 11 12 13 14 15-10

-8

-6

-4

-2

0

2

4

6

8


Time(s)

Sensorvoltagefo

rvibrationamplitude

Free vibration

PPF control



42/45


0 5 10 15 20

-20

-10

0

10

20

30

40

frequency (Hz)



free vibration




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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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)



free vibration




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



44/45


10 15 20 25 30

-80

-60

-40

-20

0

20

frequency (Hz)



free vibration


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



45/45

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:

16 part2.activevibrationcontrolpiezo fuzzy

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