APPLICATION OF ADAPTIVE FILTERING ALGORITHM IN...

Sudan Engineering Society Journal, March 2010, Volume 56 No.54

29

APPLICATION OF ADAPTIVE FILTERING ALGORITHM IN VIBRATION CONTROL

Martino Ojwok Ajangnay1, Barry W Williams

2 and Matthew W Dunnigan

3

1 Electrical Engineering Department, Sudan University of Science and Technology, Sudan,

[email protected] 2Electronic and Electrical Engineering Department, Strathclyde University, United Kingdom,

[email protected]

3 Electrical, Electronic and Computer Engineering, Herriot-Watt University, United Kingdom

[email protected]

Received Nov. 2009, accepted after revision March 2010

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ABSTRACT

This paper presents a design of Adaptive Filtering Algorithm for control of the vibration force

applied on the specimen in testing control systems. In order to reduce the computational

complexity associated with time-domain adaptive filtering method, frequency-domain adaptive

filtering scheme was adapted instead to alleviate the above mentioned complexity. In this paper

Adaptive Filtering algorithm in association with Fast Fourier Transformation (FFT) was used and

implemented in Digital Signal Processor (DSP). Our algorithm show significant reduction in

computational complexity as shown in the result given in section 6.

Keywords: Frequency-domain adaptive filtering, Shaker, Hybrid Partitioned, DSP


30


1. INTRODUCTION

Vibration design and control, aims either to

eliminate or to reduce the undesirable vibration

effects that may cause human discomfort and

hazards, structural degradation and failure,

performance deterioration and malfunction of

machinery and processes. When a mechanical

or electronic system is exposed to a vibration

force, it causes the system to vibrate, producing

an output response as a result of the vibratory

excitation force. The control objective, in such a

case, is to suppress the output response to a

level that is acceptable. In an adaptive vibration

control system, the vibration responses are

explicitly sensed through transducers. This

sensed response is fed to the controller

producing the force that counteracts the effect

of the vibration source, suppressing vibration at

the sensing location. This force is applied to the

system through the actuator. In a vibration test

setup, the vibration excitation used to simulate

the environment that the device under test will

undergo, can be generated either by using a

dropping machine, or a shaker.

In this paper the electrodynamics shaker, with a

permanent magnetic field, is used as a vibration

exciter and the control algorithm adopted is a

variant of adaptive control. The application of

adaptive control for shaker control is motivated

by the fact that some parameters of the shaker

are time-varying (e.g. coil inductance is

frequency dependent, and the coil resistance

may change with time as the result of skin

effect and temperature). Also the specimen or

load characteristic is usually unknown

beforehand and it may be nonlinear [1], [2]. The

shaker control algorithms proposed in the

literature [3], [4], [5], have been derived based

on a linear shaker model or on the assumption

that the load nonlinearity and variation of

shaker parameters with time can be neglected.

The performance of these controllers degraded

when the shaker dynamics are time-varying or

the load is highly nonlinear. Besides, the

frequency domain adaptive filtering algorithm

studied in [6] suffers from high computational

complexity and long time delay resulting from

the utilization of the block frequency domain

method. The limitations of these algorithms

were addressed in this paper by utilizing a time

and frequency block partitioning adaptive

filtering algorithm to reduce the computational

complexity, convergence time and the time

delay.

2. Electrodynamic shaker fundamentals

The electrodynamic shaker’s main function is to

deliver a force proportional to the current

applied to its armature coil. These devices are

used in such diverse activities as product

evaluation, stress screening, squeak-and-rattle

testing, and modal analysis. The shakers may be

driven by sinusoidal, random or transient

signals, depending on the application. They are

invariably driven by an audio-frequency power

amplifier and may be used ‘open loop’ (as in

most modal testing) or under closed-loop

control, where the input to the driving amplifier

is servo-controlled to achieve a desired motion

level in the device under test. There are three

major shaker types widely used (Hydraulic,

Inertial, and Electrodynamic Shakers) in

vibration testing. However, electrodynamic

shakers have many advantages compared to

the other types due to their high output

bandwidth and moderate input power

requirements. In the vibration control system

with the shaker used as an exciter, it is essential

to characterize the shaker dynamic model and

to compute the shaker mechanical and

electrical parameters that will be used in the

simulation stage. An initial experiment is

performed by monitoring the voltage input to

the power amplifier or the current delivered by

the power amplifier, and measuring the

dynamic response of the shaker (accelerometer

signal) with a bare shaker table, then with a

known table load [7], [8], [9]. The schematic in

Figure (1) shows a sectioned view of a


31

permanent magnet electrodynamic shaker with

emphasis on the magnetic circuit and the

suspended driving table. At the heart of the

shaker is a single-layer armature coil of copper

wire, suspended in a uniform radial magnetic

field.

When a current is passed through the coil, a

longitudinal force F is produced in proportion

to; the current I flowing in the coil, the length l

of the coil in the magnetic field, and the

strength B of the field flux. This force is

transmitted to the table structure to which the

device under test is attached [10], [11], [12-19].

The generated force in the armature coil is

mathematically expressed as

BlIF =

Where

)( current, coil armature theis )( field, in the coil armature oflength theis

)( density,flux magnetic theis )( force, coil armature theis

AIml

TBNF

Cu shorting

ring

Test flexure

(table)

Outer pole Outer poleInner

pole

Permanent

magnet

Magnet structure (body)

Armature

fixtures

Armature coil

Armature

Diaphragm

Figure 1: Electrodynamics shaker cross-section

3. Modelling of Electrodynamic shaker

Modal analysis is a procedure of ‘experimental

modelling’ whose primary purpose is to develop

a dynamic model for mechanical and electrical

systems. The technique is a powerful tool for

the analysis and modelling of shaker dynamics

in frequency domain. In this procedure the

experimental test is set-up to compute the

frequency response (Transfer Function) of the

shaker from which the mechanical and

electrical parameters are obtained. The

electrodynamic shaker can be expressed as a

current driven or voltage, transfer function. In

the current driven transfer function mode, the

acceleration frequency response is plotted as

current supplied by the power amplifier against

the shaker acceleration response. In this case,

the effect of electromagnetic damping is not

evidenced [13], [14], [15]. The frequency

response plot reflects only the structural

damping terms, those that could be measured

with external excitation applied to the shaker

with its drive coil un-terminated. The same low

damping factors are usually evident when a

current amplifier drives the shaker. In contrast,

the voltage driven transfer function (voltage

applied to the shaker system against the

acceleration, reflects the significant

electromagnetic damping applied by the cross-

coupling terms between the electrical and

mechanical components of the system [16]-

[20]. The force provided by the shaker is given

by F=BIl. Figures (2) and (3) show respectively,

the current–driven and voltage-driven

frequency response when the swept sine signal

of amplitude 0.7 v was applied to shaker power

amplifier.

The coil, pole plating, and table structure

combination, is called the armature assembly.

The test object is rigidly mounted to the

armature assembly (table). Therefore the test

object and armature assembly move together,

relative to the shaker body. The armature is

accurately centered in the narrow gap between

inner and outer poles via an elastic suspension

system. This allows it to move axially while

being restrained from all other motion [7], [8],

[15]. The compliant connection between the

armature assembly and the shaker body forms

an obvious spring/mass/damper vibration

system. The shaker mechanical model is

modeled by assuming the armature structure is

elastic rather than rigid. This gives the shaker


32


mechanical model three degrees-of-freedom.

This is achieved by modeling the coil and table

as separate masses connected by springs and

dampers.

Figure 2: Current driven transfer function of

the unloaded Electrodynamic shaker

Figure 3: Voltage driven transfer function of

the unloaded Electrodynamic shaker

In order to compute the mechanical and

electrical parameters of the electrodynamic

shaker and the associated load, a swept sine

test is conducted to compute the frequency

response function as the ratio between the

shaker’s output response (accelerometer

signal), and the input supply voltage (voltage

mode).

The resonance frequencies in the operating

range and the half-power points are recorded.

These are used in mathematical formulas

(Equation 1), to estimate the masses, damping

constant, spring stiffness, and electrical

impedance. Some of the parameters are tuned

using trial and error during simulation, so that

the simulated frequency response matches the

measured frequency response shape. The

mathematical Equations used to deduce the

mechanical and electrical parameters from the

frequency response, are given by [2],[3], [4]:

( )

edB

en

an

aa

e

MfD

MfK

ff

fMM

3

2

22

2

2

∆=

=

−=

π

π

(1)

where

n

dB

a

n

a

e

ff

f

fDKMM

frequency resonance bounding points (-3dB)power -half

loaded isshaker when thefrequency resonance

kHz) (6.38 load no hasshaker when thefrequency resonance

factor damping stiffness spring

load theof mass systemshaker theof mass effective

3∆

3

.1 Electrical equivalent model

The electrical model of the electrodynamic

shaker consists of the coil resistance R and

inductance L. The electrical impedance of the

shaker coil reflects the mechanical motion of

the shaker table. When the coil moves in the

magnetic field, a voltage is generated across the

coil proportional to the motion velocity

(E=Blu=αu). Thus the voltage at the coil

terminals may be written in terms of the

flowing current i and the velocity u as:

udt

diLRiv α++=

(2)

where α=Bl is constant, called the transduction

factor.

The mechanical mobility (velocity/force) of the

shaker mechanical components may be

represented by a driving-point frequency

response function Hfu, so that

FHu fu=

(3)

The coil produces an axial force, acting on the

shaker mechanical elements, in proportion to

the applied current.


33

iF α= (4)

Combining Equations (2), (3), and (4), yields the

impedance Z exhibited by the coil

fuHfLjRi

vZ

22 απ ++== (5)

The minimum coil impedance is determined by

the dc resistance, which is real-valued. The coil

inductance contributes an imaginary (900

phase-shifted) ac component that increases in

direct proportion to frequency. The mechanical

mobility contributes frequency-dependent

terms that exhibit a real maximum at each

mechanical resonance. These can significantly

increase the impedance in a narrow frequency

band. The effective resistance and inductance

of the coil can be measured by clamping the

fixture table (locked rotor test). The shaker

electrical model can be approximated by a

transformer equivalent circuit by observing

that, the moving coil and pole-plating forms a

rudimentary transformer with a shorted

secondary [2],[3], [4]. The moving coil is then a

multi-turn primary winding and the pole-plating

is a single-turn short-circuited secondary

winding. The equivalent circuit is shown in

Figure (3) where R1 and L1 are the resistance

and leakage inductance of the moving coil, R2

and L2 are the resistance and leakage

inductance of the copper pole-plating, and Lm is

the moving coil magnetizing inductance.

Using current mesh analysis, the mathematical

Equations for the shaker electrical model can be

derived from Figure (3.3) as

−++=

+

−++=

dt

di

dt

diL

dt

diLiR

dt

dl

dt

di

dt

diL

dt

diLiRv

m

m

122222

1211111

0

α

Figure 4: Moving coil T-circuit showing the

short-circuit secondary

3.2 Mechanical equivalent model

The shaker mechanical system includes a means

for storing potential energy (spring), a means

for storing kinetic energy (mass or inertia), and

a means by which energy is gradually lost

(dampers). The mechanical model of the shaker

consists of two distinct elements, the moving

coil and the fixture table. The fixture table is

suspended by a suspension flexure to the

shaker body assembly. The fixture table can be

modelled as a pair of masses 2M and 3M, with

flexure stiffness, 2K and 3K, and damping

coefficients, 2D and 3D. The moving coil of

mass M1 is adhered to the fixture table by an

adhesive bonding, which also can be

characterized by a spring with a finite stiffness

1K and a damping element with coefficient D1.

Thus Figures (3) and (4) can represent the

unloaded shaker electrical and mechanical

systems respectively. The mechanical system

mathematical Equations can be expressed as:

( )

( ) ( )

( )dt

dlDlK

dt

ldM

dt

dl

dt

dlDllK

dt

dl

dt

dlDllK

dt

ldM

dt

dl

dt

dlDllK

dt

dl

dt

dlDllK

dt

ldMi

3

3332

3

2

3

32

2322

32

23222

2

2

2

21

1211

21

12112

1

2

11

++=

−+−

−+−+=

−+−

−+−+=α

The terms in Equation 7 are defined in

Figure 5.

(7) (6)


34


Figure 5: Mechanical equivalent circuit of the

unloaded electrodynamic shaker

4. PARTIALLY HYBRID TIME-FREQUENCY

DOMAIN ADAPTIVE FILTERING ALGORITHM

In vibration testing control systems, the shock

test is conducted to simulate the effect of the

shock that a specimen is expected to be

subjected to, during its lifetime. To prevent

testing damage, it is essential that the test be

controlled such that the shaker output

converges smoothly to the intended reference

shock pulse. Usually, in shaker vibration control,

the load dynamics are not well defined before

hand [1]. Thus, the control algorithm must be

designed with the following consideration:

• The controller should be able to update its

parameters to cope with load uncertainty;

• The controller must have a fast response,

especially when the pulse used is a shock

pulse; and

• The controller must be robust, such that it

can adapt to a large range of load

variations.

The following sections give a brief description of

inverse adaptive filtering, time domain adaptive

filtered-x filtering, and hybrid time-frequency

domain adaptive filtering algorithms.

4.1 Filtered-x adaptive filtering algorithm

The filtered-x algorithm has been extensively

applied in the active control of sound and

vibration [1], [2], [3]. The design is carried out in

two phases. In the first phase, the model of the

dynamic system to be controlled is computed.

In the second phase, the controller weights are

updated, and the optimal values found are

implemented to control the dynamic system.

The main feature of the filtered-x algorithm is

that the signal used in the controller weights

adaptation, is produced by filtering the

reference input signal, via the system model

weights. Figure (6) illustrates the block diagram

of the filtered-x adaptive filtering algorithm for

the electrodynamic shaker.

Assume the model of the shaker and the

specimen has been computed. Let the number

of weights in the shaker/specimen model and

the control filter be Lc and Lp, respectively. The

output response of the FIR filter controller is

computed as a convolution of the FIR filter

weights and the input reference signal:

∑−

=

−=1

0

)()()(cL

i

i inrncnu

(8)

The FIR filter output is applied to the

shaker/specimen, generating the acceleration

output of a(n). The error signal is computed as

the difference between the delayed input signal

and the shaker/specimen acceleration output

response.

)()()( nanrne −∆−= (9)

where Δ is the time delay.

The input reference signal is filtered through

the shaker model weights to generate the

filtered signal uf(n) given by

(10a)

Using the LMS algorithm, the controller weights

are updated using

∑−

=

−=

1

0

)()()(

pL

i

if inrnwnu


35

)()()()1( nnenn fucc µ+=+ (10b)

where μ is the step-size, c(n

) is the controller weight vector, and uf(n) is the

regression vector.

The controller weight vector is given by

[ ]TLccccn 110 ,.....,,)( −=c

(11)

The regression vector is defined as

[ ]Tpffff Lnununun )1(),....,1(),()( +−−=u

(12)

r(n-∆)

a(n)

e(n)

uf(n)

u(n)

r(n)

+

Filtered-x algorithm

_ Shaker

/Specimen Filter c(n)

Filtering algorithm

Shaker

Model

w(n)

∆

Figure 6: Filtered-x adaptive filtering block

diagram

4.2 Partitioned Block Frequency Domain

Adaptive Filtering (PBFDAF) algorithm

The computational complexity and long time-

delay problems, associated with the

conventional frequency domain adaptive

filtering algorithm, can be minimized by using

the partitioned block frequency domain

adaptive method, in which the weights of the

FIR filter are sequentially split into non-

overlapping partitions. Then the frequency

domain adaptive algorithm is applied to each

partition. The main advantage of PBFDAF over

the non-partitioned algorithm is that a small

processing block size is required, consequently,

the delay of the PBFDAF is small. Figure (7)

shows the PBFDAF block diagram.

To derive the Equations that govern the PBFADF

algorithm, assume that the FIR filter has M

weights, divided into P partitions, each partition

containing N weights. Therefore the output of

the blocks of N samples is given by

∑−

=

−+−

=−+=1

0

1)1( )](),...,()[(

)]1(),...,([P

p

TNppN

Tk

kNwkNwpkA

NkNykNyy

w

here )(kA is an NxN matrix whose i,j element is

given by

1,...,1,0,),()(, −=−+= NjijikNxkA ji

The output of each partition is a circular

convolution of the partition input with the

weight vector of the partition at the kth

time,

where for example, the input of the partition

(overlap-save) and weight vector per partition

are, respectively,

TNppp

T

kNwkNwk

Npkx

NpkxNpkxpk

]0,....,0),(),...,([)(

)]1]1([..,

,.])([),...,]1([[)(

1)1( −+=

−+−

−−−=−

w

x

(15)

Thus the input and partition weight vector in

the frequency domain are defined as

1...,1,0)()(

1,...1,0)),(()(

−==

−=−=−

PpkFk

PppkFdiagpk

pwW

xX

(16)

where F denotes the DFT operator of order 2N.

For the overlap-save method of the block

sectioning, the adaptive filter output in the time

domain is

)()()(

)(]0[)]1(),...,([1

0

1

kpkkwhere

kFINkNykNy

pp

P

p

nn

T

k

WXY

Yy p

−=

=−+= ∑−

=

−

(17)

where ]0[ NN I

is Nx2N, is the output projection

matrix used to force the first element of the

output vector to zero as a result of the

application of the overlap-save sectioning on

the input block The weight update of each

partition is defined as


36


−

+=+

−)}()({

00

0

2)()1(

*1kpkF

I

Fkk

NN

NN

p

EX

WW p µ

(18)

where )(kE is the error vector in the frequency

domain and is defined as

TNkNe

kNeFkFk

)]1(..,

),.(,0,...,0,0[))(()(

−+

== eE (19)

5. Adaptive Filtering Control Algorithm design

The control objectives for a vibration control

system can be stated as: given a desired

acceleration profile (signal), it is required to

design a controller such that when the shaker is

used to generate vibration force, the output of

the shaker converges to the reference signal

smoothly (avoiding large overshoot) and at the

same time, the output tracks the reference

signal in a short time (minimal delay between

the input and output signal). In this research the

filtered-x algorithm is used to implement the

inverse control algorithm. The proposed

algorithm differs from the known filtered-x

algorithm in that the PHTFDAF is designed with

the aim of minimising the delay inherent in

previous schemes. The other objective of this

scheme is to reduce the computational

complexity associated with the standard

filtered-x algorithm.

5.1 Time domain adaptation of the first

partition’s weights

The control algorithm for the PHTFDAF is

derived splitting the controller vector’s weights

into non-overlapping partitions. Let the

controller weight length be Lc. To implement

the partition algorithm, these weights are

sequentially split into Mc non-overlapping

partitions, each containing Pc weights. The input

block formed, from the reference signal, is

defined as T

ccc NmkrNmkrNmkrmk )]1)1((,...,)(),....,)1(([)( −−−−−−=−r (20)

The weight vector at instant n is expressed as

T

P ncncncncc

)](),...,(),([)( 1−= (21)

Then the output of the first partition in the time

domain is given by

)()()( nnnuT

rc= (22)

The weight during this initial block data

collection is updated using the following Least

Mean Square (LMS) Equation

))()((

/))()(()()1(

nn

nnenn

fT

f

f

rr

r

+

+=+

ε

µcc (23)

where

)()()(1

0

inrnwnrL

i

if −=∑−

= is the filtered input,

with w(n) the shaker model.

The vector rf(n) is given by

[ ]Tffff Lnrnrnrn )1(),...,1(),()( +−−=r

5.2 Frequency domain adaptation of controller

weights

The controller input vector is defined as in

Equation (20). This vector is transformed into

the frequency domain, giving

))(()( mkFdiagmk −=− rR (24)

where F denotes the DFT operator.

The weight vector of each partition is padded

with a block of zeros before transformation into

the frequency domain. Thus, if the time domain

weight vector of each partition is given by

TckN

Nmc

ckNmNckmc

]0,...,0),(1)1(

),...,([)(

−+

= (25)

then the corresponding partitioned weight

vector in the frequency domain, will be

adaptationut througho 1,...1,0

stage initial theduring1,...2,1

)()(−=

−=

=c

c

mm Mm

Mm

kFk cC (26)


37

Each partition output is computed as

)()( kkC RUm = (27)

These partition outputs are summed to form

the controller output, which is inversly

transformed to time domain. Thus the

controller vector in the time domain is defined

as:

##)(]0[

#)(]0[)()(

)]1(),...,([)(

1

0

1

1

1

1

+

=

−+=

∑

∑−

=

−

−

=

−

M

m

mN

M

m

mNc

Tccc

kFI

kFIkck

NkNukNuk

U

U

u

χ (28)

# During the initial stage

## Throughout adaptation

The above controller output signal is applied to

a parallel to serial converter, and the output

signal is apply to the shaker system. The shaker

output response is applied to a serial-to-parallel

device to form the output vector, which is

subtracted from the desired reference vector.

The difference (output of the shaker and

reference signal) forms the error vector that is

used with the gradient estimate to update the

weights in each partition.

The error vector is expressed as

TcNckNe

ckNekkkc)]1(

),...,([)()()(

−+

=−= yre (29)

According to the overlap-save method, the

error vector is preceded with a block of zeros,

then the resultant vector is transformed into

the frequency domain to form the frequency

domain error vector as follows

TcNckNe

ckNeFkFk

)]1(

),...,(,0,...,0[)()(

−+

== ecE (30)

5.3 Controller partition weights adaptation in

the frequency domain

The weights in each partition are adaptively

updated using the filtered-x method, where the

input vector of the adaptive process is filtered

though the shaker model [1]. Thus the weights

are computed as

( )( )

−

+=+

∗−)()().(

00

0

)()1(

1kkmkF

I

F

kCkC

m

NcNc

NcNc

c

mm

EWR

µ (31)

where cµ is the convergence rate of the

controller weights.

The adaptation convergence rate is increased

by using a variable step-size, whereby the step

size is varied in each frequency bin. This is

accomplished by normalizing the bin-wise step

size by the estimate of the input power as

follows. 1)(

0−= kxrPc µµ

(32)

where xrP is the input power of the adaptive

process, and is defined as

( ) ( ){ })()()1()1(

)()()()()(

kkk

kCmkkCmkEk

rrcxrc

mmxr

XXP

RRP

∗−+−=

−−=

αα

(33)

where cα is the forgetting factor and

)()( mkkCmr −= RX is the resultant vector

after filtering the input signal via the shaker

model.

rf(k)

a(k)

r(k-∆)

e(k)

u(k)

r(k)

+

PHTFDAF control

algorithm

_

Shaker/ specimen Filter C(k)

Filtering algorithm

Shaker Model

∆ S/P

concat

FFT

Figure 7: PHTFDAF control algorithm block

diagram

6. DISCUSSION OF RESULTS AND

CONCLUSION

In this paper, the loaded shaker was

characterized using a swept sine signal,


38


generated from the HP 3562A Dynamic Signal

Analyzer (Figure 15). The signal, swept from 10

Hz to 10 kHz, was applied to the

shaker/specimen via the power amplifier. The

acceleration output response was measured via

a charge amplifier, whose output was

connected to the 2nd

channel of the Dynamic

Signal Analyzer. The frequency response was

computed as the ratio between output

acceleration measured by charge amplifier and

the input swept sine signal of 0.7v rms

amplitude. The charge amplifier sensitivity and

scale values were set to 80 pC/g and 5 g/V,

respectively. From the measured transfer

function of the unloaded shaker (Figure 1), the

upper resonance frequency occurs at 6.38 kHz

and low resonance at 54.32 Hz. The resonance

frequency values from Figure (8a) and the half-

power values were used to compute masses,

and springs and damper constants of the

mechanical system. Some electrical

components were measured. The

MATLAB/SIMULINK software was used to

simulate the linear and nonlinear model of the

loaded shaker.

Performance of the algorithms discussed in the

previous sections was investigated with the

shaker having an inertia mass of 0.273 kg. The

10 Hz to 10 kHz frequency response measured

practically for a swept sine and rectangular

shock pulse input, are shown in Figures (8a) and

(8b) respectively. Figure (9) shows the shaker

input signal and output response. The input

shock pulse has the pre and post pulses added

to comply with the condition that the velocity

and acceleration profile of the shaker response

should have zero initial and final values. From

the shaker and FIR model output responses

(Figure 9) it is evident that the shaker response

is lightly damped. The ringing is due to

discontinuity (sharp change in the rectangular

pulse derivative) in the shock pulse shape and

the shaker damping dynamics.

(a) swept sine test

(b) Rectangular shock pulse

Figure 8: 10 Hz to 10 kHz frequency

response.

Figure 9: Input signal (upper) and shaker output

response (lower)

6.1 Partially hybrid time-frequency domain

adaptive filtering results

The shaker model represented by the FIR

adaptive filter is identified using the partially

hybrid frequency domain adaptive filtering

algorithm described in Section V.1. The FIR filter

representing the model has 1024 weights. The

g/V

g/V


39

time-domain weights are divided into two

partitions, each of 512 taps. The weights of the

first partition are adapted using the non-block

time domain filtering algorithm only in the first

block input stage )511,...,1,0( =n . In subsequent

block input iterations, the weights in the first

partition and the weights of the other partitions

are updated using the frequency domain

adaptive filtering algorithm.

After the shaker model is computed, the model

weights are used in the PHTFDAF FIR controller

model, such that when the resultant FIR

controller is connected in cascade with the

shaker, the shaker output tracks the reference

input signal. That is, the control objective can

be stated as: given the desired input reference

signal and the shaker model, it is required to

compute the FIR controller model such that

when cascaded with the shaker, the output of

the shaker tracks the reference input signal. The

transition of the shaker controlled output

response to the input reference signal, should

occur in a short time (due to the shock pulse

width, usually 2-20 ms) and converge smoothly

(no overshoot). The control model is

represented by an FIR filter of 1024 weights.

The weights are divided into two partitions,

each of 512 weights, as in the case of the

system identification of the shaker model. Using

the filtered-x method in the time and frequency

domains, the shaker output tracks the

reference signal. The adaptation process

converges to the optimal values after 20 block

input iterations.

The input reference signal and shaker

controlled output responses are shown in

Figures (10) and (11) respectively. From

comparison of the desired input reference

signal (Figure 10) and the shaker-controlled

output (Figure 11), it is seen that the shaker

output tracks the input reference signal. The

control algorithm implementation results in a

reduction in the ringing of the shaker output

response. The rise time and settling time

achieved are 0.05 ms and 0.85 ms, respectively.

The controller output signal (shaker input) is

shown in Figure (12). During simulation it is

found that the PHTFDAF algorithm converges to

the desired signal faster than with the

conventional frequency domain adaptive

filtering (FDAF) method.

Figure 10: Input reference signal

Figure 11: Shaker controlled output

Figure 12: Controller output (shaker input)


40


6.2 Time domain Filtered-x results

The time domain filtered-x algorithm for

computing the controller weights is executed in

two phases. In the first phase, the model of the

electrodynamic shaker and payload is

computed. The model is then used to filter the

reference input signal to generate the signal

used in the controller weights update. The

shaker and payload model is represented by an

FIR filter of 512 taps, as is the controller FIR

filter. The adaptation process is executed for

100,000 samples with a step-size of 0.075. The

optimal FIR weights are used to model the

controller for the shaker plus payload. The

output response of the cascaded system is

shown in Figure (13), while the controller

output signal (shaker/payload input) is shown in

Figure (14).

Comparing Figure (13) and (11), it is clear that

the PHTFDAF algorithm gives better results than

the time domain filtered-x algorithm. This

shows that the PHTFDAF algorithm is superior

to the time domain filtered-x algorithm.

Figure 13: Shaker controlled output

Figure 14: Controller output (shaker input)

Table 1 shows the numerical computational

complexity of the different time/frequency-

domain filtering algorithms when the adaptive

filter weights are split into two partitions.

Computational gain for the different algorithms

is computed as the ratio of the difference

between the number of multiplications per N

samples of the time-domain LMS and the

number of multiplications per N samples of a

particular algorithm, to the number of

multiplications of the time-domain LMS

algorithm. The table shows that the non-

partitioned frequency domain adaptive filtering

(FDAF) algorithm results in a reduction of

computational complexity. But the PBFDAF

algorithm requires less computation than the

FDAF algorithm. The table also shows that the

PHTFDAF algorithm has a high computational

complexity, especially during the initial stage

but subsequently has the same computational

complexity as the PBFDAF algorithm. The

advantage of the PHTFDAF algorithm is that the

delay in the system is minimised.

From the practical results it is concluded that

the proposed PHTFDAF algorithm for system

identification and control of an electrodynamics

shaker is different from the method used in [3],

[6] in the following respects:

• Computational complexity is reduced, due

to the splitting of the long FIR weights into

partitions. This reduces the input block size

needed (see table 5.2) for implementation

of the frequency domain filtering algorithm

and

• The proposed algorithm minimises the time

delay in the system. This is achieved by

updating the first partition’s weights using a

non-block time domain filtering algorithm.

References

1. Ajangnay M O A, 2004, Hybrid Partitioned

Frequency/Time Domain Adaptive Filtering

Algorithm for Shaker Control, Proceedings


41

of the IASTED International Conference,

Circuit, Signal, and Systems, 28 Nov- 1, Dec

2004, Clearwater, FL,

2. Ajangnay M O A, 2004 Adaptive Shock

Control of Electrodynamic Shaker with

Nonlinear Loads, PhD Thesis, Heriot-Watt

University, Edinburgh, United Kingdom.

3. Karshenas A M, 1997, Random Vibration

and Shock Control of an Electrodynamic

Shaker, PhD Thesis, Heriot-Watt University-

Edinburgh.

4. Macdonald H M, 1994, Analysis and Control

of an Electrodynamic Shaker, PhD Thesis,

Heriot-Watt University, Edinburgh.

5. Karshenas A M, 1997, Random Vibration

and Shock Control of an Electrodynamic

Shaker, PhD Thesis, Heriot-Watt University-

Edinburgh.

6. Bendel Y et al, 2001, Delayless Frequency

Domain Acoustic Echo Cancellation, IEEE

Trans on Speech and Audio Proc. Vol. 9, No.

5, pp 589-597

7. Olmos S, Sornmo L, and Laguna P, 2002,

Block Adaptive Filters with Deterministic

Reference Inputs for Event-Related Signals:

BLMS and BRLS, IEEE Tran. of Sig. Proc., Vol.

50, No. 5 pp 1102-1112.

8. Li X and Jenkins K, 1996, The Comparison of

the Constrained and Unconstrained

Frequency-Domain Block-LMS Adaptive

Algorithm, IEEE Tran. On Sig. Proc. Vol. 44,

No. 7, pp 1813-1816.

9. Shynk J, 1992, Frequency-Domain and multi-

rate adaptive filtering, IEEE Sig. Proc. Mag.,

Vol. 9, No. 1, pp 14-37.

10. Lee J C and Un C K, 1989, Performance

Analysis of Frequency Domain Block LMS

Adaptive Digital Filters, IEEE Tran. On Cir

and Sys. Vol. 36 No. 2, pp 173-188.

11. Sommen P C et al, 1987, Convergence

Analysis of a Frequency-Domain Adaptive

Filter with Exponential Power Averaging and

Generalized Window Function, IEEE Tran. On

Cir and sys., Vol. Cas-24, No. 7, pp 788-798.

12. Mansour D and Gray A H, 1982,

Unconstrained Frequency-Domain Adaptive

Filter, IEEE Tran. On Acust. Speech and Sig.

Proc., Vol ASSP-30, No. 5, pp 726-733.

13. Li X and Jenkins W K, 1995, Convergence

Properties of the Frequency-Domain Block-

LMS Adaptive Algorithm, IEEE Conference

on Signals, Systems and Computers, Vol. 2,

pp 1515 – 1519.

14. George F L, 1997, Electrodynamic Shaker

Fundamentals, Sound and Vibration, Data

Physics Corporation, San Jose, California.

15. George F L and Dave S, 2001, Understanding

the Physics for Electrodynamic Shaker

Performance, Dynamic reference issue,

Sound and Vibration, Data Physics

Corporation, San Jose, California.

16. De Silva C W, 2000, Vibration fundamentals

and Practice, CRC press.

17. Rust K J, 1997, Introduction to

Accelerometers and Calibration techniques,

MTS Systems Corporation.

18. Eren H, 1999, Acceleration, Vibration, and

Shock Measurement, by CRC Press LLC.

19. The fundamentals of model testing, 1997,

Application Note 243-3, Hewlett Packard,

9/97, USA

Table 1: Numerical computation complexity when the adaptive weights are split into 2 partitions

Input Block length N 512


42


Filter weights L 1024

Number of Partitions M 2

Implementation

FFT

length

Number of

FFT/IFFT

Multiplication per

N samples

Computational

gain Percentage

LMS 2048 2097152

BLMS 2048 2097152

Constrained FDAF 2048 5 120832 1976320 94.24%

Unconstrained FDAF 2048 3 75776 2021376 96.39%

Constrained PBFDAF 1024 7 79872 2017280 96.19%

Unconstrained PBFDAF 1024 3 38912 2058240 98.14%

Constrained PHTFDAF (initial

stage) 1024 5 579584 1517568 72.36%

Constrained PHTFDAF (after the

initial stage) 1024 7 79872 2017280 96.19%

Unconstrained PHTFDAF (initial

stage) 1024 3 559104 1538048 73.34%

Unconstrained PHTFDAF (after

the initial stage) 1024 3 38912 2058240 98.14%

Figure 15: Vibration Test Control Apparatus

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