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Adaptive Signal Selection Control Based on Adaptive

FF Control Scheme and Its Applications to Sound Selection Systems

483

As shown in Fig. 10, we introduce the virtual FB canceling error ()

ek and the virtual FF

canceling error

()

ek as well as the prediction error ()

ek to update the parameters of the

four FIR adaptive filters

(,)Bzk ,

(,)Czk ,

(,)Hzk and

(,)Dzk , which are given as

ˆˆ ˆ

(,) () () ()

Dzk d kz d kz d kz

−

−−

=+++

(24)

ˆˆ ˆ

(,) () () ()

Hzk h kz h kz h kz

−

−−

=+++

(25)

ˆˆ ˆ

(,) () () ()

Bzk b kz b kz b kz

−

−−

=+++

(26)

ˆˆ ˆ

(,) () () () .

Czk c kz c kz c kz

−

−−

=+++

(27)

Let the parameter vector of each adaptive filter be defined by

()

ˆˆˆˆ

() (), (), , ()

kdkdk dk=θ 

(28)

()

ˆˆˆˆ

() (), (), , ()

khkhk hk=θ 

(29)

()

ˆˆˆ ˆ

() (), (), , ()

kbkbk bk=θ 

(30)

()

ˆˆ ˆ

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

kckck ck=θ 

(31)

Let the corresponding regressor vector be defined as:

()

( ) ( 1), ( 2), , ( )

ksk sk skL=− − −σ 

(32)

()

( ) ( 1), ( 2), , ( )

kuk uk ukL=− − −ς 

(33)

()

( ) ( 1), ( 2), , ( )

kqk qk qkL=− − −δ 

(34)

()

( ) ( 1), ( 2), , ( ) .

kxk xk xkL

=− − −

(35)

The principle of the 2DF-VE controller is not to force the error

()ek directly into zero, but to

force it indirectly by making three virtual errors

(), (), ()

ABC

ekekek zero. That is why we call

the approach as the virtual error method.

In Fig. 10,

()

ek is a prediction error, and if the signals have the PE property then the

identification of

()Pz and ()Gz can be completely done. However, the PE property is not

required but we only need the convergence of

()

ek to zero. Therefore, the parameters in

(,)Dzk

and

(,)Hzk

are adjusted adaptively so that

()

ek becomes zero. ()

ek is a virtual

FB canceling error, and the parameters of the adaptive FB controller

(,)Bzk

are updated to

Numerical Simulations of Physical and Engineering Processes

484

cancel the estimated plant noise

()nk

so that ()

ek converges to zero. ()

ek is a virtual FF

canceling error, and the parameters of the adaptive FF controller

(,)Czk are also updated so

that

()

ek converges to zero. These errors are related with the signals in Fig. 10 and can be

expressed as follows:

ˆˆˆˆˆ

() () ( )() ()

ek ek DDBHHCdk ek=−− − −

(36)

ˆˆ

() (1 )()

ek BHnk=−

(37)

ˆˆˆˆ ˆ

() ( )(),

ek DDBHCHdk=− −

(38)

then it gives that

ˆˆ

() () () () ( )().

ek ek ek ek HCCHdk++=+−

(39)

If the parameters of

(,)Hzk and

(,)Czk converge to constants, then it gives that

() () () ().

ek ek ek ek++=

(40)

Therefore, if

(), (), ()

ABC

ekekek are separately forced to zero, then it holds that ()ek

converges to zero.

To derive the adaptive algorithm which forces the errors

(), (), ()

ABC

ekekek separately to

zero, we first describe the error systems relating each error with the parameter errors as

ˆˆ

() ( )() ( )() ()

ˆˆ

() () () () ()

DD H H

ek PDsk GHuk k

kk kkk

=− −− +

 

−−−+

 

θθ σ θθ ς

(41)

() () () () () ()

BBB

ek B

kk kkk

ββ





=− + − +







θθ δ

(42)

ˆˆˆˆ

() () () () () ()

CCC

DDBH

ek Cxk k k k k

γϕγ



−



=−+− +







θθ

(43)

where

(), (),()kkk

βγ

are uncertain terms due to unmodelled dynamics, and the subscript *

denotes the true value of each adaptive filter.

As a result, we can derive the robust adaptation laws by using

-modification approach

(Narendra & Annaswarny, 1986) as follows;

() ()()

ˆˆ

(1)1 ()

()

ADA

DAD

ek ek k



+=− +





θθ

(44)

() ()()

ˆˆ

(1)1 ()

()

AHA

HAH

ek ek k



+=− +





θθ

(45)

() ()()

ˆˆ

(1)1 ()

()

BBB

ek ek k



+=− +





θθ

(46)

Adaptive Signal Selection Control Based on Adaptive

FF Control Scheme and Its Applications to Sound Selection Systems

485

() ()()

ˆˆ

(1)1 ()

()

CCC

ek ek k

μϕ



+=− +





θθ

(47)

()

(1) () max ,1

()

AmAAA

mk mk g



+= +





(48)

()

(1) () max(),1

BmBBB

mk mk g k

+= + δ

(49)

()

(1) () max(),1

CmCCC

mk mk g k

μϕ

+= +

(50)

where

01,0,0,

μγμ

<< > > is step size, and

•

denotes the vector norm.

()mk is the

normalizing signal employed to stabilize the adaptation even in the presence of unmodelled

dynamics (Ortega & Yu, 1987).

4.4 Overall structure of adaptive signal selection controller

The overall structure of the adaptive Signal Selection Controller in Fig. 4 and Fig. 5 is now

given by combining the extractor in Fig. 6 and adaptive controllers

(,)Kzk in Fig. 7~10.

Fig. 11 shows one example of overall Necs-Extraction Controller, which is Fig. 4 type

combining Fig. 6 and Fig. 10.

()zP

()zG

−

Primary path

dynamics

Adaptive

FF controller

Secondary path

dynamics

Adaptive

FB controller

−

(, )zkB

−

()kσ

()kδ

()kζ

()

()nk

()

()ek

()yk

()

()uk

()vk

()

()nk

()

()qk

()

Cancelling

error

Prediction

error

Virtual

FB cancelling

error

Virtual

FF cancelling

error

Adaptation

2DF Virtual Error

Controller

()

Disturbance

−

()ak

()bk

Necs

Extractor

()ak

()bk

() ()ak bk+

Necs-Extraction Controller

−

()Necs k

()kφ

(, )zkH

(, )zkD

(, )zkC

(, )zkD

(, )zkH

(, )zkN

(, )Kzk

Fig. 11. One example of overall structure for Necs-Extraction Controller

Numerical Simulations of Physical and Engineering Processes

486

5. Results of numerical simulations

To validate effectiveness of the proposed SSC, two applications to Sound Selection Systems

(SSS) are considered as numerical simulations.

5.1 Numerical simulation of Necs-Extraction Control

First example is an application of the Necs-Extraction Controller to a smart window system

(SWS) of a car, which can transmit only electronic siren sound of ambulance as Necs, but

block any other noises such as road noise and engine noise. Purpose of this application is to

keep car room silent and safety against car accidents at the same time.

In the simulation, we compare the results obtained by the four adaptive algorithms: (i) the

filtered-X LMS approach (FX-SWS), (ii) the 2DF filtered-X LMS approach (2DF-FX-SWS), (iii)

the Virtual Error approach (VE-SWS), and (iv) the proposed 2DF Virtual Error approach

(2DF-VE-SWS). In the two filtered-X approaches, the modeling error of 20% in the filtered

dynamics is considered. The simulation setup for the proposed method is summarized in

Table 3.

Schemes Design parameters

2DF-VE Controller

1, 0.220, 20, 0.23,

5, 0.068, 30, 1.20,

0.4, 0.6, 0.0006,

0.4, 0.6, 0.0018,

0.4, 0.6, 0.0220

dD H H

bB C C

mA A A

mB B B

mC C C

μμ

μγ

== = =

===

Extractor

max min

0.03, 16, 4, 500, 4,

20 650 ,650 1200 ,700 850 ,870 1060

Hz Hz Hz Hz

ωω

μμ μ

=====

=− − − −

Table 3. Design parameters of the 2DF-VE-SWS in simulations

In the simulation, it is assumed that the Necs

()ak is set to the siren signal of a passing

ambulance car at speed of 60km/h from 10 sec to 20 sec, the Unecs

()bk is set to a unwanted

stationary noise of 840Hz sinusoid with sound pressure level (SPL) of 90dB, and the

unknown disturbance

()nk is 100Hz sinusoid with SPL of 80dB.

Fig. 12 shows the four simulation results. Fig. 12 (a) gives

()Necs k when uncontrolled, Fig.

12 (b) plots the estimated frequency by the extractor, Fig. 12 (c)-(f) compare the obtained

profiles of

()Necs k when controlled by each control approach. As shown in Fig. 12 (c) and

(e), the FF control approaches cannot reduce the disturbance

()nk , though they can block

the Unecs

()bk and transmit the Necs ()ak . On the other hand, the proposed 2DF-VE-SWS

can successfully reduce both disturbance

()nk and Unecs ()bk and transmit only the Necs

()ak , as in Fig. 12 (f), while the 2DF-FX-SWS based on the ordinary filtered-x algorithm

cannot obtain stabilized results due to modeling errors as shown in Fig. 10 (d). Note that the

scale of the vertical axis in Fig. 12 (d) is different from other figures. Fig. 12 (b) shows that

the proposed extractor can estimate the frequencies of

() ()ak bk+ correctly, where large

variations of the two estimated frequencies around 15 sec are due to passing of the

ambulance. Consequently, it is validated that the performance of the proposed method 2DF-

VE-SWS is the best in four controllers.

Adaptive Signal Selection Control Based on Adaptive

FF Control Scheme and Its Applications to Sound Selection Systems

487

0 5 10 15 20 25

1.5

0.5

1.5

Time [s]

Sensor Voltage [V]

0 5 10 15 20 25

400

600

840

1010

810

915

735

Time [s]

Frequency [Hz]

(a) Uncontrolled Necs(k) (b) Frequency estimation

0 5 10 15 20 25

1.5

0.5

1.5

Time [s]

Sensor Voltage [V]

0 5 10 15 20 25

Time [s]

Sensor Voltage [V]

0 5 10 15 20 25

1.5

0.5

1.5

Time [s]

Sensor Voltage [V]

0 5 10 15 20 25

1.5

0.5

1.5

Time [s]

Sensor Voltage [V]

(e) Necs(k) of VE-SWS (f) Necs(k) of 2DF-VE-SWS

Fig. 12. Simulation results of four Necs-Extraction Controllers

Numerical Simulations of Physical and Engineering Processes

488

5.2 Numerical simulation of Unecs-Canceling Control

Second example is another application of the Unecs-Canceling Controller to rotating

machinery such as a compressor, which can attenuate only sound of rotating motor as Unecs

even when rotating speed is changing, but transmit some abnormal sound of the machinery.

Purpose of this application is to keep machinery silence and detectability of machine

abnormality at the same time.

In the simulation, we will examine whether the Unecs-Canceling Controller works correctly

or not. We consider the case that AC servo motor of a compressor starts to rotate from 0rpm

at 0sec, and accelerate to 3600rpm until 10sec, then rotate constantly at 3600rpm after 10sec.

Additionally, 240Hz abnormal sound of the machinery occurs from 8sec to 20sec.

And in this simulation, we consider the Unecs-Canceling Controller composed of 2-degree-

of-freedom Virtual Error Controller, because the 2DF-VE controller shows the best

performance in the simulation of Necs-Extraction Controller in section 5.1.

The simulation setup for the proposed method is summarized in Table 4.

Fig. 13 shows the simulation results. Fig. 13 (a) gives

()Necs k when uncontrolled, Fig. 13 (b)

shows the obtained profiles of

()Necs k when controlled by 2DF-VE control approach, Fig. 13

cancelling error.

Fig. 13. Simulation results of Unecs-Canceling Controller

Adaptive Signal Selection Control Based on Adaptive

FF Control Scheme and Its Applications to Sound Selection Systems

489

As shown in Fig. 13(b), the Unecs-Canceling Controller can attenuate motor sound, and

transmit only abnormal sound of machinery from 8sec to 20sec. And Fig. 13(c) shows the

good performance of the frequency estimation of the extractor. Besides, Fig. 13(d) shows

that adaptation of the 2DF-VE controller is stable and rapid enough even if the frequency of

extracting signal varies.

Consequently, it is validated that Unecs-Canceling Controller shows good performance to

attenuate only desired signals.

Schemes Design parameters

2DF-VE Controller

1, 0.220, 20, 0.23,

5, 0.068, 30, 1.20,

0.4, 0.6, 0.0006,

0.4, 0.6, 0.0018,

0.4, 0.6, 0.0220

dD H H

bB C C

mA A A

mB B B

mC C C

μμ

μγ

== = =

===

Extractor

max min

0.03, 16, 4, 500, 4,

0 650 ,200 300 ,300 600 ,600 1000

Hz Hz Hz Hz

ωω

μμ μ

=====

=− − − −

Table 4. Design parameters of the Unecs-Canceling Controller in simulations

6. Conclusion

We have proposed a novel control scheme which can transmit necessary signals (Necs) and

attenuate only unnecessary signals (Unecs) selectively. The control scheme is named as

Signal Selection Control (SSC) scheme.

Proposed control schemes are two types of the SSC; one is Necs-Extraction Controller which

transmits only signals set as Necs, and the other is Unecs-Canceling Controller which

attenuates only signals set as Unecs.

Besides, in the SSC, we have introduced four types of adaptive controller, and it is validated

that the 2-degree-of-freedom Virtual Error controller has the best performance in the four

adaptive controllers.

Consequently, effectiveness of both SSC are validated in two numerical simulations of the

Sound Selection Systems.

7. Acknowledgment

This work was fully supported by Graduate School of Integrated Design Engineering, Keio

University, Yokohama, Japan.

8. References

C. Bao and J. Pan, "Experimental study of different approaches for active control of sound

transmission through double walls," J. Acoust. Soc. Am. , Vol. 102, No. 3, pp. 1664-

1670, 1997.

J.C. Burgess, "Active adaptive sound control in a duct," J. Acoust. Soc. Am. , Vol. 70, No. 3,

pp. 715-726, 1981.

Numerical Simulations of Physical and Engineering Processes

490

A. Jakob and M. Möser, "Active control of double-glazed windows Part I: Feedforward

control," J. Applied Acoust. , Vol. 64, pp. 163-182, 2003.

A. Jakob and M. Möser, "Active control of double-glazed windows. Part II: Feedback

control," J. Applied Acoust. , Vol. 64, pp. 183-196, 2003.

O.E. Kaiser, S.J. Pietrzko and M. Morari, "Feedback control of sound transmission through a

double glazed window," J. Sound and Vibration, Vol. 263, pp. 775-795, 2003.

T. Kohno and A. Sano, "Direct adaptive active noise control algorithms in case of uncertain

secondary path dynamics, Int. J. Adapt. Contr. Signal Process. , Vol. 19, pp. 153-176, 2005.

S.M. Kuo, D.R. Morgan, Active Noise Control Systems -Algorithms and DSP Implementations-,

Wiley. Interscience, 1996.

S.M. Kuo and D.R. Morgan, "Active noise control: a tutorial review," Proc. The IEEE, Vol. 87,

No. 6, pp. 943-973, 1999.

S.O. Reza Moheimani, "A survey of recent innovations in vibration damping and control

using shunted piezoelectric transducers," IEEE Trans. Contr. Syst. Tech. , Vol. 11, No.

4, pp. 482-494, 2003.

S.O. Reza Moheimani and A.J. Fleming, Piezoelectric Transducers for Vibration Control and

Damping, Springer, 2005.

K.S. Narendra and A.M. Annaswamy, "Robust adaptive control in the presence of bounded

disturbances," IEEE Trans. Automatic Control, Vol. AC-31, No. 4, pp. 306-315, 1986.

Y. Ohta and A. Sano, "Direct adaptive approach to multichannel active noise control and

sound reproduction", Proc. 2004 American Control Conference, pp.2895- 2900, Boston,

USA, 2004.

H. Okumura, R. Emi and A. Sano, "Adaptive two degree-of-freedom vibration control for

flexible plate with piezoelectric patches," Proc. SICE Annual Conf. 2008, Tokyo,

1A09-5, pp. 218-221, 2008.

H. Okumura and A. Sano, "Adaptive Necessary Signal Extraction Controll Based on 2DF

Virtual Error Approach to Smart Window Systems," Proc. 48th IEEE Conf. on

Desision and Control held jointly with 2009 28th Chinese Control Conf. , Shanghai,

China, ThC06. 4, pp. 5446-5453, 2009.

R. Ortega and T. Yu, "Theoretical results of robustness of direct adaptive controllers," Proc.

IFAC World Congress, Munchen, 1987.

J. Pan and C. Bao, "Analytical study of different approaches for active control of sound

transmission through double walls, J. Acoust. Soc. Am. , Vol. 103, No. 4, pp. 1916-

1922, 1998.

P. Sas, C. Bao, F. Augusztinovicz and W. Desmet, "Active control of sound transmission

through a double panel partition," J. Sound and Vibration, Vol. 180, No. 4, pp. 609-

625, 1995.

Y. Shimada, Y. Nishimura, T. Usagawa and M. Ebata, "Active control for periodic noise with

variable fundamental -an extended DXHS algotithm with frequency tracking ability

-," J. Acoust. Soc. Jpn. (E), Vol. 20, No. 4, pp. 301-312, 1999.

O. Tokhi and S. Veres, Active sound and vibration control -theory and applications-, The

Institution of Electrical Engineers, 2002.

B. Widrow, D. Shur and S. Shaffer, "On adaptive inverse control," Proc. the 15th Asilomar Conf.

Circuits, Syst. Comput. , Vol. 9, No. 11, pp. 185-189, Pacific Grove, CA1981-11, 1982.

Pavel Pavlíˇcek

Palacky University, Faculty of Science, Regional Centre of Advanced Technologies and

Materials, Joint Laboratory of Optics of Palacky University and Institute of Physics of

Academy of Science of the Czech Republic

Czech Republic

1. Introduction

White-light interferometry is an established method to measure the geometrical shape of

objects. A typical setup for white-light interferometry is shown in Fig. 1.

Fig. 1. Schematic of white-light interferometry.

A Michelson interferometer is illuminated by a broadband light source (e.g. light-emitting

diode, superluminescent diode, arc or incandescent lamp). At the output of the interferometer,

a CCD camera is used as a multiple detector. The measured object is placed in one arm of the

interferometer and moved in the longitudinal direction as indicated by the arrow in Fig. 1.

The surface of the object is imaged by a telecentric optical system onto the lightsensitive area

of the CCD camera. During the moving of the object in the longitudinal direction, a series

of images is acquired. From the acquired series, the coherence function (also referred to as

correlogram or interferogram) can be extracted for each object point. The maximum of the

Measurement Uncertainty of White-Light

Interferometry on Optically Rough Surfaces

2 Will-be-set-by-IN-TECH

envelope of the correlogram is assigned to the longitudinal distance of the respective object

point (Kino & Chim, 1990; Lee & Strand, 1990). A typical white-light correlogram is shown in

Fig. 2.

Fig. 2. Typical white-light correlogram.

Unlike to classical interferometry, white-light interferometry can be used for the measurement

of the objects with an optically rough surface (Dresel et al., 1992). A surface is regarded

as being optically rough when the standard deviation of the height variations within one

resolution cell of the imaging system exceeds one-fourth of the wavelength of the used light.

The property of the surface to be optically smooth or rough depends not only on the surface

roughness but also on the wavelength of the used light and the size of the resolution cell of

the imaging system (Häusler et al., 1999). In white-light interferometry on rough surface, the

longitudinal distance of the object point is determined from the envelope of the correlogram

only. The phase of the correlogram is not evaluated because it is a random quantity. The rough

surface of the measured object implies the formation of speckle pattern in the image plane (on

the lightsensitive area of the CCD camera).

In this work, we consider two inﬂuences that cause the measurement uncertainty: rough

surface and the shot noise of the camera. The inﬂuence of rough surface on measurement

uncertainty was described in our previous work (Pavlíˇcek & Hýbl, 2008). It shows that the

measurement uncertainty caused by surface roughness depends on the roughness and the

intensity of individual speckle. The measurement uncertainty δz is given by the formula

derived by T. Dresel (Dresel, 1991)

δz

√



I

obj



obj

.(1)

Here σ

is the rms roughness of the surface, I

obj

is the local intensity and I

obj

 is the mean

intensity of the speckle pattern. The subscript obj emphasizes that the intensities I

obj

, I

obj

are

meant with the shut reference arm (only the object arm is illuminated). Equation (1) indicates

that the measurement of the longitudinal coordinate z is more precise for brighter speckles.

The intensity in the speckle pattern is distributed according to the gamma distribution (Parry,

1984)

obj

M −1

obj

I

obj



Γ(M)

exp



−

obj

I

obj



,(2)

where Γ

()is the gamma function. The shape parameter M depends on the rms roughness σ

and the coherence length l

of the used light. For a light source with a Gaussian spectrum, the

492

Numerical Simulations of Physical and Engineering Processes