acontrolsourcestructureofsingleloudspeakerandrear ...0,5f 0] [40,200] where f 0:= c 0/(12(l u −l...

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2010, Article ID 730813, 9 pagesdoi:10.1155/2010/730813

Research Article

A Control Source Structure of Single Loudspeaker and RearSound Interference for Inexpensive Active Noise Control

Yasuhide Kobayashi,1 Hisaya Fujioka,2 and Naoki Jinbo3

1 Department of Mechanical Engineering, Faculty of Engineering, Nagaoka University of Technology, Nagaoka, Niigata 940-2188, Japan2 Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University,Kyoto 606-8501, Japan

3 Graduate School of Engineering, Nagaoka University of Technology, Niigata 940-2188, Japan

Correspondence should be addressed to Yasuhide Kobayashi, [email protected]

Received 17 March 2010; Accepted 4 June 2010

Academic Editor: Marek Pawelczyk

Copyright © 2010 Yasuhide Kobayashi et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Active noise control systems of simple ducts are investigated. In particular, open-loop characteristics and closed-loop performancescorresponding to various structures of control sources are compared based on both mathematical models and experimental results.In addition to the standard single loudspeaker and the Swinbanks’ source, we propose and examine a single loudspeaker with arear sound interference as a novel structure of control source, where the rear sound radiated from the loudspeaker is interferedwith the front sound in order to reduce the net upstream sound directly radiated from the control source. The comparisons of thecontrol structures are performed as follows. First, the open-loop transfer function is derived based on the standard wave equation,where a generalized control structure unifying the three structures mentioned above is considered. Secondly, by a comparison ofthe open-loop transfer functions from the first principle modeling and frequency response experiments, it is shown that a certainphase-lag is imposed by the Swinbanks’ source and the rear sound interference. Thirdly, effects on control performances of controlsource structures are examined by control experiments with robust controllers.

1. Introduction

It is well known that the Swinbanks’ (unidirectional) source[1] has advantages against a standard single loudspeaker(bidirectional source) in not only performances but also theimplementation cost. Indeed, the performance improvementhas been reported experimentally in both adaptive androbust control setups [2, 3]. In addition, the possibilityof inexpensive implementation has been pointed out byshowing that the controller gain is lower and, as the result,lower amplitude of driving signal for loudspeakers areachieved by the Swinbanks’ source [2, 4], which enablesus to use lower power loudspeakers. Furthermore, it hasbeen shown that the advantages of Swinbanks’ source aretheoretically proved for a practical setting on the duct lengthand loudspeaker locations [5, 6].

The Swinbanks’ source is composed of two loudspeak-ers where the upstream sound radiated directly from the

downstream loudspeaker is cancelled out by the sound withthe same amplitude and the opposite phase generated bythe upstream loudspeaker. The advantage of the Swinbanks’source are mainly obtained by the extension of the timeperiod by which the upstream sound directly radiated fromthe control source travels to the reference microphone.Indeed, it is empirically known that the closed-loop perfor-mance is improved when the actuator (control source) andthe sensor (reference microphone) are well separated in space[7, 8].

The reverse-phased sound in the Swinbanks’ source is,however, also radiated from the backside of the loudspeakerswhile it is not utilized in the existing control source foractive noise control (ANC). Therefore, one can expect thata single loudspeaker can achieve a similar performanceto the Swinbanks’ source if the rear sound could beinterfered to reduce the front sound at an upstream junctionthrough some additional ducts. In this scenario, inexpensive

2 Advances in Acoustics and Vibration

implementation is also expected since the additional loud-speaker in the Swinbanks’ source is not necessary to generateopposite-phased sound. Moreover, there is no guarantee thatthe Swinbanks’ source is the best structure for an inexpensiveANC system.

In this paper, we propose a new structure of controlsource which is composed of a single loudspeaker with a rearsound interference so that the rear sound radiated from theloudspeaker is interfered with the front sound in order toreduce the net upstream sound directly radiated from thecontrol source. The validity of the proposed method will beshown by comparing to the existing two structures of controlsources, namely, the standard single loudspeaker (bidirec-tional source) and the Swinbanks’ (unidirectional) source,in terms of the open-loop characteristic and the closed-loop performances based on both mathematical models andexperimental results.

2. Experimental Apparatus

Figure 1 and Table 1, respectively, show a block diagram andinstruments of the experimental apparatus which are similarto that in [2] except that a real ventilation fan is replacedto a loudspeaker (SPK1) as a noise source, and the ductlength is shortened to simplify mathematical models. Twoloudspeakers, SPK2 and SPK3, are used as control sources.

In Figure 1, blue lines show PVC pipes of 10 cm indiameter and 7 mm in thickness, while brown thinner linesshow flexible PVC ducts which are commercial products forresidential ventilation systems and are connected to adjustthe duct length. The brown broken curve duct, which iscalled subduct and is also made by the flexible PVC duct,is used only for the proposed control source after removingSPK3. The experimental apparatus is used in three ways asfollows for each structure of the control source:

Case (a). Bidirectional source: SPK3 is turned off,that is, the driving signal v(t) is set to 0. The controlsource is called bidirectional source in this paper,since the source generates same sounds in upstreamand downstream direction.

Case (b). Swinbanks’ source [4]: SPK3 is driven tocancel out the upstream sound generated by SPK2,that is, v(t) = −u(t − τ), τ = (lu − lv)/c0, where u(t)is the driving signal for SPK2, c0 is the sound speed,and lu − lv is the distance between SPK2 and SPK3.

Case (c). Rear-sound-aided source (Proposedsource): SPK3 is removed and the subduct is attachedso that the rear sound of SPK2 is interfered with thefront sound at the junction of ducts. Note that onlySPK2 is used and hence the implementation is lessexpensive than case (b).

The delay τ in the case (b) is approximately implementedas a module of the real-time application interface (RTAI)for Linux that updates the signal v(t) at every 0.1 mswhich should be short enough to avoid an aliasing effect.

In addition, we use τ = 2 ms which exactly correspondsto 20 times of the period of the real-time module. Theeffective frequency range of the Swinbanks’ source is givenas [ f0, 5 f0] � [40, 200] where f0 := c0/(12(lu − lv)) [4],while c0 = 344 m/s in a normal temperature environment.Moreover, the length of the subduct (88 cm) was adjustedexperimentally so that the upstream sound cancellation isimproved: We can identify the length of the delay by injectingan impulse signal to SPK2 and observing the referencemicrophone output y. The delay was maximized by adjustingthe length of the subduct.

The whole system from [ wu ] to[ zy]

including dynamicsof electrical circuits, acoustic ducts, microphones, andloudspeakers, is considered as the plant transfer function

G(s) :=[Gzw(s) Gzu(s)Gyw(s) Gyu(s)

], for each case, where w is the driving

signal for SPK1, z is the error microphone signal, and y isthe reference microphone signal as depicted in Figure 1. G(s)is used for robust controller design in Section 5. Note that vis determined by u, and hence G(s) depends on the controlstructure. We will determine G(s) by frequency responseexperiments, and the experimental results will be comparedwith the first principle model in the following sections inorder to examine the effect of control sources on open-loopcharacteristics.

3. First Principle Model

In this section, frequency response functions forG(s) definedin the previous section are derived by the first principlemodeling where a generalized control structure unifying thethree structures previously mentioned is considered. Figure 2shows a model of G(s). The system is mainly composed ofa duct of length L, where a noise source SPK1, a referencemicrophone, a subduct, a control source SPK2, and an errormicrophone are located at x = 0, ly , lv, lu, and lz, respectively.The subduct is LS in length, and is terminated with anothercontrol source. H(s) is a transfer function relating twocontrol sources. The control source in Figure 2 gives a generalrepresentation including the three control sources as specialcases: Case (a) corresponds to LS = 0 and H(s) = 0; Case(b) corresponds to LS = 0 and H(s) = −e−s((lu−lv)/c0); Case (c)corresponds to LS = lu − lv and H(s) = −1.

In order to derive frequency response functions, letp• and u• denote the complex pressure and the particlevelocity, respectively, for each position as shown in Figure 2.In addition, dynamics of loudspeakers, microphones, andelectrical circuits such as low pass filters, are neglected,since those dynamics have only common contribution tothe frequency response functions to be compared for cases(a), (b), and (c). Thus, let us assume for simplicity, thatmicrophone signal is proportional to the complex pressure,and that the complex particle velocity at each loudspeakeris proportional to the driving signal, since our interest is inproportion of the gain and phase characteristics. Specifically,we assume y = py , z = pz, u f = u, and u0 = w. Then,frequency response functions to be derived are given by

G̃( jω) :=[G̃zw( jω) G̃zu( jω)

G̃yw( jω) G̃yu( jω)

]=[pz( jω)/u0( jω) pz( jω)/u f ( jω)py( jω)/u0( jω) py( jω)/u f ( jω)

].

Advances in Acoustics and Vibration 3

Pow.AMP

PCLPFPow.AMP

LPFPre−AMP

AMPAMPPow.

LPF D/A

D/A

A/D

A/D

D/A

LPF

LPF

Pre−

8cm121 cm 71 cm

88 cm

SPK2 SPK3Error mic.

158 cm

Ref.mic. SPK1

3cm

y

u

z

y

w

Figure 1: Experimental apparatus.

x L lz lu lv ly 0

(p5,u5)

(pz ,uz) z

(p4,u4)(p f ,u f )

(p3,u3)

Ls

(pr ,ur)H

(p1,u1)(p6,u6)(p2,u2)

u

(py ,uy) y (p0,u0)

w

Figure 2: Model for G(s) with a generalized control source.

In the following, only a derivation process of G̃yu willbe explained (( jω) will be omitted). The other 3 transferfunctions can be similarly derived.

It is assumed as a boundary condition that p5 = 0 holdsat the open-end, and u0 = 0 at the closed-end. In addition,the width of the loudspeaker is assumed to be sufficientlysmall. It is also assumed that there is no dissipation in soundpropagation. Then, the following relationships are obtainedby using the transfer matrix method [9] by which a pair ofcomplex pressure and particle velocity at a certain locationin the duct is described as a pair at different location withdistance l multiplied by a transfer matrix T(l) as below:

[p1

u1

]

= T(lv)

[p0

0

]

,

[p3

u3

]

= T(lu − lv)[p2

u2

]

, (1)

[0u5

]

= T(L− lu)

[p4

u4

]

,

[p6

u6

]

= T(LS)

[prur

]

, (2)

T(l) :=

⎡

⎢⎣

cos kl − jρ0c0 sin kl

− j

ρ0c0sin kl cos kl

⎤

⎥⎦, k := ω

c0, (3)

where ρ0 is the density of the medium. Moreover, thefollowing standard assumptions are posed:

p1 = p2 = p6, p4 = p3 = p f ,

u2 = u1 + u6, u4 = u3 + u f , ur = Huf .(4)

Then, the following equality holds:[

0u5

]

= T(L− lu)

[p4

u4

]

= T(L)

[p0

0

]

+ T(L− lv)[

0u6

]

+ T(L− lu)

[0u f

]

,

(5)

where the relation T(l1)T(l2) = T(l1 + l2) is used. Aftereliminating intermediate variables such as u6, pr , and ur byusing above relationships, G̃yu is obtained as shown in (6) inthe next page. The other functions are shown in (7)–(9):

G̃yu =pyu f= cos kly · G̃0u,

G̃0u := p0

u f= jρ0c0

sin k(L−lv)H+sin k(L− lu) cos kLScos kL cos kLS−sin k(L−lv)sin kLS cos klv

,

(6)

G̃zu =(

cos klz − sin k(lz − lv) sin kLS cos klvcos kLS

)G̃0u

− jρ0c0

(sin k(lz − lu) +

sin k(lz − lv) ·Hcos kLS

),

(7)

G̃yw = cos kly · G̃0w − jρ0c0 sin kly ,

G̃0w := jρ0c0sin kL cos kLS−sin k(L−lv) sin kLS sin klvcos kL cos kLS−sin k(L−lv) sin kLS cos klv

,(8)

G̃zw =(

cos klz − sin k(lz − lv) sin kLS cos klvcos kLS

)G̃0w

− jρ0c0

(sin klz +

sin k(lz − lv) sin kLS sin klvcos kLS

).

(9)

It can be easily checked that (6)–(9) is consistent withresults in [5] for cases (a) and (b) by substituting LS =0, H( jω) = 0, or H( jω) = −e− jω((lu−lv)/c0).

Note that H(s) is only appeared in G̃yu(s) and G̃zu(s)since the corresponding control source is operated by u.Furthermore, the open-loop frequency response function


Table 1: Experimental instruments.

Loudspeaker (SPK1-3) FOSTEX FE87E

Microphones Electret condenser type

Power amplifier TOSHIBA TA8213K

LPF for measurements NF ELECTRONIC INSTRUMENTS FV-664 (2 ch, 200 Hz, 24 dB/oct)

LPF for driving 500 Hz 4th order Butterworth

PC Dell PowerEdge840 (RTAI3.6.1/Linux kernel 2.6.20.21)

A/D, D/A CONTEC AD12-16(PCI), DA12-4(PCI) (12 bit, ±5 V, 10 μs)

G̃zw(s) for cases (a) and (b) is common since LS = 0 in bothcases. The situation is similar for G̃yw(s).

Note that there are two major differences between theexperimental ducts and the first principle model: (i) damp-ing or dissipation effect and (ii) number of loudspeakers incase (c). For (i), the experimental duct system might havelarge damping effect due to the flexible PVC ducts, while thefirst principle model does not depend on damping effect. For(ii), the front and rear sound from SPK2 in the experimentalapparatus are modeled by using two loudspeakers in the firstprinciple model. These points will be discussed in the nextsection.

4. Comparison of First Principle Model andExperimental Results

Figure 3 shows frequency responses of experimental resultsand the first principle model.

For the first principle model, the following parametersare used from the configuration of the experimental appara-tus in Figure 1: L = 3.61, ly = 0.03, lv = 1.61, lu = 2.32,and lz = 3.53. The parameter LS and H are set as explainedin the previous section for each control source. In addition,c0 = 344, ρ0 = 1.21 are used. We also assume k = ω/c0 −0.09 j instead of (3) in order to consider the dissipation asin [10], which implies that the first principle model G̃(s)is evaluated on a slightly-shifted imaginary axis by settings = jkc0 = jω+30 where the value 30 was chosen for a bettercomparison with the experimental result. Furthermore, gaincharacteristics of the first principle model is divided by 1000just convenience of comparison to experimental result usingthe same scales of figures. Note that the gain characteristicscalculated by the first principle model do not roll off in thehigh frequency range, since the low pass filters are neglectedin the model as mentioned in the previous section.

It can be seen that the first principle models are consistentwith the frequency response experiments in the cases (a) and(b). The gain characteristics of Gzw and Gyw have peaks atabout 24, 71, 119, 167, 214, and 262 Hz which are resonancefrequencies given by fi := (2i − 1)c0/4L (i = 1, 2, . . .). Ingain characteristic of Gzu, the gain of case (a) is lower thanthat of case (b) at the frequency ranges between f1 andf2, f2, and f3, and so on, while it is higher at the frequencyrange below f1. In gain characteristic of Gyu, the gain of case(a) has a notch at the frequency range between f3 and f4.For phase characteristic of Gyu, the phase-lag of case (b) isconsistent with a delay element −e−s(2(L−lu)/c0). Indeed it is

similar to the broken curve in magenta, which is obtainedby adding the delay to the original characteristic of case (a).Note that this fact has been already pointed out for the firstprinciple model [5]. Because of the consistencies between thefrequency responses of the experimental result and the firstprinciple model, one might conclude that the damping effectdue to the flexible PVC duct is not important to affect thecomparison result.

In addition to the cases (a) and (b), the first principlemodel also matches to the experimental results in the case(c). For example, the gain characteristics of Gzw and Gyw hasa deep around f3 compared with those of cases (a) and (b).

The purpose of the proposed method is to provide analternative method putting an additional phase-lag againstthe case (a). In the experiment, it can be seen that the phasecharacteristic of case (c) is similar to that of the case (b)in the middle frequency range, which justifies to use theproposed source for an inexpensive ANC system. Howeverthe phase-lag is not large in the first principle model. Thereason is under consideration, but we may conclude that aproper setting of H makes the phase-lag larger. In the phasecharacteristic of the first principle model G̃yu in Figure 3, thebroken curve of magenta shows that for H = −0.95 insteadH = −1, where a larger phase-lag appears. This might beconsistent to the experimental result since the rear soundcould be smaller than the front one due to some obstacle ofthe rear side of the loudspeaker. Therefore, we expect that theproposed source has a similar advantage to the Swinbanks’source. It should be noted here that the gain of Gzw and G̃zw

for case (c) is smaller than those of cases (a) and (b). Thisimplies an advantage of attaching subduct.

The effect of the proposed source on control performancewill be examined by control experiments in the next section.

5. Controller Design and Control Experiments

In this section, only outline of the design procedure isexplained since the design procedure is the same as in [2].First, for each control source, a nominal plant is obtainedby the subspace-based method from the frequency responseexperiment results shown in left-hand side of Figure 3, wherethe order is taken as 53. Secondly, in order to guaranteethe closed-loop stability against the modeling error of thenominal plant, an additive uncertainty model is introducedfor feedback-path transfer function, Gyu(s) as

Gyu(s) = Gyu(s) +W(s)δ(s), (10)


−5760

−4320

−2880

−1440

0

−80

−60

−40

−20

0

20P

has

e(d

eg)

Mag

nit

ude

(dB

)

Case (a)Case (b)Case (c)

101 102 103

Frequency (Hz)

Experiments Gzw

−5760

−4320

−2880

−1440

0

−80

−60

−40

−20

0

20

Ph

ase

(deg

)M

agn

itu

de(d

B)


101 102 103

Frequency (Hz)

First principle models G̃zw

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0

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0

20

Ph

ase

(deg

)M

agn

itu

de(d

B)


101 102 103

Frequency (Hz)

Experiments Gzu

−5760

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−1440

0−80

−60

−40

−20

0

20

Ph

ase

(deg

)M

agn

itu

de(d

B)

Case (a)Case (b)

Case (c)Case (c) (H=-0.95)

101 102 103

Frequency (Hz)

First principle models G̃zu

−5760

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0

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−40

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0

20

Ph

ase

(deg

)M

agn

itu

de(d

B)


101 102 103

Frequency (Hz)

Experiments Gyw

−5760

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−1440

0

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−40

−20

0

20

Ph

ase

(deg

)M

agn

itu

de(d

B)


101 102 103

Frequency (Hz)

First principle models G̃yw

Figure 3: Continued.


−5760

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−1440

0−80

−60

−40

−20

0

20

Ph

ase

(deg

)M

agn

itu

de(d

B)

Case (a)Case (b)

Case (c)Case (a) with a delay

101 102 103

Frequency (Hz)

Experiments Gyu

−5760

−4320

−2880

−1440

0−80

−60

−40

−20

0

20

Ph

ase

(deg

)M

agn

itu

de(d

B)

Case (a)Case (b)

Case (c)Case (c) (H=-0.95)

101 102 103

Frequency (Hz)

First principle models G̃yu

Figure 3: Frequency response experiment G and first principle model G̃.

d

α z

ζ

yG̃

v

νd−1

Wp

W

u

S Kd H

Figure 4: Robust performance problem with scalings.

where Gyu(s) is the nominal plant for Gyu(s), δ(s) is thenormalized modeling error whose H∞ norm is less than orequal to 1, and W(s) is a weighting function which is chosenso that Gyu can be recovered by δ. We take the commonweighting function for all cases as

W(s) = 0.06ω2

1

s2 + 2ζω1s + ω21

, ω1 = 2500, ζ = 0.7.

(11)

Then, sampled-data H∞ control synthesis [11] is appliedto the following digital controller design problem: find adiscrete-time controller Kd(z) which maximizes the positivescalar α so that the following conditions hold: (i) the closed-loop system of Figure 4 is internally stable; (ii) there existsa positive scalar d such that the L2 induced norm of theclosed-loop system is less than 1, where S is the sampler withsampling period h = 1 ms, H is the zero-th order hold, andWp(s) is a bandpass filter given by

Wp(s) =(

s

s + ωp1

)2(ωp2

s + ωp2

)2

,

ωp1 = 2π × 40, ωp2 = 2π × 200.

(12)

−50

0

50M

agn

itu

de(d

B)


101 102

Frequency (Hz)

−720

−360

0

360

720

Ph

ase

(deg

)


101 102

Frequency (Hz)

Figure 5: Bode plot of controllers.

Note that the closed-loop system gain is robustly mini-mized by maximizing α to improve the control performancein the pass band to attenuate the noise.

Figure 5 shows the resultant controller characteristics.It can be seen that the controller of case (b) has the flatgain characteristic in the effective frequency range of theSwinbanks’ source, which is consistent with [2]. On the otherhand, gain of the cases (a) and (c) have some peaks in thefrequency range, however, we expect similar behavior for case


Table 2: Mean square value of error mic. signal z(t).

Mean square value (V2)

case (a) case (b) case (c)

Without cont. (1) 0.00598 0.00575 0.00431

With control (2) 0.00322 0.00166 0.00229

Reduction (1)-(2) 0.00276 0.00409 0.00203

(c) as case (b), since the gain in the middle frequency rangeis low.

Figure 6 shows time responses of the error microphonesignal z, where the first 12.5 seconds is without control andthe following 12.5 seconds is with control. The disturbancesignal w(t) was generated as a 0-mean uniformly distributedpseudo-random sequence in [−0.3, 0.3] by using rand()function in the GNU C Library. The sampling periodused for the measurement is 0.5 ms. It can be seen thatthe ascending order of the peak-to-peak value is case (b)< case (c) < case (a). Namely, the case (c) achieves abetter performance than the case (a) without using multipleloudspeakers as in the case (b) with multiple loudspeakers.This can be also confirmed by evaluating mean square valueof the time response as shown in Table 2. Note that the errormicrophone signal without control of case (c) differs fromthose of cases (a) and (b) due to attaching the subduct. It isalso shown in Table 2. One might think that the performanceof case (c) could not better than case (a) since the overallnoise reduction by control source loudspeaker in case (c) issmaller than that in case (a) as shown in Table 2. However,we here remark that the case (c) might provide betterperformance than case (a) since the total noise reduction isbetter than case (a) by accounting the noise reduction givenby attaching the subduct.

Figure 7 shows the power spectral density of the errormicrophone signal z(t) where the Welch spectral estimator isused with the Hamming window, and the segment length ischosen as 2048 which corresponds to about 1 Hz in frequencyresolution since the sampling period for measurement is0.5 ms. In addition, the result of “without control” for case(a) has been shown in the same figure of the case (c), sincewe are interested in the total performance of the controlsource including subduct for case (c). It can be seen bycomparing the results for “without control” that noise levelis reduced by attaching the subduct in case (c), whichconsists with the gain characteristics of Gzw and G̃zw inFigure 3 as explained in the previous section. Furthermore,by comparing the “with control” and “without control (case(a)),” it can be seen that there are some amplifications inthe low frequency range around 50 to 100 Hz, however, case(c) has the similar advantage to case (b) that the magnitudearound 6th resonance (262 Hz) is reduced by control, andamplification at the 2nd resonance (70 Hz) is prevented.

Figure 8 shows time responses of control input u. It canbe seen that the ascending order of the peak-to-peak value iscase (b) < case (c) < case (a), which also shows an advantageof the proposed method against the bidirectional source.

0.398

−0.5

0

0.5

Ou

tpu

tz

(V)

0 5 10 15 20 25

Time (s)

Case (a)

0.286

−0.5

0

0.5

Ou

tpu

tz

(V)

0 5 10 15 20 25

Time (s)

Case (b)

0.334

−0.5

0

0.5

Ou

tpu

tz

(V)

0 5 10 15 20 25

Time (s)

Case (c)

Figure 6: Time response of z.


10−4

10−3

10−2

PSD

ofer

ror

sign

alz

(V/H

z0.

5)

101 102 103

Frequency (Hz)

Without controlWith control

Case (a)

10−4

10−3

10−2

PSD

ofer

ror

sign

alz

(V/H

z0.

5)

101 102 103

Frequency (Hz)

Without controlWith control

Case (b)

10−4

10−3

10−2

PSD

ofer

ror

sign

alz

(V/H

z0.

5)

101 102 103

Frequency (Hz)

Without control (case (a))Without controlWith control

Case (c)

Figure 7: Power spectral density of z.

4.9

−3

−2

−1

0

1

2

3

Inpu

tu

(V)

0 5 10 15 20 25

Time (s)

Case (a)

2.3

−3

−2

−1

0

1

2

3

Inpu

tu

(V)

0 5 10 15 20 25

Time (s)

Case (b)

4.1

−3

−2

−1

0

1

2

3

Inpu

tu

(V)

0 5 10 15 20 25

Time (s)

Case (c)

Figure 8: Time response of u.


6. Conclusions

In this paper, a control source which uses the rear soundinterference has been proposed for ANC system. The validityof the proposed method has been shown by comparingwith the existing bidirectional source and the the Swinbanks’source: three structures of control source have been com-pared with a simple ducts in open-loop and closed-loopcharacteristics based on first principle models and experi-mental results. First, the open-loop transfer function for ageneralized control structure unifying the three structureshas been derived. Secondly, it has been shown that the open-loop transfer functions are consistent with the frequencyresponse experiments in the cases of bidirectional and theSwinbanks’ source. Especially, the additional phase-lag in thefeedback path by the Swinbanks’ source corresponds to twiceof the distance between the control source and the open-end.In the proposed source, a similar phase characteristic as theSwinbanks’ source has been shown by frequency responseexperiments.

Finally, effects on control performances of control sourcestructures have been examined by control experimentswith robust controllers. The smaller amplitude in errormicrophone signal has been achieved with smaller amplitudein driving signal of control source by the proposed source ascompared to the bidirectional source. Although the proposedsource does not achieve a much better performance thanthe Swinbanks’ source, it has an advantage for inexpensiveimplementation since less number of loudspeakers arenecessary than the Swinbanks’ source. There might be adifficult situation to adopt case (c) because of the spacelimitation on the installation, however, for the situationwhere the space limitation is mainly on the duct length,the case (c) could be a solution as inexpensive ANC systemcompared with the case (b) since the distance between theloudspeaker and the upstream junction is the same to thedistance between two loudspeakers in case (b).

Therefore we conclude that the proposed structureof control source provides a good trade-off for a well-performed and inexpensive ANC system.

References

[1] M. A. Swinbanks, “The active control of sound propagating inlong ducts,” Journal of Sound and Vibration, vol. 27, pp. 411–436, 1973.

[2] Y. Kobayashi and H. Fujioka, “Active noise cancellation forventilation ducts using a pair of loudspeakers by sampleddataH∞ optimization,” Advances in Acoustics and Vibration, vol.2008, Article ID 253948, 8 pages, 2008.

[3] S. Kijimoto, H. Tanaka, Y. Kanemitsu, and K. Matsuda,“Howling cancellation for active noise control with twosound sources,” Transactions of the Japan Society of MechanicalEngineers, vol. 67, no. 656, pp. 52–57, 2001 (Japanese).

[4] J. Winkler and S. J. Elliott, “Adaptive control of broadbandsound in ducts using a pair of loudspeakers,” Acustica, vol. 81,no. 5, pp. 475–488, 1995.

[5] Y. Kobayashi and H. Fujioka, “Analysis for robust active noisecontrol systems of ducts with a pair of loudspeakers,” in

Proceedings of the 15th International Congress on Sound andVibration (ICSV ’08), Daejeon, South Korea, 2008.

[6] Y. Kobayashi and H. Fujioka, “Robust stability analysis foractive noise control systems of ducts with a pair of loudspeak-ers,” in Proceedings of the 37th Symposium on Control Theory(SICE ’08), pp. 21–24, 2008.

[7] D. S. Bernstein, “What makes some control problems hard?”IEEE Control Systems Magazine, vol. 22, no. 4, pp. 8–19, 2002.

[8] J. Hong and D. S. Bernstein, “Bode integral constraints,colocation, and spilover in active noise and vibration control,”IEEE Transactions on Control Systems Technology, vol. 6, no. 1,pp. 111–120, 1998.

[9] H. Isaka, K. Nisida, and K. Saito, “Active control of exhaustnoise from a muffler in consideration of the location of sec-ondary source,” Transactions of the Japan Society of MechanicalEngineers. C, vol. 66, no. 645, pp. 1502–1508, 2000 (Japanese).

[10] S. J. Elliott, Signal Processing for Active Control, Signal Process-ing and Its Applications, Academic Press, Boston, Mass, USA,2000.

[11] T. Chen and B. A. Francis, Optimal Sampled-Data ControlSystems, Springer, New York, NY, USA, 1996.

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