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

A Control Source Structure of Single Loudspeaker and Rear Sound Interference for Inexpensive... Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2010, Article ID 730813, 9 pages doi:10.1155/2010/730813 Research Article A Control Source Structure of Single Loudspeaker and Rear Sound Interference for Inexpensive Active Noise Control 1 2 3 Yasuhide Kobayashi, Hisaya Fujioka, and Naoki Jinbo Department of Mechanical Engineering, Faculty of Engineering, Nagaoka University of Technology, Nagaoka, Niigata 940-2188, Japan Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan Graduate School of Engineering, Nagaoka University of Technology, Niigata 940-2188, Japan Correspondence should be addressed to Yasuhide Kobayashi, kobayasi@vos.nagaokaut.ac.jp 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 Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Active noise control systems of simple ducts are investigated. In particular, open-loop characteristics and closed-loop performances corresponding 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 a rear sound interference as a novel structure of control source, where the rear sound radiated from the loudspeaker is interfered with the front sound in order to reduce the net upstream sound directly radiated from the control source. The comparisons of the control 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 of the open-loop transfer functions from the first principle modeling and frequency response experiments, it is shown that a certain phase-lag is imposed by the Swinbanks’ source and the rear sound interference. Thirdly, effects on control performances of control source structures are examined by control experiments with robust controllers. 1. Introduction downstream loudspeaker is cancelled out by the sound with the same amplitude and the opposite phase generated by It is well known that the Swinbanks’ (unidirectional) source the upstream loudspeaker. The advantage of the Swinbanks’ [1] has advantages against a standard single loudspeaker source are mainly obtained by the extension of the time period by which the upstream sound directly radiated from (bidirectional source) in not only performances but also the implementation cost. Indeed, the performance improvement the control source travels to the reference microphone. has been reported experimentally in both adaptive and Indeed, it is empirically known that the closed-loop perfor- mance is improved when the actuator (control source) and robust control setups [2, 3]. In addition, the possibility of inexpensive implementation has been pointed out by the sensor (reference microphone) are well separated in space showing that the controller gain is lower and, as the result, [7, 8]. The reverse-phased sound in the Swinbanks’ source is, lower amplitude of driving signal for loudspeakers are achieved by the Swinbanks’ source [2, 4], which enables however, also radiated from the backside of the loudspeakers while it is not utilized in the existing control source for us to use lower power loudspeakers. Furthermore, it has been shown that the advantages of Swinbanks’ source are active noise control (ANC). Therefore, one can expect that theoretically proved for a practical setting on the duct length a single loudspeaker can achieve a similar performance to the Swinbanks’ source if the rear sound could be and loudspeaker locations [5, 6]. The Swinbanks’ source is composed of two loudspeak- interfered to reduce the front sound at an upstream junction ers where the upstream sound radiated directly from the through some additional ducts. In this scenario, inexpensive 2 Advances in Acoustics and Vibration implementation is also expected since the additional loud- In addition, we use τ = 2 ms which exactly corresponds speaker in the Swinbanks’ source is not necessary to generate to 20 times of the period of the real-time module. The opposite-phased sound. Moreover, there is no guarantee that effective frequency range of the Swinbanks’ source is given the Swinbanks’ source is the best structure for an inexpensive as [ f ,5 f ]  [40, 200] where f := c /(12(l − l )) [4], 0 0 0 0 u v ANC system. while c = 344 m/s in a normal temperature environment. In this paper, we propose a new structure of control Moreover, the length of the subduct (88 cm) was adjusted source which is composed of a single loudspeaker with a rear experimentally so that the upstream sound cancellation is sound interference so that the rear sound radiated from the improved: We can identify the length of the delay by injecting loudspeaker is interfered with the front sound in order to an impulse signal to SPK2 and observing the reference reduce the net upstream sound directly radiated from the microphone output y. The delay was maximized by adjusting control source. The validity of the proposed method will be the length of the subduct. [ ] shown by comparing to the existing two structures of control The whole system from to y including dynamics sources, namely, the standard single loudspeaker (bidirec- of electrical circuits, acoustic ducts, microphones, and tional source) and the Swinbanks’ (unidirectional) source, loudspeakers, is considered as the plant transfer function Gzw (s) Gzu (s) in terms of the open-loop characteristic and the closed- G(s):= , for each case, where w is the driving G (s) G (s) yw yu loop performances based on both mathematical models and signal for SPK1, z is the error microphone signal, and y is experimental results. the reference microphone signal as depicted in Figure 1. G(s) is used for robust controller design in Section 5. Note that v is determined by u, and hence G(s) depends on the control 2. Experimental Apparatus structure. We will determine G(s) by frequency response Figure 1 and Table 1, respectively, show a block diagram and experiments, and the experimental results will be compared instruments of the experimental apparatus which are similar with the first principle model in the following sections in to that in [2] except that a real ventilation fan is replaced order to examine the effect of control sources on open-loop to a loudspeaker (SPK1) as a noise source, and the duct characteristics. length is shortened to simplify mathematical models. Two loudspeakers, SPK2 and SPK3, are used as control sources. In Figure 1, blue lines show PVC pipes of 10 cm in 3. First Principle Model diameter and 7 mm in thickness, while brown thinner lines show flexible PVC ducts which are commercial products for In this section, frequency response functions for G(s)defined residential ventilation systems and are connected to adjust in the previous section are derived by the first principle the duct length. The brown broken curve duct, which is modeling where a generalized control structure unifying the called subduct and is also made by the flexible PVC duct, three structures previously mentioned is considered. Figure 2 is used only for the proposed control source after removing shows a model of G(s). The system is mainly composed of SPK3. The experimental apparatus is used in three ways as a duct of length L, where a noise source SPK1, a reference follows for each structure of the control source: microphone, a subduct, a control source SPK2, and an error microphone are located at x = 0, l , l , l ,and l ,respectively. y v u z The subduct is L in length, and is terminated with another Case (a). Bidirectional source: SPK3 is turned off, control source. H (s) is a transfer function relating two that is, the driving signal v(t) is set to 0. The control control sources. The control source in Figure 2 gives a general source is called bidirectional source in this paper, representation including the three control sources as special since the source generates same sounds in upstream cases: Case (a) corresponds to L = 0and H (s) = 0; Case and downstream direction. −s((l −l )/c ) u v 0 (b) corresponds to L = 0and H (s) =−e ; Case (c) Case (b). Swinbanks’ source [4]: SPK3 is driven to corresponds to L = l − l and H (s) =−1. S u v cancel out the upstream sound generated by SPK2, In order to derive frequency response functions, let that is, v(t) =−u(t − τ ), τ = (l − l )/c ,where u(t) u v 0 p and u denote the complex pressure and the particle • • is the driving signal for SPK2, c is the sound speed, velocity, respectively, for each position as shown in Figure 2. and l − l is the distance between SPK2 and SPK3. u v In addition, dynamics of loudspeakers, microphones, and electrical circuits such as low pass filters, are neglected, Case (c). Rear-sound-aided source (Proposed since those dynamics have only common contribution to source): SPK3 is removed and the subduct is attached the frequency response functions to be compared for cases so that the rear sound of SPK2 is interfered with the (a), (b), and (c). Thus, let us assume for simplicity, that front sound at the junction of ducts. Note that only microphone signal is proportional to the complex pressure, SPK2 is used and hence the implementation is less and that the complex particle velocity at each loudspeaker expensive than case (b). is proportional to the driving signal, since our interest is in proportion of the gain and phase characteristics. Specifically, The delay τ in the case (b) is approximately implemented we assume y = p , z = p , u = u,and u = w.Then, y z f 0 as a module of the real-time application interface (RTAI) frequency response functions to be derived are given by for Linux that updates the signal v(t)atevery 0.1ms G (jω) G (jω) p (jω)/u (jω) p (jω)/u (jω) zw zu z 0 z f G(jω):= = . which should be short enough to avoid an aliasing effect.   p (jω)/u (jω) p (jω)/u (jω) G (jω) G (jω) y 0 y f yw yu Advances in Acoustics and Vibration 3 8cm 3cm 121 cm 71 cm 158 cm Error mic. SPK2 SPK3 Ref.mic. SPK1 Pow. 88 cm Pre− LPF D/A A/D LPF AMP AMP Pow. Pow. PC LPF D/A D/A LPF AMP AMP Pre− A/D LPF AMP Figure 1: Experimental apparatus. Llz lu lv ly 0 (p , u ) (p , u ) 1 1 5 5 (p , u ) 4 4 y (p , u ) (p , u ) y y 0 0 s (p , u ) 6 6 (p , u ) (p , u ) z z f f (p , u ) 2 2 (p , u ) 3 3 (p , u ) r r Figure 2: Model for G(s) with a generalized control source. In the following, only a derivation process of G will where the relation T (l )T (l ) = T (l + l ) is used. After yu 1 2 1 2 be explained ((jω) will be omitted). The other 3 transfer eliminating intermediate variables such as u , p ,and u by 6 r r functions can be similarly derived. using above relationships, G is obtained as shown in (6)in yu It is assumed as a boundary condition that p = 0holds 5 the next page. The other functions are shown in (7)–(9): at the open-end, and u = 0 at the closed-end. In addition, the width of the loudspeaker is assumed to be sufficiently G = = cos kl · G , yu y 0u small. It is also assumed that there is no dissipation in sound propagation. Then, the following relationships are obtained p sin k(L−l )H +sin k(L − l ) cos kL 0 v u S by using the transfer matrix method [9] by which a pair of G := = jρ c , 0u 0 0 u cos kL cos kL −sin k(L−l )sin kL cos kl f S v S v complex pressure and particle velocity at a certain location (6) in the duct is described as a pair at different location with distance l multiplied by a transfer matrix T (l)asbelow: sin k(l − l ) sin kL cos kl z v S v G = cos kl − G zu z 0u p p p p cos kL 1 0 3 2 = T (l ) , = T (l − l ) ,(1) v u v (7) u 0 u u 1 3 2 sin k(l − l ) · H z v − jρ c sin k(l − l ) + , 0 0 z u cos kL 0 p p p 4 6 r = T (L − l ) , = T (L ) ,(2) u S u u u u 5 4 6 r G = cos kl · G − jρ c sin kl , yw y 0w 0 0 y ⎡ ⎤ cos kl −jρ c sin kl 0 0 (8) ⎢ ⎥ sin kL cos kL −sin k(L−l ) sin kL sin kl S v S v T (l) := j , k := ,(3) ⎣ ⎦ G := jρ c , 0w 0 0 − sin kl cos kl cos kL cos kL −sin k(L−l ) sin kL cos kl S v S v ρ c 0 0 where ρ is the density of the medium. Moreover, the ( ) sin k l − l sin kL cos kl z v S v following standard assumptions are posed: G = cos kl − G zw z 0w cos kL p = p = p , p = p = p , 1 2 6 4 3 f (9) (4) sin k(l − l ) sin kL sin kl z v S v u = u + u , u = u + u , u = Hu . − jρ c sin kl + . 2 1 6 4 3 f r f 0 0 z cos kL Then, the following equality holds: It can be easily checked that (6)–(9) is consistent with 0 p results in [5] for cases (a) and (b) by substituting L = = T (L − l ) −jω((l −l )/c ) u u u v 0 5 4 0, H (jω) = 0, or H (jω) =−e . (5) Note that H (s)isonlyappearedin G (s)and G (s) yu zu p 0 0 since the corresponding control source is operated by u. ( ) ( ) ( ) = T L + T L − l + T L − l , v u 0 u u 6 f Furthermore, the open-loop frequency response function 4 Advances in Acoustics and Vibration 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, ±5V, 10 μs) G (s) for cases (a) and (b) is common since L = 0inboth similar to the broken curve in magenta, which is obtained zw S cases. The situation is similar for G (s). by adding the delay to the original characteristic of case (a). yw Note that there are two major differences between the Note that this fact has been already pointed out for the first experimental ducts and the first principle model: (i) damp- principle model [5]. Because of the consistencies between the ing or dissipation effect and (ii) number of loudspeakers in frequency responses of the experimental result and the first case (c). For (i), the experimental duct system might have principle model, one might conclude that the damping effect large damping effect due to the flexible PVC ducts, while the due to the flexible PVC duct is not important to affect the first principle model does not depend on damping effect. For comparison result. (ii), the front and rear sound from SPK2 in the experimental In addition to the cases (a) and (b), the first principle apparatus are modeled by using two loudspeakers in the first model also matches to the experimental results in the case principle model. These points will be discussed in the next (c). For example, the gain characteristics of G and G has zw yw section. adeeparound f compared with those of cases (a) and (b). The purpose of the proposed method is to provide an alternative method putting an additional phase-lag against 4. Comparison of First Principle Model and the case (a). In the experiment, it can be seen that the phase Experimental Results characteristic of case (c) is similar to that of the case (b) in the middle frequency range, which justifies to use the Figure 3 shows frequency responses of experimental results proposed source for an inexpensive ANC system. However and the first principle model. the phase-lag is not large in the first principle model. The For the first principle model, the following parameters reason is under consideration, but we may conclude that a are used from the configuration of the experimental appara- proper setting of H makes the phase-lag larger. In the phase tus in Figure 1: L = 3.61, l = 0.03, l = 1.61, l = 2.32, y v u characteristic of the first principle model G in Figure 3, the yu and l = 3.53. The parameter L and H are set as explained z S broken curve of magenta shows that for H =−0.95 instead in the previous section for each control source. In addition, H =−1, where a larger phase-lag appears. This might be c = 344, ρ = 1.21 are used. We also assume k = ω/c − 0 0 0 consistent to the experimental result since the rear sound 0.09j instead of (3) in order to consider the dissipation as could be smaller than the front one due to some obstacle of in [10], which implies that the first principle model G(s) the rear side of the loudspeaker. Therefore, we expect that the is evaluated on a slightly-shifted imaginary axis by setting proposed source has a similar advantage to the Swinbanks’ s = jkc = jω + 30 where the value 30 was chosen for a better source. It should be noted here that the gain of G and G zw zw comparison with the experimental result. Furthermore, gain for case (c) is smaller than those of cases (a) and (b). This characteristics of the first principle model is divided by 1000 implies an advantage of attaching subduct. just convenience of comparison to experimental result using The effect of the proposed source on control performance the same scales of figures. Note that the gain characteristics will be examined by control experiments in the next section. calculated by the first principle model do not roll off in the high frequency range, since the low pass filters are neglected 5. Controller Design and Control Experiments in the model as mentioned in the previous section. It can be seen that the first principle models are consistent In this section, only outline of the design procedure is with the frequency response experiments in the cases (a) and explained since the design procedure is the same as in [2]. (b). The gain characteristics of G and G have peaks at zw yw First, for each control source, a nominal plant is obtained about 24, 71, 119, 167, 214, and 262 Hz which are resonance by the subspace-based method from the frequency response frequencies given by f := (2i − 1)c /4L (i = 1, 2,...). In i 0 experiment results shown in left-hand side of Figure 3,where gain characteristic of G , the gain of case (a) is lower than zu the order is taken as 53. Secondly, in order to guarantee that of case (b) at the frequency ranges between f and the closed-loop stability against the modeling error of the f , f ,and f , and so on, while it is higher at the frequency 2 2 3 nominal plant, an additive uncertainty model is introduced range below f .Ingaincharacteristicof G , the gain of case 1 yu for feedback-path transfer function, G (s)as yu (a) has a notch at the frequency range between f and f . 3 4 For phase characteristic of G , the phase-lag of case (b) is yu ( ) ( ) ( ) ( ) −s(2(L−l )/c ) G s = G s + W s δ s , (10) u 0 yu yu consistent with a delay element −e . Indeed it is Advances in Acoustics and Vibration 5 Experiments G First principle models G zw zw 20 20 0 0 −20 −20 −40 −40 −60 −60 −80 −80 0 0 −1440 −1440 −2880 −2880 −4320 −4320 −5760 −5760 1 2 3 1 2 3 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Case (a) Case (a) Case (b) Case (b) Case (c) Case (c) Experiments G zu First principle models G zu −20 −20 −40 −40 −60 −60 −80 0 −80 −1440 −1440 −2880 −2880 −4320 −4320 −5760 1 2 3 10 10 10 −5760 1 2 3 10 10 10 Frequency (Hz) Frequency (Hz) Case (a) Case (b) Case (a) Case (c) Case (c) Case (b) Case (c) (H=-0.95) Experiments G First principle models G yw yw 20 20 0 0 −20 −20 −40 −40 −60 −60 −80 −80 −1440 −1440 −2880 −2880 −4320 −4320 −5760 −5760 1 2 3 1 2 3 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Case (a) Case (a) Case (b) Case (b) Case (c) Case (c) Figure 3: Continued. Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) 6 Advances in Acoustics and Vibration Experiments G First principle models G yu yu 20 20 0 0 −20 −20 −40 −40 −60 −60 −80 −80 0 0 −1440 −1440 −2880 −2880 −4320 −4320 −5760 −5760 1 2 3 1 2 3 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Case (a) Case (c) Case (a) Case (c) Case (b) Case (a) with a delay Case (b) Case (c) (H=-0.95) Figure 3: Frequency response experiment G and first principle model G. −1 α W y 0 −50 1 2 SK H 10 10 Frequency (Hz) Figure 4: Robust performance problem with scalings. Case (a) Case (b) Case (c) where G (s) is the nominal plant for G (s), δ(s) is the yu yu normalized modeling error whose H norm is less than or equal to 1, and W (s) is a weighting function which is chosen so that G can be recovered by δ. We take the common yu weighting function for all cases as −360 −720 ω 1 2 1 10 10 W (s) = 0.06 , ω = 2500, ζ = 0.7. s +2ζω s + ω 1 Frequency (Hz) (11) Case (a) Case (b) Case (c) Then, sampled-data H control synthesis [11]isapplied to the following digital controller design problem: find a Figure 5: Bode plot of controllers. discrete-time controller K (z) which maximizes the positive scalar α so that the following conditions hold: (i) the closed- loop system of Figure 4 is internally stable; (ii) there exists a positive scalar d such that the L induced norm of the Note that the closed-loop system gain is robustly mini- closed-loop system is less than 1, where S is the sampler with mized by maximizing α to improve the control performance sampling period h = 1ms, H is the zero-th order hold, and in the pass band to attenuate the noise. W (s) is a bandpass filter given by Figure 5 shows the resultant controller characteristics. 2 2 It can be seen that the controller of case (b) has the flat s p W (s) = , gain characteristic in the effective frequency range of the s + ω s + ω p p 1 2 (12) Swinbanks’ source, which is consistent with [2]. On the other hand, gain of the cases (a) and (c) have some peaks in the ω = 2π × 40, ω = 2π × 200. p p 1 2 frequency range, however, we expect similar behavior for case Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Advances in Acoustics and Vibration 7 0.5 Table 2: Mean square value of error mic. signal z(t). Mean square value (V ) 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 0.398 (c) as case (b), since the gain in the middle frequency range is low. Figure 6 shows time responses of the error microphone −0.5 signal z, where the first 12.5 seconds is without control and 0 5 10 15 20 25 the following 12.5 seconds is with control. The disturbance Time (s) signal w(t) was generated as a 0-mean uniformly distributed Case (a) pseudo-random sequence in [−0.3, 0.3] by using rand() function in the GNU C Library. The sampling period used for the measurement is 0.5 ms. It can be seen that 0.5 the ascending order of the peak-to-peak value is case (b) < case (c) < case (a). Namely, the case (c) achieves a better performance than the case (a) without using multiple loudspeakers as in the case (b) with multiple loudspeakers. This can be also confirmed by evaluating mean square value of the time response as shown in Table 2. Note that the error microphone signal without control of case (c) differs from 0.286 those of cases (a) and (b) due to attaching the subduct. It is also shown in Table 2. One might think that the performance of case (c) could not better than case (a) since the overall noise reduction by control source loudspeaker in case (c) is smaller than that in case (a) as shown in Table 2.However, we here remark that the case (c) might provide better performance than case (a) since the total noise reduction is −0.5 better than case (a) by accounting the noise reduction given 0 5 10 15 20 25 by attaching the subduct. Time (s) Figure 7 shows the power spectral density of the error Case (b) microphone signal z(t) where the Welch spectral estimator is used with the Hamming window, and the segment length is chosen as 2048 which corresponds to about 1 Hz in frequency 0.5 resolution since the sampling period for measurement is 0.5 ms. In addition, the result of “without control” for case (a) has been shown in the same figure of the case (c), since we are interested in the total performance of the control source including subduct for case (c). It can be seen by comparing the results for “without control” that noise level is reduced by attaching the subduct in case (c), which 0.334 consists with the gain characteristics of G and G in zw zw Figure 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 in the low frequency range around 50 to 100 Hz, however, case (c) has the similar advantage to case (b) that the magnitude around 6th resonance (262 Hz) is reduced by control, and −0.5 0 5 10 15 20 25 amplification at the 2nd resonance (70 Hz) is prevented. Time (s) Figure 8 shows time responses of control input u.Itcan be seen that the ascending order of the peak-to-peak value is Case (c) case (b) < case (c) < case (a), which also shows an advantage of the proposed method against the bidirectional source. Figure 6: Time response of z. Output z (V) Output z (V) Output z (V) 8 Advances in Acoustics and Vibration −2 −3 4.9 −1 −4 −2 −3 1 2 3 10 10 10 0 5 10 15 20 25 Frequency (Hz) Time (s) Without control Case (a) With control Case (a) −2 −3 0 2.3 −4 −1 −2 1 2 3 10 10 10 −3 Frequency (Hz) 0 5 10 15 20 25 Time (s) Without control With control Case (b) Case (b) −2 −3 4.1 −4 −1 −2 1 2 3 10 10 10 Frequency (Hz) −3 0 5 10 15 20 25 Without control (case (a)) Time (s) Without control With control Case (c) Case (c) Figure 8: Time response of u. Figure 7: Power spectral density of z. 0.5 0.5 0.5 PSD of error signal z (V/Hz ) PSD of error signal z (V/Hz ) PSD of error signal z (V/Hz ) Input u (V) Input u (V) Input u (V) Advances in Acoustics and Vibration 9 6. Conclusions Proceedings of the 15th International Congress on Sound and Vibration (ICSV ’08), Daejeon, South Korea, 2008. In this paper, a control source which uses the rear sound [6] Y. Kobayashi and H. Fujioka, “Robust stability analysis for interference has been proposed for ANC system. The validity active noise control systems of ducts with a pair of loudspeak- of the proposed method has been shown by comparing ers,” in Proceedings of the 37th Symposium on Control Theory with the existing bidirectional source and the the Swinbanks’ (SICE ’08), pp. 21–24, 2008. source: three structures of control source have been com- [7] D. S. Bernstein, “What makes some control problems hard?” pared with a simple ducts in open-loop and closed-loop IEEE Control Systems Magazine, vol. 22, no. 4, pp. 8–19, 2002. characteristics based on first principle models and experi- [8] J. Hong and D. S. Bernstein, “Bode integral constraints, mental results. First, the open-loop transfer function for a colocation, and spilover in active noise and vibration control,” generalized control structure unifying the three structures IEEE Transactions on Control Systems Technology, vol. 6, no. 1, pp. 111–120, 1998. has been derived. Secondly, it has been shown that the open- loop transfer functions are consistent with the frequency [9] H. Isaka, K. Nisida, and K. Saito, “Active control of exhaust noise from a muffler in consideration of the location of sec- response experiments in the cases of bidirectional and the ondary source,” Transactions of the Japan Society of Mechanical Swinbanks’ source. Especially, the additional phase-lag in the Engineers. C, vol. 66, no. 645, pp. 1502–1508, 2000 (Japanese). feedback path by the Swinbanks’ source corresponds to twice [10] S. J. Elliott, Signal Processing for Active Control, Signal Process- of the distance between the control source and the open-end. ing and Its Applications, Academic Press, Boston, Mass, USA, In the proposed source, a similar phase characteristic as the Swinbanks’ source has been shown by frequency response [11] T. Chen and B. A. Francis, Optimal Sampled-Data Control experiments. Systems, Springer, New York, NY, USA, 1996. Finally, effects on control performances of control source structures have been examined by control experiments with robust controllers. The smaller amplitude in error microphone signal has been achieved with smaller amplitude in driving signal of control source by the proposed source as compared to the bidirectional source. Although the proposed source does not achieve a much better performance than the Swinbanks’ source, it has an advantage for inexpensive implementation since less number of loudspeakers are necessary than the Swinbanks’ source. There might be a difficult situation to adopt case (c) because of the space limitation on the installation, however, for the situation where the space limitation is mainly on the duct length, the case (c) could be a solution as inexpensive ANC system compared with the case (b) since the distance between the loudspeaker and the upstream junction is the same to the distance between two loudspeakers in case (b). Therefore we conclude that the proposed structure of 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 in long ducts,” Journal of Sound and Vibration, vol. 27, pp. 411– 436, 1973. [2] Y. Kobayashi and H. Fujioka, “Active noise cancellation for ventilation ducts using a pair of loudspeakers by sampleddata H 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 two sound sources,” Transactions of the Japan Society of Mechanical Engineers, vol. 67, no. 656, pp. 52–57, 2001 (Japanese). [4] J. Winkler and S. J. Elliott, “Adaptive control of broadband sound 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 noise control systems of ducts with a pair of loudspeakers,” in International Journal of Rotating Machinery International Journal of Journal of The Scientific Journal of Distributed Engineering World Journal Sensors Sensor Networks Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2010 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Acoustics and Vibration Hindawi Publishing Corporation

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

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Copyright © 2010 Yasuhide Kobayashi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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10.1155/2010/730813
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2010, Article ID 730813, 9 pages doi:10.1155/2010/730813 Research Article A Control Source Structure of Single Loudspeaker and Rear Sound Interference for Inexpensive Active Noise Control 1 2 3 Yasuhide Kobayashi, Hisaya Fujioka, and Naoki Jinbo Department of Mechanical Engineering, Faculty of Engineering, Nagaoka University of Technology, Nagaoka, Niigata 940-2188, Japan Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan Graduate School of Engineering, Nagaoka University of Technology, Niigata 940-2188, Japan Correspondence should be addressed to Yasuhide Kobayashi, kobayasi@vos.nagaokaut.ac.jp 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 Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Active noise control systems of simple ducts are investigated. In particular, open-loop characteristics and closed-loop performances corresponding 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 a rear sound interference as a novel structure of control source, where the rear sound radiated from the loudspeaker is interfered with the front sound in order to reduce the net upstream sound directly radiated from the control source. The comparisons of the control 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 of the open-loop transfer functions from the first principle modeling and frequency response experiments, it is shown that a certain phase-lag is imposed by the Swinbanks’ source and the rear sound interference. Thirdly, effects on control performances of control source structures are examined by control experiments with robust controllers. 1. Introduction downstream loudspeaker is cancelled out by the sound with the same amplitude and the opposite phase generated by It is well known that the Swinbanks’ (unidirectional) source the upstream loudspeaker. The advantage of the Swinbanks’ [1] has advantages against a standard single loudspeaker source are mainly obtained by the extension of the time period by which the upstream sound directly radiated from (bidirectional source) in not only performances but also the implementation cost. Indeed, the performance improvement the control source travels to the reference microphone. has been reported experimentally in both adaptive and Indeed, it is empirically known that the closed-loop perfor- mance is improved when the actuator (control source) and robust control setups [2, 3]. In addition, the possibility of inexpensive implementation has been pointed out by the sensor (reference microphone) are well separated in space showing that the controller gain is lower and, as the result, [7, 8]. The reverse-phased sound in the Swinbanks’ source is, lower amplitude of driving signal for loudspeakers are achieved by the Swinbanks’ source [2, 4], which enables however, also radiated from the backside of the loudspeakers while it is not utilized in the existing control source for us to use lower power loudspeakers. Furthermore, it has been shown that the advantages of Swinbanks’ source are active noise control (ANC). Therefore, one can expect that theoretically proved for a practical setting on the duct length a single loudspeaker can achieve a similar performance to the Swinbanks’ source if the rear sound could be and loudspeaker locations [5, 6]. The Swinbanks’ source is composed of two loudspeak- interfered to reduce the front sound at an upstream junction ers where the upstream sound radiated directly from the through some additional ducts. In this scenario, inexpensive 2 Advances in Acoustics and Vibration implementation is also expected since the additional loud- In addition, we use τ = 2 ms which exactly corresponds speaker in the Swinbanks’ source is not necessary to generate to 20 times of the period of the real-time module. The opposite-phased sound. Moreover, there is no guarantee that effective frequency range of the Swinbanks’ source is given the Swinbanks’ source is the best structure for an inexpensive as [ f ,5 f ]  [40, 200] where f := c /(12(l − l )) [4], 0 0 0 0 u v ANC system. while c = 344 m/s in a normal temperature environment. In this paper, we propose a new structure of control Moreover, the length of the subduct (88 cm) was adjusted source which is composed of a single loudspeaker with a rear experimentally so that the upstream sound cancellation is sound interference so that the rear sound radiated from the improved: We can identify the length of the delay by injecting loudspeaker is interfered with the front sound in order to an impulse signal to SPK2 and observing the reference reduce the net upstream sound directly radiated from the microphone output y. The delay was maximized by adjusting control source. The validity of the proposed method will be the length of the subduct. [ ] shown by comparing to the existing two structures of control The whole system from to y including dynamics sources, namely, the standard single loudspeaker (bidirec- of electrical circuits, acoustic ducts, microphones, and tional source) and the Swinbanks’ (unidirectional) source, loudspeakers, is considered as the plant transfer function Gzw (s) Gzu (s) in terms of the open-loop characteristic and the closed- G(s):= , for each case, where w is the driving G (s) G (s) yw yu loop performances based on both mathematical models and signal for SPK1, z is the error microphone signal, and y is experimental results. the reference microphone signal as depicted in Figure 1. G(s) is used for robust controller design in Section 5. Note that v is determined by u, and hence G(s) depends on the control 2. Experimental Apparatus structure. We will determine G(s) by frequency response Figure 1 and Table 1, respectively, show a block diagram and experiments, and the experimental results will be compared instruments of the experimental apparatus which are similar with the first principle model in the following sections in to that in [2] except that a real ventilation fan is replaced order to examine the effect of control sources on open-loop to a loudspeaker (SPK1) as a noise source, and the duct characteristics. length is shortened to simplify mathematical models. Two loudspeakers, SPK2 and SPK3, are used as control sources. In Figure 1, blue lines show PVC pipes of 10 cm in 3. First Principle Model diameter and 7 mm in thickness, while brown thinner lines show flexible PVC ducts which are commercial products for In this section, frequency response functions for G(s)defined residential ventilation systems and are connected to adjust in the previous section are derived by the first principle the duct length. The brown broken curve duct, which is modeling where a generalized control structure unifying the called subduct and is also made by the flexible PVC duct, three structures previously mentioned is considered. Figure 2 is used only for the proposed control source after removing shows a model of G(s). The system is mainly composed of SPK3. The experimental apparatus is used in three ways as a duct of length L, where a noise source SPK1, a reference follows for each structure of the control source: microphone, a subduct, a control source SPK2, and an error microphone are located at x = 0, l , l , l ,and l ,respectively. y v u z The subduct is L in length, and is terminated with another Case (a). Bidirectional source: SPK3 is turned off, control source. H (s) is a transfer function relating two that is, the driving signal v(t) is set to 0. The control control sources. The control source in Figure 2 gives a general source is called bidirectional source in this paper, representation including the three control sources as special since the source generates same sounds in upstream cases: Case (a) corresponds to L = 0and H (s) = 0; Case and downstream direction. −s((l −l )/c ) u v 0 (b) corresponds to L = 0and H (s) =−e ; Case (c) Case (b). Swinbanks’ source [4]: SPK3 is driven to corresponds to L = l − l and H (s) =−1. S u v cancel out the upstream sound generated by SPK2, In order to derive frequency response functions, let that is, v(t) =−u(t − τ ), τ = (l − l )/c ,where u(t) u v 0 p and u denote the complex pressure and the particle • • is the driving signal for SPK2, c is the sound speed, velocity, respectively, for each position as shown in Figure 2. and l − l is the distance between SPK2 and SPK3. u v In addition, dynamics of loudspeakers, microphones, and electrical circuits such as low pass filters, are neglected, Case (c). Rear-sound-aided source (Proposed since those dynamics have only common contribution to source): SPK3 is removed and the subduct is attached the frequency response functions to be compared for cases so that the rear sound of SPK2 is interfered with the (a), (b), and (c). Thus, let us assume for simplicity, that front sound at the junction of ducts. Note that only microphone signal is proportional to the complex pressure, SPK2 is used and hence the implementation is less and that the complex particle velocity at each loudspeaker expensive than case (b). is proportional to the driving signal, since our interest is in proportion of the gain and phase characteristics. Specifically, The delay τ in the case (b) is approximately implemented we assume y = p , z = p , u = u,and u = w.Then, y z f 0 as a module of the real-time application interface (RTAI) frequency response functions to be derived are given by for Linux that updates the signal v(t)atevery 0.1ms G (jω) G (jω) p (jω)/u (jω) p (jω)/u (jω) zw zu z 0 z f G(jω):= = . which should be short enough to avoid an aliasing effect.   p (jω)/u (jω) p (jω)/u (jω) G (jω) G (jω) y 0 y f yw yu Advances in Acoustics and Vibration 3 8cm 3cm 121 cm 71 cm 158 cm Error mic. SPK2 SPK3 Ref.mic. SPK1 Pow. 88 cm Pre− LPF D/A A/D LPF AMP AMP Pow. Pow. PC LPF D/A D/A LPF AMP AMP Pre− A/D LPF AMP Figure 1: Experimental apparatus. Llz lu lv ly 0 (p , u ) (p , u ) 1 1 5 5 (p , u ) 4 4 y (p , u ) (p , u ) y y 0 0 s (p , u ) 6 6 (p , u ) (p , u ) z z f f (p , u ) 2 2 (p , u ) 3 3 (p , u ) r r Figure 2: Model for G(s) with a generalized control source. In the following, only a derivation process of G will where the relation T (l )T (l ) = T (l + l ) is used. After yu 1 2 1 2 be explained ((jω) will be omitted). The other 3 transfer eliminating intermediate variables such as u , p ,and u by 6 r r functions can be similarly derived. using above relationships, G is obtained as shown in (6)in yu It is assumed as a boundary condition that p = 0holds 5 the next page. The other functions are shown in (7)–(9): at the open-end, and u = 0 at the closed-end. In addition, the width of the loudspeaker is assumed to be sufficiently G = = cos kl · G , yu y 0u small. It is also assumed that there is no dissipation in sound propagation. Then, the following relationships are obtained p sin k(L−l )H +sin k(L − l ) cos kL 0 v u S by using the transfer matrix method [9] by which a pair of G := = jρ c , 0u 0 0 u cos kL cos kL −sin k(L−l )sin kL cos kl f S v S v complex pressure and particle velocity at a certain location (6) in the duct is described as a pair at different location with distance l multiplied by a transfer matrix T (l)asbelow: sin k(l − l ) sin kL cos kl z v S v G = cos kl − G zu z 0u p p p p cos kL 1 0 3 2 = T (l ) , = T (l − l ) ,(1) v u v (7) u 0 u u 1 3 2 sin k(l − l ) · H z v − jρ c sin k(l − l ) + , 0 0 z u cos kL 0 p p p 4 6 r = T (L − l ) , = T (L ) ,(2) u S u u u u 5 4 6 r G = cos kl · G − jρ c sin kl , yw y 0w 0 0 y ⎡ ⎤ cos kl −jρ c sin kl 0 0 (8) ⎢ ⎥ sin kL cos kL −sin k(L−l ) sin kL sin kl S v S v T (l) := j , k := ,(3) ⎣ ⎦ G := jρ c , 0w 0 0 − sin kl cos kl cos kL cos kL −sin k(L−l ) sin kL cos kl S v S v ρ c 0 0 where ρ is the density of the medium. Moreover, the ( ) sin k l − l sin kL cos kl z v S v following standard assumptions are posed: G = cos kl − G zw z 0w cos kL p = p = p , p = p = p , 1 2 6 4 3 f (9) (4) sin k(l − l ) sin kL sin kl z v S v u = u + u , u = u + u , u = Hu . − jρ c sin kl + . 2 1 6 4 3 f r f 0 0 z cos kL Then, the following equality holds: It can be easily checked that (6)–(9) is consistent with 0 p results in [5] for cases (a) and (b) by substituting L = = T (L − l ) −jω((l −l )/c ) u u u v 0 5 4 0, H (jω) = 0, or H (jω) =−e . (5) Note that H (s)isonlyappearedin G (s)and G (s) yu zu p 0 0 since the corresponding control source is operated by u. ( ) ( ) ( ) = T L + T L − l + T L − l , v u 0 u u 6 f Furthermore, the open-loop frequency response function 4 Advances in Acoustics and Vibration 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, ±5V, 10 μs) G (s) for cases (a) and (b) is common since L = 0inboth similar to the broken curve in magenta, which is obtained zw S cases. The situation is similar for G (s). by adding the delay to the original characteristic of case (a). yw Note that there are two major differences between the Note that this fact has been already pointed out for the first experimental ducts and the first principle model: (i) damp- principle model [5]. Because of the consistencies between the ing or dissipation effect and (ii) number of loudspeakers in frequency responses of the experimental result and the first case (c). For (i), the experimental duct system might have principle model, one might conclude that the damping effect large damping effect due to the flexible PVC ducts, while the due to the flexible PVC duct is not important to affect the first principle model does not depend on damping effect. For comparison result. (ii), the front and rear sound from SPK2 in the experimental In addition to the cases (a) and (b), the first principle apparatus are modeled by using two loudspeakers in the first model also matches to the experimental results in the case principle model. These points will be discussed in the next (c). For example, the gain characteristics of G and G has zw yw section. adeeparound f compared with those of cases (a) and (b). The purpose of the proposed method is to provide an alternative method putting an additional phase-lag against 4. Comparison of First Principle Model and the case (a). In the experiment, it can be seen that the phase Experimental Results characteristic of case (c) is similar to that of the case (b) in the middle frequency range, which justifies to use the Figure 3 shows frequency responses of experimental results proposed source for an inexpensive ANC system. However and the first principle model. the phase-lag is not large in the first principle model. The For the first principle model, the following parameters reason is under consideration, but we may conclude that a are used from the configuration of the experimental appara- proper setting of H makes the phase-lag larger. In the phase tus in Figure 1: L = 3.61, l = 0.03, l = 1.61, l = 2.32, y v u characteristic of the first principle model G in Figure 3, the yu and l = 3.53. The parameter L and H are set as explained z S broken curve of magenta shows that for H =−0.95 instead in the previous section for each control source. In addition, H =−1, where a larger phase-lag appears. This might be c = 344, ρ = 1.21 are used. We also assume k = ω/c − 0 0 0 consistent to the experimental result since the rear sound 0.09j instead of (3) in order to consider the dissipation as could be smaller than the front one due to some obstacle of in [10], which implies that the first principle model G(s) the rear side of the loudspeaker. Therefore, we expect that the is evaluated on a slightly-shifted imaginary axis by setting proposed source has a similar advantage to the Swinbanks’ s = jkc = jω + 30 where the value 30 was chosen for a better source. It should be noted here that the gain of G and G zw zw comparison with the experimental result. Furthermore, gain for case (c) is smaller than those of cases (a) and (b). This characteristics of the first principle model is divided by 1000 implies an advantage of attaching subduct. just convenience of comparison to experimental result using The effect of the proposed source on control performance the same scales of figures. Note that the gain characteristics will be examined by control experiments in the next section. calculated by the first principle model do not roll off in the high frequency range, since the low pass filters are neglected 5. Controller Design and Control Experiments in the model as mentioned in the previous section. It can be seen that the first principle models are consistent In this section, only outline of the design procedure is with the frequency response experiments in the cases (a) and explained since the design procedure is the same as in [2]. (b). The gain characteristics of G and G have peaks at zw yw First, for each control source, a nominal plant is obtained about 24, 71, 119, 167, 214, and 262 Hz which are resonance by the subspace-based method from the frequency response frequencies given by f := (2i − 1)c /4L (i = 1, 2,...). In i 0 experiment results shown in left-hand side of Figure 3,where gain characteristic of G , the gain of case (a) is lower than zu the order is taken as 53. Secondly, in order to guarantee that of case (b) at the frequency ranges between f and the closed-loop stability against the modeling error of the f , f ,and f , and so on, while it is higher at the frequency 2 2 3 nominal plant, an additive uncertainty model is introduced range below f .Ingaincharacteristicof G , the gain of case 1 yu for feedback-path transfer function, G (s)as yu (a) has a notch at the frequency range between f and f . 3 4 For phase characteristic of G , the phase-lag of case (b) is yu ( ) ( ) ( ) ( ) −s(2(L−l )/c ) G s = G s + W s δ s , (10) u 0 yu yu consistent with a delay element −e . Indeed it is Advances in Acoustics and Vibration 5 Experiments G First principle models G zw zw 20 20 0 0 −20 −20 −40 −40 −60 −60 −80 −80 0 0 −1440 −1440 −2880 −2880 −4320 −4320 −5760 −5760 1 2 3 1 2 3 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Case (a) Case (a) Case (b) Case (b) Case (c) Case (c) Experiments G zu First principle models G zu −20 −20 −40 −40 −60 −60 −80 0 −80 −1440 −1440 −2880 −2880 −4320 −4320 −5760 1 2 3 10 10 10 −5760 1 2 3 10 10 10 Frequency (Hz) Frequency (Hz) Case (a) Case (b) Case (a) Case (c) Case (c) Case (b) Case (c) (H=-0.95) Experiments G First principle models G yw yw 20 20 0 0 −20 −20 −40 −40 −60 −60 −80 −80 −1440 −1440 −2880 −2880 −4320 −4320 −5760 −5760 1 2 3 1 2 3 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Case (a) Case (a) Case (b) Case (b) Case (c) Case (c) Figure 3: Continued. Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) 6 Advances in Acoustics and Vibration Experiments G First principle models G yu yu 20 20 0 0 −20 −20 −40 −40 −60 −60 −80 −80 0 0 −1440 −1440 −2880 −2880 −4320 −4320 −5760 −5760 1 2 3 1 2 3 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Case (a) Case (c) Case (a) Case (c) Case (b) Case (a) with a delay Case (b) Case (c) (H=-0.95) Figure 3: Frequency response experiment G and first principle model G. −1 α W y 0 −50 1 2 SK H 10 10 Frequency (Hz) Figure 4: Robust performance problem with scalings. Case (a) Case (b) Case (c) where G (s) is the nominal plant for G (s), δ(s) is the yu yu normalized modeling error whose H norm is less than or equal to 1, and W (s) is a weighting function which is chosen so that G can be recovered by δ. We take the common yu weighting function for all cases as −360 −720 ω 1 2 1 10 10 W (s) = 0.06 , ω = 2500, ζ = 0.7. s +2ζω s + ω 1 Frequency (Hz) (11) Case (a) Case (b) Case (c) Then, sampled-data H control synthesis [11]isapplied to the following digital controller design problem: find a Figure 5: Bode plot of controllers. discrete-time controller K (z) which maximizes the positive scalar α so that the following conditions hold: (i) the closed- loop system of Figure 4 is internally stable; (ii) there exists a positive scalar d such that the L induced norm of the Note that the closed-loop system gain is robustly mini- closed-loop system is less than 1, where S is the sampler with mized by maximizing α to improve the control performance sampling period h = 1ms, H is the zero-th order hold, and in the pass band to attenuate the noise. W (s) is a bandpass filter given by Figure 5 shows the resultant controller characteristics. 2 2 It can be seen that the controller of case (b) has the flat s p W (s) = , gain characteristic in the effective frequency range of the s + ω s + ω p p 1 2 (12) Swinbanks’ source, which is consistent with [2]. On the other hand, gain of the cases (a) and (c) have some peaks in the ω = 2π × 40, ω = 2π × 200. p p 1 2 frequency range, however, we expect similar behavior for case Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Advances in Acoustics and Vibration 7 0.5 Table 2: Mean square value of error mic. signal z(t). Mean square value (V ) 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 0.398 (c) as case (b), since the gain in the middle frequency range is low. Figure 6 shows time responses of the error microphone −0.5 signal z, where the first 12.5 seconds is without control and 0 5 10 15 20 25 the following 12.5 seconds is with control. The disturbance Time (s) signal w(t) was generated as a 0-mean uniformly distributed Case (a) pseudo-random sequence in [−0.3, 0.3] by using rand() function in the GNU C Library. The sampling period used for the measurement is 0.5 ms. It can be seen that 0.5 the ascending order of the peak-to-peak value is case (b) < case (c) < case (a). Namely, the case (c) achieves a better performance than the case (a) without using multiple loudspeakers as in the case (b) with multiple loudspeakers. This can be also confirmed by evaluating mean square value of the time response as shown in Table 2. Note that the error microphone signal without control of case (c) differs from 0.286 those of cases (a) and (b) due to attaching the subduct. It is also shown in Table 2. One might think that the performance of case (c) could not better than case (a) since the overall noise reduction by control source loudspeaker in case (c) is smaller than that in case (a) as shown in Table 2.However, we here remark that the case (c) might provide better performance than case (a) since the total noise reduction is −0.5 better than case (a) by accounting the noise reduction given 0 5 10 15 20 25 by attaching the subduct. Time (s) Figure 7 shows the power spectral density of the error Case (b) microphone signal z(t) where the Welch spectral estimator is used with the Hamming window, and the segment length is chosen as 2048 which corresponds to about 1 Hz in frequency 0.5 resolution since the sampling period for measurement is 0.5 ms. In addition, the result of “without control” for case (a) has been shown in the same figure of the case (c), since we are interested in the total performance of the control source including subduct for case (c). It can be seen by comparing the results for “without control” that noise level is reduced by attaching the subduct in case (c), which 0.334 consists with the gain characteristics of G and G in zw zw Figure 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 in the low frequency range around 50 to 100 Hz, however, case (c) has the similar advantage to case (b) that the magnitude around 6th resonance (262 Hz) is reduced by control, and −0.5 0 5 10 15 20 25 amplification at the 2nd resonance (70 Hz) is prevented. Time (s) Figure 8 shows time responses of control input u.Itcan be seen that the ascending order of the peak-to-peak value is Case (c) case (b) < case (c) < case (a), which also shows an advantage of the proposed method against the bidirectional source. Figure 6: Time response of z. Output z (V) Output z (V) Output z (V) 8 Advances in Acoustics and Vibration −2 −3 4.9 −1 −4 −2 −3 1 2 3 10 10 10 0 5 10 15 20 25 Frequency (Hz) Time (s) Without control Case (a) With control Case (a) −2 −3 0 2.3 −4 −1 −2 1 2 3 10 10 10 −3 Frequency (Hz) 0 5 10 15 20 25 Time (s) Without control With control Case (b) Case (b) −2 −3 4.1 −4 −1 −2 1 2 3 10 10 10 Frequency (Hz) −3 0 5 10 15 20 25 Without control (case (a)) Time (s) Without control With control Case (c) Case (c) Figure 8: Time response of u. Figure 7: Power spectral density of z. 0.5 0.5 0.5 PSD of error signal z (V/Hz ) PSD of error signal z (V/Hz ) PSD of error signal z (V/Hz ) Input u (V) Input u (V) Input u (V) Advances in Acoustics and Vibration 9 6. Conclusions Proceedings of the 15th International Congress on Sound and Vibration (ICSV ’08), Daejeon, South Korea, 2008. In this paper, a control source which uses the rear sound [6] Y. Kobayashi and H. Fujioka, “Robust stability analysis for interference has been proposed for ANC system. The validity active noise control systems of ducts with a pair of loudspeak- of the proposed method has been shown by comparing ers,” in Proceedings of the 37th Symposium on Control Theory with the existing bidirectional source and the the Swinbanks’ (SICE ’08), pp. 21–24, 2008. source: three structures of control source have been com- [7] D. S. Bernstein, “What makes some control problems hard?” pared with a simple ducts in open-loop and closed-loop IEEE Control Systems Magazine, vol. 22, no. 4, pp. 8–19, 2002. characteristics based on first principle models and experi- [8] J. Hong and D. S. Bernstein, “Bode integral constraints, mental results. First, the open-loop transfer function for a colocation, and spilover in active noise and vibration control,” generalized control structure unifying the three structures IEEE Transactions on Control Systems Technology, vol. 6, no. 1, pp. 111–120, 1998. has been derived. Secondly, it has been shown that the open- loop transfer functions are consistent with the frequency [9] H. Isaka, K. Nisida, and K. Saito, “Active control of exhaust noise from a muffler in consideration of the location of sec- response experiments in the cases of bidirectional and the ondary source,” Transactions of the Japan Society of Mechanical Swinbanks’ source. Especially, the additional phase-lag in the Engineers. C, vol. 66, no. 645, pp. 1502–1508, 2000 (Japanese). feedback path by the Swinbanks’ source corresponds to twice [10] S. J. Elliott, Signal Processing for Active Control, Signal Process- of the distance between the control source and the open-end. ing and Its Applications, Academic Press, Boston, Mass, USA, In the proposed source, a similar phase characteristic as the Swinbanks’ source has been shown by frequency response [11] T. Chen and B. A. Francis, Optimal Sampled-Data Control experiments. Systems, Springer, New York, NY, USA, 1996. Finally, effects on control performances of control source structures have been examined by control experiments with robust controllers. The smaller amplitude in error microphone signal has been achieved with smaller amplitude in driving signal of control source by the proposed source as compared to the bidirectional source. Although the proposed source does not achieve a much better performance than the Swinbanks’ source, it has an advantage for inexpensive implementation since less number of loudspeakers are necessary than the Swinbanks’ source. There might be a difficult situation to adopt case (c) because of the space limitation on the installation, however, for the situation where the space limitation is mainly on the duct length, the case (c) could be a solution as inexpensive ANC system compared with the case (b) since the distance between the loudspeaker and the upstream junction is the same to the distance between two loudspeakers in case (b). Therefore we conclude that the proposed structure of 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 in long ducts,” Journal of Sound and Vibration, vol. 27, pp. 411– 436, 1973. [2] Y. Kobayashi and H. Fujioka, “Active noise cancellation for ventilation ducts using a pair of loudspeakers by sampleddata H 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 two sound sources,” Transactions of the Japan Society of Mechanical Engineers, vol. 67, no. 656, pp. 52–57, 2001 (Japanese). [4] J. Winkler and S. J. Elliott, “Adaptive control of broadband sound in ducts using a pair of loudspeakers,” Acustica, vol. 81, no. 5, pp. 475–488, 1995. [5] Y. Kobayashi and H. 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