A Hybrid Nonlinear Active Noise Control Method Using Chebyshev Nonlinear Filter

Transcription

1 A Hybrd Nonlnear Actve Noe Control Method Ung Chebyhev Nonlnear Flter Bn Chen,, Shuyue Yu, Yan Gao School of Automaton, Beng Unverty of Pot and elecommuncaton, Beng, 00876, Chna Key Laboratory of Noe and Vbraton Reearch, Inttute of Acoutc, Chnee Academy of Scence, Beng, 0090, Chna Invetgaton nto actve noe control (ANC) technque have been conducted wth the am of effectve control of the low-frequency noe. In practce, however, the performance of currently avalable ANC ytem degrade due to the effect of nonlnearty n the prmary and econdary path, prmary noe and louder peaker. h paper propoe a hybrd control tructure of nonlnear ANC ytem to control the non-tatonary noe produced by the rotatng machnery on the nonlnear prmary path. A fat veron of enemble emprcal mode decompoton ued to decompoe the non-tatonary prmary noe nto ntrnc mode functon, whch are expanded ung the econd-order Chebyhev nonlnear flter and then ndvdually controlled. he convergence of the nonlnear ANC ytem alo dcued. Smulaton reult demontrate that propoed method outperform the FSLMS and VFXLMS algorthm wth repect to noe reducton and convergence rate. Keyword: Nonlnear actve noe control, Chebyhev nonlnear flter, non-tatonary noe, enemble emprcal mode decompoton. Introducton Actve noe control (ANC) baed on the prncple of detructve nterference by generatng an ant-noe wth the ame ampltude but nvere phae of the noe. he lnear fnte mpule repone (FIR) flter wth the fltered-x leat mean quare (FxLMS) algorthm wdely ued for conventonal ANC ytem. However, dffculte of the nonlnearte are often encountered n the prmary and econdary path, prmary noe or the louder peaker ytem. A a reult, the performance of the ANC ytem degrade wth repect to convergence rate and noe reducton. he adaptve flter, uch a Volterra flter -6, functonal lnk artfcal neural network (FLANN) 7-, and even mrror Fourer nonlnear (EMFN) flter 3-4, have been degned and mplemented n the ANC ytem. an and ang degned a Volterra fltered-x LMS (VFXLMS) algorthm for the non-mnmum phae of the econdary path, whch acheved better performance compared to the tandard FxLMS. Snce th frt tudy, many computatonally effcent Volterra flter have been preented for nonlnear ANC ytem 3-5. o overcome the mpule noe n the nonlnear ANC ytem, Lu and Zhao 6 developed a Volterra expanon model by mnmzng the lp-norm of the logarthmc cot. he Volterra flter requre a large number of mult-dmenonal coeffcent for accurately modelng nonlnear ytem, whch ha a large computatonal complexty. Only the econd-order Volterra and thrd-order Volterra flter have been uccefully appled n practcal applcaton. o reduce the computatonal cot, Da and Panda 7 propoed a novel fltered- leat mean quare (FSLMS) algorthm ung the FLANN tructure. Some modfed FLANN flter were then developed to mprove the learnng proce, uch a reduced feedback FLANN 8, nonlnear neuro-controller baed FLANN 9, recurve FLANN 0 and convex combnaton of two FLANN flter. Neverthele, the cro-term,.e. product of ample wth dfferent tme delay were not taken nto account n ther work 7-. Scuranza and Carn propoed a generalzed FLANN (GFLANN) by addng approprate cro-term to the FLANN flter. It noted that the ba functon of FLANN and GFLANN flter do not atfy the condton of the well-known Stone-Weertra approxmaton theorem 5. Recently, the EMFN flter ha been propoed and appled to the nonlnear ANC ytem, whch can acheve better convergence rate and lower approxmaton error n preence of the trong nonlnearte 3-4. he Chebyhev nonlnear (CN) flter, deduced by Alberto and Govann 6, a product of the frt knd Chebyhev polynomal expanon of the nput ample. It atfe the requrement of the Stone-Weertra approxmaton theorem, whch utable for any caual, tme-nvarant, fnte-memory, contnuou and nonlnear ytem. h paper decrbe a hybrd control tructure of nonlnear ANC ytem. It ntroduce the CN flter to control the non-tatonary noe produced by rotatng machnery on the nonlnear prmary path. Frt of all, a fat veron of enemble emprcal mode decompoton (EEMD) 7 ued to decompoe the non-tatonary prmary noe nto ntrnc mode functon (IMF). he IMF are ndvdually expanded ung the econd-order CN flter, the weght of whch are updated by the LMS algorthm. It demontrated that the propoed algorthm robut for the non-tatonary noe and nonlnear prmary path. he remanng part of the paper organzed a follow. Secton preent the prncple of propoed algorthm baed on the real-tme EEMD and adaptve CN flter. A et of mulaton have been carred out n Secton 3 to evaluate the performance of the propoed nonlnear ANC ytem. Fnally, concluon are drawn n Secton 4. he Propoed Algorthm. Hybrd Control Structure of Nonlnear ANC Sytem EEMD Noe decompoton Nonlnear prmary path EXPANSION CN flter expanon and weght update LMS Fgure. Schematc block dagram of the propoed algorthm. he bac block dagram of the nonlnear ANC ytem hown n Fgure. It compre two procee, ncludng noe decompoton ung the real-tme EEMD, and the CN flter expanon and weght update. In the fgure, xn ( ) denote the prmary noe; un ( ) denote the output of the adaptve CN flter; en ( ) denote the error gnal; SOUND & VIBRAION/Augut 08

2 Sz () repreent the tranfer functon of the econdary path; Sz ˆ( ) repreent the etmate of the econdary path; C are the decompoed IMF; V are the fltered vector of Y econdary path etmate Sz ˆ( ) through. he prmary noe after pang through the nonlnear prmary path denoted a reference noe dn, () and the output un () after pang through the econdary path referred to a econdary noe dn. ˆ( ). Nonlnear Prmary Path Model Block-orented model, contng of a lnear tme-nvarant (LI) block and a tatc nonlnear block, the mot popular repreentaton of nonlnear ytem. h method reduce the number of coeffcent and the ze of the requred memory. Fgure how three type of Wener, Hammerten and LNL model 3. In the ANC ytem, the LI block expreed by fnte-mpule-repone (FIR) flter, whle the nonlnearty can be repreented by a polynomal wth fxed memoryle nonlnear functon. (a) (b) (c) Fgure. Nonlnear prmary path model: (a) Wener (b) Hammerten (c) LNL. A hown n Fgure (a), the Wener ytem cont of a LI block followed by a tatc nonlnear block, expreed by H z = h x( n ) () = d = f ( z) () xn () the prmary noe; zn ( ) the output of FIR flter; dn () the reference noe; H denote order of the FIR flter; h() n denote FIR flter coeffcent; f () denote a nonlnear functon. he Hammerten ytem, hown n Fgure (b), revere the order of LI and tatc nonlnear block n Wener-baed model, denoted by z = f ( x) (3) H d = h z( n ) (4) = he LNL ytem a ere connecton of the Wener and Hammerten, wth a tatc nonlnearty andwched between two LI block, hown n Fgure (c), gven by H z = h x( n ) (5), = z = f ( z ) (6) H d = h z ( n ) (7), = he nonlnear block a contnuou functon expreed by aylor expanon 8, and the output of whch gven by z = a t,0 x at, x( n ) at, mx( n m) t = t! = t = t= 0 t= 0 tm = 0 tm!( t t)! ( tm tm)! ( at,0 x ) ( at, x( n ) ) ( at, mx( n m) ) t t t t t t t t m m (8) N an order. If, the prmary path nonlnear; otherwe, t lnear. he roller bearng one of the key element n rotatng machnery. he generated noe become much more non-tatonary under varable peed, epecally for tartup and hutdown, whch can be expreed a x = Bq co( qfn q) (9) q f a rotatonal frequency of the haft; Bq and q ampltude and ntal phae of the q-th harmonc. Subttutng Equaton (9) nto the power functon () k t lead to when q=, k= B B x = co( f n ) (0) when q=, k= B 3B x = co( 3 f n 3 ) co( f n ) () 4 4 when q=, k= x = B co( 4 f n 4 ) B co( f n ) B 8 8 () he analytcal expreon are obtaned by I dk, co( ( k ) f n ( k ) ) f k = I k = x = I dk, co( ( k ) f n ( k ) ) dk,0 f k = I = (3) =,,, I, k N ; d k, are ampltude of the harmonc; d k,0 are contant term. Smlarly, when q, analytcal expreon are gven by I ' k ( ) = ' k, co( k, ) ' k,0 = x n d f n d (4) =,,, I', I ' = q k, k N ; ' k, are d are the ampltude of the harmonc; d ' k,0 are contant term. Subttutng Equaton (4) nto Equaton (8) lead to the output gnal gven by I ' z = g co( f n ) (5) k, k, = () n a contant term. Compared to the prmary noe, x ( n ), t found that the frequency component of the output gnal paed through the nonlnear prmary path contan ome hgher order harmonc..3 Noe Decompoton he EEMD a popular method n the analy of the non-tatonary prmary noe. Partton the data ere of prmary noe nto wndow, each ub-ere thu would be proceed by EEMD, equentally, from left to rght. he choce of the wndow length need to atfy the followng two condton: (a) Suffcently long to reult n reaonably tatonary IMF; (b) Short enough to enure a horter delay and fat repone to the prmary noe. Approprate length can be aduted n the experment. he procedure are a follow: () Intalze the number of tral K, the ampltude of the added whte noe, and et the tral number =. he generated whte noe ere SOUND & VIBRAION/Augut 08

3 n added to the prmary noe to obtan the -th tral by x = x n (6) 3 SOUND & VIBRAION/Augut 08 () Decompoe the noe-added gnal nto IMF wth the EMD a x = r (7) =,, =,,..., K, =,,..., ;, denote the -th IMF of the -th tral; r, a redue of the gnal; denote the total number of the decompoed IMF by = fx(log ( N )) (8) W fx() an ntegral functon; N W denote a length of the wndow. (3) Repeat the EMD decompoton K tme wth random whte noe to lead to an enemble of IMF. Fnally, the -th IMF calculated by IMF K =, = (9) K.4 Weght Update o overcome the nonlnearty extng n the prmary path, the IMF are expanded ung the CN flter baed on the Chebyhev polynomal, whch a famly of orthogonal polynomal generated by the followng recurve relaton ( x) = x ( x) ( x) (0) n n n ( x ) the Chebyhev polynomal of order n. For example, n ( x ) =, () x = x, ( x) = x, ( x) = 4x 3 3x, 0 3 x = x x and 5 3 ( x ) = 6 x 0 x 5 x. ( ) For the -th IMF C =[ c, c ( n ),, c ( n N )], the expanded gnal wth a econd-order CN flter gven by Y = y,0, y,, y,, y, Q = [ c ], [ c ( n )],, [ c ( n N )], [ c ], [ c ( n )],, [ c ( n N )], [ c ], [ c ( n )],, [ c ( n N )], [ c ] [ c ( n )],, [ c ( n N )] [ c ( n N )], [ c ] [ c ( n )],, [ c ( n N 3)] [ c ( n N )], [ c ] [ c ( n N )] () Q the length of expanded gnal. he output of -th IMF controller a convoluton of the expanded gnal and mpule repone. he um of all controller taken a an nput of the louder peaker, gven by u = W Y () = the weght vector W ( ),0( ),,( ),,, ( ) n = w n w n w Q n. he error gnal meaured at the error mcrophone a ummaton of reference noe and econdary noe by e = d dˆ (3) wth the econdary noe gven by L Q L dˆ( n ) = ( n ) u ( n ) = ( n ) u ( n l ) = w ( n l) y ( n l) l, l = 0 = = 0 Q L l l = 0 = w ( n ) y ( n l), l = = 0 l = 0 (4) n ( ) an mpule repone of the tranfer functon of the econdary path Sz () S() z = z z z z (5) 3 L 0 3 L z denote a unt delay; L an order of econdary path. Aumed the etmated econdary path Sz ˆ( ) the ame a econdary path Sz (), the error gnal mplfed a Q, = = 0 = e = d w v ( n ) = d W V (6) L v = y ( n l) and V ( n ) ( ), ( ),, ( ) = v n v n v n Q. l l = 0 he cot functon defned by the mean quare error (MSE) of error gnal by ( ) E e n = (7) E an expectaton operator. he gradent of cot functon obtaned by e e = = e = e V W W W (8) Hence, the weght vector updated ung the teepet decent method by W ( n ) = W = W e V (9) a tep-ze..5 Convergence Analy Subttutng Equaton (6) nto Equaton (9), the expectaton of the W n gven by weght vector ( ) W( n ) = W E d V E V V W (30) = Accordng to bac prncple of the EEMD, the IMF are orthogonal. hu Equaton (30) can be mplfed a ( ) W ( n ) = I R W P (3) R = E V V and P = E d() n V () n. When the ANC ytem acheve the teady tate, W ( n ) W. he optmal weght vector become W = lm W = R P (3) n Subtractng W from Equaton (3), the error of the weght vector gven by n ( n ) = I R = A I A = A I A ( 0) (33) ( ) ( ) n = W n W ; R = A A ;,,, Q a dagonal matrx compoed of the egenvalue of he convergence condton of Equaton (33) = dag,,, R. max. hu, the tep-ze atfyng the convergence and tablty of the propoed algorthm gven by 0 (34) max max the maxmum egenvalue. 3 Smulaton and Dcuon In th ecton, ome mulaton are conducted to demontrate the effectvene of the propoed nonlnear ANC tructure, n comparon wth the VFXLMS algorthm and FSLMS algorthm 7. he amplng frequency 3 khz. he memory ze of the FLANN, Volterra and

4 CN flter are choen a N = 0. In the EEMD algorthm, N = 300, K = 50, and the ampltude of the added whte noe et a 0. tme the tandard devaton of prmary gnal. he reference noe at the cancellaton pont generated baed on the followng econd-order polynomal model gven by d = z( n ) 0.8 z( n ) z( n ) 0.75 z z( n 3) (35) H( z) z 0.3z 0.z = (36) he econdary path tranfer functon gven by 3 4 S( z) = z.5z z (37) W 3. Cae he non-tatonary prmary noe generated by ung Equaton (9). It cont of three haft harmonc, wth ampltude and phae beng of B =.0, B =.5, B 3 =., = /6, = /3, 3 = /. he roller bearng experence a run-up and run-down proce wth the peed curve gven by f = [800 00n( 0. t)] 60. Random noe added wth a gnal-to-noe rato of 3.7 db, and the length of prmary noe 0. Fgure 3 plot the mxed gnal wth the haft harmonc and random noe, the SF pectrogram of whch hown n Fgure 4, wth the ordnate beng 0-00 Hz for better dplay. Clearly, three dtnct harmonc ext n the tme-frequency repreentaton, varyng wth tme n term of the approxmate ne. Fgure 3. he mulated gnal: (a) the haft harmonc; (b) the random noe; (c) the mxed gnal. Fgure 5. Spectra of the decompoed IMF. Fgure 5 how the pectra of the decompoed IMF for a computaton wndow of the prmary noe from teraton. Snce the EEMD ha a property of bnary flterng, the frequency band of the IMF narrowed down wth ncreang the decompoton level. he thrd harmonc, econd harmonc and fundamental harmonc component are eparated from the mxed gnal, approxmately locatng at the IMF4, IMF5 and IMF6 wth peak value of 5.73 Hz, 35.6 Hz and 7.58 Hz repectvely. In the decompoton proce, the IMF become more mple and tatonary. Fgure 6 how the error gnal and PSD by ung the FSLMS, -6 VFXLMS and propoed algorthm wth a tep-ze of 4 0. o better dcern the curve behavor, the power pectral denty (PSD) obtaned by averagng over 0 ndependent run and moothed wth a wndow of length equal to 0 ample. A hown n Fgure 6(a), compared to the FSLMS and VFXLMS algorthm, the propoed method exhbt the leat redual error n teady-tate, and provde a fater convergence rate nce the CN flter ha orthogonal ba functon for nput gnal. It can be een from Fgure 6(b), the FSLMS acheve a large reducton of.8 db at 0 Hz near the fundamental harmonc, but neffectve for other frequence of the prmary noe. In comparon, both the VFXLMS and propoed algorthm have a large reducton below 50 Hz n the varaton of the haft harmonc, wth the latter havng the larget reducton from 5-80 Hz. Fgure 7(a) plot the SF pectrogram of the prmary noe pang through the nonlnear prmary path. Clearly, three hgher order harmonc appear n comparon of the prmary noe. Fgure 7(b)-7(d) plot the SF pectrogram of the error gnal canceled by the FSLMS, VFXLMS and propoed algorthm repectvely. he FSLMS effectvely reduce the prmary noe at the fundamental harmonc but neffcent for hgher order harmonc. Compared to the FSLMS, the VFXLMS perform better nce t explot cro-term n the nonlnear expanon wth the econd-order Volterra flter. It obvou that the propoed method effectve at all the haft harmonc and ha the mot reducton compared to the other algorthm for controllng the non-tatonary noe. h ugget that the propoed algorthm capable of compenatng for nonlnear dtorton uch a the harmonc ntroduced by the nonlnear prmary path. Fgure 4. he SF pectrogram of the mxed gnal. SOUND & VIBRAION/Augut 08 4

5 Fgure 6. Smulaton reult for Cae : (a) error gnal; (b) PSD. Fgure 7. he SF pectrogram: (a) reference gnal; (b) FSLMS; (c) VFXLMS; (d) propoed algorthm. 3. Cae he prmary noe wa acqured from a roller bearng under run-up peed at a tet rg, a llutrated n Fgure 8, contng of an AC motor, frequency converter, haft, rollng element bearng and load controller. he haft wa drven by the AC motor, and the rotatng peed wa aduted by the frequency converter. he load wa regulated by governng the compreon of the prng. Fgure 9 how the waveform of the prmary noe wth a length of 0. Fgure 8. Expermental et-up: (a) the tet rg; (b) the rollng element bearng. 5 SOUND & VIBRAION/Augut 08

6 Fgure 9. he waveform of the non-tatonary noe. he waveform and SF pectrogram of the prmary noe hown n Fgure 0. Obvouly, the gnal vare wth tme and ha a charactertc of non-tatonarty, contanng ome dtnct haft harmonc. Moreover, thee harmonc approxmately ncreae n a contant acceleraton wth tme. Fgure 0. he SF pectrogram of the non-tatonary noe. Fgure how the reult of error gnal and PSD by ung the FSLMS, VFXLMS and propoed algorthm wth a tep-ze of he noe reducton of three method n the entre frequency band hown n able. he VFXLMS ha a reducton of 0.03 db for the non-tatonary prmary noe, whle the FSLMS exhbt a lghtly better performance at ome frequence wth a reducton of db. he propoed algorthm can cancel the non-tatonary noe to a greater extent on the nonlnear prmary path, wth more reducton of 3.07 db and db compared to the FSLMS and VFXLMS. Bede, t provde the fatet convergence rate through about 4000 teraton to reach the teady-tate. Fgure. Smulaton reult for Cae : (a) error gnal; (b) PSD. able. Noe reducton of dfferent method (db). Algorthm FSLMS VFXLMS Propoed method Overall frequency band Concluon h paper propoe a new algorthm baed on the adaptve CN flter and EEMD for actve control of the non-tatonary prmary noe on the nonlnear prmary path. It ha been hown that the noe become more table and ealy controlled by ung the real-tme EEMD decompoton. he adaptve CN flter ha been demontrated to accurately model the nonlnear prmary path than the FLANN and Volterra flter. Smulaton have been carred out to how that the propoed algorthm outperform the FSLMS and VFXLMS n noe reducton and convergence rate. Acknowledgement: he author greatly acknowledge the upport of the Natonal Natural Scence Foundaton of Chna under Grant and Reference. Schrmacher, R., Current tatu and future development of ANC ytem, Sound and Vbraton, Vol. 50, No. 9, pp. 6-9, 06.. an L., ang., Adaptve Volterra flter for actve control of nonlnear noe procee, IEEE ranacton on Sgnal Proceng, Vol. 49, No. 8, pp , Zhou, D. Y., Brunner, V. D., Effcent adaptve nonlnear flter for nonlnear actve noe control, IEEE ranacton on Crcut and Sytem, Vol. 54, No. 3, pp , Zhao, H., Zeng, X., Zhang, X., He, Z., L,. et al., Adaptve extended ppelned econd-order Volterra flter for nonlnear actve noe controller, IEEE ranacton on Audo, Speech, and Language Proceng, Vol. 0, No. 4, pp , Zhao, H. Q., Zeng, X. P., He, Z. Y., L,. R., Adaptve RSOV flter ung the FELMS algorthm for nonlnear actve noe control ytem, Mechancal Sytem and Sgnal Proceng, Vol. 34, pp , Lu, L., Zhao, H. Q., Adaptve Volterra flter wth contnuou lp-norm ung a logarthmc cot for nonlnear actve noe control, ournal of Sound and Vbraton, Vol. 364, pp. 4-9, Da, D. P., Panda, G., Actve mtgaton of nonlnear noe procee ung a novel fltered- LMS algorthm, IEEE ranacton Speech and Audo Proceng, Vol., No. 3, pp. 33-3, Zhao, H. Q., Zeng, X. P., Zhang,. S., Adaptve reduced feedback FLNN flter for actve noe control of nonlnear noe procee, Sgnal Proceng, Vol. 90, pp , Zhang, X., Ren, X., Na,., Zhang, B., Huang, H., Adaptve nonlnear neuro-controller wth an ntegrated evaluaton algorthm for nonlnear actve noe ytem, ournal of Sound and Vbraton, Vol. 39, pp , Scuranza, G. L., Carn, A., On the BIBO tablty condton of adaptve recurve FLANN flter wth applcaton to nonlnear actve noe control, IEEE ranacton on Audo, Speech, and Language Proceng, Vol. 0, No., pp , 0.. Zhao, H. Q., Zeng, X. P., He, Z. Y., Yu, S.., Chen, B. D., Improved functonal lnk artfcal neural network va convex combnaton for nonlnear actve noe control, Appled Soft Computng, Vol. 4, pp , 06.. Scuranza, G. L., Carn, A., A generalzed FLANN flter for nonlnear actve noe control, IEEE ranacton on Audo, Speech, and Language Proceng, Vol. 9, No. 8, pp. 4-47, Carn, A., Scuranza, G. L., Even mrror Fourer nonlnear flter, Proceedng of ICASSP 03, Internatonal Conference SOUND & VIBRAION/Augut 08 6

7 on Acoutc, Speech, Sgnal Proceng, Vancouver, Canada, May Patel, V., George, N. V., Partal update even mrror Fourer non-lnear flter for actve noe control, Proceedng of the 3rd European gnal proceng conference (EUSIPCO), Nce, France, Augut Rudn, W., Prncple of Mathematcal Analy, New York: McGraw-Hll, Carn, A., Scuranza, G. L., A tudy about Chebyhev nonlnear flter, Sgnal Proceng, Vol., pp. 4-3, Wu, Z. H., Huang, N. E., Enemble emprcal mode decompoton: a noe-ated data analy method, Advance n adaptve data analy, Vol., pp. -4, Luo, L., Sun,. W., Huang, B. Y., Effcent combnaton of feedforward and feedback tructure for nonlnear narrowband actve noe control, Sgnal Proceng, Vol. 8, pp , 06. he author can be reached at: bnchen@bupt.edu.cn. 7 SOUND & VIBRAION/Augut 08