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Control Design Methods for Active Sound Reduction Systems
Copyright © IFAC Design Methods of Control Systems, Zurich, Switzerl and , 199 1
CONTROL DESIGN METHODS FOR ACTIVE SOUND REDUCTION SYSTEMS N. J. Doelman TND Insitlllc of Applied Physics, P.D. Box 155, 2600 AD Del/t, The Netherlandl' and Laboratory of Acoustics and Seismics, Del/t Uni versity of Technology, Th e Netherlands
Abstract. A re\'iew is gi ve n of the curre nt ada pt i\'c filt e rin g tec hn iq ues for the active ('ontrol of sound . In addition , the Gene ra li zed :\1inimum Varia nce IGMV , co nt rul stra tegy is proposed as a n importa nt exte nsion of th e existing stra tegies. Th e ad a pt i\'e fi lte ring strategy i; id('ntifi ed as a sim plifi ed rea li zati on of G:\1V cont rol wi th feedforward . Th e limita ti ons of the ada pti\"e fil tering strategies a re di,cussed com pa ring t h is tech n ique with ori gi na l G:\1V st rategies. which are shown to perform m uch belle r bot h in theo ry and in si mu la ti on ex periments on measured time :,:c ri es. Fina ll y. the is:,ue of non·m inimum phase proces~es is addressed . It is shown that the 50 call ed met hod of weigh ted mi nimum va ri ance is vc ry att ractive fu r use in active soun d control syste ms. Key words. Acousti c controL Se lf~ tunin g regul a tors; Ad a pti\' c fil te ri ng : Feedforward controL Nonminimum -ph ased systems. T hi s a pproach a rises from phys ical cons ide rations. Th e unwanted or prima ry wave fi eld can be duplica ted in a nti- phase ba sed on th e princ iple of acou,t ica l hologra phy; see Fig. 1. Once a sound wa ve fi eld is k now n at a ce rtain surface it ca n extra polated to a nother sur face or a rea by hologra phi c mea ns; [M a ngiante, 1977]. The sound fi eld is ml'asured by a n array of sensors, e.g. microphones. A co nt roll er in between the se nso rs a nd th e sources calcula tes the req uired signa ls for t he a nt i-sound sources in order to cance l the prima ry wave fi eld . This type of control is typi cally feedforward, as the dist urha nce signa l is assumed to be com pl etely mea sura ble. The nece"a ry num be r a nd geometry of sensors a nd sources de pe nds on th e cha racte ri stics of t he prima ry sound , e.g. th e frequency band , and the acoustic a nd spat ia l features of the e ncl osure. Th e numbe r of acoustic modes of the enelosu re globall y determine the complexity of t h e mul t i-c h a nn e l aco usti c co n t rol sys t e m . Th e spatia l cha racteri st ics of the dominant acoustic modes have a large influe nce on the des ired tra nsducer pos itions. Th e acti ve reduction of sound in e nclos ures of low mod al de nsity is di scussed in [Nelson , 19871. In a duct syste m it is poss ible to reduce low-frequ ency sound employing me rely a single source a nd a single sensor . For low freque ncies only th e pla ne wave mode can propagate through the duct . We now a rrive at the system de picted in Fi g. 2.
INTRODUCTI ON In th e fi eld of noise abate ment th e active reduction of sound has hee n deve loped , whi ch now has becom e feasi ble du e to t h e fas t developments in electronic e ngin ee ring. Acti ve sound reduction is based on t he destructive in te rferpnce of sou nd fie ld s. An un wan ted sound fi eld . ca ll ed the prim a ry sound fie ld . ca n be reduced by gen erating a secon dary soun d fi eld . ,,·hic h is in a nt i-ph ase wit h prima ry fi eld. Th e use of a set of ext ra sources to generate t he second a ry wa ve fi eld ex pl ai ns the ex pressio n ael;,·" ",u,/l1d r~ducl;()n in contrast to pa ssit'e sou nd redud ;ol1 th at utili zes th e a bso rpt ion. refl ecting or diffu sing prope rty of matte r. Active sound reduct ion is very effecti ve for low freque ncies. say be low 1000 H z. whi ch makes it compl eme n ta ry to passive tec hn iq ues. In pract ice acti ve sou nd reduction is mai nly a ppli ed to duct systems le.g. blowers. engi ne exh a usts. gas turbin es and ve ntilati on systems) or enclosu res of low moda l de nsity (aircra ft a nd vehicl e cabins. headphones a nd control rooms ).
11
ADAPTIVE FILTE RI NG
Ph ys ical Background
primary source
In th e seventies th e scientifi c interest in th e active control of sound and vibra tion bega n to grow signifi ca ntl y. Thi s can be ascri bed to t he ra pid developm ents in electroni c engi neering, wh ich e na bl e t he a ppli cation of active means to reduce un wan ted sound or vibrat ions. Many pa pers a nd repor ts on th eory an d a ppli cation of active soun d control ha \'e been publis hed since t he n . Th e prin ci pa l type ofa sou nd control syste m is based on pure feedforward ; see e.g. 1Roure. 198 7 1. primary source
sensors
0) Fi g. 1. Diagram of an acti\'e acoustic holography.
sources
detection sensor
secondary source
error sensor
F ig. 2. Di agra m of a single-cha nn el syste m fo r acti ve sound control. The pl ane wa\'e sound fiel d is ge ne ra ted at the begi nning of th e duct. Th e fi eld is detected by a so-call ed detecti on se nsor near the primary source. The controlle r calcul a tes th e optima l strength for the antisou nd source . whi ch is to cance l th e prima ry sound field. The ma in object ive of t he ('o ntl'ol syste m is to reduce t he sound emitted from the d uct's mouth . As the wave fi eld is pla ne th is can be achieved by minim izing th e sound pressure at a single sensor placed a t the e nd of the duct. Th is so-ca ll ed error sensor can be used to mon itor the residua l wa \'e fi e ld a nd e \'e ntu a ll y to a dju st the controller characteri stics. In thi s way t he syste m becomes adapti ve.
region of
silence
syste m based on
575
Control Strategies Random primary sound. The ea rly methods to determine the ideal controller characteri stics were rather uncomplicated . For the system of Fig. 2 the model is gi\'en in the backward s hift ope rator q - I: ylll~
Bl q - I, C'q - II - - - u'( - r l ' + - -- tll - [, I + zlf! Alq - I I Eiq - I,
with
+ Qn.-\.q-n A
17 b )
Thi s method is more direc t than the one proposed above. The identification of the secondary acoustic process is not straightforward. As can be seen in I 11the control s ignal should be correlated with the primary s ignal. which mean s that the primary and secondary contributions to."1I I can not be se pa rated properly . In IEriksson, 19891 it is s uggested to add low level random noi se based on a Galois seq uence toth e control signal in order to find the required parameters of Band A. For s tep 3 the orders of Hand L have to be determined . As both polynomial s are to represent a fraction of polynomials, it seem s that the optimal H a nd L in (61 can only be approximated by finite order pol ynomial s. The estimation procedure required in s tep 3 will be di scu ssed below.
E (q - l ) = I + e! q ~l + .. + e"t,'q-rl E
C lq - I I ~ go + glq - I + ... + g"c q -"I' The signalsyU I, ult I and xII I are respectively the output s ignal, the control signal and the primary signal meas ured by the detection sensor. The signal z(t I represe nts extraneous noise and is assumed to be pure Gaussian for the mome nt. The measurement of the primary wave field by the detection sensor is contaminated with the secondary sound field , as the sen sor is sensitive to the sound field coming from the secondary source as well. The detection s ignal d(1 I is modelled as: FI - I I dill ~ x i I I + -q-I-1I1 1 - r, I (21
Periodic prima ry so und . In ma ny situa tions the sound generated by th e noisy so urce is essentially periodic. A diesel engine exhaust or a Roots blowe r e.g. apply to thi s case. For this type ofprimary sound the detect ion sensor need not necessa ril y be a coustic. A tachometer with an optical or magnetic ~ ensorob se rving the rotating equipment can de li ve r the fundamental frequency of the periodic sound. In fact , thi s is s ufficie nt information for the controller to generate the appropriate a nti -sound . Th" main advantage of this approach is the fact that th e aco u s ti c feedback to the de tection se nsor is eliminated, as the sensor is not acoustic. Th e system set-up for periodic primary sound is shown in Fig. 3.
A .q- I
F lq - II ~ /(J+fi(l - r , +.
B (t ! ="Autt-r] 1
1 Estimate th e secon dary acoustic process q - r'B /A. 2 Calculate th e filt e red u a nd d s ignal s: ut' and d F 3 E stimat e the controller parameters u s ing 17a I. 4 Apply con trol according to 151 .
( 11
Br q - I I ~ bo + b Iq - I + ... + b" B q - " H
with
f'
Eriksson now s uggest s the foll owing adapti\'e filtering strategy:
with A(q ~J) = I+ a lq - I+.
lI.
+("rq-"F
The poles of the acoustic feedback path from anti-sound source to detection sensor are modelled equivalent to those of the secondary acoustic process q - r'B/A. Thi s is based on the fact that both transfer function s apply to the sa me enclosure. Principally, the denominator polynomial is determined by the damping ratio and the reso nan ce frequency of the acoustic modes of the enclosure, which are the sa me for both proce sses. The primary acoustic process, q - r, C /E, is modelled with another denominator poly nomial, as this does not represent a true transfer function , but a fraction of two transfe r functions describing the dynam ic relation betwee n two sensor s ignal s. Combining (1) and 12 1, the system equati on can be rewritten a s
primary source
secondary source
error sensor
y(tl=[q-r l Blq -II _q - r, - r, C lq - I IF lq - 1I] UIII Al q-I I E lq - I IA(q - 1I +q - r, C (q - I 1dill + zlll
1.'1
E lq - II The most straight and uncomplicated way to determine a control law is neglecting the contribution of zlt 1 and setting ,v(Ii to zero: 1Roure, 1986J. This yields 11°( 11=-
qr,- r'C(q - 1IAlq - 1I B (q - I IE lq - 11_ q r,-r,- r' C (q - 1IFlq - 1I
dill
d(t)
' - - - - -... ~I controller
Fig. 3. Diagram of a control system for periodic sound.
141
The polynomial F bei ng absent in the modifi ed system equation 131 yie lds t he original Eq.1 1 1. The solution for the Hand L polynomials si mplifies accordingly
The causality of thi s control law is plausible , as based on an acoustical analysis of the propagation times it holds that [,= [,+[" The indirect control strategy can be s ummarized as
Successful laboratory experiments based on thi s type of control strategy have been reported in 1Roure , 1986 1and IDoelman, 19871. However, it is s uggested to estimate the acoustic processes off-line. This results in a fixed controller with no capability for tracking changes in the acoustical environment. These changes can be quite substantial due toe.g. a temperature gradient or a change in the duct flow rate. The need for adaptability is a lso recognized by Eriksson [1989J, who proposes an adaptive strategy described below. The controller structure is like an Infinite Impul se Respon se I IIR I filter: ult l ~ - - - I - dil l
Hlq -
yltl = q - r:
~q-r,
1 2 3 4
15 1
j
Alq - ) )
211)
181
!!.~ [lIltJ + L lq - I IdUl j + zltl Alq - II
.91
Again , a fil te red ve rs ion of the detection s ignal , df'u), is required . The strategy for thi s a dapti ve filtering system becomes
I
B (q - I I [ H 'q - I'UIlI+ L'q - I,dll, ... ___
B
~ 1IFIII + L lq-1ldFUI+ zil)
A possible solution for the Hand L polynomi a ls is, see [Eriksson, 19891. . L = q'l - r: CA 161 H -- I - q rl - r-_- r,. CF EB ' EB This solution is intrinsically presen t in •31: ylll
Q
Clearly, the required controller s tructure is nothing but a Finite Impul se Res ponse (FIR I filter . For pe riodic sound the optimal L polynomial is finite in s pite of the description in 181. The minimal number of coefficients in L equals twice the number of harmonic components contained in the pe riodic primary sound. Implementation of solution 181 in the system equation I I I gives; IDoelman, 1991bl:
Estimate the primary , the secondary and the feedback acoustic processes. 2 Apply control according to 141.
Liq - I I
L ~ q'l- r,
H ~ I:
Estimate the secondary acoustic process q -r'B/A. Calculate the filtered signa l df'((I according to (7b ). Estimate the controller param ete rs of L u sing (9), Apply control as in I Ell with H= 1.
Identification of the seconda ry aco us tic process can be accomplished by applying a seq uent ia l regression algorithm to system equation 11 I: see e.g. 1Goodwin, 19841. Problem s with the correlation between ul I Ianddlt Ican be avoided by adding a low level random noise signal to the control s ignal lilt I.
17al
576
Parame ter Es timat ion Algorithm:-:
C.
The algorithms for the estimat io n of th e con troll er pa rameters are derived from the adapti"e filtering theOl·~·. The princip le of this approach compares to the problem of s.,·,;te m idcntifica tion . Th e main differe nce bet\\'een the pruble m of syste m id ent ification an d the estimation of the feedforward co ntruller parameter" is th e presence ofa tran sfe r function in be tw een the filt e r output and the system output. The us ual algurithms for adapti" e filtering need to be modified in order to deal with seco ndar~' aco u ,;tic proce,s. Both in [Eriksson. 1989[ and [Elliot!. 198 7 [ a ver~' simple modification is proposed. The controller characteristics arc presumed to be timeinvariant compared to the timescalc of the impulse res ponse of the secondary acoustic process. Thi s modifica tion makes thl' adaptive filte ring problem di scu ssed in thi s sect ion su it a blc for mere lv the Least Mean Squares , L:\'IS , e,;tima ti on a lgorit hm: ,ee [Doe lm an. 1991a[. The update equation of thi , g rad il' nt-typ<: algorithm reads 9 1I + I I = 911 I + ~.LI
,I!
Iyl I I
911 I =
[i,,1!
I
ill! I · .
i""
primar~'
fi (
F '1I 1=[riFllld II - 11· ."F ,/ _ III.
IBDM+Q Cl
1II 1
IfI'
t 1~)
1171
r,1
l/JI/ + rll=Plq-I'yl/ ~ rl l ~ Qlq - I ' '''1 1
-
r,
I
1191
" In the case of random prim a ry sound th e detected primary signal xII 1 is not available. as the detection signal is contaminated with secondary sound . Th e re lation 12 I ha s to be impl eme nted in Eq.1151, which gives Cl/J ' I / + rl '
IL
= BDM .;. CQ _ q' l - r, - r, GFDM]/l1I I +
E
S
P.I
GADM ,(f' + - -E- - d'l + r l
-
r,1
I ~Ol
The exp ressions I 19 1and 120 I form t he basis for the development of controller estimation algorithm s.
11.11
in which ;U I is white noi se ",ith zero mea n. Thi s syste m model is more general than I l l. as the ex traneous noi se is no longer purely Gaussian. The D polynomi a l is a dded to th e denominator of the extraneous noise mode l in order to handle most acoustic si tuation s. Consider the following auxiliary fun ct ion
P = P" / Pr!
GDAM
N C
Clq I I _ - ----.;1/1 D l q -' IA l q-1 I
D lq - I, = I + d ,q - ' + ... + d ,, /)q - "lJ
with
.\'
1/1= - p,/·,Il - ----;;;-dl/T r l -
As mentioned in sect ion 1\ th e system principle differs for random a nd for periodic primary sou nd . For periodic sound the primary aco ustic process can be modelled with the id entity E=A. whereas the a bsence of acoustic feedback yie lds F=O. Thi s gives for the optimal prediction of th e auxiliar." function:
Clq - l)= I +C'lq - l + "' TCn(,q - "C
with
/I
Rando m and Periodic Prima",' Sound
The Generalized Minimum Variance IG :VIV I control strategy was proposed by Clarke and Gawthrop [ 1975 [. The inclusion offeedforward in the GMV strate~ can be found in e.g. [Allidina. 198 1 [. The strategy will be shown to be capable of cont roll ing a system give n by I
r l - r,11151
The es timation algorithm in step 2 can be selected from the methods like Rec ursive InHJ'uOl t'nta l Variables. Recursive Maximum Like lihood. Extended Lea,t Squares or ordinary Recursive Least Squares. The last algorithm also a ppli es to thi s case, as the optimal (Irl/+T , I eq ual s zero: IGoodwin. 1984 [. In [Allidina. 1985[ the problem of real -time tuning of P a nd Q is a ddressed . This technique may req uire a considerable computational e ffort. which is not favourable in act ive sound control.
Short Description of the G:vIV Control Stra tegy
£iq -I
~
Calculate the aux il ia r.,· function: Eq. 1141 2 Estimate the pa ram ete rs of the prediction of the auxiliary function: Eq. I I;;, 3 Apply control accordin g to 117 1. 4 If neccssar.v adjust polynomial s P a nd Q.
III GENERALIZED :vI I Nl:\Wyl VARIA."CE COi'
A lq - I I
E
,;ound
r'
y( ll = - - - uff - r1 ' + - - - 'tU - r, 1 +
-GDA - -M dll
Unlike. the met hod of adaptive filtering the closed-loop poles do not equa l the open-loop poles. One can c,'e n assign suitable locations to th e closed -loop pol es by tuning P and Q accordingl y. The main object ive of Q. ho\\·e,·er. is to retain the ,;ystem stability. If B is non minimum pha::;e. the pun' minimum \'a ri anct' s trategy, [.>=1 and Q=O. would result in an un stable controller. The weighting on lI1/1 ca n pre"ent the control effort to become exceed ingly large , if it is appropriatel,' chosen. More det ai ls on th is weighting issue will be di,;cu ssed in section V. In gC'ne ra l th e procedurE' for self-tuningGMV control is
It can be concluded that the two a da ptive filt e ring ,trategies for active sound reduction a re uncomplicated. The controller structure is an IIR or a FIR filter and the co ntrolle r parameters arc estimated u sing an uncomplicated a lgo rithm . Thi s makes th radapti" e filtering s trategy easy to implemen t both in ha rd ware and software form . It will be shown, however. that th e simplicity is acqui red at th e cost of a degraded performa nce. In the next sections more a d" anced control strategies and estimators a re proposed to handle a broader range of active sound reduction problems.
GI<, - I I
+
With th ese ideal controller parameters th e closed-loop behaviour will be: GAQ BDM + CQ _ \'1/1 = xll - r~) + .:;/t) 1181 . EIBP -r AQ [ DIBP + AQI'
The assumption. that th e controll e r charactl'ristic, arc onl~' s lowlv time-varying puts a restricti on on the spt-ed of convergence of th e algorithms. It appears that the mod ification or the common L:VIS algorithm for adapti"e filt e rin g to dea l ",ith a tran, fi.' r fun ction in between is not optimal. The correct modification has recent l~' been de rived and can be found in [Doelman. 1991 al.
Blq - I I
T
The optimal control law \\'hich arises from th e object to minimize the "ariance of l/J iT r,' can be found. 1('Iarkc. 1975 [. to be
T
And for the strategy for periodic
IBD.\! (q . I/1'1'
1161
in which th e vector 8~ t l con ta in : :; the parameter I:'~timate~ and the regression vector 01 t I contain:; pa:-:t :-; ig-n nl sampl{':-" For the control s trategy for ra ndom ~o u nd, ~ 7 J, thl'::'t' \'{'ctor:::; art' g-i\'l'1l b~':
clFI! - I//.
P'I
The polynomials.H and .\' a re h';"en by th e Diophantine equation
110 1
i",'1 I]
1/- ri ' = -.\' yl/l ~
IV IDENTIFICATIO:-'; OF THE ADAPTIVE FILTERING STRATEGY In this section it will be demons trated . that the adaptive filtering as extensi" ely di scussed in sec ti on 11. can be derived from the s trate~' of minimum variance. In fact, adaptive filtering is a special case of pure feedforward control. Thi s type of control comes ou t of the G:\lV s trat e~' if the detection signal dl I1 suffices to acquire the optima l prediction of t he aux ili a ry function . In other words, nothing of the contribution of ~(f , to th e a uxiliary function , PCIDA ~I!+T, I. can be predict ed by u sin g yl I I. For the adaptive filtering s tra te~·.
I\ ~ I
Q = Q" / Qd
The object of the G:\1V control s trategy is to minimize the "ariance of the a uxiliary function l/J 1+ r I. Thi s requires prediction of l/J I ~ r l I. The optimal predicti on in least squa res .~en'e ~'ields
577
strategy it i~ rl'quin'd that /) = l.(; = (). J) = 1 and (' =.-\ : tht' (,()lIt rihut ion of ~UI to th~> auxiliary function (J;~ t+r , ) i:-, pun.' whilt' JlOl....:l·. The op timal predicti on of the auxiliar.\· fUJld ion ill thi:-, ~it uati(lIl hecollll':-:
.
rB
(/J - (t + r l):::- ! ~-(I
., . . . • GF '
LA
. \ L'
(;
U!tl-~E,dlt _ :-
dampt·d (I\·eJ'-c.tll poll':' dOl':-' nllt ha\'l' a hi gh priority . A....: till' ....:t.'t-point l 'q Uti I.. .: zero in t hi....: ca..;l'. l11l'rel.\· in t i nH'-\' ar~'i ng' :-'.\" ....:t t'm....: 1'<."1 ....:1 tracking i;-; J'equirt.'d. Tlw tuning of q C;lll han' a ....:l' ri()u ~ influl~ n('l' on th e rl·....:iciual \'ari a n ct' of thl>....:y....:tl'll1 out put. which 1()l\ow:-, from (2-1 I .
- I"~
-u.:.
whi ch pe rfect ly' compan':, to 1:31, :,incl' now it huld:-: that (/). /1:::-.\' 11 '. It s hould be realized that in all ea""" ill which t h e adapu\l' filtl'ring strategy as a control ~t r a t egy i~ a ppl ied. t lw co nd it iOIl:' a:-. ll11' nt ionl,d abo\'(~ do not nl'ct'~sa ri l y hold. It I:::' ju:'t a mattt'r of ignoring th e characterist ics of thE' ext ran c()u:, noi:-:l.' . In a fl'w ca....:!':, ufact i \"l' ....:()U Ild control the extrHneou~ noise contributiun i....: ....:uch in....:ignificant. that it v·:ouId be undue to account for it. Tht' C()[b t.'qU(,Ill"t':' of thl ' l'onlrul strategy simplification in :; i tuati on~ with <:1 con~i d l'ra hll' am()unt of undetectahle nOi5(~ can bc' fou nd in tIll' ....:t·niu[l \'1.
Tht' wC'ighted minimum \'ari i:l IlCl' control ....:tra tl'gy I:=; a . . : pccia i form of minirnum \"arianl'l' cuntrol for Ilon -minimum pha....:l' ~y~tem~. E\'en !(1I ' Q=O th e ideal =-,y::'tl'Ill rl'nwin::: :,tahlt., li)J' a non-minimum pha se pulynomial H. The choice f(Jr}) ....:l·{'lll....: ob\'iou::: whcn l'xamining (24 1:
Proceeding with ( 21 10n(' a!Ti\"(,'~ at the cO lltrul procedure:-: of adapt in' filtering for both random a nd peri odic primar.\· :,ollnd a ddr(> ....:~t.·d in sec t ion II. The popu lar filtered r and filt e red 11 L'.I S ,dgOJ"ithlll" fil l' acti\'e sound control e m e rg(~ furm thl! C.\I \' control ~tn l tl'gy under t h e fo ll owing restriction::::
The syste m can be modelled b.\· an uutput ('ITor mudt·1 structure: the extrant'oll~ Iloi:-;(' i ~ Pllrply \\·h itl' noi....:l.' 2 Th e weighting I::; likt.· minimum \"(l r i;lIlCt· : P::::.l (,::::(). 3 For both th e random and tht., peri{)oic primary :::ulIlld ca;-;t· t Ill' controller polynomi a l;-; h:l\"l' to uht'.\· ....:p(·c ial !()I'm....:: ....:l't.' (b I and (8 I. Thi s results in a neCl':-\::--ary idl'lltilkat iun ofthl' ~(' c()ndary aco us tic proccss. 4 The controller p a ram e t e r ~ a re e.-.;tim .'l.tl>d ll:-,ing a ....:lowly converbring modification of thl' L~ I S nlgorithm
\ ' 1 ~1'. l l · LAT I ()"~
Thl' ....:illlulatiun experim ent...; an' to (,luc!ci;lIt· t hi.' i'l'
These fo ur requireme nt s for adapti\'e filtering with filterl'd l.'.IS a lgorithms cou ld ex hibit th0 Ilon -u~a hility ()fthi~ :-; trtlt l'gy I {OW( '\·ll!'. in many publication s. e.g, 1Elliott. 19,,71 and 1Erib,;on. 191'i9 1 satisfying resul ts \ .... ith the ~ trat eg.\· art' cl
(Jft'tJllt
con trol strategy
rol....:t r ~ltl·gil'.-.; ill ....:illlulatioI1 ex periments. estimation a lgorithm
auxiliary function
Adaptive Fi ltering 1
eq (21 )
Adaptive Filtering 2
eq, (21 )
Feedback Control
e q ,( 15 ) with L = 0
GMV
Control
eq, (15)
LMS. Eq , (10) RLS RLS RLS
~()l(' that the feedback ....:trategy i~ a G:\'lV :-;tratl:'gy. The required identificatiun of thl' :-;ccon dary acou:-:t ic prOCl'S~ fo r the adaptive filt('rin g~ tr a t ebrjel" w(lscarriedout hefol'l'il and. ~lon.'u\·l'r. th e optimal \'alul' of the st ep-size parameter.iI \\'H ::; cho~(>n f()rthc L:\'l S a lgorithm . Th (' term "res" stand....: for the :-;ljuart' of the rl'~ idu a l output signa l ~'1 o r(' ::,iOlulation experiment:i on acti\'C' sound contro l can be found in IDoelman. 1991b, 199 1c I,
Use of The We ighting Polynomiab in Ada pti\'l' Filterin " Althoug h the adaptive filtering s tr ategy' ha s bee n deri\'ed fi))· pun' minimum variance, it wil l turn out that P can be u..;cd to impro\'!: it:-; performance, A u sefu l setting is P=A a nd q=O, A",-uming that thl' degree of A is s mall e r than th e d ea d time r i • the predictio n of t h, ' aux il iary function 12 11 gi \'es
Simu la t ion 1. In thi....: ex perimellt th (' time ;=,;('r i l~:' an' measured data
yil'ld~
in Cl \'C: l1tilati on duct. The rl'mainingsl'conchu.\· and feedbuc k acoustic proc(':-;....: arc modelled <-1:-;:
For periodic prim ar:,': sound thi s wc ig hti ng
B[ 1/ 1/1 + Lel' / I [
A"I
Thi s weighting is es pecial ly effec tiYl' {(H' gradil'llt typC' l·.-.;timatioll a lgorithms. as demonstrated in IDoelman. 1991 a l, The elimination of the A poly nomi al from th e filte rin g of" and 11 re,sul t" in a i'<"ter rate of converge nce of th e controller parameter" i n the L\IS-t\'pl' algorithms. This is \'c rified in the s imulation C'xpL' r inH:'nt;..;.
V
l~_ 1'"
, 26 l W 1'" in which B contain:' all un:-:tahlt, zero."; urn and th e til dc - s ta nds for t aking thl' recip rucal polYlloJl1ii.ll. ~ I url' dl'taib ofthi....: tt'chnique can be filLllld in IGrimbl ..... 19K1l. Appli cation ol"tl1l:-' weighting require :' a factorizat ion of the H po i.nlomial. which can bl' tim e-co ns uming. Th t' rd ~lI·l' . it i....: ~uggl'~tl·d to inclucil' tiH' complt:tv /) pulynomial in the jJ \\cighl ing. Rl·::-iult.....: o{'thi.....: control ~t r atq.~.\· can a l",o ht' ti) und in the ::,im ulation l'x perimenb . p =
POLE
roc
The expression for the closed-loop beha\'iour of the ideal G'.IY controll e r re\'ea ls the important role of the weighting polynomiak vlll=
GAQ
X'I _ [,, + BD." -CQ~'l l
Ol
ci5 S a. S
12-1 ,
o
E[BP + AQJ D[BP-AQ[' Th e main u se of P and Q is to ensure closed-loop " t abilit.\ In mo,;t cases thi s means. that ifB is non -minimum phase , Q is usualh tuned in order to retain the closed-loop pol es inside th e unit circle: "''', [Goodwin, 19841, As well-known the tuning of Q has to be accurate to avoid a stead y-sta te error. Inclu s ion of an integrator in Q ll SlI Cl:Il.\· soh'es thi s problem. When ,\'11 l is periodic it is \'ery useful to include in Q a polynomlal Q I \\'ith t h e follo\\'ing feature: Ql lq - I IXlll = O
c:c
:~,o ,2
o
400
800 1200 Sample Instant
1600
2000
0
400
800
1600
2000
2
Ol
if!
S
Ef
0
,2 -' ,
QI
e qual s 1-2co51 Ii,T,C! ' 1+ q -:. ill which T i, tlll' sampling instant, :vIore accurate tunin g of P and q to obtain \\'l'IIIfxlt I is cosl lii I e,g ..
,= I
The ,am pl e frequency i, 1600 Hz, Th e re,;u lt ing ,;ou lld le l'e ls are " h o\l'n in Fig. 4, D ue to a partl y unprediClablc primary s ound. the adap ti\'e filtering controll e r ,; Iea\'l' a r elati\'ely high rbidual leve l. Both G:\I\' "tr"teh~e,; dea l \\'Ith the priman' ""un d much better,
A."m ZERO ASSIG~:\IE:\T 1:\ ,,'.1\ ' CO:\TROL
.
I
2
1200
Sar'1p:e Instant
F ig.4 COnl'd ..
578
<0
2
Adaptive Filtering 2
co
'"
if!
"5 0. "5 0
0 ·2
"5 0. "5 0
400
-n
'"
"5 0. "5 0
1200 800 Sample Instant
2 0 ·2
"5 0. "5 0
res = 0.012 400
1600
GMV Control
-n
res
=
400
1200 800 Sample tnstant
1600
2000
.-1 1'1
x 1/
100
200
300 400 500 600 Sampling Instant
700
800
900
:2 \\-it h
indit' ~ ltt, ~ thl' .... tart
fi1~t t iTll l··\·a r~· il1g di:,turhFlI1C'l'
of COl1tro l. tl)
c!P !ll () !1 ....;t r ate the U:-'l'
I, ~
1_11 . .'il 1 -
O' \ )~(I
I
- 1 1_
I ... I
n'l
I
fl
= i
f'
= I1
f) , 'I 1 1::: t
-l
1 1;
H I'I
1,'1 'I
= CO:-; 12m
~
1 1
""'£\ ((1
1
\"ar 1';1/ I1 = 11.1
I
Th l' model li)J" lht' :-,(,t'tH1dar\ atllll:-,t i t' prot'P:-;:-; indude:-; hoth Cl nonminimum pha:-'l' zero and p()orl\-d a mpe d po]e:-, " Tht, l'Un\'ergl'Ilt'1:' :-:J"H.,t'd oft hl' adapt i \ "t'
fi I tl'ri ng:-'t rateg-il':-' :-'lI rfl'r:-; from t h(, n '\"('rlwrant
chnra ct (' ri :-; tic :-;, Whl'rl'H:-; thl' (;\1\' ;.;trat eg-i p:" t'annul appl.\" pun'
minimum '"<:lrialH'p ("ont rol. The wl'ighting ill "Ad aptive Fi ltering 1A"
i" P= I and in ·'Adaptive Filtering 1 S·· P =.-I . For tilt' G:\I\' cont roller,: it hol d" tha t
<0
2 0-.----------------------------------~
~
co
u;'" "5 0. "5
4
o
o-
Adaplrve Filtering 1A
~~'I~WN ~\, ~ViV.V-~~.~~v~~~~ n
.20
0
res
=
2 48
o
200
400 600 Sampling Insta nt
800
1000
o
200
400 600 Sampling Instanl
800
1000
0
200
400 600 Sampling InSlant
800
tOOO
0
200
400 600 Sampling In slant
800
1000
0
200
400 600 Sampling In slant
800
1000
m co
·4 0
400
800 1200 Sampling Instant
'"
1600
if!
"5 0. "5
4
o
<0 co
0 <0
·4
co
0
400
800 1200 Sampling Instant
u;'"
1600
"5 0. "5
o
4
GMV Control
<0 co
"5 0. "5 0
0
0
(' 1'1 1 1 ;;; !
co
u;'"
900
t'harat'll'ri:-,lit':' a rt' Ilwdl,lll'd a:-,
<0
"5 a. "5 0
800
ofthl\ \\"pighting polyno miaL·.; in t Ilt' cont rol :-ill'<1tegil':-: . Th e :',\':-,ll'm
Btq - II = I
'"
700
Si m u l tit io/\ :L Thi :-: :-:imulati ol1 l'xpl' rinwll t i:-:
0.0015
The results in fi g. 5 demon strate that the de tection signal does not completely determine the prima ry sound in s ide the cabin. Th e G!\1V s trategy can handle t he unknown pa rt quite well.
if!
300 400 500 600 Sampling Instant
'1'I'T
Simulation 2. For thi s s imulation th e detection and prim a ry s ignal were measured in a Swed is h passe nger ca r . Th e sampling frequt'ncy equals 500 Hz . Though in s id e thi s \·ehide a multi ·channe l control system is required, th e s imulation demonstra tes the proble m,; with controlling interior ca r sound. The u su a l way to de a l with the (insta ntaneou sl harmonic sound is u sing a s:.ste m fQr pe r·i od ic ,,,u nd control as s hown in Fi g. 3. Th e secondary path is model led a,;
'"
200
Fig.5 . Rl':-:u lt of :-;illltllat ion
C I'I
"5 0. "5 0
100
:,ignal: an arrow
Fi g. 4. Results of si mul ation 1: an arrow indicate, the , ta rt o[co nt rol.
if!
0
·2
2000
,~
0
>-
·2
2
'"
6> if!
0 ·f1Il ·2
2
0
2000
Feedback Control
co
if!
1600
1
0
<0
800 1200 Sample Instant
2
co
.Q> (j)
g,
"5 0. "5 0
0
<0
C'
if!
0 <0
res = 0.136
·4 0
400
1200 800 Sampling InSlant
20
co
'"
if!
1600
"5 0. "5 0
0 ·20
fig . 5. Results of s imulati on 2 ; an arrow indicates th e start of co ntrol. 20
'" g,
The feedback s trateg:. was also tested on data measured during acceleration of th e car. In thi s case t he det ection s ignal could not he utilized by any feedforwa rd stra tegy. Th e result is s hown in f ig. 6. Again , the feedback con trolle r achie\'es a good reducti on .
if!
"5 0. "5 0
0 ·20
F ig"l. Rt' :.:uI t:o:: of:.:imu \cHinl1:3: an a rrow i ndic3tl':-: t h e :-, t<1rt of control.
579
GMV Control A: P =
HIB
Q =O
VIII AC'K.,'OWLEDGD·IENTS
GMVControIB: P =0. :i - O. 7q - 1 +0 2:iq - C -O.25q - '
T hese im'estigation in th e program of th e Fund a mental Research on :\Ia tte r I FO~I I hm'c been supported by the ;-';etherlands Technology Foundation I STW I.
Q = 0.:; + O :iq - C
As ca n be concluded from Fig . 7, theA-weightingonyll I impro\'es the pe rformance of th e adapti\'e filte ring st rategy. Al so. the weighted minimum variance st rategy performs bettcr than the G:vIV stra tegy with P an d Q weighting. :\Ioreo\·er. th e cho ice for P a nd Q in thi s s trategy is not easy to make.
IX
REFERENCES
Allidina: A.Y.. F .:\1 . Hughes and C. Tye 119811. Self-tuning control for systems employin g feedforward. lEE Prof. D , 128, pp . 283 29 l. Allidina. AY .. F.:\1 . Hughes an d T.Tahmassebi 119861. An implicit self-tuning technique with va riable time-delay. Int. J. Con trol , !H. pp . 1437-1-157. Clarke. D.W. and BA Gaw throp 119751. Self-tuning controller. lEE Proc.D, 122. pp . 929-934. Doelman. N.J .1 19871. Adapti\'e active a tte nu at ion of broad band so und . Master the.,i" Delli Unice rsity a/ Technology. Doelman. N.J .1 199 1a I. A nove l adapti \'e filt er LMS algorithm for' use in thcacti ve con t rol of'sound . S ubm itted to IEEE Tran sactions on Sigllal Processill/!. Doe lma n. N.J. 1199 1bl. A unified control stra tegy for the active reduction of sound . In Recell t Adl'On ces in Actice Control ofSollnd alld Vibratioll. Technomic Publi shing. La ncaster. Doelma n. N.J. and E.J.J. Doppenberg 11991cl. The application of se lf-tuning control stra tegies to th e act ive reduction of sound. lEE COII/erence Con trul9 1. Elliott , S.J., I.M . Stoth ers and PA Nelson 119871. A multiple e rror LMS a lgorithm a nd its a pplication to th e active control of sound a nd vibration. IEEE Tran sactiolls all ASSP, aQ, pp. 1423- 1434. Eriksson, L.J. and :VI. C. Allie 119891. Use of ra ndom noise for on-line t ra nsduce r modeling in a n ada ptive active attenuation syste m .J. Acollst. Suc. Am .. 1:lli. pp. 797-802. Goodwin. G.C. a nd K.S. Sin 119841. Adapticc Filtering, Prediction and COlltrvl. Prentice-Ha ll. Gri mble, M.J . 1198 11. A control weighted minimum-variance contro ll er for non-minimum phase systems. l ilt. J . Control, aa, pp. 75 1-762. Mangi a nte, G.A. 119771. Active sound absorption . J . Acollst. Soc. Am .. fit pp 1516-1523. Neely, S.T. a nd J.B . Ali e n 119791. Invertibility of room impul se res ponse . J. Acollst. S oc. Am ., 22. pp . 165- 169. Nelson , P.A and co-work ers 119871. the active minimization of harmonic e nclosed sound fields , part I, 11 a nd Ill. J ournal of SOl/nd alld Vibratioll. ill, pp. 1-58. Roure. A 119861. Self-a dapt ive broadband active sound control system . J ournal ofSol/nd and Vibration. 101 , pp . 429-44l.
VII CONC LUSIONS [n thi s pa per ada pti\'e tech n iques for the act ive con trol of sound h a\'c been disc ussed. Th" accustomed method of ada ptive filt e ring h as been presented a nd a na lyzed. ln addition, the st ra tegy of Generalized :v1inimum Varian ce IGl\IVI co ntrol has been proposed as a useful extension of the ex istin g acti\'e sound con trol strategies. [n fact. it can be derived that ada pti\'e filt e ring is a very special case of the GMV control strategy. The require ments for adapti ve filte ring to a ri se form the G:vIV strategy a re: The contributi on of ext ra neous noi se to the system output s ignal is jus t white noise. 2 Th e remaining part of the di stu rba nce signal at th e output can be completely predicted by e mpl oying the available detection signal. 3 Pure minimum \'a riance control is a pplied: P= 1 Q=O: t he setpoint equals zero for a ll t. In active sound control adapt ive filterin g is widely used. Th e main stra tegies as described by IElliott, 1987: Eriksson , 19891 corres pond to the followin g: 4 The controll e r polynomial s a re such that a s pecific IIR I random sound 1or a FIR I pe ri odic sound 1filter structure results . Thi s requires an addit iona l estimation procedure for the seconda ry acoust ic process . 5 The controller parameters are estimated using a non-ideal modifi cation of the LMS al gorithm. The fi ve require ments for th e GMV cont rol strategy to correspond to the cu rre nt methods of active sound co ntrol cl early revea l the limita ti ons of these me thods. In most cases of active sound control the extraneous noi se contribution is coloured le. g. a ventilation syste m or a ca r in teriOJ·1. A feedback control loop should be added to the system then . Furth ermore, it has been demon strated th a t 4 an d 5 ca n seriously degrade the performance as well. In IDoelm an, 1991b l improvements of controll er structure (4 1 and gradient type estima t or 15 1are suggested. From both theoretical derivations and the simul a tion experiments th e conclusion can be dra wn that th e GMV stra tegy can ha ndle the compli cated sound control problem s much more effecti vely. It is sh own to be a very attractive basic stra tegy to de ri ve special controll e rs from . Th e mai n draw back of GM V control compared with ada pti ve filte ring is the necessary tuning of the weighting polynomial s in the case of a non-minimum phase process. Non-minimum phase acoustic processes a re lik ely tooccur as can be found in INee ly, 1979 1. In simul a tions it has been shown that the weighted minimum variance strategy is the most attractive way to tune the weighting polynomial s. The time consuming factorization procedure can be avoided by using th e wh ole B poly nomial instead of B- . Nonminimum phase processes diminish the computational adv antage direct self-tuning stra tegies have above e.g. the indirect or the adaptive filtering meth ods. Periodically a Diophantine equati on h as to be solved IAllidin a. 19851 or the secondary acoustic process h as to be identified. Fina lly, it has bee n shown th at the performance of th e adaptive filt e ring strategy can be impro\'ed by an a ppropriate system output weighting. This makes the strategy attractive for the more simple sound control probl ems.
580
CONTROL DESIGN METHODS FOR ACTIVE SOUND REDUCTION SYSTEMS N. J. Doelman TND Insitlllc of Applied Physics, P.D. Box 155, 2600 AD Del/t, The Netherlandl' and Laboratory of Acoustics and Seismics, Del/t Uni versity of Technology, Th e Netherlands
Abstract. A re\'iew is gi ve n of the curre nt ada pt i\'c filt e rin g tec hn iq ues for the active ('ontrol of sound . In addition , the Gene ra li zed :\1inimum Varia nce IGMV , co nt rul stra tegy is proposed as a n importa nt exte nsion of th e existing stra tegies. Th e ad a pt i\'e fi lte ring strategy i; id('ntifi ed as a sim plifi ed rea li zati on of G:\1V cont rol wi th feedforward . Th e limita ti ons of the ada pti\"e fil tering strategies a re di,cussed com pa ring t h is tech n ique with ori gi na l G:\1V st rategies. which are shown to perform m uch belle r bot h in theo ry and in si mu la ti on ex periments on measured time :,:c ri es. Fina ll y. the is:,ue of non·m inimum phase proces~es is addressed . It is shown that the 50 call ed met hod of weigh ted mi nimum va ri ance is vc ry att ractive fu r use in active soun d control syste ms. Key words. Acousti c controL Se lf~ tunin g regul a tors; Ad a pti\' c fil te ri ng : Feedforward controL Nonminimum -ph ased systems. T hi s a pproach a rises from phys ical cons ide rations. Th e unwanted or prima ry wave fi eld can be duplica ted in a nti- phase ba sed on th e princ iple of acou,t ica l hologra phy; see Fig. 1. Once a sound wa ve fi eld is k now n at a ce rtain surface it ca n extra polated to a nother sur face or a rea by hologra phi c mea ns; [M a ngiante, 1977]. The sound fi eld is ml'asured by a n array of sensors, e.g. microphones. A co nt roll er in between the se nso rs a nd th e sources calcula tes the req uired signa ls for t he a nt i-sound sources in order to cance l the prima ry wave fi eld . This type of control is typi cally feedforward, as the dist urha nce signa l is assumed to be com pl etely mea sura ble. The nece"a ry num be r a nd geometry of sensors a nd sources de pe nds on th e cha racte ri stics of t he prima ry sound , e.g. th e frequency band , and the acoustic a nd spat ia l features of the e ncl osure. Th e numbe r of acoustic modes of the enelosu re globall y determine the complexity of t h e mul t i-c h a nn e l aco usti c co n t rol sys t e m . Th e spatia l cha racteri st ics of the dominant acoustic modes have a large influe nce on the des ired tra nsducer pos itions. Th e acti ve reduction of sound in e nclos ures of low mod al de nsity is di scussed in [Nelson , 19871. In a duct syste m it is poss ible to reduce low-frequ ency sound employing me rely a single source a nd a single sensor . For low freque ncies only th e pla ne wave mode can propagate through the duct . We now a rrive at the system de picted in Fi g. 2.
INTRODUCTI ON In th e fi eld of noise abate ment th e active reduction of sound has hee n deve loped , whi ch now has becom e feasi ble du e to t h e fas t developments in electronic e ngin ee ring. Acti ve sound reduction is based on t he destructive in te rferpnce of sou nd fie ld s. An un wan ted sound fi eld . ca ll ed the prim a ry sound fie ld . ca n be reduced by gen erating a secon dary soun d fi eld . ,,·hic h is in a nt i-ph ase wit h prima ry fi eld. Th e use of a set of ext ra sources to generate t he second a ry wa ve fi eld ex pl ai ns the ex pressio n ael;,·" ",u,/l1d r~ducl;()n in contrast to pa ssit'e sou nd redud ;ol1 th at utili zes th e a bso rpt ion. refl ecting or diffu sing prope rty of matte r. Active sound reduct ion is very effecti ve for low freque ncies. say be low 1000 H z. whi ch makes it compl eme n ta ry to passive tec hn iq ues. In pract ice acti ve sou nd reduction is mai nly a ppli ed to duct systems le.g. blowers. engi ne exh a usts. gas turbin es and ve ntilati on systems) or enclosu res of low moda l de nsity (aircra ft a nd vehicl e cabins. headphones a nd control rooms ).
11
ADAPTIVE FILTE RI NG
Ph ys ical Background
primary source
In th e seventies th e scientifi c interest in th e active control of sound and vibra tion bega n to grow signifi ca ntl y. Thi s can be ascri bed to t he ra pid developm ents in electroni c engi neering, wh ich e na bl e t he a ppli cation of active means to reduce un wan ted sound or vibrat ions. Many pa pers a nd repor ts on th eory an d a ppli cation of active soun d control ha \'e been publis hed since t he n . Th e prin ci pa l type ofa sou nd control syste m is based on pure feedforward ; see e.g. 1Roure. 198 7 1. primary source
sensors
0) Fi g. 1. Diagram of an acti\'e acoustic holography.
sources
detection sensor
secondary source
error sensor
F ig. 2. Di agra m of a single-cha nn el syste m fo r acti ve sound control. The pl ane wa\'e sound fiel d is ge ne ra ted at the begi nning of th e duct. Th e fi eld is detected by a so-call ed detecti on se nsor near the primary source. The controlle r calcul a tes th e optima l strength for the antisou nd source . whi ch is to cance l th e prima ry sound field. The ma in object ive of t he ('o ntl'ol syste m is to reduce t he sound emitted from the d uct's mouth . As the wave fi eld is pla ne th is can be achieved by minim izing th e sound pressure at a single sensor placed a t the e nd of the duct. Th is so-ca ll ed error sensor can be used to mon itor the residua l wa \'e fi e ld a nd e \'e ntu a ll y to a dju st the controller characteri stics. In thi s way t he syste m becomes adapti ve.
region of
silence
syste m based on
575
Control Strategies Random primary sound. The ea rly methods to determine the ideal controller characteri stics were rather uncomplicated . For the system of Fig. 2 the model is gi\'en in the backward s hift ope rator q - I: ylll~
Bl q - I, C'q - II - - - u'( - r l ' + - -- tll - [, I + zlf! Alq - I I Eiq - I,
with
+ Qn.-\.q-n A
17 b )
Thi s method is more direc t than the one proposed above. The identification of the secondary acoustic process is not straightforward. As can be seen in I 11the control s ignal should be correlated with the primary s ignal. which mean s that the primary and secondary contributions to."1I I can not be se pa rated properly . In IEriksson, 19891 it is s uggested to add low level random noi se based on a Galois seq uence toth e control signal in order to find the required parameters of Band A. For s tep 3 the orders of Hand L have to be determined . As both polynomial s are to represent a fraction of polynomials, it seem s that the optimal H a nd L in (61 can only be approximated by finite order pol ynomial s. The estimation procedure required in s tep 3 will be di scu ssed below.
E (q - l ) = I + e! q ~l + .. + e"t,'q-rl E
C lq - I I ~ go + glq - I + ... + g"c q -"I' The signalsyU I, ult I and xII I are respectively the output s ignal, the control signal and the primary signal meas ured by the detection sensor. The signal z(t I represe nts extraneous noise and is assumed to be pure Gaussian for the mome nt. The measurement of the primary wave field by the detection sensor is contaminated with the secondary sound field , as the sen sor is sensitive to the sound field coming from the secondary source as well. The detection s ignal d(1 I is modelled as: FI - I I dill ~ x i I I + -q-I-1I1 1 - r, I (21
Periodic prima ry so und . In ma ny situa tions the sound generated by th e noisy so urce is essentially periodic. A diesel engine exhaust or a Roots blowe r e.g. apply to thi s case. For this type ofprimary sound the detect ion sensor need not necessa ril y be a coustic. A tachometer with an optical or magnetic ~ ensorob se rving the rotating equipment can de li ve r the fundamental frequency of the periodic sound. In fact , thi s is s ufficie nt information for the controller to generate the appropriate a nti -sound . Th" main advantage of this approach is the fact that th e aco u s ti c feedback to the de tection se nsor is eliminated, as the sensor is not acoustic. Th e system set-up for periodic primary sound is shown in Fig. 3.
A .q- I
F lq - II ~ /(J+fi(l - r , +.
B (t ! ="Autt-r] 1
1 Estimate th e secon dary acoustic process q - r'B /A. 2 Calculate th e filt e red u a nd d s ignal s: ut' and d F 3 E stimat e the controller parameters u s ing 17a I. 4 Apply con trol according to 151 .
( 11
Br q - I I ~ bo + b Iq - I + ... + b" B q - " H
with
f'
Eriksson now s uggest s the foll owing adapti\'e filtering strategy:
with A(q ~J) = I+ a lq - I+.
lI.
+("rq-"F
The poles of the acoustic feedback path from anti-sound source to detection sensor are modelled equivalent to those of the secondary acoustic process q - r'B/A. Thi s is based on the fact that both transfer function s apply to the sa me enclosure. Principally, the denominator polynomial is determined by the damping ratio and the reso nan ce frequency of the acoustic modes of the enclosure, which are the sa me for both proce sses. The primary acoustic process, q - r, C /E, is modelled with another denominator poly nomial, as this does not represent a true transfer function , but a fraction of two transfe r functions describing the dynam ic relation betwee n two sensor s ignal s. Combining (1) and 12 1, the system equati on can be rewritten a s
primary source
secondary source
error sensor
y(tl=[q-r l Blq -II _q - r, - r, C lq - I IF lq - 1I] UIII Al q-I I E lq - I IA(q - 1I +q - r, C (q - I 1dill + zlll
1.'1
E lq - II The most straight and uncomplicated way to determine a control law is neglecting the contribution of zlt 1 and setting ,v(Ii to zero: 1Roure, 1986J. This yields 11°( 11=-
qr,- r'C(q - 1IAlq - 1I B (q - I IE lq - 11_ q r,-r,- r' C (q - 1IFlq - 1I
dill
d(t)
' - - - - -... ~I controller
Fig. 3. Diagram of a control system for periodic sound.
141
The polynomial F bei ng absent in the modifi ed system equation 131 yie lds t he original Eq.1 1 1. The solution for the Hand L polynomials si mplifies accordingly
The causality of thi s control law is plausible , as based on an acoustical analysis of the propagation times it holds that [,= [,+[" The indirect control strategy can be s ummarized as
Successful laboratory experiments based on thi s type of control strategy have been reported in 1Roure , 1986 1and IDoelman, 19871. However, it is s uggested to estimate the acoustic processes off-line. This results in a fixed controller with no capability for tracking changes in the acoustical environment. These changes can be quite substantial due toe.g. a temperature gradient or a change in the duct flow rate. The need for adaptability is a lso recognized by Eriksson [1989J, who proposes an adaptive strategy described below. The controller structure is like an Infinite Impul se Respon se I IIR I filter: ult l ~ - - - I - dil l
Hlq -
yltl = q - r:
~q-r,
1 2 3 4
15 1
j
Alq - ) )
211)
181
!!.~ [lIltJ + L lq - I IdUl j + zltl Alq - II
.91
Again , a fil te red ve rs ion of the detection s ignal , df'u), is required . The strategy for thi s a dapti ve filtering system becomes
I
B (q - I I [ H 'q - I'UIlI+ L'q - I,dll, ... ___
B
~ 1IFIII + L lq-1ldFUI+ zil)
A possible solution for the Hand L polynomi a ls is, see [Eriksson, 19891. . L = q'l - r: CA 161 H -- I - q rl - r-_- r,. CF EB ' EB This solution is intrinsically presen t in •31: ylll
Q
Clearly, the required controller s tructure is nothing but a Finite Impul se Res ponse (FIR I filter . For pe riodic sound the optimal L polynomial is finite in s pite of the description in 181. The minimal number of coefficients in L equals twice the number of harmonic components contained in the pe riodic primary sound. Implementation of solution 181 in the system equation I I I gives; IDoelman, 1991bl:
Estimate the primary , the secondary and the feedback acoustic processes. 2 Apply control according to 141.
Liq - I I
L ~ q'l- r,
H ~ I:
Estimate the secondary acoustic process q -r'B/A. Calculate the filtered signa l df'((I according to (7b ). Estimate the controller param ete rs of L u sing (9), Apply control as in I Ell with H= 1.
Identification of the seconda ry aco us tic process can be accomplished by applying a seq uent ia l regression algorithm to system equation 11 I: see e.g. 1Goodwin, 19841. Problem s with the correlation between ul I Ianddlt Ican be avoided by adding a low level random noise signal to the control s ignal lilt I.
17al
576
Parame ter Es timat ion Algorithm:-:
C.
The algorithms for the estimat io n of th e con troll er pa rameters are derived from the adapti"e filtering theOl·~·. The princip le of this approach compares to the problem of s.,·,;te m idcntifica tion . Th e main differe nce bet\\'een the pruble m of syste m id ent ification an d the estimation of the feedforward co ntruller parameter" is th e presence ofa tran sfe r function in be tw een the filt e r output and the system output. The us ual algurithms for adapti" e filtering need to be modified in order to deal with seco ndar~' aco u ,;tic proce,s. Both in [Eriksson. 1989[ and [Elliot!. 198 7 [ a ver~' simple modification is proposed. The controller characteristics arc presumed to be timeinvariant compared to the timescalc of the impulse res ponse of the secondary acoustic process. Thi s modifica tion makes thl' adaptive filte ring problem di scu ssed in thi s sect ion su it a blc for mere lv the Least Mean Squares , L:\'IS , e,;tima ti on a lgorit hm: ,ee [Doe lm an. 1991a[. The update equation of thi , g rad il' nt-typ<: algorithm reads 9 1I + I I = 911 I + ~.LI
,I!
Iyl I I
911 I =
[i,,1!
I
ill! I · .
i""
primar~'
fi (
F '1I 1=[riFllld II - 11· ."F ,/ _ III.
IBDM+Q Cl
1II 1
IfI'
t 1~)
1171
r,1
l/JI/ + rll=Plq-I'yl/ ~ rl l ~ Qlq - I ' '''1 1
-
r,
I
1191
" In the case of random prim a ry sound th e detected primary signal xII 1 is not available. as the detection signal is contaminated with secondary sound . Th e re lation 12 I ha s to be impl eme nted in Eq.1151, which gives Cl/J ' I / + rl '
IL
= BDM .;. CQ _ q' l - r, - r, GFDM]/l1I I +
E
S
P.I
GADM ,(f' + - -E- - d'l + r l
-
r,1
I ~Ol
The exp ressions I 19 1and 120 I form t he basis for the development of controller estimation algorithm s.
11.11
in which ;U I is white noi se ",ith zero mea n. Thi s syste m model is more general than I l l. as the ex traneous noi se is no longer purely Gaussian. The D polynomi a l is a dded to th e denominator of the extraneous noise mode l in order to handle most acoustic si tuation s. Consider the following auxiliary fun ct ion
P = P" / Pr!
GDAM
N C
Clq I I _ - ----.;1/1 D l q -' IA l q-1 I
D lq - I, = I + d ,q - ' + ... + d ,, /)q - "lJ
with
.\'
1/1= - p,/·,Il - ----;;;-dl/T r l -
As mentioned in sect ion 1\ th e system principle differs for random a nd for periodic primary sou nd . For periodic sound the primary aco ustic process can be modelled with the id entity E=A. whereas the a bsence of acoustic feedback yie lds F=O. Thi s gives for the optimal prediction of th e auxiliar." function:
Clq - l)= I +C'lq - l + "' TCn(,q - "C
with
/I
Rando m and Periodic Prima",' Sound
The Generalized Minimum Variance IG :VIV I control strategy was proposed by Clarke and Gawthrop [ 1975 [. The inclusion offeedforward in the GMV strate~ can be found in e.g. [Allidina. 198 1 [. The strategy will be shown to be capable of cont roll ing a system give n by I
r l - r,11151
The es timation algorithm in step 2 can be selected from the methods like Rec ursive InHJ'uOl t'nta l Variables. Recursive Maximum Like lihood. Extended Lea,t Squares or ordinary Recursive Least Squares. The last algorithm also a ppli es to thi s case, as the optimal (Irl/+T , I eq ual s zero: IGoodwin. 1984 [. In [Allidina. 1985[ the problem of real -time tuning of P a nd Q is a ddressed . This technique may req uire a considerable computational e ffort. which is not favourable in act ive sound control.
Short Description of the G:vIV Control Stra tegy
£iq -I
~
Calculate the aux il ia r.,· function: Eq. 1141 2 Estimate the pa ram ete rs of the prediction of the auxiliary function: Eq. I I;;, 3 Apply control accordin g to 117 1. 4 If neccssar.v adjust polynomial s P a nd Q.
III GENERALIZED :vI I Nl:\Wyl VARIA."CE COi'
A lq - I I
E
,;ound
r'
y( ll = - - - uff - r1 ' + - - - 'tU - r, 1 +
-GDA - -M dll
Unlike. the met hod of adaptive filtering the closed-loop poles do not equa l the open-loop poles. One can c,'e n assign suitable locations to th e closed -loop pol es by tuning P and Q accordingl y. The main object ive of Q. ho\\·e,·er. is to retain the ,;ystem stability. If B is non minimum pha::;e. the pun' minimum \'a ri anct' s trategy, [.>=1 and Q=O. would result in an un stable controller. The weighting on lI1/1 ca n pre"ent the control effort to become exceed ingly large , if it is appropriatel,' chosen. More det ai ls on th is weighting issue will be di,;cu ssed in section V. In gC'ne ra l th e procedurE' for self-tuningGMV control is
It can be concluded that the two a da ptive filt e ring ,trategies for active sound reduction a re uncomplicated. The controller structure is an IIR or a FIR filter and the co ntrolle r parameters arc estimated u sing an uncomplicated a lgo rithm . Thi s makes th radapti" e filtering s trategy easy to implemen t both in ha rd ware and software form . It will be shown, however. that th e simplicity is acqui red at th e cost of a degraded performa nce. In the next sections more a d" anced control strategies and estimators a re proposed to handle a broader range of active sound reduction problems.
GI<, - I I
+
With th ese ideal controller parameters th e closed-loop behaviour will be: GAQ BDM + CQ _ \'1/1 = xll - r~) + .:;/t) 1181 . EIBP -r AQ [ DIBP + AQI'
The assumption. that th e controll e r charactl'ristic, arc onl~' s lowlv time-varying puts a restricti on on the spt-ed of convergence of th e algorithms. It appears that the mod ification or the common L:VIS algorithm for adapti"e filt e rin g to dea l ",ith a tran, fi.' r fun ction in between is not optimal. The correct modification has recent l~' been de rived and can be found in [Doelman. 1991 al.
Blq - I I
T
The optimal control law \\'hich arises from th e object to minimize the "ariance of l/J iT r,' can be found. 1('Iarkc. 1975 [. to be
T
And for the strategy for periodic
IBD.\! (q . I/1'1'
1161
in which th e vector 8~ t l con ta in : :; the parameter I:'~timate~ and the regression vector 01 t I contain:; pa:-:t :-; ig-n nl sampl{':-" For the control s trategy for ra ndom ~o u nd, ~ 7 J, thl'::'t' \'{'ctor:::; art' g-i\'l'1l b~':
clFI! - I//.
P'I
The polynomials.H and .\' a re h';"en by th e Diophantine equation
110 1
i",'1 I]
1/- ri ' = -.\' yl/l ~
IV IDENTIFICATIO:-'; OF THE ADAPTIVE FILTERING STRATEGY In this section it will be demons trated . that the adaptive filtering as extensi" ely di scussed in sec ti on 11. can be derived from the s trate~' of minimum variance. In fact, adaptive filtering is a special case of pure feedforward control. Thi s type of control comes ou t of the G:\lV s trat e~' if the detection signal dl I1 suffices to acquire the optima l prediction of t he aux ili a ry function . In other words, nothing of the contribution of ~(f , to th e a uxiliary function , PCIDA ~I!+T, I. can be predict ed by u sin g yl I I. For the adaptive filtering s tra te~·.
I\ ~ I
Q = Q" / Qd
The object of the G:\1V control s trategy is to minimize the "ariance of the a uxiliary function l/J 1+ r I. Thi s requires prediction of l/J I ~ r l I. The optimal predicti on in least squa res .~en'e ~'ields
577
strategy it i~ rl'quin'd that /) = l.(; = (). J) = 1 and (' =.-\ : tht' (,()lIt rihut ion of ~UI to th~> auxiliary function (J;~ t+r , ) i:-, pun.' whilt' JlOl....:l·. The op timal predicti on of the auxiliar.\· fUJld ion ill thi:-, ~it uati(lIl hecollll':-:
.
rB
(/J - (t + r l):::- ! ~-(I
., . . . • GF '
LA
. \ L'
(;
U!tl-~E,dlt _ :-
dampt·d (I\·eJ'-c.tll poll':' dOl':-' nllt ha\'l' a hi gh priority . A....: till' ....:t.'t-point l 'q Uti I.. .: zero in t hi....: ca..;l'. l11l'rel.\· in t i nH'-\' ar~'i ng' :-'.\" ....:t t'm....: 1'<."1 ....:1 tracking i;-; J'equirt.'d. Tlw tuning of q C;lll han' a ....:l' ri()u ~ influl~ n('l' on th e rl·....:iciual \'ari a n ct' of thl>....:y....:tl'll1 out put. which 1()l\ow:-, from (2-1 I .
- I"~
-u.:.
whi ch pe rfect ly' compan':, to 1:31, :,incl' now it huld:-: that (/). /1:::-.\' 11 '. It s hould be realized that in all ea""" ill which t h e adapu\l' filtl'ring strategy as a control ~t r a t egy i~ a ppl ied. t lw co nd it iOIl:' a:-. ll11' nt ionl,d abo\'(~ do not nl'ct'~sa ri l y hold. It I:::' ju:'t a mattt'r of ignoring th e characterist ics of thE' ext ran c()u:, noi:-:l.' . In a fl'w ca....:!':, ufact i \"l' ....:()U Ild control the extrHneou~ noise contributiun i....: ....:uch in....:ignificant. that it v·:ouId be undue to account for it. Tht' C()[b t.'qU(,Ill"t':' of thl ' l'onlrul strategy simplification in :; i tuati on~ with <:1 con~i d l'ra hll' am()unt of undetectahle nOi5(~ can bc' fou nd in tIll' ....:t·niu[l \'1.
Tht' wC'ighted minimum \'ari i:l IlCl' control ....:tra tl'gy I:=; a . . : pccia i form of minirnum \"arianl'l' cuntrol for Ilon -minimum pha....:l' ~y~tem~. E\'en !(1I ' Q=O th e ideal =-,y::'tl'Ill rl'nwin::: :,tahlt., li)J' a non-minimum pha se pulynomial H. The choice f(Jr}) ....:l·{'lll....: ob\'iou::: whcn l'xamining (24 1:
Proceeding with ( 21 10n(' a!Ti\"(,'~ at the cO lltrul procedure:-: of adapt in' filtering for both random a nd peri odic primar.\· :,ollnd a ddr(> ....:~t.·d in sec t ion II. The popu lar filtered r and filt e red 11 L'.I S ,dgOJ"ithlll" fil l' acti\'e sound control e m e rg(~ furm thl! C.\I \' control ~tn l tl'gy under t h e fo ll owing restriction::::
The syste m can be modelled b.\· an uutput ('ITor mudt·1 structure: the extrant'oll~ Iloi:-;(' i ~ Pllrply \\·h itl' noi....:l.' 2 Th e weighting I::; likt.· minimum \"(l r i;lIlCt· : P::::.l (,::::(). 3 For both th e random and tht., peri{)oic primary :::ulIlld ca;-;t· t Ill' controller polynomi a l;-; h:l\"l' to uht'.\· ....:p(·c ial !()I'm....:: ....:l't.' (b I and (8 I. Thi s results in a neCl':-\::--ary idl'lltilkat iun ofthl' ~(' c()ndary aco us tic proccss. 4 The controller p a ram e t e r ~ a re e.-.;tim .'l.tl>d ll:-,ing a ....:lowly converbring modification of thl' L~ I S nlgorithm
\ ' 1 ~1'. l l · LAT I ()"~
Thl' ....:illlulatiun experim ent...; an' to (,luc!ci;lIt· t hi.' i'l'
These fo ur requireme nt s for adapti\'e filtering with filterl'd l.'.IS a lgorithms cou ld ex hibit th0 Ilon -u~a hility ()fthi~ :-; trtlt l'gy I {OW( '\·ll!'. in many publication s. e.g, 1Elliott. 19,,71 and 1Erib,;on. 191'i9 1 satisfying resul ts \ .... ith the ~ trat eg.\· art' cl
(Jft'tJllt
con trol strategy
rol....:t r ~ltl·gil'.-.; ill ....:illlulatioI1 ex periments. estimation a lgorithm
auxiliary function
Adaptive Fi ltering 1
eq (21 )
Adaptive Filtering 2
eq, (21 )
Feedback Control
e q ,( 15 ) with L = 0
GMV
Control
eq, (15)
LMS. Eq , (10) RLS RLS RLS
~()l(' that the feedback ....:trategy i~ a G:\'lV :-;tratl:'gy. The required identificatiun of thl' :-;ccon dary acou:-:t ic prOCl'S~ fo r the adaptive filt('rin g~ tr a t ebrjel" w(lscarriedout hefol'l'il and. ~lon.'u\·l'r. th e optimal \'alul' of the st ep-size parameter.iI \\'H ::; cho~(>n f()rthc L:\'l S a lgorithm . Th (' term "res" stand....: for the :-;ljuart' of the rl'~ idu a l output signa l ~'1 o r(' ::,iOlulation experiment:i on acti\'C' sound contro l can be found in IDoelman. 1991b, 199 1c I,
Use of The We ighting Polynomiab in Ada pti\'l' Filterin " Althoug h the adaptive filtering s tr ategy' ha s bee n deri\'ed fi))· pun' minimum variance, it wil l turn out that P can be u..;cd to impro\'!: it:-; performance, A u sefu l setting is P=A a nd q=O, A",-uming that thl' degree of A is s mall e r than th e d ea d time r i • the predictio n of t h, ' aux il iary function 12 11 gi \'es
Simu la t ion 1. In thi....: ex perimellt th (' time ;=,;('r i l~:' an' measured data
yil'ld~
in Cl \'C: l1tilati on duct. The rl'mainingsl'conchu.\· and feedbuc k acoustic proc(':-;....: arc modelled <-1:-;:
For periodic prim ar:,': sound thi s wc ig hti ng
B[ 1/ 1/1 + Lel' / I [
A"I
Thi s weighting is es pecial ly effec tiYl' {(H' gradil'llt typC' l·.-.;timatioll a lgorithms. as demonstrated in IDoelman. 1991 a l, The elimination of the A poly nomi al from th e filte rin g of" and 11 re,sul t" in a i'<"ter rate of converge nce of th e controller parameter" i n the L\IS-t\'pl' algorithms. This is \'c rified in the s imulation C'xpL' r inH:'nt;..;.
V
l~_ 1'"
, 26 l W 1'" in which B contain:' all un:-:tahlt, zero."; urn and th e til dc - s ta nds for t aking thl' recip rucal polYlloJl1ii.ll. ~ I url' dl'taib ofthi....: tt'chnique can be filLllld in IGrimbl ..... 19K1l. Appli cation ol"tl1l:-' weighting require :' a factorizat ion of the H po i.nlomial. which can bl' tim e-co ns uming. Th t' rd ~lI·l' . it i....: ~uggl'~tl·d to inclucil' tiH' complt:tv /) pulynomial in the jJ \\cighl ing. Rl·::-iult.....: o{'thi.....: control ~t r atq.~.\· can a l",o ht' ti) und in the ::,im ulation l'x perimenb . p =
POLE
roc
The expression for the closed-loop beha\'iour of the ideal G'.IY controll e r re\'ea ls the important role of the weighting polynomiak vlll=
GAQ
X'I _ [,, + BD." -CQ~'l l
Ol
ci5 S a. S
12-1 ,
o
E[BP + AQJ D[BP-AQ[' Th e main u se of P and Q is to ensure closed-loop " t abilit.\ In mo,;t cases thi s means. that ifB is non -minimum phase , Q is usualh tuned in order to retain the closed-loop pol es inside th e unit circle: "''', [Goodwin, 19841, As well-known the tuning of Q has to be accurate to avoid a stead y-sta te error. Inclu s ion of an integrator in Q ll SlI Cl:Il.\· soh'es thi s problem. When ,\'11 l is periodic it is \'ery useful to include in Q a polynomlal Q I \\'ith t h e follo\\'ing feature: Ql lq - I IXlll = O
c:c
:~,o ,2
o
400
800 1200 Sample Instant
1600
2000
0
400
800
1600
2000
2
Ol
if!
S
Ef
0
,2 -' ,
QI
e qual s 1-2co51 Ii,T,C! ' 1+ q -:. ill which T i, tlll' sampling instant, :vIore accurate tunin g of P and q to obtain \\'l'IIIfxlt I is cosl lii I e,g ..
,= I
The ,am pl e frequency i, 1600 Hz, Th e re,;u lt ing ,;ou lld le l'e ls are " h o\l'n in Fig. 4, D ue to a partl y unprediClablc primary s ound. the adap ti\'e filtering controll e r ,; Iea\'l' a r elati\'ely high rbidual leve l. Both G:\I\' "tr"teh~e,; dea l \\'Ith the priman' ""un d much better,
A."m ZERO ASSIG~:\IE:\T 1:\ ,,'.1\ ' CO:\TROL
.
I
2
1200
Sar'1p:e Instant
F ig.4 COnl'd ..
578
<0
2
Adaptive Filtering 2
co
'"
if!
"5 0. "5 0
0 ·2
"5 0. "5 0
400
-n
'"
"5 0. "5 0
1200 800 Sample Instant
2 0 ·2
"5 0. "5 0
res = 0.012 400
1600
GMV Control
-n
res
=
400
1200 800 Sample tnstant
1600
2000
.-1 1'1
x 1/
100
200
300 400 500 600 Sampling Instant
700
800
900
:2 \\-it h
indit' ~ ltt, ~ thl' .... tart
fi1~t t iTll l··\·a r~· il1g di:,turhFlI1C'l'
of COl1tro l. tl)
c!P !ll () !1 ....;t r ate the U:-'l'
I, ~
1_11 . .'il 1 -
O' \ )~(I
I
- 1 1_
I ... I
n'l
I
fl
= i
f'
= I1
f) , 'I 1 1::: t
-l
1 1;
H I'I
1,'1 'I
= CO:-; 12m
~
1 1
""'£\ ((1
1
\"ar 1';1/ I1 = 11.1
I
Th l' model li)J" lht' :-,(,t'tH1dar\ atllll:-,t i t' prot'P:-;:-; indude:-; hoth Cl nonminimum pha:-'l' zero and p()orl\-d a mpe d po]e:-, " Tht, l'Un\'ergl'Ilt'1:' :-:J"H.,t'd oft hl' adapt i \ "t'
fi I tl'ri ng:-'t rateg-il':-' :-'lI rfl'r:-; from t h(, n '\"('rlwrant
chnra ct (' ri :-; tic :-;, Whl'rl'H:-; thl' (;\1\' ;.;trat eg-i p:" t'annul appl.\" pun'
minimum '"<:lrialH'p ("ont rol. The wl'ighting ill "Ad aptive Fi ltering 1A"
i" P= I and in ·'Adaptive Filtering 1 S·· P =.-I . For tilt' G:\I\' cont roller,: it hol d" tha t
<0
2 0-.----------------------------------~
~
co
u;'" "5 0. "5
4
o
o-
Adaplrve Filtering 1A
~~'I~WN ~\, ~ViV.V-~~.~~v~~~~ n
.20
0
res
=
2 48
o
200
400 600 Sampling Insta nt
800
1000
o
200
400 600 Sampling Instanl
800
1000
0
200
400 600 Sampling InSlant
800
tOOO
0
200
400 600 Sampling In slant
800
1000
0
200
400 600 Sampling In slant
800
1000
m co
·4 0
400
800 1200 Sampling Instant
'"
1600
if!
"5 0. "5
4
o
<0 co
0 <0
·4
co
0
400
800 1200 Sampling Instant
u;'"
1600
"5 0. "5
o
4
GMV Control
<0 co
"5 0. "5 0
0
0
(' 1'1 1 1 ;;; !
co
u;'"
900
t'harat'll'ri:-,lit':' a rt' Ilwdl,lll'd a:-,
<0
"5 a. "5 0
800
ofthl\ \\"pighting polyno miaL·.; in t Ilt' cont rol :-ill'<1tegil':-: . Th e :',\':-,ll'm
Btq - II = I
'"
700
Si m u l tit io/\ :L Thi :-: :-:imulati ol1 l'xpl' rinwll t i:-:
0.0015
The results in fi g. 5 demon strate that the de tection signal does not completely determine the prima ry sound in s ide the cabin. Th e G!\1V s trategy can handle t he unknown pa rt quite well.
if!
300 400 500 600 Sampling Instant
'1'I'T
Simulation 2. For thi s s imulation th e detection and prim a ry s ignal were measured in a Swed is h passe nger ca r . Th e sampling frequt'ncy equals 500 Hz . Though in s id e thi s \·ehide a multi ·channe l control system is required, th e s imulation demonstra tes the proble m,; with controlling interior ca r sound. The u su a l way to de a l with the (insta ntaneou sl harmonic sound is u sing a s:.ste m fQr pe r·i od ic ,,,u nd control as s hown in Fi g. 3. Th e secondary path is model led a,;
'"
200
Fig.5 . Rl':-:u lt of :-;illltllat ion
C I'I
"5 0. "5 0
100
:,ignal: an arrow
Fi g. 4. Results of si mul ation 1: an arrow indicate, the , ta rt o[co nt rol.
if!
0
·2
2000
,~
0
>-
·2
2
'"
6> if!
0 ·f1Il ·2
2
0
2000
Feedback Control
co
if!
1600
1
0
<0
800 1200 Sample Instant
2
co
.Q> (j)
g,
"5 0. "5 0
0
<0
C'
if!
0 <0
res = 0.136
·4 0
400
1200 800 Sampling InSlant
20
co
'"
if!
1600
"5 0. "5 0
0 ·20
fig . 5. Results of s imulati on 2 ; an arrow indicates th e start of co ntrol. 20
'" g,
The feedback s trateg:. was also tested on data measured during acceleration of th e car. In thi s case t he det ection s ignal could not he utilized by any feedforwa rd stra tegy. Th e result is s hown in f ig. 6. Again , the feedback con trolle r achie\'es a good reducti on .
if!
"5 0. "5 0
0 ·20
F ig"l. Rt' :.:uI t:o:: of:.:imu \cHinl1:3: an a rrow i ndic3tl':-: t h e :-, t<1rt of control.
579
GMV Control A: P =
HIB
Q =O
VIII AC'K.,'OWLEDGD·IENTS
GMVControIB: P =0. :i - O. 7q - 1 +0 2:iq - C -O.25q - '
T hese im'estigation in th e program of th e Fund a mental Research on :\Ia tte r I FO~I I hm'c been supported by the ;-';etherlands Technology Foundation I STW I.
Q = 0.:; + O :iq - C
As ca n be concluded from Fig . 7, theA-weightingonyll I impro\'es the pe rformance of th e adapti\'e filte ring st rategy. Al so. the weighted minimum variance st rategy performs bettcr than the G:vIV stra tegy with P an d Q weighting. :\Ioreo\·er. th e cho ice for P a nd Q in thi s s trategy is not easy to make.
IX
REFERENCES
Allidina: A.Y.. F .:\1 . Hughes and C. Tye 119811. Self-tuning control for systems employin g feedforward. lEE Prof. D , 128, pp . 283 29 l. Allidina. AY .. F.:\1 . Hughes an d T.Tahmassebi 119861. An implicit self-tuning technique with va riable time-delay. Int. J. Con trol , !H. pp . 1437-1-157. Clarke. D.W. and BA Gaw throp 119751. Self-tuning controller. lEE Proc.D, 122. pp . 929-934. Doelman. N.J .1 19871. Adapti\'e active a tte nu at ion of broad band so und . Master the.,i" Delli Unice rsity a/ Technology. Doelman. N.J .1 199 1a I. A nove l adapti \'e filt er LMS algorithm for' use in thcacti ve con t rol of'sound . S ubm itted to IEEE Tran sactions on Sigllal Processill/!. Doe lma n. N.J. 1199 1bl. A unified control stra tegy for the active reduction of sound . In Recell t Adl'On ces in Actice Control ofSollnd alld Vibratioll. Technomic Publi shing. La ncaster. Doelma n. N.J. and E.J.J. Doppenberg 11991cl. The application of se lf-tuning control stra tegies to th e act ive reduction of sound. lEE COII/erence Con trul9 1. Elliott , S.J., I.M . Stoth ers and PA Nelson 119871. A multiple e rror LMS a lgorithm a nd its a pplication to th e active control of sound a nd vibration. IEEE Tran sactiolls all ASSP, aQ, pp. 1423- 1434. Eriksson, L.J. and :VI. C. Allie 119891. Use of ra ndom noise for on-line t ra nsduce r modeling in a n ada ptive active attenuation syste m .J. Acollst. Suc. Am .. 1:lli. pp. 797-802. Goodwin. G.C. a nd K.S. Sin 119841. Adapticc Filtering, Prediction and COlltrvl. Prentice-Ha ll. Gri mble, M.J . 1198 11. A control weighted minimum-variance contro ll er for non-minimum phase systems. l ilt. J . Control, aa, pp. 75 1-762. Mangi a nte, G.A. 119771. Active sound absorption . J . Acollst. Soc. Am .. fit pp 1516-1523. Neely, S.T. a nd J.B . Ali e n 119791. Invertibility of room impul se res ponse . J. Acollst. S oc. Am ., 22. pp . 165- 169. Nelson , P.A and co-work ers 119871. the active minimization of harmonic e nclosed sound fields , part I, 11 a nd Ill. J ournal of SOl/nd alld Vibratioll. ill, pp. 1-58. Roure. A 119861. Self-a dapt ive broadband active sound control system . J ournal ofSol/nd and Vibration. 101 , pp . 429-44l.
VII CONC LUSIONS [n thi s pa per ada pti\'e tech n iques for the act ive con trol of sound h a\'c been disc ussed. Th" accustomed method of ada ptive filt e ring h as been presented a nd a na lyzed. ln addition, the st ra tegy of Generalized :v1inimum Varian ce IGl\IVI co ntrol has been proposed as a useful extension of the ex istin g acti\'e sound con trol strategies. [n fact. it can be derived that ada pti\'e filt e ring is a very special case of the GMV control strategy. The require ments for adapti ve filte ring to a ri se form the G:vIV strategy a re: The contributi on of ext ra neous noi se to the system output s ignal is jus t white noise. 2 Th e remaining part of the di stu rba nce signal at th e output can be completely predicted by e mpl oying the available detection signal. 3 Pure minimum \'a riance control is a pplied: P= 1 Q=O: t he setpoint equals zero for a ll t. In active sound control adapt ive filterin g is widely used. Th e main stra tegies as described by IElliott, 1987: Eriksson , 19891 corres pond to the followin g: 4 The controll e r polynomial s a re such that a s pecific IIR I random sound 1or a FIR I pe ri odic sound 1filter structure results . Thi s requires an addit iona l estimation procedure for the seconda ry acoust ic process . 5 The controller parameters are estimated using a non-ideal modifi cation of the LMS al gorithm. The fi ve require ments for th e GMV cont rol strategy to correspond to the cu rre nt methods of active sound co ntrol cl early revea l the limita ti ons of these me thods. In most cases of active sound control the extraneous noi se contribution is coloured le. g. a ventilation syste m or a ca r in teriOJ·1. A feedback control loop should be added to the system then . Furth ermore, it has been demon strated th a t 4 an d 5 ca n seriously degrade the performance as well. In IDoelm an, 1991b l improvements of controll er structure (4 1 and gradient type estima t or 15 1are suggested. From both theoretical derivations and the simul a tion experiments th e conclusion can be dra wn that th e GMV stra tegy can ha ndle the compli cated sound control problem s much more effecti vely. It is sh own to be a very attractive basic stra tegy to de ri ve special controll e rs from . Th e mai n draw back of GM V control compared with ada pti ve filte ring is the necessary tuning of the weighting polynomial s in the case of a non-minimum phase process. Non-minimum phase acoustic processes a re lik ely tooccur as can be found in INee ly, 1979 1. In simul a tions it has been shown that the weighted minimum variance strategy is the most attractive way to tune the weighting polynomial s. The time consuming factorization procedure can be avoided by using th e wh ole B poly nomial instead of B- . Nonminimum phase processes diminish the computational adv antage direct self-tuning stra tegies have above e.g. the indirect or the adaptive filtering meth ods. Periodically a Diophantine equati on h as to be solved IAllidin a. 19851 or the secondary acoustic process h as to be identified. Fina lly, it has bee n shown th at the performance of th e adaptive filt e ring strategy can be impro\'ed by an a ppropriate system output weighting. This makes the strategy attractive for the more simple sound control probl ems.
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