5th AIAA/CAS Aeroacoutc Conference (3th AIAA Aeroacoutc Conference) - 3 ay 9, am, Florda AIAA 9-344 Actve Noe Control nde the Loadmater Area of a Turboprop Tranport Arcraft Kay Kochan and Delf Sachau elmut-schmdt-unverty / Unverty of the Federal Armed Force amburg, Germany and arald Bretbach 3 Arbu Deutchland, amburg, Germany The paper decrbe the degn and tet of a prototype actve noe controller for a emencloed cabn nde a turbo-prop arcraft. ere, the eght actve noe control loudpeaker and teen mcrophone are mounted at the cabn wall and celng. For the development proce, an acoutcal mock-up bult to reproduce the acoutc tuaton nde the arcraft on the ground. The acoutc requrement on uch a mock-up are decrbed and wellfounded by a couplng analy. oreover, the controller baed on a frequency doman mplementaton of the teepet decent algorthm. To acheve the optmal performance and robutne requrement, a method to adjut the mcrophone and loudpeaker weght preented. The new method alo conder the uncertanty of the ytem n the controller parameter degn proce. permental reult how the effcency of the prototype ytem for a typcal load cae. Nomenclature A = equvalent aborbng urface of a room B = dampng matr C = controller tranfer functon matr n the teady tate c = peed of ound n ar d = column matr of the prmary noe gnal at the control mcrophone d = column matr of the prmary noe gnal at the montor mcrophone, = acoutcal energy denty nde a room e e = error gnal and the column matr of the error gnal at the control mcrophone e = column matr of the error gnal at the montor mcrophone G = tranfer functon matr between the loudpeaker and error mcrophone G = tranfer functon matr between the loudpeaker and montor mcrophone = general runnng nde J = controller cot functon K k R n P = tffne matr = couplng factor = ma matr = dcrete tme = acoutc energy Scentfc Atant, Department of echatronc, oltenhofweg 85, D-43 amburg Germany. Chef of the echatronc Department, oltenhofweg 85, D-43 amburg Germany. 3 Program Coordnator A4, Cabn Acoutc, Kreetlag, D-9 amburg, Germany. Copyrght 9 by elmut-schmdt-unverty amburg Germany. Publhed by the, Inc., wth permon.
p, p ˆ Q R = preure and column matr of the preure at the node of the F meh = controller weghtng matr for the error gnal = controller weghtng matr for the loudpeaker actuaton gnal S = couplng urface between the loadmater area and the cargo hold t = contnuou tme uu, = loudpeaker actuaton gnal and loudpeaker actuaton gnal n the teady tate opt ma u opt = mamum allowed loudpeaker actuaton gnal V ˆ, yz, α Φ ρ = volume of the cargo hold or loadmater area = modal egenvector = Cartean coordnate = dampng factor = pref to ndcate the uncertanty of the accordng value = modal matr wth modal egenvector = denty of ar ω = egenfrequence = nde to ndcate the nomnal tranfer functon or prmary noe feld.,. = Frobenu norm and -norm T F I. Introducton he nteror ound feld n propeller-drven arcraft are manly affected by hgh tonal ound preure level. Th caued by the propeller blade whch pa the fuelage at certan angular frequence (BPF, blade pang frequency), ee Fg.. The vbratng fuelage tructure ecte the nteror ar volume. Inde the cabn, the ound preure level may be hgher than db(a) wthout noe treatment. The degner of an arcraft oblged to aemble pave noe nulaton materal and/or to degn an actve noe control ytem to comply wth the noe pecfcaton and regulaton, ee for eample the uropean Drectve 3/C. The actve noe control method uperor for low frequency noe, when compared to pave noe nulaton method. Such an actve noe control (ANC) ytem cont of loudpeaker and mcrophone connected to a controller whch calculate loudpeaker actuaton gnal u to mnmze the dturbng ound preure d at thee mcrophone, ee Fg.. d reference gnal u G + + Ĝ e Copyrght Arbu ltary S. L. 5 Small em-encloed Loadmater Area attached to the large cargo hold ANC Controller e error gnal d dturbance gnal u loudpeaker actuaton gnal G econdary path tranfer functon Fgure. Tranport Arcraft Fgure. Actve Noe Prncple
In the th paper, the actve noe control method employed to the acoutcal problem of the loadmater area, ee Fgure. ght loudpeaker and teen mcrophone are ntalled on the celng and the eteror wall of the loadmater area. The loadmater area a mall em-encloed volume connected to the large cargo hold. Th couplng of a mall compartment (loadmater area) wth a large compartment (cargo hold) lead to new queton for the actve noe control ytem degn: ow can the couplng be mulated n a laboratory envronment? Whch nfluence doe the couplng have on actve ytem requrement? oreover, n the fnal degn proce, the hardware and poton of the actve noe control loudpeaker and error mcrophone are already defned. But the controller oftware and parameter can tll be adapted n order to mprove the performance and robutne of the ytem. Queton that are more cloely related to ndutral applcaton are mportant here, uch a: Whch parameter etup rule can be ued to guarantee robut controller behavour? ow hould the controller parameter be et to acheve the neceary controller performance and tablty? Th paper organzed a follow: For the acoutc ground tet, the acoutc couplng between the loadmater area and the cargo hold eamned for a mplfed model ung two theoretc method. The frt deploy an energy baed method whch focued on the acoutcal energy flow n uch a coupled ytem. The econd ue a ubtructure technque where the hape of the ound feld nde a mall loadmater area are n focu. Baed on that analy, an acoutc ground tet faclty developed. ere, the room acoutc conderaton and the prototype actve noe controller are n the center of the chapter. Afterward, a controller parameter degn method ntroduced whch nvolve ndependent loudpeaker and mcrophone weght n order to acheve the neceary actve noe control performance. At the end, ome epermental actve noe control reult are preented. II. Couplng Analy of the Loadmater Area and the Cargo old A mentoned above, the pecal charactertc of the uppoed turboprop arcraft that the relatvely mall volume of the loadmater area coupled to a large cargo hold. To dentfy the bac phycal behavor of uch a coupled ytem and to defne requrement for an acoutc ground tet, a mplfed model of an encloed ound feld condered. A. odelng of the coupled Sytem 3 A llutrated n Fgure 3, the cargo hold repreented by a rectangular compartment ( volume V 45 m ). In addton, the loadmater area wth a volume V of appromately 4.5 m³ attached at the front of the cargo hold. The couplng urface S between both compartment appromately.6 m² large. The loadmater area ha a lghtly curved outer wall whch mpled by the chamfer. The reverberaton tme nde the volume aumed to be comparable to the real arcraft ( T 6 =.6). Cargo old (nde ) V = 45 m³ P P couplng urface S =.6 m² Loadmater Area (nde ) V = 4.5 m³ P Fgure 3. Smplfed model of the coupled ytem loadmater area and cargo hold Tonal ectaton (f < 3 z) of the ound feld nde the arcraft aumed to be prmarly through the fuelage tructure n the propeller plane 3. Due to the ynchrophang of the arcraft propeller, the noe entry nto the arcraft n phae on both de. Inde the arcraft, the ound wave are partly reflected and aborbed at the nteror wall. ereby the major dpaton effect nduced by the aborpton on the boundare, n comparon to 3
the mnor effect of nternal frcton n the flud. The prmary noe feld n the loadmater area ected va the couplng urface S. If the actve noe control ytem wtched on, the prmary noe feld nde the loadmater area upermpoed by a econdary noe feld. Th econdary noe feld generated by the actve noe control loudpeaker at the eteror wall and the celng of the loadmater area. In the net two ecton, we can ue the mplfed model twce. Durng the applcaton of the energy baed method, only the compartment volume and the equvalent aborbng urface of the model are of nteret. In the ubtructure baed method, the mplfed geometry only evaluated n two dmenon. Th trong mplfcaton of the real arcraft tolerable, due to the fact that only the prncpal phycal behavor and tendence are of nteret. B. Acoutcal nergy Flow n the Couplng Surface Frt of all, the energy flow n the mplfed coupled ytem of great nteret. ere, the volume and aborbng urface a hown n Fgure 3 are condered. In the theory of room acoutc, the law of conervaton of the acoutcal energy can be appled to encloed ound feld 4. The overall acoutcal energy of condered a contant. At the ytem boundare, acoutcal energy can be dpated at aborbng urface or radated by acoutcal ource. From th pont of vew, the acoutcal power balance for the cargo (nde ) and the loadmater area (nde ) can be wrtten a c c c P A S+ S = 4 4 4 c c c P A + S S =. 4 4 4 () The varable and denote the acoutcal energy denty n the cargo hold and the loadmater area. The aborbng urface n both compartment can be ummarzed to the equvalent aborbng urface A and A repectvely. A ndcated above, the urface whch eparate the loadmater area from the cargo hold decrbed a S. At th urface, both compartment are able to echange acoutcal energy. Furthermore, P = P + P and P denote the acoutcal power whch radated by the acoutc ource n the accordng compartment. The abbrevaton c denote the peed of ound. If we further ntroduce the abbrevaton A = A + S and A = A + S, equaton () can be rewrtten a 4 P A+ S = c 4 P A + S =. c () In the further analy, two cae wll be tuded. The frt one addree the cae where the prmary noe pae through the fuelage and ecte the cargo hold. The econd one deal wth the ectaton of an actve noe control loudpeaker whch placed n the loadmater area. The other correpondng compartment then only ected va the couplng urface S.. ctaton of the Prmary Noe Feld Durng the prmary ectaton, the acoutcal power of the ytem ntroduced va the prmary ource P ( P = ). The energy denty can be evaluated ung the equaton ytem (): 4 P 4 P = = c S c A + k A A + A A. (3) The denomnator how that the contrbuton of the acoutcal aborbng urface A to the energy denty reduced by the couplng factor k = S A. If the lmt of a ound oft loadmater area condered, the econd ummand n the denomnator of q. (3) change to: 4
lm A A S + S A = S. (4) On the other hand, the lmt of the couplng factor k for a ound hard loadmater area lead to S lm = A A + S. (5) Inerton of the lmt n the equaton (3) lead to the energy denty epreon for the lmt of a ound oft loadmater area 4 P = c A + S (6) h and the energy denty epreon for the lmt of a ound hard loadmater area h 4 P = c A + A. (7) quaton (6) how that the couplng urface can be een a a full aborbng urface for the lmt of a ound oft loadmater area. Furthermore, for a ound hard loadmater area the couplng large. The energy denty n the loadmater area can be obtaned by S = = k. (8) A If the couplng factor k cloe to, the acoutcal energy unformly dtrbuted n both compartment. On the other hand, f the couplng factor k mall, a gnfcant dfference between the energy denty and can be epected.. Actuaton of the Noe Control Loudpeaker In the cae of an actuaton wth the noe control loudpeaker, the acoutcal power of the ytem ntroduced va the econdary ource P ( P = ). The energy denty can agan be evaluated ung the equaton ytem (): 4 P 4 P = = c S c k A + A A + A A. (9) Correpondng to the analy above, the denomnator how that the contrbuton of the acoutcal aborbng urface A to the energy denty now reduced by the couplng factor k = S A. Therefore, the lmt of a ound oft cargo hold change equaton (9) to 4 P = c S + A () and for a ound hard cargo hold to h 4 = c A P + A 5. ()
A analyzed above, for the ound oft lmt the couplng factor k mall, the couplng to the cargo hold low. On the other hand, the couplng wll be large for a ound hard cargo hold. The rato of the energy denty can be calculated wth the ytem of equaton () S = = k. () A Wth th equaton, the rato of the energy denty can be calculated from the raton of the urface S and A. C. odal Behavor nde the Loadmater Area In ecton B, an analy wa done to undertand couplng n the ene of the energy flow. The am of the followng analy to undertand the hape of the ound feld n a mall compartment (loadmater area) whch coupled to a large compartment (cargo hold). At low frequence, the acoutc ound feld can be decrbed by the wave theory. ere, the encloed ound feld compoed by a lnear combnaton of ndependent acoutcal mode. The decrpton of the ound feld can be developed from a econd order homogenou lnear partal dfferental equaton whch known a acoutc wave equaton: (,,, ) p y z t p(, y, z, t) =. (3) c ρ t ρ ere, the varable p refer to the acoutc preure, the contant ρ decrbe the denty of ar, and = + + y z (4) tand for the Laplace-operator of the Cartean coordnate, y and z. The product c ρ pecfe the adabatc bulk modulu or room tffne. The acoutc wave equaton (3) can be dcretzed by the fnte element (F) method: where repreent the ma matr, K the tffne matr and p ˆ + Kpˆ = (5) [ p p p ] p ˆ = ˆ ˆ ˆ,,..., N (6) the column matr of the preure at the N F-node. In order to olve the homogenou dfferental equaton numercally, the ntroducton of dmenon factor m ( = m ) and k ( K = k K) a well a a non-dmenonal tme t = ω t wth ω R R = k m approprate. Th lead to the followng dfferental equaton: p ˆ+ Kp ˆ = (7) where the econd dervatve wth repect to the non-dmenonal tme abbrevated wth the large crcle above the varable. To nvolve dampng effect, the homogenou equaton (7) can be etended by a dampng matr B : p ˆ+ Bp ˆ+ Kp ˆ = (8) whch repreent the nternal frcton n the flud and uppoed to be a mall tffne proportonal dampng wth 6
B = αk. (9) In the cae of the two coupled compartment (cargo hold nde, loadmater area nde ), the equaton (8) of the encloed ound feld can be decompoed: C pˆ B B C pˆ K K ˆ C p ˆ ˆ ˆ C C p + B B p + K K C p =. () T T T T T T ˆ ˆ ˆ C C C pc B C B C B C pc K C K C K C pc For th, the preure column vector ˆp reorted to collect the degree of freedom for the ndvdual ubtructure ˆp and ˆp a well a the couplng degree of freedom ˆp C. The ma matr decompoed n the correpondng ubtructure ma matrce and a well a the couplng ma matr C. The dampng matr B and the tffne matr K are mlarly fragmented. If the couplng degree of freedom ˆp C are et to zero, the equaton () can be decompoed nto two ubtructure: p ˆ + Bp ˆ + Kp ˆ = pˆ + B pˆ + K pˆ =. () Solvng the egenvalue problem lead to the modal matrce of the coupled ytem (q. ()): and of the two ubtructure (q. ()) [ ˆ ˆ ˆ ] Φ =,,..., N () [ ˆ, ˆ,..., ˆ N ] [ ˆ, ˆ,..., ˆ ] Φ = Φ = N (3) ˆ ˆ ere, ˆ, and are the egenmode of the accordng tructure. Followng the nomenclature of Crag 5, the ubtructure mode n Φ and Φ are fed-nterface mode whch correpond to two eparate compartment wth an deal ound oft urface at S. ω = ωω of the coupled ytem: The accordng comple egenvalue contan the egenfrequence ( ) and the egenfrequence of the two ubtructure [ ] R ω, ω,..., ω N (4) [ ω, ω,..., ωn] [ ω ω ω ],,...,. N (5) In order to eplan ome prncple couplng effect of the mall loadmater area wth the cargo hold, the mple two-dmenonal model can be ued a hown n Fgure 3. Therefore, the coupled ytem of Fgure 3 wa modelled n two-dmenon wth the fnte element oftware COSOL veron 3.5. Altogether, 793 lnear-lagrange quadrlateral element were ued n the entre model; of whch 43 lnear-lagrange element were deployed n the ubtructure of the loadmater area. Th correpond to at leat 5 node per wave length up to 3 z. Ung the eport functon of COSOL, the ma matr and tffne matr K were eported to ATLAB veron 8a. Afterward, the dampng matr B wa calculated ung the tffne matr, ee q. (9). ere, the parameter α wa et to % to ntroduce a weak dampng. Wth the ma, dampng and tffne matr, the mode 7
hape of the coupled ytem and the ubtructure can be evaluated ung the equaton () to (3) and the ATLAB functon polyeg. In Fgure 4, the egenfrequence for the mplfed coupled ytem (cargo hold + loadmater area) and the ubtructure of the loadmater area are dplayed up to 3 z. A epected the amount of egenfrequence ncreae accordng to the room ze. Whle the decoupled mall loadmater area ha only egenfrequence up to 3 z, the large coupled ytem ha 4 egenfrequence. In order to compare the mode hape of the ubtructure of the loadmater area and the correpondng ubecton of the coupled ytem, the modal aurance crteron (AC) wa appled. In Fgure 4, the egenfrequence wth AC value hgher than.75 are connected va lne. The gray cale of the lne vare accordng to the AC value (AC =.75 wth lne; AC =. black lne). We can ee that the mode hape of the mall loadmater area ubtructure can be found n the mode hape of the entre coupled ytem. ereby, the larget mlarty follow a certan pattern. It can be found below and above the ubtructure egenfrequence. The couplng of the lengthwe mode (e.g. ω 3 ) tronger than the tranveral mode (e.g. ω 5 ). On the other hand, a clear allocaton between the mode hape of the cargo hold ubtructure and the coupled ytem could not be oberved. Th caued by the conderably hgher modal denty of the large cargo hold. ω = 6. z ω = 3.7 z ω 3 = 8.4 z ω 4 = 4.7 z ω 5 = 4.9 z + Frequency n z 5 5 5 3 loadmater area ubtructure 5 5 5 3 Frequency n z coupled ytem + ω = 49. z ω = 6.5 z ω 3 = 53.7 z ω 4 =. z ω 5 = 46.4 z couplng urface Fgure 4. Loadmater area ubtructure egenfrequence allocated to the egenfrequence of the coupled ytem. Detal of the preure mode hape nde the loadmater area. For a vual comparon, the ubtructure mode hape and a electon of the mode hape of the coupled ytem are alo hown n Fgure 4. Only the mode hape ubecton of the loadmater area are dplayed for the coupled ytem. When comparng the mode hape, the mlarty can alo be found wth only mall varaton cloe to the couplng urface. Th caued by the uage of fed-nterfaced mode whch ha zero preure at the couplng urface. 8
D. Conequence For the uppoed arcraft, the equvalent aborbng urface A of the cargo hold much larger than the equvalent aborbng urface A of the loadmater area. Furthermore, the couplng urface S larger than the equvalent aborbng urface A. The analy how that for the prmary ectaton the couplng factor k = S A lghtly maller than. If equaton (8) evaluated, the energy denty n both compartment wll be equal. ence, the coupled ytem can be een a one ytem from the vewpont of the prmary ectaton when the condered condton are appled. Th denote that hgh ound preure level nde the cargo hold wll propagate unmpededly n the loadmater area. From the vewpont of an ectaton wth the ANC loudpeaker, the couplng urface S can be aumed a a full aborbng urface. The acoutcal energy denty n the cargo hold wll be maller than nde the loadmater area. Therefore, depte the ound hard materal n the loadmater area, the loadmater area can be condered a a hghly damped room. Th ha two conequence for the actve noe control ytem. Frt, t tend to mooth tranfer functon between the ANC loudpeaker and the control mcrophone wth lower reonance peak and a teady lope of the phae repone. Th advantageou for a robut actve noe control ytem becaue t reduce the entvty n the etmated tranfer functon, ee Ref. 6. Secondly, due to the hgher dampng, maller reonance effect nduced by the ANC loudpeaker occur n the em-encloure. Th lead to the requrement of ANC loudpeaker wth hgh performance. The analy of the modal behavor ha hown that only a few ndependent mode hape nfluence the acoutc behavor nde the loadmater area. Thee mode hape are defned by the local geometry of the loadmater area and hould be non-entve to change n the cargo hold. The modal denty gven by the coupled ytem and wll be hgh n comparon to a decoupled loadmater area. III. permental Setup for the Acoutc Ground Tet Th chapter decrbe the acoutc ground tet faclty to mulate the acoutc tuaton nde the arcraft. Afterward, the electro-acoutc component of the epermental etup are lted. oreover, the actve noe controller mplemented n the frequency doman wll be brefly ntroduced. A. Acoutc Ground Tet ock-up Accordng to the analy above, an acoutcal mock-up can be ued for the actve noe control eperment. Th mock-up hould reproduce the gnfcant geometrc crcumtance dependng on the hortet mportant wave length. The prmary noe feld ected manly through the couplng urface. Therefore, the mock-up can be made of wood; an epenve vbro-acoutc mock-up not neceary. The cargo hold mulated by an approprate laboratory. That laboratory room hould have a mlar mode denty whch can be fulflled f the room large enough. The prmary ectaton can be realzed wth loudpeaker whch are able to ecte the room mode. The reverberaton tme hould be comparable to the real cargo hold. An epermental etup whch atfe thee requrement hown n Fgure 5. The mock-up of the loadmater area wa contructed accordng to the dgtal mock-up and wa aembled nde a laboratory room whch ha mlar dmenon to the cargo hold, ee Ref. 7. The mock-up geometry wa mplfed correpondng to manufacturng poblte and the hortet ound wave length. Afterward, the mock-up wa equpped wth a honeycomb lnng. The reverberaton tme nde the laboratory about.6 econd. At the uppoed propeller plane, two publc addre loudpeaker were placed on both de of the laboratory to ecte the prmary noe feld. 9
laboratory prmary loudpeaker vewng drecton n the photo LA mock-up prmary loudpeaker 5 m 7.5 m 4.9 m Fgure 5. The wooden mock-up of the loadmater area (LA) placed nde a laboratory. B. lectro-acoutc Component In the preent actve noe control eperment, N L = 8 loudpeaker and N = 6 mcrophone were mounted at the eteror wall and the celng of the loadmater area. The tranducer poton of the ANC ytem were algned on the reult of an optmzaton tudy whch wa performed at a former mock-up, refer to Ref. 8. The actve control algorthm wa mplemented on a rapd prototypng controller board (dspac 3). oreover, to evaluate the performance of the actve noe control ytem, two ear mcrophone of an artfcal head (AD acoutc S III) and mcrophone (Bruel&Kjaer B&K 488) were arranged to meaure the ound preure level n a montor volume around the uppoed head of a ttng peron, ee Fgure 9. The gnal were multaneouly analyzed wth a econd dspac 3 and a B&K LAN-XI front-end module. A B&K 694 6- channel DeltaTron condtonng amplfer wa ued a power upply for the N = 4 montor mcrophone. C. Actve Noe Controller For tonal actve noe control, dfferent control algorthm can be ued. For the preent eperment, a feedforward frequency doman mplementaton choen. The electon crtera are a robut behavor n the contet of uncertante, a low computatonal complety due to the large amount of tranducer, the acceblty of a reference gnal and mple parameter adjutablty. A mnor crteron n the preent applcaton the convergence performance due to the fact that frequency hft of the propeller engne wll be low n comparon to the algorthm tme contant. The controller may be able to control a fundamental frequency, the frt blade pang frequency (BPF), and two hgher harmonc frequence (BPF and 3BPF). The prncple cheme of the controller hown nde the grey bo of Fgure 6. The tme doman gnal of the reference () t and the error e () t are feed to the controller. Afterward, the error gnal e () t are tranformed n the frequency doman ung a Fat-Fourer-Tranformaton (FFT). The reference gnal ued to dcard the dle bn of the comple frequency pectrum. ere, at each dcrete tme tep n, the comple frequency bn of BPF = 3 are grouped to the accordng column matrce ( = ), BPF ( = ) and 3BPF ( ) k N ( n) e ( n),..., e ( n),..., e ( n) e =. (6) The comple loudpeaker actuaton gnal u are calculated n each controller mnmzaton of the controller cot functon J R R uppoed flght drecton C accordng to the teratve J = e Qe + u R u. (7)
The weghtng matrce Q and R n equaton (7) are dagonal matrce whch are ued to weght the mcrophone and loudpeaker ndependently. Thee weghtng have a mlar mathematcal effect a dfferent loudpeaker and mcrophone poton and change the phae and ampltude of the loudpeaker actuaton. Therefore, changng the weghtng matrce Q and R alo affect the redual noe at the montor mcrophone. Thee N + NL weghtng parameter of Q and R have to be adjuted durng the controller degn proce, ee chapter IV. The mnmzaton of the controller cot functon carred out by the teratve update equaton ( n + ) = [ µ ] ( n ) µ ( n) u I R u G Qe (8) wth the tep ze µ. The reultng comple actuaton gnal u are tranformed n the tme doman wth an nvere Dcrete-Fourer-Tranformaton (DFT). oreover, to enure a afe controller ervce n the cae of algorthm ntablty, a fal-afe functon ncluded n the controller. Th functon check at each teraton the magntude of the loudpeaker actuaton gnal and wtche the controller off f neceary. The real-tme mplementaton above wa realzed wth a ample-tme of 496 z. A ammng-wndow wth 48 coeffcent and ¾ overlappng were choen for the FFT. Th lead to a controller update rate of 8 z whch adequate for the low frequency hft of the engne. The computatonal load of the real-tme proceor wa about 7%. d( t) + G + e( t ) () t DFT FFT d + + e Fal-Safe u ( n ) u ( n) u 3 ( n) C C C 3 e ( n ) e ( n) e 3 ( n) dspac 3 Noe Source d C u opt + G G + e Fgure 6. The epermental actve noe controller mplemented on the dspac 3. Fgure 7. Frequency doman block dagram of the control ytem. IV. Controller Parameter Degn ethod The determnaton of the weghtng matrce Q and R n focu of the controller degn method. For each frequency, one et of weghtng matrce Q and R ha to be defned. Therefore, the followng method ha to be performed for each frequency. For the remander of the chapter the eplct dependency on the frequency wll be dropped for clarty. In Fgure 7, the block dagram of the control ytem hown. The redual ound preure at the error mcrophone e are the um of the prmary noe feld d and the product of the tranfer functon matr G wth the actuaton gnal u. Smultaneouly, the loudpeaker ectaton effect the ound preure at the N montor mcrophone va the tranfer functon matr G. Th lead to e= Gu+ d and e = Gu+ d. (9)
The prmary noe d and d are aumed to be ected va the ame tonal noe ource. Dfferentaton of the cot functon J R accordng to u lead to the teady tate controller tranfer functon matr C C = G QG + R G Q. (3) The problem of the degn method to affect the behavor of the controller to acheve the lowet redual error at the montor mcrophone e. ereby, the nomnal degn baed on the nomnal plant. On the other hand, the robut degn conder addtonal uncertante of the plant. A. Nomnal Controller Degn ethod In the cae of the nomnal controller degn the tranfer functon matrce a well a the prmary noe feld are aumed to be contant at ther nomnal value. Therefore, we can et G = G, = G G, d = d, d = d and uopt = u opt. (3) Wth th nomnal plant, the performance of the adaptve controller can be evaluated wth the energetc mean redual ound preure level at the N montor mcrophone. ence, the redual mean quared error defned a the performance crteron whch ha to be mnmzed: mnmze QR, log e. (3) In the ene of an optmzaton problem, Q and R are the optmzaton varable and the redual ound preure n the montor volume the objectve functon. oreover, th optmzaton ha to be performed accordng to contrant. The control tablty of the teepet decent adaptve controller can be guaranteed f the mallet egenvalue λ mn of the matr G QG + R potve 9. Th fact ued a the tablty contrant, ee q. (33) b). oreover, the loudpeaker actuaton hould not ma be larger than the mamum allowed actuaton ampltude u opt whch lead to the contrant c) n q. (33). Fnally, to acheve a conve controller cot functon J R, the weghtng matrce Q and R ha to be potve defnte. Th lead to the contrant d) and e) n q. (33) where denote the potve egenvalue of the matrce. Ung the q. (3), (3) and the contrant, the nomnal degn problem can be wrtten a an optmzaton problem: Q, R NL = ma opt ( ) mnmze log G u Q, R + d a) opt { { G QG R} } ubject to: mn eg + >, b) u NL ma u >, c) = ( ) opt() Q, d) R, e) opt = + u Q, R G QG R G Qd f). (33) Th nonlnear contraned optmzaton problem can be olved by ung a genetc algorthm. The optmal weghtng matrce Q and R are found when the contrant are atfed and the bet control performance acheved. B. Robut Controller Degn ethod The mplfed mathematcal model of the plant above are an abtracton of the dynamc behavor of real tmevarant ytem wth weak nonlnearte. It aumed that the real phycal plant vare around the nomnal plant.
The mplemented model n the controller equate the nomnal model. ence, accordng to quaton (3), we et for the phycal plant G = G + G G = G + G u = u + u opt opt opt d = d + d d = d + d (34) where the varable labeled wth repreent the addtve modelng error. ere, the modelng error are arbtrarly and only known n ther norm. Smlar to the optmzaton problem above, a robut optmzaton problem defned whch combne the robut control tablty and robut control qualty crteron. The oluton of the optmzaton problem ha to guarantee the tablty and ha to acheve the bet performance for all norm-bounded uncertan plant. In other word, the objectve functon and the contrant hould be evaluated for the wort-cae tablty and wort-cae performance. The wort-cae can be found by olvng a mamzaton problem accordng to the varable G, d, G and d. owever, uch a mnma optmzaton problem not olvable n an effcent manner. On the other hand, the wort-cae can be etmated after a longer calculaton by algebrac formula, refer to Ref.. For eample, the wort-cae actuaton can be appromated by ung the Sherman-orron-Woodbury formula for an etmaton of the varable u opt opt (, ) (, ) u Q R C d+ C Gu Q R. (35) oreover, ung the trangle nequalty and neglectng the quadratc term lead to an etmaton of the redual noe at the montor mcrophone: opt Guopt + d +... ma ( G + G ) uopt ( Q, R) + d + d ma G uopt + ma G C d +... G d ma GC Guopt + ma d G d. (36) quaton (36) can be ealy etmated f the norm of G, d, G and d are known. For eample, f the norm d known, the thrd term can be appromated n the followng manner: ma d G C d < G C d. (37) F The wort-cae tablty follow by the lower bound of the mallet egenvalue of the matr GQ( G + G) + R, ee []. Th egenvalue can be etmated by ( ) λ ( ) ˆ λ QR, = QR, G Q G (38) where λ are the egenvalue and are the egenvector of the matr G QG + R, ee Ref. 3. Accordng to the optmzaton problem (33), the robut optmzaton problem baed on the wort-cae etmaton: Guopt + d +... mnmze log ma Guopt + ma G C d +..., QR G d ma GC G uopt + ma d G d 3
ubject to opt = + + opt u G QG R G Qd u Q R (39) NL mn ˆ λ > = u ma opt NL ma u >. = opt() The optmzaton varable are tll the weghtng matrce Q and R. owever, the optmzaton now performed for the wort-cae appromaton. In addton to the nomnal plant, the norm of perturbaton of G, d, G and d are neceary. V. Actve Noe Control perment For the actve noe control eperment, one load cae condered. The fundamental frequency of the prmary noe feld et to 9 z. In addton, two hgher harmonc are ected at 84 z and 76 z. The ound preure level at the artfcal head adjuted to 75 db(a) for each frequency. Th level choen to guarantee the lnear behavor of the equpment. The tranfer functon and the prmary noe feld are meaured for the nomnal condton. To mulate the perturbaton caued by dfferent peron and object nde the cabn, the tranfer functon and prmary noe feld are perturbed wth three dfferent ound hard dffractng object upended on the celng of the loadmater area, ee Fgure 8. The meaured medan norm of the uncertanty of the prmary noe feld and the tranfer functon hown n Table for the condered load cae. A decrpton of the meaurement procedure can be found n Ref. 4. Table. Referenced medan uncertanty norm of the prmary noe feld and the tranfer functon. 9 z 84 z 76 z G G.36.6.39 F F d d.47.5.46 G G.34.49. F F d d.48.68.7 Subequently, the controller weghtng matrce Q and R are optmzed accordng to (33) and (39) ung atlab wth the Genetc Algorthm and Drect Search Toolbo. The nonlnear contrant are ncluded n the nonlnear cot functon wth the barrer functon method. Fnally, the controller parameter are coped to the controller. After the controller calbraton, the controller ready to attenuate the prmary noe feld. Fgure 8. Dffractng object upended on the celng of the loadmater area. Fgure 9. crophone n the montor volume around the artfcal head to evaluate the control performance. 4
The control proft meaured wth the montor mcrophone around the artfcal head, ee Fgure 9. The reult are hown n Fgure. ere, a gnfcant energetc mean noe reducton up to db can be acheved for 9 z and 84 z. ereby, the noe reducton of the nomnal controller degn hgher than the robut controller degn method. Only a mnor noe reducton acheved wth the nomnal degn method at 76 z. The robut controller degn method can acheve no noe reducton. owever, wth uncertante the robut controller degn method wll acheve a more relable noe reducton. 9 z Prmary Noe Feld 84 z 76 z 9 z 84 z 76 z Robut Degn Nomnal Degn Fgure. Sound feld mappng around the artfcal head for the prmary noe feld a well a the control performance acheved wth nomnal controller degn and the robut controller degn. VI. Concluon In the paper, an eperment etup to repreent the acoutc tuaton nde a turbo-prop arcraft. ere, the couplng of the mall loadmater area to a large compartment play an mportant role. The energy baed analy how that the prmary noe tranmt through the couplng urface very well. Therefore, hgh ound preure level are epected due to prmary ectaton. On the other hand, econdary noe whch generated by the loudpeaker leak to the cargo hold. Therefore, the loadmater area wll be hghly damped depte the ound hard materal n the loadmater area wall. Th ndcate that the econdary loudpeaker have to provde hgh capacte. The tranfer functon between the econdary ource and control mcrophone are mooth wth low reonance peak. Th 5
advantageou for the controll tablty. Secondly, the ubtructure baed analy how that only a mall amount of ndependent mode hape nfluence the vbraton behavor nde the loadmater area. Thee couplng analy ndcate that an acoutcal mock-up of the loadmater area placed nde a large laboratory room can mulate the mot mportant acoutc effect nde the arcraft. A prototype actve noe control ytem ntalled n the loadmater area. ere, a teepet decent algorthm ued to control the loudpeaker actuaton. oreover, two new degn method of the optmal parameterzaton of the controller weghtng matrce Q and R are preented. The frt degn method take only the nomnal cae nto account. The econd degn method alo conder the uncertanty of the prmary noe feld and the tranfer functon n the degn proce. Frt epermental reult how the capablte of thee new method. In future, the hgher robutne of the robut weghtng approach wll be hown. oreover, a non-dagonal tructure of the weghtng matr Q and R wll be analyzed. Reference uropean Parlament, Drectve 3//C of the uropean Parlament and of the Councl of 6 February 3 on the mnmum health and afety requrement regardng the epoure of worker to the rk arng from phycal agent (noe). 3, Offcal Journal of the uropean Unon. Kuo, S.. and D.R. organ, Actve Noe Control Sytem: Algorthm and DSP Implementaton. Wley Sere n Telecommuncaton and Sgnal Proceng. 996, New York: John Wley & Son, Inc. 3 eyer,. and.-g. Neumann, Phykalche und Technche Akutk. 979: Veweg Verlag. 4 Cremer, L. and.a. üller, De wenchaftlchen Grundlagen der Raumakutk. Vol. I. 978, Stuttgart: rzel Verlag. 5 5. Crag, R.R. Couplng of Subtructure for Dynamc Analye: An Overvew. n AIAA Dynamc Specalt Conference.. Atlanta, GA: AIAA Paper. 6 Kochan, K. Regelalgorthmen zur aktven Schallredukton. n DAGA 8-34. Jahretagung für Akutk. 8. Dreden, Germany. 7 Böhme, S., et al. ock-up of a Loadmater Area for Acoutc Ground Tet. n t CAS uropean Ar and Space Conference. 7. Berln, Germany. 8 Kletchkowk, T., et al., Optmzed actve noe control of emcloed arcraft nteror. Internatonal Journal of Aeroacoutc, 7. 6(): p. 6-7. 9 llott, S.J., C.C. Boucher, and P.A. Nelon, The Behavor of a ultple Channel Actve Control Sytem. I Tranacton on Sgnal Proceng, 99. 4(5). De Jong, K.A., volutonary Computaton. 6, Cambrdge, A, USA: IT Pre. Kochan, K., Robut Parameterzaton of Adaptve ult-channel Actve Noe Controller, n Department of echancal ngneerng. elmut-schmdt-unverty: amburg, (unpublhed) Golub, G.. and C.F. van Loan, atr Computaton. 3 ed. 996, Baltmore, aryland: John opkn Unverty Pre. 3 Stewart, G.W. and J.-g. Sun, 99. atr Perturbaton Theory, San Dego: Academc pre. 4 Kochan, K.,. Wezel, and D. Sachau. valuaton of the wort-cae Stablty and Performance of an uncertan Actve Noe Control Sytem. n Actve 9, 9 Internatonal Sympoum on Actve Control of Sound and Vbraton. 9. Ottawa, Canada.(unpublhed) 6