Active Compensation of Transducer Nonlinearities

Transcription

1 Actve Compensaton of Transducer Nonlneartes Wolfgang Klppel Klppel GmbH, Dresden, 01277, Germany, ABSTRACT Nonlneartes nherent n electromechancal and electroacoustcal transducers produce sgnal dstorton and lmt the mamal ampltude of the output sgnal. Assessng the large sgnal performance has been a subject of acoustcal research for many years provdng nonlnear models and new methods for the measurement of the large sgnal parameters. The dentfed model allows predcton and smulaton of the nonlnear behavor and drect comparson wth measured symptoms. The good agreement between model and realty s the bass for developng novel dgtal controllers dedcated to transducers that compensate actvely for nonlnear dstorton by nverse preprocessng of the electrcal nput sgnal. Ths paper gves a summary on the actvtes n the last 15 years and new challenges of the future. 1 Introducton All loudspeakers operated at suffcently hgh ampltudes produce sgnal dstorton whch are not generated n the small sgnal doman. Ths knd of dstorton s a symptom of nonlneartes nherent n the loudspeaker. It mpars the perceved sound qualty and ndcates that the loudspeaker s operated close to the physcal lmts. If the dstorton become audble the lstener usually attenuates the sgnal to protect the loudspeaker aganst thermal or mechancal damage. In the past the large sgnal doman s not ntended for normal operaton but s more used as a safety range or head room. However the on-gong trend to smaller loudspeakers gvng more output at hgher qualty make t necessary to eplot any unused resources and to cope wth nonlnear and thermal mechansms n loudspeakers. The large sgnal modelng and the measurement of the loudspeaker parameters reveal the causes of the dstorton and the relatonshp to the results of tradtonal dstorton measurements. Ths nformaton s crucal for the desgn of the drver and the passve loudspeaker system. However, a lnear loudspeaker s not necessarly optmal. If the sze, weght, effcency and mamal output s also a pont of concern, some of the loudspeaker nonlneartes are accepted as desred propertes and are separated as regular nonlneartes from ecessve nonlneartes caused by loudspeaker defects [42]. For eample a systematc offset of the voce col poston or an asymmetrc suspenson produces ecessve dstorton and unstable behavor. Ths can avoded or easly fed by a proper desgn and consstent manufacturng. The regular loudspeaker nonlneartes are the man target of actve lnearzaton and other forms of electrcal control. Ths paper gves a summary on the actvtes n ths feld. 2 Glossary of Symbols The followng symbols are used wthn the paper: u v a p T v z q a d R e(t v) L e M ms R ms K ms() nput current voltage at termnals voce col dsplacement velocty of voce col acceleraton sound pressure ncrease of voce col temperature state vector transformed state vector used n normal form volume velocty of the ar n the port dstorton generated n the loudspeaker model electrcal voce col resstance at DC dependng on voce col temperature voce col nductance at low frequences mechancal mass of drver daphragm assembly ncludng ar load and voce col mechancal resstance of total-drver losses mechancal stffness of drver suspenson C ms() = 1 / K ms() mechancal complance of drver suspenson Bl() B F m(,) force factor (Bl product) magnetc nducton reluctance force due to varaton of L e()

2 f s M P C B R P s=jω resonance frequency of the mechancal system acoustc mass of ar n the port, acoustc complance of ar n enclosure, acoustc resstance of port losses, Laplace operator, H (s)=x(s)/u(s) lnear transfer functon between voltage and dsplacement, H C(s) lnear system functon of the controller, K(s) loop gan n servo control H I(s) nput flter n servo control h g G (s) P A τ nonlnear subsystem n loudspeaker model nonlnear subsystem n controller lnear subsystem n controller parameter vector nonlnear states tme delay 3 Bascs of loudspeaker lnearzaton 3.1 Nonlneartes n Loudspeakers The loudspeaker may be consdered as a sngle-nput-multple-output system (SIMO) as llustrated n Fg. 1. Usng an amplfer wth low mpedance output the voltage u at the loudspeaker termnals may be consdered as the sngle nput whle the sound pressure sgnal p(t, r ) at the pont r n the sound feld s one of the multple outputs. The sgnal flow between nput and output may be modeled by a chan of lnear and nonlnear subsystem, llustrated by thn and bold blocks, respectvely, n Fg. 1. The power amplfer, passve crossover and other electronc crcuts (e.g. for protecton) may be consdered as lnear at the ampltudes used n pras. The followng electromechancal transducer s nonlnear usng a movng col n a statc magnetc feld and a mechancal suspenson system. The voce col dsplacement may be consdered as the output of ths subsystem and s also the state varable that s drectly related wth the nonlneartes nherent n ths block. Ths voce col current at the electrcal termnals reflects not only the state of the electrcal parts but also the mechancal vbraton. Radaton Sound Propagaton Room Acoustcs p(r 1 ) Audo sgnal Amplfer Crossover u Electromechancal Transducer Mechanoacoustcal Transducer Radaton Sound Propagaton Room Interference p(r 2 ) sound feld Radaton Sound Propagaton Room Interference p(r 3 ) Fg. 1: General model descrbng the basc sgnal flow n loudspeakers usng lnear (thn) and nonlnear (bold) subsystems. The mechano-acoustcal transducer uses the daphragm, the enclosure or a horn to transform the voce col dsplacement nto an ar flow. The ar n a sealed bo or the ar between daphragm and phase-plug n a horn loaded compresson drver behaves nonlnear f the varaton of the ar volume s not neglgble. In many loudspeakers the radaton of the sound has to be modeled by multple subsystem whch comprse dfferent nonlneartes and dsperse the sgnal n dfferent drecton. For eample, a drver mounted n a vented enclosure or coupled wth a passve radator has two separate sound sources. The nonlneartes of the ar flow n the port or the nonlnear suspenson of the passve radator wll produce dstorton whch are not n the sound radated from the daphragm of the actve radator. The Doppler effect depends on the radaton angle and produces phase modulaton between low and hgh frequency components. The radaton n as of the drver produces mamal Doppler dstorton because the drecton of radaton concdes wth the dsplacement of the cone. A low frequency component also produces ampltude modulaton of a hgh-frequency component f the cone moves wth respect to the acoustcal boundares (frame, enclosure, baffle) and the radaton condton changes. In horn compresson drvers the propagaton of the sound wave becomes nonlnear f the sound pressure s hgh. A pressure mamum travels faster than a pressure mnmum causng a gradual steepenng of the wave front. The nteracton wth the room, the superposton of the drect sound wth reflected waves may be modeled by a lnear system because the ar behaves suffcently lnear. 3.2 Constrants n loudspeaker lnearzaton The actve lnearzaton technques use an electrcal controller connected to the nput of the loudspeaker. If the nonlneartes are located n the subsystems connected n seres to the nput and all of the remanng subsystems n the output branches behave lnearly then the sound pressure n the complete sound feld can be lnearzed. However, f the nonlnearty n the radaton and sound propagaton depends on the path of the wave then the sound pressure can be lnearzed at one pont only. Fortunately, most of the domnant nonlneartes n the motor, suspenson and n a horn can be modeled by a sngle nput and sngle output system and can be compensated before the sgnal s dspersed. The Doppler effect s an eample where actve compensaton s lmted to a certan radaton angle. Here nonlnear 2

3 control s lmted n the same way as the equalzaton of the lnear ampltude response. 4 Analogue lnearzaton technques Snce the loudspeaker s stll an analogue devce, a controller realzed wth analogue electroncs s the frst choce for actve lnearzaton. V(s) H I (s) - U(s) H C (s) D(s) H (s) Nonlnear System s 2 X(s) A(s) The transfer functon H C(s) of the controller has to be adjusted to the transfer functon of the loudspeaker to ensure stablty. A hgh-qualty sensor s permanently requred whch montors the dsplacement, velocty or voce col acceleraton. The measurement of the sound pressure s not practcal for two reasons: Frst, a mcrophone even located n the near feld of the drver wll cause a small tme delay and the correspondng phase shft n K(s) requres an attenuaton of the K(s) at hgher frequences to preserve stablty n the feedback. Second, ambent nose montored by the mcrophone wll be compensated at the mcrophone poston but the amplfed sgnal wll also be radated to other ponts n the sound feld. Servo Controller Loudspeaker A malfuncton of the sensor or electroncs n controller may also damage the loudspeaker. A protecton system s requred to attenuate the nput sgnal when thermal or mechancal overload may damage the speaker. Fg. 2: Servo control of a loudspeaker by usng feedback of voce col acceleraton 4.1 Servo control usng output feedback The frst approach [1] - [7] to actve loudspeaker lnearzaton used negatve feedback of the measured output sgnal as llustrated n Fg. 2. The loudspeaker s modeled by a lnear path comprsng the lnear system H (s) and second-order dfferentator (represented by the squared Laplace operator s 2 ) transformng the voce col dsplacement X nto the acceleraton A. A nonlnear system models the effect of the motor and suspenson nonlneartes dependng on the voce col dsplacement. The output D may be nterpreted as nonlnear dstorton added to the nput voltage U. The acceleraton A of the cone may be measured by usng an accelerometer and suppled to the servo controller. The controller comprses a frst system H C (s) to realze a desred open loop gan K( s) s 2 = H C ( s) H ( s). The magntude of the transfer functon 2 A( s) H ( s) s = D( s) 1 + K( s) between the Laplace transformed dstorton D(s) and the acceleraton sgnal A(s) may be reduced by ncreasng the loop gan K(s) n the denomnator. Thus the dstorton can be attenuated by servo control wthout modelng the nonlnear mechansms n greater detal. The lnear system H I(s) at the controller nput s used to realze a desred lnear transfer functon A( s) H I ( s) H ( s) s = V ( s) 1 + K( s) between controller nput V(s) and acceleraton A(s). Usng H I(s)=1+K(s) the controller preserves the orgnal small sgnal behavor (resonance frequency and loss factor) but compensates for the loudspeaker nonlneartes only. However, servo control appled to loudspeakers has drawbacks: 2 (1) (2) (3) V(s) H I (s) H C (s) Servo Controller - V(s) Nonlnear Detector I(s) D(s) H (s) Nonlnear System Nonlnear System Transducer s 2 X(s) A(s) Fg. 3: Servo control of a loudspeaker by usng measured voce col current n the feedback loop 4.2 Servo control usng current feedback The electro-dynamcal transducer generates a back EMF defned by Bl()v at electrcal sde, dependng on the nstantaneous force factor Bl() and the voce col velocty v=d/dt. Connectng the transducer to a low mpedance source (voltage drve) the back EMF effects the measured voce col current. Thus an electro-dynamcal transducer may be used as actuator and sensor at the same tme. However, the relatonshp between velocty v and current s nonlnear due to the force factor varaton Bl() versus. A feedback of the dstorted current nduces addtonal dstorton nto the loudspeaker. The same problem occurs f the back EMF s separated by a brdge arrangement or montored by an addtonal sensng col [4] as long as a constant force factor Bl()=Bl(0) can not be guaranteed. Thus, usng a real loudspeaker as sensor tself requres a detector performng a nonlnear processng of the back EMF [29] as shown n Fg. 3. However, the desgn of the nonlnear detector requres a relable physcal model and means for measurng the large sgnal parameters precsely. Ths technque can not be realzed by usng analogue electroncs only and wll be dscussed n greater detal below. 4.3 Current drve Instead of drvng the loudspeaker wth a defned voltage provded by a low mpedance source t s also possble to drve the loudspeaker 3

4 wth a current source controlled by the nput sgnal [8]. The current source has a much hgher output mpedance than the nput mpedance of the loudspeaker. Thus, varatons of the voce col nductance L e() versus dsplacement have no effect on the nput current and the mechancal drvng force Bl(). However, the practcal beneft of current drve s questonable. The reluctance force F 1 dl ( ) 2 d e 2 rel =. whch s a second effect of the nonlnear nductance L e(), and other mechancal or acoustcal nonlneartes can not be compensated by current drve. The varaton of L e () versus may be reduced by usng a shortng rng at the optmal place. Ths s a cost effectve soluton whch solves the cause of the problem. 5 Lnearzaton wth generc adaptve flters The analogue lnearzaton technques fal f there s tme delay generated between loudspeaker nput and sensor output. For eample, n horn loaded compresson drvers the sound pressure sgnal measured at a pont where the wave steepenng s completed can not be used as a feedback nput wthout causng an nstablty n the loop. In ths case a lnearzaton can only be accomplshed as a feed-forward controller as shown n Fg. 4. No state varable measured at the loudspeaker s used as a feedback sgnal. The transfer behavor of the controller can be adjusted by the parameter vector P provded by an adaptve detector crcut. The detector crcut estmates the optmal controller parameters by usng the nput sgnal u and the output sgnal p. Ths arrangement guarantees self-tunng and an optmal adjustment of the controller whle reproducng an audo sgnal. However, the detector may be deactvated at any tme and the controller stays operatve lke a flter usng the stored parameters P. Both the controller and the detector use a nonlnear feed-forward model of the loudspeaker. If no nformaton about the physcs of the nonlnear mechansms s avalable then a generc model has to be used. Nonlnear control and system dentfcaton provde a large varety of possble canddates [10] - [20]. In the applcaton to loudspeakers the followng requrements have to be consdered: Stablty of the feed-forward controller and the adaptve parameter update system, handlng of substantal tme delay n the transducer/sensor system, the memory of the generc model should cover the mpulse response of the loudspeaker, modelng of hgher-order dstorton, optmal parameter adjustment preventng stallng at local mnma. A detector meetng these requrements s depcted n Fg. 4. It comprses a lnear adaptve FIR flter wth the transfer functon H ln (s) connected n parallel to a nonlnear adaptve flter wth a lnear weght network. The output of the lnear flter y = P. l T l s the scalar product of the lnear parameter vector P l and the state vector (4) (5) T = [ ut ( ), ut ( τ), ut ( 2 τ),..., ut ( t )]. comprsng the tme-delayed samples of the tme-dscrete voltage sgnal. The nonlnear flter epands the state vector nto a nonlnear state vector A by usng, for eample, a polynomal epanson such as T A = T( ) = [,,...,,,...,,..., ] (7) N N N or a neural network wth fed parameters n the hdden layers. Lnear weghtng of A wth the nonlnear parameter P n gves the output of the nonlnear flter y = PA. n To estmate the parameters n both flters the error sgnal T n e= p p p. l s calculated as the dfference between loudspeaker output p and the sum of both model outputs. Searchng for the mnmum of mean squared error n ( ()) 2 MSE J = E e. (6) (8) (9) (10) where the gradent of J versus the parameters becomes zero result n the steepest-descent algorthm provdng optmal estmates on the lnear parameter vector P( + 1) = P( ) +µ e (11) and the nonlnear parameter vector P ( + 1) = P ( ) +µ ea. n n l (12) The nonlnear controller n Fg. 4 contans a copy of the nonlnear flter provded wth the nonlnear parameters P n. The followng lnear flter wth the nverse transfer functon H ln(s) -1 compensates for the lnear transfer of the syntheszed dstorton d through the loudspeaker to accomplsh a cancellaton of the real dstorton at the loudspeaker output. Usng adaptve flters wth a feed-forward structure have some dsadvantages: 1. If the ampltude of the syntheszed dstorton d are not small then the nput sgnals v and u at the two nonlnear feed-forward system wll dffer and wll produce a msmatch between the syntheszed and real dstorton. Ths lmts the applcaton of ndrect parameter updatng to medum ampltudes. 2. The compensaton of thrd- and hgher-order dstorton requres a large number of nonlnear parameters. Snce the model has to be mplemented n the controller as well as n the detector ths approach wll be lmted to second- and thrd-order polynomal epanders to make the mplementaton feasble. 3. The lnear modelng of a loudspeaker reproducng the full audo band requres a lnear FIR wth many parameters. The 4

5 number of the nonlnear parameters n a polynomal flter of nth-order wll rse wth the nth-power of the taps of the lnear flter. 4. The parameters and state varables used n a generc structure does not drectly correspond wth the electrcal, mechancal or acoustcal mechansms n a loudspeaker and can not be used to protect a loudspeaker under overload condtons. 5. The nput-output lnearzaton based on generc structures requres that the output sgnal can be observed and measured by a sensor. Observng an nternal state varable (e.g. nput current) and lnearzng the output sgnal (e.g. sound pressure) at the same tme s not possble because the relatonshp between the nternal state and the output s unknown. v Audo sgnal ep(-τs) d u amplfer p Lnear Parameter Estmator Delay Lne Nonlnear Epander H ln -1 P l P l y l Controller Delay Lne X Nonlnear Epander A P n y n P n Detector Nonlnear Parameter Estmator Fg. 4: Adaptve ndrect control of a loudspeakers usng a generc feedforward structure both n controller and detector 6 Lnearzaton based on loudspeaker modelng H 1 (s) p(r 1 ) The results of loudspeaker modelng n the large sgnal doman revealed detaled nformaton about the dstorton generaton whch can be used to compensate loudspeaker dstorton n a most effectve way. However, ths approach leads to specal controllers dependng on the transducer prncple, the radaton ads used and ntended applcaton. 6.1 Autonomous subsystems connected n seres u 0 H 0 (s) h h N H 2 (s) p(r 2 ) y 1 y h y N y N+1 H 3 (s) p(r 3 ) The dscusson of the general sgnal flow chart n Fg. 1 revealed that the relatonshps between sngle nput and multple output can only be lnearzed f all nonlneartes may be concentrated n a nonlnear subsystems h wth = 1,... N connected n a seres as shown n Fg. 5. The relatonshp between the sgnal y N+1 and the multple outputs p(r ) n the sound feld should be lnear transfer functon H (s). The AC-coupled amplfer and the passve crossover are consdered n the lnear system H o (s) between audo sgnal u o and the sgnal y 1 at the termnals of the electro-mechancal transducer. Fg. 5: Basc modelng of a loudspeaker by usng lnear and nonlnear subsystems If the nonlneartes n the electrcal, mechancal or acoustcal doman are only related va the state varables y for =1,..N they may be separated n autonomous subsystems. For eample, the motor nonlneartes whch are part of the subsystem h 1 have to be separated from the nonlnear radaton (Doppler effect) n a successve subsystems h 2 provded wth voce col dsplacement y 2= [24]. The nonlnear sound propagaton may be modeled by the followng subsystems h 3,..., h N. The wave steepenng n horn loaded compresson drvers s modeled n [25] by such a seres of nonlnear subsystems representng successve sectons of the horn. The nonlnearty n each horn secton has a dstnct asymmetrcal characterstc and produces second-order dstorton predomnantly but the followng sectons wll transform the second-order dstorton nto thrd- and hgher-order dstorton. Contrary, the domnant nonlnertes n electro-dynamcal transducers such as force factor Bl(), voce nductance L e () and mechancal 5

6 complance C ms() nterfere wth each other va the dsplacement and can not be separated n two autonomous subsystems. A controller connected to the nput of the loudspeaker modeled n Fg. 5 gves a lnear nput-output relatonshp of the overall system f the controller has the followng propertes: 1. If the loudspeaker model comprses N nonlnear subsystem then there est for each subsystem h wth =1,..., N a correspondng nonlnear subsystem g =1,..., N and a lnear subsystem G -1 n the controller to ensure that the tme delayed output sgnal u ( t τ ) y ( t) = 1,..., N (13) = 2. The lnear subsystem G 0 ( 0 τ 0 1 s) = H ( s) ep( s( )) (14) at the output of the controller s the nverse of the lnear flter H o(s). 3. A constant tme delay τ -1 for =1,..., N may be added to the lnear system functons G -1(s) to make the system causal. Any lnear system G N(s) connected to the nput of the controller and the lnear system H N+1(s) after the nonlnear subsystems n the loudspeaker model have no nfluence on the lnearzaton of the output sgnal. of each nonlnear control system g s equal wth the correspondng nput sgnals y of the nonlnear subsystem h of the loudspeaker model. u N G N (s) g N G g G 0 (s) H 0 (s) 1 h 1... h y 1 y 2... H N+1 (s) v u N-1 (s) N... g G N u -1 (s)... v N-1 v v -1 v 1 u 1 u y y 0 y+1 h y N N N+1 p(t, r 1 ) Controller Transducer Fg. 6: Actve lnearzaton of a loudspeaker comprsng separated nonlneartes 6.2 Lnearzaton of an autonomous subsystem The comple task of lnearzng a loudspeaker wth multple nonlnartes requres the ndependent lnearzaton of each subsystem h by a nonlnear control law g whch transforms the nput sgnal v nto the output sgnal u by usng an nternal state vector '. The nternal states ' n nonlnear control subsystem g correspond wth the nternal states n the subsystem h of the loudspeaker model ' ( t τ ) = ( t) = 1,..., N (15) If there s any tme delay (τ >0) between controller output u and transducer nput y +1 for =1,..., N, then the controller states ' can not be measured drectly but have to be syntheszed by usng the controller nput u +1 or output u. The structure of the control law g and the vector ' depends on the physcal mechansms n subsystem h and the mathematcal tools used n the modelng: Integro-dfferental equaton The relatonshp between nput y and output y +1 of the system h can be descrbed by an ntegro-dfferental equaton usng natural state varables of the physcal system such as current, dsplacement, sound pressure n vector '. The nonlnear operatons are performed n the tme doman by performng multplcatons between state varables (e.g. dsplacement ) and the nonlnear parameters (e.g. force factor Bl()) whch are statc nonlnear functons wthout any memory. Some of the lnear dynamcs may be separated from the nonlnear dynamcs by usng lnear system functons whch are transformed va the nverse Laplace transform nto the tme doman. Ths representaton s very useful for modelng acoustcal processes wth tme delay and hgh number of modes (e.g. horn propagaton and vbratons on a panel). Usng ths representaton the control law g can be derved by the followng steps: 1. Representng the nonlnear subsystems between control output u and transducer output y +1 by an ntegrodfferental equaton E T 2. The desred lnear overall system between control nput v and transducer output y +1 s represented by an ntegrodfferental equaton E D. 3. Combnng equatons E T and E D gves the control law between control nput v and control output u. 4. The state varables used n the control law are generated from the control nput v or control output u dependng on the state generaton n h. Ths technque has been used for the dervaton of specal mrror flters compensatng for domnant drver nonlnartes, Doppler effect [24] and nonlnear sound propagaton [25]. Ths approach s powerful and general but requres some ntuton and eperence n representng the equaton E T and E D n an approprate form. The resultng control law comprses a mnmal number of terms whch can be mplemented n a dgtal processng system by usng lnear flters, statc nonlnartes (look-up tables, power seres epansons), adders and multplers. Each term compensates for a separate nonlnear effect and s nterpretable from a physcal pont of vew. 6

7 6.2.2 State-space representaton Suykens [27] suggested to use the standard geometrcal control technque to derve the nverse dynamcs (ID-controller) and the lnear dynamcs (ID-controller) from the state-space model of h. Instead of usng two separate controller parts to compensate for the complete dynamcs frst and then to rentroduce the desred lnear dynamcs t s preferable to derve the control law n the drect form [29]: 1. The nonlnear subsystems between control output u and transducer output y +1 s represented by a state space model S T usng the state vector '. 2. Transformaton of the state space model S T nto the normal form S N by usng a dffeomorphsm T whch transforms nto a new state vector z. 3. The desred overall system between control nput v and transducer output y +1 s represented by a lnear state space model S D n normal form usng state vector z. 4. Combnng the state space models S N and S D gves the control law n the drect form between control nput v and control output u. 5. The state varables ' can be measured at the transducer drectly f there s no tme delay τ between the control output u and system nput y (commonly known as statc state feedback control). In most loudspeaker applcatons the state vector ' has to be estmated by usng an observer or a state predctor. ' The control law s completely statc n the state varables and may be realzed by usng only statc nonlneartes, adders and multplers. Unlke the control law derved from the ntegro-dfferental equaton there are no addtonal lnear systems (dfferentators, flters). However, despte dfferences n the form both control laws are dentcal n prncple and allow perfect lnearzaton of the modeled nonlneartes. The man advantage of ths approach s that f the system h can be represented by a state space model, then formal tools can be used to derve the control law and to check ts stablty. Unfortunately, ths approach s less convenent f there s unmodelled dynamcs (e.g. vbraton on a panel) nvolved, the number of modes and state varables s hgh and tme delay has to be consdered. For ths reason ths approach has been appled to woofers only. the lnear, second-order and thrd-order system functon from the dfferental equaton. Usng ths approach a control law may be derved n the followng way: 1. The nonlnear subsystems between control output u and transducer output y +1 s modeled by a Volterra seres epanson of nth-order based on an analytcal representaton of the lnear and hgher-order system functons H (s 1,...,s ) wth =1,..., n. 2. The control g s modeled by a Volterra seres epanson of nth-order comprsng a lnear, quadratc and hgher-order homogenous power system connected n parallel. Each homogenous power system s represented by the system functons G (s 1,...,s ) wth =1,..., n. 3. The transfer functon H (s 1,...,s ) s transformed nto the nverse system functons G (s 1,...,s ) for =1,..., n by usng G ( s,..., s ) 1 H ( s,..., s ) (16) 1 = H ( s ) Based on the analytcal structure of G (s 1,...,s ) a dscrete nonlnear system s syntheszed comprsng lnear systems (flter, dfferentator) statc nonlneartes, adders and multplers. Ths approach results n a consequent feed-forward structure where the states are syntheszed mplctly form the control nput v. Ths guarantes stablty of the control system g for any choce of the parameters. However, the hgher-order systems functons comprse a large number of terms whch can not be mplemented n avalable dgtal platforms. Controllers usng only second- and thrd-order systems functons only provde an "appromatve" lnearzaton of h lmted to low and medum ampltudes. 6.3 Eample: Electro-mechancal transducer The dervaton of the control law s dscussed on the electromechancal transducer usng the electro-dynamcal prncple. Ths eample s chosen because the mrror flter desgn and classc control desgn lead bascally to the same control low gvng eact lnearzaton of the modeled nonlneartes. The man dfference n the two approaches s the generaton of the state varables X, the stablty and robustness of the controller and the sutablty for adaptve parameter adjustment resultng to a self-learnng system Volterra Model Kazer [22] suggested to model the loudspeaker under voltage and current drve by a Volterra seres and derved analytcal epresson for 7

8 F m (,) R e L e () p ' M ms 1/K ms () R ms q P /S D S D2 M p u Bl()' Bl() Bl() p A S D C B /S D 2 S D 2 R P Fg. 7: Equvalent Crcut of the vented-bo loudspeaker system Equvalent Crcut At low frequences the electro-mechancal transducer may be modeled by an equvalent crcut wth lumped elements because the wavelength s large n comparson to the geometrcal dmensons. The equvalent crcut of a vented-bo system s shown n Fg. 7. The DC-resstance R e and the voce col nductance L e () and the back EMF represent the electrcal doman whle neglectng rregulartes of the nput mpedance (para-nductance) caused by the generaton of eddy currents n the pole peces and the voce col former. For an electro-dynamcal motor wthout shortng rng or copper cap on the pole pece the nonlnear parameter L e () has a dstnct asymmetrc characterstc gvng hgher nductance for negatve voce col dsplacement (movng towards the back-plate). The force factor Bl() represents the electro-dynamcal couplng between electrcal and mechancal sde. The Bl() has a symmetrcal characterstc f the magnetc nducton B s unform n the gap, there s a symmetrcal frnge feld and the col s at optmal rest poston. The varaton of Bl() versus voce col current causes an effect called flu modulaton. However, the varaton of Bl() s neglgble as long as the statc part of the nducton B generated by the magnet s much larger than the AC part generated by the voce col current. On the mechancal sde the electro-dynamcal drvng force Bl() and electro-magnetc drvng force gvng n Eq. (4) ecte the mass-sprng system and the acoustcal resonator transformed to the mechancal doman. The stffness of the mechancal suspenson s assumed as nonlnear dependng on the voce col dsplacement. Nonlneartes of the R ms are neglected because the electrcal dampng s usually domnant over the mechancal dampng n loudspeakers connected to a voltage source State space representaton The nonlnear dfferental equaton of the analogous crcut wrtten n the general state space form s & = a( ) + b( ) y (17) 1 y = h( ) 2 where y 1=u s the voltage at the termnals of the voltage drven loudspeaker, y 2 s the output correspondng to the voce-col dsplacement. The state vector of the system = [ 1, 2, 3, 4, 5] T =[, d/dt,, q P, p A ] T comprses dsplacement and velocty v=d/dt of the voce-col, the electrcal nput current, the volume velocty q P n the port and the sound pressure p A n the acoustc system. The components a(), b() and h() are smooth nonlnear functons of the state vector specfed for the vented-bo system n [29, Eq. 3-5]. The state space model n Eq. (17) s llustrated by a sgnal flow chart depcted n Fg. 8. The nonlnear functons b() and a() and the ntegrator are part of a feedback loop generatng the state vector. The dstorton react to ts own generaton process formng a feedback loop. If the nonlneartes n a() s only a smple squarer generatng secondorder harmonc and ntermodulaton dstorton prmarly the feedback of the dstorton to the same squarer wll also produces thrd- and eventually all knds of hgher-order dstorton. The feedback also eplans nstabltes (col jump out effect), the complcated relatonshp between nput/output ampltude (nonlnear ampltude compresson) and the nteractons between dfferent drver nonlneartes. Any generc flter wth a feed-forward structure (polynomal flter) can only appromate the behavor of the electromechancal transducer n a lmted ampltude range. y 1 b(x) a(x) X Transducer Subsystem h 1 h(x) Fg. 8: State space model for the electro-mechancal transducer State space representaton n normal form To lnearze the subsystem h 1 a drect relatonshp between nput y 1 and output y 2 s requred. Followng the general approach of geometrcal control theory [31] the output y 2 s dfferentated untl the system nput y 1 appears y = L h( ) + L L h( ) y = f( ) + g( ) y (18) ( γ ) γ γ 1 2 a b a 1 1 where L ah and L bh stand for the Le dervatves of h() along the smooth vector felds a() and b(), respectvely. The number γ of dfferentaton requred to have L bl ah() 0, for =0,..., γ-2, and y 2 8

9 L bl γ-1 ah() 0 for all s called the relatve degree of the system. It corresponds wth the ecess of poles over zeros n a lnear system. () The output sgnal y 2 and ts dervatves y 2 for =1,...,γ-1 may be consdered as new state varable z 1,..., z γ. If the relatve degree γ s smaller than the order n of the system than the system has addtonal nternal dynamcs represented by addtonal state varables z γ,..., z n whch are not drectly controllable from the nput y 1 and observable at the output y 2. The old and the new state vector are related by a smooth, locally defned coordnate transformaton (dffeomorphsm) whch has also a smooth nverse. Usng the dffeomorphsm T - : z Eq. (18) s epressed n the state space representaton n normal form z& = z z& = z z& γ = f ( T ( z) ) + g( T ( z) ) y1 = f( ) + g( ) y1 z& γ + 1 = I1( z)... z& n = In γ ( z) y = z 2 usng the functon I 1 (z),... I n-γ (z) whch descrbe the nternal dynamcs. The vented-bo system has the order n=5 and a strong relatve degree γ=3 where the frst three elements of the state vector z are the dsplacement z 1=, velocty z 2=v and acceleraton z 3=a. If the nput y 1 s used to control the dsplacement and ts dervatve to zero z 1 =z 2 =z 3 =0 then the two nternal states varables z 3 =q p and z 5 =p A, volume velocty n the port and sound pressure, respectvely, descrbe the zero dynamcs n the acoustcal (Helmholz) resonator. Due to the acoustc resstance R p any vbraton n the zero dynamcs vanshes eponentally. Ths s an mportant crtera for the stablty of the system. The state space representaton of the closed bo system n normal form s llustrated by the sgnal flow chart n Fg. 9. The nonlnear functons f() and g() gven n [29, Eq. 9,10] produce scalar outputs whch are multpled wth and added to the nput sgnal gvng the acceleraton of the drver. A seres of frst-order ntegrators generate the state varables z 2 and the output sgnal y 2 =z 1 =. The feedback loop s closed by the dffeomorphsm T -1- : z transformng the state varables back nto the old parameters. The zero dynamcs s also a feedback loop separated from the nput/output path and generatng the addtonal states z 4 and z 5. 1 (19) y 1 g() f() T -1 z 3 I 1 (z) Transducer Subsystem h 1 z 4 z 5 z z 2 z 1 I 2 (z) Zero Dynamcs Fg. 9: State space model of a loudspeaker wth radaton ads (e.g. vented enclosure) n normal form Lnearzaton wth separate ID and LD- Controller Havng the transducer system transformed nto the normal state space form the system can be lnearzed by compensatng the effect of the nonlnear system g() and f() by a controller wth nverse dynamc (ID) w f( ) u1 = g( ) gvng the overall relatonshp y 1 (γ) =w 1. Thus, applyng the IDcontroller to a closed-bo system the dsplacement rses to lower frequences by 18 db per octave. An addtonal controller wth lnear dynamcs (LD) y 2 (20) w1 = v1 gd + fd( z ) (21) s connected n front of the ID-controller to lmt the dsplacement below the Helmholz resonance, to preserve a desred system algnment and to have a constant ampltude sound pressure response above the resonance frequency. Appled to a vented-bo system the control law wth the functon f D (z) and g() gven n [29, Eq. 16, 17] produces a lnear overall system equal wth the system functon H (s) usng the lnear loudspeaker parameters [29, Eq. 18,19]. 9

10 LD Controller ID Controller z 3 z 2 z 1 v 1 g D w 1 (t u 1 y 1 y 2 z 4 z 5 f D (z) -f() g() -1 g() f() z I 1 (z) I 2 (z) T Zero Dynamcs T -1 z Subsystem g 1 Transducer Subsystem h 1 Fg. 10: Perfect lnearzaton by statc state feedback wth a separate controllers for the lnear and nverse nonlnear dynamcs Control Law n Drect Form The lnearzaton of a transducer usng separate LD and ID controllers requres full access to all state varables n vector and suffcent processng capacty to transform z and to calculate the f(), g(), f D(z). Although, the compensaton of the entre lnear and nonlnear dynamcs and the replacement of the desred poles s the standard approach n nonlnear control t s advantageous to summarze the LD and ID controller n Eqs. (20) and (21) to the control law n the drect form [29] vg 1 D + fd( z) f( ) v1 β ( ) u1 = =. g( ) α( ) The control law for the vented-bo system s llustrated n Fg. 11. The transducer contans a lnear model n the ntegrator-decoupled form where the terms f D(z) and g D and the zero dynamcs I 2(z) and I 2(z) determne the desred transfer functon H (s). The nonlnear terms α() and β() n the feedback loop generate the undesred dstorton when the loudspeaker s operated n the large sgnal doman. The controller comprses only the nverse terms β() and α() -1 whch supply "syntheszed" dstorton to the control nput to compensate the real dstorton n the transducer model. The drect form of the control law n Eq. (22) has the followng advantages : If the loudspeaker s operated n the small sgnal doman and the ampltudes of the state varables n are small then (22) the control nput becomes dentcal wth the output because β() = 0 and α() -1 = 1. Ths s a nce feature for the mplementaton on nepensve sgnal processors usng a fed-pont arthmetc wth lmted resoluton. Contrary, the output w 1 of the LD-controller n Fg. 10 requres 10 bt more resoluton for the numercal representaton of a dgtal sgnal coverng the whole audo band. Whereas the ID controller requres full state measurement the drect control law for the electro-dynamcal transducer requres only the dsplacement, ts velocty v=d/dt and the nput current as state nformaton. The two nternal states varables q p and p A, volume velocty n the port and sound pressure, respectvely, and the lnear parameters C B, M p and R p of the acoustcal resonator whch are used n Eqs. (20) and (21) do not appear n Eq. (22). The control law s vald for the electro-dynamcal transducer wth domnant Bl()-, L e ()- und C ms ()- nonlnearty mounted n a vented- or closed-bo system or coupled wth any radaton ad (horn, transmsson lne, passve radator, dstrbuted mode panel). All these acoustcal elements contrbute to the zero dynamcs whch does not appear n the control law n the drect form. Thus, the lnearzaton of the output y 2 can also be accomplshed also n cases where the zero dynamcs s hard to model (e.g. hgh number of modes on a panel) or even nonlnear (e.g. nonlnear stffness of a passve radator). 10

11 z 3 z 2 z 1 v 1 u 1 y 1 g D y 2 z 4 z 5 -β() α() -1 α() β() f D (z) I 1 (z) I 2 (z) Controller Subsystem g 1 X d/dt T -1 Transducer Subsystem h 1 z Desred Dynamcs Fg. 11: Perfect lnearzaton by statc state feedback usng a control law n the drect form 7 Provdng the state nformaton 7.1 Drect measurement The drect control law n Eq. (22) requres the nstantaneous state varables dsplacement, velocty v and nput current. They may be measured drectly on the electro-dynamcal transducer by usng dsplacement sensor and a current sensor and feed back to the control law as shown n Fg. 11. However, ths approach has two major drawbacks: parameters and full modelng of the zero dynamcs (acoustcal resonator). A dfferentator as shown n Fg. 12 generates the velocty from the dsplacement. The voltage equaton only requres electrcal parameters of the transducer (R e, L e, Bl()) to predct the velocty v and then generates the dsplacement by ntegratng v. Ths method has the dsadvantage that the DC part of the dsplacement can not be generated accurately. v u y Desred Dynamcs y Whereas the measurement of the current can be accomplshed by a shunt there s no smple soluton for the measurements of the mechancal sgnals. The voltage v may be generated by dfferentatng the voce col dsplacement. Unfortunately, the dsplacement can not be derved from the velocty or acceleraton because the ntegraton can not provde the DC-component of the voce col dsplacement whch s dynamcally generated f the transducer nonlneartes are asymmetrcal. A laser dsplacement sensor usng the trangulaton prncple may be used for measurng dsplacement under labotory condtons but ths soluton s too epensve and mpractcal for common use. -β( ) X' Controller Subsystem g α( ) -1 α() β() Transducer Subsystem h 2. Statc state feedback based on drect measurement and feedback of the states varables does not allow any tme delay n the sensor path. Snce the realzaton of the nonlnear control law requres dgtal sgnal processng, specal converters (DAC and ADC) are requred. For eample, the tme delay produced n sgma-delta converters commonly used n audo applcatons may produce a phase shft between syntheszed and real dstorton whch mpars the compensaton and may also cause nstabltes n the overall system. 7.2 State Observer Beerlng et. al. [26] suggested a nonlnear model to estmate the state varables of the transducer. Snce the voce col current can easly be measured by a shunt the other state varables may be estmated by an observer based on the voltage equaton or nonlnear force equaton [35]. The force equaton requres knowledge of the mechancal X v v s Dsplacement Estmator State Observer Fg. 12: Controller based on statc state feedback controller wth state observer usng the measured voce col current However, t s more advantageous to dspense wth any drect state measurement at the transducer and to synthesze the requred state varables from the control output u 1 by usng the state space model n Eq. (17) as shown n Fg. 13. Any tme delay τ may be added between the control output u 1 and the transducer nput y 1. The observer requres full knowledge of the lnear and nonlnear transducer parameters and the zero dynamcs. Usng ths control scheme wth adaptve parameter estmaton the nternal feedback n the 11

12 observer and the eternal feedback va the control law may cause nstabltes for some parameters settngs. Even f the observer and the control law are stable as separate systems the stablty of the feedback connecton of both system may become unstable. Fortunately, the parameters both n the control g and n the observer have a common physcal bass and may be checked for plausblty (Bl() > 0, C ms() > 0, M ms > 0). v u y ep(-τ ι s) Desred Dynamcs y +1 -β( ) α( ) -1 α() β() v u ep(-τ ι s) y Desred Dynamcs y +1 Controller Subsystem g v Transducer Subsystem h H (s) s v -β( ) α( ) -1 α() β() H (s) ln Current Estmator State Predctor Controller Subsystem g Transducer Subsystem h u Fg. 14: Lnearzaton of the electro-mechancal transducer usng a feedforward controller based on state predcton (mrror flter) X v b( ) a( ) State Observer ' 8 Adaptve Control The success of the lnearzaton based on transducer modelng depends on two condtons: 1. Adequate modelng of the nonlnear mechansms n the subsystem h. Fg. 13: State feedback controller usng an observer for generatng all state nformaton from control output 7.3 State Predcton The problems related wth state observers may be avoded by generatng the requred state varables from the control nput v 1 as shown n Fg. 14. Applyng the control law g to the transducer subsystem h the overall system between the control nput v and the dsplacement y +1= becomes lnear and can be descrbed by the system functon H (s). Thus a lnear flter wth the system functon H (s) and a smple dfferentator may be used to predct the dsplacement and velocty v drectly. The lnear flter also comprses the zero dynamcs (caused by the enclosure or other radaton ads). Only the generaton of the nput current requres a nonlnear current estmator [29, Eq. 30] and a lnear flter [29, Eq. 26]. The state predctor and the control law result n a feed-forward structure whch s just the mrror mage of the transducer model comprsng the nverse elements n a feedback loop. The feed-forward structure guarantees stablty of the overall system as long as the lnear flters and the nonlnear current estmator are stable. Ths control scheme s an deal canddate for adaptve control. 2. Optmal adjustment of the free parameters of the controller to the partcular transducer subsystem h. Whereas the frst condton has to be proven once for the loudspeaker type used, the second condton has to be ensured for any unt for the full tme of usage. Adaptve updatng of the free control parameters s an nterestng way for realzng a self-learnng devce and for compensatng for the nfluence of clmate varaton (temperature and humdty) and agng of the materal. However, adaptve algorthm rases the followng requrements: Updatng should dspense wth an artfcal test stmulus but should stay operatve for most audo sgnals. A state varable of the transducer has to be montored whch s easy accessble. The sensor should be nepensve, robust and precse. The update algorthm tself should be stable for any choce of the parameters. The update algorthm should be robust n cases of parameter uncertantes and un-modeled dynamcs. The adaptve control technques may be separated nto the drect and ndrect methods: 8.1 Drect updatng Adjustng the parameters n the controller by searchng for mnmal dstorton n the transducer output y +1 and lnearzng the overall system s called drect updatng. An error sgnal e s generated by 12

13 comparng the lnear output y +1 ' wth the measured output y +1 that reflects the nstantaneous nonlnear dstorton [36]. A mnmum of the mean squared error ensures an optmal adjustment of the controller. Lnear Parameter Estmator Fg. 15 shows the drect adaptve adjustment of the parameters of the control law wth state predcton (mrror flter). The lnear parameters of the state predctor are separately adjusted by usng a lnear adaptve flter H (s) connected n parallel to the overall system and straghtforward adaptve technques [21]. The nonlnear parameters may be drectly adjusted n the control law. The control law g s parameterzed by usng a nonlnear epander generatng the nonlnear states A scaled by the weghts P gvng u T (23) 1 = v1+pa. v State Predctor ' u ep(-τ s) ep(-τ s) y H (s) P l Transducer Subsystem h y' +1 - y +1 All of the unknown parameters are collected n parameter P whle the elements of A are known nonlnear functons of, v and. Nonlnear Epander The parameters P may be updated by usng an ntermttent flteredgradent LMS algorthm [34, Eqs. 34, 35] ( h t t ) e ema Pn( + 1) = Pn( ) + µ e( t) ( τ )* A ( ) (24) < where h =L -1 {H (s)} usng the nverse Laplace transformaton L and the convoluton *. P n Nonlnear Parameter Estmator e The update process s nterrupted when the error eceeds an allowed threshold (for eample e ma= 0.1 y +1 ) and the lnear gradent s lkely to fal. At the begnnng of the nonlnear update process when β() = 0 and α() -1 = 1 then the nonlnear paramters are updated only at small and medum ampltudes of when the error e < e ma. Wth the progress of the updatng process when the parameters P approach to the optmal values P ' the control law lnearzes the overall system and the ampltude of the error decreases for all ampltude of. Fnally, the condton e < e ma holds for all and the parameters P are updated permanently. The ntermttent fltered-error algorthm [34, Eqs. 36, 37] s an alternatve technques for updatng the parameters drectly. It requres only one flter for the nverse flterng of the error sgnal e and some tme delay for each gradent sgnal n A. However, ths approach does not mnmze the error n y +1 but vrtually n the control nput v. Ths corresponds wth a spectral coloraton of the resdual error sgnal e. Fg. 15: Adaptve lnearzaton of the electro-acoustcal transducer usng a nonlnear feedforward controller and a lnear adaptve model H (s) 8.2 Indrect updatng More tradtonal approaches use ndrect technques where the loudspeaker s adaptvely modeled and the estmated parameters are transformed nto control parameters and coped nto the control law. Clearly ths approach requres more elements and processng power than drect updatng of the control law. There are three ways to realze ndrect updatng: Parallel Model Here the adaptve model s connected n parallel to the subsystem h. The electro-mechancal transducer and other loudspeaker subsystems requre a model havng a feedback structure as shown n Fg. 8 correspondng wth the state space model n Eq. (17). Snce the model parameters are not lnear n the output, the update algorthm [34, Eq. 19] mght be trapped n local mnma of the cost functon Inverse Model Alternatvely, the adaptve model may be connected n seres to the output of the transducer. It s the target to lnearze the overall system between transducer nput y and model output. The control law may be drectly used as the nverse model system. It s an advantage of the nverse model that the free parameters are lnear n the output and can be updated wth straghtforward methods [34, Eqs. 13, 14]. However, the adaptve updatng of the nverse model wll be based f the loudspeaker output s corrupted by measurement nose Impedance model A specal form of ndrect updatng can be appled to electro-dynamcal transducers. Here the nonlnear relatonshp between voltage y =u and current at the transducer termnals as shown n Fg. 16. The mpedance model s based on the voltage and force equaton [35, Eqs. 13, 14] and generates an error sgnal e [35, Eq. 18] used for the 13

14 updatng of the detector parameters P. The parameters P of the detector are closely related to the physcs of the transducer and may be nterpreted as lnear parameters (resonance frequency f s, loss factor Q ts) and as relatve nonlnear parameters Bl(/ ma )/Bl(0), C ms (/ ma )/C ms (0) and L e(/ ma) wth the mamal dsplacement ma. Clearly the mechancal parameters (M ms, C ms) and state varables (, v) can not be dentfed as absolute quanttes by performng only an mpedance measurement and wthout havng addtonal nformaton about the mechancal system. Ths technque have been used for the dynamc measurement of the large sgnal parameters of electro-dynamcal transducers [38]. The detector parameters P satsfy the free parameters n the control law f epressed n the drect form and n the state epander as used n the mrror flter n Fg. 14. Snce the parameter vector P vary slowly due to vared ambent condton and agng the feedback of parameters s mmune to any tme delay τ n the sgnal path. Montorng the mpedance at electro-dynamcal transducers has the advantage: v Inepensve sensors for voltage and current (shunt or current sensors), no addtonal wrng, mmune aganst ambent nose and acoustcal envronment, no tme delay compensaton requred. ' state predctor Control law Controller Subsystem g u ep(-τ s) P y Transducer Subsystem h Model Parameter Estmator Adaptve Parameter Detector y +1 Fg. 16: Indrect adaptve lnearzaton of an electro-mechancal transducer based on voltage and current measurement 8.3 Intal control nformaton The robustness of an adaptve control system can be ncreased by adjustng the degree of freedom of the control structure to the partcular loudspeaker system. For nstance a vented enclosure compared wth the closed-bo-system ntroduces addtonal zero dynamcs whch ncreases the order and the number of free parameters n the lnear system H (s) n the state predctor. The followng nformaton should be provded at the begnnng of adaptve learnng: type of the transducer prncple (electro-dynamc or electrostatc) and the applcaton (woofer, md-range or tweeter), the radaton ads used (sealed-bo, vented-bo, horn, panel), lmt parameters descrbng the allowed mechancal or thermal overload. e 9 Aulary Control Functonalty The control based on loudspeaker modelng provdes nformaton about the nstantaneous state and parameters whch may be used to epand the functonalty and to enable an automatc control of a varety of dfferent loudspeaker systems. The eample n Fg. 17 shows such an automatc control system for a woofer system. 9.1 Dagnostcs Snce the parameters n P dentfed by adaptve control have a physcal meanng the may be checked for plausblty. If the parameter updatng fals for any reasons (wrong transducer type or loose electrcal connecton) the control may be deactvated and a user message can be generated. The nterpretablty of the control parameters may also be eploted to ndcate a loudspeaker defect (voce col offset). Storng the ntal nonlnear complance characterstc C ms() as a reference long term varaton may be montored and also a user message may be generated f agng or fatgue requres a replacement of the drver used n professonal applcatons. 9.2 Memory Snce most of parameters P measured by adaptve control vares slowly t s useful to store the parameters n a memory. Thus, restartng the controller the old parameters may be used as ntal parameters n the adaptve control. The control law and the state predctor may also be operated as a non-adaptve controller wth frozen parameters. Ths s mportant f lmted processng power s avalable and mult-taskng s requred. Implemented n a personal computer the adaptve updatng n the loudspeaker controller may be actvated when the processor s dlng and the reproduced musc sgnal gves suffcent ectaton of the loudspeaker connected. 9.3 Protecton system Lnearzaton of a loudspeaker makes only sense f there are also means provded to protect the speaker aganst thermal or mechancal damage because reducng the dstorton n the loudspeaker output makes t also more dffcult for the user to recognze a crtcal overload stuaton n tme. The lnear, nonlnear and thermal parameters dentfed by adaptve control make an effectve protecton of the loudspeaker possble: Mechancal protecton In a passve loudspeaker wthout control the mechancal load of the suspenson and the dstorton generated n the output sgnal lmt the mamal peak dsplacement X ma and the radated sound pressure at low frequences. The actve lnearzaton relaes the lmtng factor of the dstorton. However, the mamal peak dsplacement X ma has to be lmted to avod a damage on the suspenson system and to keep the power handlng reasonable. The protecton lmts used n the large sgnal parameter measurement n the Dstorton Analyzer [38] may also be used n a controller. If the mnmal value of the rato C ms()/c ms(0) may be compared wth an allowed lmt value (e.g. C lm= 20 %) then the voce col dsplacement has to lmted by an attenuator (e.g. adjustable hgh-pass). The lnearzaton of the force factor dstorton requres addtonal power for frequences far away from the resonance frequency. Thus, t s reasonable to lmt the dsplacement f the force factor rato Bl()/Bl(0) falls below a defned lmt (e.g. Bl lm= 25 %) Thermal protecton The ncrease of the voce col temperature T v may be estmated from the changes of the nput mpedance. Whereas the adaptve mpedance 14

15 detector may provde the nstantaneous voce col temperature the nonadaptve mode requres an addtonal thermal model usng the thermal parameters of the loudspeaker. Due to the convecton coolng and eddy currents generated n the pole tps there are many nteractons between the mechancal and thermal mechansms [40]. If the voce col temperature T v eceeds a defned lmt value the nput sgnal have to be attenuated n some way. 9.4 Actve compensaton of voce col offset If the voce col has an offset from optmal rest poston the lmted voce col heght ncreases the asymmetry of the Bl()-curve whch generates addtonal dstorton and unstable behavor above the resonance frequency f s. An offset s not only caused by a defect n drver manufacturng but can also be generated by fatgue or ageng of the suspenson, wrong storage or the gravty actng on the voce col f the drver s mounted n horzontal poston (e.g. hat rest n a car). Although the control law compensates for the effect of the offset t s better to adjust the voce col poston actvely. Knowng the offset n the Bl()-characterstc a smple DC voltage s generated and suppled va the DC-coupled amplfer to the transducer. Ths becomes more and more nterestng for loudspeakers wth etremely small (sealed) enclosures where drvers wth hgh complance are preferred. 9.5 Addtonal Processng Any addtonal sgnal processng commonly appled to audo sgnal such as actve crossovers, equalzers and lmters may be appled to the nput sgnal of the controller. Thus the controller, amplfer and transducer may be consdered as unt havng a defned (lnear) transfer characterstc. Addtonal Processng (crossover) V[n] Protecton System Control Law u[n] DAC Offset compensator u Thermal Model State Predctor ADC ADC Adaptve Parameter Detector Feedforward Controller Memory Dagnostcs Detector parameter user nformaton Fg. 17: Automatc control system for woofer 10 Concluson After dscussng many techncal detals of actve loudspeaker control the general goals, techncal requrement and practcal benefts shall be summarzed: Clearly actve lnearzaton and other forms of loudspeaker control competes wth passve means avalable for loudspeaker mprovements. The man goal of actve loudspeaker control s not to compensate for loudspeaker defects (e.g. rub and buzz phenomena [42]) whch can easly be fed by mproved loudspeaker desgn or manufacturng. Actve loudspeaker control can only be justfed n applcatons requrng smaller, less epensve transducers gvng more output wth lower dstorton at hgher senstvty. Snce the domnant nonlneartes n loudspeakers can be modeled, control schemes whch eplot ths nformaton are superor over generc controllers. Very mportant s the stablty and robustness of the controller especally f the loudspeaker behaves rregular or there are any defects n the control system (loose connecton). The controller should have self-tunng capabltes to smplfy the adjustment to the partcular loudspeaker and to cope wth heatng or long term parameter varatons. A sensor system used should be nepensve and mmune aganst ambent nose. Lnearzaton and protecton of a loudspeaker are closely related wth each other. The protecton system should prevent a damage for any nput sgnal whle causng mnmal mpact on the reproduced sound qualty. The actvely controlled loudspeaker should handle any sgnal wthout causng a thermal or mechancal damage. Snce the nonlnear controller requres a defned transfer functon between control output and loudspeaker nput any form of sgnal attenuaton or flterng (equalzer) should be performed pror to the nonlnear part. Such a control system can not be realzed n the analog doman. If a platform for dgtal sgnal processng s avalable loudspeaker control can be mplemented as a software module requrng no or only mnmal addtonal hardware (e.g. current sensor). Actve loudspeaker control gves new freedom for the desgn of the passve transducers [32]. For eample, the transducer wll be optmzed n respect wth sze, weght, senstvty, bandwdth and drectvty of the reproduced sound [33] but the lnear transfer behavor and the regular loudspeaker dstorton can also be controlled actvely. Thus, there are new challenges n the 15

16 desgn of actve loudspeaker systems whch requre a close cooperaton between drver, system and controller development. 11 References [1] J.A.M. Catrysse, On the Desgn of Some Feedback Crcuts for Loudspeakers, J. Audo Eng. Soc., vol. 33, pp (1985 June). [2] J.A. Klaasen and S.H. de Konng, Motonal Feedback wth Loudspeakers, Phlps Tech. Rev., vol. 28, pp (1968 May). [3] E. De Boer, Theory of Motonal Feedback, IRE Trans. Audo (1961 Jan. - Feb.). [4] R.A. Grener and T.M. Sms, Loudspeaker Dstorton Reducton, J. Audo Eng. Soc., vol. 32, pp (1984 December). [5] S.A. Lane and R.L. Clark, "Improvng Loudspeaker Performance for Actve Nose Control Applcatons," J. Audo Eng. Soc., vol. 46, pp (1998 June). [6] P. R. Wllams, "A Dgtal Approach to Actvely Controllng Inherent Nonlneartes of Low Frequency Loudspeakers," presented at the 87 th Conventon of Audo Eng. Soc. New York, (1989 October 18-21). [7] D.J. Schrader, Servo-Controlled Amplfer and Method for Compensatng for Transducer Nonlneartes, U.S. patent (1989 Sept.). [8] P.G.L. Mlls and M.O.J. Hawksford, Dstorton Reducton n Movng-Col Loudspeaker Systems Usng Current-Drve Technology, J. Audo Eng. Soc., vol. 37, pp (1989 Mar.). [9] P.M. Larsen, "A Method of Correctng Non-Lnear Transfer Behavor n a Loudspeaker", patent applcaton WO 97/25833 (1997 January). [10] F.Y. Gao, Adaptve Lnearzaton of a Loudspeaker, presented at 93rd Conventon of the Audo Eng. Soc., October 1-4, 1992, San Francsco, preprnt [11] M. J. Reed, M. O. Hawksford, "Non-Lnear Error Correcton of Horn Transducers Usng a Volterra Flter," presented at the 102 nd Conventon of Audo Eng. Soc March 22-25, Munch, Germany, preprnt [12] T. Katayama, M. Serkawa, "Reducton of Second-Order Non- Lnear Dstorton of a Horn Loudspeaker by a Volterra Flter Real- Tme Implementaton," presented at the 103 rd Conventon of Audo Eng. Soc., 1997 September 26-29, New York, preprnt [13] U. Horbach," Desgn of Hgh-Qualty Studo Loudspeakers Usng Dgtal Correcton Technque," presented at the 109 th Conventon of Audo Eng. Soc., 2000 September 22-25, Los Angeles, USA, preprnt [14] P. Robneau, et. al., "Devce for Processng an Audo-Frequency Electrcal Sgnal," US patent #4,995,113 fled November 19, [15] V.J. Mathews, Adaptve Polynomal Flters, IEEE Sgnal Processng Magazne, pp , July (1991). [16] K. Hornk, M. Stnchcombe, and H. Whte, Multlayer Feedforward Networks are Unversal Appromators, Neural Networks, Vol. 2, pp (1989). [17] W. A. Frank, An Effcent Appromaton to the Quadratc Volterra Flter and ts Applcaton n Real-Tme Loudspeaker Lnearzaton, Sgnal Processng, vol. 45, pp , (1995). [18] S. Low and M.O.J. Hawksford, A Neural Network Approach to the Adaptve Correcton of Loudspeaker Nonlneartes, presented at the 95th Conventon of the Audo Eng. Soc., New York, 1993, October 7-10, preprnt [19] P.R. Chang, C.G. Ln and B.F. Yeh, Inverse Flterng of a Loudspeaker and Room Acoustcs usng Tme-Delay Neural Networks, J. Acoust. Soc. Am., vol. 95, pp , (June 1994). [20] H. Schurer, C.H. Slump and O.E. Herrmann, Second-order Volterra Inverses for Compensaton of Loudspeaker Nonlnearty, Proccedngs of the IEEE ASSP Workshop on applcatons of sgnal processng to Audo & Acoustcs, New Paltz, October 15-18, 1995, pp [21] S. Haykn, Adaptve Flter Theory, Prentce Hall, 1991, Englewood Clffs, New Jersey. [22] A. J. Kaser, Modelng of the Nonlnear Response of an Electrodynamc Loudspeaker by a Volterra Seres Epanson, J. Audo Eng. Soc. 35, p. 421, (1987 Jun). [23] W. Klppel, "Schaltungsanordnung zur Korrektur des lnearen und nchtlnearen Übertragungsverhaltens elektroakustscher Wandler, Offenlegungsschrft DE Deusches Patentamt, fled Aprl 9 th [24] W. Klppel, The Mrror flter - a New Bass for Reducng Nonlnear Dstorton and Equalzng Response n Woofer Systems, J. Audo Eng. Soc., Vol. 32, pp , (1992 Sept.). [25] W. Klppel, Compensaton for Nonlnear Dstorton of Horn Loudspeakers by Dgtal Sgnal Processng, J. Audo Eng. Soc., vol. 44, pp , (1996 Nov.). [26] M.A.H. Beerlng, C.H. Slump, O.E. Herrman, Reducton of Nonlnear Dstorton n Loudspeakers wth Dgtal Motonal Feedback, presented at the 96th Conventon of the Audo Eng. Soc., Amsterdam, February 26 - March 1, 1994, preprnt [27] J. Suykens, J. Vandewalle and J. van Gndeuren, Feedback Lnearzaton of Nonlnear Dstorton n Electrodynamc Loudspeakers, J. Audo Eng. Soc., Vol. 43, No. 9, pp (1995). [28] H. Schurer, C.H. Slump and O.E. Herrmann, Eact Input-Output Lnearzaton of an Electrodynamcal Loudspeaker, presented at the 101th Conventon of the Audo Eng. Soc., Los Angeles, November 8-11, 1995, preprnt [29] W. Klppel, Drect Feedback Lnearzaton of Nonlnear Loudspeaker Systems, J. Audo Eng. Soc., Vol. 46, pp (1995 June). [30] H. Schurer, C. H. Slump, O.E. Herrmann, Theoretcal and Epermental Comparson of Three Methods for Compensaton of Electrodynamc Transducer Nonlnearty, Audo Eng. Soc., Vol. 46, pp (1998 September). [31] H. Njmejer and A.J. van der Schaft, Nonlnear Dynamcal Control Systems (Sprnger, New York, 1990). [32] A. Brght, "Compensatng Non-Lnear Dstorton n an "Equal- Hung" Voce Col," presented at the 111 th Conventon of Audo Eng. Soc. New York September 21 24,

17 [33] A. Brght, "Actve Control of Loudspeakers: An Investgaton of Practcal Applcatons," PhD-thess, Techncal Unversty of Denmark, Lyngby, Denmark, 2002 [34] W. Klppel, Nonlnear Adaptve Control of Loudspeaker Systems, Audo Eng. Soc., Vol., pp (1998 November). [35] W. Klppel, "Nonlnear Adaptve Controller for Loudspeakers wth Current Sensor", presented at the 106 th Conventon of Audo Eng. Soc., 1999, May 9-11, Munch, Germany, preprnt [36] W. Klppel, U. Sedel, Measurement of Impulsve Dstorton, Rub and Buzz and other Dsturbances presented at the 114 th Conventon of Audo Eng. Soc., 2003, March 22-25, Amsterdam, The Netherlands, preprnt. [37] W. Klppel, Measurement of Large-Sgnal Parameters of Electrodynamc Transducer, presented at the 107 th Conventon of the Audo Engneerng Socety, New York, September 24-27, 1999, preprnt [38] W. Klppel, Dstorton Analyzer a New Tool for Assessng and Improvng Electrodynamc Transducer, presented at the 108 th Conventon of the Audo Engneerng Socety, Pars, February 19-22, 2000, preprnt [39] W. Klppel, "Dagnoss and Remedy of Nonlneartes n Electrodynamcal Woofers, presented at the 109 th Conventon of the Audo Engneerng Socety, Los Angeles, September 22-25, 2000, preprnt 5261 [40] W. Klppel, "Nonlnear Modelng of the Heat Transfer n Loudspeakers, " presented at the 114 th Conventon of the Audo Engneerng Socety, Amsterdam, March 22-25, 2003, preprnt. [41] W. Klppel, "Assessment of Voce Col Peak Dsplacement X ma," presented at the 112th Conventon of the Audo Engneerng Socety, 2002 May 10 13, Munch, Germany, preprnt [42] W. Klppel, U. Sedel, "Measurement of Impulsve Dstorton, Rub and Buzz and other Dsturbances," presented at the 114th Conventon of the Audo Engneerng Socety, 2003 March 22 25, Amsterdam, The Netherlands, preprnt. 17