This paper discusses a time-domain technique for synthesizing acoustic impedance at the diaphragm of a loudspeaker using a proportional-plus-derivative output feedback. The dynamics of electroacoustic transducers such as moving-coil loudspeakers can be readily controlled either by direct feedback principle on acoustic quantities, or by plugging a shunt network at the electrical terminals. Any conventional loudspeaker first intended to be a sound transmitter may then become a versatile electroacoustic resonator capable of absorbing (or of reflecting as much) the incident sound energy in a frequency-dependent way by simple electronic controls. Instead of counteracting some unwanted sound by using superposition principle, as is the case for conventional active noise control, such actuator-based strategy aims at monitoring the reaction of a loudspeaker embedded into walls so as to control the proportion of reflected sound waves on this boundary. After a short description of the dynamics of moving-coil loudspeakers giving emphasis on the advantage of electromechanical coupling reversibility, a proportional plus derivative output feedback combined to a feed-forward action is proposed for synthesizing of desired acoustic impedance. As a conclusion, the overall performance of the proposed method is presented along with computed results and general discussions on practical implementation

Boulandet, Romain

Lissek, Hervé

This paper discusses a time-domain technique for synthesizing acoustic impedance at the diaphragm of a loudspeaker using a proportional-plus-derivative output feedback. The dynamics of electroacoustic transducers such as moving-coil loudspeakers can be readily controlled either by direct feedback principle on acoustic quantities, or by plugging a shunt network at the electrical terminals. Any conventional loudspeaker first intended to be a sound transmitter may then become a versatile electroacoustic resonator capable of absorbing (or of reflecting as much) the incident sound energy in a frequency-dependent way by simple electronic controls. Instead of counteracting some unwanted sound by using superposition principle, as is the case for conventional active noise control, such actuator-based strategy aims at monitoring the reaction of a loudspeaker embedded into walls so as to control the proportion of reflected sound waves on this boundary. After a short description of the dynamics of moving-coil loudspeakers giving emphasis on the advantage of electromechanical coupling reversibility, a proportional plus derivative output feedback combined to a feed-forward action is proposed for synthesizing of desired acoustic impedance. As a conclusion, the overall performance of the proposed method is presented along with computed results and general discussions on practical implementation.LEM

Infoscience - École polytechnique fédérale de Lausanne

ACOUSTIC IMPEDANCE SYNTHESIS AT THE DI-APHRAGM OF MOVING COIL LOUDSPEAKERS USINGOUTPUT FEEDBACK CONTROLRomain Boulandet, Herve´ LissekLaboratoire d’Electromagne´tisme et d’Acoustique, Ecole Polytechnique Fe´de´rale de Lausanne,Station 11, CH-1015 Lausanne, Switzerland, e-mail: romain.boulandet@epfl.chThis paper discusses a time-domain technique for synthesizing acoustic impedance at the di-aphragm of a loudspeaker using a proportional-plus-derivative output feedback. The dynamicsof electroacoustic transducers such as moving-coil loudspeakers can be readily controlled ei-ther by direct feedback principle on acoustic quantities, or by plugging a shunt network atthe electrical terminals. Any conventional loudspeaker first intended to be a sound transmittermay then become a versatile electroacoustic resonator capable of absorbing (or of reflecting asmuch) the incident sound energy in a frequency-dependent way by simple electronic controls.Instead of counteracting some unwanted sound by using superposition principle, as is the casefor conventional active noise control, such actuator-based strategy aims at monitoring the re-action of a loudspeaker embedded into walls so as to control the proportion of reflected soundwaves on this boundary. After a short description of the dynamics of moving-coil loudspeakersgiving emphasis on the advantage of electromechanical coupling reversibility, a proportional-plus-derivative output feedback combined to a feed-forward action is proposed for synthesiz-ing of desired acoustic impedance. As a conclusion, the overall performance of the proposedmethod is presented along with computed results and general discussions on practical imple-mentation.1. IntroductionThe dynamics of electroacoustic transducers such as moving-coil loudspeakers can be readilycontrolled either direct feedback on acoustic quantities [1, 2, 3, 4], or by plugging a shunt networksat the electrical terminals [5, 6, 7]. With the help of such very basic control strategies variable acous-tic impedance can be achieved at the transducer diaphragm. A conventional loudspeaker can thenbe transformed into a versatile electroacoustic resonator capable of absorbing sound energy within alarge frequency range, typically more than a frequency decade around its natural frequency [7]. Gen-erally speaking, feedback-based techniques are viewed as a specialized section of control engineering,whereas shunt-based methods relate more to a specialized section of electrical engineering. In the areaof acoustic impedance control, both strategies aim at preventing (or at reinforcing) the reflection ofincident sound waves by controlling the dynamics of the electroacoustic transducer. A common fea-ture is that design procedures most often rely on the frequency response approach (Bode diagram)for determining the appropriate control parameters. The performance of the transient response is thenspecified in an indirect manner in terms of phase margin, gain margin, bandwidth, resonant peakICSV18, 10–14 July 2011, Rio de Janeiro, Brazil 118th International Congress on Sound and Vibration, 10–14 July 2011, Rio de Janeiro, BrazilFigure 1. Schematic of a moving-coil loudspeaker in closed-box.magnitude, etc., which depend on the ratio of the feedback gains [7]. One way of achieving a desiredacoustic impedance by direct feedback is to apply to the electric terminals a command voltage whichis proportional to a linear combination of both velocity and sound pressure at the vicinity of the di-aphragm [3, 4, 8]. With a purely proportional feedback however, one single closed-loop pole can takeon a pre-assigned value in the left hand part of the complex plane. In order to more effectively controlthe dynamics of a system, a proportional-plus-derivative (PD) output feedback may be introduced.[9]. The idea of synthesizing a desired acoustic impedance by designing a PD controller within statespace approach is the main motivation for this paper.The remainder of this paper is organized as follows. First, a brief description of the dynamics ofa moving-coil loudspeaker will be detailed, giving emphasis on the advantage of electromechanicalcoupling reversibility. The problem of controlling the dynamics of the transducer will be then formu-lated in state space. A pole placement technique will be provided for determining the output feedbackgains that satisfy a desired behavior for the closed-loop system. As a conclusion, the overall perfor-mance of the proposed method for synthesizing a desired acoustic impedance will be presented alongwith simulation results and general discussions on practical implementation.2. System dynamics2.1 Moving-coil loudspeaker modelA common description of the moving-coil loudspeaker for small displacements and below thefirst modal frequency of the diaphragm is given by the following set of linear differential equations[10]:Sp(t) = Mms v˙(t) +Rms v(t) +1Cmcξ(t)−Bl i(t)e(t) = Re i(t) + Le i˙(t)−Bl v(t)(1)where p(t) is the input sound pressure acting on the loudspeaker diaphragm, v(t) = ξ˙(t) is the di-aphragm velocity, ξ(t) is the diaphragm displacement, i(t) is the electrical current flowing throughthe voice coil, and e(t) is the input voltage applied at the electrical terminals. For the model parame-ters, S is the effective piston area, Bl is the force factor, Mms and Rms are the mass and mechanicalresistance of the moving part, Re and Le are the dc resistance and the inductance of the voice coil.Here, Cmc = (1/Cms + ρc2/Vb)−1 is the equivalent mechanical compliance accounting for both theflexible edge suspension and spider of the loudspeaker and the cabinet, where ρ and c are the den-sity and celerity of air and Vb is the volume of the cabinet. The terms Bl i(t) and Bl v(t) in Eq. (1)are the Laplace force induced by the current circulating through the coil and the back electromotiveforce (emf) induced by the motion of the voice coil within the magnetic field, respectively. They bothexpress the electromechanical coupling on the mechanical and electrical side, respectively.218th International Congress on Sound and Vibration, 10–14 July 2011, Rio de Janeiro, BrazilTable 1. Loudspeaker Visaton R© AL-170 technical data.Description Notation Value Unitdc Resistance Re 5.6 ΩVoice coil inductance Le 0.9 mHForce factor Bl 6.9 T.mMoving mass Mms 13 gMechanical resistance Rms 0.8 N.s.m−1Mechanical compliance Cms 1.35 mm.N−1Effective piston area S 133 cm22.2 Advantage of electromechanical coupling reversibilityAny conventional electroacoustic transducer can be employed either as a sound transmitter,when it converts electrical energy into acoustical energy, or as a sound receiver when operating in theopposite way [10]. When the system is driven by an auxiliary voltage source and also subjected to anexogenous sound source these conversion processes happen simultaneously. A complete descriptionof the transducer implies thus to account for the nature of the input voltage. When operating as a soundtransmitter for instance, an auxiliary voltage source es(t) is connected at the electrical terminals andthe driven voltage can be expressed as:e(t) = es(t)− Zs i(t) (2)where Zs is the internal impedance of the power source. When operating in reverse as a soundreceiver, es(t) and Zs may be tailored in view of modifying the transducer dynamics so that it moreor less reflects some sound energy in a frequency-dependent way [7].3. Problem formulation3.1 State space representationLooking at the loudspeaker as a dynamical system, the input voltage e(t) can be viewed as acommand (or controllable) input for controlling its dynamics. The output, namely the process variablewe are interested in controlling, is the diaphragm velocity. Being not directly controllable, the soundpressure p(t) may be regarded as a disturbance input that may affect the output. Likewise, the physicalquantities v(t), ξ(t) and i(t) in Eq. (1) are internal state variables since they are involved in the circu-lation of energy within the dynamical system. Introducing the state vector x(t) = (ξ(t), v(t), i(t))Tand the scalar command variable u(t) = e(t) the set of equations of Eq. (1) can be written in a stateform [11]. The general state space representation of this linear single-input-single-output (SISO)system can be expressed as:x˙(t) = Ax(t) + bu(t) + bp p(t)y(t) = cT x(t) (3)whereA =0 1 0− 1CmcMms−RmsMmsBlMms0BlLe−ReLe b =001Le bp =0SMms0 cT = [ 0 1 0 ] (4)Note that matrices A, b and bp are composed of the model parameters of the loudspeaker whereasthe output matrix cT depends on which state variable of x(t) is measured. Note also that a commandvariable u(t) can be constructed to transfer the system from any initial output y(t0) to any final output318th International Congress on Sound and Vibration, 10–14 July 2011, Rio de Janeiro, BrazilFigure 2. Block diagram of the system with proportional-derivative controller in feedback.y(t1) in a finite time interval t0 ≤ t ≤ t1 since the observability matrix[cT cTA cTA2]Tis of rank3, and the controllability matrix[cTb cTAb cTA2b]is of rank 1, meaning that the system is bothobservable and controllable [12].3.2 Output feedback lawGenerally speaking, designing a state (or output) feedback means developing an appropriatecontrol law so that the closed-loop system behaves with acceptable performance in terms of staticaccuracy, disturbance rejection and transient response [11]. Output feedback control is simply morepractical since it covers situations where all states are not accessible to measurement. A commonstrategy for implementing output feedback control relies on the pole placement techniques. The basicidea is to specify the desired location of the poles of the closed-loop system, and then determine thefeedback gains to achieve these poles. Applying a proportional-plus-derivative output feedback law,as shown in Fig. 2, can be written as [9]:u(t) = −Γy(t)− τ y˙(t) + r(t) (5)where Γ and τ are constant proportional and derivative feedback gains and r(t) is the reference value.The negative sign simply indicates a negative feedback on the output and its derivative.3.3 Pole placement problemThe problem of pole placement of Eq. (3) is of determining the output feedback gains suchthat the closed-loop system poles take on pre-assigned values Λ = {λi, i = 1, 2, 3}. Substituting thecommand variable given by Eq. (5) in the system described by Eq. (3) yields the closed-loop systemas:x˙(t) =(I + bτcT)−1 (A− bΓcT)x(t) +(I + bτcT)−1(bpp(t) + br(t)) (6)provided that I+bτcT is invertible. The transient response of the closed-loop system is determined bythe eigenvalues of the system matrix Ac =(I + bτcT)−1 (A− bΓcT). Noting that the characteristicpolynomial H(s) = |sI− Ac| may be written in terms of the feedback gains Γ and τ as:H(s) =1|I + bτcT | |sI− A + b (Γ + sτ) cT | (7)and that the desired characteristic polynomial for a third-order system can be written as:H(s) =3∏i=1(s− λi) = s3 + β2s2 + β1s+ β0 (8)418th International Congress on Sound and Vibration, 10–14 July 2011, Rio de Janeiro, Brazilthe output feedback gains are readily determined by equating Eq. (7) and Eq. (8). Substituting Eq. (4)into Eq. (7) the feedback gains can be expressed as a function of the loudspeaker’s parameters and ofcoefficients of the desired characteristic polynomial as:Γ =RmsReBl+LeCmcBl−Bl − LeMmsBlβ1τ =ReMmsBl+LeRmsBl− LeMmsBlβ2(9)Given that the closed-loop system order exceeds twice the number of inputs or outputs only twopoles of the closed-loop system can be specified with the control law given by Eq. (5) [9]. In view ofimproving the dynamic compensation of the system by tempting to assign a third pole one additionaloutput must be envisaged so that the system order does not exceed twice the number of inputs oroutputs.3.4 Improvements using feed-forward controlGenerally speaking the control system performance can be improved by combining the feedback(or closed-loop) action of the proportional-integral-derivative (PID) controller with a feed-forward(or open-loop) element. In the case of controlling the dynamics of a loudspeaker, the feed-forwardcontroller will anticipate the influence of all sound pressure disturbance on the diaphragm velocityand will deploy control actions so that the loudspeaker behaves as expected in a timely fashion.Figure 3. Block diagram of the system with proportional-derivative controllers in feedback and feed-forward.In order to obtain a broadband absorption at the loudspeaker diaphragm, it has been shown thatthe ratio of the velocity feedback gain over the sound pressure gain should equal the target acous-tic impedance value [7]. The feed-forward gains to match the impedance of the diaphragm to thecharacteristic impedance of air are simply given by:Γp =Γρcτp =τρc(10)Now combining the feed-forward action to the control law given in Eq. (5) yields a new expres-sion for the command voltage as:u(t) = −Γy(t)− τ y˙(t) + Γp p(t) + τp p(t) = e(t) (11)Substituting Eq. (11) into Eq. (1) leads, after Laplace transform and some rearrangements, tothe acoustic admittance Y (s) = V (s)/P (s) of the electroacoustic absorber, where V (s) and P (s) are518th International Congress on Sound and Vibration, 10–14 July 2011, Rio de Janeiro, Brazilthe Laplace transforms of velocity v(t) and sound pressure p(t), as:Y (s) =s2 (SLe +Blτp) + s (SRe +BlΓp)s3MmsLe+ s2 (MmsRe +RmsLe −Blτ) + s(RmsRe +LeCmc− (Bl)2 −BlΓ)+ReCmc(12)Note that the feed-forward element may alter the zeros of the acoustic admittance, thus offeringadditional degrees of freedom for attaining any desired specifications. Regarding stability, the feed-forward action should never cause oscillation into the system since it is outside the feedback loop.4. Results and discussionTo gain an understanding of the achievable performance from the PD feedback controller, theacoustic admittance of the loudspeaker diaphragm is computed for various control settings. Thespecifications of the Visaton R© AL 170 low-midrange loudspeaker mounted in a sealed cabinet, thevolume of which is Vb = 10 l, are given in Tab. 1. The different control settings considered in thissection are listed in Tab. 2.Table 2. Control settings of the different strategies investigated.Case Γ τ Γp τpV.(m/s)−1 V.(m/s2)−1 V.Pa−1 V.(Pa/s)−1Closed circuit 0 0 0 0P controller -40 0 0.1 0PD controller -40 -0.5 0.1 0.0012As shown in Fig. 4, the proportional-derivative controller combined to a feed-forward actionmakes possible to assign two poles and two zeros such that the dynamics of the loudspeaker is bet-ter controlled than with a purely proportional controller. In the frequency domain this should resultin an extension of the bandwidth where the diaphragm fully absorbs the sound energy. Working to-gether, the combined open-loop feed-forward controller and closed-loop PD controller should providea more responsive, stable and reliable control system. Implementing an electroacoustic absorber ap-pears quite straightforward and consists of first adjusting the ratio of gains for achieving a desiredacoustic resistance, and then increasing simultaneously the gains while their ratio remains constantwith respect to stability margins.Figure 4. Bode diagram of the computed acoustic impedance and computed absorption coefficient.618th International Congress on Sound and Vibration, 10–14 July 2011, Rio de Janeiro, Brazil5. ConclusionsIn this paper a time-domain method has been described for synthesizing acoustic impedanceat the diaphragm of an electroacoustic transducer by means of a proportional-plus-derivative (PD)output feedback controller combined to a feed-forward action. It as been shown that the problem ofsynthesizing an acoustic impedance can be solved by the techniques developed for pole assignmentthroughout control theory. In comparison with a purely proportional output feedback that is usuallyused for acoustic impedance matching, the PD controller introduces one additional degree of freedomfor achieving a desired acoustic impedance without increasing the order of the system. Expectedresults are a better control of the dynamics of the electroacoustic absorber in terms of rise time,maximum overshoot, and settling time, and hence some improvement on both bandwidth and stability.6. AcknowledgementsThis work was supported by the Swiss National Science Foundation under research grant 200021-116977.REFERENCES1 H.F. Olson and E.G. May, Electronic sound absorber, Journal of the Acoustical Society of America,25(6), pp. 1130-1136, (1953).2 E. De Boer, Theory of motional feedback, IRE Transaction on Audio, 9(1), pp 15-21, (1961).3 P. Darlington, Loudspeaker circuit with means for monitoring the pressure at the speaker di-aphragm, means for monitoring the velocity of the speaker diaphragm and a feedback circuit, WorldPatent No WO 1997/003536, (1997).4 X. Meynial, Active acoustic impedance control device, World Patent No 1999/059377, (1999).5 R.J. Bobber, An active transducer as a characteristic impedance of an acoustic transmission line,Journal of the Acoustical Society of America, 48(1B), pp. 317-324, (1970).6 A.J. Fleming,D. Niderberger, S.O.R Moheimani, Control of resonant acoustic sound fields by elec-trical shunting of a loudspeaker, IEEE Transaction on Control System Technology, 15(4), pp. 689-703, (2007).7 H. Lissek, R. Boulandet, R. Fleury, Electroacoustic absorbers: bridging the gap between shuntloudspeakers and active sound absorption, Journal of the Acoustical Society of America, 129(5),(2011).8 M. Furstoss and D. Thenail and M-A. Galland, Surface impedance control for sound absorption:direct and hybrid passive/active strategies, Journal of Sound and Vibration, 203(2), pp. 219-236,(1997).9 H. Seraji, M. Tarokh, Design of proportional-plus-derivative output feedback for pole assignment,Proc. IEEE, 124(8), pp. 729-732, (1977).10 M. Rossi, Audio, Presses Polytechniques et Universitaires Romandes, (2007).11 R.C. Dorf, R.H. Bishop, Modern control systems (11th Ed.), Pearson Prentice Hall, (2008).12 K. Ogata, Modern control engineering (3rd Ed.), Prentice Hall International, (1997).7

Acoustic impedance synthesis at the diaphragm of moving coil loudspeakers using output feedback control

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Acoustic impedance synthesis at the diaphragm of moving coil loudspeakers using output feedback control

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