Active Compensation of Transducer Nonlinearities W. Klippel KLIPPEL GmbH, Dresden, Germany Symposium Nonlinear Compensation of Loudspeakers Technical University of Denmark, 2003 Active Compensation, 1 Loudspeaker of the Future What are the objectives? Smaller, lighter, cheaper More output at less distortion Higher Efficiency Self-protection Active Compensation, 2
Loudspeaker of the Future... and the way? New materials New manufacturing technologies New transducer principles Improved design Active control Active Compensation, 3 Scope of the Paper Analog Techniques Digital Techniques Current Drive Servo Control Transducer oriented Generic Approach Self-learning System Practical Application Active Compensation, 8
Scope of the Paper Analog Techniques Digital Techniques Current Drive Servo Control Transducer oriented Generic Approach Self-learning System Practical Application Active Compensation, 9 Current-driven Transducer F m (x,i) x p ' M ms 1/K ms (x) R ms q P /S D S D2 M p F=Bl(x)i p A S D C B /S D 2 S D 2 R P Equivalent circuit of current driven transducer JAES Mills, Hawksford 1989 V in ~ I o compensates for variation of impedance due to Le(x) nonlinear damping due to Bl(x) fails in nonlinear excitation due to Bl(x) reluctance force Fm(x,i) stiffness Kms(x) of suspension Active Compensation, 10
Scope of the Paper Analog Techniques Digital Techniques Current Drive Servo Control Transducer oriented Generic Approach Self-learning System Practical Application Active Compensation, 11 Servo Control Using Output Feedback basic concept V(s) H I (s) - U(s) H C (s) D(s) H x (s) Nonlinear System s 2 X(s) A(s) Reference: Greiner, Schoessow 1983 Catrysse, 1985 Servo Controller Loudspeaker distortion transfer function 2 A( s) H x ( s) s = D( s) 1 + K( s) maximize K(s) ensure stability open loop gain K ( s) = H ( s) H ( s) s C x 2 Active Compensation, 13
Servo Feed-back Control Monitoring electrical impedance U(s) H P (s) - U L (s) P(s) H C (s) V E (s) Advantages: a simple theory analog technique no additional sensor Linear Detector I(s) Drawbacks: stability (voice coil temperature) linear motor (Bl(x)=const.) required restricted to Cms(x) nonlinearity Active Compensation, 14 Effect of Bl(x) nonlinearity Nonlinearity U(s) H P (s) - U L (s) q x (s) q r (s) P(s) H C (s) X(s) CONTROLLER V E (s) V(s) s Nonlinearity LOUDSPEAKER AND LINEAR DETECTOR Draw-backs: Nonlinear relationship between velocity and back EMF generates additional distortion in output Active Compensation, 15
Servo control using current feedback V(s) H I (s) - H C (s) D(s) H x (s) Nonlinear System s 2 X(s) A(s) V(s) Nonlinear Detector Servo Controller I(s) Nonlinear System Transducer Reference: Larsen 1997 linear input current I gives distorted sound pressure output Nonlinear detector required to estimate velocity V Active Compensation, 16 Scope of the Paper Analog Techniques Digital Techniques Servo Control Current Drive Transducer oriented Generic Approach Self-learning System Practical Application Active Compensation, 18
Requirements for Distortion Reduction p dist (t) Original distortion Residual distortion synthesized distortion t Equal amplitude 180 degree phase shift Active Compensation, 19 Polynomial Filter Compensation u(t) K 1 (s) u e (t) H 1 (s) p m (t) K 2 (s 1, s 2 ) H 2 (s 1, s 2 ) K 3 (s 1, s 2, s 3 ) H 3 (s 1, s 2, s 3 ) Filter Model Parameter Transfer Patent: Kaizer, 1985 Active Compensation, 20
Polynomial Filter with Generic Structure h 2 (0,0) Input z -1 z -1 h 2 (0,1) h 2 (1,2) Output h 2 (t 1,t 2 ) quadratic kernel can be synthesized by delay elements, multipliers and weights z -1 second order polynomial filter Advantages no loudspeaker model required can be used for any nonlinear system flexible, simple theory feedforward, stable inverse and parallel modeling possible Active Compensation, 21 Disadvantages fails at high amplitudes high computational load large number of parameters parameters are not interpretable special measurement technique required DSP Requirements Generic polynomfilter h 2 (t 1,t 2 ) Frank, 1995 Impulse response h(t) H(f)= Displacement X / Stimulus 50 Measured Windowed 40 30 200 ms 10000 [mm / V] 20 10 0-10 10000-20 -50 0 50 100 150 200 250 300 350 400 450 left:-10.000 Time [ms] right:436.667 Frank 1995 Applied to a woofer at 48 khz sampling: h 2 (t 1,t 2 ) 100 MIPS h 3 (t 1,t 2,t 3 ) 10 6 MIPS compensation of higher order distortion? Active Compensation, 22
Time Delay Neural Network FIR Filter v(t) Audio signal Controller d(t) Reference: State Generator Nonlinear Expander Chang 1994 Tapped Delay line Neural Network Adjustable weights in the output layer Active Compensation, 24 Time Delay Neural Network IIR Filter v(t) Audio signal Tapped Delay Line Neural Network Tapped Delay Line Controller Advantages reduced number of states, parameters reasonable computational load Problems: Stability!! Parameter Adjustment?? Active Compensation, 25
Generic Control Approach Summary Advantage Generic control structure Theory, tools available Disadvantage Many parameters and state variables Not related to physics not interpretable Active Compensation, 27 Scope of the Paper Analog Control Techniques Digital Control Techniques Servo Control Current Drive Transducer oriented Generic Approach Self-learning System Practical Application Active Compensation, 28
Design of Transducer-oriented Controller Approach: 1. Model speaker at high amplitudes Search for dominant nonlinearities Separate static nonlinearities from linear dynamics Introduce varying parameter Develop mathematical model 2. Derive control law 3. Generate state variables in controller 4. Determine optimal parameters Active Compensation, 29 1st step: Nonlinear Transducer Modeling port nonlinearity Multiple Outputs Single Input Air compression Radiation Sound Propagation Room Acoustics p(r 1 ) Audio signal Amplifier Crossover EQ u(t) Electromechanical Transducer x(t) Mechanoacoustical Transducer Radiation Sound Propagation Room Interference p(r 2 ) sound field Bl(x) Kms(x) Le(x) i(t) Radiation Doppler Effect Sound Propagation Room Interference Wave Steepening p(r 3 ) Active Compensation, 30
Criteria for dominant Nonlinearities The nonlinear mechanism limits acoustical output Generates audible distortion indicates an overload situation causes unstable behavior is related with cost, weight, volume, efficiency affects speaker system alignment Active Compensation, 31 Ranking List of Transducer Nonlinearities 1. Force Factor Bl(x) 2. Compliance C ms (x) tweeter 3. Inductance L e (x) 4. Nonlinear Sound Propagation c(p) 5. Flux Modulation L e (i) 6. Doppler Distortion τ(x) 7. Nonlinear Cone Vibration 8. Port Nonlinearity R A (v) 9. many others... woofers horns Active Compensation, 32
Linearization of a SIMO-System System Requirement H 1 (s) p(r 1 ) u 0 (t) H 0 (s) h 1...... h N H 2 (s) p(r 2 ) y 1 (t) y i (t) h i y N (t) y N+1 (t) H 3 (s) p(r 3 ) Nonlinear subsystems are connected in series at the input Parallel systems at the output are linear Active Compensation, 33 Equivalent Input Distortion H(f,r 1 ) p(r 1 ) sound field Sinusoidal sweep Distortion in Voltage u u D Nonlinear System H(f,r 2 ) p(r 2 ) H(f,r 3 ) p(r 3 ) small signal response Sound pressure measurement 3rd harmonic distortion in voltage Signal at IN1 3rd harmonics absolute Signal at IN1 0-5 nearfield 30 cm 60 cm 1 m distance KLIPPEL 90 85 1 m distance 60 cm distance 30 cm distance nearfield KLIPPEL -10 80-15 75 db - [V] -20-25 -30-35 Inverse filtering with H(f,r) db - [V] 70 65 60 55-40 50-45 45 50 100 200 500 1k Frequency [Hz] 40 50 100 200 500 1k Frequency [Hz] Active Compensation, 34
Criterium for Distortion Reduction 3rd-order EHID 0.1 m 0.5 m 1 m 10 1 [Percent] 10 0 10-1 Nonlinearities located in one-dimensional signal path 10 2 10 3 Frequency [Hz] Distortion generated in multi-dimensional domain can not be compensated by control Active Compensation, 35 Linearization of Serial Subsystems Mirror Filter Approach u ( t τ ) y ( t) i i = i = 1,..., N i G (s) i G N (s) g N G N-1 (s) g i u N u i-1 un v N v N-1...... vi g v i-1 1 v1 u 1 G 0 (s) u 0 H 0 (s) h 1... h y 1 i... y yi+1 h i y N N y 2 y N+1 H N+1 (s) p(t, r 1 ) Controller Transducer corresponding subsystems in controller and transducer successive linearization by inverse filtering state variables in controller and transducer are identical time delay may be added (causal filters) Patent Klippel 1991 Active Compensation, 36
Force Factor Bl(x) magnet Pole piece permanent flux Φ 0 Voice coil A / 5,5 5,0 4,5 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 0,0 N force factor -7,5-5,0-2,5 0,0 2,5 5,0 7,5 x[mm] Bl(x) determined by Magnetic field distribution Height and overhang of the coil Optimal voice coil position Active Compensation, 38 Stiffness Kms(x) of Suspension K(x) [N/m] 800 700 600 0 x F= K(x) x 500 400 300 200 100 Kms(x) determined by suspension geometry impragnation adjustment of spider and surround -15-10 -5 0 5 10 Displacement x [mm] Active Compensation, 39
Voice Coil Inductance L e (x) mh inductance 1,00 0,75 0,50 0,25 voice coil 0,00-7,5-5,0-2,5 0,0 2,5 5,0 7,5 x[mm] L e (x) determined by geometry of coil, gap, magnet optimal size and position of short cut ring Active Compensation, 40 Equivalent Circuit of the vented-box loudspeaker system F m (x,i) R e L e (x) x p ' M ms 1/K ms (x) R ms i q P /S D S D 2 M p u Bl(x)x' Bl(x) Bl(x)i p A S D C B /S D 2 S D 2 R P Nonlinear Parameters are not constant but depend on state variables (displacement, current) Active Compensation, 42
State space model voltage y 1 (t) h(x) y 2 (t) sound pressure Linear and Nonlinear parameter (Bl(x), Kms(x), Le(x) b(x) Active Compensation, 43 a(x) X Transducer Subsystem h 1 State variables: Displacement x Velocity v Current i... Preferred Representation Nonlinear terms Linear part u(i) H l (z) n(i) p(i) α(x) -1 β(x) STATE EXPANDER X PLANT Nonlinear part separated from linear part Scalar operation applied to the input signal Active Compensation, 44
State Space Model in Normal Form integrator-decoupled form y 1 (t) z 3 z 2 z 1 y 2 (t) input-output dynamics separated from zero dynamics z 4 z 5 g(x) f(x) x T -1 I 1 (z) z I 2 (z) Zero Dynamics Zero dynamics: Vented box system Passive radiator mechanical resonances (panel) acoustical resonances (horn) Transducer Subsystem h 1 Active Compensation, 45 Generation of Distortion Kms(x) (x)-nonlinearity Voltage distortion fs highpass pressure pass band Displacement x fs lowpass Bass tone multiplier Multiplication of displacement nonlinear distortion Active Compensation, 46
Distortion Generation Bl(x) (x)-nonlinearity (parametrical Excitation) Voltage distortion impedance fs highpass pressure current fs multiplier x lowpass fs Motor force F=Bl(x)*i Multiplication of x and i Active Compensation, 47 Distortion Generation L(x)-Nonlinearity (Variation of input impedance) Voltage fs pressure distortion differentiator impedance highpass 6dB/oct current fs multiplier L(x) x fs lowpass 1. Multiplication of x and i 2. Differentiation of distortion Active Compensation, 48
2nd step: : Derivation of the Control Law Electro-acoustical model Volterra Series Kaizer 1985 Integro-differential equation Klippel 1991 Integrator-Decoupled Form Generic Control Theory (Beerling 1994,Suykens 1995) Special Polynomial Filter Exact Control Law Active Compensation, 49 Polynomial Filter dedicated to Speakers u(t) K 1 (s) u e (t) H 1 (s) p m (t) K 2 (s 1, s 2 ) H 2 (s 1, s 2 ) K 3 (s 1, s 2, s 3 ) H 3 (s 1, s 2, s 3 ) Patent Kaizer 1986 Filter Model Approach 1. Analytical Modeling with Volterra Series 2. Inversion of the kernels 3. Synthesize kernel function Active Compensation, 50 Disadvantages restricted to low-order nonlinearities fails at high amplitudes high computational load
Mirror Filter Approach Nonlinear terms Desired function w(i) u(i) H l (z) p(i) n(i) β(x) α(x) α(x ) -1 β(x ) STATE EXPANDER Patent Klippel 1991 X CONTROLLER X PLANT Approach: 1. Nonlinear integro-differential equation 2. Desired overall transfer function Hi(z) 3. Difference between nonlinear and linear equation Advantages: perfect linearization ad-hoc solution non-minimalphase systems Active Compensation, 51 State Feedback Control separate LD and ID controllers LD Controller ID Controller z 3 z 2 z 1 v 1 (t) g D w 1 (t u 1 (t) y 1 (t) y 2 (t) z 4 z 5 f D (z) -f(x) g(x) -1 g(x) f(x) T z I 1 (z) I 2 (z) Zero Dynamics Subsystem g 1 x x T -1 z Transducer Subsystem h 1 Advantages: common control theory applicable straightforward derivation perfect linearization JAES Suykens 1995 Problems: full information on states and parameters robustness (under parameter uncertainties) Active Compensation, 53
Direct State Feedback Control v 1 (t) u 1 (t) y 1 (t) g D z 3 z 2 z 1 y 2 (t) z 4 z 5 -β(x) α(x) -1 α(x) β(x) f D (z) I 1 (z) I 2 (z) Controller Subsystem g 1 X dx/dt i X dx/dt i Transducer Subsystem h 1 Advantages: Perfect linearization control independent on internal dynamics minimal state information required (only x, dx/dt, i) minimal parameters required (nonlinear only) x T -1 z Desired Dynamics JAES Klippel 1995 Active Compensation, 54 3rd step: : Generation of State Variables State Variables (displacement Current) State Predictor Klippel 1991 State Observer Beerling 1994 State Measurement Suykens 1995 Active Compensation, 55
Feedback Control with State Measurement w Nonlinear Controller u Sensor State Vector X Current State Measurement Displacement Drawbacks: Sensors for all states (x, i) required No time delay in DAC and ADC DC displacement must be monitored optimal speaker parameters required Active Compensation, 58 State Feedback Control with Observer using the controller output voltage v i i (t) -β(x ) α(x ) -1 u i (t) exp(-τ ι s) y i (t) α(x) β(x) Desired Dynamics y i+1 (t) Advantages: avoids problems with time delay no sensor required Controller Subsystem g i Transducer Subsystem h i u i (t) Beerling 1994 Schurer 1995 X v i b(x ) a(x ) x' Problems: precise speaker parameters required observer has a feedback structure observer might become unstable State Observer Active Compensation, 60
State Feed-forward Control State Prediction v i (t) u i (t) y i (t) exp(-τ ι s) Desired Dynamics Desired Dynamics y i+1 (t) x v i -β(x ) α(x ) -1 x v i α(x) β(x) Controller Subsystem g i Advantages: Perfect linearization Stable, robust No sensor required feedforward system delay may be added Simple digital implementation Active Compensation, 63 Transducer Subsystem h i Mirror Filter Klippel Patent 1991 Probing the Signals in the Controller K MS (x)-distortion Bl(x)-distortion L(x)-distortion v i (t) - - - u i (t) exp(-τ ι s) y i (t) CD Desired Dynamics d K d Bl d L d L d Bl d K Desired Dynamics y i+1 (t) Output x v i Controller Subsystem g i Transducer Subsystem h i Control Output Active Compensation, 65
Scope of the Paper Analog Techniques Digital Techniques Servo Control Current Drive Transducer oriented Generic Approach Self-learning System Practical Concerns Active Compensation, 68 Effect of a Disagreement between synthesized and original distortion Error in magnitude Error in phase Active Compensation, 69
Adjustment of the Controller Audio Controller Voltage Sound Pressure Parameters Parameters depend on type of transducer, unit time, aging temperature, humidity stimulus (music) Adaptive Adjustment Active Compensation, 71 Problems of the Adaptive Approach Loudspeaker is a strong nonlinear system measured signals are corrupted by noise the controller is connected to speaker input Active Compensation, 75
Indirect Updating based on Inverse Modeling z -K n(i) noise - w(i) Controller u(i) Loudspeaker- Sensor-System p(i) Adaptive Model e(i) W W X Parameter Estimation Active Compensation, 76 Indirect Updating based on Inverse Modeling ADVANTAGES: Model and Update system are always stable Parameter estimation has an unique solution DISADVANTAGES: High computational complexity Transformation of parameters is required Parameter estimation is biased if measurement is corrupted by noise Active Compensation, 77
Indirect Updating based on Parallel Modeling n(i) w(i) Controller W C u(i) Loudspeaker-Sensor- System p(i) - e(i) e(i) Model Transformation X W Parameter Estimation Active Compensation, 78 Indirect updating with generic filters parameters are linear in the output v(t) Audio signal exp(-τs) d(t) u(t) amplifier p(t) Linear Parameter Estimator Delay Line Nonlinear Expander H lin -1 P l P l y l (t) Controller Delay Line X Nonlinear Expander A P n y n (t) AES Preprint Gao, 1992 P n Detector Nonlinear Parameter Estimator Problem: feedforward model is limited to small signal domain! Active Compensation, 79
Indirect Updating based on Parallel Modeling ADVANTAGES: Immune against measurement noise DISADVANTAGES: High computational complexity (two nonlinear systems, parameter transformation) Model with feedback structure is unstable Model with feedforward structure causes bias in parameter estimation Active Compensation, 80 Direct Adaptive Control H l (z) p D (i) n(i) - e(i) w(i) Controller u(i) Loudspeaker-Sensor System p(i) X W Parameter Estimation Active Compensation, 81
Direct Updating Disadvantages: Nonlinear Relationship between control parameters and error signal The state and the distortion generation of the loudspeaker depend on the control parameters Update may become unstable Special update algorithm required Advantages: Low computational complexity Optimal parameter adjustment without bias Simplified calculation of gradient signals Implementation on available DSP-systems Active Compensation, 82 Speaker as Sensor? Nonlinear audio Control Law signal - parameter vector Adaptive Detector state vector Patent: Klippel 1993 voltage current Problems: detector for EMF required effect of nonlinearities parameter variation Advantages: Robust sensor high accuracy low distortion low cost no mechanical problems low acoustical disturbances special hardware available Adaptive nonlinear system Active Compensation, 84
Detection of voice coil velocity R e L(x) v m 1/k(x) R m F m i u b(x)v b(x) b(x)i Z A mh inductance A / N force factor 5,5 5,0 Voltage current 1,00 0,75 0,50 Back EMF Bl(x)v 4,5 4,0 3,5 3,0 2,5 2,0 Mechanical signal (velocity) 1,5 0,25 1,0 0,5 0,00 0,0-7,5-5,0-2,5 0,0 2,5 5,0 7,5 x [mm] -7,5-5,0-2,5 0,0 2,5 5,0 7,5 x[mm] Active Compensation, 85 Scope of the Paper Analog Techniques Digital Techniques Servo Control Current Drive Transducer oriented Generic Approach Self-learning System Practical Application Active Compensation, 87
Klippel ControllerC Additional Processing (crossover) V[n] Protection System Control Law u[n] DAC Offset compensator u(t) i(t) Thermal Model State Predictor ADC ADC Adaptive Parameter Detector Feedforward Controller Memory Diagnostics Detector parameter user information Active Compensation, 89 Klippel Controller first evaluation system using the Distortion Analyzer amplifier Distortion Analyzer Loudspaaker system for any electrodynamical transducer mounted in closed or vented enclosures, horns,... monitors voltage + current only (no sensor) music as stimulus real time diagnostics no driver information required full protection, loudspeaker alignment Active Compensation, 90
Modes of Operation 1. Step: : Initial Identification with noise 2. Step: Predictive Control for any input Active Compensation, 91 Synergetics technology used in the Klippel Analyzer System Verification of modeling Parameter identification Learning with music Evaluation and optimal design TRF, DIS LPM, LSI PWT AUR, SIM Active Compensation, 92
Benefit: Linearization Second-order intermodulation distortion in percent ( d2 ) Signal at IN2 12,5 Control ON 0.25 V Control OFF 0.25 V Percent 10,0 7,5 5,0 Distortion reduction 2,5 2*102 3*102 4*102 5*102 6*102 7*102 8*102 9*102 Frequency f1 [Hz] Third-order intermodulation distortion in percent ( d3 ) Signal at IN2 30 Control ON 0.25 V controller OFF 0.25 V 25 Percent 20 15 10 5 Distortion reduction 0 2*102 3*102 4*102 5*102 6*102 7*102 8*102 9*102 Frequency f1 [Hz] Active Compensation, 93 Benefit: Compensation Coil Offset gives more sensitivity less distortion stable driver V[n] Offset compensator Control Law u[n] DC-coupled Amplifier!! DAC Patent, Klippel 1995 Bl [N/A] Force factor Bl(X) vs displacement Bl(X) 3,5 3,0 2,5 2,0 1,5 1,0 0,5-4 -3-2 -1-0 1 2 3 4 Displacement X [mm] Active Compensation, 94 Shifting the coil Bl [N/A] Force factor Bl(X) vs displacement Bl(X) 3,5 3,0 2,5 2,0 1,5 1,0-4 -3-2 -1-0 1 2 3 4 Displacement X [mm]
Benefit: Protection of the Driver audio signal Protection System state vector Mirror Filter - parameter vector Adaptive Detector voltage current Benefits: Access to critical state variables (displacement, temperature) automatic detection of critical limits (Xmax) full mechanical protection due to prediction of envelope minimal impact on sound quality no additional time delay Active Compensation, 95 Benefit: Speaker Diagnostics Nonlinear audio Control Law signal - parameter vector state vector Diagnostics Adaptive Detector voltage current messages Benefits: control parameters have a physical meaning monitoring aging of suspension detection of voice coil offset generation of service messages (warnings) prevention of failure during professional applications reduction of distortion Active Compensation, 97
3,0 2,5 2,0 1,5 1,0 0,5 0,0 Bl(X) -4-3 -2-1 -0 1 2 3 4 Displacement X [mm] Benefit: New Degrees of freedom Active Speaker System Controller Design Adjustment Passive Speaker Design sound quality (linear and nonlinear distortion) Active Compensation, 98 cost,weight directivity enclosure volume max. sound pressure efficiency Linearization by Passive Means Bl [N/A] 3,0 2,5 2,0 1,5 1,0 0,5 Pole plate voice coil pole piece Force factor Bl(X) vs displacement Bl(X) LPB130 coil height = 5.3 mm gap height = 4 mm R E = 3.5 Ohm M MS = 5,2 g L E = 0.2 mh L m = 92 db 0,0-4 -3-2 -1-0 1 2 3 4 Displacement X [mm] Bl [N/A] Pole plate voice coil pole piece Force factor Bl(X) vs displacement LPB130* (more overhang) coil height = 14.3 mm gap height = 4 mm R E = 8.75 Ohm M MS = 7 g L E = 0.5 mh L m = 85.4 db Sensitivity decreased by 6.6 db! Active Compensation, 99
Practical Considerations Active speaker control becomes powerful for loud speakers with small size & weight, but high output & sensitivity if driver, system and DSP design cooperates in combination with speaker protection and diagnostics Controller is realized by software only Minimal hardware platform is available Active Compensation, 101 Speaker Problems fixed by Control Active remedies are superior: Distortion due to limited voice coil height Distortion from progressive suspension Bl(x)-asymmetries caused by field geometry Voice coil offset due to aging of suspension Cms(x)-asymmetries in high-frequency driver Distortion from voice coil inductance Voice coil former hits backplate Active Compensation, 102
Speaker Problems fixed by Design Passive Remedies are superior for coping with nonlinearities causing loudspeaker instabilities nonlinearities in the multi-dimensional domain (cone break-up, radiation) loudspeaker defects (Rub & Buzz) Active Compensation, 103 Summary Transducers can be modeled at high amplitudes Physical models are superior over generic models Dominant nonlinearities can be compensated Adaptive control self-learning system Actuator may be used as sensor New degrees of freedom for passive driver design New ways for driver protection and diagnosis Active Compensation, 108