Downloaded from orbit.dtu.dk on: Dec 18, 2017 Active control of loudspeakers: An investigation of practical applications Bright, Andrew Paddock; Jacobsen, Finn; Polack, Jean-Dominique; Rasmussen, Karsten Bo Publication date: 2002 Document Version Early version, also known as pre-print Link back to DTU Orbit Citation (APA): Bright, A. P., Jacobsen, F., Polack, J-D., & Rasmussen, K. B. (2002). Active control of loudspeakers: An investigation of practical applications. 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Active Control of Loudspeakers: An Investigation of Practical Applications Andrew Bright Ørsted·DTU – Acoustic Technology Technical University of Denmark Published by: Ørsted·DTU, Acoustic Technology, Technical University of Denmark, Building 352, DK-2800 Kgs. Lyngby, Denmark, 2002 3 This work is dedicated to the memory of Betty Taliaferro Lawton Paddock Hunt (1916-1999) and to Samuel Raymond Bright, Jr. (1936-2001) 4 5 Table of Contents Preface 9 Summary 11 Resumé 13 Conventions, notation, and abbreviations 15 Conventions....................................................................................................................................... 15 Notation............................................................................................................................................. 16 Abbreviations .................................................................................................................................... 20 1. Introduction 23 1.1. Active control of loudspeakers .............................................................................................. 28 1.2. Organisation of thesis ............................................................................................................ 29 1.3. References.............................................................................................................................. 31 2. Loudspeaker models 33 2.1. Linear models of loudspeakers .............................................................................................. 33 2.1.1. Electrical dynamics........................................................................................................ 35 2.1.2. Mechanical dynamics .................................................................................................... 38 2.1.3. Electro-mechanical transduction.................................................................................... 40 2.1.4. Acoustical components .................................................................................................. 40 2.1.5. Acoustic radiation.......................................................................................................... 43 2.1.6. Linear frequency response ............................................................................................. 44 2.1.7. Response prediction of loudspeaker mounted in a sealed cabinet ................................. 44 2.1.8. Response prediction of a loudspeaker mounted in a vented-box enclosure................... 47 2.2. Nonlinear models of loudspeakers......................................................................................... 48 2.2.1. Parametric nonuniformity and causes of nonlinearity ................................................... 48 2.2.2. Nonlinear simulation...................................................................................................... 54 2.3. Discrete-time physical modelling .......................................................................................... 56 2.3.1. FIR filter for electrical admittance................................................................................. 58 2.3.2. IIR filter for receptance of an SDOF system ................................................................... 60 2.3.3. IIR filter for mobility of an SDOF system....................................................................... 62 2.3.4. Nonlinear discrete-time loudspeaker model .................................................................. 64 2.4. Parametric uncertainty ........................................................................................................... 68 2.5. References.............................................................................................................................. 69 3. Theory of active control of loudspeakers 71 3.1. Feedback control for loudspeakers ........................................................................................ 72 3.1.1. Constant-current output amplifiers ................................................................................ 72 3.1.2. Negative amplifier output impedance............................................................................ 72 3.1.3. Feedback processing using vibration measurement....................................................... 74 3.2. Feedforward controllers.........................................................................................................76 3.2.1. Linear feedforward processing ...................................................................................... 76 6 3.2.2. Nonlinear feedforward processing .................................................................................76 3.3. Feedback linearisation............................................................................................................77 3.3.1. Feedback linearisation of continuous-time systems.......................................................77 3.3.2. Example: simplified closed box loudspeaker in continuous-time..................................80 3.3.3. State observer and partial state measurement ................................................................82 3.3.4. Feedforward formulation using a ‘simulation’ state observer .......................................83 3.3.5. Feedforward formulation assuming ideal alignment.....................................................84 3.3.6. Feedback linearisation of discrete-time systems............................................................86 3.3.7. Feedback linearisation with a discrete-time loudspeaker model....................................87 3.4. Adaptive feedforward controllers ..........................................................................................88 3.5. System identification by adaptive filtering ............................................................................90 3.5.1. General adaptive algorithms ..........................................................................................90 3.5.2. Adaptive IIR filters .........................................................................................................92 3.6. References..............................................................................................................................94 4. Loudspeaker system identification 99 4.1. Overview of approach, implementation, and evaluation .......................................................99 4.1.1. Forms for developing an error equation.......................................................................100 4.1.2. Hardware implementation and system-under-test........................................................101 4.1.3. Software implementation .............................................................................................104 4.1.4. Stability triangle...........................................................................................................106 4.1.5. Tolerance quadrilateral ................................................................................................106 4.1.6. Frame-based updating..................................................................................................110 4.2. Electrical current output-error form.....................................................................................112 4.2.1. Updating Reb .................................................................................................................113 4.2.2. Updating ak...................................................................................................................114 4.2.3. Updating of σu as a feedforward coefficient ................................................................115 4.2.4. Updating of σu as a dependent variable, defined by a2 ................................................115 4.2.5. Partial derivative of εoei[n] with respect to φ0...............................................................116 4.2.6. Convergence performance ...........................................................................................117 4.2.7. Accuracy of the converged parameters for the current output-error form ...................124 4.2.8. Updating φk ..................................................................................................................126