A negative imaginary approach to the actuation of a guitar string

Abstract A feedback control approach to damp the resonances of a guitar string is considered. A guitar string is tightly stretched and fastened at both its ends. This string is placed in a magnetic field and set to vibrations by passing current through it. A displacement sensor is placed such that the actuator and the sensor are collocated. The linear dynamics characterizing the input/output (supplied current/measured displacement) relationship is identified and a model is fit. It is observed that the model satisfies the negative imaginary condition. An Integral Resonant Controller (IRC) is designed, by exploiting the negative imaginary characteristics, to damp the first five modes of the string. Signal conditioning circuitry associated with the sensor could introduce significant time delays that destroy the negative imaginary property. Here, an IRC is designed even when the system does not exhibit negative imaginary characteristics. The effectiveness these control designs are experimentally validated.

[1]  Bob L. Sturm,et al.  Proceedings of the International Computer Music Conference , 2011 .

[2]  S. O. Reza Moheimani,et al.  A Negative Imaginary Approach to Modeling and Control of a Collocated Structure , 2012, IEEE/ASME Transactions on Mechatronics.

[3]  Ian Postlethwaite,et al.  Multivariable Feedback Control: Analysis and Design , 1996 .

[4]  André Preumont,et al.  Vibration Control of Active Structures: An Introduction , 2018 .

[5]  S. O. Reza Moheimani,et al.  Integral resonant control of collocated smart structures , 2007 .

[6]  Mark J. Balas,et al.  Trends in large space structure control theory: Fondest hopes, wildest dreams , 1982 .

[7]  Ian R. Petersen,et al.  Multivariable integral control of resonant structures , 2008, 2008 47th IEEE Conference on Decision and Control.

[8]  Matti Karjalainen,et al.  Body Modeling Techniques for String Instrument Synthesis , 1996, ICMC.

[9]  Sourav Patra,et al.  Stability Analysis of Interconnected Systems With “Mixed” Negative-Imaginary and Small-Gain Properties , 2011, IEEE Transactions on Automatic Control.

[10]  Jeremy F. Alm,et al.  Time-Frequency Analysis of Musical Instruments , 2002, SIAM Rev..

[11]  B. Bhikkaji,et al.  A negative imaginary approach to control the vibrations of a string , 2012, 2012 24th Chinese Control and Decision Conference (CCDC).

[12]  Naoto Abe,et al.  Smith predictor control and internal model control - a tutorial , 2003, SICE 2003 Annual Conference (IEEE Cat. No.03TH8734).

[13]  Ian R. Petersen,et al.  A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems , 2010, IEEE Transactions on Automatic Control.

[14]  S. O. Reza Moheimani,et al.  Experimental implementation of extended multivariable PPF control on an active structure , 2006, IEEE Transactions on Control Systems Technology.

[15]  B. Bhikkaji,et al.  Integral Resonant Control of a Piezoelectric Tube Actuator for Fast Nanoscale Positioning , 2008, IEEE/ASME Transactions on Mechatronics.

[16]  Robert L. Clark,et al.  The application of smart structures toward feedback suppression in amplified acoustic guitars , 2003 .

[17]  Ian R. Petersen,et al.  Stability Robustness of a Feedback Interconnection of Systems With Negative Imaginary Frequency Response , 2008, IEEE Transactions on Automatic Control.

[18]  S. Šali,et al.  Measuring the quality of guitar tone , 2000 .

[19]  Matti Karjalainen,et al.  Towards High-Quality Sound Synthesis of the Guitar and String Instruments , 1993, ICMC.

[20]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[21]  Julius O. Smith,et al.  Inducing Unusual Dynamics in Acoustic Musical Instruments , 2007, 2007 IEEE International Conference on Control Applications.

[22]  S. O. Reza Moheimani,et al.  Spatial Control of Vibration: Theory and Experiments , 2003 .