Collocated proportional-integral-derivative (PID) control of acoustic musical instruments

Abstract

The dynamic behavior of an acoustic musical instrument can be drastically modified by closing a feedback loop around a single sensor and actuator. Proportional-Integral-Derivative (PID) control provides for a simple yet effective paradigm, whose analysis is simplest for collocated systems. The analysis is further simplified when the musical instrument is modeled by a single, lightly damped mass-spring-damper system. This example is presented as a pedagogical laboratory exercise (http://ccrma.stanford.edu/realsimple/pidcontrol) within the framework of the RealSimPLE project (http://ccrma.stanford.edu/realsimple). Students learn what physical models are, why they are useful, what the basic ideas behind feedback control are, and how they may be applied to a plucked string. A plucked string model is implemented with a digital waveguide in the Pure Data environment. Given the waveguide model and the theoretical results from the simplified mass-spring-damper model, students are asked to find PID coefficients that result in particular controlled instrument behaviors (increase in damping, decrease in pitch, etc.). At the close of the talk, a few more advanced topics are covered. In particular, a proof is presented explaining why damping with collocated velocity feedback is stable no matter how large the feedback loop gain is. [Work supported by the Wallenberg Global Learning Network.]