Feedback control of acoustic musical instruments when the number of sensors and actuators differ
Edgar J. Berdahl · The Journal of the Acoustical Society of America · 2019
Actuated instruments can be created in a variety of ways. An interesting class of actuated instruments are feedback-controlled acoustic musical instruments. A robust way to create these is to employ pairs of collocated sensors and actuators, and then to use the feedback control to simulate virtual physical systems. In the linear and time-invariant case, this means that the feedback control functions can be designed to be positive real. Accordingly, the physical properties of the instrument can become adjustable via the feedback control. An interesting case arises when the number of actuators and sensors differ. The actuators and sensors can however still potentially be collocated, which will result in the individual transfer functions from the actuators to the sensors (e.g. the mobilities) in being positive real. Under some special cases, stable feedback control can still be attained for a wide variety of feedback gains. The Feedback Guitar serves as an interesting case study for this. It has one actuator, which is approximately collocated with six piezoelectric sensors, one for each string. Using any non-negative linear combination of the sensor signals, an approximately positive real mobility can be obtained, which can enable stable feedback control for a wide variety of feedback gains.