ABabcdfghiejkl

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

Edgar Berdahl and Julius O. Smith III

Department of Electrical EngineeringCenter for Computer Research in Music and Acoustics (CCRMA)

Stanford UniversityStanford, CA, 94305

Education in Acoustics: Tools for Teaching AcousticsThursday Morning at 10:20AM, June 7th, 2007

—Special thanks to the Wallenberg Global LearningNetwork for supporting the REALSIMPLE project

ABabcdfghiejkl

Outline

Overview

Theory

Laboratory Exercise In Pure Data

ABabcdfghiejkl

The RealSimPLE Project

◮ RealSimPLE is a web-based teacher’s resource for studentlaboratory sessions in musical acoustics.

ABabcdfghiejkl

The RealSimPLE Project

◮ RealSimPLE is a web-based teacher’s resource for studentlaboratory sessions in musical acoustics.

◮ Music is a good way to interest young people in math,science, and engineering.

ABabcdfghiejkl

The RealSimPLE Project

◮ RealSimPLE is a web-based teacher’s resource for studentlaboratory sessions in musical acoustics.

◮ Music is a good way to interest young people in math,science, and engineering.

◮ Physical experiments and pedagogical computer-basedsimulations of the same systems run in parallel andinterconnected.

ABabcdfghiejkl

The RealSimPLE Project

◮ RealSimPLE is a web-based teacher’s resource for studentlaboratory sessions in musical acoustics.

◮ Music is a good way to interest young people in math,science, and engineering.

◮ Physical experiments and pedagogical computer-basedsimulations of the same systems run in parallel andinterconnected.

◮ The traditional lab bench is enhanced rather than replaced.

ABabcdfghiejkl

The RealSimPLE Project

◮ RealSimPLE is a web-based teacher’s resource for studentlaboratory sessions in musical acoustics.

◮ Music is a good way to interest young people in math,science, and engineering.

◮ Physical experiments and pedagogical computer-basedsimulations of the same systems run in parallel andinterconnected.

◮ The traditional lab bench is enhanced rather than replaced.◮ Only standard computers and some inexpensive,

easy-to-build hardware are required.

ABabcdfghiejkl

The RealSimPLE Project

◮ RealSimPLE is a web-based teacher’s resource for studentlaboratory sessions in musical acoustics.

◮ Music is a good way to interest young people in math,science, and engineering.

◮ Physical experiments and pedagogical computer-basedsimulations of the same systems run in parallel andinterconnected.

◮ The traditional lab bench is enhanced rather than replaced.◮ Only standard computers and some inexpensive,

easy-to-build hardware are required.◮ The RealSimPLE Project is a collaboration between

Stanford University and KTH in Sweden.

RealSimPLE Laboratory Assignment Dependencies

Transfer Function

PsychoacousticsLab

Harmonic Contentof a Plucked String

ExperimentsMonochord

Monochord ActivityWeighted

ControlPID

Plucked String DigitalWaveguide Model

Traveling Waves InA Vibrating String

Auditory FilterBank Lab

Illusions LabMusical

Flute LabVirtual

Virtual AcousticTube Lab

MonochordAssembly

SoundcardSetup

START

Introduction to STKand Reverberation

Time−VaryingDelay Effects

Guitar ModelElectric

and Piano ModelsAcoustic Guitar

Measurement Toolbox

ABabcdfghiejkl

Summary Of PID Control Lab Objectives

◮ Explain the basic idea behind feedback control.

ABabcdfghiejkl

Summary Of PID Control Lab Objectives

◮ Explain the basic idea behind feedback control.◮ Describe how this discipline may be applied to a vibrating

string.

ABabcdfghiejkl

Summary Of PID Control Lab Objectives

◮ Explain the basic idea behind feedback control.◮ Describe how this discipline may be applied to a vibrating

string.◮ Describe how modifying the control parameters affects the

harmonic content.

ABabcdfghiejkl

Summary Of PID Control Lab Objectives

◮ Explain the basic idea behind feedback control.◮ Describe how this discipline may be applied to a vibrating

string.◮ Describe how modifying the control parameters affects the

harmonic content.◮ Explain what instability is and how it may arise.

ABabcdfghiejkl

Summary Of PID Control Lab Objectives

◮ Explain the basic idea behind feedback control.◮ Describe how this discipline may be applied to a vibrating

string.◮ Describe how modifying the control parameters affects the

harmonic content.◮ Explain what instability is and how it may arise.◮ Experiment with a virtual controlled string using the Pure

Data software.

ABabcdfghiejkl

Summary Of PID Control Lab Objectives

◮ Explain the basic idea behind feedback control.◮ Describe how this discipline may be applied to a vibrating

string.◮ Describe how modifying the control parameters affects the

harmonic content.◮ Explain what instability is and how it may arise.◮ Experiment with a virtual controlled string using the Pure

Data software.◮ Gain experience using Pure Data.

ABabcdfghiejkl

Feedback control

Feedback control is the discipline in which system dynamics arestudied and altered by creating feedback loops.

System+r xu

Controller

Figure: Typical block diagram for a control application

ABabcdfghiejkl

Feedback control

Feedback control is the discipline in which system dynamics arestudied and altered by creating feedback loops.

System+r xu

Controller

Figure: Typical block diagram for a control application

◮ Application to cruise control

ABabcdfghiejkl

Feedback control

Feedback control is the discipline in which system dynamics arestudied and altered by creating feedback loops.

System+r xu

Controller

Figure: Typical block diagram for a control application

◮ Application to cruise control◮ Application to a vibrating string

ABabcdfghiejkl

Outline

Overview

Theory

Laboratory Exercise In Pure Data

ABabcdfghiejkl

System Model

◮ If we collocate the sensor and actuator, then we can use thefollowing model of the lowest resonance:

Figure: Lightly-damped harmonic oscillator (R is small)

ABabcdfghiejkl

System Model

◮ If we collocate the sensor and actuator, then we can use thefollowing model of the lowest resonance:

Figure: Lightly-damped harmonic oscillator (R is small)

◮ Equivalent mass m, spring with constant K , and dampingparameter R

ABabcdfghiejkl

System Model

◮ If we collocate the sensor and actuator, then we can use thefollowing model of the lowest resonance:

Figure: Lightly-damped harmonic oscillator (R is small)

◮ Equivalent mass m, spring with constant K , and dampingparameter R

◮ mx + Rx + Kx = 0

ABabcdfghiejkl

System Model

◮ If we collocate the sensor and actuator, then we can use thefollowing model of the lowest resonance:

Figure: Lightly-damped harmonic oscillator (R is small)

◮ Equivalent mass m, spring with constant K , and dampingparameter R

◮ mx + Rx + Kx = 0

◮ Pitch f0 ≈ 12π

√

Km , and the decay time constant τ = 2m

R

ABabcdfghiejkl

Proportional-Derivative (PD) Control

◮ If we implement the feedback law F = PD x + PPx , then wearrive at the following differential equationmx + Rx + Kx = PDx + PPx

ABabcdfghiejkl

Proportional-Derivative (PD) Control

◮ If we implement the feedback law F = PD x + PPx , then wearrive at the following differential equationmx + Rx + Kx = PDx + PPx

◮ The controlled dynamics aremx + (R − PD)x + (K − PP)x = 0

ABabcdfghiejkl

Proportional-Derivative (PD) Control

◮ If we implement the feedback law F = PD x + PPx , then wearrive at the following differential equationmx + Rx + Kx = PDx + PPx

◮ The controlled dynamics aremx + (R − PD)x + (K − PP)x = 0

◮ f0 ≈ 12π

√

K−PPm and decay time τ = 2m

R−PD

ABabcdfghiejkl

Integral Control

◮ Similarly the feedback law F = PI∫

x dt

ABabcdfghiejkl

Integral Control

◮ Similarly the feedback law F = PI∫

x dt◮ The controlled dynamics are third order

mx + Rx + Kx = PI∫

x dt

ABabcdfghiejkl

Integral Control

◮ Similarly the feedback law F = PI∫

x dt◮ The controlled dynamics are third order

mx + Rx + Kx = PI∫

x dt◮ The analysis is more complicated, but for small |PI |,

τ ≈ 2m

R+PI

4π2f 2

0

ABabcdfghiejkl

Integral Control

◮ Similarly the feedback law F = PI∫

x dt◮ The controlled dynamics are third order

mx + Rx + Kx = PI∫

x dt◮ The analysis is more complicated, but for small |PI |,

τ ≈ 2m

R+PI

4π2f 2

0

◮ PID Control:◮ Can control the damping with PD and PI

ABabcdfghiejkl

Integral Control

◮ Similarly the feedback law F = PI∫

x dt◮ The controlled dynamics are third order

mx + Rx + Kx = PI∫

x dt◮ The analysis is more complicated, but for small |PI |,

τ ≈ 2m

R+PI

4π2f 2

0

◮ PID Control:◮ Can control the damping with PD and PI◮ Can control the pitch (some) with PP

ABabcdfghiejkl

Outline

Overview

Theory

Laboratory Exercise In Pure Data

ABabcdfghiejkl

Pure Data

ABabcdfghiejkl

Instability

◮ Students can choose the control parameters PP , PI , and PD

over a reasonable range.

ABabcdfghiejkl

Instability

◮ Students can choose the control parameters PP , PI , and PD

over a reasonable range.◮ Some combinations of the control parameters result in

instability.

ABabcdfghiejkl

Instability

◮ Students can choose the control parameters PP , PI , and PD

over a reasonable range.◮ Some combinations of the control parameters result in

instability.◮ The patch disables sound if the level becomes too large.

ABabcdfghiejkl

Questions For Students

After students have been given the relevant backgroundinformation, they are prodded through a series of questions toconclude the following

ABabcdfghiejkl

Questions For Students

After students have been given the relevant backgroundinformation, they are prodded through a series of questions toconclude the following

1. Sensors placed at nodes of particular harmonics will notmeasure any energy at these frequencies.

ABabcdfghiejkl

Questions For Students

After students have been given the relevant backgroundinformation, they are prodded through a series of questions toconclude the following

1. Sensors placed at nodes of particular harmonics will notmeasure any energy at these frequencies.

2. In comparison with PD, PI causes lower harmonics to bedamped more quickly than higher harmonics.

ABabcdfghiejkl

Questions For Students

After students have been given the relevant backgroundinformation, they are prodded through a series of questions toconclude the following

1. Sensors placed at nodes of particular harmonics will notmeasure any energy at these frequencies.

2. In comparison with PD, PI causes lower harmonics to bedamped more quickly than higher harmonics.

3. It is not possible to change the pitch as far as the modelpredicts before instability sets in.

ABabcdfghiejkl

Questions For Students

After students have been given the relevant backgroundinformation, they are prodded through a series of questions toconclude the following

1. Sensors placed at nodes of particular harmonics will notmeasure any energy at these frequencies.

2. In comparison with PD, PI causes lower harmonics to bedamped more quickly than higher harmonics.

3. It is not possible to change the pitch as far as the modelpredicts before instability sets in.

4. This is a weakness of the simplified model.

ABabcdfghiejkl

Challenge Problem

◮ We use Pure Data because it is easy to see how thepatches work.

ABabcdfghiejkl

Challenge Problem

◮ We use Pure Data because it is easy to see how thepatches work.

◮ We challenge advanced students to modify the patch toimplement a time-varying amplitude effect.

ABabcdfghiejkl

Challenge Problem

◮ We use Pure Data because it is easy to see how thepatches work.

◮ We challenge advanced students to modify the patch toimplement a time-varying amplitude effect.

◮ They are given the following hint:

ABabcdfghiejkl

Bibliography

E. Berdahl and J. O. Smith III,PID Control of a Plucked String, Online REALSIMPLELaboratory,http://ccrma.stanford.edu/realsimple/pidcontrol,2007.

C. Besnainou,Transforming the voice of musical instruments by activecontrol of the sound radiation,International Symposium on Active Control of Sound andVibration, Fort Lauderdale, FL, 1999.

E. Berdahl, J. O. Smith III, and A. Freed,Active damping of a vibrating string,International Symposium on Active Control of Sound andVibration, Adelaide, Australia, 2006.

ABabcdfghiejkl

Thanks

Questions?