Applications of Passivity Theory to the Active Control of Acoustic

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Applications of Passivity Theory to the ActiveControl of Acoustic Musical Instruments

Edgar Berdahl, Gunter Niemeyer, and Julius O. Smith III

Acoustics ’08 Conference, Paris, France

June 29th-July 4th, 2008

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Outline

Introduction

Passivity For Linear Systems

PID Control

Other Passive Linear Controllers

Nonlinear PID Control

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Introduction

◮ Project goal: To make the acoustics of a musical instrumentprogrammable while the instrument retains its tangible form.

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Introduction


◮ We use a digital feedback controller.

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Introduction


◮ We use a digital feedback controller.◮ The resulting instrument is like a haptic musical instrument

whose interface is the whole acoustical medium.

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Introduction


◮ We use a digital feedback controller.◮ The resulting instrument is like a haptic musical instrument

whose interface is the whole acoustical medium.◮ We apply the technology to a vibrating string, but the

controllers are applicable to any passive musical instrument.

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System Block Diagram

−+r

parameters

musical instrument

Controller

G(s)

u

F v

Figure: System block diagram for active feedback control

◮ We would like the controller to be robust to changesin G(s).

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Outline

Introduction


PID Control



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s-Domain Positive Real Functions

◮ We operate in the Laplace s-domain.

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◮ We operate in the Laplace s-domain.◮ A function G(s) is positive real if

1. G(s) is real when s is real.

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1. G(s) is real when s is real.2. Re{G(s)} ≥ 0 for all s such that Re{s} ≥ 0.

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◮ Some consequences are1. |∠G(jω)| ≤ π

2 for all ω.

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2 for all ω.2. 1/G(s) is positive real.

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2 for all ω.2. 1/G(s) is positive real.3. If G(s) represents either the driving-point impedance or

driving-point mobility of a system, then the system is passiveas seen from the driving point.

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4. G(s) is stable.

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4. G(s) is stable.5. G(s) is minimum phase.

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4. G(s) is stable.5. G(s) is minimum phase.

◮ Note: The bilinear transform preserves s-domain andz-domain sense positive realness.

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Interpretation

◮ If the sensor and actuator are not collocated, then there is apropagation delay between them.

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Interpretation


◮ To minimize the delay (phase lag), we should collocate thesensor and actuator.

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Interpretation



◮ |∠G(jω)| < π

2 for all ω.

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Interpretation



◮ |∠G(jω)| < π

2 for all ω.◮ If |∠K (jω)| ≤ π

2 for all ω, then no matter how large the loopgain K0 ≥ 0 is, the controlled system will be stable.

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Interpretation



◮ |∠G(jω)| < π



◮ If we choose a positive real controller K (s), then we can turnup the loop gain K0 ≥ 0 as much as we want.

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Interpretation



◮ |∠G(jω)| < π



◮ If we choose a positive real controller K (s), then we can turnup the loop gain K0 ≥ 0 as much as we want.

◮ This property is known as unconditional stability.

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Simplest Useful Model

◮ We first model a single resonance.

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◮ We first model a single resonance.◮ Given a mass m, damping parameter R, and spring with

parameter k , we have

mx + Rx + kx = −F (1)

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mx + Rx + kx = −F (1)

◮ For F = 0, fundamental frequency f0 ≈ 12π

√

km ,

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mx + Rx + kx = −F (1)

◮ For F = 0, fundamental frequency f0 ≈ 12π

√

km ,

◮ and the envelope of the impulse response decaysexponentially with time constant τ = 2m/R.

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Outline

Introduction


PID Control



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PID Control

F ∆= PDD x + PD x + PPx (2)

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PID Control


(m + PDD)x + (R + PD)x + (k + PP)x = 0 (3)

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PID Control


(m + PDD)x + (R + PD)x + (k + PP)x = 0 (3)

◮ With control we have

τ =2(m + PDD)

R + PD(4)

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PID Control


(m + PDD)x + (R + PD)x + (k + PP)x = 0 (3)

◮ With control we have

τ =2(m + PDD)

R + PD(4)

f0 ≈1

2π

√

k + PP

m + PDD(5)

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PID Control Mechanical Equivalent

PDPP

PDD

−K(s)

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Outline

Introduction


PID Control



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◮ Observations:1. Feedback signal leads by π/2 radians

⇒ resonance frequency increases

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⇒ resonance frequency increases2. Feedback signal lags by π/2 radians

⇒ resonance frequency decreases

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⇒ resonance frequency decreases3. Feedback signal is out of phase

⇒ resonance is damped

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⇒ resonance is damped◮ Other positive real controllers:

1. Leads and lags

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1. Leads and lags2. Band pass filter

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1. Leads and lags2. Band pass filter3. Band stop filter

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1. Leads and lags2. Band pass filter3. Band stop filter4. Feedforward comb filters

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1. Leads and lags2. Band pass filter3. Band stop filter4. Feedforward comb filters5. Filter alternating between ±π/2 radians

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Bandpass Control

◮ We limit the control energy to a small frequency region.

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Bandpass Control

◮ We limit the control energy to a small frequency region.◮ If the Q is large and the center frequency ωc is aligned with

an instrument partial, this partial is damped, while otherpartials are left unmodified.

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Bandpass Control



◮ If we invert the loop gain, then we can selectively applynegative damping.

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Bandpass Control




◮ Multiple bandpass filters may placed in parallel in the signalprocessing chain.

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Bandpass Control





◮ Kbp(s) =ωcs

Q

s2+ωcs

Q +ω2c

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Bandpass Control





◮ Kbp(s) =ωcs

Q

s2+ωcs

Q +ω2c

m

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Bandpass Filter

100

101

102

103

104

105

−100

−50

0

50

100

Frequency [Hz]

Mag

nitu

de [d

B]

100

101

102

103

104

105

−2

−1

0

1

2

Frequency [Hz]

Ang

le [r

ad]

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Notch Filter Control

◮ We damp over all frequencies except for a small region.

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◮ We damp over all frequencies except for a small region.◮ Multiple band stop filters may be placed in series in the

signal processing chain.

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◮ We damp over all frequencies except for a small region.◮ Multiple band stop filters may be placed in series in the

signal processing chain.

◮ Knotch(s) =s2+

ωcsαQ +ω

2c

s2+ωcs

Q +ω2c

m

o

o

x

x

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Notch Filter

100

101

102

103

104

105

−20

−15

−10

−5

0

Frequency [Hz]

Mag

nitu

de [d

B]

100

101

102

103

104

105

−1

−0.5

0

0.5

1

Frequency [Hz]

Ang

le [r

ad]

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Alternating Filter

◮ The frequency response shown below is such that partials atn100Hz (shown by x ’s) will be pushed flat.

0 200 400 600 800 1000−50

0

50

Frequency [Hz]

Mag

nitu

de [d

B]

0 200 400 600 800 1000−2

−1

0

1

2

Frequency [Hz]

Ang

le [r

ad]

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Alternating Filter Implementation

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Im

agin

ary

axis

Real axis

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Alternating Filter Implementation (Zoomed)

0.5 0.6 0.7 0.8 0.9 1−0.3

−0.2

−0.1

0

0.1

0.2

0.3Im

agin

ary

axis

Real axis

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Wideband Idealized Frequency Response

100

101

102

103

104

105

−50

0

50

Frequency [Hz]

Mag

nitu

de [d

B]

100

101

102

103

104

105

−2

−1

0

1

2

Frequency [Hz]

Ang

le [r

ad]

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Wideband Frequency Response Including Delay

100

101

102

103

104

105

−50

0

50

Frequency [Hz]

Mag

nitu

de [d

B]

100

101

102

103

104

105

−4

−2

0

2

Frequency [Hz]

Ang

le [r

ad]

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Outline

Introduction


PID Control



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◮ Since PID control works so well, let’s try collocated nonlinearPID control by feeding back the displacement and velocity.

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◮ Given our single-resonance model, we obtain

mx + R(x , x) + K (x) = 0 (6)

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◮ Given our single-resonance model, we obtain

mx + R(x , x) + K (x) = 0 (6)

◮ There are many methods for analyzing the behavior ofsecond-order nonlinear systems.

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Linear Dashpot

R(x)

x.

.

Figure: Linear Dashpot

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Linear Dashpot

R(x)

x.

.

Figure: Linear Dashpot

◮ R(x , x) = Rx for some constant R

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Saturating Dashpot

.R(x)

x.

Figure: Saturating Dashpot

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Saturating Dashpot

.R(x)

x.


◮ Damping is passive if xR(x , x) ≥ 0 for all x and x .◮ R(x , x) > 0 for x > 0◮ R(x , x) < 0 for x < 0

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Saturating Dashpot

.R(x)

x.


◮ Damping is passive if xR(x , x) ≥ 0 for all x and x .◮ R(x , x) > 0 for x > 0◮ R(x , x) < 0 for x < 0

◮ Damping is strictly passive if xR(x , x) > 0 for all x andfor all x 6= 0 (i.e. there is no deadband).

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Spring

◮ A linear spring behaves according to K (x) = kx for someconstant k .

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Spring


x

K(x)

Figure: Stiffening Spring

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Spring


x

K(x)

Figure: Stiffening Spring

◮ The spring is strictly locally passive if xK (x) > 0 ∀x 6= 0.

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Stability

◮ The system mx + R(x , x) + K (x) = 0 is stable if both thespring and dashpot are strictly locally passive.

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Stability


◮ You can prove this using the Lyapunov function

V =1m

∫ x

0K (σ)dσ +

12

x2 (7)

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Stability


◮ You can prove this using the Lyapunov function

V =1m

∫ x

0K (σ)dσ +

12

x2 (7)

V = −R(x , x)x

m≤ 0 (8)

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Nonlinear Dashpot for Bow at Rest

.R(x)

x.

Figure: Bowing Nonlinearity

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Nonlinear Dashpot For Moving Bow

bow velocity

R(x)

x.

.


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bow velocity

R(x)

x.

.


◮ Now the dashpot is NOT passive.

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bow velocity

R(x)

x.

.


◮ Now the dashpot is NOT passive.◮ The negative damping region adds energy so that the

bowed instrument can self-oscillate.

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Thanks!

◮ Sound examples are on the websitehttp://ccrma.stanford.edu/˜eberdahl/Projects/PassiveControl

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Thanks!

◮ Sound examples are on the websitehttp://ccrma.stanford.edu/˜eberdahl/Projects/PassiveControl

◮ Questions?

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Bibliography

E. Berdahl and J. O. Smith III, Inducing Unusual Dynamics inAcoustic Musical Instruments, 2007 IEEE Conference onControl Applications, October 1-3, 2007 - Singapore.

H. Boutin, Controle actif sur instruments acoustiques, ATIAMMaster’s Thesis, Laboratoire d’Acoustique Musicale,Universite Pierre et Marie Curie, Paris, France, Sept. 2007.

K. Khalil, Nonlinear Systems, 3rd Edition, Prentice Hall,Upper Saddle River, NJ, 2002.

C. W. Wu, Qualitative Analysis of Dynamic Circuits, WileyEncyclopedia of Electrical and Electronics Engineering, JohnWiley and Songs, Inc., Hoboken, New Jersey, 1999.

Documents

Applications of Passivity Theory to the Active Control of Acoustic