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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
◮ Project goal: To make the acoustics of a musical instrumentprogrammable while the instrument retains its tangible form.
◮ We use a digital feedback controller.
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Introduction
◮ Project goal: To make the acoustics of a musical instrumentprogrammable while the instrument retains its tangible form.
◮ 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
◮ Project goal: To make the acoustics of a musical instrumentprogrammable while the instrument retains its tangible form.
◮ 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
Passivity For Linear Systems
PID Control
Other Passive Linear Controllers
Nonlinear PID Control
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s-Domain Positive Real Functions
◮ We operate in the Laplace s-domain.
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s-Domain Positive Real Functions
◮ 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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s-Domain Positive Real Functions
◮ We operate in the Laplace s-domain.◮ A function G(s) is positive real if
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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s-Domain Positive Real Functions
◮ We operate in the Laplace s-domain.◮ A function G(s) is positive real if
1. G(s) is real when s is real.2. Re{G(s)} ≥ 0 for all s such that Re{s} ≥ 0.
◮ Some consequences are1. |∠G(jω)| ≤ π
2 for all ω.
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s-Domain Positive Real Functions
◮ We operate in the Laplace s-domain.◮ A function G(s) is positive real if
1. G(s) is real when s is real.2. Re{G(s)} ≥ 0 for all s such that Re{s} ≥ 0.
◮ Some consequences are1. |∠G(jω)| ≤ π
2 for all ω.2. 1/G(s) is positive real.
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s-Domain Positive Real Functions
◮ We operate in the Laplace s-domain.◮ A function G(s) is positive real if
1. G(s) is real when s is real.2. Re{G(s)} ≥ 0 for all s such that Re{s} ≥ 0.
◮ Some consequences are1. |∠G(jω)| ≤ π
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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s-Domain Positive Real Functions
◮ We operate in the Laplace s-domain.◮ A function G(s) is positive real if
1. G(s) is real when s is real.2. Re{G(s)} ≥ 0 for all s such that Re{s} ≥ 0.
◮ Some consequences are1. |∠G(jω)| ≤ π
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.
4. G(s) is stable.
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s-Domain Positive Real Functions
◮ We operate in the Laplace s-domain.◮ A function G(s) is positive real if
1. G(s) is real when s is real.2. Re{G(s)} ≥ 0 for all s such that Re{s} ≥ 0.
◮ Some consequences are1. |∠G(jω)| ≤ π
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.
4. G(s) is stable.5. G(s) is minimum phase.
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s-Domain Positive Real Functions
◮ We operate in the Laplace s-domain.◮ A function G(s) is positive real if
1. G(s) is real when s is real.2. Re{G(s)} ≥ 0 for all s such that Re{s} ≥ 0.
◮ Some consequences are1. |∠G(jω)| ≤ π
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.
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
◮ If the sensor and actuator are not collocated, then there is apropagation delay between them.
◮ To minimize the delay (phase lag), we should collocate thesensor and actuator.
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Interpretation
◮ If the sensor and actuator are not collocated, then there is apropagation delay between them.
◮ To minimize the delay (phase lag), we should collocate thesensor and actuator.
◮ |∠G(jω)| < π
2 for all ω.
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Interpretation
◮ If the sensor and actuator are not collocated, then there is apropagation delay between them.
◮ To minimize the delay (phase lag), we should collocate thesensor and actuator.
◮ |∠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
◮ If the sensor and actuator are not collocated, then there is apropagation delay between them.
◮ To minimize the delay (phase lag), we should collocate thesensor and actuator.
◮ |∠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.
◮ 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
◮ If the sensor and actuator are not collocated, then there is apropagation delay between them.
◮ To minimize the delay (phase lag), we should collocate thesensor and actuator.
◮ |∠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.
◮ 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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Simplest Useful Model
◮ 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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Simplest Useful Model
◮ 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)
◮ For F = 0, fundamental frequency f0 ≈ 12π
√
km ,
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Simplest Useful Model
◮ 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)
◮ 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
Passivity For Linear Systems
PID Control
Other Passive Linear Controllers
Nonlinear PID Control
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PID Control
F ∆= PDD x + PD x + PPx (2)
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PID Control
F ∆= PDD x + PD x + PPx (2)
(m + PDD)x + (R + PD)x + (k + PP)x = 0 (3)
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PID Control
F ∆= PDD x + PD x + PPx (2)
(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
F ∆= PDD x + PD x + PPx (2)
(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
Passivity For Linear Systems
PID Control
Other Passive Linear Controllers
Nonlinear PID Control
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Other Passive Linear Controllers
◮ Observations:1. Feedback signal leads by π/2 radians
⇒ resonance frequency increases
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Other Passive Linear Controllers
◮ Observations:1. Feedback signal leads by π/2 radians
⇒ resonance frequency increases2. Feedback signal lags by π/2 radians
⇒ resonance frequency decreases
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Other Passive Linear Controllers
◮ Observations:1. Feedback signal leads by π/2 radians
⇒ resonance frequency increases2. Feedback signal lags by π/2 radians
⇒ resonance frequency decreases3. Feedback signal is out of phase
⇒ resonance is damped
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Other Passive Linear Controllers
◮ Observations:1. Feedback signal leads by π/2 radians
⇒ resonance frequency increases2. Feedback signal lags by π/2 radians
⇒ resonance frequency decreases3. Feedback signal is out of phase
⇒ resonance is damped◮ Other positive real controllers:
1. Leads and lags
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Other Passive Linear Controllers
◮ Observations:1. Feedback signal leads by π/2 radians
⇒ resonance frequency increases2. Feedback signal lags by π/2 radians
⇒ resonance frequency decreases3. Feedback signal is out of phase
⇒ resonance is damped◮ Other positive real controllers:
1. Leads and lags2. Band pass filter
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Other Passive Linear Controllers
◮ Observations:1. Feedback signal leads by π/2 radians
⇒ resonance frequency increases2. Feedback signal lags by π/2 radians
⇒ resonance frequency decreases3. Feedback signal is out of phase
⇒ resonance is damped◮ Other positive real controllers:
1. Leads and lags2. Band pass filter3. Band stop filter
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Other Passive Linear Controllers
◮ Observations:1. Feedback signal leads by π/2 radians
⇒ resonance frequency increases2. Feedback signal lags by π/2 radians
⇒ resonance frequency decreases3. Feedback signal is out of phase
⇒ resonance is damped◮ Other positive real controllers:
1. Leads and lags2. Band pass filter3. Band stop filter4. Feedforward comb filters
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Other Passive Linear Controllers
◮ Observations:1. Feedback signal leads by π/2 radians
⇒ resonance frequency increases2. Feedback signal lags by π/2 radians
⇒ resonance frequency decreases3. Feedback signal is out of phase
⇒ resonance is damped◮ Other positive real controllers:
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
◮ 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.
◮ If we invert the loop gain, then we can selectively applynegative damping.
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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.
◮ If we invert the loop gain, then we can selectively applynegative damping.
◮ Multiple bandpass filters may placed in parallel in the signalprocessing chain.
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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.
◮ If we invert the loop gain, then we can selectively applynegative damping.
◮ Multiple bandpass filters may placed in parallel in the signalprocessing chain.
◮ Kbp(s) =ωcs
Q
s2+ωcs
Q +ω2c
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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.
◮ If we invert the loop gain, then we can selectively applynegative damping.
◮ Multiple bandpass filters may placed in parallel in the signalprocessing chain.
◮ 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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Notch Filter Control
◮ 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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Notch Filter Control
◮ 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
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104
105
−20
−15
−10
−5
0
Frequency [Hz]
Mag
nitu
de [d
B]
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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
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101
102
103
104
105
−50
0
50
Frequency [Hz]
Mag
nitu
de [d
B]
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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
Passivity For Linear Systems
PID Control
Other Passive Linear Controllers
Nonlinear PID Control
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Nonlinear PID Control
◮ Since PID control works so well, let’s try collocated nonlinearPID control by feeding back the displacement and velocity.
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Nonlinear PID Control
◮ Since PID control works so well, let’s try collocated nonlinearPID control by feeding back the displacement and velocity.
◮ Given our single-resonance model, we obtain
mx + R(x , x) + K (x) = 0 (6)
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Nonlinear PID Control
◮ Since PID control works so well, let’s try collocated nonlinearPID control by feeding back the displacement and velocity.
◮ 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.
Figure: Saturating Dashpot
◮ 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.
Figure: Saturating Dashpot
◮ 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
◮ A linear spring behaves according to K (x) = kx for someconstant k .
x
K(x)
Figure: Stiffening Spring
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Spring
◮ A linear spring behaves according to K (x) = kx for someconstant k .
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
◮ The system mx + R(x , x) + K (x) = 0 is stable if both thespring and dashpot are strictly locally passive.
◮ You can prove this using the Lyapunov function
V =1m
∫ x
0K (σ)dσ +
12
x2 (7)
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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.
◮ 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.
.
Figure: Bowing Nonlinearity
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Nonlinear Dashpot For Moving Bow
bow velocity
R(x)
x.
.
Figure: Bowing Nonlinearity
◮ Now the dashpot is NOT passive.
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Nonlinear Dashpot For Moving Bow
bow velocity
R(x)
x.
.
Figure: Bowing Nonlinearity
◮ 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.
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