applications of passivity theory to the active control of acoustic
TRANSCRIPT
![Page 1: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/1.jpg)
ABabcdfghiejkl
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
![Page 2: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/2.jpg)
ABabcdfghiejkl
Outline
Introduction
Passivity For Linear Systems
PID Control
Other Passive Linear Controllers
Nonlinear PID Control
![Page 3: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/3.jpg)
ABabcdfghiejkl
Introduction
◮ Project goal: To make the acoustics of a musical instrumentprogrammable while the instrument retains its tangible form.
![Page 4: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/4.jpg)
ABabcdfghiejkl
Introduction
◮ Project goal: To make the acoustics of a musical instrumentprogrammable while the instrument retains its tangible form.
◮ We use a digital feedback controller.
![Page 5: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/5.jpg)
ABabcdfghiejkl
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.
![Page 6: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/6.jpg)
ABabcdfghiejkl
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.
![Page 7: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/7.jpg)
ABabcdfghiejkl
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).
![Page 8: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/8.jpg)
ABabcdfghiejkl
Outline
Introduction
Passivity For Linear Systems
PID Control
Other Passive Linear Controllers
Nonlinear PID Control
![Page 9: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/9.jpg)
ABabcdfghiejkl
s-Domain Positive Real Functions
◮ We operate in the Laplace s-domain.
![Page 10: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/10.jpg)
ABabcdfghiejkl
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.
![Page 11: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/11.jpg)
ABabcdfghiejkl
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.
![Page 12: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/12.jpg)
ABabcdfghiejkl
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.
![Page 13: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/13.jpg)
ABabcdfghiejkl
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.
![Page 14: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/14.jpg)
ABabcdfghiejkl
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.
![Page 15: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/15.jpg)
ABabcdfghiejkl
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.
![Page 16: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/16.jpg)
ABabcdfghiejkl
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.
![Page 17: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/17.jpg)
ABabcdfghiejkl
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.
![Page 18: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/18.jpg)
ABabcdfghiejkl
Interpretation
◮ If the sensor and actuator are not collocated, then there is apropagation delay between them.
![Page 19: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/19.jpg)
ABabcdfghiejkl
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.
![Page 20: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/20.jpg)
ABabcdfghiejkl
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.
![Page 21: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/21.jpg)
ABabcdfghiejkl
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.
![Page 22: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/22.jpg)
ABabcdfghiejkl
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.
![Page 23: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/23.jpg)
ABabcdfghiejkl
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.
![Page 24: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/24.jpg)
ABabcdfghiejkl
Simplest Useful Model
◮ We first model a single resonance.
![Page 25: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/25.jpg)
ABabcdfghiejkl
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)
![Page 26: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/26.jpg)
ABabcdfghiejkl
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 ,
![Page 27: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/27.jpg)
ABabcdfghiejkl
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.
![Page 28: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/28.jpg)
ABabcdfghiejkl
Outline
Introduction
Passivity For Linear Systems
PID Control
Other Passive Linear Controllers
Nonlinear PID Control
![Page 29: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/29.jpg)
ABabcdfghiejkl
PID Control
F ∆= PDD x + PD x + PPx (2)
![Page 30: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/30.jpg)
ABabcdfghiejkl
PID Control
F ∆= PDD x + PD x + PPx (2)
(m + PDD)x + (R + PD)x + (k + PP)x = 0 (3)
![Page 31: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/31.jpg)
ABabcdfghiejkl
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)
![Page 32: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/32.jpg)
ABabcdfghiejkl
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)
![Page 33: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/33.jpg)
ABabcdfghiejkl
PID Control Mechanical Equivalent
PDPP
PDD
−K(s)
![Page 34: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/34.jpg)
ABabcdfghiejkl
Outline
Introduction
Passivity For Linear Systems
PID Control
Other Passive Linear Controllers
Nonlinear PID Control
![Page 35: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/35.jpg)
ABabcdfghiejkl
Other Passive Linear Controllers
◮ Observations:1. Feedback signal leads by π/2 radians
⇒ resonance frequency increases
![Page 36: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/36.jpg)
ABabcdfghiejkl
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
![Page 37: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/37.jpg)
ABabcdfghiejkl
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
![Page 38: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/38.jpg)
ABabcdfghiejkl
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
![Page 39: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/39.jpg)
ABabcdfghiejkl
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
![Page 40: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/40.jpg)
ABabcdfghiejkl
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
![Page 41: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/41.jpg)
ABabcdfghiejkl
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
![Page 42: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/42.jpg)
ABabcdfghiejkl
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
![Page 43: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/43.jpg)
ABabcdfghiejkl
Bandpass Control
◮ We limit the control energy to a small frequency region.
![Page 44: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/44.jpg)
ABabcdfghiejkl
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.
![Page 45: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/45.jpg)
ABabcdfghiejkl
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.
![Page 46: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/46.jpg)
ABabcdfghiejkl
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.
![Page 47: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/47.jpg)
ABabcdfghiejkl
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
![Page 48: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/48.jpg)
ABabcdfghiejkl
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
![Page 49: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/49.jpg)
ABabcdfghiejkl
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]
![Page 50: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/50.jpg)
ABabcdfghiejkl
Notch Filter Control
◮ We damp over all frequencies except for a small region.
![Page 51: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/51.jpg)
ABabcdfghiejkl
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.
![Page 52: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/52.jpg)
ABabcdfghiejkl
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
![Page 53: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/53.jpg)
ABabcdfghiejkl
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]
![Page 54: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/54.jpg)
ABabcdfghiejkl
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]
![Page 55: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/55.jpg)
ABabcdfghiejkl
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
![Page 56: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/56.jpg)
ABabcdfghiejkl
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
![Page 57: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/57.jpg)
ABabcdfghiejkl
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]
![Page 58: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/58.jpg)
ABabcdfghiejkl
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]
![Page 59: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/59.jpg)
ABabcdfghiejkl
Outline
Introduction
Passivity For Linear Systems
PID Control
Other Passive Linear Controllers
Nonlinear PID Control
![Page 60: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/60.jpg)
ABabcdfghiejkl
Nonlinear PID Control
◮ Since PID control works so well, let’s try collocated nonlinearPID control by feeding back the displacement and velocity.
![Page 61: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/61.jpg)
ABabcdfghiejkl
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)
![Page 62: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/62.jpg)
ABabcdfghiejkl
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.
![Page 63: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/63.jpg)
ABabcdfghiejkl
Linear Dashpot
R(x)
x.
.
Figure: Linear Dashpot
![Page 64: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/64.jpg)
ABabcdfghiejkl
Linear Dashpot
R(x)
x.
.
Figure: Linear Dashpot
◮ R(x , x) = Rx for some constant R
![Page 65: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/65.jpg)
ABabcdfghiejkl
Saturating Dashpot
.R(x)
x.
Figure: Saturating Dashpot
![Page 66: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/66.jpg)
ABabcdfghiejkl
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
![Page 67: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/67.jpg)
ABabcdfghiejkl
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).
![Page 68: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/68.jpg)
ABabcdfghiejkl
Spring
◮ A linear spring behaves according to K (x) = kx for someconstant k .
![Page 69: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/69.jpg)
ABabcdfghiejkl
Spring
◮ A linear spring behaves according to K (x) = kx for someconstant k .
x
K(x)
Figure: Stiffening Spring
![Page 70: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/70.jpg)
ABabcdfghiejkl
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.
![Page 71: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/71.jpg)
ABabcdfghiejkl
Stability
◮ The system mx + R(x , x) + K (x) = 0 is stable if both thespring and dashpot are strictly locally passive.
![Page 72: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/72.jpg)
ABabcdfghiejkl
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)
![Page 73: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/73.jpg)
ABabcdfghiejkl
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)
![Page 74: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/74.jpg)
ABabcdfghiejkl
Nonlinear Dashpot for Bow at Rest
.R(x)
x.
Figure: Bowing Nonlinearity
![Page 75: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/75.jpg)
ABabcdfghiejkl
Nonlinear Dashpot For Moving Bow
bow velocity
R(x)
x.
.
Figure: Bowing Nonlinearity
![Page 76: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/76.jpg)
ABabcdfghiejkl
Nonlinear Dashpot For Moving Bow
bow velocity
R(x)
x.
.
Figure: Bowing Nonlinearity
◮ Now the dashpot is NOT passive.
![Page 77: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/77.jpg)
ABabcdfghiejkl
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.
![Page 78: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/78.jpg)
ABabcdfghiejkl
Thanks!
◮ Sound examples are on the websitehttp://ccrma.stanford.edu/˜eberdahl/Projects/PassiveControl
![Page 79: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/79.jpg)
ABabcdfghiejkl
Thanks!
◮ Sound examples are on the websitehttp://ccrma.stanford.edu/˜eberdahl/Projects/PassiveControl
◮ Questions?
![Page 80: Applications of Passivity Theory to the Active Control of Acoustic](https://reader036.vdocuments.us/reader036/viewer/2022071600/613d1e96736caf36b7598b0c/html5/thumbnails/80.jpg)
ABabcdfghiejkl
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.