In this paper, the problem of active sound cancellation in finite-length ducts is investigated. The closed-form solution of a one-dimensional wave equation is obtained as the plant model. The controllability, observability, and transmission zeros are discussed based on the transfer function model. For ducts with totally reflective boundaries, stabilization can be achieved by using a speaker (actuator) and a microphone (sensor). Cases of collocated and noncollocated sensors and actuators are presented. A repetitive control algorithm was developed to drive the actuator so that harmonic noises in a duct are attenuated. For a duct with partially reflective boundaries, the application of repetitive control prevents sound from leaking out of the duct at a chosen end. A simulation study demonstrating the effects of this feedback control scheme is also presented.

1.
Balas
M. J.
,
1982
, “
Trends in Large Space Structure Control Theory: Fondest Hopes, Wildest Dreams
,”
IEEE Trans. Automatic Control
, Vol.
AC-27
, pp.
15
33
.
2.
Brierley
S. D.
,
Chiasson
J. N.
,
Lee
E. B.
, and
Zak
S. H.
, “
On Stability Independent of Delay for Linear Systems
,”
IEEE Transactions on Automatic Control
, Vol.
AC-27
(
1)
Feb., pp.
252
254
.
3.
Curtis
A. R. O.
,
Nelson
P. A.
,
Elliott
S. J.
, and
Bullmore
A. J.
,
1987
, “
Active Suppression of Acoustic Resonance
,”
J. Acoust. Soc. Am.
, Vol.
81
, pp.
624
631
.
4.
Doak
P. E.
,
1973
, “
Excitation, Transmission and Radiation of Sound from Sources in Hard-walled Ducts of Finite Length (1): The Effect of Duct Crosssection Geometry and Source Distribution Space-Time Pattern
,”
J. of Sound and Vibration
, Vol.
31
, pp.
1
72
.
5.
Dohner
J. L.
, and
Shoureshi
R.
,
1989
, “
Modal Control of Acoustic Plants
,”
ASME J. Vibration, Acoustics, Stress, and Reliability in Design
, Vol.
111
, pp.
326
330
.
6.
Eghtesadi
Kh.
, and
Leventhall
H. G.
,
1981
, “
Active Attenuation of Noise: The Chelsea Dipole
,”
J. of Sound and Vibration
, Vol.
75
, pp.
127
134
.
7.
Eghtesadi
Kh.
, and
Leventhall
H. G.
,
1982
, “
Active Attenuation of Noise—The Monopole System
,”
J. Acoust. Soc. Am.
, Vol.
71
, pp.
608
611
.
8.
Elliott
S. J.
,
Joseph
P.
,
Nelson
P. A.
, and
Johnson
M. E.
,
1991
, “
Power Output Minimization and Power Absorption in the Active Control of Sound
,”
J. Acous. Soc. Am.
, Vol.
90
, pp.
2501
2511
.
9.
Eriksson, L. J., Allie, M. C., and Greiner, R. A., 1987, “The Selection and Application of an IIR Adaptive Filter for Use in Active Sound Attenuation,” IEEE Trans. ASSP, ASSP-33, No. 4.
10.
Fontaine
R. F. La
and
Shepherd
L. C.
,
1983
, “
An Experimental Study of a Broad Band Active Attenuator for Cancellation of Random Noise
,”
J. of Sound and Vibration
, Vol.
91
(
3)
, pp.
351
362
.
11.
Francis
B. A.
, and
Wonham
W. M.
,
1975
, “
The Internal Model Principle for Linear Multivariable Regulators
,”
Applied Math. and Optimization
, Vol.
2
, pp.
170
194
.
12.
Hale, J., 1977, Theory of Functional Differential Equations, New York, Springer-Verlag.
13.
Hara
S.
,
Yamamoto
Y.
,
Omata
T.
, and
Nakano
M.
,
1988
, “
Repetitive Control System: A New Type of Servo System for Periodic Exogenous Signals
,”
IEEE Transactions on Automatic Control
, Vol.
AC-33
(
7)
, July, pp.
659
668
.
14.
Hull
A. J.
,
Radcliffe
C. J.
, and
MacCluer
C. R.
,
1991
, “
State Estimation of the Nonself-Adjoint Acoustic System
,”
ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL
, Vol.
113
, No.
1
, pp.
122
126
.
15.
Jessel
M. J. M.
, and
Mangiante
G. A.
,
1972
, “
Active Sound Absorbers in an Air Duct
,”
J. of Sound and Vibration
, Vol.
23
, pp.
383
390
.
16.
Kamen
E. W.
,
1980
, “
On the Relationship Between Zero Criteria for Two-Variable Polynomials and Asymptotic Stability of Delay Differential Equations
,”
IEEE Transactions on Automatic Control
, Vol.
AC-25
(
5)
, Oct., pp.
983
984
.
17.
Lewis
R. M.
, and
Anderson
B. O.
,
1980
, “
Necessary and Sufficient Conditions for Delay-Independent Stability of Linear Autonomous Systems
,”
IEEE Transactions on Automatic Control
, Vol.
AC-25
(
4)
, Aug., pp.
735
739
.
18.
Lueg, P., 1936, “Processes of Silencing Sound Oscillations,” Patent No. 2,043,416.
19.
Mori
T.
,
1985
, “
Criteria for Asymptotic Stability of Linear Time-Delay Systems
,”
IEEE Transactions on Automatic Control
, Vol.
AC-30
(
2)
, Feb, pp.
158
161
.
20.
Morse, P. M., and Ingard, K. U., 1968, Theoretical Acoustics, Princeton University Press, Princeton, NJ.
21.
Pierce, A. D., 1981, “Acoustics: An Introduction to Its Physical Principles and Applications,” McGraw-Hill, New York.
22.
Shoureshi, R., Kubota, N., and Batta, G., 1990, “A Modern Control Approach to Active Noise Control,” ASME Winter Annual Meeting, Active Noise and Vibration Control, pp. 167–175.
23.
Swinbanks
M. A.
,
1973
, “
The Active Control of Sound Propagation in Long Ducts
,”
J. of Sound and Vibration
, Vol.
27
(
3)
, pp.
411
436
.
24.
Tomizuka
M.
,
1987
, “
Zero-Phase Error Tracking Algorithm for Digital Control
,”
ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT AND CONTROL
, Vol.
109
, pp.
65
68
.
25.
Tomizuka, M., Tsao, T. C., and Chew, K. K., 1988, “Discrete-Time Domain Analysis and Synthesis of Repetitive Controllers,” American Control Conference, pp. 860–866.
26.
Trinder
M. C. J.
, and
Nelson
P. A.
,
1983
, “
Active Noise Control in Finite Length Ducts
,”
J. of Sound and Vibration
, Vol.
89
, pp.
95
105
.
27.
Vidyasagar, M., 1978, Nonlinear System Analysis, Prentice Hall, Engelwood Cliffs, NJ.
28.
Warner J. V., Water, D. E., and Bernhard, R. J., 1988, “Digital Filter Implementation of Local Active Noise Control in a Three-Dimensional Enclosure,” ASME Winter Annual Meeting, NCA-6, pp. 1–9.
29.
Yang
B.
, and
Mote
C. D.
,
1992
, “
On Time Delay in Non-Collocated Control of Flexible Mechanical Systems
,”
ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL
, Vol.
114
, No.
3
, Sept., pp.
409
415
.
30.
Tan
C. A.
and
Yang
B.
,
1992
, “
Transfer Functions of One-Dimensional Distributed Parameter Systems
,”
ASME Journal of Applied Mechanics
, Vol.
59
Dec., pp.
1009
14
.
This content is only available via PDF.
You do not currently have access to this content.