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Research Papers

Root Locus Analysis of the Adaptation Process in Active Noise Control for Repetitive Impulses

[+] Author and Article Information
Guohua Sun

Vibro-Acoustics and Sound Quality
Research Laboratory,
Mechanical and Materials Engineering,
College of Engineering and Applied Science,
University of Cincinnati,
598 Rhodes Hall, P.O. Box 210072,
Cincinnati, OH 45221-0072
e-mail: sungh@ucmail.uc.edu

Mingfeng Li, Teik C. Lim

Vibro-Acoustics and Sound Quality
Research Laboratory,
Mechanical and Materials Engineering,
College of Engineering and Applied Science,
University of Cincinnati,
598 Rhodes Hall, P.O. Box 210072,
Cincinnati, OH 45221-0072

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 31, 2015; final manuscript received October 10, 2015; published online November 6, 2015. Assoc. Editor: Jwu-Sheng Hu.

J. Dyn. Sys., Meas., Control 138(1), 011005 (Nov 06, 2015) (12 pages) Paper No: DS-15-1052; doi: 10.1115/1.4031825 History: Received January 31, 2015; Revised October 10, 2015

The popular filtered-x least-mean squares (FxLMS) algorithm has been widely adopted in active noise control (ANC) for relatively stationary disturbances. The convergence behavior of the FxLMS algorithm has been well understood in the adaptation process for stationary sinusoidal or stochastic white noises. Its behavior for transient impulses has not received as much attention. This paper employs the root locus theory to develop a graphical tool for the analysis and design of the adaptive ANC system for repetitive impulses. It is found that there is a dominant pole controlling the stability of the adaptation process, in which the maximum step size can be determined. The analysis also observes a transient adaptation behavior in the FxLMS algorithm for repetitive impulses. In this case, the predicted step-size bound decreases as the number of repetitive impulses increases for a general secondary path. Furthermore, the dominant root tuning process is applied by incorporating a digital filter after the output of the adaptive controller, which significantly increases the step-size bound. The accuracy of the analysis was extensively validated by numerical simulation studies by assuming various secondary path models. The simulated results show an excellent agreement with analytical predictions.

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References

Figures

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Fig. 1

Feedforward control diagram configured with the FxLMS algorithm

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Fig. 2

Flow diagram for computing the maximum convergence step-size based on matlab

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Fig. 3

Magnitude and phase responses of the secondary path models (keys: thin-solid line , S1(z); dashed line , S2(z); dashed-dotted line , S3(z); bold-solid line , S4(z); and dotted line , S5(z))

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Fig. 4

Root loci for secondary path models 1–4 when  K=16 : (a) S1(z), (b) S2(z), (c) S3(z), and (d) S4(z) (keys: bold-dotted line the dominant pole branch, cross point is the pole, and circular point is the zero)

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Fig. 5

Dominant root locus starting from pole (1,0) for the secondary path S1(z)

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Fig. 6

(a) Dominant root locus starting from pole (1, 0) for the secondary path S1(z) at different number of impulses  K and (b) zoomed-in plot of (a) (keys: dotted line , K=16; bold-solid line , K=24; dashed-dotted line , K=32; dashed line , K=64; and thin-solid line , K=128)

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Fig. 7

(a) Dominant root locus starting from pole (−1, 0) for the secondary path  S5(z) at different number of impulses  K and (b) zoomed-in plot of (a) (keys: dotted line , K=16; bold-solid line , K=24; dashed-dotted line , K=32; dashed line , K=64; and thin-solid line , K=128)

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Fig. 8

Magnitude and phase responses of the primary path

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Fig. 9

Amplitude envelope of the error signal after control on for the secondary path  S5(z) (keys: solid line , amplitude envelope by Hilbert transform and dotted line , peak envelope)

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Fig. 10

Convergence behavior represented by the amplitude envelope of error signal after control on for the secondary path  S5(z): (a)  K=8; (b) K=16; (c) K=24; (d) K=32; (e) K=64; and (f) K=128

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Fig. 11

Convergence behavior represented by the amplitude envelope of error signal after control on for the secondary path  S1(z): (a)  K=8; (b) K=16; (c) K=24; (d) K=32; (e) K=64; and (f) K=128

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Fig. 12

Comparison of the maximum step size calculated by the proposed analysis and numerical simulation for different secondary path models, from top to bottom are S5(z),  z−5, and S1(z)

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Fig. 13

Root loci of the standard and tuned system for the secondary path model S1z (keys: solid line , ξ=0 and dashed line , ξ=−0.1)

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Fig. 14

Convergence behavior of the tuned system with ξ=−0.1 for the secondary path S1z: (a)  K=8; (b) K=16; (c) K=24; (d) K=32; (e) K=64; and (f) K=128

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Fig. 15

Comparison of the maximum step size of the standard and tuned FxLMS algorithm for secondary path model S1z, ξ=0 means the standard ANC system with the FxLMS algorithm and ξ=−0.1 denotes the tuned system

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Fig. 16

Responses of the last impact event before and after control using the original and tuned system with the secondary path S1z: (a) time domain and (b) frequency domain

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