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March  2022, 12(1): 1-16. doi: 10.3934/mcrf.2020055

Exact noise cancellation for 1d-acoustic propagation systems

Jérôme Lohéac 1,, , Chaouki N. E. Boultifat 2, , Philippe Chevrel 2, and  Mohamed Yagoubi 2,

1. 

Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France
2. 

IMT-Atlantique, LS2N UMR CNRS 6004, (Laboratoire des Sciences du Numérique de Nantes), F-44307 Nantes, France

* Corresponding author

Received  March 2020 Revised  August 2020 Published  March 2022 Early access  December 2020

This paper deals with active noise control applied to a one-dimensional acoustic propagation system. The aim here is to keep over time a zero noise level at a given point. We aim to design this control using noise measurement at some point in the spatial domain. Based on symmetry property, we are able to design a feedback boundary control allowing this fact. Moreover, using D'Alembert formula, an explicit formula of the control can be computed. Even if the focus is made on the wave equation, this approach is easily extendable to more general operators.

Keywords: Active noise control, noise cancellation, 1D-wave equation, boundary control, D'Alembert formula.
Mathematics Subject Classification: Primary: 35L05, 93B52, 93C20; Secondary: 93B53.
Citation: Jérôme Lohéac, Chaouki N. E. Boultifat, Philippe Chevrel, Mohamed Yagoubi. Exact noise cancellation for 1d-acoustic propagation systems. Mathematical Control and Related Fields, 2022, 12 (1) : 1-16. doi: 10.3934/mcrf.2020055
References:
[1]

M. R. Bai and H. Lin, Plant uncertainty analysis in a duct active noise control problem by using the $H_\infty$ theory, The Journal of the Acoustical Society of America, 104 (1998), 237-247.  doi: 10.1121/1.423274.

[2]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.

[3]

M. Bodson, J. S. Jensen and S. C. Douglas, Active noise control for periodic disturbances, Proceedings of the 1998 American Control Conference. ACC(IEEE Cat. No.98CH36207), 4 (1998), 2616-2620.  doi: 10.1109/ACC.1998.703109.

[4]

C. Boultifat, P. Chevrel, J. Lohéac, M. Yagoubi and P. Loiseau, One-dimensional acoustic propagation model and spatial multi-point active noise control, in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017), 2947–2952. doi: 10.1109/CDC.2017.8264088.

[5]

C. Boultifat, P. Loiseau, P. Chevrel, J. Lohéac and M. Yagoubi, FxLMS versus $H_\infty$ control for broadband acoustic noise attenuation in a cavity, IFAC-PapersOnLine (20th IFAC World Congress), 50 (2017), 9204-9210.  doi: 10.1016/j.ifacol.2017.08.1277.

[6]

C. Cattaneo and L. Fontana, D'Alembert formula on finite one-dimensional networks, J. Math. Anal. Appl., 284 (2003), 403-424.  doi: 10.1016/S0022-247X(02)00392-X.

[7]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures, Mathématiques & Applications (Berlin), Mathematics & Applications, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[8]

H. Feng and B.-Z. Guo, Observer design and exponential stabilization for wave equation in energy space by boundary displacement measurement only, IEEE Trans. Automat. Control, 62 (2017), 1438-1444.  doi: 10.1109/TAC.2016.2572122.

[9]

H. Feng and B.-Z. Guo, A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance, IEEE Trans. Automat. Control, 62 (2017), 3774-3787.  doi: 10.1109/TAC.2016.2636571.

[10]

B.-Z. Guo and F.-F. Jin, Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance, IEEE Trans. Automat. Control, 60 (2015), 824-830.  doi: 10.1109/TAC.2014.2335374.

[11]

W. Guo, Z.-C. Shao and M. Krstic, Adaptive rejection of harmonic disturbance anticollocated with control in 1d wave equation, Automatica J. IFAC, 79 (2017), 17-26.  doi: 10.1016/j.automatica.2017.01.034.

[12]

B.-Z. Guo and C.-Z. Xu, The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation, IEEE Trans. Automat. Control, 52 (2007), 371-377.  doi: 10.1109/TAC.2006.890385.

[13]

M. Gugat, Exponential stabilization of the wave equation by Dirichlet integral feedback, SIAM J. Control Optim., 53 (2015), 526-546.  doi: 10.1137/140977023.

[14]

M. Gugat and G. Leugering, Time delay in optimal control loops for wave equations, ESAIM, Control Optim. Calc. Var., 23 (2017), 13-37.  doi: 10.1051/cocv/2015038.

[15]

V. Komornik, Exact Controllability and Stabilization, The Multiplier Method., RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[16]

S. M. Kuo and D. R. Morgan, Active noise control: A tutorial review, Proceedings of the IEEE, 87 (1999), 943-973.  doi: 10.1109/5.763310.

[17]

J. le Rond D'Alembert, Recherches sur la courbe que forme une corde tendue mise en vibrations, Histoire de l'Académie Royale des Sciences et Belles Lettres (Année 1747), 3 (1747), 214–249.

[18]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

[19]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I., Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972.

[20]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. II, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182, Springer-Verlag, New York-Heidelberg, 1972.

[21]

P. Loiseau, P. Chevrel, M. Yagoubi and J. Duffal, Broadband active noise control design through nonsmooth $H_\infty$ synthesis, IFAC-PapersOnLine (8th IFAC Symposium on Robust Control Design ROCOND 2015), 48 (2015), 396-401.  doi: 10.1016/j.ifacol.2015.09.489.

[22]

P. Loiseau, P. Chevrel, M. Yagoubi and J.-M. Duffal, $H_\infty$ multi-objective and multi-model MIMO control design for broadband noise attenuation in an enclosure, in 2016 European Control Conference (ECC), (2016), 643–648. doi: 10.1109/ECC.2016.7810361.

[23]

R. T. O'Brien, J. M. Watkins, G. E. Piper and D. C. Baumann, $H_\infty$ active noise control of fan noise in an acoustic duct, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334), 5 (2000), 3028-3032.  doi: 10.1109/ACC.2000.879121.

[24]

B. Rafaely and S. J. Elliott, $H_2/H_\infty$ active control of sound in a headrest: Design and implementation, IEEE Transactions on Control Systems Technology, 7 (1999), 79-84.  doi: 10.1109/87.736757.

[25]

B. Sayyarrodsari, J. P. How, B. Hassibi and A. Carrier, An $H_{\infty}$-optimal alternative to the FxLMS algorithm, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207), 2 (1998), 1116-1121.  doi: 10.1109/ACC.1998.703585.

[26]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[27]

M. Tucsnak and G. Weiss, From exact observability to identification of singular sources, Math. Control Signals Systems, 27 (2015), 1-21.  doi: 10.1007/s00498-014-0132-z.

show all references

References:
[1]

M. R. Bai and H. Lin, Plant uncertainty analysis in a duct active noise control problem by using the $H_\infty$ theory, The Journal of the Acoustical Society of America, 104 (1998), 237-247.  doi: 10.1121/1.423274.

[2]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.

[3]

M. Bodson, J. S. Jensen and S. C. Douglas, Active noise control for periodic disturbances, Proceedings of the 1998 American Control Conference. ACC(IEEE Cat. No.98CH36207), 4 (1998), 2616-2620.  doi: 10.1109/ACC.1998.703109.

[4]

C. Boultifat, P. Chevrel, J. Lohéac, M. Yagoubi and P. Loiseau, One-dimensional acoustic propagation model and spatial multi-point active noise control, in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017), 2947–2952. doi: 10.1109/CDC.2017.8264088.

[5]

C. Boultifat, P. Loiseau, P. Chevrel, J. Lohéac and M. Yagoubi, FxLMS versus $H_\infty$ control for broadband acoustic noise attenuation in a cavity, IFAC-PapersOnLine (20th IFAC World Congress), 50 (2017), 9204-9210.  doi: 10.1016/j.ifacol.2017.08.1277.

[6]

C. Cattaneo and L. Fontana, D'Alembert formula on finite one-dimensional networks, J. Math. Anal. Appl., 284 (2003), 403-424.  doi: 10.1016/S0022-247X(02)00392-X.

[7]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures, Mathématiques & Applications (Berlin), Mathematics & Applications, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[8]

H. Feng and B.-Z. Guo, Observer design and exponential stabilization for wave equation in energy space by boundary displacement measurement only, IEEE Trans. Automat. Control, 62 (2017), 1438-1444.  doi: 10.1109/TAC.2016.2572122.

[9]

H. Feng and B.-Z. Guo, A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance, IEEE Trans. Automat. Control, 62 (2017), 3774-3787.  doi: 10.1109/TAC.2016.2636571.

[10]

B.-Z. Guo and F.-F. Jin, Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance, IEEE Trans. Automat. Control, 60 (2015), 824-830.  doi: 10.1109/TAC.2014.2335374.

[11]

W. Guo, Z.-C. Shao and M. Krstic, Adaptive rejection of harmonic disturbance anticollocated with control in 1d wave equation, Automatica J. IFAC, 79 (2017), 17-26.  doi: 10.1016/j.automatica.2017.01.034.

[12]

B.-Z. Guo and C.-Z. Xu, The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation, IEEE Trans. Automat. Control, 52 (2007), 371-377.  doi: 10.1109/TAC.2006.890385.

[13]

M. Gugat, Exponential stabilization of the wave equation by Dirichlet integral feedback, SIAM J. Control Optim., 53 (2015), 526-546.  doi: 10.1137/140977023.

[14]

M. Gugat and G. Leugering, Time delay in optimal control loops for wave equations, ESAIM, Control Optim. Calc. Var., 23 (2017), 13-37.  doi: 10.1051/cocv/2015038.

[15]

V. Komornik, Exact Controllability and Stabilization, The Multiplier Method., RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[16]

S. M. Kuo and D. R. Morgan, Active noise control: A tutorial review, Proceedings of the IEEE, 87 (1999), 943-973.  doi: 10.1109/5.763310.

[17]

J. le Rond D'Alembert, Recherches sur la courbe que forme une corde tendue mise en vibrations, Histoire de l'Académie Royale des Sciences et Belles Lettres (Année 1747), 3 (1747), 214–249.

[18]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

[19]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I., Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972.

[20]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. II, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182, Springer-Verlag, New York-Heidelberg, 1972.

[21]

P. Loiseau, P. Chevrel, M. Yagoubi and J. Duffal, Broadband active noise control design through nonsmooth $H_\infty$ synthesis, IFAC-PapersOnLine (8th IFAC Symposium on Robust Control Design ROCOND 2015), 48 (2015), 396-401.  doi: 10.1016/j.ifacol.2015.09.489.

[22]

P. Loiseau, P. Chevrel, M. Yagoubi and J.-M. Duffal, $H_\infty$ multi-objective and multi-model MIMO control design for broadband noise attenuation in an enclosure, in 2016 European Control Conference (ECC), (2016), 643–648. doi: 10.1109/ECC.2016.7810361.

[23]

R. T. O'Brien, J. M. Watkins, G. E. Piper and D. C. Baumann, $H_\infty$ active noise control of fan noise in an acoustic duct, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334), 5 (2000), 3028-3032.  doi: 10.1109/ACC.2000.879121.

[24]

B. Rafaely and S. J. Elliott, $H_2/H_\infty$ active control of sound in a headrest: Design and implementation, IEEE Transactions on Control Systems Technology, 7 (1999), 79-84.  doi: 10.1109/87.736757.

[25]

B. Sayyarrodsari, J. P. How, B. Hassibi and A. Carrier, An $H_{\infty}$-optimal alternative to the FxLMS algorithm, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207), 2 (1998), 1116-1121.  doi: 10.1109/ACC.1998.703585.

[26]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[27]

M. Tucsnak and G. Weiss, From exact observability to identification of singular sources, Math. Control Signals Systems, 27 (2015), 1-21.  doi: 10.1007/s00498-014-0132-z.

Figure 1.  Experimental setup used to derive the system (1), $ e(t) = p(t, x_c) $ is the sound pressure measured at the controlled point
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Figure 2.  Illustration of the positions assumptions for the acoustic system (2)
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Figure 3.  Plots of the control and disturbance effect on $ p(t, 0) $. (Parameter and disturbances used are given in Table 1.)
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Table 1.  Parameters and disturbance used for the numerical illustration of Figure 3
$ L $ $ \xi $ $ \omega $ $ d(t) $ $ d_0(t, x) $
$ 2 $ $ 3/4 $ $ (a, b) $ with $ \sin(5t) $ $ 10\sin(3t)(x-a)(x-b) $
$ a=-7/4 $ and $ b=-5/4 $
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$ L $ $ \xi $ $ \omega $ $ d(t) $ $ d_0(t, x) $
$ 2 $ $ 3/4 $ $ (a, b) $ with $ \sin(5t) $ $ 10\sin(3t)(x-a)(x-b) $
$ a=-7/4 $ and $ b=-5/4 $
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