# American Institute of Mathematical Sciences

January  1995, 1(1): 119-149. doi: 10.3934/dcds.1995.1.119

## Feedback control of noise in a 2-D nonlinear structural acoustics model

 1 Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212, United States 2 ICASE, NASA Langley Research Center, Hampton, VA 23681, United States

Received  October 1994 Published  October 1994

A time domain feedback control methodology for reducing sound pressure levels in a nonlinear 2-D structural acoustics application is presented. The interior noise in this problem is generated through vibrations of one wall of the cavity (in this case a beam), and control is implemented through the excitation of piezoceramic patches which are bonded to the beam. These patches are mounted in pairs and are wired so as to create pure bending moments which directly affect the manner in which the structure vibrates. Th application of control in this manner leads to an unbounded control input term and the implications of this are discussed. The coupling between the beam vibrations and the interior acoustic response is inherently nonlinear, and this is addressed when developing a control scheme for the problem. Gains for the problem are calculated using a periodic LQR theory and are then fed back into the nonlinear system with results being demonstrated by a set of numerical examples. In particular, these examples demonstrate the viability of the method in cases involving excitation involving a large number of frequencies through both spatially uniform and nonuniform exterior forces.
Citation: H. T. Banks, R.C. Smith. Feedback control of noise in a 2-D nonlinear structural acoustics model. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 119-149. doi: 10.3934/dcds.1995.1.119
 [1] Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397 [2] Sanling Yuan, Yongli Song, Junhui Li. Oscillations in a plasmid turbidostat model with delayed feedback control. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 893-914. doi: 10.3934/dcdsb.2011.15.893 [3] Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653 [4] Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 [5] Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607 [6] Junyoung Jang, Kihoon Jang, Hee-Dae Kwon, Jeehyun Lee. Feedback control of an HBV model based on ensemble kalman filter and differential evolution. Mathematical Biosciences & Engineering, 2018, 15 (3) : 667-691. doi: 10.3934/mbe.2018030 [7] Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial and Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099 [8] Pedro M. Jordan. Finite-amplitude acoustics under the classical theory of particle-laden flows. Evolution Equations and Control Theory, 2019, 8 (1) : 101-116. doi: 10.3934/eect.2019006 [9] Kazuyuki Yagasaki. Optimal control of the SIR epidemic model based on dynamical systems theory. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2501-2513. doi: 10.3934/dcdsb.2021144 [10] Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065 [11] Qiying Hu, Chen Xu, Wuyi Yue. A unified model for state feedback of discrete event systems II: Control synthesis problems. Journal of Industrial and Management Optimization, 2008, 4 (4) : 713-726. doi: 10.3934/jimo.2008.4.713 [12] Heinz Schättler, Urszula Ledzewicz. Perturbation feedback control: A geometric interpretation. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 631-654. doi: 10.3934/naco.2012.2.631 [13] Barbara Kaltenbacher. Mathematics of nonlinear acoustics. Evolution Equations and Control Theory, 2015, 4 (4) : 447-491. doi: 10.3934/eect.2015.4.447 [14] Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415 [15] K. Renee Fister, Jennifer Hughes Donnelly. Immunotherapy: An Optimal Control Theory Approach. Mathematical Biosciences & Engineering, 2005, 2 (3) : 499-510. doi: 10.3934/mbe.2005.2.499 [16] Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063 [17] Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197 [18] Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085 [19] Daniel Franco, Chris Guiver, Phoebe Smith, Stuart Townley. A switching feedback control approach for persistence of managed resources. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1765-1787. doi: 10.3934/dcdsb.2021109 [20] K.F.C. Yiu, K.L. Mak, K. L. Teo. Airfoil design via optimal control theory. Journal of Industrial and Management Optimization, 2005, 1 (1) : 133-148. doi: 10.3934/jimo.2005.1.133

2021 Impact Factor: 1.588