**ACTIVE STRUCTURAL-ACOUSTIC ** **CONTROL OF INTERIOR NOISE IN **

**VIBRO-ACOUSTIC CAVITIES **

**ASHOK KUMAR **

**DEPARTMENT OF MECHANICAL ENGINEERING ** **INDIAN INSTITUTE OF TECHNOLOGY DELHI **

**JUNE 2016 **

**©Indian Institute of Technology Delhi (IITD), New Delhi, 2016**

**ACTIVE STRUCTURAL-ACOUSTIC ** **CONTROL OF INTERIOR NOISE IN **

**VIBRO-ACOUSTIC CAVITIES **

*by *

**ASHOK KUMAR **

### Department of Mechanical Engineering

*Submitted *

### in fulfillment of the requirements of the degree of Doctor of Philosophy

*to the *

### Indian Institute of Technology Delhi

**June 2016**

**Dedicated ** **to **

### My Supervisor Dr. S.V. Modak

### &

### My wife Sonika

### My daughter Yashika Bagha

**Certificate **

**This is to certify that the thesis entitled Active Structural-Acoustic Control of ** **Interior Noise in Vibro-Acoustic Cavities being submitted by Mr. Ashok ** **Kumar to the Indian Institute of Technology Delhi for the award of degree of ** **Doctor of Philosophy is a record of bonafide research work carried out by him ** under my supervision and guidance. The thesis work, in my opinion has reached the requisite standard fulfilling the requirements for the degree of Doctor of Philosophy. The results contained in this thesis have not been submitted in part or in full, to any other University or Institute for the award of any Degree or Diploma.

**Dr. S.V. Modak ** Associate Professor, Date: Department of Mechanical Engineering,

### New Delhi Indian Institute of Technology Delhi

**Acknowledgements **

I wish to express my deepest gratitude to my supervisor Dr. S.V. Modak for guiding in my research endeavour. His constant support and guidance from choosing the right research topic to the writing of this manuscript has been very valuable. Without persistent encouragement and motivation provided by Dr. S.V. Modak this research work would not have been possible.

It was due to his initiatives and valuable instructions that this work got accomplished. It was his innovative ideas, wisdom, experience and goodness that kept this research going. I will forever remain indebted to you Sir for your encouragement, interest and supportive guidance throughout this research period.

I take this opportunity also to thank my other Student Research Committee (SRC) members- Prof. S.P. Singh and Prof. J.K. Dutt (of Mechanical Engineering Department) and Prof.

Santosh Kapuria (of Department of Applied Mechanics), for a critical review of this research work and for providing several constructive suggestions.

This is also time to express my sincere thanks to Mr. Ravi Sharma, Dr. Vineet Prabhakar, Prof. Rakesh Chandra, Dr. Subhash Chander for their guidance and support throughout my academic carrier.

I would also like to thank my fellow PhD scholars in no particular order Sky lab Bhore, Sharad K. Pradhan, Manoj Chouskey, Dipak V. Nehete, Dinesh Kumar, Sukesh Babu, Sachin, Faisal Rahmani and Amrita Puri who made every day easier by helping me in various ways. Special thanks to Sharad K. Pradhan for valuable scientific discussions and utterly selfless help.

I am thankful to Mr. S. Babu, Mr. D. Jaitly, Mr. K.N. Madasundaran, Mr. Ayodhya Prashad and Mr. Rishi Lal for their support and cooperation in the experimental work carried out in Design Research Lab.

I am thankful to my grandparents and parents for their encouragement and support to carry out this research work. I would like to thank my wife, Sonika, and my daughter, Yashika, for their patience, understanding, unconditional and immeasurable support during the course of this work. I am thankful to my wife who managed all other affairs, so that I may be free for the research analysis, and for sharing all my successes and disappointments over these years.

I can always count on her to make the best in any situation, without you it would not be possible for me to complete this or any other task of life.

I am thankful to IIT Delhi for providing a family accommodation in Nalanda Hostel.

**Ashok Kumar **

**Abstract **

Active noise control offers an effective alternative for control of low frequency interior noise in cavities like automobile passenger compartments, aerospace interiors, helicopters, marine vehicles, launch vehicles and enclosed spaces. Disturbances acting on the surrounding elastic structure are a major contributor to the generated noise. This thesis addresses the development of an active structural-acoustic control (ASAC) system using feedback control strategies. It also addresses development of methods for virtual sensing including incorporation of system identification in the control system design process. The feedback control strategies based on direct output feedback, linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) control are explored for ASAC. Objective is also to achieve sensing and actuation through piezoelectric structural sensors and actuators. Numerical and experimental studies are performed on a 3-D rectangular box cavity with a flexible plate to evaluate and compare the performance of the proposed sensing and the control methods.

A numerical model of the 3-D rectangular box cavity with the flexible plate (glued with piezoelectric patches) and with other five surfaces treated rigid is developed using finite element (FE) method. Collocated pairs of piezoelectric patches are used for sensing the vibrations and applying control forces on the structure. A finite element (FE) based numerical model of the piezo-structural-acoustic system is developed from which state space models for predicting the structural and the acoustic response in the physical as well

as in the modal domain are developed. Modal analysis of the cavity for single and multiple pairs of collocated piezoelectric patches is performed. The selection of optimal location of piezoelectric sensors and actuators based on observability and controllability grammians matrix is carried out.

The thesis proposes a new strategy for virtual sensing of the acoustic potential energy inside a vibro-acoustic cavity with the objective of achieving global active control of noise. It is based on the established concept of ‘radiation modes’ and does not add too many states to the order of the system. Sensing of radiation modes is carried out through two separate filters. A Kalman filter is used to sense the modal amplitudes of the radiation modes and a frequency-weighting filter is designed to model their radiation efficiencies.

The results of virtual sensing based on the proposed strategy are also compared with sensing based on the direct estimation of the structural modal velocities from the outputs of the piezo sensors and also with sensing based on an acoustic filter.

The thesis considered five feedback control strategies for ASAC. These are a) a control strategy based on direct output feedback (AVC-DOFB) b) a control strategy based on LQR to reduce structural vibrations (AVC-LQR) c) an LQR control strategy with a weighting scheme based on structural-acoustic coupling coefficients (ASAC-LQR) d) an LQG control strategy with an acoustic filter as an observer and e) an LQG control strategy with the proposed virtual sensing filter.

The first two strategies i.e., AVC-DOFB and AVC-LQR are indirect strategies in which noise reduction is achieved through active vibration control (AVC). The third strategy (ASAC-LQR) is an active structural-acoustic control (ASAC) strategy proposed in this work. This strategy is an LQR based optimal control strategy in which the information

about the coupling between the various structural and the acoustic modes is used to design the controller. The last two strategies are based on virtual sensing methods based on acoustic and radiation filters. The performance of these strategies is compared for active noise control in the 3-D rectangular box cavity.

An experimental evaluation of the ASAC system performance is carried out on a 3-D rectangular cavity with a flexible plate. The virtual sensing and the control system are implemented on a dSPACE controller board. A methodology to incorporate system identification in the development of LQG controllers is presented. System identification based on modal testing and finite element model updating is proposed to develop an accurate state observer in the form of a Kalman filter. An LQG controller is built by combining the Kalman filter based on the updated model and the LQR controller.

The results of the numerical and the experimental studies carried out using the feedback control strategies and the virtual sensing strategies show that the active structural- acoustic control system developed in this work is effective in suppressing the interior noise in cavities.

**Contents **

**Certificate ……….. i **

**Acknowledgements ………... ii **

**Abstract ………. iv **

**Contents ………. vii **

**List of Figures ……… xi **

**List of Tables ………. xvi **

**List of Symbols ……….. xvii **

**Chapter 1. Introduction and literature Survey ………... …. 1**

1.1 Introduction ……… 1

1.2 Literature Survey ……….... 6

1.2.1 Analytical/ Numerical modeling of ASAC systems…...………. 8

1.2.2 Sensors and actuators for ASAC systems……….... 14

1.2.3 Optimal location of sensors and actuators……….... 16

1.2.4 Coupling between structural and acoustic modes………... 18

1.2.5 Radiation modes………... 19

1.2.6 Virtual sensing………... 20

1.2.7 Control approaches………... 22

1.2.7.1 Feedforward control...………... 24

1.2.7.2 Feedback control...………... 29

1.2.8 System identification………... 34

1.3 Concluding remarks about literature survey ……….. 35

1.4 Objectives of the thesis ……….. 37

1.5 Overview of the thesis ……… 37

**Chapter 2. Numerical simulation of piezo-structural-acoustic ** ** cavity system for active structural-acoustic control…..….. ** **41**

2.1 Introduction ………... 41

2.2 Numerical model of the piezo-structural-acoustic system……… 42

2.3 State space model of the structural domain of the cavity………. 51

2.3.1 State space model of the plant in physical domain……….. 52

2.3.2 State space model of the plant in modal domain...……….. 55

2.4 State space model of the acoustic domain of the cavity………... 57

2.5 Numerical study ……… 58

2.5.1 Details of the case study ……….. 58

2.5.2 Modal analysis of the piezo-structural and the acoustic systems...….. 66

2.5.3 Modal analysis of the piezo-structural system for other locations of piezoelectric patches...….... 72

2.5.4 Choice of location of the piezoelectric patches based on mode shapes. 76 2.5.5 Optimal location of the piezoelectric patches …………...…………... 79

2.6 Conclusions...……….... 82

**Chapter 3. Feedback control strategies in active structural-acoustic ** ** control of a vibro-acoustic cavity……….. 83 **

3.1 Introduction ………... 83

3.2 Active vibration control using direct output feedback (AVC-DOFB)... 84

3.2.1 Open and closed loop system model...………… 85

3.2.2 Design of direct output feedback gain ………... 89

3.2.3 Results of closed loop control.……….. 92

3.3 Active vibration control using an LQR controller (AVC-LQR)……… 96

3.4 Active structural-acoustic control using an LQR controller (ASAC-LQR)... 100

3.5 Results of closed loop control using LQR controllers...………... 101

3.6 Comparative results...………... 109

3.7 Conclusions...………... 113

**Chapter 4. Virtual sensing in a vibro-acoustic cavity………... 115**

4.1 Introduction ………. 115

4.2 Virtual sensing using radiation filter... 117

4.2.1 Description of the radiation filter...………….. 117

4.2.2 Mathematical formulation of the radiation filter...……… 120

4.2.2.1 Frequency-weighting filter………... 120

4.2.2.2 Kalman filter………..………... 124

4.2.2.3 State space model of the radiation filter... 128

4.3 Virtual sensing using acoustic filter... 128

4.4 Virtual sensing using direct estimation... 131

4.5 Estimation of the exact acoustic potential energy... 133

4.6 Numerical study………... 133

4.6.1 Radiation modes of the flexible plate of the cavity...…….………….. 133

4.6.2 Results and discussion……….………. 137

4.6.2.1 Frequency response of the sensing system...………... 137

4.6.2.2 Sensing performance under impulsive and broadband disturbances... 143

4.6.2.3 Time response under impulsive and broadband disturbance... 147

4.7 Conclusions...………... 153

**Chapter 5. Active structural-acoustic control using virtual sensing ** ** strategies ………... 155 **

5.2 Global active noise control using radiation filter...……….. 156

5.3 Global active noise control using acoustic filter...………... 160

5.4 Numerical study ………... 162

5.4.1 Frequency response of the control system...…………... 162

5.4.1.1 Using single pair of collocated piezoelectric patches for

sensing and actuation... 162

5.4.1.2 Using multiple pairs of collocated piezoelectric patches for sensing and actuation... 164

5.4.1.3 Effect on modal damping factors... 166

5.4.2 Control performance under broadband disturbances……….. 167

5.4.3 Time response of the control system...……….. 171

5.5 Conclusions...………... 178

**Chapter 6. Experimental studies in active structural-acoustic ** ** control of a vibro-acoustic cavity …... 179 **

6.1 Introduction ………. 179

6.2 Active structural-acoustic control using an LQG controller incorporating system identification...…. 180

6.2.1 System identification...……… 181

6.2.1.1 FE model of the cavity structure...………... 181

6.2.1.2 Modal testing…………..………... 183

6.2.1.3 Finite element model updating of the plant... 184

6.2.2 State observer...……… 186

6.2.3 Linear quadratic gaussian (LQG) controller...……… 189

6.3 Experimental study...…………...………….. 193

6.4 Conclusions... ………... 213

**Chapter 7. Conclusions and research contributions ………... 215**

7.1 Concluding remarks ……… 215

7.2 Research contributions ……… 220

**References ……… 221**

**Papers published and communicated ……….... 233 **

**Bio-data ………. 234 **

*List of figures *

**List of figures **

** Figure number and title Page no.**

**1.1. An active structural-acoustic control system ** 7

**1.2. Active noise control system by Leug (1936) ** 23

**1.3. Active noise control by Olson and May (1953) ** 24

**1.4. Principle of feedforward control ** 25

**1.5. Principle of feedback control ** 30

**2.1. Flexible plate with two collocated piezoelectric patches (piezo-structural **
**system) **

43
**2.2. One way coupling between piezo-structural-acoustic cavity systems ** 51
**2.3. Short circuit electric boundary condition for sensor ** 52
**2.4. Schematic representation of a rigid-wall rectangular box 3-D cavity with a **
**flexible plate and a pair of collocated piezoelectric patches **

59
**2.5. FE mesh of the plate with single pair of collocated piezoelectric patches ** 62
**2.6. FE mesh of the plate with multiple pair of collocated piezoelectric patches ** 63

**2.7. FE mesh of the acoustic domain of the cavity ** 64

**2.8. FE mesh of piezo-structural-acoustic domain of the vibro-acoustic cavity **
showing excitation point (node no. 103) on structure and acoustic response point
**(node 1941) inside the cavity **

65

**2.9. The plots of five structural modes of the plate with piezos in the range of 0-**
**400 Hz **

70
**2.10. The plots of first two acoustic modes of the cavity ** 71
**2.11. Alternative location of piezoelectric patches for modal analysis ** 72
**2.12. An alternative choice for the locations of multiple pairs of piezoelectric **
**patches **

75

*List of figures *

**2.13. Mode shapes of structural modes 1, 2, 3 and 4 ** 77

**2.14. Mode shapes of structural modes 5, 6, 7 and 8 ** 78

**3.1. AVC system using direct output feedback control ** 85

**3.2. Pole-zero plots at G***d** = 0 and G**v* = 0.08 (blue: open loop poles; red: closed
loop poles)

90

**3.3. Actuator voltage (G***d** =0 and G**v*** = 0.08) ** 93

**3.4. Open and closed loop sensor output voltage (G***d** =0 and G**v*** = 0.08) ** 93
**3.5. Open and closed loop structural displacement response at node number 103 ** 94
**3.6. Open and closed loop acoustic pressure at node number 1941 inside the cavity ** 95
**3.7. Open and closed loop instantaneous SPL in dB at node number 1941 inside **
**the cavity **

96

**3.8. AVC system using an LQR control strategy ** 97

**3.9. Structural modal velocity (a) first mode (b) seventh mode ** 106
**3.10. Acoustic nodal pressure with and without control at node number 1941 **
**inside the cavity **

107

**3.11. Actuator voltage ** 108

**3.12. Comparison of frequency response function of the acceleration of the **
**structure in dB with and without control **

110
**3.13. Comparison of frequency response function (Pressure in Pa/ Force in N) of **
**acoustic nodal pressure in dB with and without control **

111
**3.14. Comparison of frequency response function of the actuation voltage in dB ** 112
**4.1. Virtual sensing of acoustic potential energy using radiation filter. Symbols **
used are: disturbance (1), control signal (2), sensor output (3), measurement noise
**(4), measured output (5), acoustic pressure (p), acoustic potential energy (Ep) **

118

**4.2. The Kalman filter model ** 125

**4.3. Virtual sensing using direct estimation ** 132

**4.4. The radiation efficiencies (or singular values) of the first four radiation modes **
**of the structure **

134
**4.5. Radiation mode shapes of the flexible plate at 545 Hz (a) first mode (b) **
**second mode (c) third mode (d) fourth mode **

136
**4.6. Frequency response of modal velocity of the first radiation mode estimated by ** 138

*List of figures *

**the Kalman filter with a single pair of piezoelectric patches **

**4.7. Frequency response of sensing of global acoustic potential energy with a **
**single pair of piezoelectric patches **

139
**4.8. Frequency response of modal velocity of first radiation mode estimated by the **
**Kalman filter with multiple pairs of piezoelectric patches **

141
**4.9. Frequency response of sensing of global acoustic potential energy with **
**multiple pairs of piezoelectric patches **

142
**4.10. Comparison of modal velocity of first structural mode estimated by the **
Kalman filter and estimated directly from the output of multiple pairs of
**piezoelectric patches in the presence of an impulsive disturbance **

144

**4.11. Comparison of modal velocity of first radiation mode estimated by the **
Kalman filter and estimated directly from the output of multiple pairs of
**piezoelectric patches in the presence of an impulsive disturbance **

145

**4.12. Sensing of global acoustic potential energy with multiple pairs of **
piezoelectric patches for a broadband disturbance with measurement noise SNR of
**60dB **

146

**4.13. Comparison of modal velocity of first radiation mode estimated by the **
**Kalman filter and its true value with single pair of piezoelectric patches **

148
**4.14. Comparison of modal velocity of first radiation mode estimated by the **
**Kalman filter and its true value with multiple pairs of piezoelectric patches **

149
**4.15. Comparison of modal velocity of first radiation mode estimated by the **
Kalman filter and its true value with single pair of piezoelectric patches at SNR of
**60dB **

150

**4.16. Comparison of modal velocity of first radiation mode estimated by the **
Kalman filter and its true value with multiple pairs of piezoelectric patches at SNR
**of 60dB **

151

**4.17. Comparison of modal velocity of first radiation mode estimated by the **
Kalman filter and its true value with single pair of piezoelectric patches at SNR of
**6dB **

152

**4.18. Comparison of modal velocity of first radiation mode estimated by the **
Kalman filter and its true value with multiple pairs of piezoelectric patches at SNR
**of 6dB **

153

**5.1. Global active structural-acoustic control using radiation filter. Symbols used **
are: disturbance (1), control signal (2), sensor output (3), measurement noise (4),
**measured output (5), acoustic pressure (p), acoustic potential energy (Ep) **

157

**5.2. Global active structural-acoustic control using acoustic filter. Symbols used **
are: disturbance (1), control signal (2), sensor output (3), measurement noise (4),

160

*List of figures *

**measured output (5), acoustic pressure (p), acoustic potential energy (Ep) **

**5.3. Global acoustic potential energy with and without control with a single pair of **
**piezoelectric patches **

163
**5.4. Open and close loop frequency response of global acoustic potential energy **
**with multiple pairs of piezoelectric patches **

165

**5.5. Random disturbance acting on the structure ** 168

**5.6. Control of global acoustic potential energy with multiple pairs of piezoelectric **
**patches for a broadband disturbance with measurement noise SNR of 60dB **

169

**5.7. Control of global acoustic potential energy with multiple pairs of piezoelectric **
**patches for a broadband disturbance with measurement noise SNR of 6dB **

170
**5.8. Control of acoustic nodal pressure at node number 1941 with single pair of **
**piezoelectric patches **

171
**5.9. Control of acoustic nodal pressure at node number 1941 with multiple pairs of **
**piezoelectric patches **

172
**5.10. Control of acoustic nodal pressure at node number 1941 with single pair of **
**piezoelectric patches at SNR of 60dB **

173
**5.11. Control of acoustic nodal pressure at node number 1941 with multiple pairs **
**of piezoelectric patches at SNR of 60dB **

174
**5.12. Control of acoustic nodal pressure at node number 1941 with single pair of **
**piezoelectric patches at SNR of 6dB **

175
**5.13. Control of acoustic nodal pressure at node number 1941 with multiple pairs **
**of piezoelectric patches at SNR of 6dB **

176

**6.1. A stochastic state observer ** 187

**6.2. Linear quadratic Gaussian (LQG) controller for ASAC ** 190

**6.3. Experimental vibro-acoustic cavity ** 194

**6.4. Experimental set-up for modal analysis (a) Modal hammer (b) FFT analyzer **
(c) Accelerometer (d) PZT patch (e) Flexible plate (f) Clamping screws (g)
Acoustic cavity (h) Microphones (i) Sound pressure measuring holes

196

**6.5. Torsional spring stiffness updating parameters for the plate ** 200
**6.6. Comparison of the overlays of inertance FRF-H**161, 84 using IESM with the
**measured FRF for FE model (a) before updating (b) after updating **

202

**6.7. Experimental setup used for ANC (a) vibro-acoustic cavity (b) ENDEVCO **
signal conditioner (c) dSPACE controller board (d) analog low pass filter (e)
**Controller (f) Piezoelectric voltage amplifier (g) CRO **

205

*List of figures *

**6.8. Schematic of the experimental setup used for ANC ** 206

**6.9. Kalman filter performance evaluation at L1 and L2 location ** 208
**6.10. Comparison of the estimated acceleration with the measured acceleration at **
**location L1 **

209
**6.11. Comparison of the estimated acceleration (L2) with the measured **
**acceleration at location L1 **

210
**6.12. Frequency response function with and without control at a microphone **
**position in the cavity **

211
**6.13. Open and closed loop frequency response function at a node inside the vibro-**
**acoustic cavity **

212

*List of tables *

**List of tables **

** Table number and title** ** ** **Page no. **

**2.1. Natural frequencies of the steel plate and plant and the acoustic cavity ** 66
**2.2. Natural frequencies of the steel plate with single pair of piezoelectric **
**patches **

74

**2.3. Natural frequencies of the steel plate with multiple pair of piezoelectric **
**patches **

75
**2.4. Observability grammians eigenvalues for first few modes of structure ** 81
**3.1. Characteristics of various closed loop poles as a function of G***v* 91
**3.2. Modal structural-acoustic coupling coefficients (C**_{ij}**) ** 104
**3.3. Modal damping factors with different control strategies ** 113
**5.1. Modal damping factors with single pair of collocated piezoelectric patches ** 166
**5.2. Modal damping factors with multiple pairs of collocated piezoelectric **

**patches **

167

**5.3. Comparison of RMS value of acoustic nodal pressure (in dB) in case of **
broadband disturbance. The values inside the brackets representing reductions
**after the control **

177

**6.1. Experimental natural frequencies and damping factors of the cavity structure **
**and the acoustic cavity **

197
**6.2. MAC values and FE model and experimental natural frequencies ** 198
**6.3. The intial and the values of the updating parameters after updating ** 201
**6.4. Comparison of the correlation between the measured and the updated natural **

**frequencies and the mode shapes **

201
**6.5. Modal structural-acoustic coupling coefficients (C**_{ij}**) ** 204

*List of symbols and abbreviations *

**List of Symbols and Abbreviations **

**Nomenclature **

**a** Modal amplitude of radiation modes

**ˆa ** Estimated modal velocity amplitudes of radiation modes
**A*** _{CL}* Closed loop state matrix for LQR controller

**A**,**B*** _{a}*,

**B**

_{g}**C**,

**D**State space matrices of the plant

**A**A, **B**_{A}, **C**_{A}, **D**_{A} State space matrices for the acoustic filter

**A**f ,**B**_{f} , **C**_{f},**D**_{f } State space matrices for the frequency-weighting filter
**A**K,**B**_{aK}, **B**_{gK} , **C**_{K}, **D**_{K} State space matrices for the Kalman filter

**A**R, **B**_{R}, **C**_{R}, **D**_{R} State space matrices for the radiation filter

**A**Z, **B**_{Z}, **C**_{Z}, **D**_{Z} State space matrices for the frequency-domain filter
**A ,**u **B , **^{u}_{a} **B ,**_{g}^{u} **C**^{u}, **D **^{u} State space matrices of the updated Kalman filter model
**b** State vector of frequency-weighting filter

*C* Capaciatnce of the charge amplifier

*List of symbols and abbreviations *

**C**A Acoustic viscous damping matrix

**C*** _{AS}* Structural-acoustic coupling matrix in the modal domain

**C**

*Elastic constant matrix at constant electric field*

^{E}**C*** _{new}* Modified damping matrix due to effect of control damping
matrix

**C**S Structural viscous damping matrix

u

**C**T Updated damping matrix of the cavity structure
**d ** State vector of the frequency-domain filter

**D** Electrical displacement

**e** Piezoelectric stress coefficients

**E** Electrical field vector

e

*E**p* Acoustic potential energy for an acoustic element

*E**p* True global acoustic potential energy in frequency domain
ˆ '

*E**p,R* Square root of the acoustic potential energy estimated by the
radiation filter

'

ˆ A

*E**p,* Square root of the acoustic potential energy estimated by the
acoustic filter

ˆ*p*

*E* Estimated global acoustic potential energy

**g** Vector of the random disturbances acting on the cavity
Gd Displacement gain correponding to the sensor output voltage
Gv Velocity feedback gain correponding to the rate of the sensor

output voltage

*List of symbols and abbreviations *

**G** LQR gain matrix due to structural modal velocities and
radiation filter states

**G**A LQR gain matrix due to structural modal velocities and acoustic
filter states

**I** Identity matrix

**K**T Combined structural and piezoelectric stiffness matrix

a

**K**w_{φ} Electro-mechanical coupling matrices between the structure and
the actuators

s

**K**w_{φ} Electro-mechanical coupling matrices between the structure and
the sensors

**K**A Acoustic stiffness matrix

**K**a_{φφ} Electric capacitance matrix for actuators
**K**s_{φφ} Electric capacitance matrix for sensors
**K*** _{str}* Structural stiffness matrix

**K*** _{sen}* Piezoelectric sensor stiffness matrix

**K**

*Piezoelectric actuator stiffness matrix*

_{act}**K** Full state feedback LQR gain matrix

**K*** _{new}* Modified stiffness matrix due to effect of control stiffness
matrix

u

**Κ**T Updated stiffness matrix of the cavity structure

**L** Steady state Kalman gain matrix

**L **u Updated Kalman filter gain

**M**T Combined structural and piezoelectric mass/inertia matrix

*List of symbols and abbreviations *

**M**A Acoustic mass/inertia matrix

**M**act Piezoelectric actuator mass/inertia matrix
**M**sen Piezoelectric sensor mass/inertia matrix

**M**str Structural mass matrix

u

ΜT

Μ Μ

Μ Updated mass matrix of the cavity structure

*na * Number of acoustic modes

* ns * Number of structural modes

**p** Vector of the nodal acoustic pressures

*p * *Acoustic pressure inside the element *

**p**e Vector of elemental nodal pressures

*q*a Vector of the electric charges at the actuator electrodes
*q*s Vector of the electric charges at the sensor electrodes
**q**d Modal displacement gain for LQR controller

**q**v Modal velocity gain for LQR controller
**Q**Vib Weighting matrix for structural states
**r ** State vector of the radiation filter
**r**A State vector of the acoustic filter

**R** Weighting matrix for the control input

g, v

*s s * Process and measurement noise covariance respectively

**S** Structural-acoustic coupling matrix

*S**ca* *Gain of the charge amplifier in mV/pC *

*List of symbols and abbreviations *

**U** Orthonormal matrix

**v** Measurement noise vector

**V** Acoustic volume matrix

**V **m Diagonal acoustic modal volume matrix

**w** Vector of structural transverse and rotational degrees of
freedom

**w**ɺɺ Vector of physical acceleration of the cavity structure
ˆu

**w**ɺɺ Estimate of the physical nodal acceleration

**W**o Diagonal matrix of controllability grammians eigenvalues
**y****s** Sensor output voltage corrupted with measurement noise
**y** Measured acceleration

u

**y**m Estimate of the measured acceleration

**Z**a Structure to acoustic modal frequency response function matrix

**Greek Symbols **

**ζ*** ^{S}* Piezoelectric dielectric matrix at constant mechanical strain
φa Vector of voltages on the piezoelectric patches used as the

actuators

φs Vector of voltages on the piezoelectric patches used as the sensors

ρ* _{act}* density of the piezoelectric actuator

*List of symbols and abbreviations *

ρ*str* density of the structure

ρ*sen* density of the piezoelectric sensor

ψ*S*

ψ ψ

ψ *Mass normalized in-vacuo eigenvectors of the structural-piezo *
system

ξ*S* Viscous modal damping factors of the structure

λ* _{S}* Square root of the eigen values of the structure-piezo system

ψ*A*

ψ ψ

ψ Mass normalized eigenvectors of the acoustic system

ξ*A* Viscous modal damping factors of the acoustic domain
λ* _{A}* Square root of the eigen values of the acoustic system

2

λλλλ*S* Eigenvalue matrix of the structure-piezo system
ΛS

ΛΛ

Λ Diagonal modal damping matrix of structure-piezo system
representing Λ_{S}* _{ii}*= 2= 2= 2= 2ξ

_{S}

*λ*

_{i}_{S}

_{i}2

λλλλ*S* Eigenvalue matrix of the acoustic system
ΛA

ΛΛ

Λ Diagonal modal damping matrix of acoustic system

representing Λ_{A}*ii* = 2= 2= 2= 2ξ_{A}*i*λ_{A}*i*

ηηηη*S* Vector of structural modal displacements
ɺ_{S}

ηηηη Vector of structural modal velocities
ɺɺηηηη*S* Vector of structural modal accelerations
**β**1 Vector of structural modal displacements
**β**2 Vector of structural modal velocities

**β** State vector of structural modal displacement and velocities
**η**A Vector of acoustic modal pressure amplitudes

*List of symbols and abbreviations *

**α**A Acoustic frequency response function matrix in the modal
domain

**σ** Matrix of singular values

ˆK

**β** Estimated structural modal vector by the Kalman filter

ˆ2K

**β**ɺ Estimated structural modal velocity amplitudes by the Kalman
filter

ψR

ψψ

ψ Radiation mode shape matrix

φca Sensor output voltage passing through charge amplifier ω Excitation frequency

α^{ } Weight on the acoustic potential energy

λλλλ_{X} Experimental eigenvalues of the cavity structure

ψψψψ_{X}^{ } Experimental eigenvectors of the cavity structure

ξX Experimental viscous modal damping factors of the cavity structure

u

λλλλS Updated eigenvalues of the cavity structure

u

ψS

ψ ψ

ψ Updated eigenvectors of the cavity structure

u

ΛS

ΛΛ

Λ Updated modal viscous damping matrix of the cavity structure

*List of symbols and abbreviations *

**Subscripts **

a, act Actuator

A Acoustic

ca Charge amplifier

CL Closed

f, z Filter

K Kalman

L Laminate

N-L Non-laminate

nu Number of updating parameters

R Radiation

s, sen Sensor

S, str Structure

T Total

**Superscripts **

a Actuator

E Constant electric field

e Element

S Constant strain

s Sensor

*List of symbols and abbreviations *

T Transpose

u Updated

**Abbreviations **

ANC Active Noise Control

ASAC Active Structural-Acoustic Control

AVC Active Vibration Control

DOFB Direct Output Feedback

DOFs Degrees of freedom

FE Finite element

FRFs Frequency Response Functions IESM Inverse Eigen-Sensitivity Method

LMS Least Mean Square

LQG Linear Quadratic Gaussian

LQR Linear Quadratic Regulator

MAC Modal Assurance Criterion

PVDF Polyvinylidene Fluoride

SNR Signal to noise ratio