**ACTIVE CONTROL OF NOISE ** **IN **

**VIBRO-ACOUSTIC CAVITIES **

**AMRITA PURI **

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

**MARCH 2019 **

**©Indian Institute of Technology Delhi (IITD), New Delhi, 2019**

**ACTIVE CONTROL OF NOISE ** **IN **

**VIBRO-ACOUSTIC CAVITIES **

### by

**AMRITA PURI **

### Department of Mechanical Engineering

### Submitted

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

**INDIAN INSTITUTE OF TECHNOLOGY DELHI **

**MARCH 2019 **

**Dedicated to **

**My parents **

i

**Certificate **

### This is to certify that the thesis entitled "Active Control of Noise in Vibro-

**acoustic Cavities" being submitted by Ms. Amrita Puri 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 her 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.

**Prof. Subodh V. Modak Prof. Kshitij Gupta **

### Department of Mechanical Engineering Department of Mechanical Engineering Indian Institute of Technology Delhi Indian Institute of Technology Delhi New Delhi - 110016 New Delhi - 110016

ii

**Acknowledgements **

I would like to express my sincere and deepest gratitude to my thesis advisor **Prof. Subodh **
**V. Modak for his consistent guidance and encouragement. The regular discussions with him **
deepened my understanding in the fields of vibro-acoustic system, active noise control and
modal analysis. His ingenuity and hard work inspired me. Without his technical expertise and
invaluable suggestions, this thesis would not have been in the same shape as it is now.

I would like to express my earnest and humble gratitude to my thesis advisor **Prof. Kshitij **
**Gupta for his guidance, supervision and motivation. The discussions with him always gave **
me an informed perspective and pushed me to think critically. I felt privileged and blessed to
be under the tutelage of an experienced researcher like him.

I want to thank my SRC (Student Research Committee) members, Prof. Naresh Tandon, Prof.

S. P. Singh and Prof. J. K. Dutt for their valuable questions and comments during the regular presentations of my research work.

I am thankful to Mr. S. Babu and Mr. K. N. Madanasundaran for their support and cooperation during the course of experimental work carried out at Vibration and Instrumentation (V&I) Lab, IIT Delhi. I want to specially thank Mr. Kadeem to assist me for various works related to fabrication of experimental set-up. I would also like to thank Mr.

Suresh Chand Sharma, Mr. Ayodhya Prashad and Mr. Ashok Kumar for providing me suitable instruments and allowing me to use facilities available at central workshop, IIT Delhi. I would also like to extend my thanks to Mr. Davesh, Mr. Ankit and Mr. Naveen for their assistance at the central workshop and V&I Lab.

I am thankful to IIT Delhi HPC facility for computational resources. I also want to extend my thanks to Dr. Manish Agarwal and Mr. Puneet Singh to familiarise me with the usage of HPC facility.

iii

I am thankful to Dr. Deepak Nehete and Dr. Ashok Kumar for the effective research discussions during initial years of my Ph. D. I am grateful to my fellow researchers Nikhil, Vikas and Abhinav for helping me in the experimental work.

I am thankful to my dear friends Parwinder, Sushma, Vaishali, Deepika, Ali, Banpreet and many more for making my stay comfortable and joyful.

I would like to express my heartfelt gratitude to my parents, Mrs. Sudesh Puri and Mr. Ranbir Kumar Puri, to support me throughout my life. Your unconditional love gives me strength to do good work. I also want to thank my siblings Shabnam and Shekhar to be there for me during my highs and lows.

At last, I would like to thank each and every person who directly or indirectly helped me during the course of my Ph. D.

**Amrita Puri **

iv

**Abstract **

This thesis deals with development of efficient methods for global active noise control (ANC) in vibro-acoustic cavities, such as automotive vehicles, aircrafts and other transportation equipment, under the presence of both the acoustic and structural disturbances.

First, optimal control inputs to the acoustic and structural control sources for the maximum reduction in acoustic potential energy in a vibro-acoustic cavity under the presence of both the acoustic and structural disturbances are derived and studied. Then, an adaptive feedforward control algorithm using acoustic sensing, which is applicable to irregular-shaped vibro-acoustic cavities under the presence of both acoustic and structural disturbances, is developed to realise the optimal control. Numerical study in a car-like vibro-acoustic cavity shows that the noise reduction obtained using the proposed global ANC method using acoustic sensing is very close to the maximum reduction possible with the optimal control. A simultaneous use of both acoustic and structural control sources is required for a maximum global noise reduction under the presence of both acoustic and structural disturbances but this leads to an undesirable effect of increase in kinetic energy of the enclosing flexible structure.

To address this issue, a constrained minimisation problem, in which the objective is to minimise the acoustic potential energy in the cavity subject to a constraint that the kinetic energy of the structure does not increase after control, is formulated. An adaptive feedforward algorithm, based on exterior penalty function approach in which the solution of the constrained problem is sought as the converged solution of a sequence of unconstrained minimisation problems, is developed to realise optimal control inputs in real-time. Numerical

v

studies for tonal as well as multi-tonal noise indicate that the proposed adaptive feedforward control method ensures active noise control without increasing structural vibrations.

Modal based ANC strategies reported in the literature are in the framework of feedback control or internal modal control architecture. This work presents development of a modal based feedforward control algorithm, named as 'Modal Filtered-x Least Mean Square algorithm', for global ANC in a vibro-acoustic cavity. This is achieved by formulating the conventional FxLMS algorithm in the modal domain of the cavity which then allows focussing attention on reduction of contributions of specific acoustic modes to the acoustic potential energy. The formulation introduces concepts of 'modal secondary paths' and 'modal filtered reference signals'. A numerical study in an irregular-shaped vibro-acoustic cavity shows that the Modal FxLMS algorithm is capable of reducing contributions of the chosen acoustic modes. A variable step-size FxLMS algorithm is developed for active control of noise of continuously varying frequency. This algorithm makes an advance prediction of the continuously varying noise and uses this information for optimally adjusting the step-size.

Numerical studies show an improved ANC performance with the developed algorithm over the conventional FxLMS algorithm for various rates of frequency variation. Global ANC algorithm using acoustic sensing as well as the Modal FxLMS algorithm developed in this work are validated experimentally on a 3D rectangular box cavity with a flexible plate. The ANC methods are implemented on a dSPACE controller board. Results of the experimental study of global feedforward ANC using acoustic sensing show that the proposed method is effective in achieving global noise control under the presence of both the structural and acoustic disturbances. Results of the experimental study of Modal FxLMS method show that the method is able to reduce modal contributions of the chosen acoustic modes to the acoustic potential energy.

यह थीसिि ध्वसनक और िंरचनात्मक दोनों गड़बड़ी की उपस्थथसत के तहत, मोटर-वाहन, सवमान और अन्य पररवहन नों जैिे वाइब्रो-ध्वसनक गुहाओं में वैसिक िसिय शोर सनयंत्रण के सिए कुशि

तरीकों के सवकाि िे िंबंसधत है। िबिे पहिे, ध्वसनक और िंरचनात्मक गड़बड़ी दोनों की उपस्थथसत के तहत एक वाइब्रो-ध्वसनक गुहा में ध्वसनक स्थथसतज ऊजाा में असधकतम कमी के सिए ध्वसनक और

िंरचनात्मक सनयंत्रण स्रोतों के सिए इष्टतम सनयंत्रण इनपुट व्युत्पन्न और अध्ययन सकया है। सिर, ध्वसनक िंवेदन का उपयोग करते हुए एक अनुकूिी िीडिॉवाडा कंटरोि एल्गोररथ्म को इष्टतम सनयंत्रण को करने के सिए सवकसित सकया है। यह एल्गोररथ्म ध्वसनक और िंरचनात्मक गड़बड़ी की

उपस्थथसत के तहत असनयसमत आकार के वाइब्रो-ध्वसनक गुहाओं है।

कार वाइब्रो-ध्वसनक गुहा में िंख्यात्मक अध्ययन िे पता चिता है सक ध्वसनक िंवेदन का उपयोग करते हुए प्रस्तासवत वैसिक िसिय शोर सनयंत्रण सवसध का उपयोग करके प्राप्त शोर में कमी इष्टतम सनयंत्रण के िाथ िंभव असधकतम कमी के करीब है। दोनों ध्वसनक और िंरचनात्मक सनयंत्रण स्रोतों का

एक िाथ उपयोग ध्वसनक और िंरचनात्मक दोनों गड़बड़ी की उपस्थथसत के तहत असधकतम वैसिक शोर में कमी के सिए आवश्यक है, िेसकन इििे िंिग्न िंरचना की गसतज ऊजाा में वृस्ि का अवांछनीय प्रभाव होता है। इि िमस्या को हि करने के सिए एक सववश न्यूनतम िमस्या तैयार की गई है सजिमें

उद्देश्य एक बाधा के अधीन गुहा में ध्वसनक स्थथसतज ऊजाा को कम करना है। बाधा के िंरचना

की गसतज ऊजाा वैसिक िसिय शोर सनयंत्रण के नहीं । बाहरी दंड काया दृसष्टकोण पर आधाररत एक अनुकूिी िीडिॉवाडा एल्गोररथ्म को वास्तसवक िमय में इष्टतम सनयंत्रण आदानों का

करने के सिए सवकसित सकया गया है। इि एल्गोररथ्म में न्यूनता िमस्याओं के

अनुिम के converged िमाधान के रूप में सववश िमस्या का िमाधान सकया गया है।

शोर और शोर के सिए िंख्यात्मक अध्ययन िे िंकेत समिता है सक प्रस्तासवत अनुकूिी िीडिॉवाडा सनयंत्रण पिसत िंरचनात्मक कंपन को बढाए सबना िसिय शोर सनयंत्रण िुसनसित करती है।

िासहत्य में ररपोटा की गई मॉडि आधाररत िसिय शोर सनयंत्रण रणनीसत िीडबैक सनयंत्रण या आंतररक मोडि (Model) सनयंत्रण वास्तुकिा के ढांचे में हैं। थीसिि में एक मॉडि (Modal) आधाररत

िीडिॉवाडा कंटरोि एल्गोररथ्म के सवकाि को प्रस्तुत सकया गया है। यह एल्गोररथ्म वाइब्रो-ध्वसनक गुहा

में वैसिक िसिय शोर सनयंत्रण के सिए है औ 'मॉडि सिल्टडा-एक्स सिस्ट मीन स्क्वायर (Modal FxLMS) एल्गोररदम' के नाम सकया है। यह पारंपररक एि एक्स एि एम एि

(FxLMS) एल्गोररथ्म को गुहा के मॉडि डोमेन में तैयार करके प्राप्त सकया है। यह एल्गोररथ्म ध्वसनक स्थथसतज ऊजाा के सिए सवसशष्ट ध्वसनक मोड के योगदान को कम करने पर ध्यान केंसित करने

की देता है। मॉडि एि एक्स एि एम एि (Modal FxLMS) एल्गोररदम का 'मॉडि

िेकेंडरी पाथ्ि' और 'मॉडि सिल्टडा रेिरेंि सिग्नल्स' की अवधारणाओं का पररचय देता है। असनयसमत आकार के वाइब्रो-ध्वसनक गुहा में एक िंख्यात्मक अध्ययन िे पता चिता है सक मॉडि एि एक्स एि

एम एि (Modal FxLMS) एल्गोररथ्म चुने हुए ध्वसनक मोड के योगदान को कम करने में िक्षम है।

िगातार बदिती आवृसि के शोर के िसिय सनयंत्रण के सिए एक एि

एक्स एि एम एि (FxLMS) एल्गोररथ्म सवकसित सकया गया है। यह एल्गोररथ्म िगातार शोर की आवृसि की असिम भसवष्यवाणी करता है और इि जानकारी का उपयोग को

बेहतर ढंग िे िमायोसजत करने के सिए करता है। िंख्यात्मक अध्ययन आवृसि सभन्नता की सवसभन्न दरों के सिए पारंपररक एि एक्स एि एम एि (FxLMS) एल्गोररथ्म सवकसित एल्गोररथ्म का बेहतर िसिय शोर सनयंत्रण प्रदशान गया है। इि काया में सवकसित ध्वसनक िंवेदन ग्लोबि

िसिय शोर सनयंत्रण एल्गोररथ्म और मॉडि एि एक्स एि एम एि (FxLMS) एल्गोररथ्म का उपयोग करते हुए ग्लोबि िसिय शोर सनयंत्रण को 3D आयताकार बॉक्स गुहा एक िचीिी प्लेट है, पर प्रयोगात्मक रूप िे मान्य सकया गया है। िसिय शोर सनयंत्रण सवसधयों को dSPACE सनयंत्रक बोडा पर िागू सकया गया है। ध्वसनक िंवेदन का उपयोग करते हुए वैसिक िीडिॉवाडा िसिय शोर सनयंत्रण के प्रायोसगक अध्ययन के पररणाम बताते हैं सक प्रस्तासवत सवसध िंरचनात्मक और ध्वसनक गड़बड़ी दोनों की उपस्थथसत के तहत वैसिक शोर सनयंत्रण प्राप्त करने में प्रभावी है। मॉडि एि एक्स एि एम एि (Modal FxLMS) सवसध के प्रायोसगक अध्ययन के पररणाम बताते हैं सक सवसध ध्वसनक स्थथसतज ऊजाा के सिए चुने हुए ध्वसनक मोड के मॉडि योगदान को कम करने में िक्षम है।

vi

**Contents **

**Certificate... i **

**Acknowledgements... ii **

**Abstract... iv **

**Contents... vi **

**List of Figures... xii **

**List of Tables... xxiii **

**List of Symbols and Abbreviations... xxviii **

**Chapter 1. Introduction and literature review... 1**

1.1 Introduction... 1

1.2 Literature review... 6

1.2.1 Global active noise control in cavities... 6

1.2.2 Control algorithms for active noise control... 15

1.2.3 Sensing and actuation... 23

1.3 Concluding remarks of the literature review... 36

1.4 Objectives of the thesis... 37

1.5 Overview of the thesis... 37

**Chapter 2. A variable step-size FxLMS algorithm for active ** ** control of continuously varying noise in acoustic ** ** cavities... 41 **

2.1 Introduction... 41

2.2 A simulated study of the FxLMS algorithm... 42

vii

2.2.1 The FxLMS algorithm... 43

2.2.2 Study of location of secondary source... 46

2.2.3 Study of location of error sensor... 52

2.2.4 Study of phase error in secondary path model... 54

2.2.5 Study of choice of convergence coefficient... 58

2.3 The proposed variable step-size FxLMS algorithm for active control of continuously varying noise... 67

2.3.1 Variable step size FxLMS algorithm (VSS-FxLMS)... 68

2.3.2 Frequency domain FxLMS algorithm... 74

2.3.3 Numerical study of the proposed VSS-FxLMS algorithm.... 76

2.3.4 Comparison of the proposed VSS-FxLMS algorithm with the frequency domain FxLMS algorithm... 84

2.4 Conclusion... 89

**Chapter 3. Optimal feedforward control laws for global active ** ** noise control in vibro-acoustic cavities under ** ** structural and acoustic disturbances... 91 **

3.1 Introduction... 91

3.2 Numerical model of vibro-acoustic cavities... 93

3.3 Optimal control inputs for the maximum minimisation in acoustic potential energy... 95

3.3.1 Computation of acoustic potential energy and kinetic energy... 95

3.3.2 Optimal control inputs... 97

3.4 Numerical study... 100

3.4.1 Control using only acoustic control source... 103

3.4.2 Control using only structural control source... 115

3.4.3 Control using both acoustic and structural control sources.. 120

3.5 Conclusion... 125

viii

**Chapter 4. Global feedforward active noise control in vibro- **

** acoustic cavities using acoustic sensing... 127 **

4.1 Introduction... 127

4.2 Adaptive feedforward global active noise control using acoustic sensing of the acoustic potential energy... 128

4.2.1 Description of global active noise control system... 128

4.2.2 Estimation of acoustic potential energy in an irregular cavity using acoustic sensing... 130

4.2.3 Adaptive feedforward algorithm for minimisation of acoustic potential energy for global active noise control... 132

4.3 Numerical study... 134

4.3.1 Performance of acoustic potential energy estimator... 139

4.3.2 Performance of the proposed adaptive feedforward control method... 142

4.4 Conclusion... 153

**Chapter 5. Modal FxLMS algorithm for global active noise ** ** control in vibro-acoustic cavities... 155 **

5.1 Introduction... 155

5.2 Modal filtered-x least mean square algorithm... 156

5.2.1 Formulation of modal filtered-x least mean square algorithm... 157

5.2.2 Identification of modal secondary paths... 160

5.2.3 Estimation of modal states of the cavity... 163

5.2.4 Description of global active noise control using Modal Filtered-x LMS algorithm... 164

5.3 Numerical study... 166

5.3.1 Modal secondary paths... 167

5.3.2 Performance of Modal FxLMS algorithm... 170

5.4 Conclusion... 180

ix

**Chapter 6. Optimal global active noise control in vibro-acoustic **

** cavities without increasing structural vibrations... 182 **

6.1 Introduction... 182

6.2 Optimal global active noise control without increasing structural vibrations... 183

6.2.1 Computation of acoustic potential energy and structural kinetic energy... 183

6.2.2 Formulation of optimal global active noise control without increasing structural vibrations... 184

6.2.3 Solution of the constrained optimization problem... 185

6.3 Numerical Study... 186

6.3.1 Acoustic disturbance... 186

6.3.2 Structural disturbance... 205

6.3.3 Acoustic and structural disturbances... 208

6.3.4 Comparison of optimal active noise control solution with and without the kinetic energy constraint... 211

6.4 Conclusion... 214

**Chapter 7. Global feedforward active noise control in vibro- ** ** acoustic cavities without increasing structural ** ** vibrations... 217 **

7.1 Introduction... 217

7.2 Adaptive feedforward technique for global active noise control without increasing structural vibrations... 217

7.2.1 Iterative method for solution of the constrained optimization problem... 218

7.2.2 Adaptive feedforward implementation of the iterative method for solution of the constrained optimization problem... 219 7.3 Numerical study of active noise control without increasing

structural vibrations using the proposed adaptive feedforward

x

technique... 225 7.3.1 At 568 Hz... 226 7.3.2 At 412 Hz... 229 7.3.3 Frequency response of active noise control without

increasing structural vibrations using the proposed

adaptive feedforward technique... 230 7.4 Global active noise control without increasing structural

vibrations for multi-tonal noise... 235 7.5 Conclusion... 240

**Chapter 8. Experimental studies in active noise control in vibro- **

** acoustic cavities... 242 **

8.1 Introduction... 242 8.2 Description of experimental set up... 243 8.3 Experimental modal analysis of the rectangular vibro acoustic

cavity... 246 8.4 Active noise control in the presence of acoustic disturbance... 252

8.4.1 Results of proposed global active noise control using

acoustic sensing... 253 8.4.2 Results of the global active noise control using the

proposed Modal FxLMS algorithm... 266 8.5 Active noise control under the presence of both acoustic and

structural disturbances... 279 8.5.1 Results of proposed global active noise control using

acoustic sensing... 280 8.5.2 Results of ANC using the proposed Modal FxLMS

algorithm... 282 8.6 Active noise control using two control sources... 285

8.6.1 First and third acoustic modes control using the proposed

Modal-FxLMS method... 285 8.6.2 First five acoustic modes control using the proposed Modal

xi

FxLMS method... 286

8.7 Conclusion... 288

**Chapter 9. ** **Conclusions, research contributions and future work.** **290**

9.1 Conclusions... 290
9.2 Research contributions... 295

9.3 Future scope of work... 296

**References... 297 **

**List of papers published and communicated... 307 **

**Bio-data...**

**...**

**...** **308 **

xii

**List of Figures **

** **

**Fig. No. ** **Figure Title ** **Page No. **

**1.1 ** Internal model control architecture (Kuo and Morgan (1999))... 23
**2.1 ** Block diagram of the FxLMS algorithm... 44
**2.2 ** Simulated 3D box cavity (P : Location of the primary source, S:

Location of the secondary source, , E1: Location of error microphone, , and E2, E3 and E4: Locations of other

microphones)... 46
**2.3 ** Plots of primary noise, secondary noise and residual noise at

four different locations in the cavity at 70Hz when secondary source is placed at the location S (Red: Primary noise, Blue:

Secondary noise and Black: Residual noise)... 48
**2.4 ** Plots of primary noise, secondary noise and residual noise at

four different locations in the cavity at 189Hz when secondary source is placed at the location S (Red: Primary noise, Blue:

Secondary noise and Black: Residual noise)... 49
**2.5 ** Plots of primary noise, secondary noise and residual noise at

four different locations in the cavity at 308Hz when secondary source is placed at the location S (Red: Primary noise, Blue:

Secondary noise and Black: Residual noise)... 50
**2.6 ** Plots of primary noise, secondary noise and residual noise at

four different locations in the cavity at 70Hz when secondary source is placed at the location P (Red: Primary noise, Blue:

Secondary noise and Black: Residual noise)... 51
**2.7 ** Plots of primary noise, secondary noise and residual noise at

four different locations in the cavity at 308Hz when secondary source is placed at the location P (Red: Primary noise, Blue:

Secondary noise and Black: Residual noise)... 52

xiii

**2.8 ** Plots of primary noise, secondary noise and residual noise at
four different locations in the cavity at 189Hz when microphone
at location E4 acts as error sensor (Red: Primary noise, Blue:

Secondary noise and Black: Residual noise)... 53
**2.9 ** Plots of primary noise, secondary noise and residual noise at

four different locations in the cavity at 189Hz when microphone at location E3 acts as the error sensor (Red: Primary noise, Blue:

Secondary noise and Black: Residual noise)...

54
**2.10 ** Plots of primary noise, secondary noise and residual noise at the

location of error microphone, E1, (a) with secondary path model (b) without secondary path at 176.7 Hz (Red: Primary noise,

Blue: Secondary noise and Black: Residual noise)... 56
**2.11 ** Plots of primary noise, secondary noise and residual noise at the

location of error microphone, E1, (a) with secondary path model (b) without secondary path at 181.1 Hz (Red: Primary noise,

Blue: Secondary noise and Black: Residual noise)... 56
**2.12 ** Plots of primary noise, secondary noise and residual noise at the

location of error microphone, E1, (a) with secondary path model (b) without secondary path model at 183.2 Hz (Red: Primary

noise, Blue: Secondary noise and Black: Residual noise)... 57
**2.13 ** Plots of primary noise, secondary noise and residual noise at the

location of error microphone, E1, (a) with secondary path model (b) without secondary path model at 204 Hz (Red: Primary

noise, Blue: Secondary noise and Black: Residual noise)... 57
**2.14 ** Plots of primary noise, secondary noise and residual noise at the

location of error microphone, E1, (a) with secondary path model (b) without secondary path model at 236.2 Hz (Red: Primary

noise, Blue: Secondary noise and Black: Residual noise)... 58
**2.15 ** Secondary path model (a) Magnitude versus frequency, (b)

Phase versus frequency, (c) Impulse response and (d) Unwrap

phase versus frequency... 63
**2.16 ** (a) Performance of the conventional FxLMS algorithm, (b)

xiv

Performance of the conventional FxLMS algorithm with optimum choice of step-size, (c) Zoom view of Fig. 2.16 (a), and (d) Zoom view of Fig. 2.16 (b) (Red: Primary noise, Blue:

Secondary noise, Black: Residual Noise)... 65
**2.17 ** Block diagram of the proposed variable step-size FxLMS

algorithm... 70
**2.18 ** (a) Linear fit in time-frequency data to predict frequency at

future time, t and (b) Several line-fits approximating a curve of

frequency variation with time... 71
**2.19 ** Schematic diagram showing sequential steps of the FFT and line

fitting in the proposed algorithm... 73
**2.20 ** Flow chart of the procedure used to obtain optimum value of the

convergence coefficient, µ... 74
**2.21 ** Block Diagram of the frequency-domain FxLMS algorithm... 76
**2.22 ** Comparison of actual frequency and estimated frequency for (a)

0.5 Hz/sec, (b) 50 Hz/sec and (c) Mixed Sweep Rates... 78
**2.23 ** Variation of primary noise with respect to time for sweep rate of

1 Hz/sec (a) Instantaneous value of acoustic pressure (b) dB

values of acoustic pressure with reference pressure of 20µ Pa... 80
**2.24 ** Comparison of performances of the conventional FxLMS and

the proposed VSS-FxLMS for different sweep rates (a) 0.5 Hz/sec, (b) 1 Hz/sec, (c) 2 Hz/sec, (d) 3 Hz/sec, (e) 10 Hz/sec, (f) -10 Hz/sec, (g) 50 Hz/sec, and (h) Mixed sweep rates (Magenta: Primary noise, Red: Residual noise with the

conventional FxLMS, Black: Residual noise with the proposed

VSS-FxLMS)... 83
**2.25 ** Comparison of performances of conventional FxLMS and

proposed VSS-FxLMS for different sweep rates (a) 0.5 Hz/sec, (b) 1 Hz/sec for 200 seconds duration (Magenta: Primary noise, Red: Residual noise with the conventional FxLMS, Black:

Residual noise with the proposed VSS-FxLMS)... 84
**2.26 ** Performance of frequency-domain FxLMS with 512 adaptive

weights at (a) 159 Hz and (b) 160 Hz... 86

xv

**2.27 ** Performance of frequency-domain FxLMS with 2048 adaptive

weights at (a) 159 Hz and (b) 160 Hz... 86
**2.28 ** Performance of frequency-domain FxLMS with 4096 adaptive

weights at (a) 159 Hz and (b) 160 Hz... 86
**2.29 ** Performance of the frequency-domain FxLMS algorithm for

different sweep rates (a) 0.5 Hz/sec (Instantaneous values of acoustic pressure), (b) 0.5 Hz/sec (dB values of acoustic

pressure), (c) 10 Hz/sec (dB values of acoustic pressure) and (d)

Mixed sweep rates (dB values of acoustic pressure)... 89
**3.1 ** Rectangular box vibro-acoustic cavity (Magenta dot: Acoustic

disturbance source, Cyan dot: Acoustic control source, Yellow dot: Structural disturbance force, Green dot: Structural control

source)... 102
**3.2 ** Comparison of (a) acoustic potential energy and (b) kinetic

energy before and after control with only acoustic control source

when both, acoustic and structural, disturbances are present... 104
**3.3 ** Modal amplitudes before and after control with only acoustic

control source when both, acoustic and structural, disturbances act at 569 Hz a) rigid-walled acoustic modes b) in-vacuo

structural modes... 106
**3.4 ** Modal coupling coefficients of third rigid-walled acoustic mode

with first twenty two in-vacuo structural modes... 107
**3.5 ** Mode shapes of the modes excited by the acoustic disturbance at

569 Hz (which is third cavity-controlled resonance) (a) figure of the vibro-acoustic cavity (red colour showing the flexible panel), (b) third acoustic mode and [(c), (d), (e) and (f)]

structural modes... 109
**3.6 ** Modal amplitudes before and after control with only acoustic

control source when both, acoustic and structural, disturbances act at 116 Hz (a) rigid-walled acoustic modes and (b) in-vacuo

structural modes... 112
**3.7 ** Mode shapes of the acoustic modes excited by acoustic

disturbance at 116 Hz... 113

xvi

**3.8 ** Modal coupling coefficients of third in-vacuo structural mode

with first fifteen rigid-walled acoustic modes... 113
**3.9 ** Mode shapes of the modes excited by the structural disturbance

at 116 Hz (third panel-controlled resonance) (a) third in-vacuo

structural mode and [(b), (c), (d), (e)] Acoustic modes... 114
**3.10 ** Comparison of (a) acoustic potential energy and (b) kinetic

energy before and after control with only structural control source when both, acoustic and structural, disturbances are

present... 117
**3.11 ** Modal amplitudes before and after control with only structural

control source when both, acoustic and structural, disturbances act at 569 Hz (a) rigid-walled acoustic modes and (b) in-vacuo

structural modes... 118
**3.12 ** Modal amplitudes before and after control with only structural

control source when both, acoustic and structural, disturbances act at 116 Hz (a) rigid-walled acoustic modes (b) in-vacuo

structural modes... 119
**3.13 ** Comparison of (a) acoustic potential energy and (b) kinetic

energy before and after control with both acoustic and structural control sources when both, acoustic and structural, disturbances

are present... 122
**3.14 ** Modal amplitudes before and after control with both acoustic

and structural control sources when both, acoustic and

structural, disturbances act at 569 Hz (a) rigid-walled acoustic

modes and (b) in-vacuo structural modes... 123
**3.15 ** Modal amplitudes before and after control with both acoustic

and structural control sources when both, acoustic and

structural, disturbances act at 116 Hz (a) rigid-walled acoustic

modes and (b) in-vacuo structural modes... 124
**4.1 ** Adaptive feedforward active noise control of acoustic potential

energy estimated using acoustic sensing... 129
**4.2 ** 3D irregular vibro-acoustic cavity (Magenta dot: Acoustic

disturbance source, Cyan dot: Acoustic control source, Yellow

xvii

dot: Structural disturbance force, Green dot: Structural control

source, Black dots: Pressure sensors)... 135
**4.3 ** Plot of acoustic potential energy versus frequency (Star: Cavity

controlled resonances and Circle: Panel controlled resonances)... 136
**4.4 ** Time histories of the acoustic potential energy without and with

control using the proposed method when both the acoustic and structural disturbances act at 568 Hz and both the acoustic and

structural control sources are used... 144
**4.5 ** Time histories of the acoustic potential energy without and with

control using the proposed method when both the acoustic and structural disturbances act at 412 Hz and both the acoustic and

structural control sources are used... 145
**4.6 ** Time histories of the acoustic potential energy without and with

control using the proposed method when both the acoustic and structural disturbances act at 532 Hz and both the acoustic and

structural control sources are used... 147
**4.7 ** Car cavity (Black dots: Pressure sensors, Red dots: Acoustic

control sources, Magenta dot: Acoustic disturbance, Green dot:

Structural disturbance)... 150
**4.8 ** Acoustic potential energy before and after control with optimal

control inputs to two acoustic control sources... 151
**5.1 ** Block diagram of Modal Filtered-x Least Mean Square

Algorithm (M Filtered-x LMS) for minimisation of acoustic

potential energy inside a vibro-acoustic cavity... 165
**5.2 ** Plots of first six acoustic modal impulse responses

corresponding to the acoustic control source... 168
**5.3 ** Plots of first six acoustic modal impulse responses

corresponding to the structural control source... 169
**5.4 ** Comparison of amplitudes of acoustic modal responses before

and after control at 568 Hz (a) when only one acoustic mode (5th) is reduced (b) when two acoustic modes (4th and 5th) are

reduced and (c) when all the 12 acoustic modes are reduced... 172
**5.5 ** Comparison of amplitudes of acoustic modal responses before

xviii

and after control at 412 Hz (a) when five acoustic modes (1st, 2nd, 3rd, 4th and 5th) are reduced and (b) when all the 12

acoustic modes are reduced... 174
**5.6 ** Comparison of amplitudes of acoustic modal responses before

and after control at 532 Hz (a) when three acoustic modes (3rd, 4th and 5th) are reduced and (b) when all the 12 acoustic modes

are reduced... 176
**6.1 ** Plots of (a) acoustic potential energy and (b) kinetic energy

before and after control when only acoustic disturbance acts on

the cavity and only acoustic control source is used... 189
**6.2 ** Acoustic and structural modal responses when only acoustic

disturbance act on the cavity at 568 Hz and the acoustic control

source is used... 190
**6.3 ** Acoustic and structural modal responses when only acoustic

disturbance act on the cavity at 412 Hz and the acoustic control

source is used... 191
**6.4 ** Acoustic and structural modal responses when only acoustic

disturbance act on the cavity at 532 Hz and the acoustic control

source is used... 192
**6.5 ** Plots of (a) acoustic potential energy and (b) kinetic energy

before and after control when only acoustic disturbance act on

the cavity and structural control source is used... 195
**6.6 ** Acoustic and structural modal responses when only acoustic

disturbance act on the cavity at 568 Hz and the structural control

source is used... 196
**6.7 ** Acoustic and structural modal responses when only acoustic

disturbance act on the cavity at 412 Hz and the structural control

source is used... 197
**6.8 ** Acoustic and structural modal responses when only acoustic

disturbance act on the cavity at 532 Hz and the structural control

source is used... 198
**6.9 ** Plots of (a) acoustic potential energy and (b) kinetic energy

before and after control when only acoustic disturbance act on

xix

the cavity and both acoustic and structural control sources are

used... 201
**6.10 ** Acoustic and structural modal responses when only acoustic

disturbance act on the cavity at 568 Hz and both the acoustic

and structural control sources are used... 202
**6.11 ** Acoustic and structural modal responses when only acoustic

disturbance act on the cavity at 412 Hz and both the acoustic

and structural control sources are used... 203
**6.12 ** Acoustic and structural modal responses when only acoustic

disturbance act on the cavity at 532 Hz and both the acoustic

and structural control sources are used... 204
**6.13 ** Plots of acoustic potential energies before and after control

when only structural disturbance act on the cavity and both

acoustic and structural control sources are used... 206
**6.14 ** Acoustic and structural modal responses when only structural

disturbance act on the cavity at 568 Hz and both acoustic and

structural control sources are used... 207
**6.15 ** Plots of acoustic potential energies before and after control

when both acoustic and structural disturbance act on the cavity

and both acoustic and structural control sources are used... 209
**6.16 ** Acoustic and structural modal responses when both acoustic and

structural disturbance act on the cavity at 568 Hz and both

acoustic and structural control sources are used... 210
**7.1 ** Block diagram of the proposed adaptive feedforward active

control system for noise reduction without increasing structural

vibrations... 224
**7.2 ** Frequency response of the acoustic potential energy before and

after control with the kinetic energy constraint when only acoustic disturbance acts on the cavity and both, acoustic and structural, control sources are used to reduce acoustic potential energy (a) with the proposed adaptive feedforward method (b)

with optimal control... 232
**7.3 ** Frequency response of the structural kinetic energy before and

xx

after control with the kinetic energy constraint when only acoustic disturbance acts on the cavity and both, acoustic and structural, control sources are used to reduce acoustic potential energy (a) with the proposed adaptive feedforward method (b)

with optimal control... 233
**8.1 ** Experimental set-up to implement global active noise control

schemes

[(1) Rectangular box cavity with one flexible plate, (2) Primary loudspeaker, (3) Power amplifier for the primary loudspeaker, (4) Electrodynamic shaker, (5)Power amplifier for the

electrodynamic shaker (6) Function generator, (7) First control loudspeakers, (8) Second control loudspeaker, (9) Microphones

(Nos. 8)]... 244
**8.2 ** Schematic of the experimental set-up to implement global active

noise control schemes... 245
**8.3 ** (a) Experimental set-up for modal analysis of the plate and (b)

Schematic of the experimental set-up for modal analysis of the plate

[(1) Aluminium plate backed by the rectangular box acoustic cavity, (2) Modal hammer, (3) Accelerometer, (4) Signal

conditioner and (5) FFT analyser]... 246
**8.4 ** First seven experimental mode shapes of the plate backed by the

rectangular cavity... 247
**8.5 ** (a) Experimental set-up for modal analysis of the cavity and (b)

Schematic of the experimental set-up for modal analysis of the cavity

[(1) Loudspeaker to generate sweep sine signal, (2)

Microphone, (3) Power amplifier for loudspeaker, (4) Signal

conditioner for the microphone and (5) FFT analyser]... 248
**8.6 ** First five experimental mode shapes of rectangular cavity of

dimensions (0.524 m × 0.600 m × 1.37 m)... 249
**8.7 ** Location of eight microphones in the rectangular box vibro-

acoustic cavity... 252

xxi

**8.8 ** Acoustic pressure at eight microphones at 131 Hz before and
after control with global ANC technique based on acoustic

sensing... 255
**8.9 ** Estimated modal amplitudes of first five acoustic modes at 131

Hz before and after control with global ANC technique based on

acoustic sensing... 256
**8.10 ** Plots of (a) estimated acoustic potential energy and (b) sum of

squares of acoustic pressures at 131 Hz before and after control

with global ANC technique based on acoustic sensing... 257
**8.11 ** Acoustic pressure at eight microphones at 70 Hz before and

after control with global ANC technique based on acoustic

sensing... 260
**8.12 ** Estimated modal amplitudes of first five acoustic modes at 70

Hz before and after control with global ANC technique based on

acoustic sensing... 261
**8.13 ** Plots of (a) Estimated acoustic potential energy and (b) sum of

squares of acoustic pressures at 70 Hz before and after control

with global ANC technique based on acoustic sensing... 262
**8.14 ** Acoustic pressure at eight microphones at 131 Hz before and

after control of only first acoustic mode with the proposed

Modal FxLMS... 270
**8.15 ** Estimated modal amplitudes of first five acoustic modes at 131

Hz before and after control of only first acoustic mode with the

proposed Modal FxLMS... 271
**8.16 ** Plots of (a) estimated acoustic potential energy and (b) sum of

squares of acoustic pressures at 131 Hz before and after control

of only first acoustic mode with the proposed Modal FxLMS... 272
**8.17 ** Acoustic pressure at eight microphones at 170 Hz before and

after control of first five modes with the proposed Modal

FxLMS... 277
**8.18 ** Estimated modal amplitudes of first five acoustic modes at 170

Hz before and after control of first five modes with the proposed

Modal FxLMS... 278

xxii

**8.19 ** Plots of (a) estimated acoustic potential energy and (b) sum of
squares of acoustic pressures at 170 Hz before and after control

of first five modes with the proposed Modal FxLMS... 279
**8.20 ** Application of structural disturbance on the cavity... 280

xxiii

**List of Tables **

**Table No. ** **Table Title ** **Page No. **

**2.1 ** Frequency of primary noise... 47
**2.2 ** Frequency of primary noise... 50
**2.3 ** Frequency of primary noise and corresponding phase... 55
**2.4 ** Step changes in the frequency of primary noise at different

time intervals... 63
**2.5 ** Comparison of amount of reduction in sound pressure levels

(SPL) at different time instants with the FxLMS algorithm with constant convergence coefficient and with optimum

choice of convergence coefficient... 66
**2.6 ** Block size required for different sweep rates... 77
**4.1 ** First twelve diagonal elements of observability Gramain

matrices at the chosen 18 sensor locations... 138
**4.2 ** Comparison of estimated modal amplitudes of first 12

acoustic modes and acoustic potential energy with its exact

values at 568 Hz... 140
**4.3 ** Comparison of estimated modal amplitudes of first 12

acoustic modes and acoustic potential energy with its exact

values at 412Hz... 141
**4.4 ** Comparison of estimated modal amplitudes of first 12

acoustic modes and acoustic potential energy with its exact

values at 532Hz... 142
**4.5 ** Comparison at 568 Hz of reduction in exact acoustic potential

energy with the proposed adaptive feedforward method with

the maximum reduction possible with optimal control forces... 144
**4.6 ** Comparison at 412 Hz of reductions in exact acoustic

potential energy with the proposed adaptive feedforward

xxiv

method with the maximum possible reductions with the

optimal control forces... 145
**4.7 ** Comparison at 532 Hz of reduction in exact acoustic potential

energy with the proposed adaptive feedforward method with

the maximum reduction possible with optimal control forces... 147
**4.8 ** Comparison of reduction in acoustic potential energy after

control with the proposed method in the presence of

measurement noise at 568 Hz... 149
**4.9 ** Comparison of reductions in acoustic potential energy before

and after control with the proposed method with the maximum

possible reduction applying optimal control inputs... 152
**5.1 ** Comparison of reduction in acoustic potential energy obtained

using M Filtered-x LMS algorithm with the maximum possible reduction obtained using optimal control inputs for different cases of modal control at 568 Hz, 412 Hz and 532

Hz when acoustic control source is used alone... 173
**5.2 ** Comparison of reduction in acoustic potential energy obtained

using M Filtered-x LMS algorithm with the maximum possible reduction obtained using optimal control inputs at

other frequencies... 178
**5.3 ** Comparison of reduction in acoustic potential energy obtained

using M Filtered-x LMS algorithm with the maximum

possible reduction obtained using optimal control inputs in the

presence of measurement noise... 179
**5.4 ** Comparison of reduction in acoustic potential energy obtained

using M Filtered-x LMS algorithm with the maximum possible reduction obtained using optimal control inputs for different cases of modal control at 568 Hz, 412 Hz and 532 Hz when both the acoustic and the structural control sources

are used... 180
**6.1 ** Comparison of acoustic potential energy and kinetic energy

before and after optimal control for disturbances at 568 Hz for minimization of acoustic potential energy with and without

xxv

the kinetic energy constraint... 212
**6.2 ** Comparison of acoustic potential energy and kinetic energy

before and after optimal control for disturbances at 412 Hz for minimization of acoustic potential energy with and without

the kinetic energy constraint... 213
**6.3 ** Comparison of acoustic potential energy and kinetic energy

before and after optimal control for disturbances at 532 Hz for minimization of acoustic potential energy with and without

the kinetic energy constraint... 214
**7.1 ** Comparison of reduction in acoustic potential energy and

increase in kinetic energy after control with the proposed adaptive feedforward control method with and without

constrained minimisation for disturbances at 568 Hz... 228
**7.2 ** Comparison of reduction in acoustic potential energy and

increase in kinetic energy after control with the proposed adaptive feedforward control method with and without

constrained minimisation for disturbances at 412 Hz... 230
**7.3 ** Comparison of reduction in acoustic potential energy and

increase in kinetic energy after control with kinetic energy constraint using the proposed adaptive feedforward scheme

with and without acoustic feedback at 568 Hz... 235
**7.4 ** Reduction in acoustic potential energy and increase in kinetic

energy with optimal control on the basis of constrained minimisation using fmincon for multi-tonal noise comprising

of two frequencies 568 Hz and 519 Hz... 238
**7.5 ** Reduction in acoustic potential energy and increase in kinetic

energy after control with the proposed adaptive feedforward method with constrained minimisation for multi-tonal noise

comprising of two frequencies 568 Hz and 519 Hz... 239
**8.1 ** Experimental coupled natural frequencies of the vibro-

acoustic cavity... 251
**8.2 ** Reduction in acoustic pressures at locations of eight

microphones, estimated modal amplitudes, estimated acoustic

xxvi

potential energy and sum of squares of acoustic pressures at 131 Hz before and after control with global ANC technique

based on acoustic sensing... 257
**8.3 ** Reduction in acoustic pressures at locations of eight

microphones, estimated modal amplitudes, estimated acoustic potential energy and sum of squares of acoustic pressures at 70Hz before and after control with global ANC technique

based on acoustic sensing... 262
**8.4 ** Reduction in acoustic pressures at locations of eight

microphones, estimated modal amplitudes, estimated acoustic potential energy and sum of squares of acoustic pressures at 170 Hz before and after control with global ANC technique

based on acoustic sensing... 264
**8.5 ** Reduction in acoustic pressures at locations of eight

microphones, estimated modal amplitudes, estimated acoustic potential energy and sum of squares of acoustic pressures at 250 Hz before and after control with global ANC technique

based on acoustic sensing... 265
**8.6 ** Reduction in acoustic pressures at locations of eight

microphones, estimated modal amplitudes, estimated acoustic potential energy and sum of squares of acoustic pressures at 131 Hz before and after control of different modes with the

proposed Modal FxLMS... 268
**8.7 ** Reduction in acoustic pressures at locations of eight

microphones, estimated modal amplitudes, estimated acoustic potential energy and sum of squares of acoustic pressures at 170 Hz before and after control of different modes with the

proposed Modal FxLMS method... 275
**8.8 ** Reduction in acoustic pressures at locations of eight

microphones, estimated modal amplitudes, estimated acoustic potential energy and sum of squares of acoustic pressures at 131 Hz before and after control using only second control loudspeaker when both acoustic and structural disturbances

xxvii

act on the cavity... 282
**8.9 ** Reduction in acoustic pressures at locations of eight

microphones, estimated modal amplitudes, estimated acoustic potential energy and sum of squares of acoustic pressures at 170 Hz before and after control of different modes with the

proposed Modal FxLMS... 284
**8.10 ** Comparison of reduction estimated modal amplitudes at 170

Hz before and after control of the first and third acoustic modes with the proposed Modal FxLMS for the three cases of

control sources... 286
**8.11 ** Comparison of reduction in acoustic pressures at locations of

eight microphones, estimated modal amplitudes, estimated acoustic potential energy and sum of squares of acoustic pressures at 170 Hz before and after control of the first five acoustic modes with the proposed Modal FxLMS for the three

cases of control sources... 287

xxviii

**List of Symbols and Abbreviations **

**Symbols **

Weights of the adaptive filter

Convergence coefficient or step-size

Error noise

Filtered reference signal

̂ A constant value of convergence coefficient

Leakage factor

Objective function

Structural modal velocity vector

Orthonormal matrix

Transformed modal velocity vector

Primary noise

Secondary noise

Secondary path impulse response Reference signal

̂ Impulse response of secondary path model

P Primary source

S Secondary source

xxix E1 First error microphone E2 Second error microphone E3 Third error microphone E4 Fourth error microphone

Maximum stable of convergence coefficient or step-size

̅̅̅ Mean square value of filtered reference signal
*N * Number of weights of adaptive filter

Number of samples delayed

Amplitude of the reference signal Gain of secondary path

Power of filtered reference signal

Power of reference signal

*Q * Length of secondary path impulse response vector
n^{th} time sample

Optimum value of convergence coefficient
*f * Frequency in Hertz

*t * Time

Magnitude of secondary path FRF Phase of secondary path FRF

Power of filtered reference signal Sampling frequency

Weights of the adaptive filter in frequency domain FFT of error signal

FFT of filtered reference signal

xxx

Magnitude of secondary path FFT FFT of reference signal

m^{th} element of convergence coefficient vector in frequency domain
FxLMS

Phase of reference signal

Radian frequency

Rate of variation of frequency Frequency in Hertz at initial time

Vector of nodal acoustic pressures Vector of structural degrees of freedom

Vector of volume velocity of acoustic sources Vector of structural forces

Acoustic mass matrix Acoustic stiffness matrix Structural mass matrix Structural stiffness matrix

Structural-acoustic coupling matrix Acoustic viscous damping matrix Structural viscous damping matrix Density of fluid

Mass-normalised acoustic mode shape matrix

Mass-normalised in-vacuo structural mode shape matrix Vector of modal acoustic pressures

xxxi

Vector of modal structural degrees of freedom Modal coupled mass matrix

Modal coupled stiffness matrix Modal coupled damping matrix Acoustic eigenvalue matrix Structural eigenvalue matrix

Complete mode shape matrix comprising acoustic and structural mode shapes

Acoustic viscous damping factor matrix Structural viscous damping factor matrix Acoustic radian natural frequency matrix Structural radian natural frequency matrix

Vector of amplitudes of acoustic modal pressures

Vector of amplitudes of structural modal degrees of freedom Modal dynamic stiffness matrix

Vector of amplitudes of acoustic volume velocity sources and structural forces

Vector of amplitudes nodal acoustic pressures

Vector of amplitudes of structural degrees of freedom Acoustic dynamic stiffness matrix

Structural dynamic stiffness matrix

Time averaged acoustic potential energy of a finite element

Speed of sound

Volume of a finite element

xxxii Magnitude of pressure variation

Vector of acoustic elemental shape functions

Vector of complex amplitudes of nodal acoustic pressures of a finite element

Elemental volume matrix

Global volume matrix

Time averaged structural kinetic energy of a finite element of flexible structure

Density of flexible structure Thickness of flexible structure

Area of a finite element of flexible structure ̇ Magnitude of velocity at a point

Vector of structural elemental shape functions

̇_{ } Vector of complex amplitudes of structural degrees of freedom at
nodes of a finite element

Elemental structural mass matrix

Time averaged acoustic potential energy of complete acoustic cavity

Vector of complex amplitudes of disturbances

Vector of complex amplitudes of control inputs

Vector of complex amplitudes of acoustic disturbances

Vector of complex amplitudes of structural disturbances

Vector of complex amplitudes of acoustic control inputs

Vector of complex amplitudes of structural control inputs

Time averaged acoustic potential energy of complete acoustic cavity

xxxiii before control

Time averaged acoustic potential energy of complete acoustic cavity after control

Time averaged kinetic energy of flexible structure before control

Location matrix

Vector of complex amplitudes of control inputs

Real part of

_{ } Imaginary part of

**, ** ** and ** **A (Matrix), b (vector) and c (constant term) of quadratic Hermitian **
form of acoustic potential energy

Real part of ** matrix **

Imaginary part of ** matrix **

Real part of ** vector **

Imaginary part of ** vector **

Vector of complex amplitudes of optimal control inputs

Time averaged acoustic potential energy of the cavity after optimal control

Time averaged kinetic energy of flexible structure after optimal control

Disturbance structural force

Disturbance volume velocity of a point acoustic source Control structural force

Volume velocity of a control point acoustic source

xxxiv

̅̅̅ Estimate of acoustic potential energy
**S(z) ** Secondary path model matrix

Instantaneous acoustic potential energy of a finite element Instantaneous acoustic pressure

Instantaneous nodal acoustic pressure vector for a finite element Instantaneous acoustic potential energy of complete cavity

̅ Vector of measured nodal acoustic pressures

̅ Vector of modal acoustic pressures

̅ Modal matrix of chosen rigid walled acoustic modes for some chosen locations

*M' * Number of chosen acoustic modes
*N**a * Number of acoustic sensors

̅ Vector of acoustic pressures at sensors locations contributed by disturbances

̅ Vector of acoustic pressures at sensors locations contributed by control inputs

Secondary path from control source to k^{th }sensor

Filtered reference signal corresponding to k^{th }secondary path
Observability Gramian matrix

Output matrix of state space model of a vibro-acoustic cavity State matrix of state space model of a vibro-acoustic cavity

Modal amplitude of the k^{th}* acoustic mode *
*M * Number of acoustic modes in the matrix

xxxv

*N**d * Number of total nodes in FE model

Acoustic pressure vector contributed by primary sources
Acoustic pressure vector contributed by secondary sources
Acoustic pressure at k^{th}* FE node *

Primary path impulse response corresponding to k^{th} node
Secondary path impulse response corresponding to k^{th} node
Primary source input vector

Secondary source input vector

Matrix of primary path impulse responses corresponding to all nodes of FE model

Matrix of secondary path impulse responses corresponding to all nodes of FE model

Matrix of modal primary path impulse responses

Matrix of modal secondary path impulse responses

k^{th} modal primary path

k^{th} modal secondary path
k^{th} acoustic modal pressure

k^{th} modal filtered reference signal

̂_{ } Model of k^{th} modal secondary path

Time averaged kinetic energy of flexible structure after control
**, ** **, ** **A (Matrix), b (vector) and c (constant term) of quadratic Hermitian **

form of structural kinetic energy

Real part of ** matrix **

xxxvi

Real part of ** vector **

Unconstrained objective function

Penalty parameter

Pressure at k^{th} sensor
Number of velocity sensors

Number of estimates used for averaging

Acoustic pressure at k^{th} acoustic sensor at i^{th} sample

Filtered reference signal corresponding to k^{th} microphone

Filtered reference signal corresponding to j^{th} velocity sensor

Experimentally obtained acoustic mode shape matrix Experimentally obtained physical secondary paths

Experimentally obtained modal secondary paths

**Abbreviations**

ANC Active noise control

ASAC Active structural acoustic control DSAS Discrete structural acoustic sensing

DSVAS Discrete structural volume acceleration sensing EE-FxLMS Eigenvalue equalisation filtered-x least mean square

FE Finite element

FFT Fast Fourier transform FIR Finite impulse response

xxxvii

FLANN Function link artificial neural network FRF Frequency response function

FxAP Filtered-x affine projection FxLMS Filtered-x least mean square

FxRLS Filtred-x recursive least square IESM Inverse eigen sensitivity method

IFFT Inverse fast Fourier transform IIR Infinite impulse response

LMS Least mean square

LQG Linear quadratic Gaussian

MFxLMS Modal filtered-x least mean square MIMO Multi-input multi-output

NFxLMS Normalized filtered-x least mean square PVDF Polyvinylidene fluoride

PZT Lead zirconate titanate

RMS Root mean square

SISO Single input and single output SPL Sound pressure level

SQP Sequential quadratic programming VSS Variable step-size