active control of sound in structural-acoustic coupled

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UNIVERSITY OF SOUTHAMPTON FACULTY OF ENGINEERING AND APPLIED SCIENCE INSTITUTION OF SOUND AND VIBRATION RESEARCH

ACTIVE CONTROL OF SOUND IN STRUCTURAL-ACOUSTIC COUPLED SYSTEMS

by

Sang-Myeong Kim

A thesis submitted for the degree of Doctor of Philosophy April 1998

UNIVERSITY OF SOUTHAMPTON ABSTRACT Faculty of Engineering and Applied Science Institution of Sound and Vibration Research Doctor of Philosophy

Active control of sound in structural-acoustic coupled systems by Sang-Myeong Kim The thesis is concerned with the active control of harmonic and random sound in structuralacoustic coupled systems. The principal application is to the active control of sound transmission into an enclosure when it is excited by a harmonic or stationary random sound field. The main objective of this thesis is to clarify the role of acoustic and structural actuators in the global active control of harmonic and random sound transmission into structuralacoustic coupled systems. The objective is tackled by way of investigations into i) the physics of structural-acoustic coupling and ii) the physical limitations and control mechanisms of the active control of harmonic and random sound. The investigations are performed by employing three concepts; the concept of impedance and mobility for the analysis of structural-acoustic coupled systems, the concept of control spillover for the active control of harmonic sound, and the constraint of causality for the active control of random sound. A new approach using the concept of impedance-mobility is proposed to establish a profound theoretical background to the analysis of structural-acoustic coupled systems. The criterion of weak coupling is presented and discussed. In order to provide an intuitive understanding of structural-acoustic coupling, a series of analogous mechanical models representing various cases of structural-acoustic coupling are presented using lumped parameter systems. For the active control of harmonic sound, a theoretical model using both acoustic and structural actuators is presented. In particular, the role of structural and acoustic actuators is investigated using the concept of control spillover. The hybrid use of structural and acoustic actuators is proposed as an effective configuration to control sound in the coupled system whose response is generally governed by both plate-controlled and cavity-controlled modes. For the active control of random sound, the classical Wiener filter is extensively employed in various application examples; a one degree-of-freedom vibrating system, an acoustic duct, and finally the problem of sound transmission into an enclosure. It is demonstrated that the causally unconstrained Wiener filter is the optimal controller for harmonic sound while causally constrained Wiener filter is the optimal controller for stationary random sound. The constraint of causality should be satisfied for a Wiener filter to be physically realisable, and may be the key to understand the restrictions and mechanisms of the active control of random sound. A comprehensive theoretical model for the active control of stationary random sound in multi-input-multi-output(MIMO) control systems is presented by extending the classical Wiener filter theory. It is demonstrated that the hybrid use of both acoustic and structural actuators is again a desirable configuration for the active control of random sound transmission. The theoretical models for the analysis and for the active control of harmonic and random sound in structural-acoustic coupled systems are validated by experiments.

i

ACKNOWLEDGEMENTS

I would like to thank my supervisor Dr. Michael Brennan for all his help and advice during the course of this project. I also extend my appreciation to all the academic staff at the ISVR including my examiners, Prof. Steve Elliott, Dr. Roger Pinnington, and Dr. Trevor Sutton. In particular, I would like to thank Prof. Steve Elliott for his comments on Chapter 2 of this thesis, and thank Prof. Phil Nelson and Prof. Joe Hammond for their advice on Chapter 5. Special thanks is given to Dr.(soon) Seong-ho Yoon for his help on my experimental rig design. Scholarships from the department and the faculty of engineering and applied science are most gratefully acknowledged. I would like to thank all the Korean fellows and their wives in the university for their encouragement with innumerable dinners. Finally, I would like to thank my family in Korea for their love and support.

ii

베풀어 주신 큰 사랑에 감사하며 부모님께

(To My Parents):







(Kim, Taek-Soo)







(Ko, Kwang-Koom)

iii

CONTENTS

1 1. Introduction 1.1 Thesis objective and background 1.2 Thesis structure

1 5

Part I: Analysis of Structural-Acoustic Coupled Systems

8

Chapter 2. Impedance-Mobility Approach for the Analysis of StructuralAcoustic Coupled Systems

9

2.1 Introduction 2.2 Basic theory of the impedance-mobility approach 2.2.1 Force excitation 2.2.2 Excitation by an acoustic source 2.2.3 Both force and acoustic excitation 2.3 structural-acoustic coupling theory in modal co-ordinates 2.3.1 Co-ordinate systems and assumptions 2.3.2 General theory 2.3.3 Impedance-mobility approach 2.3.3.1 Structural excitation 2.3.3.2 Acoustic excitation 2.3.3.3 Both structural and acoustic excitation 2.3.3.4 The criteria for weak coupling 2.4 Structural-acoustic coupling theory in physical co-ordinates 2.4.1 Basic assumptions and system representation 2.4.2 Impedance-mobility approach 2.4.2.1 Structural excitation 2.4.2.2 Acoustic excitation 2.4.2.3 Both structural and acoustic excitation 2.5 Conclusion

iv

9 12 13 14 16 19 19 20 24 24 27 29 30 33 33 36 36 39 40 41

Chapter 3. Mechanical Analogy of Structural-Acoustic Coupling

42

3.1 Introduction 3.2 Mechanical analogy in modal co-ordinates 3.2.1 Impedance and mobility representation 3.2.1.1 Coupling between a single structural mode and a single acoustic mode 3.2.1.2 Coupling between a single structural mode and several acoustic modes 3.2.1.3 Coupling between a single acoustic mode and several structural modes 3.2.2 Dynamics of a 2 d.o.f lumped mass system 3.2.3 Mechanical analogy 3.2.3.1 Coupling between a single structural mode and a single acoustic mode 3.2.3.2 Coupling between a single structural mode and several acoustic modes 3.2.3.3 Coupling between a single acoustic mode and several structural modes 3.3 Mechanical analogy in physical co-ordinates 3.3.1 Description of the example system 3.3.2 Structural excitation 3.3.3 Acoustic excitation 3.3.4 Both structural and acoustic excitation 3.4 Conclusion

42 44 44 44 45 47 49 51 51

Part 2. Active Control of Harmonic Sound

54 54 56 56 57 59 61 63

64

Chapter 4. Active Control of Harmonic Sound Transmission Using Both Acoustic and Structural Actuators

65

4.1 Introduction 4.2 Theory 4.2.1 Assumptions and co-ordinate systems 4.2.2 Structural-acoustic coupled response 4.2.3 Minimisation of the acoustic potential energy 4.3 Active control of sound transmission into a rectangular enclosure 4.3.1 Model description and system characteristics 4.3.2 Active minimisation of the acoustic potential energy 4.3.2.1 Control using a single force actuator 4.3.2.2 Control using a single acoustic source 4.3.2.3 Control using both the piston source and the structural actuator 4.4 Conclusion Appendix 4A: Equation for the analysis of the rectangular enclosure

65 67 67 68 72 74 74 77 77 79 81 84 85

v

Part 3. Active Control of Random Sound

Chapter 5. Active Control of Random Sound in SISO Systems

88

89

5.1 Introduction 5.2 Wiener filter theory 5.2.1 Continuous Time Domain Representation 5.2.2 Laplace Domain Representation 5.2.3 Discrete Time Domain Representation 5.2.4 z-Domain Representation 5.3 Feedforward and Feedback Control of the SISO system 5.3.1 Feedforward Control 5.3.1.1 Analogue control 5.3.1.2 Digital control 5.3.2 Feedback Control 5.3.2.1 Internal Model Control 5.3.2.2 Analogue control 5.3.2.3 Digital control of minimum phase systems 5.3.3 Numerical Analysis Model for Digital Control 5.4 Active Control of a Minimum Phase System 5.4.1 General Description of the System 5.4.2 Feedforward Control 5.4.2.1 Analogue controller 5.4.2.2 Digital controller 5.4.3 Feedback Control 5.4.3.1 Analogue controller 5.4.3.2 Digital controller 5.5 Active control of non-minimum phase systems 5.5.1 Phase angle difference function 5.5.2 Examples 5.5.2.1 Minimum phase system 5.5.2.2 Non-minimum phase system 5.6 Conclusion Appendix 5A Calculation of the sub-Wiener filter Appendix 5B Calculation of the minimum mean square error Appendix 5C Impulse acceleration response of a one d.o.f vibration system

89 92 92 95 98 99 102 102 104 109 112 113 115 117 118 121 121 124 124 127 131 131 132 134 134 135 135 136 139 140 142 144

Chapter 6. Active Control of Random Sound in MIMO Systems

147

6.1 Introduction

147 vi

6.2 Wiener filter theory 6.2.1 SIMO and TISO systems 6.2.2 MIMO systems 6.2.2.1 Wiener-Hopf equation for MIMO systems 6.2.2.2 Causally unconstrained Wiener filter 6.2.2.3 Causally constrained Wiener filter 6.3 Optimal control of harmonic and random sound fields 6.3.1 Feedforward and feedback control 6.3.1.1 Feedforward control 6.3.1.2 Feedback control 6.3.2 Optimal control of harmonic and random sound fields 6.3.2.1 Global feedforward control of harmonic sound fields 6.3.2.2 Global feedforward control of stationary random sound fields 6.3.2.3 Local control of stationary random sound fields 6.4 Active control of sound fields inside a duct 6.4.1 System description 6.4.2 Global control of harmonic and random sound fields 6.4.2.1 Control of the acoustic potential energy 6.4.2.2 Effect of a time delay 6.4.2.3 Effect of system damping 6.4.2.4 Control of the approximated acoustic potential energy 6.4.3 Local control of stationary random sound fields 6.4.3.1 Feedforward control 6.4.3.2 Feedback control 6.5 Conclusion Appendix 6A. Wiener filter for the Two-Input-Single-Output (TISO) system

149 149 152 152 154 156 159 159 159 161 164 164 167 169 170 170 172 172 174 177 179 180 180 184 186 187

Chapter 7. Active Control of Random Sound Transmission

190

7.1 Introduction 7.2 Optimal minimisation of the acoustic potential energy 7.2.1 Active control of harmonic sound transmission 7.2.2 Active control of random sound transmission 7.3 Active control of sound transmission into a rectangular enclosure 7.3.1 Model description and system characteristics 7.3.2 Control of harmonic sound 7.3.3 Control of random sound 7.3.3.1 Data processing 7.3.3.2 Results 7.4 Experiment 7.4.1 Experimental set-up 7.4.2 Characteristics of the system 7.4.3 Off-line control of sound transmission 7.5 Conclusion

190 192 193 198 202 202 205 209 209 214 218 218 224 228 233

vii

Chapter 8. Conclusions and Recommendations for Future Work

234

8.1 Conclusions 8.2 Recommendations for future work

234 239

References

240

viii

GLOSSARY OF TERMS English letters

an (ω ) An(ω) a A A bm (ω ) Bm(ω) b b B c co Cn,m C n′ ,m C ( jω ) C C′ C ( jω ) d(t) D(jω) dp ds d(t) Df

Dq D(ω) e(t) Ek Ep e(t) E(ω) f (y,ω ) F Fs F(yl) fc F

Complex amplitude of the n-th acoustic mode Resonance term(shape function) of the n-th acoustic mode Complex acoustic modal amplitude vector Shape function matrix of the acoustic system Auto-correlation matrix of the received signal Complex amplitude of the m-th structural mode Resonance term(shape function) of the m-th structural mode Complex vibrational modal amplitude vector Cross-correlation vector between the received and desired signals Shape function matrix of the structural system Damping constant Speed of sound Geometric coupling coefficient between the n-th acoustic and m-th structural mode shapes Normalised term of Cn,m Feedback controller Structural-acoustic mode shape coupling coefficient matrix Normalised matrix of C Feedback controller vector Desired signal Fourier transform of the desired signal d(t) Vector showing coupling between the primary source and the acoustic modes Vector showing coupling between the secondary source and the acoustic modes Desired signal vector Matrix showing coupling between the point force locations and the structural modes Matrix showing coupling between the acoustic source locations and the acoustic modes Fourier transform of d(t) Error signal Vibrational kinetic energy of a flexible structure Acoustic potential energy of a cavity Error signal vector Fourier transform of e(t) Force distribution function Force Sampling frequency Equivalent point force of the l-th element Generalised force vector due to the structural control force actuators Force vector ix

g d (t ) g p (t )

Force vector to the acoustic system Force vector to the structural system M-th mode generalised force due to the control forces Generalised modal force of the m-th structural mode Impulse response of acceleration Impulse response of displacement Impulse response of the primary plant

g s (t ) g v (t ) G p (s)

Impulse response of the secondary plant Impulse response of velocity Laplace transform of g p (t )

G+(s) G-(s) G p ( jω )

Causal stable minimum phase term of a power spectrum Anti-causal stable term of a power spectrum Fourier transform of g p (t )

G p (e jω )

Fourier transform of the discrete version of g p (t )

G p ( z)

Z-transform of the discrete version of g p (t )

g ga gp g p (t )

Generalised force vector Generalised force vector acting on the acoustic system Generalised force vector due to the primary plane wave excitation Impulse response vector or matrix of the primary plants

g s (t ) Gp(jω)

Impulse response vector or matrix of the secondary plants Fourier transform of g p (t )

G f ( jω )

Transfer function vector from a structural actuator to the error vector

G q ( jω )

Transfer function vector from an acoustic source to the error vector

h h(t) ho(t) Ho(jω) Hq(jω) Hf(jω) H o′ (s ) Huo(jω) ho h(t) H uo ( jω ) I I or I J J’ Ji Jo k K Ka

Thickness of a plate Filter Optimal filter or Wiener filter Causally constrained Wiener filter or Wiener filter Acoustic optimal controller for an acoustic source Structural optimal controller for a force actuator Sub-Wiener filter Causally unconstrained Wiener filter Wiener filter coefficient vector Filter vector Causally unconstrained Wiener filter vector Number of filter coefficients for a FIR filter for control Identity matrix Cost function or Number of filter coefficients for a FIR filter for modelling Normalised cost function Initial value of the cost function J Minimum value of the cost function J Spring constant Number of the force actuators(Part 2) or number of the control filters(Part 3) Bulk stiffness of an acoustic system

FA FS gc,m gm g a (t )

x

L

L0 m M Ms N p pmic p(yl) p (y , ω ) p ext (y, ω ) p int (y, ω ) pinc (r, ω ) p qn q(y) Q QA QS Qp(ω) Qs(ω) Q(yl) QS(yl) q qc qs Q QS r(t) R(jω) Rrr (τ ) Rrd (τ ) r r(t) R(ω) s s ( x, ω ) Sf So S rr (ω ) S rd (ω )

Number of the whole elements(Part 1) or number of the acoustic control sources(Part 2) or number of error sensors(Part 3) Length of a duct Mass Number of the structural modes considered Mass of a flexible structure Number of the acoustic modes considered Acoustic pressure Reference microphone response Pressure of the l-th element Acoustic pressure distribution on the flexible boundary Exterior acoustic pressure distribution on the flexible surface Sf of an enclosure Interior acoustic pressure distribution on the flexible surface Sf of an enclosure Incident plane wave Acoustic pressure vector in physical co-ordinates Generalised acoustic source strength of the n-th acoustic mode Acoustic source distribution on boundaries y Source strength Effective source strength acting on the acoustic system Induced structural source strength Primary source Secondary control source Excitation source strength on the l-th element Induced structural source strength on the l-th element Generalised acoustic source strength vector Complex source strength vector of the acoustic control sources Induced generalised source strength vector on the structural system Equivalent source strength vector Induced source strength vector Received signal Fourier transform of the received signal r(t) Auto-correlation function of r(t) Cross-correlation function between r(t) and d(t) Co-ordinate system for the external sound field Received signal matrix Fourier transform of r(t) complex variable in the Laplace domain Acoustic source strength density function Area of the flexible structure Cross section area of a duct Auto-power spectrum of r(t) Cross-power spectrum between r(t) and d(t)

S t T

Diagonal matrix consisting of the area of each element ΔSl Time Sampling period xi

Ta u u(y, ω ) u(yl) V w(t) x y(t) Y YA YCS y Ys YS Ycs YCS z Z ZS ZCA Za ZA Zca ZCA

Time constant of the first acoustic mode velocity Normal velocity of the flexible structure Normal velocity of the l-th element Volume of the cavity White noise Co-ordinate system for the acoustic field inside a cavity Estimated signal Mobility Uncoupled acoustic mobility Coupled structural mobility Co-ordinate system for a flexible structure Uncoupled structural modal mobility matrix Uncoupled structural mobility matrix Coupled structural modal mobility matrix Coupled structural mobility matrix Variable in the z-domain Impedance Uncoupled structural impedance Coupled acoustic impedance Uncoupled acoustic modal impedance matrix Uncoupled acoustic impedance matrix Coupled acoustic modal impedance matrix Coupled acoustic impedance matrix

Greek letters

δ (t) δ ′(t ) Δ Δ(e jω ) ΔSl

ζm ζn η θ

Θ(e jω )

ρo ρs σ w2 τ τ′ φ m (y ) Φ Φj

Dirac delta function or the unit impulse Unit doublet function Number of samples delayed Discrete Fourier transform of a delay function Area of the l-th element Damping ratio of the m-th structural mode Damping ratio of the n-th acoustic mode Hysteric damping coefficient Angle of incidence Phase angle difference function Density of air Density of a structure Mean square error of white noise w(t) Arbitrary time variable Normalised time delay Mode shape function of the m-th structural mode Structural mode shape vector Structural mode shape vector at yj xii

[Φ ]

ϕ ψ n (x) Ψ Ψj

ΨL [Ψ]

ω ωd ωm ωn

Structural mode shape matrix in physical co-ordinates Biased angle Mode shape function of the n-th acoustic mode Acoustic mode shape vector Acoustic mode shape vector at yj Acoustic mode shape matrix at the L sensor locations Acoustic mode shape matrix in physical co-ordinates Frequency Resonance frequency Natural frequency of the m-th structural mode Natural frequency of the n-th acoustic mode or the natural frequency

Special symbols

E[ • ] L (•) AH AT A+

Ensemble average of [ • ] Laplace transform of (•) Hermitian transpose of a matrix A Transpose of a matrix A Minimum phase component of a signal A in the frequency domain when A = A+ Aap where Aap is an allpass filter

A+ *

Causal part of a signal A in the frequency domain Convolution integral Complex conjugate

*

Abbreviations

ARMA dB d.o.f FIR FFT F-u ifft IIR IMC LTI MIMO p-Q ref. SIMO SISO

Auto-Regressive-Moving-Average Decibel Degree of freedom Finite-Impulse-Response Fast-Fourier-Transform Force-velocity Inverse Fast-Fourier-Transform Infinite Impulse Response Internal Model Control Linear-Time-Invariant Multi-Input-Multi-Output Pressure-source strength Reference Single-Input-Multi-Output Single-Input-Single-Output

TISO

Two-Input-Single-Output

xiii

LIST OF FIGURES

Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7

Figure 2.8 Figure 2.9

Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8

Figure 4.1 Figure 4.2 Figure 4.3

Figure 4.4

Impedance representation of the conceptual system subjected to force excitation p-Q representation of the conceptual system in acoustic source strength excitation F-u and p-Q representation of the conceptual system for both force and acoustic excitation F-u and p-Q representation for weak coupling A structural acoustic coupled system with the volume V surrounded by a flexible structure and an acoustically rigid wall. Equivalent impedance and mobility representation of the structural-acoustic coupled system in either structural or acoustic excitation Equivalent F-u and p-Q representation of the structural-acoustic coupled system in modal co-ordinates when both structural and acoustic excitations are acting together F-u and p-Q representation for weakly coupling in modal co-ordinates when both structural and acoustic excitations are acting The structural acoustic coupled system with the Lth-divisioned structure including both the flexible structure and the rigid wall.

Equivalent F-u and p-Q representation of the coupling between a single acoustic mode an and a single structural mode bm Equivalent F-u representation when a single structural mode bm couples with several acoustic modes Equivalent p-Q representation when a single acoustic mode an couples with several structural modes Equivalent F-u representation of a 2 d.o.f lumped mass system Mechanical analogy of modal coupling between a single structural mode and a single acoustic mode Mechanical analogy of general modal coupling Schematic diagram of the driver-pipe system Mechanical analogy of the driver-pipe system subjected to both force and acoustic source excitations

A structural acoustic coupled system with the volume V and its flexible boundary surface Sf. The rectangular enclosure with one simply supported plate on the surface Sf on which external plane wave is incident with the angles of (ϕ = 0°) and (θ=45°). Effects of minimising the acoustic potential energy using a point force actuator ( solid line : without control, dashed line : with control ), where ‘*’ and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively. Effects of minimising the acoustic potential energy using an acoustic piston xiv

Figure 4.5

Figure 4.6

Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.11 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19 Figure 5.20

source ( solid line : without control, dashed line : with control ), where ‘*’ and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively. Effects of minimising the acoustic potential energy using both a point force actuator and an acoustic piston source ( solid line : without control, dashed line: with control ), where ‘*’ and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively. Comparison of control efforts of the three control strategies; using each actuator separately ( solid line ) and using both the force actuator and the piston source (dashed line ) , where ‘*’ and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively. Wiener’s problem of finding the optimal filter H(ω) in order to minimise the mean square error when the desired and received signal are stationary random. The optimal realisable filter Feedforward control of random vibration field when the primary force fp(t) is stationary random. Feedforward control in the continuous time domain by Wiener filter theory An arbitrary signal d (t ) = g p (t1 + τ ) and only the solid line part when t ≥ 0 is used for Laplace transform Feedforward control in the discrete time domain by Wiener filter Feedback control of a random vibration field when the primary force fp(t) is stationary random. Internal model control Internal model control of the transient impulse response with a pure time advance on the primary path Wiener filter using the inverse model Active control of a 1 d.o.f system The impulse response (a) and the pole-zero plot in z-plane (b) of the 1 d.o.f system Feedforward control of the 1 d.o.f plant The normalised mean square error J ′ according to the normalised time delay τ ′ = τ Tn and damping ratio. Vibration responses before(solid line) and after feedforward control(dashed line) according to the sample delay Δ = 1, 5, and 30 in (a), and the Wiener filter when Δ = 1 in (b). The normalised mean square error J ′ according to the sample delay where the solid line is obtained from the numerical solution and the dashed line is from the approximate solution in equation (5.56b). Vibration response before(solid line) and after feedforward control(dashed line) with the sample delay Δ = 1 when the input is white-like noise instead of the unit impulse. Vibration responses before(solid line) and after feedback control(dashed line) according to the sample delay Δ = 1, 5, and 30 in (a), and the Wiener filter when Δ = 1 in (b). Phase angle difference functions in feedforward and feedback control of the mobility according to the electric time delay Pole-zero plots for the receptance and inertance of the one d.o.f system xv

Figure 5.21

Performance and phase difference of displacement feedback control according to the time delay Figure 5.22 Performance and phase difference of acceleration feedback control according to the time delay Figure 5A.1 A right triangle for calculating trigonometric functions of angle ϕ

Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6 Figure 6.7 Figure 6.8

Figure 6.9 Figure 6.10

Figure 6.11

Figure 6.12 Figure 6.13

Figure 6.14

Figure 6.15 Figure 6.16 Figure 6.17

Figure 7.1 Figure 7.2

Wiener filter in a general SIMO system TISO(two-input-single-output) system Wiener filters for a MIMO(multi-input multi-output) system Block diagram of feedforward control for MIMO systems Feedback control of MIMO systems ) Internal model control with perfect plant modelling G s ( jω ) = G s ( jω ) . Feedforward and feedback control of sound in a duct Acoustic potential energy; the solid line in the subplot (a) denotes before control, the dotted line shows after the harmonic sound controller using the unconstrained Wiener filter as the feedforward controller, and the dashed line shows after the random sound controller using the constrained Wiener filter. In the subplots (c) and (d) the dotted and dashed lines are the unconstrained and constrained Wiener filters, respectively. Optimal filter responses according to the sample delay Performance of feedforward control according to a time delay for its digital processing (dot line : harmonic controller, dash line : delay = 15 samples, dashdot line : delay = 35 samples) Acoustic potential energy with a time constant for the first mode 0.1 s and damping ratio 0.05. ( solid line: before control, dotted line: harmonic sound feedforward control, and dashed line: random sound feedforward control) Acoustic potential energy ( solid line: before control, dotted line: harmonic sound feedforward control, and dashed line: random sound feedforward control) Performance of local feedforward control at x1=0. (a) Sound pressure before( solid line) and after(dash line) control, (b) Phase angle difference function between the desired and received signals, (c) the constrained Wiener filter, and (d) Frequency components of the causally constrained(dash line) and unconstrained(dot line) Wiener filters The Wiener filter at x1=Lo when it is used as a feedforward controller. (a) the constrained Wiener filter and (b) Frequency components of the causally constrained(dash line) and unconstrained(dot line) Wiener filters Performance of feedforward control according to the location of the error sensor which is normalised by the length of the duct. Performance of feedback control according to the location of the error sensor which is normalised by the length of the duct. Phase angles of the desired(solid line ) and received(dash line) signals at x1=0.75 Lo.

Feedforward control of sound transmission using both structural and acoustic actuators A block diagram of the feedforward control of sound transmission using both xvi

Figure 7.3 Figure 7.4

Figure 7.5

Figure 7.6

Figure 7.7 Figure 7.8 Figure 7.9

Figure 7.10

Figure 7.11

Figure 7.12

Figure 7.13 Figure 7.14 Figure 7.15 Figure 7.16 Figure 7.17

Figure 7.18

Figure 7.19

an acoustic and a structural actuators Feedforward control of sound transmission Comparison of responses with and without the weakly coupling assumption where solid line denotes the fully coupled modelling and dashed line denotes the weakly coupled modelling. The predicted acoustic potential energies from the fully coupled modelling before(solid line) and after feedforward control according to the actuators used(dashed); the acoustic actuator(top), the structural force actuator(middle), and the hybrid use of both actuators(bottom). The frequency responses of each plant whose error is the first acoustic modal response; The input values to the primary plant(solid line), the acoustic secondary plant(dash line), and the structural secondary plant(dot line) are the external sound pressure of 1 Pa, the source strength 10 −3 m 3 s , and the force 1 N , respectively. The phase angle difference function in each Wiener filter for the acoustic actuator(solid) and the structural actuator(dashed). The impulse responses of the plants shown in Figure 7.6; the primary plant(a), the acoustic secondary plant(b), and the structural secondary plant(c). the frequency and time responses of plants for the microphone error sensor located at x1 = 4 L1 10 ; (a) Frequency responses of the primary(solid), acoustic secondary(dash), and structural secondary plants(dot), (b) Impulse responses of the primary(top), acoustic secondary(middle), and structural secondary plants(bottom) The control performance using the acoustic actuator and the response of the constrained feedforward optimal controller; the original cost function(solid) and after using the constrained(dashed) and unconstrained(dotted) optimal controllers. The control performance of using the structural actuator and the response of the constrained feedforward optimal controller; the original cost function(solid) and these after using the constrained(dashed) and unconstrained(dotted) optimal controllers. The control performance of using the both actuators and the response of the constrained feedforward optimal controllers; (a) the original cost function(solid) and these after using the constrained(dash) and unconstrained optimal controllers, (b) acoustic (top) and structural (bottom) actuators Experimental set-up The simply supported plate; in (a) ① Al plate 5 mm, ② Steel strip(Shimstock), ③ Baffle, ④ Adapter(steel), ⑤ Al angle, and ⑥ Plywood 25 mm. Filter characteristics(dB ref. = 1 V) Input signal characteristics of the external speaker(a), the internal speaker(b) and the force actuator(c). (dB ref. = 1 V) Acoustic transfer impedances at the microphone location 5 subject to the internal acoustic excitation; experiment(solid line) and theory(dashed line) (dB ref. = 20 μ Pa) Structural point mobility and the acoustic pressure response at the microphone location 5 subject to the force excitation; experiment(solid line) and theory with the fully coupled modelling(dashed line) (dB ref. = 20 μ Pa) Acoustic pressure response at the microphone location 5 subject to the incident xvii

Figure 7.20

Figure 7.21

Figure 7.22

Figure 7.23

plane wave excitation; experiment(solid line) and theory with the fully coupled modelling(dashed line) (dB ref. = 20 μ Pa) Modelled impulse responses of each plant at the microphone location 5; the primary path(top), and acoustic(middle) and structural(bottom) secondary plants. The control performance of using the acoustic actuator and the response of the constrained optimal controller; the original cost function(solid) and these after using the constrained(dash) and unconstrained optimal controllers. The control performance of using the structural actuator and the response of the constrained feedforward optimal controller; the original cost function(solid) and these after using the constrained(dash) and unconstrained optimal controllers. The control performance of using the both actuators and the response of the constrained feedforward optimal controllers; (a) the original cost function(solid) and these after using the constrained(dash) and unconstrained optimal controllers, (b) acoustic (top) and structural (bottom) actuators

xviii

LIST OF TABLES Table 4.1 Table 4.2

Material properties The natural frequencies and geometric mode shape coupling coefficients of each uncoupled system

Table 5.1

Table 5C.1

Wiener filter and the minimum mean square error in the continuous and discrete time domains Wiener filter and the minimum mean square error in feedforward and feedback optimal controllers Impulse responses and transfer functions of the one d.o.f vibration system

Table 6.1 Table 6.2 Table 6.3

The acoustic duct data Performance of feedforward control according to the sample delay Performance of feedforward control according to the system damping

Table 7.1 Table 7.2

Material properties The natural frequencies and geometric mode shape coupling coefficients of each uncoupled system Comparison of performances according to the use of each and both actuators Experimental equipment A comparison of natural frequencies of the structural-acoustic coupled system Modelling error of the primary and secondary plants Comparison of performances according to the use of each and both actuators

Table 5.2

Table 7.3 Table 7.4 Table 7.5 Table 7.6 Table 7.7

xix

Chapter 1. Introduction

CHAPTER 1 INTRODUCTION

1.1 Thesis objective and background In the world of waves, one plus one does not equal two. When two linear sound waves with same amplitude superpose, for example, the amplitude of the resulting wave can be twice or zero or some other value depending on the phase difference of two waves. When we apply the well known superposition principle to the addition of two waves in a linear medium, phase difference as well as amplitude difference should be accounted. Active control uses destructive wave interference between an original wave field and a secondary wave field generated by control power sources. In this sense simply speaking, active control may be a subject which studies how to make one plus one as close as to zero. Since the concept was first patented by [Lueg(1936)], it has been applied to the active control of various sound fields, for example, in the acoustic free field [Nelson et al(1987)], in a duct [Curtis et al(1987)], and in an acoustic cavity [Bullmore et al(1987)]. The simple discussion above is in the case of two harmonic waves in the acoustic free field. However, the active control of sound in structural-acoustic coupled systems subject to random excitation is far more complicated, and is thus exciting and attractive to explore.

1

Chapter 1. Introduction

This thesis is concerned with the active control of harmonic and random sound in structuralacoustic coupled systems. The principal application is to the active control of sound transmission into an enclosure when it is excited by a harmonic or stationary random sound field. The main objective of this thesis is to clarify the role of acoustic and structural actuators in the global active control of harmonic and random sound transmission into structuralacoustic coupled systems. The objective is tackled by way of investigations into i) the physics of structural-acoustic coupling and ii) the physical limitations and control mechanisms of the active control of harmonic and random sound. The thesis is divided into three parts; Part 1(Analysis of structural-acoustic coupled systems), Part 2(Active control of harmonic sound), and Part 3(Active control of random sound). The investigations are performed by employing three concepts; the concept of impedance and mobility for the analysis of structural-acoustic coupled systems in Part 1, the concept of control spillover for the active control of harmonic sound in Part 2, and the constraint of causality for the active control of random sound in Part 3.

Part 1 considers analysis of structural-acoustic coupled systems. The aim of this part is to investigate the physics of structural-acoustic coupling through the visualisation of coupling using mechanical analogy to lumped parameter systems. The concept of impedance and mobility is extensively employed for the visualisation as well as the analysis of structuralacoustic coupled systems. In particular, weak coupling is focused upon, and the criterion for weak coupling is presented and discussed.

A comprehensive theoretical model for coupled responses in a structural-acoustic coupled system has been presented by [Dowell et al(1977)]. The classical theory by Dowell et al(1977) is re-examined from the impedance-mobility point of view, and a general method of structural-acoustic coupling analysis is presented. The impedance-mobility approach is well known to electrical engineers and physicists, and is particularly applicable to the description of linear dynamic systems of high complexity which are composed of several independent element systems. The approach was adapted by mechanical engineers, and was widely applied to mechanical vibration problems in 1950s[Crandall(1958)]. A general theory of the approach and application examples to mechanical systems can be found[Hixon(1987)]. Furthermore, the approach has been successfully applied to various sound and vibration problems, such as 2

Chapter 1. Introduction

the coupling between actuators and sub-structures, sound radiation from a plate to open space, and wave propagation through media with different physical properties as can be found in many textbooks[Cremer and Heckl(1987), Kinsler et al(1982), and Fahy(1985)]. Since a mode of an uncoupled system can be regarded as a one d.o.f(degree-of-freedom) lumped parameter system in modal co-ordinates, the use of mechanical analogy to structural-acoustic modal coupling has been often suggested especially in the statistical energy analysis(SEA) field[Lyon and Maidanik(1962)]. Although it has been discussed for several decades, no simple mechanical analogy using lumped parameters( mass, spring, and damper) has been available to date.

Part 2 considers the active control of harmonic sound in structural-acoustic coupled systems. The aim of this part is to investigate the role of structural and acoustic actuators for the active control of harmonic sound. Control spillover is a helpful concept to clarify the control mechanisms and physical limitations of each actuator.

Analytical studies of vibro-acoustic systems have been conducted by many investigators to achieve physical insight so that effective active control systems can be designed. It is well established that a single point force actuator and a single acoustic piston source can be used to control well separated vibration modes in structures and well separated acoustic modes in cavities, respectively, provided that the actuators are positioned to excite these modes[Fuller et al(1996) and Nelson and Elliott(1992)]. Active control has also been applied to structuralacoustic coupled systems for example, the control of sound radiation from a plate[Wang et al(1991),

Johnson

and

Elliott(1995),

Fuller

et

al(1991),

and

Meirovitch

and

Thangjitham(1990)] and the sound transmission into a rectangular enclosure[Snyder and Tanaka(1993) and Pan et al(1990)]. Meirovitch and Thangjitham(1990), who discussed the active control of sound radiation from a plate, concluded that more control actuators resulted in better control effects. Pan et al(1990) used a point force actuator to control sound transmission into an enclosure, and discussed the control mechanism in terms of plate and cavity-controlled modes.

3

Chapter 1. Introduction

Part 3 considers active control of random sound in structural-acoustic coupled systems. The aim of this part is to investigate the role of structural and acoustic actuators for the active control of random sound. The Wiener filter provides an optimal controller of random sound, and it should satisfy the constraint of causality to be causal stable( interpreted as ‘physically realisable’ in the literature). The constraint of causality may be the key to understanding the mechanisms and physical limitations of the active control of random sound.

Wiener filter theory was first published by a mathematician [Wiener(1950)], and its engineering applications and analytical solutions are well established in many textbooks, for example, by [Van Trees(1968) and Popoulius(1984)] for analogue filters, and by [Orfanidis(1988)] for digital filters. As reviewed by [Kailath(1977)], the applications of the theory covers the areas of signal estimation, communication, and stochastic control theories, etc. Nelson et al(1990) firstly introduced the Wiener filter theory to the active control of stationary random sound in the acoustic free field and in a duct. They introduced the Wiener filter technique for presenting a formal analytical basis for the calculation of a causally constrained optimal controller to minimise stationary random sound in the mean square sense. In fact, the optimal controller is the Wiener filter. The theory behind this filter has been one of the most influential theories in the areas of signal processing and control. A numerical model using the Wiener filter has also been presented by [Joplin and Nelson(1990)] who investigated the minimisation of acoustic potential energy inside a cavity excited by white noise. The controller in this work was a digital controller operating in the discrete time domain, and its analytic solution procedure was later published by [Nelson(1996)] using the spectral factorisation technique.

4

Chapter 1. Introduction

1.2 Thesis structure The thesis is divided into three parts. The first part(Pat 1: Chapter 2 and 3) considers analysis of structural-acoustic coupled systems by using the impedance-mobility approach. The second part(Part 2: Chapter 4) considers the active control of harmonic sound transmission into an enclosure using both acoustic and structural actuators. The third part(Part 3: Chapter 5, 6, and 7) considers the active control of stationary random sound using the Wiener filter with applications to various sound and vibration control problems including the one d.o.f(degreeof-freedom) vibration system, the one dimensional acoustic duct, and finally the active control of random sound transmission into a structural-acoustic coupled system.

In Chapter 2, the classical theory by Dowell et al(1977) is re-examined from the impedancemobility point of view, and a general method of structural-acoustic coupling analysis is presented. The basic theory of the impedance-mobility approach is considered with a conceptual structural-acoustic coupled system. The conceptual system is the starting point of general coupling, and in fact represents the coupling between a single structural mode and a single acoustic mode. The approach is generalised to analyse structural-acoustic coupled systems in both modal and physical co-ordinates, and a comprehensive theoretical formulation is presented in both co-ordinate systems. Special attention is given to the degree of coupling, and the criterion of weak coupling is presented and discussed. A series of useful mechanical terms and analysis methods are newly introduced through the formulation procedure.

In Chapter 3, The theory developed in Chapter 2 is extensively used for the mechanical analogy of various cases of modal coupling in order to provide an intuitive way to understand the structural-acoustic coupling. The discussion starts with an exact visualisation of modal coupling using the F-u(force-velocity) and p-Q(pressure-source strength) representations. A review of the dynamics of a two d.o.f system is followed, and a series of mechanical analogous models consisting of lumped parameters( mass, spring, damper) are presented to represent various cases of modal coupling. The driver-pipe system, which is often referred as an example of the structural-acoustic system due to its simplicity, is represented by an analogous lumped parameter system. In Chapter 4, a theoretical investigation into the active control of sound transmission in a 5

Chapter 1. Introduction

‘weakly coupled’ structural-acoustic system using both structural and acoustic actuators is considered. A general formulation for the active control of sound in structural-acoustic coupled systems is presented, and it is applied to a rectangular enclosure having one flexible plate through which external noise is transmitted. Three active control systems classified by the type of actuators are compared using computer simulations. They are; i) a single force actuator, ii) a single acoustic piston source, and iii) simultaneous use of both the force actuator and the acoustic piston source. For all three control systems the acoustic potential energy inside the enclosure is adopted as the cost function to minimise, and perfect knowledge of the acoustic field is assumed. The effects of each system are discussed and compared, and a desirable control system is suggested.

In Chapter 5, basic Wiener filter theory, which is one of the most influential theories in signal processing and control fields, is re-examined from the active control point of view for a Single-Input-Single-Output(SISO) system. The filter offers the realisable optimal solution which minimises the mean square error in stationary random fields. The theory is employed extensively in the active control of random sound fields. Analytical and numerical solution procedures are presented with a one d.o.f vibration system which is possibly the simplest application example. Control performance is investigated in terms of the electrical time delay in both feedforward and feedback control. Furthermore, the influence of having a nonminimum phase mechanical plant on control performance is investigated, and a practical way of predicting control performance is proposed.

In Chapter 6, Wiener filter theory originally developed for SISO systems is extended to MultiInput-Multi-Output(MIMO) cases. A comprehensive theoretical Wiener filter model for MIMO systems is presented in both continuous and discrete time domains, and is employed to the feedforward and feedback control of MIMO systems. The simple one-dimensional acoustic duct, which has been investigated by many researchers due to its simplicity, is adopted as the model for the application of Wiener filter theory. Acoustic potential energy is used as the global measure of control performance, and influences of the electric time delay and system damping are demonstrated. It is also demonstrated that the Wiener filter without the constraint of causality is the optimal controller for harmonic sound. Local control of acoustic pressure is also considered, to examine the influence of having a non-minimum phase mechanical plant on control performance. 6

Chapter 1. Introduction

Chapter 7 considers the active control of stationary random sound inside a structural-acoustic coupled system whose sound field is formed by transmission of an external plane wave through a flexible wall of an enclosure. This chapter is the concluding chapter of this thesis, and the acoustics and control theories developed in the previous chapters are extensively applied. The general theoretical model for MIMO active feedforward control of random sound presented in Chapter 6 is rearranged to be suitable for the sound transmission control systems considered here. Like the control of harmonic sound transmission in Chapter 4, three active control systems classified by the type of actuators used are compared. They are; i) a single acoustic actuator, ii) a single structural actuator, and iii) simultaneous use of both the acoustic and the structural actuators. Instead of a weakly coupled structural-acoustic model as in Chapter 4, a fully coupled system is considered as an application model to draw general conclusions. The acoustic potential energy is taken as the cost function to minimise, and the control theory is applied to a fully coupled rectangular enclosure. The effects of each system are discussed, and each control performance for stationary random sound is compared with that for harmonic sound. Lastly, the theoretical models for the analysis of coupling and for the active control of harmonic and random sound are validated by experiments.

The original contributions of the work reported in this thesis are

1. An investigation into the criterion of weak coupling through a comprehensive analysis of structural-acoustic coupled systems based on the impedance-mobility approach(Chapter 2) 2. Provision of a new intuitive way to understand the structural-acoustic coupling using mechanical analogous lumped parameter systems (Chapter 3) 3. Provision of a comprehensive theoretical model, based on the classical Wiener filter theory, for the active control of stationary random sound in single-input-single-output(SISO) and multi-input-multi-output(MIMO) control systems (Chapter 5 and 6) 4. Investigation into the role of structural and acoustic actuators for the active control of sound in structural-acoustic coupled systems(Chapters 4 and 7) 5. Experimental validation of the theoretical models for analysis and control (Chapter 7)

7

Chapter 2. Impedance-Mobility Approach

Part 1. Analysis of Structural-acoustic coupled Systems

8

Chapter 2. Impedance-Mobility Approach

CHAPTER 2

IMPEDANCE-MOBILITY APPROACH FOR THE ANALYSIS OF STRUCTURALACOUSTIC COUPLED SYSTEMS

2.1 Introduction This chapter is concerned with coupling analysis in structural-acoustic coupled systems. The concepts of impedance and mobility are applied to establish a general method of analysis for structural-acoustic coupled systems.

The impedance-mobility approach is well known to electrical engineers and physicists, and is particularly applicable to the description of linear dynamic systems of high complexity which are composed of several independent element systems. Each system at the connection point is characterised by impedance and mobility, and the dynamics of a coupled system can be described at some or all of the points of interest. They are particularly useful concepts to judge the degree of coupling when two or more systems are connected, and are widely used for the analysis of electric systems[Ogata(1970) p.80-81]. The approach was adapted by mechanical engineers, and was widely applied to mechanical vibration problems in 1950s [Crandall(1958)]. Impedance in mechanical systems is simply defined as the ratio of force to velocity at a point while mobility is defined as its inverse. A general theory of the approach and application examples to mechanical systems can be found[Hixon(1987)]. Furthermore, the approach has been successfully applied to various sound and vibration problems, such as 9

Chapter 2. Impedance-Mobility Approach

the coupling between actuators and sub-structures, sound radiation from a plate to the acoustic free field, and wave propagation through media with different physical properties, as can be found in many textbooks[Cremer and Heckl(1987), Kinsler et al(1982), and Fahy(1985)].

The advantages of the approach may be three fold; i) an easier analysis of coupling, ii) power flow is calculated directly, and iii) use of intuitive electrical and mechanical analogous systems. Firstly, by simply looking at the impedance and mobility at connection points before and after connection, the degree of coupling can be easily judged. Although the coupling is a global property, it is not necessary to observe the whole dynamics of a coupled system in all points. Secondly, the approach also offers a direct measure of power since power is described in terms of impedance or mobility. Power is an important property which determines energy transmission between systems. Thirdly, the approach may provide an intuitive way of understanding structural-acoustic coupling using simple mechanical or electrical analogous systems consisting of lumped parameters; mass, spring, and damper in mechanical systems.

Analysis of coupling in structural-acoustic systems has been of great interest during the last half a century, as reviewed by [Pan et al(1990)] and [Hong and Kim(1996)]. A comprehensive theoretical model for coupled responses in a structural-acoustic coupled system has been presented by [Dowell et al(1977)]. They provided solutions for coupled responses in terms of modal characteristics of each uncoupled system, i.e. the uncoupled structural and acoustic systems which compose the coupled system.

In this chapter, the classical theory by Dowell et al(1977) is re-examined from the impedancemobility point of view, and a general method of structural-acoustic coupling analysis is presented. The basic theory of the impedance-mobility approach is considered in Section 2.2 with a conceptual structural-acoustic coupled system. The conceptual system is the starting point of a description of the general coupling, and in fact represents the coupling between a single structural mode and a single acoustic mode that will become clear in Chapter 3. The approach is generalised in Section 2.3 and 2.4 to analyse structural-acoustic coupled systems in both modal and physical co-ordinates, respectively, and a comprehensive theoretical formulation is presented in both co-ordinate systems. Special attention is paid to the degree of 10

Chapter 2. Impedance-Mobility Approach

coupling, and the criterion for weak coupling is presented and discussed. A series of useful mechanical terms and analysis methods are introduced. Based on the theoretical background developed in this chapter, a series of mechanical analogous systems to aid the understanding of structural-acoustic coupling are considered in Chapter 3.

11

Chapter 2. Impedance-Mobility Approach

2.2 Basic theory of the impedance-mobility approach Impedance and mobility are particularly applicable to the analysis of coupled mechanical systems which are composed of several individual systems. If we know the velocity u and the force F at a point in a single input system, the mobility YS and impedance ZS are given by[Hixon(1987)]

YS =

u , F

ZS =

F u

(2.1a,b)

where the subscript S denotes a structural system. It can be seen that in this case impedance is simply the inverse of mobility. When acoustic systems are considered, and if we know the acoustic pressure p and the source strength Q at a point in a single input system, the impedance and mobility used are defined as[Kinsler et al(1982)]

ZA =

p , Q

YA =

Q p

(2.2a,b)

where the subscript A denotes an acoustic system. The choice, whether impedance or mobility is used to define a system, is the users’ preference. Throughout this chapter all final equations are arranged using mobility for structural systems and impedance for acoustic systems, because these correspond to conventional transfer functions which are represented as the ratio of output response to input excitation.

It should be noted that the dimensions of impedance in structural and acoustic systems are different; the dimension of structural impedance is [Ns/m] and the dimension of acoustic impedance is [Ns/m5]. Likewise, structural and acoustic mobilities also have different dimensions. This fact makes the whole following theoretical formulation of structuralacoustic coupling different from general impedance and mobility analysis in mechanical systems[Hixon(1987)], as will be described later.

12

Chapter 2. Impedance-Mobility Approach

A coupled system consisting of a structural and an acoustic system is considered, and it is assumed that each component can be represented by the scalar quantity of mobility or impedance as in the single point input systems. This is a simplified conceptual model since impedance and mobility are of vector or matrix form for general multi-d.o.f(degree of freedom) systems. However, the conceptual coupled system is considered to establish the basic theory for the analysis of structural-acoustic coupled systems before we consider general cases in Section 2.3 and 2.4. As we will see later in Chapter 3, the conceptual model represents the coupling between a single acoustic mode and a single structural mode. The following sub-sections consider the theory for three different excitation conditions, i.e. force excitation, acoustic excitation, and both force and acoustic excitation.

2.2.1 Force excitation Consider a single known structural force F exciting the conceptual structural-acoustic coupled system. It can be represented by the parallel connection of each impedance i.e. ZS and ZCA as shown in Figure 2.1(a). The impedance ZS is defined as the uncoupled structural impedance and is the ratio of the effective force applied to the structure FS to the velocity u. The impedance ZCA represents the acoustic reaction force FA to the structural input velocity u and is defined as the coupled acoustic impedance. The word ‘coupled’ is used since it relates the acoustic response as a result of a structural input, not an acoustic input. These impedances can be expressed mathematically as (see Figure 2.1(b))

ZS =

FS , u

ZCA =

FA u

(2.3a,b)

Using the force equilibrium condition, F = FS + FA , then we get

u=

YS 1 F= F Z S + Z CA 1 + YS Z CA

FA = Z CAu =

YS Z CA F 1 + YS Z CA

13

(2.4a)

(2.4b)

Chapter 2. Impedance-Mobility Approach

where YS =

1 . ZS

When FA is negligibly small compared to F, the system is regarded as weakly coupled. In other words, if ZCA is negligibly small compared to ZS in equation (2.4a), i.e. ZS >> ZCA , the structural response u can be calculated without considering the acoustic system. In this case, a criterion of weak coupling is YS Z CA > YCS , then the system can be regarded as weakly coupled. The criterion is Z A YCS 0, and the function G − (s) and its inverse 1 G − ( s ) are analytic for s0, its inverse Laplace transform is zero when τ < 0. Whereas if it is analytic for s 0 [ Papoulis(1984)]. Since the Fourier transform representation of Srr(ω) is real and even, the pole-zero plot of Srr(-js) is symmetric about both the real(σ) and imaginary(jω) axes. The term G+(s) contains all poles and zeros in the left half of the s-plane, and the other term G − (s) is its mirror image about the imaginary axis. [Ch. 6 in Van trees(1968)]. Thus, the function G+(s) and its inverse 1/ G+(s) are causal, stable, and minimum phase. On the other hand, the function G − (s) and its inverse 1 G − ( s ) are anticausal stable.

Substituting equation (5.9) into (5.8), we get

H uo ( s)G + ( s) = −

S rd (− js ) G − ( s)

(5.10)

If the optimal filter Huo(s) satisfies the constraint of causality, the other term on the left side G+(s) is also realisable since its inverse Laplace transform is zero when τ < 0. Thus, by taking only the causal part on the right side and by replacing s by jω to get the Fourier domain representation, the causally constrained optimal controller is given by

H o ( jω ) = −

⎡ S rd (ω ) ⎤ 1 G ( jω ) ⎢⎣ G − ( jω ) ⎥⎦ + +

(5.11)

where [•]+ denotes the causal part, and the filter Ho(jω) is called the causally constrained Wiener filter or simply Wiener filter. Unlike the filter in equation (5.8), the Wiener filter with the constraint of causality in equation (5.11) is guaranteed to be causal so that it is physically realisable.

The optimal realisable filter in equation (5.11) can be represented by the block diagram shown in Figure 5.2 where the signal w(t) after the filter 1/G+(jω) in the figure is white noise 96

Chapter 5. Wiener Filter in SISO Systems

and the filter is called as the whitening filter. The entire optimum filter is just a cascade of the whitening filter and H o′ ( jω ) . For all rational spectra, there exists a realisable time invariant linear filter whose output is a white process and whose inverse is a realisable linear filter[Van trees(1968)].

Since the received signal for the sub-Wiener filter H o′ ( jω ) is white noise, from the spectral factorisation of equation (5.6) the filter is given by H o′ ( jω ) = −[S wd (ω )]+

(5.12)

Using the relation in equation (5.12), the filter H o ( jω ) can also be expressed as

H o ( jω ) = −

1 [Swd (ω )]+ G ( jω )

(5.13)

+

The minimum mean square error can be calculated by first taking the inverse Fourier transform of the above equation, and by substituting the resulting time response into equation (5.5). Note the time domain expression of equation (5.13) is now guaranteed to be causal.

r(t)

1 G ( jω )

w(t)

H o′ ( jω )

+

) d(t)

⎡ S (ω ) ⎤ H o′ ( jω ) = − ⎢ −rd ⎥ ⎣ G ( jω ) ⎦ +

H o ( jω )

Figure 5.2 The optimal realisable filter

97

Chapter 5. Wiener Filter in SISO Systems

5.2.3 Discrete Time Domain Representation

Discrete time domain analysis is necessary when the filter is implemented by a digital system rather than an analogue system. This representation is also required for computer simulations of active control where all characteristics of signals and systems are discritized. The discrete form of the Wiener-Hopf equation, which is compatible to equation (5.4) for the continuous time domain, is given by[Haykin(1996)] ∞

∑R

rr

[n − k ]ho [k ] = − Rrd [n ],

n≥0

(5.14)

k =0

In contrast to the continuous time Wiener-Hopf equation, equation (5.14) can be solved in a closed form when a linear transverse filter, or Finite Impulse Response(FIR) filter, is used. Suppose the filter is implemented by I number of filter coefficients, then equation (5.14) is represented as a compact matrix form and given by Aho = −b

(5.15)

where the I×I auto-correlation matrix A is a Toeplitz matrix, and is given by

Rrr (1) L Rrr ( N − 1) ⎤ ⎡ Rrr (0) ⎢ R (−1) Rrr (0) L Rrr ( N − 2)⎥⎥ rr A=⎢ ⎢ ⎥ M M M ⎢ ⎥ Rrr (0) ⎦ ⎣ Rrr (− N + 1) Rrr (− N + 2) L

(5.16a)

and the I×1 Wiener filter coefficient vector ho and the I×1 cross-correlation vector b are given by

ho = {ho (0) ho (1) LLLLL ho ( N − 1)}

T

(5.16b)

b = {Rrd (0) Rrd (1) LLLLL Rrd ( N − 1)}

T

The Wiener filter can be solved simply using matrix inversion and is given by 98

(5.16c)

Chapter 5. Wiener Filter in SISO Systems

ho = − A −1 b

(5.17)

The matrix A is real-valued and symmetric since Rrr (τ ) = Rrr (−τ ) because of symmetry of the autocorrelation function. When the received signal r(t) is spectrally rich like in random processes, the matrix A is guaranteed to be positive definite, and hence it must be invertible.[Nelson and Elliott(1992)].

The mean square error corresponding to equation (5.3) for continuous time can be expressed in term of the arbitrary filter coefficient vector h as J = Rdd (0) + 2hT b + hT Ah

(5.18)

The optimal state can be obtained by substituting equation (5.17) into (5.18), which gives J o = Rdd (0) − bT A −1 b

(5.19)

The above direct method is a practical way of solving the discrete form of the Wiener-Hopf equation. When the input received signal is white noise, the correlation matrix A equals the (I×I)dimension identity matrix. The Wiener filter solution in equation (5.17) will be ho = −b , and thus the minimum mean square error is given by

J o = Rdd (0) − b T b = Rdd (0) − hoT ho

(5.20)

5.2.4 z-Domain Representation

To solve the discrete form of the Wiener-Hopf equation in equation (5.14) in an analytical way, the z-domain representation is considered. The z-transform of a discrete signal is analogous to the Laplace transform of a continuous signal. We again consider the solution for equation (5.14) with and without the constraint of causality.

99

Chapter 5. Wiener Filter in SISO Systems

If there were no causality constraint in equation (5.14) i.e. n ≥ 0 , its z-transform representation is obtained by taking z transforms of both sides, and be given by

H uo ( z ) = −

S rd ( z ) S rr ( z )

(5.21)

where Huo(z) is the z-transform of huo[n], and Srd(z) and Srr(z) are the z-transforms of Rrd[n] and Rrr[n], respectively. As mentioned earlier, there is no guarantee whether the filter huo[n] is physically realisable or not. In order to design a causal discrete optimal filter, we again use the factorisation technique with the constraint of causality. Suppose the discrete auto-spectrum of the received signal Srr(z) has a rational spectrum, then it can be factored into a product Srr ( z ) = G ( z )G (1 z )

(5.22)

where the function G(z) and its inverse are analytic for z > 1 , and the function G (1 z ) and its inverse are analytic for z < 1 . The term G(z) contains all poles and zeros inside the unit circle, and the other term G (1 z ) is its mirror image about the unit circle. If a function in the z-domain is analytic for z > 1 , its inverse z-transform is causal so that it is zero when n < 0. Whereas if it is analytic for z < 1 , its inverse z-transform is anti-causal so that it is zero when n > 0 [Ch. 10 in Papoulis(1984) and Ch. 6 in Van trees(1968)]. Substituting equation (5.22) into (5.21) gives

H uo ( z )G ( z ) = −

S rd ( z ) G (1 z )

(5.23)

Using the same method as in the continuous domain, the optimal discrete filter with the causality constraint Ho(z) is given by

Ho ( z) = −

1 ⎡ Srd ( z ) ⎤ G ( z ) ⎢⎣ G (1 z ) ⎥⎦ +

100

(5.24)

Chapter 5. Wiener Filter in SISO Systems

The optimal discrete causal filter in the above equation can also be represented by a block diagram which will have a similar representation as in Figure 5.2. Like in the continuous system, the discrete signal w(n) after the discrete whitening filter 1/G(z) will be white noise. When the received signal is white noise, the optimal discrete causal filter corresponding to equation (5.12) for continuous systems can be obtained similarly. The discrete expression corresponding to equation (5.13) can be written as

Ho ( z) = −

1 [Swd ( z )]+ G( z )

(5.25)

The mean square error with a Wiener filter can be calculated by first taking the inverse ztransform of the above equation, and by substituting the resulting time response into equation (5.18). Important results obtained from this section for the design of analogue and digital Wiener filters are tabulated in Table 5.1 to emphasise the similarity in the solutions for the Wiener filter and the minimum mean square error.

Table 5.1 Wiener filter and the minimum mean square error in the continuous and discrete time domains

Analogue Wiener filter Wiener filter Minimum mean square error

H o ( jω ) = −

Digital Wiener filter

1 [Swd (ω )]+ (5.13) G ( jω ) +



J o = Rdd (0) − ∫ ho2 (τ 1 )dτ 1 0

(5.7)

101

Ho ( z) = −

1 [Swd ( z )]+ G( z )

J o = Rdd (0) − hoT ho

(5.25) (5.20)

Chapter 5. Wiener Filter in SISO Systems

5.3 Feedforward and Feedback Control of the SISO system

The basic Wiener filter theory has been reviewed in four different domains; continuous and discrete time domains, and Laplace and Z-domains for their analytical solutions. To demonstrate their use in the active control of random sound and vibration, we consider a simple Single-Input-Single-Output(SISO) control system where a single actuator is used to minimise the mean square of the signal from a single error sensor. The error sensor measures the response of a mechanical system which is assumed to be excited by a single primary actuator. A stationary random signal is assumed as the input to the primary actuator, and all systems considered in this thesis are assumed to be Linear-Time-Invariant(LTI). Both feedforward and feedback techniques are considered, and their analytical and numerical models are presented.

5.3.1 Feedforward Control

Feedforward control is commonly used for active control of sound and vibration fields. The characteristic of the primary field, whether it is a single frequency component or stationary random field, is important for designing active controllers. With harmonic sound fields, future signals are a repetition of the past signals, so that causality does not emerge as a constraint. For the control of random fields, a statistical approach is generally necessary so that the constraint of causality is considered.

A SISO feedforward control system is shown in Figure 5.3(a) where the mechanical system is excited by a single stationary random primary force fp(t), and the response of the system measured by the error signal e(t) is aimed to be minimised in mean square sense by the secondary force actuator fs(t). In feedforward control a signal related to the primary force is assumed to be measurable, thus this signal is used to operate the secondary force actuator via the controller H(s). For mathematical convenience, a pure time delay τ is assumed to be the 102

Chapter 5. Wiener Filter in SISO Systems

model of electrical system, and is expressed by its Laplace domain expression i.e., e − sτ . The time delay is the time taken to process the signals in the controller, and is impossible to avoid especially in digital controllers where anti-aliasing and reconstruction filters are used. The block diagrams of the primary and secondary paths are shown in Figure 5.3(b) where G p (s ) and Gs (s ) are the transfer functions of mechanical plants on the primary and secondary paths, respectively. Since the controller and the mechanical plant are assumed to be LTI, the two blocks can be commutative as shown in Figure 5.3(b). The whole block diagram is shown in Figure 5.3(c) by combining the primary and secondary paths. The optimal controller is dependent on the spectral characteristics of the error signal which is excited by the primary source. Thus if the spectral distribution of the primary source changes, the corresponding optimal controller should be redesigned. To cope with this input signal dependency for the analytical study here, white noise with zero mean and variance σ w2 = 1 is used as the input to the primary source. White noise is an idealised random process whose power spectral density is constant. It is a useful concept for analysis rather than an actual process, and is difficult to generate in practice. As mentioned in Section 5.2.1, the autocorrelation function of white noise is the unit impulse function δ (t ) . The use of a white noise process is parallel to that of an unit impulse input in the analysis of linear systems[p.198 in Van Trees(1968)]. This is because the correlation between white noise and itself or the response of a LTI system excited by white noise are the same as the correlation between an impulse and itself or an impulse response of the system. Therefore, when white noise is assumed as the primary force input signal, it can be equivalently replaced by an impulse.

Now the problem becomes the active control of an impulse response. The advantages of this approach are; firstly, an easier systematic approach based on the deterministic linear filter theory can be used instead of the general stochastic optimal control theory, and as a consequence, impulse responses replace correlation functions for the solutions. Secondly, the simulation time is greatly reduced since it is not required to generate a long length random data for the solution of the discrete form of Wiener-Hopf equation. In Figure 5.3(c), thus, the primary force fp(t) is replaced by an impulse δ (t ) . Thus, analogue and digital controllers for minimising the impulse response of the LTI system will be analysed next sections.

103

Chapter 5. Wiener Filter in SISO Systems

f p (t )

e − sτ

d (t )

Primary path

Error Signal e(t )

f p (t )

f s (t )

f p (t )

G p (s )

H(s)

e − sτ

Gs (s )

e f p (t )

H(s)

f s (t )

− sτ

y (t ) Gs (s )

y (t )

r (t ) H(s)

Secondary path (a) Feedforward control of a SISO system

(b) Block diagram representations of each path

G p (s ) f p (t )

d (t )

e

− sτ

Gs (s )

r (t )

H(s)

y (t )

e(t )

(c) Block diagram representation of the whole system

Figure 5.3 Feedforward control of random vibration field when the primary force fp(t) is stationary random.

5.3.1.1 Analogue control It is easier for the following analysis if we redraw the LTI system block diagram by moving the time delay term e − sτ into the time advance term e sτ on the primary path as shown in Figure 5.4(a). As discussed earlier, an impulse instead of white noise is used as the primary force signal. Thus it can be seen in Figure 5.4(a) that the problem now becomes a Wiener filter design problem where deterministic signals r(t) and d(t) are available and the mean square error is to be minimised by the controller H(s). In other words, it is to find a realisable filter which approximates the time advanced impulse response of the primary path as close as possible in mean square sense.

For the analytical solution of the Wiener filter, the spectral factorisation method can be used and its block diagram is shown in Figure 5.4(b). The method can also be called the whitening 104

Chapter 5. Wiener Filter in SISO Systems

filter or inverse model method since the whitening filter is the realisable inverse model of the secondary plant system.

The signals for the Wiener filter are r (t ) = g s (t )

and

d (t ) = g p (t1 + τ ) where g s (t ) and g p (t ) are the impulse responses of Gs ( s ) and G p ( s ) ,

respectively. The time variable t1 is an arbitrary variable used for distinguishing it from the present time t.

e sτ

G p (s )

δ (t )

d (t ) Gs (s )

r (t )

y (t ) H(s)

e(t )

Wiener Filtering (a) Feedforward controller subject to the unit impulse input

d (t ) = g p (t1 + τ )

H o (s ) r (t ) = g s (t )

1 G (s) + s

r ′(t )

H o′ (s )

y (t )

e(t )

(b) Wiener filter using the inverse model

Figure 5.4 Feedforward control in the continuous time domain by Wiener filter theory

105

Chapter 5. Wiener Filter in SISO Systems

Thus using the Wiener filter solution in equation (5.11) and replacing G + (s) by Gs+ (s ) , the optimal controller is given by

H o ( s) =

1 H o′ ( s ) G ( s) + s

(5.26)

where H o (s ) and H o′ (s ) are the Laplace transforms of the impulse responses ho (t ) and ho′ (t ) , respectively.

When the secondary plant is minimum phase i.e. Gs ( s ) = G s+ ( s ) , a further simplification of the solution is straightforward. Collocation control of analogue systems is in general minimum phase. A safe method to check minimum phase of a system expressed as a rational function is to draw a pole-zero diagram in the Laplace or z-domains. In the Laplace domain, no pole or zero should be in the right half plane, whereas in the z-domain all poles and zeros are inside the unit circle with the same number of poles and zeros. A sample delay system has one pole inside the unit circle, for example, but its reciprocal is a sample advance system which is not realisable. The driving point mobility of a structure is minimum phase, and the transfer mobility is generally non-minimum phase. In the case studied here, the impulse signal δ (t ) passing through Gs (s ) and then its inverse 1 Gs ( s ) in Figure 5.4(b) will remain the same i.e. r ′(t ) = δ (t ) . Otherwise if the plant is nonminimum phase, the received signal r ′(t ) for the sub-Wiener filter is an allpass filter[Neely and Allen(1979)]. Magnitude responses of allpass filters are unity for all frequencies like white noise, but they have approximately linear phase characteristics [Ch. 5 in Blinchikoff and Zverev(1976)]. Under the assumption of minimum phase for the secondary plant, from equation (5.12) the sub-Wiener filter H o′ (s ) can be written as

H o′ ( s ) = −

[L ( R

wd

]

(τ )) + = −[L (g p (t1 + τ ) )]+

106

(5.27)

Chapter 5. Wiener Filter in SISO Systems

where Rwd (τ ) = d (t ) = g p (t1 + τ ) since the correlation between δ (t ) and a signal is the signal itself. The operator

L(•) denotes Laplace transform, and the factorised term [L (g p (t1 + τ ))]+

denotes the Laplace transform of the time-shifted impulse response g p (t1 + τ ) only for t = t1 + τ ≥ 0 . It can also be written as

[L (g

(t1 + τ ) )]+ = ∫ g p (t1 + τ )e − st dt ∞

p

0

(5.28)

It is graphically shown in Figure 5.5 for an arbitrary signal where d (t ) = g p (t1 + τ ) and only the causal part(solid line) when t ≥ 0 is used for the Laplace transform.

Thus, the Wiener filter for the feedforward collocation control of a SISO system can be written as

H o ( s) = −

[L (g

p

(t1 + τ ) )]+

Gs ( s )

(5.29)

and since it is guaranteed to be causal, its time response is obtained from the simple inverse Laplace transform of the equation. When τ = 0 , then G p ( s ) = [L (g p (t1 ) )]+ so that the Wiener filter is H o ( s ) = − G p ( s ) Gs ( s ) .

107

Chapter 5. Wiener Filter in SISO Systems

d (t )



0

t

Figure 5.5 An arbitrary signal d (t ) = g p (t1 + τ ) and only the solid line part when t ≥ 0 is used for Laplace transform

The minimum mean square error J o can be obtained from the general result for the Wiener filter given in equation (5.5). For the minimum phase Gs (s) , the equation can be further developed using equation (5.7) with substituting ho′ (τ1 ) instead of ho (τ1 ) , and J o is given by



J o = Rdd (0) − ∫ ho′2 (τ1 )dτ1

(5.30a)

0

ho′ (τ 1 ) =

where

L− 1 (H 0′ ( s ) )





−∞

−∞

,

and

Rdd (0) = ∫ d 2 (τ 1 )dτ 1 = ∫ g p (τ 2 + τ )dτ 1 , and 2

τ1 = τ 2 + τ

set





0

,

then



ho′2 (τ 1 )dτ 1 = ∫ g 2p (τ 2 + τ )dτ 1 . 0

It is convenient to express the minimum mean square error in non-dimensional form as the ratio J o J i , where Ji is the initial mean square error due to the primary force only and can be expressed as J i = Rdd (0) , then it gives

Jo = 1− Ji

∫ ∫



0 ∞

−∞

g 2p (τ 2 + τ )dτ 1 g 2p (τ 2 + τ )dτ 1

108

(5.30b)

Chapter 5. Wiener Filter in SISO Systems

Since physical systems are causal i.e. g p (t ) = 0 for t < 0 , it can be written as



Jo =1− Ji

∫τ ∫



0

g 2p (t )dt

(5.30c)

g 2p (t )dt

When τ = 0 in equation (5.30c), the normalised minimum mean square error J o J i becomes exactly zero. As the value τ increases, the ratio gets bigger and it will finally become zero since the non-zero signal on the positive time part in Figure 5.5 tends to zero and then there is no control effect. It should be noted that, as long as the system Gs (s) on the secondary path is minimum phase, the performance is determined by the time delay τ.

Now suppose the excitation is harmonic, the constraint of causality does not occur in this case and the causally unconstrained Wiener filter solution in equation (5.8) can be used. The solution becomes H uo ( s ) = − G p ( s ) Gs ( s ) which is the same as the causally constrained Wiener filter in equation (5.29) having a minimum phase secondary plant and no control time delay. Thus, it can be said that, when a feedforward controller with zero time delay has a minimum phase secondary plant, the realisable filter for random sound can be easily obtained from the causally unconstrained Wiener filter solution. In this case perfect error minimisation is achieved with the feedforward control of harmonic sound in SISO linear systems.

5.3.1.2 Digital control Based on the discrete Wiener filter and analogue feedforward control, an analytical study of the digital optimal controller is conducted in the z-domain. The unit sample delay operator z-1 is used such that, for example, z −1 g (n) = g (n − 1) . The discrete time series g (n) is written as g (n ) = a0δ ( n ) + a1δ ( n − 1) + a2δ ( n − 2) + L where the unit impulse function in discrete signal δ (n) = 1 when n = 0 , and its z-transform is written as G ( z ) = a0 + a1 z −1 + a2 z −2 + L . The block diagram for the digital filter corresponding to Figure 5.4 is shown in Figure 5.6 where the continuous time t is replaced by the discrete time n, and the assumed processing 109

Chapter 5. Wiener Filter in SISO Systems

time delay τ is replaced by the number of samples delayed Δ. To get the analytical expression for the optimal controller, we again use the inverse model method and its block diagram is given Figure 5.6(b).



G p (z )

δ (n)

d (n) Gs (z )

r (n)

y (n )

H(z)

e(n)

Wiener Filtering (a) Feedforward controller subject to the unit impulse input

d (n) = g p (n + Δ )

H o (z ) r ( n) = g s ( n )

1 Gs ( z )

r ′(n)

y (n )

H o′ (z )

e(n)

(b) Wiener filter using the inverse model

Figure 5.6 Feedforward control in the discrete time domain by Wiener filter

The analytical solution for general non-minimum phase secondary systems can be obtained using the general solution presented in Section 5.2, following the procedure shown for the analogue controller in the previous section. In this section, minimum phase is assumed for the secondary plant Gs (z ) which allows a further development of the Wiener filter equation easily. A benefit of the assumption is that it guarantees the received signal to the sub-Wiener filter H o′ (z ) to be an impulse i.e. r ′(n) = δ (n) . Thus the controller H o′ ( z ) = − D( z ) , and can be written as

[

]

H o′ ( z ) = − z Δ G p ( z ) +

110

(5.31)

Chapter 5. Wiener Filter in SISO Systems

If the impulse response of the primary plant can be modelled as a J-length Finite Impulse Response(FIR)

Filter

G p ( z ) = b0 + b1 z −1 + L + bΔ z − Δ + L + bJ z − J

i.e.

,

then

H o′ ( z ) = −bΔ − bΔ +1 z −1 − bΔ + 2 z −2 − L − bJ z − J + Δ .

From equation (5.25), the causal Wiener filter for discrete feedforward control can be given by

[z G ( z )] H ( z) = − Δ

p

o

where

the

+

Gs ( z )

secondary

, Ho ( z) = −

plant

is

bΔ + bΔ +1 z −1 + bΔ + 2 z −2 + L + bJ z − J + Δ a0 + a1 z −1 + a2 z − 2 + L + a J z − J

also

modelled

by

a

J-length

(5.32a,b)

FIR

Filter,

Gs ( z ) = a0 + a1 z −1 + L + a J z − J . If the filter is implemented by a I-length FIR filter, the

analytical equation in equation (5.32) is the same as the discrete Wiener filter ho in equation (5.17). When there is no time delay involved in the control process, then it becomes

Ho ( z) = −

Gp ( z) , Gs ( z )

ho ( n ) = −

g p (n) g s (n)

(5.33a,b)

Since the input to the sub-filter H o′ (z ) is the unit impulse, from equation (5.20) the mean square error after control is given by

J

J

j =0

j =Δ

J o = ∑ b 2j − ∑ b 2j

(5.34)

Again, the sample delay Δ determines the performance. Note the performance equation is the same as equation (5.20) for the white noise input in the discrete time domain representation.

111

Chapter 5. Wiener Filter in SISO Systems

5.3.2 Feedback Control

In this section, the application of Wiener filter theory is extended to the case of feedback control. Figure 5.7(a) shows a typical configuration where the secondary actuator is operated by the error signal via a controller − C (s ) . The reason for the minus sign on the controller C (s ) is to be compatible with the well developed negative feedback theory, and we basically

follow the sign convention in the work by [Elliott and Sutton(1996)] developed for comparison of performances between feedforward and feedback systems for active control. Figure 5.7(b) shows the block diagrams for systems with and without control. The error signal is given by

E (− js ) =

1 D(− js ) 1 + Gs ( s )C ( s )

(5.35)

where E (− js ) and D(− js ) are the Laplace transforms of e(t) and d(t), respectively, and -js is used to distinguish these signals from the variable s for systems.

f p (t )

G p (s )

Without control

Error Signal e(t ) -C(s) f p (t )

f s (t )

d (t ) = e(t )

d (t ) f s (t )

Gs (s )

y (t )

e(t )

-C(s) With control (a) Feedback control

(b) Block diagram representations

Figure 5.7 Feedback control of a random vibration field when the primary force fp(t) is stationary random. 112

Chapter 5. Wiener Filter in SISO Systems

5.3.2.1 Internal Model Control In this work, the Internal Model Control (IMC) structure is used as an alternative to the classical feedback structure. It is widely used for active control and a comprehensive discussion can be found in the textbook by [Morari and Zafiriou(1989)]. The idea of this structure is to model the secondary plant inside the controller − C (s ) in order to turn the feedback structure into a open loop feedforward structure. Its main advantage is that it allows use of the Wiener filter theory developed in feedforward control. ) The IMC structure is shown in Figure 5.8(a) where an estimate of the secondary plant Gs (s)

is inside the controller − C (s ) , and the controller is given by

− C ( s) =

H (s) ) 1 + H ( s )Gs ( s )

(5.36)

Substituting equation (5.36) into (5.35) with the assumption of perfect modelling of the ) secondary plant Gs ( s ) = Gs ( s ) , the control system is transformed into a feedforward one and is written as E ( − js ) = [1 + Gs ( s ) H ( s )]D( − js )

(5.37)

and when the perfect knowledge of the input signal is assumed, its corresponding block diagram is shown in Figure 5.8(b). IMC starts from a very idealised assumption that rarely exists in practice; no model uncertainty and no unknown extra inputs. Thus robust IMC considering the robust stability due to model and inputs uncertainty is generally required for practical implementation[Rafaely(1997)]. However the emphasis of this work is on the understanding of control mechanisms rather than practical implementation. To have knowledge of the best performance that can be achieved is also of great practical interest since no practical controller will provide a better performance than one in the idealised case.

113

Chapter 5. Wiener Filter in SISO Systems

d (t )

H(s)

f s (t )

Gs (s )

+ e(t )

y (t )

+

+

) Gs (s )

-

-C(s) (a) Internal model control

d (t )

f p (t )

G p (s )

Gs (s )

r (t )

H(s)

y (t )

+ e(t )

+ Wiener filtering

) (b) Equivalent block diagram if Gs ( s ) = Gs ( s )

Figure 5.8 Internal model control

A pure time delay is assumed in the secondary path for the electric control process as we did in feedforward control, and it can be transferred to the primary path as the time advance term

e sτ . When the primary force is white noise, we can replace it by the unit impulse function and the problem becomes active control of an impulse response. The whole block diagram is shown in Figure 5.8(b), and the only one difference from the feedforward control block diagram is the location of the primary plant. In feedforward control, only the secondary plant is considered for constructing the inverse model inside the Wiener filter to make the resultant signal white noise, whereas in feedback control both primary and secondary plants should be considered for the inverse model. To obtain a simple analytical expression of the Wiener filter and the mean square error, we thus assume the primary as well as the secondary plants are minimum phase. Wiener filter design for general non-minimum phase plants can be solved numerically in Section 5.3.3, and its influence on the performance will be discussed in Section 5.5.

114

Chapter 5. Wiener Filter in SISO Systems

d (t ) = g p (t1 + τ )

e sτ

δ (t )

G p (s )

r (t )

Gs (s )

H(s)

y (t )

+ e(t )

+ Wiener filtering

Figure 5.9 Internal model control of the transient impulse response with a pure time advance on the primary path

5.3.2.2 Analogue control Under the assumption of minimum phase in both primary and secondary plants, the analysis is straightforward and similar to the feedforward control case. Using the inverse model method as shown in Figure 5.10, the Wiener filter can be written as

H o ( s) =

1 H o′ ( s ) G p ( s )Gs ( s )

(5.38)

where the plants G p ( s ) and Gs ( s ) are minimum phase so that G p ( s ) = G p+ ( s ) and Gs ( s ) = Gs+ ( s ) . The input to the Wiener filter r (t ) can be expressed as the convolution

integral of g p (t ) and g s (t ) i.e., r (t ) = g p (t ) ∗ g s (t ) . Due to the assumption of minimum phase in the primary and secondary plants, the signal after the inverse system is an impulse. Thus H o′ (s ) is the same as in equation (5.28) for feedforward control, and the Wiener filter can be written as H o ( s) = −

[L (g

p

(t1 + τ ) )]+

G p ( s )Gs ( s )

115

(5.39)

Chapter 5. Wiener Filter in SISO Systems

d (t ) = g p (t1 + τ )

H o (s ) r (t ) = g p (t ) ∗ g s (t )

1 G p ( s )Gs ( s )

δ (t )

H o′ (s )

y (t )

e(t )

Figure 5.10 Wiener filter using the inverse model

If we compare the block diagram in Figure 5.10 with that for feedforward control given in Figure 5.4(b), both sub-Wiener filters H o′ (s ) do the same work; that is to estimate the future signal of the impulse response of the primary plant. Thus, the mean square error after control is the same as that for feedforward control given in equation (5.30). It is because the problem again becomes that of finding the sub-Wiener filter H o′ (s ) which minimises the error in the mean square sense with an impulse as the received signal. Thus the same conclusion that the time delay τ is the most important factor for control performance can be drawn for feedback control using IMC.

It is interesting to investigate the control performance when the time delay is zero. The mean square error for no time delay will be zero from equation (5.30). Thus, perfect control of a disturbance seems to be achieved under the unlikely situation of the minimum phase primary and secondary plants with the idealised assumption of perfect modelling of the secondary plant. However, a problem occurs for the construction of the physical analogue controller − C (s ) in equation (5.36). The Wiener filter for this case is the inverse system of the plant

Gs (s ) i.e., H o ( s ) = −1 Gs ( s ) since G p ( s ) = [L (g p (t1 + τ ) )]+ in equation (5.39). If we substitute it into equation (5.36), it turns out the denominator becomes zero. Thus perfect control using feedback control is not possible as pointed out by [p.47-50 in Morari and Zafiriou(1989)]. Nevertheless, it is of great theoretical and practical interest to know the ideal solution from the point of view of how closely it can be approached in real situations.

116

Chapter 5. Wiener Filter in SISO Systems

5.3.2.3 Digital control of minimum phase systems

Again minimum phase is assumed for the primary and secondary plants, thus analysis for the digital controller is straightforward from the knowledge of analogue feedback control and digital feedforward control. An analogous block diagram to the analogue controller in Figure 5.10 can be easily drawn and so is omitted here. From equation (5.38) the Wiener filter is given by

Ho ( z) =

1 H o′ ( z ) G p ( z )Gs ( z )

(5.40)

and using equation (5.31) it is rewritten as

Ho

[z (z) = −

Δ

]

G p ( z) +

G p ( z )Gs ( z )

(5.41)

The mean square error after control is the same as equation (5.34) for feedforward control. When the sample delay Δ = 0 , we can again see that perfect control is not possible.

Important results obtained from feedforward and feedback techniques of analogue and digital controllers are tabulated in Table 5.2 for the minimum phase primary and secondary plants, G p ( s ) and Gs (s ) . As can be seen, only G p ( s ) term is additionally added to the denominator

of the Wiener filter for feedback control compared with that for feedforward control. Both techniques result in the same minimum mean square error, but note that the optimal feedback controller with no time delay is unstable as discussed in the previous section.

117

Chapter 5. Wiener Filter in SISO Systems

Table 5.2 Wiener filter and the minimum mean square error in feedforward and feedback optimal controllers for minimum phase G p ( s ) and Gs (s ) when G p ( s ) is excited by white noise.

Analogue Wiener filter Feedforward

H o ( s) = −

[L (g

p

Digital Wiener filter

(t1 + τ ) )]+

Gs ( s )

(5.29) Feedback

[z G ( z )] Δ

Ho ( z) = −

p

+

G p ( z)

(5.32)

[L (g H ( s) = − o

p

(t1 + τ ) )]+

G p ( s )Gs ( s )

[z G ( z )] H ( z) = − Δ

(5.39)

where τ is a time delay

p

o

+

G p ( z )Gs ( z )

(5.41)

where Δ is the number of samples delayed

Minimum mean square error





J o = ∫ g 2p (t )dt − ∫ g 2p (t )dt 0

τ

(5.30)

J

J

j =0

j =Δ

J o = ∑ b 2j − ∑ b 2j

(5.34)

5.3.3 Numerical Analysis Model for Digital Control

The analytical models for analogue and digital controllers discussed earlier are useful for obtaining physical insight into the control mechanism involved with the Wiener filter. However the analytical procedure is quite tedious and frequently impossible to solve, especially when the plant is non-minimum phase or is expressed as a high order polynomial rational function. As a practical method of solving discrete type Wiener filter problems for general minimum and non-minimum phase plants, the numerical method discussed in Section 5.2.3 can be employed. To construct the matrix equation for the discrete type of Wiener-Hopf equation given in equations (5.15) and (5.16), only the correlation functions Rrr (n) and Rrd (n) are required. In the following description, they are obtained from the inverse Fourier transform of each power spectrum functions of r(n) and d(n) are required.

118

for which the frequency response

Chapter 5. Wiener Filter in SISO Systems

Feedforward control of a SISO system subject to a white noise primary force input is considered first, and its block diagram is shown in Figure 5.4(a). The discrete Fourier transform of desired and received signals for the Wiener filter in Figure 5.4(b) are D(e jω ) = G p (e jω ) ,

R(e jω ) = Gs (e jω )Δ(e jω )

(5.42)

where the functions Gs (e jω ) , G p (e jω ) , and Δ(e jω ) are the Fourier transform of a sampled signals, g s (n) , g p (n) , and the Δ-samples delay function δ (n − Δ) , respectively. The sampling period dependency is omitted for brevity, otherwise when the sampling period is T, the term is e jωT instead of e jω for the frequency domain and nT instead of n for the time domain can be used[Nelson and Elliott(1992)]. Unlike the block diagram in Figure 5.4(a), the delay term Δ(e jω ) in equation (5.42) is on the secondary path. The result is the same in both cases, however, it is used for the convenience of numerical analysis since it can be easily generated by the simple Fourier transform of the Δ-samples delay function δ (n − Δ) .

For the feedback control case shown in Figure 5.9, the discrete Fourier transform of desired and received signals for the Wiener filter are given by D(e jω ) = G p (e jω ) ,

R(e jω ) = G p (e jω )Gs (e jω )Δ(e jω )

(5.43)

Using the systematic approach in linear systems and signals theory, the difference between feedforward and feedback control are clearly seen in equations (5.42) and (5.43). Note the desired signals are the same in both cases, but there is an additional term G p (e jω ) in the received signal for feedback control.

The discrete Fourier transform of the error signal is the addition of the desired and estimated signals, and is given by E (e jω ) = D( e jω ) + Y (e jω )

119

(5.44)

Chapter 5. Wiener Filter in SISO Systems

The discrete power spectra required are given by

[

]

[

E R * (e jω ) ⋅ R (e jω ) E R * (e jω ) ⋅ D(e jω ) jω S rr (e ) = , S rd (e ) = Fs2 Fs2 jω

where the sign

*

]

(5.45)

denotes the complex conjugate of the signal, and Fs denotes the sampling

frequency for discrete data. The spectra are divided by the square of the sampling frequency to avoid their amplitude dependency on the sampling frequency. The unconstrained Wiener filter can be directly calculated from equation (5.8) using the spectrum expressions.

The auto- and cross-correlation functions can be obtained by taking the inverse Fourier transforms of the power spectra in equation (5.45), and thus the matrix and vectors required can be constructed from equation (5.16). By Parseval’s theorem the mean square value can be obtained from the power spectrum[p.134 in Bendat and Piersol(1986)], and the normalised performance can be written as

J J ′ = o = 1− Ji

∫ ∫



−∞ ∞

−∞

S ee (e jω )dω S dd (e jω )dω

(5.46)

where S dd (e jω ) and S ee (e jω ) denote the discrete power spectra of the desired and error signals. The power spectrum of the error signal can be obtained from the relation in equation (5.45) with the error signal given in equation (5.44).

It should be noted that, if we use the deterministic approach by simply replacing white noise input by an impulse, no time consuming averaging process in equation (5.45) is required. A numerical validation of this replacement is considered in Section 5.4.2.

120

Chapter 5. Wiener Filter in SISO Systems

5.4 Active Control of a Minimum Phase System The system considered here is assumed to be LTI and minimum phase so that its realisable inverse system equals the inverse of the original system in the Laplace or z-domains. Although physical systems are generally non-minimum phase, it simplifies the analytical processes a great deal, which may provide better insight into the control mechanisms. Both feedforward and feedback approaches for the simple SISO minimum phase system are considered, and their analytical and numerical results are presented.

5.4.1 General Description of the System

A one d.o.f(degree-of-freedom) system under control is shown in Figure 5.11 where m, k, and c are the mass, spring constant, and damping of the system, respectively. The system is excited by the primary force fp(t) which is assumed to be white noise with zero mean and variance σ w2 = 1 , and the velocity response of the system measured by the error signal e(t) is aimed to be minimised in the mean square sense by using the secondary force actuator fs(t). Since the primary force signal is assumed to be measurable in feedforward control, this signal is fed to the controller H(s) to control the secondary force actuator fs(t), whereas in feedback control the error signal is used to control the actuator fs(t) via the controller − C (s ) .

Error Signal e(t ) m

Error Signal e(t ) m

f s (t )

f p (t )

k

c

− C (s )

f p (t )

f s (t )

k

e − sτ

c

H (s )

(a) Feedforward control

(b) Feedback control

Figure 5.11 Active control of a 1 d.o.f system

121

Chapter 5. Wiener Filter in SISO Systems

The plant is single d.o.f so that the primary and secondary plants are the same, and is denoted by G (s ) . The velocity response to an impulse force is given by [Tse et al(1978)]

g (t ) = −

ω n −ςω t e sin (ω d t − ϕ ), mω d n

t≥0

(5.47)

where the damped natural frequency ω d = ω n 1 − ς 2 , the natural frequency ω n = k m , the damping coefficient ς =

1−ς 2 c , and the phase angle ϕ = tan −1 . ς 2 mk

Its Laplace transform is

G (s) =

1 s ⋅ 2 m s + 2ςω n s + ω n2

(5.48)

The physical plant in continuous time is causal, stable, and minimum phase with one zero at the origin of the s-plane and poles at (−ςω n + jω d ) and (−ςω n − jω d ) as a conjugate pair.

Using the impulse invariant transform[Blinchikoff and Zverev(1976)], the z-domain representation of the impulse response can be written as[Ogata(1970)]

G( z) = −

(

ω n z − z 1 − ς 2 + e −ςω T sin(ω d T + ϕ ) ⋅ mω d z 2 − 2 ze −ςω T cos ω d T + e − 2ςω T n

n

n

)

(5.49)

where Δ is the sampling period.

For the simulation of this section, the physical parameters of the one d.o.f plant are set to be mass m = 1 kg , the natural frequency f n = 200 Hz where f n = ω n 2π , and the damping ratio

ζ = 0.05 . For the numerical solution in the discrete time domain using equation (5.15), the sampling period is T = 10 −3 s , the plant is modelled by a J-length FIR filter where J = 1024 , and the controller consists of a I-length FIR filter where I = 64 . The velocity response due to an impulse force is shown in Figure 5.12(a). From the pole-zero plots in the z-plane shown in

122

Chapter 5. Wiener Filter in SISO Systems

Figure 5.12(b) where the sign ‘×’ denotes poles and ‘Ο’ denotes zeros, the plant is assured to be minimum phase.

Feedforward and feedback control of random vibration excited by white noise are considered in the one d.o.f plant which may be the simplest possible example, and for the analytical solution the theory in Section 5.3 is fully utilised.

1.2

1 0.8

0.8

0.6

0.6

0.4

Imaginary part

Velocity(m/s)

1

0.4 0.2 0

0.2 0 −0.2 −0.4

−0.2

−0.6 −0.4

−0.8 −0.6

−1 −0.8 0

0.2

0.4

0.6 Time(s)

0.8

1

−1

1.2

(a)

−0.5

0 Real part

0.5

1

(b)

Figure 5.12 The impulse response (a) and the pole-zero plot in z-plane (b) of the 1 d.o.f system

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Chapter 5. Wiener Filter in SISO Systems

5.4.2 Feedforward Control

5.4.2.1 Analogue controller

As discussed in Section 5.3.1, white noise can be equivalently replaced by an impulse function. The plant in both paths is the same, so we bring it to the common path as shown in Figure 5.13(a). The problem is to design a Wiener filter which will optimally predict the future signal of its input signal r (t ) so to minimise the error in mean square sense. Since the received signal is the impulse response of a minimum phase system, the solution procedure is straightforward and solved by the inverse model method as shown in Figure 5.13(b).

r (t )

e sτ d (t )

δ (t )

y (t )

r (t )

e(t )

H(s)

G (s )

Wiener Filtering (a) Feed forward controller subject to white noise input

d (t ) = g (t + τ )

H o (s )

r (t ) = g (t )

1 G (s)

δ (t )

H o′ (s )

y (t )

e(t )

(b) Feed forward controller subject to the unit impulse input

Figure 5.13 Feedforward control of the 1 d.o.f plant

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Chapter 5. Wiener Filter in SISO Systems

Following the procedure in Section 5.3.1.1, the optimal controller H o (s) in equation (5.26) is given by H o ( s) =

1 H o′ ( s ) G( s)

(5.50)

Since the signal after the inverse model is a perfect impulse and ho′ (t ) = d (t ) , from equation (5.47) ho′ (t ) is written as

ho′ (t ) =

ω n −ςω e mω d

n

( t +τ o )

sin(ω d (t + τ o ) − ϕ ),

t≥0

(5.51)

The Laplace transform of the causal part of the equation is not simple since it is time shifted to the negative time axis. Through detailed algebra shown in Appendix 5A, the Laplace domain representation is given by

H o′ ( s ) =

1⎞ ⎛ 2 ⎜ − g (τ ) + ω n g d (τ ) ⎟ ( s + ςω n ) + ω ⎝ s⎠ s

2

(5.52)

2 d

where g (τ ) and g d (τ ) are the velocity and displacement responses at time τ respectively due the impulse force, and are given by g d (τ ) =

g (τ ) = −

ω n −ςω τ e sin (ω dτ − ϕ ) and mω d n

1 −ςω nτ e sin (ω dτ ) . mω d

Combining equations (5.48),(5.50), and (5,52), then the Wiener filter is

H o ( s ) = − mg (τ ) + mω n2 g d (τ )

1 s

(5.53)

where the controller is constructed by both proportional and integral operators, and its inverse Laplace transform gives ho (t ) = − mg (τ )δ (t ) + mω n2 g d (τ )

125

(5.54)

Chapter 5. Wiener Filter in SISO Systems

As can be expected, when τ = 0 , H o ( s ) = −1 and ho (t ) = −δ (t ) that means the secondary force is acting with the opposite sign of the primary signal. In this trivial case, the error becomes exactly zero. Using equations (5.51) and (5.30) through the integral calculation in Appendix 5B, the normalised minimum mean square error is given by

J′ =

⎞ ⎛ 2ςω n2 Jo = 1 − e − 2ςω nτ ⎜⎜1 + sin (ω d τ )sin (ω d τ − ϕ )⎟⎟ 2 Ji ωd ⎠ ⎝

(5.55)

The first term e −2ςω nτ dominates the performance, and the second term inside the brackets shows oscillation. Introducing the normalised delay to the natural period τ ′ = τ Tn where the period of the impulse response Tn = 2π ω n and using the relation ω d = ω n 1 − ς 2 , it can be rewritten as ⎛ ⎞ 2ς J ′ = 1 − e − 4πςτ ′ ⎜⎜1 + sin(2π 1 − ς 2 τ ′) sin(2π 1 − ς 2 τ ′ − ϕ ) ⎟⎟ 2 ⎝ 1−ς ⎠

(5.56a)

Figure 5.14 shows the normalised mean square error J ′ according to the normalised time delay τ ′ = τ Tn and damping ratio. It is interesting to note that perfect control is achievable for the system with no damping, regardless of the delay. It is because the impulse response of this system is of a sinusoidal signal lasting forever, i.e., a harmonic wave. As can be expected, the oscillation term only has a small affect on the performance. If we ignore this term, the above equation can be simplified to J ′ ≅ 1 − e −4πςτ ′

(5.56b)

The equation is a function of the multiplication of the normalised delay and damping ratio

ςτ ′ , and thus from the control performance point of view the effect of each term is the same. For example, the performance due to doubling the value ς is the same as due to doubling the value τ ′ . Thus, it can be said for a given system with fixed damping, the normalised time

126

Chapter 5. Wiener Filter in SISO Systems

delay τ ′ determines the control performance as we have drawn from the general analysis in Section 5.3.

Normalised mean square error

1 0.8 0.6 0.4 0.2 0 0.1 0.08

10 0.06

8 6

0.04

4

0.02 Damping ratio

2 0

0

Normalised time delay

Figure 5.14 The normalised mean square error J ′ according to the normalised time delay τ ′ = τ Tn and damping ratio.

5.4.2.2 Digital controller

The digital Wiener filter can be obtained by following the analytical procedure described from equation (5.31) to (5.34), however its analytical expression is rather long and complicated to describe. Thus to the system given by equation (5.49) we use the numerical method discussed in Section 5.3.3 whose results are the same as for the analytical method. The discrete impulse response is obtained from the z-domain representation of impulse response in equation (5.49) with the sampling period T = 10 −3 s . From the impulse response modelled with the 1024-length FIR filter, its frequency response is calculated by the discrete Fourier transform and the result is used for constructing the input and desired signals in equation (5.42). The delay term Δ(e jω ) is generated by the discrete Fourier transform of the Δ-samples delay function δ (n − Δ) . The Fast Fourier Transform (FFT) algorithm is widely used for such transform. After the auto- and cross-spectrum functions in equation (5.45) are 127

Chapter 5. Wiener Filter in SISO Systems

calculated, they are transformed back to the time domain and construct the elements for the matrix equation given in equation (5.17). The error signal can be calculated after the convolution between the input signal and the Wiener filter coefficients, and its result is used for the power spectrum of error in equation (5.46).

Using the numerical simulation procedure, some results for various sample delays are shown in Figure 5.15 where the vibration amplitudes before and after control, and the Wiener filter when Δ = 1 is shown in Figure 5.15(a) and (b), respectively. If there is no sample delay i.e., Δ = 0 , from equation (5.33) the desired and received signals are exactly same in this case so

that the controller will be act as an sign inverter and a perfect error control is achieved. From Figure 5.15(b), it shows a FIR Wiener filter with less than 10-length is satisfactory for the control of system with the sample delay Δ = 1 . The mean square error J ′ can be calculated from the vibration response in the frequency domain in Figure 5.15(a) using equation (5.46). It is simulated according to the increase of the sample delay and its result is shown in Figure 5.16 where the solid line is obtained from the numerical simulation and the dashed line is from the approximate solution in equation (5.56b). Good agreement can be seen between the two graphs, and it demonstrates that a better performance can be achieved when there is a smaller sample delay in the digital control processing.

140 0.8

135 0.6

125

0.4

120

Filter coeficients

Velocity amplitude(dB)

130

delay=1

115

delay=30

0.2

0

110 −0.2

105 delay=5 100 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

−0.4 0

10

20

30

40

50

60

70

Samples

(b) the Wiener filter when Δ = 1

(a) vibration amplitude(dB ref.=10-9 m/s)

Figure 5.15 Vibration responses before(solid line) and after feedforward control(dashed line) according to the sample delay Δ = 1, 5, and 30 in (a), and the Wiener filter when Δ = 1 in (b).

128

Chapter 5. Wiener Filter in SISO Systems

1 0.9

Normalised mean square error

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

10

20

30 40 delay(number of samples)

50

60

70

Figure 5.16 The normalised mean square error J ′ according to the sample delay where the solid line is obtained from the numerical solution and the dashed line is from the approximate solution in equation (5.56b).

In order to prove the validity of using an impulse instead of white noise, we use a generated white noise as the input force for the system with the sample delay Δ = 1 and its results are compared. Since white noise is an idealised process which is difficult to generate in practice, a white-like noise with zero mean and unit variance is generated with 10000 samples by a random number generator. The simulation results are shown in Figure 5.17 where vibration responses before and after control are shown in both the frequency and time domains. For estimating the power spectrum in Figure 5.17(a), the Welch's averaged periodogram method with Hanning window is used[Krauss et al(1994) and Kay(1988)]. Except some noise effects due to not having the perfect white noise, good agreement in the filter coefficients of the Wiener filter as well as the performance has been achieved compared with the results in Figure 5.15 obtained from an impulse input. It demonstrates that the utility of an impulse is parallel to that of white noise for the active control of linear systems. For the theoretical study presented here white noise is assumed as the system input and it is replaced by an impulse so that a complicated statistical averaging process is not necessary. However, inputs to mechanical systems are not generally white noise, and thus in practice the statistical approach is required to design a Wiener filter for a real implementation.

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Chapter 5. Wiener Filter in SISO Systems

140 0.8

135 0.6

125

0.4

Filter coeficients

Velocity amplitude(dB)

130

120

115

0.2

0

110 −0.2

105

100 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

−0.4 0

500

10

6

6

4

4

2

2

Error signal(m/s)

Desired signal(m/s)

8

0

−2

−6

−6

2 Time(s)

2.5

60

70

−2

−4

1.5

50

0

−4

1

40

(b) the Wiener filter

8

0.5

30 Samples

(a) Velocity (dB ref.=10-9 m/s)

−8 0

20

3

3.5

(c) before control in the time domain

4

−8 0

0.5

1

1.5

2 Time(s)

2.5

3

3.5

4

(d) after control in the time domain

Figure 5.17 Vibration response before(solid line) and after feedforward control(dashed line) with the sample delay Δ = 1 when the input is white-like noise instead of the unit impulse.

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Chapter 5. Wiener Filter in SISO Systems

5.4.3 Feedback Control

5.4.3.1 Analogue controller

Feedback control using IMC is considered in this section for the one d.o.f plant shown in Figure 5.11(b). Analogue control is considered first, and its analytical expressions can be developed from the theory in Section 5.3.2. Since the primary and secondary plants in this case are the same and minimum phase, equation (5.38) can be rewritten as

H o ( s) =

1 H o′ ( s ) G ( s)

(5.57)

2

Because the received signal to the sub-Wiener filter in Figure 5.10 is an impulse, the impulse response ho′ (t ) = d (t ) , and thus is the same as for the feedforward control in equation (5.51). Substituting the sub-Wiener filter H o′ (s ) in equation (5.52) and the plant G ( s ) in equation (5.48) into equation (5.57), the Wiener filter is

H o ( s ) = A1s + A0 + A−1

where

the

constants

A1 = − m 2 g (τ )

1 1 + A−2 2 s s

,

(5.58)

(

A0 = m 2 ω n2 g d (τ ) − 2ςω n g (τ )

)

,

A−1 = m 2 (2ςω n3 g d (τ ) − ω n2 g (τ ) ) , and A− 2 = m 2ω n4 g d (τ ) , and its impulse response is

ho (t ) = A1δ ′(t ) + A0δ (t ) + A−1 + A−2t

(5.59)

where the unit doublet function δ ′(t ) is a singularity function and defined at t = 0 . As discussed in Section 5.3.2.2, the controller for the delay τ = 0 cannot be realised exactly since it makes the denominator of equation (5.36) zero. Since the plant is minimum phase, the normalised minimum mean square error for feedback control is the same as given in equation (5.55).

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Chapter 5. Wiener Filter in SISO Systems

5.4.3.2 Digital controller

Now, the digital Wiener filter is calculated by the practical numerical method presented in Section 5.3.3. The same procedure under the same condition explained in the last section for feedforward control are used, but with the different input frequency response R(e jω ) = G p (e jω )Gs (e jω )Δ(e jω ) in equation (5.43). Figure 5.18(a) shows vibration responses before and after control according to the delay Δ = 1, 5, and 30 , and the Wiener filter when Δ = 1 is shown in Figure 5.18(b). It shows the frequency distribution of the mean square error

after control is the same as those in Figure 5.15 for feedforward control. Due to the difference of the inverse models, however, the Wiener filter is different from that for feedforward control. The normalised mean square error according to the sample delay is also the same as the feedforward control case, so is omitted.

140

1000

135

800

130

125

120

Filter coeficients

Velocity amplitude(dB)

600

delay=1

115

400

200

delay=30 0

110 −200

105 delay=5 100 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

−400 0

10

20

30

40

50

60

70

Samples

(b) the Wiener filter when Δ = 1

(a) vibration amplitude(dB ref.=10-9 m/s)

Figure 5.18 Vibration responses before(solid line) and after feedback control(dashed line) according to the sample delay Δ = 1, 5, and 30 in (a), and the Wiener filter when Δ = 1 in (b).

132

Chapter 5. Wiener Filter in SISO Systems

Throughout the section, only minimum phase plants have been considered and the influence of the electric time delay to the performance has been investigated. In minimum phase plants, the electric time delay for signal processing is the factor determining the performance and should be avoided as much as possible, regardless of feedforward or feedback control. This fact can be taken granted since the assumption of minimum phase in mechanical plants means no time delay is taken for transmitting a wave from the input position to the output position. In the next section, general non-minimum phase plants are considered.

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Chapter 5. Wiener Filter in SISO Systems

5.5 Active control of non-minimum phase systems

5.5.1 Phase angle difference function

Regardless of whether feedforward or feedback technique are used to control minimum or non-minimum phase plants, it is a Wiener filter design problem and its performance is determined by the inputs of the filter; the desired and received signals. Suppose they are impulse responses of arbitrary plants where as an easy example, the desired signal travels faster than the received signal at all frequencies, it is a prediction problem and the performance is dependent on how much faster the desired signal is. If the desired signal is the same or slower than the received signal at all frequencies, it becomes a filtering problem and perfect filtering can be achieved theoretically. However, such a simple discussion is only possible for non-dispersive systems because in dispersive systems wave speeds depend on frequencies and show complicated non-linear phase characteristics[Cremer and Heckl(1987)].

Thus, the phase angle difference function is used to show the wave speed difference between the primary and secondary systems which is defined as Θ(e jω ) = ∠D(e jω ) − ∠R(e jω )

(5.60)

where the variable e jω denotes frequency dependency of sampled signals and ∠ denotes the phase angle. From equation (5.42) and (5.43) the discrete Fourier transforms of desired and received signals are D(e jω ) = G p (e jω ) , R(e jω ) = Gs (e jω )Δ(e jω )

in feedforward control

D(e jω ) = G p (e jω ) , R(e jω ) = G p (e jω )Gs (e jω )Δ(e jω ) in feedback control

In feedforward control the relative phase angle of the secondary plant is compared with that of the primary plant and determines the phase angle difference function. Whereas in feedback control, equation (5.60) can be simplified to

134

Chapter 5. Wiener Filter in SISO Systems

Θ(e jω ) = −∠Gs (e jω )Δ (e jω )

(5.61)

The primary plant G p (e jω ) is not shown in the above equation, and the phase difference function is solely determined by the absolute phase angle of the secondary plant, when there is no electric time delay.

Since control performance of a random sound controller mainly depends on the time delay whether it is time taken to pass through electrical or mechanical plants, the phase angle difference function showing the time delay can be used as an indirect measure of control performance. The effects of the phase angle difference function on control performance is investigated in this section with using the one d.o.f system as an example.

5.5.2 Examples

5.5.2.1 Minimum phase system

The mobility of the one d.o.f system in the discrete time domain is minimum phase as known from the pole-zero plot shown in Figure 5.12. We re-examine the system considering the phase angle difference function described in equation (5.60). Figure 5.19 shows the phase angle difference function of the desired and received signals according to the electric time delay in both feedforward and feedback control. In feedforward control with zero delay shown in Figure 5.19(a), the input signals are the same so that perfect control is achieved. As the electric time delay increases, the Wiener filter needs to further predict the desired signal and the control performance deteriorates as shown in Figure 5.15.

Figure 5.19(b) shows the phase angle difference plot for feedback control. Like feedforward control, it shows the same trend with increasing electric time delay except for the non-linear phase characteristics. As the prediction amount increases, the performance shown in Figure 5.18 gets worse.

135

Chapter 5. Wiener Filter in SISO Systems

14

14

12

12

delay=5

delay=5 10 Angle(Radian)

Angle(Radian)

10 8 6 4

8 6 4

2

delay=1

2

delay=1

0

0 delay=0

delay=0 −2 0

−2 50

100

150

200 250 300 Frequency(Hz)

350

400

450

(a) Feedforward control

500

0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

(b) Feedback control

Figure 5.19 Phase angle difference functions in feedforward and feedback control of the mobility according to the electric time delay

5.5.2.2 Non-minimum phase system

Consider again the one d.o.f plant of mass, spring, and damper system in Section 5.4. Its receptance, mobility, and inertance in the Laplace and z-domains are tabulated in Table 5C.1. In the Laplace domain these frequency response functions are all minimum phase since no zero is located in the right half plane. However, it is interesting that the receptance and inertance in the discrete time domain are non-minimum phase as can be seen from their polezero plots shown in Figure 5.20. It is because the receptance has lower order of zeros than poles, and the inertance has one zero outside the unit circle. Digital control working in the zdomain of the receptance and inertance is considered next.

136

Chapter 5. Wiener Filter in SISO Systems

4

1 0.8

3

0.6

2

0.2

Imaginary part

Imaginary part

0.4

0 −0.2

1

0

−1

−0.4

−2

−0.6 −0.8

−3

−1 −1

−0.5

0 Real part

0.5

(a) Receptance

1

−4 −9

−8

−7

−6

−5

−4 −3 Real part

−2

−1

0

1

(b) Inertance

Figure 5.20 Pole-zero plots for the receptance and inertance of the one d.o.f system

In feedforward control of the receptance and inertance of the one d.o.f system, the phase angle difference function shows the same linear phase graph as in Figure 5.19(a). It is because the desired and received signals are the same in the one d.o.f since G p ( z ) = Gs ( z ) . Thus their performances are similar to the performance plot of mobility control in Figure 5.16 which starts from a perfect control at zero delay. However as the general case if the primary and secondary plants are different, G p ( z ) ≠ Gs ( z ) , the performance is affected by the phase difference of the two signals. Feedback control of displacement is shown in Figure 5.21 where the phase angle difference function and performance are shown according to the electric sample delay in (a) and (b), respectively. Even when there is no sample delay, the phase difference is non-minimum phase and thus perfect minimisation cannot be achieved. In feedback control of acceleration, its phase difference functions shown in Figure 5.22(a) give the same trend except its starting point at zero frequency. As the delay increase, we also see the slope of graph increases. Control performance according to the zero, one, five samples delay is shown in Figure 5.22(b). From the discussion, in feedforward control the phase angle difference between the primary and secondary plant transfer functions determines the control performance, whereas, only the secondary plant is important in feedback control. For better performance with both 137

Chapter 5. Wiener Filter in SISO Systems

feedforward and feedback control, the electric processing time delay should be avoided as much as possible. Note the phase angle difference function is not a direct measure of performance, and its numerical algorithm can be fooled by sparse, rapidly changing phase values[Ch. 12 in Oppenheim and Schafer(1989) and the command ‘unwrap’ in Krauss et al(1994)]. The graph calculated from either theoretically or experimental data allows us to approximate the wave propagation time delay of mechanical plants. This method would be helpful for searching the location of the control source before a real controller is implemented.

140

14 delay=5

12

130 delay=5

Displacement amplitude(dB)

Angle(Radian)

10 8 6 delay=1

4

delay=0

2 0

delay=1 120

delay=0

110

100

90

−2 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

80 0

500

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

(b) Performance(dB ref.=10-12 m)

(a) Phase difference

Figure 5.21 Performance and phase difference of displacement feedback control according to the time delay

145

delay=5

14

140

12

135

8

Acceleration amplitude(dB)

Angle(Radian)

10

delay=1

6 delay=0 4 2

130

delay=5 delay=1

125 120 delay=0 115 110 105

0

100

−2 0

50

100

150

200 250 300 Frequency(Hz)

(a) Phase difference

350

400

450

500

95 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

(b) Performance(dB ref.=10-6 m/s2)

Figure 5.22 Performance and phase difference of acceleration feedback control according to the time delay 138

Chapter 5. Wiener Filter in SISO Systems

5.6 Conclusion The chapter has considered the active control of stationary random sound and vibration in SISO systems. An easy systematic deterministic method for the active control of random sound was proposed based on two facts. Firstly, the Wiener filter with the constraint of causality is the optimal controller of stationary random sound, and secondly, white noise is equivalently replaced by an impulse. Thus, as long as the input primary source is white noise, the active control of stationary random sound transforms to the active control of an impulse response.

Both feedforward and feedback techniques have been applied for the optimal control of SISO systems. To obtain simple analytical expressions for the Wiener filter and the mean square error, the secondary plant is assumed to be minimum phase in feedforward control, and both primary and secondary plants are assumed to be minimum phase in feedback control. The proposed deterministic method has been applied to the vibration control of a one d.o.f system as an application example. Conclusions obtained from theoretical and numerical analysis in this chapter are 1. It has been shown with a numerical example that white noise can be equivalently replaced by an impulse for the purpose of active control of random sound. The advantages of the replacement are; Firstly, an easy systematic deterministic approach can be applied to the optimal control of stochastic signals. Secondly, the calculation time is greatly reduced since no time consuming averaging process is required. 2. When a feedforward controller has zero time delay, the realisable filter for random sound is the same as the causally unconstrained Wiener filter. 3. Perfect control can be achieved in feedforward control, whereas in feedback control it can never be achieved due to stability problems. 4. The control performances in both feedforward and feedback control are adulterated as the electric time delay and system damping increase(see Figure 5.14). 5. The phase angle difference function suggested can be used as a practical way of predicting control performance. In the active control of SISO systems, a bigger function to the plus angle(phase lead) brings a poorer control performance.

139

Chapter 5. Wiener Filter in SISO Systems

Appendix 5A. Calculation of the sub-Wiener filter The impulse response ho′ (t ) of the sub-Wiener filter can be rewritten as

ho′ (t ) =

ω n −ςω τ −ςω t e e sin (ω d t + ω dτ − ϕ ), mω d n

n

t≥0

(5A.1)

From the theorem sin( A + B ) = sin A cos B + cos A sin B , where A and B are angles, the above equation is again rewritten as

ho′ (t ) =

ω n −ςω τ −ςω t {sin(ω d t ) cos(ω dτ − ϕ ) + cos(ω d t ) sin(ω dτ − ϕ )}, t ≥ 0 (5A.2) e e mω d n

n

The trigonometric functions of angle ϕ are easily calculated from Figure 5A1.

ωn ϕ

ωd

ςω n

Figure 5A.1 A right triangle for calculating trigonometric functions of angle ϕ

Using the one-sided Laplace transforms which start integration from zero instead of minus infinity, we get the relationships[Spiegel(1968)]





0

e − at sin(bt )e − st dt =

b , (s + a)2 + b2

where a and b are arbitrary constants. 140





0

e − at cos(bt )e − st dt =

s+a (5A.3) (s + a)2 + b2

Chapter 5. Wiener Filter in SISO Systems

Thus, the Laplace transform of ho′ (t ) can be written as H o′ ( s ) =

1 2 ⎛ ⎞ ⎜ − g (τ ) + ω n g d (τ ) ⎟ ( s + ςω n ) + ω ⎝ s ⎠ s

2

2 d

(5A.4)

where the impulse velocity g (τ ) and displacement g d (τ ) responses at time τ are [Tse et al(1978)]

g (τ ) = −

ω n −ςω τ 1 −ςω τ e sin (ω dτ − ϕ ) , g d (τ ) = e sin (ω dτ ) mω d mω d n

n

141

(5A.5)

Chapter 5. Wiener Filter in SISO Systems

Appendix 5B. Calculation of the minimum mean square error From equation (5.30), the minimum mean square error for the feedforward control of the one d.o.f system is given by ∞

g 2 (t )dt ∫ Jo τo = 1− ∞ J1 g 2 (t )dt

(5B.1)



0

and from equation (5.47) the square of impulse response function can be written as

⎛ ω g (t ) = ⎜⎜ n ⎝ mω d 2

Set

t − ϕ ω d = τ1 ,

⎛ ω A = ⎜⎜ n ⎝ mω d

then

it

can

2

⎞ −2ςω nt 2 ⎟⎟ e sin (ω d t − ϕ ) ⎠

be

written

as

g 2 (τ 1 ) = Ae aτ1 sin 2 (bτ 1 )

(5B.2)

where

2

⎞ 2ςω n ϕ ω d ⎟⎟ e , a = −2ςω n , and b = ω d .Thus, equation (5B.1) can be written as ⎠



g 2 (τ 1 )dτ 1 ∫ Jo τ o −ϕ ω d = 1− ∞ J1 g 2 (τ )dτ

∫ϕ ω −

1

(5B.3)

1

d

Using the relationship[Spiegel(1968)]

at 2 ∫ Ae sin btdt =

⎛ at A 2b 2 at ⎞ ⎜ ( ) e bt a bt b bt sin sin 2 cos − + e ⎟⎟ a 2 + 4b 2 ⎜⎝ a ⎠

We thus get

142

(5B.4)

Chapter 5. Wiener Filter in SISO Systems

⎛ 2ςω n2 ⎞ Jo sin (ω dτ ) sin (ω dτ − ϕ )⎟⎟ = 1 − e − 2ςω nτ ⎜⎜1 + 2 Ji ωd ⎝ ⎠

(5B.5)

Using the impulse response expressions at t = τ in equation (5A.5), equation (5.B5) becomes Jo = 1 − e − 2ςω nτ o + m 2 2ςω n g d (τ o ) g (τ o ) Ji

143

(5B.6)

Chapter 5. Wiener Filter in SISO Systems

Appendix 5C. Impulse acceleration response of a one d.o.f vibration system The impulse response for a one d.o.f vibration system can be found in the basic vibration text written by [Tse et al(1978)]. They considered the displacement response due to the unit impulse when t > 0 , however, the impulse response at t = 0 is also required for the analysis of this work. When the one d.o.f plant shown in Figure 5.11 is subject to the unit impulse δ (t ) , the equation of motion is m&x& + cx& + kx = δ (t )

(5C.1)

where the variable x is also a function of t, but the time dependency is abbreviated. Since δ (t ) is applied at t=0, it is valid for 0 − ≤ t ≤ 0 + . The system is assumed to be at rest before the unit impulse is applied, thus x (0 − ) = x& (0 − ) = &x&(0 − ) = 0 . The displacement at t = 0 + can be obtained by integrating equation (5C.1) twice for the period 0 − ≤ t ≤ 0 + , and

the velocity and acceleration are from integrating the equation once and from the equation itself, respectively[Tse et al(1978)]. Thus, the system responses at t = 0 + are given by

x (0 + ) = 0,

x& (0 + ) =

1 , m

&x&(0 + ) =

δ (t ) m



c m

(5C.2)

Now the non-homogenous equation in equation (5C.1) becomes an initial value problem with a homogenous equation given by m&x& + cx& + kx = 0

144

(5C.3)

Chapter 5. Wiener Filter in SISO Systems

and the initial conditions are given equation (5C.2). The solutions are given in equation (5C.6) where the variable x(t) is replaced by g a (t ) where the subscript a denotes acceleration. It should be noted that the acceleration impulse response g a (t ) includes the unit impulse

δ (t ) which influences the response only at t = 0 . The singular function term cannot be obtained from taking the simple derivative of the velocity impulse response g v (t ) . The other impulse responses and transfer functions of the one d.o.f vibration system which are used for the content are tabulated for quick reference in Table 5C.1.

145

Chapter 5. Wiener Filter in SISO Systems

Table 5C.1 Impulse responses and transfer functions of the one d.o.f vibration system

Time Domain displacement

g d (t ) =

velocity

g v (t ) = −

acceleration

g a (t ) =

1 −ςω nt e sin ω d t , mω d

t≥0

ω n −ςω t e sin (ω d t − ϕ ), mω d n

δ (t ) m

+

(5C.4) t≥0

ω n2 −ςω t e sin (ω d t − 2ϕ ), mω d

(5C.5) t≥0

n

(5C.6)

where the natural frequency ω n = k m , the damped natural frequency c ω d = ω n 1 − ς 2 , the damping coefficient ς = , and the phase 2 mk angle ϕ = tan

−1

1−ς 2

ς Laplace Domain

displacement

Gd ( s ) =

1 1 ⋅ 2 m s + 2ςω n s + ω n2

(5C.7)

velocity

Gv ( s ) =

1 s ⋅ 2 m s + 2ςω n s + ω n2

(5C.8)

Ga ( s ) =

1 s2 ⋅ 2 m s + 2ςω n s + ω n2

(5C.9)

acceleration

Z(Discrete Time) Domain (using the impulse invariant method) displacement

velocity

acceleration

ze −ςω nT sin ω d T 1 Gd ( z ) = ⋅ mω d z 2 − 2 ze −ςω nT cos ω d T + e −2ςω nT Gv ( z ) = −

(

(5C.10)

ω n z − z 1 − ς 2 + e −ςω T sin(ω d T + ϕ ) ⋅ mω d z 2 − 2 ze −ςω T cos ω d T + e −2ςω T n

n

(

)

ω n2 z − z 2ς 1 − ς 2 + e −ςω nT sin(ω d T + 2ϕ ) 1 Ga ( z ) = + ⋅ m mω d z 2 − 2 ze −ςω nT cos ω d T + e − 2ςω nT

where T is the sampling period

146

(5C.11)

n

)

(5C.12)

Chapter 6. Wiener Filter in MIMO Systems

CHAPTER 6.

ACTIVE CONTROL OF RANDOM SOUND IN MIMO SYSTEMS

6.1 Introduction The classical Wiener filter theory was considered from the active control point of view in Chapter 5. It was shown that the Wiener filter with the constraint of causality offers the optimal controller of random sound, and a comprehensive analytical and numerical method to design the Wiener filter was presented for the active control of stationary random sound in SISO(single-input-single-output) control systems. Using the fact that white noise can be equivalently replaced by an impulse, a systematic and deterministic approach was conducted instead of conventional approaches to stochastic optimal control.

The same theoretical background is used in this chapter, but the classical Wiener filter theory originally developed for SISO systems is extended to cover MIMO(multi-input-multi-output) systems. A thorough theoretical formulation of the Wiener filter for MIMO systems is presented in both continuous and discrete time domains in Section 6.2, after dealing with that of the SIMO(single-input-multi-output) and TISO(two-input-single-output) systems. The theory is employed on the active control of random sound in MIMO systems in Section 6.3. It is shown that active control of MIMO feedforward and feedback systems are the same as the MIMO Wiener filter problems. In Section 6.4, the simple one-dimensional acoustic duct which has been investigated by many researchers due to its simplicity is taken as the 147

Chapter 6. Wiener Filter in MIMO Systems

application model. Both global and local control configurations are considered, which correspond to the Wiener filter design problem for SIMO and SISO systems, respectively. The acoustic potential energy is adopted as the global measure of control performance, and influences of the electric time delay and the system damping are demonstrated. It is shown that the Wiener filter solution without the constraint of causality is the optimal controller for harmonic sound while the filter with the constraint is the optimal controller for stationary random sound. The local control of sound at a point is considered to show the use of the phase angle difference function introduced in Chapter 5 for an approximate prediction of performance in the real acoustic system. In Chapter 7, the Wiener filter theory for MIMO systems developed in this chapter is applied to the active control of random sound transmission into an enclosure.

148

Chapter 6. Wiener Filter in MIMO Systems

6.2 Wiener filter theory Classical Wiener filter theory was considered in Chapter 5 to design an optimal controller for stationary random sound. In this section, the theory originally developed for SISO systems is further developed to cover general MIMO(multi-input-multi-output) systems. As it is necessary to understand the classical Wiener filter theory for the active control of SISO systems, the Wiener filter theory for MIMO systems is necessary for the active control of MIMO systems which will be described in Section 6.3. The following formulation starts with considering SIMO(single-input multi-output) and TISO(two-input single-output) systems as stepping stones to reach the Wiener filter theory for MIMO systems.

6.2.1 SIMO and TISO systems

In this section, the classical Wiener filter theory considered in Chapter 5 is extended to the SIMO and TISO cases. A Wiener filter in a general SIMO system is shown in Figure 6.1(a). The single filter h(t) is used to minimise the sum of several mean square errors under different received and desired signals acting at each filter. When the system is linear, the positions of the filter and the input signal vector can be exchanged as shown in Figure 6.1(b) as a vector flow block diagram form. In the name SIMO, the SI(single-input) represents the single Wiener filter and MO(multi-output) represents multiple error responses. Since Lnumber of error sensors are used in total, each signal flow is of a L-length vector. The cost function in this case of multi-error signals can be written as

⎡L ⎤ J = E ⎢∑ el2 (t )⎥ ⎣ l =1 ⎦

(6.1)

and the Wiener-Hopf equation for this case is given by [Nelson et al(1990)]





0

L

L

l =1

l =1

ho (τ 2 )∑ Rrr ,l (τ 1 − τ 2 )dτ 2 = −∑ Rrd ,l (τ 1 ),

149

τ1 ≥ 0

(6.2)

Chapter 6. Wiener Filter in MIMO Systems

where ho(τ2) is the impulse response of the Wiener filter, and the auto- and cross-correlation functions Rrr ,l (τ ) = E [rl (t )rl (t + τ )] and Rrd ,l (τ ) = E [rl (t )d l (t + τ )] . The integral equation is subject to the constraint of causality τ 1 ≥ 0 and implicitly τ 2 ≥ 0 as can be seen at the lower

limit of integral where τ1 and τ2 are arbitrary time variables. The optimal filter ho(t) is physically realisable and called the constrained Wiener filter or simply Wiener filter.

The minimum mean square error for the SIMO system is written as

L



L

J o = ∑ Rdd ,l (0) + ∫ ho (τ 1 )∑ Rrd ,l (τ 1 )dτ 1 0

l =1

(6.3)

l =1

[

]

where the initial mean square error of the l-th sensor Rdd ,l (0) = E {d l (t )}2 .

The Wiener filter for SIMO systems was first formulated by [Nelson et al(1990)] to control stationary random sound inside a duct. They used an analytical method in the continuous time domain to design a Wiener filter which minimises the acoustic potential energy inside a duct.

d L (t )

rL (t )

eL (t )

h(t )

h(t ) r (t ) d 2 (t ) r2 (t )

h(t ) : a scalar d (t ) : L − length vector

d1 (t ) r1 (t )

h(t )

d (t ) e (t )

r (t ) : L − length vector

e2 (t )

h(t )

+

+

e1 (t )

e (t ) : L − length vector

(a) detailed block diagram

(b) equivalent block diagram

Figure 6.1 Wiener filter in a general SIMO system

150

Chapter 6. Wiener Filter in MIMO Systems

The TISO system is considered and its block diagram is shown in Figure 6.2. Two filters,

h1 (t ) and h2 (t ) , are used to minimise the mean square error of the single error sensor. The

[

]

mean square error is chosen as the cost function, and is written as J = E e 2 (t ) where the error e(t ) = d (t ) + y (t ) . The estimated signal y (t ) can be expressed by the convolution integral of the received signals and the impulse responses of the filters, and written as ∞

y (t ) = ∫ {h1 (τ ) r1 (t − τ ) + h2 (τ )r2 (t − τ )}dτ where the filter impulse responses are assumed to 0

) be causal. Substituting d (t ) into J, the cost function can be explicitly expressed as a

functional of the filter impulse responses h1 (t ) and h2 (t ) . Using the calculus of variations for finding its minimum described in detail in Appendix 6A, the Wiener-Hopf equation for the TISO system is given by

∫ ∫



0 ∞

0



ho1 (τ 2 ) R11 (τ 1 − τ 2 )dτ 2 + ∫ ho 2 (τ 2 ) R12 (τ 1 − τ 2 )dτ 2 = − R1d (τ 1 ), 0



ho1 (τ 2 ) R21 (τ 1 − τ 2 )dτ 2 + ∫ ho 2 (τ 2 ) R22 (τ 1 − τ 2 )dτ 2 = − R2 d (τ 1 ), 0

τ1 ≥ 0

(6.4)

τ1 ≥ 0

where h1 (t ) and h2 (t ) are the impulse responses of the optimal filters, and R11 (τ ) = E[r1 (t )r1 (t + τ )] ,

R1d (τ ) = E [r1 (t )d (t + τ )]

R12 (τ ) = E [r1 (t )r2 (t + τ )] ,

,

and

R2 d (τ ) = E[r2 (t )d (t + τ )] . From equation (6A.13), the minimum mean square error is given

by ∞



0

0

J o = Rdd (0) + ∫ ho1 (τ 1 ) R1d (τ 1 )dτ 1 + ∫ ho 2 (τ 1 ) R2 d (τ 1 )dτ1

[

]

where the initial mean square error of the l-th sensor Rdd ,l (0) = E {d l (t )}2 .

151

(6.5)

Chapter 6. Wiener Filter in MIMO Systems

r1 (t ) r2 (t )

h1 (t )

y1 (t )

h2 (t )

y2 ( t )

y (t ) +

+ d (t ) e(t )

Figure 6.2 TISO(two-input-single-output) system

6.2.2 MIMO systems

6.2.2.1 Wiener-Hopf equation for MIMO systems Now, consider a general MIMO system with multiple-filters whose block diagram is shown in Figure 6.3(a). The input signals and filters can be expressed as vector-matrix form given by

⎡ r11 (t ) r12 (t ) ⎢ r (t ) r (t ) 22 r (t ) = ⎢ 21 ⎢ ⎢ ⎣rL1 (t ) rL 2 (t )

r1K (t ) ⎤ r2 K (t )⎥⎥ ⎥ ⎥ rLK (t )⎦

h(t ) = {h1 (t ) h2 (t ) L hK (t )}

T

(6.6a)

(6.6b)

where in rl ,k (t ) the first variable in the subscript denotes the error location and the next variable denotes the filter location.

Using matrices expression, Figure 6.3(a) can be simplified by using signal vector flows as shown in Figure 6.3(b). Note the locations of h(t) and r(t) are exchanged in order to be consistent with the general linear algebra notation for vectors and matrices. Thus, the L-length column vector y(t ) can be written as the convolution integral between the L×K matrix r(t) and the K-length column vector h(t ) i.e., y(t ) = r (t ) ∗ h(t ) where the sign ∗ denotes the convolution integral. This is a MIMO case with K inputs and L outputs which correspond to the numbers of the filters and the error signals, respectively.

152

Chapter 6. Wiener Filter in MIMO Systems

rL ,1 (t ) rL , 2 (t )

rL , K (t )

r1,1 (t ) r1, 2 (t )

h1 (t )

d L (t ) y L (t )

h2 (t )

eL (t )

hK (t )

r (t )

h1 (t ) y1 (t )

+

r (t ) : L × K matrix

d1 (t )

h2 (t )

d (t ) e (t )

y(t ) +

h(t )

e1 (t )

h(t ) : K − length vector d (t ) : L − length vector

r1, K (t )

e (t ) : L − length vector

hK (t )

(a) detailed block diagram

(b) equivalent block diagram

Figure 6.3 Wiener filters for a MIMO(multi-input multi-output) system

The Wiener-Hopf equation for MIMO systems can be obtained similarly using the method given in Appendix 6A together with that for SIMO given in equation (6.2). Thus the WienerHopf equation for a (L×K)-number inputs and L-number outputs system can be written as Knumber of equations given by

K

∑∫ k =1

0

K

∑∫ k =1





0

L

L

l =1

l =1

L

L

l =1

l =1

L

L

l =1

l =1

hok (τ 2 )∑ R1k ,l (τ 1 − τ 2 )dτ 2 = −∑ R1d ,l (τ 1 ), hok (τ 2 )∑ R2 k ,l (τ 1 − τ 2 )dτ 2 = −∑ R2 d ,l (τ 1 ),

τ1 ≥ 0 τ1 ≥ 0

M K

∑∫ k =1



0

hok (τ 2 )∑ RKk ,l (τ 1 − τ 2 )dτ 2 = −∑ RKd ,l (τ 1 ),

153

τ1 ≥ 0

(6.7)

Chapter 6. Wiener Filter in MIMO Systems

where

hok (τ 2 )

is

a

optimal

filter,

and

[

Rxy ,l (τ ) = E rlx (t )rly (t + τ )

]

and

Rkd ,l (τ ) = E [rlk (t )d l (t + τ )] .

The minimum mean square error is written as

L

K

l =1

k =1

L



J o = ∑ Rdd ,l (0) + ∑ ∫ hok (τ 1 )∑ Rkd ,l (τ 1 )dτ 1 0

(6.8)

l =1

6.2.2.2 Causally unconstrained Wiener filter It is interesting to solve equation (6.7) without the constraint of causality, then the problem becomes extremely simple. By taking Fourier transform of the equation (6.7), the unconstrained optimal solution can be written as A(ω ) H uo ( jω ) = −b(ω )

(6.9)

where, H uo ( jω ) is used to distinguish it from the Fourier transform of the causal solution Ho(jω), and the variable jω inside the brackets is used for systems, and ω is for signals.

The input spectrum matrix is Hermitian and given by

⎡ S11,l (ω ) S12,l (ω ) L S1K ,l (ω ) ⎤ L ⎢S S 22,l (ω ) L S 2 K ,l (ω ) ⎥⎥ 21,l (ω ) A(ω ) = ∑ ⎢ ⎢ M ⎥ M M l =1 ⎢ ⎥ ⎣ S K 1,l (ω ) S K 2,l (ω ) L S KK ,l (ω )⎦

(6.10a)

where, for example, S xy ,l (ω ) is the Fourier transform of Rxy ,l (τ ) and the K-length vectors are

H uo ( jω ) = {H uo1 ( jω ) H uo 2 ( jω ) LLLLL H uoK ( jω )}

(6.10b)

b(ω ) = ∑ {S1d ,l (ω ) S 2 d ,l (ω ) LLLLL S Kd ,l (ω )}

(6.10c)

T

L

T

l =1

154

Chapter 6. Wiener Filter in MIMO Systems

where S kd ,l (ω ) is the Fourier transform of Rkd ,l (τ ) . When SISO or SIMO systems where the number of Wiener filters K = 1 , equation (6.9) becomes a scalar equation consisting of complex numbers. Letting R(ω) and D(ω) be the Fourier transforms of r(t) and d(t), respectively, the variables A(ω) and b(ω) can also be expressed as

[

]

A(ω ) = E R H (ω ) R(ω ) ,

[

b(ω ) = E R H (ω ) D(ω )

]

(6.10d,e)

From equation (6.9), the unconstrained Wiener filter is given by H uo ( jω ) = − A −1 (ω )b(ω )

(6.11)

Although such a filter is not physically realisable for random signal fields, the solution is known to be of great theoretical and practical importance because it offers a useful guideline to how much control can be achieved in the ideal state of ignoring the constraint of causality. Performance of a constrained Wiener filter is generally poorer than that of the unconstrained one.

Combining the Fourier transform of equation (6.8) with equation (6.11), the minimum mean square error can be written as

N

J o (ω ) = ∑ S dd ,n (ω ) − b H (ω ) A −1 (ω )b(ω )

(6.12a)

n =1

N

This can be normalised by its initial mean square error ∑ S dd ,n (ω ) , and thus the normalised n =1

mean square error can be written as

155

Chapter 6. Wiener Filter in MIMO Systems

J o′ (ω ) = 1 −

b H (ω ) A −1 (ω )b(ω ) N

∑S

dd ,n

(6.12b)

(ω )

n =1

The unconstrained Wiener filter plays a great role in active control of harmonic sound fields where only a single frequency is dominantly excited. As long as the system is linear time invariant(LTI) and is subject to harmonic excitation, causality does not occur as a constraint. Note in harmonic fields the received and desired signals shown in Figure 6.3 are sinusoidal rather than stationery random and the constraint of causality can be ignored in the WienerHopf equation.

6.2.2.3 Causally constrained Wiener filter For the solution of the causally constrained Wiener filter, an analytical method using the spectral factorisation method is extremely complicated. Thus we use the numerical method of the discrete form of the Wiener-Hopf equation in the discrete time domain as described in Chapter 5. The discrete form of equation (6.7) can be written as

I

K

L

L

∑∑∑ hok (i) R1k ,l (n − i) = −∑ R1d ,l (n), i =0 k =1 l =1 I

K

n≥0

l =1

L

L

∑∑∑ hok (i) R2k ,l (n − i) = −∑ R2d ,l (n), i =0 k =1 l =1

n≥0

l =1

(6.13)

M I

K

L

L

∑∑∑ hok (i) RKk ,l (n − i) = −∑ RKd ,l (n), i =0 k =1 l =1

n≥0

l =1

and it can be rewritten as a vector-matrix form written as Aho = −b

where the input correlation matrix A is written as

156

(6.14)

Chapter 6. Wiener Filter in MIMO Systems

⎡ A11,l L ⎢A 21 ,l A = ∑⎢ ⎢ M l =1 ⎢ ⎣ AK1,l

A12 ,l A22 ,l M AK2 ,l

A1K ,l ⎤ A2K ,l ⎥⎥ M ⎥ ⎥ L AKK ,l ⎦ L L

(6.15a)

The total matrix consists of several element matrices which is written as

Axy ,l

Rxy ,l (1) L Rxy ,l ( I − 1) ⎤ ⎡ Rxy ,l (0) ⎢ R (−1) Rxy ,l (0) L Rxy ,l ( I − 2)⎥⎥ xy ,l ⎢ = ⎢ ⎥ M M M ⎢ ⎥ Rxy ,l (0) ⎥⎦ ⎢⎣ Rxy ,l (− I + 1) Rxy ,l (− I + 2) L

[

(6.15b)

]

where Rxy ,l (i ) = E rl,x (n)rl,y (n + i ) . Although the element matrix Axy ,l itself is not symmetric due to Rxy (i ) ≠ Rxy (−i ) , the total matrix A becomes symmetric due to Axy = ATyx . Unlike the case with a SISO system considered in Chapter 5, the matrix A is no longer a Toeplitz matrix. It can be easily checked by considering only the TISO(two-input single output) case where the matrix A is a 2I×2I matrix. When in SIMO systems, A is of a (I×I) Toeplitz matrix. It should be emphasised that the present formulation based on the classical Wiener filter theory results in the symmetric auto-correlation matrix unlike the former result presented by [Elliott and Sutton(1996)]. The (I×K)-length column vectors required are written as

{

T ho = ho1

L

{

b = ∑ b1T,l l =1

T T ho2 LLLLL hoK

}

T

b2T,l LLLLL bKT ,l

(6.15c)

}

T

(6.15d)

where, for example, hok = {hok (0) hok (1) LLLLL hok ( I − 1)}

T

bk ,l = {Rkd ,l (0)

Rkd ,l (1) LLLLL Rkd ,l ( I − 1)} where, Rkd ,l (i ) = E[rlk (n)d l (n + i )] . T

The Wiener filter can be solved simply using the inverse of the real symmetric matrix A and is given by 157

Chapter 6. Wiener Filter in MIMO Systems

ho = − A −1 b

(6.16)

and the corresponding normalised mean square error is

J ′ = 1−

bT A −1 b L

∑R

dd ,l

(6.17)

(0)

l =1

where the denominator denotes the total initial mean square error.

A Wiener filter formulation for general MIMO systems has been presented both with and without the constraint of causality. In the next section, the Wiener filter theory developed here is used for the active control of both harmonic and random sound fields.

158

Chapter 6. Wiener Filter in MIMO Systems

6.3 Optimal control of harmonic and random sound fields 6.3.1 Feedforward and feedback control

6.3.1.1 Feedforward control Feedforward control of harmonic sound fields is considered first. For the convenience of mathematical formulation, only a single primary source Qp(ω) is assumed to exist and be perfectly measured. In addition, all plants considered are assumed to be linear time invariant (LTI). Feedforward control of the general MIMO system is shown in Figure 6.4 where the primary source Qp(ω) generates the L-length error responses E(ω) via the L-length primary plants Gp(jω). To optimally minimise the mean square of the error responses, the K-length filter set is used to operate the K-length secondary sources Qs(ω) and controls the error responses via the (L×K)-size secondary plants Gs(jω).

The error vector can be written as E (ω ) = G p ( jω )Q p (ω ) + G s ( jω ) H ( jω )Q p (ω )

[

(6.18)

]

The mean square error is defined as E E H (ω ) E (ω ) , however, the averaging process E can be omitted under the assumption that the whole process is deterministic as generally used in the analytical study of active control of harmonic sound fields. Thus, the mean square error becomes the Hermitian quadratic form of the complex variable vector H(ω). The optimal solution for harmonic fields is well known and is given by

(

H uo (ω ) = R H ( jω ) R( jω )

)

−1

R H ( jω ) D ( jω )

(6.19)

where R(ω ) = G s ( jω )Q p (ω ) ,

D(ω ) = G p ( jω )Q p (ω )

159

(6.20)

Chapter 6. Wiener Filter in MIMO Systems

The above solution turns out to be the same as the unconstrained Wiener filter solution H uo ( jω ) in equation (6.11) into which equation (6.10d,e) are substituted. Thus, it is clear that the unconstrained Wiener filter is the optimal controller for harmonic sound.

G p ( jω ) q p (t )

d (t )

H ( jω )

q s (t )

G s ( jω )

y(t )

e (t )

Figure 6.4 Block diagram of feedforward control for MIMO systems

The constrained Wiener filter solution is straightforward. In this case the primary source signal is stationary random, and the problem is to find the Wiener filter set which minimises the error responses in mean square sense. To construct elements for the matrix and vectors in equation (6.15), the L-length desired signal vector d(t) and the (L×K) input signal matrix r(t) are required. They are of the simple convolution integral form of equation (6.20), and written as d (t ) = g p (t ) ∗ q p (t ) ,

r (t ) = g s (t ) ∗ q p (t )

(6.21)

where the sign ∗ denotes the convolution integral. The plant impulse responses can be calculated using a time domain formulation of the system. The correlation functions in equation (6.15) can be calculated by various signal smoothing methods, for example, the Welch’s averaged periodogram method[Kay(1988) and Krauss et al(1994)].

If the primary source is white noise with zero mean and unit variance, it was proved in Chapter 5 that white noise can be equivalently replaced by an impulse. As a consequence, it 160

Chapter 6. Wiener Filter in MIMO Systems

transforms the optimal control of stochastic signals into the optimal control of impulse responses. In this case, equation (6.21) become r (t ) = g s (t ) and d (t ) = g p (t ) . The corresponding primary source strength which is compatible to the white impulse is one i.e., Qp(ω) = 1.

6.3.1.2 Feedback control In this section, classical Wiener filter theory is applied to feedback control of MIMO systems in both harmonic and stationary random sound fields. As in feedforward control, only a single primary source is assumed. A general arrangement of the MIMO feedback control system is shown in Figure 6.5(a) where block diagrams with and without control are drawn. In the case of without control, the single primary source qp(t) excites the system through the L-length primary plants Gp(jω), and their responses d(t) are measured by the L-number of error sensors e(t). As the (K×L)-size controllers − C ( jω ) are operated, the error signals are fed to the Klength secondary sources qs(t) through the controllers, and then those controlled sources excite the (L×K)-size secondary plants matrix Gs(jω) and their responses are superimposed with the original error signals due to the primary source. The error signal vector in the frequency domain is given by[Elliott and Sutton(1996)] E (ω ) = ( I + G s ( jω )C ( jω ) ) D(ω ) −1

(6.22)

where the desired signal vector D(ω ) = G p ( jω )Q p (ω ) and the matrix I is the identity matrix.

Internal Model Control (IMC) is used for feedback control and it is shown in Figure 6.5(b) ) where an estimated model of the secondary plants G s ( jω ) is included inside the controller − C( jω ) . In this case, the (K×L)-size controller matrix is given by

) − C ( jω ) = I + H ( jω )G ( jω )

(

161

)

−1

H ( jω )

(6.23)

Chapter 6. Wiener Filter in MIMO Systems

q p (t )

d (t ) = e (t )

G p ( jω )

d (t )

q s (t )

G s ( jω )

+

y(t )

e (t )

+

Without control

+

d (t ) q s (t )

G s ( jω )

+ e(t )

y(t )

) G s ( jω )

+

-

H ( jω )

− C ( jω )

− C ( jω )

With control (a) A MIMO feedback control system

(b) Internal Model Control (IMC)

Figure 6.5 Feedback control of MIMO systems

Substituting equation (6.23) into (6.22) with the assumption of perfect modelling of the ) secondary plants G s ( jω ) = G s ( jω ) gives E (ω ) = D(ω ) + G s ( jω ) H ( jω ) D(ω )

(6.24)

Now, the problem becomes a feedforward one and its block diagram is shown in Figure 6.6. To obtain the optimal solution as in feedforward control, the (K×L)-size matrix H ( jω ) is required to be changed into a vector form H ′( jω ) . Then it can be rearranged as E (ω ) = D(ω ) + R(ω ) H ′( jω )

(6.25)

where D(ω ) = G p ( jω )Q p (ω )

(6.26a)

R(ω ) = [G s (ω ) D1 (ω ) G s (ω ) D2 (ω ) L G s (ω ) DL (ω )]

(6.26b)

[

H ′(ω ) = H 1T (ω ) H 2T (ω ) L H LT (ω )

{

}

T

where H l ( jω ) = H 1,l ( jω ) H 2,l ( jω ) L H K ,l ( jω ) . 162

]

T

(6.26c)

Chapter 6. Wiener Filter in MIMO Systems

The mean square error for equation (6.25) becomes the Hermitian quadratic form of the complex variable vector H ′( jω ) , and the solution procedure is the same as in the feedforward control of harmonic fields.

d (t ) q p (t )

y(t )

G p (t )

H (t )

G s (t )

+

e (t )

+

) Figure 6.6 Internal model control with perfect plant modelling G s ( jω ) = G s ( jω ) .

The constrained Wiener filter solution is obtained from the convolution integral form of equation (6.26) which can be written as d (t ) = g p (t ) ∗ q p (t )

r (t ) = [g s (t ) ∗ d1 (t )

(6.27a) g s (t ) ∗ d 2 (t ) L g s (t ) ∗ d L (t )]

[

h′(ω ) = h1T (t ) h2T (t ) L hLT (t )

{ noise, then r (t ) = [g

]

T

(6.27b) (6.27c)

}

T

where hl ( jω ) = h1,l (t ) h2,l (t ) L hK ,l (t ) . If the single primary is assumed to be white s

(t ) ∗ g p1 (t )

]

g s (t ) ∗ g p 2 (t ) L g s (t ) ∗ g pL (t ) since d (t ) = g p (t ) .

Note the number of Wiener filters is determined by the numbers of error sensors and secondary sources. The number of control sources is important since it determines the computation time for solving the inverse of the auto-correlation matrix A in equation (6.16). For example, when L-number of error sensors and K-number of control sources are used and each filter is modelled as a FIR filter with I-coefficients, then the matrix A is of

(( K + L) ⋅ I ) × (( K + L) ⋅ I ) -size. In feedforward control, however, the size is solely determined by the number of control sources, when a single primary source exists. Due to the high computation burden, it is inefficient to use the proposed Wiener filter design method for the

163

Chapter 6. Wiener Filter in MIMO Systems

purpose of the global feedback control of sound fields where many error sensors are used in general.

6.3.2 Optimal control of harmonic and random sound fields

A general optimal control model of harmonic sound in structural-acoustic coupled systems has been presented in Chapter 4. In this section, the control plant is restricted to a purely acoustic system. To control both harmonic and random sound, the Wiener filter theory developed in the previous section is applied to SISO and SIMO control systems. In order to understand the basic theory of Wiener filter, application examples of this chapter are restricted to simple cases whose characteristics are well known. Applications to the active control of random sound in a structural-acoustic coupled system with a MIMO Wiener filter configuration will be considered in Chapter 7.

6.3.2.1 Global feedforward control of harmonic sound fields

Global control of harmonic sound fields is considered first. Following the work in Chapter 4, the acoustic potential energy given by equation (4.5) is again used as the cost function as a global measure of control performance, and is repeated here for clarity

Ep =

V 4ρ c

2 o o

aHa

(6.28)

When the system is subject to a single primary qp and a single secondary source qs is used for control, the general expression for the complex amplitude of acoustic modal pressure vector a given in equation (4.8) can be rewritten as a = Z a (d p q p + d s q s )

(6.29)

where the N-length vectors dp and ds determine coupling between the primary and secondary acoustic source locations and acoustic modes, respectively. 164

Chapter 6. Wiener Filter in MIMO Systems

The uncoupled acoustic modal impedance is defined as Za =

ρ o co2 V

A and the matrix A is a

(N×N) diagonal matrix whose (n,n) diagonal term An(ω) is given by

A1 (ω ) =

1 , 1 Ta + jω

when n=1

(6.30a)

where the time constant Ta = Ra Ca . The term Ca quantifies the acoustic compliance of the cavity and is equal to V ρ o co2 , and the term Ra quantifies the acoustic resistance and defines the rate at which air can escape from the cavity via the “leak” of cavity walls[Nelson and Elliott(1992)]. For other modes, it is written as

An (ω ) =

jω , when n ≠ 1 ω − ω + j 2ζ nω nω 2 n

2

(6.30b)

where ω n and ζ n are the natural frequency and damping ratio of the n-th acoustic mode, respectively. If equation (6.30b) is used for the first mode n=1, then A1 (ω ) = 1 jω where the modal damping cannot be accounted. The difficulties of using the function are that it is not defined at zero frequency, and also its time domain response becomes A1 (t ) = 1 which can not be modelled as a FIR(Finite Impulse Response) filter. Equation (6.30a) for the first acoustic mode is in fact a better representation of the physics, and can be modelled by a FIR filter that is required for the random sound control simulation considered in the next section.

The optimal source strength of the secondary source can be obtained by substituting equation (6.29) into (6.28). The resulting equation becomes the Hermitian quadratic form of the complex variable qs, the solution can be obtained from equation (6.19). It can also be solved by the unconstrained Wiener filter solution in equation (6.11) with the desired and received signal vector defined as R(ω ) = Z a d s q p and D(ω ) = Z a d p q p

165

(6.31a)

Chapter 6. Wiener Filter in MIMO Systems

where the transfer functions of primary and secondary plants can be defined as G s ( jω ) = Z a d s

G p ( jω ) = Z a d p

(6.31b,c)

The optimal secondary source strength q s = H uo q p , where Huo is the unconstrained Wiener filter. From the control point of view, the problem is a SIMO system with a single actuator and N-number of error sensors which measure the modal amplitudes of each mode an .

In practice, modal sensors measuring acoustic modal amplitudes are not available. As an alternative approach, microphones are used to measure acoustic pressure fluctuations at a discrete number of sensor locations, and the sum of the square pressure amplitudes at these locations are adopted as the cost function. The approximation of acoustic potential energy is given by[Bullmore et al(1987)] ) Ep =

L V 2 p(x l , ω ) ∑ 2 4 ρ o co L n=1

(6.32)

where p(x l ,ω ) is the complex pressure amplitude at the lth sensor location. The equation can be written as ) Ep =

V p Hp 4 ρ o co2 L

(6.33)

where the vector p is the L-length vector whose lth component is p(x l ,ω ) , and it given by[Bullmore et al(1987)]

p = ΨLT a

(6.34)

where the (L×N)-size matrix ΨLT shows the N modal amplitudes at L sensor locations. Now, it becomes feedforward control of a SIMO system with a single actuator and L error sensors. In this case, the desired and received signal vectors are defined as

166

Chapter 6. Wiener Filter in MIMO Systems

D(ω ) = ΨLT Z a d p q p

R(ω ) = ΨLT Z a d s q p

and

(6.35a)

where the transfer functions of primary and secondary plants are defined as

G p ( jω ) = ΨLT Z a d p

G s ( jω ) = ΨLT Z a d s

and

(6.35b,c)

The optimal filters for both cases can be obtained from equation (6.19) by substituting equation (6.31) and (6.35), respectively.

6.3.2.2 Global feedforward control of stationary random sound fields In this section, the global feedforward control of stationary random sound is considered. White noise with zero mean and unit variance is assumed as the primary source signal. From the white noise assumption as discussed in Section 6.3.1.1, the Wiener filter problem can be solved in a deterministic way by replacing it by an impulse. Thus the problem becomes a Wiener filter design problem to minimise the impulse responses of the system.

Provided the modal sensors are again available and a single actuator is used, the acoustic potential energy is of the same form of the SIMO case in equation (6.1) and given by

J=

N

V 4ρ c

∑a

2 o o n =1

2 n

(t )

(6.36)

where an(t) is the inverse Fourier transform of the acoustic modal amplitude of nth mode an(ω). The above equation can be of the same form as in equation (6.28) but with the time dependent modal amplitude vector a whose n-th element is written as an (t ) = d n ( t ) + rn (t ) ∗ ho (t )

The desired and received signals are written as

167

(6.37a)

Chapter 6. Wiener Filter in MIMO Systems

d n (t ) =

ρ o co2 V

rn (t ) =

An (t )d p ,n

ρ o co2 V

An (t )d s ,n

(6.37b,c)

and from equation (6.30), the time domain representation of the acoustic modal resonance terms An(t) is given by[ see Table 5C.1]

A1 (t ) = e



An (t ) = −

t Ta

ω n −ςω t e sin (ω d t − ϕ ) ωd n

when n=1

(6.38a)

when n ≠ 1

(6.38b)

where the angular resonance frequency ω d = ω n 1 − ς n2 , and the phase angle

ϕ = tan

−1

1 − ς n2

ςn

.

From the time domain formulation, the received and desired signals are used for the design of a constrained Wiener filter as explained in Section 6.3.2.3.

When L-number of microphones are used as error sensors, the desired and received signals are given by[see equation (4.3)]

d l (t ) =

ρ o co2 V

N

∑ψ n (x l ) An (t )d p,n

rl (t ) =

n =1

ρ o co2 V

N

∑ψ

n

(x l ) An (t )d s ,n

(6.39a,b)

n =1

and the optimal filter can be similarly obtained.

Feedback global control using multi-error sensors is not considered in this work since the proposed method based on the classical Wiener filter theory results in an auto-correlation matrix of very large dimension as discussed in Section 6.3.1.2. Its solution procedure demands high computation power and long computation time, and is thus very ineffective with the current computing processors.

168

Chapter 6. Wiener Filter in MIMO Systems

6.3.2.3 Local control of stationary random sound fields The meaning of ‘local’ is confined to a single error sensor response in this section, and a single actuator is used to minimise the error. When the primary field is harmonic and the system is linear, then perfect reduction of the error can be achieved at each frequency. However, when the primary field is random, the perfect reduction is not possible in general as discussed in Chapter 5. This section considers active control of a single error sensor using a single acoustic source. It is a SISO Wiener filter problem, and due to its simplicity both feedforward and feedback control configurations can be easily used.

Either a modal sensor or a microphone can be used as the error sensor, and the required responses for solving the constrained Wiener filter solution are given in equation (6.37) and (6.39), respectively. When a modal sensor is used and no time delay is involved in digital processing, the time dependent term An(t) in the desired and received signals are minimum phase so that perfect error reduction can be achieved theoretically in both feedforward and feedback controls as discussed in Chapter 5. If a microphone response is used as the error sensor, the desired and received signals in equation (6.39) are non-minimum phase in general. The performance of each controller according to the position of the error sensor is considered in the next section using a simple one-dimensional acoustic duct as the plant.

169

Chapter 6. Wiener Filter in MIMO Systems

6.4 Active control of sound fields inside a duct

6.4.1 System description

As a numerical example, a one-dimensional duct which has been investigated by many researchers due to its simplicity is taken as the application model. The acoustic duct of length L0 with a square cross-section of area S0 is assumed to be LTI and only plane wave propagation is admissible along the length of the duct. The physical dimensions and material properties of the acoustic duct are shown in Table 6.1 where the cut-off frequency denotes the upper limit frequency of plane wave propagation. A feedforward configuration is shown in Figure 6.7(a) where a primary source qp is located at the left-hand end of the duct and its input signal is fed to the secondary control source qs installed at the other end of the duct via the control filter H(jω). The filter can be classified as either a constrained or unconstrained Wiener filter depending upon the frequency characteristics of the primary source signal. Global or local control can be further classified by the number of error sensors used, and modal or physical sensors can be used as the error sensor. The feedback control configuration is shown in Figure 6.7(b) where the primary source signal is unknown and the output from the single error sensor is minimised.

The working frequency range of the primary source should be less than the cut-off frequency to satisfy the one dimensional acoustic field, and here the frequency range of excitation up to 500 Hz is considered. To achieve this, a low passed version of white noise with the filter cutoff frequency 500 Hz is assumed as the input to the primary source. The time domain representation of the low passed white noise can be roughly regarded as a step pulse with some time duration dependent on the filter cut-off frequency[Ch. 3 in Blinchihoff and Zverev(1976)]. For discrete modelling of the time responses given in equations (6.37) and (6.39), they are sampled at a frequency of 1000 Hz up over 2s and thus modelled as 2000length FIR(Finite Impulse Response) filters. With both harmonic and random excitation, only the first ten acoustic modes up to the (9,0,0) mode(450 Hz) are assumed to be excited and the 170

Chapter 6. Wiener Filter in MIMO Systems

eleventh mode at 500 Hz is ignored to avoid aliasing. It is assumed the primary source strength is 10-5 m/s3 throughout the frequency range of interest for harmonic sound control, and the same amplitude of pulse input is used for the low-passed white noise input.

In the next section feedforward control of acoustic potential energy is considered first, and is followed by feedforward and feedback controls of a single microphone.

Table 6.1 The acoustic duct data Length

Area

Density

Sound speed

Cut-off frequency

Low pass filter

Lo (m)

So(m2)

σo(kg/m3)

co (m/s)

(Hz)

frequency (Hz)

3.4

0.1×0.1

1.21

340

1700

500

qp

qs

Lo

x1

H(jω)

(a) Feedforward Control

qp

qs Microphone

p H(jω)

(b) Feedback Control Figure 6.7 Feedforward and feedback control of sound in a duct 171

Chapter 6. Wiener Filter in MIMO Systems

6.4.2 Global control of harmonic and random sound fields

6.4.2.1 Control of the acoustic potential energy Feedforward control as shown in Figure 6.7(a) is considered first using acoustic potential energy as the cost function. Modal sensors are assumed to exist and a total of ten sensors are used to measure the first ten acoustic modes. The system thus becomes a SIMO Wiener filter problem with a single actuator and ten error signals. From the time domain representation of the system given in equations (6.37) and (6.38), discrete time data is generated and it is used for both harmonic and random sound control simulations. Since the primary source strength is assumed to be 10-5 m/s3 throughout the frequency range, the same amount is multiplied to get the desired and received time responses in equation (6.37b,c). From the discrete time signals, the auto- and cross-spectra in equation (6.10) can be obtained and combined to give the unconstrained Wiener filter which is used for harmonic sound control. For random sound control, the correlation matrix and vector A and b in equation (6.14) can be constructed again from the discrete signals, and the causally constrained Wiener filter can be obtained from the matrix equation (6.16).

For system damping, a time constant for the first mode of 0.2s is assumed and the damping ratio of 0.01 are assumed for the other modes. The results are shown in Figure 6.8(a) where the acoustic potential energies before and after control are shown up to 500 Hz. The solid line denotes the original acoustic potential energy, and the dotted and dashed lines are after implementing causally unconstrained and constrained Wiener filters, respectively. For harmonic sound control which is shown in dot line, the result agrees well with the work done from the frequency domain formulation presented by [Elliott et al(1991)] and it shows good performance near resonance and poor between resonance. The frequency characteristics of the filter are shown by the dotted line in Figure 6.8(c,d) which shows a negative cosine-like function as the real part with zero imaginary part and agree well with the analytical solution published by [Curtis et al(1987)]. For the harmonic sound control simulation using the time domain formulation, however, care should be taken for choosing the sampling frequency and time data length to avoid aliasing errors. It should be noted that the unconstrained Wiener 172

Chapter 6. Wiener Filter in MIMO Systems

filter offers the optimal solution for harmonic sound control, and its time response is generally non-causal so that it is not valid for controlling random sound fields.

When the primary source emits a low-passed white noise, the acoustic potential energy can be minimised by the constrained Wiener filter which is modelled by a 50-coefficients FIR filter. It is a SIMO Wiener filter design problem whose block diagram is shown in Figure 6.1(a) and inputs of each Wiener filter block are minimum phase since the modal sensors are used. It has been shown in Chapter 5, that if both desired and received signals of a constrained Wiener filter are minimum phase, the performance when using an constrained Wiener filter should be the same as that of using an unconstrained filter. However, the whole performance denoted by the dash line in Figure 6.8(a) is not as good as that with the unconstrained filter. The discrepancy is of cause since here we have only a single Wiener filter which minimises multierror sensors, not a single error sensor. The discrete time and frequency components of the constrained Wiener filter are shown in Figure 6.8(b,c,d) and show an impulse-like response at the sample number n = 10 . It is interesting to compare this numerical solution with the analytical result by [Nelson et al(1990)]. They presented an analytical solution for white noise input as the primary source signal inside an idealised undamped one-dimensional acoustic field up to infinite frequency, which is given by ho (t ) = −δ (t − τ o ) where τo is the wave propagation time from the primary source to the secondary source. For the current simulation model it takes 0.01s and, since the sampling time between samples is 0.001 s, it is compatible to a 10 sample distance. The optimal filter for the low passed white noise shown in Figure 6.9(b) agrees reasonably well with the analytic solution. Due to the influence of damping in the system between the original primary pulse and the secondary source position, the amplitude of the optimal filter is slightly less than one. In addition, the use of a finite number of modes and the sampling process may result in some mismatches from the analytical solution worked out in the continuous time domain.

173

Chapter 6. Wiener Filter in MIMO Systems

65

60

0

Optimal Filter Response

Acoustic Potential Energy(dB)

55

50

45

40

−0.2

−0.4

−0.6

35 −0.8

30

25 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

−1 0

500

(a) Acoustic potential energy

5

10

15

20

25 Samples

30

35

40

45

(b) Optimal filter response 1.5

1

1

Real

Amplitude

10

0

10

0.5 0 −0.5

−1

10

−1 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

−2

10

0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500 1.5 1

Imaginary

Phase(Radian)

0

−10

0.5 0 −0.5

−20 −1 0

−30 0

50

100

150

200 250 300 Frequency(Hz)

350

(c) Power spectrum of filter signals

400

450

500

(d) Real and imaginary parts of filter signals

Figure 6.8 Acoustic potential energy; the solid line in the subplot (a) denotes before control, the dotted line shows after the harmonic sound controller using the unconstrained Wiener filter as the feedforward controller, and the dashed line shows after the random sound controller using the constrained Wiener filter. In the subplots (c) and (d) the dotted and dashed lines are the unconstrained and constrained Wiener filters, respectively.

6.4.2.2 Effect of a time delay An electric time delay is assumed for the digital processing of signals, and control performance due to the delay is shown in Table 6.2. The overall energy ratio for harmonic and random sound control are calculated from the normalised mean square error in equation (6.12b) and (6.17), respectively, and they are also expressed in Decibels. The overall energies 174

Chapter 6. Wiener Filter in MIMO Systems

for the harmonic sound controller Huo(jω) are calculated by summing each acoustic potential energy over the frequency range, and is independent on the time delay. The overall reductions are similar until 10 samples delay, and reduce by a factor of about two from the 15 to 30 samples delay, and then become smaller again for the 35 samples delay. The optimal filters for the 5 and 10 samples delay are shown in Figure 6.9(a,b). If we compare Figure 6.8(b) and 6.9(a,b), it is clear that the pulse at sample number 10 moves forward as much as the delay takes. After the 10 samples delay, there is a big jump in the overall reduction every 20 samples which is the time taken for a pulse starting from the secondary source position to return back. A physical interpretation of this wave cancellation mechanism has been presented by [Curtis et al(1987)]. They showed that the causal global optimal controller acts as a perfectly absorbing termination at its secondary source position. Thus as the delay increases first to 10 samples and then every 20 samples steps, the original pulse takes longer to arrive at the secondary source position so that it increases the residual energy inside the system. Figure 6.10 shows the residual acoustic potential energies for 15 and 35 samples delay.

Table 6.2 Performance of feedforward control according to the sample delay

Samples delayed

Overall energy ratio Jo/Ji (%)

Overall reduction (dB)

Huo(jω)

8.82

-10.5

0 sample

15.75

-8.0

5 samples

15.86

-8.0

10 samples

16.06

-7.9

15 samples

37.26

-4.3

20 samples

37.58

-4.3

25 samples

37.63

-4.2

30 samples

38.13

-4.2

35 samples

51.47

-2.9

Ho(jω)

175

Chapter 6. Wiener Filter in MIMO Systems

0

Optimal Filter Response

Optimal Filter Response

0

−0.2

−0.4

−0.6

−0.8

−1 0

−0.2

−0.4

−0.6

−0.8

5

10

15

20

25 Samples

30

35

40

(a) delay = 5 samples

−1 0

45

5

10

15

20

25 Samples

30

35

40

(a) delay = 10 samples

Figure 6.9 Optimal filter responses according to the sample delay

65

60

Acoustic Potential Energy(dB)

55

50

45

40

35

30

25 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

Figure 6.10 Performance of feedforward control according to a time delay for its digital processing (dot line : harmonic controller, dash line : delay = 15 samples, dash-dot line : delay = 35 samples)

176

45

Chapter 6. Wiener Filter in MIMO Systems

6.4.2.3 Effect of system damping Control performance was investigated with heavier damping and the resulting residual energies after the use of the unconstrained and constrained Wiener filters are shown in Figure 6.11(a). In Figure 6.11(b), because of heavier damping the constrained Wiener filter shows a smaller pulse at the sample number 10 than for the filter for more lightly damped system which was shown in Figure 6.8(b). This is because of heavier damping; the original pulse from the primary source is distorted and damped when it arrives at the secondary source position. Table 6.3 shows a comparison of the performance of controllers with different system damping. The performance gets poorer as the damping increases. Similar results have been reported by [Joplin and Nelson(1990)] in a three-dimensional rectangular cavity. However, their method is different from the proposed method for two reasons; firstly, instead of using an impulse they generated white noise and performed a series of time consuming averaging process which is not necessary for this theoretical study. Secondly, instead of using the time domain formulation they used the inverse Fourier transform to obtain the system impulse responses which can be easily contaminated by non-causal error particularly in highly damped cases. Non-causal error is caused by the use of the inverse Fast Fourier Transform(FFT) of a causal system ignoring its higher frequency components. It is the same phenomenon that the inverse Fourier transform of white noise is causal while the inverse Fourier transform of low-pass filtered white noise is non-causal[Blinchikoff and Zverev(1976)]. A clearer explanation of the non-causal error will be described in Chapter 7 with examples.

177

Chapter 6. Wiener Filter in MIMO Systems

65

60

0

Optimal Filter Response

Acoustic Potential Energy(dB)

55

50

45

40

−0.2

−0.4

−0.6

35 −0.8

30

25 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

−1 0

500

(a) Acoustic potential energy

5

10

15

20

25 Samples

30

35

40

45

(b) Optimal filter

Figure 6.11 Acoustic potential energy with a time constant for the first mode 0.1 s and damping ratio 0.05. ( solid line: before control, dotted line: harmonic sound feedforward control, and dashed line: random sound feedforward control)

Table 6.3 Performance of feedforward control according to the system damping

Damping Time constant To(s) 0.2 0.1 0.02

damping ratio(ζ) 0.01 0.05 0.1

Energy ratio Jo/Ji (%) Harmonic Huo(jω) 8.82 31.02 60.57

178

Random Ho(jω) 15.75 42.11 72.29

Overall reduction (dB) Harmonic Huo(jω) -10.5 -5.1 -2.2

Random Ho(jω) -8.0 -3.8 -1.4

Chapter 6. Wiener Filter in MIMO Systems

6.4.2.4 Control of the approximated acoustic potential energy Microphones instead of modal sensors are used as the error sensors and the performance of control is simulated. A total of 21 microphones are used for the approximation of the acoustic potential energy and they are assumed to be located at every Lo 20 along the direction of the duct. The desired and received signals for the Wiener filter solution are given in equation (6.39). Figure 6.12(a) shows the performance of harmonic and random sound control, and the constrained Wiener filter is shown in Figure 6.12(b). With the use of 21 microphones responses, the same results are achieved as with the modal sensors which was shown in Figure 6.8. It demonstrates that instead of the modal sensors, a multi-number of microphones can be successfully used for the global control of both harmonic and random sound fields.

65

60

0

Optimal Filter Response

Acoustic Potential Energy(dB)

55

50

45

40

−0.2

−0.4

−0.6

35 −0.8

30

25 0

50

100

150

200 250 300 Frequency(Hz)

(a) Acoustic potential energy

350

400

450

−1 0

500

5

10

15

20

25 Samples

30

35

40

45

(b) Optimal filter

Figure 6.12 Acoustic potential energy ( solid line: before control, dotted line: harmonic sound feedforward control, and dashed line: random sound feedforward control)

179

Chapter 6. Wiener Filter in MIMO Systems

6.4.3 Local control of stationary random sound fields

Local control of a single error sensor is considered in this section, and its feedforward and feedback configurations were shown in Figure 6.7(b). It is a SISO Wiener filter problem to design an optimal filter to minimise the error sensor in mean square sense. In the SISO problem, harmonic sound control is trivial and not considered since it always offers a perfect minimisation of error as long as the system is linear. The use of modal sensor to control random sound is also trivial because the desired and received signals generated by an impulse input are minimum phase. In this case, a perfect minimisation of error is possible in both feedforward and feedback control configurations. As discussed in Chapter 5, perfect minimisation from feedback control is not possible physically due to the unrealisable controller in equation (6.23). Thus only a microphone sensor is considered as the error and control performance is investigated according to the location of the error sensor along the length of the duct. For the numerical analysis, the time constant of the first acoustic mode 0.1s and the damping ratio for other modes 0.05 are used, and the remaining conditions are left unchanged.

6.4.3.1 Feedforward control From the feedforward configuration in Figure 6.7(a), the controller acts to minimise a single error sensor located at a point along the duct. For the case when the error sensor is located at x1 = 0 , the control performance and both constrained and unconstrained optimal filters are shown in Figure 6.13. Figure 6.13(b) shows the phase angle difference function between the desired and received signals which was introduced in Chapter 5. It was shown that the phase difference function can be used as an indirect measure to show the approximate time delay in both electrical and mechanical systems. In this case, no electrical time delay is assumed so that the phase difference is caused purely due by the mechanical plants. The desired signal in feedforward control can be interpreted as the impulse response from the primary source to the error sensor on the primary path, and the received signal is the impulse response from the secondary source to the error sensor on the secondary path. There is no time delay in the primary plant since it is minimum phase. However the secondary plant is a transfer response 180

Chapter 6. Wiener Filter in MIMO Systems

and is thus non-minimum phase. From Figure 6.13(b) the phase angle difference function shows approximately linear phase characteristic with the positive angle(phase advance), which means a prediction problem. Time delay, whether it is due to the mechanical secondary plant or an electrical digital processing, deteriorates the performance significantly as shown in Figure 6.13(a). As expected from the phase difference function, the optimal filter has a strong pulse at sample number 10 as shown Figure 6.13(c). It is compared with the unconstrained Wiener filter in the frequency domain in Figure 6.13(d), and a large difference is found

100

30

95

25

90

20

Angle(Radian)

Sound Pressure(dB)

between the causally constrained and unconstrained filters.

85

15

80

10

75

5

70 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

0 0

500

50

100

150

(a)

200 250 300 Frequency(Hz)

350

400

450

500

(b)

0.2

10

0.1

Real

5

0

−0.1

−5 0

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

−0.2

6

−0.3

4 Imaginary

Optimal Filter Response

0

−0.4

−0.5

2 0 −2

−0.6 0

20

40

60

80

100 120 Samples

140

160

180

200

(c)

−4 0

(d)

Figure 6.13 Performance of local feedforward control at x1=0. (a) Sound pressure before( solid line) and after(dash line) control, (b) Phase angle difference function between the desired and received signals, (c) the constrained Wiener filter, and (d) Frequency components of the causally constrained(dash line) and unconstrained(dot line) Wiener filters

181

Chapter 6. Wiener Filter in MIMO Systems

If the sensor is located at the other end of the duct where the secondary source is placed(collocation control), the desired signal and received signals in the former case are swapped around so that it gives the negative phase difference function(phase lag) of Figure 6.13(b). It is a filtering problem and a perfect minimisation is achieved in this case. The optimal filter is shown in Figure 6.14(a) and is compared with the unconstrained filter. It shows that both constrained and unconstrained filters in this case are the same so that the causal Wiener filter can be obtained simply from the inverse Fourier transform of the unconstrained Wiener filter. The fact simplifies the design of the causally constrained Wiener filter significantly without dealing with correlation functions from the complicated time domain formulation. It should be noted that as long as the secondary plant is minimum phase as is generally the case in collocation control, the inverse Fourier transform of the unconstrained Wiener filter can be directly used as the constrained Wiener filter for random noise control.

0.1

1 0.5

Real

0

Optimal Filter Response

−0.1

0 −0.5

−0.2

−1 0

−0.3

1

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

0.5

Imaginary

−0.4

−0.5

0 −0.5

−0.6 0

20

40

60

80

100 120 Samples

140

160

180

200

−1 0

Figure 6.14 The Wiener filter at x1=Lo when it is used as a feedforward controller. (a) the constrained Wiener filter and (b) Frequency components of the causally constrained(dash line) and unconstrained(dot line) Wiener filters

182

Chapter 6. Wiener Filter in MIMO Systems

Figure 6.15 shows the performance of feedforward control according the locations of the sensor which are decided to be the same locations for the global control considered earlier but is now plotted against the normalised length of the duct. The normalised mean square error(overall energy ratio) calculated from equation (6.17) is used as the measure of performance. At normalised locations less than 0.5, poor performance is observed since a pulse takes longer to travel from the secondary source to the error sensor than from the primary source. At the mid-position, both phase characteristics are the same and thus perfect minimisation is achieved. At normalised locations greater than 0.5 where the error sensor is closer to the secondary source, excellent performance is achieved, but not perfect minimisation between the positions 0.55 and 0.9. It is because due to added damping, the wave propagation inside the duct is dispersive[Ch. 7 in Kinsler et al(1982)] which means the phase speed changes as the frequency changes so that strictly speaking it is not appropriate to interpret it by a pure time delay. Such frequency dependent phase speed characteristics of the desired and received signals make the performance variance between 0 and 0.45 and between 0.55 and 0.9. At the positions 0.95 and 1, there is collocated control so that perfect minimisation is achieved. At the positions 0.95 and 1, they are collocation control so that perfect minimisation is achieved.

1 0.9

Normalised mean square error

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 Normalised location of the error sensor

0.9

1

Figure 6.15 Performance of feedforward control according to the location of the error sensor which is normalised by the length of the duct. 183

Chapter 6. Wiener Filter in MIMO Systems

6.4.3.2 Feedback control Feedback control shown in Figure 6.7(b) is considered using a microphone response as the error sensor. It is a SISO Wiener filter problem, and its solution procedure is explained in Section 6.3.1.2. The performance according to the location of the sensor is shown in Figure 6.16 where the performance get gradually better as the sensor gets closer to the secondary source location at Lo. As the error sensor gets closer to the secondary source position in the block diagram for feedback control shown in Figure 6.6, the time delay on the secondary path reduces. At the points 0.95 and 1, there is collocated control so that the error becomes zero. However, the physical feedback controller for this cannot be directly implemented in practice, as discussed in Chapter 5. To explain the performance difference between feedforward and feedback control, when the error sensor is located at the position 0.75, Figure 6.17 shows the phase angle difference functions in both feedforward and feedback control. In feedforward control, the phase angle difference function decreases(phase lag) as the frequency increases so that the performance in Figure 6.15 show a nearly perfect minimisation. In feedback control, the phase difference function increases(phase advance) and results in a poorer performance as shown in Figure 6.16. It is demonstrated that the phase angle difference function is closely related with control performance. Thus, it can be used as an indirect measure of predicting performance approximately before any real implementation is taken.

184

Chapter 6. Wiener Filter in MIMO Systems

1 0.9

Normalised mean square error

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 Normalised location of the error sensor

0.9

1

0

7

−2

6

−4

5

−6

4

Angle(Radian)

Angle(Radian)

Figure 6.16 Performance of feedback control according to the location of the error sensor which is normalised by the length of the duct.

−8

−10

3

2

−12

1

−14

0

−16 0

50

100

150

200 250 300 Frequency(Hz)

(a) Feedforward control

350

400

450

−1 0

500

50

100

150

200 250 300 Frequency(Hz)

350

400

450

500

(b) Feedback control

Figure 6.17 Phase angles of the desired(solid line ) and received(dash line) signals at x1=0.75 Lo .

185

Chapter 6. Wiener Filter in MIMO Systems

6.5 Conclusion A comprehensive treatment of Wiener filter theory for MIMO systems has been reviewed in both continuous and discrete time domains, and its applications to active control of random sound have been presented for both feedforward and feedback techniques for MIMO systems. It has been shown that the solutions of causally unconstrained and constrained Wiener filters are the optimal controllers for harmonic and random sound fields, respectively. From the application example with an acoustic duct, numerical models for global and local feedforward control of random sound have been presented using the time domain formulation.

The results obtained from the application example demonstrate 1. A finite number of microphones can be successfully used for the global control of harmonic and random sound fields. 2. It has been shown by a numerical example with various time delays that the global active control of random sound in purely acoustic systems can be interpreted as the wave absorbing mechanism on the control speaker. 3. As system damping increases, the global overall reduction for the feedforward control of both harmonic and random sound is decreased. 4. It has been shown with numerical examples that the phase difference function can be used as an approximate measure of the control performance of a random sound controller in SISO systems.

The theoretical framework developed in this chapter will be used in Chapter 7, where causally unconstrained and constrained Wiener filters are designed for an experimental set-up to control sound transmission into an enclosure.

186

Chapter 6. Wiener Filter in MIMO Systems

Appendix 6A. Wiener filter for the Two-Input-SingleOutput (TISO) system The TISO system shown in Figure 6.2 is considered. The mean square error is given by

[

J = E e 2 (t )

]

(6A.1)

Using e(t ) = d (t ) + y (t ) , it can be rewritten as

[

]

[

J = E d 2 (t ) + 2 E [d (t ) y (t )] + E y 2 (t )

]

(6A.2)

The estimated signal y (t ) = y1 (t ) + y2 (t ) and the convolution integral relationship between input and impulse responses gives ∞

y (t ) = ∫ {h1 (τ ) r1 (t − τ ) + h2 (τ )r2 (t − τ )}dτ 0

(6A.3)

where impulse responses of the filters are assumed to be causal, and τ is an arbitrary variable. Substituting equation (6A.3) into (6A.2), we get ∞ J = Rdd (0) + 2 E ⎡d (t )⎛⎜ ∫ {h1 (τ 1 )r1 (t − τ 1 ) + h2 (τ 1 )r2 (t − τ 1 )}dτ 1 ⎞⎟⎤ ⎢⎣ ⎝ 0 ⎠⎥⎦ (6A.4) ∞ ∞ ⎡ ⎤ + E ∫ {h1 (τ 1 )r1 (t − τ 1 ) + h2 (τ 1 )r2 (t − τ 1 )}dτ 1 ∫ {h1 (τ 2 )r1 (t − τ 2 ) + h2 (τ 2 )r2 (t − τ 2 )}dτ 2 ⎢⎣ 0 ⎥⎦ 0

where τ 1 and τ 2 are arbitrary variables. The last term on the right before averaging can be rewritten as ∞ ∞

∫∫ 0

0

h1 (τ 1 )h1 (τ 2 )r1 (t − τ 1 )r1 (t − τ 2 ) + h1 (τ 1 )h2 (τ 2 )r1 (t − τ 1 )r2 (t − τ 2 )

h2 (τ 1 )h1 (τ 2 )r2 (t − τ 1 )r1 (t − τ 2 ) + h2 (τ 1 )h2 (τ 2 )r2 (t − τ 1 )r2 (t − τ 2 )dτ 1dτ 2

Taking the mean value in equation (6A.4), we get

187

Chapter 6. Wiener Filter in MIMO Systems



J = Rdd (0) + 2∫ {h1 (τ 1 ) R1d (τ 1 ) + h2 (τ 1 ) R2 d (τ 1 )}dτ 1 0

+∫





0



h1 (τ 1 )h1 (τ 2 ) R11 (τ 1 − τ 2 ) + h1 (τ 1 )h2 (τ 2 ) R12 (τ 2 − τ 1 )

0

(6A.5)

h2 (τ 1 )h1 (τ 2 ) R21 (τ 1 − τ 2 ) + h2 (τ 1 )h2 (τ 2 ) R22 (τ 1 − τ 2 )dτ 1dτ 2

+

[

]

where Rdd (0) = E d 2 (t ) , and the second and third terms in the double integral are the same since R12 (τ ) = R21 (−τ ) . Thus we get



J = Rdd (0) + 2∫ h1 (τ 1 ) R1d (τ 1 ) + h2 (τ 1 ) R2 d (τ 1 )dτ 1 0

+∫



0





0

h1 (τ 1 )h1 (τ 2 ) R11 (τ 1 − τ 2 ) + 2h1 (τ 1 )h2 (τ 2 ) R12 (τ 1 − τ 2 ) + h2 (τ 1 )h2 (τ 2 ) R22 (τ 1 − τ 2 )dτ 1dτ 2 (6A.6)

Now the functional J is expressed in terms of functions h1 (t ) and h2 (t ) , and we can get optimal

functions

h1 (t )

for

and

h2 (t )

that

make

the

functional

a

minimum

[Hilderbrand(1965)]. Following the analysis given by Nelson et al(1990) for a SISO system, assume h1 (τ ) = ho1 (τ ) + ε1hε 1 (τ ), h2 (τ ) = ho 2 (τ ) + ε 2 hε 2 (τ )

(6A.7)

where ho1 (τ ) and ho 2 (τ ) are the optimal functions which minimise the functional, and ε1 and

ε 2 are arbitrary real parameters, and hε 1 (τ ) and hε 2 (τ ) are arbitrary continuous differentiable functions representing deviations from the optimal functions.

Substituting equation (6A.7) into (6A.6), then ∞

J = Rdd (0) + 2 ∫ (ho1 (τ 1 ) + ε1hε 1 (τ 1 ) )R1d (τ 1 ) + (ho 2 (τ 1 ) + ε 2 hε 2 (τ 1 ) )R2 d (τ 1 )dτ 1 0





0

0





0

0

∫ ∫ (h + 2 ∫ ∫ (h +

+





0

0

∫ ∫ (h

o1

(τ 1 ) + ε 1hε 1 (τ 1 ) )(ho1 (τ 2 ) + ε1hε 1 (τ 2 ) )R11 (τ 1 − τ 2 )dτ 1dτ 2

o1

(τ 1 ) + ε1hε 1 (τ 1 ) )(ho 2 (τ 2 ) + ε 2 hε 2 (τ 2 ) )R12 (τ 1 − τ 2 )dτ 1dτ 2

o1

(τ 1 ) + ε1hε 1 (τ 1 ) )(ho 2 (τ 2 ) + ε 2 hε 2 (τ 2 ) )R22 (τ 1 − τ 2 )dτ 1dτ 2

188

(6A.8)

Chapter 6. Wiener Filter in MIMO Systems

Once ho1 (τ ) , ho 2 (τ ) , hε 1 (τ ) , and hε 2 (τ ) are assigned, the functional is then a function of ε1 and ε 2 . The necessary conditions for the functional to take on its minimum value are ∂J = 0 when ε1 = 0, ∂ε1

and

∂J = 0 when ε 2 = 0 ∂ε 2

(6A.9)

Using the first condition gives

∂J ∂ε1 ε

∞ ∞ = 2 ∫ hε 1 (τ 1 )⎛⎜ R1d (τ 1 ) + ∫ ho1 (τ 2 ) R11 (τ 1 − τ 2 ) + ho 2 (τ 2 ) R12 (τ 1 − τ 2 )dτ 2 ⎞⎟dτ 1 = 0 0 0 ⎝ ⎠

1 =0

(6A.10) A necessary and sufficient condition for this to hold for any value of hε 1 (τ ) is that





0



ho1 (τ 2 ) R11 (τ 1 − τ 2 )dτ 2 + ∫ ho 2 (τ 2 ) R12 (τ 1 − τ 2 )dτ 2 = − R1d (τ 1 ), 0

τ1 ≥ 0

(6A.11)

τ1 ≥ 0

(6A.12)

Following the same process, the second condition gives





0



ho1 (τ 2 ) R21 (τ 1 − τ 2 )dτ 2 + ∫ ho 2 (τ 2 ) R22 (τ 1 − τ 2 )dτ 2 = − R2 d (τ 1 ), 0

By substituting equation (6A.11) and (6A.12) into (6A.6), the minimum mean square error is written as





0

0

J o = Rdd (0) + ∫ ho1 (τ 1 ) R1d (τ 1 )dτ 1 + ∫ ho 2 (τ 1 ) R2 d (τ 1 )dτ 1

189

(6A.13)

Chapter 7. Active Control of Random Sound Transmission

Chapter 7. Active control of random sound transmission

7.1 Introduction This chapter considers the active control of stationary random sound inside a structuralacoustic coupled system whose sound field is formed by transmission of an external plane wave through a flexible wall of an enclosure. This chapter is the concluding chapter of this thesis, and we apply acoustics and control theories developed in the previous chapters.

Analysis of structural-acoustic coupled systems differs from that of uncoupled systems. The dynamics of uncoupled components are analysed independently, from which a dynamic equation of the whole coupled system is formed in accordance with a coupling law. A new approach to the theory of structural-acoustic coupling has been developed in Chapter 2 based on the impedance-mobility approach. The theory was used for visualising structural-acoustic coupling in Chapter 3, and a number of lumped parameter analogy models for various coupling were presented in Chapter 4. Based on the theory, a model for the active control of harmonic sound transmission has been presented in Chapter 4. In particular, the role of structural and acoustic type actuators was investigated with a simple application example of a weakly coupled rectangular enclosure. It was demonstrated that a structural actuator is effective in controlling well separate plate modes while an acoustic actuator is effective in controlling well separate cavity modes. Since the coupled acoustic response consists of both plate and cavity controlled modes, it was concluded that the hybrid use of both type actuators

190

Chapter 7. Active Control of Random Sound Transmission

can offer the best performance with the least number of actuators in the weakly coupled structural-acoustic system studied.

Random sound differs from harmonic sound in that future sound amplitude signals are not a simple repetition of past signals. Due to this physical difference, there are many differences in designing random optimal controllers for random sound fields. In Chapter 5 and 6, comprehensive theoretical models for the active control of stationary random sound have been presented for SISO(single-input-single-output) and MIMO(multi-input-multi-output) control systems. Classical Wiener filter theory, which offers a realisable optimal controller for minimising the mean square error, was extensively applied to the theoretical models. It was demonstrated that the causally unconstrained Wiener filter is the optimal harmonic sound controller while the constrained Wiener filter is the optimal random sound controller. Furthermore, a systematic and deterministic approach to the optimal control of a stochastic signal has been presented when white noise is used as the input excitation signal. To explore the field of random sound control, applications of SISO and SIMO control systems have been investigated with simple examples of a one d.o.f(degree-of-freedom) system and a purely acoustic duct, respectively.

Active control of random sound transmission inside a fully coupled structural-acoustic system is considered in this chapter. The general theoretical model for MIMO active feedforward control of random sound presented in Chapter 6 is rearranged in Section 7.2 to be suitable for the sound transmission control systems considered here. Like the control of harmonic sound transmission in Chapter 4, three active control systems classified by the type of actuators used are compared. They are; i) a single acoustic actuator, ii) a single structural actuator, and iii) simultaneous use of both the acoustic and the structural actuators. The acoustic potential energy inside the coupled enclosure is taken as the cost function to minimise, and the control theory is applied to a fully coupled rectangular enclosure in Section 7.3. The effects of each system are discussed, and each performance for stationary random sound is compared with that for harmonic sound. Lastly, the theoretical model is validated by an experiment in Section 7.4.

191

Chapter 7. Active Control of Random Sound Transmission

7.2 Optimal minimisation of the acoustic potential energy A general optimal control model of harmonic sound transmission into enclosures has been presented in Chapter 4. The work presented in this chapter has the same theoretical background and configurations described in Chapter 4, but to the control of random sound transmission. A feedforward control configuration for a general sound transmission problem is shown in Figure 7.1 where both an acoustic source and a structural actuator are used as the control actuators. A plane wave, whether it is random or harmonic, is incident on a flexible structure and the resulting vibration of the structure radiates sound inside the cavity. A microphone outside the enclosure denoted pmic is used to measure the incident wave, and it is used as the reference signal to operate both a single acoustic source and a single structural force actuator via the feedforward controllers, Hq(jω) and Hf(jω), respectively. The acoustic potential energy inside the cavity is adopted as the measure of global performance, and control techniques for both harmonic and random sound transmission are considered.

Three separate sets of co-ordinates systems are used; Co-ordinate x is used for the acoustic field in the cavity, co-ordinate y is used for the vibration of the structure, and co-ordinate r is used for the sound field outside the enclosure. Assumptions made for the theoretical model in Chapter 4 are; 1) Weak coupling is assumed between the external sound field and the structure. Thus the external sound field is governed by the incident wave, and the radiated sound from structural vibration is neglected. 2) The coupled responses of the system are assumed to be described by finite summations of the uncoupled acoustic and structural modes; first N number of acoustic modes and first M number of structural modes are used. 3) All plants are Linear-Time-Invariant(LTI). 4) For the theoretical study, perfect knowledge of signals and systems is assumed with no measurement noise.

192

Chapter 7. Active Control of Random Sound Transmission

incident plane wave H f ( jω )

fc(y f,ω )

reference signal, p mic

flexible structure, S f

H q ( jω )

volume V q c(xq ,ω ) y

’

r x rigid boundary

Figure 7.1 Feedforward control of sound transmission using both structural and acoustic actuators

7.2.1 Active control of harmonic sound transmission

Following the work of harmonic sound transmission control in Chapter 4, the acoustic potential energy Ep inside the cavity volume V is adopted as the global measure of control performance. When the acoustic mode shape functions are normalised to be V = ∫ ψ 2n ( x ) dV , V

then the acoustic potential energy is given by [see equation (4.5)]

Ep =

V 4ρ c

2 o o

aHa

(7.1)

where ρo and co denote the density and the speed of sound in air, respectively, and the acoustic modal amplitude vector a consists of the complex amplitude of the acoustic pressure modes 193

Chapter 7. Active Control of Random Sound Transmission

an (ω ) . Since N-number of acoustic modes are used, the vector a is of length N and the superscript H denotes Hermitian transpose. When a harmonic plane wave denoted by pinc (r, ω ) is incident, the acoustic modal amplitude vector a can be expressed as a function of control variables qc and fc, and is given by

a = (I + Z a Ycs ) Z a (d q qc + CYs g p + CYs d f f c ) −1

(7.2)

where Za and Ys are the uncoupled modal acoustic impedance and uncoupled modal structural mobility, respectively, and the matrix Ycs is the coupled structural modal matrix and the matrix C is the structural-acoustic mode shape coupling matrix. The vector gp is the generalised modal force vector due to the primary plane wave excitation, and its m-th element is given by g p ,m = ∫ φ m (y ) p ext (y,ω )dS where the pressure p ext (y, ω ) is measured by the Sf

reference microphone denoted by pmic . The N-length vector dq determines the coupling between acoustic mode shapes and the location of the acoustic source at xq, and its n-th element is given by d q ,n =

1 Sq

∫ψ V

n

(x q )dV , whereas, the M-length vector df determines the

coupling between structural mode shapes and the location of the force actuator at yf, and its m-th element is given by d f ,m = ∫ φ m (y )δ (y − y f )dS . Sf

Substituting equation (7.2) into (7.1) gives the Hermitian quadratic form of complex variables qc and fc. The solution, which gives the minimum acoustic potential energy, is well known as described in Chapter 4. As also shown in Chapter 6, the solution can also be obtained from the causally unconstrained Wiener filter.

194

Chapter 7. Active Control of Random Sound Transmission

G p ( jω ) pmic

H q ( jω )

qc

H f ( jω )

fc

+

G q ( jω )

+

a

G f ( jω )

(a) Feedforward control of sound transmission using both actuators

d (t ) rq (t )

r f (t )

+

H q ( jω )

+

e (t )

H f ( jω )

(b) 2 Input N Output Wiener filter problem

Figure 7.2 A block diagram of the feedforward control of sound transmission using both an acoustic and a structural actuators

The feedforward controller can be redrawn as a block diagram form shown in Figure 7.2(a) where two linear-time-invariant controllers Hq(jω) and Hf(jω) are used separately to operate the two control actuators qc and fc in order to minimise the N-length error vector a in the mean square sense. The N-length vector G p ( jω ) denotes the plant response on the primary path, and the plant responses on the secondary paths Gq ( jω ) and G f ( jω ) denote the transfer function vectors from the acoustic and structural actuators to the error vector, respectively. All systems are assumed to be LTI, thus it can be redrawn as a 2-Input-N-Output Wiener filter problem as shown in Figure 7.2(b) where the frequency transforms of rq (t ) and r f (t ) are Rq (ω ) = Gq ( jω ) pmic and R f (ω ) = G f ( jω ) pmic , respectively. Here, note that modal

sensors measuring the complex amplitude of N-number of acoustic modes are intrinsically assumed to exist. To solve the unconstrained Wiener filters for H q ( jω ) and H f ( jω ) , the

195

Chapter 7. Active Control of Random Sound Transmission

frequency representations of the desired and received signal vectors are required. From equation (7.2) they can be written as D(ω ) = (I + Z a Ycs ) Z a CYs g q

(7.3a)

Rq (ω ) = (I + Z a Ycs ) Z ad q pmic

(7.3b)

−1

−1

R f (ω ) = (I + Z a Ycs ) Z aCYs d f pmic −1

(7.3c)

and the primary plant response vector is G p (ω ) = D(ω ) pmic and secondary plant response

vectors are written as Gq (ω ) = Rq (ω ) pmic and G f (ω ) = R f (ω ) pmic . The solution of the unconstrained Wiener filter for MIMO(multi-input-multi-output) systems was presented in Chapter 6, and is obtained from the matrix equation given by A(ω ) H uo ( jω ) = −b(ω )

(7.4)

where the input spectrum matrix is Hermitian and is given by N ⎡S S qf ,n (ω )⎤ qq ,n (ω ) A(ω ) = ∑ ⎢ S ff ,n (ω ) ⎥⎦ n =1 ⎣ S fq ,n (ω )

where

the

[ (ω ) = E [R

elements

* f ,n

(ω ) R f ,n

the

matrix

are

[ (ω )]

S qq ,n (ω ) = E Rq*,n (ω ) Rq ,n (ω )

]

,

] , S (ω ) = E [R (ω ) R , and (ω )] where the sign * denotes the complex conjugate. Note if all

S qf ,n (ω ) = E Rq*,n (ω ) R f ,n (ω ) S ff ,n

of

(7.5a)

fq ,n

* f ,n

q ,n

signals and systems are deterministic, the averaging process E [•] can be omitted. The unconstrained Wiener filter vector and the cross-spectrum vector are given by ⎧ H uoq ( jω )⎫ H uo ( jω ) = ⎨ ⎬, ⎩ H uof ( jω ) ⎭

196

N ⎧S qd , n (ω ) ⎫ b(ω ) = ∑ ⎨ ⎬ n =1 ⎩ S fd , n (ω ) ⎭

(7.5b,c)

Chapter 7. Active Control of Random Sound Transmission

[

]

[

]

where, S qd ,n (ω ) = E Rq*,n (ω ) Dn (ω ) and S fd ,n (ω ) = E R *f ,n (ω ) Dn (ω ) . If either the acoustic or structural actuator is used, the cross-spectrum term in equation (7.5a) disappears and then A(ω ) becomes a scalar value.

From Equation (7.4), the unconstrained Wiener filter is given by H uo ( jω ) = − A −1 (ω )b(ω )

(7.6)

and the optimal source strength of actuators qc and fc can be obtained from the relations H uoq ( jω ) = qc p mic and H uof ( jω ) = f c pmic , respectively. Replacing the performance

variable Ep by J to be consistent to the notation for Wiener filter, the minimum acoustic potential energy is given by

N

J o (ω ) = ∑ S dd ,n (ω ) − b H (ω ) A −1 (ω )b(ω )

(7.7a)

n =1

[

]

and it can be normalised by its initial mean square error S dd ,n (ω ) = E Dn* (ω ) Dn (ω ) , and that gives

J o′ (ω ) = 1 −

b H (ω ) A −1 (ω )b(ω ) N

∑S

dd ,n

(7.7b)

(ω )

n =1

When L-number of microphones instead of the modal sensors are used as the error sensors, then an approximation of the acoustic potential energy becomes the cost function showing global control performance. A larger number of microphones generally brings a closer approximation of the acoustic potential energy. As presented by[Bullmore et al(1987)], the cost function can be written as a vector equation as ) Ep =

V p Hp 4 ρ o co2 L

197

(7.8)

Chapter 7. Active Control of Random Sound Transmission

where the vector p is the L-length vector whose l-th component is the complex pressure amplitude at the l-th sensor location p(x l ,ω ) . The pressure vector measured at microphone positions is given by

p = ΨLT a

(7.9)

where the (L×N)-size matrix ΨLT shows the N modal amplitudes at L sensor locations. It is a Wiener filter problem of 2-Input-L-output, and the solution procedure is similar to the former case for the modal sensors, and thus is omitted here.

7.2.2 Active control of random sound transmission

When the incident wave is stationary random, the optimal controller must satisfy the constraint of causality to be physically realisable. As described in Chapter 6, the optimal controller for random sound is no more than the Wiener filter solution satisfying the constraint of causality. A theoretical model of the Wiener filter for MIMO systems has been presented in Chapter 6, and here we follow the solution procedure. When white noise is the incident random sound, then it can be equivalently replaced by an impulsive wave which is measured as an impulse by the reference sensor. The validity of the replacement has been demonstrated in Chapter 5 with a numerical example of a one d.o.f vibrating system. The problem is now transformed to the active control of an impulsive sound transmission using both acoustic and structural actuators. For solving the constrained Wiener filter problem, the time domain representation of desired and received signal vectors is required. They can be obtained by taking the inverse Fourier transform of equation (7.3), and thus we get

198

Chapter 7. Active Control of Random Sound Transmission

d (t ) = ifft{D(ω )} ,

rq (t ) = ifft {Rq (ω )},

r f (t ) = ifft {R f (ω )}

(7.10)

where the sign ifft denotes the inverse Fourier transform of a frequency signal vector. Since the frequency domain representations in equation (7.3) are the transfer functions of each plant, their time domain representations are the impulse responses of corresponding plant. To get the impulse responses, there are two ways in general; analytical and numerical inverse Fourier transforms. The first method can be called the time domain formulation, and has been used in Chapter 5 and 6 for the cases of purely structural or acoustic systems. For the sound transmission problem into the structural-acoustic coupled system, however, the time domain formulation is extremely complicated. As a numerical way to obtain the discrete impulse responses, thus, the frequency domain formulation using the inverse of the popular FastFourier-Transform(FFT) method is used from the transfer functions of plants available at discrete frequencies. This is a practical and efficient way since the complicated time domain formulation is no longer required.

Care should be taken, however, to reduce numerical errors such as the non-causal error and the alias error which are determined by the bandwidth of the signal and the sampling frequency. An example of the non-causal error will be given in Section 7.3.3. To avoid these errors, theoretically infinite number of frequency data covering all frequencies are required. However such digital processing is not possible, a good compromise should be made between the sampling frequency and the number of data. The frequency domain formulation using ifft was successfully used for the purpose of global control of sound by Joplin and Nelson, and thus this chapter here considers only global control of sound.

Following the solution procedure in Chapter 6, the constrained Wiener filter can be obtained from the matrix equation given by

Aho = −b

where the input correlation matrix A is given by

199

(7.11)

Chapter 7. Active Control of Random Sound Transmission

N ⎡A qq ,n A = ∑⎢ A n =1 ⎣ fq ,n

Aqf ,n ⎤ A ff ,n ⎥⎦

(7.12a)

which is a matrix consisting of element matrices that is expressed as, for example,

Aqf ,n

Rqf ,n (1) L Rqf ,n ( I − 1) ⎤ ⎡ Rqf ,n (0) ⎢ R (−1) Rqf ,n (0) L Rqf ,n ( I − 2)⎥⎥ qf , n ⎢ = ⎢ ⎥ M M M ⎢ ⎥ Rqf ,n (0) ⎥⎦ ⎢⎣ Rqf ,n (− I + 1) Rqf ,n (− I + 2) L

[

(7.12b)

]

where Rqf ,n (i ) = E rn,q (n)rn,f (n + i ) . Although the element matrix Aqf ,n itself is not symmetric due to Rqf ,n (i ) ≠ Rqf ,n (−i ) , it is important to note that the total matrix A becomes symmetric due to Axy = ATyx as mentioned in Chapter 6. The impulse response of each transfer function is assumed to be modelled by a J-length FIR(Finite Impulse Response) filter, and each Wiener filter is assumed to be implemented by a I-length FIR filter. The (2I)-length column vectors are given by ⎧⎪hoq ⎫⎪ ho = ⎨ ⎬ , ⎪⎩hof ⎪⎭

where,

for

bqd ,n = {Rqd ,n (0)

N ⎧b ⎪ qd ,n ⎫⎪ b = ∑⎨ ⎬ n =1 ⎪ ⎩b fd ,n ⎪⎭

(7.12c,d)

hoq = {hoq (0) hoq (1) L hoq ( I − 1)}

T

example,

[

and

]

Rqd ,n (1) L Rqd ,n ( I − 1)} where Rqd ,n (i ) = E rn,q (n)d n (n + i ) . If only T

N

[

]

one actuator is used, matrix A becomes either A = ∑ Aqq ,n for the acoustic actuator or n =1

N

[

]

A = ∑ A ff ,n for the structural actuator. n =1

The Wiener filter can be solved using matrix inversion and is given by ho = − A −1 b

200

(7.13)

Chapter 7. Active Control of Random Sound Transmission

and the normalised mean square error corresponding to Equation (7.7b) is written as

J′ =1−

b T A −1 b

(7.14)

N

∑R

dd ,n

(0)

n =1

[

]

where the initial mean square error of n-th acoustic mode Rdd ,n (0) = E {d n (t )}2 .

When L-number of microphones are used as the sensor set, it is a constrained Wiener filter design for 2-Input-L-Output and its solution procedure is similar to the above case, so is omitted.

201

Chapter 7. Active Control of Random Sound Transmission

7.3 Active control of sound transmission into a rectangular enclosure 7.3.1 Model description and system characteristics

As an application model, a duct-like rectangular enclosure surrounded by five rigid walls and a simply supported flexible plate on top is considered as shown in Figure 7.3. The duct-like enclosure was chosen for the convenience of analysis as well as for an easier experimental work, which is considered next section, without losing any generality of the physics of sound transmission. Three co-ordinates systems, x, y, r are used for specifying the cavity, the plate and the sound field outside the cavity, respectively. The top plate is assumed to be on the surface of an infinite baffle which is not shown for clarity. The dimensions of the cavity are L1× L2 × L3 where L1 = 1.5m , L2 = 0.3m , and L3 = 0.4m , and the thickness of the aluminium plate is 5 mm. A plane wave is incident on the top plate with angles of (ϕ = 0°) and (θ = 45°) where ϕ is the angle from the co-ordinate y1 on the plate and θ is the angle of incidence from the plate. The incident wave is measured with the reference microphone located at the right edge of the plate and its signal is used to operate both the acoustic and structural actuators via the controllers H q ( jω ) and H f ( jω ) , respectively.

As the global measure of control performance, the acoustic potential energy and its approximation are used separately corresponding to the error sensors used. A number of acoustic modal sensors or microphones can be used as the error sensors, and Figure 7.3 shows 11-number of microphones located in equi-distance along the centre line of the duct. A loudspeaker whose radius 0.15 m was installed at the left end with its centre at (0, L2 2 , L3 2 ), and a point force was used as the structural actuator and is located at ( 9 L1 20 , L2 2 ) on the plate. The material properties of air and aluminium(Al) used in the simulations are listed in Table 7.1. The modal damping ratios of the plate and the cavity are assumed to be 0.01, and the time constant of the first acoustic mode is 0.2 s.

202

Chapter 7. Active Control of Random Sound Transmission

Incident plane wave

θ

H of (ω )

Structural actuator

Ref. Mic. y2

Acoustic actuator

y1

x2

L3

Error microphones

L2

x3 x1

L1

H oq (ω )

Figure 7.3 Feedforward control of sound transmission

A total of 4 acoustic and 6 structural modes are assumed to contribute to the coupled responses within the frequency range of interest. Although an infinite number of modes are required to express the response at all frequencies in principle, we have to truncate higher modes for calculation. The assumption gives a reasonable approximation up to the highest frequency of the modes in a lightly damped system. Table 7.2 shows the natural frequencies of both uncoupled systems and their geometric mode shape coupling coefficients normalised by their maximum value. The (m1, m2) and (n1, n2, n3) indicate the indices of the m-th plate mode and the n-th cavity mode. Equations for the analysis of the rectangular enclosure are given in Appendix 4A in Chapter 4.

203

Chapter 7. Active Control of Random Sound Transmission

Table 7.1 Material properties

Density

Phase speed

Young’s modulus

Poisson’s

Damping

(kg/m3)

(m/s)

(N/m2)

ratio (ν)

ratio (ζ)

Air

1.21

340

-

-

0.01

Al

2770

-

71×109

0.33

0.01

Material

Table 7.2 The natural frequencies and geometric mode shape coupling coefficients of each uncoupled system Order

Plate Type Freq.

1

2

3

4

5

6

(1,1)

(2,1)

(3,1)

(4,1)

(5,1)

(6,1)

141 Hz 157 Hz 184 Hz 222 Hz

270 Hz

330 Hz

Cavity 1

(0,0,0) 0 Hz

1.0000

0

0.3333

0

0.2000

0

2

(1,0,0) 113 Hz

0

0.9428

0

0.3771

0

0.2424

3

(2,0,0) 227 Hz -0.4714

0

0.8485

0

0.3367

0

4

(3,0,0) 340 Hz

-0.5657

0

0.8081

0

0.3143

0

To show the degree of coupling of the system, the acoustic potential energy and the vibrational kinetic energy of the fully coupled system are compared in Figure 7.4 with those with the weakly coupling assumption. On each graph, uncoupled natural frequencies of the plate and the cavity are marked ‘*’ and ‘o’, respectively. It can be seen that there are noticeable differences on the responses so the fully coupled analysis is used for the application model. Details on the coupling analysis were presented in Chapter 3, and the analysis of sound transmission was given in Chapter 4.

204

60

50

55

45

50

40

Vibrational Kinetic Energy(dB)

Acoustic Potential Energy(dB)

Chapter 7. Active Control of Random Sound Transmission

45 40 35 30 25

35 30 25 20 15

20

10

15

5

10 0

50

100

150

200 250 Frequency(Hz)

300

350

(a) Acoustic potential energy

0 0

400

50

100

150

200 250 Frequency(Hz)

300

350

400

(b) Vibrational kinetic energy

Figure 7.4 Comparison of responses with and without the weakly coupling assumption where solid line denotes the fully coupled modelling and dashed line denotes the weakly coupled modelling.

7.3.2 Control of harmonic sound

A theoretical model for global control of harmonic sound transmission was presented in Chapter 4, and it was applied to a weakly coupled rectangular enclosure model. In this section, the theory is applied to the duct-like fully coupled enclosure shown in Figure 7.3, and the role of structural and acoustic actuators is investigated. A harmonic wave of pressure amplitude 1 Pa is assumed to incident, and system responses are calculated from 0 Hz to 512 Hz with the frequency division 0.5 Hz. For the acoustic potential energy, four modal sensors measuring the first four acoustic modes are assumed to be used, while a total of eleven microphones shown in Figure 7.3 are used for the approximation of the acoustic potential energy.

The modal sensors are used as the error sensors, and the harmonic controller can be obtained by following the solution procedure given in Section 7.2.1. The acoustic potential energies before(solid line) and after control are shown in Figure 7.5 up to 400 Hz according to the actuators used; the acoustic actuator(top), the structural force actuator(middle), and the hybrid

205

Chapter 7. Active Control of Random Sound Transmission

use of both actuators(bottom). As shown in Chapter 4, the acoustic actuator is effective in controlling the cavity controlled modes while the structural force actuator is effective in controlling the plate controlled modes. The hybrid use of both actuators again offers excellent control performance of the fully coupled system.

In general, control performance is dependent on various factors given for a specific system considered, such as the locations of actuators, the degree of coupling, and the geometric shape of enclosures. Thus it does not seem to allow any general discussion on the role of each actuator in fully coupled systems. However, it should be noted that the hybrid use of both type of actuators offers the best performance with least number of actuators. In other words, it is superior to the separate use of multi-structural or acoustic actuators. It may be best explained by referring to the analogous mechanical model of modal coupling between structural and acoustic modes in Chapter 3. In Figure 3.6(a), we presented a lumped parameter model consisting of both upper and lower mass systems which represents coupling between a single structural mode and multi-acoustic modes. In the sound transmission problem, our target is to reduce the cavity acoustic response which is generated by the vibration of plates. The acoustic response, which can be represented as the spring forces of upper mass systems in the analogous model, can be huge if either one of the lower(plate controlled mode) or upper(cavity controlled mode) mass systems is resonant at a coupled resonance frequency.

Consider the cavity controlled mode case first so that one of the upper mass systems is resonant. Then, the question becomes which type of actuator is the best way to control the coupled acoustic response. In general, many structural modes contribute to the response of a single acoustic mode in an arbitrary frequency, regardless of the degree of coupling. The acoustic actuator, which directly acts on upper mass systems, is the best way to control the cavity controlled mode without bringing high ‘control spillover’, as far as well separated structural and acoustic modes are concerned. As discussed in Chapter 4, from the mode shapes point of view a use of discrete actuators working in discrete locations inherently induces control spillover. The terms of cavity and plate controlled modes defined in Chapter 4 are also available for analysing fully coupled systems as long as two uncoupled natural frequencies are different. Likewise, a structural actuator is the best way to control plate control modes, regardless of the degree of coupling. 206

Chapter 7. Active Control of Random Sound Transmission

When, instead of the modal sensors, the eleven number of microphones are used as the error sensors, an approximation of the acoustic potential energy can be used as the global measure of performance. The summation of the square pressure responses at eleven equi-distanced microphone positions gives an indistinguishable approximation of the acoustic potential energies for both before and after control, so the results are omitted.

207

Chapter 7. Active Control of Random Sound Transmission

60

Acoustic Potential Energy(dB)

55 50 45 40 35 30 25 20 15 10 0

50

100

150

200 250 Frequency(Hz)

300

350

400

50

100

150

200 250 Frequency(Hz)

300

350

400

50

100

150

200 250 Frequency(Hz)

300

350

400

60

Acoustic Potential Energy(dB)

55 50 45 40 35 30 25 20 15 10 0

60

Acoustic Potential Energy(dB)

55 50 45 40 35 30 25 20 15 10 0

Figure 7.5 The predicted acoustic potential energies from the fully coupled modelling before(solid line) and after feedforward control according to the actuators used(dashed); the acoustic actuator(top), the structural force actuator(middle), and the hybrid use of both actuators(bottom).

208

Chapter 7. Active Control of Random Sound Transmission

7.3.3 Control of random sound

7.3.3.1 Data processing For the control of stationary random sound, a low-passed version of white noise under 512 Hz is assumed as the incident plane wave. It can be equivalently replaced by a low-passed version of an impulse, and the inverse Fourier transform of the frequency responses can be used as the transient responses of the system subject to the impulse as discussed in Chapter 6. Due to the difficulty of the time domain formulation of sound transmission, the simulation follows the procedure explained in Section 7.2.2 using the inverse Fourier transform of the plant frequency responses. The frequency responses calculated for the control of harmonic sound in the last section are known at discrete frequencies up to 512 Hz with 0.5 Hz division. To reduce the non-causal error, only first four acoustic and first six structural uncoupled modes whose maximum is 340 Hz as listed in Table 7.2 are used to calculate coupled responses. When the acoustic modal sensors are used as the error sensors using both actuators, it is a 2-input-4-output Wiener filter problem whose block diagram was shown in Figure 7.2. The amplitude and phase responses of each plant are known from the frequency domain formulation used for harmonic sound control. They are shown in Figure 7.6, as an example, when the error is the first acoustic modal response. The primary plant response denoted by solid line is the first acoustic modal response to the incident wave, and the acoustic(dash) and structural(dot) secondary plant responses are the modal responses to the control inputs of acoustic source strength and force, respectively.

209

Chapter 7. Active Control of Random Sound Transmission

140

2

0

120

Phase(radian)

Sound Pressure(dB)

−2 100

80

−4

−6

60 −8

40

20 0

−10

50

100

(a) Amplitude

150

200 250 Frequency(Hz)

300

350

−12 0

400

50

100

150

200 250 Frequency(Hz)

300

350

400

(b) Phase

Figure 7.6 The frequency responses of each plant whose error is the first acoustic modal response; The input values to the primary plant(solid line), the acoustic secondary plant(dash line), and the structural secondary plant(dot line) are the external sound pressure of 1 Pa, the source strength 10 −3 m 3 s , and the force 1 N , respectively.

For the four Wiener filters corresponding to the first four acoustic modes, Figure 7.7 shows the phase angle difference functions from the primary plant to the acoustic(solid) and structural(dashed) secondary plants, respectively. Its usefulness for an approximate prediction of control performance in SISO systems was demonstrated in Chapter 6. Furthermore, since the function is dependent on actuator location, the function can be used for searching for the optimal location of control actuator in an early stage of control task before real time control is implemented. For the given primary plants, the plots demonstrates that the phase difference functions for the acoustic actuator give bigger phase lag than those to the structural actuator. It means that it takes a shorter time for the acoustic source to control the cost function, which is the acoustic potential energy, than the force actuator does.

210

Chapter 7. Active Control of Random Sound Transmission

2

0

−2

0

−4

Phase(radian)

Phase(radian)

−2

−4

−6

−8

−6 −10

−8

−10 0

−12

50

100

150

200 250 Frequency(Hz)

300

350

(a) the 1st acoustic mode 2

2

0

0

−2

−2

−4

−6

−8

−8

50

100

150

200 250 Frequency(Hz)

(a) the 3rd acoustic mode

100

150

200 250 Frequency(Hz)

300

350

400

300

350

400

−4

−6

−10 0

50

(a) the 2nd acoustic mode

Phase(radian)

Phase(radian)

−14 0

400

300

350

−10 0

400

50

100

150

200 250 Frequency(Hz)

(a) the 4th acoustic mode

Figure 7.7 The phase angle difference function in each Wiener filter for the acoustic actuator(solid) and the structural actuator(dashed).

Before the inverse FFT transform, the other half part of frequency response data, from Fs 2 to Fs where the sampling frequency Fs = 1024 Hz , should be generated in a way to produce real time data as following the properties of FFT[Ch. 8 in Oppenheim and Schafer(1989)]. The Fourier transform of a real discrete time sequence is of a complex sequence whose real sequence should be symmetric and whose imaginary sequence is anti-symmetric to the frequency Fs 2 . Finally using the well-known Discrete Fourier Transform(DFT), the impulse response sequences in equation (7.10) are prepared. Each sequence is of 2048 data samples with the sampling frequency 1024 Hz and the record period 2s.

211

Chapter 7. Active Control of Random Sound Transmission

Figure 7.8 shows the impulse responses of the plants shown in Figure 7.6. The response gradually increasing near the recording time(2s) is defined as the non-causal error[see, especially Figure 7.6(b)], and is due to the use of the inverse Fourier transform without a full inclusion of the original frequency response band. In practice, a non-causal error is impossible to occur, and thus, we neglect the physically meaningless error from the impulse responses for the simulation presented in this chapter. For a small non-causal error, this brings a small mismatch of frequency response amplitude from the original near the folding frequency, Fs 2 . However, the rest of frequency range will remain similar to the original so that a reasonable analysis can be performed.

Microphones can also be used as the error sensor, and the data processing procedure is the same as the above for the modal sensor. Figure 7.9, as an example, shows the frequency and time responses of plants for the microphone error sensor located at x1 = 4 L1 10 . Non-causal errors are also assumed to be negligible, and are excluded from impulse responses. Calculating the impulse responses for the other error sensors, the plant responses in equation (7.10) are determined, and used to construct the matrix and vector in equation (7.12).

212

Chapter 7. Active Control of Random Sound Transmission

30

20

Pressure(Pa)

10

0

−10

−20

−30 0

0.2

0.4

0.6

0.8

1 Time(s)

1.2

1.4

1.6

1.8

2

1.6

1.8

2

1.6

1.8

2

(a) the primary plant 900 800 700

Pressure(Pa)

600 500 400 300 200 100 0 −100 0

0.2

0.4

0.6

0.8

1 Time(s)

1.2

1.4

(b) the acoustic secondary plant 150

100

Pressure(Pa)

50

0

−50

−100

−150 0

0.2

0.4

0.6

0.8

1 Time(s)

1.2

1.4

(c) the structural secondary plant

Figure 7.8 The impulse responses of the plants shown in Figure 7.6; the primary plant(a), the acoustic secondary plant(b), and the structural secondary plant(c).

213

Chapter 7. Active Control of Random Sound Transmission

Pressure(Pa)

140 130 120

50

0

−50 0

0.2

0.4

0.6

0.8

1 Time(s)

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

0.8

1 Time

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

0.8

1 Time

1.2

1.4

1.6

1.8

2

100

4000

Pressure(Pa)

Sound Pressure(dB)

110

90 80

2000 0 −2000 0

Pressure(Pa)

70 60 50 40 0

50

100

150

200 250 Frequency(Hz)

300

350

(a) Frequency responses

500

0

−500 0

400

(b) Impulse responses

Figure 7.9 the frequency and time responses of plants for the microphone error sensor located at x1 = 4 L1 10 ; (a) Frequency responses of the primary(solid), acoustic secondary(dash), and structural secondary plants(dot), (b) Impulse responses of the primary(top), acoustic secondary(middle), and structural secondary plants(bottom)

7.3.3.2 Results

The impulse responses of the plants determined in the last section are used to calculate correlation functions required for the matrix A and vector b in equation (7.12). Excluding the non-causal error near the recording time, only the first 1800 sample data are used for the simulation. Another 1800 sample data with zero value is added to each impulse response in order to avoid an error due to using the circular convolution in the frequency domain[ Oppenheim and Schafer(1989)].

The case of using the acoustic actuator to control the acoustic potential energy is considered first. In this case, the modal sensors are intrinsically assumed to be available. Figure 7.10(a) shows the acoustic potential energy with and without control. The initial acoustic potential energy(solid line) is due to the original primary incident wave, and the potential energies with control are shown for both random(dashed) and harmonic(dotted) waves. As explained earlier, the random and harmonic sound controllers are the causally constrained and unconstrained Wiener filters, respectively. Note that control performance for harmonic sound is better than that for random sound at all frequencies. The constrained Wiener filter is 214

Chapter 7. Active Control of Random Sound Transmission

modelled by a 400 coefficient FIR filter and its response is shown in Figure 7.10(b). In this case, the constrained Wiener filter is the optimal controller for a low-pass version of white noise or equivalently a low-passed impulsive wave as the incident plane wave, and the unconstrained Wiener filter is the optimal controller for the harmonic incident wave. Overall performances of each controller are tabulated in Table 7.3 where very small performance difference is shown between the harmonic and random noise cases. It is due to the large phase lag in the phase angle difference functions shown in Figure 7.7. For weakly coupled systems, in fact, the systems from the acoustic actuator to each acoustic mode are minimum phase, which means no time delay exists in each secondary plant.

Figure 7.11 shows control performance for the structural actuator(a) and the constrained optimal filter response(b), and its overall performance is shown in Table 7.3. Compare to the performance difference(0.4 dB) between harmonic and random sound cases for the acoustic actuator, a bigger difference(about 3 dB) is shown for the structural actuator. It is because the time delay from the structural actuator to acoustic responses is longer than that from the acoustic actuator. However, it still offers good performance with the overall reduction 5.7 dB for random sound.

When both the acoustic and structural actuators are used, the performance and optimal filters are shown in Figure 7.12. As can be seen from Table 7.3, excellent performance over than 10 dB for both harmonic and random sound cases are achieved for the given lightly damped system. In both harmonic and random sound cases, thus, the control performance for the hybrid use is superior to those for each type actuator. In addition, the case of using both actuators does not bring a large performance difference between harmonic and random sound cases.

215

Chapter 7. Active Control of Random Sound Transmission

0.015

60 0.01

50 0.005

45

Optimal Filter(Hoq)

Acoustic Potential Energy(dB)

55

40 35 30

0

−0.005

25 −0.01

20 15 10 0

50

100

150

200 250 Frequency(Hz)

300

350

−0.015 0

400

(a) Control performance

50

100

150

200 Samples

250

300

350

400

(b) Response of the constrained optimal controller

Figure 7.10 The control performance using the acoustic actuator and the response of the constrained feedforward optimal controller; the original cost function(solid) and after using the constrained(dashed) and unconstrained(dotted) optimal controllers.

0.15

60 0.1

50 0.05

45

Optimal Filter(Hof)

Acoustic Potential Energy(dB)

55

40 35 30

0

−0.05

25 20

−0.1

15 10 0

50

100

150

200 250 Frequency(Hz)

(a) Control performance

300

350

−0.15 0

400

50

100

150

200 Samples

250

300

350

400

(b) Response of the constrained optimal controller

Figure 7.11 The control performance of using the structural actuator and the response of the constrained feedforward optimal controller; the original cost function(solid) and these after using the constrained(dashed) and unconstrained(dotted) optimal controllers.

216

Chapter 7. Active Control of Random Sound Transmission

0.02

0.015

60 0.01

Optimal Filter(Hoq)

50 45

0.005

0

−0.005

−0.01

−0.015

40 −0.02 0

35

50

100

150

200 Samples

250

300

350

400

300

350

400

0.2

0.15

30 0.1

25

Optimal Filter(Hof)

Acoustic Potential Energy(dB)

55

20 15

0.05

0

−0.05

−0.1

−0.15

10 0

50

100

150

200 250 Frequency(Hz)

300

(a) Control performance

350

400

−0.2 0

50

100

150

200 Samples

250

(b) Response of the constrained optimal controllers

Figure 7.12 The control performance of using the both actuators and the response of the constrained feedforward optimal controllers; (a) the original cost function(solid) and these after using the constrained(dash) and unconstrained optimal controllers, (b) acoustic (top) and structural (bottom) actuators

Table 7.3 Comparison of performances according to the use of each and both actuators

Use of the actuators

Energy ratio J o J i (%)

Overall reduction(dB)

Harmonic

Random

Harmonic

Random

Acoustic source

39.13

42.94

-4.1

-3.7

Point force

13.16

26.69

-8.8

-5.7

Both actuators

4.6

7.0

-13.4

-11.5

217

Chapter 7. Active Control of Random Sound Transmission

7.4 Experiment 7.4.1 Experimental set-up

To show the validity of the analytical model, an experiment was performed with the same configuration of the simulation model considered in the last section. Figure 7.13(a) shows the arrangement of the experiment where an elevated large speaker generates the acoustic incident wave which then excites the Al(aluminium) plate fitted in a baffle. The experimental work was undertaken in the University of Southampton anechoic chamber whose cut-off frequency is approximately 100 Hz. The speaker was set at about 2 m from the plate and was facing down at an angle of approximately 45°. In Figure 7.13(b), a reference microphone measuring the incident wave was facing down and located at the right-hand edge of the plate. A wood panel of the area 3.5m × 0. 9m with the thickness 12 mm was fitted into the plate as a model of the acoustic baffle. The AVC Instrumentation Series 712 piezoelectric actuator, which uses piezoceramic materials in a sealed titanium housing, having a dummy mass 150 g on its top was bolted to the plate in order to generate a point force to the plate. Since the unsupported piezoelectric actuator can be modelled as a 1 d.o.f(degree of freedom) massspring system, the transmitted force to the plate is the multiplication of the dummy mass and its acceleration[Piezoelectric Actuator Series 712 Operation Guide]. Thus, the transmitted force can be calculated by the response of an accelerometer located on the top of the mass. The duct-like rectangular enclosure was constructed with the same dimensions of the simulation model considered in the last section(the cavity with dimensions of L1×L2×L3 where L1 = 1.5m , L2 = 0.3m , and L3 = 0.4m , and the thickness of aluminium plate 5 mm), and is clearly shown in Figure 7.13(c) and (d). In Figure 7.13(c), a speaker is installed at the left end of the enclosure to control the transmitted sound inside the enclosure. To make the acoustically rigid boundary condition, the wood walls were made of Plywood with thickness 25 mm and they were surrounded by 75 mm depth sand layers packed by the extra container shown in Figure 7.13(d). To make the simply support boundary condition for the plate, steel strips with 1.25 mm thickness are bolted around the peripheral of the plate as shown in Figure 7.14(a) and (b). The concept of the design uses the characteristics of the thin strip that are

218

Chapter 7. Active Control of Random Sound Transmission

relatively rigid to in-plane motions and flexible to rotation. The rotational flexibility can be controlled by changing the height of the strip.

(a)

(b)

(c) Figure 7.13 Experimental set-up

‘ ’

(d)

“ ” • –

(a) Drawing

(b) Photo

Figure 7.14 The simply supported plate; in (a) ‘ Al plate 5 mm, ’ Steel strip(Shimstock), “ Baffle, ” Adapter(steel), • Al angle, and – Plywood 25 mm. 219

Chapter 7. Active Control of Random Sound Transmission

The experimental work was mainly to identify the system responses of the primary and secondary plants, with which off-line optimal control of harmonic and random sound transmission is performed in Section 7.4.3. A moving microphone was used to measure the sound response at eleven equi-distanced positions along the duct-like enclosure whose distance is L1 10 . The primary plant represents the inside error microphone response to the incident wave measured by the outside reference microphone. The acoustic and structural secondary plants were identified by the error microphone response to the input signals of the speaker and the force actuator, respectively.

A low-passed version of random noise was used to excite the three actuators for the system identification of the plants; the external speaker for the primary plant, the internal speaker for the acoustic secondary plant and the force actuator for the structural secondary plant. Figure 7.15(a) shows the power spectrum of the original random noise, and its low-passed version is shown in (b) after passing through a low pass filter whose frequency responses are shown in (c) and (d). Due to the gradual transition band of the filter, the cut-off frequency of the passband was set to 300 Hz. The merits of using the low-passed version of random signal to excite each actuator are that it reduces excitation of higher acoustic modes as well as aliasing error on sampling. Figure 7.16 shows the power spectra of each actuator output signal; the spectrum of the external speaker(a) was measured by the reference microphone, that of the internal speaker(b) was measured by an accelerometer attached to the cone surface of the speaker, and that of the force actuator(c) was measured by an accelerometer attached on top of the unsupported force actuator. Due to the inherent mechanical dynamics of the speakers, they show poor performance at low frequencies. Gradual reduction at high frequencies is the influence of the lowpass filter. In Figure 7.16(c), it is interesting that the acceleration response which is proportional to the plate input force shows resonance characteristics. It is due to strong coupling between the actuator and the mounting plate. Experimental equipment used are listed in Table 7.4 with their configurations.

220

Chapter 7. Active Control of Random Sound Transmission

10

4 3

0

2

Power Spectrum(dB)

Power Spectrum(dB)

1 0 −1 −2

−10

−20

−30

−3 −4

−40

−5 −6 0

50

100

150

200 250 Frequency(Hz)

300

350

−50 0

400

(a) Random noise

50

100

150

200 250 Frequency(Hz)

300

350

400

350

400

(b) Low-passed random noise

5

1 0

0

−1 −2 Phase(Radian)

Amplitude(dB)

−5

−10

−3 −4 −5

−15

−6 −7

−20

−8 −25 0

50

100

150

200 250 Frequency(Hz)

300

350

400

(c) Amplitude of the filter

−9 0

50

100

150

200 250 Frequency(Hz)

300

(d) Phase of the filter

Figure 7.15 Filter characteristics(dB ref. = 1 V)

221

Chapter 7. Active Control of Random Sound Transmission

10

0

Power Spectrum(dB)

−10

−20

−30

−40

−50

−60 0

50

100

150

200 250 Frequency(Hz)

300

350

400

300

350

400

300

350

400

(a) −10

Power Spectrum(dB)

−20

−30

−40

−50

−60

−70 0

50

100

150

200 250 Frequency(Hz)

(b) 20

10

Power Spectrum(dB)

0

−10

−20

−30

−40

−50

−60 0

50

100

150

200 250 Frequency(Hz)

(c) Figure 7.16 Input signal characteristics of the external speaker(a), the internal speaker(b) and the force actuator(c). (dB ref. = 1 V)

222

Chapter 7. Active Control of Random Sound Transmission

Table 7.4 Experimental equipment and their configurations

Analyser

Hewlett Packard Frequency Response Analyser HP 3566A (Sampling frequency : 1024 Hz, Frequency span : 0.5 Hz)

Low-pass filter

Kemo Dual Variable Filter, (cut-off 300 Hz)

Acoustic equipment

Power Supplier: B&K 2633 × 2 Microphone: B&K 4135 × 2 , 1/4” dia. Calibrator : LARSON-DAVIS Laboratories. (114 dB 250 Hz) Speakers : 165mm 50W Bass Driver YN25C, (45 Hz-7 kHz) 400mm (cut-off 45 Hz) Speaker Amplifier : Professional Mos-Fet Power Amplifier DVA 900, Audio management

Vibration equipment

Accelerometer : B&K 4393 × 2 B&K 4374 Pre-Amplifier : B&K 2635 × 2 Piezo Actuator : PCB AVC Series 712. + 150g mass Amplifier : 790 Series Power Amplifier, AVC A Division of PCB Piezotronics Inc.

223

Chapter 7. Active Control of Random Sound Transmission

7.4.2 Characteristics of the system

In this section, experimental measurement results are compared with theoretical ones. The impedance and mobility approach for analysing structural-acoustic coupled systems was considered in Chapter 2, and a theoretical model for sound transmission problems were shown in Chapter 4.

The internal speaker shown in Figure 7.13(c) was used as the acoustic source, and an accelerometer was attached on its surface to measure its acceleration. The acceleration response was used for calculating the source strength of the acoustic source assuming that the motion of the speaker is uniform all over its surface. The source strength can be calculated from the integral of acceleration which, in the frequency domain, is simply a jω where a is the measured acceleration. Figure 7.17 shows a comparison between weakly(a) and fully(b) coupled models at the microphone location 5, i.e., x1 = 4 L1 10 , and both experimental(solid line) and theoretical(dashed line) results are shown. As in the simulation model in Section 7.3, the same damping ratio of 0.01 for both plate and cavity was used with 0.2 s for the time constant of the first acoustic mode. In the weakly coupled modelling, the theoretical result was obtained by assuming the plate is a hard wall i.e., the coupling matrix Za Ycs = 0 in equation (7.2), whereas the experimental result was obtained by evenly distributing additional heavy masses on the plate. In the fully coupled modelling, both theoretical and experimental results clearly show structural coupling effects near the structural natural frequencies. It demonstrates the difference between weak and full coupling, and proves the validity of experimental and theoretical models.

With the force actuator shown in Figure 7.13(b) exciting the plate, the point mobility at the excitation point and the acoustic pressure response at the microphone location 5 are compared with their analytical results in Figure 7.18(a) and (b), respectively. The point mobility was calculated from measuring responses of two accelerometers located at the top mass of the actuator and at the excitation position on the plate, respectively. It shows a good agreement in both amplitude and phase responses between experimental and theoretical results above about 80 Hz. In Figure 7.18(a), the small peak after the 4th structural mode at 221.6 Hz is due to the

224

Chapter 7. Active Control of Random Sound Transmission

strong coupling with the 2nd acoustic mode at 226.6 Hz. A comparison of theoretical and analytical natural frequencies in the structural-acoustic coupled system is shown in Table 7.5 where the uncoupled acoustic and structural modes are listed. For the experimental structural modes, coupled modes are listed for the comparison with the theoretical uncoupled ones.

140

3

130

2 120

Phase(Radian)

Sound Pressure(dB)

1 110

100

90

0

−1

80

−2 70

60 0

50

100

150

200 250 Frequency(Hz)

300

350

−3 0

400

50

100

150

200 250 Frequency(Hz)

300

350

400

150

200 250 Frequency(Hz)

300

350

400

(a) Weakly coupled modelling 140

3

130

2 120

Phase(Radian)

Sound Pressure(dB)

1 110

100

90

0

−1

80

−2 70

60 0

50

100

150

200 250 Frequency(Hz)

300

350

400

−3 0

50

100

(b) Fully coupled modelling Figure 7.17 Acoustic transfer impedances at the microphone location 5 subject to the internal acoustic excitation; experiment(solid line) and theory(dashed line) (dB ref. = 20 μ Pa)

225

Chapter 7. Active Control of Random Sound Transmission

160

3

150

2

140

Phase(Radian)

Sound Pressure(dB)

1 130

120

0

−1 110

−2 100

90 0

50

100

150

200 250 Frequency(Hz)

300

350

−3 0

400

50

100

150

200 250 Frequency(Hz)

300

350

400

50

100

150

200 250 Frequency(Hz)

300

350

400

(a) Structural mobility 120

3

110

2

100

Phase(Radian)

Sound Pressure(dB)

1 90

80

0

−1 70

−2 60

50 0

50

100

150

200 250 Frequency(Hz)

300

350

−3 0

400

(b) Acoustic pressure response Figure 7.18 Structural point mobility and the acoustic pressure response at the microphone location 5 subject to the force excitation; experiment(solid line) and theory with the fully coupled modelling(dashed line) (dB ref. = 20 μ Pa)

Table 7.5 A comparison of natural frequencies of the structural-acoustic coupled system Acoustic Mode

Theory

Experiment

Error(%)

(1,0,0) (2,0,0) (3,0,0) Structural Model

113.3 226.6 340

113 227.5 339

0.3 0.4 0.3

(1,0) (2,0) (3,0) (4,0) (5,0) (6,0)

140.5 156.7 183.7 221.6 270.2 329.7

143 160 180.5 222.5 271 328.5

1.7 2.1 1.8 1.7 0.3 0.3

226

Chapter 7. Active Control of Random Sound Transmission

Figure 7.19 shows the acoustic pressure response at the microphone location 5 subject to the incident plane wave excitation which is measured by the external microphone shown in Figure 7.13(b). The noisy response at frequencies below about 100 Hz was due to poor coherence between the external and internal microphones, and the peaks over about 100 Hz are at the natural frequencies of the coupled system. Furthermore, due to the physical difficulty of realising the plane wave with a fixed incident angle, the measurement result is somewhat different from the theoretical prediction. In the external sound field whose arrangement is shown in Figure 7.13(a), the external speaker would emit a spherical wave so that its incident angle on the plate would change according to the position of the plate. However, the external microphone is able to measure a time advanced version of the incident wave so that a good control performance is expected for the off-line experiment of global feedforward control which is considered in the following section.

110

3

100

2

90

Phase(Radian)

Sound Pressure(dB)

1 80

70

0

−1 60

−2 50

40 0

50

100

(a) Amplitude

150

200 250 Frequency(Hz)

300

350

−3 0

400

50

100

150

200 250 Frequency(Hz)

300

350

400

(b) Phase

Figure 7.19 Acoustic pressure response at the microphone location 5 subject to the incident plane wave excitation; experiment(solid line) and theory with the fully coupled modelling(dashed line) (dB ref. = 20 μ Pa)

227

Chapter 7. Active Control of Random Sound Transmission

7.4.3 Off-line control of sound transmission

For the identification of each plant, a FIR filter was used for modelling each impulse response. Since mean square error is adopted as the criterion of modelling performance, the modelling of each plant is a SISO(single-input-single-output) Wiener filter problem considered in Chapter 5. General discussions on system identification problems are available, for example, see [Ljung(1987)]. In this case, it is convenient to define the error as the difference between the desired and approximated signals. The desired signal can be measured by the internal microphone, and the approximated signal is a modelled response after passing through the filter. The received signal of the filter can also be measured and is the input signal of each plant whose frequency component is shown in Figure 7.16(a,b,c). Both desired and received signals were measured simultaneously for a time record 40 s so that each has a 40960 number of data, and a FIR filter of 1024 coefficients was used for the identification of each plant. Figure 7.20 shows the impulse responses of primary and secondary plants at the microphone location 5 i.e. x1 = 4 L1 10 . Instead of the source strength, the derivative of source strength which is measurable directly from the accelerometer attached on the speaker was used as the input signal in Figure 7.20(b). It is because the method of multiplying jω used for obtaining the acoustic response subject to the source strength input in Figure 7.17 causes a large non-causal error in the time domain. A direct use of a velocity type sensor would be a way to avoid this problem. A longer length filter is desirable for modelling such a lightly damped system, but it imposes a heavy computational burden since the identification method involves a matrix inversion of the large auto-correlation matrix. To reduce the number of filter coefficients, the infinite-impulse-response(IIR) filter using an autoregressive-moving-average(ARMA) model may be preferable for lightly damped systems as suggested by [Vipperman, Burdisso and Fuller(1993)]. Whereas the FIR filter is useful for modelling highly damped systems such as inside a car or a room.

Table 7.6 shows the modelling error of each plant that is defined as the variance ratio of error and desired signals. Except the microphone position 1 of the acoustic secondary path where the locations of speaker and microphone are the same, very accurate modelling with error values less than 1 % are achieved all over the plants. In general, poor coherence between the received and desired signals induces large modelling error. 228

Pressure(Pa)

Chapter 7. Active Control of Random Sound Transmission

100

0

Pressure(Pa)

−100 0

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

5

0

−5 0

Pressure(Pa)

0.1

500

0

−500 0

Figure 7.20 Modelled impulse responses of each plant at the microphone location 5; the primary path(top), and acoustic(middle) and structural(bottom) secondary plants

Table 7.6 Modelling error of the primary and secondary plants

Position

Primary plant (%)

1 2 3 4 5 6 7 8 9 10 11

0.63 0.28 0.29 0.46 0.28 0.21 0.47 0.54 0.62 0.43 0.44

Acoustic secondary plant (%) 2.88 0.17 0.21 0.25 0.16 0.11 0.17 0.20 0.30 0.09 0.16

229

Structural secondary plant (%) 0.09 0.09 0.04 0.09 0.14 0.06 0.04 0.07 0.10 0.05 0.07

Chapter 7. Active Control of Random Sound Transmission

Using the impulse responses identified from the experiment we apply the theoretical optimal control model developed in Section 7.2. The off-line control procedure is straightforward, and is in fact the same as Section 7.3.3 except using the experimentally identified impulse responses. Figure 7.21, 7.22, and 7.23 show control performance results and constrained optimal filter responses according to the type of actuators used; the acoustic actuator, the structural actuator, and the hybrid use of both actuators, respectively. The approximated acoustic potential energy less than 100 Hz are highly noise-contaminated due to poor sensitivity between the external and internal microphones at low frequencies. Compared with the analytical result in Figure 7.10, differences at higher frequencies are due to the practical difficulty of generating a plane wave with a fixed incident angle.

The performances of harmonic sound controllers denoted by dotted lines in Figures 7.21(a) and 7.22(a) show that the acoustic actuator is effective in controlling cavity-controlled modes(113 Hz, 227.5 Hz and 339 Hz), whereas the structural actuator is effective especially in controlling plate-controlled modes(at 143 Hz, 160 Hz, 180.5 Hz, 271 Hz and 328.5 Hz) although it also offers large reductions at the cavity-controlled modes in this model studied. By using the both actuators in Figure 7.23(a), a larger reduction was achieved especially at the first cavity-controlled mode at 113 Hz. The performance of the random sound controller using the acoustic actuator almost reaches that of the harmonic controller(Figure 7.21a), and the difference gets bigger when the structural actuator is used(Figure 7.22a). As shown in Figure 7.23(a), the hybrid use of both actuators does not make the difference any larger than the structural actuator does in Figure 7.22(a). The overall control performance of the above three cases are tabulated in Table 7.7. Again as emphasised in the simulation, the hybrid use of both type actuators offers a better performance over the separate uses of each actuator in both harmonic and random sound fields. Unlike for the simulation case discussed in Section 7.3.3, the constrained Wiener filter responses have longer tails because of the poorly correlated responses at low frequencies.

The aim of this off-line experiment was to show the validity of theoretical models for both harmonic and random sound transmission. As discussed above, the experimental results show a very good agreement with the theoretical results in Section 7.3.3. Thus the theoretical model using Wiener filter in Section 7.2 can be applied to the design of optimal controllers for both harmonic and random sound transmission. 230

Chapter 7. Active Control of Random Sound Transmission

8

60

6

55

2

45

Optimal Filter(Hoq)

Acoustic Potential Energy(dB)

4

50

40 35 30

0 −2 −4 −6

25 20

−8

15

−10

10 0

50

100

150

200 250 Frequency(Hz)

300

350

−12 0

400

(a) Control performance

50

100

150

200 Samples

250

300

350

400

(b) Response of the constrained optimal controller

Figure 7.21 The control performance of using the acoustic actuator and the response of the constrained optimal feedforward controller; the original cost function(solid) and these after using the constrained(dash) and unconstrained optimal controllers.

0.1

60

0.05

50 45

Optimal Filter(Hof)

Acoustic Potential Energy(dB)

55

40 35 30

0

−0.05

25 −0.1

20 15 10 0

50

100

150

200 250 Frequency(Hz)

(a) Control performance

300

350

−0.15 0

400

50

100

150

200 Samples

250

300

350

400

(b) Response of the constrained optimal controller

Figure 7.22 The control performance of using the structural actuator and the response of the constrained optimal feedforward controller; the original cost function(solid) and these after using the constrained(dash) and unconstrained optimal controllers.

231

Chapter 7. Active Control of Random Sound Transmission

8

6

60 4

Optimal Filter(Hoq)

50 45

2

0

−2

−4

−6

40 −8 0

35

50

100

150

200 Samples

250

300

350

400

0.08

0.06

30 0.04

25

Optimal Filter(Hof)

Acoustic Potential Energy(dB)

55

20

0.02

0

−0.02

15

−0.04

10 0

50

100

150

200 250 Frequency(Hz)

300

350

(a) Control performance

400

−0.06 0

50

100

150

200 Samples

250

300

350

400

(b) Response of the constrained optimal controllers

Figure 7.23 The control performance of using the both actuators and the response of the constrained optimal controllers; (a) the original cost function(solid) and these after using the constrained(dash) and unconstrained optimal controllers, (b) acoustic (top) and structural (bottom) actuators

Table 7.7 Comparison of performances according to the use of each and both actuators

Use of the actuators

Energy ratio J o J i (%)

Overall reduction(dB)

Harmonic

Random

Harmonic

Random

Acoustic source

30.25

33.63

-5.2

-4.7

Point force

13.58

21.18

-8.7

-6.7

Both actuators

7.93

12.18

-11.0

-9.1

232

Chapter 7. Active Control of Random Sound Transmission

7.5 Conclusion The active control of random sound transmission has been considered in a fully coupled structural-acoustic system, and its control performance has been compared with that from the active control of harmonic sound transmission. A theoretical model to design optimal controllers for harmonic and random sound has been presented using the theoretical basis of the Wiener filter. In particular, the role of acoustic and structural actuators for the control of harmonic and random sound has been investigated. Both theoretical and experimental results demonstrate 1. In harmonic sound control, the acoustic actuator is effective in controlling cavity-controlled modes while the structural actuator is effective in controlling plate-controlled modes. 2. In random sound control, since the wave propagation delay in the structural secondary plant is longer than that in the acoustic secondary plant, the overall reduction difference between harmonic and random sound using the structural actuator is bigger than the difference using the acoustic actuator(see Tables 7.3 and 7.7). 3. the hybrid use of both acoustic and structural actuators is a desirable configuration for the active control of both harmonic and random sound transmission into a general coupled system whose response is governed by plate and cavity controlled modes.

233

Chapter 8. Conclusion

CHAPTER 8

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK

8.1 Conclusions

The active control of sound in structural-acoustic coupled systems has been considered. Part 1 of the thesis(Chapter 2 and 3) considered analysis of structural-acoustic coupled systems, and Part 2(Chapter 4) and Part 3(Chapter 5, 6, and 7) considered the active control of harmonic and random sound, respectively.

Chapter 2 introduced the impedance-mobility approach into the analysis of structural-acoustic coupling by reformulating the traditional description of the structural-acoustic systems, and a comprehensive theoretical model was presented in both modal and physical co-ordinates systems. The formulations were expressed in terms of vectors and matrices that are convenient for numerical computations. Due to the mismatch of dimensions of impedance and mobility

234

Chapter 8. Conclusion

between structural and acoustic systems, new mechanical terms were introduced for the coupled system analysis; the coupled acoustic impedance and the coupled structural mobility. These two terms are necessary whenever two different physical systems are coupled. The Fu(force-velocity) and p-Q(pressure-source strength) representations were also introduced for the impedance and mobility representations of a complete coupled system. The criterion of weak coupling was of particular interest through the theoretical formulations. The criterion presented shows that weak coupling is an inherent characteristic of systems studied and that this criterion does not depend on the type of actuator or excitation location. Overall, it was demonstrated that the approach is well suited for the analysis of structural-acoustic coupled systems as well as for evaluating the degree of coupling. It is anticipated that the impedance-mobility approach may be applicable to general structurestructure coupled systems problems with the appropriate introduction of new terms to the case.

Based on the impedance-mobility approach, Chapter 3 considered a new intuitive way to understand the physics of structural-acoustic coupling by way of the visualisation of coupling. Through the F-u and p-Q representations of modal coupling based on the impedance-mobility approach, a series of analogous mechanical models consisting of lumped parameters( mass, spring, damper) were presented to represent various cases of modal coupling; the coupling between a single structural mode and a single acoustic mode, the coupling between a single structural mode and several acoustic modes, and the coupling between a single acoustic mode and several structural modes. The analogy was further extended to a simple physical system, a driver-pipe system, and an intuitive way of looking at structural-acoustic coupling phenomena in the physical co-ordinates was provided. It is expected that the impedance-mobility approach developed in this work may be extended to the analysis of structure-structure coupled systems by defining appropriate terms to the case.

Chapter 4 considered the active control of the sound transmission into a ‘weakly coupled’

235

Chapter 8. Conclusion

structural-acoustic system. The mechanism of controlling sound transmission into a rectangular enclosure was investigated, and the role of each type actuator was studied using the concept of control spillover. The results obtained demonstrated that a single point force actuator is effective in controlling well separated plate-controlled modes, whereas, a single acoustic piston source is effective in controlling well separated cavity-controlled modes. By using the hybrid approach with both structural and acoustic actuators, improved control effects on the plate vibration, further reduction in sound transmission, and reduced control efforts of the actuators can be achieved. Since the acoustic behaviour is governed by both plate and cavity resonances, the hybrid control approach can be desirable in controlling sound transmission in a ‘weakly coupled’ structural-acoustic system.

Chapter 5 considered the active control of stationary random sound and vibration in SISO systems. An easy systematic deterministic method for the active control of random sound was proposed based on two facts. Firstly, the Wiener filter with the constraint of causality is the optimal controller of stationary random sound, and secondly, white noise is equivalently replaced by an impulse. Thus, as long as the input primary source is white noise, the active control of stationary random sound transforms to the active control of an impulse response. Both feedforward and feedback techniques were applied for the optimal control of SISO systems. To obtain simple analytical expressions for the Wiener filter and the mean square error, the secondary plant was assumed to be minimum phase in feedforward control, and both primary and secondary plants were assumed to be minimum phase in feedback control. The proposed deterministic method was applied to the vibration control of a one d.o.f system as an application example. Conclusions obtained from theoretical and numerical analysis in this chapter were 1. It was shown with a numerical example that white noise can be equivalently replaced by an impulse for the purpose of active control of random sound. The advantages of the replacement are; Firstly, an easy systematic deterministic approach can be applied to the optimal control of stochastic signals. Secondly, the calculation time is greatly reduced since no time consuming averaging process is required.

236

Chapter 8. Conclusion

2. When a feedforward controller has zero time delay, the realisable filter for random sound is the same as the causally unconstrained Wiener filter. 3. Perfect control can be achieved in feedforward control, whereas in feedback control it can never be achieved due to stability problems. 4. The control performances in both feedforward and feedback control are adulterated as the electric time delay and system damping increase(see Figure 5.14). 5. The phase angle difference function suggested can be used as a practical way of predicting control performance. In the active control of SISO systems, a bigger function to the plus angle(phase lead) brings a poorer control performance.

Chapter 6 described a comprehensive treatment of Wiener filter theory for MIMO systems in both continuous and discrete time domains, and its applications to active control of random sound were presented for both feedforward and feedback techniques for MIMO systems. It was shown that the solutions of causally unconstrained and constrained Wiener filters are the optimal controllers for harmonic and random sound fields, respectively. From the application example with an acoustic duct, numerical models for global and local feedforward control of random sound were presented using the time domain formulation. The results obtained from the application example demonstrated 1. A finite number of microphones can be successfully used for the global control of harmonic and random sound fields. 2. It was shown by a numerical example with various time delays that the global active control of random sound in purely acoustic systems can be interpreted as the wave absorbing mechanism on the control speaker. 3. As system damping increases, the global overall reduction for the feedforward control of both harmonic and random sound is decreased. 4. It was been shown with numerical examples that the phase difference function can be used as an approximate measure of the control performance of a random sound controller in SISO systems.

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Chapter 8. Conclusion

Chapter 7 is the concluding chapter of this thesis, and the acoustics and control theories developed in the previous chapters were applied. The chapter considered the active control of random sound transmission in a fully coupled structural-acoustic system, and its control performance was compared with that from the active control of harmonic sound transmission. A theoretical model to design optimal controllers for harmonic and random sound was presented using the theoretical basis of the Wiener filter. In particular, the role of acoustic and structural actuators for the control of harmonic and random sound was investigated. Both theoretical and experimental results demonstrated 1. In harmonic sound control, the acoustic actuator is effective in controlling cavity-controlled modes while the structural actuator is effective in controlling plate-controlled modes. 2. In random sound control, since the wave propagation delay in the structural secondary plant is longer than that in the acoustic secondary plant, the overall reduction difference between harmonic and random sound using the structural actuator is bigger than the difference using the acoustic actuator(see Tables 7.3 and 7.7). 3. the hybrid use of both acoustic and structural actuators is a desirable configuration for the active control of both harmonic and random sound transmission into a general coupled system whose response is governed by plate and cavity controlled modes.

Overall, the thesis has shown that the hybrid use of both acoustic and structural actuators is a desirable configuration for the active control of both harmonic and random sound transmission into a general coupled system.

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Chapter 8. Conclusion

8.2 Recommendations for future work It is anticipated that the impedance-mobility approach presented in Part 1 may be applicable to the general structure-structure coupled systems problems with some appropriate introduction of new terms to the case. The corresponding mechanical analogy for the case would also be possible. It is recommended that since the proposed impedance-mobility approach is suitable for the power calculation, power flow analysis between a structure and an acoustic system would be helpful to understand the structural-acoustic coupling.

The acoustic potential energy adopted as the cost function of this theoretical study is not practical since it requires many microphones to estimate. For an efficient practical implementation of active control of harmonic and random sound, analytical studies of a local control configuration to achieve global reduction are required. The local control configuration can be, for example, using self-sensing actuators without using remote microphones.

For the active control of random sound, the use of ARMA(auto-regressive-moving average) models for plants and filters should be investigated. Robust control considering modelling uncertainty and non-linearity should be conducted for the real implementation of feedback control. Since the proposed active control method based on the classical Wiener filter theory presented in Part 3 is not practical for MIMO feedback control systems with the current computing processors, a new computationally efficient technique is required.

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