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I University

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This is to certify that the

dissertation entitled

Global Active Noise Control of a One-Dimensional Acoustic
Duct Using a Feedback Controller

presented by

Andrew James Hull

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mechanical Engineering

 

 

 

Mm, 41 we
/ 7

MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

rm 0 712951.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Affirmative Action/Equal Opportunity Institution

 

GLOBAL ACTIVE NOISE CONTROL OF A ONE-DIMENSIONAL
ACOUSTIC DUCT USING A FEEDBACK CONTROLLER

By

Andrew James Hull

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1990

 

geax

e05

ABSTRACT

GLOBAL ACTIVE NOISE CONTROL OF A ONE-DIMENSIONAL
ACOUSTIC DUCT USING A FEEDBACK CONTROLLER

By

Andrew James Hull

Active noise control of acoustic enclosures is a classical engineering problem. The
active noise control of a one-dimensional hard-walled duct with a partially dissipative
boundary condition is addressed in this dissertation. Previous techniques have attacked
this problem by developing adaptive filters to decrease the noise level at a single
measurement location; they ignore the problem of noise reduction at other locations in the
duct. The work presented here applies classical control theory to reduce noise levels in a
one-dimensional acoustic enclosure. Classical control theory provides a basis to reduce the
noise levels in the duct globally, rather than at a single location. This is accomplished by
adding a response measurement microphone and a control speaker to the open loop system.
Pressure measurements are taken at a single location and passed to an observer, which
provides state estimates of the system. Once the state estimates are known, a pole
placement control algorithm is used to lower the noise level. Pole placement produces
noise control globally, rather than at a single location. Experimental result obtained here
show that the noise level in the duct can be reduced by 55% when the system is excited by

random noise excitation.

Copyright

Copyright by
Andrew James Hull

1990

Certification
This is to certify that the dissertation entitled

Global Active Noise Control of a One-Dimensional Acoustic
Duct Using a Feedback Controller

presented by

Man/W §//§/€0

Andrew J. 11 Date
DOCTOR PHILOSOPHY Candidate

 

has been accepted towards fulfillment of the requirements for a
DOCTOR OF PHILOSOPHY DEGREE in Mechanical Engineering at

Michigan State University

by

  
  

 

 

 

Clark J. Radcliffe 7 Date
Associate Professor, Disse on Advisor

W. . W REA/‘3‘ 5/ 1 Ar/ q 5
Charles R. MacCluer Date

Professor, Dept. of Mathematics

MW WW gflQ/TO

Alan G. Haddow / / Date
Assistant Professor, Dept. of Mechanical Engineering

M M/ / W.) . 374/90

H. Roland Zapp Date
Associate Professor, Dept. of Electrical Engineering

Acknowledgements

I would like to express my sincere appreciation to my advisor Clark Radcliffe, for
helping me all the way through the dissertation. His guidance was always positive even in
the darkest of dissertation times. Additionally I would like to thank Chuck MacCluer for
his help and insight in the theoretical areas of the dissertation.

I would like to thank my committee members Roland Zapp and Alan Haddow for
their comments concerning the dissertation. Also I would like to thank Milan Miklavcic,
J.C. Kurtz, Erik Goodman and Robert Soutas-Little, all of whom were helpful in one way
or another. I would like to thank Bob Rose and Leonard Eisele who made the complicated
fixtures necessary for the experiments in this work.

I would like to thank my mother, brother and sister for their encouragement when I
was in graduate school.

I would like to thank my officemate, Steve Southward for his friendship over the
years and his understanding of this problem. I would like to thank Mark Barlow for
helping me with the development of the digital signal processing code. Finally I would like
to thank my friends from around the East Lansing area over the past five years. They
include: Bashar AbdulNour, Guy Allen, Jeff Beranek, Teresa Brown, Dave Butcher,
Yossi Chait, Greg Fell, Eva-Marie Fadel, Jane Hawkins, Jon Hickey, Dave Miner, Jim

Oliver, Ace Sannier, and Shawna Sannier among others.

Table of Contents

List of Tables
List of Figures
Chapter 1 - Introduction
1.1 Overview
1.2 A New Control Technique
1.3 Chapter Summary
Chapter 2 - State Space Model Representation
2.1 Introduction
2.2 System Model
2.3 Separation of Variables
2.4 Series Solution
2.5 A New Inner Product
2.6 State Space Formulation
2.7 Frequency Response
2.8 Accuracy of Truncated Solution
2.9 Transient Response
Chapter 3 - Experimental Verification of the State Space Model
3.1 Model Verification Experiment
3.2 Pressure Excitation, Constant Acoustic Impedance
3.3 Mass Flow Excitation, Constant Acoustic Impedance
34 Pressure Excitation, Nonconstant Acoustic Impedance

3.5 Transient Response

vi

OOQUIUINN

10
15
17
18
19
21
26
26
26
29
29
34

Chapter 4 - Acoustic Impedance Measurement

4.1 Introduction
4.2 System Model and System Eigenvalues
4.3 Acoustic Impedance Computation

4.4 Acoustic Impedance Measurement Experiment

Chapter 5 - State Estimation

5.1 Introduction

5.2 Observer Equations

5.3 Observer Gain Placement

5.4 Closed Form Solution

5.5 Numerical Simulations for Continuous Time Observers

5.6 Numerical Simulations for a Discrete Time Observer

Chapter 6 - Active Noise Control

6.1 Active Noise Control Using Pole Placement
6.2 Controller Equations

6.3 Control Gain Placement

6.4 Frequency Domain Simulations

6.5 Time Domain Simulations

6.6 Actuator Dynamics

6.7 Active Noise Control Experiment

Chapter 7 - Conclusions

7.1 Dissertation Summary

7.2 Directions for Future Work

Appendix. Wave Equation Derivation
Bibliography

vii

38
38
39
41

53
53
55
56
58
61
65
68
68
68
69
70
76
78
80
86
86
87
89
91

List of Tables

Table Page
3.1 Calculated Acoustic Impedance. 32
4.1 Measured Duct Eigenvalues for the Foam End. 49
4.2 Calculated Acoustic Impedance for the Foam End 49
4.3 Measured Duct Eigenvalues for the Capped End. 51

4.4 Calculated Acoustic Impedance for the Capped End. 51

viii

Figure

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

List of Figures

Third Eigenfunction for Duct Particle Displacement with End Impedances:
K = .5+Oi and K = 0+0i.

Third Eigenfunction for Duct Particle Displacement with End Impedances:
K = O+3.0i and K = 0+0i.

Third Eigenfunction for Duct Particle Displacement with End Impedances:
K = 0.8+0.5i and K = 0+0i.

End-Point (x=O) Pressure Excited Steady State Frequency Response with an
Impedance Constant of K = O.3+O.2i at x = 0.4267 m.

Frequency Response Truncation Error of an 11 Term Model with Impedance
Constant of K = O.3+O.2i.

Interior Point Mass Flow Excited Steady State Frequency Response with an

Impedance Constant of K=0.3+O.2i at x=0.4267 m. Excitation at xi=1.22 m.

Harmonic Time Response of 5 and 51 Term Duct Models with an Impedance
Constant of K = O.3+O.2i.

Pulse Time Response of a 1.5 m Duet with an End Impedance of K = O.3+0i.

Laboratory Configuration.

Frequency Response of Duet with End Pressure Excitation.
Frequency Response of Duet with Interior Mass Flow Excitation.
Frequency Response of Duct with Nonconstant Acoustic Impedance.
System Pressure Input at x=O.

Transient Response of Duct at x=0.792 m with 51 Terms.

Transient Response of Duet with 21 Terms.

Transient Response of Duet with 15 Terms.

ix

Page

11

12

13

22

23

24

25
25
27
28
30
33
36
36
37
37

4.1

4.2
4.3
4.4
5.1
5.2
5.3

5.4
5.5

5.6

5.7
6.1
6.2
6.3
6.4
6.5

6.6
6.7
6.8
6.9
6.10
6.11
6.12
A.l

System Eigenvalues An, for Constant K.

Laboratory Configuration.

Frequency Response of Duet with Foam End at x=0.792 m.
Frequency Response of Duct with Closed End at x=0.792 m.
Observer and Duct Diagram (Chen, 1984).

Duet with Observer and Input Speaker.

Closed Form Solution of Duct Pressure Calculated by Observer Compared
with Steady State Duct Model Response.

Integrated Observer Solution and Model Duct Response.

Integrated Observer Solution and Model Duct Response. Nonzero Initial
Conditions Showing Robustness of Observer Design.

State Estimate a1(t).

Discrete Time Observer Solution and Model Duct Response.

Active Noise Control System.

Plant and Control System.

Closed and Open Loop Frequency Response, Compensator at x=3.56 m.
Closed and Open LoOp Frequency Response, Compensator at x=1.37 m.

Closed and Open Loop frequency Response, Showing Change of Natural
Frequency of System.

Time Domain Response with Modeled Disturbance.
Time Domain Response with Unmodeled Disturbance.
Actuator Magnitude and Phase Plots.

Experimental Frequency Response, First Mode.
Experimental Frequency Response, Second Mode.
First Mode Response.

Second Mode Response.

Duet with Section dx.

42

45
50
52
54
55

62

63

67
69
70
73
74

75
77
78
79
83
84
85
85
90

That which does not kill us will surely make us strong

Nietzsche, F., circa 1800

xi

Chapter 1. Introduction

1.1. Overview

Active noise control can produce quieter environments that are safer, more
productive, and more comfortable. In work areas such as power plants, factories, and
offices that contain long duct assemblies, reduction of sound energy in the ducts generated
by fans and other machinery is often required. The advantages of reducing noise levels is
evident, and recently more communities have passed legislation to limit excessive noise.
Passive techniques such as the use of absorbent materials have little effect on low
frequency noise, and then active noise controllers are required. Passive techniques are also

difficult if not impossible to install in many acoustic environments.

An active noise control system in a duct usually consists of one or more cancellation
speakers driven by an algorithm designed to reduce noise levels. The duct normally has
one or more signal microphones at some location while noise is driven by excitation
through one end. A variety of different active noise suppression schemes have been
deve10ped in recent years (Swinbanks, 1973; Ross, 1981; Trinder and Nelson, 1983;
LaFontaine and Shepherd, 1983; Tichy et al., 1984; Roure, 1985; M0110 and Bernhard,
1987; Eriksson et al., 1988; Manjal and E1iksson, 1988; Warner et al., 1988). The control
techniques used in these studies are some type of adaptive filter which cancels noise at a
specific measurement location. The response at areas other than the measurement location
is either ignored or not measured and typically is increased. These studies have not used
classical feedback control theory and the wealth of design and stability theorems this theory

provides. Feedback control can provide global noise reduction to the system, unlike

previous studies whose noise reduction has been limited to one point or a small region of

the duct.

1.2. A New Control Technique

The objective of the research discussed here is the application of classical pole
placement to achieve active noise control in a duct. Pole placement modifies the
eigenstructure of the system to increase the dissipation of the 'duct and attenuate duct noise.
This dissertation develops a control technique for a one-dimensional hard-walled duct with
a control speaker at some location, a totally reflective entrance boundary condition, and a
partially absorptive termination boundary condition. The effect of the partially absorptive
boundary condition is to allow propagating and standing wave responses to exist in the
duct simultaneously. These two wave characteristics yield eigenvalues with eigenfunctions
that are not orthogonal on the interval of the length of the duct with respect to the ordinary
inner product. This is unlike previous discretized acoustic models which considered only

idealized reflecting boundary conditions.

Analogous problems which have propagating and standing wave characteristics
coexisting in the system include circular saw blades, large scale space structures, turbine
wheels, robot arms and electrical transmission lines with resistive loads. Understanding
the acoustic problem stated here can lead to a better understanding of these analogous
systems. Additionally, the first step in building any generalized three-dimensional control

system is a thorough understanding the one-dimensional control problem.

1.3. Chapter Summary

Chapter 2 develops the state space model for the acoustic duct. State space model

representation for the system is required to apply classical pole placement to the duct. The

3

model is derived from separation of variables. Due to the nonself-adjoint boundary
condition, traditional methods of orthogonal mode shapes cannot be applied to this
problem. The separation of variables problem is transferred onto a different interval, where
the time and space modes are decoupled. The problem is then transferred back onto the

original interval, where the state space model results.

Chapter 3 describes an experimental verification of the state space model.
Experimental verification of the state space model is essential if real time control is to be
applied to the duct. Steady state verification of end excitation and domain excitation are
presented for a frequency constant termination end. The problem of frequency dependent
terminations is discussed and an experiment incorporating this behavior is shown. The

transient solution to the duct is also experimentally verified.

Chapter 4 develops a measurement technique to determine the boundary condition
on the termination end of the duct. This is accomplished by taking the system eigenvalue
equation and solving the inverse problem for the acoustic impedance. This measurement
technique is experimentally verified and the stability of the measurement method is

discussed.

Chapter 5 develops a state estimation method using observer theory. This is
necessary since the feedback control system needs a knowledge of the system states, which
are not measurable directly. They can be estimated, however, by measuring the pressure at
some location in the duct which is used to drive a state estimator. The state estimator
requires real time integration, and the effects of four different integration routines is

benchmarked and discussed.

Chapter 6 develops a pole placement control algorithm for the duct. Frequency and
time domain simulations are shown. The effects of nonideal control actuators is discussed.

Active noise control is demonstrated experimentally. The effect of control from a

4

frequency domain standpoint is shown. Global noise attenuation is verified. System

stability and instability is discussed.

Chapter 7 summarizes this dissertation. Directions for future research are

discussed.

Chapter 2. State Space Model Representation‘

2.1. Introduction

The dynamic response of an enclosed acoustic system is determined by both the
governing differential equations and associated boundary conditions. The modeling of
one-dimensional acoustic response is a classical engineering problem (Rayleigh, 1878).
The response of hard-walled duets with idealized reflecting and/or nonreflecting
terminations to point source excitation has been developed previously (Snowdon, 1971;
Doak, 1973a; Swinbanks, 1973; Trinder and Nelson, 1983; Tichy et al., 1984). Models of
ducts with idealized totally reflective boundary conditions yield self-adjoint differential
operators and are easily discretized from their mutually orthogonal modes. Models of duets
with totally absorbent boundary conditions do not resonate, so wave propagation models

are used.

Actual acoustic systems have non-idealized, partially reflective boundary
conditions, yielding some combination of propagating and standing wave components in
their acoustic pressure response (Davis et al., 1954; Spiekermann, 1986; Spiekermann and
Radcliffe, 1988a, 1988b). Nonreflecting terminations in linear systems result in
propagating wave response while reflecting terminations result in standing wave response.
Currently available analytical model solutions do not take into account the possibility that
the acoustic response could be a combination of standing and propagating waves nor do

they consider the effect of partially absorptive boundary conditions on duct models. The

 

* This chapter is based on the paper “State Space Representation of the Nonself-Adjoint Acoustic Duct
System ,” accepted for publication in the ASME Journal of Vibration and Acoustics.

5

6

partially absorptive boundary condition produces a nonself-adjoint differential operator.
Traditional methods of orthogonal mode shape discretization for this class of problems
cannot be applied because the eigenfunctions are not orthogonal with the conventional inner
product over the domain of the operator and the conventional eigenfunction inner product
does not decouple the model's state equations. Nonself-adjoint operators may yield non-
conjugate, complex eigenvalues. Physically, the nonself-adjoint model results from energy

propagation down the duct and out the end.

This chapter develops an infinite order, diagonal, state space model of a duct with a
partially absorptive boundary condition. The model is intended for future development of a
time domain control theory since time domain control of high order systems frequently uses
state observers designed from state space system models. The analysis diagonalizes the
state space model; past work (Chait et al., 1988) has shown that diagonalized system
equations provide more accuracy when truncated than models which include off-diagonal
terms. The results here may also contribute to methods for evaluating duct end point

impedances and other issues in the development of duct designs.

2.2. System Model

The system model is of a one-dimensional hard-walled duct excited by a pressure
input at one end, a partially reflective boundary condition at the other end, and an arbitrary
number of mass flow inputs in the domain. This partially reflective boundary condition
allows the acoustic response model to include standing and propagating wave responses
simultaneously. The partially reflective condition in the duct allows some energy to be
dissipated out the end while the rest is reflected back into the system producing a bounded

complex system response from a nonself-adjoint differential operator.

The linear second order wave equation modeling particle displacement in a hard-

walled, one-dimensional duct is (Seto, 1971; Doak, 1973b)

9.22%; 23211014) __ 3 ammo _ k _ 3 M,(t)
atZ C 3X2 — ax p glfio‘ Xi)]at pS (2.1)

 

where u(x,t)=particle displacement (m), c=wave speed (tn/S), x=spatial location (m),

t=time (s), p=density of the medium (kg/m3), Mi(t)=mass flow input in the domain (kg/s),

xi=location of mass flow input (m), S=speaker area driving the mass flow input (m2),

P(t)=pressure excitation at x=0 (N/mz), and 8(x)=the Dirac delta function. The wave
equation assumes an adiabatic system, no mean flow in the duct, uniform duct cross
section and negligible air viscosity effects. The hard-wall assumption models the duct as
having dissipation only at the termination end. The one-dimensional assumption requires
the diameter of the duct to be small compared to its length. The duct's mean flow Mach

number is assumed to be much less than one.

The partially reflective boundary condition model at location x=L is the relationship
between the spatial gradient and the time gradient of particle displacement and is expressed

as (Seto, 1973; Pierce, 1981; Spiekermann and Radcliffe, 1988a)

93(14): _K(l)§}l(1,,t) K at 0+0i, 1+Oi, oo (22)
8x c at

where K=complex impedance of the termination end (dimensionless). Implicit in (2.2) is
the acoustic analogy with electrical systems in which volume velocity is analogous to
current and duct pressure is analogous to voltage. A second formulation called the
reciprocal acoustic mobility analogy is also sometimes used; and if applied to this system,
the parameter K in (2.2) would be the acoustic admittance. When Re(K) equals zero or is
infinity, the termination end of the duct reflects all the acoustic energy and the response is

composed of standing waves only. All other values of K yield some combination of

8

propagating and standing wave response (Spiekermann and Radcliffe, 1988a). When
K=1+0i the termination end of the duct absorbs all the acoustic energy and the response is
composed of propagating waves only. In general, the reflection coefficient (1-K)/(1+K)
gives the relative magnitude of the reflected pressure wave. The real part of K (acoustic
resistance) is associated with energy dissipation and is sometimes also called a loss
coefficient, as it is a measure of the amount of energy leaving the duct. The imaginary part
of K (acoustic reactance) is associated with conservative fluid compliance and/or inertia

effects.

The duct end at x=0 is modeled as a totally reflective, open end. This boundary

condition is
330.0 = 0 . (2.3)

This corresponds to an open duct end (or an electrical short circuit). The acoustic pressure

of the system is related to the spatial gradient of the particle displacement by (Seto, 1973)

P(x,t) = —pcziu-(x,t) . (2.4)
fix

The above four equations represent a mathematical model of the duct.

2.3. Separation of Variables

A decoupled series of ordinary differential equations in state space form which
represent the wave equation are now developed. They will incorporate the boundary

conditions (2.2-2.3) as well as initial conditions in the duct.

The eigenvalues and eigenfunctions of the model are found by applying separation

of variables to (2.2-2.3) and the homogeneous version of (2.1). Separation of variables

 

9

assumes each term of the series solution is a product of a function in the spatial domain
multiplied by a function in the time domain:

new 4? ) (233» (2.5)

e

Substituting (2.5) into the homogeneous version of (2.1) produces two independent

ordinary differential equations, each with complex valued separation constant 1., namely

 

 

2
x
d dxg") - fixer) = 0 (2. 6)
and
2
d :2“) - Azczrm = 0 . (2.7)

The separation constant 71:0 is a special case where X(x)=T(t)=1 to satisfy (2.2) and (2.3).
The spatial ordinary differential equation (2.6) is solved for 7&0 using the boundary
condition (2.3) yielding

X(x) = e“ + 6’“ . (2.8)

The time dependent ordinary differential equation yields the following general solution
T(t)=Aem+Be"‘°‘ .‘ (2.9)
Applying boundary condition (2.2) to (2.8) and (2.9) yields B=0 and the separation

constant

1 l—K ntti
7. =—lo — —— , n=0,i1,i:2, ..... 2.10
“ 2L g°[l+K] L ( )

Inserting the separation constant into (2.8) produces complex valued spatial eigenfunctions

(pn(x) where

(9.100 = 6“" + 6'1"“ . (2.11)

 

10

(“1
I
Unless the acoustic impedance K is zero or infinity, the eigenfunctions are not mutually 1

orthogonal, conventional modal analysis of the original model (2.1) is not possible, and the

time response cannot be found.

A typical eigenfunction for K real is shown in Figure 2.1. The solid line is the third
(n=3) eigenfunction for a partially reflective termination (K=0.5+0i), and a duct length of
1.524 m (5 ft) used in previous studies (Spiekermann, 1986; Spiekermann and Radcliffe,
1988a, 1988b). The dashed line is the eigenfunction for the third (n=3) mode with fully
reflective duct termination (K=0+Oi). Note the real value of the eigenfunction with
idealized reflecting termination, K=0+0i, and the complex value of the eigenfunction for
K¢O (or 1+0i). Increasing the real part of K causes a shift in the magnitude of the
eigenfunction near the termination and a rounding of the phase angles associated with
increasing dissipation. A typical eigenfunction for K imaginary is shown in Figure 2.2.
Increasing the imaginary part of K causes a spatial shift of the magnitude and the phase
angle towards the termination. The relative magnitude of the eigenfunction remains
unchanged. Figure 2.3 is an eigenfunction plot for K complex valued. It exhibits both

characteristics of K real and K imaginary.

2.4. Series Solution

Traditional methods of orthogonal mode shapes cannot be applied here due to the
nonself-adjoint operator. However, by extending the problem definition onto a virtual
duct, and then redefining the eigenfunctions over [-L,L], the time and space modes will

decouple and a solution to the problem can be found. This technique is explained below.

The solution to the forced wave equation is now written as a series solution plus a

time dependent term arising from the i=0 eigenvalue

 

 

 

 

11
1.25 _
Relative — .
1.00— " 1_ '\
0.75 l ‘ .' :3 Ill '1'
: 1‘. r: 3|. fl
0.50— '1. i f
j :2 . l ,1:
I "~. .' K=0+0i——-.‘g
0.25:- l. :1' l. 3'
_ . t, r .5
0.00 II I I'lll I I I I I‘.Ij I I I l Ilhl II I
0.0 0.3 0.6 0.9 1.2 1.5
Distance x, (meters)
200
Phase
(degrees)
100

 

 

 

 

 

 

0.6

I I I I 1 I I I I I j I I j I I—I I
0.3

I

 

Figure 2.1. Third Eigenfunction for Duct Particle Displacement with End Irnpedances
K = 0.5+Oi and K = 0+0i.

I I I I
0.9

 

1.2 1.5
Distance x, (meters)

12

 

1.25

Relative _ -
Magnitude _ K=0.0+3.01

—
1 m— a p
e " Q - “
o . .
-
t s ,
I

I
'I’
I
I
o

_- a ‘
s--\

 

 

 

0.00 I I I I’ I I I I I I I I’I I In I I I I I I I
0.0 0.3 0.6 0.9 1.2 1.5
- Distance x, (meters)

 

 

Phase
(degrees)

 

 

 

 

 

h-—-------.c us. 0....
.-

 

 

 

'200 I I l T_I V I l I I I I I l I I I I I I I I I I
0.0 0.3 0.6 0.9 1.2 1.5
Distance x, (meters)

Figure 2.2. Third Eigenfunction for Duct Particle Displacement with End Impedances:
K = 0+0i and K = 0.0+3.0i.

l3

 

 

 

 

 

 

 

 

 

 

1.25
Relative : K— -
Magnitude _ -0.8+0.51
1.(X)—" -‘ r'
0.75—— ‘,
: ". f. . 'J ‘z‘ "I
0.50: I. Ii ;; "‘3 i:
0.25:- ' ' ‘3 _"
: . I' K :0 0 o 't‘ :1 2'. j:
_. if :3. Ian.
0'00 IIII'.IIITIIIIEIIIIIIIIJIIII
0.0 0.3 0.6 0.9 1.2 1.5
Distance x, (meters)
200
Phase 4
(degrees) ‘
100—
0....
I
-100._
_I
-200 I I I I I I I I T I I I I I I I I I I I I I I T
0.0 0.3 0.6 0.9 1.2 1.5

Distance x, (meters)

Figure 2.3. Third Eigenfunction for Duct Particle Displacement with End Impedances:
K = 0+0i and K = 0.8+0.31.

14

n=—°O

u(x I): G(t)-I- LXM") (2.12)

where G(t) and an(t) are state variables (generalized coordinates) and (pn(x) are the

eigenfunctions defined in (2.11). The coordinate G(t) will not contribute to the pressure
response in the duct because the pressure response is only dependent on the spatial partial
derivative. To derive the state space representation, the time derivative of the wave

equation is expressed in two different forms. The first uses (2. 12) giving

—(x, I): G(t)+ 2an(t)tpn(x).. (2.13)

nz—oo

The second arises using (2.5) and (2.9) as

2110(3): icxnanfimdx) - (2-14)

nz—oo
Equating (2.13) and (2.14) results in

G(t)+ 21a (t)-— c1, a n(t)]tpn(x)= 0 . (2.15)

nz—oo
The assumption is now made that the differentiation will distribute over the summation.
Success with decoupling will validate this assumption. Evaluating the forced wave
equation (2.1) by inserting the second spatial derivative of (2.12) and the first time

derivative of (2. 14) produces

+oo .
21am)—ck.a.(t>1cx.<o.<x>=—— a —[-5———(")P(°]- 216(11— wig-1V (”I . (2.16)
n=—oo x i=1 pS

 

15
2.5. A New Inner Product

Independent ordinary differential equations for each generalized coordinate an(t) are

found by differentiating (2.15) with respect to the spatial variable x and multiplying by c.
The result is then added to equation (2. 16) to form

+oo
Elan“)‘CMMOIZCMCW='— a i[8(—X__)P(t)]"k 2150‘- x)]_[M (0]
n='°° x i=1 pS

x e [0,L] (2.17)

and subtracted from equation (2.16) to yield

 

*°°- _ -}.nx__ _8_ 8(x)P(t) _ _ _(_t)
HEP”) cxnanunzcxne _ Bxl: p ]- i=21[8(xx x9] :[pis]

x e [0,L] (2.18)

respectively. The interval of (2.18) is then changed to [-L,0] by substitution of -x for x
yielding

+°°- _ I...“ 3 8(—x)P(t) k _ (t)
n;£a“(t) ctnan(t)]2c71ne _ BXI p ]+;[8(—x- 1%)]:[éM—S-I

x e [—L,O]. (2.19)

To guarantee the same effect over [-L,0] as [0,L] the mass flow input term
[8(x — xi)] 3Mi(t) / at must be extended as an odd function on [-L,L] because it interacts

with an odd local pressure. For this reason, it undergoes a sign change in (2.19). The
pressure gradient B[8(x)P(t)] / Ex is an even function for the above mentioned reason, so
the sign remains unchanged. Combining (2.17) and (2.19) using (2.10) and breaking the
exponential into terms that contain the index n and terms that do not contain the index 11

results in

 

16

:Xla, (t)— at a Mamet e L =
r c[_ _1log_l__- KIX 5 _ k
2L °1+K ( X)P(t) _ _ (t) _
I ail p I?“ x x‘)] BIM pS H ”[120]
I (2.20)

1

—1 —l(
eileg°1+K)[_ aaxIL—(X)P(°]_ 2[8(x- xinéfl p4?” xe[0,L].

An exponential ei‘m‘x/L is now multiplied on both sides of (2.20) and the resulting

 

\

equation is integrated from -L to L. The functions on the left hand side of the integral

equation are orthogonal and have a nonzero value only when indices n=m giving

L ' —m1rx imttx
I [am(t)—c3\.nanm(t)]2c7tne ‘- e L dx=
-L

[5mm - cknam(t)]4anL n = m
I (2.21a)

0 n¢m

Using the reflection property of integrals, the right hand side of (2.20) becomes

0 mm c[__ --l-—10g I- -K]X 8
21. “1+K <— x)P(t) _ (t)
Ire—L— I a—aXI—]+2[5(+n>1§t[—ps HM +

p

L mm 11 l-K k
T— [2T °g°l—':K) 8<x)P<t) _ _ _ (t)
I e [- aa—x-I p I 2[5(x x,)1 :[ht—S—dx

 

0 i=1

k
—2P<t> + EEI.M._1(‘>]I_1_I_d‘Pn(Xi) . (2.21b)
p i=1 at pS 1'11 (12‘

Equations (2.21a) and (2.21b) can be equated to form

 

 

\ “($4
1. K p
f ) . ”'1 H
r" \‘ A ’

17 ’/
-P(t) 333 Mi(t) ago.)
20)»an i=1 at 4ClinS dX

 

 

(inn) — cknan] = (2.22)

 

This is the ordinary differential equation for each of the state variables an(t).

2.6. State Space Formulation

The infinite order state space formulation is taken directly from (2.22) which is

rewritten in matrix form as

a(t) = Aa(t) + Bm(t) + bpp(t) (2.23)

 

m‘hwfi

where a(t) = the vector of modal wave amplitudes an(t),
A = the diagonal matrix [cln],

B = the matrix 21 dcpn(xi) ,
4ckanS dx

m(t) = the vector of mass flow inputs B[M1 .......... Mk]T / at,

 

 

bp = the vector —1 , and
2c7tan

p(t) = the pressure input at x = 0,

P(xm,t) = cTa(t), ' (2.24)

where P(xm,t) = the pressure in the duct at x = xm,

2 d¢n(xm)].

and c = the vector [—pc
dx

A is diagonal because the result of the inner product (2.21) is nonzero only when n=m
along the diagonal. The resulting state space representation is extremely useful since the
individual modes are decoupled from each other and each state's input from each excitation

is explicit.

18

The initial conditions on the an(t) coordinates are found using a similar method to

the derivation (2.12) - (2.22), which produces

L
= 1 _ _ dtpn(x)
an(0) 4L7an 0 aau(x, O)(pn(x)dx 4111;?“ Li: :X( ,———0) dx ———dx . (2.25)

 

 

2.7. Frequency Response

The general form of the transient time response to harmonic pressure excitation at

x=0 is found by solving for each of the states an(t) as

 

 

an(t)= [an (0)-iw d“ n]e°l“‘+ d“ eicot (2.26)

— c)» in) - ckn

P .
ithd = 0 andPt =-P cm".
W n 2&an ( ) O

 

The steady state response of (2.26) is

a (t) — Poem (2 27)
n ' 2canp(im—cxn) ' '

 

Finally inserting the an(t) back into the original series solution and using (2.4) to evaluate

the pressure response produces the following series solution

P(x, t)- — —pc2 :an(t)—— d¢“(x) . (2.28)
“z-.. .
Evaluating (2.28) with a finite number of symmetric terms yields the following truncated

series solution

PN(X t)— _ _pC2 +ZNEH(t)—_ (kpdnxO‘)

n=-N

(2.29)

 

19
2.8. Accuracy of Truncated Solution

The exact, infinite-order transient response to this duct problem has not been
previously available. The exact series form for the transient solution for pressure excitation
and/or mass flow excitation must be truncated to a finite number of terms for engineering
use. The effect of this truncation on both steady state and transient forms of the exact
series solution will now be examined. The effect of truncation on a control system using a
truncated model is to introduce observation and control spillover which occurs whenever an
unmodeled mode in the system interacts with the control system action. This spillover
typically degrades performance and in extreme cases lead to closed-loop control system

instability.

Quantitative information on the accuracy of a truncated, steady state, series solution
(2.29) is found here by comparing it to an exact, steady state, frequency response. The
exact steady state response for the system described in (2.1-2.3) can be independently
calculated (Spiekermann, 1986; Spiekermann and Radcliffe, 1988) for the special case of

harmonic pressure excitation only with a closed form solution (2.29) as

(0

K 1 iv L)+ K+1 4%”)
P(x,t)=P0( )6 ,0, ( )c ,0, cm” . (2.30)
C

 

~l—X 1 X
(K—l)e ° +(K+1)e
This exact steady state response model is only valid for the special case of pressure
excitation at x=0. It can not model the mass flow excitation in the domain nor does it
model transient responses. It is used here only for comparison with the more general result

(2.28).

Figure 2.4 is a frequency response of a 1.524 m (5 ft) duct at location x=0.4267 m
(1.4 ft) from the pressure excitation at x=O. The impedance is K=0.3+0.2i and a truncated

series model with eleven terms is used to approximate the solution. The solid line is the

 

 

20

truncated steady state series solution (2.29) and the dotted line is the exact solution (2.28,
2.30). The mean relative error up to the fifth duct resonance is 3.2% (-30 dB). Numerical
simulations suggest the state space model requires one state to model zero frequency
response plus two states per duct resonance. The model yields acceptable accuracy up to

the highest duct resonance modeled

The error of the truncated solution to this acoustic problem is periodic and bounded.
This situation can be demonstrated (Figure 2.5) with the eleven term model (Figure 2.4).
This is unlike typical mechanical systems (Chait et al., 1988) that have a vanishing bound
as the forcing frequency approaches infinity. The impact on a control system is that the
higher frequency modal magnitudes are not small and cannot be ignored. The truncated
state space solution for system frequency response approaches zero as the driving
frequency approaches infinity. The state space solution is exponentially stable
(Kwakernaak and Sivan, 1972). Exponential stability is guaranteed since the real parts of

all the eigenvalues are negative and B and b are bounded.

Figure 2.6 is a frequency response of a 1.524 m duct at location x=0.4267 m from

the end. In this example, the excitation has been moved from x=0 to xi=1.22 m (4 ft)

where it is now a mass flow excitation. The mass flow excitation term is M(t) = -M0e1‘m.

The impedance is K=0.3+0.2i and fifteen terms are used to form the solution. The
previous steady state model (2.30) cannot be compared to this frequency response since it
is only valid for pressure excitation at x=0 and frequency (0. The resonance peaks still
occur at the same frequency as (2.30). This is predicted behavior since the modes are not

dependent on excitation location.

 

21
2.9. Transient Response

The solution to a transient problem is shown in Figure 2.7. The input into the duct
is a harmonic pressure excitation with a frequency of 150 Hertz initiated at t=0. The input
location is x=0 and the response is calculated at location x=0.914 (3 ft). The solid line is
the five term solution and the dotted line is a 51 term solution. The impedance constant is
K=0.3+0.2i. The duct acoustic pressure and particle velocity are zero at time t=0 yielding
zero initial conditions for all model states. Since the state model is diagonal, the response
of the generalized coordinates (states) are independent. Adding additional terms to the state
model does not alter the response of each of the previous states. This behavior does not
occur for coupled state space models with nondiagonal A matrices. Because the states are
decoupled, modal truncation error is reduced and predicted response with only 5 terms is

quite similar to that for 51 terms.

The solution to another transient problem is shown in Figure 2.8. The input into
the duct is a pressure pulse of magnitude one and width of 0.001 seconds. The input
location is x=0 and the response is calculated at location x=0.610 m (2 ft). The impedance
constant is K=0.3+0i. The solid line is the 51 term solution which approximates the exact
solution very well and the dashed line is the 11 term solution which has the same response
in each of the states but lower resolution due to the reduction in state number. The exact
solution pulse magnitude which can be calculated for this simple special case equals one for ‘
the first pulse. Every successive pulse then decreases by 54.8% due to energy escaping
from the end of the duct. The exact solution pulse width is 0.001 seconds and the first one
occurs at t=0.00183 seconds. The truncated series solution has these characteristics.
Because the forcing function is a pulse with infinite Fourier series, adding additional states

in this example helps the accuracy of the solution.

 

22

 

 

 

Exact Solution —\

11 term series solution

 

 

4
Magnitude —
.3. :
P —
0 3—
2...
1 _
fl
0
O
200

  

I ‘ I ' l '
200 400 600
Frequency (Hertz)

 

800

 

 

 

 

 

l I l
600
Frequency (Hertz)

 

800

Figure 2.4. End-Point (x=0) Pressure Excited Steady State Frequency Response
with an Impedance Constant of K = O.3+O.2i at x = 0.4267 m.

Magnitude
‘ P-PN

 

0

23

 

 

 

 

2 —-t
1 _
O l I I l l l I I I I l | l I I l I I l I l | I I
0 500 1000 1500 2000 2500
Frequency (Hertz)

Figure 2.5. Frequency Response Truncation Error of an 11 Term

Model with Impedance Constant of K = O.3+O.2i.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Magmmde 15 term series solution
'1
1
(m)(sec)
2 _.
1 ._
G I I I I I I
0 200 400 600 800
Frequency (Hertz)
200
Phase of ‘
l 100 —
M0 :
(degrees) -
0 _
- 100 “" W
-200 d . I . I I '
0 200 400 600 800
Frequency (Hertz)

Figure 2.6. Interior Point Mass Flow Excited Steady State Frequency Response with
an Impedance Constant of K = O.3+O.2i at x = 0.4267 m. Excitation at xi = 1.22 m.

2
x = 0.91 m
Real P(x,t) —
1— 1| \ ' ,—
—1 'l
0‘ ”I :5
'1_ X 5 terms
: 51 terms
‘2' I I I I I I I I I I I I I I I I I I I I | I I I
O .01 .02 .03 .04

25

 

 

 

 

.05
Time (Seconds)

Figure 2.7. Harmonic Time Response of 5 and 51 Term Duct Models
with an Impedance Constant of K = O.3+O.2i.

2
Real P(x,t)

 

x=0.61 m

 

 

 

I I I I I I I I

.005

 

I I I I I I I I I I I
.010 .015
Time (Seconds)

.020

Figure 2.8. Pulse Time Response of a 1.5 m Duct with an
End Impedance of K = 0.3+0i.

Chapter 3. Experimental Verification of the State Space Model“

3.1. Model Verification Experiment

The state space model developed in Chapter 2 is experimentally verified for four
different test cases. Open loop verification of the model is essential if real time control is to
be successfully applied to the duct. In all the experiments, the impedance at the end of the
duct was calculated from the experimental system transfer functions. This measurement

technique is discussed in Chapter 4.

3.2. Pressure Excitation, Constant Acoustic Impedance

The first experiment involved pressure excitation at x=0 with a constant, frequency
invariant, acoustic impedance in the termination end. The impedance was produced by
inserting a flat piece of packing foam which had a nearly constant impedance at all
frequencies of interest. The steady state response of (2.23) to harmonic excitation using
the pressure excitation term is
P0 eion

—— 3.1
2c2tan(iw — ckn) ( )

an(t) =
where the pressure is evaluated using (2.4), (2.5) and (2.11). The experiment used a 76
mm (3 in) circular PVC schedule 40 duct that was 2.60 m (8.52 ft) long driven by a 254
mm (10 in) diameter speaker (Realistic 40-1331B). Speaker input pressure was measured

in the exit plane of the input speaker with a Bruel and Kjaer Type 4166 half inch

 

* This chapter is based on the paper “Experimental Verification of the Nonself-Adjoint State Space Duct
Model,” submitted for publication to the ASME Journal of Vibration and Acoustics.

26

27

microphone attached to a Hewlett Packard 5423A digital signal analyzer. At a location of
x=0.792 m (2.60 ft) from the speaker, the response of the tube was measured with another
Bruel and Kjaer Type 4166 half inch microphone. The output of the response microphone
was then connected to the signal analyzer (Figure 3.1). Both microphones were calibrated
using a Bruel and Kjaer Type 4230 Sound Level Calibrator. The packing foam had a ~
measured acoustic impedance of approximately K=O.285+0.079i from zero to 400 Hertz.

 

 

 

 

 

 

 

 

 

 

 

 

L .—
X
Excitation Speaker
Duct
Z
’ o
In ut Response Acoustrc
12.5mm Measurement Impedance K
Micr0phone Microphone at the. .
P(O t) P(x,t) Termination
, End
Fast F
. requency
$211116me p Response

 

 

 

Figure 3.1. Laboratory Configuration.

The results of the experiment are shown in Figure 3.2. The measured responses
are marked by X’s and the theoretical response with the measured end impedance is
denoted by a solid line. There is a high degree of accuracy in the magnitude and the phase
data. The disagreement between the experimental data and the theory are possibly due to a

slight nonlinearity of the packing foam impedance.

28

 

 

 

 

 

 

 

 

 

 

4.0
Magnitude _ Expenment X
3 - a I.
I; 3.0— ’ a
— I / Theory
200— I
1 t
1.0— :
0.0 I I I I I I I I I I I I I I I I I
0 100 200 300 400
Frequency (Hertz)
200
Phase of : I x” ’
P -1 I t)
_ l
I?) 100'“ . . It.
‘ I s \‘
(degrees) J I t. “
_ ' .
I
0— : 9 A I
_. \ I
_ l
. I
_. ., i
" I
-100—‘ I
— ‘\ ’I |
_ \ |
- ‘1 9 -’ I
“2% I I I I 1 I I I I f I I I I I I I I
O 100 200 300 400
Frequency (Hertz)

Figure 3.2. Frequency Response of Duct with End Pressure Excitation.

 

29
3.3. Mass Flow Excitation, Constant Acoustic Impedance

The experiment was rerun with mass flow excitation in the domain. The steady
state response of (2.23) for mass flow excitation is

_Moei0)t I-d‘pn(xi):l
4cAfiLpsam — can)L dx

 

a,,(t) = (3.2)

where the pressure is evaluated using the same equations as in experiment 1. A Realistic
102 mm (4 in) speaker was located in the wall of the duct at x1=1.58 m (5.17 ft) using a
schedule 40 test tee. The input signal was measured using a Bruel and Kjaer Type 3544
helium neon laser velocity measurement system attached to the signal analyzer. The laser
measured the velocity of the speaker cone which is proportional to the mass flow. The test
tee had a plexiglass window inserted in its side so the laser could shine on the speaker cone
face to measure it’s velocity. The length of the duct tested was 4.42 m (14.5 ft) and the
response was measured at x=0.762 m (2.50 ft). The foam used in the first experiment was

again used in this experiment. The results of the experiment are shown in Figure 3.3.

3.4. Pressure Excitation, Nonconstant Acoustic Impedance

The acoustic impedance of the systems discussed here is given in (2.2). Equation
(2.2) makes the assumption that the acoustic impedance is constant. For some termination

ends, the impedance is frequency dependent. For these systems, (2.2) is written as

a—u(L,t) = —K(co)(l)§ll(L,t) K at 0+0i, 1+0i, oo . (3.3)
ax c at
Although the separation of variables in Chapter 2 is for K constant, termination ends where

K is a function of frequency can be approximated by

fl(L,t) = —Kn(l)§(L,t) K ¢ 0+0i,1+0i,oo, n = 0,1,2,... (3.4)
EX C at

 

 

w=wn (n=mn

30

 

 

 

 

 

 

 

 

 

 

0.125 _
Magnitude -
P I ‘
— 0.100—
“O :
_ Experiment
1 0.075:
(m)(seC) _ [Theory
0.050—
0.025L
0.000 — 1 I I—I I I I T I I I I I I I I I I I I I
0 50 100 150 200 250
Frequency (Hertz)
200
Phase of :
‘1: 100 _
Mo _
(degrees)
0—
:x
-100_
-200 IIHIIIlelj—I’IIIIIIIIIII
0 50 100 150 200 250

Frequency (Hertz)

Figure 3.3. Frequency Response of Duct with Interior Mass Flow Excitation.

31

Using this relationship, the state space model derived in Chapter 2 can be used to

approximate systems with frequency dependent terminations. An example is given below.

The third experiment involved pressure excitation at x=0 with a nonconstant
acoustic impedance in the termination end. The nonconstant termination was produced by
placing a hemisphere of foam with a diameter equal to the duct’s diameter in the end of the
duct. The resulting acoustic wave was effected by the foam material as well as the shape of
the material. This produced a nonconstant acoustic impedance. The values are listed in
Table 3.1. They were found by obtaining a frequency response of the system from zero to
800 Hertz and then solving the inverse problem for K at each duct eigenvalue (Chapter 3).
The length of the duct was 1.59 m (5.22 ft) and the response was measured at x=1.09 m
(3.56 ft). The state space model was then assembled using the individual acoustic
impedance measurements of K at each resonant frequency rather than a single value for K.
The eigenvalues of the system are non-conjugate complex values since K is complex. The
results are shown in Figure 3.4. Figure 3.4 demonstrates that for a nonconstant impedance
end the linear state space model is reasonably accurate and can predict resonant peak
locations as well as system phase angles. The errors tend to be minimized near the natural
frequencies and maximized between the natural frequencies. This is because the modal

impedances, Kn, were measured at the natural frequencies. The model does not account

for varying values of the impedance between the modes.

 

32

Table 3.1. Calculated Acoustic Impedance.

 

 

 

 

 

 

Eigenvalue (n) (on (Hertz) RC (Kn) Im (Kn)
1 104.8 0.599 0.066
2 213.8 0.585 0.054
3 314.6 0.594 0.206
4 424.2 0.522 0.198
5 533.5 0.491 0.182
6 645.0 0.508 0.104
7 754.6 0.459 0.081

 

 

33

 

 

 

 

 

 

 

 

 

 

2.5
Magnitude : Experiment
iI —
2. —
I?) 0 ' X
I " Theory
P f a
1.5— "
: 4
1.0: '20; I :3
e X '
3 Xxx “'
0.5— \v
0.0 _ I T I I I I
0 200 400 600 800
Frequency (Hertz)
200
Phase of -
§ 100—
0 _
(degrees)
0—
-100—
-200 I I I I I I
0 200 400 600 800
Frequency (Hertz)

Figure 3.4. Frequency Response of Duet with Nonconstant Acoustic Impedance.

34
3.5. Transient Response

The pressure excitation steady—state responses presented previously can be
computed from previous steady—state solutions (Davis et al., 1954, Spiekermann, 1986) to
the wave equation as well as the state space model investigated here. The general transient
response to pressure and flow excitation has not been available previously and the transient
response of a state model is required to implement modern control methods. This transient

response was verified in the set of experiments below.

The experiment was initiated by attaching a signal generator to the excitation
speaker and caused one cycle of sine wave excitation at approximately 500 Hertz (Figure
3.5). The system input pressure and response pressure were then measured by Bruel and
Kjaer Type 4166 half inch microphones attached to an Apple Computer, Inc. Macintosh lIx
computer running National Instruments Labview software and an NB-MIO-16L analog to
digital converter. The measured time domain experimental data was then compared to
theoretical model response. The length of the duct in this test was 2.44 m (8.00 ft) and the
response was measured at x=0.792 m (2.60 ft). The packing foam used in the first
experiment was also used in this experiment to provide a nonzero acoustic impedance in the

end of the duct.

Figure 3.6 is a plot of the experimental data and theoretical state space model. The
solid line is the experimental data and the dashed line is the theoretical model. The
theoretical model response was computed using a fifth order Adams’ integration method
with 51 states. The forcing function in the Adams’ integration routine was the measured
system input at x=0. The integration step size was At=0.000035 which matched the
Labview sampling rate of 28571 Hertz. There is an excellent match between the theoretical
model and experimental data from t=0 to t=0.015 seconds. After that, the experimental
data and theoretical prediction deviate because the propagating pressure pulse reflects off

the now inactive speaker and is effected by the speakers impedance. The theoretical model

35

does not account for impedance" at the speaker, however, Figures 3.1 and 3.3 illustrate that

an active speaker has little effect on the impedance at the source end.

Truncation effects of the state space model are show in Figures 3.7 and 3.8. Figure
3.7 is the experimental data compared to a 21 term state space model and Figure 3.8 is the
experimental data compared to a 15 term state space model. The 51 term state space model
has an error at the pulse peak of less than 1% that increases to an error of 15% for the 21
term state space model and further increases to 40% for the 15 term state space model.
Models with less than 15 terms do not have the bandwidth required to give a reasonable
solution accuracy for the approximately 500 Hertz transient input used in this experiment.
In general, more terms used in the model produces a more accurate solution because the
model bandwidth is increased with the addition of each mode. Most real-time control
schemes cannot incorporate a large number of model terms due to the speed required to
control the system. There is always a trade off between control complexity (model size)
and bandwidth (speed of response) in any control system. These issues are especially

important in distributed parameter control systems (Chait et al.,-1988).

 

36

 

 

 

 

 

 

 

 

 

 

 

1.0
Magnitude :
P _
Max(P) 0.5—
0.0"-
-0.5—-
-1.01IIII1FTIIIIIIIIIII
0 0.005 0.010 0.015 0.020
Time (seconds)
Figure 3.5. System Pressure Input at x=0.
1.0
Magmtude _ Theory
P -I
M _
“(13) 05— - \i‘.
1 ' i
0.0— - “ - E
'1 i '
-05—1 ~
Experiment
-1.0IIIIIIIIITIIIIIIII
0 0.005 0.010 0.015 0.020

Time (seconds)

Figure 3.6. Transient Response of Duct at x=0.792 m with 51 Term Model.

 

37

 

 

 

 

 

 

 

 

 

 

1.0
Magnitude ' -
P -'
MaX(P) -
0.5—
- Theory—\
0 O_ - ’ i r". 'J. 3 -‘u’ "\ 9_J- ‘ ‘o u, 'I u . ' I
_0 5; i i Experiment '
‘1.0llTlIllIlllIllIIIT‘FIIleIllll
0 0.005 0.010 0.015
Time (seconds)
Figure 3.7. Transient Response of Duet with 21 Terms.
1.0
Magnitude -
P I
MaX(P) - g
0.5— .' ‘- Theory\
0.0—I -". E. . | c \ ‘- I ,' | 'I \
-0.5—- ' Experiment
“1.0llllwllllIIIIHIIIIIIIHITIIII
0 0.005 0.010 0.015
Time (seconds)

Figure 3.8. Transient Response of Duet with 15 Terms.

 

Chapter 4. Acoustic Impedance Measurement”

4.1. Introduction

Measuring the acoustic impedance of a boundary is important since the acoustic
response of any acoustic system is governed by the acoustic impedance of its boundaries.
Accurate mathematical models of acoustic systems require accurate measurements of
acoustic impedance. The acoustic impedance of boundaries determines the magnitude and

frequency of resonant peaks and the spatial distribution of acoustic response.

A variety of acoustic impedance measurement techniques have been developed in
the past. The first techniques used an impedance tube and a single microphone (Hall,
1939; Beranek, 1940; Morse and Ingard, 1968; Dickinson and Doak, 1970; Pierce, 1981).
They require measurement of maximum and minimum sound pressure levels at an acoustic
resonance in an impedance tube and their spatial locations. These locations and magnitudes
are then used to calculate the corresponding impedance (ASTM Standard C 384, 1.985a).
Identifying the location of maximum and minimum sound pressure levels in an impedance
tube is normally difficult and requires physical changes in microphone position. Two of
the impedance tube measurement methods (Hall, 1939; Beranek, 1940) use approximate

formulas for computing impedance which can also lead to impedance measurement error.

A recent acoustic impedance measurement technique utilizes a two microphone
system (Seybert and Ross, 1977; Chung and Blaser, 1980a, 1980b). This technique

requires two similar, phase calibrated, microphones at some location in the tube with a

 

* This chapter is based on the paper “An Eigenvalue Based Acoustic Impedance Measurement Technique,”
accepted for publication in the ASME Journal of Vibration and Acoustics.

38

39

known distance between them. The acoustic wave response is then mathematically
decomposed into its reflected and incident components using a transfer function between
the acoustic pressure at the two microphone locations. The decomposition allows the
computation of acoustic impedance (ASTM Standard E 1050, 1985b). ASTM 1050 E,
although better than ASTM C 384, requires measurement of the exact distance from the test
sample to the center of the nearest microphone and the exact spacing of the microphones.
Both these physical dimensions can be difficult to measure accurately. The two
microphone method works best with two phase matched microphones and a source whose
transfer function has constant magnitude around the frequency of interest. If the
microphones are not phase matched, then a correction must be included in the computation
of acoustic impedance. These measurement requirements can lead to errors when

measuring acoustic impedance using the two microphone technique.

This chapter develops a method for calculating the acoustic impedance based on the
eigenvalues of a tube with unknown end impedance. A Fast Fourier analyzer is used to
measure complex frequency response from which the eigenvalues of the system are
extracted. Acoustic impedance at each resonance is then computed from these eigenvalues.
The eigenvalue measurement is independent of microphone position and the location of the
response microphone in the tube is arbitrary. The computation of the acoustic impedance
from the duct eigenvalues is a closed form solution based on the same plane wave
assumptions present in previous methods. The only physical constants required are duct

length and the speed of sound in the duct.

4.2. System Model and System Eigenvalues

The system model is of a one-dimensional hard-walled duct excited by a pressure
input at one end and a partially reflective boundary condition at the other end represented by

a complex boundary impedance (Chapter 2). The termination end impedance is a ratio

40

between the pressure and the particle velocity at x=L and is expressed as (Seto, 1971;

Pierce, 1981; Spiekermann and Radcliffe, 1988)

23¢» = —K(%)%%(L,t) . (4.1)
Implicit in (4.1) is the acoustic analogy with electrical systems in which volume velocity is
analogous to current and duct pressure is analogous to voltage. A second formulation
called the reciprocal acoustic mobility analogy is also sometimes used; and if applied to this
system, the parameter K in (4.1) would be the acoustic admittance. The linear second
order wave equation modeling particle displacement in a hard-walled, one—dimensional duct
is (Seto, 1971; Doak, 1973)

82u(x,t) = a [ammo]

2
a U(X,t) _ c2 _
P

BIZ 8x2 3x

 

 

(4.2)

The one—dimensional assumption is usually valid when f<0.586(c/d) where f is the
frequency (Hertz) and dis the diameter of the tube (m) (Annual Book of ASTM Standards,
1985a; 1985b). The duct end at x=0 is modeled as a totally reflective, open end. This
boundary condition is (Seto, 1971)

§(0,t) = 0 . (4.3)

This corresponds to an open duct end (or electrical short circuit). Equation (4.3) along
with the right hand side of (4.2) model the speaker as a pressure source at x=0. Implicit in
(4.3) is the assumption the source impedance is negligible. If the source impedance is not
small, it can be incorporated into the model (Swanson, 1988). The acoustic pressure of the

system is related to the spatial gradient of the particle displacement by (Seto, 1971)

P(x,t) = -pcza—u(x,t) . (4.4)
EX

41

The separation constants of the model were found in Chapter 2. They are

1 1—K ntti
7t =—lo — —— , n=0,il,i2, ..... . 4.5
“ 2L ge[1+K] L ( )

The system eigenvalues An are equal to the separation constant multiplied by the wave
speed c (An=c?tn). An eigenvalue plot is shown in Figure 4.1. These eigenvalues are each
functions of acoustic impedance, K. The inverse function will allow impedance, K, to be

computed from measured eigenvalues.

4.3. Acoustic Impedance Computation

The acoustic impedance K of the end can be determined at each duct resonance from
the eigenvalue at that resonance. This computation assumes the eigenvalues of the system
are known. Measuring these duct system eigenvalues is discussed in the next section.

Directly solving for K in terms of A is very difficult, therefore an intermediate variable [3 is

introduced to simplify the acoustic impedance computation. The variable Bn is related to

the nth eigenvalue An from equation (4.5) as

nrtci

Re(An) + i Im(An) = Z—i-logJReGin) + i Im(Bn)] — T (4. 6)

where Re( ) denotes the real part, Im( ) denotes the imaginary part, and the subscript “n”
denotes the nth term. Equation (4.6) is now broken into two parts, one equating the real
coefficients and the other equating the imaginary coefficients. The complex logarithm on

the right hand side is rewritten as

loge[Re(I3n) + i Im(Bn)] = logelfinl+i arg(|3n) (4.7)

where IBnl is the magnitude of [in and arg(Bn) is the argument of Bu, i.e., the arctangent of

[III] (Bn)/Re(Bn)l-

42

 

 

 

[Imaginaw
0 l- K 301: .
__ +—
X [ZLar-:g(1 K] L]1
0 1—K 201:
— — __ +_ '
X [ZLarg[1+K] L]1
c l—K 01:
—-I +-— '
X [2_L mink] Lil
0 1—K
>< - l-argI—Jli
2L 1+K Real
.-
£10;K|1-
2L oge 1+K
' 2L g1+K L 1
0 1—K 201:
X [2L [1+K] Li1
0 1-K 301:
X [2L [1+K] Lil
V

Figure 4.1. System Eigenvalues An, for Constant K.

43

The intermediate variable fin is now solved for in terms of the real and imaginary

parts of the eigenvalues. The real part of [in is

l

_ .5
exp( 4L Re(An))

0

Re(Bn) = i-
1+ tan2(——2Ld“)

 

(4.8a)

C

 

 

-

where c1n = Im(An)+ 31°55- .

The sign of Re(Bn) in (4.8a) is determined by

+1 if 0 SIAIS 0.25

sgn[Re(Bn)] = (4%)
—1 if 0.25 <IA|S 0.50

(a

If the value of A is less than -O.5 or greater than 0.5, the eigenvalue index n is incorrect and

 

where A =

corresponds to an eigenvalue other than the nth one. The value, it, must then be changed to

produce a A between -0.5 and 0.5 which will correspond to the correct eigenvalue index.

Once Re(Bn) is found, Im(Bn) is found by the equation

 

Im(Bn) = Re(Bn) m(ztd“) (4.9)

where Re(Bn) is given in (4.8).

The term (1-K)/(1+K) is now equated to the intermediate variable [3n using (4.5)

and (4.6) as

 

44
1- Re(Kn) — i Im(KQ
1+ Re(Kn) + i Im(Kn)

 

Re(B.) +i Im(B.) = (4.10)

where Re(Kn) is the real part of K and Im(Kn) is the imaginary part of K for the nth

eigenvalue. Breaking (4.10) into two equations, and solving for Kn as a function of [in

 

 

yields the acoustic impedance as
2 2
Re(Kn) ___ 1- [R6030]2 - [Im(l3n)l2 (4.11)
[Re(Bn) + 1] + [Im(Bnn
_ '2 Im(Bn) .
MK“) ’ [Re(Bn)+ 1]2 + 11mm“)? (4'12)

Acoustic impedance measurement Kn represents the acoustic impedance at the nth resonant

frequency.

4.4. Acoustic Impedance Measurement Experiment

The viability of the above acoustic impedance method was investigated through
laboratory tests. The test used a 0.0762 m (3 in) circular PVC schedule 40 duct that was
2.94 m (9.65 ft) long driven by a 0.254 m (10 in) diameter speaker (Realistic 40-1331B).
The impedance of a piece of 30 mm thick packing foam inserted in the termination end was
tested. The packing foam will be shown to have acoustic impedance which is nearly
constant with frequency (Table 4.2), allowing for frequency response comparison to
known theory. Speaker input pressure was measured in the exit plane of the input speaker
with a Bruel and Kjaer Type 4166 half inch microphone (input reference microphone)
attached to a Hewlett Packard 5423A digital signal analyzer. The response of the tube was
measured at various locations with another Bruel and Kjaer Type 4166 half inch
microphone (response measurement microphone) attached to the signal analyzer (Figure
4.2). Both microphones were calibrated using a Bruel and Kjaer Type 4230 Sound Level
Calibrator.

45

.-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I X
Excitation Speaker
Duct
Lg

Input I Response Acoustic
Reference Measurement Impedance
Microphone Microphone to be
P(0,t) P(x,t) Measured

r------ ------------------- 1

' P X,(D '

i F i Hm» = P200); i 7:

I as . ’ Curve I'l Impedance

' Pom” "" Fit I " Calculation "" Kn

| Transform Frequency I

: Response :

I. __________________________ J

HP5423A Structural Dynamics Analyzer

Figure 4.2. Laboratory Configuration.

The impedance measurement technique developed here does not require phase
matched microphones nor does it require compensation for phase mismatched
microphones. Phase mismatch in the microphones is neglected since the measurements are
made at a duct resonant frequency, i.e. the measurements are made when the system phase
angles are changing rapidly through 180 degrees. Microphones operating under 500 Hertz
rarely have phase error greater than 5 degrees (Bruel and Kjaer, 1982). Steady state
eigenvalue measurements are amplitude dominated. The distance between the microphones
is not critical, since the duct eigenvalues are independent of measurement location. This is

unlike previous methods (Seybert and Ross, 1977; Chung and Blaser, 1980a, 1980b)

46

where microphone spacing is a required parameter in the analysis and phase matched
microphones (or a compensation function) are necessary because wave propagation across
the microphones is detected. Errors in the method developed here are only a function of
errors associated with measuring the eigenvalues of the duct, the duct length, and the speed
of sound. The computation of acoustic impedance from duct eigenvalues is a closed form
solution. The method uses the input microphone as an amplitude reference and the
excitation speaker does not require a flat response around the frequency of interest, because
the response is normalized by the pressure input reference when the Fast Fourier

Transform is computed.

The 5423A Structural Dynamics Analyzer used here is capable of providing a
number of real time analyses including determining the transfer function (frequency
response) of a system and calculating the corresponding eigenvalues. The 5423A
Structural Dynamics Analyzer does this by curve fitting a single mode vibration model (two
first order states) to the experimental data using the following equation (Hewlett Packard,

1979)

H(w)= Re(An)+iIm(An) + Re(An)—i1m(An)
in) — Re(An) — i Im(An) ico - Re(An) + i Im(An)

 

 

+Blto+B0 (4.13)

where H((o) = the transfer function,
A = the system residue, and
B1 and B0 = compensation constants for overlapping modes.

Included in the single mode vibration model is compensation for other modes which may
be overlapping at that particular frequency. During the curve fitting process, the real and
imaginary parts of the eigenvalues are calculated. It is beyond the scope of this paper to
describe this process, however, there exist additional alternative methods to extract modal

parameters from the transfer function of a system (Hewlett Packard, 1979; Structural

47

Dynamics Research Corporation, 1983). Eigenvalue extraction is a common function of

commercial Fast Fourier analyzers.

The first part of the experiment measured the transfer function of the duct with the
foam end impedance. The 5423A Structural Dynamics Analyzer does this by sending a
random noise signal to the speaker and then computing the ratio of the Fast Fourier
Transforms of the input and response signals. Once the transfer function was known, the
eigenvalues of the duct were found using the curve fitting process above (4.13). From the
eigenvalues, the acoustic impedance of the foam was determined using equations (4.5) -

(4.12).

The mean and standard deviation of the measured eigenvalues of the system with
the foam end impedance are shown in Table 4.1. These values are derived from five
independent sets of measurements at x=0.792 m (2.60.ft) through x=1.42 m (4.67 ft) at
0.157 m (0.52 ft) increments. Each individual eigenvalue was measured from a transfer
function composed of 20 averaged Fast Fourier transforms. The calculated acoustic
impedance of the foam is shown in Table 4.2. In this case, the real part of the acoustic
impedance dominates the response. Figure 4.3 shows the measured frequency response at
x=0.792 m (2.60 ft) compared to the theoretical frequency response for K=0.273+0.034i
at the same point. The value of P/PO is the ratio of the response to the input and the
measured responses are marked by X's while the theoretical response is denoted by a solid
line. The impedance, K, used in the theoretical response is the average of the six individual
acoustic impedance measurements taken at different locations along the duct. Figure 4.3
demonstrates that the theoretical model using the measured acoustic impedance of a material
can accurately predict duct response. There is a high degree of accuracy in both the

magnitude and the phase angles.

48

The acoustic impedances of the above experiment were calculated by increasing the
magnitudes of both the real and imaginary parts of the measured eigenvalues by one, two
and three standard deviations from their mean values. After these changes, the magnitude
of the calculated impedance K only changed by an average of 1.7%, 3.2%, and 5.0%
respectively. This shows the high stability of the measurement technique, it’s resistance to

error propagation, and the accuracy of acoustic impedances determined using it.

The experiment was repeated for a capped end. The results are shown in Tables
4.3 and 4.4. An ideal closed end would have an impedance of infinity, however, the real
material used here has some absorption. The large impedances shown in Table 4.4 indicate
this trend and the variation of impedance with frequency in this case. Figure 4.4 shows the
measured frequency response compared to a theoretical frequency response calculated
using the measured impedances. The theoretical frequency response was produced by
assembling a state space model which used the measured acoustic impedances at each
eigenvalue. As in Figure 4.3, there is a high degree of accuracy in both the magnitude and

phase angles.

It is important to monitor the value of A when testing extremely reflective ends. It
is possible for eigenvalue computation errors to yield a A greater than 0.50 with the correct
index n for large impedances. From the above measurements, a value of coefficient,
A=.51 was calculated for the one of the eigenvalues. Because the measurements were only
accurate to two significant figures, the coefficient was rounded to A=0.5. For most

materials the reflectivity is not large enough for this to be a concern.

 

Table 4.1. Measured Duct Eigenvalues for the Foam End.

49

 

 

 

Eigenvalue Re(An) R“A19 1m(An) 1m(An)
(n) mean, x std. dev., s mean, x std. dev., s

(HertZ) (Hera)

1 -4.955 0.033 57.89 0.045

2 -5.204 0.124 116.3 0.055

3 -5.713 0.091 174.7 0.152

4 -5.755 0.131 233.8 0.259

5 -4.231 0.122 291.7 0.164

6 -5.414 0.117 350.5 0.192

 

 

 

 

 

Table 4.2. Calculated Acoustic Impedance for the Foam End.

 

 

 

 

Eigenvalue (n) Re(Kn) Im(Kn)
1 0.260 0.032

2 0.273 0.037

3 0.298 0.042

4 0.300 0.014

5 0.224 0.046

6 0.283 0.031
mean, x 0.273 0.034
std. dev., 8 0.028 0.011

 

 

 

 

 

50

 

 

 

 

 

 

 

 

 

 

4.0
Ma 'tude —
gm _ I x
3 _ a Experiment ‘ I
I; 3.0— ’ i 5
_ ‘ Theory
_ .~ / .
2.0— I.
- s
_ b ‘
1.0— .,
I ’ »
_ . 9
0.0 I I I I I I I I I I I I I I I I I
0 100 200 300 400
Frequency (Hertz)
200
Phase of : I
3 — I
It 10°: I
(degrees) -— i
I I
0— i
I
_ I
- l
T I
-100— I
_ I
_ i
'200 l I I l T l l I l I l l I l l l l
0 100 200 300 400
Frequency (Hertz)

Figure 4.3. Frequency Response of Duct with Foam End at x=0.792 m.

51

Table 4.3. Measured Duct Eigenvalues for the Capped End.

 

 

 

Eigenvalue (n) Re(An) Im(An) (Hertz)
1 -0.723 29.1
2 -0.847 88.4
3 -1.010 147.2
4 -1.279 206.0
5 -1.419 264.6
6 —0.9 14 324.1
7 -1.038 381.9

 

 

 

Table 4.4. Calculated Acoustic Impedance for the Capped End.

 

 

 

Eigenvalue (n) Re(Kn) Im(Kn)
1 24.37 -5.88
2 15.06 - 10.22
3 11.06 -9.03
4 8.54 -7. 16
5 8.07 -6.38
6 3.32 -7.47
7 6.81 -8.68

 

 

 

 

 

52

 

 

 

 

 

 

 

 

 

 

 

 

10
Magnitude _ ) : 2,; Theory X
a g_ ’/
P Experiment > I
0 _ >S< [F
6—
- X
X ,
4 _ >~
_ < >
2 __
O I I I I I | I I
0 100 200 300 400
Frequency (Hertz)

 

 

 

 

 

Frequency (Hertz)

Figure 4.4. Frequency Response of Duct with Closed End at x=0.792 m.

Chapter 5. State Estimation“

5.1. Introduction

An active noise control system in a duct usually consists of one or more cancellation
speakers driven by an algorithm designed to reduce noise levels in the duct. The duct
normally has one or more signal microphones at some location while noise is driven by
excitation through one end. The goal of the research discussed here is the application of
classical pole placement to the problem of active noise control in a duct. Pole placement
modifies the eigenstructure of the system to increase the dissipation of the duct and
attenuate duct noise. This requires knowledge of the values of the system states. In the
previously developed state space duct model (Chapter 2), the system states in the duct are

not measurable, and a state estimator must be developed.

The observer is a computer model that runs in parallel with the actual duct system
(Luenberger, 1964; Luenberger, 1966) whose input is the input into the actual system plus
an error feedback designed to drive the computer model states to values approaching the
actual system states. The output of the observer are state estimates and a system pressure
estimate. This chapter develops the state estimator (Figures 5.1 and 5.2) for the one-
dimensional hard-walled duct with a speaker at some location, a totally reflective entrance
boundary condition, and a partially absorptive termination boundary condition. The

excitation is a speaker with input signal v(t) that is independent of the dynamics of the duct.

 

"' This chapter is based on “State Estimation of the Nonself-Adjoint Acoustic Duct System,” accepted for
publication in the ASME Journal of Dynamic Systems. Measurement, and Control.

53

54

The next chapter will develop a state space controller to generate v(t) as a noise control

based on state estimates.

 

 

 

 

State Estimate II

Figure 5.1. Observer and Duct Diagram (Chen, 1984).

55

—> State
E-flm

Input
P(xm,t) Signal v(t)

Microphone é Speaker

Acoustic
O Duct +Energy
Flux

 

 

Totally Partially
Reflective ReflCCthC
End End
(x=0) (x=L)

Figure 5.2. Duct with Observer and Input Speaker.

5.2. Observer Equations

Typical observer structure is shown in the flow chart in Fig. 5.1. The thick lines
represent vector quantities and the thin lines represent scalar quantities. The system input is

a mass flow excitation at location x=xi in the duct or a pressure excitation at x=0. The error

feedback is the comparison between the actual noise field and the estimated noise field at
location x=xm. The state estimates, once available, can be passed to a controller to drive a
compensator. The control objective is to change the eigenvalues of the system so that the

duct is less responsive to external pressure excitation at certain design frequencies.

The duct is an infinite dimensional system. The observer is a finite dimensional
system whose order is based on the truncated state space model. When the state space
formulation is truncated, the truncated model response only converges to the exact response
over a limited frequency range (Chapter 2). Increasing the number of terms 11 in the
truncated solution increases the frequency bandwidth over which the truncated solution

converges to the exact solution. Although many methods have been developed recently to

 

56

suppress the errors associated with model truncation (Chait et al., 1988), observation
spillover error will always occur when an unmodeled mode in the infinite dimensional
system produces an error in the state estimator. The effect of truncation-induced

observation spillover errors will be examined below.

The observer equations for a single input are

a(t) = [A — lcT]a(t) + bv(t) + lP(xm,t) (5.1)

Pest(x,t) = cTa(t) (5.2)

where Pest(x,t)=the observer estimate of the pressure in the duct, v(t)=the system input,

b=the input vector (different depending on if the excitation is in the domain or at the end), I
the observer gain vector, and the superscript T denotes the matrix transpose. Equations
(5.1) and (5.2) are the observer equations for the duct used to provide for a state estimate in

the duct when the states are not measurable.

5.3. Observer Gain Placement

The observer gains form the vector l. Conventional practice is to choose them so
that the real part of the eigenvalues of A-lcT are two to four times the magnitude of the
imaginary part of the eigenvalues of A (Franklin and Powell, 1980). The imaginary part of
the eigenvalues typically remains unchanged. This practice allows the observer to resonate
at the same frequencies as the duct. The algorithm to place the observer gains is listed

below.

1. Find the characteristic equation of A. This is the determinant of [sI-A] and is

written as 5n + Oils".1 + 0L2s“_2 + a3s“_2+. . . .+0tn_ls+ an where the 0t's will be used to

57

determine l. The (1's are related to the eigenvalues of A by the elementary symmetric
functions
11
a1: 211i = traceA (Ai = cli)
1:1
a2 = iZIIAiAj = tracezA

i<j

n
(13: zAiAjAk = trace3A
1:

i<j<k

it
an = l—IAi = detA = tracenA
i=1

2. Find the characteristic equation of [A—clT]. This is written as
sn + l31 s"_1 + stn‘2 + I33 sn‘3+. . . .+Bn_ls + [3,, = 0 The [3's are related to the eigenvalues

of [A-clT] in the same manner as the (it's are related to the eigenvalues of A above.

3. Define the intermediate observer vector IIT using the ori and Bi derived above.

lIr=[Bn_an Bn_1-an_1 132—0t1 [SI—a1].

4. Form the modal intermediate observation matrix T. The columns of T are
t,rl = c

tn1=Ac+0tlc

__ 2
tn_2 — A c+a1Ac+a20

_ n-1 n—2
tl—A 0+0tlA c+....+0tn_1Ac+0tnc
and

58

5. Invert T and find the observer gains by IT = [IT—1.

Due to the spectral bandwidth of the A's, the matrix T is algorithmically singular when n>5
(L=l.524 m) for matrix inversion using the International Math and Statistical Libraries
(IMSL) linear equation solver with double precision constants. The matrix is column—wise
ill conditioned. Using this knowledge, the matrix can be preprocessed (for 5<n<12) by
dividing each member of each column by a complex number that has the largest real and
imaginary values in that particular column. The modified matrix is then passed to the IMSL
linear equation solver and inverted. The inverted matrix is now rescaled, with the rows
divided by the above complex number. Using this technique, the matrix T can be inverted
for n=1 1. For n>11, the characteristic equations cannot be generated and the matrix T
cannot be constructed on a MicroVAX H. If a computer with more precision is available,

the problem can be solved for larger values of n.

5.4. Closed Form Solution

A closed form solution to (5.1) and (5.2) can be derived by transforming the
equations into orthogonal space. This starts by making the substitution

a(t) = Uz(t) (5. 3)

where U=the eigenvectors of [A-lcT] arranged by columns sorted in order of smallest to
largest imaginary value of the corresponding eigenvalue. Using this substitution, (5.1) and

(5.2) become

U in) = [A — lcT]Uz(t) + bv(t) + IP(xm,t) (5.4)

Pest(x,t) = cTUz(t) . (5.5)

 

59

To transform the equations into an orthogonal space, (5.4) is multiplied on the left side by
the matrix V*, where V*=the conjugate transpose of the eigenvectors of the conjugate
transpose of [A-lcT] arranged by columns sorted in order of largest to smallest imaginary
value of the corresponding eigenvalue. The magnitudes of U and V* are based on the

orthonormal relationship V*U=I. The observer equations in orthogonal space are

V‘U in) = v‘IA — IcTIUztt) + V*[bv(t) + [P(xm,t)] (5. 6)

or
. A
Iz(t) = Dz(t) + b(t) (5.7)
where D = the diagonal matrix [2“],
Zn 2 the observer eigenvalues,
A
and b(t) = the vector {V‘[bv(t) + lP(xm,t)]}.

Equation (5.7) is now composed of decoupled coordinates. The pressure estimate of the

observer is obtained by solving for the zn(t)'s and inserting them into (5.5). The state

estimates are the an(t)'s.

Equations (5.7) and (5.5) were assembled and solved for the case of harmonic

excitation. The zn(t)‘s can be solved for independently as

- _i_ znl _EIL— 10)!
zn(t)—Izn(0) 003—211)} +|:(im—Zn)Ie , (5.8)

where I) = the vector {V*[b + lPs]}
and P5 = steady state amplitude.

The result for a test case of a 5 term observer (—2 s n S 2) and a duct length of

1.52 m (5 ft) is shown in Fig. 5.3. The driving speaker has harmonic excitation at a

60
frequency of 92.1 Hertz (579 radls) and is at location xi=0.305 m (1 ft). This corresponds
to resonance at the n=1 natural frequency. The error feedback is measured at location,
x=0.610 m (2 ft). The initial conditions on the z(t) variables are all zero. The solid line is
the real part of the observer pressure and the dashed line is the real part of the duct pressure
(equation 5.2) at location 0.762 m (2.5 ft) from the end of the duct. The real part of the
pressure is the complex pressure fields image on the real axis. It represents the response a
microphone would measure. The duct termination end impedance is K=0.25+0.5i. Figure
5.3 shows the observer nearly converging to steady state system values after only 3 cycles.

This response demonstrates that the observer is operating correctly.

 

A
P . Estimated Res nse
Re 81 (76,0 - po

I
N
I

.
1 I
.

\- Steady State Response

I I I I I I I I I
0 .02 .04 .06 .08 .10
Time (s)

 

 

.IA

 

Figure 5.3. Closed Form Solution of Duct Pressure Calculated by Observer
Compared with Steady State Duct Model Response.

61
5.5. Numerical Simulations for Continuous Time Observers

Numerical simulations were performed on (5.7) and (5.5) to test for robustness and
sensitivity to various integration algorithms. Implementation of a real-time observer
requires a stable, noise immune algorithm. Numerical simulations of different algorithms
were used to compare the response of three different integration algorithms: the Euler
Method, the Run ga-Kutta Method, and the Hamming Predictor Corrector General Method.
The observer equations were integrated from 0 to 0.1 seconds with varying integration step
sizes. The observer parameters were the same as those listed in the Closed Form Solutions
section except zero initial conditions were set on acoustic states, a(t)'s. The numerical

integrations were done on a MicroVAX 11 running a VMS version 4.7 operating system.

The fastest and most stable integration routine was the fourth order Runga—Kutta
Integration scheme. Stability was determined by the largest time step the integration
algorithm would converge with in this particular problem. The largest step size needed to
get a converging value was 0.001 seconds or 100 integration points on the interval
0 S t s 0.1 seconds. The integration required 4.5 seconds from the central processing unit
(CPU seconds). Figure 5.4 shows the results of this integration. Although similar to
Figure 5.3, Figure 5.4 is an actual system numerically integrated rather than a closed form
solution. The solid line is the real part of the observer pressure at x=0.762 m while the
dashed line is the real part of the duct pressure at the same location. The integrated
observer simulation converges to the correct values in the same number of cycles as the
closed form solution. The Hamming Predictor Corrector General Method was the second
most stable integration routine. It converged to the correct values for a maximum step size
of 0.0002 seconds or 500 integration points on the interval. This required 12.4 CPU
seconds. The least robust integration routine turned out to be the Euler Method. It required
a step size of 0.00002 seconds or 5000 integration points on the interval and took 51.3

CPU seconds. This was an expected result since the Euler Method is a first order method.

 

62

The observer was next tested for response to initial conditions. This is important if
a control scheme is going to attenuate transient noise. The initial conditions of all the
a(t)'s were changed from zero to an(0)=1+li. The results are shown in Figure 5.5. The
integration scheme used was a Fourth Order Runga-Kutta Method with a step size of 0.001
seconds. The observer converges to the system values after 0.1 seconds. The response
changes by over 4 magnitudes in approximately 100 integration steps. Increasing the

values of the initial conditions by 10 (an(0)=10+10i) requires only two additional cycles for

the observer to converge to the system values.

 

 

 

 

4 _
P(.76,t) I
Real _ : Integrated Estimated Response

I It "

2‘. '2
: .5 I.
: I "

of 5
:5
I

-2—.2 s
j k Steady State Response

-4 _ l I l I l I I I l I I I l I I l I I I I I l I l I l I l l l
0 0.25 0.50 0.75 0.10

Time (s)

Figure 5.4. Integrated Observer Solution and Model Duct Response.

63

 

    
     

 

 

 

 

Real 15000 _
P(.76,t) 10000_:
“0 :
: Observer Solution
5000:
of -----
E Duct Response
-5000—_
-10000—:
-15(XX) _()l I I I I I I I I l IOIZSTj T Ifi IT I I I l0l5()l I I I l l I I I rjo
. . Time (s) . 75
Real °
P(.76,t)

llllllllJllLLllilllllllllllllll

 

 

 

 

0

-4

'8 ‘I’ l I l l I l I fifi T T l l I l l l l I I rfi'
.075 .100 .125

Time (s) ‘150

Figure 5.5. Integrated Observer Solution and Model Duct Response.
Nonzero Initial Conditions Showing Robustness of Observer
Design.

64

Numerical simulations suggest the observer requires one state to model zero
frequency response plus two states per duct resonance. For the simulations shown
previously, a three term observer will give the same response since the excitation frequency
is at n=1. Nonresonant excitation behaves in a similar manner. If the duct were excited
between the n=1 and the n=2 natural frequencies, then a five term observer is typically

necessary for reasonable accuracy.

A state estimate is shown in Fig. 5.6. This corresponds to integrated solutions for

zero initial conditions on the an(t)'s. The integration method used is the fourth order
Runga-Kutta scheme. The state estimate shown is a1(t). The solid line is the real part and
the dashed line is the imaginary part. The first state was chosen here because the model is
being forced at its first natural frequency and the first state is dominating the response. The

first state reaches steady state response after approximately 0.025 seconds.

 

 

 

 

 

.0010 _
Magnitude E Re(a](t)) 'f' :1. :,:| _: I‘ll | I:
: .3‘. 3' 1: IE ? 3| I E I
.0005; '5, ; l ,s i ,3 '5 i I
‘ ‘ . -, | ' : . ‘- I .' J -
: I '~ :‘ ' ,' " 1 '. E :
: I 2' :" ' I, :’ 5 .5 I l
_ ' - I . 1 I 5 I ,5 i s
— ;' i ’ I i E 5 f i i .
0‘- : . I i -. s I 5 ~ ' I
I i' =. ' '1 l i 3 s I 5 l
: . i . . z i - . . I; ‘-
"0005’: -: ‘ .’ ' 3' § 5 ' 2 - :‘
3 '=-I a: I I; '. I i; =..=‘
: J 3" I" 3i
: Im(a1(t))
’.0010 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
0 .025 .050 .075 , .1
Tune (5)

Figure 5.6. State Estimate a1(t).

65

Truncation induced spillover for the observer and duct only occur if the frequency
of the excitation speaker is greater than the bandwidth of the observer. The bandwidth of
the observer is equal to the imaginary part of the largest eigenvalue. When this occurs, it
produces an unmodeled mode in the duct and as a result the observer cannot converge to
the duct values. This problem is avoided if the bandwidth of the observer is larger than the
excitation frequency. This is desirable behavior by the observer since it is explicit when the
observer will and will not converge. This result is derived from the diagonal state space

model formulation (Chait et al., 1988).

5.6. Numerical Simulations for 3 Discrete Time Observer

A faster observer than the continuous observers listed above is a discrete time
observer. This was investigated since any real-time control system will have to operate
extremely fast to control noise propagating at the speed of sound. To get the discrete time
equations, the continuous time equations (5.1) have to be transformed into discrete time.
The transformation of equation (5.1) into discrete time is (Ogata, 1970)

a(k + 1) = F(T)a(k) + g(T)v(t) + h(T)P(xm,t) (5.8)

where the matrix F(T) and the vectors g(T) and h(T) are dependent on the sampling
period T. When the sampling period is fixed, F(T), g(T), and h(T) are constant. The
matrices F, g, and h are found by solving the continuous time differential equations then

transforming the solution into discrete time (Ogata, 1970). They are

F(T) = eAT (5.9)

g(T) = UOTeA‘dtjb (5.10)

 

66
h(T) = UJeAtdtjl (5.11)

where A, b, and l are from (5.1). Equations (5.9), (5.10) and (5.11) can be solved for on

 

 

a digital computer by
— _ -1_ °° Ame ~ M Ame
F(T)—InvLapl(sI A) ]_n;[ m! J~I§0£ m! j (512)
°° Ame+l M Ame+1
T = _ .-.-. .
g( ) mgtl (1131+1)l)b "12:11 (m+1)g]b (513)

h(T): Z —— I: 2 -————-—l (5.14)
m=0 (m+l)! m=0 (m+1)!
where InvLap denotes the inverse Laplace Transform, and M+1 is the number of truncated

terms used to form the solution. Once the discrete time equations are solved, the pressure

at time t is found by the equation

Pes,(x,t) = cTa(k + 1) . (5.15)

A discrete time observer simulation was performed on the system described in
Section 5.5. Figure 5.7 is a plot of the results. The estimated pressure nearly matches the
actual pressure, except for a slight phase shift (time delay) associated with the sampling
rate. The simulation took 1.1 CPU seconds on a MicroVAX 11 using a step size of
T=0.001 seconds, which is 75% faster than the fastest continuous observer. The faster
time is a result of the integration being done before the matrices are formed, i.e., the

integration doesn’t have to be executed on-line or in real time. Changing the initial

conditions to an(0)=1+1i produces results similar to Figure 5.5.

 

67
Discrete Observer Estimated Response

\I~
II lllllllllll c
- -- S
n
IIIIIIIIIIIIIIII o
- --- P
l \ s
I IIIII I \ I e
- R
I I I II I I I I I m
\- - w
I 1 I ------- I .I- S.
- - m
- t -- IV. R
- - S

 

 

.10

Time (s)

 

Figure 5.7. Discrete Time Observer Solution and Model Duct Response.

Chapter 6. Active Noise Control‘

6.1. Active Noise Control Using Pole Placement

The control objective is to change the eigenvalues of the system so that the duct is
less responsive to external pressure excitations (disturbances) at certain design frequencies.
This is done by placing the closed loop system poles to the left of the open loop system
poles in the complex plane to enhance the systems stability. Guaranteeing the stability of
an infinite dimensional system over an infinite disturbance bandwidth is not possible with
truncated state space models. When controlling a finite number of modes in an infinite
dimensional system, the problem of an uncontrolled mode going unstable arises. This
instability is called control spillover. The closed loop eigenvalues of the acoustic duct can
only be placed a reasonable distance to the left of the open loop eigenvalues before an

unmodeled mode causes observation or control spillover.

6.2. Controller Equations

The control system of the duct is shown in Figure 6.1 and Figure 6.2. The state

space equations for the duct with a disturbance speaker and a control speaker are

é-(t) = Aa(t) + bld(t) + b2v(t) (6.1)

P(xm,t) = cTa(t) (6.2)

 

* This chapter is based on the paper “Global Active Noise Control of a One-Dimensional Acoustic Duct
Using a Feedback Controller,” manuscript in preparation. to be submitted to ASME Journal of Dynamic
Systems, Measurement, and Control.

68

69

where d(t) is the system disturbance and v(t) is the feedback signal. Placement of the
system disturbance on the end is not necessary, however, this corresponds to most
industrial problems where a disturbance source excites the duct from one of the ends. The
feedback signal is of the form

v(t) = —ka(t) (6. 3)

where k is the control gain vector. When the observer is added to the system (Figure 6.2),

the a(t) becomes the observer output, awest-

6.3. Control Gain Placement

The controller gains form the vector k. They are chosen using the same algorithm

T

used to determine the observer gains 1 (Section 5.3) except that the vector c is replaced

with the vector b (Equation 2.23). (The input vector b is related to the control gains k in

T

the same way the output vector c is related to the observer gains l.)

 

 

 

 

 

 

Duct
V
é
Response Nonzero
Control ~
- Measurement Acoustic
Drsturbance Microphone Speaker Impedance
Speaker P(x t) v(t)
d(t) m’
¥ Observer
’ and
Controller

 

 

 

Figure 6.1. Active Noise Control System

70

Plant

 

Disturbance d(t) _
" a(t) = Aa(t) +bld(t)+b2v(t) Pressure P(x,t)

 

Feedback v(t) > P(xm,t)=cTa(t)

 

 

 

 

 

|
l
Observer :
|
I

a(K + 1 )esI = F(T)a(K)est + g(T)v(t) + h(T)[P(x,t) — Pm(x,t)]
P(x,t)

v(t) T

——-— Pest(x,t) = c a (t )est

 

 

 

 

 

State Estimate

‘I
I
I
I
I
a(t)est
I

Controller II

 

 

v(t) = —ka(t)est
I Control System _-

Figure 6.2. Plant and Control System.

6.4. Frequency Domain Simulations

Once the control gains are known, frequency domain simulations can be calculated
for (6.1) and (6.2). Frequency domain simulations predict the closed loop steady state '
response of the system for sinusoidal excitation. The time response of the closed loop

system is found by inserting (6.3) into (6.1) which produces

d(t) = [A — b2k]a(t) + b1d(t) . (6.4)

Computing the Laplace transfer of (6.4) produces
sa(s) = [A — b2 k]a(s) + b1d(s) (6. 5)

71

where (6.5) is in the complex 3 plane. Equation (6.5) is solved for at each frequency by

letting s=ic0, and using the following two definitions

d(ico) = Poem” (6. 6)

and

a(iw) = Xeimt (67)

where X is, in general, a complex valued vector. Solving for X yields

X = {i031 - [A — bzk]}'lblP0 . (6.8)

The steady state pressure can then be calculated by inserting (6.8) into (6.2). It represents
a closed loop system whose eigenstructure has been modified through feedback. The
solution to (6.8) assumes that the disturbance dynamics are known and measurable. In
general, there is no way to determine the excitation of the duct. The effect of unknown
disturbances is discussed in the next section. Additionally, the observer is not included in

these frequency domain simulations.

A frequency response of a controlled system is shown in Figure 6.3. The length of
the duct is 3.66 m (12.0 ft), the response is measured at x=0.792 m (2.60 ft), and the
control speaker is at x=3.56 m (11.69 ft). The acoustic impedance at the end is K=0.3+0i.
The number of terms used to form the model was 11. The solid line is the closed loop
response and the dashed line is the open loop response. The real part of the closed loop
eigenvalues are twice the magnitude of the open loop eigenvalues. The imaginary part of
the eigenvalues remain unchanged. Note that the resonant peaks are all reduced by
approximately 50 percent. This is not a generalized result for the placement of the control
speaker at any location. Figure 6.4 is a frequency response of the above system with the
control speaker at x=1.37 m (4.50 ft). A control speaker placed at this location has little

effect on the first mode of the system. Varying the location of the control speaker in

72

relation to the reSponse measurement and disturbance input has different effects on the
system. Numerical simulations show that locating the control speaker near the termination
end is the best location for first mode control and a good location for second mode control.
They also show that locating the control speaker near the disturbance is a poor location.
The eigenstructure of the system can still be modified by placing the control speaker near
the disturbance, however, the resulting change in closed loop magnitude response is

minimal.

The closed loop poles of the system can also be altered by changing the imaginary
part of the open loop eigenvalues. This is useful if the disturbance is occurring at or near
an open loop natural frequency. The above system has imaginary eigenvalues at multiples
of 47.1 Hertz (296 Rad/sec) for five duct resonances. If there were a disturbance that had
harmonics around 141 Hertz (888 Rad/sec), the third mode of the duct would be excited
and the response of the system would be large. Figure 6.5 is a plot of the closed loop
system with the imaginary part of the third eigenvalue placed at 169 Hertz (1060 Rad/sec).
The closed loop pole has increased in amplitude in addition to having its frequency content
shifted. If this were a problem, then the real and imaginary parts of the eigenvalues could
both be placed. Although the theoretical closed loop eigenvalues can be placed in any
location, the actual closed loop eigenvalues can only be moved a reasonable distance from
their corresponding open loop values. The further a closed loop pole is moved away from
it’s open loop value, the more likely an unmodeled eigenvalue will create system instability
because the high control gains required for a large eigenvalue shift may destabilize

unmodeled modes.

73

 

 

3 _
P ' :1

Open Loop Response

 

Closed Loop Response

 

 

IFIIIIIIIIII

 

250

 

 

 

 

 
 

 

I
200
Frequency (Hertz)

 

I I

250

Figure 6.3. Closed and Open Loop Frequency Response, Compensator at 3.56 m.

74

 

 

 

 

 

0'0IllllllllllIIIIIIIIIIIII

0 50 100 150 200 250
Frequency (Hertz)

200

 

 
  

Phase of ‘

 

 

 

 

-200 I‘F I 1 I I l I I l I I I I l I I I I I I I l
0 50 100 150 200 250
Frequency (Hertz)

Figure 6.4. Closed and Open Loop Frequency Response, Compensator at 1.37 m.

75

 

     

 

 

 

 

 

5.0
_ — Closed Loop Response
Magmtude -
3‘ :
4.0—
I?)
: Open Loop
_ Response
3.0—
2.0——
1.0——
0-0 — I I l I I I I I I I I l | I I I I I I I | I I |
0 50 100 150 200 250
Frequency (Hertz)
200
Phase of - M

 

 

 

 

 

A

‘200 I I I | I I I I I I I I I I I I I I I I I I | I
0 50 100 150 200 250
Frequency (Hertz)

 

 

 

Figure 6.5. Closed and Open Loop Frequency Response, Showing Change of
Natural Frequency of System.

76
6.5. Time Domain Simulations

Time domain simulations predict the response of the system over a finite time.
They are useful in determining the transient response of the controller and the effects of
unmodeled disturbances on the system response. The frequency domain simulations can
not include these effects. A computer program modeling the plant and control system
(Figure 6.2) was written. It allowed the observer the option to model the disturbance or to
leave the disturbance unmodeled. In general, the disturbance is rarely a measurable
quantity available to any control system. Analyzing the effects of modeled and unmodeled
dynamics in simulation does offer additional insight on system performance and control

effectiveness.

A typical time domain simulation is shown in Figures 6.6 and 6.7. The length of
the duct is 3.66 m (12 ft), the response is measured at x=0.792 m, (2.60 ft) and the control
speaker is at 3.56 m (11.7 ft). The impedance at the termination end was K=0.05+0i.
Five terms were used in the model simulations. The simulation time step size was 0.00005
seconds. The simulation excitation was a sine wave at 47.1 Hertz (296 Rad/sec) which
corresponds to the ducts first natural frequency. The observer and controller were both
activated at time t=0.05 seconds. Figure 6.6 is the simulation with the disturbance modeled
while Figure 6.7 is the simulation with the disturbance unmodeled. In both figures, the
solid line is the model duct response while the dashed line is the estimated pressure by the
discrete time observer. The steady state value of the simulation with the disturbance known
is 6.5 and the steady state value of the simulation with the disturbance unknown is 8.7.
Simulations produce similar response for the second mode. These results show that the

control action is not as effective if the disturbance is unmodeled.

 

77

 

 

 

 

 

20 —0 792
— x— . m
w ‘ Model Duct Response
P .-
0 _
10
0 — ‘ . .
-10 . l ‘ .
: Estimated Pressure Response
_20 I I I I I I I I I I I I I I I I I I I I I I I M
0 0.1 0.2 0.3 0.4 0.5

Time (Seconds)

Figure 6.6. Time Domain Response with Modeled Disturbance

 

 

 

 

20
x=0.792m
W Model Duct Response
P _.
0 _
10:‘ I I . i I : I : | I I ‘ I I
0— """ 5‘ -' 2 I ' i
‘ I I E I ' '
-10-—- v I I '
: Estimated Pressure Response
‘20 IIIIIIIIIIIIlI—FIIIIIIII
0 0.1 0.2 0.3 0.4 0.5

Time (Seconds)

Figure 6.7. Time Domain Response with Unmodeled Disturbance

78
6.6. Actuator Dynamics

The frequency and time domain simulations presented above assume that there are
no actuator dynamics. This is not a realistic assumption for control compensators of
acoustic systems. The speaker/amplifier actuator system add significant dynamic effects to
the control system. This was not a problem in earlier open loop experiments since the input
was not the voltage to the speaker, but rather the pressure at the speaker face. It is not
possible to ignore the actuator dynamics in the closed loop system since the control system

generates a signal that is to be converted into an (unaltered) mass flow.

Figure 6.8 is the magnitude and phase plots of a Bruel and Kjaer Type 2706 power
amplifier attached to a 101 mm (4 in) diameter (Realistic 40-1022A) speaker. The mass
flow of the speaker was measured with a Bruel and Kjaer Type 3544 helium neon laser
velocity measurement system. The velocity of the speaker face is proportional to mass
flow. The laser was attached to a Hewlett Packard 5423A digital signal analyzer which
measured the values shown in Figure 6.8. The excitation signal was random noise. The
X’s in the figure are the measured transfer function values and the dashed line is the
transfer function for an ideal actuator. Figure 6.8 shows that the transfer function of the

actuator is not close to an ideal actuator transfer function.

There are several options to overcome poor actuator dynamics. The first is to
design a control system for the actuator itself, using a separate controller. The second is to
obtain a number of amplifiers and speakers and test them to determine which has the most
ideal characteristics. The third method is to design the closed loop experiment in a way
such that actuator dynamics are minimized. This is the method used in the experiment in
the next section, choosing a duct length where the first duct natural frequency is around 50
Hertz, an operating region where the actuator has nearly zero phase. This will also provide

an actuator bandwidth wide enough for control of the second mode. Control of the third

79

 

 

 

 

 

 

 

 

 

6.0
Magnitude
[MI _ ><><><>< Actual
V X XXXX
X
4.0— >§< XX
X
k xX ><><><X
g XXX
(sec)(volt) _ X
X
2.0— X ' Ideal
x f
_X _ ___ -__._.___._._.___ -__--_ -____ -_
0 G I I I I I | l I I I I l I I I I I r
O 50 100 150 200
Frequency (Hertz)
200
Phase of —
M L
V 100 _ X
(degrees)
0 _ XXXXX
I ”WWW
-100—
'200 I I I l I l l I I I I I I I I I I I I
50 100 150 200
Frequency (Hertz)

Figure 6.8. Actuator Magnitude and Phase Plots.

80

mode will not be possible due to the phase angle associated with the actuator at the ducts

third natural frequency.

6.7. Active Noise Control Experiment

The duct and active noise control system shown in Figure 6.1 were built using 76
mm (3 in) circular PVC schedule 40 tube that was 3.66 m (12.0 ft) long. The actuator
described in section 6.6 was located at 3.56 m (11.7 ft) and the response measurement
microphone was a Bruel and Kjaer Type 4166 half inch microphone located at 0.792 m
(2.60 ft) from the excitation end. The excitation source was a 254 mm (10 in) diameter
speaker (Realistic 40-1331B). The worst case control scenario is an open duct end since it
allows the highest pressure levels in the duct of any termination, therefore an open end was
chosen. At the open end of the duct, the acoustic impedance was measured to be

K=0.04+0i.

The real time numerical calculations (control system) were run on a Spectral
Innovations, Inc. MacDSP64KC digital signal processing (DSP) board that resided in a
Macintosh IIx computer. The DSP board is benchmarked at 24 million floating point
operations per second (MFLOP). Running a five mode model takes 0.00014 seconds for
each data sample, state estimation, control calculation and control output. Attached to the
DSP board was a 128000 Hertz data acquisition card with a digital to analog converter and
an analog to digital converter. The control system program was written in the C
programming language using an AT&T DSP32C C language compiler. Implicit in the
compiler is a translation from C to assembly language which allows the DSP board to
operate at extremely fast speeds (24 MFLOP) compared to conventional computers such as

a MicroVAX or SUN IV.

 

81

Acoustic measurements were made with two additional Bruel and Kjaer
microphones. The first one was a Type 4166 fixed at the exit plane of the speaker and
measured disturbance input pressure. The second one was a Type 4155 and was moved to
various locations to measure system response. Both microphones were attached to a
Hewlett Packard 5423A digital signal analyzer which measured the transfer function of the
system. The excitation of the system was a random noise signal provided by the 5423A
signal analyzer. Although periodic signals are available to excite the system, random noise
is typical of industrial duct noise. The entire measurement and excitation procedures were
independent from the control system. This insured the controller would not be exciting

modeled disturbances, which is not a realistic control system.

Due to the actuator dynamics, it is not possible to place the experimental closed loop
system poles exactly. However, it is possible to move their real values to the left in the
complex plane and obtain a good estimate on where they were placed. This is done by
choosing k such that the closed loop poles are twice the magnitude of the open loop poles.
The system model used in the experiment is a five term model (two modes). The gain on
the actuator amplifier is then increased to a point where the third mode almost lies on the
imaginary axis, or very close to the onset of system instability. The system transfer
function is then measured, and the location of the placed pole is calculated from the peak
values of the closed loop resonant peaks. This method also shows the maximum peak

reduction possible for the described system.

Figures 6.9 and 6.10 are experimental frequency responses at x=2.81 m (9.21 ft).
Figure 6.9 is the open and closed loop transfer functions near the first mode and Figure
6.10 is the open and closed loop transfer functions near the second mode of the duct. The
boxes denote the open loop response and the X’s denote the closed loop response.

Although the slight frequency shift of the peaks was not modeled, it has been observed in

82

the control of mechanical systems (Chait, 1988). The small peak in Figure 6.9 near 55
Hertz is a result of the actuators natural frequency being detected by the measurement
instrumentation. Figures 6.11 and 6.12 are plots of the maximum experimental transfer
function values versus the length of the duct. Since the method described here is a global
noise reduction algorithm, these figures are included to verify that the noise reduction is
present everywhere. Figure 6.11 is the open and closed loop maximum values of the first
mode. Measurements at 20 locations yield an average attenuation of 58% along the entire
duct. Figure 6.12 is the open and closed loop maximum values of the second mode. The
average attenuation of the second mode was 55%. From these values, the real part of the
closed loop poles were moved approximately three times their open loop values to the left

in the complex plane (including compensation for unmodeled dynamics).

83

 

 

 

 

30
Magnitude _ Open Loop Response El
1| ‘
it) ' \iii
20— DE]
_ [:1 Ci
_. D D
— Closed Loop Response [:1 [j
10— E]
G EQQQQQQQ Qggéfimmmmu
I I I I I l l I I I l l I I I I I I l I I I l l
35 40 45 50 55 60
Frequency (Hertz)

 

 

 

 

IlllIlITfi

40

I l I

45

 

I l I I

50

I l I I I I I I
55 60
Frequency (Hertz)

Figure 6.9. Experimental Frequency Response, First Mode.

84

 

40
ngmx 3
I}: ‘ Open Loop Response
P0 30; —\i
I S”
20; Closed Loop Response E E]
El

.0; $ng
I §§§ Emma
EEEEEQQQQ Emma

 

 

 

 

 
  

 

 

 

O I I I r 85I I I l I 9IO I I l 1 9'5] I l I I l I
80 100 105
Frequency (Hertz)
200
J
_I
Phase of "
% 1m—
0 _
(degrees)
0- El E! D [:1 Cl [3 E] D [j I'l'l'l'll'rlr:.,-
:XXXXxxxxX"" '
400—: CP§<
: fifiaaamfim
‘200 I I l I I l I I l I I I l l I l l l I T I l l I
80 85 90 95 100 105
Frequency (Hertz)

Figure 6.10. Experimental Frequency Response, Second Mode.

 

 

85

 

 

 

 

 

 

 

 

Magnitude : Open Loop Response
LI — /
P0 40; /'BB BE.
I 1343’ El
_ dIZI BE
30— . '\
: 121 1%
: g? I Closed Loop Response \.
20— fl X X- 7 I3
_ 121/ Xxx X X XXX 5
. x—X“ ‘X \
10— / ,X' X E]
_ [ZXX X‘
2 X x
0 I I I I *Ii I I I T I I
0 0.610 1.22 1.83 2.44 3.05 3.66
Duct Location, x (meters)
Figure 6.11. First Mode Response.
, 4G
Magmtude _
_P_ ..
1;) ‘ R R
30‘. Ba I31. Era \‘\
/ I ‘
- El. [,2 El
” Q E? ,’ ‘\
20 I 'I/ '\ ', LT"
“ . t
: EH ‘1 93 'x
/ . .’ . \_
.} XXXX \. ’ X'X-X \ I
10 X'X- \x \l\ m x X [3
_ x \ I
‘ .X ‘2‘ \ I. .X Xx
q ' “x E] [Ix X
_ x1;
0 I I | I I I I I I I I
0 0.610 1.22 1.83 2.44 3.05 3.66

Duct Location, x (meters)
Figure 6.12. Second Mode Response.

Chapter 7. Conclusions

7.1. Dissertation Summary

The dynamic response of an enclosed acoustic system with point impedance on one
end can be represented in state space form. A series solution can be implemented to
approximate the exact dynamic response. Furthermore, the state space model is diagonal
so only a small number of terms are required to get a solution that converges to the exact
solution over a limited frequency range. Steady state experiments of pressure excitation
and mass flow excitation correlated well with theory. The theory can be used to
approximate duct ends that have nonconstant acoustic impedance. The transient response
model predicts the measured transient experimental data accurately. The accuracy of the
model’s transient response is predictably reduced by truncating the model’s number of

states.

Calculation of the acoustic impedance of a duct end from experimentally obtained
impedance tube eigenvalues is developed in this dissertation. These eigenvalues are easily
determined from a measured duct transfer function by commercially available Fast Fourier
analyzers. This method has the advantage of stationary microphone positioning at any
location in the impedance tube. The computational step from eigenvalue to acoustic
impedance is a closed form solution. Errors in measured impedance can arise only from
errors in measured system eigenvalues, duct length, and the speed of sound. Experimental

results show that the method is both accurate and insensitive to measurement errors.

An active noise control technique is developed to globally reduce noise levels.

Frequency domain simulations predict that the method will attenuate noise. Time domain
86

87

simulations also predict that the technique will attenuate noise and they show the effects of
unmodeled disturbances on the system. The dynamic response of control actuators is
discussed and the actuators effects are related to the acoustic duct described here. An active
noise experiment is demonstrated in which the ducts noise level is reduced globally by 58%

for the first mode and 55% for the second mode.

7.2. Directions for Future Work

There are several areas for future work or improvement in the method stated in this
dissertation. One of the most obvious extensions would be to move this work into three-
dirnensional enclosures, since most acoustic systems are three-dimensional rather than one-
dimensional. Three-dimensional noise reduction would be extremely useful in aircraft,
factories, and other enclosures that requires human habitation. This is a difficult problem
since the effects of nonuniform boundary conditions are more prevalent than in a one-

dimensional acoustic enclosure.

A topic of improvement in the one-dimensional system is in the area of actuator
dynamics. Better actuator dynamics will give better noise control characteristics. A better
actuator will also increase the control bandwidth, making control of the third and
subsequent modes possible. It might be best to retain the current actuator, and build a
separate control unit for the amplifier and speaker which delivers nearly ideal

characteristics.

Adding additional sensors and control speakers might also enhance the closed loop
systems performance for both the one-dimensional and three-dimensional noise control
problem. It would be possible place the system eigenfunctions, rather than its’
eigenvalues. This may give better noise attenuation and enhance the closed loop systems

stability. One final method noise control method that could be considered is to combine the

88

pole placement technique with one of the adaptive filter techniques mentioned earlier. The

two noise control techniques working together might provide better noise attenuation.

APPENDIX

 

 

 

Appendix. Wave Equation Derivation

The derivation of the wave equation is reproduced here. This set of equations can
be found in any good acoustics textbook, however some authors formulate the problem

using pressure as the independent variable rather than particle displacement.

Consider the one-dimensional duct shown in Figure A.1 of length L and uniform
cross-sectional area A. The air in the duct is considered to be an ideal gas with density p.
The ideal gas assumption requires the temperature to be constant throughout the tube and
the air viscosity effects to be negligible. When the duct is excited, the density of the air in
any area (of any section) is time variant. Additionally, the density of the air in the duct
changes spatially. When the duct is excited, the initial section dx and instantaneous section
(dx+du) always contain the same mass of air. This results in

Apdx = A(p + dp)(dx + du) (A.1)

where u is the instantaneous displacement of any cross section dx of the enclosure, (p+dp)
is the instantaneous density of the air, and (dx+du) is the instantaneous length of the

section of air.

Rewriting (A.1) and neglecting the higher order term dpdu yields

d
dp = mg“ . (A2)

The change of pressure due to the change of volume is expressed as

89

90

dP=—Bfl

(A.3)
where P is the pressure and B is the bulk modulus of the air. Using (A.2), equation (A.3)
can be rewritten as
dp = _Bill , (A. 4)
dx
When the duct is excited, pressure changes indicated in (A4) will exert dynamic forces on

the section dx. Balancing the inertia force and the pressure forces on the section results in

2 ' 2
Apdxd—:=A(P—Bd—u)—A P—Bd—u—Bd—‘Z‘dx . (A.5)
dt dx dx dx

Equation (A.5) can be simplified, and since the second derivatives of u are both functions

of spatial location x and time t, the derivatives are changed to partial derivatives. This

equation is
azu Zazu
Btz :0 8x2 (A6)

where c2=B/p which is the square of the wave speed in the duct. The excitation the

speakers exert on the system can be added to (A5) on the right hand side as forcing

ll.

functions on the section.

0
L

 

*iH

Figure A.1. Duct with Section dx.

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