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thesis entitled

THE ACTIVE NOISE CONTROL OF A
ONE-DIMENSIONAL DUCT

presented by

Young-Su Lee

has been accepted towards fulﬁllment
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THE ACTIVE NOISE CONTROL OF A ONE-DIMENSIONAL DUCT
By

Young-Sn Lee

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

MASTER OF SCIENCE

Department of Mechanical Engineering

1 997

ABSTRACT
THE ACTIVE NOISE CONTROL OF A ONE-DIMENSIONAL DUCT
By

Young-Sn Lee

Active noise control can reduce acoustic noise in situations where passive
material sound absorption is ineffective. The speciﬁc problem studied is the active noise
control of a one-dimensional, hard-walled duct with a partially dissipative boundary
condition. Feedforward control technique have been used to cancel noise measured at a
speciﬁc location. The control technique presented here is the modern state feedback
control to globally reduce noise in a one-dimensional duct.

A state-space model of a one-dimensional acoustic duct was derived using the
ﬁnite element method, which discretized the acoustic duct model continuum. A duct can
be assumed to be one-dimensional model when its length is relatively long compared to
cross-sectional diameter. The state-space model allows the computation of the system
response and truncation of the system model.

State feedback control with a state estimator are developed from the truncated
model to reduce noise level in a duct. The state estimator observes the system state
through a pressure measurement microphone in the duct. The state feedback control is
implemented to change the eigenstructure of the system and globally reduce noise
through the control speaker. The estimator gains are determined by linear, quadratic
optimum control theory. The results are discussed and the simulation is demonstrated in

this paper.

Copyright by
Young-Sn Lee
1997

To my parents

iv

ACKNOWLEDGMENTS

I wish to express my sincere appreciation and gratitude to my major professor

Dr. Clark J. Radcliffe for his valuable guidance and understanding during course of this
study.

I also would like to recognize the guidance and encouragement of Dr. Larry J.
Segerlind to me. His comments and advice were greatly appreciated.

Special thanks to my parents, brother and sister for their constant support and
encouragement they have given me the entire time I have been in school.

Finally, I would like to thank Choongmin Jung, Jaeman Cho, Soonsung Hong,
Christopher Hause, Yihong Zhang, Mink, and I-Ping Liu for their friendship and

encouragement.

TABLE OF CONTENTS

LIST OF FIGURES .......................................................................................................... vii
1. INTRODUCTION .......................................................................................................... 1
2. SYSTEM MODEL .......................................................................................................... 3
2-1. Physical System Model ............................................................................................ 3
2-2. Finite Element Model ............................................................................................... 5
3. STATE SPACE REPRESENTATION ........................................................................... 9
4. TRUNCATION ............................................................................................................. 11
5. STATE FEEDBACK CONTROL .............................................................................. 14
5-1. State Estimator ....................................................................................................... 14
5-2. State Feedback Control with Optimized Feedback Gain ..................................... 16
5-3. Frequency Domain Simulation ............................................................................... 18
5-4. Time Domain Simulation ....................................................................................... 22
6. CONCLUSIONS ........................................................................................................... 24
BIBLIOGRAPHY .............................................................................................................. 25

vi

LIST OF FIGURES

Figure 1 Physical acoustic duct model ................................................................................ 4
Figure 2 The frequency response of 3.66m duct by ﬁnite element solution (solid line) and
analytical solution (dashed line). Impedance constant K=O.3, input location x=Om,

and output location x=O.762m. Number of element for ﬁnite solution is 50. ......... 8

Figure 3 The frequency response of 7 term (solid line) and 62 term (dashed line) duct
model with impedance constant K=O.3+O.2i. Total length of duct L=3.66m, input
location x=Om, and output location x=O.792m. ..................................................... 12

Figure 4 The time response of 7 term (solid line) and 62 term (dashed line) duct model to
the harmonic pressure input with a frequency of 150 Hz initiated at t = 0 second.

Total length of duct L=3.66m and impedance constant K=O.3+O.2i. Input location

=0m, and output location x=O.792m. ................................................................... 13

Figure 5 The time response of 7 term (solid line) and 62 term (dashed line) duct model to
the pulse pressure input with a magnitude of 1 N /m2 and width of 0.005 second

initiated at t=0 second. Total length of duct L=3.66m and impedance constant

=0.3+0.2i. Input location x=0m and output location x=O.792m. ....................... 13
Figure 6 The schematic diagram of active noise control system ...................................... 19
Figure 7 The block diagram of active noise control system ............................................. 20

vii

Figure 8 The frequency response of closed (solid line) and open (dashed line) loop
system of 3.66m duct. Disturbance input location x=Om, states measurement
location x=O.792m, K=O.3+O.2i. Number of truncation term for observer is 6. ...20

Figure 9 Pole locations of closed (0) and open (x) loop systems of a 3.66m duct.
Disturbance input location x=Om, states measurement location x=O.792m, and
control input location x=3.56m. Impedance constant at the termination end
K=O.3+O.2i. Number of truncation term for observer is 6. ................................... 21

Figure 10 The time response of closed (solid line) and open (dashed line) loop systems
of a 3.66m duct to the harmonic pressure input with a magnitude of 1 N / m2 and
a frequency of 150 Hz initiated t=0 second. Harmonic pressure input location
x=Om, state measurement location x=O.792m, control input location x=3.56m.
Impedance constant of the termination end K=O.3+O.2i and number of truncation

terms used for observer is 6. .................................................................................. 23

viii

1. INTRODUCTION

Indoor noise reduction has increasingly become an important issue. The growth
of noise in the workplace containing long duct assemblies is mostly due to the increase in
the usage of high speed fans and complex machinery. This threatens the health of people
and adversely affects the productivity. This problem can be solved by either passive or
active noise control.

Passive noise control is control that uses noise absorption materials. These noise
absoption materials counteract the undesirable effects of sound reﬂection by the hard,
rigid, interior surfaces which they cover or replace. The advantage of passive noise
control is simplicity and stability. Effective passive noise absorption materials need to
have a thickness at least 10% of the noise wave length (Mankovsky, 1971). However,
they are not suitable to reduce noise at low frequencies. The noise absorption might not
be effective when the system or the operating conditions are changed because the noise
absorption materials are designed for the speciﬁc systems or operating conditions.

The active noise control uses the intentional superposition of acoustic waves to
reduce noise, and is able to adapt for system changes. In active noise control, low
frequency noise reduction can be obtained. In feedforward control, the system inputs are
measured and the cancellation sound, which contains exactly the same frequency,

magnitude, and phase components as the noise to be cancelled, is generated. In this case,

the noise cancellation is limited to a small region where noise is measured and cancelled.
At the other locations away from the measurement location, noise is often increased
because errors of frequency, magnitude, and phase are eveloped. In the feedback control,
the system states are measured or estimated and the pole locations are changed to reduce
but not eliminate the noise level at all positions in the duct.

The objectives of the research presented here are modeling of a one-dimensional
hard-walled acoustic duct with a partially dissipative boundary condition, and the
application of the state feedback control to achieve active noise control in the duct. This
paper uses the ﬁnite element method to develop a discretized state space model of the
duct. The state feedback control is applied to obtain noise reduction in the duct. The

spillover problem is also quantiﬁed in this paper.

2. SYSTEM MODEL

2-1. Physical System Model

The system to modeled (Fig. 1) is a one-dimensional, hard-walled duct. The
length of duct is longer enough that the wave propagation in the cross sectional direction
can be ignored. The wall of duct is assumed to be rigid that the energy dissipation occurs
only at the ends. The gas in the duct is assumed to be uniform density p, and
temperature T.

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

2 d“(x’t)

P(x,t) = —pc dc

(1)

The duct model is governed by a linear wave equation for the particle

displacement u as a function of time, and the spatial location x (Kinsler and Frey, 1962).

é’2u(x,t) _ c2 ﬂzu(x,t)

a2 _ a; (2)
This model equates the spatial gradient of pressure the particle acceleration.
The boundary condition at x=0 is related to the pressure input.
L40”) = _—1—2— P(0,t) (3)
63: xx

The partially reﬂective boundary condition at x = L is the relationship between the spatial

 

 

 

 

 

Figure 1 Physical acoustic duct model

gradient and the time gradient of the particle displacement. This is expressed as (Seto,
1971)

d1(L,t) _ _£ d1(L,t)
dc c 0’?

, K at o, 1, oo (4)

where K is dimensionless complex impedance of the termination end. The real part of K
is associated with energy dissipation. The imaginary part of K is associated with
conservative ﬂuid compliance and inertial effects. When real part of K equals zero, or
inﬁnity, the termination end of the duct reﬂects all the acoustic energy and the response is
composed of standing waves only. When K=1 , the termination end of duct absorbs all the
acoustic energy and the response is composed of propagating waves only. All other

values of K yield some combination of propagating and standing wave response

(Spiekennann and Radcliﬂ‘e, 1988).

One or more control inputs associated with mass ﬂow are allocated in the domain
to control noise in the duct. The system equation with these inputs is nonhomogeneous.
The linear second order wave equation modeling particle displacement in a hard-walled,

one-dimensional duct is (Seto, 1971)

3’ ,t_ 6255.82 ,t ”a M, t
0” i=1 105

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

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

(kg/s), x, = location of mass ﬂow input (m), S = speaker area driving the mass ﬂow input

(m2 ), P(t) = pressure excitation at x = O (N/ m2 ), and 6 (x) = the Dirac delta ﬁmction.

2-2. Finite Element Model

The ﬁnite element method can change this partial differential equation for the
continuous medium in the duct to a system of ordinary differential equation for a
discretized medium. Finite element analysis based on Galerkin’s method uses weighting
functions to approximate equation solutions. This method uses piecewise smooth
functions, and the Galerkin residual integral formulation to generate a system of algebraic
equations (Segerlind, 1984). The Galerkin’s residual integral with respect to the space

coordinates for ﬁxed instant of time is

 

_" ,(c azuoc t) azu(x t) M(t) _
xlwc (2:2 -§[6(x— x.)1—{— pg Ddx-o

(6)

where W T is the Galerkin weighting functions which consists one dimensional shape
functions associated with speciﬁc nodes in the ﬁnite element mesh. This residual integral
can be solved for the system of differential equations for the particle displacement vector,
u.

The ﬁrst term on the right hand side yields the element stiffness term and

interelement term If). The stiffness element is generated by conventional integration.
The interelement term is a row vector which contains the element contributions. This
vector is deleted except when derivative boundary conditions are speciﬁed (Segerlind,
1984). In the case of duct, the interelement term contains only the boundary conditions
for the left and right end. The boundary condition on the right end can be separated from
interelement vector, because it can be expressed as damping element. The remaining
interelement vector only the left boundary condition. The second term on the right hand
side of the Galerkin’s residual integral turns out to be a mass element, and the third term
yields point source vector which contains the mass ﬂows at system input location. This
point source vector and the interelement vector which contains the left boundary
condition consist the force vector.
Mii+D|i+Ku=fp(t)+fM(t) (7)

where M = TN TNdx

‘o‘N’o‘N
K=j01 Zdx

X

 

D = If;

_ (e)
fp _ Ix=O

k .
i=1

x=x,

Dimension of each matrix, n is determined by number of elements. Each matrix row

represents a speciﬁc location in the duct. The boundary conditions are incorporated into

D and f p (t). D is derived from 15; which represents the boundary condition of the

right end. fp (t) is consist of I“)

,3, term that contains the boundary condition of the left
end, and fM (t) is consist of the summation term which represents the mass ﬂow inputs,
M,(t) at x = x,.

The frequency response of 3.66m duct (Fig. 2) show comparison between ﬁnite
element solution (solid line) and analytical solution (dashed line). The response is
measured at x=0 m. The impedance at the termination end is K=O.3 and 50 elements are

used for ﬁnite element solution. The ﬁnite element solution is quite similar to the

analytical solution.

 

PhueoHPl
O

 

 

 

 

 

4 I I l I I I l I I
I

Es- I .
3
32- _

1

o l 1 l l l l l I l

0 60 100 160 200 250 300 350 400 450 500
Freqremy (Hz)

m I I I I I I I l I

100-

.1” _

 

 

 

 

0 50 100 150 200 250 NO 350 400 460 500

 

 

 

Figure 2 The frequency response of 3.66m duct by ﬁnite element solution (solid line) and
analytical solution (dashed line). Impedance constant K=O.3, input location
x=Om, and output location x=O.762m. Number of element for ﬁnite solution is
50.

3. STATE SPACE REPRESENTATION

The state space representation used in the control system observer and simulation

can be derived from the ﬁnite element form of governing equation. Deﬁning that the
state vector, 1r1 represents the vector of particle displacements, u in equation (7) and X,
represents the vector of particle velocities, in , the second order differential equation of the

system can be changed to the ﬁrst order differential equation. Letting x, = u , x2 = u ,

and x = [xl,x2]T gives

X(t) = Ax(t) + Bpfp(t) + BMfM(t) (8-1)
P = ch(t) (3'2)
l ° ’ l
where A = -1 -1
-M K -M D

A is 2n x 2n matrix which contains the characteristic of physical duct system. B has

dimension 2n x n and plays role of gain of force vectors 1', and {M in state space. Force
vectors f p and f M are the disturbance input and control input, respectively. 2n x1

matrix, c contains the spatial gradient of particle displacement which is expressed in

equation (1).

10

A can be diagonalized using modal transformation, x = (Dz. (D is modal matrix
and z is the state vector in modal coordinate. Substituting x = (Dz and premultipling

inverse of modal matrix gives
i=ADZ+Bprp+Bwa (9'1)
P = cDTz (9-2)

where A D = (V‘ACD

BMD = (D-IBM
cDT = cTCD

A D is diagonal matrix and has the same characteristic equation and eigenvalues as what
A has, because A D is mathematically similar to A. This state space representation

allows the individual modes to be decoupled from each other.

4. TRUNCATION

A truncated ﬁnite element model is needed because the exact solution to the
acoustic duct problem requires an inﬁnite number of elements which is not available. A
truncated model can be obtained by simply truncating high frequency modes of the state
space model shown in equation (9), because the state space model has decoupled
individual modes. This truncated model allows only dominant modes to be observed and
controlled.

Spillover is generated by truncation. Spillover is system contamination by
unmodeled modes (Balas, 1978). Spillover occurs when unmodeled modes in the system
interacts with the control system action. This Spillover degrades control performance and
can cause instability.

The frequency response of a 3.66m duct (Fig. 3) shows the effect of truncation.
The response is measured at x=O.792m with the pressure excitation at x=Om. The
impedance is K=O.3+O.2i and a truncated model with 7 term is used to approximate the
solution (solid line). This truncated model with 7 term has only 3 dominant modes while
a 62 term model (dashed line) includes higher ﬁ'equency modes.

Time response of the ﬁnite element model (Fig. 4) shows the effect of truncation
in time domain. The input into the duct is a harmonic pressure excitation with a

frequency of 150 Hertz initiated at t=0 second. The input location is x=0 and the

11

12

response is calculated at location x=O.792m. The impedance constant K=O.3+O.2i is used.
This time response shows the predicted response with only 7 term (solid line) is quite
similar to that for 62 term (dashed line) because truncated error is reduced by decoupling
the states.

Another time response of the ﬁnite element model (Fig. 5) shows the response to

the pulse pressure input with 7 term model (solid line) and 62 term (dashed line). The

magnitude of pulse input is 1 N / m2 and width is 0.005 second. The input location is
x=0 and the response is calculated at x=O.792m. The impedance constant is K=O.3+O.2i.
The 7 term ﬁnite element model shows the same response in each of the states but lower

resolution due to the reduction in state number.

 

 

Wdlﬁ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3 The frequency response of 7 term (solid line) and 62 term (dashed line) duct
model with impedance constant K=O.3+O.2i. Total length of duct L=3.66m,
input location x=Om, and output location x=O.792m.

 

 

 

 

 

 

 

Figure 4 The time response of 7 term (solid line) and 62 term (dashed line) duct model to
the harmonic pressure input with a frequency of 150 Hz initiated at t = 0 second.
Total length of duct L=3.66m and impedance constant K=O.3+O.2i. Input
location x=Om, and output location x=O.792m.

 

 

 

 

 

———1——A_——L_—I—A—i
‘0'. (I 0.01 0.” 0.03 0.04 0.06 9.00 0.07 0.“ 0.“ 0.1

 

 

 

Figure 5 The time response of 7 term (solid line) and 62 term (dashed line) duct model to
the pulse pressure input with a magnitude of 1 N/ m2 and width of 0.005 second initiated

at t=0 second. Total length of duct L=3.66m and impedance constant K=O.3+O.2i. Input

location x=0m and output location x=O.792m.

5. STATE FEEDBACK CONTROL

5-1. State Estimator

State feedback control through an appropriate state feedback gain matrix can
change the eigenstructure of the physical duct system to the desired one. In this case, all
states need to be measurable and available. However, none of states variables of the
physical duct are directly measurable, therefore a state variable estimator is needed. A
state observer is a device or computer program that runs in parallel with the actual system
and estimates the state variables based on the measurements of input and output from
physical system. An estimate of the inﬁnite number of state variables of the physical
duct is not possible. So a truncated state Observer is used.

The state observer equations for single disturbance are

2 = Am +BM~fM +Bp~fp +g~(P- 1") (10-1)

13 = cfviN (IO-2)

where A N is a truncated matrix which contains the N dominant dynamic characteristics

of the duct system, 1'5 is the estimated sound pressure by observer, f M is control input
vector which is mass ﬂow and located at z ¢ 0 , B W is control input matrix, f p is
disturbance input which is a pressure excitation and located at the left end, B m is the

disturbance input matrix, g N is the observer gain vector, and P is the pressure

14

15

measurement output from the actual duct. 2,, designates the observed state vector which
approximates the actual state vector, 2”. The last term on the right side of equation (10-
1) is a correction term. This correction term involves the difference between actual P and
estimated P , and is weighted by g N . This Observer gain vector, g N determines the error
dynamics between the actual states and the estimated states.
The observer error equation can be expressed by
e=Aee+Bmfp (11)

where e = 2N — a” that is the error vector which designates the difference between the
actual states and the estimated states, and A, = A N — g Ne; . The poles of observer error
equation determine the rate of convergence of the error to zero. If matrix A, is a stable
matrix, error vector will converge, which means the estimated states will converge to the

actual states. Hence, if the eigenvalue of matrix A is chosen such that the dynamic

behavior of the error vector is asymptotically stable and adequately fast, then any error
will converge with adequate speed. However, the error equation is driven by any
unknown input, f p. The desired poles of matrix A, can be located by determining the

observer gain matrix, g N . This pole placement can be done using Ackermann’s formula.

The Ackermann’s formula for an observer is (Phillips and Harbor, 1996)

I- -I-1I— -

 

 

 

 

c” O
c A E

gN zae(AN)l N- N 0 (12-1)
_¢~A'i«-IJ -1-

a,(AN) = A7,, + a"_,A',§,"+---+a,AN + aoI = 0 (12-2)

16

where a¢(AN) is the desired characteristic polynomial whose roots are eigenvalues for
error system. The observer gain is recommended to be chosen so that the dynamic speed
of the observer is twice or fourth times faster than those of the system (Phillips and

Harbor, 1996).

5-2. State Feedback Control with Optimized Feedback Gain

Pole location determines the speed and damping of the dynamic response.
Changing pole location gives different eigenstructures of the system. Pole location is
generally desired to be as far to the left of the origin as possible, because they have faster
dynamic response. However, locating poles far to the left of the origin yields high
control gain and signals which are too large to be produced within the system power
limitations. Such high gain systems also have signal noise problems, and limiting the
speed of response is necessary to avoid saturation and noise problems. The pole location
for the speed limitation of response can be determined by Ackermann’s formula

expressedas
kac=[0 0 IIBPN ANBPN ANn-IBPN]ac(AN) (13)

where kw is the state feedback gain vector and ac(A N) is the desired characteristic
polynomial. The state feedback gain matrix can be decided by the desired poles and can
be used for control law f M = —kxi,,.

Another technique to determine the state feedback gain vector is linear quadratic
optimal control technique. This optimal control technique generates control which

minimize both system error and the input required to a device. These system error and

17

the input can be expressed by performance index. Performance index in this paper
consists the integral of a quadratic form of the truncated state vector, 2,, plus an another

quadratic form of the control input, f M , that is

J = Taft)“ + fMRfM)dt (14)
0

The ﬁrst term on the right side of the equation (14) represents a penalty on the deviation
of the N dominant state vector , 2,, from the origin, while the second term represents the
control effort. Q is a state weighting matrix which determines the relative importance of
the each state variable, and assume to be positive deﬁnite Herrnitian or real symmetric
matrix. In this paper, Q is selected as a diagonal matrix which has the different diagonal
elements to weight each state variable differently. R is a matrix which contains the
weighting value for control vector, and chosen 1 x 1 matrix with weighting value 1.

The optimal control law which minimize the performance index, J in equation

(14) can be expressed as

fl" = -kapi~ (15)
where kqp is a optimal feedback gain vector. Therefore, performance index can be
minimized by this optimal control law, and such an optimal control law can be obtained
by a proper vector kap. The equation that gives the optimal k 0,, is given by

k0,, = R"B,,,‘P (16)
where matrix P is the symmetric, positive deﬁnite solution of the Riccati equation

expressed by (Ogata, 1970)

18

AN‘P + PAN - PBPNR"B,N'P + Q = 0 (17)

5-3. Frequency Domain Simulation

The schematic and block diagram of the control system for duct is shown in Fig. 6
and Fig. 7. The state estimator measures pressure output ﬁ'om the physical system and
the feedback input. The disturbance input is unknown because, in general, there is no
way to determine the disturbance excitation of the duct.

A frequency response simulation of the controlled system is shown in Fig. 7. The
length of the duct is 3.66m, the response is measured at x=O.792m, and the control
speaker is at x =3.56m. The acoustic impedance at the end is K=O.3+O.2i. The number of
terms used to form the model is 7. The state feedback gain is chosen to minimize the

performance index J with

r- '1

oococo
ooocgo
oococo
cccoco
ogoooo

cases

6000

 

 

R = [1]
The frequency response (Fig. 7) shows that the fourth closed loop resonance tends to
increase in amplitude as the closed loop resonances in the lower frequency range (solid
line). The more magnitudes of closed loop resonances in frequency range of modeled
dominant poles are decreased, the more likely magnitudes of closed loop resonances in

frequency range of unmodeled poles are increased.

19

The pole locations of these system in complex plane (Fig. 8) also shows control

spillover. Placing the dominant poles to the left from their original place causes some

unmodeled poles to move to the right from their original place. In extreme case, those

unmodeled poles move to the positive real plane, and cause the instability of the system.

 

 

 

 

 

   

 

 

 

 

 

Observer
and
Controller
Response ["1 Control
. Measurement Speaker
Drsturbanee Microphone
Speaker
\ Nonzem
Acoustic Duct Acoustic
Impedance

 

 

 

Figure 6 The schematic diagram of active noise control system

20

 

 

 

 

 

Disturbance Input
mm“ ‘M ‘0 to) = Ana) + an, (r) + sumo I Pressure pom)
P = 3:0) I '
State Estimator
‘2” (r) = (AN - w; )2” (r) + nmruu) + g”?

 

 

 

 

 

 

 

 

Control System

ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

Figure 7 The block diagram of active noise control system

 

10

 

W“
neon

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 8 The ﬁ'equency response of closed (solid line) and open (dashed line) loop
system of 3.66m duct. Disturbance input location x=Om, states measurement
location x=O.792m, K=O.3+O.2i. Number of truncation term for observer is 6.

21

 

 

0.8

0.4

0.2

-0.2

-o.4

-0.8

-0.8

 

 

 

 

 

 

 

Figure 9 Pole locations of closed (0) and open (x) loop systems of a 3.66m duct.
Disturbance input location x=Om, states measurement location x=O.792m, and
control input location x=3 .56m. Impedance constant at the termination end
K=O.3+O.2i. Number of truncation term for observer is 6.

22

5-4. Time Domain Simulation

Time domain simulations predict the response of the system over a ﬁnite time.
They are useful in determining the transient response of the controller. A time domain
simulation is performed with the plant and the control system shown in Fig. 7.

A time domain simulation with 3.66m duct is shown in Fig. 10. The response is
measured at x =0.792m and the control speaker is at x =3.56m. The impedance at the
termination end is K=O.3+O.2i. 6 truncated term was used for state estimator. The
simulation step size is 0.0002 seconds. The simulation excitation is a sine wave with unit

magnitude at 150 Hertz. The state estimator and controller are both activated at t=0
second. The steady state amplitude of closed loop system (solid line) is 0.9 N / m2 while

that of open loop system (dashed line) is 1.4 N / m2 .

23

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.:

“-—
~

 

 

 

0.04 0.05 0.00 0.07
M

0.“ 0.00

Figure 10 The time response of closed (solid line) and open (dashed line) loop systems
of a 3.66m duct to the harmonic pressure input with a magnitude of l N / m2
and a frequency of 150 Hz initiated t=0 second. Harmonic pressure input
location x=Om, state measurement location x=O.792m, control input location
x=3.56m. Impedance constant of the termination end K=O.3+O.2i and number of
truncation terms used for observer is 6.

 

 

 

6. CONCLUSIONS

A one-dimensional hard-walled acoustic duct with a partially dissipative boundary
condition was modeled using the ﬁnite element method. This paper showed the ﬁnite
element method discretize the continuum system and approximate the exact dynamic
response of the system with a convenient numerical computation.

Active noise control technique was developed to reduce noise levels. Frequency
and time domain simulations are presented. These simulations showed the active
feedback control techniques globally reduced noise in the duct instead of canceling the
noise in particular region. Active feedback control generated spillover which degrades
control performance and caused instability.

There are several areas for ﬁrture work or improvement in the method. One
obvious extension is to move this work into three dimensional enclosures, since most
acoustic system are three-dimensional rather than one-dimensional. Three-dimensional
noise reduction would be extremely useful in aircraﬁ and factories.

A topic of improvement in the one-dimensional system is in the area of ﬁlters.
Noise reduction by feedback control has limitation because of spillover problem.
However implementing the ﬁlters in order to minimize the spillover problem will allow

more noise reduction by feedback control.

24

BIBLIOGRAPHY

BIBLIOGRAPHY

Balas, M. J ., 1978, “Feedback Control of Flexible Systems”, IEEE Transactions on
Automatic Control, Vol. AC-23, No. 4, pp. 673-679.

Hull, A. J., Radcliffe, C. J., Miklavic, M., and MacCluer, C. R., 1990, “State Space -"
Representation of the Nonself-Adjoint Acoustic Duct System”, Journal of Vibration and

Acoustics, Vol. 112, No. 4, pp.483-488.

Hull, A. J ., Radcliffe, C. J ., and Southward, S. C., 1993, “Global Active Noise Control of

a One-Dimensional Acoustic Duct Using a Feedback Controller”, Transaction of the

ASME, Vol. 115, pp. 488-494.

Mankovesky, V. S., 1971, Acoustics of Studios and Auditoria, Communication Arts
Books, NY.

Ogata, K., 1970, Modern Control Engineering, Prentice-Hall, Englewood Cliffs, NJ.
Phillips, C. L., 1996, Feedback Control Systems, Englewood Cliffes, NJ.

Segerlind, L. J ., 1984, Applied Finite Element Analysis, John Wilely & Sons, Inc., NY
Seto, William W., 1971, Theory and Problems of Acoustics, McGraw-Hill, NY.
Spiekennan, C. E. and Radcliffe, C. J ., 1988, “Decomposing One-Dimensional total

Acoustic Pressure Response into Propagating and Standing Waves”, Joumal of
Acoustical Society of America, Vol. 84, No. 4, pp. 1536-1541

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