NASA Technical Memorandum 110281

U. S. Army Research Laboratory Technical Report 1226

"///

Performance of Optimized Actuator and

Sensor Arrays in an Active Noise Control

System

D. L. Palumbo and S. L. Padula

Langley Research Center, Hampton, Virginia

K. H. Lyle and J. H. Cline

Vehicle Structures Directorate

U.S. Army Research Laboratory

Langley Research Center, Hampton, Virginia

R. H. CabeU

Virginia Polytechnic Institute and State University, Blacksburg, Virginia

September 1996

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-0001

https://ntrs.nasa.gov/search.jsp?R=19970001270 2017-09-11T14:10:14+00:00Z

PERFORMANCE OF OPTIMIZED ACTUATOR AND SENSOR

ARRAYS IN AN ACTIVE NOISE CONTROL SYSTEM

D. L. Palumbo and S. L. Padula

NASA Langley Research Center, Hampton, VA

K. H. Lyle and J. H. Cline

U. S. Army Vehicle Structures Directorate-ARL, Hampton, VA

R. H. Cabell

Virginia Polytechnic Institute and State University

Blacksburg, VA

Abstract

Experiments have been conducted in NASA

Langley's Acoustics and Dynamics Laboratory to

determine the effectiveness of optimized actuator/sensor

architectures and controller algorithms for active control

of harmonic interior noise. Tests were conducted in a

large scale fuselage model - a composite cylinder which

simulates a commuter class aircraft fuselage with three

sections of trim panel and a floor. Using an

optimization technique based on the component transfer

functions, combinations of 4 out of 8 piezoceramic

actuators and 8 out of 462 microphone locations were

evaluated against predicted performance. A

combinatorial optimization technique call tabu search

was employed to select the optimum transducer arrays.

Three test frequencies represent the cases of a strong

acoustic and strong structural response, a weak acoustic

and strong structural response and a strong acoustic and

weak structural response. Noise reduction was obtained

using a Time Averaged/Gradient Descent (TAGD)

controller. Results indicate that the optimization

technique successfully predicted best and worst case

performance. An enhancement of the TAGD control

algorithm was also evaluated. The principal

components of the actuator/sensor transfer functions

were used in the PC-TAGD controller. The principal

components are shown to be independent of each other

while providing control as effective as the standard

TAGD.

Introduction

This work is part of a continuing effort at NASA's

Langley Research Center to better understand the

methods and techniques that can be brought to bear on

the problem of active control of aircraft interior noise.

A primary concern in this effort has been harmonic

noise of the type that is created by propeller blades and

other rotating machinery. Of specific interest in this

paper is the integration of arrays of actuators and

sensors into an effective active structural acoustic

control (ASAC) system.

Initial experiments and analytical studies using

ASAC to control interior noise focused on actuation of

the primary shell of the structure 1'2. These efforts

demonstrated that control sufficient to achieve

substantial reduction (-12 dB) in interior noise levels

could be achieved. However, concerns have been raised

in the aircraft industry over any additional structural

fatigue that an ASAC system might introduce. If the

ASAC system did indeed cause increased structural

fatigue, then the cost of certifying and maintaining the

aircraft would increase, thereby compromising any cost

advantages that ASAC may provide.

An alternative approach has been investigated which

uses force actuators bonded to the trim panels 3 In this

previous effort, several piezoelectric actuators were

bonded to the surface of a trim panel which was hard

mounted to the ring frames of a composite fuselage

model. Modal decomposition of the acoustic field was

used to select actuator/sensor sets that were well

matched to the primary response (excited by an external

acoustic source). Best results were obtained when the

actuator modal response did indeed match the primary,

but even then only moderate noise reduction was

possible (-5 dB).

This paper reports efforts to improve the

effectiveness of aim panel ASAC by using

optimization techniques to select best case

actuator/sensor architectures and controller algorithms.

The performance of the actuator/sensor arrays were
testedusingaTime Average/Gradient Descent controller

(TAGD). To better judge how well the optimization

procedure predicted the system response, worst case

actuator/sensor arrays were also computed and tested.

These results are compared to results obtained with

actuator/sensor arrays chosen using modal

decomposition procedures 3. Finally, a modified version

of the TAGD controller was tested in which the

principal components of the actuator/sensor transfer

functions were used in the gradient descent algorithm.

The benefit of this approach is that the principal

components with their corresponding error components

are independent of each other. The result is a control

algorithm in which each channel of control is

orthogonal to every other channel. This provides more

robust convergence during adaptation, and enables

parallel implementation of the algorithm.

The next sections provide an overview of the

optimization techniques. This is followed by a

description of the experiment procedure. The paper

concludes with the details of implementing the

optinaization techniques and the associated results.

Architecture Optimization Overview

Figure 1 is a block diagram of a generic noise

control problem. It is desired to reduce the acoustic

noise field, ej, which is produced by the primary source,

p, by applying control through the actuator array, ck.

The coupling of the primary and control sources with

the acoustic field is given by their respective transfer

functions, H v and H c. The relationship of these

components is written as

= H c

ej ___ kCk + HP p,

k=l

(1)

where N c is the number of actuators in the control array.

A measure of the total sound field of a particular control

solution, Jr, is the sum of the squared pressures over the

N, points which define the response field.

Jr = E eft;, (2)

j=l

where * is the complex conjugate. The control is

most effective (the performance is optimized) when Jr

is minimized.

Given a wide choice of actuator and sensor locations,

the goal of optimization is to identify a subset of the

actuators and sensors which provides the best

performance, i.e. has the potential to reduce the acoustic

field to its lowest level. The optimization methodology

Primary Source Sensor Array Actuator Array

\ oo...o / mm...m

,,o ...o...? [ .ram...m ck

T/+0..." \ " .:."

q

Figure 1. A generic noise control system

which was used is conceptually simple. From a given

starting state, perturbations of sensors and actuator

settings are simulated, those combinations achieving

the best performance levels are retained and refined until

a global optimum is found. As basis for the

simulation, broad band transfer functions between the

noise sources and the acoustic field are acquired.

At a frequency of interest, equation (1) is solved for

the control output and the resulting response using a

least squares method which minimizes J .+ For a total

number of possible actuators, No, there are No

combination N c possible subsets of control arrays. The

combination operator is given by,

(3)

An exhaustive search of all possible control array

subsets would uncover one array at which the response

is minimized globally. For the optimum control array,

c °, a corresponding optimum sensor array, s °, must be

found. Given N, elements of the sensor subset, there

are N, combination N s possible subsets of the response

field at which sensors may be located. Equations (I)

and (2) become

c o

Si = EH_c k q" HPp (4)

k=l

and

N,

J, = T_m.;]. (5)

/=I

A search similar to the one performed to locate c° is

performed to find s°.
Themethoddescribed above uses exhaustive search

to find the optimum actuator and sensor subsets. For

practical applications, this may not be possible.

Consider a test matrix for the actuators on a cylindrical

shell where actuators are placed every 10° from -90 ° to

0° along the circumference and every 1/3 meter over 3

meters along the axis for a total of 100 possible

actuator locations. If the best 10 actuators sites are to

be selected, then over 10 t3 combinations would need to

be considered. Clearly an alternative search procedure is

required.

The search space can be narrowed using techniques

of integer and combinatorial optimization s. These

techniques rely on the evaluation of a cost for each

state. Having this information, the search through the

state space can be directed along those paths which

promise to reduce the cost the most. The expressions

for Jr and J, in equations (2) and (5) respectively are

ideal cost metrics. One drawback of combinatorial

optimization is that the search may converge to a local

minimum, i.e., a state in which the cost is not the

global minimum, yet, each step to a nearby state

increases the cost. Combinatorial searches may also

visit the same state repeatedly. To reduce the likelihood

of these drawbacks, a method referred to as tabu search

was employed 6. Tabu search alternately focuses and

diversifies its search; first considering states very

similar to the current state, then considering states very

different. In addition, a history of the states already

visited is kept so that a state, once visited, is not

evaluated again.

By manipulating the parameters of the optimization

process, it was possible to obtain different sensor sets,

for example s l °, s2 °, s3 °, which return practically

identical noise reductions, i.e., J,i--J,2--'J,3. A further test

of the solution quality has been suggested 7'8where

advantage is taken of the similarities which exist

between adaptive noise control algorthms and linear

regression algorithms. The variance of a regression

coefficient 9 is one such measure of solution quality and

can be written in the context of the noise control

problem as

V,c , ia,kfO2 "cH"cl 't

(6)

where diag k denotes the k _ diagonal, oa is the variance

of the measurement noise, and H cn is the Hermitian

of H c. The measurement noise is estimated by Elst°l

which is the error of the noise control solution 7'8.

Large variances in the regression coefficients

(control forces, cO can be related to dependencies

(colinearities) in the transfer functions 8"9. These

dependencies can produce erroneous solutions, i.e.,

where J, is not minimized, and can increase the control

algorithm's sensitivity to measurement noise. It is,

therefore, desirable to select sensor solution sets with

low control force variance.

Modal Decomposition Overview

The sound pressure field, ej, can be decomposed into

modal components to identify the acoustic modes 3. The

decomposition is based on:

v Io,)=j'j'j" e( ,o lx)dV

vol

where the mode shapes, _m(X), are the hardwall cavity

modes and are orthogonal. Win(co) is the contribution

of the m 'h mode to the acoustic pressure, P(x,o_), as a

function of frequency, co. V is the volume of the

acoustic space. For the modal decomposition, the

pressure field was measured experimentally and the

mode shapes were derived using NASTRAN. In

reference 3, 40 mode shapes were derived. The 40 modes

had 7 different circular cross-section distributions as

shown in figure 2. Node lines are shown in figure 2 as

dashed lines. The axial modes varied as cosn_x/L where

n varied from 0 for a uniform axial distribution to 7.

For the modes of interest in this work, the

correspondence between mode number, the circular

cross-sectional distribution and the axial mode number

are given in Table 1.

A preferred actuator set is selected by inspection of

the modal decompositions of the primary source and the

individual actuator responses at a particular frequency.

Dominant modes in the primary response are matched

with dominant modes in the actuator responses. An

actuator set that is well matched to the primary will

return good noise reduction with little spillover. A

sensor set is chosen by inspection of the pressure field.

Sensors are placed at nodes and nulls to stimulate

control and discourage spillover. This process becomes

more difficult when the primary and actuator have

mutiple strong modes. The process as a whole is

heuristic and several iterations may be required to get

the best results.
Table 1. Mode number (m) vs. circular

cross-section mode shape (Y), axial

mode number (n) and (f)

m I Y I n I f

12 C 2 200.9

20 D 2 260.2

21 C 4 261.0

22 E 2 265.6

23 B 5 270.9

28 D 4 309.0

29 E 4 313.6

A B C

D E F

G

Figure 2. Circular cross-section mode shapes for

cylinder interior

Algorithm Optimization Overview

The basic Time Averaged/Gradient Descent (TAGD)

algorithm is straightforward. It is assumed that the

mean of the squared error sensor signals (MSE) is

proportional to the overall interior noise level. The

controller is, therefore, designed to reduce the MSE.

The controller MSE value is equivalent to the mean of

the cost metric, J,, used by the optimization procedure.

It has been shown t° that the MSE is a quadratic function

of the controller outputs, ck, thus having only one

minimum. Using a process of gradient descent, the

TAGD controller perturbs the magnitude and phase of

each actuator control signal to evaluate the gradient.

The actuator signal is then moved in the direction that

reduces the MSE. To keep the phase of the internal

reference constant with respect to the primary noise

disturbance, the controller is phase locked to the

primary signal. The controller's interrupt rate is kept

well above the Nyquist frequency by multiplying the

phase locked signal by 8. A distinguishing feature of

the TAGD algorithm is that, unlike the filtered-X LMS

algorithm _°,TAGD does not require a model of the

controller subsystem.

Several parameters are available to fine tune the

TAGD controller's behavior. Table 2 contains

parameter values that provided dependable, but slow,

noise reduction. A delay is installed between the time

that an actuator is adjusted and the controller begins to

sample the MSE. The delay is necessary to avoid

sampling the transients which are caused by the abrupt

setting of the actuator to a new value. A delay of 200

cycles (or interrupts) is equivalent to about 100 ms at a

primary frequency of 230 Hz. A two coefficient FIR

digital filter is used to set the magnitude and phase of

each controller output. The algorithm adjusts one filter

weight at a time by an amount equivalent to

_r.Peek*MSE to determine the direction of decreasing

MSE. When the correct direction is known, the

controller will set the filter weight to a value of

sign(Peek)*Step*MSE. It was found that Step must

be kept less than Peek for stable operation of the

controller.

Table 2. TAGD

Delay (cycles)

Averages

Peek _%MSE)

Step (%MSE)

)arameters

200

400

1.0

0.8

Given the quadratic shape of the cost function, it is

possible, in theory, to enhance the performance of the

basic TAGD. A parabola can be fit to a slice through

the surface, the slice being defined by 3 or more data

points. The minimum of the parabola can then be

computed and the control set to the projected minimum.

This takes more computation for a given step, but this

single step should produce very good results. In

practice, good single step performance is not observed.

The modified algorithm searches for the surface

minimum as before yet with the added burden of

increased computation. This poor performance can be

attributed to the dependencies which exist between the

actuators that alters the topology of the MSE function

for one actuator as another is stepped. This shifting is

illustrated in figures 3a and 3b.

Figure 3a is a surface plot of the MSE of one

actuator over a small range of the two controller filter

weights. Figure 3b is the MSE surface for the same

actuator, over the same range of filter weights when a

second actuator has been turned on. The MSE

relationship for the first actuator has changed

4
14,

10

10

10 0

Figure 3a. MSE surface of one actuator

14.

10.

8.

6_

4:,

0

10 0

Figure 3b. MSE surface of actuator with interference

dramatically due to the influence of the second actuator.

This phenomena can cause poor convergence and erratic

performance. The observed shifting demostrates that the

actuators are not linearly independent.

Principal Components. The actuator array can be

decomposed into a matrix of independent (orthogonal)

principal components using singular value

decomposition IL_2. In effect, n dependent actuators are

transformed into n independent nxl vectors. To design

the control algorithm in terms of principal components,

equation (4) is first written in matrix form and without

the primary disturbance.

s=Hc

(8)

Where s is a N,x 1 vector of sensor readings, e is a Ncx 1

vector and H is the N,xNc transfer function matrix. For

the single frequency case, the elements of H are

complex values. Applying singular value

decomposition, equation (8) becomes,

s= USV'c

(9)

where S (a N,xN, matrix) contains real singular values

on its diagonal and both U (a N,xN, matrix) and V (a

NcxNc matrix) are unitary matrices, i.e., UUH=I.

Equation (9) is rearranged to obtain

Spc = Scpc (10)

where s_ = UHs and c_ = Vhc. The norm of s_ is the

same as the norm of s, as in equation (5), owing to the

unitary nature of U.

The next section describes the experimental

procedure employed to evaluate the optimization

methodologies.

Ex_rimen_Procedure

An array of 8 piezoceramic patches installed on a

trim panel in the composite fuselage model served as a

test case for the optimization methodology. In general,

the procedure followed the optimization methodology as

outlined above. Broad band transfer function data in the

range of 150 Hz to 650 Hz was obtained for the primary

disturbance and each of the actuators by sweeping a

microphone boom throughout the interior of the model.

A total of 462 transfer functions was obtained from the

sweep. In addition, structural data was obtained from a

set of accelerometers installed on the trim panels. The

acoustic and structural data were used to select 3

frequencies of interest. These were 210 Hz (strong

acoustic mode, strong structural mode), 230 Hz (weak

acoustic mode, strong structural mode) and 275 Hz

(strong acoustic mode, weak structural mode).

The optimization procedure was then used to select

the best 4 out of 8 actuators and 8 out of 462

microphone positions for each of the 3 frequencies. The

performance of the selected actuators and sensors was

tested by configuring the model accordingly and using

the TAGD controller to minimize the mean square error

of the sensor microphones. The interior of the cylinder

was then surveyed to obtain overall SPL. Noise

reduction for the interior is determined with respect to a

baseline, no-control case. These results are compared

with an architectural configuration derived using the

modal decomposition technique described by Lyle 3.
Thefollowingsectionsdescribe the composite

model test facility, details of the optimization process

and the controller design.

The Composite Fuselage Model. The composite

fuselage model is shown in figure 4. The cylinder is

3.6 m long and 1.68 m in diameter. The outer shell is

a 9 layer filament wound graphite epoxy composite.

Total skin thickness is 1.7 ram. The cylinder takes on

fuselage attributes in that it is stiffened with composite

stringers and ring frames. A plywood floor is supported

on the ring frames 0.544 m above the bottom of the

cylinder. To complete the fuselage model, 3 inner trim

panel sections are hard mounted to the ring frames. The

trim panels have a 6.35 mm honeycomb core with 0.64

2nd Control Cross

Section at x--2.17m

1st Control Cross

Section at x=1.41m

PZT Locations

Figure 5. Sketch of instrumentation layout

Figure 4. The composite model fuselage

mm graphite epoxy laminate sheet bounded to either

side of the core. Refer to Lyle 3for a more detailed

description of the model.

Measurement Instrumentation. The measurement

instrumentation consisted of the primary source,

microphone boom traverse and related acquisition

equipment. The primary source was a 100 watt, baffled,

electrodynamic loudspeaker. The speaker was mounted

at a radial position of 1.1 m and angle of -90 ° relative

to the cylinder's centerline, 0.3m from the exterior

sidewall. In the interior, six equally spaced 12.7 mm

microphones were mounted on a boom between the

radial distances of r=-0.13 m and r=0.73 m. The boom

could be swept from azimuthal position 0=-108 ° to

0=108 ° . The axial traversing mechanism could translate

the boom from x=0.356 m to x=3.59 m. See figure 5

for a sketch of the layout.

Data were taken at 11 azimuthal positions and 7

axial locations for a total of 462 points (7xl Ix6) within

the interior cylinder volume. The azimuthal positions

were 0_ {+108 °, +84 °, +60 °, -1-36°, +12 °, 0°}. The

axial locations were x_ {0.36m, 0.88m, 1.49m, 1.79m,

2.1m, 2.71m, 3.23m}. A stationary reference

microphone was mounted at (x=0, r=1.0m, 0=108 °) as

an acoustic reference for diagnostic purpose. These 7

channels plus the primary signal which drove the

speaker were acquired at each position in the matrix.

The coherence and transfer function with respect to the

primary signal were computed and monitored in real

time during each survey.

Controller Instrumentation. Controller

instrumentation consisted of the actuator array, 8 error

sensors and the controller digital signal processor. The

actuators were lead zirconate titanate (PZT5)

piezoceramic patches measuring 3.81 cm by 7.62 cm.

The patches were machined to match the curvature of

the trim panel and were bonded to its outer surface. The

patches were installed in pairs, 16 patches yielding 8

actuators. The actuators were mounted at 2 axial

locations, 4 actuators at x=1.41 m and 4 actuators at

x=2.17m. These axial locations are on the center trim

panel and equidistant from the midpoint. Each pair of

patches were spaced 6.35 mm apart and centered at

angles 0_ {0% -30 °, -60 °, -90°}. The actuator locations

are shown in figure 5. The error sensors were 4.8 mm

electret-condenser microphones. They were mounted on

stands and positioned in the interior according to the

optimization specification. The control algorithm

executed in a PC based digital signal processor

(TMS320C30). The controller had the capability to

acquire 8 channels and output 6 channels. See figure 6

for a diagram of the controller.
Architecture Optimization

Based on measured transfer functions, preliminary

actuator optimization runs were made selecting the best

sets of 3, 4 and 5 actuators for 8 frequencies. Increasing

the number of actuators from 3 to 4 actuators improved

the noise reduction on average 0.65 dB, but, increasing

Cq

© @

Figure 6. Controller instrumentation

the number from 4 to 5 actuators improved the noise

reduction only 0.28 dB. Based on these results, the best

of 4 out of 8 actuators were chosen at each frequency of

interest (210 HZ, 230 Hz, 275 Hz).

Results from the sensor optimization showed it was

possible to obtain sensor solutions where the derived

control forces had high variances (see discussion of

variance in the Architecture Optimization Overview

section). It was believed that those sensor solutions

which caused high control force variances would be hard

to control. To test this hypothesis, sensor solutions

with both high and low actuator force variances were

calculated for the 3 frequencies of interest. For each

case, the best 8 out of 462 sensor locations were found

for each case.

The potential benefit of actuator/sensor optimization

can be seen in figures 7a through 7c where the predicted

noise reduction for 1000 samples of 8 randomly selected

sensors is plotted in a histogram for both the best case

and worst case set of 4 actuators (the best case actuators

return the greatest noise reduction measured at all 462

microphone locations, the worst case microphones

return the least). Observe that the best case

distributions are similar for the three frequencies. The

best case distributions indicate thatfair noise reduction

can be obtained with 50% of the randomly selected

Random Sensor Locations at 275 Hz

[_..=c.. - wo_c,..]

3OO

_ 150

1 O0

O - : : : : : : : : :n

NoIIm ReckJCflon (db)

Figure 7a. Noise reduction at 275 Hz

Random Sensor Locations at 210 Hz

Ioe,ac_ owomc_, I

300 ,

_ 150

IO0

0 : : : : : : :

N_lse Reduction {db)

Figure 7b. Noise reduction at 210 Hz

RandomLocationsat 230Hz

toB_Ca_ IWo=tCaM,I

300

150

100

50

, , , , , .

v

Noltm Reduclton ((Jb)

Figure 7c. Noise Reduction at 230 Hz

sensor locations. The distributions also indicate that

that fewer than 5% of the randomly selected sensor

locations return the greatest reduction. The maximum

reduction for 210 Hz and 275 Hz was approximately

-5.5 dB. The maximum reduction for 230 Hz was

-3.8 dB. Notice also that the shape of the distributions

change dramatically for the worst case actuators at

275 Hz and 230 Hz, yet remains basically the same for

210 Hz. This may be due to the single acoustic mode

which dominates at 210 Hz. Both 230 Hz and 275 Hz

have multiple modes of approximately equal amplitude.

These distributions seem to indicate that it should be

easier to get fair performance from a noise field with a

single dominant mode vs. one with multiple,

7
competing, modes. The modal decomposition of the

data is discussed in more detail in the following.

Results. Tables 3 through 5 show the results

obtained for the 3 test frequencies. At each frequency the

4 criteria for locating the error microphone sources

were, control force-high variance (HV), control forcelow variance (LV), worst case actuator set (WC) and

modal decomposition (MD). For 230 Hz, the modal

response was too weak to support the decomposition

analysis making the actuator selection almost random.

The microphone locations were selected by inspection

of the primary response field (PR). The sum of the

variance values are printed along side their respective

symbols. Note that the 275 Hz HV case has a low

variance when compared with the variances of 210 Hz

and 230 Hz HV cases. For the error microphone

configurations both the predicted noise reduction and

reduction actually obtained with TAGD are shown. The

TAGD results are given in terms of both the relative

amount of noise reduction and the actual controlled

SPL/uncontrolled SPL values.

Table 3. Optimization results at 210 Hz

Stron_ Acoustic/Stron_ Structural

Error Mic.

Criteria

HV(47.2)

LV(13.2)

WC

MD

Noise Reduction (dB)

Predicted TAGD

-5.3 -2.5 73.7176.2

-5.3 -4.4 72.2/76.6

-3.5 -3.1 73.5/76.6

NA -2.1 72.3/74.4

Table 4. Optimization results at 275 Hz

Strong Acoustic/Weak Structural

Error Mic,

Criteria

HV(6.9)

LV(2.6)

WC

MD

Noise Reduction (dB)

Predicted TAGD

- 5.7 -6.4 74.1/80.5

- 5.7 -3.9 75.1/79.0

- 0.5 -0.4 78.6/79.0

NA -2.7 76.4/79.1

The sensor/actuator arrays selected through

combinatorial optimization (LV cases) returned greater

noise reduction than the modal decomposition cases

(MD). However, at 210 Hz where the modal spectrum

is dominated by a single acoustic mode (12), both the

LV and MD cases reduced the noise to approximately

the same level (-72.2 dB). The major difference in the

noise reduction is due to differences in the uncontrolled

SPL. The discrepancy in uncontrolled SPL is thought

to be caused by changes in environmental conditions

which modified the structural/acoustic response of the

cylinder. Small changes in temperature, for example,

may significantly alter the structural/acoustic coupling.

The MD case used 2 actuators and 6 microphones and is

a reproduction of the 210 Hz case reported in reference

3.

Table 5. Optimization results at 230 Hz

Weak Acoustic/Stron_ Structural

I

Error Mic. Noise Reduction (dB)

Crtena

HV(22.6)

LV(S.6)

WC

Predicted TAGD

- 0.5

PR NA

- 3.7 -0.1 72.6/72.7

- 3.8 -2.5 70.7/73.5

+0.2 73.7/73.5

-1.2 70.8/72.0

The 275 Hz acoustic response is dominated by

several modes (20, 21, 22, 23, 28 and 29). These

modes span 4 cross-sectional shapes (see table 1 and

figure 2). Unlike 210 Hz, the uncontrolled SPL

remained relatively constant. This is thought to be due

to the large number of acoustic modes with which the

structure can couple. The LV case had better noise

reduction than the MD case indicating that the

complexity of the modal response makes it more

difficult to select actuators using modal decomposition.

In general the results indicate that selecting the right

combination of actuators and sensors significantly

improves noise control performance. The LV, MD and

PR cases outperformed the WC case. The LV cases

returned better noise reduction than the I-IV cases

demonstrating that microphone arrays that have

corresponding control forces of low variance are easier

to control. The relative performance of the LV and WC

responses are well modeled by the distributions in

figures 7a through 7c. A large difference exists between

the LV and WC cases for 230 Hz and 275 Hz while the

difference between the 210 Hz LV and WC cases is

about 1 dB.

Principal Component Control Algorithm

The TAGD control algorithm was redesigned to use

the principal components of the actuator and sensor

arrays. The algorithm perturbed one element of the

principal component vector, e w, at a time, then

transformed the result into the physical actuator settings

using e = Vcr_. The subsequent sensor readings, s, were

then transformed into s_ using so¢ = UHs. The

component of so, corresponding to the originally

perturbed component of %c was used to evaluate the

8
effect of the change. It should be noted that only the

first N¢ terms of sp¢ are controllable, i.e., any cost value

in terms other than the first N¢ cannot be reduced. For

these tests, N¢ = 4 and N, = 8.

Results. The principal component TAGD

controller, PC-TAGD, performed as theory predicted.

Individual components could be perturbed without

effecting the others. Figures 8a and 8b show the error

surface of a principal component in the absence and in

the presence of a second principal component. Notice

that in contrast to the single actuator case (figures 3a

and 3b), the shape of the principal component error

surface has not changed at all. The curve in 8b is

elevated relative to figure 8a due to the presence of the

added noise power from the second principal component.

The PC-TAGD controller was run at all 3

frequencies for the low variance configuration. The

performance of PC-TAGD relative to the basic TAGD

is shown in Table 6. In general PC-TAGD performed

25,

20,

15-

5-

0

2 10

2

10 0

Figure 8a. Single principal component

25-

20.

15,

tO.

5,

10

10 0

Figure 8b. Principal component with interference

as well as TAGD. The 210 Hz case stands out as an

exception. The reduction in PC-TAGD performance

(-4.4 vs. -2.6) for the 210 Hz LV case may be due to

excess effort to reduce a principal component which is

poorly suited to the control problem t3. The excellent

partitioning provided by the principal components in

effect raises each component to the same level of

observability, regardless of the component's relative

ability to control the noise field.

Table 6. PC-TAGD performance

Freq. Noise Reduction (dB)

TAGD PC-TAGD

210 -4.4 72.2/76.6 -2.6 73.4/76.0

230 -2.5 70.7/73.5 -2.4 70.9/73.3

275 -3.9 75.1/79.0 -4.1 74.8/78.9

This phenomenon is illustrated in figure 9 where a

plot of SPL vs. relative MSE is shown for two actuator

placements, an optimum case and a random case. The

SPL decreases with MSE for the optimum case as

expected. The increase in SPL for the random case

occurred as the controller tried to reduce the MSE of the

last principal component. This demonstrates that the

control effort required to reduce the error of the last

principal component caused spillover of control energy

and an increase in the global SPL. An optimumly

selected actuator/sensor set may reduce the chance that a

principal component is ill-behaved. However, as the

210 Hz case demonstrates, using an optimally

actuator/sensor set may not be sufficient to guarantee

the effectiveness of all principal components.

PrincipalComponentPerformance

Randomvs. OptimumActuators

Random --If- Optlmurn I

811

78

;;t ,

1 0.75 0.5 0.25 0

Relative MSE

Figure 9. Two cases of principal component

performance

9
Concluding Remarks References

Two aspects of optimization have been explored.

One, actuator/sensor architecture optimization, deals

with the number and placement of the actuators and

sensors. The other, algorithm optimization, is

concerned with how to best design a control algorithm

to work with arrays of actuators and sensors. It has

been found that for frequencies where multiple acoustic

modes are present, that it is necessary to perform some

type of optimization of the number and placement of

the actuators and sensors. It is also evident that modal

decomposition methods break down when the modal

spectrum gets too complex. The combinatorial

optimization technique used herein performed well in all

cases. A disadvantage of the technique is that the

transfer functions of all the actuator locations under

consideration must be produced either experimentally or

analytically. It is unlikely that a transfer function of

sufficient quality can be produced analytically. The

alternative experimental approach has limits in the

number of actuator placements that can be tested. For

example, a comprehensive study of a 10xl0 area would

require that 100 actuators be attached to the surface. If

this were done, questions concerning the affect of the

large number of actuators on the structure's response

would arise. The means of performing this type of

comprehensive test needs to be developed before this

optimization technique can be used to its fullest

potential.

If arrays of actuators and sensors are to be controlled,

then a principal component based controller has the

distinct advantage of creating a virtual set of independent

actuators and sensors. With principal component

control in place other optimization techniques which

require a stationary cost function can be used. The

increased overhead of computing optimized solutions

can be addressed by operating the principal component

controller in parallel. The principal component

algorithm depends on the actuator transfer functions.

Thus, the PC-TAGD controller may require some kind

of on-line system identification to function properly.

The dependency of principal component based control

on the accuracy of its transfer function models remains

to be evaluated.

1. Fuller, C.R., Snyder, S., Hansen, C. and Silcox,

R.J." "Active Control of Interior Noise in Model

Aircraft Fuselages using Piezoceramic Actuators",

AIAA Journal, Vol 30, no 11, pp 2613-2617,

November 1992.

2. Silcox, R.J., Lefebvre, S., Metcalf, V.L., Beyer,

T.B. and Fuller, C.R.: "Evaluation of Piezoceramic

Actuators for Control of Aircraft Interior Noise",

Proceedings of DGLR/AIAA 14th Aeroacoustics

Conference, vol 2, pp 542-551, May 1992.

3. Lyle, K.H. and Silcox, R.J.: "A Study of Active

Trim Panels for Noise Reduction in an Aircraft

Fuselage", paper no. 951179 in SAE General,

Corporate and Regional Aviation Meeting and

Exposition, May 1995.

4. Silcox, R.J., Fuller, C.R., and Lester,

H.C.:"Mechanisms of Active Control in Cylindrical

Fuselage Structures", AIAA Journal, vol. 28, no. 8, pp.

1397-1404, August 1990.

5. Padula, S.L. and Kincaid, R.K.: "Aerospace

Applications of Integer and Combinatorial

Optimization", NASA Technical Memorandum 110210,

October 1995.

6. Glover, F.: "Tabu Search: A Tutorial",

Interfaces, vol. 20, pp. 74-94, 1990.

7. Snyder, S.D. and Hansen, C.H.: "Using

Multiple Regression to Optimize Active Noise Control

System Design", Journal of Sound and Vibration, vol.

148, no. 3, pp. 537-542, 1991.

8. Ruckman, C.E. and Fuller, C.R.: :A Regression

Approach for Simulating Feedforward Active Noise

Control", Journal of Acoustical Society of America,

vol. 97, no. 5, pp.2901-2918, 1995.

9. Draper, N.R. and Smith, H.: "Applied

Regression Analysis", John Wiley and Sons, 2nd

edition, 1981.

10. Elliot, S.J., Stothers, I.M., ans Nelson, P.A.:

"A Multiple Error LMS Algorithm and its Application

to the Active Control of Sound and Vibration", IEEE

Transactions on Acoustics, Speech and Signal

Processing, pp. 1423-1434, 1987.

11. Strang, G.: "Linear Algebra and Its

Applications", Third Edition, Harcout Brace

Jovanovich, 1988.

12. Jolliffe, I.T.: "Principal Component Analysis",

Springer-Verlag, 1986.

13. Elliot, S.J. and Boucher, C.C.: "The Behavior

of a Multiple Channel Active Control System. IEEE

Transcations on Signal Processing, vol .40, no. 5, May

1992.

10

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September 1996 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Performance of Optimized Actuator and Sensor Arrays in an Active Noise WU 538-03-14-01

Control System

s. AUTHOR(S)

D. L. Palurnbo, S. L. Padula, K. H. Lyle, J. H. Cline, and R. H. Cabell

7. PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-0001 and

Vehicle Structures Directorate

U.S. Army Research Laboratory

NASA Langley Research Center, Hampton, VA 23681-0001

9. SPONSORINGI MONITORINGAGENCYNAME(S)ANDADDRESS(F.S)

National Aeronautics and Space Administration

Washington, DC 20546-0001 and

U.S. Army Research Laboratory

Adelphi, MD 20783-1145

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REPORT NUMBER

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NASA TM 110281

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i11.SUPPLEMENTARYNOTES

pep_ was presentedot the 2rid A[/_ Aeroacousdcs Co_-cacc (17_ AIAA AeroacotBdcsCoat'ex_ncc),

May 6-8, 1996, State College, PA.

Palumbo and Psdula: langley Research Center; Ly]e and C]ine: US A]U.,-VSD; CabeIl, VPI&SU.

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13. ABSTRACT (Maximum 200 words)

Experiments have been conducted in NASA Langley's Acousticsand Dynamics Laboratoryto determine the effectiveness of

optimized actuator/sensor architectures and controlleralgorithmsfor active controlof harmonic interior noise. Tests were

conducted in a large scale fuselage model - a composite cylinder which simulatesa commuter class aircraft fuselage with

three sections of trim panel and a floor. Using an optimizationtechniquebased on the component transferfunctions,

combinations of 4 out of 8 piezoceramic actuators and 8 out of 462 microphone locationswere evaluated against predicted

pedormance. A combinatorial optimizationtechnique called tabusearch was employed to select the optimum transducer

arrays. Three test frequencies represent the cases of a strongacoustic and strongstructuralresponse, a weak acoustic and

strong structuralresponse and a strong acoustic and weak structuralresponse. Noise reductionwas obtained using a Time

Averaged/Gradient Descent (TAGD) controller. Resultsindicate that the optimizationtechnique successfullypredicted best

and worst case performance. An enhancement of the TAGD controlalgorithmwas also evaluated. The principal

components of the actuator/sensor transferfunctionswere used in the PC-TAGD controller. The principal components are

shown to be independent of each other while providingcontrolas effective as the standard TAGD.

14. SUBJECT TERMS

Active Control, Noise Control, Optimization, Actuator Arrays, Sensor Arrays

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