Smart Structural Systems
Daniel J.Kolepp andRalph C. Smith 1
CenterforResearchinScientic Computation
Department ofMathematics
North Carolina StateUniversity
Raleigh,NC 27695-8205
Abstract
Thisinvestigationfocusesonthedevelopmentofnarrowbandcontrolstrategieswhichareeective
for structural systems subjected to bothharmonicexogenous forcesand stochastic uncertaintiesin
themodelormeasurements. Acantileverbeamwithsurface-mountedpiezoceramicactuatorsisused
asaprototypicalstructuresinceitexhibitsphysicalattributesofavarietyofvibratingstructuresbut
issuÆcientlysimpleto facilitateinitialcontroldesign. A PDEmodelforthestructureisdiscretized
using a spline-based Galerkin technique to obtain a semi-discrete system that is appropriate for
nite-dimensionalcontrol design. A narrowbandcontrol method,which can be tuned to attenuate
eithernatural frequenciesorspecic frequenciesintheexogenous input,isdeveloped and compared
withstandard LQRtechniquesthroughnumericalexamples.
1
Piezoceramicelementshaveprovenhighlysuccessfulinawiderangeofstructural,structuralacoustic,
and uid-structure systems dueto theirdual actuator and sensor capabilities and theirbroadband
response. Theseattributesareaugmentedbythefactthatpiezoceramicpatchesarelightweight,and
hence donot signicantlyalterthe passivedynamics of theunderlyingstructure, and are relatively
inexpensivetofabricate. Incombination,thesepropertiescanbeutilizedto designcontrollerswhich
areeectiveforalargerangeofbothstochasticandharmonicstructuraldynamics. Furthermore,by
utilizinginherent coupling between structural vibrations and adjacent elds, PZT transducers can
beemployed forstructuralacoustic controldesignorcontrolof adjacent ows.
In thispaper,wefocusonthedevelopment ofnarrowbandcontroldesignsforstructural systems
whichexhibitstrongharmonicresponsesoraredrivenbyperiodicexogenousforces. Suchexogenous
forces include engine noise, electromagnetic inputs, or disturbances due to rotating components.
The goal of attenuating harmonicstructural responses has been augmented bythe development of
composites materials which are lightweight and provide minimaldamping. Finally,all systems are
subject to some degree of stochastic uncertainty due to either broadband or stochastic exogenous
forces(e.g.,turbulence)orlimitationsinthemodels,numericalapproximations,orcontrolhardware.
Hence the narrowband control designs will ultimately be combined with feedback mechanisms to
providesuÆcient robustnessfora range ofapplications.
Toprovidearegimewhichfacilitatesmodeldevelopment,analysis,andnumericalapproximation
while retaining attributes of a range of structural systems, we consider the design of narrowband
control techniquesin the context of a cantilever beam with surface-mountedpiezoceramic patches.
This illustratesa number of theassociatedmodelingand numerical issuesand illustratesthe
capa-bilitiesof the controltechniques fora prototypical system. These techniquescan then be extended
tomorecomplexphysicalsystemsoncethesystemsarequantiedbynitedimensionalmodels(e.g.,
niteelement ormodal).
Appropriate modelsand numericaltechniqueswillbedevelopedinSection2and controldesigns
comprisedofnarrowbandandfeedback componentswillbe discussedinSection3. Abriefsummary
of existing control methods for smart structural systems will also be provided in that discussion.
Numerical examplesillustratingthecontroltechniqueswillbeprovidedinSection4.
2 Model and Numerical Approximation Techniques
We illustratethedevelopment ofnarrowbandcontroltechniquesinthecontext of acantileverbeam
with surface-mounted piezoceramic patches as depicted in Figure 1. For modeling purposes, the
beamis assumed to have length`, thicknessh
b
,and widthb, and is oriented along the x-axiswith
(a)
(b)
Figure 1. (a) Thin beam with surface-mountedpiezoceramic patches. (b) Bending moments
Voigt damping coeÆcients for the beam are respectively denoted by b ;E b and c D and analogous
quantiesforthePZTpatcharedenotedby
pzt ;E
pzt ;c
pzt
. ThecoeÆcientforair(viscous)damping
isdenotedby andexogenous forcesarerepresentedbyg. Finally,thepatchesareassumedto have
thicknessh
pzt
and coverthe regionp
1
xp
2 .
As detailedin [3 ],forceand moment balancing yield thePDEmodel
(x) @ 2 w @t 2 + @w @t + @ 2 M int @x 2
=g(t;x)+ @ 2 M pzt @x 2
w(t;0) = @w
@x
(t;0)=0
M
int (t;`)=
@M
int
@x
(t;`)=0
(1)
wherethecompositedensityis
(x)= ( b h b b+2
pzt h
pzt
b ; p
1
xp
2
b h
b
b ; otherwise:
The internalbendingmoment is
M
int
(t;x)=EI(x) @ 2 w @x 2 +c D I(x) @ 3 w @x 2 dt (2)
wherethespatiallyvarying stinessand damping parametersaredened by
EI(x)= 8 > > > < > > > : E b h 3 b b 12 + 2b 3 E pzt a 3 ; p 1
xp
2 E b h 3 b b 12 ; otherwise c D I(x)=
8 > > > < > > > : c D h 3 b b 12 + 2b 3 c pzt a 3 ; p 1
xp
2 c D h 3 b b 12 ; otherwise wherea 3 =(h b =2+h
pzt ) 3 h 3 b
=8 (detailsregardingtheformulationoftheseparameterscanbefound
in[3, 12 ]). In applications, thepiecewise constant parameters (x);EI(x) and c
D
I(x) aretypically
estimated througha leastsquarest to data.
Under theassumptionof linearpiezoceramicresponses,theexternalbendingmoment generated
throughtheapplication ofdiametricallyout-of-phasevoltages u(t)to the patchesis givenby
M
ext
(t;x)=E
pzt d 31 (h b +h pzt )u(t) pzt
(x): (3)
Here d
31
denotesthe piezoelectric couplingcoeÆcientand the characteristic function
pzt (x)=
(
1 ; p
1
xp
2
0 ; otherwise
isolates inputs to the region covered by thepatches. We note that the relation (3)is valid onlyat
when quantifyingexternalmoments.
Thestrongformofthemodel(1)requiresthederivativesofdiscontinuousmaterialparametersand
patch coeÆcients withthelatter yielding highlyunboundedcontrolinputs(derivativesof theDirac
distribution). Furthermore,numericalapproximationof thestrongformof themodel wouldrequire
theuseofhigh-orderbasisfunctionsforGalerkinmethodsduetothefourth-orderspatialderivatives.
To provide a formulation which is amenable to approximation and subsequent nite-dimensional
controldesign,itisadvantageoustoconsideraweakformulationofthemodelinwhichthestatespace
isX =L 2
(0;`)and thespace oftest functionsistaken to be V =f2H 2
(0;`) j(0)= 0
(0)=0g.
As detailed in [3 ], either integration by parts or the application of Hamiltonian energy principles
thenyieldsthecorresponding weak formof themodel
Z
`
0
(x)ydx + Z ` 0 _ wdx+ Z ` 0 EI(x)w 00 00 dx+ Z ` 0 c D I(x)w_
00 00 dx = Z ` 0
g(t;x)dx+ Z ` 0 M pzt 00 dx (4)
which mustbe satisedforall2V.
To obtainacorresponding semi-discrete systemappropriateforcontrol implementation,we
em-ploy Galerkin approximations in space with temporally-varying coeÆcients. To satisfy
smooth-ness requirements while minimizing system size, the basis is comprised of cubic B-splines
modi-ed to satisfy the essential boundary conditions. For a uniform partition of [0;`] with gridpoints
x
j
=jh;h=`=N;j=0;;N;thebasisfunctions f
j g
N+1
j=1
are denedby
1
(x)= 2s
1
(x)+s
0 (x) 2s 1 (x) j
(x)=s
j
(x) ; j=2;;N +1
(5) where s j (x)= 1 h 3 8 > > > > > > > > > < > > > > > > > > > : (x j+2 x) 3 ; x j+1
<xx
j+2 3(x j+1 x) 3 +3h(x j+1 x) 2 +3h 2 (x j+1
x)+h 3
; x
j
<xx
j+1
3(x x
j 1 )
3
+3h(x x
j 1 ) 2 +3h 2 (x x j 1 )+h
3
; x
j 1
<xx
j (x x j 2 ) 3 ; x j 2
<xx
j 1
0 ; otherwise
(see [9 ]). It should be noted that with this choice of basis, H N
= spanf
i
g V. Approximate
solutionsare thenspeciedas
w N
(t;x)= N+1 X j=1 w j (t) j
(x): (6)
To obtain a nite dimensional system, the weak form of the model is projected into the nite
forall
i
2H . Employingmatrixnotation, thesecond-order systemcan beformulatedas
M
~ w+C
_
~
w+Kw~ = ~
bu(t)+ ~
f (7)
where w(t)~ =[w
1
(t);;w
N+1
(t)]. The mass, dampingand stiness matrices,along with the force
and controlinputvectors, have the components
[M]
ij =
Z
`
0 (x)
i
j dx
[C]
ij =
Z
`
0
i
j dx+
Z
`
0 c
D I(x)
00
j
00
i dx
[K]
ij =
Z
`
0
EI(x) 00
i
00
j dx
[ ~
f]
i =
Z
`
0
g(t;x)
i dx
[ ~
b]
i =K
b Z
p2
p
1
00
i dx
(8)
whereK
b =E
pzt d
31 (h
b +h
pe
). Letting y(t)=[~w(t); _
~ w(t)]
T
and
A= "
0 I
M 1
K M
1
C #
; F(t)= "
0
M 1
~
f(t) #
; B(t)= "
0
M 1~
b(t) #
;
thesecond-order system(7)can be posed astherst-order system
_
y(t)=Ay(t)+Bu(t)+F(t)
~
y(0)=~y
0 ;
(9)
where the 2N 1 vector ~y
0
denotes the projection of the initial conditions into H N
. The nite
dimensionalmodel(9)can thenbe usedforsubsequent controldesign.
Wepoint outthatformore complexphysicalsystems,modalorniteelement techniquescan be
employed to construct nite dimensional models of the form dened in (9). Once such models are
constructed, thecontroltechniquesdescribed inthenext sectioncan be employed.
3 Control Design
There exista largenumberof classical, adaptive, optimal, and robustcontroltechniqueswhichcan
beemployed to specifycontrols ufor thesystem(9). Toachieve thegoal of ultimatelydetermining
optimal capabilities of certain smart structural designs, we focus here on certain optimal control
techniques. In addition to determining design capabilities, an important criterion is to develop
methodswhich aresuÆcientlyeÆcientand robustto permit real-timeimplementation.
Evenwithintherealmofoptimalcontroldesign,thereexistanumberoffeasibleapproaches. We
summarizeheretwoclassicalformulationsbasedontimedomaincostfunctionalsandthenconsidera
narrowbanddesignbasedonfrequencydomaincriteria. Thersttwoapproachesarewelldocumented
in theliterature forsmart material control designand we summarizehere only those detailswhich
nationof temporallyperiodicfunctions with commonperiod. We thenconsidertheminimization
of thequadraticfunctional
J(u)= 1
2 Z
0
f hQy;yi+hR u;ui gdt (10)
subjectto
_
y(t)=Ay(t)+Bu(t)+F(t)
y(0)=y():
(11)
Here h;i denotestheEuclideaninnerproductinlR N+1
. The matrix Qwhichweights thestates is
oftentaken to be
Q= "
d
1
K 0
0 d
2 M
#
;
where M and K are the mass and stiness matrices dened in (8) since this is the identity with
regardto theproductspaceinnerproduct. The controlweightR istypicallyascalar,orinthecase
of multiple inputs, a scalar multiple of an identity matrix having the correct dimensions. Further
detailsregarding theconstructionofbothmatricescanbefoundin[1]. Asdetailedin[3 ,Chapters8
and 9],or[5 ] fortheinnitedimensionalproblem,theoptimalcontrolwhichminimizes(10) isgiven
infeedbackform by
u(t)= R 1
B T
[z(t) r(t)] (12)
wherethe-periodictrajectoryz(t) isobtainedbysolvingtheclosed loop system
_ z(t)=
h
A BR 1
B T
i
z(t) BR 1
B T
r(t)+F(t)
z(0)=z():
(13)
The Riccativariable and trackingvariabler(t)respectivelysatisfytheequations
A T
+A BR
1
B T
+Q=0 (14)
and
_ r(t)=
h
A BR
1
B T
i
T
r(t)+F(t)
r(0)=r():
(15)
Inpractice,thesolutionoftheboundaryvalueproblem(15)istypicallysimpliedbyconsidering
the nal time condition r() = 0 and time-marching the solution forward in negative time in lieu
of enforcing the boundary condition r(0) = r(). In spite of this approximation, however, the
determination of an optimal control u is costly since it requires the solution, storage, and possible
updatingoftrackingtrajectoriesr(t). Furthermore,thedetermination of r isoften performed prior
to closingthe loopwhichproduces robustnessissuesdueto potentiallyunincorporatedphaseshifts.
Hence, while the control law (12) has been experimentally implemented [2 ], the robustness issues
and complexity associated with the determination of r(t) render this approach more feasible as a
theoretical ornumericaldesigntoolthan analgorithm to beimplementedinphysicalsystems.
A second approach is to design a feedback law for the unforced system to obtain suboptimal
attenuationinthepresenceofanexogenousforce. Thisisaccomplishedbyminimizingthefunctional
J(u)= 1
2 Z
1
0
_
y(t)=Ay(t)+Bu(t)
y(0)=y
0 :
(17)
The controlinthiscaseis
u(t)= R 1
B T
y(t) (18)
where again solvesthe algebraic Riccati equation (ARE) (14). The use of thefeedback law (18)
intheoriginalsystem(9)willattenuatestochastic disturbancesormodeluncertaintiesbutwillhave
less eect on harmonic disturbances or steady state system dynamics due to periodic exogenous
forces.
To develop a feedback law which targets either natural frequencies for the system, or specic
frequencies in theexogenous force,it is advantageous to considera frequency-domainfunctional of
theform
J(u)= 1
2 Z
1
0 [y
(i!)Q(i!)y(i!)+u
(i!)R u(i!)]d! (19)
where
represents the complexconjugate transpose and Q(i!);R (i!) again denote design
param-eters. In order to transform thisfrequency domain cost functional into an equivalent time domain
cost functional,considertheinput/outputrelationgiven bythetransferfunction P(s)
e
y(s)=P(s)y(s): (20)
IfP(s) isa ratioofpolynomialsand thenumberof zeros ofP doesnotexceedthenumberofpoles,
then(20) can beexpressedas thelinearsystem
_ z=
e
A z+ e
By
~ v=
e
Cz+ e
Dy:
(21)
To determine appropriate choices for e
A; e
B; e
C; e
D, we note that minimization of the alternative
frequencydomaincost functional
J(u) = 1
2 Z
1
0 [~y
(i!)y(i!)~ +u
(i!)R u(i!)]d!
= 1
2 Z
1
0 [y
(i!)P
(i!)P(i!)y(i!)+u
(i!)R u(i!)]d!
(22)
isequivalent to minimizing(19)when Qis denedby
Q(i!)=P
(i!)P(i!):
Dening Q(i!) in this manner satises the necessary condition that equation (22) yields a
non-negative realnumber. The choice ofP usedhere is
P(i!)=
! 2
r i!
! 2
r !
2
+2!
r i!
(23)
where is a design parameter and !
r
is the lowest known resonant frequency [7 , 11 ]. Note that
Q(i!)penalizessystemresponses tofrequencies inaneighborhoodof thelowestresonant frequency.
The sizeof the frequency range is dictated by the design parameter . Figure 2 demonstrates the
behaviorof Q(i!) forthree valuesof anda resonant frequencyof !
r
0
100
200
300
400
500
600
700
0
20
40
60
80
100
120
140
160
180
Frequency (Radians)
0
100
200
300
400
500
600
700
0
20
40
60
80
100
120
140
160
180
Frequency (Radians)
0
100
200
300
400
500
600
700
0
20
40
60
80
100
120
140
160
180
Frequency (Radians)
(a) (b) (c)
Figure 2. Prolesof Q(i!) for(a) =0:1; (b) =0:5 and (c) =1.
From (20)and (21) it follows thatan associateddierential equation systemforthe choice (23)
forP is
" z r 1 z r 2 # = " 0 1 ! 2 r 2! r #" z r 1 z r 2 # + " 0 e B 1 # v(t) (24) where e B 1
is arow matrixthatweights theelements ofv.
Asillustratedin[7 ],minimizationof(19)isequivalenttominimizingthecostfunctionalassociated
withthe augmented system
2 6 6 4 y z r 1 z r 2 3 7 7 5 = 2 6 6 6 4
A 0 0
0 0 1
e B 1 ! 2 r 2! r 3 7 7 7 5 2 6 6 4 y z r 1 z r 2 3 7 7 5 + 2 6 6 4 B 0 0 3 7 7 5 u(t)+ 2 6 6 4 F(t) 0 0 3 7 7 5 : (25)
Iftheaugmented state isdenoted byy
1 =[y T ;z r 1 ;z r 2 ] T
,(25) can be formulatedas
_ y 1 =A 1 y 1 +B 1
u(t)+F
1
(t) (26)
where the denitions of A
1 ;B
1
and F
1
(t) follow from (25). By considering the system (26), LQR
techniques can be employed to compute frequency-shaped Riccati solutions and corresponding
feedbackgains.
Inasimilarmanner,thesystemresponsetotwofrequencies!
r and!
d
can beminimizedthrough
considerationof theaugmentedsystem
2 6 6 6 6 6 6 6 6 6 4 y z r 1 z r 2 z d 1 z d 2 3 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 4
A 0 0 0 0
0 0 1 0 0
e B 1 ! 2 r 2! r r 0 0
0 0 0 0 1
e
B
2
0 0 !
2 d 2! d d 3 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 4 y z r 1 z r 2 z d 1 z d 2 3 7 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 6 4 B 0 0 0 0 3 7 7 7 7 7 7 7 7 7 5 u(t)+ 2 6 6 6 6 6 6 6 6 6 4 F(t) 0 0 0 0 3 7 7 7 7 7 7 7 7 7 5 (27) where e B 2
is asecond designparameter. Fory
2 =[y T ;z r 1 ;z r 2 ;z d 1 ;z d 2 ] T
,the systemcan beposedas
_ y 2 =A 2 y 2 +B 2
u(t)+F
2
(t): (28)
temsinthetimedomain,one obtainsafeedbackformulationthatisno morecomplexto implement
than standard LQR laws but targets specied system or exogenous frequencies. This signicantly
enhancesthefeasibilityofexperimentallyimplementingthemethodinphysicalsystemswhile
achiev-ing the control authority required when employing PZT actuators in smart structures. Numerical
examplesillustratingattributesofthisnarrowbandcontrolapproachareprovidedinthenextsection.
Finally, we note that this narrowband design can be combined with standard LQR feedback
techniques to augment the eectiveness of the controller for attenuating unmodeled dynamics or
stochastic disturbances. The development of suchhybridcontrol approaches hasbeenillustratedin
[4 ] in the context of narrowband adaptive techniques and similar hybrid techniques are currently
beingdeveloped which utilizethenarrowbandcontrolsdiscussedhere.
4 Numerical Examples
To illustrate the performance of the narrowband control method, we considerthe response of the
prototypical system to a 600 Hz exogenous force; hence g(t;x) = sin(2600t) in (4) and the
correspondingnitedimensionalmodel. Thelengthofthebeamwastakentobe`=:4573mandthe
coordinatesforthepatchpairwerep
1
=0:2mand p
2
=0:25m. Thedensity,stiness,dampingand
controlinput parameters were taken to be
b
=0:093 kg/m,
pzt
=0:443 kg/m, EI
b
=0:491Nm 2
,
EI
pzt
= 0:793 Nm 2
, c
D
= 6:4910 6
sNm 2
, c
pzt
= 1:225 10 5
sNm 2
, =0:013 sN/m 2
and
K
b
=1:74610 2
Nm/V.Forthissetofdimensionsandparameterchoices, thebeamhasaprimary
resonant frequencyof 5.5Hz and a secondaryresonant frequencyof 32Hz.
The uncontrolleddisplacement at thepointx=3`=5and theclosedloopdisplacementobtained
with theLQR law (18)are compared inFigure 3a whilethe corresponding velocitiesare plottedin
Figure 3b. The control wasinitiated at 0.74 seconds and forthe presented simulations, the design
parameters were speciedas
Q=5 "
K 0
0 M
#
; R=110 4
: (29)
Itisobservedthatwhilethegeneralfeedbacklawreducesthedisplacementsbyanorderofmagnitude
within1.5secondsofcontrolimplementation,itdidnotreducethehighfrequency600Hzcomponent
reected in the velocity. This is further illustrated in the FFT of the beam response from 1.25 to
2.5 seconds which is plotted in Figure 3c. This latter gure illustratesthat the 600 Hz component
of the beam responseis actually unaected by the feedback law (18). This is to be expected since
this formulation does not incorporate any information about the exogenous force into the penalty
functionalorfeedback law.
We next illustratethe performanceof thenarrowbandcontrollerwhichis designed to attenuate
either specic natural orexogenous frequencies. We considerrst a penalty functionalthat targets
the 5.5 Hz primary frequency. The parameters in the augmented system (26) were taken to be
!
r
=5:5(2) and =1. Thedesignparameters fortheaugmentedsystem were speciedas
Q
1 =
"
Q 0
0 I #
; R
1
=0:001
whereQ isspeciedin(29).
The displacement obtainedin thiscase is compared with that resultingfrom the LQR law (18)
0
0.5
1
1.5
2
2.5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x 10
−3
Displacement (m)
Time (sec)
Uncontrolled
LQR feedback
0
0.5
1
1.5
2
2.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time (sec)
Velocity (m/s)
(a) (b)
0
100
200
300
400
500
600
700
800
0
0.01
0.02
0.03
0.04
Frequency, (Hz)
Closed loop
0
100
200
300
400
500
600
700
800
0
0.01
0.02
0.03
0.04
Open Loop
6 Hz
6 Hz
32 Hz
(c)
Figure 3. (a) Open and closed loop displacements obtained withthe feedback law (18); (b)Open
and closedloop velocities;(c) FFTof velocitiesoverthe timeinterval[1:25;2:5].
theprimary5.5Hz responsebutthere isessentially noattenuationofthesteadystate responsedue
to theexogenous force sincethisinformationis notincludedinthe controlformulation.
Tominimizeboththetransient responseandthe 600Hz steady state response,wesubsequently
employed the formulation (28) with !
r
= 5:5 and !
d
= 600 is employed along with the design
parameters R=0:001 and
Q
2 =
2
6
6
4
Q 0 0
0 10I 0
0 0 I
3
7
7
5 :
The resultingdisplacementand velocityareplotted inFigure 5. It is observed thatby1.5seconds,
bothstates areeectivelyattenuated. This isreiteratedintheFFTof thedisplacement(not shown
here) whichindicatesnegligibleenergyat600 Hz after1.5 seconds. Thisprovides initialverication
0
0.5
1
1.5
2
2.5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x 10
−3
Time (sec)
Displacement (meters)
LQR
Res. Shaped
0
0.5
1
1.5
2
2.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time (sec)
Velocity (m/s)
(a) (b)
0
100
200
300
400
500
600
700
800
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Frequency (Hz)
(c)
Figure 4. (a) Closedloop displacement obtained with the feedback law (18) and LQR design for
theaugmented system(26) employing!
r
=5:5; (b) Closedloopvelocities; (c)Fft of velocitiesover
thetime interval[1:25;2:5].
5 Concluding Remarks
Anarrowbandoptimalcontrolmethodologyforsmartstructureshasbeenoutlinedandillustratedfor
aprototypical structure withpiezoceramicactuators. Thecriteria forthismethod weretwofold: (i)
to provideeective attenuation inthepresence of exogenous disturbancesand (ii) to besuÆciently
eÆcient and robustto permit real-timeimplementationin physical systems. Therst criterion can
beachievedthroughtheformulationoffeedbacklawswhichinclude trackingvariables;however, the
complexityoftheseformulationsleadstorobustnessissuesandoftenprecludesdirectimplementation.
The controlapproachconsideredhere employsa frequencydomainfunctionalto target specied
frequencies. Forimplementationpurposes,acorrespondingaugmentedsystem isformedwhich
con-tainstheoriginal linearmodelasa subsystem. Becausestandard LQRtechniquescan be employed
to compute gains for the augmented system (which in turn contains the necessary informationfor
0
0.5
1
1.5
2
2.5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x 10
−3
Time (sec)
Displacement (m)
LQR
2−Freq. Shaped
0
0.5
1
1.5
2
2.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time (sec)
Velocity (m/s)
(a) (b)
Figure 5. (a) Closedloop displacement obtained with the feedback law (18) and LQR design for
theaugmentedsystem (28)employing!
r
=5:5 and!
d
=600;(b)Closedloopvelocities.
Initial verication regarding the validity of the method is provided by numerical examples.
Cur-rent investigationsare focused on the experimental implementation of the method to ascertain its
eectiveness forphysicalsystems.
Acknowledgements
ThisresearchwassupportedinpartbytheAirForceOÆceofScienticResearchunderthegrant
AFOSR-F49620-01-1-010 7.
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