Active control of a clamped beam equipped with piezoelectric actuator and sensor using generalized predictive control

Active Control

of a

clamped

beam

equipped

with

piezoelectric

actuator

and

sensor

using

Generalized Predictive

Control

Julien

RICHELOT1, Joel BORDENEUVE-GUIBE2,

and

Valerie

POMMIER-BUDINGER;3

i2.3Ai\-(;,

.s andSystemDepartment, ENSICA. I PlaceEmileBlouin, 31056Toulouse Cedex5,

France

e-mail: (julien.richelot 1, joel.bordeneuve 2,valerie.pommier3)%ensica.fr

Abstrict- A predictive control method to perform the ac-tivedampingofaflexiblestructureisherepresented. The stud-ied structure isa clamped-free beamequipped withcollocated piezoelectric actuator/sensor. Piezoelectric transducers

advan-tages lie in theirs compactness and reliability, making them

commonly used in aeronautic applications, context in which our study fits. Theirs collocated placement allow the use of well-known control strategies with guaranteed stability. First an analytical model of thisequipped beam is given, using the

Hamilton's principleand theRayleigh-Ritzmethod. Aftera

re-viewof theexperimentalsetup(and notablvof thepiezoelectric transducers), two controllaws aredescribed. Thechosen one -Generalized Predictive Control (GPC)- will becomparedto a typical control law in the domain of flexible structures, the Positive Position Feedback, one of the control lam mentioned above. Majorsbenefits ofGPC lie in itsrobustness in front of model uncertaintiesand othersdisturbances. The resultsgiven

come from experiments onthestructure, performed thanks to a DSP. GPC appears to suit for the consideredstudy'scontext (i.e. dampingofthe first vibrationmode). Someimprovements may,be reached. Amongthem, amorecomplexstructure with morethanasingle modetodamp,and moreuncertainties may beconsidered.

Inder Terns-Active Control - Piezoelectric transducers

-Predictive Control-Flexible structures. I. INTRODUCTION

Thereduction of vibrationsin flexible structuresfindsone of its most important applications in avionics. Indeed,

con-sidering thevibratory environment in this domain, we can

distinguish two kinds onvibration: the transient vibrations whicharerelatedtoimpulsesorgusts and thepermanent vi-brations dues torepetitive efforts or turbulences. Transient

vibrations, inparticularly, are located around the wings and

the fuselage, appears in low frequency and may excite the

modesof the structure (See details in [7]).

In this paper, a Generalized Predictive Control (GPC)

is used to perform the active damping of a clamped beam

equipped withpiezoelectric sensor and actuator. The pur-poseshereareon theonehand toperform the damping of the structurei.e. toperform asefficiently as possible the vibra-torydisturbances reject and on the other hand to evaluate the performances of such a control law, in the domain of flexible structures.

The flexible structure considered has its eigenfunctions

in low frequency - around 20 Hz -just like avionics

struc-tures. Thisequipped beam will be described in section

Il-A,

where moredetails

concerming

the

piezoelectric

components

and the experimental environment are given. An analytical

state model follows this

descriptioni, taking

into account the full control

loop

i.e.

including

the

piezoelectric

actuator and

sensor. The section IVdeals with the control law. The

ba-sis andthe main features of thechosen control law arehere

explained. Finally,

the

efficiency

ofsuch a control law is

studied in the sectionV. These results will allow

appreciat-ingthe interest of

GPC,

accordingto moreclassical control laws.

II. EXPERIMENTAL SETUTP

A. Thecontrolloop

The wholecontrol

loop

isrepresented Fig. 1.

The studied structure in an aluminium

clamped-free

beam,

whose characteristics will be given in Table I. This beam is

equipped

with

piezoelectric

sensorandactuator

(cf.

next

section).

bounded in its clamped side. In case of

vi-brations,

the sensor is

subject

to a

bending

moment which appearsas acharge

variation,

then as avoltage, thanksto a

charge

amplifier.

In an

analog

way, the actuatorinduce

ilnto

the structure a

bending

moment proportional to the

voltage

applied

toit.

This flexible structure has its eigenfunctions in low

fre-quency-around 20 Hz-justlikeavionicsstructures, domain inwhichour

study

fits.

The output andcontrolsignalsarerespectivelythestresses

measured

by

the sensor andprovided bythe actuator atthe

clamped

side ofthebeam. The full control loop alsoincludes

a

charge amplifier

andavoltage amplifier.

Fig.1. Controlloop

Using such a charge amplifier allow us to measure a

charge

quantityinsteadofavoltageandthus to avoid

ties inherent tothe cablesimpedance. This amplifiersetthe tensionbetweenthe sensor'selectrodestozero. IThe charges are conveyed toa capacityinsidethecharge amplifier. The

quantity of charges is determined by measuring thetension

atthecapacity'sterminals.

Thecontrol lawx is computed byaDSP, adSpaceboard.

TABLEI BEAMCHARACTERISTICS

B. Thepiezoelectric devices

The piezoelectric transducers allow the conversion

be-tweenmechanical and electricalenergy. Thankstothe

piezo-electric effect, bending moments can be measured and

pro-vided(becausetheyareproportionaltothedelivered and

pro-vided voltages).

These patchesarebondedoneach side of thebeam. They

arecollocated (c.f the 'Modelling' section) Itisshown that

the most efficient location for them is on the clamped side

of thestructure. Thechosenactuatormaterial isPZT

(Lead-Zirconate-Titanate). This actuatorwillbe glueonthebeam

usinganepoxy adhesive. The beambeing used as ground,

the sensor's electrode has itspotential settozero. Acableis gluedontheother electrode.

The sensorwill be cut in a PVDF (Pohlvinvlidene fltio-ride) sheet. Itis placedontheotherface of thebeam, using double-faceadhesive,withone cableperelectrode.

The main benefits ofusing piezoelectric transducers lie

in the fact that this is nowadays a mature technology (see

[1], [2]). Theirslinearity, compactnessandreliabilityshould makethemparticularly efficient for suchastructure.

More-overpiezoelectric patches arecommonlyusedin aeronautic

applications (preciselybecause oftheirs above advantages),

context inwhichourstudyfits.

Main characteristics of thepiezoelectrictransducers will begiveninTable 7.

III. MODELING

Thepurpose ofthis section is togive an analytical state

modeloftheequipped beam.

The firststepistoconsider the Hamilton's principle (see (1)), thanks towlhich the relation between the conservative forces's work and the kinetic andpotential energies canbe expressed.

It2

tft2

6

(T

-

tJ)dt

+y 6Tdt= 0l

where:

* r: conservativeforce'swork

TABLEII

PIEZOELECTRIC TRANSDUCERSCHARACTERISTICS

T: kineticenergy

t

l: potentialenergy

This continuous problem is solved using an

approxima-tion

method,

like the Rayleigh-Ritz (or variational) one,

where the considered generalized coordinates q, will be

the nodes displacements. The deflectioni tip will be

ap-proximated by the sum ofthese generalized displacements

weightedby the shlape

functions

';,asdescribed in(2)

zc(x, y. z.t) = E l(X y. z).q(t) -'14/ (2) The Hamilton's principle isrewritten, using (2) and isused

jointlywith theLagrangeequations,definedin(3).

d (dT\UOT PUt

K -- - F (3)

dt

04)

O1 +

q)

where Frepresentt generalized

forces.

This leadtothedynamicsequations(4):

Alij+

CqXj+-Kq

F (4)

where Al and K arerespectively themasvsandst.iffness

ma-triv. C isthedampingmatrix. It hasbeenaddedtoimprove

the model.

Because the aim isto compute acontrol law, thenotions

ofinput andoutput areaddedtotheaboveequations.

Mtj1(t)

+CC(t) +Kq(t) ut(t) (

yQt) = Cdq(t) +cj?q(t) +c0tj(t) where uandyarerespectivelytheinputandoutput.

Thus astate-space representation of the structure canbe

written. However the flexible structures domain deals with thedynamic behaviorofthesestructures. Arealmodal

anal-ysis must be performed sothe modal base is chosento

ex-press the state-space representation, where the

eigenfunc-tionswillbe the mode shapes of the beam.

Thestate-spacerepresentationinsuchabasewill be

writ-tenas..

| q'i L

_24t

I0 w; i 0b

lY = [0 c,t,

].:yqi,]+ .

(6) Thesensorandtheactuatorwerepreviously called

'collo-cated'. Thistermisused whenastructure isequipped with

anactuatorapplying angeneralized effort and withasensor

measuring the corresponding degree offreedom. Thiscase

results in avery particular allure of the Bode diagram (see

material Aluminium length L= 30cm width I= 2cm? thickness h = 2mnmn Youngmodulus 7CGPa density 2970 Actuator/Sensor material PZT

length

L,

=

L,

=2.5cm-i width &-

1,

=2

cm-Ci

thickness It,-= 0.5tnmm

le

-

25.

1()

-m

6??

Young modulus

E0

G0GPa

EC

F 60GPa

piezo.

coef

d30

210.1

j-12;Jpj

foractivecontrol offlexiblestructures, namedPPF(Position

PositiveFeedback). The second one is (of course)theGPC

(Generalized Predictive Control).

A. PPF

The interests and

beniefits

of using collocated

actua-tor/senisor

couples are already been remarked. A class of control law that guarantee the asymptotic stabilityhas been used and

successfully

tested.

Fig.

3. SISOloop Fig.2. collocatedsystem (4 firstmodes)

Fig. 2)and inanalternating pole-zero pattern near the imagi-nary axisintherootlocus. These properties aren'tanecdotic,

they have animportant significance intermofstability.

In-deed, such a pattern guaranitees the

asymptotic

stability

of somewell-known conltrol laws('andallowstoavoid the pole-zeroflipping phenomenonfor

example).

Oneof thesetypical

controllaws will be reviewedinthenextsection.

The state-space model musttake accountinto the

piezo-electric devices. Bundling the actuator and sensor into the

modelleadstoconsider thepiezoelectricequations (7).

S = s-6a_{S s%x+dtc}+dc{;

~~~~~(7)

{D dc&T+ ec

Thosebind electrical (electric field _- and displacement D) to

mechanical values (stress S and strain aT).

(This

notationi

is

on/v

vlYaid/for

this

equation

set). Thankstothese equations,

the relations between the voltages (applied by the actuator

and delivered by the senisor) and the

bending

moments

cor-respondingcanbe written. The expressions of b, andc; can

also bewritten,infunction of piezoelectric variables (see de-tailsin[2]).

Note: Another

war1;

liesin adding the energetic contrih-tioni ofthlepiezoelectrie patches intotheLagrange equations. Thefinalmodel include each element quoted above. The last stepinthemodelling process is the model reduction. In-deed, on the one hand a too large number of states in the

modelwouldmake the control almost impossible toperfonn

experimentally and onthe other hand, the choice has been made tofocus the control on the first vibration mode. So the

model hasbeen reduced to the second order.

IV. CONTROLLAW

The two chosen control laws will be here described. As

been saidabove, the control focus on the first flexible mode ofthe beam. So thedisturbance used during the tests will be chosentosuitthis aim.

To beable toevaluate the contribution ofGPC in the study

domain,weneedtocompare it to a standard. The 'reference' control law selected is one of the typical control laws used

Such control laws are built on considering

independent

SISO

loops g.Go(p).D(p)

(see

Fig.3),

where g is a scalar

gain,

GC

the sensor/actuator transfer and D the controller.

Amongthese controllaws -andbecause of the natural

input

andoutput ofourstructurei.e. stresses-thePositivePosition Feedback(PPF) appears asparticularlyappropriate(see

[3]).

Its starting idea is that some

roll-offt

must exist in

g.Gn'(p).D(p)

todeflect thephaselag duetotheactuator

dy-namic orthe digital sampling for example. If the structure

(GC)

doesn't include enough

roll-off,

it must appearin the

controller. Let's consideratypical structure to damp. The

governing equations arealmostsimilartoasecond order

fil-ter:

{ .H+ J-+

2+x

= att

t

B~~T'

(8)

The controllaw vr will be definedby:

{r;

+ 2

wf

jtf

K/!+

jf

c, =

f fI2 ₍₉₎

(,-,

isintroducedtomakegnon-dimensional).

The structure todamp hastwo morepoles than zeros (see

(8) andtwo extrapolesarebrought by the controller. Thus,

as seen in Fig.4, four asymptotes exist in the root locus.

An analysis shows that stability of the closed loop system

isguaranteed for0 <g < 1.

Notes: Th-eaim is todamp the

first

mode, so these

equa-tionshav?ebeengiv.en itn themono-dimensionalcase.

B. GPC

Each Predictive control methods involve three steps. First of all comestheOutput ptediction. AsPredictive controlis

a model-based control law, this model allow to predict the

future behavior of the system output from the actual data.

Then occurs theControlcalculation, when the controlsignal is compute to make the predictedoutput as close as possi-bletothedesired futureoutput. The last step lies inclosing

thfiefeedback

loop, duringwhich thecurrentvalue of the pro-posed futurecontrol signal isapplied to thesystem. Because

these three steps happen at each sample instant, predictive

control isknownasareceding horizonstrategy.

EmeDagarns~

A %

Im(p)

stabilitylimit

(reachedforg=1)

Fig.

4. PPF-root locus

Generalized Predictive Control (GPC) is an interesting

and later approach among the differents Predictive Control

methods(see [4], [5], [6]).

It uses a CARIMA model of the considered plant

(in-stead oftheimpulseorstep responsemodelspreviouslyused,

which bring difficulties to handle unstable systems).

Con-sideraplant of the form:

A(q-

hg(t)

= q 'B(ql

u)(t)

10)

Where y, ai arerespectively the output and

inlput

ofthis plant.

Thelinear CARIMA model will bewritten as:

A(q ')Ay(t) q

'-1B(q-1)Autt(t

- 1) +

$Q(t)

( 1) Where$isthe the disturbance.

The control law is computedby minimizing a cost

func-tion (see('12)), whichinvolves thepredictedoutput

y*

anda reference set r.

I

[y(t

+

j)

-

7r(t

+±j)]2

+ E A

[Au(t

+j-1)]2

j=hi j=1

(12) We candistinguish in thiscostfunction the minimumand

maximumprediction horizon(h1 and h2, respectively), the

controlhorizon

(h,

)and the control increment weighting(A).

Thesefour valuesarethedesign parameters of GPC.

4(t

+j)is the predicted output attimet+j. This predic-tionisperformedthanks to the availableinput/outputdataat time t, as shown in(13).

y't+

j)

=

Fj

(q-

)y(t)

+

Ej

(q

-I)B(q-l

)Au(k

+

.11)

(13)

With theDiophantineequation (14):

1 =

Ej(q- )A(q- ).(1

-q-

)

+

q-'Fj(q-

)

(14)

Theprediction of thesystem output y is based on two

dif-ferent components, named

free

andforced responses. The

free response represents the predicted behavior of the output

y(t +

jjt)

(in the range fromt+ 1 tot+N), basedon pre-vious outputsy(t-

ilt)

andinputs

u(t

-

jlt),

considering no future control. Theforced responseistheadditional compo-nent of the output computed fromtheoptimizationcriterion.

Thetotalprediction isthesumof bothcomponents (for

lin-earsystems). Together with the knownl referenice values the

future errors canbe calculated. Caused bythese future

er-rors, ifuture control signalsarecalculatedto set theoutput to the desiredreference values.

The following step is to separate the forces and free

re-sponses,using (15):

y(t+j) -)(t ± jif) +C(q hAu(k ±j - 1) (15)

(15) canalsobe written:

ii = G.&-t+f where and where9i Weobtai *Y "q

f

(16)

[q(t

+

bt1)

*

.i(t

+

h2)]

[qt(t +b1

it)

I.

.* +

1u2It)]t

(17)

[Au(t)

...

'\u(t

+

-h,X

-

1)lt

91 9 ... 0

G9=

. ... (18) Y-h 1h2 9(1;-1.-I )

arethe polynomial elementsofG(q-

1).

n:

adopt

= (GtG+A.Id)

'.G(r-

f) _('19)

GPC belongs to the group of 'long-range predictive

con-trollers" and generates a set of future controlsignals in each sampling interval, but only the first element of the control sequenceisappliedtothesysteminput, asdescribed in(20).

utt(t) o

f(t

- )+gt(r- ) (20)

Withy, beingthe first row of

(ICG

+ A.Id) 1

G1.

Inadditionto its well-known good controlperfonrance the

robustness properties makes GPCinteresting and realizable

for practical control applications. For this purposes GPC

offers a compact control strategy in terms of model

mis-matches, variable dead time and disturbances.

V. RESULTS

Asmentioned above, the aim of the control law is to damp

thefirstvibration mode. The tests mustreflect this purpose. Release tests have thus been performed: the beam's tip is de-viated fromits equilibrium position and released. This kind of disturbance allows to excite almost exclusively the first

mode of the beam. Both of the control laws have been

ap-pliedtothe structure. Two criteria will be used to allow the

quantitative comparisonbetween them (seebelow). The

re-sult figures willprovidethe temporal responses curves using each ofthe controlstrategies anda"5%zone", corresponding

to5% of the maximummagnitude of theresponse.

The first criterion used is a temporal one. Based on the

"5% zone",itwillindicatethe,contribution of the considered controller, interms ofvelocity for returning to equilibrium state. Concretely, this criterion represent the response times ratiowith andwithout the controller.

The second criterion is an

energetic

one. It express the correspondingenergyof theresponse signal, i.e. the

cumu-latedsumof the response'ssquare (normalized bytheenergy for the uncontrolled structure).

0.3 - - -. 0.2 --- -. ... 0.1 r-- -ref 1- PPF L --- GPCO

j

GPC-tr,,=14.3167% _ --uncontrolled 15.7758°, k 0.3 0.2 0.1 0 K

--

i: EI -0.1D -0.2 -0.3 -0.o1 -0.2 0.5 1.S 0

Fig.

7.

Genieralized

Predictive

Contrul

-0 3

L

0 0.5 I1

Fio.i. controllawscomparisoni

First ofall, we canmake a qualitative remark about the

responses curveenvelops. The response ofthesystem

con-trolled byPPF has the most "roped' curve: the "5% zonie"

is reached very quickly. This behavior is noticeable, when

observingtheexperiment, thevibration reductionis particu-larlyimpressive. HoweverLthe control isn'tvery efficient at

thebeginningofthetest.

ThePPFisnoticeably effective accordingtothe temporal criterion. Asremarkedabove,the"50o zone"ismorequickly

reached thanwhenGPC isused. For thatmatter. testsusing

the PPF controllerarethemostvisuallyimpressive.

PPF- =12.0281%

03I

.|---uncontrolled 0.3L -S --- 40.9139% I''

O.~~~~~~~~~~~~~~~~~~~~~~~

... ... I ~~~~~~~~; ~I'

0.1

{

~~~~~~~~~~~~~~~~~~~~~~~----

.''',:-:'::1

---

...

...I

I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~~~~~~ . i ,.... 0 0.5 1 1

Fig.6. Positive Position Feedback

However, according to the energetic criterion, GPC is far

moreefficient. Indeed, as described in Table III theenergy correspondingto theresponsefor the system controlled with

GPC is nearfrom the value of16% (versus about 40% for

PPF

control.).

Wecan

explain

such

large

differencesby

look-ingatthestructurebehavior

during

the

beginning

oftests(see

Fig.

5 and7).

Indeed,

the GPC controller

sensibly

damps

vibrations since the first overshoot. In return, as damping is

particu-lary

efficient in the

beginning

ofthe test, the "5% zone" is

reachedlaterthan withPPFcontrol.Thismeansless

impres-sive

performances

according

tothetemporal

criterion.

controllaw

temporal

criterion

energetic

criterion

PPF

12.0:3%

40.917%C

CGPC 14.31'c

15.7`8%c

I

-- --

~~~~I

TABLE III

COMPARATIVEANALYSIS

VI. CONCLUSION AND PERSPECTIVES

Thisstudy'spurposewastoevaluatehowefficient

predic-tive control couldbe, whenappliedtotheflexiblestructures

domain. For the considered structureanid context (i.e.

con-trol of the firstvibrationmode ofaclamped-freebeam),GPC

seemsto suit. However, thisstudyrepresent afirst step and

fits in a larger context. Indeed, the choice has been made

to focusonithe firstvibrationmode andthe tests

perfonned

havebeendesignedinthisperspective.

Some improvements have to be made to complete the

presentresults. Amongthem,there is thesensorchoice. The

actual sensoris aPVDFpiezoelectric patch. The choice of

thismaterial has beenmotivatedbyitsnatureitself. PVDFis

apolymer(instead ofaceramic, likePZT,). PVDFis

notice-ably flexible, somaybe muchthinnerand largerthanPZT.

Inreturn, someproblemsofnoiseinthesignalmeasuredby

thesensoroccurduring experiments, problems whichcanbe avoidedbytheuseofaPZTpatch,thinnerthanoneusedfor theactuation.

Alsoconcerningthetransducers,itwouldbeusefultotake

f.

I

1

1 1 1 1 I

alook at thedimensioning. Thisstudyis atwork,toevaluate thepotentialperforniancesof the actuator.

At least, the full potential ofthe predictive conitrol can't

beexploited when controlling a unique mode. Indeed, this

conitrol law is classified as a globalcontrol law. It means

that such acontrol law damps notonly the mode for which

she wasdesigned. In fact, aglobal control lawdamps each

modes of the structure, andmainly the mode it is designed

for.

These improvements refer to a followinig study, using a

morecomplexstructure. This newexperimental support calls

upon several modes coupled in morethananlunique

dimen-sion and it needs a particularly robust control

law,

because of some additionaluncertainties dueto fluid/structure inter-actionswithinit.

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