DTIC ADA451951: H infinity Control Design for a Magnetostrictive Transducer

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by  Defense Technical Information Center

H
Control Design for a Magnetostrictive Transducer
1
James Nealis and Ralph C. Smith
Center for Research in Scientiflc Computation, North Carolina State University, Raleigh, NC 27695
Abstract model [9] and a corresponding approximate inverse al-
gorithmwhichlendsitselftoreal-timeimplementation.
Magnetostrictive transducers are becoming increas-
To focus the discussion, we consider an industrial
inglyprevalentinindustrialapplicationsincludinghigh
application requiringhighspeed, highaccuracymilling
speed milling and hybrid motor design due to their
using the prototypical magnetostrictive actuator illus-
broadband,largeforcecapabilities. Toachievethelevel
trated in Figure 1. The objective of this application
ofperformancerequiredbysuchapplications, however,
is milling out-of-round objects at speeds of 3000 rpm
these transducers must operate in nonlinear and hys-
within a tolerances of 1-2 microns. Stresses and dis-
tereticregimes. Toaccommodatethisnonlinearbehav-
placements are provided by the Terfenol-D rod in re-
ior, models and control laws must incorporate known
sponse to applied flelds generated by the surrounding
physics and be su–ciently robust to operate under re-
alistic operating conditions. We develop here an H1 solenoid. Apermanentmagnetencasingthetransducer
provides a biased fleld to achieve bidirectional strains
robust control design for a prototypical magnetostric-
as well as providing a mechanism for (cid:176)ux shaping.
tive transducer but note that the design is su–ciently
As detailed in [1, 2], these transducers possess the
general to be utilized for several commonly used smart
ability to generate broadband, high force responses.
materials including piezoceramics and shape memory
However, they also exhibit hysteresis and nonlineari-
alloys. Theperformanceofthecontrolstrategyisillus-
ties in the relation between the input fleld H and the
trated through a numerical example.
magnetization M generated in the Terfenol-D rod. We
focus on this magnetostrictive transducer in this paper
1. Introduction but note that the hysteresis model and control design
are su–ciently general to permit direct extension to
Applications utilizing piezoceramic (PZT), magne-
analogous piezoelectric and shape memory alloy mod-
tostrictive and shape memory alloy (SMA) compounds
els as developed in [3, 10, 11].
range from nanopositioning stages in an atomic force
Whereas employing an inverse fllter attenuates the
microscope (AFM) to vibration suppression systems in
primary efiects of the nonlinear and hysteretic behav-
buildings. However, the material attributes which pro-
ior of the smart materials, there are still disturbances
vide these compounds with unique control capabilities
intheprocessduetodiscretizationandmodelingerrors
also produce hysteresis and constitutive nonlinearities as denoted by d in Figure 2. In realistic applications,
which must be accommodated in models and control the errorsd are not the only disturbance to the control
algorithms to meet the stringent design criteria associ-
system which must be accommodated. Sensor noise is
ated with these applications.
also present in all applications and can cause a degra-
The method we employ to attenuate this nonlinear
dation in the transducer performance if unaccommo-
and hysteretic behavior is to utilize an approximate dated. Therefore, we consider an H1 design capable
inverse of the hysteresis model as a fllter to mitigate
of rejecting both classes of disturbances.
theseefiects. Thiscontrolstrategyrequiresahysteresis
modelwhichadmitsaninversethatcanbeemployedin
real-time. Filtersofthistypehavebeenconstructedus-
P
ingadomainwallmodel[6,8]aswellasPreisachmod- u u+d
els [12]. Here we utilize a free-energy based hysteresis
Figure 2. Approximate model inverse employed as a
Compression Bolt Terfenol−D Rod
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) SWparisnh(cid:0)(cid:0)(cid:1)(cid:1)ge(cid:0)(cid:0)(cid:1)(cid:1)r CHuettaindg Milled Object fllter for robust control design.
2. Hysteresis Model
Wound Wire Solenoid
Permanent Magnet
We summarize here the relevant aspects of the hys-
Figure 1. Terfenol-D transducer. teresis model presented in [9]. This model quantifles
the energy required to reorient moments in combina-
1
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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
H1 Control Design for a Magnetostrictive Transducer
5b. GRANT NUMBER
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North Carolina State University,Center for Research in Scientific REPORT NUMBER
Computation,Raleigh,NC,27695-8205
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Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
y
tion with stochastic homogenization techniques to ac- (M)=G (0,M)
G (H1,M) G (H 2,M)
commodate variations in coercive and efiective flelds.
Weassumeflxedtemperaturesforthisdevelopmentand
themodelignoreslossesduetoeddycurrentsandthere-
foreshouldbeemployedforoperatingregimesinwhich M0 MIMR M M M
eddy current losses are negligible. A further assump-
tionisthatthespringwasherinthetransducerprovides (a)
su–cientprestresstodominatecrystallineanisotropies.
Weflrstquantifytheinternalenergyduetotheinter- M M MR M
action of moments through the Helmholtz energy. As
detailedin[9],undertheassumptionthatmomentsori- MI
H H H
ent either with the applied fleld or diametrically oppo- Hc
sitetoit,areasonableformoftheHelmholtzenergyis
8
ˆ =>><>>: ···222(((MMMI+¡¡MMMRRR)))22‡MMI2 ¡MR· ;;;jMMMj•‚<M¡MMIII: (1) FGmiafgogurnreietnicz3rae.taios(ainn)gMHfleeollmndshtohHlet.zfl((ebeb)lnd)erDHgyepaˆetntadhneednlcaGettiobifcbetshleeevnleeolrcgianyl
the absence of thermal activation.
As depicted in Figure 3, MR and MI respectively de-
note the point at which the minimum of ˆ occurs and
and the initial moment orientation is
the in(cid:176)ection point.
ergIynitshgeivpernesebnyce of an applied fleld H, the Gibbs en- 8>>< H· ¡MR ;H(0)•¡Hc
G =ˆ(M)¡HM: (2) [M(H;Hc;»)](0)=>>: » ;¡Hc <H(0)<Hc
WenotethemagnetostaticenergyisE =„0HM,where H· +MR ;H(0)‚Hc :
„0 denotes the magnetic permeability, so the Gibbs re- For a description of models which incorporate thermal
lation(2)canbeinterpretedasincorporating„0 intoˆ. activation, the reader is referred to [9].
Magnetoelasticcouplingcanbeincorporatedthrough
Therelation(3)isderivedundertheassumptionthat
the extended Helmholtz relation
the lattice structure is homogeneous which is overly
1 simplisticinthatitneglectsmaterialdefects, polycrys-
ˆe(M;")=ˆ(M)+ YM"2¡YM(cid:176)"M2
2 tallinityandmaterialnonuniformities. Also,themodel
and the corresponding Gibbs energy assumes that the efiective fleld He at the domain level
is the applied fleld H. To incorporate these efiects,
1
G(H;M;")=ˆ(M)+ YM"2¡YM(cid:176)"M2¡HM¡(cid:190)": stochasticdistributionsareemployedtodevelopabulk
2 magnetization model for nonhomogeneous Terfenol-D
Here, ˆ isspecifledby(1), YM istheYoung’smodulus sampleswithnonconstantefiectiveflelds(see[9]forde-
at constant magnetization, (cid:176) is a magnetoelastic cou- tails).
pling coe–cient, (cid:190) is the applied stress and " denotes To include the efiects of material nonhomogeneities,
the strains. we consider a distribution of free energy proflles to ac-
For operating regimes in which thermally activated commodatethenonhomogeneouslatticestructures. We
processes are negligible, the local average magnetiza- assume that the local coercive fleld Hc =·(MR¡MI)
tion can be quantifled from the necessary condition is normally distributed with mean Hc. Hence the co-
@G = 0. In the absence of applied stresses, the lo- ercive fleld has the density
@M
cal model thus predicts a linear relationship between
H and M with slope 1. As detailed in [9], the local f(Hc)=C1e¡(Hc¡Hc)2=b
·
magnetization in this limiting case is given by
8 where the parameters C1 and b are positive.
>< [M(H;Hc;»)](0) The second extension entails the incorporation of
[M(H;;Hc;»)](t)=>: H· ¡MR (3) vthaaritatitioinssninortmhaelleyfiedcitsitvreibfluetleddtahbroouugthththeeaapspsluiemdpfltieolnd
H· +MR H. Therefore, the efiective fleld has the density
for the respective cases f¿(t) = ;g, f¿(t) 6= ; and
H(max¿(t)) = ¡Hcg, f¿(t) 6= ; and H(max¿(t)) = fb(He)=C2e¡(H¡He)2=b
Hcg. The transition points are specifled as where C2 and b are positive. Combination of the coer-
¿(t)=ft2(0;Tf] j H(t)=¡Hc or H(t)=Hcg cive and efiective fleld distributions yields the magne-
2
tization relation 4. Transducer Model
Z Z
1 1 Inthissection,weemploytherelations(4)and(5)to
M(H)=C [M(H;Hc;»)]f(Hc)fb(He)dHedHc: quantify the displacements generated by applied flelds
0 ¡1
(4) in the prototypical transducer depicted in Figure 1.
The model for the transducer is based on the model
The integrals are approximated using a compos-
developed in [2].
ite quadrature rule for the numerical implementation
One end of the rod (x = 0) is assumed to be flxed
of the free energy model thus allowing the hysteresis
whiletheotherend(x=L)isconstrainedbyadamped
model to be computed algebraically and hence at low
oscillatorandhasanattachedpointmassasdepictedin
cost. Because of the exponential decay of the distri-
Figure 5. The internal damping coe–cient and density
butions, they can be truncated to speed computation.
oftheTerfenol-DrodaredenotedbycM and‰andthe
Themodelisthussu–cientlye–cienttoallowthepos-
sibility of real time implementation. pointmassisdenotedML. Theendspringhasstifiness
The elastic constitutive relation is provided by the kL and damping coe–cient cL.
@G The constitutive relation (5) is for an undamped
equilibrium condition =0 which yields
@"
material. To incorporate Kevin-Voigt damping, the
(cid:190) =YM"¡YM(cid:176)M2: (5) stresses at any point x, 0 • x • L, are assumed to be
proportional to a linear combination of strain, strain
The coupled constitutive relations (4) and (5) quantify rate and squared magnetization which yields
the magnetization and stresses for an undamped mag-
@w @2w
netostrictive material. These relations are employed (cid:190) =YM +cM ¡YM(cid:176)M2
@x @x@t
in Section 4 when quantifying the displacements pro-
duced by the magnetostrictive transducer depicted in where the linear strain relation "= @w is employed.
@x
Figure 1.
Balancing forces yields
@2w @N
3. Inverse Hysteresis Model ‰A = tot (6)
@t2 @x
The monotonicity and e–ciency of the hysteresis where A is the cross sectional areaR of the Terfenol-D
model are exploited to construct an approximate in- rod and the force resultant Ntot = A(cid:190)dA is given by
verse. Todeterminetheappliedfleldrequiredtocreate
adesiredmagnetization,thehysteresismodelisusedto @w @2w
incrementthemagnetizationuntilthedesiredmagneti- Ntot =YMA@w +cMA@x@t ¡YMA(cid:176)M2:
zationissurpassed. Thentheappliedfleldiscomputed
To obtain appropriate boundary conditions, we flrst
bylinearinterpolation. Thecomputationalspeedofthe
inversecompensatordependsonthesizeofthestep¢H note that w(t;0) = 0. As detailed in [2], balancing
forces at x=L gives
taken in advancing the hysteresis model. Larger steps
will increase the speed while decreasing the accuracy @w @2w
of the inverse compensator. The relative linearization Ntot(t;L)=¡kLw(t;L)¡cL @t (t;L)¡ML@t@t(t;L):
error for an input signal with a frequency of 1 Hz and
a step size of ¢H =1 employed in the inverse model is Initial conditions are taken to be w(0;x) = 0 and
plotted in Figure 4. A successful control design must @w(0;x)=0.
@t
be able to reject this error to the input of the plant. The model can be implemented utilizing the weak
form and a Galerkin flnite element approximation. To
further simplify our system, we assert that the magne-
0.02 tization in the Terfenol-D rod can be taken as uniform
0.01
0
or
ve Err−0.01 +w kLw
Relati−0.02 Rod Ntot ML
−0.03 x=L C dw
L dt
−0.04
0 0.5 1 1.5 2
Time
Figure 4. Relative error created by inverse flltering. Figure 5. Boundary conditions for the rod model.
3
e u d
We Wu Wd
r +
Wr K P
− e u u+d y
n
W
n
Figure 6. System representation including input disturbances d and sensor noise n.
overthelengthoftherod. Thisisreasonableinpresent The signal n represents noise in the sensor the mea-
actuator designs since (cid:176)ux shaping via the surround- surements of y. For the simulation results presented in
ing magnet is used to minimize end efiects in the rod. Section 6, we assume sensor noise with a frequency of
Hence,sinceeachsectionoftherodreactsidenticallyto 60Hz. Theoutputsignalsebandubdenotetheweighted
the uniform magnetic fleld, the elements act uniformly tracking error and weighted output of the controller
and therefore the transducer dynamics can be mod- K, respectively. The weighting functions Wu, Wd, We,
eledasadampedspring-mass. Toachievebidirectional Wr and Wn are chosen to maximize the performance
strains, we linearize the magnetostrictive relationship of the controller utilizing a priori knowledge regarding
aboutabiasingmagnetizationlevelofMs=2. Thisbias the characteristics of the signals.
canbeachievedmymeansofthepermanentmagnetas We now construct the transfer function representa-
depicted in Figure 1. For more details on this model tionoftheopen-loopsystem. Thetransfermatrixfrom
development see [2, 5]. the inputs r, d, n and u to the outputs eb, ub and v is
The dynamics of the Terfenol-D transducer can be specifled by
now represented by 2 3
W W ¡W PW ¡W W ¡W P
e r e d e n e
x˜+kx_ +cx=”M(t) G=4 0 0 0 Wu 5:
(7)
x(0)=x0 ; x_(0)=x_0: Wr ¡PWd ¡Wn ¡P
The scalars k, c, and ” are determined by fltting the The control system can then be represented as a linear
model (7) to the Galerkin approximation of (6) or to fractionaltransformationasshowninFigure7. Details
data from the physical device. For our sample, k = regardingthesystemformulationcanbefoundin[5,7].
7:8899£103, c = 6:4251£107 and ” = 1:3724£10¡2 To proceed, we partition G as
yielded accurate model flts. Note that the hysteresis 2 3
inherent to the Terfenol-D rod is still present in (7) A B1 B2
6 7
through(4)whichquantiflesthehystereticrelationship G(s)=4 C1 0 D12 5 (8)
between H and M.
C2 D21 0
5. Robust Control Design where
" # " #
Inthissection,wepresentarobustcontroldesignfor A B A B
the Terfenol-D transducer depicted in Figure 1. The G11 = 1 ; G12 = 2 ;
control system incorporates the presence of external C1 0 C1 D12
" # " # (9)
disturbances, such as the errors cause by the inverse A B A B
fllterandsensornoise, andminimizes theirefiectswith G21 = 1 ; G22 = 2 ;
respect to the H1 norm C2 D21 C2 0
kTk1 = sup(cid:190)[T(j!)] respectively represent the transfer functions from w to
!2R z, u to z, w to v, and u to v, (see Figure 7). To en-
where (cid:190)[T(j!)] denotes the maximum singular values sure the existence of a H1 sub-optimal controller, the
following assumptions are made regarding the system:
of the closed-loop map T. Analogous H2 control laws
are developed in [5, 7].
1. (A, B1) is controllable and (C1, A) is observable,
Figure 6 illustrates the block diagram of the system
to be controlled. In the diagram, P represents the 2. (A, B2) is stabilizable and (C2, A) is detectable,
transducermodelgivenbythedifierentialequation(7).
The signal to be tracked and the position of the tip of 3. D1⁄2D12 >0 and D21D2⁄1 >0,
the Terfenol-D rod are respectively denoted by r and " #
y. The signal d represents the error in the lineariza- 4. A¡j!I B2 has full column rank for all !,
tion of the input by the inverse fllter (see Figure 4). C1 D12
4
z−−− eu dr −−−w 5.2. H1 Sub-Optimal Control Design
G n Employing the notation deflned in (9), the design of
a sub-optimal H1 controller which gives kTk1 < (cid:176)
incorporates two Riccati equations
e u
¡ ¢
K A⁄X +XA+X (cid:176)¡2B1B1⁄¡B2B2⁄ X +C1⁄C1 =0
(10)
and
Figure 7. Linearfractionaltransformation(LFT)rep-
¡ ¢
resentation of transducer. AY+YA⁄+Y (cid:176)¡2C1⁄C1¡C2⁄C2 Y+B1B1⁄ =0: (11)
" #
A¡j!I B The following theorem from [13] guarantees the exis-
5. C D1 has full row rank for all !. tence of an H1 sub-optimal control.
2 21
Theorem1: Thereexistsanadmissiblecontrollersuch
Details regarding the validity of these assumptions are that kTk1 <(cid:176) if and only if
provided in [5, 7].
1. There exists a positive deflnite solution X to (10),
5.1. Weighting Functions 2. There exists a positive deflnite solution Y to (11),
The selection of the weighting functions is critical 3. ‰(XY)<(cid:176)2.
to the performance of the robust control design. A
discussion of the choice of these function can be found The H1 optimal controller is subsequently given by
in[4,5,7,13]. Bydesign,thefrequencyofthereference
" #
signal r is taken to be 1 Hz so we construct the pass- A ¡ZL
bandfllterWr tohaveabandwidthof1Hzcenteredat K · F 0 (12)
1 Hz. The frequency of the noise n can be accurately
determinedforthedevicemeasuringthepositionofthe
with
tip of the Terfenol-D rod. For the simulation results,
noise with a frequency of 60 Hz is added to the system A·A+(cid:176)¡2B1B1⁄X +B2F +ZLC2;
soWnistakentobeapass-bandfllterwithabandwidth F ·¡B⁄X; L·¡YC⁄:
2 2
of 10 Hz centered at 60 Hz.
To determine the weighting function Wd, a signal The control given in (12) is sub-optimal in that it
with the same frequency as the reference signal was provides a closed-loop system with an H1-norm less
flltered by the inverse compensator and then fed into than (cid:176). The hinfsyn command in Matlab can be uti-
the hysteresis model. The output is deflned to be the lized to decrease (cid:176) until an assumption of Theorem 1
desired control signal plus the disturbance d (see Fig-
is violated. This yields a control design method which
ure 2). The power spectrum of d indicates that the
is close to optimal.
most signiflcant frequencies in d lie below 400 Hz so
Wd istakenasalow-passfllterwithacut-ofifrequency
of 400 Hz. 5.3. Numerical Example
Theweightingfunctionontheerrorsignalwastaken
Wepresenthereanumericalexampleillustratingthe
to be We = s+(cid:176)e†e with (cid:176)e =4£106 and †e =1£10¡8. H1 robust control law. The results were computed
An integrator was chosen to prevent the error from using a noise signal n with a magnitude of 1£10¡5,
achieving steady state at a nonzero value and the pole
whichis10%ofthereferencesignal,andafrequencyof
wasshiftedslightlyofizerotoensurethatthecontroller
60Hz. Theinversecompensatorwascomputedusinga
designisrealizable. Wespecifledtheweightonthecon- step size of ¢H = 1 in the approximate inverse model
troller output to be Wu = 5£10¡6. Since we do not to ensure a simulation of the control process which has
experience any problems with saturation or other such
the potential for real-time implementation. Figure 8a
efiects,weminimallyweightubtofocusthecontrolleron
illustrates the controller’s ability to track the reference
tracking and disturbance rejection. It is important to
signal and reject the noise and disturbance signals. A
notethatincreasingtheorderoftheflltersincreasesthe
tracking error less that 2 microns is achieved after a
number of states in the controller design. To facilitate
short period, as illustrated in Figure 8b. Additional
real-timeimplementation,aminimalcontrolrealization
examples illustrating the performance of the method
wasutilizedtolimitthenumberofstatesinthecontrol
are provided in [7].
gain K.
5
x 10−4 x 10−6
1.5 10
Reference
Simulated
1 8
6
0.5
osition (m) 0 Error 24
P
−0.5
0
−1 −2
−1.5 −4
0 0.5 1 1.5 2 0 0.5 1 1.5 2
Time Time
(a) (b)
Figure 8. (a) H1 tracking performance and (b) tracking error.
6. Concluding Remarks [5] J. Nealis, Model-Based Robust Control Designs for
High Performance Magnetostrictive Transducers,
An H1 robust control design for a magnetostrictive Ph.D Dissertation, North Carolina State Univer-
systemhasbeensummarizedinthispaper. Itisshown
sity, Raleigh, NC, 2003.
that the H1 control design is capable of maintaining
accuratetrackingwhilerejectingsensornoiseandadis- [6] J. Nealis and R.C. Smith, \Partial inverse com-
turbance due to an inexact inverse fllter. The same pensation techniques for linear control design
methodologycanbeusedtoincludeotherdisturbances in magnetostrictive transducers," Proceedings of
to the system if necessary. Finally, while the methods the SPIE, Smart Structures and Materials, 2001,
were illustrated in the context of a magnetostrictive Vol. 4326, pp. 462-473, 2001.
transducer,theyaresu–cientlygeneraltobeappliedto
[7] J. Nealis and R.C. Smith, \Model-based robust
systems utilizing piezoceramic or shape memory alloys
controldesignformagnetostrictivetransducersop-
due to the unifled modeling framework used to quan-
eratinginhystereticandnonlinearregimes,"CRSC
tify hysteresis and constitutive nonlinearities inherent
Technical Report CRSC-TR03-25; IEEE Transac-
to all of these materials [10].
tions on Automatic Control, submitted.
[8] R.C. Smith, C. Bouton and R. Zrostlik, \Par-
Acknowledgments
tial and full inverse compensation for hysteresis in
This research was supported in part by the Air Force smart material systems," Proceedings of the 2000
O–ce of Scientiflc Research under the grant AFOSR- ACC,Chicago,IL,June28-30,pp2750-2754,2000.
F49620-01-1-0107.
[9] R.C. Smith, M.J. Dapino and S. Seelecke, \A free
energy model for hysteresis in magnetostrictive
transducers," Journal of Applied Physics, 93(1),
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6