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Magnetostrictive Models
R. Venkataraman, P. S. Krishnaprasad
Low dimensional models
Presentation to Dr. Randy Zachery, ARO May 25, 2004, Harvard University
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Magnetostrictive models
Experimental data from
actuators
• coupled PDE’s representing magnetic and mechanical dynamic equilibrium. • material properties appear through shape of potential functions. • eddy-current losses modeled via Maxwell’s equations.
• coupled ODE’s or integro- differential equations
representing magnetic and mechanical equilibrium.
• material properties appear through constitutive equations.
• eddy-current losses modeledvia a resistance.
valid
ation
model model
validation
model
simulation
Low-dimensional modelsMicromagnetic model
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• Derivation of the bulk magnetostriction model.
• Parameter estimation algorithm.
• Validation of the model.
• Discussion of results.
• Current and future directions.
Organization of the talk
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Low Dimensional Magnetostrictive Models
• W.F. Brown derived expressions for work done by a battery in changing the magnetization of a magneto-elastic body. The body was considered to be a continuum.
• Jiles and Atherton postulated expressions for magnetic hysteresis losses in a ferromagnet. This lead to an ODE with 5 parameters for the evolution of the average magnetization in a thin ferromagnetic rod.
• Sablik and Jiles extended this result to a quasi-static magnetostriction model.• Hom, Shankar et al. have a model for electrostriction that includes inertial effects. But hysteresis
was not modeled.
Our Work• Our model takes account of ferromagnetic hysteresis, magnetostriction, inertial effects, mechanical
damping and eddy-current effects. It is low-dimensional with 4 continuous states and 12 parameters.• We proposed parameter estimation algorithms that are easy to implement. • We have experimentally verified the structure of our model.• Current work involves inverting the hysteresis nonlinearity and design of a robust controller.
Background
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Derivation of the bulk magnetostriction model
Langevin’s theory of Paramagnetism
Consider a collection of N atomic magnetic moments under the influence of an external magnetic field . Then the average magnetic moment of the ensemble is given by:
m H
NmsMkTmHz
zzsMzsMM
;;1cothL
kT
emH
kTMHmz
zzsMM
;1coth
Weiss’ theory of FerromagnetismWeiss postulated that an additional “molecular field” experienced by an individual moment in an ideal ferromagnet, where is the average magnetic moment of the ferromagnet. Suppose an external field is applied in the direction of . Then the magnitude of is given by:
Weiss considered an ideal ferromagnet without losses. In particular, the curve in the plane is anhysteretic.
MmH HM
M M
MH ,
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Derivation of the bulk magnetostriction model
The anhysteretic magnetization curve
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Derivation of the bulk magnetostriction model
Jiles and Atherton’s assumptions for a lossy ferromagnet
• The average magnetization is composed of reversible and irreversible components:
• The losses during a magnetization process occur due to the change in the irreversible component: where and are constants with
and .
• The reversible and irreversible magnetizations are related to the anhysteretic magnetization as: .
• Further
irrrev MMM
c k
10 c 0k
irranrev MMcM
irrmag dMckL 1Hsign
00
HH
0dH
dM irrandand 0
0
MMMM
an
anif
Principle of conservation of energy
elmag LL mechbat WW elmagelmag WWW K
Change in external input Change in internal energy losses Change in kinetic energy
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Derivation of the bulk magnetostriction model
W. F. Brown’s expression for work done by the battery
mechbat WW HdM0 Fdx
where F is the external force, x is the displacement of the tip of the actuator, H is the average external magnetic field, and M is the average magnetization in the actuator.
Adding the integral of any perfect differential over a cycle does not change the value on the left hand side.
mechbat WW HdM0 Fdx MdM
Magnetoelastic energy density (following Landau) =
Elastic energy = ; Kinetic energy =
xbM 2
2
21 xmeff 2
21 dx
Expressions for some of the energy terms :
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The bulk magnetostriction model
The model equations
Magnetic dynamic equilibrium equations
Mechanical dynamic equilibrium equation
H
bxdHdMckMMk
MMdHdMck
M
e
ananM
anMe
an
000
0
2
1
e
esean H
aa
HcothMHM 2
MbxHHe
0
2
3
Hsign 4
100
M:::
00
HH
00
MMMM
an
an
otherwiseandand
5
FbMdxxcxmeff 21 6
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The bulk magnetostriction model
Schematic diagram of the bulk magnetostriction model with eddy current effects included
Eddy currents losses are modeled by a resistor in parallel
Voltage source
displacement output
Mechanical system transfer function
Rate-independent hysteresis operator
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The bulk magnetostriction model
Sufficient condition on parameters
7
8
9
Analytical result
Theorem : Consider the system of equations (1 - 6). Suppose the matrix A =
has eigenvalues with negative real parts and the parameters satisfy conditions (7 - 9). Suppose the input is given by and the initial state is at the origin. Then there exists a such that if then the limit set of the solution trajectory is a periodic orbit.
effeff mc
md 1
10
0B Bb || tUtHtu cos)()( max
0223 000
bXM
acMkk
ss
123 0
bX
acM s where xX sup
10 c
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Parameter Estimation
The parameters to be found are :
- electrical circuit parameter (includes lead resistance of the magnetizing coil).R
edR - eddy current parameter.
sMab ,,, - magnetic parameters not pertaining to hysteresis.
kc, - magnetic hysteresis pertaining to hysteresis.
1c - mechanical dynamic losses parameter.
effm - inertia parameter.
d - elasticity parameter.
F - prestress parameter.
Three step algorithm for parameter identification
Step 1 : Apply a sinusoidal current input of a very low frequency (0.5 Hz) and measure the voltage and displacement of the actuator as a function of time. This leads to the identification of . Repeat the same experiment, for higher frequencies (200Hz, 350 Hz, 500Hz). This leads to identification of .
kc,edRc ,1
Step 2 : Obtain the anhysteretic displacement curve of the actuator. This leads to the identification of ,,,, abdF .
Step 3 : Apply a swept sine wave current signal to the actuator and record the displacement versus the frequency. This leads to the identification of .effm
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Parameter Estimation
Result of step 2 :
Input current waveform
Output displacement versus current
Result of parameter estimation
Parameter Value
sM 9000
F 7104.2
(in CGS units)
R 7105.7
1c 3101.5
edR 8104
a 4.187
4109.1
b 1.2
d 10109.1
k 2.48
c 3.0
effm 3106.2
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Experiment versus Simulation
500 Hz
Amps-1.5 1.5
200 Hz
Amps-1.5 1.5
45 m
icro
ns
50 m
icro
ns
100 Hz
Amps-1.5 1.5
60 m
icr o
ns
50 Hz
Amps-1.5 1.5
80 m
icro
n s
480 Hz
Amps
45 m
icro
ns
-1.5 1.5
100 Hz
Amps12
0 m
icro
ns
-1.5 1.5
240 Hz
Amps
100
mic
rons
-1.5 1.5
50 Hz
Amps
80 m
icro
ns
-1.5 1.5
Frequency(Hz) Peak-Peak current (A) Peak-Peak displacement ( m) 0.25 2.5 53 1 2.5 53 10 2.5 54 50 2.5 54 100 2.5 51 200 2.5 58 350 2.5 66 500 2.3 44
Frequency(Hz) Peak-Peak current (A) Peak-Peak displacement ( m) 1 2.13 71 10 2.26 71 50 2.17 63 100 2.22 54 150 2.19 50 200 2.28 42 350 2.11 54 500 2.39 45
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Validation of the structure of the model
Original goal : Trajectory tracking by means of an non-identifier based adaptive controller.Why adaptive control?
Reasons : (1) Transient effects are unmodeled in our model. (2) System parameters may change with time due to heating
heating etc.
Basic idea of universal adaptive stabilization :
are unknown and tcxty
tbutaxtx
, ;00 xx 0,,, xcba .0b
Suppose
.0 k tytktu ;2tytk and
Then .exp 00
xdscbskatxt
txTherefore is monotonically increasing as
decays exponentially andHence for ,*tt tx .lim ktkt
.0 cbtka Hence .00 ** cbtkat .020 kcxttk long as
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Validation of the structure of the model
Universal Adaptive Stabilization result for relative degree one systems.
Consider a class of nonlinearly-perturbed, single input, single output, linear systems with nonlinear actuator characteristics :
. , ,
; ; ,,
; ,,
00
tytuwtw
tcwtytututgtv
twtdtvtwtfbtAwtw
n
t
Assumptions: (2) The linear system is minimum phase. is a Caratheodory function and has the (3) nn : d. 0 cb(1)
, wc 1 , wtd for almost all and allt . wproperty that for some scalar (4) : n f
and has the property that, for some scalar and known continuous function
wc w wt,f , 0, : n
is a Caratheodory
for almost all and allt . w
(5) There exists a map uGu
condition below and such that every actuator characteristic is contained in the graph of
in the following sense:G 2 , t and every : u with
Gutg t ,, where
satisfying the
, tu
tu denotes the restriction of u to . , t
G is a continuous map from to compact intervals of with the property that, for
some scalars 0 and ., \ , , sign ,0 2112 G
condition below and such that
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Validation of the structure of the model
Universal Adaptive Stabilization (contd.)
The class R of reference signals is the Sobolev space , ,1 W with norm . ,1
Adaptive Strategy (assuming ) :
0 cb
1
, 1
, )(
ytetetk
tesignytektu
ttyte
Theorem (Ryan) : Let n0 , : ,, tkewx be a maximal solution of the initial value problem.
1.
2.
3.
4.
.
x is bounded.
tkt lim exists and is finite.
. as 0 tte
Then
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Validation of the structure of the model
Simulation example for Morse-Ryan controller
2
4
20104
s
sG
Reference and output trajectories
Input non-linearity Gain evolution
Morse-Ryan controller design for relative degree 2 systems
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Validation of the structure of the model
Experimental Setup :
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Validation of the structure of the model
Result of trajectory-tracking experiment
Reference (sinusoids) vs. actual displacements
Control current
seconds0 0.05
amps
seconds0 10
amps
seconds0 0.05
mic
rons
seconds0 0.5
mic
rons
seconds0 0.4
mic
rons
seconds0 10
mic
rons
1 Hzseconds0 0.4
amps
50 Hzseconds
0 0.1
amps
200 Hz 500 Hz
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Discussion of Results
1. We derived a low dimensional model for a thin magnetostrictive actuator that is phenomenology based and models the magneto-elastic effect; ferromagnetic hysteresis; inertial effects; eddy current effects; and losses due to mechanical motion.
The model has 12 parameters, 4 continuous states and can be thought to be composed of magnetic and mechanical sub-systems that are coupled.
2. Analytically we showed that for initial conditions at the origin and periodic inputs, the system equations have a unique solution trajectory that is asympotically periodic. This models experimentally observed phenomena.
3. We have proposed a simple parameter estimation algorithm and estimated the parameters for a commercially available actuator. Simulation results show trajectories that are comparable to the actual.
4. We have also validated the structure of the model by designing a trajectory tracking control-law for relative degree 2 systems with input non-linearity. The closed loop system remained stable for all frequencies from 0 to 750 Hz, thus showing that our model structure is correct.
There are some differences in the size of the peak-peak displacement predicted by the simulation and actual results.
In particular, the predicted peak-peak displacement is larger than the actual for low frequencies while it is smaller than the actual for high frequencies.
This can be explained by an eddy-current resistance value that is slightly smaller than the estimated value.
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Current Work and Future Directions
Drawback of the present model :The rate-independent nonlinearity is defined only for periodic signals. Therefore it is unsuitable for the development of a controller.
Solution :Replace the rate-independent nonlinearity by a moving Preisach operator, that is defined as follows :
ddtHtM e ,
where for continuous inputs
u is defined as , u
. if , if 1 , if 1
tutu
tutu
tu
i
where is a partition such that
is monotonic in each sub-interval.
N110 t 0 ii tttt ufor 1 , ii ttt
Facts about the Preisach operator: 1. The Preisach operator is Lipschitz continuous and its definition can be extended to the space of functions over the real line that are bounded with integrable derivatives over compact intervals.
2. The Preisach operator is rate-independent and models properties that are observed in bulk ferromagnetic hysteresis like minor-loop closure and saturation. 3. It is invertible in the space of functions defined in 1, under some mild conditions on the measure ,
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Current Work and Future Directions
1. We are currently working on an algorithm for the inversion of the Preisach operator, so that we can approximately linearize the rate-independent nonlinearity.
2. Once this is achieved, we can utilize methods from robust control of linear systems for controller design.
Current Work
Future Directions
While designing complex magnetostrictive systems, one can obtain low dimensional models from the numerical results obtained from PDE model.
This will enable us to short circuit the implementation step and design controllers without actual experimental data.