magnetostrictive models

Harvard - Boston University - University of Maryland

Magnetostrictive Models

R. Venkataraman, P. S. Krishnaprasad

Low dimensional models

Presentation to Dr. Randy Zachery, ARO May 25, 2004, Harvard University


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


• Derivation of the bulk magnetostriction model.

• Parameter estimation algorithm.

• Validation of the model.

• Discussion of results.

• Current and future directions.

Organization of the talk


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


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 ,



The anhysteretic magnetization curve



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



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 :


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



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



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


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


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


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


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



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



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



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



Experimental Setup :



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


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.


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 ,


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.

magnetostrictive models

Documents