Oral Preliminary Exam

Jon Ernstberger

Advisor: Ralph C. Smith

April 5, 2007

Outline• Motivation

• Model Description

• Past Work

• Parameter Estimation for the AMS

• Conclusions

• Future Work

Motivation-Active Machining System• ETREMA Products, Inc.• Active Mat.: Terfenol-D• High-Speed Milling

(4,000 RPM)Courtesy of http://www.etrema-usa.com/

Motivation

• Sound Spoofing

• Atomic Force Microscopy

• Towed Sonar Arrays

• Consumer Audio Products

Introduction to Model

• Know Terfenol-D exhibits a physical lengthening in response to magnetic field.

• Goal is to implement accurate controller in the use of machining.

• Requires an accurate model representation.• Need to model how rod tip displacement

(assume uniform strain) varies nonlinearly in response to magnetic field.

EnergiesGibbs EnergyHelmholtz Energy

w. neg. thermal relaxation

Local Hysteron from

Thermal Relaxation

Moment Fraction Evolution:

Local Avg. Magnetization:

Expected Magnetization:

Switching Likelihood:

Boltzmann Relation:

Homogenized Energy Model

Subject to:

Where:

Dissipativity of HEM

• If a system is dissipative, it loses energy.• “The energy at final time is less than or equal to

initial energy plus input energy.”• Showed dissipativity of

– Preisach model– HEM with negligible thermal relaxation for supply rates

and – HEM with thermal relaxation for same supply rates

• Statement of stability and helps design controllers

HM MH

Lumped Rod Model

Balance rod forces σA with restoring mechanism

or

Previous Work and Future Directions• Prior work

– Homogenized Energy and Lumped Rod Model– Parameter Estimation with HEM using normal/lognormal

and general densities.– Modeling of temperature dependence.

• Direction– Drastically reduce parameter estimation time for the

HEM/LRM.– Enforce density shapes and incorporate modeling of

physical behaviors.– Deliver a nearly black box parameter estimation routine.

Parameter ID-Initial Estimate

Parameter ID-Initial Estimate (2)

From PZT5H Data and manufactured results

Density Choice-Normal/Lognormal Densities

Only 100 Hz Data

Top-Right: Fit to one data set. Bottom: Fit to multiple data sets.

Densities-Galerkin ExpansionsUse Galerkin Expansion to Approximate General Densities

Advantages: 1. Smaller parameter space (8+3(N+1)/2 vs. 8+6N) 2. Decrease in Runtime. 3. Smoother den. approx. Better for controls.

Cubic Galerkin Expansion-No Density Constraints

Left: Displacement vs. field. Center: Interaction field density. Right: Coercive Field Density. N=7, 4 Pt. Gaussian Quadrature.Example of how close fit to displacement can be obtained while violating physical density behavior.

Constraints-Linear Expansion

Sequential Quadratic Programming

Newton Update:

QP Subproblem:

Constrained Optimization

Problem:

Results-SQP/SQP

100 Hz 200 Hz

300 Hz 500 Hz

•N=8 Intervals•4 Pt. Gauss. Quad.•Linear Expansion•2000 SQP Fcn Evals•Runtime: 164.7s

Genetic Algorithms

1. Initialize Population1. Evaluate

Population

1. Iteratea. Selectb. Crossoverc. Mutated. Evaluate

Algorithm:

Fix-Seeded GA• Normal/

Lognormal used to seed GA

• Ran enough iterations to get close to a local minima.

Results-SQP/GA/SQP

100 Hz 200 Hz

300 Hz 500 Hz

•N=8 Intervals•Linear Expansion•4pt. Gauss. Quad.•250 GA Generations/ 33250 Fcn Evals•1000 SQP Fcn Evals•Runtime 2,449.3s•No Crossover

Simulated AnnealingBoltzmann Distribution:

Ratio:

Results: SQP-SA

100 Hz 200 Hz

300 Hz 500 Hz

•N=8 •4-Pt. Gauss. Quad.•Many fcn evals •Soft constraints

Alternate Optimization Routines

• Differential Evolution– Evolution Alg./No initial population– Only simple constraint bounds (upper,lower)– Creates new population by differencing

alternate members and adding other member.• Patternsearch

– Direct Search/initial population– Allows for constraints– Choose optimal search via a specified pattern.

Temperature Dependence

Terfenol-D behavior as temperature rises:• Reduction of saturation magnetization• Decrease in hysteresis in H-M relation• Change to anhysteretic H-M relation

through the freezing temperature

How may we incorporate this behavior into the HEM for ferromagnetics?

Temperature DependenceUsing a Helmholtz Energy which incorporates Temperature

from which are yielded

through the relation

Temperature Dependence

Top: Terfenol-D Data M vs. H data taken at 292 and 363 K.Bottom: Fits to data using estimated parameters.

Conclusions

• Showed the HEM to be dissipative.• Greatly reduced the time required to

perform accurate parameter estimation.• Begin optimization with more accurate

initial estimates (in some cases).• Constrained densities to have quasi-

physical characteristics.• Incorporated temperature dependence into

model.

Future Work• Investigate external optimization routines

(donlp2 and NLPQLP)– Current Model Evaluations is a C implementation.– MATLAB interface for optimization routine comes

at a cost of function handles, calls, and passes.– Data sorting occurs at every model call.– Preliminary results support the robustness of

several of these “industrial-grade” solvers.

Future Work

• Obtain better initial parameter estimates– We can obtain good initial estimates for HEM

given magnetization vs. field data.– Can we obtain LRM parameters given strain

and magnetization data?– Can we obtain model parameters if only given

strain vs. field data?

Future Work

• Data exhibits behavior which may be attributed to 90-degree switching

• Obtain estimates using a model which incorporates 90-degree switching.

• Parameters will be more difficult to estimate and bound with this model.

Future Work

• In too many cases, we’ve had to use constraints to enforce desired density behavior.– Identify other densities which do not require

these bulk constraints.– Hope to reduce number of parameters to

estimate for densities.

Future Work

• Combine work into black-box parameter estimation scheme for the AMS.– This work must be executable on the AMS

when delivered.– "Hands- off" implementation required for

general users.

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