Efficient Parameter Estimation Techniques forHysteresis Models

Jon M. Ernstberger1, Ralph C. Smith2

March 9, 2009

This research was supported in part by theAir Force Office of Scientific Research

through the grantsAFOSR-FA9550-04-1-0203 and AFOSR FA9550-08-1-0348

1LaGrange College, jernstberger@lagrange.edu2NC State, CRSC, rsmith@eos.ncsu.edu

Outline

I Motivation/Application

I Polarization Models

I Density Representations

I Gradient-based Numerical Optimization Results

Motivation/Applications: PZT-Actuated Devices

I Micro-/nano-positioningI Atomic Force Microscope

(AFM)I Optical Storage

I THUNDER Actuator

I MEMS Switches

Figure: AFM Schematic

Figure: faceinternational.com

Figure: AFM image from sciencegl.com.

Modeling: Internal EnergiesI Helmholtz Energy

ψ(P) =η

2

(P + PR)2 P ≤ −PI

(P − PR)2 P ≥ PI

(P − PI )

(P2

PI− PR

)|P| < PI

I Gibbs Energy G (E ,P) = ψ(P)− EP

I Local Polarization P (E + EI ; Ec) =E + EI

η+ δ (E + EI ; Ec) PR

G(E,P)

PR

PR

P I

P I

(a) (b)

P

E

Modeling: Incorporating Thermal Relaxation

Boltzmann Energy µ(G ) = Ce−G(E ,P)V/kT

Switching Likelihood p+− =1

T (t)

∫ PI

PI−ε e−G(E ,P)V/kTdP∫∞PI−ε e−G(E ,P)V/kTdP

Dipole Fraction Evolution x+ = −p+−x+ + p−+x−

Expected Polarization 〈P+〉 =

∫∞PI

PeG(E ,P)V/kTdP∫∞PI

eG(E ,P)V/kTdP

Local Average Polarization P = x+〈P+〉+ x−〈P−〉

Homogenized Energy Model: Macroscopic Polarization

[P(E )] (t) =

∫ ∞0

∫ ∞−∞

ν1(Ec)ν2(EI )[P(E + EI ; Ec)

](t)dEIdEc

1.) ν1(x) is only defined for x > 0;

2.) ν2(−x) = ν2(x) for density symmetry;

3.) |ν1(x)| ≤ c1e−a1x and |ν2(x)| ≤ c2e

−a2|x|.

We may discretize [P(E )](t) via

[P(E )] (t) ≈Ni∑i=1

Nj∑j=1

ν1

(E i

c

)ν2

(E j

I

) [P(E + E j

I ; E ic

)](t)viwj .

Galerkin Expansions w. Families of Normal/LognormalBasis Elements

ν1(Ec) =

Ni∑i=1

αi φi

(Ec ;µi

c , σic

)and ν2(EI ) =

Nj∑j=1

βj φj

(EI ;σ

jI

)I Lognormally distributed elements for the coercive field density

φi

(Ec ;µi

c , σic

)=

1√2πEcσi

c

e−[ln(Ec )−µic ]

2/2(σi

c)2

σic = σc − vσc +

(2σcv

Ni

)i for i = 0 . . .Nσ

I Normally distributed elements for the interaction field density.

φj

(EI ;σ

jI

)=

1√

2πσjI

e−E 2I /2(σj

I )2

I Vary mean and standard deviation of bases for ν1(Ec) and standarddeviation of bases of ν2(EI ).

Parameter Identification w. Negligible Relaxation

Est. PR , η, α1 . . . α12, and β1 . . . β7 (21 params total).

−1 −0.5 0 0.5 1−0.25

−0.2−0.15

−0.1−0.05

0

0.05

0.10.15

0.20.25

Electric Field (MV/m)

Pol

ariz

atio

n (C

/m )2

DataModel

−1 −0.5 0 0.5 1−0.25

−0.2−0.15

−0.1

−0.050

0.05

0.10.150.2

0.250.25

Electric Field (MV/m)

Pol

ariz

atio

n (C

/m )2

DataModel

−1 −0.5 0 0.5 1−0.25

−0.2

−0.15−0.1

−0.05

00.05

0.10.15

0.20.250.25

Electric Field (MV/m)

Pol

ariz

atio

n (C

/m )2

DataModel

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 x 10−5

Coercive Field (MV/m)0 2000 4000

0

0.5

1

1.5

2

2.5 x 10−3

Interaction Field (V/m)

Parameter Identification w. Thermal Relaxation

Est. PR , η, τ(T ), γ, α1 . . . α12, and β1 . . . β7 (23 params total).

−1 −0.5 0 0.5 1

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Electric Field (MV/m)

Pol

ariz

atio

n (C

/m )2

DataModel

−1.5 −1 −0.5 0 0.5 1 1.5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Electric Field (MV/m)

Pol

ariz

atio

n (C

/m )2

DataModel

−1 −0.5 0 0.5 1

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Electric Field (MV/m)

Pol

ariz

atio

n (C

/m )2

DataModel

5 10 15x 100

2

4

6

8

x 10−6

Coercive Field (V/m)−4 −2 0 2 4x 104

2

4

6

8

10

12

x 10−5

Interaction Field (V/m)

5

Concluding Remarks

I We adapted the homogenized energy model to incorporate densitiesconstructed using Galerkin expansions with normal and lognormalbasis elements.

I These bases are used to construct decaying densities for modelcharacterizations that are less computationally intense and improveaccuracy in comparison to previous quantifications.

I Using this new model representation, we successfully characterizePZT behavior utilizing measured field inputs in comparison topolarization data from a PZT-driven stack actuator [9].

References

1 K. Mossi, Z. Ounaies, and S. Oakley, Optimizing energy harvestingof a composite unimorph pre-stressed bender, Sixteenth TechnicalConference of the American Society for Composites, vol. 2001, 2001.

2 R. Smith, A. Hatch, T. De, M. Salapaka, R. del Rosario, and J.Raye, Model development for atomic force microscope stagemechanisms, SIAM Journal on Applied Mathematics, vol. 66, no. 6,pp. 1998 2026, 2006.

3 K. Mossi, Z. Ounaies, R. Smith, and B. Ball, Prestressed curvedactuators: characterization and modeling of their piezoelectricbehavior, Smart Structures and Materials 2003: Active Materials:Behavior and Mechanics. Edited by Lagoudas, Dimitris C.Proceedings of the SPIE,, vol. 5053, pp. 423435, 2003.

4 B. Ball, R. Smith, and Z. Ounaies, A dynamic hysteresis model forTHUNDER transducers, Smart Structures and Materials 2003:Modeling, Signal Processing, and Control. Proceedings of the SPIE,vol. 5049, pp. 100111, 2003.

References, cont.

5 R. Smith, Smart Material Systems: Model Development.Philadelphia: Society for Industrial and Applied Mathematics, 2005.

6 R. Smith, S. Seelecke, Z. Ounaies, and J. Smith, A free energymodel for hysteresis in ferroelectric materials, Journal of IntelligentMaterial Systems and Structures, vol. 14, 2003.

7 J. Ernstberger and R. Smith, High-speed parameter estimationalgorithms for nonlinear smart materials, in Proceedings ofSPIEVolume 6523: Modeling, Signal Processing, and Control forSmart Structures 2007, April 2007. 6523OS.

8 R. Smith, A. Hatch, B.Mukherjee, and S. Liu, A homogenizedenergy model for hysteresis in ferroelectric materials: general densityformulation, Journal of Intelligent Material Systems and Structures,vol. 16, no. 9, p. 713, 2005.

9 S. Seelecke and A. York, Experimental investigation ofrate-dependent inner hysteresis loops in pzt, Materials ResearchSociety Symposium Proceedings, vol. 881, p. 48, 2005.