ERNSTBERGER, JON M. High Speed Parameter Estimation for a Homogenized Energy Model. (Under the direction of Professor R. C. Smith).
Industrial, commercial, military, biomedical, and civilian uses of smart materials are increasingly investigated for high performance applications. These compounds couple applied field or thermal energy to mechanical forces that are generated within the mate-rial. The devices utilizing these compounds are often much smaller than their traditional counterparts and provide greater design capabilities and energy efficiency.
The relations that couple field and mechanical energies are often hysteretic and nonlinear. To accurately control devices employing these compounds, models must quantify these effects. Further, since these compounds exhibit environment-dependent behavior, the models must be robust for accurate actuator quantification.
by
Jon M. Ernstberger
A dissertation submitted to the Graduate Faculty of North Carolina State University
in partial fullfillment of the requirements for the Degree of
Doctor of Philosophy
Computational Mathematics
Raleigh, North Carolina
2008
APPROVED BY:
Dr. K. Ito Dr. M. Haider
DEDICATION
BIOGRAPHY
Jon M. Ernstberger was born in Murray, KY in 1979. His parents, Mike and Debbie, raised him in a home with strong Christian influences. In 1997 he graduated from Calloway County High School and received scholarships to Murray State University. In 2001, Jon graduated from Murray State University with a Bachelor of Science in Applied Mathematics. He received his Master of Science in Mathematics from Murray State in 2003 at which time he was accepted to the graduate program in Applied Mathematics at North Carolina State University. Jon and his fianc´ee, Stacey, then moved to Raleigh in August, 2003 to being the doctoral program. On January 3, 2004, Jon and Stacey were happily married. On June 1st, 2007, their son, Lucas Hall, was born.
ACKNOWLEDGMENTS
My first thanks must go to Professor Ralph C. Smith. His advice and leadership have been continually present in all aspects of my graduate work. I could not have finished without him. I greatly appreciate Professors Mansoor Haider, Kazufumi Ito, and Hien Tran. As members of my committee I am thankful to them for the gift of their valuable time and their wonderful suggestions.
I have made many friends who helped make my years at NCSU enjoyable. From the time of our arrival, and especially in the summer of 2005, Jeff and Heidi Hood were wonderful friends. I made a great friend of Nathan Gibson in the office, the gym, and outdoors. I was blessed to get to know Tom, Sarah, and Matthew Braun. Tom became a great friend to me and helped immensely by acting as a sounding board for much of this research. Sarah (Grove) Muccio, already a close friend to Stacey, became a close friend to me in the summer of 2005 (my second qualifier summer). I consider Sarah to be part of our family.
The members of the Cary church of Christ have been a wonderful Christian family to us. Edna and Laddie Munger, two wonderful people, have shown us extraordinary kindness by weekly opening their hearts and home to Lucas.
Without my family I could not have finished this degree. Mom and Dad always taught me that I was able to and should follow my dreams. They love and give so freely in so many different ways. My awesome sister, Krysta, is always there for me. The visits, phone calls, and memories with all my grandparents are cherished and have helped keep me sane. Ma and Pa Lawler, and the rest of the family into which I married, have given wonderful support to me while in North Carolina. I am truly blessed with each member of my family.
My wife and son have, however, been most crucial to my success. Lucas has only been with us for a year, but has unknowingly helped to keep my priorities straight and has shown me the important things in life. I am thankful to God for such an amazing blessing as my son. I owe so many thanks to my wife, Stacey. She has given selflessly and held me up when I was uncertain of my self. She is undoubtedly my best friend and my biggest fan. I love her inexpressibly and thank God for her daily.
TABLE OF CONTENTS
LIST OF TABLES . . . viii
LIST OF FIGURES . . . x
1 Introduction . . . 1
1.1 Ferromagnetic Materials . . . 3
1.2 Ferroelectric Materials . . . 5
1.3 Outline . . . 6
2 Models . . . 10
2.1 Homogenized Energy Model . . . 10
2.1.1 Ferromagnetic Materials . . . 10
2.1.2 Ferroelectric Materials . . . 13
2.1.3 90◦-Switching . . . . 16
2.1.4 Temperature-Dependence . . . 19
2.2 Lumped Rod Model . . . 21
3 Gradient-Based Parameter Estimation Algorithms . . . 23
3.1 Gradient-Based Routines . . . 23
3.1.1 Sequential Quadratic Programming . . . 24
3.1.2 Gradient and Hessian Approximations . . . 26
3.1.3 Newton Trust-Region Routines . . . 27
3.2 Parameterized Densities . . . 28
3.3 Initial Parameter Estimates for the Homogenized Energy Model Using Pa-rameterized Densities . . . 29
3.3.1 Initial Estimates Using Magnetization/Polarization . . . 30
3.3.2 Initial Estimates Using Strain/Displacement Data . . . 34
3.4 General Densities . . . 38
3.5 Galerkin Expansion Formulation of the Homogenized Energy Model . . . 41
3.5.1 Galerkin Expansions: Linear and Cubic Bases . . . 43
3.5.2 Galerkin Expansions: Normal/Lognormal Families . . . 48
3.6 Temperature-Dependent Parameter Estimation . . . 63
4 Parameter Estimation Using Stochastic and Direct Searches . . . 67
4.1 Genetic Algorithms . . . 67
4.1.1 Differential Evolution . . . 68
4.2 Patternsearch . . . 70
4.3 Simulated Annealing . . . 71
5 Conclusion . . . 80
Bibliography . . . 82
Appendices . . . 87
A Dissipativity of the Homogenized Energy Model . . . 88
A.1 Dissipativity . . . 88
A.2 Dissipativity of the Homogenized Energy Model . . . 90
A.2.1 Kernel Dissipativity . . . 90
A.2.2 Macroscopic Model Dissipativity . . . 95
B Graphical User Interface for Parameter Estimation . . . 98
B.1 Choice of Data . . . 98
B.2 Specification of Initial Estimates . . . 99
B.3 Density Choice . . . 101
LIST OF TABLES
Table 3.1 Parameters corresponding with results of Figures 3.1-3.2 when employing parameterized densities.. . . 30 Table 3.2 Initial estimates of parameters η, PR, σI, σc and µc using the techniques
discussed in Section 3.3.1 which generated the results in Figure 3.6. Table rows (a)-(d) correspond to subplots (a)-(d) in Figure 3.6. . . 34 Table 3.3 Number of quadrature intervals required for composite quadrature to
con-verge for the coercive field density within a given tolerance (denoted at the top) using a fixed number of basis elements (left) and four-point Gaussian quadrature when using the lognormally distributed basis elements. . . 51 Table 3.4 Number of quadrature intervals required for composite quadrature to
con-verge for the interaction field density within a given tolerance (denoted at the top) using a fixed number of basis elements (left) and four-point Gaussian quadratures for the normally distributed basis elements. . . 52 Table 3.5 Runtimes and objective functional values on Computer 1 with the trust-region
parameter estimation routine. . . 53 Table 3.6 Runtimes and objective functional values on Computer 1 with the SQP
pa-rameter estimation routine. . . 53 Table 3.7 Runtimes and objective functional values on Computer 2 with the trust-region
optimization routine. . . 53 Table 3.8 Runtimes and objective functional values on Computer 2 with the SQP
op-timization routine. . . 53 Table 3.9 Lumped rod and kernel parameter estimates corresponding to the results of
Figure 3.20. MR is fixed at 5×105 A/m. . . 55 Table 3.10 Lumped rod and kernel parameter estimates corresponding to the results of
Figure 3.22. MR is fixed at 5×105 A/m. . . 56 Table 3.11 Parameter estimates using gradient-based routines and the families of normals
for both probability density functions. . . 59 Table 3.12 Estimated kernel parameters using gradient-based routines that correspond
Table 3.13 Estimated kernel parameters using gradient-based routines that correspond to results of Figure 3.31. . . 63 Table 3.14 Initial and final parameter estimates corresponding to the results of
Fig-ure 3.33. . . 66
LIST OF FIGURES
Figure 1.1 (a) Image of the AMS from http://www.etrema.com. (b) Cross-sectional schematic of the AMS. . . 2 Figure 1.2 (a) Schematic of an atomic force microscope. (b) A THUNDER actuator
with a clamped side which may be used for a variety of actuation purposes. Image from http://www.faceinternational.com.. . . 3 Figure 1.3 Magnetic domains in iron. Image from http://www.ien.it/.. . . 4 Figure 1.4 (a) In the absence of an applied field, magnetic domains exhibit a
minimiza-tion of internal energy via differing moment orientaminimiza-tions. (b) Reconfiguraminimiza-tion of magnetic domains as a magnetic field is applied. (c) Domains with dipole moments oriented along theeasy axis. (d) Moment orientations in the direction of the applied field. . . 5 Figure 1.5 (a) Material strainε0as magnetic domains are misaligned. (b) Increased
ma-terial strain resulting in magnetostriction as domain moment orientations become aligned in direction of the applied field. . . 5 Figure 1.6 Measured Terfenol-D rod actuator data from [10]. (a) Magnetization versus
magnetic field. (b) Rod tip displacement versus magnetic field. . . 6 Figure 1.7 Polarization of a ferroelectric material when only 180◦-switching occurs.
Image depicts dipole orientation at (a) a positive field value E1, (b) a greater
applied field generating internal strains, and (c) a return to field valueE1. As the
field value decreases below the negative coercive field value, (d) the dipole changes to a negative orientation, and as field becomes more negative, (e) the structure has greater internal strain. . . 7 Figure 1.8 Data from a PZT stack actuator [34]. (a) Polarization versus electric field.
(b) Actuator tip displacement versus electric field. . . 7 Figure 1.9 PZT data from [44] with applied compressive stresses of 16, 8 and 1 MPa,
respectively. . . 8
Figure 2.1 (a) Gibbs energy for 180◦-switching. (b) Local magnetization hysteron
Figure 2.2 (a) Position of the Titanate ion with positively oriented polarization. (b) Re-orientation of the Titanate ion into a negative 90◦ orientation due to applied
com-pressive stresses. (c) Polarization orientation mapping for±180◦ and ±90◦
direc-tions. . . 17 Figure 2.3 Use of the Terfenol-D rod in a simple actuator model.. . . 21
Figure 3.1 Optimized model results to displacement data from [30, 31] collected at (a) 100 Hz, (b) 200 Hz, (c) 300 Hz, and (d) 500 Hz input currents. The model employed the densities of Figure 3.2. . . 29 Figure 3.2 Parameterized densities employed in the model to generate results of
Fig-ure 3.1. . . 30 Figure 3.3 Depiction of of densities ν1(Ec) and ν2(EI) near regions of saturation in
relation to the E-P hysteresis curve.. . . 32 Figure 3.4 (a) Width of density ν1(Ec) in terms of mean and standard deviation of
the underlying normal distribution. (b) Width of the density ν2(EI) in terms of the maximally applied field value and the minimum field value in the region of saturationEL. . . 33 Figure 3.5 “Point and click” methods are displayed for choosing (a) susceptibility η,
(b) interaction field standard deviationσI, (c) remanencePR, and (d) the lognormal coercive field density mean ¯Ec. Data taken from [34]. . . 34 Figure 3.6 Unoptimized model results using GUI-obtained parameter estimates in
com-parison to PZT data from [34] . The model employs the negligible relaxation kernel. 35 Figure 3.7 Initial estimate to displacement data using the GUI-based interface and
tech-niques referenced in this section. Data collected at 100 Hz current and referenced from [30, 31]. . . 38 Figure 3.8 Optimized model fits initiated via GUI-based techniques. Data taken from
[30, 31].. . . 39 Figure 3.9 Densities corresponding to optimized results of Figure 3.8. . . 40 Figure 3.10 Optimized model fits to displacement data employing penalized generalized
densities. Data taken from [30, 31] and collected at current input frequencies of (a) 100 Hz, (b) 200 Hz, (c) 300 Hz, and (d) 500 Hz. . . 41 Figure 3.11 General coercive and interaction field densities corresponding to results from
Figure 3.12 Optimized model results to displacement data from [30, 31] collected at (a) 100 Hz, (b) 200 Hz, (c) 300 Hz, and (d) 500 Hz input currents. Model results are generated using the constrained general density formulation. . . 43 Figure 3.13 Constrained generally distributed coercive and interaction field densities
employed in the model output of Figure 3.12. . . 44 Figure 3.14 (a) Fit to 100 Hz Terfenol-D displacement data from [30, 31] using linear
Galerkin expansion densities. (b) Coercive and interaction field densities exhibiting lack of density decay and unimodality. . . 44 Figure 3.15 Optimized fits to data using a constrained linear Galerkin expansion with
eight quadrature intervals. Data published in [30, 31] and collected at current input rates of (a) 100 Hz, (b) 200 Hz, (c) 300 Hz, and (d) 500 Hz. . . 47 Figure 3.16 Constrained linear Galerkin expansion coercive and interaction field
densi-ties. Two-point Gaussian quadrature employed with corresponding results in Fig-ure 3.15. . . 48 Figure 3.17 Optimized fits to Terfenol-D displacement data from [30, 31] collected at
current frequencies of (a) 100 Hz, (b) 200 Hz, (c) 300 Hz, and (d) 500 Hz. The model employs constrained cubic B-spline density expansions. . . 49 Figure 3.18 Coercive and interaction field densities generated using constrained cubic
B-spline expansions. . . 50 Figure 3.19 Lognormal (a) single-mean and (b) multi-mean coercive field basis elements.
These elements were constructed from a previous identification of parameterized densities to Terfenol-D data. . . 50 Figure 3.20 Optimized model fits to displacement data published in [30, 31] and collected
at (a) 100 Hz, (b) 200 Hz, (c) 300 Hz, and (d) 500 Hz input currents. Corresponding densities for these results are found in Figure 3.21. . . 54 Figure 3.21 Coercive and interaction field densities using families of normal and
single-mean lognormal bases. . . 54 Figure 3.22 Optimized model results to displacement data from [30, 31] collected at
(a) 100 Hz, (b) 200 Hz, (c) 300 Hz, and (d) 500 Hz. Corresponding densities constructed using the multi-mean Galerkin normal/lognormal basis are shown in Figure 3.23.. . . 55 Figure 3.23 Coercive and interaction field densities using families of normal and
multi-mean lognormal bases. . . 56 Figure 3.24 Optimized negligible relaxation model fits to PZT (a) minor loop, (b) creep,
elements used to construct the employed coercive and interaction field densities. . . 57 Figure 3.25 Optimized model fits to PZT (a) minor loop, (b) creep, and (c) major loop
data from [34]. The model employed (d) densities constructed via the multi-mean Galerkin normal/lognormal basis and incorporated thermal relaxation. . . 58 Figure 3.26 Optimized model fits to PZT data from [34] using the multi-mean Galerkin
normal/lognormal basis and employing the kernel with negligible thermal relaxation and 90◦-switching. . . 59
Figure 3.27 (a) Coercive and interaction field densities used to obtain fits to data shown in Figure 3.26. (b) Coercive and interaction field basis elements used to generate coercive and interaction field densities of (a). . . 60 Figure 3.28 Comparison of the homogenized energy model with 90◦ and 180◦-switching
kernels without thermal relaxation to two different data sets. . . 60 Figure 3.29 Optimized model fits to PZT (a) major loop, (b) minor loop, and (c) creep
data from [34]. Model incorporates 90◦-switching and thermal relaxation. (d)
Den-sities used to generate comparisons (a)-(c). . . 61 Figure 3.30 Optimized model fits to major loop PZT data from [44] with (a) 16 MPa,
(b) 8 MPa, and (c) 1 MPa of compressive stress applied. (d) Densities were con-structed employing the Galerkin normal/lognormal basis. . . 62 Figure 3.31 Optimized model fits to major loop PZT data from [44] with (a) 16 MPa,
(b) 8 MPa, and (c) 1 MPa applied compressive stresses. Employed densities are displayed in Figure 3.32. . . 63 Figure 3.32 (a) Coercive and interaction field densities corresponding to results of Figure
3.31. (b) Single-mean coercive and interaction field expansion elements used to construct densities (a). . . 64 Figure 3.33 Temperature-dependent model fits to Terfenol-D data from [10] at (a) 298◦K,
(b) 317◦ K, (c) 336◦ K, and (d) 355◦ K. . . 65
Figure 3.34 Coercive and interaction field densities corresponding to Figure 3.33(a). . . . 66
Figure 4.1 Genetic algorithm outline. . . 69 Figure 4.2 Optimized fits to Terfenol-D data from [30, 31] collected at (a) 100 Hz,
(b) 200 Hz, (c) 300 Hz, and (d) 500 Hz. Model employed a constrained genetic algorithm as described in Chapter 4.1 with constraints imposed from Chapter 3.5.1. 70 Figure 4.3 Densities corresponding to results of Figure 4.2 employing a linear Galerkin
Figure 4.4 Optimized fits to Terfenol-D data from [30, 31] collected at (a) 100 Hz, (b) 200 Hz, (c) 300 Hz, and (d) 500 Hz using the Differential Evolution algorithm of Chapter 4.1.1. . . 72 Figure 4.5 Nonphysical densities corresponding to Figure 4.4 which employ a cubic
Galerkin expansion. . . 73 Figure 4.6 Pattern search algorithm as defined in [25]. . . 73 Figure 4.7 Optimized fits to Terfenol-D data using a SQP/PS/SQP combination with
imposed constraints of Chapter 3.5.1. Data published in [30, 31] and collected at (a) 100 Hz, (b) 200 Hz, (c) 300 Hz, and (d) 500 Hz. . . 74 Figure 4.8 Nonphysical coercive and interaction field densities corresponding to the
results of Figure 4.7. . . 75 Figure 4.9 Fits to PZT (a) minor loop, (b) creep, and (c) major loop data collected
by York [34]. Simulated annealing was employed for optimization of the model incorporating 180◦-switching and thermal relaxation. (d) Densities employed use
families of lognormally and normally distributed basis elements. . . 76 Figure 4.10 Optimized model fit to PZT (a) major loop, (b) minor loop, and (c) creep
data from [34]. The model employs the negligible relaxation kernel with 90◦
-switching and the densities of Figure 4.11(a). . . 77 Figure 4.11 (a) Densities employed to generate model fits of Figure 4.10. (b) Coercive
field and interaction field basis elements used to generate the densities of Fig-ure 4.11(a). . . 78 Figure 4.12 Optimized model fit to PZT data from [44] with (a) 16 MPa, (b) 8 MPa, and
(c) 1 MPa applied compressive stress. (d) Densities employed to generate model fits (a)-(c).. . . 79
Figure B.1 GUI-based choice of data to which parameter estimation will be performed. 99 Figure B.2 GUI-based choice of initial estimates to begin model optimization. . . 99 Figure B.3 (a) If default parameters initiate identification, the user must choose the
ma-terial most similar to that which drives their actuator. (b) If the user is not picking bounds on the parameter estimation, pre-defined bounds must also be selected.. . . 100 Figure B.4 (a) If GUI-based identification methods are invoked, the user must choose
Figure B.5 (a) The user must choose the struct field which relates the externally applied field. (b) For fits to displacement, the user must similarly choose the struct field which relates the collected displacement data. A similar dialog box exists when choosing magnetization/polarization.. . . 102 Figure B.6 Use of GUI-based tools to estimate (a) remanencePR, (b) susceptibilityη,
(c) the mean of the coercive field density ¯Hc, and (d) the standard deviation of the interaction and coercive field densitiesσI and ¯σc. . . 103 Figure B.7 GUI-based methods to approximate model parameters if fitting strain or
displacement data. . . 104 Figure B.8 (a) Manual specification of kernel and density parameters using an input
dialog. (b) User option to specify parameter bounds. . . 104 Figure B.9 Constrained parameter identification (a) lower and (b) upper bound
speci-fication dialog. . . 105 Figure B.10 Flowchart relating GUI paradigm for specifying field densities. . . 106 Figure B.11 Choice of number of quadrature intervals for the discretization of (2.15). . 106 Figure B.12 The user must specify the (a) interaction and (b) coercive field density
formulation. . . 107 Figure B.13 If a Galerkin expansion density representation is chosen, the class of basis
element must be chosen. . . 107 Figure B.14 If either density is chosen to be constructed using a Galerkin expansion, the
user must specify (a) the number of basis elements in the expansion. If employing the Galerkin normal/lognormal basis elements, then (b) the user must specify the number of means (coercive field density only) and standard deviations. . . 107 Figure B.15 (a) GUI to choose kernel incorporating or neglecting thermal relaxation.
Chapter 1
Introduction
“Smart” or active materials, are compounds that exhibit unique transduction ca-pabilities due to electromechanical or magnetomechanical coupling. These materials provide actuator capabilities due to their ability to transform field energy (electric, magnetic, or heat) into mechanical energy (electrostriction for ferroelectric materials or magnetostric-tion for ferromagnetic materials). Secondly, they provide sensing capabilities due to the materials’ ability to transform mechanical energy into field energy.
As early as the 1940’s, piezoelectric compounds (primarily ceramics) were used as sonar transducers on submarines. Today, active materials are used for a wide variety of military, industrial, biomedical and commercial purposes.
Various smart materials are presently under investigation for military applications. For example, the Navy is considering the use of the ferromagnetic compound Terfenol-D (Tb.73Dy.93Fe2) as a transducing material in towed sonar arrays for submarines. DARPA is
investigating the equipment of infantry soldiers with energy-harvesting ferroelectric mate-rials in boot heels [27] that would recharge manually-transported battery-powered devices while increasing field mobility. This is motivated by the goal of removing unnecessarily large batteries resulting in lower package weights.
After objects cast in a die-mold have finished cooling (such as piston heads), they must be machined to meet design specifications. The machining process requires the precise posi-tioning of the milling head to ensure adherence to tolerances. The main drive of the AMS is shown in Figure 1.1(a). The cross-sectional view of the AMS in Figure 1.1(b) displays a pre-stressed Terfenol-D rod surrounded by a wound-wire solenoid that is used to generate magnetic fields. The Terfenol-D rod is coupled via a spring-loaded mechanism to the milling head which is positioned by inducing magnetostriction in the Terfenol-D rod.
Another commercial use of smart materials is the nano-positioning stage used for atomic force microscopy as detailed in [36] and depicted in Figure 1.2(a). The stage is posi-tioned using PZT (PbTiO3) stacked actuators which are driven by a laser-guided feedback
control law. Another application of PZT is the THUNDER (THin LayerUNimorph Ferro-electricDrivER and Sensor) transducer (pictured in Figure 1.2(b) and detailed in [2, 28]). In this construct, the PZT is bonded to an aluminum, brass, or stainless steel substrate. Depending on current directions, the actuator will flatten or create a larger arching pattern. THUNDER are presently being investigated for pumps in industrial or externally affixed biomedical applications as well as energy harvesting purposes. One motivation for smart materials is their solid state nature which permits them to replace gear-driven hydraulically-actuated devices of substantial mass presently in service. In contrast, the devices employing active materials for use in actuators may be smaller, more energy efficient, and require less setup time.
An important aspect in the use of these materials in devices is the need for accurate models and control laws. Depending on the operating regime, these materials may exhibit highly hysteretic and/or nonlinear behavior. Models and model-based control laws must
Milling Head
Object to Be Milled Spring
Permanent Magnet Compression Bolt Wound Wire Solenoid Terfenol−D Rod
(a) (b)
Feedback Law
Photodiode
AFM Piezoceramic Stage
Laser
Sample Micro−Cantilever
(a) (b)
Figure 1.2: (a) Schematic of an atomic force microscope. (b) A THUNDER actuator with a clamped side which may be used for a variety of actuation purposes. Image from http://www.faceinternational.com.
be able to quantify the nonlinear and hysteretic relations between inputs and outputs. A related aspect is the estimation of model parameters to fit data. To prompt a desired “smart” actuator response, a control law must reference a model that accurately quantifies measurable behaviors such as magnetization or polarization and strain or displacement.
1.1
Ferromagnetic Materials
Materials such as iron are ferromagnetic. Ferromagnetic materials have a domain structure in which magnetic moments have similar orientations. While the temperature of the material remains below the Curie point, there is a spontaneous magnetization of the material resulting in a hysteretic relationship between an externally applied magnetic field and the magnetization of the material itself. As the material temperature increases above the Curie point, the field-magnetization relationship becomes anhysteretic, but retains non-linearity.
If we examine the material at a microscopic scale, we see that the material is divided into magnetic domains (as displayed for iron in Figure 1.3). When the material is demagnetized, these domains create a net magnetization of zero as shown in Figure 1.4(a). However, as a greater magnetic field is applied, the domains change size and shape to minimize the total magnetization as shown in Figure 1.4(b). As larger magnetic fields are applied, the magnetic moments will eventually all switch into the direction of theeasy axis as shown in Figure 1.4(c). Finally, as the magnetic field increases the moments will all switch into the direction of the magnetic field as represented in Figure 1.4(d).
Examining material strains, we see that individual magnetic domains exhibit the greatest strain in the direction of their magnetic moment orientation. However, with no externally applied field, the moments remain situated in their reduced energy configuration and the material has an original strain value ε0 as shown in Figure 1.5(a). As the applied magnetic field increases, the magnetic moments (and consequently the domains) will align into a parallel direction to the applied field yielding the largest possible strains as seen in Figure 1.5(b). Figure 1.6 illustrates data collected from a Terfenol-D rod actuator at ETREMA. Figure 1.6(a) displays the macroscopic magnetization versus applied magnetic field of the material while Figure 1.6(b) displays the relation between displacement and magnetic field.
(a) H=0
(b) H >0
3
(d) H >H
2(c) H >H
2 11
Figure 1.4: (a) In the absence of an applied field, magnetic domains exhibit a minimization of internal energy via differing moment orientations. (b) Reconfiguration of magnetic domains as a magnetic field is applied. (c) Domains with dipole moments oriented along the easy axis. (d) Moment orientations in the direction of the applied field.
1.2
Ferroelectric Materials
The mechanisms for ferroelectricity are sufficiently similar to those of ferromag-netism so that we may model phenomena peculiar to each with the same phenomenological model. However, there are differences between these materials and their ferromagnetic counterparts. We will not discuss these but instead briefly describe the phenomena and
0
ε
(a)
(b)
H=0
H>0
ε>ε0
Figure 1.5: (a) Material strain ε0 as magnetic domains are misaligned. (b) Increased
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x 105
−6 −4 −2 0 2 4 6
x 105
Magnetic Field (A/m)
Magnetization (A/m)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x 105
0 100 200 300 400 500 600 700 800 900
Magnetic Field (A/m)
Displacement (
µ
m)
(a) (b)
Figure 1.6: Measured Terfenol-D rod actuator data from [10]. (a) Magnetization versus magnetic field. (b) Rod tip displacement versus magnetic field.
give illustration. Details regarding the ferroelectric material behavior are provided in [35]. As an external electric fieldEis applied, an internal ion (in PZT, the Titanate ion) changes position inside the molecule and will change the dipole into a positive orientation. This results in a positive polarization as the field value passes acoercive field valueEc. As a larger magnitude field is applied, the ion continues to displace itself further resulting in a strain of the molecule. These phenomenon are illustrated in Figures 1.7(a) and (b). As the applied electric field value is decreased, the displacement of the ion recedes to previous positions (Figure 1.7(c)). When the electric field is decreased past the negative coercive field value, the ion switches position changing the orientation of the polarization as seen in Figure 1.7(d). The molecule exhibits similar strain in the negative direction as shown in Figure 1.7(e). To illustrate with data, we display polarization in a PZT stack actuator, collected by Alex York, in Figure 1.8 (see [34]). Figure 1.8(a) shows the macroscopic polar-ization of the PZT given an input of electric field. Figure 1.8(b) displays the relationship between electric field and strain/displacement. When compressive stresses are applied, we see results such as shown in Figure 1.9.
1.3
Outline
1
E >0 E > E2 1 E >0 1 E <−E 3 c E < E4 3
(b)
(a) (c) (d) (e)
Figure 1.7: Polarization of a ferroelectric material when only 180◦-switching occurs. Image
depicts dipole orientation at (a) a positive field valueE1, (b) a greater applied field gener-ating internal strains, and (c) a return to field valueE1. As the field value decreases below
the negative coercive field value, (d) the dipole changes to a negative orientation, and as field becomes more negative, (e) the structure has greater internal strain.
of the kernel, the function that relates magnetic switching at a microscopic level, with or without thermal relaxation is related for both 180◦ and 90◦-switching. These kernels are derived from minimizing the Gibbs energy with respect to magnetization/polarization or by balancing the thermal and Gibbs energies. Then, homogenization techniques are employed to incorporate variances in the interaction and coercive fields by relating these as manifes-tations of underlying densities. A final modification of the homogenized energy model is summarized which incorporates temperature-dependence into the remanence value and the coercive field density.
Chapter 3 describes the gradient-based routines used to estimate parameters for
−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
Polarization (C/m
2)
Electric Field (MV/m)
−1.5 −1 −0.5 0 0.5 1 1.5
0 10
20
x 10−5
Electric Field (MV/m)
Displacment (m)
(a) (b)
−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) 16 MPa
8 MPa 1 MPa
Polarization (C/m )
2
Figure 1.9: PZT data from [44] with applied compressive stresses of 16, 8 and 1 MPa, respectively.
the nonlinear programming problem which is characterized by the weighted square of the`2
residual of the model versus collected data. Past efforts via density formulations are related for parameter estimations with respect to the homogenized energy model. Benefits and weaknesses of these previous density formulations are discussed and results are presented. Modifications are imposed upon a previous formulation to create densities which exhibit de-cay and unimodality. A new set of density formulations for the homogenized energy model are set forth that reduce computational load but may yield non-physical densities. Again constraints are described and imposed which generate unimodality and exponentially de-caying densities and, again, results follow. A final variant is presented which requires only simple bound constraints and further reduces computational loads for the parameter estima-tion problem. Results are presented for these new formulaestima-tions with respect to Terfenol-D and PZT data in terms of magnetization/polarization and strain/displacement. Finally, re-sults corresponding to the temperature-dependent homogenized energy model are presented along with final parameter estimates.
Chapter 4 details deterministic and stochastic searches employed to identify model parameters. In many cases, gradient-based routines will only converge to local minima. In contrast, stochastic and direct searches often have global convergence properties. Distinct advantages and disadvantages to these routines are discussed. Results using the models of Chapter Two are presented using each of the examined routines with focus given to those results found with simulated annealing.
con-clusion are two appendices. Appendix A gives a brief summary of the energy criteria known as dissipativity and the proof of dissipativity of the homogenized energy model, with and without thermal relaxation, when only 180◦-switching is modeled. Appendix B details the
Chapter 2
Models
In this chapter, the models for magnetization and polarization are summarized using an applied field as input. The model is developed in variants that incorporate 180◦
and 90◦-switching, both with and without thermal relaxation. Next, the homogenized energy model with an incorporated temperature-dependence is summarized. Finally, the model for strain/displacement is developed using magnetization or polarization as input.
2.1
Homogenized Energy Model
We begin by characterizing the internal energies for ferromagnetic materials when 180◦-switching is modeled. Local magnetization is then discerned via internal energy
re-lations. The macroscopic magnetization is subsequently constructed by homogenization techniques to incorporate material inhomogeneities and effects from internally generated fields. Homogenized energy models characterizing 90◦ and 180◦-switching are then
devel-oped in the context of polarization.
2.1.1 Ferromagnetic Materials
As detailed in [35, 38], we employ the piecewise quadratic relation
ψ(M) = η 2
(M +MR)2 M ≤ −MI
(M −MR)2 M ≥MI
(MI−MR)
³
M2
MI −MR ´
|M|< MI
G(H,M)
MR
MR
MI
MI
(a) (b)
M
H
Figure 2.1: (a) Gibbs energy for 180◦-switching. (b) Local magnetization hysteron
neglect-ing thermal relaxation.
to quantify the Helmholtz energy due to moment reorientation. Here, MRis the remanent magnetization,η is the reciprocal of the slope of theH−M relation, andMI is the positive inflection point of (2.1). The Gibbs energy
G(H, M) =ψ(M)−HM (2.2)
incorporates elastic and electrostatic work and is depicted in Figure 2.1(a). By minimizing the Gibbs energy (2.2) with respect to magnetization,
∂G
∂M = 0, (2.3)
we construct a relation for the local average magnetization ¯
M(H+HI;Hc) = H+ηHI +δ(H+HI;Hc)MR (2.4) in the absence of thermal activation. Here
δ(H+HI;Hc) =
1 H+HI ≥Hc
−1 H+HI ≤ −Hc
(2.5)
quantifies the switching mechanism. The kernel (2.4) is depicted in Figure 2.1(b). This particular kernel is initialized using the relation
M(H0+HI;Hc) =
H0+HI
η −MR H0≤0 H0+HI
η +MR H0>0
for an initial field value H0.
To incorporate thermal effects, we employ the Boltzmann relation
µ(G) =Ce−G(H,M)V /kT (2.7)
which balances the Gibbs energyG(H, M) and the relative thermal energykT /V where k
is Boltzmann’s constant,V is volume, andC is chosen to ensure integration to one. The likelihoods of a moment switching from a positive to negative or negative to positive orientation are respectively specified by the relations
p+− = T1(t)
RMI
MI−²e
−G(H,M)V /kTdM
R∞
MI−²e
−G(H,M)V /kTdM (2.8)
and
p−+=
1
T(t)
R−MI+²
−MI e
−G(H,M)V /kTdM
R−MI+²
−∞ e−G(H,M)V /kTdM
(2.9)
where²is a small positive constant andT(t) is the reciprocal of the frequency at which mo-ments attempt to switch. Letx+ andx− denote the fraction of total moments respectively having positive and negative orientations. These ratios are governed by the differential equations
˙
x+ =−p+−x++p−+x− (2.10) ˙
x− =−p−+x−+p+−x+ (2.11)
wherex++x−= 1. Given (2.7), we express the expected magnetizations hM+i and hM−i
as
hM+i=
R∞
MIM e
G(H,M)V /kTdM
R∞
MIe
G(H,M)V /kTdM (2.12)
and
hM−i=
R−MI
−∞R M eG(H,M)V /kTdM −MI
−∞ eG(H,M)V /kTdM
. (2.13)
For an applied field H, the local average magnetization is then given by ¯
The macroscopic magnetization of the material can be expressed as [M(H)](t) =
Z ∞
0 Z ∞
−∞
ν1(Hc)ν2(HI)
£
M(H+HI;Hc)
¤
(t)dHIdHc (2.15) when we assume that material inhomogeneities and variable effective fields are manifesta-tions of underlying distribumanifesta-tions. We denote the densities associated with these distribumanifesta-tions asν1(Hc) and ν2(HI) and impose the restrictions
1.) ν1(x) is only defined for x >0;
2.) ν2(−x) =ν2(x) for density symmetry; (2.16)
3.) |ν1(x)| ≤c1e−a1x and |ν2(x)| ≤c2e−a2|x|.
The third requirement in (2.16) guarantees density non-negativity and exponential decay as distance from the density mean increases whilec1,c2,a1 and a2 remain positive. Equation
(2.15) can be discretized as
[M(H)](t)≈
Ni X i=1 Nj X j=1 ν1 ¡ Hci¢ν2
³ HIj
´ h
¯
M ³
H+HIj;Hci ´i
(t)viwj (2.17)
using a composite quadrature rule whereHi
candHIjare the quadrature nodes,viandwj are the quadrature weights associated with the quadrature nodes, and Ni, Nj are the number of quadrature nodes; see [35] for details. The implementation algorithms for the model are additionally described in [4].
2.1.2 Ferroelectric Materials
In a manner similar to the previous discussion for magnetization, we characterize polarization as a function of applied electric field using the same homogenized energy model.
The Gibbs energy for polarization is
G(E, P) =ψ(P)−EP (2.18)
where
ψ(P) = η 2
(P+PR)2 P ≤ −PI (P−PR)2 P ≥PI (PI−PR)
³
P2
PI −PR ´
|P|< PI
is the Helmholtz energy. Solving
∂G
∂P = 0 (2.20)
forP yields the local relation for polarization ¯
P(E+EI;Ec) = E+ηEI +δ(E+EI;Ec)PR (2.21) given an interaction fieldEI, remanent polarization PR, coercive field valueEc, and
η= dE
dP (2.22)
whereE denotes the applied field. The termδ is given by
δ(E+EI;Ec) =
1 E+EI ≥Ec
−1 E+EI ≤ −Ec
. (2.23)
Another local polarization kernel includes thermal relaxation by balancing thermal and Gibbs energies via the Boltzmann relation
µ(G) =Ce−G(E,P)V /kT. (2.24)
Similar to magnetic dipole moments, polarization dipoles maintain positive or negative orientations and can be expressed as
x++x−= 1. (2.25)
The value x+ is the fraction of total dipoles oriented in a positive direction andx− is the
fraction oriented in the negative orientation. The evolution equations for x+ and x− are
quantified by the differential equations
˙
x+ =−p+−x++p−+x− (2.26)
˙
x− =−p−+x−+p+−x+ (2.27)
or instead
˙
By solving (2.28) coupled with (2.25), we have representations forx+andx−. The switching
likelihoodsp−+andp+−are the likelihoods of a dipole switching orientation at a given Gibbs
energy and are given by
p+−= T1(t)
RPI
PI−²e
−G(E,P)V /kTdP
R∞
PI−²e
−G(E,P)V /kTdP (2.29)
and
p−+= 1
T(t)
R−PI+²
−PI e
−G(E,P)V /kTdP
R−PI+²
−∞ e−G(E,P)V /kTdP
(2.30)
for² >0. The expected polarizations of the positively oriented dipoleshP+iand negatively oriented dipoleshP−i are given as
hP+i=
R∞
PI P e
G(E,P)V /kTdP
R∞
PI e
G(E,P)V /kTdP (2.31)
and
hP−i=
R−PI
−∞ P eG(E,P)V /kTdP
R−PI
−∞ eG(E,P)V /kTdP
. (2.32)
We then express the local average magnetization as ¯
P =x+hP+i+x−hP−i. (2.33)
The macroscopic polarization of a material is then given by [P(E)] (t) =
Z ∞
0 Z ∞
−∞
ν1(Ec)ν2(EI)
£¯
P(E+EI;Ec)
¤
(t)dEIdEc (2.34)
whereν1(Ec) andν2(EI) are the manifestations of the underlying distributions of the coer-cive and interaction fields which adhere to the restrictions (2.16). Again, we may discretize (2.34) as
[P(E)] (t)≈
Ni X
i=1
Nj X
j=1
ν1¡Eci¢ν2 ³
EIj ´ h
¯
P ³
E+EIj;Eci ´i
(t)viwj (2.35)
2.1.3 90◦-Switching
In the previous polarization model discussion, we presumed only 180◦ switching. However, the crystalline structure of many active materials allows switching to occur in other directions including 90◦ orientations. This may be caused by the application of an
external compressive stress or an external field. The phenomenon of an external stress resulting in polarization change is known as ferroelasticity. We use PZT as the active material to exhibit these behaviors and to estimate parameters using techniques presented in Chapters 3 and 4.
When we only model switching in the ±180◦ direction we see that an ion (in PZT,
the Titanate ion) switches orientation as the applied field passes the coercive field value. However, when 90◦ switches are taken into account, the Titanate ion may switch into±90◦
orientations as depicted in Figure 2.2. Similar to previous work in [1], it may be shown that the free Helmholtz energy due to polarization reorientation may be approximated as
ψP(P) =
η(P+PR)2 P ≤ −PI
η1(P+PM)2/2 +β −PI < P <−P90I
η2P2/2 + ∆ |P|< P 90I
η1(P−Pm)2/2 +β P90I < P < PI
η(P−PR)2/2 P ≥PI
(2.36)
where
Pm = η(PηI(−PPR)P90I−η2P90IPI
I−PR)−η2P90I (2.37)
η1 =ηPPI−PR
I−Pm
(2.38)
β = η
2(PI−PR)
2−η1
2 (PI−Pm)
2 (2.39)
∆ = η1
2 (P90I−Pm)
2+β−η2
2P90I. (2.40)
The values P90I and η2 are the inflection point and reciprocal slope of the polarization
relation for which 90◦switches occur. As before, we describe a moment fraction relationship usingx+, x−, andx90to denote the respective number of dipoles oriented into the designated orientations. We combine the moment fractions of both±90◦ orientations intox
90to model
the polarization as one-dimensional. All moment fractions sum to one; i.e.,
P
σ
σ
(a)
(b)
P
−90
−180 +90 +180
(c)
Figure 2.2: (a) Position of the Titanate ion with positively oriented polarization. (b) Re-orientation of the Titanate ion into a negative 90◦ orientation due to applied compressive
stresses. (c) Polarization orientation mapping for ±180◦ and ±90◦ directions.
To incorporate energies due to strain, we employ the energy relation
ψes(P, ε) =−q1εP −q2εP2 (2.42)
and elastic energy is incorporated by
ψel(ε) = 12Y ε2. (2.43)
Here q1 and q2 are piezoelectric and electrostrictive coupling coefficients and Y is Young’s
modulus. The total Helmholtz free energy is then
ψ(P, ε) =ψP(P) +ψes(P, ε) +ψel(ε) (2.44)
and the corresponding Gibbs energy is
G(E, P, σ, ε) =ψ(P, ε)−EP−σε. (2.45)
In the absence of thermal relaxation, enforcing ∂G
∂E = 0 yields ε=Y−1¡σ+q1P+q2P2
¢
(2.46)
which may be substituted into the Gibbs energy to yield the relation
To determine a relation for the local polarization we must solve the nonlinear minimization problem
¯
P = arg min
P G(E, P, σ). (2.48)
Another method of determining ¯P is to analytically solve ∂G
∂P = 0. We may also use this
minimization to find coercive field values in terms of an applied stress given an inflection point. Further, we may write a coercive field value in the 90◦ direction as
Ec90(σ) =−Y−1¡σq1+P90 ¡
q21−η2Y + 2q2σ ¢
+ 3q1q2P902 + 2q22P903 ¢
. (2.49)
To incorporate thermal relaxation, we employ the Boltzmann relation
µ(G(E, P, σ)) =Ce−G(E,P,σ)V /kT. (2.50) The switching likelihoods can be approximated by
p−90= 1
τ(T)
e−G(E+EI;−PI)V /kT R−PI
−∞ e−G(E+EI;P)V /kTdP p+90= τ(1T) e
−G(E+EI;PI)V /kT R∞
PI e
−G(E+EI;P)V /kTdP (2.51) p90+= τ(1T) e
−G(E+EI;P90I) RP
90I −P90Ie
−G(E+EI;P)dP
p90−= 1
τ(T)
e−G(E+EI;P−90I) RP90I
−P90Ie
−G(E+EI;P)dP
where the factor 1/² from endpoint approximates to the integrals are incorporated into 1/τ(T). The expected polarizations are denoted as
hP+i=
R∞
PI P e
−G(E,P,σ)V /kTdP
R∞
PI e
−G(E,P,σ)V /kTdP
hP−i=
R−PI
−∞ P e−G(E,P,σ)V /kTdP
R−PI
−∞ e−G(E,P,σ)V /kTdP
(2.52)
hP90i=
RP90I
−P90IP e
−G(E,P,σ)V /kTdP
RP90I
−P90Ie
−G(E,P,σ)V /kTdP and the dipole fraction evolutions are then given by
˙
x90=p+90x++p−90x−−(p90++p90−)x90
˙
x−=−p−90x−+p90−x90 (2.53)
˙
Substitution of the relation (2.41) into (2.53) yields ˙
x−=−p−90x−+p90−(1−x+−x−)
=−(p−90+p90−)x−−p90−x++p90− (2.54)
˙
x+=−(p+90+p90+)x+−p90+x−+p90+. (2.55)
We can rewrite this as the system of ordinary differential equations
d dt
x− x+
=
−p90+ −(p−90+p90−)
−p90+ −(p+90+p90+)
x− x+
+
p90− p90+
(2.56)
which may be solved using the implicit Euler method. The local average polarization is given via the weighted sum of expected polarizations
¯
P =x+hP+i+x−hP−i+x90hP90i. (2.57)
Note that the limiting cases of the expected polarizations are
hP+i= E+EI
η +PR
hP−i= E+ηEI −PR (2.58)
hP90i= E+η EI
2 .
The macroscopic polarization model may once again be given modeled by
P(E) =
Z ∞
0 Z ∞
−∞
ν1(Ec)ν2(EI)
£¯
P(E+EI;Ec)
¤
(t)dEIdEc (2.59) where both the coercive field valueEcand the interaction field values are manifestations of underlying densities.
2.1.4 Temperature-Dependence
We begin by defining the temperature-dependent Helmholtz energy relation
ψ(T, M) =HhMs 2 " 1− µ M Ms
¶2#
(2.60)
+HhT 2Tc
" Mln
µ
M +Ms
Ms−M ¶
+Msln
Ã
1−
µ M Ms
¶2!#
(2.61) whereHh is a bias field and Msis the saturation magnetization. The temperature is given by T and the Curie temperature is denoted byTc. The Gibbs energy is
G(H, M) =ψ(H, M)−HM. (2.62)
The first derivative of (2.62) yields the relation
Hc(T) = MHh s
MI(T)−HThT c
arctanh(MI(T)/Ms) (2.63) for coercive field values. The second derivative of (2.62) is
∂2G ∂M2 =−
Hh
Ms +
HhT
MsTc
µ
1−
³
M Ms
´2¶. (2.64)
Equating (2.64) to zero yields the temperature-dependent inflection magnetization points
±MI(T) =±Ms
p
1−T /Tc. (2.65)
Substitution of (2.65) into (2.63) yields the relation
Hc(T) =Hh
p
1−T /Tc−HThT
c arctanh
p
1−T /Tc (2.66)
=Hh(1−T /Tc)3/2
·
1− T
3Tc +. . . ¸
. (2.67)
Recall the relation
(MR(T)−MI(T))η =Hc(T) (2.68) which implies
MR(T) = Hηh (1−T /Tc)3/2
·
1− T
3Tc +. . .
¸
+Ms
p
1−T /Tc. (2.69) To maintain a uniform formulation, we now write
¯
Hc(T) = ˜Hc(1−T /Tc)p1 (2.70) ¯
MR(T) = ˜MR(1−T /Tc)p2. (2.71)
2.2
Lumped Rod Model
In Section 2.1, we have in detail related how the processes of magnetization or polarization will be modeled for the research discussed in this dissertation. However, most applications of active materials utilize the magnetization or polarization of a material to create internal strains. These effects are known as magnetostriction and electrostriction. Consequently, by exploiting magneto/electrostrictive properties, materials can be used for actuation purposes.
Here, we characterize the behavior of the Terfenol-D rod as connected to the cutting mechanism of the milling device shown in Figure 1.1 by the simplified model depicted in Figure 2.3. We employ a simple one-dimensional lumped rod model formulation to relate magnetization to strains. However, similar discussions have been applied to other applications such as atomic force microscopy (as shown in [36]).
The Terfenol-D rod of length ` is coupled to some device which is modeled as a spring-mass-dashpot system with damping c`, stiffness k` and mass m` at the end of the Terfenol-D rod. The Terfenol-D rod has cross-sectional areaA and densityρ. Consider the functionu(t, x) which denotes the displacement of the rod at any pointx∈[0, `] at timet.
As detailed in [35], we can define the constitutive relation for stress in terms of strain and magnetization as
σ =Y ε+Cε˙−a1(M(H)−M0)−a2(M(H)−M0)2 (2.72)
where M is input generated by (2.15), Y is the Young’s modulus, a1 and a2 are coupling
coefficients, andC is the Kelvin-Voigt damping coefficient. Due to a fixed-end position we enforce the constraintu(t,0) = 0. We assume that strain is uniform along the length of the rod so that strains have the form
ε(t) = u`(t)
` (2.73)
k
u m
c l
x=0 x=l
l l
where u`(t) is rod tip position at time t and ` is the rod length. Balancing the forces of the rod σA (where A is cross-sectional area and ρ is density) with those of the restoring mechanism yields
ρA`u¨`(t) +CA` u˙`(t) +Y A` u`(t) (2.74) =−m`u¨`(t)−c`u˙`(t)−k`u`(t) +Aa1[M(H)−M0] +Aa2[M(H)−M0]2.
This can be reformulated as the vector system
d dt
u` ˙
u`
=
0 1
−c2 −c1
u` ˙
u`
+
0
1
a˜1[M(H)(t)−M0] + ˜a2[M(H)(t)−M0]2 (2.75)
where
c1 = CA
` +c`
ρA`+m` (2.76)
c2 =
Y A ` +k` ρA`+m`
(2.77) ˜
a1 = ρA`Aa+1m
`
(2.78) ˜
a2 = ρA`Aa+2m
`. (2.79)
Note thatm`, c`, and k` are material properties associated with the attachment of the rod end as shown in Figure 2.3. Initial conditions are then specified to be
u(0, `) =u`(0) (2.80)
ut(0, `) = ˙u`(0). (2.81)
Chapter 3
Gradient-Based Parameter
Estimation Algorithms
This chapter is devoted to gradient-based routines and their use in identifying parameters for the homogenized energy/lumped rod model pair. We begin with a discus-sion of several gradient-based constrained optimization routines. Past modeling techniques are presented with results. New modeling techniques are then revealed and comparisons between results using these new techniques and collected data are presented along with identified parameters.
3.1
Gradient-Based Routines
The gradient-based routines that are employed use a standard core of iterations based upon a Newton step. Due to their heritage, they have a common set of qualities:
1. For good initial estimates, convergence of approximately q-superlinear order is guar-anteed if gradient and Hessian approximates are sufficiently accurate.
2. Results are repeatable due to the deterministic nature of the routines.
3.1.1 Sequential Quadratic Programming
Sequential quadratic programming (SQP) addresses the nonlinear programming problem
min
q∈QJ(q) (3.1)
subject to the constraints
a(q) =0 (3.2)
b(q)≤0. (3.3)
Given the constraints (3.2) and (3.3), vectors of Lagrange multipliersuandvare introduced to form the augmented objective function
L(q,u,v) =J(q) +aT (q)u+bT (q)v (3.4) termed the Lagrangian. Appending slack variablesz to the constraint b(q)≤0 yields the new equality constraint
b(q) +z=0 (3.5)
which, when incorporated into the Lagrangian, generates the functional
L(q,u,v,z) =J(q) +aT(q)u+ (b(q) +z)T v (3.6) subject to (3.2) and (3.5). The second order Taylor series approximation for L(q,u,v,z) at iterationk is
L(q,u,v,z)≈L(qk,uk,vk,zk) +∇qL(qk,uk,vk,zk)Tsk+1 2
³
sk
´T
∇qqL(qk,uk,vk,zk)sk
where∇qL(q,u,v,z) denotes the gradient ofL(q,u,v,z) and ∇qqL(q,u,v,z) is the
Hes-sian ofL(q,u,v,z). Then the minimization problem
min
q∈QL(q,u,v,z) (3.7)
is approximated by min
sk
·
L(qk,uk,vk,zk) +∇TqL(qk,uk,vk,zk)sk+ 1 2
³
sk
´T
∇qqL(qk,uk,vk,zk)sk
¸
which is an iterated quadratic programming problem. The value L(qk,uk,vk,zk) is a constant and may be neglected to generate a simpler quadratic programming subproblem
min
sk
1 2
³
sk
´T
∇qqL(qk,uk,vk,zk)sk+∇qL(qk,uk,vk,zk)Tsk. (3.9)
A minimum occurs at sk when
∇qqL(qk,uk,vk,zk)sk+∇qL(qk,uk,vk,zk) = 0 (3.10)
implying that
sk=−∇−qq1L(qk,uk,vk,zk)∇qL(qk,uk,vk,zk) (3.11)
where the QR Factorization algorithm is used to compute the inverse of the Hessian matrix. This is clearly a step in the Newton iteration with respect toqas∇qqL(qk,uk,vk,zk) is a square matrix of full rank. The updateszfor the slack variables is a result of the solution
to the quadratic programming subproblem. The updates for the Lagrange multipliers are also generated sequentially from the QP subproblem.
The constraints in the QP subproblem can directly be derived from the first-order Taylor approximation:
1. ai(q) ≈ ai(qk) +¡∇
qai(qk)
¢T
sk
q for i = 1. . . NA (the length of a(q)). Then, if ai(q) = 0, it follows that
³
∇qai(qk)
´T
skq=−ai(qk) (3.12)
2. bj(q) +zj ≈³bj(qk) +¡zk¢j´+¡∇
qbj(qk)
¢T
sk
q+
¡
sk
z
¢j
forj= 1. . . Nb (the length of b(q)) which implies that
³
∇qbj(qk)
´T
skq+
³
skz
´j
=−
µ
bj
³
qk
´
+
³
zk
´j¶
(3.13)
3. z≈zk+skz≥0⇒skz≥ −zk.
The parameter estimate is then updated using the iteration
qk+1=qk+αksk (3.14)
3.1.2 Gradient and Hessian Approximations The gradient is defined as
∇qL(q,u,v,z) =
· ∂L ∂q1,
∂L ∂q2, . . . ,
∂L ∂qn
¸T
(3.15) and the Hessian matrix is defined as
[∇qqL(q,u,v,z)]ij =
·
∂2L(q,u,v,z) ∂qi∂qj
¸
(3.16)
∂L ∂qi ≈
L(q+eihkqk,u,v,z)−L(q,u,v,z)
hkqk (3.17)
whereek is thekthelementary vector. The Hessian matrix is often approximated with the first-order approximate credited to Broyden, Fletcher, Goldfarb, and Shanno. The BFGS Hessian approximation algorithm for an objective functional L(q,u,v,z) is:
1. H0=I.
2. Solve QP subproblem forsk
q.
3. Updateqk+1=qk+αksk
q.
4. yk= ∇qL(q
k+1,uk+1,vk+1,zk+1)− ∇
qL(qk,uk,vk,zk)
αk
5. Using the Sherman-Morrison formula, create the low-memory version of the approxi-mated Hessian inverse
Hk−+11 =Hk−1+
³ sk q ¡ sk q
¢T´ ³¡
sT
q
¢T
yk+yT
kHk−1yk
´ ³¡ sk q ¢T yk
´2 −H
−1
k yk
¡
sk
q
¢T
+¡sk
q
¢T
yT kHk−1
¡ sk q ¢T yk (3.18) 6. Iterate steps two through five until convergence as determined by stopping criteria
with the Newton Step has occurred.
