MULTI-RESOLUTION METHODS FOR HIGH FIDELITY MODELING AND CONTROL ALLOCATION

325 Pages·2006·23.55 MB·English

MULTI-RESOLUTION METHODS FOR HIGH FIDELITY MODELING AND
CONTROL ALLOCATION IN LARGE-SCALE DYNAMICAL SYSTEMS
A Dissertation
by
PUNEET SINGLA
Submitted to the Office of Graduate Studies of
Texas A&M University
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
May 2006
Major Subject: Aerospace Engineering
MULTI-RESOLUTION METHODS FOR HIGH FIDELITY MODELING AND
CONTROL ALLOCATION IN LARGE-SCALE DYNAMICAL SYSTEMS
A Dissertation
by
PUNEET SINGLA
Submitted to the Office of Graduate Studies of
Texas A&M University
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
Approved by:
Chair of Committee, John L. Junkins
Committee Members, Srinivas R. Vadali
John E. Hurtado
Goong Chen
Kamesh Subbarao
Head of Department, Helen Reed
May 2006
Major Subject: Aerospace Engineering
iii
ABSTRACT
Multi-Resolution Methods for High Fidelity Modeling and Control Allocation in
Large-Scale Dynamical Systems. (May 2006)
Puneet Singla, B.Tech, Indian Institute of Technology, Kanpur, India;
M.S., Texas A&M University
Chair of Advisory Committee: Dr. John L. Junkins
This dissertation introduces novel methods for solving highly challenging model-
ing and control problems, motivated by advanced aerospace systems. Adaptable, ro-
bust and computationally efficient, multi-resolution approximation algorithms based
on Radial Basis Function Network and Global-Local Orthogonal Mapping approaches
are developed to address various problems associated with the design of large scale
dynamical systems. The main feature of the Radial Basis Function Network approach
is the unique direction dependent scaling and rotation of the radial basis function via
a novel Directed Connectivity Graph approach. The learning of shaping and rota-
tion parameters for the Radial Basis Functions led to a broadly useful approximation
approach that leads to global approximations capable of good local approximation
for many moderate dimensioned applications. However, even with these refinements,
many applications with many high frequency local input/output variations and a
high dimensional input space remain a challenge and motivate us to investigate an
entirely new approach. The Global-Local Orthogonal Mapping method is based upon
a novel averaging process that allows construction of a piecewise continuous global
family of local least-squares approximations, while retaining the freedom to vary in
a general way the resolution (e.g., degrees of freedom) of the local approximations.
These approximation methodologies are compatible with a wide variety of disciplines
such as continuous function approximation, dynamic system modeling, nonlinear sig-
iv
nal processing and time series prediction. Further, related methods are developed
for the modeling of dynamical systems nominally described by nonlinear differential
equations and to solve for static and dynamic response of Distributed Parameter Sys-
tems in an efficient manner. Finally, a hierarchical control allocation algorithm is
presented to solve the control allocation problem for highly over-actuated systems
that might arise with the development of embedded systems. The control allocation
algorithm makes use of the concept of distribution functions to keep in check the
“curse of dimensionality”. The studies in the dissertation focus on demonstrating,
through analysis, simulation, and design, the applicability and feasibility of these ap-
proximation algorithms to a variety of examples. The results from these studies are
of direct utility in addressing the “curse of dimensionality” and frequent redundancy
of neural network approximation.
v
To my parents for their love, encouragement and belief in me
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ACKNOWLEDGMENTS
Firstofall, Iwouldliketoexpressmyheartfeltthankstomyadvisor, Dr. JohnL.
Junkins for introducing me to the diverse field of modeling and control of dynamical
systems and for giving me complete freedom in selecting my research topic. His
invaluable suggestions and guidance have helped me greatly in achieving my goals
at Texas A&M and will continue to do so throughout my career. His commitment
and dedication to work have often been my source of inspiration. Working with him
has not only been a great learning experience but also an exhilarating one due to the
open and cordial atmosphere he provided in all our interactions and discussions.
I also thank my committee members, mentors and collaborators Dr. John Hur-
tado, Dr. Kamesh Subbarao, Dr. Srinivas R. Vadali, Dr. Goong Chen, Dr. Daniele
Mortari and Dr. John Valasek for many constructive conversations on various top-
ics directly/indirectly related to my dissertation work. Special thanks are due to Dr.
Subbarao for not only providing the constructive feedback that significantly increased
the quality and robustness of the results presented in this dissertation but also for
helping me out as a friend during my stay at Texas A&M. The opportunity to work
with all of you has been a very pleasant and inspiring experience. I am also grate-
ful to Lisa Willingham for her cooperation in conducting my final examination and
helping me with many administrative hurdles during my stay at Texas A&M. I am
indebted to Dr. Junkins for giving me this opportunity to work with a most inspiring,
ambitious and helpful group of people.
It is my pleasure to acknowledge the support of my friends, Praveen Bhojwani,
Sarbjyot Singh Dali, Hement Mahawar, Sreekanth Reddy, Gaurav Singhal, Rohit
Singhal, Arun Surendaran, Sesha Sai Vaddi, Veera and several others, who have
made my stay at College Station a memorable event in my life. I also remember the
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great company of my deceased friends Kishore Naik and his wife Garishma during my
initial stay at TAMU. They lived together a happy and compassionate life. Christian
Brucccoleri, Anup Katake, Waqar Malik, and Prasenjit Sengupta all had the misfor-
tune of getting involved in many ill-conditioned technical discussions with me over a
cup of coffee. My CHAAPUCTOP friends from IITK have been a good source of sup-
port and encouragement throughout my graduate and under-graduate studies. Their
close friendship and belief in my abilities provided me strength whenever I found my
confidence drooping. Thanks are also due to Troy Henderson and Mrinal Kumar for
helping me out in generating some nice figures for the dissertation.
I also offer my gratitude to my parents for providing me the best of opportunities
in school and college education. I have great admiration for my parents who have
sacrificed a lot to help me in achieving my goals. Coming from a rural area which had
no good schools, I needed to stay away from home at an early age. This dissertation
would have not been possible without their support, love and encouragement. I am
grateful to my elder sister Preeti, brother-in-law Anoop Goyal and my cousin brothers
Rajeev Singla and Sanjeev Singla for their support and encouragement and letting
me concentrate on my studies only, while I was away from my home for my education.
Finally, I acknowledge the support of Texas Institute for Intelligent Bio-Nano
Materials and Structures (TIIMS) for Aerospace Vehicles - a NASA University Re-
search, Engineering and Technology Institute (URETI) for funding the main part of
this research work. The technical liaison of Dr. Tom Gates is warmly acknowledged.
This research in part was also supported by Lockheed Martin and Harris under the
DefenseAdvancedProjectsAgencysInnovativeSpace-BasedRadarAntennaTechnol-
ogy (ISAT) program and by NASA Langley under the EO-3 Geostationary Imaging
Fourier Transform Spectrometer (GIFTS) mission contract. The collaboration of Mr.
Thomas Depkovich of Lockheed Martin for the ISAT program and Mr. Michael Jacox
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for the GIFTS program is highly appreciated.
Any opinions, findings, and conclusions or recommendations expressed in this
dissertation are those of the authors and do not necessarily reflect the views of the
funding agencies.
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TABLE OF CONTENTS
CHAPTER Page
I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II DIRECTIONDEPENDENTLEARNINGAPPROACHFOR
RADIAL BASIS FUNCTION NETWORKS . . . . . . . . . . . 9
A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 9
B. Direction Dependent Approach . . . . . . . . . . . . . . . 14
C. Directed Connectivity Graph . . . . . . . . . . . . . . . . . 20
1. Estimation Algorithm . . . . . . . . . . . . . . . . . . 24
2. Cayley Transformation . . . . . . . . . . . . . . . . . 25
3. Additive Decomposition of the “Covariance” Ma-
trix, R . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4. Cholesky Decomposition of “Covariance” Matrix, R . 29
D. Modified Minimal Resource Allocating Algorithm
(MMRAN) . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
E. Numerical Simulations and Results . . . . . . . . . . . . . 37
1. Test Example 1: Function Approximation . . . . . . 38
2. Test Example 2: 3 Input- 1 Output Continuous
Function Approximation . . . . . . . . . . . . . . . . . 47
3. Test Example 3: Dynamical System Identification . . 51
4. Test Example 4: Chaotic Time Series Prediction
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5. Test Example 5: Benchmark Against the On-line
Structural Adaptive Hybrid Learning (ONSAHL)
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 58
F. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 63
III GLOBAL LOCAL ORTHOGONAL POLYNOMIAL MAP-
PING (GLO-MAP) IN N-DIMENSIONS: APPLICATIONS
TO INPUT-OUTPUT APPROXIMATION . . . . . . . . . . . . 65
A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 65
B. Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . 67
C. Approximationin1-,2-andN-DimensionsUsingWeight-
ing Functions . . . . . . . . . . . . . . . . . . . . . . . . . 71
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CHAPTER Page
D. OrthogonalApproximationin1-, 2-andN-Dimensional
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
1. One Dimensional Case . . . . . . . . . . . . . . . . . . 85
2. Two Dimensional Case . . . . . . . . . . . . . . . . . 87
3. N- Dimensional Case . . . . . . . . . . . . . . . . . . 92
E. Algorithm Implementation . . . . . . . . . . . . . . . . . . 94
1. Sequential Version of the GLO-MAP Algorithm . . . . 96
F. Illustrative Engineering Applications . . . . . . . . . . . . 100
1. Function Approximation . . . . . . . . . . . . . . . . . 100
2. Synthetic Jet Actuator Modeling . . . . . . . . . . . . 106
a. Experimental Set up . . . . . . . . . . . . . . . . 107
3. Space Based Radar (SBR) Antenna Shape
Approximation . . . . . . . . . . . . . . . . . . . . . . 112
4. PorkchopPlotsApproximationforMissiontoNear
Earth Objects (NEOs) . . . . . . . . . . . . . . . . . . 117
G. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 125
IV MULTI-RESOLUTION ALGORITHM . . . . . . . . . . . . . . 126
A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B. Multi-Resolution Learning Algorithm . . . . . . . . . . . . 127
C. Choice of Basis Function and Approximation Error . . . . 131
1. Probabilistic Analysis of the GLO-MAP Algorithm . . 139
D. Adaptive Multi-Resolution Algorithm . . . . . . . . . . . . 146
E. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 148
1. Calibration of Vision Sensors . . . . . . . . . . . . . . 148
2. Simulation and Results . . . . . . . . . . . . . . . . . 150
3. DCG Approximation Result . . . . . . . . . . . . . . . 153
4. Local Approximation Results . . . . . . . . . . . . . . 153
F. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 156
V ROBUST NONLINEAR SYSTEM IDENTIFICATION AL-
GORITHMS USING AN ORTHOGONAL POLYNOMIAL
NETWORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B. Problem Statement and Background . . . . . . . . . . . . 159
C. Novel System Identification Algorithm . . . . . . . . . . . 163
1. Linear System Identification . . . . . . . . . . . . . . 166
2. State Variable Estimation . . . . . . . . . . . . . . . . 171