Improved Filtering And Estimation Methods For Aerospace Problems

210 Pages·2015·3.54 MB·English

Improved Filtering And Estimation
Methods For Aerospace Problems

This thesis is submitted to the Jadavpur University in partial
fulfillment of the requirements for the degree of
Doctor of Philosophy

By
SHOVAN BHAUMIK

Department of Electrical Engineering
Jadavpur University
Kolkata, India
2008
Names, Designations and Affiliations of
Supervisors

1. Dr. SMITA SADHU
Reader
Department of Electrical Engineering
Jadavpur University
Kolkata-700 032
India

2. Dr. TAPAN KUMAR GHOSHAL
Professor
Department of Electrical Engineering
Jadavpur University
Kolkata-700 032
India
Certificate from Supervisors

This is to certify that the candidate Shri SHOVAN BHAUMIK was registered on
…………………., and fulfilled the residence and other requirements for submitting the
thesis for the Ph.D. (Engg) degree of this University as per rules.
The thesis is a genuine piece of research carried out by the candidate under our
supervision and neither his thesis nor any part of part of the thesis has been submitted for
any degree / diploma or any other academic award anywhere before. In our opinion this is
a fit piece of work for submission for the Ph.D. degree.

1. -------------------------------------------- 2.---------------------------------------

Signature of the supervisor and Signature of the supervisor and
date with Office Seal date with Office Seal
ABSTRACT

This dissertation deals with nonlinear filtering and estimation methods for systems with
parametric uncertainties. The work focuses primarily on risk-sensitive estimators, which are
known to be robust in the presence of uncertainties in the system parameters. Risk sensitive filters
(RSF), which are based on risk-sensitive control law, minimize the expected value of an
exponential of a convex function of the estimation error. A designer-chosen parameter, called the
risk-sensitive parameter, provides a tool for design trade-off between the filtering performance
for the nominal model and the robustness to model uncertainty. Though the work was motivated
by the need to evolve improved methods of filtering for aerospace systems, the scope of the thesis
has been broadened to include non-aerospace problems as well. Thus the dissertation deals with
aerospace problems and other nonlinear problems with uncertain system model, un-modeled
biases, unknown inputs and non-nominal noise covariance.
Though existing literature on risk-sensitive filters claims that these filters are robust, the
implementation of these filters for nonlinear problems had been practically impossible as these
filters admit closed form expressions only for a very limited class of models including finite state-
space Markov chains and linear Gaussian models. Until very recently, the only method of
implementation of risk-sensitive filters for nonlinear problems was by using the extended Risk
sensitive filter (ERSF), which uses an EKF-based linearization method. However, all the
shortcomings of the EKF are inherited by the ERSF.
In this dissertation, some novel filters have been proposed, which significantly extend the
range of applications of risk-sensitive techniques to nonlinear, non-Gaussian systems. The
proposed filters would enable one to study the properties of risk-sensitive filters and to explore
the possibility of application of RSFs for a wide variety of problems including aerospace
problems like target tracking and navigation. Some simple test problems have been selected for
characterizing the existing filters and testing the applicability of developed filters. Though it
cannot be claimed that RSFs have been found to be superior to other existing (risk-neutral)
nonlinear filters, this dissertation makes an effort to summarize the results of comparison of the
various filters for each chosen application area.
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The salient contributions of this research work are as follows:
• Development of Risk sensitive unscented Kalman filter (RSUKF), which is a novel
method for non-linear risk-sensitive estimation based on the unscented Kalman filter.
(Paper accepted for publication in IET Control Theory and
Applications, UK, 2008.)
• Development of Central difference risk-sensitive filter (CDRSF), which is a novel
numerically efficient algorithm for risk-sensitive filters of nonlinear plants, using
central difference approximation.
(Paper published in IEEE Signal Processing Letters, 2007.)
• Development of Risk sensitive particle filter (RSPF), which is a novel particle
implementation of Risk-sensitive filters (RSF) for nonlinear, non-Gaussian state-
space models based on a probabilistic re-interpretation of the RSF recursions.
(Paper accepted for publication in Elsevier Journal of Signal
Processing and available online since 1st October, 2008. One IEEE
INDICON conference papers published in 2006 have been cited and
critically appreciated by international researchers.)
• Development of Adaptive grid risk sensitive filter (AGRSF), which is a novel
adaptive grid based method employing the point-mass approximation for
computation of risk-sensitive state estimates in non-linear non-Gaussian problems.
(Work presented and published in the proceedings of the IEEE
Nonlinear Statistical Signal Processing Workshop, Cambridge, UK, 2006
and IEEE INDICON, 2005.)
• Formulation of an approach for preliminary design of guidance system for homing
missiles by using the Cramer Rao Bound (CRB) to provide a quantitative
understanding of the influence of model parameters and instrumentation/signal
processing capabilities of the tracking filter performance, without going into the
specifics of filter design and elaborate Monte Carlo simulation for performance
analysis.
(Work presented and published in the proceedings of the IEEE
International Symposium on Signal Processing and Information
Technology (ISSPIT), Rome, Italy, 2004, paper published in the
Journal of the Institution of Engineers, India, 2006.)
• Comparative performance analysis of existing and proposed filters for the Bearing
only tracking problem with different measurement models.
• Application of the filters to a self-alignment problem for Inertial Navigation System.
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ACKNOWLEDGEMENTS

My supervisors Dr. Smita Sadhu and Professor Tapan Kumar Ghoshal deserve my deepest
gratitude. This thesis is, to a large extent, a result of their skillful meticulous guidance and
encouraging support to my work. I would also like to acknowledge them for their valuable
comments suggestions during work and review of manuscript at different stages that have lifted
the thesis to a level I never would have reached on my own.
I would like to thank my co-worker M Srinivasan for research cooperation, comparing simulation
results and fruitful discussions.
The work has been partly sponsored by Aeronautical Research and Development Board (India)
and Council of Scientific & Industrial Research (CSIR). I thankfully acknowledge their support.
I have great pleasure in expressing my thanks to, Prof. Shyamal Kumar Goswami, Prof. Kalyan
Kumar Dutta and Prof Samar Bhattacharya, Jadavpur University, for their encouragement. I also
wish to thank the successive Heads of Electrical Engineering and the Coordinators of Center for
Knowledge Based System for providing research infrastructure.
Lastly I would like to express my heart-felt gratitude to my parents, family members and friends
for their love, affection, support and encouragement.

Shovan Bhaumik
Jadavpur University, October 2008

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LIST OF PUBLICATION

International Journal (Published/Communicated)
1. Shovan Bhaumik, Smita Sadhu, T.K.Ghoshal “Risk Sensitive Formulation of Unscented
Kalman Filter”, accepted for publication to IET Control Theory & Applications, 2008.
2. Smita Sadhu, M. Srinivasan, S. Bhaumik, T. K. Ghoshal, Central Difference Formulation
of Risk-Sensitive Filter, IEEE Signal Processing Letters, Vol 14, issue 6, pp 421-424,
June 2007.
3. Smita Sadhu, Shovan Bhaumik, Arnaud Doucet, T.K.Ghoshal “Risk Sensitive Filtering
by Particle Methods” accepted for publication in Elsevier Journal of Signal Processing,
2008.
4. S.Bhaumik, S.Sadhu, M.Srinivasan, T.K.Ghoshal, “Adaptive grid method for nonlinear
risk sensitive estimation problems”, submitted to International journal of modelling and
simulation.

National Journal (Published)
5. S.Bhaumik, S.Sadhu and T.K. Ghoshal “Parametric Performance Analysis of Tracking
System Using Posterior Cramer Rao Lower Bounds” Journal of Institution of Engineers
(India), electrical engineering division, issue 3, vol 47, December 2006.

Papers Presented/Published in International Conferences
6. S.Bhaumik, M. Srinivasan, S. Sadhu, T.K.ghoshal “A Risk Sensitive Estimator for
Nonlinear Problems using the Adaptive Grid Method” in Proceedings of Nonlinear
Statistical Signal Processing Workshop- University of Cambridge, UK 2006.
7. S. Sadhu, S. Bhaumik, and Prof. T. K. Ghoshal “Homing Guidance Configuration using
Cramer Rao Bounds”- in Proceedings of 4th IEEE international symposium on signal
processing and information technology, Rome, Italy, pp 437-440, Dec 2004.
8. M.Srinivasan, S.Bhuamik, S.Sadhu, T.K.Ghoshal, “Exo-Atmospheric IR Tracking of
Ballistic Objects”, in Proceedings of International Conference of IEEE INDICON 2005,
IIT Madras, Chennai, India, Dec 12-13, 2005
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9. S.Bhaumik, S. Sadhu, T.K.Ghoshal “Risk Sensitive Estimators for Inaccurately Modelled
Systems”, in Proceedings of International Conference of IEEE INDICON’2005, IIT
Madras, Chennai, India, Dec 12-13, 2005.
10. S.Bhuamik, M.Srinivasan, S.Sadhu, T.K.Ghoshal, “Adaptive Grid Solution of Risk
Sensitive Estimator Problems”, in Proceedings of International Conference of IEEE
INDICON 2005, IIT Madras, Chennai, India, Dec 12-13, 2005.
11. S.Bhaumik, S.Sadhu, and T.K.Ghoshal, “Alternative Formulation for Risk Sensitive
Particle Filter (Posterior)”, in Proceedings of International Conference of IEEE
INDICON’2006, New Delhi, India, Sep 15-17, 2006.

Papers Presented/Published in National Conferences
12. S. Bhaumik, S. Sadhu “Auxiliary Particle filter for Bearing Only Tracking” in
proceedings of 28th national systems Conference, Vellore institute of technology, pp
263-267, Dec 2004.
13. S.Bhaumik, S. Sadhu and T.K.ghoshal “Formulation of LMI Based Risk Sensitive
Estimator”, in proceedings of International Conference on “Emerging Trends in
Electrical Engineering” Kolkata, January 12-14, 2007.

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CONTENTS
Abstract i
Acknowledgement iii
List of Publication iv
Content vi
List of Figures xi
List of Table xiv

Chapter 1 Introduction
1.1Motivation 1
1.2Thesis Objective 3
1.3Background on aerospace Problem Domain 3
1.3.1 Target Tracking 3
1.3.2 Navigation and Its Use 5
1.4Background in Solution Domain 6
1.4.1 Background on Filters and State Estimators 7
1.5Problem Statement and Scope 8
1.6The Approach and Methodology 9
1.7Contribution 10
1.7.1 Publication Generated from this Work 11
1.7.2 Credits to Co-Workers 11
1.8Organization and Content of the Thesis 12

Chapter 2 Literature Survey
2.1Introduction to Literature Survey 13
2.2Problem Domain Literature Survey 13
2.2.1 Tracking of Maneuvering Target 13
2.2.1.1 Bearing Only Tracking 14
2.2.1.2 Ballistic Object Tracking 16
2.2.2 Navigation 19
2.2.2.1 Estimation Problems in Inertial Navigation 21
2.2.2.2 Literature on Error Models 22
2.2.2.3 Alignment Problem 23
2.2.2.3.1 Self Alignment: Coarse 24
2.2.2.3.2 Self Alignment: Fine 25
2.2.2.3.3 Transfer and Aided Alignment 27
2.2.2.3.4 In Flight Alignment 28
2.2.2.3.5 GPS Based Alignment 29
2.2.2.4 GPS INS Estimation and Data Fusion 29
2.2.2.4.1 GPS INS Estimation Using Kalman Filter 30
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2.2.2.4.2 GPS INS Estimation Using Post Kalman Filter 33
2.3Literature Survey Solution Domain 34
2.3.1 Linear Regression Kalman Filter (LRKF) Techniques 35
2.3.1.1 Unscented Kalman Filter (UKF) 35
2.3.1.2 Central Difference Filter (CDF) and Divided Difference 36
Filter (DDF)
2.3.2 Pseudo Measurement Filter 37
2.3.3 Approximate Grid Based Filter 38
2.3.3.1 Adaptive Grid Filter 38
2.3.4 Particle Filter 39
2.3.5 Robust Filtering 41
2.3.6 Robust H2 and H∞ Filter 41
2.3.6.1 Riccati Equation Based Approach 41
2.3.6.2 LMI Based Approach 43
2.3.7 Risk Sensitive (RS) Filtering 44
2.3.7.1 Nonlinear Risk sensitive Filter 46
2.3.8 Cramer Rao Lower Bound (CRLB) 47
2.4Discussion and Conclusion 48

Chapter 3 Test Problems
3.1 Introduction 49
3.2First Order Linear System (L1) 49
3.3Second Order Linear (L2) 49
3.4First Order Nonlinear Problem (N1-1) 50
3.5Another First Order Nonlinear System (N1-2) 51
3.62nd Order Bearing Only Tracking (BOT) Problem (non-linear) 52
3.6.1 BOT Problem Formulation 53
3.6.2 Measurement Model I 54
3.6.3 Measurement Model II 56
3.6.4 Measurement Model III 57
3.6.5 Filter Initialisation 58

Chapter 4 Risk Sensitive Estimator
4.1 Introduction 59
4.2 Formulation of Risk Sensitive Estimation Problem 60
4.3 Recursive Solution of Risk Sensitive Estimation 61
4.3.1 Posterior Solution 61
4.3.2 Prior Solution 62
4.4 Risk Sensitive Estimation For Linear Gaussian Problem 63
4.4.1 Posterior Risk Sensitive Kalman Filter 63
4.4.2 Prior Risk Sensitive Kalman Filter 64
4.5 LMI Based Formulation of RSKF 65
4.5.1 LMI Based Kalman Filter Formulation 65
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