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Title of Thesis

Adaptive Estimation Using State Space Methods

Author(s)

Maj Muhammad Salman

Institute/University/Department Details
College of Electrical and Mechanical Engineering / National University of Sciences and Technology, Rawalpindi
Session
2009
Subject
Engineering Electrical
Number of Pages
129
Keywords (Extracted from title, table of contents and abstract of thesis)
Spectrum, Development, Leakage, Adaptive, Methods, State, Prediction, Space, Using, Mathematical, Estimation

Abstract
This thesis focuses on the development of new variants of adaptive filters. Built around state-space framework, the proposed filters are especially suitable for applications like tracking, output feedback control and recursive spectrum estimation. They operate without prior knowledge of process and observation noise statistics and exhibit good stability properties.The development in this thesis can broadly be classified into statespace least mean square (SSLMS) and finite memory least-squares filters.
SSLMS is a generalization of the well-known least mean square (LMS) filter. Incorporating linear time-varying state-space model of the underlying environment, SSLMS exhibits marked improvement in its tracking performance over the standard LMS. An extension of SSLMS is SSLMS with adaptive memory (SSLMSWAM).SSLMSWAM iteratively tunes the step-size parameter by stochastic gradient method in an attempt to yield its most appropriate value.This filter is useful for situations where a suitable value of step-size parameter is difficult to obtain beforehand.
Recursive nature of an adaptive filter brings with it stability issues. The concept of finite memory (or receding horizon) for an adaptive filter is appealing because it ensures stability.This motivates the development of finite memory filters, both for unforced and forced systems. Finite impulse response (FIR) adaptive filter, built around structure of an unforced system, uses weighted observations on a finite interval. Uniform weighting of the observations results in rectangular RLS (RRLS). Additional flexibility is achieved by developing an adaptive memory variant of FIR adaptive filter.Similar to SSLMSWAM, the data window size is iteratively tuned so as to minimize the prediction error. For the forced system case, a useful solution in the form of receding horizon state observer is obtained. It finds utility in output feedback control of linear time-varying systems. An insight into convergence properties of finite memory based filters is provided by the convergence analyses.
Spectrum update with the arrival of new data is a desirable feature in real-time spectrum estimation applications.The mathematical equivalence of RRLS resonator bank and recursive discrete Fourier transform (DFT) gives the rationale for using the newly developed filters for recursive spectrum estimation. A symmetric windowed variant of RRLS called ‘truncated exponential RLS (TERLS)’ is useful for reducing spectral leakage.Same is true for an SSLMS resonator, which has an attractive feature that spectral side levels and main lobe width may be reduced simultaneously by reducing the step-size parameter. The higher order resonator (HOR), constructed from several SSLMS resonators, exhibits close resemblance to an ideal (rectangular) frequency bin, thus minimizing spectral leakage and increasing resolution.

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4,578 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

viii
90 KB
2

1

INTRODUCTION


1.1 Estimation
1.2 Tracking
1.3 Feedback Control Systems
1.4 Spectrum Estimation
1.5 The Need for New Developments
1.6 Overview of the Thesis

1
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3 2 STATE-SPACE LEAST MEAN SQUARE


2.1 State-Space Model
2.2 State Estimator
2.3 Observer Gain
2.4 Steady-State Solution
2.5 Initialization
2.6 Analogy with the Standard LMS
2.7 Example (Sinusoid Tracking)

14
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4 3 STATE-SPACE LEAST MEAN SQUARE WITH ADAPTIVE MEMORY

3.1 Memory Length
3.2 Adaptation of Step-Size Parameter
3.3 Initialization
3.4 Example (Tracking Van der Pol Oscillations)
3.5 Computational Complexity

25
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5 4 FINITE IMPULSE RESPONSE ADAPTIVE FILTER

4.1 Batch Processed Solution
4.2 Recursive Solution
4.3 Alternate Forms of Estimator
4.4 Transfer Function Representation
4.5 Stability
4.6 Initialization
4.7 Equivalence of FIR Adaptive Filter and State-Space RLS
4.8 Convergence Analysis
4.9 Example (Tracking Van der Pol Oscillations)
4.10 Adaptive Memory Variant of FIR Adaptive Filter

38
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6 5 RECEDING HORIZON STATE OBSERVER FOR LINEAR TIMEVARYING SYSTEMS

5.1 Problem Statement
5.2 Batch Processed Least Squares State Estimate
5.3 Recursive Solution
5.4 Time Invariant Form
5.5 Stability
5.6 Initialization
5.7 Convergence Analysis
5.8 Example (Gyro Motion Control)

62
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7 6 NON-PARAMETRIC RECURSIVE LEAST SQUARES SPECTRUM ESTIMATION

6.1 RRLS Resonator
6.2 Equivalence of RRLS Resonator and DFT
6.3 Truncated Exponential RLS.
6.4 SSLMS and SSRLS Based Resonators
6.5 Higher Order Resonator (HOR)
6.6 Example (Spectrum Estimation of Non-Orthogonal Tone)

77
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8 7 CONCLUSIONS AND FUTURE SUGGESTIONS

7.1 Conclusions
7.2 Future Suggestions

101
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9 8 REFERENCES AND APPENDIX

105
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