estimation-based adaptive filtering and control - CiteSeerX

ESTIMATION-BASED ADAPTIVE FILTERING AND CONTROL

a dissertation submitted to the department of electrical engineering and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy

Bijan Sayyar-Rodsari July 1999

c Copyright by Bijan Sayyar-Rodsari 1999

All Rights Reserved

ii

I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Professor Jonathan How (Principal Adviser)


Professor Thomas Kailath


Dr. Babak Hassibi


Professor Carlo Tomasi

Approved for the University Committee on Graduate Studies:

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Abstract Adaptive systems have been used in a wide range of applications for almost four decades. Examples include adaptive equalization, adaptive noise-cancellation, adaptive vibration isolation, adaptive system identification, and adaptive beam-forming. It is generally known that the design of an adaptive filter (controller) is a difficult nonlinear problem for which good systematic synthesis procedures are still lacking. Most existing design methods (e.g. FxLMS, Normalized-FxLMS, and FuLMS) are adhoc in nature and do not provide a guaranteed performance level. Systematic analysis of the existing adaptive algorithms is also found to be difficult. In most cases, addressing even the fundamental question of stability requires simplifying assumptions (such as slow adaptation, or the negligible contribution of the nonlinear/time-varying components of signals) which at the very least limit the scope of the analysis to the particular problem at hand. This thesis presents a new estimation-based synthesis and analysis procedure for adaptive “Filtered” LMS problems. This new approach formulates the adaptive filtering (control) problem as an H∞ estimation problem, and updates the adaptive weight vector according to the state estimates provided by an H∞ estimator. This estimator is proved to be always feasible. Furthermore, the special structure of the problem is used to reduce the usual Riccati recursion for state estimate update to a simpler Lyapunov recursion. The new adaptive algorithm (referred to as estimation-based adaptive filtering (EBAF) algorithm) has provable performance, follows a simple update rule, and unlike previous methods readily extends to multi-channel systems and problems with feedback contamination. A clear connection between the limiting behavior of the EBAF algorithm and the classical FxLMS (Normalized-FxLMS) iv

algorithm is also established in this thesis. Applications of the proposed adaptive design method are demonstrated in an Active Noise Cancellation (ANC) context. First, experimental results are presented for narrow-band and broad-band noise cancellation in a one-dimensional acoustic duct. In comparison to other conventional adaptive noise-cancellation methods (FxLMS in the FIR case and FuLMS in the IIR case), the proposed method shows much faster convergence and improved steady-state performance. Moreover, the proposed method is shown to be robust to feedback contamination while conventional methods can go unstable. As a second application, the proposed adaptive method was used for vibration isolation in a 3-input/3-output Vibration Isolation Platform. Simulation results demonstrate improved performance over a multi-channel implementation of the FxLMS algorithm. These results indicate that the approach works well in practice. Furthermore, the theoretical results in this thesis are quite general and can be applied to many other applications including adaptive equalization and adaptive identification.

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Acknowledgements This thesis has greatly benefited from the efforts and support of many people whom I would like to thank. First, I would like to thank my principle advisor Professor Jonathan How. This research would not have been possible without Professor How’s insights, enthusiasm and constant support throughout the project. I appreciate his attention to detail and the clarity that he brought to our presentations and writings. I would also like to acknowledge the help and support of Dr. Alain Carrier from Lockheed Martin’s Advanced Technology Center. His careful reading of all the manuscripts and reports, his provocative questions, and his dedication to meaningful research has greatly influenced this work. I would like to gratefully acknowledge members of my defense and reading committee, Professor Thomas Kailath, Professor Carlo Tomasi, and Dr. Babak Hassibi. It was from a class instructed by Professor Kailath and Dr. Hassibi that the main concept of this thesis originated, and it was their research that this thesis is based on. It is impossible to exaggerate the importance of Dr. Hassibi’s contributions to this thesis. He has been a great friend and advisor throughout this work for which I am truly thankful. My thanks also goes to Professor Robert Cannon and Professor Steve Rock for giving me the opportunity to interact with wonderful friends in the Aerospace Robotics Laboratory. The help from ARL graduates, Gordon Hunt, Steve Ims, Stef Sonck, Howard Wang, and Kurt Zimmerman was crucial in the early stages of the research at Lockheed. I have also benefited from interesting discussions with fellow ARL students Andreas Huster, Kortney Leabourne, Andrew Robertson, Heidi Schubert, and Bruce Woodley, on both technical and non-technical issues. I am forever thankful for their invaluable friendship and support. I also acknowledge the camaraderie of more vi

recent ARL members, Tobe Corazzini, Steve Fleischer, Eric Frew, Gokhan Inalhan, Hank Jones, Bob Kindel, Ed LeMaster, Mel Ni, Eric Prigge, and Luis Rodrigues. I discussed all aspects of this thesis in great detail with Arash Hassibi. He helped me more than I can thank him for. Lin Xiao and Hong S. Bae set up the hardware for noise cancellation and helped me in all experiments. I appreciate all their assistance. Thomas Pare, Haitham Hindi, and Miguel Lobo provided helpful comments about the research. I also acknowledge the assistance from fellow ISL students, Alper Erdogan, Maryam Fazel, and Ardavan Maleki. I would like to also name two old friends, Khalil Ahmadpour and Mehdi Asheghi, whose friendship I gratefully value. I owe an immeasurable amount of gratitude to my parents, Hossein and Salehe, my sister, Mojgan, and my brother, Bahman, for their support throughout the numerous ups and downs that I have experienced. Finally, my sincere thanks goes to my wife, Samaneh, for her gracious patience and strength. I am sure they agree with me in dedicating this thesis to Khalil.

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Contents Abstract

iv

Acknowledgements

vi

List of Figures

xii

1 Introduction

1

1.1

Motivation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

An Overview of Adaptive Filtering (Control) Algorithms . . . . . . .

6

1.4

Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.5

Thesis Outline

12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Estimation-Based adaptive FIR Filter Design

14

2.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2

EBAF Algorithm - Main Concept

. . . . . . . . . . . . . . . . . . .

16

2.3

Problem Formulation

. . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.4

2.3.1

H2 Optimal Estimation

. . . . . . . . . . . . . . . . . . . . .

19

2.3.2

H∞ Optimal Estimation . . . . . . . . . . . . . . . . . . . . .

20

H∞ -Optimal Solution

. . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.4.1

γ-Suboptimal Finite Horizon Filtering Solution . . . . . . . .

21

2.4.2

γ-Suboptimal Finite Horizon Prediction Solution . . . . . . .

22

2.4.3

The Optimal Value of γ . . . . . . . . . . . . . . . . . . . . .

23

2.4.3.1

23

Filtering Case

. . . . . . . . . . . . . . . . . . . . . viii

2.4.3.2 2.4.4

Prediction Case . . . . . . . . . . . . . . . . . . . .

27

Simplified Solution Due to γ = 1 . . . . . . . . . . . . . . . .

29

2.4.4.1

Filtering Case: . . . . . . . . . . . . . . . . . . . . .

29

2.4.4.2

Prediction Case: . . . . . . . . . . . . . . . . . . . .

30

2.5

Important Remarks

. . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.6

Implementation Scheme for EBAF Algorithm . . . . . . . . . . . . .

32

2.7

Error Analysis

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.7.1

Effect of Initial Condition . . . . . . . . . . . . . . . . . . . .

35

2.7.2

Effect of Practical Limitation in Setting y(k) to sˆ(k|k) (ˆ s(k))

36

2.8

Relationship to the Normalized-FxLMS/FxLMS Algorithms . . . . . 2.8.1

Prediction Solution and its Connection to the FxLMS Algorithm

2.8.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Filtering Solution and its Connection to the Normalized-FxLMS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.9

38

Experimental Data & Simulation Results

40

. . . . . . . . . . . . . . .

41

2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3 Estimation-Based adaptive IIR Filter Design

58

3.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.2

Problem Formulation

. . . . . . . . . . . . . . . . . . . . . . . . . .

61

Estimation Problem . . . . . . . . . . . . . . . . . . . . . . .

63

Approximate Solution . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.2.1 3.3

3.3.1

γ-Suboptimal Finite Horizon Filtering Solution to the Linearized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.2

γ-Suboptimal Finite Horizon Prediction Solution to the Linearized Problem

3.3.3

66

. . . . . . . . . . . . . . . . . . . . . . . . .

Important Remarks

66

. . . . . . . . . . . . . . . . . . . . . . .

66

3.4

Implementation Scheme for the EBAF Algorithm in IIR Case . . . .

67

3.5

Error Analysis

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

3.6

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

3.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

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4 Multi-Channel Estimation-Based Adaptive Filtering 4.1 4.2

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.1.1

79

4.4

Multi-Channel FxLMS Algorithm

. . . . . . . . . . . . . . .

Estimation-Based Adaptive Algorithm for Multi Channel Case

. . .

81

. . . . . . . . . . . . . . . . . . . . . .

85

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.3.1

Active Vibration Isolation . . . . . . . . . . . . . . . . . . . .

86

4.3.2

Active Noise Cancellation . . . . . . . . . . . . . . . . . . . .

89

4.2.1 4.3

78


Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Adaptive Filtering via Linear Matrix Inequalities

104

5.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2

LMI Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2.1

Including H2 Constraints

. . . . . . . . . . . . . . . . . . . . 110

5.3

Adaptation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6 Conclusion

121

6.1

Summary of the Results and Conclusions

6.2

Future Work

. . . . . . . . . . . . . . . 121

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A Algebraic Proof of Feasibility A.1 Feasibility of γf = 1

126

. . . . . . . . . . . . . . . . . . . . . . . . . . . 126

B Feedback Contamination Problem

128

C System Identification for Vibration Isolation Platform

132

C.1 Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

C.2 Identified Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

C.2.1 Data Collection Process . . . . . . . . . . . . . . . . . . . . . 133 C.2.2 Consistency of the Measurements . . . . . . . . . . . . . . . . 134 C.2.3 System Identification

. . . . . . . . . . . . . . . . . . . . . . 137 x

C.2.4 Control design model analysis . . . . . . . . . . . . . . . . . . 140 C.3 FORSE algorithm

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Bibliography

155

xi

List of Figures 1.1

General block diagram for an FIR Filterm . . . . . . . . . . . . . . . . . .

13

1.2

General block diagram for an IIR Filter

. . . . . . . . . . . . . . . . . . .

13

2.1

General block diagram for an Active Noise Cancellation (ANC) problem . . . .

46

2.2

A standard implementation of FxLMS algorithm . . . . . . . . . . . . . . .

47

2.3

Pictorial representation of the estimation interpretation of the adaptive control problem: Primary path is replaced by its approximate model

. . . . . . . . .

47

2.4

Block diagram for the approximate model of the primary path

. . . . . . . .

48

2.5

Schematic diagram of one-dimensional air duct . . . . . . . . . . . . . . . .

48

2.6

Transfer functions plot from Speakers #1 & #2 to Microphone #1

. . . . . .

49

2.7

Transfer functions plot from Speakers #1 & #2 to Microphone #2

. . . . . .

49

2.8

Validation of simulation results against experimental data for the noise cancellation problem with a single-tone primary disturbance at 150 Hz. The primary disturbance is known to the adaptive algorithm. The controller is turned on at t ≈ 3 seconds.

2.9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

Experimental data for the EBAF algorithm of length 4, when a noisy measurement of the primary disturbance (a single-tone at 150 Hz) is available to the adaptive algorithm (SNR=3). The controller is turned on at t ≈ 5 seconds.

. . . . . .

51

2.10 Experimental data for the EBAF algorithm of length 8, when a noisy measurement of the primary disturbance (a multi-tone at 150 and 180 Hz) is available to the adaptive algorithm (SNR=4.5). The controller is turned on at t ≈ 6 seconds.

.

52

2.11 Experimental data for the EBAF algorithm of length 16, when a noisy measurement of the primary disturbance (a band limited white noise) is available to the adaptive algorithm (SNR=4.5). The controller is turned on at t ≈ 5 seconds.

xii

.

53

2.12 Simulation results for the performance comparison of the EBAF and (N)FxLMS algorithms. For 0 ≤ t ≤ 5 seconds, the controller is off. For 5 < t ≤ 20 seconds both adaptive algorithms have full access to the primary disturbance (a singletone at 150 Hz). For t ≥ 20 seconds the measurement of Microphone #1 is used as the reference signal (hence feedback contamination problem). The length of the FIR filter is 24.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

2.13 Simulation results for the performance comparison of the EBAF and (N)FxLMS algorithms. For 0 ≤ t ≤ 5 seconds, the controller is off. For 5 < t ≤ 40 seconds both adaptive algorithms have full access to the primary disturbance (a band limited white noise). For t ≥ 40 seconds the measurement of Microphone #1 is used as the reference signal (hence feedback contamination problem). The length of the FIR filter is 32.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

55

2.14 Closed-loop transfer function based on the steady state performance of the EBAF and (N)FxLMS algorithms in the noise cancellation problem of Figure 2.13.

3.1

. .

56

General block diagram for the adaptive filtering problem of interest (with Feedback

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

3.2

Basic Block Diagram for the Feedback Neutralization Scheme . . . . . . . . .

72

3.3

Basic Block Diagram for the Classical Adaptive IIR Filter Design . . . . . . .

73

3.4

Estimation Interpretation of the IIR Adaptive Filter Design

. . . . . . . . .

73

3.5

Approximate Model For the Unknown Primary Path . . . . . . . . . . . . .

74

3.6

Performance Comparison for EBAF and FuLMS Adaptive IIR Filters for Single-

Contamination)

Tone Noise Cancellation. The controller is switched on at t = 1 second. For 1 ≤ t ≤ 6 seconds adaptive algorithm has full access to the primary disturbance. For t ≥ 6 the output of Microphone #1 is used as the reference signal (hence feedback contamination problem). . . . . . . . . . . . . . . . . . . . . . .

3.7

75

Performance Comparison for EBAF and FuLMS Adaptive IIR Filters for MultiTone Noise Cancellation. The controller is switched on at t = 1 second. For 1 ≤ t ≤ 6 seconds adaptive algorithm has full access to the primary disturbance. For t ≥ 6 the output of Microphone #1 is used as the reference signal (hence feedback contamination problem). . . . . . . . . . . . . . . . . . . . . . .

xiii

76

4.1

General block diagram for a multi-channel Active Noise Cancellation (ANC) problem

4.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Pictorial representation of the estimation interpretation of the adaptive control

. . . . . . . . .

91

4.3

Approximate Model for Primary Path . . . . . . . . . . . . . . . . . . . .

92

4.4

Vibration Isolation Platform (VIP)

. . . . . . . . . . . . . . . . . . . . .

92

4.5

A detailed drawing of the main components in the Vibration Isolation Platform

problem: Primary path is replaced by its approximate model

(VIP). Of particular importance are: (a) the platform supporting the middle mass (labeled as component #5), (b) the middle mass that houses all six actuators (of which only two, one control actuator and one disturbance actuator) are shown (labeled as component #11), and (c) the suspension springs to counter the gravity (labeled as component #12). Note that the actuation point for the control actuator (located on the left of the middle mass) is colocated with the load cell (marked as LC1). The disturbance actuator (located on the right of the middle

. . . . . . . . . . . . . . . . . .

93

4.6

SVD of the MIMO transfer function . . . . . . . . . . . . . . . . . . . . .

94

4.7

Performance of a multi-channel implementation of EBAF algorithm when distur-

mass) actuates against the inertial frame.

bance actuators are driven by out of phase sinusoids at 4 Hz. The reference signal available to the adaptive algorithm is contaminated with band limited white noise (SNR=3). The control signal is applied for t ≥ 30 seconds.

4.8

95

Performance of a multi-channel implementation of FxLMS algorithm when simulation scenario is identical to that in Figure 4.7.

4.9

. . . . . . . . . .

. . . . . . . . . . . . . . .

96

Performance of a multi-channel implementation of EBAF algorithm when disturbance actuators are driven by out of phase multi-tone sinusoids at 4 and 15 Hz. The reference signal available to the adaptive algorithm is contaminated with band limited white noise (SNR=4.5). The control signal is applied for t ≥ 30 seconds.

97

4.10 Performance of a multi-channel implementation of FxLMS algorithm when simulation scenario is identical to that in Figure 4.9.

. . . . . . . . . . . . . . .

98

4.11 Performance of a Multi-Channel implementation of the EBAF for vibration isolation when the reference signals are load cell outputs (i.e. feedback contamination exists). The control signal is applied for t ≥ 30 seconds.

xiv

. . . . . . . . . . .

99

4.12 Performance of the Multi-Channel noise cancellation in acoustic duct for a multitone primary disturbance at 150 and 200 Hz. The control signal is applied for t ≥ 2 seconds.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.13 Performance of the Multi-Channel noise cancellation in acoustic duct when the primary disturbance is a band limited white noise. The control signal is applied for t ≥ 2 seconds.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.14 Closed-loop vs. open-loop transfer functions for the steady state performance of the EBAF algorithm for the simulation scenario shown in Figure 4.13.

. . . . 102

5.1

General block diagram for an Active Noise Cancellation (ANC) problem . . . .

5.2

Cancellation Error at Microphone #1 for a Single-Tone Primary Disturbance

5.3

Typical Elements of Adaptive Filter Weight Vector for Noise Cancellation Problem in Fig. 5.2

. 116

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.4

Cancellation Error at Microphone #1 for a Multi-Tone Primary Disturbance

5.5

Typical Elements of Adaptive Filter Weight Vector for Noise Cancellation Problem in Fig. 5.4

B.1

. 118

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Block diagram of the approximate model for the primary path in the presence of the feedback path

C.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Magnitude of the scaling factor relating load cell’s reading of the effect of control actuators to that of the scoring sensor . . . . . . . . . . . . . . . . . . . .

C.2

146

Magnitude of the scaling factor relating load cell’s reading of the effect of disturbance actuators to that of the scoring sensor after diagonalization . . . . . . .

C.5

145

Magnitude of the scaling factor relating load cell’s reading of the effect of control actuators to that of the scoring sensor after diagonalization . . . . . . . . . .

C.4

144

Magnitude of the scaling factor relating load cell’s reading of the effect of disturbance actuators to that of the scoring sensor . . . . . . . . . . . . . . . . .

C.3

115

147

Comparison of SVD plots for the transfer function to the scaled/double-integrated load cell data

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

C.6

Comparison of SVD plots for the transfer function to the actual load cell data .

C.7

Comparison of SVD plots for the transfer function to the scoring sensors

C.8

Comparison of SVD plots for the transfer function to the position sensors colocated with the control actuators

148

. . . 149

. . . . . . . . . . . . . . . . . . . . . . . . . 149 xv

C.9

Comparison of SVD plots for the transfer function to the position sensors colocated with the disturbance actuators

. . . . . . . . . . . . . . . . . . . . . . . 150

C.10 The identified model for the system beyond the frequency range for which measurements are available . . . . . . . . . . . . . . . . . . . . . . . . . . .

151

C.11 The final model for the system beyond the frequency range for which measurements are available . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152

C.12 The comparison of the closed loop and open loop singular value plots when the controller is used to close the loop on the identified model

. . . . . . . . . . 153

C.13 The comparison of the closed loop and open loop singular value plots when the controller is used to close the loop on the real measured data

xvi

. . . . . . . . . 154

Chapter 1 Introduction This dissertation presents a new estimation-based procedure for the systematic synthesis and analysis of adaptive filters (controllers) in “Filtered” LMS problems. This new approach uses an estimation interpretation of the adaptive filtering (control) problem to formulate an equivalent estimation problem. The adaptation criterion for the adaptive weight vector is extracted from the H∞ -solution to this estimation problem. The new algorithm, referred to as Estimation-Based Adaptive Filtering (EBAF), applies to both Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) adaptive filters.

1.1

Motivation

Least-Mean Squares (LMS) adaptive algorithm [51] has been the centerpiece of a wide variety of adaptive filtering techniques for almost four decades. The straightforward derivation, and the simplicity of its implementation (especially at the time of limited computational power) encouraged experiments with the algorithm in a diverse range of applications (e.g. see [51,33]). In some applications however, the simple implementation of the LMS algorithm was found to be inadequate. Subsequent attempts to overcome its shortcomings have produced a large number of innovative solutions that have been successful in practice. Commonly used algorithms such as normalized

1

1.1. MOTIVATION

2

LMS, correlation LMS [47], leaky LMS [21], variable-step-size LMS [25], and FilteredX LMS [35] are the outcome of such efforts. These algorithms use the instantaneous squared error to estimate the mean-square error, and often assume slow adaptation to allow for the necessary linear operations in their derivation (see Chapters 2 and 3 in [33] for instance). As Reference [2] points out: “Many of the algorithms and approaches used are of an ad hoc nature; the tools are gathered from a wide range of fields; and good systematic approaches are still lacking.” Introducing a systematic procedure for the synthesis of adaptive filters is one of the main goals of this thesis. Parallel to the efforts on the practical application of the LMS-based adaptive schemes, there has been a concerted effort to analyze these algorithms. Of pioneering importance are the results in Refs. [50] and [23]. Reference [50] considers the adaptation with LMS on stationary stochastic processes, and finds the optimal solution to which the expected value of the weight vector converges. For sinusoidal inputs however, the discussion in [50] does not apply. In [23] it is shown that for sinusoidal inputs, when time-varying component of the adaptive filter output is small compared to its time-invariant component (see [23], page 486), the adaptive LMS filter can be approximated by a linear time-invariant transfer function. Reference [13] extends the approach in [23] to derive an equivalent transfer function for the Filtered-X LMS adaptive algorithm (provided the conditions required in [23] still apply). The equivalent transfer function is then used to analytically derive an expression for the optimum convergence coefficients. A frequency domain model of the so-called filtered LMS algorithm (i.e. an algorithm in which the input or the output of the adaptive filter or the feedback error signal is linearly filtered prior to use in the adaptive algorithm) is discussed in [17]. The frequency domain model in [17] decouples the inputs into disjoint frequency bins and places a single frequency adaptive noise canceler on each bin. The analysis in their work utilizes the frequency domain LMS algorithm [11] and assumes a time invariant linear behavior for the filter. Other important aspects

1.1. MOTIVATION

3

of the adaptive filters have also been extensively studied. The effect of the modeling error on the convergence and performance properties of the LMS-based adaptive algorithms (e.g. [17,7]), and tracking behavior of the LMS adaptive algorithm when the adaptive filter is tuned to follow a linear chirp signal buried in white noise [5,6], are examples of these studies∗ . In summary, existing analysis techniques are often suitable for analyzing only one particular aspect of the behavior of an adaptive filter (e.g. its steady-state behavior). Furthermore, the validity of the analysis relies on certain assumptions (e.g. slow convergence, and/or the negligible contribution of the nonlinear/time-varying component of the adaptive filter output) that can be quite restrictive. Providing a solid framework for the systematic analysis of adaptive filters is another main goal of this thesis. The reason for the difficulty experienced in both synthesis and analysis of adaptive algorithms is best explained in Reference [37]: “It is now generally realized that adaptive systems are special classes of nonlinear systems . . . general methods for the analysis and synthesis of nonlinear systems do not exist since conditions for their stability can be established only on a system by system basis.” This thesis introduces a new framework for the synthesis and analysis of adaptive filters (controllers) by providing an estimation interpretation of the above mentioned “nonlinear” adaptive filtering (control) problem. The estimation interpretation replaces the original adaptive filtering (control) synthesis with an equivalent estimation problem, the solution of which is used to update the weight vector in the adaptive filter (and hence the name estimation-based adaptive filtering). This approach is applicable (due to its systematic nature) to both FIR and IIR adaptive filters (controllers). In the FIR case the equivalent estimation problem is linear, and hence exact solutions are available. Stability, performance bounds, transient behavior of adaptive FIR filters are thus precisely addressed in this framework. In the IIR case, however, only an approximate solution to the equivalent estimation problem is available, and ∗

The survey here is intended to provide a flavor of the type of the problems that have captured the attention of researchers in the field. The shear volume of the literature makes subjective selection of the references unavoidable.

1.2. BACKGROUND

4

hence the proposed estimation-based framework serves as a reasonable heuristic for the systematic design of adaptive IIR filters. This approximate solution however, is based on realistic assumptions, and the adaptive algorithm maintains its systematic structure. Furthermore, the treatment of feedback contamination (see Chapter 3 for a precise definition), is virtually identical to that of adaptive IIR filters. The proposed estimation-based approach is particularly appealing if one considers the difficulty with the existing design techniques for adaptive IIR filters, and the complexity of available solutions to feedback contamination (e.g. see [33]).

1.2

Background

The development of the new estimation-based framework is based on recent results in robust estimation. Following the pioneering work in [52], the H∞ approach to robust control theory produced solutions [12,24] that were designed to meet some performance criterion in the face of the limited knowledge of the exogenous disturbances and imperfect system models. Further work in robust control and estimation (see [32,46] and the references therein) produced straightforward solutions that allowed in-depth studies of the properties of the robust controllers/estimators. The main idea in H∞ estimation is to design an estimator that bounds (in the optimum case, minimizes) the maximum energy gain from the disturbances to the estimation errors. Such a solution guarantees that for disturbances with bounded energy, the energy of the estimation error will be bounded as well. In the case of an optimal solution, an H∞ -optimal estimator will guarantee that the energy of the estimation error for the worst case disturbance is indeed minimized [28]. Of crucial importance for the work in this thesis, is the result in [26] where the H∞ optimality of the LMS algorithm was established. Note that despite a long history of successful applications, prior to the work in [26], the LMS algorithm was regarded as an approximate recursive solution to the least-squares minimization problem. The work in [26] showed that instead of being an approximate solution to an H2 minimization, the LMS algorithm is the exact solution to a minmax estimation problem. More

1.2. BACKGROUND

5

specifically, Ref. [26] proved that the LMS adaptive filter is the central a priori H∞ optimal filter. This result established a fundamental connection between an adaptive control algorithm (LMS algorithm in this case), and a robust estimation problem. Inspired by the analysis in [26], this thesis introduces an estimation interpretation of a far more general adaptive filtering problem, and develops a systematic procedure for the synthesis of adaptive filters based on this interpretation. The class of problems addressed in this thesis, commonly known as “Filtered” LMS [17], encompass a wide range of adaptive filtering/control applications [51,33], and have been the subject of extensive research over the past four decades. Nevertheless, the viewpoint provided in this thesis not only provides a systematic alternative to some widely used adaptive filtering (control) algorithms (such as FxLMS and FuLMS) with superior transient and steady-state behavior, but it also presents a new framework for their analysis. More specifically, this thesis proves that the fundamental connection between adaptive filtering (control) algorithms and robust estimation extends to the more general setting of adaptive filtering (control) problems, and shows that the convergence, stability, and performance of these classical adaptive algorithms can be systematically analyzed as robust estimation questions. The systematic nature of the proposed estimation-based approach enables an alternative formulation for the adaptive filtering (control) problem using Linear Matrix Inequalities (LMIs), the ramifications of which will be discussed in Chapter 5. Several researchers (see [18] and references therein) in the past few years have shown that elementary manipulations of linear matrix inequalities can be used to derive less restrictive alternatives to the now classical state-space Riccati-based solution to the H∞ control problem [12]. Even though the computational complexity of the LMI-based solution remains higher than that of solving the Riccati equation, there are three main reasons that justify such a formulation [19]: (a) a variety of design specifications and constraints can be expressed as LMIs, (b) problems formulated as LMIs can be solved exactly by efficient convex optimization techniques, and (c) for the cases that lack analytical solutions such as mixed H2 /H∞ design objectives (see [4], [32] and [45] and references therein), the LMI formulation of the problem remains tractable (i.e. LMIsolvers are viable alternatives to analytical solutions in such cases). As will be seen

1.3. AN OVERVIEW OF ADAPTIVE FILTERING (CONTROL) ALGORITHMS

6

in Chapter 5, the LMI framework provides the machinery required for the synthesis of a robust adaptive filter in the presence of modeling uncertainty.

1.3

An Overview of Adaptive Filtering (Control) Algorithms

To put this thesis in perspective, this section provides a brief overview of the vast literature on adaptive filtering (control). Reference [36] recognizes 1957 as the year for the formal introduction of the term “adaptive system” into the control literature. By then, the interest in filtering and control theory had shifted towards increasingly more complex systems with poorly characterized (possibly time varying) models for system dynamics and disturbances, and the concept of “adaptation” (borrowed from living systems) seemed to carry the potential for solving the increasingly more complex control problems. The exact definition of “adaptation” and its distinction from “feedback”, however, is the subject of long standing discussions (e.g. see [2,36,29]). Qualitatively speaking, an adaptive system is a system that can modify its behavior in response to changes in the dynamics of the system or disturbances through some recursive algorithm. As a direct consequence of this recursive algorithm (in which the parameters of the adaptive system are adjusted using input/output data), an adaptive system is a “nonlinear” device. The development of adaptive algorithms has been pursued from a variety of view points. Different classifications of adaptive algorithms (such as direct versus indirect adaptive control, model reference versus self-tuning adaptation) in the literature reflect this diversity [2,51,29]. For the purpose of this thesis, two distinct approaches for deriving recursive adaptive algorithms can be identified: (a) stochastic gradient approaches that include LMS and LMS-Based adaptive algorithms, and (b) least-squares estimation approaches that include adaptive recursive least-squares (RLS) algorithm. The central idea in the former approach, is to define an appropriate cost function that captures the success of the adaptation process, and then change the adaptive


7

filter parameters to reduce the cost function according to the method of steepest descent. This requires the use of a gradient vector (hence the name), which in practice is approximated using instantaneous data. Chapter 2 provides a detailed description of this approach for the problem of interest in this Thesis. The latter approach to the design of adaptive filters is based on the method of least squares. This approach closely corresponds to Kalman filtering. Ref. [44] provides a unifying state-space approach to adaptive RLS filtering. The main focus in this thesis however, is on the LMS-based adaptive algorithms. Since adaptive algorithms can successfully operate in a poorly known environment, they have been used in a diverse field of applications that include communication (e.g. [34,41]), process control (e.g. [2]), seismology (e.g. [42]), biomedical engineering (e.g. [51]). Despite the diversity of the applications, different implementations of adaptive filtering (control) share one basic common feature [29]: “an input vector and a desired response are used to compute an estimation error, which is in turn used to control the values of a set of adjustable filter coefficients.” Reference [29] distinguishes four main classes of adaptive filtering applications based on the way the desired signal is defined in the formulation of the problem: (a) identification: in this class of applications an adaptive filter is used to provide a linear model for an unknown plant. The plant and the adaptive filter are driven by the same input, and the output of the plant is the desired response that adaptive filter tries to match. (b) inverse modeling: here the adaptive filter is placed in series with an unknown (perhaps noisy) plant, and the desired signal is simply a delayed version of the plant input. Ideally, the adaptive filter converges to the inverse of the unknown plant. Adaptive equalization (e.g. [40]) is an important application in this class. (c) prediction: the desired signal in this case is the current value of a random signal, while past values of the random signal provide the input to the adaptive filter. Signal detection is an important application in this class. (d) interference canceling: here adaptive filter uses a reference signal (provided as input to the adaptive filter) to cancel unknown interference contained in a primary signal. Adaptive noise cancellation, echo cancellation, and adaptive beam-forming are applications that fall in this last class. The estimation-based adaptive filtering algorithm in this thesis is presented in the context of adaptive noise cancellation, and


8

therefore a detailed discussion of the fourth class of adaptive filtering problems is provided in Chapter 2. There are several main structures for the implementation of adaptive filters (controllers). The structure of the adaptive filter is known to affect its performance, computational complexity, and convergence. In this thesis, the two most commonly used structures for adaptive filters (controllers) are considered. The finite impulse response (FIR) transversal filter (see Fig. 1.1) is the structure upon which the main presentation of the estimation-based adaptive filtering algorithm is primarily presented. The transversal filter consists of three basic elements: (a) unit-delay element, (b) multiplier, and (c) adder, and contains feed forwards paths only. The number of unit-delays specify the length of the adaptive FIR filter. Multipliers weight the delayed versions of some reference signal, which are then added in the adder(s). The frequency response for this filter is of finite length (hence the name), and contains only zeros (all poles are at the origin in the z-plane). Therefore, there is no question of stability for the open-loop behavior of the FIR filter. The infinite-duration impulse response (IIR) structure is shown in Figure 1.2. The feature that distinguishes the IIR filter from an FIR filter is the inclusion of the feedback path in the structure of the adaptive filter. As mentioned earlier, for an FIR filter all poles are at the origin, and a good approximation of the behavior of a pole, in general, can only be achieved if the length of the FIR filter is sufficiently long. An IIR filter, ideally at least, can provide a perfect match for a pole with only a limited number of parameters. This means that for a desired dynamic behavior (such as resonance frequency, damping, or cutoff frequency), the number of parameters in an adaptive IIR filter can be far fewer than that in its FIR counterpart. The computational complexity per sample for adaptive IIR filter design can therefore be significantly lower than that in FIR filter design. The limited use of adaptive IIR filters (compared to the vast number of applications for the FIR filters) suggests that the above mentioned advantages come at a certain cost. In particular, adaptive IIR filters are only conditionally stable, and therefore some provisions are required to assure stability of the filter at each iteration. There are solutions such as Schur-Cohn algorithm ([29] pages 271-273) that monitor


9

the stability of the IIR filter (by determining whether all roots of the denominator of the IIR filter transfer function are inside the unit circle). This however requires intensive on-line calculations. Alternative implementations of adaptive IIR filters (such as parallel implementation [48], and lattice implementation [38]) have been suggested that provide simpler stability monitoring capabilities. The monitoring process is independent of the adaptation process here. In other words, the adaptation criteria do not inherently reject de-stabilizing values for filter weights. The monitoring process detects these de-stabilizing values and prevents their implementation. Another significant problem with adaptive IIR filter design stems from the fact that the performance surface (see [33], Chapter 3) for adaptive IIR filters is generally non-quadratic (see [33] pages 91-94 for instance) and often contains multiple local minima. Therefore, the weight vector may converge to a local minimum only (hence non-optimal cost). Furthermore, it is noted that the adaptation rate for adaptive IIR filters can be slow when compared to the FIR adaptive filters [33,31]. Early works in adaptive IIR filtering (e.g. [16]) are for the most part extensions to Widrow’s LMS algorithm of adaptive FIR filtering [51]. More recent works include modifications to recursive LMS algorithm (e.g. [15]) that are devised for specific applications. In other words, existing design techniques for adaptive IIR filters are application-specific and rely on certain restrictive assumptions in their derivation. Our description of the Filtered-U recursive LMS algorithm in Chapter 3 will further clarify this point. Furthermore, as [33] points out: “The properties of an adaptive IIR filter are considerably more complex than those of the conventional adaptive FIR filter, and consequently it is more difficult to predict their behavior.” Thus, a framework that allows a unified approach to the synthesis and analysis of adaptive IIR filters, and does not require restrictive assumptions for its derivation would be extremely useful. As mentioned earlier, this thesis provides such a framework. Finally, for a wide variety of applications such as equalization in wireless communication channels, and active control of sound and vibration in an environment where the effect of a number of primary sources should be canceled by a number of control (secondary) sources, the use of a multi-channel adaptive algorithm is well justified. In general, however, variations of the LMS algorithm are not easy to extend to

1.4. CONTRIBUTIONS

10

multi-channel systems. Furthermore, the analysis of the performance and properties of such multi-channel algorithms is complicated [33]. As Ref. [33] points out, in the context of active noise cancellation, the successful implementation of multi-channel adaptive algorithms has so far been limited to cases involving repetitive noise with a few harmonics [39,43,49,13]). For the approach presented in this thesis, the syntheses of single-channel and multi-channel adaptive algorithms are virtually identical. This similarity is a direct result of the way the synthesis problem is formulated (see 4).

1.4

Contributions

In meeting the goals of this research, the following contributions have been made to adaptive filtering and control: 1. An estimation-interpretation for adaptive “Filtered” LMS filtering (control) problems is developed. This interpretation allows an equivalent estimation formulation for the adaptive filtering (control) problem. The adaptation criterion for adaptive filter weight vector is extracted from the solution to this equivalent estimation problem. This constitutes a systematic synthesis procedure for adaptive filters in filtered LMS problems. The new synthesis procedure is called Estimation-Based Adaptive Filtering (EBAF). 2. Using an H∞ criterion to formulate the “equivalent” estimation problem, this thesis develops a new framework for the systematic analysis of Filtered LMS adaptive algorithms. In particular, the results in this thesis extend the fundamental connection between the LMS adaptive algorithm and robust estimation (i.e. H∞ optimality of the LMS algorithm [26]) to the more general setting of filtered LMS adaptive problems. 3. For the EBAF algorithm in the FIR case: (a) It is shown that the adaptive weight vector update can be based on the central filtering (prediction) solution to a linear H∞ estimation problem, the existence of which is guaranteed. It is also shown that the maximum

1.4. CONTRIBUTIONS

11

energy gain in this case can be minimized. Furthermore, the optimal energy gain is proved to be unity, and the conditions under which this bound is achievable are derived. (b) The adaptive algorithm is shown to be implementable in real-time. The update rule requires a simple Lyapunov recursion that leads to a computational complexity comparable to that of filtered LMS adaptive algorithms (e.g. FxLMS). The experimental data, along with extensive simulations are presented to demonstrate the improved steady-state performance of the EBAF algorithm (over FxLMS and Normalized-FxLMS algorithms), as well as a faster transient response. (c) A clear connection between the limiting behavior of the EBAF algorithm and the existing FxLMS and Normalized-FxLMS adaptive algorithms has been established. 4. For the EBAF algorithm in the IIR case, it is shown that the equivalent estimation problem is nonlinear. A linearizing approximation is then employed that makes systematic synthesis of adaptive IIR filter tractable. The performance of the EBAF algorithm in this case is compared to the performance of the Filtered-U LMS (FuLMS) adaptive algorithm, demonstrating the improved performance in the EBAF case. 5. The treatment of feedback contamination problem is shown to be identical to the IIR adaptive filter design in the new estimation-based framework. 6. A multi-channel extension of the EBAF algorithm demonstrates that the treatment of the single-channel and multi-channel adaptive filtering (control) problems in the new estimation based framework is virtually the same. Simulation results for the problem of vibration isolation in a 3-input/3-output vibration isolation platform (VIP) prove feasibility of the EBAF algorithm in multi-channel problems. 7. The new estimation-based framework is shown to be amenable to a Linear Matrix Inequality (LMI) formulation. The LMI formulation is used to explicitly

1.5. THESIS OUTLINE

12

address the stability of the overall system under adaptive algorithm by producing a Lyapunov function. It is also shown to be an appropriate framework to address the robustness of the adaptive algorithm to modeling error or parameter uncertainty. Augmentation of an H2 performance constraint to the H∞ disturbance rejection criterion is also discussed.

1.5

Thesis Outline

The organization of this thesis is as follows. In Chapter 2, the fundamental concepts of the estimation-based adaptive filtering (EBAF) algorithm are introduced. The application of the EBAF approach in the case of adaptive FIR filter design is also presented in this chapter. In Chapter 3, the extension of the EBAF approach to the adaptive IIR filter design is discussed. A multi-channel implementation of the EBAF algorithm is presented in Chapter 4. An LMI formulation for the EBAF algorithm is derived in Chapter 5. Chapter 6 concludes this dissertation with a summary of the main results, and the suggestions for future work. This dissertation contains three appendices. An algebraic proof for the feasibility of the unity energy gain in the estimation problem associated with adaptive FIR filter design (in Chapter 2) is discussed in Appendix A. The problem of feedback contamination is formally addressed in Appendix B. A detailed discussion of the identification process is presented in Appendix C. The identified model for the Vibration Isolation Platform (VIP), used as a test-bed for multi-channel implementation of the EBAF algorithm, is also presented in this appendix.

1.5. THESIS OUTLINE

13

x(k − 1)

x(k) z −1

W0

x(k − 2)

x(k − N )

z −1

W1

z −1

WN −1

W2

WN

+

u(k)

Fig. 1.1: General block diagram for an FIR Filterm

x(k)

u(k)

r(k) a0

+

+

z −1 a1

b1 z −1

a2

b2

r(k − 2)

z −1 bN

aN

Fig. 1.2: General block diagram for an IIR Filter

Chapter 2 Estimation-Based adaptive FIR Filter Design This chapter presents a systematic synthesis procedure for H∞ -optimal adaptive FIR filters in the context of an Active Noise Cancellation (ANC) problem. An estimation interpretation of the adaptive control problem is introduced first. Based on this interpretation, an H∞ estimation problem is formulated, and its finite horizon prediction (filtering) solutions are discussed. The solution minimizes the maximum energy gain from the disturbances to the predicted (filtered) estimation error, and serves as the adaptation criterion for the weight vector in the adaptive FIR filter. This thesis refers to the new adaptation scheme as Estimation-Based Adaptive Filtering (EBAF). It is shown in this chapter that the steady-state gain vectors in the EBAF algorithm approach those of the classical Filtered-X LMS (Normalized Filtered-X LMS) algorithm. The error terms, however, are shown to be different, thus demonstrating that the classical algorithms can be thought of as an approximation to the new EBAF adaptive algorithm. The proposed EBAF algorithm is applied to an active noise cancellation problem (both narrow-band and broad-band cases) in a one-dimensional acoustic duct. Experimental data as well as simulations are presented to examine the performance of the new adaptive algorithm. Comparisons to the results from a conventional FxLMS algorithm show faster convergence without compromising steady-state performance 14

2.1. BACKGROUND

15

and/or robustness of the algorithm to feedback contamination of the reference signal.

2.1

Background

This section introduces the context in which the new estimation-based adaptive filtering (EBAF) algorithm will be presented. It defines the adaptive filtering problem of interest and describes the terminology that is used in this chapter. A conventional solution to the problem based on the FxLMS algorithm is also outlined in this section. The discussion of key concepts of the EBAF algorithm and the mathematical formulation of the algorithm are left to Sections 2.2 and 2.3, respectively. Referring to Fig. 2.1, the objective in this adaptive filtering problem is to adjust the weight vector in the adaptive FIR filter, W (k) = [w0 (k) w1 (k) ... wN (k)]T (k is the discrete time index), such that the cancellation error, d(k) −y(k), is small in some appropriate measure. Note that d(k) and y(k) are outputs of the primary path P (z) and the secondary path S(z), respectively. Moreover, 1. n(k) is the input to the primary path, 2. x(k) is a properly selected reference signal with a non-zero correlation with the primary input, 4

3. u(k) is the control signal applied to the secondary path (generated as u(k) = [x(k) x(k − 1) · · · x(k − N)] W (k)), 4. e(k) is the measured residual error available to the adaptation scheme. Note that in a typical practice, x(k) is obtained via some measurement of the primary input. The quality of this measurement will impact the correlation between the reference signal and the primary input. Similar to the conventional development of the FxLMS algorithm however, this chapter assumes perfect correlation between the two. The Filtered-X LMS (FxLMS) solution to this problem is shown in Figure 2.2 where perfect correlation between the primary disturbance n(k) and the reference signal x(k) is assumed [51,33]. Minimizing the instantaneous squared error, e2 (k), as

2.2. EBAF ALGORITHM - MAIN CONCEPT

16

an approximation to the mean-square error, FxLMS follows the LMS update criterion (i.e. to recursively adapt the weight vector in the negative gradient direction) µ 2 ∇e (k) 2 e(k) = d(k) − y(k) = d(k) − S(k) ⊕ u(k)

W (k + 1) = W (k) −

where µ is the adaptation rate, S(k) is the impulse response of the secondary path, and “⊕” indicates convolution. Assuming slow adaptation, the FxLMS algorithm then approximates the instantaneous gradient in the weight vector update with 4 T ∇e2 (k) ∼ = −2 [x0 (k) x0 (k − 1) · · · x0 (k − N)] e(k) = −2h0 (k)e(k)

(2.1)

4

where x0 (k) = S(k) ⊕ x(k) represents a filtered version of the reference signal which is available to the LMS adaptation (and hence the name (Normalized) Filtered-X LMS). This yields the following adaptation criterion for the FxLMS algorithm W (k + 1) = W (k) + µh0 (k)e(k)

(2.2)

A closely related adaptive algorithm is the one in which the adaptation rate is normalized with the estimate of the power of the reference vector, i.e. W (k + 1) = W (k) + µ

h0 (k) e(k) 1 + µh∗ 0 (k)h0 (k)

(2.3)

where ∗ indicates complex conjugate. This algorithm is known as the NormalizedFxLMS algorithm. In practice, however, only an approximate model of the secondary path (obtained via some identification scheme) is known, and it is this approximate model that is used to filter the reference signal. For further discussion on the derivation and analysis of the FxLMS algorithm please refer to [33,7].

2.2

EBAF Algorithm - Main Concept

The principal goal of this section is to introduce the underlying concepts of the new EBAF algorithm. For the developments in this section, perfect correlation between

2.2. EBAF ALGORITHM - MAIN CONCEPT

17

n(k) and x(k) in Fig. 2.1 is assumed (i.e. x(k) = n(k) for all k). This is the same condition under which the FxLMS algorithm was developed. The dynamics of the secondary path are assumed known (e.g. by system identification). No explicit model for the primary path is needed. As stated before, the objective in the adaptive filtering problem of Fig. 2.1 is to generate a control signal, u(k), such that the output of the secondary path, y(k), is “close” to the output of the primary path, d(k). To achieve this goal, for the given reference signal x(k), the series connection of the FIR filter and the secondary path must constitute an appropriate model for the unknown primary path. In other words, with the adaptive FIR filter properly adjusted, the path from x(k) to d(k) must be equivalent to the path from x(k) to y(k). Based on this observation, in Fig. 2.3 the structure of the path from x(k) to y(k) is used to model the primary path. The modeling error is included to account for the imperfect cancellation. The above mentioned observation forms the basis for an estimation interpretation of the adaptive control problem. The following outlines the main steps for this interpretation: 1. Introduce an approximate model for the primary path based on the architecture of the adaptive path from x(k) to y(k) (as shown in Fig. 2.3). There is an optimal value for the weight vector in the approximate model’s FIR filter for which the modeling error is the smallest. This optimal weight vector, however, is not known. State-space models are used for both FIR filter and the secondary path. 2. In the approximate model for the primary path, use the available information to formulate an estimation problem that recursively estimates this optimal weight vector. 3. Adjust the weight vector of the adaptive FIR filter to the best available estimate of the optimal weight vector. Before formalizing this estimation-based approach, a closer look at the signals (i.e. information) involved in Fig. 2.1 is provided. Note that e(k) = d(k) − y(k) + Vm (k), where

2.3. PROBLEM FORMULATION

18

a. e(k) is the available measurement. b. Vm (k) is the exogenous disturbance that captures the effect of measurement noise, modeling error, and the initial condition uncertainty in error measurements. c. y(k) is the output of the secondary path. d. d(k) is the output of the primary path. Note that unlike e(k), the signals y(k) and d(k) are not directly measurable. With u(k) fully known, however, the assumption of a known initial condition for the secondary path leads to the exact knowledge of y(k). This assumption is relaxed later in this chapter, where the effect of an “inexact” initial condition in the performance of the adaptive filter is studied (Section 2.7). The derived measured quantity that will be used in the estimation process can now be introduced as 4

m(k) = e(k) + y(k) = d(k) + Vm (k)

2.3

(2.4)

Problem Formulation

Figure 2.4 shows a block diagram representation of the approximate model to the primary path. A state space model, [ As (k), Bs (k), Cs (k), Ds (k) ], for the secondary path is assumed. Note that both primary and secondary paths are assumed stable. The weight vector, W (k) = [ w0 (k) w1 (k) · · · wN (k) ]T , is treated as the state vector capturing the trivial dynamics, W (k + 1) = W (k), that is assumed for the FIR filter. With θ(k) the state variable for the secondary path, then ξ T =

W T (k) θT (k)

is

the state vector for the overall system. The state space representation of the system is then #" # " # " 0 W (k) 4 W (k + 1) I(N +1)×(N +1) = Fk ξk = Bs (k)h∗ (k) As (k) θ(k) θ(k + 1)

(2.5)


19

where h(k) = [x(k) x(k − 1) · · · x(k − N)]T captures the effect of the reference input x(·). For this system, the derived measured output defined in Eq. (2.4) is # " h i W (k) 4 m(k) = Ds (k)h∗ (k) Cs (k) + Vm (k) = Hk ξk + Vm (k) θ(k)

(2.6)

A linear combination of the states is defined as the desired quantity to be estimated " # h i W (k) 4 s(k) = L1,k L2,k = Lk ξk (2.7) θ(k) For simplicity, the single-channel problem is considered here. Extension to the multichannel case is straight forward and is discussed in Chapter 4. Therefore, m(k) ∈ R1×1 , s(k) ∈ R1×1 , θ(k) ∈ RNs ×1 , and W (k) ∈ R(N +1)×1 . All matrices are then of appropriate dimensions. There are several alternatives for selecting Lk and thus the variable to be estimated, s(k). The end goal of the estimation based approach however, is to set the weight vector in the adaptive FIR filter such that the output of the secondary path, y(k) in Fig. 2.3, best matches d(k). So s(k) = d(k) is chosen, i.e. Lk = Hk . Any estimation algorithm can now be used to generate an estimate of the desired quantity s(k). Two main estimation approaches are considered next.

2.3.1

H2 Optimal Estimation

Here stochastic interpretation of the estimation problem is possible. Assuming that ξ0 (the initial condition for the system in Figure 2.4) and Vm (·) are zero mean uncorrelated random variables with known covariance matrices " # # " h i ξ0 0 Π 0 ∗ E = ξ0∗ Vm (j) 0 Qk δkj Vm (k)

(2.8)

4

sˆ(k|k) = F (m(0), · · · , m(k)), the causal linear least-mean-squares estimate of s(k), is given by the Kalman filter recursions [27]. There are two primary difficulties with the H2 optimal solution: (a) The H2 solution is optimal only if the stochastic assumptions are valid. If the external disturbance


20

is not Gaussian (for instance when there is a considerable modeling error that should be treated as a component of the measurement disturbance) then pursuing an H2 filtering solution may yield undesirable performance; and (b) regardless of the choice for Lk , the recursive H2 filtering solution does not simplify to the same extent as the H∞ solution considered below. This can be of practical importance when the real-time computational power is limited. Therefore, the H2 optimal solution is not employed in this chapter.

2.3.2

H∞ Optimal Estimation

To avoid difficulties associated with the H2 estimation, we consider a minmax formulation of the estimation problem in this section. Here, the main objective is to limit the worst case energy gain from the measurement disturbance and the initial condition uncertainty to the error in a causal (or strictly causal) estimate of s(k). More specifically, the following two cases are of interest. Let sˆ(k|k) = Ff (m(0), · · · , m(k)) denote an estimate of s(k) given observations m(i) for time i = 0 up to and including 4

time i = k, and let sˆ(k) = sˆ(k|k − 1) = Fp (m(0), · · · , m(k − 1)) denote an estimate of s(k) given m(i) for time i = 0 up to and including i = k − 1. Note that sˆ(k|k) and sˆ(k) are known as filtering and prediction estimates of s(k), respectively. Two estimation errors can now be defined: the filtered error ef,k = sˆ(k|k) − s(k)

(2.9)

ep,k = sˆ(k) − s(k)

(2.10)

and the predicted error

Given a final time M, the objective of the filtering problem can now be formalized as finding sˆ(k|k) such that for Π0 > 0 M X

sup Vm , ξ0

e∗f,k ef,k

k=0

ˆ (ξ0 − ξˆ0 )∗ Π−1 0 (ξ0 − ξ0 ) +

M X k=0

≤ γ2 ∗ Vm (k)Vm (k)

(2.11)

2.4. H∞ -OPTIMAL SOLUTION

21

for a given scalar γ > 0. In a similar way, the objective of the prediction problem can be formalized as finding sˆ(k) such that M X

sup Vm , ξ0

e∗p,k ep,k

k=0

ˆ (ξ0 − ξˆ0 )∗ Π−1 0 (ξ0 − ξ0 ) +

M X

≤ γ2

(2.12)

∗ Vm (k)Vm (k)

k=0

for a given scalar γ > 0. The question of optimality of the solution can be answered by finding the infimum value among all feasible γ’s. Note that, for the H∞ optimal estimation there is no statistical assumption regarding the measurement disturbance. Therefore, the inclusion of the output of the modeling error block (see Fig. 2.3) in the measurement disturbance is consistent with H∞ formulation of the problem. The elimination of the “modeling error” block in the approximate model of primary path in Fig. 2.4 is based on this characteristic of the disturbance in an H∞ formulation.

2.4


For the remainder of this chapter, the case where Lk = Hk is considered. Referring to Figure 2.4, this means that s(k) = d(k). To discuss the solution, from [27] the solutions to the γ-suboptimal finite-horizon filtering problem of Eq. (2.11), and the prediction problem of Eq. (2.12) are drawn. Finally, we find the optimal value of γ and show how γ = γopt simplifies the solutions.

2.4.1

γ-Suboptimal Finite Horizon Filtering Solution

Theorem 2.1: [27]Consider the state space representation of the block diagram of Figure 2.4, described by Equations (2.5)-(2.7). A level-γ H∞ filter that achieves (2.11) exists if, and only if, the matrices 0 0 Hk Ip Ip Rk = and Re,k = + Pk Hk∗ L∗k (2.13) 2 2 0 −γ Iq 0 −γ Iq Lk (here p and q are used to indicate the correct dimensions) have the same inertia for all 0 ≤ k ≤ M, where P0 = Π0 > 0 satisfies the Riccati recursion ∗ Pk+1 = Fk Pk Fk∗ − Kf,k Re,k Kf,k

(2.14)


where Kf,k =

22

Fk Pk

Hk∗ L∗k

−1 Re,k

If this is the case, then the central H∞ estimator is given by ξˆk+1 = Fk ξˆk + Kf,k m(k) − Hk ξˆk , ξˆ0 = 0 −1 sˆ(k|k) = Lk ξˆk + (Lk Pk Hk∗ ) RHe,k m(k) − Hk ξˆk

(2.15)

(2.16) (2.17)

−1 with Kf,k = (Fk Pk Hk∗ ) RHe,k and RHe,k = Ip + Hk Pk Hk∗ .

Proof: see [27].

2.4.2

γ-Suboptimal Finite Horizon Prediction Solution

Theorem 2.2: [27]For the system described by Equations (2.5)-(2.7), level-γ H∞ filter that achieves (2.12) exists if, and only if, all leading sub-matrices of Lk −γ 2 Ip 0 −γ 2 Ip 0 p p (2.18) Rk = and Re,k = + Pk L∗k Hk∗ 0 Iq 0 Iq Hk have the same inertia for all 0 ≤ k < M. Note that Pk is updated according to Eq. (2.14). If this is the case, then one possible level-γ H∞ filter is given by ξˆk+1 = Fk ξˆk + Kp,k m(k) − Hk ξˆk , ξˆ0 = 0 (2.19) sˆ(k) = Lk ξˆk where Kp,k

(2.20)

−1 ∗ ∗ ˜ ˜ = Fk Pk Hk I + Hk Pk Hk

(2.21)

and P˜k = I − γ −2 Pk L∗k Lk

−1

Pk ,

(2.22)

Proof: see [27]. Note that the condition in Eq. (2.18) is equivalent to I − γ −2 Pk L∗k Lk > 0,

for k = 0, · · · , M

(2.23)

and hence P˜k in Eq. (2.22) is well defined. P˜k can also be defined as P˜k−1 = Pk−1 − γ −2 L∗k Lk ,

for k = 0, · · · , M

(2.24)


23

which proves useful in rewriting the prediction coefficient, Kp,k in Eq. (2.21), as follows. First, note that −1 −1 Fk P˜k Hk∗ I + Hk P˜k Hk∗ = Fk P˜k−1 + Hk∗ Hk Hk∗

(2.25)

and hence, replacing for P˜k−1 from Eq. (2.24) Kp,k = Fk Pk−1 − γ −2 L∗k Lk + Hk∗ Hk

−1

Hk∗

(2.26)

Theorems 2.1 and 2.2 (Sections 2.4.1 and 2.4.2) provide the form of the filtering and prediction estimators, respectively. The following section investigates the optimal value of γ for both of these solutions, and outlines the simplifications that follow.

2.4.3

The Optimal Value of γ

The optimal value of γ for the filtering solution will be discussed first. The discussion of the optimal prediction solution utilizes the results in the filtering case. 2.4.3.1

Filtering Case

2.4.3.1.1 γopt ≤ 1: First, it will be shown that for the filtering solution γopt ≤ 1. Using Eq. (2.11), one can always pick sˆ(k|k) to be simply m(k). With this choice sˆ(k|k) − s(k) = Vm (k), for all k

(2.27)

and Eq. (2.11) reduces to M X

sup Vm ∈ L2 , ξ0

Vm (k)∗ Vm (k)

k=0

ˆ (ξ0 − ξˆ0 )∗ Π−1 0 (ξ0 − ξ0 ) +

M X

(2.28) Vm (k)∗ Vm (k)

k=0

which can never exceed 1 (i.e. γopt ≤ 1). A feasible solution for the H∞ estimation problem in Eq. (2.11) is therefore guaranteed when γ is chosen to be 1. Note that it is possible to directly demonstrate the feasibility of γ = 1. Using simple matrix


24

manipulation, it can be shown that for Lk = Hk and for γ = 1, Rk and Re,k have the same inertia for all k. 2.4.3.1.2 γopt ≥ 1: To show that γopt is indeed 1, an admissible sequence of disturbances and a valid initial condition should be constructed such that γ could be made arbitrarily close to 1 regardless of the filtering solution chosen. The necessary and sufficient conditions for the optimality of γopt = 1 are developed in the course of constructing this admissible sequence of disturbances. T T T ˆ Assume that ξˆ0 = W is the best estimate for the initial condition of the θˆ0 0 system in the approximate model of the primary path (Fig. 2.4). Moreover, assume that θˆ0 is indeed the actual initial condition for the secondary path in Fig. 2.4. The actual initial condition for the weight vector of the FIR filter in this approximate model is W0 . Then, h m(0) = H0 ξˆ0 =

h

Ds (0)h∗ (0) Cs (0) Ds (0)h∗ (0) Cs (0)

i i

" "

W0 θˆ0 ˆ0 W θˆ0

# + Vm (0)

(2.29)

#

where m(0) is the (derived) measurement at time k = 0. Now, if ∗ ˆ ˆ Vm (0) = Ds (0)h (0) W0 − W0 = KV (0) W0 − W0

(2.30)

(2.31)

then m(0) − H0 ξˆ0 = 0 and the estimate of the weight vector will not change. More specifically, Eqs. (2.16) and (2.17) reduce to the following simple updates ξˆ1 = F0 ξˆ0

(2.32)

sˆ(0|0) = L0 ξˆ0

(2.33)

which given L0 = H0 generates the estimation error ef,0 = sˆ(0|0) − s(0) = L0 ξˆ0 − L0 ξ0 ˆ 0 − W0 = Ds (0)h∗ (0) W = Vm (0)

(2.34)


25

Repeating a similar argument at k = 1 and 2, it is easy to see that if ˆ 0 − W0 = KV (1) W ˆ 0 − W0 Vm (1) = [Ds (1)h∗ (1) + Cs (1)Bs (0)h∗ (0)] W

(2.35)

and

ˆ Vm (2) = [Ds (2)h (2) + Cs (2)Bs (1)h (1) + Cs (2)As (1)Bs (0)h (0)] W0 − W0 ˆ 0 − W0 = KV (2) W (2.36) ∗

∗

∗

then m(k) − Hk ξˆk = 0,

for k = 1, 2

(2.37)

Note that when Eq. (2.37) holds, and with Lk = Hk , Eq. (2.17) reduces to sˆ(k|k) = Lk ξˆk = Hk ξˆk

(2.38)

and hence ef,k = sˆ(k|k) − s(k) = sˆ(k|k) − [m(k) − Vm (k)] = Hk ξˆk − [m(k) − Vm (k)] h i = Hk ξˆk − m(k) + Vm (k) = Vm (k)

for k = 1, 2

(2.39)

Continuing this process, KV (k), for 0 ≤ k ≤ M can be defined as 

KV (0)





Ds (0)

0

0

0

···

0



h(0)



      KV (1)   Cs (1)Bs (0) Ds (1) 0 0 ··· 0   h(1)          C (2)A (1)B (0) C (2)B (1) D (2) 0 · · ·   0 (2) K h(2) =  V s s s s s   s        .. .. .. ..        . . .     .  .. KV (M) h(M) . ··· Ds (M) 4

= ∆M ΛM

(2.40)


26

such that Vm (k), ∀ k, is an admissible disturbance. In this case, Eq. (2.11) reduces to M X

sup ξ0

Vm (k)∗ Vm (k)

k=0

ˆ (ξ0 − ξˆ0 )∗ Π−1 0 (ξ0 − ξ0 ) +

M X

Vm (k)∗ Vm (k)

k=0 M X

#

" =

ˆ 0 − W0 )∗ (W

sup ξ0

KV∗ (k)KV (k)

k=0

"

∗ ˆ ˆ (ξ0 − ξˆ0 )∗ Π−1 0 (ξ0 − ξ0 ) + (W0 − W0 )

M X

ˆ 0 − W0 ) (W # KV∗ (k)KV (k)

ˆ 0 − W0 ) (W

k=0

(2.41) From Eq. (2.40), note that M X

KV∗ (k)KV (k) = Λ∗M ∆∗M ∆M ΛM = k ∆M ΛM k22

(2.42)

k=0

and hence the ratio in Eq. (2.41) can be made arbitrarily close to one if lim k∆M ΛM k2 → ∞

M →∞

(2.43)

Eq. (2.43) will be referred to as the condition for optimality of γ = 1 for the filtering solution. Equation (2.43) can now be used to derive necessary and sufficient conditions for optimality of γ = 1. First, note that a necessary condition for Eq. (2.43) is lim kΛM k2 → ∞

M →∞

(2.44)

or equivalently lim

M →∞

M X

h∗ (k)h(k) → ∞

(2.45)

k=0

The h(k) that satisfies the condition in (2.45) is referred to as exciting [26]. Several sufficient conditions can now be developed. Since k∆M ΛM k2 ≥ σmin (∆M ) kΛM k2

(2.46)


27

one sufficient condition is that σmin (∆M ) > ,

∀ M, and > 0

(2.47)

Note that for LTI systems, the sufficient condition (2.47) is equivalent to the requirement that the system have no zeros on the unit circle. Another sufficient condition is that h(k)’s be persistently exciting, that is # " M 1 X lim σmin h(k)h∗ (k) > 0 M →∞ M k=0

(2.48)

which holds for most reasonable systems. 2.4.3.2

Prediction Case

The optimal value for γ can not be less than one in the prediction case. In the previous section we showed that despite using all available measurements up to and including ˆ 0 − W0 for time k, the sequence of the admissible disturbances, Vm (k) = KV (k) W k = 0, · · · , M (where KV (k) is given by Eq. (2.40)), prevented the filtering solution from achieving γ < 1. The prediction solution that uses only the measurements up to time k (not including k itself) can not improve over the filtering solution and therefore the energy gain γ is at least one. Next, it is shown that if the initial condition P0 is chosen appropriately (i.e. if it is small enough), then γopt = 1 can be guaranteed. Referring to the Lyapunov recursion of Eq. (2.65), the Riccati matrix at time k can be written as: # " ! !∗ k−1 k−1 Y Y I 0 Pk = Fj P0 Fj , Fj = Bs (j)h∗ (j) As (j) j=0 j=0

(2.49)

Defining ΨjA = As (j)As (j − 1) · · · As (0)

(2.50)

Eq. (2.49) can be written as # " #∗ " I 0 I 0 Pk = Pk−1 j P0 Pk−1 j (2.51) ∗ k ∗ k j=0 ΨA Bs (j)h (k−1−j) ΨA j=0 ΨA Bs (j)h (k−1−j) ΨA


28

From Theorem 2.2, Section 2.4.2, the condition for the existence of a prediction solution is (I − γ −2 Pk L∗k Lk ) > 0, or equivalently (γ 2 − Lk Pk L∗k ) > 0

(2.52)

Note that Lk = [ Ds (k)h∗ (k) Cs (k) ], and therefore Eq. (2.52) can be re-written as # " h i ∗ h(k)D (k) s γ 2 − Ds (k)h∗ (k) Cs (k) Pk >0 (2.53) ∗ Cs (k) Replacing for Pk from Eq. (2.51), and carrying out the matrix multiplications, Eq. (2.53) is equivalent to γ2

"

− "

h(k)Ds∗ (k) +

Pk−1

∗j ∗ ∗ j=0 h(k−1−j)Bs (j)ΨA Cs (k) ∗ Ψ∗k A Cs (k)

h(k)Ds∗ (k) +

Pk−1 j=0

∗ h(k−1−j)Bs∗ (j)Ψ∗j A Cs (k)

∗ Ψ∗k A Cs (k)

#∗ × P0 × # >0

(2.54)

Introducing 0∗

∗

h (k) = Ds h (k) +

k−1 X

Cs (k)ΨjA Bs (j)h∗ (k−1−j)

(2.55)

j=0

as the filtered version of the reference vector, h(k), Eq. (2.54) can be expressed as # " i h 0 h (k) γ 2 − h0 ∗ (k) Cs (k)ΨkA P0 >0 (2.56) ∗ Ψ∗k C (k) A s Selecting the initial value of the Riccati matrix, without loss of generality, as # " µI 0 (2.57) P0 = 0 αI and the Eq. (2.56) reduces to ∗

∗ γ 2 − µh0 (k)h0 (k) − αCs (k)ΨkA Ψ∗k A Cs (k) > 0

(2.58)

It is now clear that a prediction solution for γ = 1 exists if ∗ 1 − αCs (k)ΨkA Ψ∗k A Cs (k) µ< h0 ∗ (k)h0 (k)

(2.59)

Equation (2.59) is therefore the condition for optimality of γopt = 1 for the prediction solution.


29

2.4.4

Simplified Solution Due to γ = 1

2.4.4.1

Filtering Case:

The following shows that with Hk = Lk and γ = 1, the Riccati equation (2.14) is considerably simplified. To this end, apply the matrix inversion lemma, (A + BCD)−1 = A−1 − A−1 B[C −1 + DA−1 B]−1 DA−1 , to " # " # h i 0 Ip Hk Re,k = + Pk Hk∗ Hk∗ 0 −Iq Hk " with A =

Ip

0

#

" ,B=

Hk

#

h

, C = I, and D = Pk

(2.60)

i Hk∗ Hk∗ . It is easy to

0 −Iq Hk −1 verify that the term DA B is zero. Therefore # " # " h i Hk 0 Ip −1 ∗ ∗ − Pk Hk −Hk Re,k = −Hk 0 −Iq

(2.61)

In which case, h

∗ Kf,k Re,k Kf,k =

Fk Pk

Hk∗ Hk∗

i

" −1 Re,k

Hk

#

Hk

! Pk Fk∗

=0

for γ = 1 and for all k. Thus the Riccati recursion (2.14) reduces to the Lyapunov recursion Pk+1 = Fk Pk Fk∗ with P0 = Π0 > 0. Partitioning the Riccati matrix Pk in block matrices conformable with the block matrix structure of Fk , (2.14) yields the following simple update   P11,k+1 = P11,k , P11,0 = Π11,0           P = P A (k) + P h(k)B ∗ (k), P =Π 12,k+1

12,k

s

11,k

s

12,0

12,0

     ∗  P22,k+1 = Bs (k)h(k)∗ P11,k h(k)Bs∗ (k) + As (k)P12,k h(k)Bs∗ (k)+      Bs (k)h∗ (k)P12,k A∗s (k) + As (k)P22,k A∗s (k), P22,0 = Π22,0 The filtering solution can now be summarized in the following theorem:

(2.62)

2.5. IMPORTANT REMARKS

30

Theorem 2.3: Consider the system described by Equations (2.5)-(2.7), with Lk = Hk . If the optimality condition (2.43) is satisfied, the H∞ -optimal filtering solution achieves γopt = 1, and the central H∞ -optimal filter is given by ξˆk+1 = Fk ξˆk + Kf,k m(k) − Hk ξˆk , ξˆ0 = 0 (2.63) −1 sˆ(k|k) = Lk ξˆk + (Lk Pk Hk∗ ) RHe,k m(k) − Hk ξˆk (2.64) −1 with Kf,k = (Fk Pk Hk∗ ) RHe,k and RHe,k = Ip + Hk Pk Hk∗ , where Pk satisfies the Lyapunov recursion

Pk+1 = Fk Pk Fk∗ , P0 = Π0 .

(2.65)

Proof: follows from the discussions above. 2.4.4.2

Prediction Case:

Referring to Eq. (2.26), it is clear that for γ = 1 and for Lk = Hk , the coefficient Kp,k will reduce to Fk Pk Hk∗ . Therefore, the prediction solution can be summarized as follows: Theorem 2.4: Consider the system described by Equations (2.5)-(2.7), with Lk = Hk . If the optimality conditions (2.43) and (2.59) are satisfied, and with P0 as defined in Eq. (2.57), the H∞ -optimal prediction solution achieves γopt = 1, and the central filter is given by ˆ ˆ ˆ ξk+1 = Fk ξk + Kp,k m(k) − Hk ξk , ξˆ0 = 0 (2.66) sˆ(k) = Lk ξˆk

(2.67)

with Kp,k = Fk Pk Hk∗ where Pk satisfies the Lyapunov recursion (2.65). Proof: follows from the discussions above.

2.5

Important Remarks

The main idea in the EBAF algorithm can be summarized as follows. At a given time k, use the available information on; (a) measurement history, e(i) for 0 ≤ i ≤ k, (b) control history, u(i) for 0 ≤ i < k, (c) reference signal history, x(i) for 0 ≤ i ≤ k, (d)

2.5. IMPORTANT REMARKS

31

the model of the secondary path and the estimate of its initial condition, and (e) the pre-determined length of the adaptive FIR filter to produce the best estimate of the actual output of the primary path, d(k). The key premise is that if d(k) is accurately estimated, then the inputs u(k) can be generated such that d(k) is canceled. The objective of the EBAF algorithm is to make y(k) match the optimal estimate of d(k) (see Fig. 2.3). For the adaptive filtering problem in Fig. 2.1 , however, adaptation algorithm only has direct access to the weight vector of the adaptive FIR filter. Because of this practical constraint, the EBAF algorithm adapts the weight vector in the adaptive FIR filter according to the estimate of the optimal weight vector given by Eqs. (2.63) or (2.66) (for the filtering, or prediction solutions, respectively). Note T T T ˆ (k) θˆ (k) . The error analysis for this adaptive algorithm is discussed that ξˆ = W k

in Section 2.7. Now, main features of this algorithm can be described as follows: 1. The estimation-based adaptive filtering (EBAF) algorithm yields a solution that only requires one Riccati recursion. The recursion propagates forward in time, and does not require any information about the future of the system or the reference signal (thus allowing the resulting adaptive algorithm to be real-time implementable). This has come at the expense of restricting the controller to an FIR structure in advance. ∗ 2. With Kf,k Re,k Kf,k = 0, Pk+1 = Fk Pk Fk∗ is the simplified Riccati equation,

which considerably reduces the computational complexity involved in propagating the Riccati matrix. Furthermore, this Riccati update always generates a non-negative definite Pk , as long as P0 is selected to be positive definite (see Eq. (2.65)). 3. In general, the solution to an H∞ filtering problem requires verification of the fact that Rk and Re,k are of the same inertia at each step (see Eq. (2.13)). In a p similar way, the prediction solution requires that all sub-matrices of Rkp and Re,k

have the same inertia for all k (see Eq. (2.18)). This can be a computationally expensive task. Moreover, it may lend to a breakdown in the solution if the condition is not met at some time k. The formulation of the problem eliminates

2.6. IMPLEMENTATION SCHEME FOR EBAF ALGORITHM

32

the need for such checks, as well as the potential breakdown of the solution, by providing a definitive answer to the feasibility and optimality of γ = 1. 4. When [ As (k), Bs (k), Cs (k), Ds (k) ] = [ 0, 0, 0, I ] for all k, (i.e. the output of the FIR filter directly cancels d(k) in Figure 2.1), then the filtering/prediction results reduce to the simple Normalized-LMS/LMS algorithms in Ref. [26] as expected. 5. As mentioned earlier, there is no need to verify the solutions at each time step, so the computational complexity of the estimation based approach is O(n3 ) (primarily for calculating Fk Pk FK∗ ), where n = (N + 1) + Ns

(2.68)

where (N +1) is the length of the FIR filter, and Ns is the order of the secondary path. The special structure of Fk however reduces the computational complexity to O(Ns3 + Ns N), i.e. cubic in the order of the secondary path, and linear in the length of the FIR filter (see Eq. (2.62)). This is often a substantial reduction in the computation since Ns N. Note that the computational complexity for FxLMS is quadratic in Ns and linear in N.

2.6

Implementation Scheme for EBAF Algorithm

Three sets of variables are used to describe the implementation scheme: 1. Best available estimate ofa variable: Referring to Eqs. (2.16) and (2.19), and ˆ T (k) θˆT (k) , W ˆ (k) can be defined as the estimate noting the fact that ξˆT = W k

ˆ of the weight vector, and θ(k) as the secondary path state estimate in the approximate model of the primary path. 4 ˆ (k) as the 2. Actual value of a variable: Referring to Fig. 2.1, define u(k) = h∗ (k)W

actual input to the secondary path, y(k) as the actual output of the secondary path, and d(k) as the actual output of the primary path. Note that d(k) and


33

y(k) are not directly measurable, and that at each iteration the weight vector ˆ (k). in the adaptive FIR filter is set to W 3. Adaptive algorithm’s internal copy of a variable: Recall that in Eq. (2.4), y(k) is used to construct the derived measurement m(k). Since y(k) is not directly available, the adaptive algorithm needs to generate an internal copy of this variable. This internal copy (referred to as ycopy (k)) is constructed by applying u(k) (the actual control signal) to a model of the secondary path inside the adaptive algorithm. The initial condition for this model is θcopy (0). In other words, the derived measurement is constructed as follows m(k) = e(k) + ycopy (k)

(2.69)

θcopy (k + 1) = As (k)θcopy (k) + Bs (k)u(k)

(2.70)

ycopy (k) = Cs (k)θcopy (k) + Ds (k)u(k)

(2.71)

where

ˆ (k), Given the identified model for the secondary path and its input u(k) = h∗ (k)W the adaptive algorithm’s copy of y(k) will be exact if the actual initial condition of the secondary path is known. Obviously, one can not expect to have the exact knowledge of the actual initial condition of the secondary path. In the next section, however, it is shown that when the secondary path is linear and stable, the contribution of the initial condition to its output decreases to zero as k increases. Therefore, the internal copy of y(k) will converge to the actual value of y(k) over time. Now, the implementation algorithm can be outlined as follows; ˆ = 0 as the initial guess for the state vector in the ˆ (0) = 0 and θ(0) 1. Start with W approximate model of the primary path. Also assume that θcopy (0) = 0, and h(0) = [ x(0) 0 · · · 0 ]T . The initial value for the Riccati matrix is P0 which is chosen to be block diagonal. The role of P0 is similar to the learning rate in LMS-based adaptive algorithms (see Section 5.3.2).


34

2. If 0 ≤ k ≤ M (finite horizon): (a) Form the control signal ˆ (k) u(k) = h∗ (k)W

(2.72)

to be applied to the secondary path. Note that applying u(k) to the secondary path produces y(k) = Cs (k)θ(k) + Ds (k)u(k)

(2.73)

at the output of the secondary path. This in turn leads to the following error signal measured at time k: e(k) = d(k) − y(k) + Vm (k)

(2.74)

which is available to the adaptive algorithm to perform the state update at time k. (b) Propagate the state estimate and the internal copy of the state of the secondary path as follows  " #  ˆ W (k + 1)     = ˆ + 1) θ(k   θcopy (k + 1) i h " # h ˆ Kf,k Cs (k) Fk + Kf,k [0 − Cs (k)] Kf,k W (k)       θ(k)  ˆ   + h i θcopy (k) As (k) 0 (Bs (k)h∗ (k) 0

i  e(k)  (2.75)

where e(k) is the error sensor measurement at time k given by Eq. (2.74), and Kf,k = Fk Pk Hk∗ (I + Hk Pk Hk∗ )−1 (see Theorem 2.3). Note that for the prediction-based EBAF algorithm Kf,k should be replaced with Kp,k = Fk Pk Hk∗ .

2.7. ERROR ANALYSIS

35

(c) update the Riccati matrix Pk using the Lyapunov recursion " # P11 P12,k+1 = ∗ P12,k+1 P22,k+1 " #" #" I 0 P11 P12,k I Bs (k)h∗ (k) As (k)

∗ P12,k P22,k

0

#∗

Bs (k)h∗ (k) As (k) (2.76)

Pk+1 will be used in (2.75) to update the state estimate. 3. Go to 2.

2.7

Error Analysis

In Section 2.6, it is pointed out that the proposed implementation scheme can deviate from an H∞ -optimal solution for two main reasons: 1. The error in initial condition of the secondary path which can cause ycopy to be different from y(k). 2. The additional error in the cancellation of d(k) due to the fact that y(k) can not be set to sˆ(k|k) (or sˆ(k)). All one can do is to set the weight vector in the ˆ (k). adaptive FIR filter to be W Here, both errors are discussed in detail.

2.7.1

Effect of Initial Condition

As earlier discussions indicate, the secondary path in Fig. 2.1 is assumed to be linear. For a linear system the output at any given time can be decomposed into two components: the zero-input component which is associated with the portion of the output solely due to the initial condition of the system, and the zero-state component which is the portion of the output solely due to the input to the system.

2.7. ERROR ANALYSIS

36

For a stable system, the zero-input component of the response will decay to zero for large k. Therefore, any difference between ycopy (k) and y(k) (which with a known input to the secondary path can only be due to the unknown initial condition) will go to zero as k grows. In other words, exact knowledge of the initial condition of the secondary path does not affect the performance of the proposed EBAF algorithm for sufficiently large k.

2.7.2

Effect of Practical Limitation in Setting y(k) to sˆ(k|k) (ˆ s(k))

As pointed out earlier, the physical setting of the adaptive control problem in Fig. 2.1 ˆ (k). only allows for the weight vector in the adaptive FIR filter to be adjusted to W In other words, the state of the secondary path can not be set to a desired value at each step. Instead, θk evolves based on its initial condition and the control input, u(k), that we provide. Assume that θ(k) is the actual state of the secondary path at time k. The actual output of the secondary path is then ˆ (k) + Cs (k)θ(k) y(k) = Ds (k)h∗ (k)W

(2.77)

which leads to the following cancellation error ˆ (k) + Cs (k)θ(k) d(k) − y(k) = d(k) − Ds (k)h∗ (k)W

(2.78)

ˆ to For the prediction solution of Theorem 2.4, adding the zero quantity ±Cs (k)θ(k) the right hand side of Equation (2.78), and taking the norm of both sides, ˆ ˆ (k) + Cs (k)θ(k) ± Cs (k)θ(k) kd(k) − y(k)k = k d(k) − Ds (k)h∗ (k)W k ˆ ˆ − θ(k) k ˆ (k) − Cs (k)θ(k) + Cs (k) θ(k) = k d(k) − Ds (k)h∗ (k)W Therefore,

ˆ − θ(k) k k Cs (k) θ(k)

kd(k) − y(k)k k d(k) − sˆ(k) k ≤ + M M M X X X ∗ ∗ −1 ˜ ∗ ∗ −1 ˜ ∗ ˜0 + ˜ ˜ V (k)V (k) Π + V (k)V (k) Π + Vm (k)Vm (k) ξ˜0∗ Π−1 ξ ξ ξ ξ ξ m 0 m 0 0 m 0 0 m 0 0 k=0

k=0

k=0

(2.79)

2.7. ERROR ANALYSIS

37

where ξ˜0 = (ξ0 − ξˆ0 ) and ξk is defined in Eq. (2.5). Note that the first term in the right hand side of Eq. (2.79) is the prediction error energy gain (see Eq. (2.12)). Therefore, the energy gain of the cancellation error with the prediction-based EBAF exceeds the error energy gain of the H∞ optimal prediction solution by the second term on the right hand side of Eq. (2.79). It can be shown that when the primary inputs h(k) are persistently exciting (see Eq. (2.48)), the dynamics for the state estimation error, ˆ θ(k)−θ(k), are internally stable which implies that the second term on the right hand side of Eq. (2.79) is bounded for all M, and in the limit when M → ∞∗ . When Ds (k) = 0 for all k, an implementation of the filtering solution that utilizes the most recent measurement, m(k), is feasible. In this case, the filtering solution in Eqs. (2.16)-(2.17) can be written as follows: ξˇk|k = ξˆk + Pk Hk∗ (Ip + Hk Pk Hk∗ )−1 m(k) − Hk ξˆk

(2.80)

ξˆk+1 = Fk ξˇk|k

(2.81)

sˆ(k|k) = Lk ξˇk|k

(2.82)

where the weight vector update in the adaptive FIR filter follows Eq. (2.80). With a derivation identical to the one for prediction solution, it can be shown that the performance bound in this case is

ˆ k Cs (k) θ(k|k) − θ(k) k

kd(k) − y(k)k k d(k) − sˆ(k|k) k ≤ + M M M X X X ∗ −1 ˜ ∗ ∗ −1 ˜ ∗ ∗ −1 ˜ ∗ ˜ ˜ ˜ ξ0 Π0 ξ0 + Vm (k)Vm (k) ξ0 Π0 ξ0 + Vm (k)Vm (k) ξ0 Π0 ξ0 + Vm (k)Vm (k) k=0

k=0

k=0

(2.83) An argument similar to the prediction case shows that the second term on the right hand side has a finite gain as well. ∗

Reference [24] shows that if the exogenous disturbance is assumed to be a zero mean white noise process with unit intensity, and independent of the initial condition of the system ξ0 , then the terminal state estimation error variance satisfies E(ξk − ξˆk )(ξk − ξˆk )∗ ≤ Pk

2.8. RELATIONSHIP TO THE NORMALIZED-FXLMS/FXLMS ALGORITHMS

2.8

38

Relationship to the Normalized-FxLMS/FxLMS Algorithms

In this section, it will be shown that as k → ∞, the gain vector in the predictionbased EBAF algorithm converges to the gain vector in the classical Filtered-X LMS (FxLMS) algorithm. Thus, FxLMS is an approximation to the steady-state EBAF. The error terms in the two algorithms are shown to be different (compare Eqs. (2.89) and (2.2)). Therefore it is expected that the prediction-based EBAF demonstrate superior transient performance compared to the FxLMS algorithm. Simulation results in the next section agree with this expectation. The fact that the gain vectors asymptotically coincide, agrees with the fact that the derivation of the FxLMS algorithm relies on the assumption that the adaptive filter and the secondary path are interchangeable which can only be true in the steady state. Similar results are shown for the connection between the filtering-based EBAF and the Normalized FxLMS adaptive algorithms. For the discussion in this section, the secondary path is assumed, for simplicity, to be LTI, i.e. [ As , Bs , Cs , Ds ]. Note that for the LTI system, ΨkA in Eq. (2.50) reduces to Aks . The Riccati matrix Pk in Eq. (2.51) can then be rewritten as " # " #∗ I 0 I 0 Pk = Pk−1 j P0 Pk−1 j ∗ k ∗ k j=0 As Bs h (k−1−j) As j=0 As Bs h (k−1−j) As

(2.84)

Eq. (2.84) will be used in establishing the proper connections between the filtered/predicted solutions of Section 2.4 and the conventional Normalized-FxLMS/FxLMS algorithms.

2.8.1

Prediction Solution and its Connection to the FxLMS Algorithm

To study the asymptotic behavior of the state estimate update, note that for an stable secondary path Aks → 0 as k → ∞. Therefore, using Eq. (2.84) # " #∗ " I 0 I 0 Pk → Pk−1 j P0 Pk−1 j as k → ∞(2.85) ∗ ∗ j=0 As Bs h (k−1−j) 0 j=0 As Bs h (k−1−j) 0


" which for P0 = " Pk → Pk−1

P11 (0) P12 (0)

# results in

P21 (0) P22 (0) I

j ∗ j=0 As Bs h (k−1−j)

39

#

"

P11 (0) Pk−1

I

j ∗ j=0 As Bs h (k−1−j)

#∗ as k → ∞ (2.86)

Selecting P11 (0) = µI as in Eq. (2.57), and noting the fact that Kp,k = Fk Pk Hk∗ (Theorem 2.4), it is easy to see that as k → ∞ # " !∗ k−1 X I ∗ j ∗ Kp,k → µ Pk Ds h (k) + Cs As Bs h (k−1−j) j ∗ j=0 As Bs h (k−j) j=0 # " I h0 (k) → µ Pk (2.87) j ∗ A B h (k−j) s j=0 s and therefore the state estimate update in Theorem 2.4 becomes # " # " #" ˆ (k+1) ˆ (k) W I 0 W = + ˆ ˆ θ(k+1) Bs h∗ (k) As θ(k) " # h0 (k) ∗ ˆ ˆ µ Pk m(k) − Ds h (k)W (k) − Cs θ(k) (2.88) j ∗ 0 j=0 As Bs h (k−j)h (k) Thus,the following update for the weight vector is derived ˆ ˆ (k+1) = W ˆ (k) + µh0 ∗ (k) m(k) − Ds h∗ (k)W ˆ (k) − Cs θ(k) W

(2.89)

Note that m(k) = e(k) + ycopy (k) (see Eq. (2.69)), and hence the difference between the limiting update rule of Eq. (2.89) (i.e. the prediction EBAF algorithm), and the classical FxLMS algorithm of Eq. (2.2) will be the error term used by these algorithms. More specifically, e(k) in the FxLMS algorithm is replaced with the following modified error (using Eq. (2.71)): ˆ = e(k) + Cs θcopy (k) − Cs θ(k). ˆ ˆ (k) − Cs θ(k) e(k) + ycopy (k) − Ds h∗ (k)W Note that if y(k) is directly measurable, then the modified error will be h i ˆ ˆ (k) − Cs θ(k) e(k) + y(k) − Ds h∗ (k)W

(2.90)


40

The condition for optimality of γ = 1 in the prediction case (see Eq. (2.59)), can also be simplified for stable LTI secondary path as k → ∞. Rewriting the optimality condition for the prediction solution, Eq. (2.59), as µ
0, then the central H∞ -optimal prediction solution to the linearized problem is obtained for γp = 1, and is given by ξˆk+1 = Fk ξˆk + Kp,k m(k) − Hk ξˆk , ξˆ0 = 0

(4.25)

ˆs(k) = Lk ξˆk

(4.26)

with Kp,k = Fk Pk Hk∗ where Pk satisfies the Lyapunov recursion (4.24).

4.3

Simulation Results

The implementation scheme for the EBAF algorithm in multi-channel case is identical to the implementation scheme in the single-channel case (see Chapter 2 for the FIR case, and Chapter 3 for the IIR case), and therefore it is not repeated here.

4.3. SIMULATION RESULTS

4.3.1

86

Active Vibration Isolation

The Vibration Isolation Platform (VIP) (see Figure 4.4 and 4.5) is an experimental set up which is designed to capture the main features of a real world payload isolation and pointing problem. Payload isolation refers to the vibration isolation of payload structures with instruments or equipments requiring a very quiet mounting [1]. VIP is designed such that the base supporting the payload (middle mass in Figure 4.5) can emulate spacecraft dynamics. Broadband as well as narrowband disturbances can be introduced to the middle mass (emulating real world vibration sources such as solar array drive assemblies, reaction wheels, control moment gyros, crycoolers, and other disturbance sources that generate on-orbit jitter) via a set of three voice coil actuators. The positioning of a second set of voice coil actuators allows for the implementation of an adaptive/active isolation system. More specifically, the Vibration Isolation Platform consists of the following main components: 1. Voice-Coil Actuators: 6 voice-coil actuators are mounted on the middle mass casing. Three of these actuators (positioned 120 degrees apart on a circle of radius 4.400 inches) are used to shake the middle mass and act as the source of disturbance to the platform. They can also be used to introduce some desired dynamics for the middle mass that supports the payload. As shown in Figure 4.5, these actuators act against the ground. The other three actuators (placed 120 degrees apart on a circle of radius 4.000 inches) act against the middle mass and are used to isolate the payload (top mass) from the motions of the middle mass. Note that the two circles on which control and disturbance actuators are mounted are concentric, and one set of actuators is rotated in the horizontal plane by 60 degrees with respect to the other. 2. Sensors: VIP is equipped with two sets of sensors, (a) Position Sensors: Each actuator is equipped with a colocated position (gap) sensor which is physically inside the casing of the actuator. Three additional position sensors are used as truth sensors (Figure 4.5) to measure the displacement of the payload in the inertial frame.


87

(b) Load Cells: Three load cells are used to measure the interaction forces between the middle mass and the payload. These sensors are colocated with the point of contact of the control actuators and the payload. It is important to note that any interaction force between the payload and the rest of the VIP system is transfered via these load cells. A state-space model for the VIP platform is identified using the FORSE system identification software developed at MIT [30]. The detailed description of the identification process, the identified model, and the model validation process is discussed in Appendix C. Figure 4.6 shows the singular value plots for the MIMO transfer functions from control ([u])/disturbance ([d]) actuators to the load cells ([lc]) and scoring sensors ([sc]). In all simulations that follow, the length of the adaptive FIR filters, i.e. L in Equation 4.2, is 4 (unless stated otherwise). This length is found to be sufficient for an acceptable performance of the adaptive algorithms. The sampling frequency for all the simulations in this section is 1000 Hz. Furthermore, all measurements are subject to band limited white noise with power 0.008. Figures 4.7 and 4.8 show the reading of the scoring sensors (i.e. the variations of the payload from the equilibrium position in the inertial frame) for the multi-channel implementation of the EBAF and the FxLMS adaptive algorithms, respectively. Disturbance actuators apply sinusoidal excitation of amplitude 0.1 Volts at 4 Hz to the middle mass. The phase for the excitation of the disturbance actuator #1 is assumed to be zero, while disturbance actuators #2 and #3 are 22.5 and 45 degrees out of phase with the first actuator. Only a noisy measurement of the primary disturbance is assumed to be available to the adaptive algorithms. The signal to noise ratio for the available reference signal is 3.0. For simulations in Figures 4.7 and 4.8 the control signal starts at t = 30 seconds. Figure 4.7 shows that the amplitude of the transient vibrations of the payload under the EBAF adaptive algorithm (for 30 ≤ t ≤ 60) does not exceed that of the open loop vibrations. In contrast, the amplitude of the transient vibrations under the FxLMS, Figure 4.8, exceeds twice the amplitude of the open loop vibrations in the system. For a smaller amplitude during transient


88

vibrations, the adaptation rate for the FxLMS algorithm should be reduced. This will result in an even slower convergence of the adaptive algorithm. Note that, for the results in Figure 4.8, the adaptation rate is 0.0001. Even with this adaptation rate, FxLMS algorithm requires approximately 20 more seconds (compared to the EBAF case) to converge to its steady state value. In the steady state, the EBAF algorithm achieves a 20 times reduction in the amplitude of the payload vibrations. For the FxLMS algorithm in this case, the measured reduction is approximately 16 times. Figures 4.9 and 4.10 show the reading of the scoring sensors when the primary disturbances are multi-tone sinusoids. The primary disturbance consists of sinusoidal signal of amplitudes 0.1 and 0.2 volts at 4 and 15 Hz, respectively. As in the single tone case, both components of the excitation for the disturbance actuator #1 are assumed to have zero phase. Each sinusoidal component of the excitation for the disturbance actuator #2 (#3) is assumed to have a phase lag of 22.5 (45) degrees with respect to the corresponding component of the excitation in actuator #1. Figures 4.9 and 4.10 demonstrate a trend similar to that discussed for the single tone scenario. For the FxLMS algorithm, a trade off between the amplitude of the transient vibrations of the payload and the speed of the convergence exists. The adaptation rate here is the same as the single tone case. Slower adaptation rates can reduce the amplitude of the transient vibrations at the expense of the speed of the convergence. For the EBAF algorithm, however, better transient behavior and faster convergence are observed. In the steady state, the EBAF algorithm provides a 15 times reduction in the amplitude of the vibrations of the payload. For the FxLMS algorithm a 9 times reduction is recorded. In Fig. 4.11 the effect of feedback contamination in the performance of the EBAF algorithm is examined. As in the previous simulations, control actuators are switched on at t = 30 seconds. The reference signal available to the adaptation algorithm is the output of the load cells which measure the forces transfered to the payload. Obviously, load cell measurements contain the effect of both primary disturbances (single tone sinusoids at 4 Hz for this example), and control actuators (and hence the classical feedback contamination problem). Here, no special measure to counter feedback contamination is taken. Figure 4.11 shows that an average of 4 times reduction in the


89

magnitude of the vibrations transfered to the payload is achieved (i.e. a degraded performance when compared to the case without feedback contamination). The EBAF algorithm, however, maintains its stability in the face of feedback contamination without any additional measures, hence exhibiting robustness of the algorithm to the contamination of the reference signal. Note that for the FxLMS adaptive algorithm, with the adaptation rate similar to that in Figure 4.10, feedback contamination leads to an unstable behavior. The adaptation rate should be reduced substantially (hence extremely slow convergence of the FxLMS algorithm), in order to recover the stability of the adaptation scheme. The simulations in this section have shown that the multi-channel implementation of the estimation-based adaptive filtering algorithm provides the same advantages observed for the single channel case. More specifically, the multi-channel EBAF algorithm achieves desirable transient behavior and fast convergence without compromising steady-state performance of the adaptive algorithm. It also demonstrates robustness to feedback contamination. It is important to note that, the above mentioned advantages are achieved by an approach which is essentially identical to the single-channel version of the algorithm.

4.3.2

Active Noise Cancellation

Consider the one dimensional acoustic duct shown in Figure 2.5. Here, disturbances enter the duct via Speaker #1. The objective of the multi-channel noise cancellation is to use both available speakers to simultaneously cancel the effect of the incoming disturbance at Microphones #1 and #2. The control signal is supplied to each speaker via an adaptive FIR filter (i.e. in the case of Speaker #1 added to the primary disturbance). Figure 4.12 shows the output of the microphones when the primary disturbance (applied to Speaker #1) is a multi-tone sinusoid with 150 Hz and 200 Hz frequencies. The length of each FIR filter in this simulation is 8. For the first two seconds the controller is off (i.e both adaptive filters have zero outputs). At t = 2.0 seconds the controller is switched on. The initial value for the Riccati matrix is P0 = diag(0.0005I2(N +1) , 0.00005INs ) (where N + 1 is the length of each FIR filter,


90

and Ns is the order of the secondary path). It is clear that the error at Microphone #2 is effectively canceled in 0.2 seconds. For Microphone #1 however the cancellation time is approximately 5 seconds. A 30 times reduction in disturbance amplitude is measured at Microphone #2 in approximately 10 seconds. For Microphone #1 this reduction is approximately 15 times. Note that the distance between Speaker #2 and Microphone #1 (46 inches) is much greater than the distance between Speaker #1 and Microphone #1 (6 inches). Due to this physical constraint, Speaker #2 alone, is not enough for an acceptable noise cancellation at both microphones. The experimental data in Figures 2.9 and 2.10 in which the result of a single-channel implementation of the EBAF algorithm, aimed at noise cancellation at Microphone #2, was shown, confirm this observation. Using a multi-channel approach however, allows for a substantial reduction in the amplitude of the measured noise at both microphones. Nevertheless, noise cancellation at the position of Microphone #1 tends to be slower than the noise cancellation at the position of Microphone #2. A similar scenario with band limited white noise as the primary disturbance is shown in Fig. 4.13. Here the length of each FIR filter is 32. The performance of the adaptive multi-channel noise cancellation problem in the frequency domain is shown in Fig. 4.14. Once again the cancellation at Microphone #2 is superior to that at Microphone #1.


Primary Source ref (k)

91

J

Vm (k)

M

Primary Path

d(k) +

+ + −

Feedback Path

e(k) +

+

− y(k) Adaptive Filter

K

Secondary Path

u(k)

x(k)

Fig. 4.1: General block diagram for a multi-channel Active Noise Cancellation (ANC) problem

Primary Path Modeling Error x(k)

A Copy Of Secondary Path

An FIR Filter J

K Secondary Path

Adaptive FIR Filter u(k)

Vm (k) d(k) + + M + −

+

e(k)

y(k)

Adaptation Algorithm

Fig. 4.2: Pictorial representation of the estimation interpretation of the adaptive control problem: Primary path is replaced by its approximate model


x(k)

A Copy Of Adaptive Filter

92

A Copy Of Secondary Path

Vm (k)

Ds (k) A Matrix Of Adaptive Filters

Bs (k)

+

Z −1

Cs (k)

As (k)

Fig. 4.3: Approximate Model for Primary Path

Fig. 4.4: Vibration Isolation Platform (VIP)

+

+ d(k)

m(k)


93

Fig. 4.5: A detailed drawing of the main components in the Vibration Isolation Platform (VIP). Of particular importance are: (a) the platform supporting the middle mass (labeled as component #5), (b) the middle mass that houses all six actuators (of which only two, one control actuator and one disturbance actuator) are shown (labeled as component #11), and (c) the suspension springs to counter the gravity (labeled as component #12). Note that the actuation point for the control actuator (located on the left of the middle mass) is colocated with the load cell (marked as LC1). The disturbance actuator (located on the right of the middle mass) actuates against the inertial frame.


94

SVD for [d]−>[lc]

SVD for [u]−>[lc]

Volts/volts

0

0

10

10

−5

−5

10

10 0

2

10

0

10 SVD for [d]−>[sc]

10 SVD for [u]−>[sc]

0

Volts/volts

2

10

0

10

10

−5

−5

10

10 0

2

10

10 Frequency (Hz)

0

2

10

10 Frequency (Hz)

Fig. 4.6: SVD of the MIMO transfer function


95

Scoring Sensor Readouts - EBAF

e(1)

10 0

−10 0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60

80

100

120

e(2)

10 0

−10

e(3)

10 0

−10

Time (sec.)

Fig. 4.7: Performance of a multi-channel implementation of EBAF algorithm when disturbance actuators are driven by out of phase sinusoids at 4 Hz. The reference signal available to the adaptive algorithm is contaminated with band limited white noise (SNR=3). The control signal is applied for t ≥ 30 seconds.


96

Scoring Sensor Readouts - FxLMS

e(1)

10 0

−10 0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60

80

100

120

e(2)

10 0

−10

e(3)

10 0

−10

Time (sec.)

Fig. 4.8: Performance of a multi-channel implementation of FxLMS algorithm when simulation scenario is identical to that in Figure 4.7.


97

Scoring Sensor Readouts - EBAF

e(1)

10 0

−10 0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60

80

100

120

e(2)

10 0

−10

e(3)

10 0

−10

Time (sec.)

Fig. 4.9: Performance of a multi-channel implementation of EBAF algorithm when disturbance actuators are driven by out of phase multi-tone sinusoids at 4 and 15 Hz. The reference signal available to the adaptive algorithm is contaminated with band limited white noise (SNR=4.5). The control signal is applied for t ≥ 30 seconds.


98

Scoring Sensor Readouts - FxLMS

e(1)

10 0

−10 0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60

80

100

120

e(2)

10 0

−10

e(3)

10 0

−10

Time (sec.)

Fig. 4.10: Performance of a multi-channel implementation of FxLMS algorithm when simulation scenario is identical to that in Figure 4.9.


99

Scoring Sensors Readouts for Single-Tone at 4 Hz With Feedback Contamination

e(1)

5

0

−5 0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60

80

100

120

e(2)

5

0

−5

e(3)

5

0

−5

Time (sec.)

Fig. 4.11: Performance of a Multi-Channel implementation of the EBAF for vibration isolation when the reference signals are load cell outputs (i.e. feedback contamination exists). The control signal is applied for t ≥ 30 seconds.


100

Multi-Channel Active Noise Cancellation in Acoustic Duct Microphone #1 (Volts)

4 2 0 −2 −4 −6

0

1

2

3

4

0

1

2

3

4

5

6

7

8

9

10

5

6

7

8

9

10

Microphone #2 (Volts)

4

2

0

−2

−4

Time (sec.)

Fig. 4.12: Performance of the Multi-Channel noise cancellation in acoustic duct for a multi-tone primary disturbance at 150 and 200 Hz. The control signal is applied for t ≥ 2 seconds.


101

Multi-Channel Active Noise Cancellation in Acoustic Duct Microphone #1 (Volts)

15 10 5 0 −5 −10

0

1

2

3

4

0

1

2

3

4

5

6

7

8

9

10

5

6

7

8

9

10

Microphone #2 (Volts)

10

5

0

−5

−10

Time (sec.)

Fig. 4.13: Performance of the Multi-Channel noise cancellation in acoustic duct when the primary disturbance is a band limited white noise. The control signal is applied for t ≥ 2 seconds.


102

Speaker #1 → Microphone #1

Oloop Cloop

Magnitude

0

10

−1

10

−2

10

2

3

10

10

Speaker #1 → Microphone #2

Oloop Cloop

Magnitude

0

10

−1

10

−2

10

2

10

3

10

Fig. 4.14: Closed-loop vs. open-loop transfer functions for the steady state performance of the EBAF algorithm for the simulation scenario shown in Figure 4.13.

4.4. SUMMARY

4.4

103

Summary

The estimation-based synthesis and analysis of multi-channel adaptive (FIR) filters is shown to be identical to the single-channel case . Simulations for a 3-input/3-output Vibration Isolation Platform (VIP), and a multi-channel noise cancellation in the one dimensional acoustic duct are used to demonstrate the feasibility of the estimationbased approach. The new estimation-based adaptive filtering algorithm is shown to provide both faster convergence (with an acceptable transient behavior), and improved steady state performance when compared to a multi-channel implementation of the FxLMS algorithm.

Chapter 5 Adaptive Filtering via Linear Matrix Inequalities In this chapter Linear Matrix Inequalities (LMIs) are used to synthesize adaptive filters (controllers). The ability to cast the synthesis problem as LMIs is a direct consequence of the systematic nature of the estimation-based approach to the design of adaptive filters proposed in Chapters 2 and 3 of this thesis. The question of internal stability of the overall system is directly addressed as a result of the Lyapunovbased nature of the LMIs formulation. LMIs also provide a convenient framework for the synthesis of multi-objective (H2 /H∞ ) control problems. This chapter describes the process of augmenting the H∞ criterion that serves as the center piece of the estimation-based adaptive filtering algorithm with H2 performance constraints, and investigates the characteristics of the resulting adaptive filter. As in Chapters 2 and 3, an Active Noise Cancellation (ANC) scenario is used to study the main features of the proposed formulation.

5.1

Background

A detailed discussion of the estimation-based approach to the design of adaptive filters (controllers) is presented earlier in Chapters 2 and 3. The discussion here will 104

5.1. BACKGROUND

105

be kept brief and serves more as a notational introduction. Figure 5.1 is a block diagram representation of the active noise cancellation problem, originally shown in Figure 2.1. Clearly, the objective here is to generate a control signal u(k) such that the output of the secondary path, y(k), is close to the output of the primary path, d(k). In Chapter 2, it is shown that an estimation interpretation of the adaptive filtering (control) problem can be used to formulate an equivalent estimation problem. It is this equivalent estimation problem that admits LMIs formulation. To mathematically describe the equivalent estimation problem, state space models for the adaptive filter and the secondary path are needed. As in Chapter 2, [As (k), Bs (k), Cs (k), Ds (k)] is the state space model for the secondary path. The state variable for the secondary path is θ(k). For the adaptive filter the weight vector, W (k) = [ w0 (k) w1 (k) · · · wN (k) ]T , will be treated as the state variable. The state space description for the approximate model of the primary path can then be described as:

"

W (k + 1)

#

" =

θ(k + 1)

I(N +1)×(N +1) Bs (k)h∗k

0

#"

As (k)

W (k)

#

θ(k)

ξk+1 = Fk ξk

(5.1)

where h(k) = [x(k) x(k − 1) · · · x(k − N)]T

(5.2)

captures the effect of the reference input x(k). Note that in Figure 5.1 e(k) = d(k) − y(k) + Vm (k)

(5.3)

where e(k) is the available error measurement, Vm (k) is the exogenous disturbance that captures measurement noise, modeling error and uncertainty in the initial condition of the secondary path, and y(k) is the output of the secondary path. To formulate the estimation problem, a derived measurement for the output of the primary path is needed. Rewriting Eq. (5.3) as 4

m(k) = e(k) + y(k) = d(k) + Vm (k)

(5.4)

the right hand side, i.e. d(k) + Vm (k), can be regarded as the “noisy measurement” of the primary path output. Note that on the left hand side of Eq. (5.4) only e(k) is

5.1. BACKGROUND

106

directly measurable. In general, an internal copy of the output of the secondary path, referred to as ycopy (k), should be generated by the adaptive algorithm. Section 2.7 discusses the ramifications of the introduction of this internal copy in detail, and shows that for the stable linear secondary path the difference between ycopy (k) and y(k) will decay to zero for sufficiently large k. Now, the derived measured output for the equivalent estimation problem can be defined as # " h i W (k) m(k) = + Vm (k) Ds (k)h∗k Cs (k) θ(k) = Hk ξk + Vm (k)

(5.5)

where m(k) is defined in Equation (5.4). Noting the objective of the noise cancellation problem, s(k) = d(k) is chosen as the quantity to be estimated: # " h i W (k) s(k) = Ds (k)h∗k Cs (k) θ(k) = Lk ξk

(5.6)

Note that m(·) ∈ R1×1 , s(·) ∈ R1×1 , θ(·) ∈ R1×1 , and W (·) ∈ R(N +1)×1 . All matrices are then of appropriate dimensions. ξkT = W T (k) θT (k) is clearly the state vector for the overall approximate model of the primary path. 4

The following H∞ criterion can be used to generate sˆ(k) = F (m(0), · · · , m(k − 1)) (the prediction estimate of the desired quantity s(k)) such that the worst case energy gain from the measurement disturbance and the initial condition uncertainty to the error in the causal estimate of s(k) is properly limited, i.e. M X

sup Vm , ξ0

[s(k) − sˆ(k)]∗ [s(k) − sˆ(k)] ≤ γ2

k=0

ξ0∗ Π−1 0 ξ0

+

M X

(5.7)

∗ Vm (k)Vm (k)

k=0

The Riccati-based solution to this problem is discussed in Chapter 2 in detail. It is sometimes desirable, however, to have the adaptive filter meet some H2 performance criteria in addition to the H∞ constraint in Equation (5.7). Linear matrix inequalities

5.2. LMI FORMULATION

107

offer a convenient framework for formulating the mixed H2 /H∞ synthesis problem. Furthermore, numerically sound algorithms exist that can solve these linear matrix inequalities very efficiently. Therefore, next section pursues a first principle derivation of the LMI formulation for the design of adaptive filters.

5.2

LMI Formulation

Assume the following specific structure for the estimator ˆ ˆ ˆ ξk+1 = Fk ξk + Γk m(k) − Hk ξk sˆ(k) = Lk ξˆk

(5.8) (5.9)

in which Γk is the design parameter to be chosen such that the H∞ criterion is met. Now, the augmented system can be defined as follows " # " #" # " # ξk+1 0 Fk ξk 0 = + Vm (k) (5.10) ξˆk+1 ξˆk Fk Hk Fk − Γk Hk −Γk " # i ξ h 4 k Zk = s(k) − sˆ(k) = (5.11) Lk −Lk ˆ ξk 4 Introducing a new variable ξ˜k = ξk − ξˆk , the augmented system can be described as # " #" # " # " 0 Fk ξk 0 ξk+1 = + Vm (k) 0 Fk − Γk Hk Γk ξ˜k+1 ξ˜k

ηk+1 = Φk ηk + Ψk Vm (k)

(5.12)

with 4

Zk = s(k) − sˆ(k) =

i

h 0 Lk

"

ξk ξ˜k

# = Ωk ηk

(5.13)

The LMI solution for the design of adaptive filters finds a Lyapunov function for the augmented system in (5.12)-(5.13) at each step. In other words, at each time step, an infinite horizon problem is solved, and the solution is implemented for the next step.


108

Introducing the quadratic function V (ηk ) = ηk T P ηk (where P > 0), it is straight forward [8] to show that for the augmented system at time k, (5.7) holds if V (ηk+1 ) − V (ηk ) < γ 2 Vm (k)T Vm (k) − ZkT Zk

(5.14)

Note that the inclusion of the energy of the initial condition error will only increase the right hand side of the inequality in Eq. (5.14). Replacing for Zk and ηk+1 from (5.12)-(5.13), and after some elementary algebraic manipulations the inequality in (5.14) can be written as " #" # h i ΦT P Φ − P + ΩT Ω T η Φ P Ψ k k k k k k k T 0 I

0

#

Θk . The solution to (5.22) provides estimator gain, Γk , as 0 Rs well as the Lyapunov matrix P which ensures that the quadratic cost V (ηk ) decreases

over time. It is shown in Chapter 2 that for the Riccati-based solution to Eq. (5.7) the optimal value of γ is 1. In the absence of H2 constraints, γ in Eq. (5.22) can be set to 1. This reduces the LMI solution to a feasibility problem in which S > 0, Rs > 0, and T should be found.


5.2.1

110

Including H2 Constraints

Augmenting the above mentioned H∞ objective with appropriate H2 performance constraints (such as the H2 norm of the transfer function from exogenous disturbance Vm (k) to the cancellation error Zk in the augmented system described by (5.12)(5.13)) is also straight forward. Recall that [8] k TZVm k22 = Tr Ωk Wc ΩTk

(5.23)

where Wc satisfies ΦTk Wc Φk − W c + Ψk ΨTk = 0

(5.24)

To translate this into LMI constraints [10], note that bounding the H2 norm by ν 2 is equivalent to

 " # T  Φ W Φ − W Ψ  c k c k k 