self-organizing radial basis function networks for adaptive flight control ...

SELF-ORGANIZING RADIAL BASIS FUNCTION NETWORKS FOR ADAPTIVE FLIGHT CONTROL AND AIRCRAFT ENGINE STATE ESTIMATION DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Praveen Shankar, B.E., M.S. ***** The Ohio State University 2007

Dissertation Committee:

Approved by

Rama K. Yedavalli, Adviser Andrea Serrani M.-H. Herman Shen Meyer J. Benzakein

Adviser Graduate Program in Aeronautical and Astronautical Engineering

ABSTRACT

The performance of nonlinear control algorithms such as feedback linearization and dynamic inversion is heavily dependent on the fidelity of the dynamic model being inverted. Incomplete or incorrect knowledge of the dynamics results in reduced performance and may lead to instability. Augmenting the baseline controller with approximators which utilize a parametrization structure that is adapted online reduces the effect of this error between the design model and actual dynamics. However, currently existing parameterizations employ a fixed set of basis functions that do not guarantee arbitrary tracking error performance. To address this problem, we develop a self-organizing parametrization structure that is proven to be stable and can guarantee arbitrary tracking error performance. The training algorithm to grow the network and adapt the parameters is derived from Lyapunov theory. In addition to growing the network of basis functions, a pruning strategy is incorporated to keep the size of the network as small as possible. This algorithm is implemented on a high performance flight vehicle such as F-15 military aircraft. The baseline dynamic inversion controller is augmented with a Self-Organizing Radial Basis Function Network (SORBFN) to minimize the effect of the inversion error which may occur due to imperfect modeling, approximate inversion or sudden changes in aircraft dynamics. The dynamic inversion controller is simulated for different situations including control surface failures, modeling errors and external disturbances with and without the ii

adaptive network. A performance measure of maximum tracking error is specified for both the controllers a priori. Excellent tracking error minimization to a pre-specified level using the adaptive approximation based controller was achieved while the baseline dynamic inversion controller failed to meet this performance specification. The performance of the SORBFN based controller is also compared to a fixed RBF network based adaptive controller. While the fixed RBF network based controller which is tuned to compensate for control surface failures fails to achieve the same performance under modeling uncertainty and disturbances, the SORBFN is able to achieve good tracking convergence under all error conditions.

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Dedicated to my loving wife Sharmila and my son Dhruva

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ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Rama K. Yedavalli, for his guidance and support throughout my graduate study. His confidence in my abilities and encouragement has been a strong motivation for me to pursue my advanced studies. I would also like to thank Dr. Andrea Serrani who along with Dr. Yedavalli has been a role model. I am grateful to Dr. Herman Shen and Dr. Meyer Benzakein for serving on my dissertation committee. I would also like to acknowledge the financial support and research expertise from the various sources during my graduate study including Collaborative Center of Control Science, Air Force Research Labs, NASA Glenn and NASA Dryden Flight Research Center. I want to thank my colleagues Ms. Sooyoung Hong, Ms. WenFei Li, Ms. Nagini Devarakonda, Ms. Hsun-Hsuan Huang and Mr. Rohit Belapurkar for their discussions and suggestions, and for creating an excellent working environment. I would like to thank my parents Mr. Udayashankar and Mrs. Vimala Shankar who have been a tremendous source of encouragement. I owe a great deal to my brother, Prithvi for many of my life’s achievements and I truly believe that he considers my success as important as his own. Most of all, I would like to acknowledge and dedicate all my success to my loving wife, Sharmila who despite the demands of her career has been an incredible support and constant source of encouragement. v

VITA

May 18, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Bangalore, India 1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bachelor of Engineering - Mechanical Engineering 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Master of Science - Aeronautical and Astronautical Engineering

FIELDS OF STUDY Major Field: Aeronautical and Astronautical Engineering Studies in Flight Dynamics and Control: Rama K. Yedavalli

vi

TABLE OF CONTENTS

Page Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapters: 1.

2.

Introduction and Problem Statement . . . . . . . . . . . . . . . . . . . .

1

1.1 1.2 1.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 7

Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1 2.2 2.3

2.4 2.5 2.6 2.7 2.8

Stability Concepts . . . . . . . . . . . . Lyapunov’s Direct Method . . . . . . . . Invariance Theory . . . . . . . . . . . . 2.3.1 LaSalle’s Invariance Principle . . 2.3.2 Barbalat’s Lemma . . . . . . . . Stable in the Mean Squared Sense . . . Strictly Positive Real Transfer Functions Additional Results . . . . . . . . . . . . Uniform Complete Observability (UCO) Radial Basis Functions . . . . . . . . . . vii

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9 11 12 12 13 13 14 14 15 16

3.

Background and Literature Review . . . . . . . . . . . . . . . . . . . . .

17

3.1 3.2

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17 17 19 19 25 27

Preliminary Studies in Growing and Pruning RBFN . . . . . . . . . . . .

31

4.1

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31 31 33 36 36 40 47

Self-Organizing RBF Network Based Adaptive Control . . . . . . . . . .

50

5.1

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51 52 53 57 58 58 60 60 62 63 77

Application to Flight Control . . . . . . . . . . . . . . . . . . . . . . . .

78

6.1 6.2

78 78 79 82 83 84 86

3.3 3.4 4.

4.2

4.3 5.

Introduction to Adaptive Control . . . . . . . . . . . . . . Approaches to Adaptive Control . . . . . . . . . . . . . . 3.2.1 Parametric Adaptive Control . . . . . . . . . . . . 3.2.2 Adaptive Approximation Based Control . . . . . . Adaptive Flight Control . . . . . . . . . . . . . . . . . . . Structurally Varying Approximators for Adaptive Control

6.3

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Weight and Center Update . . . . . . . . . . . . . . . . . 4.1.2 Growing and Pruning Algorithm . . . . . . . . . . . . . . Neural Network Augmented Dynamic Inversion Flight Controller 4.2.1 Application of Controller to Flight Vehicles . . . . . . . . 4.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Self-Organizing Adaptive Approximation 5.1.1 Case 1 . . . . . . . . . . . . . . . 5.1.2 Case 2 . . . . . . . . . . . . . . . 5.1.3 Growing . . . . . . . . . . . . . . 5.1.4 Pruning . . . . . . . . . . . . . . 5.2 Stability Analysis for Self-Organization . 5.2.1 Boundedness of N(t) . . . . . . . 5.2.2 Ultimate Bound on x˜ . . . . . . 5.3 Motivational Example . . . . . . . . . . 5.3.1 Simulation Results . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . 6.

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Introduction . . . . . . . . . . . . . . . Equations of Motion of Flight Vehicles . 6.2.1 F-15 Linear Dynamics . . . . . . Baseline Dynamic Inversion Control Law 6.3.1 Reference Model Generation . . . 6.3.2 Proportional-Integral Controller . 6.3.3 Control Mixer . . . . . . . . . . viii

Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.4

6.5 6.6 7.

Simulation Results and Discussion 7.1 7.2 7.3

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7.5 8.

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87 87 88 89 91 94 95

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96

Disturbance Rejection . . . . . . . . . . . . . . . . . Uncertainty in Rolling Moment Stability Derivatives Control Surface Failures . . . . . . . . . . . . . . . . 7.3.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . Comparison to a Fixed Structure RBF Network . . . 7.4.1 Modeling Uncertainty . . . . . . . . . . . . . 7.4.2 Disturbance . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . .

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A Neural Network Based Observer For Turbine Engine State Estimation 8.1 8.2 8.3

8.4

8.5 9.

Dynamic Inversion Errors . . . . . . . . . 6.4.1 Uncertainty in Stability Derivatives 6.4.2 External Disturbances . . . . . . . 6.4.3 Control Surface Failures . . . . . . Adaptive Controller . . . . . . . . . . . . 6.5.1 Implementation . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Neural Network Based State Observer . . . . . . . . . . . Application To Aircraft Engine State Estimation . . . . . 8.3.1 Aircraft Engine Parameters . . . . . . . . . . . . . 8.3.2 Implementation of Neural Network Based Observer 8.3.3 Training . . . . . . . . . . . . . . . . . . . . . . . . Simulation Results . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Case 1: No Deterioration in Engine . . . . . . . . . 8.4.2 Case 2: Deteriorated Engine . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

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99 103 108 108 112 115 119 119 122 124

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124 126 128 130 132 134 136 137 137 137

Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 151 9.1 9.2 9.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

ix

LIST OF TABLES

Table

Page

7.1

Proportional and Integral Gains

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98

8.1

Engine Model Variables

8.2

Sensor Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . 133

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x

LIST OF FIGURES

Figure

Page

4.1

Radial Basis Function Network . . . . . . . . . . . . . . . . . . . . .

32

4.2

Growing And Pruning Algorithm . . . . . . . . . . . . . . . . . . . .

34

4.3

Block Diagram of RBFN Augmented Control Law . . . . . . . . . . .

36

4.4

Pilot Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.5

Neural Network Adaptation: Right Stabilator is stuck at 4o below trim from 10 seconds . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.6

Lateral States: Right Stabilator is stuck at 4o below trim from 10 seconds 42

4.7

Longitudinal States: Right Stabilator is stuck at 4o below trim from 10 seconds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Control Surface Deflection with RBFN on: Right Stabilator is stuck at 4o below trim from 10 seconds . . . . . . . . . . . . . . . . . . . .

43

Control Surface Deflection with RBFN off: Right Stabilator is stuck at 4o below trim from 10 seconds . . . . . . . . . . . . . . . . . . . .

44

4.10 Number of Neurons in Hidden Layer . . . . . . . . . . . . . . . . . .

45

4.11 Neural Network Adaptation: Right Stabilator is stuck at 4o above trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . .

45

4.12 Lateral States: Right Stabilator is stuck at 4o above trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.8

4.9

xi

4.13 Longitudinal States: Right Stabilator is stuck at 4o above trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.14 Control Surface Deflection with RBFN on: Right Stabilator is stuck at 4o above trim from 10 seconds to 50 seconds . . . . . . . . . . . .

47

4.15 Control Surface Deflection with RBFN off: Right Stabilator is stuck at 4o above trim from 10 seconds to 50 seconds . . . . . . . . . . . .

48

4.16 Number of Neurons in Hidden Layer . . . . . . . . . . . . . . . . . .

48

5.1

Self-Organizing Algorithm (Growing) . . . . . . . . . . . . . . . . . .

59

5.2

Rate Limited Step Reference: Actual and Desired State . . . . . . . .

64

5.3

Rate Limited Step Reference: Actual and Approximated Function . .

64

5.4

Tracking and Function Approximation Error of Dynamic Inversion with SORBFN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

5.5

Evolution of RBF Network Organization with V0 (t) . . . . . . . . . .

65

5.6

Comparison of Tracking and Function Approximation Error: Dynamic Inversion and Dynamic Inversion with SORBFN . . . . . . . . . . . .

66

5.7

Sinusoidal Reference: Actual and Desired State . . . . . . . . . . . .

67

5.8

Tracking and Function Approximation Error of Dynamic Inversion with SORBFN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Sinusoidal Reference: Actual and Approximated Function . . . . . . .

68

5.10 Evolution of RBF Network Organization with V0 (t) . . . . . . . . . .

69

5.11 Comparison of Tracking and Function Approximation Error: Dynamic Inversion and Dynamic Inversion with SORBFN . . . . . . . . . . . .

70

5.12 Sinusoid and Rate Limited Step Reference: Actual and Desired State

70

5.13 Sinusoid and Rate Limited Step Reference: Actual and Approximated Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.9

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71

5.15 Tracking and Function Approximation Error of Dynamic Inversion with SORBFN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5.16 Comparison of Tracking and Function Approximation Error: Dynamic Inversion and Dynamic Inversion with SORBFN . . . . . . . . . . . .

73

5.17 Sinusoidal Reference with Ramp Disturbance: Actual and Approximated Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

5.18 Tracking and Function Approximation Error of Dynamic Inversion with SORBFN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

5.19 Sinusoidal Reference with Ramp Disturbance: Actual and Desired State 76 5.20 Comparison of Tracking and Function Approximation Error: Dynamic Inversion and Dynamic Inversion with SORBFN . . . . . . . . . . . .

76


77

6.1

PI Control with Integral Anti-Windup . . . . . . . . . . . . . . . . .

85

6.2

Adaptive Flight Control Architecture . . . . . . . . . . . . . . . . . .

91

7.1

Pilot Roll Command . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

7.2

Roll rate tracking with sinusoidal disturbance in roll channel . . . . . 100

7.3

Pitch rate tracking with sinusoidal disturbance in roll channel . . . . 100

7.4

Yaw rate tracking with sinusoidal disturbance in roll channel . . . . . 101

7.5

Control surface deflection with SORBFN and sinusoidal disturbance in roll channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.6

Control surface deflection without SORBFN and sinusoidal disturbance in roll channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xiii

7.7

Adaptation of SORBFN in the presence of sinusoidal disturbance in the roll channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.8

Roll rate tracking performance with uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.9

Pitch rate tracking performance with uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.10 Yaw rate tracking performance with uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.11 Control surface deflection without SORBFN and uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . 106 7.12 Control surface deflection with SORBFN and uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.13 Adaptation of SORBFN with uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.14 Stabilator and Aileron deflection with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . 108 7.15 Roll rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.16 Pitch rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.17 Yaw rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.18 Rudder and Canard deflection with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . 111 7.19 Adaptation of SORBFN with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . 111 7.20 Stabilator and Aileron deflection with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . 112 xiv

7.21 Pilot Roll and Pitch Command . . . . . . . . . . . . . . . . . . . . . 113 7.22 Roll rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.23 Pitch rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.24 Yaw rate tracking with right Stabilator stuck at 4o above trim from 10 seconds and corrected at 50 seconds . . . . . . . . . . . . . . . . . . . 115 7.25 Rudder and Canard deflection with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . 116 7.26 Adaptation of SORBFN with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . 116 7.27 Roll rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.28 Pitch rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.29 Yaw rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.30 Roll rate tracking performance with uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.31 Pitch rate tracking performance with uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.32 Yaw rate tracking performance with uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.33 Roll rate tracking performance with uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.34 Pitch rate tracking performance with uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 xv

7.35 Yaw rate tracking performance with uncertainty in base aerodynamic rolling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.1

Neural Network Based State Observer . . . . . . . . . . . . . . . . . . 129

8.2

NN Based Observer- Application to Aircraft Engine . . . . . . . . . . 135

8.3

HPT Inlet Temperature And Thrust - No Deterioration . . . . . . . . 138

8.4

Stall Margins - No Deterioration . . . . . . . . . . . . . . . . . . . . . 139

8.5

Pressures - No Deterioration . . . . . . . . . . . . . . . . . . . . . . . 140

8.6

Temperatures - No Deterioration . . . . . . . . . . . . . . . . . . . . 141

8.7

States - No Deterioration . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.8

States - No Deterioration . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.9

HPT Inlet Temperature And Thrust - Deteriorated Engine . . . . . . 144

8.10 Stall Margins - Deteriorated Engine . . . . . . . . . . . . . . . . . . . 145 8.11 Pressures - Deteriorated Engine . . . . . . . . . . . . . . . . . . . . . 146 8.12 Temperatures - Deteriorated Engine . . . . . . . . . . . . . . . . . . . 147 8.13 States - Deteriorated Engine . . . . . . . . . . . . . . . . . . . . . . . 148 8.14 States - Deteriorated Engine . . . . . . . . . . . . . . . . . . . . . . . 149

xvi

CHAPTER 1

INTRODUCTION AND PROBLEM STATEMENT

1.1

Introduction

Flight vehicles such as highly maneuverable military aircraft and hypersonic vehicles pose a significant challenge to control design due to the complex nonlinearities associated with the dynamics. These vehicles require control laws that are adaptable not only to the various operating conditions, but also as in the case of hypersonic vehicles, those that account for changing geometry of the vehicle during re-entry into the atmosphere and other uncertainties in the environment. Subsystem interactions also play a major role in defining the guidance and control laws for these flight vehicles. For example, hypersonic gliders with powerplants (Rocket, Scramjet) are typically poorly modeled powered vehicles that have strong system-wide coupling between propulsion and aerodynamics, and have thin performance margins necessitating operation near system constraints to achieve efficiency. The development of adaptive control which can permit controlled flight without a detailed aerodynamic model or with deviations away from the model (unplanned departures from aerodynamic geometry, accidents or failures) is very important for these vehicles. Linearized equations of motion that are used currently for designing flight controllers 1

fall short of achieving the desired performance when the dynamics vary widely with operating conditions. A method used by the flight control community to overcome this problem is to use dynamic inversion controllers to feedback linearize the system. The performance of a dynamic inversion controller is however, heavily dependent on accurate knowledge of the vehicle dynamics. Flight control laws are generally designed to close the loop around the rotational dynamics. These dynamics consist of the following roll(p), pitch(q) and yaw(r) equations of motion [1] which can be written as ω˙ = −I −1 (ω × Iω) + I −1 M where ω =

p q r

(1.1)

and M is the generalized moment given by M = Maero + Mδ

(1.2)

Maero is the moment generated due to the base aerodynamics of the aircraft. Mδ =G(P)δ is the moment generated due to control surface deflections and G(.) is the control effectiveness function that is dependent on parameters (P ) such as mach number, angle of attack, etc. The dynamic inversion control law for the rotational dynamics is given by δ = G−1 (P )(I ω˙ d + ω × Iω − Maero )

(1.3)

The moments generated due to the base aerodynamics need to be determined since they are required for inversion. They are generally estimated from tables of aerodynamic data obtained from extensive experimental testing. Additionally, in order to calculate the control surface deflections, control derivatives (G(P)) corresponding to the control surfaces such as elevator and rudder need to be estimated. These are again obtained from tables of experimental data. There is generally some amount 2

of uncertainty involved in estimating these values and it is one of the factors that contributes to errors in dynamic inversion. Other factors that may cause dynamic inversion errors include external disturbances or failures in the control surfaces. Augmenting the baseline controller with approximators which utilize a parametrization structure that is adapted online reduces the effect of these inversion errors [2]. The following section briefly discusses the adaptive approximation based dynamic inversion (feedback linearization) control law and addresses the issues in utilizing fixed structure approximators. Finally, the problem being addressed in this thesis is stated.

1.2

Problem Statement

Consider the nonlinear system of the form: x˙ = f (x) + g(x)u

(1.4)

where x, u ∈ 0. However, the above control methodology works under the assumption that the designer knows f(x) and g(x) exactly and g(x) is invertible. In actuality, the designer has access to an approximation of these functions fˆ(x) and gˆ(x). These approximated functions are in the form of a parametrization chosen by the designer. The control 3

law is implemented such that the parameters associated with this parametrization are adapted continuously to achieve the performance requirement. This is generally referred to as adaptive approximation based control [2]. With the approximation, the control law can be written as u = gˆ(x)−1 (x˙ d − fˆ(x) − k˜ x)

(1.7)

Now the closed loop system is of the form x˙ = f (x) + g(x)ˆ g (x)−1 (x˙ d − fˆ(x) − k˜ x)

(1.8)

It is evident from the above equation that stability is heavily dependent on the approximating functions. In the following analysis, we will assume that g(x) is known exactly and is invertible. Let us choose an approximation structure fˆ(x) = θˆfT φf (x)

(1.9) (1.10)

Since the original function f(x) is assumed to be continuous, it can be parameterized as follows: f (x) = θf∗T φf (x) + εf (x) θf∗T = arg min(f (x) − θfT φ(x)) θf

(1.11) (1.12)

εf (x) is the Minimum Function Approximation Errors (MFAE) for f(x) and θf∗ , φf (.) ∈ 0 and any to > 0, there exists a δ(, t0 ) > 0 such that kx(t0 )k < δ(, t0 ) =⇒ kx(t)k < for all t ≥ t0 9

Uniformly Stable if for any > 0 and any to > 0, there exists a δ() > 0 such that kx(t0 )k < δ() =⇒ kx(t)k < for all t ≥ t0 Unstable if it is not stable Asymptotically Stable if it is stable and for any to > 0 there exists η(t0 ) > 0, such that kx(t)k < η(t0 ) =⇒ kx(t)k −→ 0 as t −→ ∞ Uniformly Asymptotically Stable if it is uniformly stable and there exists δ > 0 independent of t such that ∀ > 0 there exists T () > 0 such that kx(t0 )k < δ =⇒ kx(t)k < for all t > t0 + T () Exponentially Stable if for any > 0, there exist some δ() > 0 such that kx(t0 )k < δ =⇒ kx(t)k < e−α(t−t0 ) for all t > t0 ≥ 0 for some α > 0 In certain cases, it is not possible to prove stability of the origin and the analysis is generally limited to boundedness of the signal. This is especially true in the case of adaptive approximation based control where lack of proof for parameter convergence limits the analysis to boundedness of the parametric error. Definition In such a case, the equilibrium xe = 0 is Uniformly Ultimately Bounded if there exists positive constants R, T(R), b such that kx(t)k < R implies that kx(t)k < b for all t > t0 + T Globally Uniformly Ultimately Bounded if it is uniformly ultimately bounded and R = ∞ where b is called the ultimate bound.

10

2.2

Lyapunov’s Direct Method

Definition Let B(r) denote an open set containing the origin. 1. A continuous function V(x) is positive definite on B(r) if V(0) = 0 and V(x) > 0 ∀ x ∈ B(r) such that x 6= 0 2. A continuous function V(x) is positive semidefinite on B(r) if V(x) ≥ 0 ∀ x ∈ B(r) such that x 6= 0 3. A continuous function V(x) is negative (semi-)definite on B(r) if -V(x) is positive (semi-)definite 4. A continuous function V(x) is radially unbounded if V(0) = 0, V(x) > 0 on 0. The dynamic inversion controller with the adaptive approximation algorithm based on SORBFN is given by u = (0.5 + 4.5sin(

x −1 ˆ )) (−k˜ x − x + x˙ d − ∆(x)) 47.5

(5.15)

It is evident that the approximator needs to approximate the unknown function given by x-sin(x) accurately for the state to track the desired signal x0 (t). 62

5.3.1

Simulation Results

The dynamic inversion control law is simulated for various input signals with and without the SORBFN. The choice of proportional control gain k is the same for both the controllers under similar inputs. The following simulation results show the ability of the SORBFN to not only approximate the unknown function(x-sin(x)) under different types of reference signals but also under the effect of a ramping disturbance signal. Step Reference Signal In this example, the reference signal is a rate limited step input which ramps up from zero to a maximum value of 20 over a period of 100 seconds. The proportional gain chosen for this example had a magnitude of 5 and the desired tracking error was 0.05. The reference signal is plotted with the actual state of the controlled system under the influence of the dynamic inversion controller with SORBFN in Figure 5.2. The actual and approximated function are plotted in Figure 5.3. The errors in function approximation and tracking is plotted in Figure 5.4. It can be seen that the tracking error converges to 0.05 asymptotically and the function approximation error is limited to 2.5 which is nothing but the product of the tracking error and the proportional gain. The evolution of the number of nodes in the network is plotted along with the evaluation function V0 (t) in Figure 5.5 where it can be seen that when V0 (t) increases continuously for a pre-specified number of time steps, a new node is added to the network. It can also be noticed that unwanted nodes are pruned at about 120 seconds and the final number of nodes converges to about 60. The comparison of the pure dynamic inversion controller and dynamic inversion with SORBFN is illustrated

63

10

9

Actual Reference

8

7

State

6

5

4

3

2

1

0

20

40

60

80

100 Time

120

140

160

180

200

Figure 5.2: Rate Limited Step Reference: Actual and Desired State

1.5 Actual Approximation

1

f(x)

0.5

0

-0.5

-1

0

20

40

60

80

100 Time

120

140

160

180

200

Figure 5.3: Rate Limited Step Reference: Actual and Approximated Function

64

0.1 0.09 0.08

Tracking Error

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100 Time

120

140

160

180

200

0.5 0.45 0.4

Approximation Error

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Figure 5.4: Tracking and Function Approximation Error of Dynamic Inversion with SORBFN

1

0

V (t)

0.5

0

-0.5

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100 Time

120

140

160

180

200

70

60

Number of Nodes

50

40

30

20

10

0

Figure 5.5: Evolution of RBF Network Organization with V0 (t)

65

in Figure 5.6. The tracking error for the dynamic inversion controller is ≈ 1.5 and

1.6 1.4

Tracking Error

1.2 1 0.8 With SORBFN Without SORBFN

0.6 0.4 0.2 0

0

20

40

60

80

100

120

140

160

180

200

8 7

Approximation Error

6 With SORBFN Without SORBFN

5 4 3 2 1 0

0

20

40

60

80

100 Time

120

140

160

180

200

Figure 5.6: Comparison of Tracking and Function Approximation Error: Dynamic Inversion and Dynamic Inversion with SORBFN

function approximation error is ≈ 7.5. A larger proportional gain will reduce the errors in the tracking but will increase the magnitude of the control input. Since the comparison between the two controllers are performed under the same proportional gain, it is evident that the dynamic inversion with SORBFN has better tracking and function approximation convergence. Sinusoidal Reference Signal A sinusoidal reference signal with an amplitude of 1.5 is applied to the system. The proportional gain chosen for this example is 1 and the desired tracking error is 0.05. The performance of the dynamic inversion controller with SORBFN is illustrated in 66

Figure 5.7 where the actual state tracks the desired sinusoidal signal accurately. The

1.5

1

State

0.5

0

-0.5 Actual Reference

-1

-1.5

0

20

40

60

80

100 Time

120

140

160

180

200

Figure 5.7: Sinusoidal Reference: Actual and Desired State

tracking error is plotted in Figure 5.8 along with the function approximation error. It can be noted that the desired tracking error is approximately 0.05 and the function approximation error is also 0.05 since the proportional gain was 1. A plot of the actual and approximated function is seen in Figure 5.9. The evolution of the number of nodes in the network can be seen in Figure 5.10 along with the evaluation function V0 (t). The addition of neurons is especially aggressive in the beginning of simulation since it is adapting to the changing state of the system. After the evolution of the state through the first sinusoid, the node addition reduces and in fact a few of the nodes are deleted. The number of nodes becomes constant as the simulation proceeds but this does not affect the performance of the controller since there are sufficient 67

0.12

0.1

Tracking Error

0.08

0.06

0.04

0.02

0

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100 Time

120

140

160

180

200

0.12

Approximation Error

0.1

0.08

0.06

0.04

0.02

0



1

f(x)

0.5

0

-0.5

-1

-1.5

0

20

40

60

80

100 Time

120

140

160

180

200

Figure 5.9: Sinusoidal Reference: Actual and Approximated Function

68

1

0

V (t)

0.5

0

-0.5

0

20

40

60

80

100

120

140

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180

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0

20

40

60

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100 Time

120

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200

25

Number of Nodes

20

15

10

5

0


nodes in the network to approximate the unknown function and the desired state is cyclic (sinusoid). The comparison of the dynamic inversion controller with the dynamic inversion controller and SORBFN can be seen in Figure 5.11. Combined Rate Limited Step and Sinusoidal Reference Signal To show the ability of the controller to track complex signals, a reference input that combines a sinusoidal signal with a rate limited step signal is applied to the system. A proportional gain of 1 and desired tracking error of 0.05 is considered. The tracking performance of the dynamic inversion with SORBFN can be seen in Figure 5.12 and the function approximation ability of the controller is illustrated in Figure 5.13. The evolution of the number of nodes in the network is shown in Figure 5.14 where it can be seen that the number of nodes increases continuously

69

0.35 With SORBFN Without SORBFN

0.3

Tracking Error

0.25

0.2

0.15

0.1

0.05

0

0

20

40

60

80

100

120

140

160

180

200

0.35 With SORBFN Without SORBFN

0.3

Approximation Error

0.25

0.2

0.15

0.1

0.05

0

0

20

40

60

80

100 Time

120

140

160

180

200


12

10

Actual Reference

State

8

6

4

2

0

0

20

40

60

80

100 Time

120

140

160

180

200

Figure 5.12: Sinusoid and Rate Limited Step Reference: Actual and Desired State

70


1

f(x)

0.5

0

-0.5

-1

-1.5

0

20

40

60

80

100 Time

120

140

160

180

200

Figure 5.13: Sinusoid and Rate Limited Step Reference: Actual and Approximated Function

1

0

V (t)

0.5

0

-0.5

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

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100 Time

120

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140

120

Number of Nodes

100

80

60

40

20

0


71

with the signal ramping up from 0 to 20 and also because of the embedded sinusoidal signal. However, once the step signal reaches its maximum value, the number of nodes remains constant. The comparison between the two controllers can be seen in Figure 5.15. The tracking error is significantly high for the purely dynamic inversion

0.12

0.1

Tracking Error

0.08

0.06

0.04

0.02

0

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100 Time

120

140

160

180

200

0.12

Approximation Error

0.1

0.08

0.06

0.04

0.02

0


controller. A better picture of the tracking and approximation error for the dynamic inversion with SORBFN controller can be seen in Figure 5.16. The tracking error is initially varying but is approximately bounded by the pre-specified desired value of 0.05. The function approximation can also be seen to be highly varying during the initial part of the simulation when the reference signal is ramping up but eventually converges approximately to 0.05.

72

7 6

Tracking Error

5 4 3 2

With SORBFN Without SORBFN

1 0 0

20

40

60

80

100

120

140

160

180

200

7

Approximation Error

6 5 4 3 2


1 0 0

20

40

60

80

100 Time

120

140

160

180


Sinusoidal Reference Signal with External Ramp Disturbance The previous examples showed the ability of the controller to approximate the unknown function under the influence of different reference signals. Here, we will consider that in addition to the unknown function x-sin(x), an external disturbance in the form of a time varying signal affects the system. The open loop system can now be written as x˙ = sin(x) + (0.5 + 4.5sin(

x ))u + ud (t) 47.5

where ud (t) is a disturbance signal which we have considered to be a ramp with slope of 0.01 units. The desired tracking error is 0.05 and the proportional gain of the system is 5. The closed loop system with the dynamic inversion control law and 73

SORBFN is given by ˆ x˜ = −k˜ x + ∆(x) − ∆(x) where ∆(x) = x − sin(x) + ud (t) Therefore, the SORBFN approximator is required to approximate the above function ∆(x) accurately to achieve the pre-specified tracking error. It can be seen that the approximated function closely matches the actual function (Refere Figure 5.17. )

40

Actual Approximation 35

30

f(x)

25

20

15

10

5

0

0

20

40

60

80

100 Time

120

140

160

180

200

Figure 5.17: Sinusoidal Reference with Ramp Disturbance: Actual and Approximated Function

The function approximation error and tracking error are illustrated in Figure 5.18. It can be seen that the tracking error does not converge to 0.05. The controller 74

0.18 0.16 0.14

Tracking Error

0.12 0.1 0.08 0.06 0.04 0.02 0

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100 Time

120

140

160

180

200

1 0.9 0.8

Approximation Error

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0


is developed under the assumption that the function that is being approximated is bounded in the region of interest but the ramp is not a bounded signal. Despite this property, the ability of the controller to track the signal with a bounded error not very different from the desired tracking error can be seen in Figure 5.19. The purely dynamic inversion controller is unable to bound the tracking error as can be noted from Figure 5.20 where the tracking and function approximation performance of the two controllers are compared. The evolution of the network structure is illustrated along with V0 (t) in Figure 5.21.

75

2 Actual Reference

1.5

1

State

0.5

0

-0.5

-1

-1.5

0

20

40

60

80

100 Time

120

140

160

180

200

Figure 5.19: Sinusoidal Reference with Ramp Disturbance: Actual and Desired State

7

6


Tracking Error

5

4

3

2

1

0

0

20

40

60

80

100

120

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160

180

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35

30


Approximation Error

25

20

15

10

5

0

0

20

40

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100 Time

120

140


76

1

0

V (t)

0.5

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30

Number of Nodes

25

20

15

10

5

0


5.4

Summary

This chapter addressed the disadvantages of a fixed structure network in adaptive approximation based control and discussed the advantages of a structurally adaptive network. The development of a self-organizing radial basis function network and its application to adaptive control is presented along with the training algorithm and the stability analysis of the closed loop. A motivational example in the form of a first order system is thoroughly investigated to understand the implementation and advantages of the SORBFN based control law.

77

CHAPTER 6

APPLICATION TO FLIGHT CONTROL

6.1

Introduction

The previous chapter addressed the development of a self-organizing radial basis function network based adaptive control law. In this chapter, we discuss the application of this adaptive control to the flight dynamics of a high performance military aircraft. A baseline dynamic inversion controller that is designed to close the loop around the rotational dynamics of the aircraft is augmented with the SORBFN to compensate for various errors that occur due to modeling uncertainties, failures and disturbances.

6.2

Equations of Motion of Flight Vehicles

The nonlinear rotational dynamics of the aircraft are given by [1]: p˙ = c1 rq + c2 pq + c3 L + c4

(6.1)

q˙ = c5 pr + c6 (p2 − r2 ) + c7 M

(6.2)

r˙ = c8 pq + c2 rq + c4 L + c9 N

(6.3) (6.4)

78

where 2 ]/Γ c1 = [(Jy − Jz )Jz − Jxz

c2 = (Jx − Jy + Jz )Jxz ]/Γ c3 = Jz /Γ c4 = Jxz /Γ c5 = (Jz − Jx )/Jy c6 = Jxz /Jy c7 = 1/Jy 2 c8 = [(Jx − Jy )Jx + Jxz ]/Γ

c9 = Jx /Γ 2 Γ = Jx Jz − Jxz

In this simulation model, the rotational dynamics are inverted to obtain the control moments required to achieve the desired tracking. The control moments are then allocated as control surface deflections using a control mixer. The controller is simulated on the linearized dynamics of an F-15 aircraft.

6.2.1

F-15 Linear Dynamics

The aircraft simulation model represents a modified F-15 military aircraft with control surfaces that include right and left ailerons (δa ), stabilators (δs ), rudders (δr ) and canards (δc ). The plant model of the F-15 aircraft is a linear state space system obtained by considering perturbations around the trim conditions of the aircraft. The model can be simulated at different flight conditions based on mach number and altitude. 79

Longitudinal Dynamics The states of the longitudinal model are the perturbations (from trim condition) in • Linear velocity (V) • Angle of attack (α) • Pitch rate(p) • Pitching angle(Θ) • Altitude (H) The control effectors to this linear model are • Average of the right and left ailerons:

δal +δar 2

• Average of the right and left stabilators: • Average of the right and left canards:

δsl +δsr 2

δcl +δcr 2

The outputs of the longitudinal dynamics include • Linear velocity (V) • Angle of attack (α) • Pitch rate (q) • Pitching angle (Θ) • Altitude (H) • Vertical load factor (Nz ) 80

Lateral/Directional Dynamics The states of the longitudinal model are the perturbations (from trim condition) in • Sideslip (β) • Roll rate (p) • Yaw rate (r) • Rolling angle (Φ) • Yawing angle (ψ) The control effectors to the lateral dynamics model are • Difference between the right and left ailerons: δar − δal • Difference the right and left stabilators: δsr − δsl • Average of the right and left rudder:

δrl +δrr 2

• Difference between the right and left canards: δcr − δcl The outputs of the lateral dynamics include • Sideslip (β) • Roll rate (p) • Yaw rate (r) • Rolling angle (Φ) 81

• Yawing angle (ψ) • Horizontal loading factor (Ny ) In this simulation model, all actuators and sensors are assumed to have first order dynamics with the same time constant.

6.3

Baseline Dynamic Inversion Control Law

The rotational dynamics of the aircraft can be written as ω˙ = −I −1 (ω × Iω) + I −1 M where ω =

p q r

(6.5)

and M is the generalized moment given by M = Maero + Mδ

(6.6)

I is the moment of inertia matrix given by 



Jx Jxy Jxz   I =  Jxy Jy Jyz  Jxz Jyz Jz Maero is the moment generated due to the base aerodynamics of the aircraft and is dependent on the flight condition. It consists of the rolling (LBAE ), pitching (MBAE ) and yawing (NBAE ) moments that are given by b b Clp p + Cl r) 2V 2V r c¯ = q¯S¯ c(Cmα α + Cmq q + Cmc δc ) 2V b b = q¯Sb(Cnβ β + Cnp p + Cn r) 2V 2V r

LBAE = q¯Sb(Clβ β +

(6.7)

MBAE

(6.8)

NBAE

(6.9)

The deflections of the canards are scheduled according to the flight condition and angle of attack. Therefore it’s effect on the pitching moment is included in the base 82

aerodynamics itself. Mδ =G(P)δ is the moment generated due to control surface deflections and G(.) is the control effectiveness function that is dependent on parameters (P ) such as mach number, angle of attack, etc. Rewriting the rotational dynamics, we have ¯ )δ ω˙ = f (ω) + MA + G(P

(6.10)

where f (ω) = −I −1 (ω × Iω) MA = I −1 Maero ¯ ) = I −1 G(P ) G(P The dynamic inversion control law for the rotational dynamics can now be written as ¯ −1 (P )(ω˙ d − f (ω) − MA ) δ=G

6.3.1

(6.11)

Reference Model Generation

The pilot commands to the aircraft are in the form of roll and pitch stick combined with a yaw pedal command. These pilot commands are converted to roll, pitch and yaw command signals (pcmd ,qcmd ,rcmd ) by multiplying with appropriate gains. The desired roll,pitch and yaw signals along with it’s derivatives are generated using reference models for each of the 3 channels. Longitudinal Reference Model The longitudinal reference model is designed to characterize the second order short period mode of the aircraft and it is given by qd qcmd

=

ωq2 (s + Lα ) s2 + 2ζq ωq s + ωq2

where Lα = 1.2, ωq = 3.5 degrees/second and ζ = 1.05. 83

(6.12)

Lateral Reference Model The lateral reference model represents the roll subsidence mode of the aircraft and it is a first order model given by pd 1/τp = pcmd s + 1/τp

(6.13)

where τp = 0.3333 seconds. Directional Reference Model The directional reference model is implemented to characterize the second order dutch roll mode of the aircraft. It is given by rd rcmd

=

ωr2 s2 + 2ζr ωr s + ωr2

(6.14)

where ωr = 3.0 degrees/second and ζr = 0.8.

6.3.2

Proportional-Integral Controller

The closed loop system when the dynamic inversion controller given by Equation 6.11 is implemented to close the loop around Equation 6.10 is given by ω˙ = ω˙ d

(6.15)

If the approximation of the base aerodynamic moments is accurate, then the closed loop dynamics follows the desired dynamics exactly. However, to meet performance specifications and pilot handling qualities, a proportional integral controller is implemented for each of the channels of roll, pitch and yaw. The gains are chosen so as to satisfy pilot handling qualities. An integral anti-windup algorithm is included to back off the actuator when it is saturated. The block diagram of the PI Control with 84

p-pd

Kpp

∫

Gp

sum

uppi

Kip

Antiwindup Gain = 0 or 1

Aileron/Stabilator saturates

q-qd

Set Gp = 0

Kpq

∫

Gq

sum

uqpi

Kiq


Canard/Stabilator saturates

r-rd

Set Gq = 0

Kpr

∫

Gr

sum

urpi

Kir


Canard/Rudder saturates

Set Gr = 0

Figure 6.1: PI Control with Integral Anti-Windup

the anti-windup logic is shown in Figure 6.1. The inner loop dynamic control law and outer loop PI controller can be assumed to decouple the closed loop dynamics which

85

is then given by p˙ = p˙d − Kpp (p − pd ) − Kip

Z Z

q˙ = q˙d − Kpq (q − qd ) − Kiq r˙ = r˙d − Kpr (r − rd ) − Kir

Z

p − pd

(6.16)

q − qd

(6.17)

r − rd

(6.18)

Therefore "

e˙ p = "

e˙ q = "

e˙ r =

0 1 −Kip −Kpp

#

0 1 −Kiq −Kpq

#

0 1 −Kir −Kpr

#

ep

(6.19)

eq

(6.20)

er

(6.21)

where

6.3.3

ep =

h R

(p − pd ) p − pd

eq =

h R

(q − qd ) q − qd

er =

h R

(r − rd ) r − rd

iT iT iT

Control Mixer

A control mixer is implemented to distribute the required aerodynamic moments among the actuators. The aerodynamic moments that need to be allocated among the actuators are given by I

−1

Mδ = I

−1

¯ )δ = ω˙ d − f (ω) − MA − Kp ω − Ki G(P )δ = G(P

Z

ω

(6.22)

G(P) is a matrix of control derivatives that is dependent on the flight condition. Since the canards are scheduled according to the flight condition (mach number and altitude) and the angle of attack, the control mixer allocates the moments among 86

the aileron, rudder and stabilator. The deflections are calculated using the following equation      δ=    

δar (t) δal (t) δsr (t) δsl (t) δrr (t) δrl (t)

    Z  ¯ )−1 (ω˙ d − f (ω) − MA − Kp ω − Ki ω)  = G(P    

(6.23)

¯ )−1 is the generalized pseudo-inverse. where G(P

6.4

Dynamic Inversion Errors

The dynamic inversion controller ideally decouples the closed loop into roll, pitch and yaw channels. The stability of these decoupled dynamics is dependent on the proportional and integral gain. The analysis in the previous section made an assumption that knowledge of the moments due to the base aerodynamics (Maero ) is accurate. However, uncertainties in the stability derivatives such as Clβ causes errors in the dynamic inversion. Other factors that may cause errors in the inversion are external disturbances and control surface failures. In this section, we discuss the effect of the various errors, failures and disturbances on the closed loop system.

6.4.1

Uncertainty in Stability Derivatives

Consider the dynamic inversion control law given by Equation 6.11. Assuming the stability derivatives (See Equation 6.7) are uncertain, we can rewrite the dynamic inversion control law as ¯ −1 (P )(ω˙ d − f (ω) − M ˆ A − Kp ω − Ki δ=G

87

Z

ω)

(6.24)

The closed loop system is then given by ˆ A − Kp ω − Ki ω˙ = ω˙ d + MA − M

Z

ω

(6.25)

Assuming that the dynamic inversion decouples the roll, pitch and yaw equations and implementing the PI controller independently in each of the 3 channels, we have the closed loop "

e˙ p = "

e˙ q = "

e˙ r =

0 1 −Kip −Kpp

#

0 1 −Kiq −Kpq

#

0 1 −Kir −Kpr

#

ˆ BAE ) + c4 (NBAE − N ˆBAE ) (6.26) ep + c3 (LBAE − L ˆ BAE ) eq + c7 (MBAE − M

(6.27)

ˆ BAE ) + c9 (NBAE − N ˆBAE ) (6.28) er + c4 (LBAE − L

˜ BAE + c4 N ˜BAE ) e˙ p = Ap ep + Bp (c3 L

(6.29)

˜ BAE e˙ q = Aq eq + Bq c7 M

(6.30)

˜ BAE + c9 N ˜BAE ) e˙ r = Ar er + Br (c4 L

(6.31)

where Bp , Bq and Br =

0 1

T

.

LBAE , MBAE and NBAE are dependent on α, β, p, q, r and V. Therefore we can rewrite the previous equation as

6.4.2

e˙ p = Ap ep + Bp Ξp (ω, β, V )

(6.32)

e˙ q = Aq eq + Bq Ξq (ω, α)

(6.33)

e˙ r = Ar er + Br Ξr (ω, β, V )

(6.34)

External Disturbances

Consider the open loop rotational dynamics ¯ )δ + Bξ ξ(t) ω˙ = f (ω) + MA + G(P 88

(6.35)

where ξ(t) is a continuous bounded disturbance affecting the system. The closed loop system with the inner loop dynamic inversion controller and outer loop PI controller is now given by

6.4.3

e˙ p = Ap ep + Bp bξp ξ(t)

(6.36)

e˙ q = Aq eq + Bq bξq ξ(t)

(6.37)

e˙ r = Ar er + Br bξr ξ(t)

(6.38)

Control Surface Failures

In this thesis, we discuss mainly stuck at faults of the control surfaces. The control surface deflection is given by Equation 6.23. Consider the situation when one of the control surfaces such as the right stabilator δsr is stuck at a particular position δ¯sr . The closed loop system is given by     ¯ )  ω˙ = f (ω) + MA + G(P    


         

(6.39)

where the actuator deflections are determined by Equation 6.23. The control effec¯ ) is given by tiveness matrix G(P 

¯ ) =  G(P  

¯ ) =  G(P 

¯ p (P )  G ¯ q (P )  G  ¯ Gr (P ) ¯ par G ¯p G ¯ psr G ¯p G ¯ prr G ¯p  G al sl rl ¯ qar G ¯q G ¯ qsr G ¯q G ¯ qrr G ¯q  G al sl rl  ¯ ¯ ¯ ¯ ¯ ¯ Grar Gral Grsr Grsl Grrr Grrl

89

Considering that the right stabilator is stuck we can rewrite the equation as      ¯ ω˙ = f (ω) + MA + G(P )     






       ¯p   −G        






       ¯p   +G        

δar (t) δal (t) δ¯sr δsl (t) δrr (t) δrl (t)

         

(6.40)

The closed loop system is now given by 

e˙ p

    ¯ = Ap ep + Bp (−Gp (P )      

e˙ q

    ¯ = Aq eq + Bq (−Gq (P )      

e˙ r

    ¯ = Ar er + Br (−Gr (P )     












       ¯ p (P )   +G         

       ¯ q (P )  +G          

       ¯ r (P )  +G         










    )    

    )    

    )    

(6.41)

(6.42)

(6.43)

Therefore, we have ¯ psr (δ¯sr − δsr (t)) e˙ p = Ap ep + Bp G

(6.44)

¯ qsr (δ¯sr − δsr (t)) e˙ q = Aq eq + Bq G

(6.45)

¯ rsr (δ¯sr − δsr (t)) e˙ r = Ar er + Br G

(6.46)

This can concisely be written as e˙ p = Ap ep + Bp Ωp (ω, δ)

(6.47)

e˙ q = Aq eq + Bq Ωq (ω, δ)

(6.48)

e˙ r = Ar er + Br Ωr (ω, δ)

(6.49)

90

6.5

Adaptive Controller

In this section, we will discuss the method of augmenting the baseline controller with adaptive approximators to minimize the effect of the error on the closed loop. The block diagram of the control architecture can be seen in Figure 6.2 The adaptive

Figure 6.2: Adaptive Flight Control Architecture

approximator implemented is the self-organizing RBFN discussed in Chapter 5. Since, we assumed that the dynamic inversion control law decouples the three rotational dynamics of motion, we will discuss the implementation of the SORBFN on only the roll channel. It is assumed that the same analysis can be extended to the pitch and yaw channels. Consider the closed loop equation of the roll channel e˙ p = Ap ep + Bp ∆p (ω)

(6.50) (6.51)

91

where 0 1 −Kip −Kpp

Ap = Bp =

0 1

!

T

and ∆p (ω) is the dynamic inversion error. Consider an adaptive approximator that is introduced into the dynamic inversion controller such that ¯ −1 (P )(ω˙ d − f (ω) − MA − Kp ω − Ki δ=G

Z

ˆ ω − ∆(ω))

(6.52)

where ˆ ∆(ω) =

ˆ p (ω) ∆ ˆ q (ω) ∆ ˆ r (ω) ∆

The closed loop for the roll channel with the adaptive approximator is given by ˆ p (ω) e˙ p = Ap ep + Bp ∆p (ω) − Bp ∆

(6.53)

Let us consider a Lyapunov function given by Vp (t) = V0 (t) + VNp (t) = eTp P ep +

Np X

γ θ˜i2

(6.54)

i=1

where γ > 0 and P is a positive definite matrix such that ATp P + P Ap = −Q, Q > 0 since Ap is Hurwitz. Consider the situation when there are no errors in the dynamic inversion controller. Then, ∆p (ω) = 0. With no nodes in the network (Np = 0), the Lyapunov function is reduced to V0 (t) . Evaluating its derivative we have V˙ 0 (t) = eTp P e˙ p + e˙ Tp P ep V˙ 0 (t) = eTp P Ap ep + (Ap ep )T P ep V˙ 0 (t) = eTp (P Ap + ATp P )ep V˙ 0 (t) = −eTp Qep 92

Since Q is a positive definite matrix, we have the Lyapunov derivative to be negative definite which implies that the error signal ep asymptotically converges to zero. Now consider the situation in which the error in dynamic inversion ∆p (ω) 6= 0. In this case the Lyapunov derivative is given by V˙ 0 (t) = eTp P e˙ p + e˙ Tp P ep V˙ 0 (t) = eTp P (Ap ep + BP ∆p (ω)) + (Ap ep + Bp ∆p (ω))T P ep V˙ 0 (t) = eTp (P Ap + ATp P )ep + eTp P BP ∆p (ω) + ∆Tp (ω)BpT P ep V˙ 0 (t) = −eTp Qep + 2eTp P Bp ∆p (ω) V˙ 0 (t) ≤ −kep kT (λmax (Q)kep k − 2kP Bp ∆p (ω)k) Therefore, the derivative of the Lyapunov function is dependent on the magnitude of the dynamic inversion error. If the dynamic inversion is small, i.e. kep k >

2kP Bp ∆p (ω)k λmax (Q)

then, there is no necessity for an approximating function to cancel the effect of the dynamic inversion error. However, if the above condition is not satisfied, the effect of the dynamic inversion error is to increase the Lyapunov function. This is one of the conditions used to add a node to the network. Consider the situation when Np nodes have been added to the network. Then, we will evaluate the derivative of the Lyapunov function Vp (t) to analyze the stability of the closed loop system. Assume that the unknown dynamic inversion error ∆p (ω) can be ideally parameterized with Np basis functions as ∆p (ω) =

Np X

θi∗ φi (ω) + εNp (ω)

i=1

93

Let ˆ p (ω) = ∆

Np X

θˆi φi (ω)

i=1

θ˜i = θi∗ − θˆi kεNp (ω)k ≤ ε¯Np Evaluating the derivative of the Lyapunov function Vp (t), we have V˙ p (t) = eTp P e˙ p + e˙ Tp P ep +

Np X

2γ θ˜i θ˜˙ i

i=1 Np

V˙ p (t) = −eTp Qep + 2eTp P Bp

X

θ˜i φi (ω) + 2

V˙ p (t) = −eTp Qep + 2

X

γ θ˜i θ˜˙ i + 2eTp P Bp εNp (ω)

i=1

i=1 Np

Np X

θ˜i (φi (ω)BpT P ep + γ θ˜˙ i ) + 2eTp P Bp εNp (ω)

i=1

Choosing the adaptation law as θ˜˙ i = −γ −1 φi (ω)BpT P ep we have the Lyapunov derivative as V˙ p (t) = −eTp Qep + 2eTp P Bp εNp (ω) V˙ p (t) ≤ −λmax (Q)kep k2 + 2kεNp (ω)kkBpT P kkep k V˙ p (t) ≤ −kep k(λmax (Q)kep k − ε¯Np kBpT P k) The negative (semi) definiteness of the Lyapunov derivative is dependent on the condition that kep k ≥

6.5.1

ε¯Np kBpT P k λmax (Q)

Implementation

The SORBFN is implemented as the adaptive approximator in each of the angular velocity channels. The training algorithm for the SORBFN is similar to the one 94

presented in Chapter 5. Nodes are added to the individual networks (roll, pitch or yaw) based on the Lyapunov function V0 (t) evaluated in that channel. The Lyapunov matrix P is calculated for each channel by considering Q to be an identity matrix. The training algorithm is independent of each other in the roll, pitch and yaw channels. The inputs to the 3 networks include all the angular velocities (p, q, r ) while the update of weights is based on the error in that channel. Thus the network is trained with information from the entire state space but adapted only to error in the particular channel. In the simulation results presented in Chapter 7, the free parameters i.e. learning rate, node addition threshold (Nφa ), pruning threshold (Nφd ) and variances (σ) are chosen to be the same for all three networks.

6.6

Summary

In this chapter, we discuss the details of the F-15 simulation model. The baseline dynamic inversion control law that is designed to close the loop around the rotational dynamics (angular velocities) of the aircraft has been presented. We also analyze the effect of different types of dynamic inversion errors including modeling uncertainties, disturbances and control surface failures on the closed loop. The implementation of the SORBFN to augment the dynamic inversion controller is discussed. In the next chapter, we will present some of the simulation results of this implementation and compare the performance of the SORBFN to the dynamic inversion controller as well as a dynamic inversion controller augmented with a fixed structure RBF network.

95

CHAPTER 7

SIMULATION RESULTS AND DISCUSSION

In this chapter, we present some of the results of the implementation of the SORBFN adaptive controller on the F-15 simulation model discussed in Chapter 6. The controller is simulated for the flight condition at altitude 25000 feet and 0.9 mach. The linear model of the longitudinal and lateral/directional dynamics for the F-15 aircraft at this operating condition is given below. x˙ long = Along xlong + Blong ulong ylong = Clong xlong + Dlong ulong x˙ lat = Alat xlat + Blat ulat ylat = Clat xlat + Dlat ulat The states, inputs and outputs of the linear model have been discussed in Chapter 6. 

Along

   =     

Blong

   =    

−0.0272 −0.0340 0.0000 −0.5597 −0.0067 −1.1596 1.0000 0.0001 0.0975 9.0710 −0.9505 0.0000 0.0000 0.0000 1.0000 0.0000 −0.0017 −15.9683 0.0000 15.9683 

−0.2318 −0.1106 0.0000  −0.1438 −0.0342 0.0000   −15.3290 7.7584 0.0000   0.0000 0.0000 0.0000   0.0000 0.0000 0.0000 96

0.0000 0.0001 0.0003 0.0000 0.0000

       



Clong

    =      

Dlong

    =     



Alat

   =     

Blat

   =     

Clat

     =        

Dlat

     =       

1.0000 0.0000 0.0000 0.0000 0.0000 0.0042 0.0000 0.0000 0.0000 0.0000 0.0000 −0.0631

0.0000 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.6551 −0.0083 0.0000 0.0000 0.0000 0.0000 0.0000 0.0852

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 1.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000

         

         



−0.2417 0.0244 −0.9997 0.0351 −0.0001  −33.0800 −2.8838 1.3973 0.0000 0.0001   5.2961 0.0024 −0.5063 0.0000 0.0000   0.0000 1.0000 0.0244 0.0000 −0.0001   0.0000 0.0000 1.0003 0.0000 0.0000 

−0.0153 0.0159 −0.0025 0.0358  16.9344 −2.5422 13.0453 3.9551   0.9261 1.8366 −0.0769 −4.7667   0.0000 0.0000 0.0000 0.0000   0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 −0.2417 −0.0660

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0244 −0.9997 0.0351 −0.0001 0.0007 −0.0048 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −0.0033

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0246 −0.0048 −0.0250

            

            

The PI control parameters chosen for this operating condition is listed in Table 7.1 The simulation is executed for the pilot roll input command shown in Figure 7.1. 97

Axis Proportional Gain Roll 7.07 Pitch 18.021 Yaw 3

Integral Gain 25.0 14.44 0.1

Table 7.1: Proportional and Integral Gains

1

Roll Stick

0.8 0.6 0.4 0.2 0

0

10

20

30

40

50

60

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

1

Pitch Stick

0.5

0

-0.5

-1

1

Yaw Pedal

0.5

0

-0.5

-1

Figure 7.1: Pilot Roll Command

98

In the following sections, we will discuss the performance of the SORBFN augmented dynamic inversion controller under different conditions of modeling uncertainties, disturbances and actuator failures and compare it with the dynamic inversion control law. It will be shown that under certain circumstances, the dynamic inversion controller loses stability while the adaptive approximation based control is able to maintain bounded tracking error. The SORBFN controller is expected to maintain a bounded tracking error of 0.01o /second in the three angular velocity channels.

7.1

Disturbance Rejection

Here, we will consider the ability of the controller to satisfactorily track the reference signal in the presence of a disturbance signal. It is assumed that a sinusoidal disturbance of a specified amplitude enters the rotational dynamics through the roll channel. The SORBFN is initialized to zero nodes and appropriate learning rates (γ), node addition threshold (Nφa ) and prune threshold (Nφd ) are chosen. It can be noticed from Figure 7.2 that the rolling rate for both the dynamic inversion as well as the adaptive approximation based controller is sinusoidal in nature due to the influence of the disturbance input. However, on closer examination of the Figure 7.2(b), we can conclude that the amplitude of the error in tracking is actually reducing asymptotically for the SORBFN based controller and achieves a minimum error amplitude of < 1o /second in the simulation time. The dynamic inversion controller, however fails to reduce the amplitude of the tracking error signal and achieves a minimum error amplitude of 8o /second. Since the coupling between the roll and pitch channels is weak, we see that the pitch rate tracking error for the two controllers are similar even though the SORBFN converges faster (See Figure 7.3). The yaw rate

99

60

50 Actual Reference

Actual Reference

50

40

40 30

p o/s

p o/s

30 20

20 10 10 0

0

-10

0

10

20

30

40

50

-10

60

10

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

4

8

3

6 2

Tracking Error o/s

Tracking Error o/s

4 2 0 -2

1 0 -1

-4 -2 -6 -3

-8 -10

0

10

20

30 Time (s)

40

50

-4

60

(a) Without SORBFN

(b) With SORBFN.

Figure 7.2: Roll rate tracking with sinusoidal disturbance in roll channel

3

2 Actual Reference

Actual Reference

2.5 1.5 2 1

q o/s

q o/s

1.5

1

0.5

0.5 0 0

-0.5

0

10

20

30

40

50

-0.5

60

3

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

2

2.5 1.5

Tracking Error o/s

Tracking Error o/s

2

1.5

1

1

0.5

0.5 0 0

-0.5

0

10

20

30 Time (s)

40

50

-0.5

60

(a) Without SORBFN

(b) With SORBFN.

Figure 7.3: Pitch rate tracking with sinusoidal disturbance in roll channel

100

tracking error with the adaptive approximation based controller is indicative (Figure 7.5 ) of the superiority of the SORBFN based controller to the baseline dynamic in-

4.5

0.3 Actual Reference

4 0.25 3.5 0.2

3

0.15

2

r o/s

r o/s


1.5

0.1

1

0.05

0.5 0 0 -0.5

0

10

20

30

40

50

-0.05

60

4.5

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

0.3

4 0.25 3.5 0.2

Tracking Error o/s

Tracking Error o/s

3 2.5 2 1.5

0.15

0.1

1

0.05

0.5 0 0 -0.5

0

10

20

30 Time (s)

40

50

-0.05

60

(a) Without SORBFN

(b) With SORBFN

Figure 7.4: Yaw rate tracking with sinusoidal disturbance in roll channel

version controller. In fact, the yaw rate tracking error meets the tracking specification of 0.01 degrees/second that is specified apriori. Figure 7.5 describe the time histories of the actuator deflection with the adaptive approximation. While the adaptive approximation controller causes the actuators to deflect more initially, it can be seen that the magnitude is continuously decreasing with time. In the case of the dynamic inversion controller, the control surface deflections (Figure 7.6 ) are initially smaller in magnitude but are continuously increasing with time. The number of nodes allocated in the RBF network and the contribution of the adaptive

101

20

Right Left

30 15 20 10 5

Stabilator

Rudder

10

0 Right Left

-10

0 -5 -10

-20 -15 -30 -20 0

10

20

30

40

50

60

0

40

10

10

5

Aileron

Canard

30

40

50

60

Right Left

15

20

0

0

-10

-5

-20

-10 -15

-30 -40

20

20

Right Left

30

10

-20 0

10

20

30 Time (s)

40

50

60

0

(a) Rudder and Canard Deflection

10

20

30 Time (s)

40

50

60

(b) Stabilator and Aileron Deflection

Figure 7.5: Control surface deflection with SORBFN and sinusoidal disturbance in roll channel

20

Right Left

30

Right Left

15 20 10 5

Stabilator

Rudder

10

0

0 -5

-10

-10 -20 -15 -30 -20 0

10

20

30

40

50

60

0

40

40

50

60

Right Left

10

10

5

Aileron

Canard

30

15

20

0

0

-10

-5

-20

-10 -15

-30 -40

20

20

Right Left

30

10

-20 0

10

20

30 Time (s)

40

50

60

0


10

20

30 Time (s)

40

50

60


Figure 7.6: Control surface deflection without SORBFN and sinusoidal disturbance in roll channel

102

40

500

35

400

30

300

25

200

Roll

Roll

approximator can be seen in Figure 7.7. Since the effect of the disturbance on the

20

100

15

0

10

-100

5

-200

0

0

10

20

30

40

50

-300

60

7

0

10

20

30

40

50

60

0

10

20

30

40

50

60

0

10

20

30 Time

40

50

60

40 35

6

30 25

Pitch

Pitch

5 4

20 15

3

10 2 1

5 0

10

20

30

40

50

0

60

30

100

25

80

20

Yaw

Yaw

60 15

40 10 20

5 0

0

10

20

30 Time

40

50

0

60

(a) Node Allocation

(b) Network Contribution

Figure 7.7: Adaptation of SORBFN in the presence of sinusoidal disturbance in the roll channel

pitch channel of the dynamics is low, the number of nodes allocated in that channel is much lower than the roll and yaw channels.

7.2

Uncertainty in Rolling Moment Stability Derivatives

The dynamic inversion control law is given by ¯ −1 (P )(ω˙ d − f (ω) − M ˆ A − Kp ω − Ki δ=G where



ˆ A = I −1 M ˆ aero M

ω)

(7.1)



ˆ BAE L    ˆ = I −1   MBAE  ˆBAE N

103

Z

(7.2)

q¯Sb(Cˆlβ β + 2Vb Cˆlp p + 2Vb Cˆlr r)   ˆ A = I −1  q¯S¯ M c(Cmα α + 2Vc¯ Cmq q + Cmc δc )  q¯Sb(Cnβ β + 2Vb Cnp p + 2Vb Cnr r) 



(7.3)

In the above equation, we assume that the uncertainty exists only in the rolling moment control derivatives. Therefore ˆ BAE 6= LBAE L ˆ BAE = MBAE M ˆBAE = NBAE N In this example, a sufficiently large uncertainty has been considered to show the approximation capabilities of the SORBFN. Since the dynamic inversion controller distributes the rolling moment between all the actuators, the effect of the uncertainty can be noticed among all the angular velocities trajectories. It is evident from Figures 7.8(a) ,7.9(a) and 7.10(a) that the baseline dynamic inversion controller is unable to compensate for the uncertainty in the stability derivatives. In fact, the pitch rate signal goes unbounded in the simulation time. The SORBFN based controller on the other hand is able to track the desired signal in all three rotational motion channels and achieve tracking error specification of 0.01o /second in all the three channels (Figures 7.8(b)7.9(b)7.10(b)). It can be seen from Figures 7.11 that both the aileron and the rudder hit the saturation limits consistently in the case of the dynamic inversion control law. However, from Figures 7.12 we see that the SORBFN is able to keep the actuators away from the saturation limits. The number of nodes allocated in the network and its contribution is shown in Figures 7.13.

104

1000

70 Actual Reference

800

Actual Reference

60

600 50 400 40

p o/s

p o/s

200 0 -200

30 20

-400 10 -600 0

-800 0

10

20

30

40

50

-10

60

1000

30

800

25

600

20

400

15

Tracking Error o/s

Tracking Error o/s

-1000

200 0 -200

20

30

40

50

60

5 0 -5

-600

-10

-800

10

10

-400

-1000

0

X: 47.63 Y: 0.007786

-15 0

10

20

30 Time (s)

40

50

-20

60

0

10

(a) Without SORBFN

20

30 Time (s)

40

50

60

(b) With SORBFN

Figure 7.8: Roll rate tracking performance with uncertainty in base aerodynamic rolling moment

23

2

x 10

2 Actual Reference

0

1.5

-2 1

q o/s

q o/s

-4 Actual Reference

-6

0.5

-8 0 -10

-12

0

10

20

30

40

50

-0.5

60

23

2

x 10

0

10

20

30

40

50

60

2

0

1.5

Tracking Error o/s

Tracking Error o/s

-2

-4

-6

1

0.5 X: 55.09 Y: -0.0003296

-8 0 -10

-12

0

10

20

30 Time (s)

40

50

-0.5

60

(a) Without SORBFN

0

10

20

30 Time (s)

40

50

60

(b) With SORBFN

Figure 7.9: Pitch rate tracking performance with uncertainty in base aerodynamic rolling moment

105

400


0.4 300

Actual Reference

0.3 0.2

200

r o/s

r o/s

0.1 100

0 -0.1

0

-0.2 -0.3

-100

-0.4 -200

0

10

20

30

40

50

-0.5

60

400

0

10

20

30

40

50

60

0.5 0.4

300

0.3 0.2

Tracking Error o/s

Tracking Error o/s

200

100

0

0.1 0 X: 50.95 Y: 0.00994

-0.1 -0.2 -0.3

-100

-0.4 -200

0

10

20

30 Time (s)

40

50

-0.5

60

0

10

(a) Without SORBFN

20

30 Time (s)

40

50

60

(b) With SORBFN

Figure 7.10: Yaw rate tracking performance with uncertainty in base aerodynamic rolling moment

20

Right Left

30 10

20

0

0

Stabilator

Rudder

10

Right Left

-10

-10 -20 -20 -30

-30 0

10

20

30

40

50

60

0

40

30

40

50

60

Right Left

15 Right Left

20

10

10

5

Aileron

Canard

20

20

30

0

0

-10

-5

-20

-10 -15

-30 -40

10

-20 0

10

20

30 Time (s)

40

50

60

0


10

20

30 Time (s)

40

50

60


Figure 7.11: Control surface deflection without SORBFN and uncertainty in base aerodynamic rolling moment

106

20

Right Left

30 15 20 10 5

Stabilator

Rudder

10 Right Left

0

0 -5

-10

-10 -20 -15 -30 -20 0

10

20

30

40

50

60

0

40

40

50

60

Right Left

10

10

5

Aileron

Canard

30

15

20

0

0

-10

-5

-20

-10 -15

-30 -40

20

20

Right Left

30

10

-20 0

10

20

30 Time (s)

40

50

60

0


10

20

30 Time (s)

40

50

60


Figure 7.12: Control surface deflection with SORBFN and uncertainty in base aerodynamic rolling moment

16

3000

14

2500

12

2000

Roll

Roll

10 8 6

1500 1000 500

4 2

0

0

-500

0

10

20

30

40

50

60

0

10

20

30

40

50

60

6 40 5

35

Pitch

Pitch

30 4 3

25 20 15

2 1

10 5 0

10

20

30

40

50

60

0

20

10

20

30

40

50

50 40

15

Yaw

Yaw

30 10

20 10

5 0 0

0

10

20

30 Time

40

50

-10

60

(a) Node Allocation

0

10

20

30 Time

40

50

60


Figure 7.13: Adaptation of SORBFN with uncertainty in base aerodynamic rolling moment

107

7.3

Control Surface Failures

In this section, we will consider stuck at faults of the control surfaces. From extensive simulation, it has been noticed that stuck at faults of the stabilators tend to have the most adverse effect on the aircraft. We will assume stuck at faults of the right stabilator which is considered to get stuck at 10o below trim at 10 seconds into the simulation. The fault is assumed to to be corrected at 50 seconds. The trim value of the stabilator for the given flight condition is 2.86o .

7.3.1

Case 1

The controller is simulated for a pilot roll command as seen in Figure 7.1. The control surface failure can be seen in the time histories of the stabilator deflection in Figure 7.14. The effect of the fault on the angular velocity tracking can be seen in

20

Right Left

15

15

10

10

5

5

Stabilator

Stabilator

20

0

-5

-10

-10

-15

-15

-20

-20 0

10

20

30

40

50

20

60

0

10

20

30

40

50

60

20

Right Left

15

15

10

10

5

5

Aileron

Aileron

Right Left

0

-5

0

-5

-10

-10

-15

Right Left

0

-5

-15

-20

-20 0

10

20

30 Time (s)

40

50

60

0

(a) Without SORBFN Deflection

10

20

30 Time (s)

40

50

60

(b) With SORBFN

Figure 7.14: Stabilator and Aileron deflection with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds

108

Figures 7.15 , 7.16 and 7.17. Both the controllers are able to stabilize the system.

50

50 Actual Reference

Actual Reference 40

40

30 30

p o/s

p o/s

20 20

10 10 0 0

0

10

20

30

40

50

-20

60

50

15

40

10

30

5

Tracking Error o/s

Tracking Error o/s

-10

-10

20

10

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

0

-5

0

-10

0

-10

0

10

20

30 Time (s)

40

50

-15

60

(a) Without SORBFN

(b) With SORBFN

Figure 7.15: Roll rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds

However, the dynamic inversion with SORBFN controller is able to achieve the prespecified tracking error of 0.01o /second in each of the 3 channels (Figures 7.15(b), 7.16(b), 7.17(b)). The dynamic inversion controller eventually recovers from the fault but there is initially a large tracking error in all three channels (Figures 7.15(a), 7.16(a), 7.17(a)). Also, from Figure 7.14 and 7.18 , we notice that the dynamic inversion controller causes the control surfaces to continuously change magnitude to compensate for the failure while the SORBFN controller is able to achieve a steady deflection of all the actuators after a short period of time. The number of nodes allocated to the network and network contribution are plotted in Figure 7.19. The ability of the network to ”back-off” can be seen with the network 109

9

4 Actual Reference

8

Actual Reference

3

7 2 6 1

q o/s

q o/s

5 4 3

0 -1

2 -2 1 -3

0 -1

0

10

20

30

40

50

-4

60

9

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

4

8

3

7 2

Tracking Error o/s

Tracking Error o/s

6 5 4 3

1 0 -1

2 -2 1 -3

0 -1

0

10

20

30 Time (s)

40

50

-4

60

(a) Without SORBFN

(b) With SORBFN

Figure 7.16: Pitch rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds

14

2 Actual Reference

12

1.5

10 1

r o/s

r o/s

8 0.5

6 0 Actual Reference

4

-0.5

2

0

0

10

20

30

40

50

-1

60

14

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

2

12

1.5

Tracking Error o/s

Tracking Error o/s

10

8

6

1

0.5

0 4 -0.5

2

0

0

10

20

30 Time (s)

40

50

-1

60

(a) Without SORBFN

(b) With SORBFN

Figure 7.17: Yaw rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds

110

30

30

20

20

10

0

Rudder

Rudder

10

Right Left

-10

-20

-20

-30 10

20

30

40

50

60

0

40

10

20

30

40

50

60

40 Right Left

30

Right Left

30 20

10

10

Canard

20

0

0

-10

-10

-20

-20

-30 -40

Right Left

-30 0

Canard

0

-10

-30

0

10

20

30 Time (s)

40

50

-40

60

0

10


20

30 Time (s)

40

50

60

(b) With SORBFN

Figure 7.18: Rudder and Canard deflection with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds

14

350

12

300 250 200

8

Roll

Roll

10

6

50

2 0

150 100

4

0 0

10

20

30

40

50

-50

60

12

0

10

20

30

40

50

60

0

10

20

30

40

50

60

0

10

20

30 Time

40

50

200

10 150

Pitch

Pitch

8 6

100

4 50 2 0

0

10

20

30

40

50

0

60

16

100

14 80

12

Yaw

Yaw

10 8

60 40

6 4

20

2 0

0

10

20

30 Time

40

50

60

(a) Node Allocation


Figure 7.19: Adaptation of SORBFN with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds

111

contribution reducing as soon as the fault is removed at t=50 seconds.

7.3.2

Case 2

In the previous example, we showed the ability of the adaptive approximator to compensate for a stuck at fault of the actuator when the pilot input is a roll command. In this section, we will show the ability of the approximator to compensate for the same fault (See Figure 7.20) when the pilot introduces a pitch command (Figure 7.21)

20

Right Left

15

15

10

10

5

5

Stabilator

Stabilator

20

0

-5

-10

-10 -15

-20

-20 0

10

20

30

40

50

20

60

0

10

20

30

40

50

20

Right Left

15

15

10

10

5

5

Aileron

Aileron

0

-5

-15

Right Left

0

Right Left

0

-5

-5

-10

-10

-15

60

-15

-20

-20 0

10

20

30 Time (s)

40

50

60

0


10

20

30 Time (s)

40

50

60

(b) With SORBFN

Figure 7.20: Stabilator and Aileron deflection with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds

in addition to the roll command during the occurrence of the fault. As in the previous case, both the controllers are able to stabilize the system, but only the dynamic inversion with SORBFN achieves the pre-specified tracking error. The tracking performance of the controller can be seen in Figures 7.22 , 7.23 and 7.24.

112

1

Roll Stick

0.8 0.6 0.4 0.2 0

0

10

20

30

40

50

60

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

1

Pitch Stick

0.8 0.6 0.4 0.2 0

1

Yaw Pedal

0.5

0

-0.5

-1

Figure 7.21: Pilot Roll and Pitch Command

113

50

50 Actual Reference 40

30

30

p o/s

p o/s

Actual Reference 40

20

20

10

10

0

-10

0

0

10

20

30

40

50

-10

60

50

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

15

40

10

30

Tracking Error o/s

Tracking Error o/s

0

20

5

0

10

-5

0

-10

0

10

20

30 Time (s)

40

50

-10

60

(a) Without SORBFN

(b) With SORBFN


10

6 Actual Reference

Actual Reference

5

8 4 6

3 2

q o/s

q o/s

4

2

1 0 -1

0

-2 -2 -3 -4

0

10

20

30

40

50

-4

60

10

4

8

3

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

2

Tracking Error o/s

Tracking Error o/s

6

4

2

1 0 -1

0 -2 -2

-4

-3

0

10

20

30 Time (s)

40

50

-4

60

(a) Without SORBFN

(b) With SORBFN


114

Once again, we notice that the magnitude of the other control surface deflections

14

2 Actual Reference

12

1.5

10 1

r o/s

r o/s

8 0.5

6 Actual Reference

0

4 -0.5

2

0

0

10

20

30

40

50

-1

60

14

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

2

12

1.5

Tracking Error o/s

Tracking Error o/s

10

8

6

1

0.5

0 4 -0.5

2

0

0

10

20

30 Time (s)

40

50

-1

60

(a) Without SORBFN

(b) With SORBFN

Figure 7.24: Yaw rate tracking with right Stabilator stuck at 4o above trim from 10 seconds and corrected at 50 seconds

is not only higher when with the baseline dynamic inversion controller but is also continuously changing to compensate for the failure. Figure 7.25 shows the rudder and canard deflections for both the controllers. Figures 7.26 describe the evolution of the number of nodes in each of the 3 channels.

7.4

Comparison to a Fixed Structure RBF Network

In the previous sections, we demonstrated the ability of the SORBFN controller to satisfy tracking performance constraints as compared to the baseline dynamic inversion controller. In this section we will compare its performance to a fixed structure RBF network. The fixed structure RBF network is implemented similar to the 115

30

20

20

10

10

Rudder

Rudder

30

0

Right Left

-10

-20

-30 0

10

20

30

40

50

60

0

40

10

20

30

40

50

60

40 Right Left

30

Right Left

30 20

10

10

Canard

20

0

0

-10

-10

-20

-20

-30 -40

Right Left

-20

-30

Canard

0

-10

-30

0

10

20

30 Time (s)

40

50

-40

60

0

10

(a) Without SORBFN

20

30 Time (s)

40

50

60

(b) With SORBFN

Figure 7.25: Rudder and Canard deflection with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds

20

350 300

15

250

Roll

Roll

200 10

150 100

5

50 0 0

10

20

30

40

50

-50

60

20

200

15

150

Pitch

Pitch

0

10

5

0

0

10

20

30

40

50

60

0

10

20

30

40

50

60

0

10

20

30 Time

40

50

60

100

50

0

10

20

30

40

50

0

60

20

120 100

15

Yaw

Yaw

80 10

60 40

5 20 0

0

10

20

30 Time

40

50

0

60

(a) Node Allocation


Figure 7.26: Adaptation of SORBFN with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds

116

SORBFN in the three channels of the rotational dynamics. Each channel network contains 36 radial basis functions whose centers are equally distributed on the state space that the flight dynamics are perceived to operate. The dynamic inversion with fixed structure RBF is simulated for the pilot roll command under right stabilator stuck at fault as described earlier. The comparative tracking abilities of the fixed RBF controller with the SORBFN controller can be seen from Figures 7.27 , 7.28 and

50

50 Actual Reference

Actual Reference

30

20

20

p o/s

40

30

p o/s

40

10

10

0

0

-10

-10

-20

0

10

20

30

40

50

-20

60

20

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

15

15

10

Tracking Error o/s

Tracking Error o/s

10

5

0

5

0

-5 -5 -10

-10

-15

0

10

20

30 Time (s)

40

50

-15

60

(a) With Fixed RBF

(b) With SORBFN


7.29 where it is evident that both the controller perform equally well. In Chapter 5, we stated that the performance of a fixed structure approximator need not be equally good across different types of faults. Now we will study the ability of the same fixed structure RBF in compensating for modeling uncertainties and external disturbances in comparison to the SORBFN. 117

5

4 Actual Reference

4

Actual Reference

3

3

2

2

q o/s

q o/s

1 1

0

0 -1 -1 -2

-2

-3

-3 -4

0

10

20

30

40

50

-4

60

5

4

4

3

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

2

2

Tracking Error o/s

Tracking Error o/s

3

0

1 0

1 0 -1

-1 -2

-2

-3

-3 -4

0

10

20

30 Time (s)

40

50

-4

60

(a) With Fixed RBF

(b) With SORBFN


3

2 Actual Reference

2.5

Actual Reference 1.5

2 1

1

r o/s

r o/s

1.5

0.5

0.5 0

0

-0.5 -0.5 -1 -1.5

0

10

20

30

40

50

-1

60

3

0

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

2

2.5 1.5

Tracking Error o/s

Tracking Error o/s

2 1.5 1 0.5 0

1

0.5

0

-0.5 -0.5 -1 -1.5

0

10

20

30 Time (s)

40

50

-1

60

(a) With Fixed RBF

(b) With SORBFN

Figure 7.29: Yaw rate tracking with right Stabilator stuck at 10o below trim from 10 seconds to 50 seconds

118

7.4.1

Modeling Uncertainty

An uncertainty in the base aerodynamic rolling moment is considered. The comparative tracking abilities of the fixed RBF and SORBFN based controller in the 3 channels can be seen in Figures 7.30 ,7.31 and 7.32. The fixed structure RBF is

300

70 Actual Reference

60

Actual Reference

200

50 100

p o/s

p o/s

40 0

30 20

-100 10 -200 0

0

5

10

15

20

25

30

35

-10

40

250

30

200

25

150

20

100

15

Tracking Error o/s

Tracking Error o/s

-300

50 0 -50

20

30

40

50

60

5 0 -5

-150

-10

-250

10

10

-100

-200

0

X: 47.63 Y: 0.007786

-15 0

5

10

15

20 Time (s)

25

30

35

-20

40

(a) With Fixed RBF

0

10

20

30 Time (s)

40

50

60

(b) With SORBFN


unable to compensate for the modeling uncertainty and in fact the simulation fails at about 35 seconds.

7.4.2

Disturbance

A disturbance in the roll channel is considered. The comparative tracking abilities of the 2 controllers in the 3 channels can be seen in Figures 7.33 ,7.34 and 7.35. Even though the performance of the fixed RBF augmented dynamic inversion controller is 119

1000

2 Actual Reference

0 1.5

-1000 -2000

q o/s

q o/s

1 -3000 Actual Reference

-4000

0.5

-5000 -6000

0

-7000 -8000

0

5

10

15

20

25

30

35

-0.5

40

1000

0

10

20

30

40

50

60

2

0 1.5

-2000

Tracking Error o/s

Tracking Error o/s

-1000

-3000 -4000

1

0.5

-5000 X: 55.09 Y: -0.0003296

-6000

0

-7000 -8000

0

5

10

15

20 Time (s)

25

30

35

-0.5

40

0

10

(a) With Fixed RBF

20

30 Time (s)

40

50

60

(b) With SORBFN


200



150

0.3 0.2

r o/s

0.1

r o/s

100

50

0 -0.1 -0.2

0

-0.3 -0.4

-50

0

5

10

15

20

25

30

35

-0.5

40

200

0

10

20

30

40

50

60

0.5 0.4

150

0.3

Tracking Error o/s

Tracking Error o/s

0.2 100

50

0.1 0 X: 50.95 Y: 0.00994

-0.1 -0.2

0

-0.3 -0.4

-50

0

5

10

15

20 Time (s)

25

30

35

-0.5

40

(a) With Fixed RBF

0

10

20

30 Time (s)

40

50

60

(b) With SORBFN


120

60

50 Actual Reference

Actual Reference

50

40

40 30

p o/s

p o/s

30 20

20 10 10 0

0

0

10

20

30

40

50

-10

60

4

4

3

3

2

2

Tracking Error o/s

Tracking Error o/s

-10

1 0 -1 -2

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

1 0 -1 -2

-3 -4

0

-3

0

10

20

30 Time (s)

40

50

-4

60

(a) With Fixed RBF

(b) With SORBFN


2.5

2 Actual Reference

Actual Reference

2

1.5

1.5

q o/s

q o/s

1 1

0.5 0.5

0

0

-0.5

0

10

20

30

40

50

-0.5

60

2.5

10

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

2

2

1.5

1.5

Tracking Error o/s

Tracking Error o/s

0

1

1

0.5

0.5

0

0

-0.5

0

10

20

30 Time (s)

40

50

-0.5

60

(a) With Fixed RBF

(b) With SORBFN


121

0.6


Actual Reference

0.5

0.25

0.15

r o/s

0.2

0.3

r o/s

0.4

0.2

0.1

0.1

0.05

0

-0.1

0

0

10

20

30

40

50

-0.05

60

0.6

0.3

0.5

0.25

0.3

0.2

0.1

20

30

40

50

60

0

10

20

30 Time (s)

40

50

60

0.15

0.1

0.05

0

-0.1

10

0.2

Tracking Error o/s

Tracking Error o/s

0.4

0

0

0

10

20

30 Time (s)

40

50

-0.05

60

(a) With Fixed RBF

(b) With SORBFN


significantly better than the baseline dynamic inversion control law, we can see that it does not match the capabilities of the SORBFN based controller. The minimum amplitude error achieved by the fixed RBF in the roll channel is close to 4o /second.

7.5

Summary

In this chapter, we demonstrated the ability of the SORBFN to compensate for errors in dynamic inversion that are caused due to modeling uncertainties, disturbances or control surface failures and compared it to the performance of a baseline dynamic inversion controller. Simulations were performed under uncertainties in rolling moment stability derivatives and a pilot roll command. An external sinusoidal disturbance in the roll channel was shown to be effectively compensated by the SORBFN controller. Additionally stuck at faults of the right stabilator (control surface failure) 122

was simulated under 2 types of pilot input commands. Finally, the SORBFN was compared to a fixed structure RBF that was tuned to compensate for control surface failures. While the fixed structure RBF did not perform well under uncertainties and external disturbances, the SORBFN was able to achieve pre-specified tracking error convergence in all conditions.

123

CHAPTER 8

A NEURAL NETWORK BASED OBSERVER FOR TURBINE ENGINE STATE ESTIMATION

8.1

Introduction

Accurate estimation of aircraft engine performance parameters for different levels of degradation is a significant challenge since parameters such as thrust, stall margin and HPT inlet temperature cannot be measured explicitly. These parameters are controlled (monitored) using estimated values that introduces uncertainty in engine operation thus resulting in conservative safety margins. Accurate estimation of these parameters enhances efficiency while increasing flight safety. Estimation of performance parameters of an engine is an extensively researched topic with varied approaches. Kobayashi et.al [32] investigate an approach based on Constant Gain Extended Kalman Filter (CGEKF) and an on-board engine model for in-flight estimation of parameters such as thrust and stall margins. This estimator requires the engine model to run simultaneously as a part of the observer. Simon et.al [33] develop an analytic method of incorporating state variable inequality constraints in the Kalman filter and apply it to a turbofan engine. The resultant filter is a combination of a standard Kalman filter and a quadratic programming problem. 124

Dewallef et.al [34] present the development of an unscented Kalman filter for engine diagnostics. Litt [35] discusses a linear point extended Kalman filter based design technique to enable the optimal estimation of thrust in an aircraft engine. Generally, the mathematical model of a jet engine is either unavailable or too complex to apply standard estimation methods. Physics based models such as in [36] are very limited in their ability to represent the complexity of the engine. However computer simulations that approximate the physical engine with high accuracy are available. Neural networks have the ability to approximate the nonlinearity when trained with engine simulation data. A number of observers based on neural networks have been discussed in literature. Passino et.al [37] discuss a neural network based method for estimation of immeasurable states in an engine simulation model. The paper focusses on the importance of input selection for the estimator and it’s influence on the accuracy of estimation. Volponi [38] describes the use of hybrid engine modeling using multi-layer perceptrons for performance tracking in turbine engines. Observers using neural networks have been reported for various other applications. Vargas and Hemerly [39] describe an observer based on linearly parameterized neural networks for estimation of unknown general nonlinear systems. Lainiotis et.al [40] discuss an estimator based on dynamic neural networks. Radial Basis Function Networks (RBFN) have been used for state estimation in electric power networks [41]. Chen et.al [42] discuss the application of neural networks for state estimation in active vibration control. In the following sections, we discuss a preliminary study conducted on the design of a neural network based observer to estimate the states of an aircraft engine using growing and pruning radial basis function networks. This technique is based on the 125

philosophy similar to [38] where a hybrid model of an engine is implemented using a physics based model augmented with an empirical model consisting of a pre-trained neural network. We, however take advantage of the function approximation properties of a Radial Basis Function Network as compared to a multi-layer perceptron used in [38]. The RBFN is trained using the algorithm discussed in Chapter 4 where hidden layer neurons are added or deleted based on the inputs arriving at the network. This method was first presented by Shankar and Yedavalli in [43]. The RBFN is trained using the simulation data from a turbine engine simulation model. The advantage of this approach is that the observer does not require an on board engine model that runs simultaneously thus decreasing computational burden. Also, since the network can be trained using training data that consists of various degraded conditions of the engine, the health parameters need not be estimated to get an accurate estimate of the performance parameters at different engine conditions.

8.2

Neural Network Based State Observer

Consider a nonlinear system of the form x˙ = f (x, u)

(8.1)

y = h(x, u) ' Cx + Du

(8.2)

Let the linearized model of the above nonlinear system around the equilibrium point be given by x˙ = Ax + Bu

(8.3)

y = Cx + Du

(8.4)

126

such that the pairs (A,B) and (A,C) are controllable and observable respectively. Since (A,C) is observable, we can design an observer with a gain L that is given by xˆ˙ = Aˆ x + Bu + L(y − yˆ)

(8.5)

yˆ = C xˆ + Du

(8.6)

When we close the loop around the linear system using the observer we get the error dynamics that is given by e˙ = (A − LC)e

(8.7)

where e=x-ˆ x. By designing a gain L such that A-LC is stable, we can ensure that the error between the actual state and estimated state goes to zero asymptotically. However when we close the loop around the nonlinear system using the linear observer (Equation 8.5), we have x˙ − xˆ˙ = f (x, u) − Aˆ x − Bu − L(y − yˆ)

(8.8)

The estimation error goes to zero only when the linear model is a close approximation of the nonlinear system. This is possible only at the equilibrium points of the nonlinear system around which it has been linearized. We can always rewrite the nonlinear system and the closed loop as x˙ = f (x, u) + Ax − Ax + Bu − Bu e˙ = (A − LC)e + ∆(x, u) where ∆(x, u) = f (x, u) − Ax − Bu

127

(8.9)

is the Linearization Error. Let ψ(y,u) be a neural network component to approximate the linearization error. Let the equation of the observer be xˆ˙ = Aˆ x + Bu + L(y − yˆ) + ψ(y, u)

(8.10)

The closed loop is then given by e˙ = (A − LC)e + ∆(x, u) − ψ(y, u)

(8.11)

If the neural network component can be implemented to approximate the linearization error, we can ensure that the error between the actual and estimated states goes to zero asymptotically. The schematic of the observer can be seen in Figure 8.1. The neural network component of the observer is composed of a Radial Basis Function Network (RBFN). The RBFN is trained offline to approximate the linearization error. The training algorithm for this network was presented in Chapter 4, Section 4.1.

8.3

Application To Aircraft Engine State Estimation

The engine model is a nonlinear simulation of an advanced high-bypass turbofan engine. This engine model has been constructed as a Component Level Model (CLM), which consists of the major components of an aircraft engine and is similar to the engine model used in [35]. The CLM represents highly complex engine dynamics and simulates real-time data. It is also possible to simulate the engine under degraded conditions. In this study we simulate the engine from idle to take-off. The nonlinear dynamics of the engine is given by x˙ = f (x, u)

(8.12)

y = Cx + Du

(8.13)

128

Outputs

Nonlinear System

Controller u

States

∫

+ -

+ + Φ(y,u)

Linear Observer Dynamics Neural Network

Figure 8.1: Neural Network Based State Observer

129

where x ∈

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