adaptive control augmentation for a hypersonic glider380238/s42837297_phd... · adaptive control...

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Adaptive Control Augmentation for a Hypersonic

Glider

Sanchito Banerjee

Bachelor of Engineering – Aerospace Avionics (Honours)

A thesis submitted for the Degree of Doctor of Philosophy at

The University of Queensland in 2015

School of Mechanical and Mining Engineering

ABSTRACT

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ABSTRACT

The aim of this research is to investigate control strategies to carry out a pull out of a

ballistic trajectory (initially the same as the sub-orbital ballistic HyShot and HIFiRE

trajectories) and a pulsed roll angle bank manoeuvre for a hypersonic glider. This study

investigates the performance of two different control methodologies in the presence of

aerodynamic, gravimetric and actuator uncertainties: pole placement control (PPC) (as the

baseline) and adaptive control (to augment the PPC).

Through simulations, it is shown that the PPC carries out the pull up manoeuvre (by

tracking the flight path angle, ) on a scaled Generic Hypersonic Aerodynamics Model

Example (GHAME model). Once the pull-up manoeuvre is carried out the

lateral/directional PPCs perform satisfactorily in tracking the commanded roll angle,

and maintaining a sideslip angle, , of zero degrees. The PPCs presented are all Single

Input Single Output (SISO) controllers. Pole-zero plots are utilised to highlight the stability

properties of the PPC. All the controllers are stable. The differences in the performance

and robustness for the various uncertainty cases are highlighted through the tracking error

norm and the time delay margin (TDM). The performance of the PPC significantly

worsens in the presence of uncertainties. This deterioration is quantified using tracking

error norms, error dynamics acceleration and a control surface metric.

adaptive control is employed as a control strategy to augment the PPC as it allows the

decoupling of the control and the estimation loop. The control augmentation design is

employed for both the longitudinal and the lateral/directional channels in the presence of

matched and unmatched uncertainties. A piecewise constant adaptive law (to estimate the

uncertainties) is adopted for all the channels. An additional challenge present in this

research is that the flight path angle dynamics of the system display non-minimum phase

behaviour. The control theory requires the inversion of the system dynamics, which

renders the control law, which cancels the unmatched uncertainties, unstable. An inverse

ABSTRACT

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DC gain method is presented and tested. Using this modification, the inversion of the

system dynamics is avoided. The main step forward that has been taken in this thesis is

the extension of the application of theory and its application to a non-minimum phase

state feedback Linear Time Varying (LTV) systems. This modified law prevents a

reformulation of the control problem, for example computing an alternative representation

of the state estimator in the augmented controller in order to have all the uncertainties

come in through the matched channel of the system or having a virtual inertial

measurement unit (IMU) in order to make the measured values minimum phase. The

augmented controller is able to cancel the uncertainties and is able to restore the

performance of the controller and bring the system closer to the desired system

performance. The improvement provided by the augmented controller is demonstrated

using the tracking error norm and the control surface metric. The augmented controller,

however, shows a reduction in the robustness parameter, which is the time-delay margin.

This result is consistent with that observed in adaptive control as there is a trade-off

between performance and robustness.

Declaration by author

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This thesis is composed of my original, and contains no material previously published or

written by another person except where due reference has been made in the text. I have

clearly stated the contribution by others to jointly-authored works that I have included in

my thesis.

I have clearly stated the contribution of others to my thesis as a whole, including statistical

assistance, survey design, data analysis, significant technical procedures, professional

editorial advice, and any other original research work used or reported in my thesis. The

content of my thesis is the result of work I have carried out since the commencement of

my research higher degree candidature and does not include a substantial part of work

that has been submitted to qualify for the award of any other degree or diploma in any

university or other tertiary institution. I have clearly stated which parts of my thesis, if

any, have been submitted to qualify for another award.

I acknowledge that an electronic copy of my thesis must be lodged with the University

Library and, subject to the policy and procedures of The University of Queensland, the

thesis be made available for research and study in accordance with the Copyright Act 1968

unless a period of embargo has been approved by the Dean of the Graduate School.

I acknowledge that copyright of all material contained in my thesis resides with the

copyright holder(s) of that material. Where appropriate I have obtained copyright

permission from the copyright holder to reproduce material in this thesis.

.


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Publications during Candidature

[1]. Banerjee, S., Boyce, R., Wang, Z., Baur, B., and Holzapfel, F. "L1 Augmented Controller for a Lateral/Directional Manoeuvre of a Hypersonic Glider," Submitted for Review at Journal of Aircraft, 2015.

[2]. Banerjee, S., Wang, Z., Baur, B., Holzapfel, F., Che, J., and Cao, C. "L1 Adaptive Control Augmentation for the Longitudinal Dynamics of a Hypersonic Glider," Journal of Guidance, Control, and Dynamics, 2015, pp. 1-17. doi: 10.2514/1.G001113

[3]. Banerjee, S., Vanyai, T., Wang, Z., Baur, B., and Holzapfel, F. "L1 Augmentation Configuration for a Lateral/Directional Manoeuvre of a Hypersonic Glider in the Presence

of Uncertainties," (AIAA 2014-2533) 19th AIAA International Space Planes and Hypersonic Systems and Technologies Conference. American Institute of Aeronautics and Astronautics, 2014.

[4]. Banerjee, S., Creagh, M., Boyce, R., Baur, B., Wang, Z., and Holzapfel, F. "L1 Adaptive Control Augmentation Configuration for a Hypersonic Glider in the Presence of

Uncertainties," (AIAA 2014-0453) AIAA Guidance, Navigation, and Control Conference. American Institute of Aeronautics and Astronautics, 2014.

[5]. Banerjee, S., Creagh, A. M., and Boyce, R. R. "An Alternative Attitude Control Strategy for

SCRAMSPACE 1 Experiment," (AIAA 2014-1475) AIAA Guidance, Navigation, and Control Conference. American Institute of Aeronautics and Astronautics, 2014.

[6]. Preller, D., Banerjee, S., Smart, M., and Creagh, A. M. "Longitudinal Control Strategy for

Hypersonic Accelerating Vehicles," (AIAA 2013-5238) AIAA Guidance, Navigation, and Control (GNC) Conference. American Institute of Aeronautics and Astronautics, 2013.


vi

Publications included in this Thesis

Sanchito Banerjee, Zhongjie Wang, Bernhard Baur, Florian Holzapfel, JiaxingChe and Chengyu Cao, "

Adaptive Control Augmentation for Longitudinal Dynamics of a Hypersonic Glider," AIAA Journal of Guidance,

Control and Dynamics, (2015).doi: http://arc.aiaa.org/doi/abs/10.2514/1.G001113. - incorporated in

Chapter 2, 4, 5 and 7.

Contributor Statement of contribution

Sanchito Banerjee (Candidate) Designed experiments (80%), Wrote the

paper (100%)

Bernhard Baur Designed experiments (5%), Edited paper

(35%)

Zhongjie Wang Designed experiments (5%), Edited paper

(35%)

Professor Florian Holzapfel Designed experiments (5%)

Jiaxing Che Designed experiments (5%), Edited paper

(20%)

Chengyu Cao Edited paper (10%)

Contributions by Others to the thesis

No contributions by others

Statement of parts of the thesis submitted to qualify for the award of another degree

None

http://arc.aiaa.org/doi/abs/10.2514/1.G001113

Acknowledgements

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Acknowledgements

There are a lot of people around me who have helped me during the course of my research

endeavours. However, it is only possible to give a few of them a mention here.

First and foremost, I would like to thank my wonderful family. My mum (Mita Banerjee),

my dad (Shantanu Banerjee) and my awesome sister (Bornika Banerjee). Without their

support, it would have been impossible to finish the work. I am thankful not only for their

words of encouragement but also for their great patience at all time. For the help and

advice that they have provided to me, a mere thank you does not suffice. Therefore, I

would like to dedicate this whole body of work to them. This is for them and them only.

I would like to acknowledge the support of my advisory team at University of Queensland

(UQ) and University of New South Wales (UNSW): Professor Russell Boyce, Professor

David Mee, Dr Michael Creagh and Dr Michael Kearney. They have taught me how good

research writing is done. I appreciate all their contributions of time and ideas to make my

PhD experience productive and stimulating. They meticulously read the entire draft,

sometimes thrice. Their criticisms, suggestions, encouragements, and questions, enabled

me to sharpen and strengthen the analyses, logic, and arguments in this study and to

eliminate errors.

I would like to extend my gratitude towards the group at Technical University Munich

(TUM). Firstly, I would like to thank Professor Florian Holzapfel (Director of the Institute

for Flight System Dynamics) for giving me the opportunity to carry out a substantial

portion of my research work in Munich. Completing this work would have been difficult

without the support, help and friendship of Bernhard Baur, Zhongjie Wang, Miguel Leitao

and Florian Peter just to name a few. I am indebted to them for their help and patience

when discussing all matter concerning my thesis and also discussing matters that had

nothing to do with my thesis.

Keywords

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Keywords

Hypersonic glider flight control, pole placement control, adaptive control, longitudinal

and lateral/directional control, descent trajectory, adaptive augmented controller,

nonminimum phase system, linear time varying system, full state feedback control, time-

delay margin.

Australian and New Zealand Standard Research Classifications

ANZSRC Code: 090101, Aerodynamics (excl. Hypersonic Aerodynamics), 10%

ANZSRC Code: 090104, Aircraft Performance and Flight Control Systems, 60%

ANZSRC Code: 090106, Flight Dynamics, 20%

ANZSRC Code: 090107, Hypersonics Propulsion and Hypersonic Aerodynamics, 10%

Fields of Research (FoR) Classification

FoR code: 0901, Aerospace Engineering, 100%

Table of Contents

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Table of Contents

ABSTRACT .............................................................................................................................................. II DECLARATION BY AUTHOR .................................................................................................................... IV ACKNOWLEDGEMENTS......................................................................................................................... VII KEYWORDS ......................................................................................................................................... VIII TABLE OF CONTENTS ............................................................................................................................. IX LIST OF FIGURES ................................................................................................................................... XII LIST OF TABLES .................................................................................................................................... XVI TABLE OF ACRONYMS ........................................................................................................................ XVII TABLE OF SYMBOLS ............................................................................................................................. XIX 1 INTRODUCTION ................................................................................................................................ 1 1.1 OVERVIEW OF RESEARCH ..................................................................................................................... 1 1.1.1 CHALLENGES OF CONTROLLING AND NEED OF HYPERSONIC VEHICLES ................................................................. 1 1.1.2 GENERIC HYPERSONIC AERODYNAMIC MODEL EXAMPLE (GHAME) .................................................................. 2 1.2 MOTIVATION .................................................................................................................................... 3 1.3 BACKGROUND TO THE RESEARCH ........................................................................................................... 5 1.4 RESEARCH GAP ................................................................................................................................. 9 1.5 RESEARCH PROBLEM ........................................................................................................................ 10 1.6 SCOPE AND FOCUS OF THE STUDY ........................................................................................................ 12 1.7 DELIMITATIONS OF THE STUDY ............................................................................................................ 12 1.8 METHODOLOGY............................................................................................................................... 13 1.9 CONTRIBUTION OF DISSERTATION ........................................................................................................ 14 1.10 ORGANISATION OF THE RESEARCH ...................................................................................................... 15 2 LITERATURE REVIEW ....................................................................................................................... 18 2.1 INTRODUCTION ............................................................................................................................... 18 2.2 STATE OF THE ART IN FLIGHT CONTROL APPROACHES ............................................................................... 18 2.3 COMPLICATIONS WITH HYPERSONIC CONTROL AND MODELLING ................................................................. 21 2.4 CONTROL METHODOLOGY OF HYPERSONIC VEHICLE – LONGITUDINAL DYNAMICS............................................ 26 2.4.1 LINEAR CONTROL ..................................................................................................................................... 27 2.4.2 LINEAR PARAMETER VARYING CONTROL ...................................................................................................... 32 2.4.3 NONLINEAR CONTROL............................................................................................................................... 34 2.4.4 ADAPTIVE CONTROL ................................................................................................................................. 37 2.4.5 TRAJECTORY CONTROL OF HYPERSONIC GLIDERS ........................................................................................... 40 2.5 CONTROL OF HYPERSONIC VEHICLE – LATERAL/DIRECTIONAL DYNAMICS ...................................................... 44 2.6 ADAPTIVE CONTROL LITERATURE REVIEW ......................................................................................... 45 2.7 SUMMARY OF LITERATURE REVIEW – THE GAPS IN THE FIELD...................................................................... 50 2.8 CONCLUSION .................................................................................................................................. 51 3 SIMULATION MODEL ...................................................................................................................... 52 3.1 INTRODUCTION ............................................................................................................................... 52 3.2 AIRCRAFT DYNAMICS ........................................................................................................................ 52 3.2.1 REFERENCE FRAMES ................................................................................................................................. 53 3.2.2 EQUATIONS OF MOTION ........................................................................................................................... 54 3.3 MODEL BASED FORCES AND MOMENTS DESCRIPTION .............................................................................. 62 3.3.1 WEIGHT MODEL ...................................................................................................................................... 63 3.3.2 ENVIRONMENT MODEL ............................................................................................................................. 63 3.4 AERODYNAMIC MODEL ..................................................................................................................... 64 3.4.1 MODEL PARAMETERS ............................................................................................................................... 65 3.4.2 GENERIC HYPERSONIC AERODYNAMIC MODEL EXAMPLE (GHAME) MODEL ..................................................... 70

Table of Contents

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3.5 ACTUATOR MODEL .......................................................................................................................... 71 3.6 SENSOR MODEL .............................................................................................................................. 73 3.7 SOURCES AND MODELLING OF UNCERTAINTIES ....................................................................................... 79 3.7.1 CONTROL SURFACE UNCERTAINTIES ............................................................................................................ 80 3.7.2 AERODYNAMIC AND GRAVIMETRIC UNCERTAINTIES ....................................................................................... 80 3.8 MONTE CARLO SIMULATIONS ............................................................................................................. 85 3.9 SIMULATION PLATFORM .................................................................................................................... 85 3.10 CONCLUSION ................................................................................................................................ 86 4. THEORETICAL FRAMEWORK AND IMPLEMENTATION ..................................................................... 87 4.1 INTRODUCTION ............................................................................................................................... 87 4.2 POLE PLACEMENT CONTROL ............................................................................................................... 89 4.2.1 BASELINE CONTROLLER – LONGITUDINAL CONTROLLER .................................................................................. 92 4.2.2 BASELINE CONTROLLER – LATERAL/DIRECTIONAL CONTROLLER (SISO) ............................................................. 93 4.2.3 STABILITY AND ROBUSTNESS ANALYSIS OF BASELINE CONTROLLER ................................................................... 95 4.3 SURVEY OF ADAPTIVE CONTROL .......................................................................................................... 95 4.4 ADAPTIVE CONTROL .................................................................................................................... 95 4.4.1 BACKGROUND ......................................................................................................................................... 96 4.4.2 PROBLEM FORMULATION .......................................................................................................................... 97 4.4.3 REFERENCE SYSTEM .......................................................................................................................... 101 4.4.4 CLOSED LOOP CONTROL STRUCTURE ......................................................................................................... 106 4.4.5 AUGMENTED CONTROLLER DERIVATION ............................................................................................... 111 4.5 PERFORMANCE METRIC (COMPARISON OF CONTROLLERS) ....................................................................... 117 4.6 TIME DELAY MARGIN ..................................................................................................................... 118 4.7 STABILITY AND PERFORMANCE BOUNDS - AUGMENTED CONTROLLER .................................................... 120 4.8 VALIDATION OF MODIFIED AUGMENTED CONTROL LAW ..................................................................... 121 4.8.1 NOMINAL CASE - STEP AND RAMP INPUT ................................................................................................... 122 4.8.2 UNCERTAINTIES - STEP INPUT ................................................................................................................... 123 4.8.3 UNCERTAINTIES - RAMP INPUT ................................................................................................................. 126 4.9 CONCLUSION ................................................................................................................................ 127 5. CONTROLLER PERFORMANCE COMPARISON ................................................................................ 128 5.1 INTRODUCTION ............................................................................................................................. 128 5.2 BASELINE CONTROLLER ................................................................................................................... 128 5.2.1 LONGITUDINAL CONTROLLER ................................................................................................................... 128 5.2.2 LATERAL/DIRECTIONAL CONTROLLER ......................................................................................................... 129 5.3 LONGITUDINAL CONTROL – BASELINE AND AUGMENTATION .................................................................... 131 5.3.1 RESULTS AND ANALYSIS ........................................................................................................................... 132 5.4 LATERAL AND DIRECTIONAL CONTROL ................................................................................................. 154 5.4.1 RESULTS AND ANALYSIS ........................................................................................................................... 154 5.4.2 BEHAVIOUR OF ADDITIONAL LATERAL/DIRECTIONAL AIRCRAFT STATES ........................................................... 166 5.5 UNCERTAINTIES IN THE SYSTEM ......................................................................................................... 170 5.5.1 LONGITUDINAL DYNAMICS ....................................................................................................................... 170 5.5.2 LATERAL LATERAL/DIRECTIONAL DYNAMICS ............................................................................................... 172 5.6 TIME DELAY MARGIN ..................................................................................................................... 174 5.7 CONCLUSION ................................................................................................................................ 176 6. FLIGHT PARAMETER ANALYSIS ..................................................................................................... 177 6.1 FLIGHT PARAMETERS - LONGITUDINAL DYNAMICS ................................................................................. 177 6.1.1 MACH NUMBER AND ALTITUDE ................................................................................................................ 178 6.1.2 AERODYNAMIC CHARACTERISTICS ............................................................................................................. 181 6.2 FLIGHT PARAMETERS - LATERAL/DIRECTIONAL DYNAMICS ....................................................................... 186 6.2.1 MACH NUMBER AND ALTITUDE ................................................................................................................ 186

Table of Contents

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6.2.2 AERODYNAMICS CHARACTERISTICS ............................................................................................................ 188 6.3 CONCLUSION ................................................................................................................................ 193 7. CONCLUSIONS AND FUTURE WORK .............................................................................................. 194 7.1 CONCLUSIONS ............................................................................................................................... 194 7.2 FURTHER RESEARCH ....................................................................................................................... 197 REFERENCES ....................................................................................................................................... 199 APPENDIX A ....................................................................................................................................... 218 FRAMES OF REFERENCE AND COORDINATE TRANSFORMATION .......................................................... 218

List of Figures

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List of Figures

Chapter 1:

Fig 1.10. GHAME Vehicle [6] .......................................................................................................... 2

Fig 1.11. Mission Profile – Descent Trajectory .............................................................................. 4

Fig 1.12. Research Positioning ......................................................................................................... 7

Fig 1.13. Framework depicting the research direction................................................................. 8

Fig 1.14. Flow Chart of Thesis Chapters ...................................................................................... 17

Chapter 2:

Fig 2.1. X-31A Post Stall Experiment Aircraft [92] ..................................................................... 19

Fig 2.2. Flight Control Architecture [94] ...................................................................................... 19

Fig 2.3. NDI Controller of VAAC [88] .......................................................................................... 20

Fig 2.4. Step Response of Non-minimum Phase System [110] ................................................. 24

Fig 2.5 Control Moment Derivative as a function of Mach Number and Altitude ............... 26

Fig 2.6. -Controller Performance - Worst Case Scenario [32] ................................................. 28

Fig 2.7. Longitudinal Controller Block Diagram [86] ................................................................. 29

Fig 2.8. Lateral Controller Block Diagram [86] ........................................................................... 29

Fig 2.9. Inertial Angle of Attack from Engine Test through Recovery Manoeuvre [86] ....... 30

Fig 2.10. Inertial Sideslip Angle from Engine Test through Recovery Manoeuvre [86] ....... 30

Fig 2.11. Comparison of In-flight Altitude and Pre-flight Altitude Predictions [86] ............ 31

Fig 2.12. Inertial Angle of Attack during Descent Phase [86] .................................................. 31

Fig 2.13. Velocity and Velocity Tracking Error [115] ................................................................. 32

Fig 2.14. Angle of Attack and Angle of Attack Error [115] ....................................................... 33

Fig 2.15. Parameter Space for Controller Design [87] ................................................................ 33

Fig 2.16. Commanded Angle of Attack (left) and Angle of Attack Error (right) [116] ......... 34

Fig 2.17. System States of Hypersonic Vehicle during Inlet Unstart [69] ............................... 36

Fig 2.18. Control and Adaptive Weight Responses during Inlet Unstart [69] ....................... 37

Fig 2.19. Augmented Controller Architecture [31] ..................................................................... 38

Fig 2.20. Typical X-15 Trajectory [121] ......................................................................................... 38

Fig 2.21. Comparison of Performance of Original X-15-3 Adaptive Controller with Modern

Day Adaptive Controller [122] ...................................................................................................... 40

Fig 2.22. Flat Top Glider [19] ......................................................................................................... 41

Fig 2.23. Flat Bottom Glider [19] ................................................................................................... 41

Fig 2.24. Flow Diagram - Adaptive Control Setup ..................................................................... 42

Fig 2.25. The Common Aerospace Vehicle (CAV) [125] ............................................................ 43

Fig 2.26. L1 Augmented Controller Performance Comparison [65] ........................................ 48

List of Figures

xiii

Chapter 3:

Fig 3.1.Typical Pole Positions for Longitudinal Motion ............................................................ 55

Fig 3.2. Relationship between Angle of Attack, Pitch and Flight Path Angle ........................ 58

Fig 3.3. Typical Pole Position for the Lateral/Directional Motion ............................................ 60

Fig 3.4. Motion Variables [156] ...................................................................................................... 67

Fig 3.5. Lift Coefficient as a Function of Mach Number and Angle of Attack ....................... 68

Fig 3.6. Drag Coefficient ................................................................................................................. 68

Fig 3.7. Pitching Moment ............................................................................................................... 69

Fig 3.8. GHAME Model [6] ............................................................................................................ 70

Fig 3.9. Second Order Actuator Model ........................................................................................ 72

Fig 3.10. Principle of Operation of INS [1] ................................................................................... 73

Fig 3.11. Propagation of errors [1]................................................................................................. 76

Chapter 4:

Fig 4.1. Autopilot Structure for Hypersonic Glider ................................................................... 88

Fig 4.2. Block Diagram State Feedback for Stability ................................................................... 89

Fig 4.3. Adaptive Controller Model ...................................................................................... 112

Fig 4.4. Filter with Constant Cut-Off Frequency ...................................................................... 116

Fig 4.5. System Output - Step Response .................................................................................... 122

Fig 4.6. System Output - Ramp Input ......................................................................................... 123

Fig 4.7. Response to Step Input with Uncertainties ................................................................. 123

Fig 4.8. Performance of System - Matched Component of Adaptive Law ........................... 124

Fig 4.9. Adaptive Signal Magnitude ........................................................................................... 124

Fig 4.10. System Performance - Unmatched Component Turned On ................................... 125

Fig 4.11. Adaptive Signal Contribution (Original and Modified) .......................................... 125

Fig 4.12. System Performance ...................................................................................................... 126

Chapter 5:

Fig 5.1. Pole Zero Plot of Closed Loop Dynamics - Longitudinal Dynamics ....................... 129

Fig 5.2. Pole Zero Plot of Closed Loop Lateral Dynamics ....................................................... 130

Fig 5.3. Pole Zero Plot of Closed Loop Directional Dynamics ............................................... 130

Fig 5.4. Benchmark Case (S1 and S2) .......................................................................................... 132

Fig 5.5. Reduced Elevon Functionality ....................................................................................... 133

Fig 5.6. LTV Error in (S9 and S10) ....................................................................................... 135

Fig 5.7. LTV Error in (S9 and S10) ........................................................................................ 136

Fig 5.8. LTV error in (S9 and S10) ....................................................................................... 137

Fig 5.9. LTV Error in (S9 and S10) ....................................................................................... 138

Fig 5.10. LTV Error in (S9 and S10) ..................................................................................... 139

List of Figures

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Fig 5.11. LTV Error in (S9 and S10) .................................................................................... 140

Fig 5.12. Gravimetric Errors (S3 and S4) .................................................................................... 142

Fig 5.13. Aerodynamic Uncertainties ......................................................................................... 143

Fig 5.14. Combination of Errors .................................................................................................. 144

Fig 5.15. Benchmark Scenario (S1 and S2) ................................................................................. 147

Fig 5.16. Reduced Actuator Functionality (S5 and S6) ............................................................. 147

Fig 5.17. Time Varying Uncertainty .................................................................................... 148

Fig 5.18. Error Dynamics and Acceleration ...................................................................... 148

Fig 5.19. Error Dynamics and Acceleration ....................................................................... 149




Fig 5.23. Gravimetric Uncertainties ............................................................................................ 151

Fig 5.24. Aerodynamic Uncertainties ......................................................................................... 151

Fig 5.25. Combination of Uncertainties ...................................................................................... 152

Fig 5.26. Roll Angle Tracking - Benchmark Case (L1 and L7) ................................................ 155

Fig 5.27. Reduced aileron deflection (L3 and L9) ..................................................................... 156

Fig 5.28. Asymmetric Elevon Deflection (L3 and L9) .............................................................. 157

Fig 5.29. Gravimetric Errors (L4 and L10) ................................................................................. 158

Fig 5.30. Aerodynamic uncertainties (L5 and L11) ................................................................... 159

Fig 5.31. Rudder Failure: -7.5 to 7.5 degrees (L2 and L8) ........................................................ 160

Fig 5.32. Combination of errors (L6 and L12) ........................................................................... 161

Fig 5.33. Behaviour of Additional Aircraft States (Rudder is functioning) .......................... 167

Fig 5.34. Behaviour of Aircraft States - Presence of Uncertainties ......................................... 168

Fig 5.35. Estimate of Uncertainties - Nominal Conditions ...................................................... 171

Fig 5.36. Estimate of Uncertainties – Presence of Uncertainties ............................................. 172

Fig 5.37. Estimate of Uncertainties - Nominal Conditions ...................................................... 173

Fig 5.38. Estimate of Uncertainties –Uncertainties ................................................................... 173

Chapter 6:

Fig 6.1. Mach Number Altitude Envelope of the Vehicle – Longitudinal ............................ 179

Fig 6.2. Dynamic Pressure (Longitudinal Manoeuvre) ........................................................... 179

Fig 6. 3. Mach Number, Altitude and Angle of Attack Time Histories................................. 180

Fig 6.4. Angle of Attack - Longitudinal ...................................................................................... 182

Fig 6.5. Drag Coefficient for the Pull Up Trajectory ................................................................. 183

Fig 6.6. Lift Coefficient for Pull Up Trajectory .......................................................................... 183

Fig 6.7. Lift to Drag Ratio for Longitudinal Test Case ............................................................. 184

List of Figures

xv

Fig 6.8. Pitching Moment Coefficient - Longitudinal............................................................... 184

Fig 6.9. Pitching Moment Coefficient as a function of Time ................................................... 185

Fig 6.10. Mach Number Altitude Map for Second Part of Trajectory .................................... 187

Fig 6.11. Dynamic Pressure (Lateral/Direction Manoeuvre) .................................................. 187

Fig 6.12. Angle of Attack for Lateral Part of the Trajectory .................................................... 188

Fig 6.13. Lift to Drag Ratio for Lateral Part of Trajectory ........................................................ 189

Fig 6.14. Rolling Moment Coefficient ......................................................................................... 189

Fig 6.15. Yawing Moment Coefficient ........................................................................................ 190

Fig 6.16. Lateral Dynamics Derivatives ..................................................................................... 191

Fig 6.17. Directional Dynamics Derivatives .............................................................................. 192

Appendix A

Fig A. 1. Frames of Reference - Aircraft ..................................................................................... 218

Fig A. 2.Body and Stability Axis of Aircraft .............................................................................. 220

List of Tables

xvi

List of Tables

Chapter 3:

Table 3.1. Overview of Motion Variables of an Airborne Platform ......................................... 67

Table 3. 2. GHAME Mass Properties ............................................................................................ 71

Table 3.3. Actuator model Parameters ......................................................................................... 73

Table 3.4. Summary of Equations ................................................................................................. 76

Table 3.5. Performance Definition of ADIS16480 - Accelerometer .......................................... 78

Table 3.6. Performance Definition of ADIS16480 – Gyroscope ................................................ 78

Table 3.7. Error Bounds for Error Vector - Time Invariant ....................................................... 81

Table 3.8. Errors Bounds of Error Vector (Time Invariant) ....................................................... 83

Chapter 5:

Table 5.1. Simulation Scenarios ................................................................................................... 131

Table 5.2. Simulation Parameters: Initial Conditions .............................................................. 131

Table 5.3. Uncertainties Analysis Monte Carlo Simulation (S3 and S4, S7 and S8, S11 and

S12) .................................................................................................................................................. 141

Table 5.4. Tracking Error Norm – Monte Carlo Runs .............................................................. 145

Table 5.5. Tracking Error Norm for Time Varying Errors ....................................................... 145

Table 5.6. Control Surface Metric ................................................................................................ 153

Table 5.7. Simulation Scenarios – Lateral/Directional ............................................................. 154

Table 5.8. Conditions - Trajectory Control ................................................................................ 154

Table 5.9. Roll Tracking Error Norm .......................................................................................... 163

Table 5.10. Altitude Mean and Standard Deviation (Lateral Test Cases) ............................. 165

Table 5.11. Control Surface Metric .............................................................................................. 165

Table 5.12. Time Delay Margin Comparison ............................................................................ 175

Table of Acronyms

xvii

Table of Acronyms

Acronym Description

ADF Automatic Direction Finder

ADI Attitude Director Indicator

ANSI American National Standards Institute

BIBO Bounded Input Bounded Output

CADAC Computer Aided Design of Aerospace Concepts

CAV Common Aerospace Vehicle

CFD Computational Fluid Dynamics

CLV Crew Launch Vehicle

DI Dynamic Inversion

DPPC Dynamic Pole Placement Controller

DoF Degrees of Freedom

ECG Extended Command Governor

EoM Equations of Motion

FSD Flight System Dynamics

GHAME Generic Hypersonic Aerodynamics Model Example

GHV Generic Hypersonic Vehicle

GPS Global Positioning System

GTM Generic Transport Model

HCV Hypersonic Cruise Vehicle

HGV Hypersonic Gliding Vehicle

HIL Hardware in the Loop

HRE Hypersonic research Engine

HSV Hypersonic Vehicle

INS Inertial Navigation System

LFT Linear Fractional Transformation

LPV Linear Parameter Varying

LQG Linear Quadratic Gaussian

LQI Linear Quadratic Integral

LQR Linear Quadratic Regulator

LTI Linear Time Invariant

LTR Linear Transfer Recovery

LTV Linear Time Variant

MATLAB Matrix Laboratory

MEMS Microelectromechanical systems

MIMO Multi Input Multi Output

Table of Acronyms

xviii

Acronym Description

MOI Moment of Inertia

MPC Model Predictive Control

MRAC Model Reference Adaptive Control

MRAS Model Reference Adaptive System

NACA National Advisory Committee for Aeronautics

NASA National Aeronautics and Space Administration

NDI Nonlinear Dynamic Inversion

NDO Nonlinear Disturbance Observer

NN Neural Networks

ODE Ordinary Differential Equations

OOP Object Oriented Programming

PDQLF Parametric Dependent Quadratic Lyapunov Function

PI Proportional Integral

PPC Pole Placement Controller

PwC Piecewise Constant

RCS Reaction Control System

SAS System Augmentation System

SCRAMJET Supersonic Combusting Ramjet

SDRE State Dependent Riccati Equation

SISO Single Input Single Output

SSTO Single Stage to Orbit

STR Self-Tuning Regulator

TDM Time Delay Margin

TUM Technical University Munich

UAV Unmanned Aerial Vehicle

VAAC Vectored thrust Aircraft Advanced Control

Table of Symbols

xix

Table of Symbols

Latin Letters

Symbol Unit Description

Body, inertial coordinates

Roll, pitch and yaw principle moments of inertia

Component of moment of inertia matrix

Sonic speed

State matrix, control matrix, output, feedforward matrix

Uncertainties in system

Closed loop system matrix

Body acceleration with respect to inertial frame

Characteristic length

Reference length

Aerodynamic or control derivative

Drag force

Derivative operator

Error vector

Tracking error dynamics

Nonlinear state dynamic function

Specific force

Reference frames

Transition matrix – Kalman Filtering

Force vector

Nonlinear control distribution function

Control coefficient matrix

Gravitational acceleration

Noise gain matrix

Height above earth

Observation Matrix

Cost function

PI Gains

Gain matrix

Kalman gain matrix

Lift force

Table of Symbols

xx

Latin Letters


Rolling moment

Lift coefficient slope

Rolling moment derivate as a function of

Mass of vehicle

Mach number

Pitching moment

Pitching moment derivative

Dimensional normal force slope derivative as a function of

Yawing moment derivative as a function of

Yaw moment

Roll rate

Pole location

Pressure

Covariance matrix

Prior covariance

Posterior covariance

Pitch rate

State weighting matrix

Process noise covariance

Dynamic pressure

Radius

Yaw rate

Gas constant

Control input weighting matrix, controllability matrix

Reynolds Number

Rotation tensor of body frame with respect to inertial frame

Measurement covariance

Platform area

Ricatti gains matrix

Laplace operator

Displacement of body with respect to inertial point

Thrust vector

Temperature

Time constant of mode A

Control input

Table of Symbols

xxi

Latin Letters


Velocity vector

Velocity of body with respect to the inertial frame

Commanded signal

X-force derivative with respect to

State vector

State estimate vector

Prior estimate

Output variable

Posterior estimate

Output estimate

Side force derivative as a function of

Force in the Y-direction

Force in the Z-direction

Table of Symbols

xxii

Greek Letters


Angle of attack

Side slip angle

Control input

Elevator deflection

Aileron deflection

Rudder deflection

Thrust equivalence ratio

Error vector

Flight path angle

Ratio of specific heat

Damping of a linear second order system

Damping ratio

Pitch angle

Rate of change of pitch angle

Parametric uncertainty error

Adaptive estimate of parametric uncertainty

Input disturbances

Adaptive estimate of input disturbances

Density

Trajectory parameters

Roll angle

Natural frequency

Angular velocity vector of vehicle frame w.r.t the inertial frame

Probability density function

Table of Symbols

xxiii

Indices

Symbol Description

Variable related to angle of attack

Variable related to sideslip angle

Variable related to aileron deflection

Variable related to elevator deflection

Variable related to rudder deflection

Freestream, steady state

Actuator

Body fixed frame

Desired rates

Geodetic frame

Dutch Roll

Earth

Earth fixed frame

Feedforward

Inertial frame, integral gain

Variable as a function of altitude, gain of difference in altitude

Local value

Static stability derivative

Pitching damping derivative

Variable related to pitch moment

Variable related to roll rate

Variable related to pitch rate

Variable related to yaw rate

Rolling mode

Spiral model

Stability axis frame

Variable as a function of velocity

INTRODUCTION

1

1 INTRODUCTION

This chapter presents the rationale underlying the study of the control of hypersonic

gliders. The chapter lays the foundation for the thesis by providing the background, the

research objectives, the research questions, scope and focus of this research, contributions

and limitations of the study. It also provides an outline of the structure of the thesis.

1.1 Overview of Research

1.1.1 Challenges of Controlling and Need of Hypersonic Vehicles

Hypersonic vehicles potentially provide a cost effective means of accessing space due to

their reusable nature [1, 2]. As a result, over the past few decades, a considerable amount

of funds have been directed to the research and development of hypersonic vehicles by the

U.S. Air Force and NASA.

The main advantage that hypersonic vehicles present is that they are air breathing vehicles

as opposed to fuel based rockets [3, 4], thus reducing the operating cost of the vehicle, as

less fuel needs to be carried and at the same time increasing the payload capacity.

Furthermore, the vehicle itself becomes significantly more efficient from a propulsion

point of view and makes the vehicle reusable. Another driving force into hypersonic

research is the commercialisation of hypersonic powered flights. In the event that this

becomes a reality, flight times would be significantly reduced.

Some of the main challenges of a hypersonic vehicle program from a control point of view

are: (i) to generate a model of the HSV and (ii) design a controller for the cruise phase of

the flight. The difficulties and challenges associated with the above two tasks consist of

modelling the dynamics of the vehicle which involve a strong coupling between the

aerodynamic, structural and the propulsion sub-systems. The long slender bodies of the

hypersonic vehicle during flight cannot be considered as rigid and the flexible effects need

to be taken into account when modelling and designing controllers for different flight

INTRODUCTION

2

phases. To ensure that the vehicle operates with high aerodynamic efficiency, the

designers are forced to design the vehicle to be open loop unstable. This means that

without a well-designed flight stability augmentation system (or commonly referred to as

an autopilot) the platform would be inoperable [5].

1.1.2 Generic Hypersonic Aerodynamic Model Example (GHAME)

The model utilised in this research is a simulation platform that has been designed by the

Dryden Flight Research Facility. The Generic Hypersonic Aerodynamic Model Example

(GHAME) was developed to enable the research and development in the field of control

and guidance and trajectory optimization [6]. The geometry of the simplified GHAME

model is shown in Fig 1.1.

Fig 1.1. GHAME Vehicle [6]

The GHAME model that has been used in this research, was developed for a single stage-

to-orbit (SSTO) flight regime that would be encountered by the National Aerospace Plane

(NASP). Studies like the one presented in [6-8] show the importance of this model and the

advantages of having such a model in helping carrying out feasibility studies along entry

trajectories. The model itself is a based upon a combination of flight test data from the

Space Shuttle and the X-24C and theoretical data from a swept delta configuration using

modified Newtonian Impact Flow method [6-8].

INTRODUCTION

3

The gravimetrics of the original GHAME model are changed to represent the HIFiRE 4

flight experiment vehicle, which is a reentry glider control experiment [9]. The gravimetric

properties utilised in the force and moment equations are obtained from [10]. The platform

area is, , the characteristic length is and the reference length is,

. A scaled down version of the GHAME model that weights 93.1kg and with the

following product and moments of inertias:

and .

A more detailed look at the GHAME model is shown in Section 3.4.2.

1.2 Motivation

The flight trajectory shown in Fig 1.2 is one proposed for launching a cruising SCRAMJET

vehicle and then recovering the vehicle by gliding it back to earth after the mission is

complete. The trajectory is initially the same as the HyShot [11] and HIFiRE [12] missions

(a sub-orbital ballistic trajectory) but on the way back through the atmosphere, the control

surfaces of the unpowered vehicle are used to pull out of the ballistic trajectory. The

SCRAMJET engine is not used during the ascent stage so a rocket is used to get the

scramjet up to 400 km altitude. During the descent phase, the vehicle pulls out of the

ballistic trajectory and goes into a horizontal path to fire and test the SCRAMJET and

cruise at a certain altitude. The focus of this study is this pull-up manoeuvre and the glide

stage after the vehicle has cruised under SCRAMJET power. This study does not include

the firing of the SCRAMJET engine. As a result this study considers how to control a

hypersonic glider over a wide range of conditions, from Mach 6 to 8 down to subsonic

conditions. Thus the study would apply to (1) the pull-up manoeuvre and (2) the glide

back towards the ground.

For this scenario, the pull-up manoeuvre is required to alter the flight path but not to lose

too much speed in the process. So after the manoeuvre, the vehicle is flying at around

INTRODUCTION

4

Mach 7 at an altitude suitable for the scramjet engine to fire (See Chapter 6 for results).

Therefore, this study is a step towards this goal of bringing an unpowered SCRAMJET

vehicle out of a ballistic trajectory, using aerodynamics to alter its path and speed and then

gliding it back towards earth. The next stage of the research will include the powered

section of flight and then glide it back to ground after that.

Fig 1.2. Mission Profile – Descent Trajectory

Once the vehicle lands, it can be maintained and brought to a state where it can be reused.

Therein lies the main motivation of this study; this dissertation presents a control strategy

to carry out a pull up manoeuvre after the hypersonic vehicle has carried out the re-

orientation (about the apogee) and descent manoeuvre down to 55km altitude. This in

turn facilitates the landing phase and thereafter the re-use of a hypersonic vehicle.

By the time the vehicle is ready to glide towards the earth, all of the fuel has been used; the

vehicle that is descending towards the Earth is therefore a glider. This significantly

increases the complexity of the control objective. The complexity and the demands on the

controller increase significantly along a pull up trajectory. Chapter 2 shows that there is

considerable literature that has been published in the field of robust and adaptive control

of hypersonic vehicles. The basic idea behind adaptive control is to estimate uncertain

INTRODUCTION

5

plant/controller parameters on-line using measured system signals. These estimated

parameters are used to compute the control input. The objective of an adaptive controller

is to maintain performance of a system in the presence of uncertainty and variations in

plant parameters. The fundamental difference between robust and adaptive control is that

adaptive controller is superior in dealing with uncertainties in constant and slowly-

varying parameters. However, robust control is superior in dealing with disturbances,

quickly varying parameters and unmodelled dynamics. Therefore, a control strategy

which has two control objectives is implemented; the robust baseline is implemented to

carry out the tracking objective and the adaptive augmentation is implemented to cancel

the effects of the uncertainties.

Flight testing of hypersonic vehicles is an expensive endeavour. As a result, high-fidelity,

six-degree-of-freedom (6-DoF) simulations play an important role in the development of

the system [13, 14]. It is within this context that such simulation platforms and study are

used in trade studies, preliminary design, hardware-in-the-loop (HIL) testing, flight

testing and training. The added motivation is that this study acts as a tool to thoroughly

test the stability, robustness and performance characteristics of the controllers without

having carried out a single test flight. Consequently, this provides a certain baseline level

of confidence for the stability, performance and robustness of the controllers.

1.3 Background to the Research

The study of control systems for hypersonic vehicles (HSV) has remained a challenging

research area owing to the characteristic of the vehicle dynamics; being highly nonlinear,

subject to uncertainties and coupling [10, 15-21]. The aerodynamics, structure, propulsion

system, and control features are dynamically inter-linked over a wide range of frequencies

due to the flexible airframe/engine integrated configuration [22-25]. Such an intricate

nature of a HSV is due to the fact that these vehicles are highly sensitive to changes in

atmospheric conditions, physical and aerodynamic parameters [26-29]. Further

INTRODUCTION

6

complications in the dynamic features of the HSV arise because of the varying flight

conditions, fuel consumption and thermal effects on the structure [25]. As a result of the

overwhelming complexity in vehicle dynamics, the control research of HSV a has become

a topic of importance.

Earlier studies on control research have primarily concentrated on flight control design to

deal with the complex dynamics in an attempt to achieve vehicle stability [1, 17, 26, 30-32].

Research on the models of the vehicle dynamics with increasing complexity mainly

focused on the longitudinal plane, as is evidenced in the model developed by Bolender

and Doman [1, 27, 33, 34]. However, the longitudinal vehicle model is known to be

unstable, exhibiting non-minimum phase behaviour and is not robust to dynamic

uncertainty [27, 34]. In order to moderate the effect of structural flexibility on vehicle

stability and control performance, a linearised model is formed around a trim point for a

nonlinear, dynamically coupled simulation model of the HSV [27, 35-37]. As an

alternative, nonlinear adaptive controllers have been employed in the HSV configuration

domain [10, 16, 22, 33, 37-43]. In the research positioning matrix of Fig 1.3, quadrants A

and B represent the studies undertaken for linear and nonlinear control methodologies to

stabilise powered HSV.

Thus, while in the area of control research a large body of scholarly work exists, the extent

of research is directed at linear and nonlinear control of powered vehicles [1, 30, 44-46].

Compared with the studies in the control of powered HSVs, limited research is available

on the development of controllers for unpowered HSVs [10, 16, 19, 47-51]. Consequently,

this distinct focus on developing control methodologies for powered HSV leaves a large

gap in the literature relating to development of control systems for unpowered HSVs once

the SCRAMJET has been fired and tested. Addressing, this gap would enable the reuse of

SCRAMJET powered vehicles to reduce flight times for global travel and make the

deployment of payloads into orbit more routine and affordable.

INTRODUCTION

7

This study addresses the neglect of controllers for unpowered HSVs along a descent

trajectory. The study into unpowered hypersonic gliders has attracted the use of both

linear and nonlinear control methodologies [10, 19, 47, 48, 51]. This research work also

uses the baseline controller, which is a linear controller, and the augmentation of the

baseline controller is through a nonlinear adaptive controller. As a result, the positioning

of this research builds on the scholarly works pertaining to both the quadrants of C and D

in Fig 1.3.

Fig 1.3. Research Positioning

Compared with the large amount of literature in the field of longitudinal control of HSVs,

few sources present the lateral-directional dynamics [29, 52-58]. The studies presented

highlight the complexities pertaining to the lateral dynamics on a re-entry trajectory. The

vehicle models used are either the space shuttle [58] or a space shuttle like vehicle [55].

Heller in [57] has presented complexities with respect to the lateral dynamics of a

hypersonic vehicle travelling at up to Mach 7. These difficulties include unstable Dutch

roll, a very high roll yaw coupling and weak roll damping.

INTRODUCTION

8

Consequently, an additional axis is added to Fig 1.3 to add depth to the current study of

hypersonic gliders. The third axis indicates whether the study undertaken concentrates

only on the longitudinal dynamics or incorporates the lateral-directional dynamics.

Having outlined the key research areas in HSV control, an in depth positioning of the

research direction is included in Fig 1.4. This figure highlights that this thesis concentrates

on a 6-DoF study of the hypersonic glider. In doing so, it fills in a key research gap.

Fig 1.4. Framework depicting the research direction

adaptive control is utilised as part of the augmented controller. The initial development

of adaptive control theory was motivated by the need to certify adaptive flight critical

systems with a more affordable verification and validation process [22, 59]. The key

feature of adaptive control is the decoupling of adaptation and the control loops. This

in turn leads to guaranteed robustness in the presence of fast adaptation [60-64]. The

architecture allows for large learning gains (adaptation rates) which is beneficial for both

performance and robustness. This is possible with the introduction of a low pass filter into

the controller.

Hypersonic Vehicle Control

Powered - SCRAMJET

Longitudinal Study Only

Lateral-Directional Dynamics Study

Unpowered - Glider

Longitudinal Study Lateral-Directional

Dynamics Study

INTRODUCTION

9

Despite numerous adaptive control architectures presented for a wide range of

applications in state feedback [62, 63, 65-68], there are very few papers that address control

of linear time varying (LTV) state feedback systems. The studies in [65] and [63] present an

augmented controller for the X-29 and X-48 vehicles. However, the controller presented is

for a linear time invariant (LTI) reference model. The adaptive controller presented uses a

control law with two low pass filters: each for the matched and the unmatched adaptive

estimates. The adaptive controller presented by Prime in [68] controls a hypersonic glider

along the same descent trajectory as the one presented in Fig 1.2, however, once again the

reference model is LTI. As the flight parameters changed significantly, an LTV reference

model might be able to better meet the different performance specifications at different

points of the flight trajectory, thus leading to overall better performance.

1.4 Research Gap

Relative to the studies presented in A and B, there has been fewer research papers on

linear and nonlinear adaptive control of hypersonic gliders, C and D (see Fig 1.3).

Furthermore, a comprehensive study of the stability and robustness characteristics of

controllers for gliders in the presence of various uncertainties has not been undertaken.

This study attempts to address the gap of control of hypersonic gliders along a decent

trajectory. This research is, therefore, positioned, as depicted in Fig 1.3 and Fig 1.4.

Significant research effort has been dedicated to the area of hypersonic flight in order to

provide a more economic and environmentally friendly access to space as compared to the

traditional rocket [17, 34, 57, 69]. Preller in [70] has presented a small access to space

systems which integrates a scramjet powered second stage as part of the launch sequence.

Once the ascent trajectory is complete and the payload has been delivered to orbit, it is

important to make sure that the vehicle (payload delivery component of the launch

system) carried out the descent manoeuvre in a controlled manner to re-use. Along the

descent trajectory, the vehicle will only have reaction jets, its inherent aerodynamic

INTRODUCTION

10

properties and control surfaces to control its descent and subsequent manoeuvres.

Another possible application is the use of the SCRAMJET engine along the descent portion

of sub-orbital ballistic trajectory. To carry this out a controller is needed to orient the

vehicle in a manner to allow the right initial conditions for the SCRAMJET to start

operating. It is within this context, the positioning of this research is an important addition

to the research field.

1.5 Research Problem

There are few studies concentrating on the control of hypersonic gliders along a descent

trajectory (as shown in Fig 1.2); HiFire 4 is one of the few examples of a program trying to

carry out a pull up manoeuvre similar to the one shown in Fig 1.2 [12]. The control along

the descent trajectory forms a precursor to landing a hypersonic glider. Carrying out a pull

up manoeuvre would facilitate and promote the affordability of HSVs for launching

payloads into orbit. In light of this research gap, this study concentrates on research

directed towards a 6-DoF study of a hypersonic glider along a descent trajectory.

Furthermore, the controller that is used in this research is different from the ones

developed for the X-15, X-43 and the space shuttle. This is largely due to the fact that for

the X-15, although an adaptive controller was used, the vehicle was piloted. The X-43 used

a PID controller at a constant Mach number and altitude (details of which are given in

Section 2.4.1). The re-entry aerodynamics of the Space Shuttle (which is a bulky flat

bottomed re-entry vehicle) are very different from those of a slender winged hypersonic

vehicle intended to fly like an aircraft [71-74]. As a result, the literature review in Chapter

2 concentrates on the studies that has been carried out for slender hypersonic vehicles.

Furthermore, the comparison of the performance of controllers is also very limited [49, 50,

63, 75-78]. The studies that present a comparison between a baseline and an augmented

baseline controller are mainly for the case of missiles, helicopters and supersonic aircraft.

INTRODUCTION

11

In this context, this thesis addresses the problem of applying and comparing two control

methodologies with respect to robustness, performance and stability.

To tackle the research problem, a baseline controller is implemented to carry out the pull

up manoeuvre along the descent trajectory for a scaled GHAME (Generic Hypersonic

Aerodynamics Model Example) model. Once the pull up manoeuvre has been carried out,

the lateral/directional baseline controllers carry out a roll-manoeuvre while keeping the

sideslip angle constant at zero. The baseline controller is thereafter augmented with an

adaptive controller [66]. The augmented controller employed in this study is similar to

those presented in [59, 61, 78-82]. This study addresses the research problem by

developing a nonlinear high-fidelity simulation model.

An additional objective of this study is to apply the augmented adaptive controller to

an LTV system. In the process modified theory is applied to an LTV system, which has not

been done before. The aim of this research is reflected in the following specific research

questions that are addressed in this thesis:

I. How does the baseline controller perform for a descent trajectory in the presence of

sensor, aerodynamic, gravimetric and control surface uncertainties and failures?

II. How does the augmented controller assist in cancelling the uncertainties present

in the system and improve the performance of the baseline controller?

III. What are the performance characteristics of the adaptive augmented controller

in the presence of uncertainties?

IV. What are the robustness properties of the augmented controller compared to the

baseline controller?

V. Is this control methodology suitable for carrying out a pull up manoeuvre along the

descent phase of a suborbital ballistic trajectory to fire and test a reusable

SCRAMJET engine?

INTRODUCTION

12

1.6 Scope and Focus of the Study

The simulation and conceptual framework of this study is based on three main fields: (a)

pole placement control (PPC) (baseline controller) (b) adaptive control, and (c) control

of longitudinal and lateral/directional dynamics of a scaled GHAME hypersonic glider

along a descent trajectory. These fields provide the theoretical and conceptual framework

for this study and nonlinear high fidelity framework to test the robustness, performance

and stability characteristics of the controllers. Consequently, this research is built on the

following elements:

I. Creating a simulation environment for simulating hypersonic vehicle flight. This is

carried out in CADAC++ [13, 14]. The trajectories of the test cases are defined in this

environment.

II. Controller synthesis for the longitudinal and lateral/directional dynamics. Testing

of the baseline controller in the presence of aerodynamic uncertainties, imperfect

sensor feedback, control surface failures and gravimetric uncertainties.

III. Controller synthesis of the adaptive controller that will augment the baseline

controllers. This adaptive controller is implemented as a tool to cancel the

uncertainties that are present in the system and to improve the performance of the

baseline controller.

IV. Investigation of the robustness properties of adaptive controller as compared to

the baseline controller for a descent trajectory of a hypersonic glider.

1.7 Delimitations of the Study

As this study considers hypersonic gliders, scramjet powered HSVs are not considered.

Another aspect of hypersonic flight that isn't considered in this study is the structural

design of the system. It is also important to note that, above a certain altitude, control

authority is minimal due to low density [83-85]. In fact the density gets so low that the

INTRODUCTION

13

phenomena that leads to the need for controllers does not apply. It is under these

conditions that a Reaction Control System (RCS) is utilised to control the attitude of the

flying platform. As a result, the application of a RCS for the test case considered in this

study is deemed to be beyond the research scope.

1.8 Methodology

The tools to carry out high-fidelity, 6-DoF simulations are provided to modelling and

simulation engineers by CADAC. CADAC was created in 1978 and was written in

FORTRAN [13, 15]. Due to the fact that FORTRAN lacks the power of object oriented

programming, the latest version of CADAC has been implemented in C++. This new

architecture is called CADAC++ which helps with the conceptualisation and simulation of

aerospace vehicles. More importantly, high fidelity simulations use random variables to

model noise, disturbances and uncertain phenomenon. As the presence of uncertainties

forms the edifice of this project, the Monte Carlo simulation capability of CADAC++

makes this tool the primary choice for this study. The uncertainties that have been

included in the model are the imperfect sensor feedback and uncertainties in the

aerodynamic-look up tables, control surface and gravimetric uncertainties.

The baseline controller is responsible for stabilising the system and tracking the desired

commands under nominal conditions. The augmentation is utilised to cancel the

uncertainties in the system. As the baseline controller, a pole placement controller (PPC) is

implemented [15]. This baseline control strategy does not consider uncertainties or

nonlinearities. It is assumed that the plant dynamics are exactly known when the baseline

control law is designed. However, that is not the case. There are always assumptions made

such as neglecting some minor influences (like certain nonlinearities) depending on

parameters which only can be identified with much effort to increase the fidelity of the

model. Therefore, in the presence of uncertainties and nonlinearities, an adaptive

augmentation strategy is employed to improve the performance of the PPCs and ensure

INTRODUCTION

14

that the control objectives are met. As a result, this study forms the basis of the first

application of a PPC and an augmented PPC along a descent trajectory of a hypersonic

glider. Furthermore, the application of these two controllers to the longitudinal and the

lateral/directional dynamics of a hypersonic glider also acts as one of the main

contributions of this research.

1.9 Contribution of Dissertation

The present study addresses knowledge gaps in the hypersonic vehicle flight control

research and makes several contributions to both theory and practice. The following

contributions are made by this study to hypersonic control literature:

I. This study presents a controller that can be used to carry out a pull up manoeuvre

along the descent phase of a suborbital ballistic trajectory to bring the vehicle to the

right initial conditions and fire and test the SCRAMJET engine.

II. From a practical perspective, the study contributes by presenting longitudinal and

lateral/directional control strategies for a hypersonic glider at high and low speeds

and compares the performance and robustness of the baseline and the augmented

controller using tracking error and control surface norms and time delay margins

(TDMs).

III. From a practical perspective, this study includes the effects of imperfect sensor

feedback, aerodynamic, actuator, gravimetric uncertainties and time delays. The

disturbance rejection qualities of the controllers are presented in a high fidelity

simulation environment and form one of the main contributions of this research.

IV. The thesis provides a comparative study between the baseline and the

augmented PPC. There are very few studies comparing a baseline controller with

an adaptive controller. Although several studies present a comparative study, these

studies only consider the longitudinal dynamics of the vehicles and the results are

only presented around a constant altitude and Mach number [52, 86, 87]. The

INTRODUCTION

15

comparisons consist of the performance and the robustness comparison between

the baseline and the augmented controller.

V. The study applies a modified piecewise constant update law to an LTV system to

cancel matched and unmatched uncertainties for an augmented controller.

VI. The study presents an application of the modified implementation of an

adaptive augmented controller. This modification avoids the inversion of the

system dynamics and simplifies the control law implementation for a LTV system.

VII. The study provides a new insight into the design of the bandwidth of the low pass

filter, which forms an integral part of the controller. The design of the filter is

dependent on the characteristics of the uncertainties present and how they affect

the performance, robustness and stability of the system.

VIII. The study presents an implementation of the PPC and the augmented controller

which does not require too many resources to run on hardware. This simplifies the

transition from simulation to the flight hardware.

IX. The study shows that it is feasible to loosen one of the assumptions that is placed on

the plant dynamics when trying to realise a control objective while applying the

piecewise constant adaptation method for an LTV system.

1.10 Organisation of the Research

A description of the content of each chapter is given below:

Chapter 2: Literature Review and Conceptual Framework provides a review of the

hypersonic flight control literature and theories in related disciplines. This chapter sets out

the main theories used in this study and provides the key theoretical ideas of the research

leading to the conceptual framework for this study.

Chapter 3: Simulation Model presents the implementation of the simulation model of the

hypersonic glider. This chapter includes the description of the aerodynamic model,

INTRODUCTION

16

actuator model and sensor implementation and also provides a presentation of the vehicle

dynamics of the hypersonic glider which is used for controller synthesis.

Chapter 4: Theoretical Framework and Implementation develops the theoretical

framework of this thesis. This chapter presents the relevant theory with respect to PPC

and the adaptive control. This chapter presents the theoretical and mathematical

background of the baseline and the augmented controller. This chapter provides the

performance and stability bounds for the augmented controller.

Chapter 5: Controller Performance Comparison details the processes used to analyse and

test the performance, robustness and stability properties of the baseline and the

adaptive controller.The performance and the robustness characteristics of the controller

are tested in the presence of uncertainties. The sources of these uncertainties are also

outlined in this chapter. A detailed comparison of the PPC and the adaptive controller

is presented and the key differences in performances and robustness are highlighted.

Chapter 6: Physical Flight Parameter Analysis outlines the reasons for the controller

performances presented in Chapter 5. It further outlines the behaviour of the aerodynamic

coefficients of the vehicle in an attempt to gain an insight into what happens in a control

problem pertaining to a high speed vehicle entering the earth's atmosphere along a

descent trajectory.

Chapter 7: Conclusions and Implications presents the conclusions and a summary of the

contributions of the research together with implications for theory, methodology,

practices, and researchers. The chapter also offers some thoughts on future research and

the possible applicability of the presented control methodologies for real flight

experiments.

Fig 1.5 depicts a flow chart of the thesis outlining the connections between the various

chapters.

INTRODUCTION

17

Fig 1.5. Flow Chart of Thesis Chapters

1. Introduction

2. Literature Review

3. Simulation Model

4. Theoretical Framework and Implementation

5. Controller Performance Comparison

6. Conclusions and Future Work

6. Physical Flight Parameters Analysis

LITERATURE REVIEW

18

2 LITERATURE REVIEW

2.1 Introduction

This chapter presents the background knowledge relevant to this thesis and reviews the

control methodologies that have been employed to control hypersonic vehicles. It presents

the unique challenges that are present when dealing with the control of hypersonic

gliders. As adaptive control forms one of the main pillars of the project, a literature

review is provided which highlights the main research effort in the field of adaptive

control theory applied to airborne platforms. The main outcome of this chapter is

identification of gaps in literature.

2.2 State of the Art in Flight Control Approaches

Boeing in 1934 develop an autopilot for its flight control system for the B247. From then

on, flight controllers have moved on from their analog predecessors. Current state of the

art in terms of flight control implementation involves a combination of

electrical/electromechanical/hydraulic components. In terms of flight control laws, there is

a great wealth of literature that presents control methodologies that are used in industry

and in research [88, 89]. The industry standard in presented here.

In Europe, Airbus for its A320 uses load factor demands as its longitudinal controller. The

controller itself is a proportional integral (PI) controller (demanded control actuation

depends on error and integral of error). The commanded variables for the lateral-

directional axes are the roll rate, sideslip and bank angle. A classical PI controller is

utilised with a gain matrix for stability and roll rate/sideslip decoupling. The A340 uses a

similar structure however the controller is increased in size and flexibility which is

required as a result of considering structural vibrations in the modelling and

implementation process [90]. The controller for the A380 is further expanded to include

airframe flexibility during the design process. The flight control laws are implemented to

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achieve desired handling qualities and flexible mode damping requirements leading to

extended controller bandwidth [91].

Fig 2.1. X-31A Post Stall Experiment Aircraft [92]

The X-31A (Fig 2.1) was a post stall experimental which was a joint project between

Germany and the USA. The main aim of this program was to test the thrust vectoring

technology. A requirement of the flight control system was to maintain controlled flight at

high angles of attack where conventional aircraft would stall [93]. A block diagram of the

flight control architecture of the X-31A is shown in Fig 2.2.

Fig 2.2. Flight Control Architecture [94]

The commanded variables for the controller were the roll, pitch and yaw rates. The sensor

provides a feedback of the body rates, angle of attack and the sideslip angle. The controller

methodology used is the LQ digital regulator (the background for this controller is in

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Section 2.2.4). The scheduling variables for the feedback gains are angle of attack, Mach

number and altitude [93].

The Eurofighter developed by Germany, England, Spain and Italy controls the attitude

or pitch angle along the longitudinal axis and bank and heading angle along the lateral

axis. The baseline controller is designed using classical control tools. Nichols/Bode plots

and linear time responses were used to analyse the performance of the controllers. The

controller structure was changed with nonlinear elements [95-97].

The Vectored thrust Aircraft Advanced Control (VAAC) program was a program in the

UK that started in 1984. The inner loop control is a Nonlinear Dynamic Inversion (NDI)

controller (background of which is outlined in Section 2.2.3). The nonlinear dynamics of

the aircraft are used to invert the nonlinearities and a PI controller is designed to track the

pitch rate command. Additionally, the pilot commands are filtered which are the inputs to

the NDI controller [98].

Fig 2.3. NDI Controller of VAAC [88]

Furthermore, LPV control techniques are used for controller synthesis for the VAAC. The

system dynamics are written as LTI models whose coefficients were a function of

scheduling parameters. LPV loop shaping used a LPV flight model to directly

synthesize a scheduled LPV controller. This controller technique was successfully

implemented between 1995 and 1998 [99, 100].

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Boeing used multivariable control as its primary control methodology for its aircraft fleet.

LQR/LQG based controller design has been utilised in order to achieve the desired

performance and robustness. During this process, there is a direct trade-off between

command response, control activity, disturbance rejection and loop bandwidth. The basic

design approach of the LQR controllers is outlined in Section 2.2.4. Furthermore, the

feedforward gains of the controllers are adjusted for flying qualities [101].

Looking at the history of controller development for civilian application, much has

changed in recent times compared to 15 years ago. Previously, the majority of the

controllers were designed using classical control approaches. However, now multivariable

control techniques are the standard. Furthermore, dynamic inversion is the most widely

used control design technique in the aircraft industry [88].

2.3 Complications with Hypersonic Control and Modelling

Over the past few decades, research into the control of HSVs, especially hypersonic

gliders, has become a distinctive part of research into the control of re-entry vehicles [10,

19, 47-51, 102]. Initial studies mainly focused on the static stability of hypersonic gliders

[19] followed by studies of control along a re-entry trajectory of the shuttle [58, 103, 104].

However, recently, the focus has shifted back to smaller hypersonic vehicles due to their

affordability in delivering payloads into orbit [70].

The challenges facing hypersonic vehicle control are many. One of the main challenges in

the hypersonic domain is the highly nonlinear time-varying vehicle dynamics. Another

source of complication are the uncertainties related to the airframe, atmosphere and the

control elements [17, 23, 30]. The main tasks, therefore, become designing, implementing

and testing a controller that is robust to all the uncertainties and complexities that are

present during flight. Therefore, before presenting the specific control methodologies that

have been utilised to control hypersonic vehicles, it is important to discuss the relevant

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phenomena that are encountered in high speed flight that make control and modelling of

these vehicles a challenge.

Aerothermodynamics

The main aim behind the design of any airborne platform is to reduce the drag. The drag

is reduced by making the body more slender [105, 106]. However, this comes with its

disadvantages when it comes to hypersonic vehicles. The design of the body can lead to

heating problems. E.g.at the stagnation point at the nose the heating is inversely

proportional to the nose radius. Although drag is an issue, fundamentally hypersonic

vehicles are heat driven as in the hypersonic regime heating varies cubicly with speed,

whereas drag varies quadratically [105].

Hypersonic Flow Phenomenon

Hypersonic flow is characterized by certain physical variables becoming progressively

more important as Mach number is increased. The equation

determines the growth

of the boundary layer [3, 106, 107]. As a result, aerodynamically the body appears to be

thicker than it is due to large boundary layer displacement thickness. This leads to

stronger interaction of the boundary layer with the inviscid field. This interaction is called

viscous interaction (as outlined in Section 1.1.2). This interaction has an impact on the

pressure distribution, lift, drag, stability, skin friction and heat transfer of the vehicle [105,

108].

High Temperature Gas Effects

There are several high temperature gas dynamics effects that increase the difficulty from a

modelling and simulation point of view. The difficulties arise in the creation of the look up

tables that form the backbone of any hypersonic vehicle simulation model. They are [106]:

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Vibrational excitation is observed around Mach 3 and fully excited around Mach

7.5 [pp. 460-461].

Caloric imperfection begin at around 800K and Mach 3.5 (specific heats change as a

function of temperature) [pp. 18-19].

dissociation at approximately 2000K at around Mach 7.5-8.5. The dissociation of

the oxygen is complete at around 4000K at around Mach 15 [pp. 460-461].

Scramjet Propulsion

For integrated hypersonic airbreathing propulsion, the whole under section of the body

forms part of the propulsion system. The forebody of the vehicle generates compression

lift. Furthermore, the aft of the body acts as the upper half of the expansion nozzle. The

forebody section of the vehicle, therefore, becomes a significant design aspect. For

example, the X-43A used three compression ramps to make the free-stream capture area as

large as possible relative to the engine intake area.

Aerodynamics-Propulsion Coupling

The "engineframe" is a concept that is unique to hypersonic vehicles. As opposed to

subsonic and supersonic aircraft, the propulsion and the aerodynamic characteristics of a

hypersonic vehicle are highly coupled. This is due to the fact that the forebody and the aft-

body of the vehicle form an integral part of the engine design. The aerodynamic

characteristics of the body determine the shape of the bow shock [105]. This in turn

influences the inlet conditions. This in turn influences thrust, lift, draft, external nozzle

conditions and the pitching moment.

Aero-Thermo-Structural-Propulsion Coupling

The slender body, in order to reduce the drag of the vehicle, leads to heating problems

because of the sharper leading edges and thinner sections. Bolender [17] notes that the

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slender shapes of vehicles designed for sustained hypersonic flight, such as engine-

integrated scramjet-powered vehicles, can be subject to structural vibrations. This is more

likely than for blunter hypersonic vehicles designed to spend as little time in the

hypersonic regime as possible. Structural vibrations can then lead to variations in the

aerodynamic loads on surfaces and this can lead to perturbations in lift, drag and pitching

moment. This in turn can affect the thrust provided by the engine, stability and the

achievable performance of the vehicle [17]. This coupling between the different aspects of

a hypersonic vehicle further complicates the modelling and control task.

Non-Minimum Phase Dynamics

The elevator to flight path dynamics transfer function has a non-minimum phase (right

half plane) zero associated with the transfer function [17, 109]. This behaviour causes an

initial undershoot (as depicted in Fig 2.4) and the direction reverses in the step response

and also limits the closed loop performance. This behaviour of a hypersonic vehicle is

further investigated in this work.

Fig 2.4. Step Response of Non-minimum Phase System [110]

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The aero-thermo-structural-propulsion effects are not being considered as part of the

GHAME model that is used in this research. This research however presents a solution for

tackling the non-minimum phase behaviour of the flight path angle dynamics.

Behaviour and Uncertainty of the Aerodynamic Parameters

Keeping the aforementioned difficulties in modelling and controlling hypersonic vehicles

the following subsection outlines some of the contributions made in the field of hypersonic

control. The modelling and controller complexities are presented under the headings of

linear control, linear parameter varying (LPV) control, nonlinear control, adaptive control,

trajectory control of gliders and the control of lateral/directional dynamics. This is

followed by a review of the literature pertaining to the application of adaptive control

to airborne platforms.

During the controller design process, in research oriented papers like the one authored by

Wang et. al [111] and Aksaray and Mavris [112], a time invariant model of B747 around a

trim point is used to design the controller. For example, the value of the control moment

derivative is at Mach 0.65 at an altitude of ft. Uncertainties of up to

30% are considered. However, this method cannot be used for a hypersonic glider study.

Along the nominal trajectory, it is seen in Fig 2.5 that changes too much for it to be

assumed as a constant values for the purpose of controller design.The ranges of Mach

numbers that are depicted in Fig 2.5 are significant in terms of the speed regimes that the

vehicle experiences. These include hypersonic all the way down to sub-sonic speeds. At a

speed of Mach 6, the vehicle descends through 30km. This is where the controller has full

authority. This fact leads to a sharp change in the control moment derivative, signifying

that the controller itself has to work in speed regimes that range from hypersonic to sub-

sonic.

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Fig 2.5 Control Moment Derivative as a function of Mach Number and Altitude

In Fig 2.5 the nominal value of is shown as a black line. Whereas, the dashed blue lines

form an error envelope ( ). The maximum nominal value of is -0.2357 and

the minimum nominal value is -22.98. There is a two orders of magnitude difference

between the minimum and the maximum values. The other derivatives of the hypersonic

glider display similar behaviour in terms of changing significantly as a function of time.

Due to this reason, controller design and testing becomes a challenging task especially

because the plant model of the vehicle is highly time varying along the trajectory. It is this

fundamental difference between the B747 and the hypersonic glider regarding the change

in the aerodynamic and control derivatives, which drives the need for a baseline controller

where the desired dynamics are a function of a trajectory parameter.

2.4 Control Methodology of Hypersonic Vehicle – Longitudinal Dynamics

The main area of focus of hypersonic control has involved the investigation of control

methodologies for an air-breathing hypersonic vehicle [18, 24, 25, 29, 33, 34, 36, 37, 113,

114]. There are many difficulties associated with modelling hypersonic vehicle control and

flight tests associated with testing the reliability of control methodologies. A summary of

the difficulties are as follows [17, 18, 20, 23, 24, 26-29, 33, 47, 48, 65, 108]:

0 1 2 3 4 5 6 7-30

-20

-10

0Plot of Control Moment Derivative as a function of Mach Number

Mach NumberMe

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

x 104

-30

-20

-10

0Plot of Control Moment Derivative as a function of Altitude

Altitude (m)

Me

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I. Modelling of hypersonic vehicles involves taking into account large altitude-

velocity flight envelopes, large parameter changes and complicated environments.

These lead to strongly coupled flight dynamics characteristics, fast time varying

parameters and highly nonlinear behaviour.

II. Complications arise with the integration of the propulsion system, guidance and

control systems and the structural modes of the vehicle.

III. The modelling of hypersonic vehicles becomes even more important if it is not

possible to sustain hypersonic flight for an extended period of time.

IV. When considering hypersonic gliders, the challenges include uncertainties

(aerodynamic and gravimetric), strict state and control constraints, low control

authority and a changing flight environment.

2.4.1 Linear Control

Anderson in [1] and Wilcox et. al. in [35] present a robust controller for a wing-cone air-

breathing hypersonic vehicle. Uncertainties such as propulsion system perturbation,

flexible body bending, and uncertainty in control effectiveness are included in the early

stages of the fixed order controller design. Changes in due to the thrust-pitch coupling

are presented. The test case that is presented in both the papers is steady level flight at

86,000ft (26km) at Mach 8. A and a controller are presented. It is deduced that the

controller is too conservative. However, it is able to maintain control authority and

maintain altitude.

The focus of [32] and [36] is to carry out control synthesis for steady level flight in the

presence of uncertainties in the plant model. [36] introduces aerothermoelastic effects into

the model and presents a controller that incorporates the aerothermoelastic effects as state-

varying uncertainties into the plant model. The bounds of the error signals are provided as

a way of proving the suitability of this type of controller.

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Gregory in [32] develops a control law for hypersonic vehicles using control and

synthesis. The control law presented is for a single stage to orbit (SSTO) air breathing

vehicle. The challenges considered in this paper are atmospheric turbulence, high

sensitivity of the propulsion system to inlet flow conditions and large uncertainties of the

parameters. An integrated flight/propulsion model is linearised as the vehicle accelerates

through Mach 8.However, the controllers, designed using modern control theory, led to

higher order controllers. This led to an increased cost of the controllers. A 9th, 7th and 5th

fixed order controllers were applied to a hypersonic vehicle and tested. The reduced order

controllers were shown to be feasible in terms of controlling hypersonic vehicles and were

able to achieve robust performance. More specifically, when control effectiveness

uncertainty is introduced into the system, there is a noticeable degradation in the

performance of the controller. A 20% uncertainty in the control effectiveness leads to

instability for the controller. On the other hand the controller is able to handle the

worst case control effectiveness scenario and still meet the performance requirements. The

performance of the controller is shown in Fig 2.6. For the case presented, the vehicle is

commanded a change in velocity of 1000ft/sec and a change in altitude of 1000ft. Although

there is significant uncertainty in the system, the deflection of the elevon (shown on the y-

axis) is less than , which was also one of the performance requirements.

Fig 2.6. -Controller Performance - Worst Case Scenario [32]

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The linear control technique for the X-43A flight is outlined in detail by Bahmet. al. in [86].

The controller design of the X-43A was based on sequential loop closure root locus

methods. The look up table of the gains was scheduled as a function of Mach number and

angle of attack, with dynamic pressure compensation. For the longitudinal and lateral

control loops gain and phase margins of 6dB and were designed for all the flight

conditions. Lower values were accepted for the descent part of the trajectory. The

guidance loop ran at 25Hz and the controller operated at 100Hz [105]. The longitudinal

and lateral controller loops are shown in Fig 2.7 and Fig 2.8.

Fig 2.7. Longitudinal Controller Block Diagram [86]

Fig 2.8. Lateral Controller Block Diagram [86]

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A detailed comparison of the Monte Carlo simulations and flight data is presented by

Bahmet. al. but some of the important results are included here to highlight the

importance of Monte Carlo simulations in terms of modelling and simulation of

hypersonic vehicles. For all of the results presented, the simulations created an envelope of

results within which the flight test data appeared. This is significant due to the fact that

this paper not only presented simulation results, but the authors were also able to validate

their methodology by comparing the results with data from the Mach 7 flight.

Fig 2.9. Inertial Angle of Attack from Engine Test through Recovery Manoeuvre [86]

Fig 2.10. Inertial Sideslip Angle from Engine Test through Recovery Manoeuvre [86]

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Fig 2.11. Comparison of In-flight Altitude and Pre-flight Altitude Predictions [86]

Fig 2.12. Inertial Angle of Attack during Descent Phase [86]

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2.4.2 Linear Parameter Varying Control

The LPV technique has been employed to define a hypersonic vehicle model. The control

strategy used for a LPV framework has concentrated on controlling the longitudinal

dynamics of a flexible air-breathing hypersonic vehicle [34, 115]. For the stability criteria,

the convergence characteristic of the output tracking error is carried out with the help of

Lyapunov stability analysis. The controller presented demonstrates that in the presence of

aerothermoelastic effects, the controller is effective in maintaining performance and robust

stability. An example of the performance of the controller is shown through the velocity

tracking error and the angle of attack tracking error (Fig 2.13 and Fig 2.14). This study was

carried out at an altitude of 80,000ft with the vehicle travelling at Mach 8.

Fig 2.13. Velocity and Velocity Tracking Error [115]

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Fig 2.14. Angle of Attack and Angle of Attack Error [115]

LPV techniques have been used in conjunction with robust controllers ( controllers) to

formulate a framework to account for the parametric uncertainties present in the

modelling of hypersonic vehicles including aerodynamic uncertainties and structural

flexibility [87]. A parameter space involving the Mach number and altitude is utilised for

controller synthesis (shown in Fig 2.15). The results presented show (using the tracking

errors in angle of attack and velocity) that the controller is robust to disturbances,

parametric uncertainties, and modelling errors for the tracking and regulation states.

Fig 2.15. Parameter Space for Controller Design [87]

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Cai et. al. in [116] presents an angle of attack tracking control system for a hypersonic re-

entry vehicle. The LPV theory based on a Linear Fractional Transformation (LFT) model is

applied during the controller synthesis process. A damping feedback loop is applied to

improve the system's damping characteristics and static stability. The linear dynamics

along with the damping loop is considered as the controlled plant for the LPV control

system design. A LPV reference model is defined to characterise the transient behaviour of

the controller. Using the proposed method, the controller is able to follow a step command

in angle of attack. When this controller is commanded to follow a nonlinear angle of attack

command, it tracks the angle of attack; however there is a small steady state error. The

commanded angle of attack and the corresponding steady state error is shown in Fig 2.16.

Fig 2.16. Commanded Angle of Attack (left) and Angle of Attack Error (right) [116]

2.4.3 Nonlinear Control

Studies outlining nonlinear control of hypersonic vehicles are presented in [33] and [113].

These studies concentrate on the development of a Lyapunov function to prove the

convergence of the output tracking error. These papers concentrate on presenting a model

that includes the propulsion module, and also outlines the coupling of different

components of the vehicle; for example: aerodynamics of the vehicle, the propulsion

system, the structural modes and also the controller. In [38] a nonlinear control law which

ensures good stability property is presented. The complexities that arise due to the

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propulsion system have been taken into account. There are some potential limits of the

technique presented in this paper: (i) backstepping is an efficient way to extend the

Lyapunov function and globally control the vehicle. However it reintroduces complexity

and does not take advantage of a possible stability of rotational motion; (ii) backstepping

is inefficient when it comes to dealing with actuator saturation; (iii) the controller does not

show asymptotic convergence to the reference signal which leads to the conclusion that it

lacks quantitative robustness.

Another nonlinear control methodology is dynamic inversion (DI) [69, 113, 117, 118]. A

dynamic inversion control law is design for a conceptual hypersonic vehicle and is shown

in [117] and [118]. The control laws require an on-board model that cancels the nonlinear

dynamics of the vehicle so that the desired command responses of the closed loop system

are achieved. The dynamic inversion law design methodology is especially attractive for

application to a hypersonic cruise vehicle to avoid complex gain scheduling which is

usually required to cover a large flight envelope from subsonic to hypersonic speeds and

from sea level to high altitudes. Through robust analysis the dynamic inversion controller

is shown to maintain stability over multiple flight conditions in the presence of

uncertainties in the aerodynamic derivatives. The inherent features of dynamic inversion

are utilised to easily improve lateral-directional controller robustness through an

adjustment to the desired dynamics. The robustness is also improved by adjusting loop

gains using knowledge gained through sensitivity analysis of the controller. The resulting

controller demonstrates improved robustness, especially at critical Mach numbers.

Through the use of dynamic inversion, a minimal amount of tuning is required to

maintain stability and robustness to uncertainties as the Hypersonic Cruise Vehicle (HCV)

continues through the design process.

A dynamic inversion controller for a Generic Hypersonic Vehicle (GHV) is shown in [69].

The control laws are robust to a decrease in control surface effectiveness (multiplicative

gains of 0.15 of the nominal control surface deflection), changes in system parameters, and

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time delays of 0.04s or less. Additionally, the control architecture is able to tolerate an inlet

unstart1 and maintain nominal tracking along a specified flight path angle trajectory.

Therefore, it is concluded that this approach of nonlinear adaptive dynamic inversion

control works well as a control architecture for the GHV. The commanded flight path

angle and the corresponding system states during an inlet unstart are shown in Fig 2.17.

Fig 2.18 shows that, after the inlet unstart, the thrust equivalence ratio is commanded to be

the maximum value. Although this is the case, thrust is not generated by the vehicle until

seconds. The vehicle still tracks the commanded flight path angle even though the

performance is degraded.

Fig 2.17. System States of Hypersonic Vehicle during Inlet Unstart [69]

1A violent breakdown of supersonic flow in the inlet and the establishment of subsonic flow through the inlet and out

in front of it [78].

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Fig 2.18. Control and Adaptive Weight Responses during Inlet Unstart [69]

2.4.4 Adaptive Control

Various studies have presented results with regard to adaptive control of hypersonic

vehicles [5, 10, 31, 33, 38-40, 119, 120]. A small fraction of these papers and the important

results from those studies are outlined here. Somanath and Annaswamy [38] present an

adaptive controller for systems with partially known input matrix . The controller

presented is globally asymptotically stable when additional information regarding the

structure of is known. If is unknown, then the adaptive control law presented yields

a controller that is only locally stable. Xu et. al. [40] presents a study of a control of

hypersonic vehicle employing the sliding model controller. Although the longitudinal

controller performs well in the presence of parametric uncertainty, it does so at the

expense of high gains and control chattering. Yang et.al. [120] present an adaptive fuzzy

controller for the velocity and altitude systems. This paper provides the effectiveness of an

online adaptive controller with the presence of numerous disturbances such as

aerodynamic interference, measurement noise, elevator disturbance and parametric

uncertainty.

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Wiese et. al. in [31] presents an adaptive augmented, gain scheduled baseline LQR-PI

controller applied to a Road-Runner 6-DoF generic hypersonic model. The uncertainties

introduced into the model include control effectiveness, longitudinal centre of gravity

location, aerodynamic coefficients, sensor bias and noise and input time delays. A block

diagram of the controller architecture is shown in Fig 2.19.

Fig 2.19. Augmented Controller Architecture [31]

The performance of the baseline controller is compared with the performance of two

different MRAC designs; a classical open loop reference model design, and the modified

closed loop reference model design. Both the augmentation configurations perform better

than the baseline controller in the presence of uncertainties. However, the closed loop

augmentation model offers the best performance.

The X-15 was an aircraft built and tested in the 1950s with the aim of investigating the

capability of a platform flying out of the atmosphere, re-entering the atmosphere and

landing horizontally [121]. The X-15 trajectory is shown in Fig 2.20.

Fig 2.20. Typical X-15 Trajectory [121]

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For flight within the atmosphere a combination of rolling tail and three-axis dual

redundant stability augmentation system (SAS) was used for landing and re-entry. For

exo-atmospheric flight, hydrogen peroxide thrust rockets were used as bang-bang reaction

controls. Thrusters on the nose were used for pitch and yaw control and on the wingtips

for roll control [121]. However, the most publicised of the X-15-3 flights had onboard the

Honeywell designed dual redundant self-adaptive control system [121-123]. Dydek et. al.

in [123] presents the results from the failed flight of 1967. The adaptive controller was

designed to achieve a high level of performance throughout the entire flight envelope. By

the end of the design process, it was observed that rapid changes in the feedforward gain

would be required and the controller gains had to be near critical values during all phases

of the flight. As a result, there was a reason that the gains had to be constantly changed

and adjusted. In order to accomplish this, the system output in the frequency range, at

which instability was expected to occur, was monitored. This meant that when there were

hints of instability in the system, the gains were reduced in order to maintain the stability.

On the other hand, when no instability was observed, the gains were increased. This

resulted in the gains being adjusted to their maximum possible values while maintaining

system stability throughout the entire flight envelope [122, 123]. In 1967 after reaching

peak altitude, the vehicle began a sharp descent and entered a Mach 5 spin. Although the

pilot was able to recover from the spin, the adaptive controller was unable to reduce the

pitch angle and the airplane continued to dive. Due to rapidly increasing dynamic

pressure the aircraft broke up approximately 65000ft above sea level. However, using

modern adaptive controller and knowing what is known about limit cycles (one of the

causes of the dysfunctional adaptive controller), robustness and stability properties, the

baseline and the adaptive controllers have been redesigned and the flight has been

recreated. It is shown in Fig 2.21 [122] that a modern adaptive controller would have

successfully followed the nominal flight trajectory even in the presence of system failures.

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Fig 2.21. Comparison of Performance of Original X-15-3 Adaptive Controller with Modern Day Adaptive

Controller [122]

The control architecture of the X-15-3 flight was a MH-96 controller and consisted of an

inner loop and a pilot model. The inner loop controls the quickly varying states, like the

longitudinal and lateral states. The pilot model controls the slowly varying state, like

altitude and speed. The controller used in the initial flight, although it was flexible and

has built in adaptive features, was unable to regain control which then ultimately led to

the crash. The Lyapunov stability based adaptive controller explicitly accounts for actuator

dynamics, parametric uncertainties, actuator saturation, unmodeled dynamics, time

delays, nonlinearities and "unknown unknowns". This fundamental difference in the

control synthesis process highlights the advancement of adaptive control as compared to

what was state of the art in the 1960s. However, it is unfortunate that it took the loss of a

pilot for researchers to understand the issues in terms of robustness and stability of

adaptive controllers.

2.4.5 Trajectory Control of Hypersonic Gliders

There is limited literature that presents the trajectory analysis of hypersonic gliders.

Methods such as extended command governor (ECG) and sliding mode control have been

employed to control the trajectory of gliders [48, 51]. More recently, pole placement

control (PPC) and adaptive control has been used to control hypersonic vehicles along a

glide trajectory [10, 16]. All of the previous studies, however, have concentrated on

controlling the longitudinal dynamics of the glider.

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Rainey [19] is one of only a few papers that covers the stability issues for both the

longitudinal and lateral dynamics of a hypersonic glider. The two main configurations

that are presented in the paper are shown in Fig 2.22 and Fig 2.23.

Fig 2.22. Flat Top Glider [19]

Fig 2.23. Flat Bottom Glider [19]

At low angles of attack, both configurations are found to be either self-trimming or can

obtain trim. However, at higher angle of attacks the lateral and directional static stability

of the glider (irrespective of the configuration) cannot be guaranteed. One of the

recommendations of the paper is to investigate this very phenomenon.

Trajectory control studies have been carried out in [51] and [124]. These two papers

attempt to control the altitude of the hypersonic vehicle using a flight path angle

controller. The guidance loop is provided with a desired altitude. Using Eq.2. 1, this

desired altitude is converted to a desired inner loop flight path angle.

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2. 1

If and are chosen and the flight path angle is controlled to follow the

commanded signal, , the altitude error then exponentially is regulated to zero. The test

case that is used to analyse this controller is very limited in its application as the desired

altitude is only 100m above the current position of the hypersonic vehicle.

Tournes and Hanks in [51] present a sliding mode controller for a hypersonic glider for

cruising at altitudes of 40-50km. This paper shows that without powered manoeuvres this

vehicle has a very wide reaching domain. A second-order sliding mode angle of attack

controller is employed to track a desired flight path angle. It is also shown that

considerable matched disturbances can be introduced in the model without the

deterioration of the performance and stability of the controller.

Creagh in [10] presents an adaptive controller for the longitudinal dynamics of a Mach 8

Hypersonic Glider. The system architecture is a three tier system to obtain the estimates of

model aerodynamics parameters. The process is outlined in Fig 2.24.

Fig 2.24. Flow Diagram - Adaptive Control Setup

Linear Least Squares

•used to obtain the state transition methods and control matrix parameter estimates from a second order plant model.

Fusion Algorithm

•weights preloaded lookup table parameters and the least squares estimated parameters to obtain a fused estimate.

•The least-squares estimates are favoured when system excitation is present, while lookup table parameters are favoured when system excitation is negligible. The selection of measurement variances, lookup table parameter variances and sampling data provides the system tuning input.

Update Lookup Tables

•The third tier of the adaptive control strategy is to update the lookup tables with a multiplying factor, which is calculated with a first-order filter. This enables the parameter estimates to retain trends learnt from earlier in the mission. Thus, for slow and consistent parameter changes, the closed-loop vehicle response is seen to improve slightly

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However, as Creagh points out, due to the inherent lag of the system outlined above, this

system is not suitable in its current form for an up-and-over trajectory profile. It may be

suitable for vehicles that visit the same flight conditions multiple times (i.e. constant

altitude & Mach number).

Prime in [16] presents a adaptive controller for a re-entry vehicle configuration. The

controller is shown to be an effective control methodology for re-entry vehicles. Reduced

performance is exhibited by the controller for some of the off-nominal conditions, however

the vehicle remains stable. This paper concentrates on a time invariant reference system. A

more detailed summary of the results presented by Prime is in Section 4.5.

Goa et. al in [125] looks at trajectory optimisation of hypersonic gliders using swarm

intelligence algorithms. The algorithms are developed for the Common Aerospace Vehicle

(CAV). This vehicle is shown in Fig 2.25.

Fig 2.25. The Common Aerospace Vehicle (CAV) [125]

[125] concentrates on providing the methodology for optimizing the guidance law for the

angle of attack command. The guidance law is formulated using the minimization of a cost

function. This means that the guidance law presented is an offline design. However, an

application of this methodology is deemed to be useful if a certain guidance law can

control a family of trajectories and maintain and achieve a desired glide trajectory. One of

the terminal conditions for this glide trajectory is the altitude. This altitude in this paper is

set to 20km. The algorithm presented in [125] is successful in tracking the desired

trajectory which has reduced the value of the cost function. However, all the simulations

are carried out with perfect sensor feedback and in the absence of aerodynamic

LITERATURE REVIEW

44

uncertainties. Although the algorithm presented is a good baseline for getting an idea

about the complexities of glider trajectory study, the added layer of uncertainties makes

the control laws presented very limited in their application.

Qian et. al. in [126] outlines a flight control system based on a nonlinear disturbance

observer (NDO) for a hypersonic gliding vehicle (HGV). The type of vehicle presented in

this paper can be used to deliver a variety of payloads inside the atmosphere. The NDO is

used to estimate the unknown disturbances. The flight controller itself is a terminal sliding

mode controller. The sliding mode controller along with the NDO is effective in cancelling

the influence of the disturbances.

Baldwin and Kolmanovsky in [47] present a model predictive control (MPC) approach to

hypersonic glider flight management. This kind of control methodology can be applied to

the guidance of a vehicle with path constraints “fly to” and “fly through” phases are

utilised to check the functionality of the guidance system. The one advantage of using

MPC is its ability to adjust the trajectory in real time in response to changes such as

appearance/disappearance of exclusion zones, disturbances or failure modes. Strategies for

on-board implementations have also been proposed.

Baldwin and Kolmanovsky in [48] propose an augmentation of an inner-loop LQ-PI

controller with an ECG to handle constraints faced during hypersonic flight. The

simulation results presented demonstrate the capability of the ECG to enforce the

constraints while minimizing the degradation in the vehicle responsiveness.

2.5 Control of Hypersonic Vehicle – Lateral/Directional Dynamics

There is very limited literature on the control of lateral/directional dynamics of hypersonic

vehicles, especially hypersonic gliders. Therefore, it is important to draw on the available

literature, recognise the gaps in the literature and address them accordingly. The limited

literature on the lateral dynamics presented in [52, 54, 56, 57, 127] are summarised here.

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45

Heller et. al. in [57] outlines a robust lateral controller for a hypersonic vehicle. A Multi-

Model Eigenstructure Assignment is applied to design a controller for a projected

hypersonic test vehicle featuring ramjet propulsion. The main aims of the controller design

are as follows: (i) provision of adequate flying qualities, especially in terms of reduction of

the high roll-yaw coupling, (ii) securing of robustness against multiple model

uncertainties and parameter deviations (robust stability) but also against disturbances and

failure cases, and (iii) low controller complexity (low order), high reliability and easy

feasibility. The uncontrolled system highlights the main challenges with respect to

controlling the lateral/directional dynamics of the vehicle. Issues include low damping of

all modes, instability of the Dutch roll and the high roll-yaw coupling. These issues in the

lateral/direction axis are removed using a highly augmented control system. The fixed

gain multi-mode eigenstructure assignment method yields robustness against a

considerable range of uncertainties. The controller shows good performance even in the

presence of considerable time delays, nonlinear effects like actuator deflection and rate

limitations, model uncertainties and multiple disturbances.

A six-degree of freedom study of the lateral-directional dynamics of the Space Shuttle

Orbiter is presented in [58]. Off-nominal aerodynamic characteristics are introduced into

the system by applying a multiplier to the aerodynamic parameter or by adding an

increment. Two control system modifications are presented to reduce the system’s

sensitivity to degraded aerodynamics: (i) an increase in the yaw-jet augmentation of the

rudder and (ii) a change in the angle of attack for switching to an aircraft type rudder-

aileron control system.

2.6 Adaptive Control Literature Review

adaptive control was initially developed to apply to flight control. The initial concept of

adaptive control was presented in [79, 80, 82, 128-131]. The main aim of adaptive

control is to decouple the estimation and the control loops. This allows for fast adaptation

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46

without loss of robustness [22, 41, 42, 61, 63, 65, 132]. Literature depicting the application

of adaptive control are presented in [16, 22, 37, 42, 61-63, 65-67, 75, 77, 78, 114, 131, 133-

146] and are summarised here.

One of the practical applications of a full adaptive controller is outlined in [62], wherein

a controller is designed for the NASA AirSTAR flight test vehicle. The adaptive

controller is applied to a subscale turbine powered Generic Transport Model (GTM). The

controller presented directly compensates for matched as well as unmatched uncertainties.

The main motive for this program was to test the performance of the aircraft and the pilots

under adverse conditions. These conditions include unusual attitudes, surface failures and

structural damage. One of the main outcomes of this program was to test the performance

and the evaluation of the controller beyond the edge of the normal flight envelope. This is

particularly important as beyond the pre-defined flight envelope the risk of vehicle loss is

high due to limited knowledge of nonlinear aerodynamics beyond stall and the potential

for high structural loads. The corresponding controller is designed at a single operating

point at the centre of the nominal flight envelope. The analysis is carried out for variable

dynamics and damage cases in highly nonlinear areas of the flight envelope, in both

piloted and non-piloted evaluations. As a result of these simulations, a theoretical

extension of the adaptive controller that accounts for the matched as well as unmatched

dynamic uncertainties is presented. Two different sets of simulations are carried out: batch

simulations and piloted simulations. The batch simulations are carried out on the nominal

aircraft, the case where the entire rudder is missing, the case where left outboard trailing

edge flap missing, loss of outboard left wing tip, loss of entire elevator from left stabilizer

and loss of entire left stabilizer. For the case of piloted simulations: a static stability

degradation, roll damping degradation, simultaneous static and roll degradation, high

angle of attack capture task, sudden asymmetric thrust, and pilot induced upset and

recovery cases are investigated. For all the aforementioned cases, the adaptive

controller performed well.

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47

Griffin et. al in [65] presents a adaptive control augmentation system to control the

lateral/directional dynamics of X-29. This study presents a comparative study between the

baseline Linear Quadratic Integral (LQI) controller and the augmented controller for the

Single Input Single Output (SISO) and the Multi Input Multi Output (MIMO) setup. The

augmentation system is once again utilised in the presence of unmatched uncertainties

which may exhibit significant cross-coupling effects. An additional advancement is the

addition of a high fidelity actuator model to the architecture to reduce the uncertainties

in the state predictor design. A predictable outcome of this study is the cancellation of the

cross coupling effects when the MIMO approach is used. It is also demonstrated that the

state predictor can be designed to match the nominal closed loop behaviour given a

certain baseline controller. As expected under nominal conditions the contribution of the

adaptive element is minimal. However, in the presence of a failure, the augmentation

system proves effective in regaining the nominal performance at the same time

eliminating the undesired sideslip-to-roll coupling. The failure scenarios covered in this

study are jammed rudder, jammed elevon and the reduction of the effect of cross coupling.

A comparison of the SISO and the MIMO augmented systems in the presence of failures is

shown in Fig 2.26.

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48

Fig 2.26. L1 Augmented Controller Performance Comparison [65]

A output feedback output controller architecture has been utilised to control the

longitudinal dynamics of a missile and also for a flexible space Crew Launch Vehicle

(CLV) in [22] and [137]. Kharisov et.al in [22] outlines the functionality and applicability of

an adaptive controller for a launch vehicle which operate often in very unforgiving and

occasionally highly uncertain environments. The output feedback controller

architecture is utilised to control the low frequency structural modes. The paper also

proposes steps to validate the adaptive controller performance. The theoretical

performance bounds of adaptive controller are verified through the nonlinear

simulations. The results presented demonstrate that a single adaptive output controller

is able to handle statically unstable flexible plant with large parametric variations without

addition of notch filters, without re-tuning for different flight conditions along the first

stage trajectory and with guaranteed transient performance.

The output controller utilised to control the longitudinal dynamics of the missile autopilot

is outlined in [137]. The controller has satisfactory performance in the presence of

parametric uncertainties and time-varying disturbances. A comparative study between the

Linear Quadratic Regulator (LQR) and the Linear Quadratic Gaussian (LQG) with Loop

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Transfer Recovery (LTR) design with the adaptive controller is presented. And the

simulations results present the clear benefits of using the controller. A further study of

the application of an adaptive controller to a missile is outlined in [75]. An control

augmentation is designed based on a novel autopilot structure for a highly, agile, tail-

controlled missile. The autopilot is designed to control the pitch and yaw accelerations

while maintaining a desired roll angle. The adaptive augmentation is implemented and

utilised to compensate for the undesirable effects of modelling uncertainties. This

augmentation is successful in increasing the robustness of the baseline controller in the

presence of significant plant uncertainties. This paper is successful in showing that,

despite the reduced complexity, the augmented controller is able to increase the robust

performance when compared to an Model Reference Adaptive Control (MRAC)

augmentation outlined in [76].

Leman in [63] presents the augmented controller for the longitudinal and the

lateral/directional dynamics for a X-48B aircraft. The piecewise constant adaptive law is

utilised to estimate the matched and unmatched uncertainties. SISO channels are utilised

on all the channels: longitudinal, lateral and directional. The performance of the baseline

controller is compared with the augmented controller in the presence of control surface

failures. An additional conclusion to be drawn from this study is the lack of high

frequency components in the control surface deflections. The augmented controller

improves the performance of the baseline controller under nominal conditions across the

entire envelope. In the event of a failure, the adaptive controller adapts to recover

desired aircraft performance and provide a predictable response to pilot inputs.

A full adaptive controller is presented by Prime in [16] to control the longitudinal

dynamics of a waverider class hypersonic vehicle where the aerodynamics are based on

NASA’s Generic Hypersonic Aerodynamics Model Example (GHAME). To test the

robustness and the performance properties of the adaptive controller, pull-up trajectory

simulations are performed using several different simulation scenarios with no change to

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the controller configuration. The simulation scenarios covered in the paper are: (i)

Nominal conditions, (ii) Ramp reduction from nominal elevator effectiveness, to

starting at an altitude of 28km over 5 seconds, (iii) Step reduction of the elevator

effectiveness at an altitude of 28km, (iv) Reduced static margin, (v) Controller processing

delay of 10ms, and (vi) Controller processing delay of 20ms. Projection based adaptive

laws are used to compensate for the matched and the unmatched uncertainties. The

adaptive controller is shown to be an effective control methodology for a re-entry vehicle.

However, for some of the off-nominal conditions there was reduced performance.

Based on a review of the literature presented in this chapter, an controller has been

chosen as the augmentation for the following reasons:

Disturbance rejection properties within the bandwidth of the controller in the

presence of fast adaptation.

High frequency content like noise does not affect the control channel when the low

pass filter is properly designed.

piecewise-constant control law can be interpreted as a linear control law, in turn,

expediting the certification process.

The controller with the piecewise-constant update law is easy to implement.

2.7 Summary of Literature Review – the Gaps in the Field

The literature review and ideas developed herein provide the basis for the conceptual

framework for this study. Reviewing the hypersonic control literature, it is clear that there

are many studies for test cases that focus on controller performance at a fixed altitude and

Mach number in the presence of uncertainties. There are few studies presenting a control

strategy for a pull out manoeuvre of a SCRAMJET powered vehicle along the descent

phase of a suborbital ballistic trajectory. Addressing this gap will lead to an increased

interest in using SCRAMJET powered vehicles as a means of transportation to reduce

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51

travel times and deliver payload into orbit (contribution number I in Section 1.9). From a

control point of view, there is limited literature that incorporates the performance and

robustness of controllers for the longitudinal and lateral/directional dynamics of a

hypersonic vehicle (contribution number II, III and IV in Section 1.9). From an adaptive

control point of view, up to this point, the adaptive and augmented controllers have

either been designed at a single operating point or the state estimator has been scheduled

at different operating points. Application of the piecewise constant adaptation method

to a LTV nonminimum phase state feedback system is presented in this study

(contribution number V and VI in Section 1.9).

2.8 Conclusion

Chapter 2 has established the main contributions made in the field of hypersonic control.

This chapter started with a brief introduction to the theoretical foundation of the most

common control methodologies presented in the literature to control airborne platforms.

Thereafter a background on typical hypersonic trajectories is presented, followed by a

discussion on the state of art of control methodologies for subsonic aircraft. An outline of

the specific difficulties associated with modelling and controlling hypersonic vehicles and

the several different longitudinal and lateral-directional control strategies utilised to

control the dynamics of a hypersonic vehicle are presented. The gaps in the literature are

outlined and, consequently, the direction of the thesis is identified.

Chapter 3 presents the simulation model utilised to set up the test cases. It also outlines

the definition and the implementation of the uncertainties.

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3 SIMULATION MODEL

This chapter presents the simulation model used in this project. Thereafter, the vehicle

model utilised for controller synthesis is presented. A description of the nonlinear second

order actuator model and the sensor model are presented. A description of the

environment and the aerodynamic model is presented along with the uncertainties and its

implementation. This chapter forms the foundation of the simulation model that is

implemented in CADAC [14]. The main aim of this research is to investigate the

performance (and the underlying reasons for that performance) of control methodologies

for hypersonic vehicle control. In order to carry out this task, the model of the hypersonic

vehicle, the aerodynamics, sensors and the actuators are all needed.

3.1 Introduction

Flight control laws are employed to change the dynamic behaviour of an aircraft.

However, before the control laws can be presented, an accurate description of the glider’s

behaviour is needed. This chapter presents the mathematical model used to describe the

behaviour of the hypersonic glider. Section 3.2 presents the linearised representation of the

glider dynamics which is used for control synthesis. Section 3.3 provides a description of

the models used to model and determine the forces and moments acting on the platform.

Section 3.4 presents the aerodynamic forces and moments. The actuator model is

presented in Section 3.5. Details of the sensor model are given in Section 3.6. Section 3.7

outlines the modelling of the actuator, gravimetric and aerodynamic uncertainties present

in the simulations. Section 3.8 presents the CADAC simulation platform that is utilised to

carry out the nonlinear simulations. The chapter ends with conclusions in Section 3.9.

3.2 Aircraft Dynamics

The dynamic behaviour of an aircraft, under the influence of forces and moments, can be

depicted using the equations of motion of an aircraft. Since the motion of the vehicle only

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has meaning in a certain frame of reference, reference frames are utilised to represent and

derive these equations. Zipfel’s implementation is used to derive the equations of motion.

In [15], Zipfel employs tensor mathematics to represent the equations. With this

formulation, the equations are independent of the frame of reference. Section 3.2.1 outlines

the frames of reference that are utilised in the nonlinear simulations.

3.2.1 Reference Frames

The frames of references utilised to carry out the 6-DoF nonlinear simulations are as

follows (the detailed relationship between these reference frames are outlined in [147]):

The inertial reference frame, , is a non-accelerating, non-rotating frame of

reference. The origin of this system is at the centre of the earth. The Z axis points

North. The X axis points towards the Vernal Equinox. The Y axis is perpendicular to

both these axes. Its direction is determined using the right hand rule.

The World Geodetic System 1984, , is widely used in GPS-based navigation. In

this reference frame, the earth is represented as an ellipsoid. The position of a plane

is represented by longitude, latitude and height.

The Earth fixed reference frame, , rotates with the earth around the spin axis.The

Z axis points north. The X axis passes through the interception point of the prime

meridian and the equator. The Y axis is perpendicular to both these axes. Its

direction is determined using the right hand rule.

The body fixed reference frame, , is attached to the vehicle. The origin of this

frame of reference is the centre of gravity of the aircraft. The X axis lies in the

symmetry plane and points forward. The Z axis lies in the symmetry plane and

points downwards. The Y axis is once again determined using the right hand rule

(this results to the Y axis pointing towards the right wing of an aircraft).

The stability axis reference frame, is similar to the body fixed reference system. It

is rotated by an angle about the y axis and an angle about the z axis.

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In CADAC [13, 15] the transformations between the frames of reference are carried out

using transformation matrices. The basics of these transformations are presented in [15]. A

small introduction to these transformations and the corresponding diagrams of the

various frames of reference are given in Appendix C.

3.2.2 Equations of Motion

The motion of an aircraft is decoupled into the longitudinal and the lateral equations [148,

149].

Longitudinal Dynamics

The full nonlinear longitudinal dynamics are presented in [15]. The full linearisation of the

nonlinear model is not shown in this document. The linearization is outlined in [148, 149].

The state space representation of the linearised equations of motion for a powered

hypersonic vehicle can be written as

3. 1

where is the system matrix, is the control gain matrix, is the

state vector, is the control input, is the number of states and is the number of

control inputs. The full linearised state space representation of the longitudinal dynamics

is:

3. 2

When considering the dynamics of the vehicle without any control input, the linear system

is a homogenous system. The system matrix, , has four eigenvalues: two pairs of complex

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55

conjugate pairs (typical positions of these poles for a civilian aircraft are shown in Fig 3.1).

The longitudinal motion of a flight vehicle mainly consists of two modes:

Angle of Attack Dynamics (A): the short period mode of the longitudinal dynamics

is relatively quick and well damped. The short period dynamics are represented

using the angle of attack and the pitch rate.

Phugoid Mode (P): this mode is relative slow and is poorly damped and consists

mainly of the flight path movement. It consists of the exchange between potential

and kinetic energy of an aircraft.

Fig 3.1.Typical Pole Positions for Longitudinal Motion

In order to calculate the eigenvalues for the longitudinal dynamics it is assumed that the

fast and the slow dynamics operate at significantly different rates. This allows for the

longitudinal dynamics to be sub-divided and considered separately. An example

demonstrating the difference between the two modes in shown in Fig 3.1.

Angle of Attack Dynamics

The fourth order longitudinal system can be written as

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56

3. 3

includes the effects of phugoid perturbation on the phugoid motion. includes the

effects of short period perturbations on phugoid motion, consists of the effects of

phugoid perturbation on short period motion and consists of the effects of the short

period perturbations on the short period motion.

can be set to a null matrix as the influence of the velocity subsystem is relatively much

slower than the influence of the angle of attack and pitch rate. Based on this simplification,

the angle of attack dynamics can be expressed using just the top left section ( .

Therefore, the second order system that explains the short period dynamics is

3. 4

The characteristic equation is

3. 5

3. 6

Comparing Eq. 3.6 with the second order equation of the form leads to:

3. 7

3. 8

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The stiffness of the angle of attack dynamics are determined mainly through the value of

. For a reasonable answer from the above equations which means that .

This leads to the definition of static stability. The damping of the short period dynamics is

determined by , and consequently by and

. The stability of the short period

dynamics depends on both and and not just from any conclusions drawn from

.

From a controller point of view the model utilised does not consider the velocity and the

pitch angle dynamics. Therefore, the short period dynamics, as presented in Eq. 3.11, are

used. The pitch dynamics presented in Eq.3. 4 is extended to include the flight path angle

dynamics in case that is the control variable. The states needed to describe these state

equations are the flight path angle , pitch rate and its integral, the pitch angle . Using

the relation (shown in Fig 3.2), the basis of the state space representation of the

flight path angle becomes

3. 9

3. 10

Combining Eq. 3.9 and Eq. 3.10 and the fact that the pitch angle is an integral of the pitch

rate, yields the following state space model of the flight path angle dynamics:

3. 11

where the dimensional derivatives are calculated from the non-dimensional derivatives

according to:

3. 12

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The plant models are used to carry out the gain calculations for the dynamic pole

placement controllers. The non-dimensional derivatives are stored in look-up tables

(figures of which are outlined and discussed in Section 3.4.1) and are converted to the

dimensional values in real time.

Fig 3.2. Relationship between Angle of Attack, Pitch and Flight Path Angle

Lateral/Directional Dynamics

The nonlinear lateral equations of motions are presented in [15]. However, the equations

for lateral/directional dynamics utilised for controller synthesis are shown in Eq.3. 13. The

states of this system are yaw rate , sideslip angle , roll rate and its integral, roll angle

. The control variables are the aileron deflection , and rudder deflection .

3. 13

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The system matrix- and the control input gain matrices are populated using dimensional

derivatives. These values are dimensionalised in real time using the non-dimensional

derivatives that are stored in look up tables. These are defined as:

3. 14

The lateral/directional dynamics have once again been written in state space form

. As is the case with the longitudinal dynamics, the behaviour

of the glider dynamics is considered without any control input. This is called the

homogeneous system. The matrix has four eigenvalues. Typically the lateral dynamics

has two real eigenvalues and a complex conjugate pair (typical position of these pole

positions is shown in Fig 3.3). The following different modes are observed in the

lateral/directional dynamics:

Roll Dynamics (R): The rotation about the roll axis. It’s aperiodic, relative quick and

is stable. As the vehicle rolls, the wing goes down and has an increased . The

opposite effect is felt by the other wing. As a result there is a difference in the lift

generated by the wings (more lift generated by the wing that goes down). The

differential lift creates a moment that restores the equilibrium ( ). If there is a

disturbance in this channel, the roll rate builds up in an exponential manner until

the restoring moment balances the disturbing force. This leads to a steady roll.

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Spiral Mode (S): the spiral motion is relatively slow and can also be unstable. A

disturbance in the roll channel results in a small side-slip ( ). The vertical stabiliser

of the aircraft now hits the oncoming air at an angle of , which causes an extra tail

lift (positive yawing moment). This yawing moment creases a positive yaw rate

which creates a positive rolling moment ( ). This increases the roll angle and

tends to increase the sideslip, thus making things worse.

Dutch Roll (D): The Dutch roll dynamics is represented using a conjugate pair. It is

a damped oscillation that couples into the roll channel. The frequency of the Dutch

roll dynamics is similar to that of the longitudinal short period mode. This mode is

a roll and yaw to the right, followed by a restoration towards the equilibrium point.

This is followed by an overshoot towards the left and then another restoration.

Fig 3.3. Typical Pole Position for the Lateral/Directional Motion

The characteristic equation for the lateral dynamics is

3. 15

where is the roll time constant, is the spiral model time constant, and and are

the damping and natural frequency of the Dutch roll dynamics. In order to simplify and

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61

decouple the system, the coupling effects of the roll dynamics on the yaw dynamics

are not considered. As a result and

are zero in the model. Consequently, the

state space representation of the lateral dynamics is

3. 16

The dynamics of the entire system, if coupling is not taken into account, is described using

the diagonal sub matrices represented through the and the matrices.

The roll dynamics of the system is represented using the matrix. The simplified

representation of the roll dynamics leads to

3. 17

The characteristic equation for this system is:

3. 18

Comparing Eq. 3.18 with

, the time constant for the roll dynamics is

.

The spiral mode degrades due to the integrator and also due to the simplification of the

system.

The top left section of matrix in Eq. 3. 13, describes the Dutch roll dynamics. The state

space representation of the Dutch roll dynamics is

3. 19

The characteristic equation for this subsystem is

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62

3. 20

Comparing the coefficients with the standard form leads to

3. 21

3. 22

The natural frequency for the Dutch roll dynamics is determined through the derivative

. This means that for stability and therefore consequently it follows that .

It is therefore concluded that the vehicle is statically stable about the lateral axis. This

means that, due to a disturbance, if there is a sideslip angle, a counter moment is

generated to restore the value to its trim value.

3.3 Model Based Forces and Moments Description

The motion of the glider is obtained by solving the equations of motion presented in

Section 3.2 and reflects only the actual motion of the vehicle when the modelled forces and

moments also approximate the actual forces and moments on the vehicle. Increasing the

details in the forces and moments is not the best approach, as the required computation

time increases to simulate the vehicle dynamics and the implementation process is time

consuming. This section discusses the forces and moments that act on the vehicle and the

way in which they are modelled. The aerodynamic forces and moments are treated

separately in Section 3.4.

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3.3.1 Weight Model

Gravity acts on the body in the Z-direction and the magnitude is , where is the mass

of the glider and the gravitational acceleration. The influence of gravity in the body is

expressed as

3. 23

As the eight acts through the centre of gravity of the glider it generates no moment. The

gravitational acceleration is modelled as a function of the glider’s altitude. It is written

as:

3. 24

where m is the mean radius of the earth. This value is in accordance with the

International Union of Geodesy and Geophysics [150].

Traditionally the mass of an aircraft is split into several parts, for example

and . As the vehicle considered in this study is a glider, and

. This means that the moments of inertia, and , and the products

of inertia, , under nominal conditions do not change for the duration of the simulation.

3.3.2 Environment Model

Aerodynamic forces and moments are included in the simulation model.

Gas Properties: The gas properties are modelled as defined by the international standard

atmosphere, ISA, or ISO 2533 [151, 152]. Within the troposphere the temperature and the

pressure are modelled by

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64

3. 25

3. 26

where is the temperature measured in Kelvin, is the altitude (in meters) and is the

pressure measured in Pascal. The temperature and the pressure relationships between an

altitude of 11km and 80km are:

3. 27

3. 28

The density , sonic speed , dynamic pressure and Mach number are given by

3. 29

3. 30

3. 31

3. 32

where is the speed of the glider and is the gas constant ( is ratio of specific

heats for air).

The equations presented in this section are valid under the assumption that the gas being

considered is a perfect gas. As real gas effects such as vibrational oscillation will be small

at Mach 8, the perfect gas assumption is appropriate for this research.

3.4 Aerodynamic Model

Aerodynamic forces and moments depend on variables such as airspeed, air density,

rotational rates of the aircraft, angle of attack, angle of sideslip and the deflection of the

control surfaces. The aerodynamic forces and moments are described using

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65

3. 33

3. 34

The gravimetric properties utilised in the force and moment equations are obtained from

[10]. The platform area is, meters, the characteristic length is meters

and the reference length is, meters.

The most common way to parameterise the aerodynamic model is with the use of Taylor

expansions. Multivariate splines [153] are also used to define the model. However, splines

are used when the resulting model needs to include large nonlinearities. The model

employed in this study is linear and therefore Taylor expansions are used. As an example,

the pitching moment is expanded as

3. 35

where

, etc. are non-dimensional coefficients. A more detailed method for

calculating the values of these parameters is presented in Section 3.4.1. These values are

also called the stability and the control derivatives of the system [148, 154].

3.4.1 Model Parameters

Different methods may be used to develop the aerodynamic data (the control and stability

derivatives) including CFD computations, wind tunnel tests, handbook methods based on

empirical data, or actual flights. The GHAME aerodynamic model is developed using

actual data from vehicles such as the Space Shuttle Orbiter, lifting body type aircraft, as

well as theories such as the modified Newtonian impact flow method. The lateral and

directional derivatives for Mach numbers higher than 8 are taken exclusively from Space

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Shuttle data. Below this speed, the lateral/directional derivatives are taken as a

combination of the space shuttle and a swept double delta configuration [7, 155].

Expanding the aerodynamic force and moment coefficients using the Taylor expansion,

only the linear terms are taken into account. The derivatives are a tabular function of Mach

number and the angle of attack. Therefore, the linearised parameterisation of the

aerodynamic forces for a conventional aircraft are

3. 36

However, the GHAME model uses the lift and the drag as force components in the X and

the Z direction respectively. Therefore, the corresponding linearised equations for the

aerodynamic forces for the GHAME model are

3. 37

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67

The equations presented for the forces and the moments, although the Taylor expansion of

the model, are not a representation of a linear model. For example, is a function of

and is once again multiplied by . The same is the case for the drag coefficient. The other

control and stability derivatives are linear in terms of remaining states but are nonlinear

with respect to Mach Number and . The lookup tables of the GHAME model has been

used as a representative set of coefficients for the hypersonic glider vehicle presented in

this research. The aerodynamic model with all of the above mentioned moments and the

forces is shown in Fig 3.4.

Fig 3.4. Motion Variables [156]

A table of the motion variables is shown in Table 3.1.

Table 3.1. Overview of Motion Variables of an Airborne Platform

Variable Symbol Coordinate System

Velocity Body fixed

Rotational Rate Body fixed

Position Earth fixed

Attitude Body fixed with respect to

geodetic system

Force Body fixed, stability

Moment Body fixed

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Every model needs to be validated. There have been multiple studies in validating the

aerodynamic model of the GHAME vehicle [7, 15]. Therefore, the model validation section

is not presented in this study and the aerodynamic look-up tables are used as a

representative set of tables for the hypersonic glider. The graph for the lift coefficient is

shown in Fig 3.5 in order to show an example of the different flow features that are seen in

the aerodynamic coefficients of a hypersonic vehicle throughout the whole flight envelope.

Fig 3.5. Lift Coefficient as a Function of Mach Number and Angle of Attack

Fig 3.6. Drag Coefficient

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Fig 3.7. Pitching Moment

Some of the key features of the lift coefficient that are evident in all of the aerodynamic

coefficients are as follows:

High Mach number independence: As the freestream Mach number increases, the

governing equations determine the coefficient of pressure, which can be integrated

over the surface to find the lift and drag coefficients. This means that the

coefficients are independent of Mach number when the value is sufficiently high.

This behaviour is seen in the pitching moment (Fig 3.7), drag (Fig 3.6) and also the

lift coefficient (Fig 3.5).

In the subsonic and supersonic regime, it is seen that the lift coefficient is nonlinear

in nature and the values are changing considerably as functions of Mach number

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and angle of attack. This is consistent with characteristics at the lower speed

regimes.

However, it is important to note that the discontinuities in the pitching moment

and drag coefficients that arise from shock induced separations are not included in

the aerodynamic model.

3.4.2 Generic Hypersonic Aerodynamic Model Example (GHAME) Model

The Generic Hypersonic Aerodynamic Model Example (GHAME) model is for a

hypothetical aircraft that is to be used for research purposes in order to carry out trajectory

studies and control systems design. The main aim behind the development of the GHAME

model was to model a single stage to orbit (SSTO) [6]. The GHAME model is shown in Fig

3.8. The fuselage of the GHAME model is a cylinder 20 feet in diameter and 120 feet long.

The control surfaces on the vehicle are modelled using 10 degree half cones. These

surfaces are connected to the fuselage at the halfway point.

Fig 3.8. GHAME Model [6]

The GHAME model is based on flight test data from the Space Shuttle, the X-24C and

theoretical data from a swept double-delta configuration and a 6 degree half-angle cone

using Newtonian Impact Flow method. The model consists of realistic data of

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aerodynamic coefficients for the hypersonic flight regime. The scaled down version of the

GHAME version that represents the glider in this research had the following mass

properties:

Table 3. 2. GHAME Mass Properties

Platform area

Characteristic length

Reference length

Mass

3.5 Actuator Model

For the control surfaces, rudder and elevens, second order actuators with rate and

deflection limits are included in the simulation model. The transfer function for the

actuator model is:

3. 38

where, is the natural frequency of the actuator model, is the damping of the

actuator. The SIMULINK block diagram for the second order nonlinear actuator model is

shown in Fig 3.9.

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Fig 3.9. Second Order Actuator Model

In Fig 3.9 is the generated controller and, due to the nonlinear actuator model,

the actual control deflection is given by . The figures associated with the actuator

model are outlined in Table 3.3. The physical nature of the model is understood from the

point of view of the integrators and their corresponding limitations. The right integrator

limits the actuator position. This property of the model is necessary due to the subcritical

damping. This is due to the fact that even if the commanded actuator position is within the

limit, a transient overshoot might occur due to oscillations. The limit of 'Integrator 1'

represents the actuator velocity. This attempts to mirror reality as the rotation speed of the

electric motor of the servo is subjected to constraints. This rate limitation also has an effect

on the closed loop dynamics. This is particularly the case when there are high amplitudes.

In these cases, the rate limit could lead to instability for high amplitude manoeuvres, while

small amplitude manoeuvres can be stabilised very well. Taking the rate and deflection

limits into consideration it should be noted that the behaviour of the actuator model is

nonlinear and the transfer function presented in Eq. 3.38, except for small amplitudes,

does not fully represent the actuator model shown in Fig 3.9. The aerodynamic and control

derivatives are indirectly dependent on the control surface deflections. The control surface

deflections change the angle of attack and in turn the Mach number of the vehicle. The

forces and moments the vehicle experiences change as a result of the change in the angle

of attack and Mach number.

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Table 3.3. Actuator model Parameters

Parameter Unit Perturbation

Elevon deflection limit [deg] -20 to 20

Rudder deflection limit [deg] -20 to 20

Elevon rate limit [deg/s] -400 to 400

Rudder rate limit [deg/s] -400 to 400

Damping ratio 0.7

Natural frequency [rad/s] 150

3.6 Sensor Model

Global Positioning System (GPS) and Inertial Navigation System (INS) are used for

positioning and attitude determination for hypersonic vehicles [157]. The governing

equations of INS are based on Newton’s second law. The INS implementation presented in

this report is a system wherein the GPS is also integrated. The principle of operation of an

INS is shown in Fig 3.10.

Fig 3.10. Principle of Operation of INS [1]

The equations used to implement the INS are covered in more detail in Zipfel [1]. A brief

overview of the implementation of the sensor equations is presented in this section.

The governing equations of INS navigation are based on Newton’s Laws of Motion. The

velocity of the vehicle centre of mass (c.m.) with respect to the inertial coordinates

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and also the vehicle's position with respect to an inertial reference point I, in inertial

coordinates are represented as

3. 39

3. 40

where

is the specific force measured by the accelerometers, and is the

gravitational acceleration in inertial coordinates. The integration of the angular velocity of

the vehicle with respect to the inertial frame, expressed in inertial coordinates , helps

to determine the rotation tensor of the body frame with respect to the inertial frame

. These equations of operations of the INS are outlined in Fig 3.10. In total, six

differential equations are solved for the basic navigation solution. An additional four

quaternion differential equations calculate the attitude angles.

The INS errors are denoted as . indicates the perturbations and the caret indicates the

computed values. The inertial acceleration experienced by a body is equal to the

specific force acting on the vehicle plus the gravitational acceleration . This is

expressed as

3. 41

The inertial coordinate frame is chosen. However, it is important to note that the forces

that are measured are measured in the body frame.

3. 42

The values in Eq. 3. 42 are the true values. These values that are provided to the guidance

computer are the computed variables (corrupted with INS errors). These errors are also

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called perturbations, and are the difference between computed and true values. The

component perturbation of a vector is given by

3. 43

where is the computed value of the vector and is the true value of the variable. The

errors in position, velocity, specific force and gravitational acceleration is modelled using

Eq. 3.43. is the tilt tensor of the INS. This tensor expresses the relationship between the

true inertial frame and the computed inertial frame of reference, . All the information

provided by the INS is presented in the computed frame of reference. Therefore, after

introducing these coordinate system conventions, the component perturbation is written

as:

3. 44

where

3. 45

A tilt tensor under small perturbation consists of a unit tensor and a skew symmetric

tensor:

3. 46

The perturbation tensor of rotation is expressed using small angle components:

3. 47

This skew symmetric matrix can be reduced to a tilt vector

. The

attitude perturbation is represented using a tilt vector.

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3. 48

The relationship between the transformation matrix perturbation and the tilt rotation

tensor is

3. 49

3. 50

The error equations are summarised in Table 3.4.

Table 3.4. Summary of Equations

Variable Perturbation

Velocity

Position

Specific force

Gravitational acceleration

Coordinate transformation

A diagram of the equations in Table 3.4 is shown in Fig 3.11.

Fig 3.11. Propagation of errors [1]

An in-depth derivation of the individual error equations is presented in [15] and is

therefore not covered here in detail.

Additional instrument errors need to be modelled if the effects of the INS system are to be

correctly represented in the high fidelity nonlinear simulation. The primary sources of

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errors that remain after factory and prelaunch calibrations have been included are the

random bias and noise, scale factor error and misalignment. These errors also apply to

gyroscopes.

3. 51

where is the random bias and noise,

is the scale factor error matrix and

is the misalignment matrix. Therefore, the output of the accelerometer is the measured

specific force in body coordinates.

3. 52

which is a combination of the true value

and the instrumenterror

. The errors

associated with the gyroscope take a similar form.

3. 53

where

is the random bias and noise vector,

is the diagonal scale factor error

matrix,

is the skew symmetric misalignment matrix and

defines the coupling

of the specific force. The output of the gyroscope is the measured angular rate in body

coordinates in the form

3. 54

which is comprised of the true value and the instrument error term .

Along with the error equations, it is also important to correctly initialise the INS errors.

Imperfections at the beginning of a simulation introduce initial uncertainties into the nine

states of position, velocity and attitude. These transfer alignment errors are known using

their covariance matrix , a 9 x 9 matrix of the variances and covariances of the nine-state

vector . The following equations helps initialise the initial INS error state.

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3. 55

where defines the unit standard deviation.

Sensors are compared with each other on the basis of certain performance factors, such as

bias and scale factor stability and repeatability or noise (e.g. random walk) [158, 159]. One

of the main determining factors in choosing sensors is often the budget of the project. For

this study, the sensor that is implemented is the ten degree of freedom inertial sensor –

ADIS16480 [160]. The particular sensor is chosen so as to represent a sensor module. This

is not to say that this particular sensor would be on-board a hypersonic glider vehicle. The

characteristics of this sensor are given in Table 3.5 and Table 3.6. The values presented are

1-sigma values.

Table 3.5. Performance Definition of ADIS16480 - Accelerometer

Misalignment

(rad)

Scale Factor (parts) Bias (g)

1*10-4 0.0017 16mg

Table 3.6. Performance Definition of ADIS16480 – Gyroscope

Axis to Axis

Misalignment

(rad)

Random

Walk

(rad/√s)

Scale

Factor

(parts)

Bias

(rad/s)

8.73*10-4 2.42*10-7 0.0017 3.03*10-5

For the trajectory study it is assumed that the Global positioning System (GPS) will

augment the INS. This is important as it helps in correcting the sensors errors in real time

and therefore makes sure that the position and the velocity errors in the states remain

bounded. Some of the main criteria that differentiate sensors are described as follows:

Gyro Bias: If the system operates unaided, the gyro bias indicates the increase of angular

error with time (in deg/h or deg/s).

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Gyro Scale Factor Error: This is an indication of the angular error which occurs during

rotation. E.g. with 300 ppm scale factor error (=0.03%) the angular error is in the vicinity of

after each revolution.

Misalignment: A misalignment between the gyro axes causes a cross-coupling between

the measured axes. A misalignment of 0.1 mrad inside of the system leads to a roll error of

during one revolution around the yaw axis.

Accelerometer Offset: an offset on the accelerometer leads to an error during alignment,

i.e. determination of initial roll and pitch angle.

Random Walk: this value is given in deg/sqrt(hr), shows the noise of the gyro. The higher

the noise the more noise is measure on the angular rates and on the angles.

The use of sensors introduces time latencies into the system. Therefore, sensor delays are

added into the simulation model used as a part of this study. A 22ms sensor delay is

added in the study carried out by Gregory in [66] (6ms on surface commands and 16ms on

sensor outputs). This was for the NASA AirSTAR program. A15ms delay is included to

model sensor delays in the present study.

Estimates of sensors error can be made using a Kalman filter. The Kalman Filter is mainly

used to estimate states that can only be observed indirectly or inaccurately by the system

itself. The Kalman filter works well in practice as well as in theory. It is a filter that

minimizes the variance of the estimation error [161, 162].

3.7 Sources and Modelling of Uncertainties

The controllers need to perform well in the presence of uncertainties that the vehicle

encounters over the entire trajectory and the operating environment. It is however

impossible to accurately model all the uncertainties that are present, therefore uncertainty

models play an important role in controller design and analysis.

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3.7.1 Control Surface Uncertainties

The trajectory of the glider studied in this thesis covers altitudes lower than 55km. This

means that this part of the flight occurs when the vehicle has re-entered the earth’s

atmosphere. It is important to note that once the vehicle has gone through the extreme

conditions of the sub-orbital ballistic trajectory, it is difficult to estimate the state of the

control surfaces. The reduced control system effectiveness corresponds to a possible

failure or degradation of the control surfaces, possibly due to component failures

experienced by the vehicle during ascent and descent. In light of this, it is important to test

the robustness of the controllers to various uncertainties. The different uncertainties that

are considered are:

I. The elevons are only being able to deflect to 30% of the commanded deflection.

II. There may be an uneven deflection of the elevons – the reduction in the

functionality of the control surfaces will not be identical for both the right and the

left elevons. Hence, it is important to consider the case wherein there is a different

level of functionality of the control surfaces.

III. Rudder failure – this test case investigates the scenario wherein the rudder is stuck

at a specific angle of deflection.

3.7.2 Aerodynamic and Gravimetric Uncertainties

Longitudinal

The aerodynamic errors that are considered for this analysis are the ones that occur in real

time and are unpredictable in nature. These errors may stem from structural degradation

(as a result of the mission and its trajectory), increasing radius of the leading edges or

thermal expansion effects which change the flow field enough to affect the model

parameters and structural distortion [10]. These errors are classified as parametric

uncertainties. There is also the case that the errors in the aerodynamic coefficients stem

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from the errors associated with CFD or wind tunnel measurements. This encompasses

differences between the computed aerodynamic coefficients and the real time

aerodynamic coefficients experienced by the glider. This class of errors can also arise as a

result of atmospheric disturbances along the descent trajectory. It is therefore important to

test the robustness of the controllers in the presence of these uncertainties.

For testing the performance boundaries of the PPC and the adaptive controller, the

uncertainties of the coefficients are put into an error vector [21, 119, 163]:

3. 56

where

are the multiplicative uncertainty of the longitudinal

stability of the model, the lift coefficient, elevator control derivative, the pitch damping,

the uncertainty in the drag coefficient and the elevator pitching moment derivative of the

model. The nominal values of the error vector are given by

3. 57

In the case of uncertainties being present, each element in the error vector is set to a

minimum and maximum value. These values are given in Table 3.7.

Table 3.7. Error Bounds for Error Vector - Time Invariant

Component of Error

Vector

Error Bounds

These errors are time invariant errors. The error margin is defined at the start of the

simulation and the error margin remains constant for the entire trajectory.

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Time varying changes to the aerodynamic coefficients are changing flight to test the

robustness of the controllers to real time uncertainties. Over the duration of the

simulation , Eqs.3. 58-3. 63 govern the time dependent changes in the coefficients.

3. 58

3. 59

3. 60

3. 61

3. 62

3. 63

where is the simulation time, and is the total simulation time. This simulation assesses

the system against slow-moving time-based aerodynamic errors and, as a result, provides

a further test case for the system.

Lateral

This section outlines the implementation of the corruption of the lateral aerodynamic

coefficients. For testing the performance boundaries of the PPC and the augmented

controller, the uncertainties of the coefficients are defined using an error vector [5, 119]:

3. 64

where are the multiplicative uncertainties of the aerodynamic and control derivatives.

The nominal values of the error vector are given by

3. 65

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Each element in the error vector is set to a minimum and maximum value. These values

are given in Table 3.8 below. The distribution that is presented in Table 3.8 is a uniform

distribution and the bounds that are presented are the maximum and minimum values of

the errors. The error bounds are defined based on the guidelines presented in [21].

Table 3.8. Errors Bounds of Error Vector (Time Invariant)

Component of Error

Vector

Error Bounds

These errors are time invariant errors which are defined at the start of every simulation

and remain constant for the whole simulation. The error margin is defined at the start of

the simulation and the error margin remains constant for the entire trajectory.

Gravimetric Errors

Before the pull up manoeuvre starts the gravimetric state of the vehicle cannot be

guaranteed, but can only be estimated. The structural load experienced by the vehicle can

cause changes that need to be taken into account when analysing the robustness of the

baseline and the adaptive controller. The errors considered here are:

I. Moment of Inertial Uncertainties -

II. Mass uncertainty - up to reduction in mass (the variation in the mass is 0.9 to

1.0 times the original mass of the vehicle)

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These uncertainties are defined at the start of the Monte Carlo simulations with a uniform

distribution. The 20% errors in the moment of inertia tensor along with the 10% mass error

are large. For civilian aircraft, such large changes are not expected due to the fact that

structural damage to the aircraft can have a severe effect on the ability of the aircraft to

stay airborne. Whereas, for the hypersonic case, after the ascent and descent phase there is

no way to definitively predict the state of the vehicle. Therefore, from a modelling and

control point of view, the task of designing a controller for a flight envelope becomes

significantly more challenging. As an example, in [164], for a transport class vehicle, in this

case a B747-100/200, moment of inertia uncertainties are added in the model. 5%

uncertainties are considered, which is four times less what has been considered in this

study.

Combination of Errors

The aforementioned uncertainties do not necessarily occur individually during flight.

Therefore the effects of a combination of errors and the controller’s robustness to these

uncertainties are tested. The following errors are present simultaneously in the simulation:

I. Moment of Inertia Uncertainties

II. Mass Uncertainty

III. Aerodynamic Uncertainties

IV. Rudder Failure-leading to the lack of tracking ability in the -channel.

V. Reduced deflection of the elevons.

VI. Asymmetric deflection of the elevons

The results presented for this scenario are obtained using Monte Carlo Simulations.

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3.8 Monte Carlo Simulations

The main aim of the Monte Carlo analysis is to test and analyse the controller

performance, stability and robustness to uncertainties and control surface failures during

the pull up and the roll angle manoeuvre. Statistical information is gathered after the

testing in order to be analysed to assess controller performance. The result that comes as a

result of these simulation runs is a comparative study between the baseline and the

augmented controller. Based on the results, either the performance of the baseline or that

of the augmented controller is more desirable.

Monte Carlo simulations become of paramount importance when there are various

sources of stochastic errors. Examples of sources of stochastic errors are INS errors (like

random walk, random bias and white noise), aerodynamic uncertainties, control surface

uncertainties and gravimetric uncertainties. To initialise these uncertainty variables, two

distributions are utilised: Gaussian [165] and uniform distributions [166].

3.9 Simulation Platform

Once the controllers have been implemented it is important to choose the right platform to

test the controller. This subsection introduces and outlines the main features of the

simulation framework Computer Aided Design of Aerospace Concepts (CADAC). As

outlined in [13, 14], the main reason for choosing CADAC++ written in C++ is that it

outperforms Matlab/SIMULINK© based simulations in programming, execution speed

and cost.

CADAC was jointly developed by the U.S. Air Force and the University of Florida.

CADAC++ has been used as a test bed for missiles, aircraft: unmanned aerial vehicles

(UAVs) and spacecraft. Its modular structure enables the reuse of subsystem models and

its well defined interfaces allow integration into higher level simulations.

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High fidelity simulations use random variables to model noise, disturbances and uncertain

phenomena. This means that the simulation platform should provide Monte Carlo

capabilities. This is needed so as to carry out a stochastic analysis, noting that many

repetitive runs need to be executed, each drawing a different value from the distribution.

To initialise a Monte Carlo analysis, the keyword MONTE is used in the input file. Along

with the key word, a random seed is also provided for the initialisation of the random

variables.

3.10 Conclusion

This chapter presents the six degrees-of-freedom nonlinear model of the dynamic

behaviour of the hypersonic glider. Thereafter, the linearised model is presented. This is

used for controller synthesis for the remainder of this thesis. This chapter also presents the

weight model, environment model and the aerodynamic model. A brief outline of the

simulation platform utilised is presented.

The control laws can change the dynamic behaviour of the glider. The model presented in

this chapter serves as a basis for the flight control law design. Performance and robustness

of these control laws designed are tested in CADAC++.

The mathematical frameworks and the derivation of the baseline and the augmented

controller are given in chapter 4.

THEORETICAL FRAMEWORK AND IMPLEMENTATION

87

4. THEORETICAL FRAMEWORK AND IMPLEMENTATION

This chapter presents the theoretical background about the pole placement controller

(PPC) and adaptive control. Full state feedback control architectures are presented for

the longitudinal and lateral/directional dynamics of the glider. This is followed by the

presentation of the derivation of an adaptive controller. The proofs of stability of the

full adaptive control are mostly excluded, but can be found in [41, 82, 128, 133]. Section

4.1 presents a generic autopilot structure that has been employed in this study. Section 4.2

explains the underlying theory of the PPC. This is followed by theory on adaptive

controller. Section 4.3 presents a survey of adaptive control, followed by Section 4.4 which

presents the derivation of the adaptive augmented controller that accounts for the

matched and unmatched uncertainties in the system. The performance and the robustness

metrics are outlined in Section 4.5 and 4.6 respectively. Section 4.7 presents the

performance and the stability bounds of the augmented controller. Section 4.8 presents

the validation of the modified augmented controller.

This chapter presents the first application of a modified augmented control law to a

hypersonic glider configuration, which forms the edifice of this research. The modified

control law is used to cancel out uncertainties in a LTV state feedback non-minimum

phase system.

4.1 Introduction

The general autopilot structure to control the longitudinal and lateral/directional

dynamics of the glider is shown in Fig 4.1. The overall block diagram consists of two main

subsections: (i) an outer guidance loop which provides the controller with the commanded

signal based on the desired trajectory and (ii) an inner loop stability augmentation and

attitude controller to track the guidance commands and reject disturbances. The block


88

diagram shown in Fig 4.1 shows a generic autopilot structure utilised to control the

longitudinal and lateral/directional dynamics of the hypersonic glider.

Fig 4.1. Autopilot Structure for Hypersonic Glider

The components of the autopilot structure are:

Desired Trajectory: The desired trajectory block is a function of the trajectory of the

vehicle during descent and also depends on the mission that is being carried out.

For the case of the glider considered in this thesis, the test trajectory chosen for this

research is shown in Fig 1.2. This study focuses on trajectory control from the point

where the pull up manoeuvre followed by a pulsed roll angle manoeuvre.

Guidance System: The role of the guidance system is to convert the desired

trajectory into a commanded flight path angle, , for the longitudinal controller.

For the case of the lateral/directional dynamics, the guidance loop converts the

lateral trajectory into a commanded bank, , and sideslip angle, .

Controller: This part of the autopilot consists of the inner loop controllers. It

expresses a functional relationship between the error signals (vehicle state offset)

relative to the commanded signal, and the actuation command to the elevons and

the rudder. The controller in this study operates at 100Hz. The sampling frequency

of the inner loop controller is a function of the natural frequency in the control

channels. Sampling frequency is chosen to be significantly higher than the natural


89

frequency in the flight path channel, which has a maximum frequency value of

15Hz.

Plant Model: This part models the vehicle. In this case it is the hypersonic glider

modelled using a representative set of gravimetrics and aerodynamics using look

up tables.

4.2 Pole Placement Control

State feedback PPC is a control methodology employed to stabilise the system [167, 168]

(see Fig 4.2) The state space representation of this system is (without the feedforward

matrix, )

4. 1

where for the longitudinal case for the roll and sideslip angle

dynamic is the system matrix, for the longitudinal case for the roll

and sideslip angle dynamic is the control gain matrix, for the longitudinal

case for the roll and sideslip angle dynamic is the output gain matrix,

for the longitudinal case for the roll and sideslip angle dynamic is the

state vector, is the control input, is the system output, is the number of states of the

system and is the number of control inputs. The plants considered in this thesis are all

Single Input Single Output (SISO) systems.

Fig 4.2. Block Diagram State Feedback for Stability


90

In order to stabilise the system, a feedback gain matrix is calculated. With the inclusion

of the feedforward gain, , the control law is:

4. 2

where is the commanded signal. Substituting the control law into Eq.4. 1, the state space

representation of the closed loop system is:

4. 3

This controller structure is used as the baseline controller to track the flight path , roll ,

and sideslip angle . The derivations of the feedback and feedforward gains are presented

next. The feedforward gain for the state feedback is defined in a way as to achieve steady

state tracking. In steady state, . This leads to the following equations:

4. 4

Now in order for the steady state error, to be zero the following equation needs to be

satisfied:

4. 5

which leads to:

4. 6

The one assumption that needs to be fulfilled is that the numbers of inputs are equal to the

number of outputs. Therefore the matrix is square. As the matrices and

are full rank matrices, solving for leads to the following equation for the feedforward

gain:

4. 7


91

Before implementing a PPC, the system must be controllable. The basic equation for the

controllability test is as follows:

4. 8

The system is considered controllable when the matrix is full rank.

where is the order of the system. Controllability of a system plays a key role in control

problems. Some of these problems include the stabilisation of unstable systems using

feedback control or optimal control. The controllability of a system means that an external

input has the ability to move of a system from any initial state to any other final state in a

finite time interval [169]. On the other hand, if any of the states are independent of the

control signal, then is becomes impossible to control this state variable and therefore the

system is uncontrollable. As controllability plays an important role in pole placement

control it is important to check for controllability of the mathematical model of the system

[167].

For the case of the longitudinal dynamics the controllability matrix is


92

is a full rank matrix, signifying controllability. The same procedure is carried out for

the roll and sideslip angle controller. The corresponding controllability matrices are full

rank.

4.2.1 Baseline Controller – Longitudinal Controller

Linearised plant models are used to describe the longitudinal dynamics. This plant model

is used to carry out the gain calculations for PPC. The aerodynamic and control

dimensional derivatives of the model are the pitch damping , static longitudinal

stability , normal force dependency on the angle of attack and the control moment

derivative . The non-dimensional derivatives are stored in look-up tables and are

converted to the dimensional values in real time during flight.

The flight path angle dynamics are outlined in Section 3.2.2 (see Eq. 3. 11). For the case of

controlling the trajectory, a flight path angle controller is implemented. The flight path


93

angle controller structure is the structure shown in Fig 4.2. This controller uses pitch rate,

pitch angle and flight path angle as the states. Substituting the plant model (see Eq. 3. 11)

into Eq. 4. 9 in order to calculate the feedback gains leads to

4. 9

This equation leads to the gain formula for the flight path angle controller

4. 10

where

The flight path angle dynamic is of third order. Therefore, the desired dynamic of this

system are defined using and , which are scheduled as a function of dynamic

pressure. is the desired closed loop natural frequency, is the damping ratio and is

one real pole location which is required to calculate the dynamic gains of the controller.

The gains are calculated at the controller loop frequency in the simulation or during flight

real time.

4.2.2 Baseline Controller – Lateral/Directional Controller (SISO)

This section of the report outlines the derivation of the pole placement controllers for the

lateral and directional dynamics. A general state space model has the following form as in

Eq. 4.1. The block diagram of controllers follows the controller structure (feedback plus


94

feedforward) shown in Fig 4.2.The gains of the PPC for the yaw and roll controllers are

determined using Eq. 4. 9. The systems (see Eq. 3. 17 and Eq. 3. 19) are of second order, so

the desired dynamics are represented using the second order equations:

4. 11

where represents the desired damping ratio of the system and is the desired natural

frequency. Substituting the state space models for the roll and the yaw controllers, the

feedback gains are presented. For the case of the glider as the conditions are changing

rapidly the desired characteristics are a function of a trajectory parameter; in this case the

dynamic pressure (see Fig 6.11). The feedback gains for the side slip controller are

4. 12

4. 13

and the feedback controller gains of the roll controller are

4. 14

4. 15

The feed-forward gain for both the controllers is

4. 16

where is invertible and the closed loop system matrix.


95

4.2.3 Stability and Robustness Analysis of Baseline Controller

As the baseline controller used to control the hypersonic glider is a PPC, linear tools of

analysis are utilised to analyse the stability of the system. At every simulation time step,

the controller gains are calculated. This means that at every time step the controller is

valid and the system matrices, and can be read. Consequently, for every set of

controller gains and the linear representation of the system, pole zero locations are

calculated. The pole positions characterise the stability of the system. For the robustness

analysis, the time delay margin (see Section 4.6 for details) is used.

4.3 Survey of Adaptive Control

An adaptive controller can change its behaviour in response to changes in the dynamics of

the process and the disturbances. Research into adaptive control was very active in the

1950s. The research was motivated as a way to design autopilots for high performance

aircrafts. The 3 main schemes for parameter adaptive control: gain scheduling, model

reference adaptive control and self-tuning regulators. A summary can be found in [170-

172]. The key problem is to find a convenient way of changing the regulator parameters in

response to changes in process and disturbance dynamics. The schemes differ only in the

way the parameters of the regulator are adjusted. The shortcomings of these three

adaptive schemes are addressed using adaptive control [41]. The main advantages of

utilizing adaptive control are outlined in Section 4.4.

4.4 Adaptive Control

When designing flight control systems, it is essential to provide and guarantee closed-loop

stability, adequate command tracking performance, as well as robustness to model

uncertainties, control surface uncertainties and environmental disturbances. The previous

section presented the baseline controller. This baseline controller stabilises the system.

Therefore, the task of the adaptive controller is to cancel the uncertainties present in the


96

system. In the presence of uncertainties, the performance of the baseline controller will

deteriorate. However the key question is: can the closed loop performance of the baseline

controller be restored, while operating in the presence of uncertainties? In this domain, adaptive

controllers are useful.

4.4.1 Background

adaptive control has been developed with aerospace applications in mind and has been

successfully applied for flying applications [42, 61, 62]. The use of adaptive control

allows for fast adaptation while at the same time maintaining robustness. Fast adaptation

leads to control chattering, however, the output of the controller is limited in amplitude

and frequency. This is due to the use of the low pass filter placed at the output of the

controller, and is utilised to make the controller act within the control channel bandwidth

[132].

A modified control law that compensates for both matched2 and unmatched uncertainties

has been implemented for the hypersonic glider model. The presence of unmatched

uncertainties signifies that the uncertainties lie outside the span of a linear system’s

control input matrix, Examples of unmatched uncertainties include [173-175]:

High frequency unmodelled dynamics

Actuator dynamics

Structural vibrations

Measurement noise

Round off error and truncation

Sampling delay

Cross-coupling

2 Matched uncertainties are uncertainties that enter through the control channel


97

Therefore, the compensation of the matched and unmatched uncertainties present in the

system is critical to achieve the desired performance and robustness of the hypersonic

glider. Benefits of the adaptive control methodology are its fast and robust adaptation.

This allows for desired performance achievement in both transient and steady state

tracking. The high estimation gains, previously mentioned that are part of adaptive

control, allow for the compensation of the undesirable effects of rapidly varying

uncertainties and significant changes in system dynamics. High estimation rates are also

critical to achieve guaranteed transient performance without resorting to gain scheduling

of the control parameters, persistency of excitation or control reconfiguration. Therefore,

in sum, this adaptive control scheme is chosen for the following reasons [41, 65, 176, 177]:

Decoupling of the rate of adaption and robustness

Guaranteed fast adaption, limited only by hardware constraints.

Guaranteed, bounded away from zero time delay margin

Guaranteed transient performance for a system’s input and output signal,

without high gain feedback or enforcing persistent excitation type assumptions.

Uniform scaled transient response dependent on changes in initial conditions,

unknown parameters, and reference input.

Suitable for development of theoretically justified verification and validation

tools for feedback systems.

4.4.2 Problem Formulation

The formulation of the basic derivation of the adaptive controller closely follows the

derivation of Mr Zhongjie Wang presented as part of the subject Nonlinear and Adaptive

Flight Control held at the Technical University Munich (TUM) – Institute for Flight System

Dynamics (FSD), Munich, Germany [178]. Before the controller design is carried out, a

general formulation for a nonlinear model is needed. The general formulation for a

nonlinear system is as follows:


98

4. 17

4. 18

where is the state vector, is the reference command, is the controller parameters, is

the plant parameters and is the time. In order to better represent and analyse nonlinear

systems, it is important to convert the system representation according to the following

guidelines:

1) Separate the linear part and the nonlinear part

2) Determine the matched and unmatched components of the nonlinear part.

3) Resolve the linear component into an assumed part plus an unknown uncertainty

4) Separate the input into input affine and non-affine part

5) Separate the input-and output disturbances

6) Parameterize control effectiveness

Taking into consideration these guidelines, the new general formulation of a general

nonlinear state space model is as shown in Eq. 4. 19. The equivalent representation that

exactly describes the system is:

4. 19

where:

= assumed system matrix

= system matrix uncertainties

= assumed input matrix

= input matrix uncertainties

= control effectiveness degradation

= matched input affine nonlinearity

= unmatched nonlinearities

= disturbances

= output nonlinearity


99

= assumed output matrix

= output matrix uncertainties

= matched state nonlinearity

= input offset

Due to the lack of general solution to nonlinear systems, restrictions are introduced to

illuminate the class of nonlinear systems that are controllable when using the Piecewise

Constant Update Law. The restrictions are as follows:

Uncertainties in the output equation are not considered.

Input matrix is known, i.e. .

Unmodelled internal dynamics of the system should be included in the

considerations to represent the controlled plant more completely and accurately.

With the aforementioned considerations, and some derivations, the system is

4. 20

where represents the matched component of the uncertainties and are the

unmatched uncertainties. Once the plant dynamics has been defined as in the equation

above, the control objective can be defined. The control objective is to render the output of

the stated plant to follow the desired system dynamics (to cancel out the matched and the

unmatched uncertainties):

4. 21

where is given as bounded piecewise continuous reference signal given to the control

system from the guidance computer.

To realise such a control objective, some further assumptions about the uncontrolled plant

dynamics are made, while applying the piecewise constant adaptation method. Current


100

research effort is aimed at relaxing these limitations in order to apply the theory to more

general nonlinear systems. The assumptions placed on the system are:

ASSUMPTION 1: Boundedness of

, such that holds for all ( only

assumes limited values typically fulfilled for stable mechanical systems; is an

upper bound for the uncertainties – matched and unmatched).

ASSUMPTION 2: Semiglobal Lipschitz Condition

For arbitrary that:

for all

uniformly in .

ASSUMPTION 3: Stability of Unmodelled Dynamics

The - dynamics are BIBO stable both with respect to initial conditions and the

input , i.e.: , such that for all ,

ASSUMPTION 4: Partial Knowledge of the System Input Gain

(i) is assumed to be unknown (non-singular) strictly row-diagonally

dominant matrix with known, i.e. .

(ii) Also assume that there exists a known compact convex set , such that

, and the nominal system input gain is known. By knowing we

know the rough direction in the state space which are produced by the inputs.

Using the effectiveness of the inputs acting in the direction of are defined. This

effectiveness is unknown, but assumed to be within a known interval given by

which also contains the nominal effectiveness .

However, the main assumption that is being relaxed as a result of this study is:

ASSUMPTION 5: Stability of Matched Transmission Zeroes


101

The transmission zeroes of the transfer matrix lie in the

open left half plane (the system is minimum phase the initial response to inputs

has the same direction as the steady state response).

With the above restrictions and assumptions, the control problem is accurately and

completely stated. In the next subsection, an adaptive control theory is presented. The

estimation scheme utilised is called the Piecewise Constant Adaptation Method and it is

used to solve the control problem for the nonlinear system with uncertainties that have

been stated thus far.

4.4.3 Reference System

In order to carry out the design of the adaptive controller, it is initially assumed that all

the uncertainties are known. This is assumed in order to calculate the performance and the

stability bounds of the system [41]. When all the uncertainties of the system are known,

this system is called the reference system. The ideal desired reference system is

4. 22

Defining as transfer function from to , leads to:

4. 23

The reference system is:

4. 24


102

4. 25

(the derivation of the control laws are presented in the subsequent equations)

The closed loop reference system “ ” with the ideal control signal is

equivalent to the ideal desired reference system “ ”. When a low pass filter is

introduced in the control signal , it cuts off high frequency content of the high gain

estimation algorithm. This leads to a closed loop reference system which describes the

reduced control target. As the unknown parameters are part of the control law, the

reference system is not implementable, but is only used for assessment, such as

performance bounds, transient performance and system stability. Reference system and

reference control signal is

4. 26

Rewriting the reference system in frequency domain leads to

4. 27

where is the Laplace transform of: and defining

, whose Laplace transform is to depict the initial

conditions. This leads to

4. 28


103

4. 29

To make is more readable the following intermediate variables are introduced:

Using these intermediate variables leads to:

4. 30

4. 31

As is known, a prefilter can be chosen to render , which leads

to

. Therefore the ideal reference

output is as follows:

4. 32

In order to derive the formula for the influence of the initial condition is omitted. It

is shown that non-zero initial condition only leads to an exponentially decaying additional

contribution. This leads to:

4. 33

4. 34


104

4. 35

4. 36

As discussed previously, it is ideal to avoid high frequency content in the control channel,

which is key to control theory. Therefore, a bandwidth limiting low-pass filter is

multiplied with the control signal for the reference system:

4. 37

Substituting the above control signal back into the plant dynamics leads to the closed loop

reference system. This is as following:

4. 38

Once again a few intermediate variables are introduced for the ease of reading:

With the above definition, the closed loop reference system is


105

4. 39

where

The extensive proofs of the full adaptive controller are not shown here (see [41]), however

the bounds of the states and the control signal of reference system are as follows:

4. 40

where and are system dependent positive constants. The mathematical derivation of

these two uniform bounds also leads to the norm condition, which is the essential core

of the whole theory by representing how to design and assess the low-pass filter .

These bounds together with the normal condition guarantee the stability of the closed-

loop reference system, as introducing the low pass filter implies that the uncertainties

cannot be completely cancelled. In order to satisfy the norm condition the is

choosen in a way such that

4. 41

If the norm condition is met, the closed-loop reference system is stable with bounded

reference states and bounded reference control signal, i.e. BIBO stable.


106

4.4.4 Closed Loop Control Structure

Once the analysis of closed-loop reference system has been carried out, the next step is

to design and implement the closed loop control structure, using the piecewise constant

adaptation method. The main aim is to render the controlled plant arbitrarily close to the

closed loop reference system both in transient and steady state period with derivable

stability and performance bounds. The state predictor is presented next.

State Predictor

The plant dynamics are

4. 42

In contrast to the reference system, the state predictor depends on estimations of the

system input gain matrix and the uncertainties within and outside the span of , i.e.

and . But in the approach with piecewise constant adaptiation law, there is no explicit

online estimation of . Instead, , the best a priori available estimation of , replaces the

system input gain matrix and novelly defined combines the estimation of the error

( and the uncertainty in the span of , i.e. together. Based on this idea, we

firstly rewrite the plant dynamics as

4. 43

All the uncertainties within the span of is considered as a part of and all the

uncertainties that are within the span of are considered within . Therefore, the plant

dynamics is further rewritten as:


107

4. 44

And once the plant dynamics are reformulated in this way, the modified state predictor is

implemented as following:

4. 45

Therefore, the state predictor has a structure, with only two uncertainties and ,

which is estimated by the piecewise constant adaptation law.

Control Law

The plant dynamics are:

4. 46

The control objective is:

4. 47

Similar to the analysis of reference system, to realise such a control objective, the design

of the control law is as follows. However, before that is done, both the systems needs to be

written in the frequency domain:

4. 48

4. 49


108

Equating the two above expressions results in the derivation of the control law:

4. 50

Introducing the low pass filter in order to cut off the high frequency components

leads to

4. 51

In Eq. 4. 51 it is assumed that the values of the uncertainties are known. But only the

estimates of the uncertainties are available. Therefore Eq. 4. 51 is changed to

4. 52

The low pass filter is designed to carry out a trade-off between performance and

robustness and how the uncertainties affect the performance on a particular channel.

Adaptive Law

The next step is to derive the piecewise constant update law for the control law. The

actual plant dynamics and the corresponding state predictor dynamics are

Plant:

The plant dynamics are seen in Eq. 4. 46 and the state predictor is seen in Eq. 4. 48 The

aforementioned plant and state predictor dynamics leads to the following error dynamics:


109

4. 53

Solving Eq. 4. 53 as an ordinary differential equation leads to:

4. 54

where is the sampling time. Based on the idea of the piecewise constant adaptation law,

and are constants for and so on. As:

4. 55

4. 56

Substituting Eqs.4. 55 and 4. 56 into 4. 54 leads to:

4. 57

4. 58

where


110

4. 59

where . At this stage it is important to note the change in the domain. Further

defining certain terms as:

4. 60

4. 61

Leads to the following representation:

4. 62

With the new notation, the prediction error at the sampling time is

4. 63

where:

4. 64

4. 65

Therefore, it is clear from the derivation that the initial conditions for the integral is the

previous time step and is independent of . The equations above consist of a recursive

term, . The adaptive update parameter is derived by cancelling out this

recursive component.


111

Therefore the piecewise constant adaptive law is:

4. 66

The detailed derivation of the stability criteria of the controller via the norm

condition and the corresponding proofs of the transient and steady-state performance of

the reference system is not given in this thesis. The derivation can be found in Chapter 5

of [41].

4.4.5 Augmented Controller Derivation

A typical controller has the architecture as depicted in Fig 4.3. The main components of

an adaptive controller are the state predictor, the feedback control system and the

adaptation law. Also an important aspect of this scheme of control is the low pass filter

which helps to decouple fast adaptation and guaranteed robustness.


112

Fig 4.3. Adaptive Controller Model

Vehicle Plant Dynamics:

The actual closed plant dynamics of the vehicle is

4. 67

where closed loop plant model which includes the baseline controller. is the control

input matrix. is chosen so that and are the control gain matrices for

the matched and the unmatched uncertainties. includes the matched uncertainties and

includes the unmatched uncertainties.

is the feedforward gain. is

the adaptive control signal. The main control objective is to design an augmentation law

that compensates for the matched and unmatched uncertainties on the output of the

system and ensures that the output tracks the desired output , given a certain

reference signal . This is to occur both in transient and steady state while all signals

remain bounded.


113

Desired Plant Dynamics:

The desired closed loop plant dynamics is shown in Eq. 4. 68. The plant model presented

here is a SISO plant model

4. 68

State Predictor:

4. 69

where and are the estimates of the matched and the unmatched uncertainties

respectively, is a matrix used to assign faster poles for the prediction error dynamics

and is . The estimates of these uncertainties are provided by the piecewise wise

constant update law.

Control Law:

is the adaptive control signal and the matching condition in the presence of

uncertainties is:


114

4. 70

where and

. is the output of the actual

plant and is the output of the desired plant. As the values of the uncertainties are

not known, the estimates of the uncertainties are used to formulate the control law.

Furthermore, as only certain frequencies of this uncertainty can be cancelled in Eq.4. 70, a

low pass filter is added as part of the adaptive control signal. The uncertainties in a system

comprise of low and high frequency components. In an attempt to cancel out all the

frequency components of the uncertainty, control chattering would be introduced. This

would lead to control saturation and degrade the physical system. Therefore, the adaptive

law is modified as:

4. 71

However, for a nonminimum phase system is unstable. Scheduling

can lead to the switching itself being unstable. Therefore, it needs to be carried out

carefully. However, the scheduling the pole-zero cancellation (to cancel the unstable pole)

can lead to unpredictable closed loop dynamics. Therefore, the inverse of the DC gain is

utilised to carry out the LTV implementation of the transfer function. This method was

first proposed by Che et. al. in [179].

An inherent property of transfer function is utilised to formulate the modified

implementation of the adaptive augmentation signal. The property of transfer functions is:

the inverse of the dc gain of is equal to the steady state value of . Using this method

the new adaptive signal is

4. 72

In non-matrix form:


115

The main advantage of this implementation is that the inversion of the system dynamics is

avoided. This means that this form of the equation can be implemented for a

nonminimum phase LTV system. As the system is time varying, the implementation in Eq.

4. 72 cannot be directly used. is implemented as a state space model. This

modification of the control law is formulated for SISO systems, as the inverse of

is guaranteed. At no point during the trajectory, is the value of

equal to zero; for example: The minimum value of is 0.0197 for the

longitudinal case.

Low Pass Filter :

For the filter used in the control structure to be a low pass filter, has to be strictly

proper. This means that the numerator degree must be lower than the denominator

degree. Furthermore, a prerequisite on this filter is that it needs to be stable. As a result the

structure of the filter is

4. 73

where is a transfer function that leads to be stable and proper with a DC gain of

one; . There are conditions placed on in order to obtain the desired . The

transfer function needs to have a relative degree of greater than zero. The

implementation of this filter is shown in Fig 4.4. For the case of a first order filter,

. This leads to a structure shown in Eq. 4.82.

The structure of the low pass filter is therefore given by

4. 74

where is the bandwidth of the low pass filter. In the case of the hypersonic glider two

first order filters of different bandwidths are implemented for the matched and the


116

unmatched uncertainties. In order to suppress the effect of uncertainties in the control

channel that have a higher frequency content, the bandwidth of is less than

which is a direct result of the nature of uncertainties that make up the matched and

unmatched uncertainties.

Fig 4.4. Filter with Constant Cut-Off Frequency

Piecewise Constant Update Law

The piecewise constant adaptation law is for the modified state predictor is given by

4. 75

The augmentation for the PPC, is implemented in SIMULINK© and compiled into C

code. The version of Matlab/SIMULINK© used is R2012b.

ωC

D(s) u uL1

-

+

C(s)


117

4.5 Performance Metric (Comparison of Controllers)

In order to compare the baseline to the augmented controller, it is important to select

performance metrics. This subsection outlines the performance metrics utilised to compare

the PPC with the augmented controller.

The analytical performance of an adaptive controller is mainly limited to the tracking error

norms. For a full adaptive controller these bounds can be obtained from the Lyapunov

stability proof. However, these analytical bounds are too conservative [180]. The

performance of an adaptive controller can be analysed during and after a simulation run

using various quantities, for example:

The settling time after a step reference command

A Fourier analysis to evaluate the spectral content of the tracking error or the

control signal.

The largest load factor etc.

There are several other metrics that have been utilised to analyse the performance of

adaptive controller but the metrics that are utilised to compare the controllers in this study

are as follows:

A performance metric is the norm of the tracking error. The tracking error norm is

defined as:

4. 76

where . The normalised norm is:

4. 77


118

The benchmark case (only baseline) in the absence of uncertainties is taken as the reference

system. The deflection of the control surface of this reference system is labelled as .

The baseline control signal in the presence of uncertainties is denoted via and the

control signal in the augmentation configuration is denoted via . Therefore, the energy

of the deviation between the reference system and the control signal in the presence of

uncertainties is defined as a normalised metric for the control signal [180]. This metric is

calculated as follows:

4. 78

4.6 Time Delay Margin

The time delay margin (TDM) is the maximum amount of time-delay that the system can

experience before instability occurs [181, 182]. For adaptive control, the adaptive gain

needs to be carefully selected in order to not reduce the robustness of the closed loop

system. This reduction at some stage might render the closed loop system unstable. This

robustness metric is known as the TDM. For a given adaptive gain of a closed loop system,

there is a corresponding input delay for which the system is on the verge of instability

[183].

The robustness of a feedback control system is an important issues especially when it

comes to adaptive control. In the field of robust control it is a well know fact that high gain

in the feedback loops leads to increased effort in the control channel and consequently a

reduction in phase margin [184]. Within the scope of adaptive control, high adaptive

gains, which lead to fast adaptation, can cause undesired high-frequency oscillations in the

control channel and sensitivity to time delays [182]. Phase margins cannot be used for

robustness analysis as adaptive controllers are nonlinear. Therefore, TDM is used to test

for the system robustness.


119

For the case of a MRAC type controller, when the adaptation gain is too high, this type of

controller displays next to no time delay margin. This is not the case for an adaptive

control. By incorporating a time delay in the relevant transfer functions, the TDM of the

adaptive controller is guaranteed to stay bounded away from zero in the presence of fast

adaptation [65]. The low pass filter in the control architecture is what helps to decouple

robustness and adaption, and thus decouple control from estimation. The bandwidth of

this low pass filter can adjust the robustness margins of the closed loop system. Reducing

the bandwidth of the low pass filter increases the TDM at the cost of reduced performance.

On the other hand, increasing the bandwidth of the low pass filter improves the

performance of the closed loop system, however with a reduced TDM. The effects of

varying bandwidth on the TDM is presented by Leman in [63] and by Bierling in [181].

The main issue with TDM is the lack of an analytical method to determine this value for a

system. Pade approximation [185] and Razumikhin Method [186, 187] has been used to

approximate the TDM. For the case of the Pade approximation, the lower bounds of the

final result it deemed to be too conservative. The Razumikhin Method provides reasonable

results for scalar system, however that still leads to conservative values for the TDM. As a

result, within the scope of this study, the TDM is calculated through simulation.

The state space of a system with no delays has the following representation:

4. 79

In the presence of delays in the input channel, the corresponding state space

representation is as follows:

4. 80

where is the delay that is introduced into the system. For all of the cases presented in

literature, and especially for LTI systems, the time delay margin is calculated by dividing


120

the phase margin with the gain crossover frequency [188]. However, for a LTV system, the

phase margin and the gain crossover frequency changes along the trajectory. Therefore,

from an analysis point of view, input delays have been added in till the point where the

system shows signs of instability.

4.7 Stability and Performance Bounds - Augmented Controller

When a piecewise constant controller is used, the following theoretical bounds can be

established [41]:

4. 81

4. 82

4. 83

4. 84

4. 85

4. 86

where , and refer to the variables and states associated with the ideal

reference system, and the constants and are positive constants. The derivation of

the bounds are not shown and can be found in [41] and [189]. To derive the bounds, the

sampling time needs to be sufficiently small. The sampling time is related to the

processing rate of the CPU. The aforementioned bounds are subject to the norm

condition:

4. 87

To carry out the proofs of stability and derive the performance bounds, the low pass filter

needs to make sure that for a given certain , there exists a so that the norm


121

condition holds, with the definitions:

, and

.

4.8 Validation of Modified Augmented Control Law

This section of the controller design looks at the validation of the modified augmented

control law as applied to a simplified case. This test is carried out to test the performance

of the controller for a simple SISO LTI test case before being applied to the hypersonic

vehicle. This is an important step in the design process as it gives confidence in the fact

that the controller does not just work for an isolated case. Rather, this section acts as a way

of proving that the controller can be applied to various classes of linear systems. There are

three different avenues to validate a controller. They are:

Flight Test

Hardware-In-the-Loop Simulations

Simulation based validation

This project considers a simulation based validation for preparation for future work.

Results in this section gives the confidence before the modified control law is applied to

the hypersonic test case. The SISO system considered as a test case is

The system considered is the roll dynamics at one point along the descent trajectory (at an

altitude of 2km). The system considered for this validation test is of the following form

4. 88


122

For this case is . is a step command and a ramp command with a gradient of

one. The uncertainties added to the system are static uncertainties: and

. The values of the matched and unmatched uncertainties are arbitrarily set for

testing purposes. The performance of the system without the augmentation, with the

original implementation of the augmentation (shown in Eq. 4. 71) and the modified

augmented control law (shown in 4. 72).

4.8.1 Nominal Case - Step and Ramp Input

The case presented here outlines the performance without the augmentation for the step

and the ramp command in the roll channel. The matched and the unmatched uncertainties

are set to zero. Both the step and the ramp command is introduced into the system at

. For both the cases, the system tracks the input without any steady state error. For

the case of the ramp input, there is a slight delay in the tracking variable. This is consistent

with systems tracking time varying commands.

Fig 4.5. System Output - Step Response


123

Fig 4.6. System Output - Ramp Input

4.8.2 Uncertainties - Step Input

The performance of the system in the presence of matched and unmatched uncertainties is

presented. The behaviour of the tracking variable is shown in Fig 4.7. Although stable, the

performance is severely degraded due to the presence of uncertainties for the case when

the augmentation is turned off.

Fig 4.7. Response to Step Input with Uncertainties

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Time(s)

Magnitude

Output Variable Plot y_p

Input Signal

Baseline Only Response

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

Time(s)

Magnitude



124

To determine the transient behaviour of the modified augmented control law as compared

to the original, only the matched component of the augmentation law is utilised to test the

performance and compare it to the complete augmentation setup. Therefore, for the next

test case the adaptive control law is . The adaptive signal magnitude is shown in

Fig 4.9 which depicts the control signal compensating for the uncertainties in the system.

However, it is seen in Fig 4.8, that the performance of the system is not restored. This is

due to the unmatched uncertainties being present in the system, and the unmatched

component of the adaptive control law being turned off.

Fig 4.8. Performance of System - Matched Component of Adaptive Law

Fig 4.9. Adaptive Signal Magnitude

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time(s)

Magnitude


0 1 2 3 4 5 6 7 8 9 10-0.05

-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

Time(s)

Adaptive C

ontr

ol S

ignal

Adaptive Control Commanded Signal


125

As the unmatched contribution of the control law is not considered in the above diagram,

there is a steady state error that is introduced into the tracking variable. The plots of the

original control law (which involves inverting the system dynamics), and the modified

control law (which involves using the inverse DC gain of the system dynamics) are shown

in Fig 4.10 and Fig 4.11. The modified control law exhibits a slightly larger overshoot and a

slightly slower response as compared to the original adaptive control law. However, the

advantage of avoiding the inversion of the system dynamics is seen as enough of an

advantage to allow this slight worsening in the performance.

Fig 4.10. System Performance - Unmatched Component Turned On

Fig 4.11. Adaptive Signal Contribution (Original and Modified)

0 1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time(s)

Tra

ckin

g V

ariable

Performance Comparison

Original Control Law

Modified Control Law

0 1 2 3 4 5 6 7 8 9 10-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Time(s)

Tra

ckin

g V

ariable

Adaptive Signal Contribution

Original Control Law

Modified Control Law


126

The steady state behaviour of both the adaptive control laws is identical. However, the

transient behaviour shows a difference. The modified controller with its inverted DC gain

of the system dynamics shows a slower response. This can be seen by the behaviour of the

adaptive signal. The red line in Fig 4.11 reaches it steady state value a bit later than the

black line.

4.8.3 Uncertainties - Ramp Input

In the presence of uncertainties, the performance of the system to a ramp input is shown in

Fig 4.12. The black line in the figure is the commanded signal. Once again, although the

response is stable, the performance is significantly degraded due to the presence of

uncertainties without the augmentation.

The blue and the green lines show the performance of the system with the original and the

modified adaptive control law. Once again the steady state performance for both the

augmentation is identical. The transient behaviour is similar as to the once seen with the

step input. The transient behaviour is only slightly worse at the very start of the

simulation.

Fig 4.12. System Performance

0 1 2 3 4 5 6 7 8 9 10-2

0

2

4

6

8

10

12

Time(s)

Magnitude

Ramp Input Test with Uncertainties

Input Signal

Output without Augmentation

Original Implementation of Augmentation

Modified Implementation of Augmentation


127

4.9 Conclusion

This chapter has outlined the theory that underlines the two control methodologies that

are implemented and tested during this research project. The underpinning concepts of

the PPC and the adaptive controller are outlined in this chapter. This chapter also

outlines the underlying mathematical concepts that govern the two control methodologies.

Furthermore, the modified adaptive control law is validated using a simulation based

study. The modified augmented controller has been presented along with a simulation

based validation of the augmented controller. An augmented controller for an LTV

nonminimum phase state feedback system has been presented. Using this modification

eliminates ASSUMPTION 5 (see Section 4.4.2).

The following chapter outlines the test cases and scenarios used to test the robustness and

performance of the PPC and the augmented adaptive controller. Sources of

uncertainties are also defined in Chapter 5. Chapter 5 also presents the performance and

robustness analysis of the controllers.

CONTROLLER PERFORMANCE COMPARISON

128

5. CONTROLLER PERFORMANCE COMPARISON

5.1 Introduction

This chapter presents the results obtained through nonlinear simulations of the SISO

longitudinal and SISO lateral/directional baseline and the augmented controllers.

Section 5.2 contains the analysis of the baseline controller. The pole-zero plots of the

baseline controllers at various points along the trajectory are analysed for the stability and

property. Section 5.3 presents the comparative results between the PPC and the

augmented controllers for the longitudinal dynamics. Section 5.4 presents the results of the

lateral/directional control strategies employed to control the glider once the pull up

manoeuvre has been carried out. A detailed comparison of the performance of the PPC

and the augmented controllers is given in these sub sections. Section 5.5 presents an

analysis of the uncertainties in the systems and presents a fundamental difference between

the controllers that leads to a difference in performance. Section 5.6 presents a TDM

analysis of the longitudinal and lateral controllers. Section 5.7 presents the main

conclusions drawn from this chapter.

5.2 Baseline Controller

The baseline controller is a dynamic pole placement controller, wherein the desired

dynamics of the closed loop system are defined as a function of the dynamic pressure.

Therefore, it is important to ensure that the poles of the closed loop system are stable.

5.2.1 Longitudinal Controller

The longitudinal controller carries out the pull-up manoeuvre. This manoeuvre is

extremely aggressive and demanding for the controller. Therefore, during the manoeuvre

it is important to ensure that the synthesis process has produced a controller that it

stabilises the plant. In order to determine the stability of the baseline controller, the pole

zero plot is utilised. This is depicted in Fig 5.1. All of the poles are placed on the left hand


129

side of the imaginary axis. This signifies stability. One of the zeros of the system is on the

right hand side of the imaginary axis. This is the nonminimum phase behaviour eluded to

in Section 2.3.

Fig 5.1. Pole Zero Plot of Closed Loop Dynamics - Longitudinal Dynamics

5.2.2 Lateral/Directional Controller

The pole-zero plots is utilised to depict the stability roll and sideslip dynamics. Both the

pole-zero plots show that the systems are stable for the duration of the flight. All of the

poles are on the left hand side on the imaginary plane which means that the systems are

stable (see Fig 5.2 and Fig 5.3). The directions of the arrow is the direction of the poles take

as the vehicle descends and the controller carries out the lateral/directional manoeuvre.

-50 -40 -30 -20 -10 0 10 20 30 40 50-6

-4

-2

0

2

4

60.80.950.9780.99

0.995

0.998

0.999

1

1

0.80.950.9780.99

0.995

0.998

0.999

1

11020304050

Pole-Zero Map

Real Axis (seconds-1)

Imagin

ary

Axis

(seconds

-1)


130

Fig 5.2. Pole Zero Plot of Closed Loop Lateral Dynamics

Fig 5.3. Pole Zero Plot of Closed Loop Directional Dynamics

The robustness analysis of the lateral and the longitudinal controllers is outlined in Section

5.6.

-8 -7 -6 -5 -4 -3 -2 -1 0-6

-4

-2

0

2

4

60.120.240.360.480.620.76

0.88

0.97

0.120.240.360.480.620.76

0.88

0.97

12345678

Pole-Zero Map


Imagin

ary

Axis

(seconds

-1)

-6 -5 -4 -3 -2 -1 0-3

-2

-1

0

1

2

30.160.340.50.640.760.86

0.94

0.985

0.160.340.50.640.760.86

0.94

0.985

123456

Pole-Zero Map


Imagin

ary

Axis

(seconds

-1)


131

5.3 Longitudinal Control – Baseline and Augmentation

To evaluate and establish the performance of the pole placement and augmented

controllers, the descent trajectory analysis is performed using several different scenarios. It

is important to note that the implementation of the adaptive controller is not changed for

the different scenarios. The different test scenarios are outlined in Table 5.1.

Table 5.1. Simulation Scenarios

Nominal conditions. The aerodynamic parameters as per the GHAME database: PPC S1

Nominal conditions. The aerodynamic parameters are not changed and are used as

per the GHAME database: augmented controller

S2

PPC with gravimetric errors S3

augmented controller with gravimetric errors S4

PPC with reduced actuator functionality S5

augmented controller with reduced actuator functionality S6

PPC with aerodynamic uncertainties. S7

augmented controller with aerodynamic uncertainties. S8

PPC with time varying aerodynamic uncertainties S9

augmented controller with time varying aerodynamic uncertainties S10

PPC with a combination of errors S11

augmented controller with a combination of errors S12

All the scenarios are analysed using Monte Carlo simulations in CADAC++. In flight, all

experiments contain an INS module. Therefore, by default the sensor module has been

turned on. The benchmark scenario is in the presence of sensor dynamics and no

uncertainties. Table 5.2 contains the initial conditions for the trajectory control simulation

case.

Table 5.2. Simulation Parameters: Initial Conditions

Altitude

(km)

Pitch

Angle

(deg)

Angle of

Attack

(deg)

Velocity

(m/s)

Roll

Angle

(deg)

Heading

angle

(deg)

Flight Path

angle (deg)

55 -47 23 2381 (Mach

6.60)

0 0 -70


132

5.3.1 Results and Analysis

The benchmark test case (S1 and S2) of the flight path angle controller is presented in Fig

5.4. The pull up manoeuvre is carried out using the flight path angle command. The flight

path angle controller is able to perform the pull manoeuvre and tracks the desired flight

path angle. The controller has authority when the error between the commanded and the

actual flight path angle starts reducing. This is also the point where no further spikes are

seen by the augmented controller. This signifies, that once control authority is present, the

estimator states in the controller follow the actual vehicle states. According to Fig 5.4

and Fig 5.15 the error starts reducing at around 9 seconds at an altitude of 38km. At this

altitude the density is high enough for the control surfaces to create enough control

moment for the vehicle to start tracking the commanded variable.

Fig 5.4. Benchmark Case (S1 and S2)

0 10 20 30 40 50 60-70

-60

-50

-40

-30

Time (s)

(d

eg)

Flight Path Angle Tracking

Commanded

Baseline

Augmentation

0 10 20 30 40 50 600

2

4

6x 10

4

Time (s)

Altitude (

m)

Altitude Profile

Trajectory - Baseline

Trajectory - Augmentation

0 10 20 30 40 50 60-40

-20

0

20

40

Time (s)

(

deg)

Control Surface Deflection

- Baseline

- Augmentation


133

Fig 5.5 shows the performance of the PPC and the augmented controller in the presence

of reduced control surface deflection. The test case describes a scenario when the elevens

can only deflect 30% of the commanded deflection. It is seen that the augmentation

configuration improves the performance of the baseline controller even with reduced

control surface deflection. From the control deflection it is seen that the augmented

controller reduced control surface oscillation and as a result there are reduced oscillations

in the tracking variable . This happens at around at an altitude of 38km. The

improved tracking ability is a result of the estimation of uncertainties in the Piecewise

Update Law and the reduction in oscillations is due to the low pass filter in the adaptive

controller structure.

Fig 5.5. Reduced Elevon Functionality

0 10 20 30 40 50 60-80

-60

-40

-20

Time (s)

(d

eg)


Commanded

Baseline

Augmentation

0 10 20 30 40 50 600

2

4

6x 10

4

Time (s)

Altitude (

m)

Altitude Profile



0 10 20 30 40 50 60-10

-5

0

5

10

Time (s)

(

deg)


- Baseline

- Augmentation


134

Fig 5.6 – Fig 5.11 show the performance of the baseline and the augmented controllers

in the presence of time varying errors (S9 and S10). This scenario describes a situation

where in the aerodynamic properties of the glider changes along the descent trajectory.

This can happen as the structure may be in a reduced state of functionality towards the

end of the suborbital ballistic trajectory (see Fig 1.2) and as the atmospheric conditions

change it may lead to further degradation of the aerodynamic features of the vehicle. Once

again the augmented controller is able to improve the performance of the baseline

controller from a control oscillation point of view and also from the point of view of the

oscillation in the tracking variable. The two main cases where the time varying

aerodynamic uncertainties effect the performance of the baseline controller are (Fig

5.6) and (Fig 5.10). The tracking variable along with the control surface oscillate in the

presence of time varying uncertainties in these two aerodynamic coefficients. The

augmented controller is able to reduce the oscillation in both the tracking variable and also

the control surface deflection. For the remainder of the test cases, the behaviour of the

augmented controller and the PPC controller are similar to cases S1 and S2.

For the case when the drag coefficient is corrupted, it has an impact on the lift to drag ratio

of the vehicle. And a change in the drag characteristics leads to a change in the angle of

attack, which in turn leads to a change to the lift properties of the vehicle. And this change

in the drag coefficient and the subsequent change in the lift coefficient deviates from the

stored values of the lift coefficient in look up tables which are used to calculate the gains of

the controllers during the course of the simulation. This deviation causes the gains to take

on values that causes the fluctuations in the tracking variable.

For the case of the uncertainties in , it is a measure of the tendency of the vehicle to fly

nose first. The static stability or the longitudinal static margin of a vehicle is in direct

relation to the position of the centre of gravity and the neutral point. The static margin of a

vehicle can be estimated by calculating . From Fig 5.6, although the moment


135

derivative was decreased, the corresponding decrease in the lift coefficient lead to an

increase in the static margin. This is analogous to a forward move in the centre of gravity.

This leads to the aircraft becoming more stable. However, the disadvantage is that the

drag increases, which is known as trim drag, and vehicle manoeuvrability suffered as can

be clearly seen from the figure below. The decrease in manoeuvrability is depicted

through the sluggish response and the tracking ability of the flight path angle of the

vehicle.

Fig 5.6. LTV Error in (S9 and S10)

0 10 20 30 40 50 60-70

-60

-50

-40

-30

Time (s)

(d

eg)


Commanded

Baseline

Augmentation

0 10 20 30 40 50 600

2

4

6x 10

4

Time (s)

Altitude (

m)

Altitude Profile



0 10 20 30 40 50 60-40

-20

0

20

40

Time (s)

(

deg)


- Baseline

- Augmentation


136


0 10 20 30 40 50 60-70

-60

-50

-40

-30

Time (s)

(d

eg)


Commanded

Baseline

Augmentation

0 10 20 30 40 50 600

2

4

6x 10

4

Time (s)

Altitude (

m)

Altitude Profile



0 10 20 30 40 50 60-40

-20

0

20

40

Time (s)

(

deg)


- Baseline

- Augmentation


137

Fig 5.8. LTV error in (S9 and S10)

0 10 20 30 40 50 60-70

-60

-50

-40

-30

Time (s)

(d

eg)


Commanded

Baseline

Augmentation

0 10 20 30 40 50 600

2

4

6x 10

4

Time (s)

Altitude (

m)

Altitude Profile



0 10 20 30 40 50 60-40

-20

0

20

40

Time (s)

(

deg)


- Baseline

- Augmentation


138


0 10 20 30 40 50 60-70

-60

-50

-40

-30

Time (s)

(d

eg)


Commanded

Baseline

Augmentation

0 10 20 30 40 50 600

2

4

6x 10

4

Time (s)

Altitude (

m)

Altitude Profile



0 10 20 30 40 50 60-40

-20

0

20

40

Time (s)

(

deg)


- Baseline

- Augmentation


139


0 10 20 30 40 50 60-70

-60

-50

-40

-30

Time (s)

(d

eg)


Commanded

Baseline

Augmentation

0 10 20 30 40 50 600

2

4

6x 10

4

Time (s)

Altitude (

m)

Altitude Profile



0 10 20 30 40 50 60-40

-20

0

20

40

Time (s)

(

deg)


- Baseline

- Augmentation


140


0 10 20 30 40 50 60-70

-60

-50

-40

-30

Time (s)

(d

eg)


Commanded

Baseline

Augmentation

0 10 20 30 40 50 600

2

4

6x 10

4

Time (s)

Altitude (

m)

Altitude Profile



0 10 20 30 40 50 60-40

-20

0

20

40

Time (s)

(

deg)


- Baseline

- Augmentation


141

Table 5.3 presents the results of S3 and S4, S7 and S8 and S11 and S12. Table 5.3 outlines

the mean altitude and flight path angle and the corresponding standard deviations. The

most important test case to be analysed is the combination of errors. In this case the

augmented controller is successful in reducing the standard deviation of the final altitude

and also reduces control surface oscillations.

Table 5.3. Uncertainties Analysis Monte Carlo Simulation (S3 and S4, S7 and S8, S11 and S12)

At 60 seconds (Baseline) At 60 seconds (Baseline +

Augmentation)

Mean Standard

Deviation

Mean Standard

Deviation

Gravimetric

Uncertainties

Altitude (m) 6319.72 210.60 6166.90 20.11

Flight Path

Angle (deg)

-55.67 0.02 -54.80 0.0058

Aerodynamic

Uncertainties

Altitude (m) 6252.78 309.35 5926.57 501.68

Flight Path

Angle (deg)

-55.69 0.13 -54.84 0.06

Combination

of

Uncertainties

Altitude (m) 5643.33 1527.48 5529.40 1352.92

Flight Path

Angle (deg)

-55.88 0.18 -56.04 0.12

Fig 5.12 depicts the performance of the PPC and the augmented controller in the

presence of gravimetric errors. The changes in the moments and products of inertia

represents a change in the position of the centre of gravity, which leads to differences in

the values of the aerodynamic and control coefficients as compared to the nominal case.

This difference is the cause of the oscillations. These differences and oscillations are also

consistent in the case of aerodynamics uncertainties and also when there are a

combination of uncertainties present. The augmented controller is able to cancel these

oscillations. Having cancelled out these oscillations, the augmented controller is able to

track the flight path angle. The performance of the augmented controller is

consequently better than the PPC.


142

Fig 5.12. Gravimetric Errors (S3 and S4)

0 10 20 30 40 50 60-80

-60

-40

-20

Time(s)

(d

eg)


0 10 20 30 40 50 600

2

4

6x 10

4

Time(s)

altitude (

m)

Altitude Profile

0 10 20 30 40 50 60-40

-20

0

20

40

Time(s)

(

deg)


Commanded

Baseline

Augmentation



- Baseline

- Augmentation


143

Fig 5.13. Aerodynamic Uncertainties

Fig 5.13 shows the performance of the PPC and augmented controller in the presence of

aerodynamic errors. Once again the augmented controller is able to cancel the oscillations

in the tracking variable and also the control surface deflections. This alleviates the load on

the structure of the glider. So the augmented controller proves to be advantageous from

a control and structural point of view. As seen in Fig 5.6, uncertainties with can cause

the response of the flight path angle to be more sluggish. The differences initial responses

in Fig 5.13 is also due to the uncertainties in .

0 10 20 30 40 50 60-80

-60

-40

-20

Time(s)

(d

eg)


0 10 20 30 40 50 600

2

4

6x 10

4

Time(s)

altitude (

m)

Altitude Profile

0 10 20 30 40 50 60-40

-20

0

20

40

Time(s)

(

deg)


Commanded

Baseline

Augmentation



- Baseline

- Augmentation


144

Fig 5.14. Combination of Errors

Fig 5.14 shows the performance of the PPC and the augmented controller in the

presence of aerodynamic, gravimetric and actuators uncertainties. This test case is the

most realistic test case as it is most likely that the errors that occur won’t occur

individually. Therefore, it is important to test the robustness of the controller for multiple

uncertainties in the system. The augmented controller significantly improves the

performance of the baseline controller in cancelling oscillations in the flight path angle and

also the control surface deflection.

0 10 20 30 40 50 60-80

-60

-40

-20

Time(s)

(d

eg)


0 10 20 30 40 50 600

2

4

6x 10

4

Time(s)

altitude (

m)

Altitude Profile

0 10 20 30 40 50 60-10

-5

0

5

10

Time(s)

(

deg)


Commanded

Baseline

Augmentation



- Baseline

- Augmentation


145

– Norm of Tracking Error

The tracking error norm is employed to compare the performance of the baseline to

the augmented controller. The tracking error norm is

5. 1

For the nominal case for the baseline controller, is 6.51. The

for the

augmented set up for the nominal case is 8.06. Table 5.4 outlines the for the case for

time invariance errors.

Table 5.4. Tracking Error Norm – Monte Carlo Runs

Baseline Augmentation Percentage

Improvement (%)

Gravimetric 7.68 7.33 4.56

Aerodynamic 7.58 6.78 10.55

Combination 14.89 12.78 14.22

The for the time varying case is shown in Table 5.5.

Table 5.5. Tracking Error Norm for Time Varying Errors


Improvement (%)

8.88 5.76 35.06

15.67 14.07 10.21

12.56 10.68 14.97

12.64 12.76 -0.95

15.68 13.21 15.75

14.14 14.11 0.21


146

From Table 5.4 and Table 5.5 the following conclusions are drawn:

For the cases where there is a change in of less than 10%, it is regarded

that the PPC and the augmented controllers give no difference in

performance.

The two main cases where the augmentation improves the performance of the

baseline is time varying changes of and .

The augmentation significantly decreases in the presence of gravimetric,

aerodynamic uncertainties and combination of errors. It is important to point out

that the combination of errors is the most important test case, as that test case

most accurately describes what could happen in a real test flight. Therefore, the

improvement of the performance of the tracking error norm is particularly

important and promising with the augmented controller.

Tracking Error and Tracking Error Acceleration

The error dynamics and its acceleration is utilised as a performance metric to compare the

performance of the PPC and the augmented controller. It is seen that for all the cases of

uncertainties (Fig 5.15 – Fig 5.25), the rate of change of the error dynamic settles at zero

quicker for the augmented controller as compared with the baseline controller. This

means that the augmented controller improves the settling time of the tracking

variable. Another important fact to note is that the steady state error is significantly

improved when the augmentation is turned on. This is evident from the steady state value

of .


147

Fig 5.15. Benchmark Scenario (S1 and S2)

Fig 5.16. Reduced Actuator Functionality (S5 and S6)

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20

Time(s)

eP (

trackin

g e

rror

- deg)

Output Variable Tracking Error

eP - Baseline

eP - Augmentation

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20

Simulation Time Steps

e

P /

t

Error Dynamics Acceleration

eP / t - Baseline

eP / t - Augmentation

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20

Time(s)

eP (

trackin

g e

rror

- deg)


eP - Baseline

eP - Augmentation

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20


e

P /

t


eP / t - Baseline



148

Fig 5.17. Time Varying Uncertainty

Fig 5.18. Error Dynamics and Acceleration

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20

Time(s)

eP (

trackin

g e

rror

- deg)


eP - Baseline

eP - Augmentation

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20


e

P /

t


eP / t - Baseline


0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20

Time(s)

eP (

trackin

g e

rror

- deg)


eP - Baseline

eP - Augmentation

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20


e

P /

t


eP / t - Baseline



149



0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20

Time(s)

eP (

trackin

g e

rror

- deg)


eP - Baseline

eP - Augmentation

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20


e

P /

t


eP / t - Baseline


0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20

Time(s)

eP (

trackin

g e

rror

- deg)


eP - Baseline

eP - Augmentation

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20


e

P /

t


eP / t - Baseline



150



0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20

Time(s)

eP (

trackin

g e

rror

- deg)


eP - Baseline

eP - Augmentation

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20


e

P /

t


eP / t - Baseline


0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20

Time(s)

eP (

trackin

g e

rror

- deg)


eP - Baseline

eP - Augmentation

0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20


e

P /

t


eP / t - Baseline



151

Fig 5.23. Gravimetric Uncertainties

Fig 5.24. Aerodynamic Uncertainties

0 2000 4000 6000 8000 10000 12000-15

-10

-5

0

5

10

15


eP (

trackin

g e

rror)


0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20


e

P /

t


eP - Baseline

eP - Augmentation

eP / t - Baseline


0 2000 4000 6000 8000 10000 12000-15

-10

-5

0

5

10

15


eP (

trackin

g e

rror)


0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20


e

P /

t


eP - Baseline

eP - Augmentation

eP / t - Baseline



152

Fig 5.25. Combination of Uncertainties

Control Signal Metric

The benchmark case (only baseline) in the absence of uncertainties is taken as the reference

system. The deflection of this reference system (nominal system without uncertainties) is

labelled as . The baseline control signal in the presence of uncertainties is denoted via

and the control signal in the augmented controller is denoted via . The energy of

the deviation between the reference system and the control signal in the presence of

uncertainties is defined as a normalised metric for the control signal [180]. The control

surface deflection for the different test cases deviates from the maximum deflection limit

of at different times. Consequently, the norm (the maximum overshoot from the

reference deflection) is not utilised as the value for all the cases would be . As there is a

distinct transition from the point in time the controller stops working in open loop

(maximum control deflection) and starts tracking the output variable, this transition from

open to closed loop could is best captured using the norm as it best highlights the

oscillations of the augmented and baseline control surface deflections as compared to the

nominal case. It also provides a measure of both actuator position and rate limits [180].

In the presence of uncertainties if ; this signifies a reduction in the energy

usage of the control surfaces when the augmented controller is used. This in turn means

0 2000 4000 6000 8000 10000 12000-5

0

5

10

15

20


eP (

trackin

g e

rror)


0 2000 4000 6000 8000 10000 12000-20

-15

-10

-5

0

5

10

15

20


e

P /

t


eP - Baseline

eP - Augmentation

eP / t - Baseline



153

that the augmented controller reduces the load on the hinge joints and the subsequent

structure. The control surface metric is shown in Eq. 5. 2. From the results it is seen that

once the augmented controller is turned on, the control effort that is added by the

adaptive controller attempts to bring the system’s behaviour closer to the desired

behaviour to that of the reference system. Hence, the adaptation successfully carries out its

intended task.

5. 2

Table 5.6 outlines and for the test cases.

Table 5.6. Control Surface Metric

Baseline ( Augmentation Improvement (%)

(LTV) 5.364 5.142 4.32

(LTV) 1.389 0.139 90.0

Actuator

Uncertainties

0.246 0.223 9.35

Gravimetric

Errors

0.232 0.097 58.2

Aerodynamic

Uncertainties

1.283 0.877 31.6

Combination

of Errors

0.680 0.49 27.9

Most of the uncertainty test cases presented in Table 5.6 shows a reduction in the control

metric when the augmented controller is turned on. For the test cases not presented in

Table 5.6, there was a minimal change in the metric. It is seen that for most of the cases

< . This means that the augmentation not only cancels out the uncertainties and

improves the performance of the baseline controller, it also manages to decrease the

amount of energy usage by the control surface. This in turn reduces the oscillations in the

tracking variable and also alleviates the structural load.


154

5.4 Lateral and Directional Control

In order to evaluate and establish the performance and robustness of the pole placement

and adaptive controllers, the manoeuvres are performed using several different

scenarios. It is important to note that the implementation of the controllers are not

changed for the different scenarios. The test cases are

Table 5.7. Simulation Scenarios – Lateral/Directional

Nominal Conditions: perfect state feedback. Baseline controller L1

Pole Placement Controller with rudder failure L2

Pole Placement Controller with reduced aileron functionality L3

Pole Placement Controller with gravimetric errors L4

Pole Placement Controller in the presence of aerodynamic uncertainties L5

Pole Placement Controller in the presence of combination of L2-L5. L6

Augmented Controller under nominal conditions L7

Augmented Controller with rudder failure L8

Augmented Controller with reduced aileron functionality L9

Augmented Controller with Gravimetric Errors L10

Augmented Controller in the presence of aerodynamic uncertainties L11

Augmented Controller in the presence of combination of L2-L5 L12

5.4.1 Results and Analysis

Table 5.8 presents the initial conditions for the trajectory control simulation case.

Table 5.8. Conditions - Trajectory Control

Altitude

(km)

Pitch

Angle

(deg)

Angle of

Attack

(deg)

Velocity

(m/s)

Roll

Angle

(deg)

Heading

angle

(deg)

Flight Path

angle (deg)

6.00 -53.83 1.86 221.02

(Mach

0.65)

0 0 -55.70

A pulsed roll-manoeuvre (3 pulses) is carried out on the descent trajectory to display the

effectiveness of the controllers in varying atmospheric conditions. This tests the


155

performance and robustness of the scheduled baseline controller where the desired

dynamics are a function of dynamic pressure (see Fig 6.11 for the dynamic pressure).

Fig 5.26. Roll Angle Tracking - Benchmark Case (L1 and L7)

In the benchmark case, the PPC is able to track the desired roll angle. In the presence of no

uncertainties, the augmented controller is unable to improve the performance of the

baseline controller. This indicates that the baseline controller has stabilised the system

under nominal conditions and perform well under nominal conditions. This point is

0 10 20 30 40 50 60-10

0

10

20

30

Time (s)

(

deg)

Roll Angle Tracking

Commanded

Baseline

Augmentation

0 10 20 30 40 50 601000

2000

3000

4000

5000

6000

7000

Time (s)

Altitude (

m)

Altitude Profile



0 10 20 30 40 50 60-6

-4

-2

0

2

4

Time (s)

(

deg)

Control Surface Deflection - aileron

a - Baseline

a - Augmentation


156

further highlighted by the difference in the tracking error norm in the nominal case as

presented in Table 5.9.

Fig 5.27. Reduced aileron deflection (L3 and L9)

In the case of reduced aileron function, the baseline controller in Fig 5.27 shows

significantly less oscillations as compared to the augmented controller. Once the

augmentation is operational, although the steady state tracking ability of the controller

improves, the oscillations increase in the transient period.

0 10 20 30 40 50 60-10

0

10

20

30

40

Time (s)

(

deg)

Roll Angle Tracking

Commanded

Baseline

Augmentation

0 10 20 30 40 50 601000

2000

3000

4000

5000

6000

7000

Time (s)

Altitude (

m)

Altitude Profile



0 10 20 30 40 50 60-3

-2

-1

0

1

2

Time (s)

(

deg)


a - Baseline

a - Augmentation


157

Fig 5.28. Asymmetric Elevon Deflection (L3 and L9)

The asymmetric deflection is defined as 10%, which means that there is a 10% difference in

the deflection between the left and right elevon. The baseline controller for the first two

commanded pulses shows inability to track the desired value in the form of a steady state

error. However, the augmented controller from the very start of the simulation is able

to track the desired roll angle. The importance of this test case lies in the realistic nature of

the uncertainty. Uncertainties are not likely to occur in a symmetric manner. Therefore, the

improved performance with the augmented controller is especially encouraging.

0 10 20 30 40 50 60-10

0

10

20

30

40

Time (s)

(

deg)

Roll Angle Tracking

Commanded

Baseline

Augmentation

0 10 20 30 40 50 601000

2000

3000

4000

5000

6000

7000

Time (s)

Altitude (

m)

Altitude Profile



0 10 20 30 40 50 60-6

-4

-2

0

2

4

Time (s)

(

deg)


a - Baseline

a - Augmentation


158

Fig 5.29. Gravimetric Errors (L4 and L10)

Fig 5.29 highlights that the gravimetric uncertainties do not have a profound impact on the

performance of the PPC and the augmented controller. However, the augmented

controller does improve the steady state tracking ability and the settling time of the PPC

controller.

0 10 20 30 40 50 60-10

0

10

20

30

40

Time(s)

(

deg)

Roll Angle Tracking

0 10 20 30 40 50 601000

2000

3000

4000

5000

6000

7000

Time(s)

altitude (

m)

Altitude Profile

0 10 20 30 40 50 60-6

-4

-2

0

2

4

Time(s)

a (

deg)


Commanded

Baseline

Augmentation



a - Baseline

a - Augmentation


159

Fig 5.30. Aerodynamic uncertainties (L5 and L11)

In the presence of aerodynamic uncertainties, once again the differences between the PPC

and the augmented controller aren’t too pronounced. However, it does seem that the

augmentation performs better in the transition region and also has a reduced steady state

error as compared to the baseline. It can also be seen from the diagram, that the steady

state value is reached quicker when the augmentation as compared to the baseline. This

means that the augmentation improves the overall performance of the baseline controller

in the presence of aerodynamic uncertainties.

0 10 20 30 40 50 60-10

0

10

20

30

40

Time(s)

(

deg)

Roll Angle Tracking

0 10 20 30 40 50 601000

2000

3000

4000

5000

6000

7000

Time(s)

altitude (

m)

Altitude Profile

0 10 20 30 40 50 60-6

-4

-2

0

2

4

Time(s)

a (

deg)


Commanded

Baseline

Augmentation



a - Baseline

a - Augmentation


160

Fig 5.31. Rudder Failure: -7.5 to 7.5 degrees (L2 and L8)

The rudder failure case presented in this research is when the rudder is stuck between -7.5

and 7.5 degrees. The PPC fails (although stable) to track the desired value. Despite the

severe failure the augmented controller is able to track the desired roll angle. The

aileron deflection increases for the augmented controller in order to correct for this error.

0 10 20 30 40 50 60-20

0

20

40

60

80

Time(s)

(

deg)

Roll Angle Tracking

0 10 20 30 40 50 601000

2000

3000

4000

5000

6000

7000

Time(s)

altitude (

m)

Altitude Profile

0 10 20 30 40 50 60-6

-4

-2

0

2

4

Time(s)

a (

deg)


Commanded

Baseline

Augmentation



a - Baseline

a - Augmentation


161

Fig 5.32. Combination of errors (L6 and L12)

The most realistic case in terms of dealing with uncertainties is shown in Fig 5.32. Before

the pull out phase of the trajectory the glider will most likely have a multitude of

uncertainties. None of the uncertainties mentioned up to this point occur as an isolated

event. The uncertainties that are introduced into the system include aerodynamic

uncertainties, gravimetric uncertainties, actuator uncertainties (asymmetric elevon

0 10 20 30 40 50 60-60

-40

-20

0

20

40

60

Time(s)

(

deg)

Roll Angle Tracking

0 10 20 30 40 50 600

2000

4000

6000

8000

Time(s)

altitude (

m)

Altitude Profile

0 10 20 30 40 50 60-6

-4

-2

0

2

4

Time(s)

a (

deg)


Commanded

Baseline

Augmentation



a - Baseline

a - Augmentation


162

deflection) and rudder failure (-7.5 to 7.5 degrees). As shown in Fig 5.32, the PPC,

although stable, is unable to track the desired roll angle. Under some conditions, the value

of the roll angle reaches as high as and as low as . The augmented controller is

able to track the commanded roll angle even in the presence of such uncertainties and

rudder failure.

Main Conclusions from Roll Angle Tracking

The main conclusions that are drawn from the test cases presented in Table 5.7 (lateral test

cases) from a performance point of view are:

The augmented controller is able to marginally improve the performance of the

baseline controller in the presence of gravimetric and aerodynamic uncertainties.

The augmented controller is able to improve the tracking ability and in turn

reduce the steady state error in the presence of asymmetric elevon deflection.

The augmented controller significantly improves the performance of the baseline

controller when there is major rudder failure (Fig 5.31). The PPC is unable to follow

the commanded roll angle, whereas the augmented controller displays minimal

steady state error when there is a rudder failure.

There is a significant improvement in the performance and tracking ability of the

controller when the augmentation is turned on in the presence of multiple

uncertainties. As seen in Fig 5.38 the augmentation is able to track the commanded

roll angle even in the presence of significant uncertainties.

For all the test cases, there is an increase in the aileron deflection (therefore in turn

the elevon deflection) when compensation for the errors that are present in the

system. This leads to an increase in the control surface metric in the

lateral/directional case as seen in Table 5.11.


163

Table 5.9. Roll Tracking Error Norm


Improvement

(%)

Baseline 26.42 26.51 -0.342

Rudder Failure (-7.5 to 7.5 deg) 71.73 25.37 64.64

Reduced elevon function 36.25 31.63 12.74

Asymmetric elevon deflection 30.87 25.93 16.02

Gravimetric uncertainties 26.21 25.37 3.24

Aerodynamic uncertainties 26.53 25.65 3.34

Combination of errors 112.20 26.99 75.94

The for the side slip angle showed minimal change therefore it has not been

presented here. The commanded sideslip angle is . There are oscillations in the sideslip

angle only when the roll controller is carrying out the roll manoeuvre. The tracking error

norm for the roll angle is shown in Table 5.9. The main conclusions and the corresponding

explanations of the behaviour of the controller are as follows:

For the benchmark case, the augmented controller is able to reduce the

tracking error norm; however the change is not significant. This leads to the

conclusion that the baseline controller has been tuned (in this research) and

satisfactorily carries out the stabilisation and tracking job in the nominal case.

The most pronounced changes in the tracking error norm are the rudder failure

cases. When there is a rudder failure, the norm is significantly worse when only

the PPC is operating. The augmentation is able to significantly reduce this

tracking error to values close to the benchmark case. This is due to the secondary

effect of deploying the rudder [190]. The rudder deflection induces a sideslip

angle (yawing motion). This leads to faster airflow over one of the elevons,

leading to this elevon generating more lift and in turn inducing a rolling

moment. And when only the baseline controller is working, it is unable to


164

account for it because it is unable to account for the difference between the

desired and the actual performance of the controller.

In the case of reduced elevon function, where the deflection is limited to only ,

although the tracking error norm is reduced, it is not reduced to the value that is

the case in the benchmark scenario.

In the case of asymmetrical elevon deflection, the performance of the controller is

similar to that of a failed rudder case. The different levels of deflection of the

elevons cause a difference in the local angle of attack. This leads to increased lift

on elevon with a higher deflection. This inequality in the amount of lift

generated by the control surfaces leads to the roll angle controller being unable

to track the commanded value. However, once again, the augmented controller

corrects the error and merges the actual and the desired performance.

The most important improvement is when there is a combination of errors. The

tracking error norm when only the baseline is operating is 112.20. However,

when the augmented controller is operating, the tracking error norm is 26.99.

This result is particularly encouraging and shows that in the presence of vast

amounts of uncertainties and system failures, the augmented controller gives

extremely promising improvements in terms of the tracking performance.

These differences in the performance of the baseline and the augmented controller are due

to the uncertainties that are present in the system. Under nominal conditions, the values

for the aerodynamic and control derivatives that are provided to the controller are from a

certain point in the look up tables. These values mirror what the vehicle experiences.

However, in the presence of uncertainties, the values that the flight controller are fed are

different to the ones the vehicle experiences. This difference leads to degradation in the

performance of the baseline controller. And it is this difference that the augmentation can

estimate and compensate and restore the controller behaviour close to nominal behaviour.


165

The same control surface metric is utilised to analyse the deflection of the elevons and the

rudder for the lateral/directional dynamics.

Table 5.10 presents the mean and the standard deviation of the altitude for the failure

cases for the lateral/directional dynamics. For the case where there are aerodynamic and

gravimetric errors, there is little improvement in terms of the standard deviation of the

altitude when the augmented controller is running. However, in the presence of rudder

failure and also a combination of errors the augmented controller significantly

improves the standard deviation. The mean altitudes for both the configurations do not

vary considerably.

Table 5.10. Altitude Mean and Standard Deviation (Lateral Test Cases)

Baseline – Mean

(Standard Deviation) –

meters

Augmentation – Mean

(Standard Deviation) -

meters

Rudder Failure

( )

1418.35 (53.81) 1437.15 (2.92)

Gravimetric uncertainties 1463.94 (39.63) 1466.56 (39.79)

Aerodynamic uncertainties 1076.92 (2.29) 1079.72 (2.28)

Combination of errors 933.70 (42.06) 957.69 (14.88)

Table 5.11. Control Surface Metric

Baseline ( Augmentation

Percentage

Change (%)

Rudder Failure

( )

1.70 2.03 19.41

Reduced elevon function 0.77 0.89 15.58

Asymmetric elevon

deflection

0.21 0.68 223.81

Gravimetric uncertainties 0.11 0.81 636.36

Aerodynamic uncertainties 0.12 0.67 458.33

Combination of errors 2.01 2.15 6.97


166

The main conclusions drawn from Table 5.11 are

All of the values for were greater than . As the system was decoupled, the

ailerons, and therefore in turn the elevons, have to be deflected more to account for

the uncertainties in the roll channel.

High values for are seen for the rudder failure cases, where the steady state

errors were the highest.

The highest value for is when there is a combination of errors present in the

system.

These increases in the control effort are expected, as the difference between the

performance of the baseline and augmented controllers different. The augmented

controller is able to recognise the difference in the actual and desired performance and

takes corrective measures in the form of added control deflection in the aileron channel.

5.4.2 Behaviour of Additional Lateral/Directional Aircraft States

For some of the test cases presented in this paper the rudder is inoperable. For this case,

the system loses one of its input channels and it is not possible to control both the bank

angle and sideslip angle (the two control objectives). Fig 5.33 depicts the behaviour of the

sideslip angle , roll rate and yaw rate . The behaviour shown in Fig 5.33 is for a

functioning rudder which indicates control authority in both the channels. The sideslip

angle, along with yaw and roll rate, deviate from their commanded and nominal values of

zero when the glider is commanded to carry out a roll manoeuvre.


167

Fig 5.33. Behaviour of Additional Aircraft States (Rudder is functioning)

Fig 5.33 with its time histories of the variables shows the effect of a roll manoeuvre on the

other aircraft states. The secondary effect of rolling is yawing. A change in the roll angle

induces a sideslip angle and consequently a change in causes a change in the yaw rate.

However due to the presence of control authority in the channel, the rudder is able to

cancel out this deviation in and null the rates.

0 10 20 30 40 50 60-2

-1

0

1

2

Time (s)

(

deg)

Sideslip Angle

0 10 20 30 40 50 60-100

-50

0

50

100

Time (s)

p (

deg/s

ec)

Roll Rate

0 10 20 30 40 50 60-20

-10

0

10

20

Time (s)

r (d

eg/s

ec)

Yaw Rate


168

Fig 5.34. Behaviour of Aircraft States - Presence of Uncertainties

In Fig 5.34, the plot represents the time histories of the additional aircraft states, in the

event that there is a rudder failure and is inoperable at a particular angle. As the system is

represented using two SISO models, it results in the loss of control authority in the

channel and consequently leads to the inability of the controller to track the desired

sideslip angle. This means that the ailerons, which is still functioning, has no control

authority in the channel. As a result, although the roll angle tracks the desired value

with the operating augmentation, it does not have an impact on and therefore, a non

zero value is seen in Fig 5.34.

0 10 20 30 40 50 60-4

-2

0

2

4

Time (s)

(

deg)

Sideslip Angle

0 10 20 30 40 50 60-100

0

100

200

Time (s)

p (

deg/s

ec)

Roll Rate

0 10 20 30 40 50 60-40

-20

0

20

40

Time (s)

r (d

eg/s

ec)

Yaw Rate


169

The main limitation of SISO controllers is that the functioning ailerons do not have control

authority in the directional channel when there is a rudder failure. This aspect of the

behaviour of the controller synthesis is highlighted in Fig 5.34. Even with the

augmentation on, the sideslip angle is not tracked in the case of a rudder failure.

Therefore, for future work the application of a MIMO baseline controller to control the

lateral/directional dynamics is to be considered. With a MIMO controller, the oscillations

and steady state errors in the sideslip angle channel could be compensated when a roll

angle is commanded. This is due to the fact that the control surfaces have control authority

on the tracking variables, as can be seen in Eq. 5.3. Furthermore, when there is a rudder

failure, the functioning aileron cannot cancel out the effects that the failed rudder has on

the sideslip angle. This is due to the limitations introduced at the modelling stage as a

result of the decoupling. The decoupling and its corresponding limitations are presented

next.

The fourth order MIMO representation of the lateral dynamics have been discussed in

Chapter 3 (Section 3.2.2 Eq. 3. 13). The decoupled lateral/directional dynamics are shown

in Eq. 3. 17 and 3. 19. In the MIMO case, the sideslip angle can be controlled through the

aileron , and the rudder . This means that even if the rudder experiences a failure, the

sideslip angle can be controlled using an aileron deflections. In the SISO representation of

the sideslip angle dynamics, the only control input is the rudder. Therefore, when the

rudder is stuck at a particular angle, as shown in Fig 5.34, the sideslip angle takes on a

particular offset value based on the angle of the rudder position. Although the ailerons

would not be able to overcome severe rudder failure, they would be able to reduce the

errors and make the value of the sideslip angle closer to the commanded variable.

The roll angle is an integral of the roll rate. It can be controlled via an aileron and rudder

deflection in the MIMO case. However, in the SISO case, the only input is the aileron

deflection. When the rudder fails it induces a sideslip angle. This means that the oncoming

flow, one side of the aircraft (and consequently one elevon) is faster than the other one.


170

The elevon with faster airflow, generates more lift which leads to the aircraft to roll. This

behaviour exhibits the secondary nature of deploying the rudders in flight and also why

baseline controller needs to be a MIMO controller in order to cancel out the cross-channel

coupling effects. It is clear to see that the baseline controller is not able to deal with the

cross coupling issues due to the lack of cross-channel control authority on the respective

channels. On the other hand, the adaptive augmentation recognises the difference in the

desired and the actual behaviour and corrects for this error by adding to the aileron

deflection to correct for the roll angle steady state error. This increase in the aileron

deflection is particularly visible in Fig 5.32.

5.5 Uncertainties in the System

Additional to just presenting, what the controller improves in the system, it is also

important to present at a fundamental level, how and why the augmented controller

improves the performance of the baseline controller. The analysis of the matched and the

unmatched uncertainties estimated by the state estimator in the Piecewise Constant

Update Law are presented here.

5.5.1 Longitudinal Dynamics

The longitudinal dynamics of the hypersonic glider are represented using a third order

system. There is only one control input which means that the matched uncertainties are a

scalar number and the unmatched uncertainties are represented using a matrix,

. The entire estimated uncertainties vectors is as follows: .

The uncertainties that are present in the system for the nominal case are shown in Fig 5.35.

As the uncertainties that are estimated are close to zero, the adaptive control signal

contribution is minimal. Therefore the performance between the nominal and the

augmented controller is also not significant, as can be seen from Fig 5.4.


171

For the non-nominal case, the estimated uncertainties are shown in Fig 5.36. As the

estimated uncertainties are more, there is more of a contribution to the control effort from

the augmentation. This contribution is what leads to the augmented controller cancelling

out these estimated uncertainties and improve the performance of the baseline controller.

It is evident that the baseline controller is unable to cancel out the uncertainties in the

initial part of the manoeuvre. This is due to the fact that the baseline control synthesis does

not take into account the uncertainties that could be present in the system. The adaptive

controller adds the following components to the control law:

. This additional component explicitly

includes the estimated uncertainties (matched and unmatched), that are shown in Fig 5.35

and Fig 5.36. The aim of the control law therefore, becomes cancelling these uncertainties

in the system and restoring and improving the performance of the baseline controller,

which is succeeds in achieving (as shown by the blue line in Fig 5.36).

Fig 5.35. Estimate of Uncertainties - Nominal Conditions

0 10 20 30 40 50 60-2

-1

0

1

Time(s)

1(e

stim

ate

)

Matched Certainties Estimates

Baseline

Augmentation

0 10 20 30 40 50 60-0.2

0

0.2

0.4

0.6

Time(s)

2(e

stim

ate

21)

Unmatched Certainties Estimates

Baseline

Augmentation

0 10 20 30 40 50 60-0.08

-0.06

-0.04

-0.02

0

Time(s)

2(e

stim

ate

22)


Baseline

Augmentation


172

Fig 5.36. Estimate of Uncertainties – Presence of Uncertainties

5.5.2 Lateral Lateral/Directional Dynamics

The uncertainties that are present in the lateral and directional channels are estimated

using a state estimator of the augmented controller. For the nominal case, the estimated of

the uncertainties are shown in Fig 5.37 and for the case with uncertainties in Fig 5.38.

0 10 20 30 40 50 60-4

-2

0

2

4

Time(s)

1 (

estim

ate

)

Matched Uncertainties Estimates

0 10 20 30 40 50 60-4

-2

0

2

Time(s)

2 (

estim

ate

21)


0 10 20 30 40 50 60-0.4

-0.2

0

0.2

0.4

Time(s)

2 (

estim

ate

22)


Baseline

Augmentation

Baseline

Augmentation

Baseline

Augmentation


173

Fig 5.37. Estimate of Uncertainties - Nominal Conditions

Fig 5.38. Estimate of Uncertainties –Uncertainties

0 10 20 30 40 50 60-0.05

0

0.05

0.1

0.15

Time(s)

1(e

stim

ate

)

Matched Certainties Estimates

Baseline

Augmentation

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

Time(s)

2(e

stim

ate

)


Baseline

Augmentation

0 10 20 30 40 50 60-0.4

-0.2

0

0.2

0.4

Time(s)

1(e

stim

ate

)

Matched Uncertainties Estimates

0 10 20 30 40 50 60-20

-10

0

10

20

Time(s)

2(e

stim

ate

)


Baseline

Augmentation

Baseline

Augmentation


174

The reasoning for the improvement of the augmented controller as compared to the

baseline controller is similar to the longitudinal controller. The augmented control law

explicitly includes the estimated uncertainties. As a result, it is able to cancel the matched

and unmatched uncertainties in the system whereas the baseline controller is not, as it

does not consider the existence of uncertainties is the system (control law of the baseline

controller from Section 4.2: ).

This is the fundamental difference between the controllers that leads to a difference in

performance and also robustness. The difference is a result of the control law and the

assumptions that are made. For the baseline, controller, it is assumed that there are no

uncertainties in the system. The augmented controller however, takes into account the

uncertainties in the system, estimates them and then compensate them.

5.6 Time Delay Margin

The robustness of the system has been tested through the time delay margin (TDM). This

sub-section presents the TDMs of the longitudinal and the lateral controllers and whether

the values presented here agree with theory is also discussed.

As previously mentioned, the bandwidth of the filter has an effect on the time delay

margin and also the performance of the controller. By reducing the bandwidth of the low

pass filter, the TDM can be increased at the cost of reduced performance. On the other

hand, increasing the bandwidth leads to improved performance at the cost of reduced

robustness (lower TDM) [63]. The adaptive controller is able to improve the

performance of the baseline controller significantly in the presence of uncertainties.

However, from a robustness point of view, there is a marked reduction. The TDM is still

bounded away from zero, although the margins are reduced. For a filter architecture,

which is of the first order, Table 5.12 outlines the time delay that can be introduced into

the input channel before the system becomes marginally stable.


175

Table 5.12. Time Delay Margin Comparison

Longitudinal Baseline Augmentation

Nominal 79ms 30ms

Reduced Control Surface

Deflection

215ms 84ms

Time Varying Error ( ) 64ms 23ms






Gravimetric uncertainties 71ms 25ms

Aerodynamic uncertainties 73ms 26ms

Combination of errors 117ms 67ms

Lateral (Roll) Baseline Augmentation

Nominal 162ms 74ms

Rudder failure (1.0 deg) 154ms 64ms

Rudder Failure (-7.5 to 7.5

deg)

146ms 60ms

Reduced elevon function 113ms 51ms

Asymmetric elevon

deflection

149ms 67ms

Gravimetric uncertainties 158ms 71ms

Aerodynamic uncertainties 161ms 73ms

Combination of errors 133ms 52ms

A reduction of the TDM for the adaptive case is consistent with what has been presented

in theory and especially by Leman in [63]. The reduction in TDM is a result of the trade off

between performance and robustness. The performance of the adaptive controller is better

than the baseline controller. This results in a reduction in the robustness margin. In the

event that this project has TDM requirements, this can be changed using the bandwidth of

the low pass filter in the architecture in order to meet the requirement.


176

5.7 Conclusion

This chapter has presented the simulation results for the baseline controller and the

augmented controller. Stability and performance metrics have been utilised to show that

the augmented controller performs better and improves the performance of the baseline

controller in the presence of uncertainties. In the case of rudder failure, the augmentation

is able to significantly improve the performance of the PPC. The main conclusion is the

successful implementation of an augmented controller to a descent trajectory of a

hypersonic glider. The application and the performance of the adaptive controller show

the viability of such a controller for such a trajectory. This is especially important as the

piecewise constant update law has not been applied to a LTV state feedback case before.

Additionally, the augmented controller improved the performance of the baseline

controller over a wide velocity range. These speeds ranged from velocities in the

hypersonic (Mach > 5) to sub-sonic (Mach < 1.0) range.

An additional important conclusion from this chapter, as a con0sequence of applying SISO

controllers, is that a MIMO controller should be considered for the baseline controller to

control the lateral/directional dynamics of the hypersonic glider. This point is most

evident when there is a rudder failure and the controllers have no authority to change the

error introduced in the sideslip angle channel. A TDM analysis has carried out for the

longitudinal and lateral cases. The TDM was significantly reduced when the augmentation

was turned on. However, the analysis did prove what the theory claims in terms of

bounded away from zero TDM.

FLIGHT PARAMETER ANALYSIS

177

6. FLIGHT PARAMETER ANALYSIS

The performance metrics have proven that the augmented controller improves the

performance of the baseline controller. This is concluded through the tracking error norm,

control surface norm and additionally the standard deviation of the final altitude at the

end of the longitudinal test cases and the lateral test cases. However, it is important to

connect the performance of the controller with the physical parameters during flight, for

example Mach number and altitude. Some of the reasons for the behaviour of the tracking

variable are presented in Chapter 5, however a deeper look into the physical behaviour of

the vehicle is presented here. It is also important to assess the behaviour of the

aerodynamic coefficients during nominal flight and also in the presence of uncertainties.

The data presented in this chapter forms an envelope along the trajectory within which the

controller has been able to stabilise the system and carry out the control objective.

Furthermore, the results presented illustrate the suitability of the controllers to bring a

vehicle to the right initial condition to fire and test a SCRAMJET engine.

6.1 Flight Parameters - Longitudinal Dynamics

Flight parameters such as the altitude indicate what the vehicle is experiencing during

flight. The physical parameters also give a deeper insight into understanding the

behaviour of the vehicle and as to why the controller performed a certain way.

Appreciating the envelope within which the controller has been tested further helps

understand the limits of the controllers presented in this study and also the scope of its

application in the hypersonic area. Furthermore, it helps generate an understanding of

how the performance of the controller is degraded in the presence of uncertainties. The

physical and operational environment together with the maths help form an overall

picture of what the vehicle is experiencing.


178

6.1.1 Mach Number and Altitude

The Mach number vs. altitude profile (Fig 6.1) shows the conditions experienced by the

vehicle. Based on the flight conditions shown in this diagram, a suitable baseline controller

methodology is selected. This helps in deciding upon a scheduling parameter, if necessary.

In this case, because the flight parameters including Mach number and altitude change

significantly (see Fig 6.1), the scheduling of the baseline controller is necessary. A fixed

gain controller, although stable from a performance point of view, would prove ineffective

for the entire flight envelope. This is due to the desired closed loop dynamics being

unscheduled as a function of the flight parameters. Dynamic pressure (see Fig 6.2) is used

as a scheduling variable when defining the desired dynamics of the pole placement

baseline controller. The first point on the trajectory (top right of Fig 6.1) is where the

vehicle is at an altitude of 55km and is travelling at approximately Mach 6.8. As the

vehicles descends at an altitude of approximately 35km, the vehicle begins to decelerate as

a result of the higher air density and the corresponding increase in the drag coefficient (as

highlighted in Fig 6.5). As the vehicle continues to travel through the atmosphere and

descends, it decelerates to subsonic speeds. Note that the blue and the red lines

correspond to the envelopes generated by the baseline controller and the augmented

controller. Within the envelope not all of the trajectories overlap. As a result of the errors

and uncertainties being estimated by the adaptive controller, the trajectory of the

vehicle changes. As the error is estimated and a corresponding adaptive correction is

added to the control deflection, this leads to the change in the final altitude and the

standard deviation after the pull up phase and the bank angle manoeuvre.

The way the Mach number varies with the altitude very much depends on the

uncertainties that are present in the system. Therefore, for various uncertainties the Mach

number and altitude profile vary (as is clear from the red and the blue lines). Based on the

Monte Carlo Simulations, a range of Mach numbers and altitudes have been determined

within which the controller needs to function and carry out the control objective. From Fig


179

6.1 it is clear to see that, although the augmentation is not able to significantly narrow the

flight envelope, at no stage of the envelope, is the envelope widened when the

augmentation setup is used. The main point to note from Fig 6.1 is that both the

augmented and baseline controller, in the presence of uncertainties, are able to stabilise the

vehicle and maintain the vehicle within a sensible flight envelope.

Fig 6.1. Mach Number Altitude Envelope of the Vehicle – Longitudinal

Note also that the controller needs to control the dynamics of the glider from a hypersonic

flight regime (Mach > 5) to sub-sonic flight regimes (Mach < 1).

Fig 6.2. Dynamic Pressure (Longitudinal Manoeuvre)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6x 10

4

Mach Number

Altitude(m

)

Mach number and Altitude profile for flights

Augmentation Results

Baseline Results

Nominal

0 10 20 30 40 50 600

2

4

6

8

10

12

14x 10

4

Time(s)

Dynam

ics P

ressure

(P

ascal))

Dynamics Pressure Plot for Longitudinal Manoeuvre

Dynamic Pressure limit of 20 kPa

Mach Number limit of 5


180

Fig 6. 3. Mach Number, Altitude and Angle of Attack Time Histories

Fig 6.2 and Fig 6. 3 outline the times histories of the dynamic pressure, Mach number,

altitude and the angle of attack. This helps is understanding the state of the vehicle when

it comes to the initial conditions for introducing the SCRAMJET powered section into the

mission and trajectory (marked as "OW" in Fig 6.2 and Fig 6. 3). Siddiqui and Ahmed in

[191] show usable ranges of dynamic pressure are 20 to 200 kPa for a SCRAMJET (see Fig

6.2). SCRAMJET operation depends on the design and geometry of the engine. If the

0 10 20 30 40 50 600

2

4

6

8M

ach N

um

ber

Time(s)

Mach Number Profile

0 10 20 30 40 50 600

2

4

6x 10

4

Altitude (

m)

Time(s)

Altitude Profile

0 10 20 30 40 50 60-10

0

10

20

30

(

deg)

Time(s)

Angle of Attack








181

vehicle is flying too slow, it becomes a RAMJET [192]. Consequently, for this work, a

freestream Mach number greater than 5 is considered as the lower limit for SCRAMJET

operation (highlighted in Fig 6. 3). Although the vehicle never reaches a dynamic pressure

of (maximum dynamic pressure of ), it does cross the Mach 5 limit. The

dynamic pressure and Mach number conditions are seen in Fig 6. 3 and depict the window

within which the SCRAMJET engine could be started. As stated in the motivation in

Section 1.2, the controller can accomplish a pull up to allow the SCRAMJET to operate.

The angle of attack is approximately at at a speed of

. These results indicate that this control methodology can be used to reach certain

initial conditions at a certain altitude to enable the SCRAMJET engine to operate.

Although the parameter values mentioned above are not ideal initial conditions for the

engine, this study is a step towards the goal of integrating the powered section of the

mission into a sub-orbital ballistic trajectory.

6.1.2 Aerodynamic Characteristics

An important parameter to understand the behaviour of the vehicle is the angle of attack.

Fig 6.4 shows the angle of attack as a function of Mach number, altitude and simulation

time. For the nominal case (the dotted green line), there are much smaller oscillations in

the angle of attack than in other cases. The lack of oscillations carries over to the behaviour

of the lift to drag (L/D) ratio, the drag coefficient, the lift coefficient and the moment

coefficient.


182

Fig 6.4. Angle of Attack - Longitudinal

When uncertainties are present in the system, both the baseline and the augmented

controller show oscillations in the presence of uncertainties. The oscillations are either in

the region of the atmosphere where the control authority is minimal or when the vehicle is

transitioning into a region of higher density at an altitude of approximately 35km. Note

that the oscillations take place in a region where the Mach number is high (Mach > 6).

However, once the vehicle is flying in denser air and starts slowing down, the oscillations

are reduced significantly in the case of both the baseline and augmented controllers. The

initial oscillations in the angle of attack in the presence of uncertainties cause the lift and

drag properties of the vehicle to change. This in turn changes the nature of the physical

interaction the vehicle has with the atmosphere. These uncertainties lead to oscillations in

the angle attack, which causes oscillations in the drag and lift forces of the vehicle, and this

has an impact on the vehicle flight trajectory as seen in Fig 6.1. Consequently, the lift to

drag (L/D) ratio also oscillates (as shown in Fig 6.7).

0 10 20 30 40 50 60-20

-15

-10

-5

0

5

10

15

20

25

30

Time(s)

Angle of Attack

Augmentation

Baseline

Nominal


183

Fig 6.5. Drag Coefficient for the Pull Up Trajectory

Fig 6.6. Lift Coefficient for Pull Up Trajectory

0 10 20 30 40 50 600.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Time(s)

cD

Drag Coefficient

Augmentation

Baseline

Nominal

0 10 20 30 40 50 60-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time(s)

Cl

Lift Coefficient

Augmentation

Baseline

Nominal


184

Fig 6.7. Lift to Drag Ratio for Longitudinal Test Case

In the case where there are various uncertainties present in the system, it is not just the lift

and drag forces that are affected. As all of the aerodynamic coefficients are stored in look

up tables and are functions of Mach number and angle of attack, a fluctuation in the angle

of attack leads to fluctuations in all of the force and moment coefficients. This includes the

pitching moment coefficient (shown in Fig 6.8)

Fig 6.8. Pitching Moment Coefficient - Longitudinal

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

Simulation Time

Lift

of

Dra

g R

atio

Lift to Drag Ratio for Entire Trajectory

Augmentation

Baseline

Nominal

0 10 20 30 40 50 60-10

-8

-6

-4

-2

0

2

4

6

8x 10

-3

Time(s)

cm

Pitching Moment Coefficient

Augmentation

Baseline

Nominal


185

The pitching moment shown in Fig 6.8 is the total pitching moment coefficient of the

vehicle. The pitching moment consists of four parts: the pitch damping derivative,

longitudinal static stability derivative, reference moment coefficient and the control

moment derivative. Fig 6.9 (plotted as a function of time), shows that, due to the

oscillations in the angle of attack, the individual parts of the moment coefficient also

oscillate. Most importantly the change in the longitudinal static stability and the control

moment coefficients have the greatest impact in terms of the behaviour of the vehicle itself

and control authority. When the effects of all of these fluctuations are combined, this leads

to a degradation in the performance. Consequently, as the physical behaviour deteriorates,

at the same time the look-up tables at that corresponding point of the flight are telling the

baseline controller that is it flying under different conditions to what the vehicle is

experiencing in reality. This is the primary reason that leads to the degradation in

performance.

Fig 6.9. Pitching Moment Coefficient as a function of Time

These differences and also the uncertainties are all estimated by the state estimator in the

augmented controller. The corresponding equations read as follows:

0 10 20 30 40 50 60-5

-4

-3

-2

-1

Time(s)

Pitch D

am

pin

g D

erivative

0 10 20 30 40 50 60-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6x 10

-3

Time(s)

Sta

tic S

tabili

ty D

erivative

0 10 20 30 40 50 60-0.03

-0.02

-0.01

0

0.01

0.02

Time(s)

Refe

rence P

itchin

g M

om

ent

0 10 20 30 40 50 60-6

-5

-4

-3

-2

-1

0x 10

-4

Time(s)Contr

ol S

urf

ace P

itchin

g M

om

ent

Derivative

Augmentation

Baseline

nominal

Augmentation

Baseline

nominal

Augmentation

Baseline

nominal

Augmentation

Baseline

nominal


186

.

and are the estimated uncertainties that are provided by the piecewise constant

update law. The update law is able to estimate these uncertainties and restore the

performance of the controller close to nominal performance. The terms and are

not present as part of the control law when only the baseline controller is switched on as

opposed to the augmented control law,

. The baseline control law, as derived in

Section Pole Placement Control, is . This control law does not take into

account the uncertainties that could be present in the aerodynamic and control derivatives.

As the baseline controller does not directly incorporate the presence of uncertainties in the

system, they cannot be compensated for, resulting in worsened performance. For the

same reason, a similar behaviour of the controller occurs in the roll channel.

6.2 Flight Parameters - Lateral/Directional Dynamics

For the second part of the trajectory, a similar approach is utilised to analyse the

aerodynamic variables, as is the case with the longitudinal dynamics. Once again the

aerodynamic behaviour of the lateral/directional dynamics is used to represent the flight

envelope. This assists with controller design.

6.2.1 Mach Number and Altitude

The flight envelope is shown in Fig 6.10 for the aircraft during the lateral/directional

manoeuvre. The manoeuvre consists of carrying out three roll angle manoeuvres of

while holding the sideslip angle at . Although there is less variation in the Mach number

and altitude, there is a enough of a change from start to finish for there to be a need for a

scheduled baseline controller. As a result, once again the baseline controller on the roll and

sideslip angle channels are scheduled as a function of dynamic pressure. Furthermore, at

this stage of the trajectory, the speed at which the vehicle is travelling is no longer


187

hypersonic speed. This means that the vehicle needs to be able to fly at sub-sonic speeds as

well during this manoeuvre.

Fig 6.10. Mach Number Altitude Map for Second Part of Trajectory

Fig 6.11. Dynamic Pressure (Lateral/Direction Manoeuvre)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70

1000

2000

3000

4000

5000

6000

7000

Mach Number

Altitude(m

)

Mach Number Altitude Envelope

Augmentation

Baseline

Nominal


188

6.2.2 Aerodynamics Characteristics

Fig 6.12 and Fig 6.13 show the angle of attack and the lift to drag ratio for the second part

of the trajectory. The increase in the angle of attack is due to the commanded flight path

angle. This increase in angle of attack leads to an increase in the lift coefficient which in

turn increases L/D. The variations in the angle of attack and consequently L/D ratio occur

exactly when a roll angle is commanded. This is as a result of the change in the local angle

of attack when the vehicle executes the roll manoeuvre. The rolling action induces an

increase in the angle of attack on the up-going wing, and an decrease in angle of attack on

the down-going wing. This change in the angle of attack is seen in Fig 6.12. This creates an

opposing aerodynamic moment, which is called the roll damping derivative which is

shown in Fig 6.14.

Fig 6.12. Angle of Attack for Lateral Part of the Trajectory

0 10 20 30 40 50 600

2

4

6

8

10

12

Time(s)

(

deg)

Angle of Attack

Augmentation

Baseline

Nominal


189

Fig 6.13. Lift to Drag Ratio for Lateral Part of Trajectory

The rolling and the yawing moments of the vehicle seen in Fig 6.14 and Fig 6.15 show

minimal change during the second part of the flight. However, the variations in the

aerodynamic coefficients of the plant model for the lateral and directional dynamics (see

Fig 6.17 and Fig 6.16) are relatively large. These variations are signification enough that a

baseline controller by itself would not be sufficient. This is confirmed using the results

presented in Section 5.4.

Fig 6.14. Rolling Moment Coefficient

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

Time(s)

L/D

Lift to Drag Ratio

Augmentation L/D

Baseline L/D

Nominal Case

0 10 20 30 40 50 60-12

-10

-8

-6

-4

-2

0

2

4

6

8x 10

-3

Time(s)

Rolli

ng M

om

ent

Rolling Moment Coefficient

Augmentation

Baseline

Nominal


190

Fig 6.15. Yawing Moment Coefficient

The control and aerodynamic coefficients for the roll dynamics are shown in Fig 6.16. The

envelope is extremely wide for the case of the roll damping derivative. This is caused by

the failure of the rudder. Note that it is physical changes in the forces and moments that

the vehicle experiences being plotted here.

The look up tables that feed the controller the aerodynamic and control derivatives are

different from those that the vehicle is experiencing. This difference in where the glider is,

compared to where the aircraft thinks it is, is the reason for the degradation in the

controller's performance. The figures shown in this chapter capture the range of values

that the glider experiences. However, the range of values that the flight controller is given

is considerably narrower. Using a piecewise constant adaptive law this difference is

estimated in the form of uncertainties and is corrected for the adaptive controller using

a control input in the roll channel.

0 10 20 30 40 50 60-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time(s)

Yaw

ing M

om

ent

Yawing Moment Coefficient

Augmentation

Baseline

Nominal


191

Fig 6.16. Lateral Dynamics Derivatives

0 10 20 30 40 50 60-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06roll damping derivative

Time(s)

CLp

0 10 20 30 40 50 602.35

2.4

2.45

2.5

2.55

2.6x 10

-3

Time(s)

CLa

Aileron Roll Control Derivative

Augmentation

Baseline

Nominal

Augmentation

Baseline

Nominal


192

Fig 6.17. Directional Dynamics Derivatives

0 10 20 30 40 50 60-0.15

-0.1

-0.05Yaw Damping Derivative

Time(s)

cln

r

0 10 20 30 40 50 600

0.005

0.01Beta Yawing Moment Derivative

Time(s)

cln

0 10 20 30 40 50 60-0.019

-0.018

-0.017Beta Side Force Derivative

Time(s)

CY

0 10 20 30 40 50 60-1.5

-1

-0.5x 10

-3 Rudder Yawing Moment Derivative

Time(s)

clnr

0 10 20 30 40 50 600

2

4x 10

-3

Time(s)

CYr

Rudder Side Force Derivative

Augmentation

Baseline

Nominal


193

6.3 Conclusion

This chapter presents the aerodynamic characteristics experienced by the vehicle during

the trajectory. The aerodynamic envelope of the vehicle helps in controller design,

selecting the scheduling parameters and also scheduling the desired dynamics of the

closed loop plant. Furthermore, the envelope formed as a result of the uncertainties

present in the system are further evidence that an augmentation is needed along this

trajectory in order to improve the performance. At the fundamental level, the augmented

control law directly accounts for the uncertainties in the system with the presence of

and whereas the baseline controller does not. Additionally, the controller pull ups

the vehicle to being the vehicle to the initial conditions for future studies to include a

SCRAMJET stage at part of the descent trajectory.

The next chapter concludes the study and also presents scope for future work. Chapter 7

outlines the main contributions of this study and the main conclusions that are drawn

from Chapter 5 and Chapter 6 in terms of the performance of the controllers.

CONCLUSIONS AND FUTURE WORK

194

7. CONCLUSIONS AND FUTURE WORK

This chapter presents the main conclusions drawn from the study. It presents the

theoretical implications of the work conducted and its potential application to real flight

systems. It also presents the answer to the main research problem that is posed at the

beginning of this thesis. This chapter concludes by outlining the potential future direction

of this work.

7.1 Conclusions

The motivation of the study was to investigate methodologies to control a hypersonic

glider over a wide range of conditions, from Mach 6 down to subsonic speed. In order to

carry out this research, detailed Monte Carlo simulations were carried out in order to

compare the performance and robustness properties of the baseline and the augmented

controller (contribution number II - IV in Section 1.9). During the course of the research,

gaps in the existing adaptive control theory were identified. The gaps in theory are

related to the control of a LTV nonminimum phase state feedback system. Using the

results presented in this research, the aforementioned gaps in the theory and its

application have been addressed and filled.

A modified augmented control law is presented. This enables the application of the

piecewise constant adaptive update law to an LTV state feedback system. As the new

control law does not require an inversion of the system dynamics, the implementation is

suitable for application to nonminimum phase systems (contribution number V and VI in

Section 1.9). The main implication of this modification is the elimination of ASSUMPTION

5 in Section 4.4.2 (stability of matched transmission zeros). As a result of this modification,

it is suggested that future studies use the modification that has been presented in Section 0

(contribution number IX in Section 1.9).


195

Some general guidelines are proposed for the design of the low pass filter. Some rules

with respect to the bandwidth design of the low pass filter are presented for the cases

where minimal information is known about the system. This makes the design process

simpler and makes the design dependent on the nature of the system (contribution

number VII in Section 1.9).

An analysis of the flight parameters in Chapter 6 demonstrates the applicability of the

baseline and augmented controllers over a wide range of speed regimes (from hypersonic

to subsonic) (see Section 6.1.1 and 6.2.1). Furthermore, the ability of the controllers to carry

out the objectives over a wide range of values of the aerodynamic and control coefficients

(see Section 6.1.2 and 6.2.2) highlights the suitability of a PPC and an augmented PPC

to pull the vehicle of its ballistic trajectory and go into a horizontal path to fire the

SCRAMJET and cruise at a certain altitude (as stated in Section 1.2) (contribution number I

- IV in Section 1.9).

The systems, along with the baseline controller, presented in this thesis are implemented

in C++. The adaptive controller is presented in SIMULINK©. Once the SIMULINK© code

is compiled into C code, the resulting code is simply a series of matrix manipulations. This

implementation of the systems and the adaptive controller makes the transition of this

study from the simulation world to the real world seamless (contribution number VII in

Section 1.9).

The answers to the research problems posed in Section 1.5 are as follows:

I. How does the baseline controller perform for a descent trajectory in the presence of

sensor, aerodynamic, gravimetric and control surface uncertainties and failures?

Pole placement controllers are used to stabilise the system (see Section 4.2 and Section 5.2).

However, in the presence of uncertainties and failures, the performance of the baseline

controllers deteriorate (see Section 5.3.1 and 5.4.1). The synthesis of the baseline controllers


196

assume nominal conditions. Therefore, in the presence of uncertainties and system failures,

degradation in performance is expected.

II. How does the augmented controller assist in cancelling the uncertainties present

in the system and improve the performance of the baseline controller?

In the presence of uncertainties, although stable, the performance of the baseline controllers

worsen (see Section 5.3.1). This is due to the plant models of the longitudinal and

lateral/directional dynamics. These models do not include the uncertainties present in the

system. The piecewise constant update law in the adaptive controller estimates these

uncertainties in the system (see Section 5.5). The resulting control law (see Section 0)

accounts for the estimates of the uncertainties in the system and introduces a control signal

that attempts to cancel out the matched and unmatched uncertainties in the system under a

certain frequency (within the bandwidth of the low pass filter).

III. What are the performance characteristics of the adaptive augmented controller

in the presence of uncertainties?

The tracking error norm (see Section 5.3.1 and 5.4.1) demonstrates that the augmentation

cancels out the effects of the uncertainties and failures in the system. This helps to bring the

performance of the augmented controller closer to the nominal closed loop behaviour. For the

test case where a combination of errors are present, the tracking error norm improves by

14.22% for the longitudinal dynamics and by 75.94% for the lateral dynamics.

IV. What are the robustness properties of the augmented controller compared to the

baseline controller?

The robustness analysis of the baseline and augmented controllers are demonstrated using

the Time Delay Margin (TDM) (see Section 5.6). The adaptive controller demonstrates a

reduced TDM margin (bounded away from zero, as the theory predicts, seen in Section 4.4).

This research shows simulation results for the longitudinal and lateral controllers for an

LTV state feedback system. For the test case where a combination of errors are present, the

TDM reduces from 117ms to 67ms for the longitudinal dynamics, and from 133ms to 52ms

for the lateral dynamics.


197

V. Is this control methodology suitable for carrying out a pull up manoeuvre along the

descent phase of a suborbital ballistic trajectory to fire and test a reusable

SCRAMJET engine?

The baseline and the augmented controllers carry out the pull up manoeuvre and bring the

vehicle to the right initial conditions (as seen in Chapter 6, Fig 6.2 and Fig 6. 3) to test a

SCRAMJET engine. The controllers showed that even in the presence of uncertainties and

failures, the vehicle, in future studies, can be bought to the right Mach number, angle of

attack and dynamic pressure to fire the engine.

7.2 Further Research

As a consequence of this project, there are several areas in which this work can be further

progressed, as outlined next.

Landing the Glider: No study has yet performed a full trajectory simulation study to

include the landing of a hypersonic glider. One of the experimental requirements for the

HiFire 4 vehicle presented in [12] is to “land” the vehicle. In this context landing means

impact the ground at a low speed and low rate of descent. However, a study can be

undertaken for a slightly larger glider to include landing gear in order to land the glider

on a runway (similar to the Space shuttle landing).

Full Adaptive Controller: Prime [3] presents a full adaptive controller, however it was

only for the longitudinal dynamics. The reference model utilised in the controller was

linear time invariant (LTI). An important future direction would be to implement a LTV

reference model of the adaptive controller to take full advantage of changing

atmospheric conditions to cancel the matched and the unmatched uncertainties.

Filter Design for the adaptive controller: Although control theory has been

successfully applied in this study, the lack of a general design strategy for makes an

adaptive controller difficult to implement for certain systems. Although there are


198

guidelines that are provided regarding the design of the bandwidth of the filter, no

concrete methodology is outlined. Future work should concentrate on formulating

guidelines for the design of the low pass filter.

Baseline Controller: The baseline controllers presented in this study are SISO controllers.

Due to the inherent coupling of the dynamics of a glider it is important to implement a

MIMO controller as the baseline. However difficulties arise from the fact that the system

presented is a LTV system. Three possible options that can be considered for control

strategies for the baseline controller are State Dependent Riccati Equation (SDRE), NDI

and MIMO PPC (using the place function in Matlab to calculate the feedback gains).

Exo-Atmospheric Control: A reaction control system can be included as part of the initial

lateral/directional control in the upper atmosphere. With the implementation of a RCS, the

whole descent trajectory from an altitude of 100km can be tested. The RCS system can also

be implemented to control the rates of the vehicle along all the 3 axes of rotations. This is

important in order to have the desired initial conditions at the point in the trajectory when

the control surfaces have control authority.

Full trajectory analysis: A combination of endo-atmospheric control, adaptive control,

operation of the SCRAMJET and the landing of the hypersonic glider can be combined to

form a study which involves studying the complexities, robustness, performance and

stability properties of the controller over the entire suborbital ballistic trajectory.

References

199

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[2] G. E. Chamitoff, "Robust intelligent flight control for hypersonic vehicles," 0572199

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[3] J. J. Bertin and R. M. Cummings, "Fifty years of hypersonics: where we've been,

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Appendix A

218

Appendix A

Frames of Reference and Coordinate Transformation

The graphical representations of the various frames of references utilised in this research

are shown below.

Fig A. 1. Frames of Reference - Aircraft

Fig A. 1 shows all of the frames of references used in flight mechanics. The red line that is

labelled reference meridian is the prime meridian running through the London

Observatory. The figure above depicts the various frames of reference explained and

outlined in Section 3.2.1.

Every transformation is a linear mapping that is represented using a matrix. For example

if a frame of reference is rotated about the z-axis at an angle , then the matrix used to

describe this rotation is as follows:


219

C. 1

The corresponding rotations about the other axes are as follows:

C. 2

C. 3

The matrices and are the matrices that are used for all sequences of

rotations. An example of a series of three rotations is presented next. The rotation

sequence is as follows:

C. 4

The above equation consists of the following rotations:

1. Rotation about the z-axis through an angle of

2. Rotation about the "new" y-axis through an angle of

3. Rotation about the "new" x-axis through an angle of

The multiplication of the transformations matrices leads to the following solution.

C. 5

A more specific example is presented next. The transformation from the stability to the

body fixed axis is presented next. The transformation when written in short hand is

written as

C. 6


220

where is the angle of attack and is the sideslip angle. Expanding the above equation

leads to:

C. 7

C. 8

A graphical representation of the stability and the body fixed axis is depicted in Fig A. 2.

Fig A. 2.Body and Stability Axis of Aircraft

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