Stall Prevention Control of Fixed-Wing Unmanned Aerial Vehicles by Matthys Michaelse Basson Thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Engineering at Stellenbosch University Supervisor: Dr. Iain K. Peddle Department of Electrical & Electronic Engineering March 2010 Declaration By submitting this thesis electronically, I declare that the entirety of the work con- tained therein is my own, original work, that I am the owner of the copyright thereof (unless to the extent explicitly otherwise stated) and that I have not previously in its entirety or in part submitted it for obtaining any qualification. March 2010 Copyright © 2010 Stellenbosch University All rights reserved. i Abstract This thesis presents the development of a stall prevention flight control subsystem, which can easily be integrated into existing flight control architectures of fixed-wing unmanned aerial vehicles (UAV’s). This research forms an important part of fault- tolerant flight control systems and will ensure that the aircraft continues to operate safely within its linear aerodynamic region. The focus of this thesis was the stall detection and prevention problem. After a thor- ough literature study on the topic of stall, a model based stall prevention control al- gorithm with feedback from an angle of attack sensor was developed. This algorithm takes into account the slew rate and saturation limits of the aircraft’s servos and is able to predict when the current flight condition will result in stall. The primary con- cernwasstallduringwings-levelflightandinvolvedthepreventionofstallbyutilising only the elevator control surface. A model predictive slew rate control algorithm was developed to override and dynamically limit the elevator command to ensure that the angleofattackdoesnotexceedapredefinedlimit. Thestallpreventioncontrolsystem was designed to operate as a switching control scheme, to minimise any restrictions imposed on the existing flight control system. Finally, software in the loop simulations were conducted using a nonlinear aircraft model and realistic sensor noise, to verify the theoretical results obtained during the development of this stall prevention control strategy. A worst-case performance analysis was also conducted to investigate the robustness of the control algorithms against model uncertainties. ii Uittreksel Hierdie tesis handel oor die ontwikkeling van ’n staak voorkomings-vlugbeheer sub- stelsel wat maklik geïntegreer kan word in bestaande vlugbeheer argitektuur van onbemande vaste-vlerk lugvaartuie. Hierdie tesis vorm ’n belangrike deel van fout- tolerante vlugbeheertegnieke en sal verseker dat die vliegtuig slegs binne sy lineêre aerodinamiese werksgebied bly. Diefokusvanhierdietesisisdiestaakopsporingenvoorkomingsprobleem. Naafloop van ’n deeglike literatuurstudie oor die onderwerp van staak, is ’n model gebaseerde staak voorkomings-beheertegniek ontwikkel, wat terugvoer van ’n invalshoek sensor ontvang. Hierdie algoritme neem die sleur tempo en defleksie limiete van die vlieg- tuig se servos in ag en is in staat om staak te voorspel. Die primêre oorweging was staak tydens simmetriese vlugte en behels slegs die voorkoming van staak deur ge- bruik te maak van die hei beheer oppervlak. ’n Model voorspellings sleur tempo beheeralgoritmeisontwikkelomdiehei-roerdinamiestebeperksodatdieinvalshoek nie’nsekerevoorafbepaaldelimietoorskrynie. Diestaakvoorkomingsbeheerstelsel is ontwerp om te funksioneer as ’n skakel beheer skema om die beperkings op die bestaande vlugbeheerstelsel te minimaliseer. Laastenswassagteware-in-die-lussimulasiesgebruikomdieteoretieseresultate,wat verkry is tydens die ontwikkeling van hierdie staak voorkomings beheer-strategie, te kontroleer. Om die robuusthied van hierdie beheeralgoritmes teen model onseker- hede te ondersoek, is ’n ergste-geval prestasie analise ook uitgevoer. iii Acknowledgements Iwouldliketoextendmymostsinceregratitudetothefollowingpeople/organisations for their contribution towards this thesis, • Dr. I.K. Peddle, for all your guidance, advice and support through the course of thisproject. Thankyouforprovidingmewithadeeperunderstandingofaircraft dynamics and control, your knowledge and insight is inspiring. • Prof. T. Jones and Mr. J.A.A. Engelbrecht, for your valued input during our re- search meetings. Your experience and insight in the field of unmanned aerial vehicles are invaluable. • The National Aerospace Centre of Excellence (NACoE), for their financial sup- port and making this research possible. • Mylovelyfiancé,AncollineWright,yourendlessunconditionallovehavecarried me throughout this project. Thank you for your support and understanding. • My family, for your love and understanding throughout this project. You pro- vided me with the necessary environment and support network to reach my full potential. You allowed me to achieve all that I have today! • AM de Jager and Chris Jaquet, for your friendship, helpful advice and ideas throughout this project. Your technical assistance during the write-up of this thesis is much appreciated. • Wihan Pieterson and Simon Pauck, for the much needed lunch-time gaming dis- tractions. I highly value your friendship, thank you for making the time in the lab enjoyable! • Reinhart Fourie, for your invaluable friendship and support throughout this project. • And all my friends in the Electronic Systems Lab, especially Raun de Hart, Rudi Gaum and Deon Blaauw for your valued inputs, advice and always being there to share ideas. iv Idedicatethisthesistomyparents andmylovelywifetobe. M.M.Basson v Contents Declaration i Abstract ii Uittreksel iii Acknowledgements iv Contents vi Nomenclature x List of Figures xiv List of Tables xviii 1 Introduction 1 1.1 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Project Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Structure and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Fundamentals of Stall 5 2.1 Theory of Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Formal Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Geometric Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Wing and Aerofoil Geometry . . . . . . . . . . . . . . . . . . . . . 8 2.2.2 Aerofoil Reference Centres . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Aerofoil Stall Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Wing Planform Stall Characteristics . . . . . . . . . . . . . . . . . . . . . 15 2.5 Types of Stalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.1 Stall Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.2 Load Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5.3 Ground Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Classical Stall Prevention and Recovery . . . . . . . . . . . . . . . . . . . 22 vi CONTENTS vii 2.6.1 Stall Warning and Safety Devices . . . . . . . . . . . . . . . . . . 23 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Mathematical Aircraft Model and Control 26 3.1 Axis Systems and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.1 Earth Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.2 Aircraft Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.3 Aircraft Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Numerical Aerodynamic Analysis . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 Simulation Results from Analysis . . . . . . . . . . . . . . . . . . 29 3.3 Aircraft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.1 Longitudinal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 Aircraft Automatic Flight Control System . . . . . . . . . . . . . . . . . . 34 3.4.1 Linear Decoupled Longitudinal Dynamics. . . . . . . . . . . . . . 34 3.4.2 Normal Specific Acceleration Controller . . . . . . . . . . . . . . 36 3.5 Stall Prevention Control Augmentation Strategy . . . . . . . . . . . . . . 38 3.5.1 Angle of Attack State Saturation . . . . . . . . . . . . . . . . . . . 38 3.5.2 Switching Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5.3 Stall Prevention Control . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4 Preliminary Methodology 43 4.1 Conceptual Phase Plane Approach . . . . . . . . . . . . . . . . . . . . . . 43 4.1.1 Phase Plane State Trajectories . . . . . . . . . . . . . . . . . . . . 45 4.1.2 Description of Stall Prevention Control . . . . . . . . . . . . . . . 46 4.2 Forward State Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2.1 State Transition Equation . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Zero Input Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3.1 State Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3.2 Zero Input State Trajectory . . . . . . . . . . . . . . . . . . . . . . 52 4.3.3 Stall Detection Envelope . . . . . . . . . . . . . . . . . . . . . . . 55 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5 Stall Prevention Control 58 5.1 Active Stall Prevention . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.1.1 Stall Prevention State Trajectory . . . . . . . . . . . . . . . . . . 59 5.2 Zero State Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.2.1 Velocity Step Component . . . . . . . . . . . . . . . . . . . . . . . 62 5.2.2 Constant Step Component . . . . . . . . . . . . . . . . . . . . . . 64 5.3 Complete State Transition Equation . . . . . . . . . . . . . . . . . . . . . 66 5.3.1 Stall Prevention State Trajectory . . . . . . . . . . . . . . . . . . 67 CONTENTS viii 5.4 AoA Peak Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.4.1 Peak angle of attack as a result of the velocity step input . . . . . 71 5.4.2 Peak angle of attack as a result of the constant step input . . . . 73 5.4.3 Linear Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 Recursive Model Predictive Slew Rate Control . . . . . . . . . . . . . . . 77 5.5.1 Modified False Position Method . . . . . . . . . . . . . . . . . . . 78 5.5.2 Algorithm Verification . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6 Angle of Attack Regulating Control 83 6.1 Stall Prevention Control Strategy Description . . . . . . . . . . . . . . . 83 6.1.1 Finite State Machine . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2 AoA Reference Tracking Controller . . . . . . . . . . . . . . . . . . . . . 85 6.2.1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2.2 Linear Closed Loop Analysis and Pole Placement . . . . . . . . . 91 6.2.3 System Delays and Anti-Windup . . . . . . . . . . . . . . . . . . . 93 6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7 Simulation and Analysis 96 7.1 Linear Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.2 Nonlinear SIL Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.2.1 Slow Speed Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.2.2 Steep Pull-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.2.3 Reactivating the NSA Control System . . . . . . . . . . . . . . . . 102 7.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.3.1 Investigating Parameter Uncertainty . . . . . . . . . . . . . . . . 104 7.3.2 Level of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.4 Worst-Case Performance Simulation . . . . . . . . . . . . . . . . . . . . . 106 7.4.1 Detection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.4.2 Stall Prevention Control System . . . . . . . . . . . . . . . . . . . 110 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8 Conclusions and Recommendations 114 8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8.1.1 Computational Efficiency . . . . . . . . . . . . . . . . . . . . . . . 116 8.1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.2.1 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 List of References 119 Appendices 121 CONTENTS ix A Mathematical Derivations 122 A.1 Aircraft Trim Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 A.2 Longitudinal Stability Characteristics . . . . . . . . . . . . . . . . . . . . 123 A.2.1 Vertical speed stability . . . . . . . . . . . . . . . . . . . . . . . . 123 A.2.2 Angle of attack stability . . . . . . . . . . . . . . . . . . . . . . . . 124 A.3 Static Contribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A.4 Variation of Elevator Trim Angle with Velocity . . . . . . . . . . . . . . . 129 A.5 Elevator Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 B Aircraft Data 134 B.1 Geometric, Inertial and Propulsion Properties . . . . . . . . . . . . . . . 134 B.2 Trim Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 B.3 Aerodynamic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 B.3.1 Analysis with AVL . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 B.4 Numeric Aerodynamic Investigation . . . . . . . . . . . . . . . . . . . . . 137 B.4.1 XFOIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 B.4.2 XFLR5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 B.5 Typical AoA Sensors for Practical Implementation . . . . . . . . . . . . . 143 B.5.1 Fixed Differential Pressure Probes. . . . . . . . . . . . . . . . . . 143 B.5.2 Pivoted Vane Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 144 B.5.3 Null-Seeking Differential Pressure Probes . . . . . . . . . . . . . 145