Flight Dynamics, Simulation, and Control For Rigid and Flexible Aircraft [2 ed.] 9781032210032, 9781032210049, 9781003266310

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Flight Dynamics, ­Simulation, and Control Flight Dynamics, Simulation, and Control of Aircraft: For Rigid and Flexible Aircraft explains the basics of non-linear aircraft dynamics and the principles of control-configured aircraft design, as applied to rigid and flexible aircraft, drones, and unmanned aerial vehicles (UAVs). Addressing the details of dynamic modelling, simulation, and control in a selection of aircraft, the book explores key concepts associated with control-configured elastic aircraft. It also covers the conventional dynamics of rigid aircraft and examines the use of linear and non-linear model-based techniques and their applications to flight control. This second edition features a new chapter on the dynamics and control principles of drones and UAVs, aiding in the design of newer aircraft with a combination of propulsive and aerodynamic control surfaces. In addition, the book includes new sections, approximately 20 problems per chapter, examples, simulator exercises, and case studies to enhance and reinforce student understanding. The book is intended for senior undergraduate and graduate mechanical and aerospace engineering students taking Flight Dynamics and Flight Control courses. Instructors will be able to utilise an updated Solutions Manual and figure slides for their course.

Flight Dynamics, ­Simulation, and Control For Rigid and Flexible Aircraft Second Edition

Ranjan Vepa

MATLAB ® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB ® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB ® software.

Second edition published 2023 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2023 Ranjan Vepa First edition published by CRC Press in 2015 Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright. com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-1-032-21003-2 (hbk) ISBN: 978-1-032-21004-9 (pbk) ISBN: 978-1-003-26631-0 (ebk) DOI: 10.1201/9781003266310 Typeset in Times by codeMantra

To my brother, Kosla, who has always been an inspiration

Contents List of Acronyms.....................................................................................................xix Preface.....................................................................................................................xxi Author................................................................................................................... xxiii Chapter 1 Introduction to Flight Vehicles..............................................................1 1.1 1.2

1.3

1.4

1.5 1.6 1.7 1.8 1.9

Introduction................................................................................1 Components of an Aeroplane.....................................................2 1.2.1 Fuselage.........................................................................2 1.2.2 Wings............................................................................2 1.2.3 Tail Surfaces or Empennage.........................................2 1.2.4 Landing Gear................................................................2 Basic Principles of Flight...........................................................3 1.3.1 Forces Acting on an Aeroplane.....................................3 1.3.2 Drag and Its Reduction..................................................4 1.3.3 Aerodynamically Conforming Shapes: Streamlining����������������������������������������������������������������� 5 1.3.4 Stability and Balance....................................................6 Flying Control Surfaces: Elevator, Ailerons and Rudder...........7 1.4.1 Flaps, High-Lift and Flow Control Devices................ 11 1.4.2 Introducing Boundary Layers..................................... 12 1.4.3 Spoilers........................................................................ 14 Pilot’s Controls: The Throttle, the Control Column and Yoke, the Rudder Pedals and the Toe Brakes........................... 15 Modes of Flight........................................................................ 15 1.6.1 Static and In-Flight Stability Margins........................ 15 Power Plant............................................................................... 17 1.7.1 Propeller-Driven Aircraft............................................ 17 1.7.2 Jet Propulsion.............................................................. 18 Avionics, Instrumentation and Systems................................... 18 1.8.1 Autonomous Navigation.............................................. 19 Geometry of Aerofoils and Wings........................................... 21 1.9.1 Aerofoil Geometry...................................................... 21 1.9.2 Chord Line.................................................................. 21 1.9.3 Camber........................................................................ 21 1.9.4 Leading and Trailing Edges........................................ 22 1.9.5 Specifying Aerofoils................................................... 23 1.9.6 Equations Defining Mean Camber Line.....................24 1.9.7 Aerofoil Thickness Distributions................................24 1.9.8 Wing Geometry...........................................................26

vii

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Chapter Highlights..............................................................................26 Exercises.............................................................................................. 30 Answers to Selected Exercises............................................................ 31 References........................................................................................... 32 Chapter 2 Basic Principles Governing Aerodynamic Flows............................... 33 2.1 Introduction.............................................................................. 33 2.2  Continuity Principle................................................................. 33 2.2.1  Streamlines and Stream Tubes.................................... 33 2.3  Bernoulli’s Principle.................................................................34 2.4  Laminar Flows and Boundary Layers......................................34 2.5  Turbulent Flows........................................................................34 2.6  Aerodynamics of Aerofoils and Wings.................................... 35 2.6.1  Flow around an Aerofoil............................................. 35 2.6.2 Mach Number and Subsonic and Supersonic Flows........................................................ 36 2.7  Properties of Air in the Atmosphere........................................ 38 2.7.1 Composition of the Atmosphere: The Troposphere, Stratosphere, Mesosphere, Ionosphere and Exosphere........................................... 38 2.7.2  Air Density.................................................................. 38 2.7.3  Temperature................................................................. 38 2.7.4  Pressure....................................................................... 39 2.7.5  Effects of Pressure and Temperature.......................... 39 2.7.6  Viscosity...................................................................... 39 2.7.7  Bulk Modulus of Elasticity.........................................40 2.7.8 Temperature Variations with Altitude: The Lapse Rate............................................................40 2.8 International Standard Atmosphere (from ESDU 77021, 1986)........................................................40 2.9  Generation of Lift and Drag..................................................... 43 2.10  Aerodynamic Forces and Moments.......................................... 45 2.10.1  Aerodynamic Coefficients...........................................48 2.10.2  Aerofoil Drag.............................................................. 51 Aircraft Lift Equation and Lift Curve Slope.............. 52 2.10.3  2.10.4  Centre of Pressure....................................................... 55 2.10.5  Aerodynamic Centre................................................... 55 2.10.6  Pitching Moment Equation.......................................... 55 2.10.7  Elevator Hinge Moment Coefficient............................ 58 Chapter Highlights.............................................................................. 58 Exercises..............................................................................................60 Answers to Selected Exercises............................................................ 62 References........................................................................................... 62

Contents

ix

Chapter 3 Mechanics of Equilibrium Flight........................................................ 63 3.1 Introduction.............................................................................. 63 3.2  Speeds of Equilibrium Flight...................................................66 3.3  Basic Aircraft Performance...................................................... 68 3.3.1  Optimum Flight Speeds.............................................. 68 3.4  Conditions for Minimum Drag................................................. 71 3.5  Stability in the Vicinity of the Minimum Drag Speed............. 72 3.6  Range and Endurance Estimation............................................ 72 3.7  Trim.......................................................................................... 73 3.8  Stability of Equilibrium Flight................................................. 76 3.9  Longitudinal Static Stability.................................................... 78 3.9.1  Neutral Point (Stick-Fixed).......................................... 79 3.9.2  Neutral Point (Stick-Free)............................................ 79 3.10  Manoeuvrability.......................................................................80 3.10.1  Pull-Out Manoeuvre....................................................80 3.10.2  Manoeuvre Margin: Stick-Fixed................................. 81 3.10.3  Manoeuvre Margin: Stick-Free................................... 82 3.11  Lateral Stability and Stability Criteria..................................... 83 3.12  Experimental Determination of Aircraft Stability Margins....84 3.13  Summary of Equilibrium- and Stability-Related Equations.... 85 Chapter Highlights.............................................................................. 88 Exercises..............................................................................................90 Answers to Selected Exercises............................................................ 95 References...........................................................................................96 Chapter 4 Aircraft Non-Linear Dynamics: Equations of Motion........................97 4.1  Introduction..............................................................................97 Aircraft Dynamics....................................................................97 4.2  4.3  Aircraft Motion in a 2D Plane.................................................. 98 4.4  Moments of Inertia................................................................. 102 Euler’s Equations and the Dynamics of Rigid Bodies........... 104 4.5  Description of the Attitude or Orientation............................. 108 4.6  Aircraft Equations of Motion................................................. 112 4.7  Motion-Induced Aerodynamic Forces and Moments............. 114 4.8  4.9 Non-Linear Dynamics of Aircraft Motion and Stability Axes......................................................................... 117 4.9.1 Equations of Motion in Wind Axis Coordinates, V T, α and β................................................................. 122 4.9.2 Reduced-Order Modelling: The Short-Period Approximations......................................................... 128 4.10  Trimmed Equations of Motion............................................... 130 4.10.1  Non-Linear Equations of Perturbed Motion............. 133 4.10.2  Linear Equations of Motion...................................... 133

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Chapter Highlights............................................................................ 135 Exercises............................................................................................ 135 References......................................................................................... 136 Chapter 5 Small Perturbations and the Linearised, Decoupled Equations of Motion.......................................................................... 137 5.1 Introduction............................................................................ 137 5.2 Small Perturbations and Linearisations................................. 137 5.3 Linearising the Aerodynamic Forces and Moments: Stability Derivative Concept.................................................. 139 5.4 Direct Formulation in the Stability Axis................................ 143 5.5 Decoupled Equations of Motion............................................. 150 5.5.1 Case I: Motion in the Longitudinal Plane of Symmetry.................................................................. 150 5.5.2 Case II: Motion in the Lateral Direction, Perpendicular to the Plane of Symmetry.................. 151 5.6 Decoupled Equations of Motion in terms of the Stability Axis Aerodynamic Derivatives.............................................. 152 5.7 Addition of Aerodynamic Controls and Throttle................... 155 5.8 Non-Dimensional Longitudinal and Lateral Dynamics......... 164 5.9 Simplified State-Space Equations of Longitudinal and Lateral Dynamics................................................................... 170 5.10 Simplified Concise Equations of Longitudinal and Lateral Dynamics................................................................... 171 Chapter Highlights............................................................................ 173 Exercises............................................................................................ 173 Reference........................................................................................... 175 Chapter 6 Longitudinal and Lateral Linear Stability and Control.................... 177 6.1 6.2 6.3

Introduction............................................................................ 177 Dynamic and Static Stability.................................................. 177 6.2.1 Longitudinal Stability Analysis................................ 177 6.2.2 Lateral Dynamics and Stability................................ 186 Modal Description of Aircraft Dynamics and the Stability of the Modes............................................................ 191 6.3.1 Slow–Fast Partitioning of the Longitudinal Dynamics.................................................................. 192 6.3.2 Slow–Fast Partitioning of the Lateral Dynamics...... 195 6.3.3 Summary of Longitudinal and Lateral Modal Equations...................................................................205 6.3.3.1 Phugoid or Long Period.............................205 6.3.3.2 Short Period...............................................206 6.3.3.3 Third Oscillatory Mode.............................206 6.3.3.4 Roll Subsidence.........................................206

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6.3.3.5 Dutch Roll������������������������������������������������� 206 6.3.3.6 Spiral��������������������������������������������������������� 207 6.4 Aircraft Lift and Drag Estimation����������������������������������������� 208 6.4.1 Fuselage Lift and Moment Coefficients��������������������210 6.4.2 Wing–Tail Interference Effects���������������������������������211 6.4.3 Estimating the Wing’s Maximum Lift Coefficient����������������������������������������������������������211 6.4.4 Drag Estimation��������������������������������������������������������212 6.5 Estimating the Longitudinal Aerodynamic Derivatives��������216 6.6 Estimating the Lateral Aerodynamic Derivatives����������������� 222 6.7 Perturbation Analysis of Trimmed Flight����������������������������� 228 6.7.1 Perturbation Analysis of Longitudinal Trimmed Flight������������������������������������������������������������������������ 228 6.7.2 Perturbation Analysis of Lateral Trimmed Flight����232 6.7.2.1 Control Settings for Steady Sideslip����������233 6.7.2.2 Control Settings for Turn Coordination and Banking���������������������������������������������� 234 6.7.3 Perturbations of Coupled Trimmed Flight����������������239 6.7.4 Simplified Analysis of Complex Manoeuvres: The Sidestep Manoeuvre�������������������������������������������239 Chapter Highlights��������������������������������������������������������������������������� 241 Exercises������������������������������������������������������������������������������������������� 244 Answers to Selected Exercises��������������������������������������������������������� 254 Note�������������������������������������������������������������������������������������������������� 255 References���������������������������������������������������������������������������������������� 255 Chapter 7 Aircraft Dynamic Response: Numerical Simulation and Non-Linear Phenomenon������������������������������������������������������������������257 7.1 7.2 7.3

7.4 7.5 7.6

7.7 7.8

Introduction����������������������������������������������������������������������������257 Longitudinal and Lateral Modal Equations���������������������������257 Methods of Computing Aircraft Dynamic Response�������������261 7.3.1 Laplace Transform Method���������������������������������������262 7.3.2 Aircraft Response Transfer Functions����������������������262 7.3.3 Direct Numerical Integration����������������������������������� 266 System Block Diagram Representation��������������������������������� 269 7.4.1 Numerical Simulation of Flight Using MATLAB®/Simulink®����������������������������������������������271 Atmospheric Disturbance: Deterministic Disturbances���������272 Principles of Random Atmospheric Disturbance Modelling������������������������������������������������������������������������������ 282 7.6.1 White Noise: Power Spectrum and Autocorrelation��������������������������������������������������������� 282 7.6.2 Linear Time-Invariant System with Stochastic Process Input������������������������������������������������������������ 283 Application to Atmospheric Turbulence Modelling�������������� 287 Aircraft Non-Linear Dynamic Response Phenomenon���������291

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7.8.1

Aircraft Dynamic Non-Linearities and Their Analysis..................................................................... 292 7.8.2 High-Angle-of-Attack Dynamics and Its Consequences............................................................ 295 7.8.3 Post-Stall Behaviour.................................................. 296 7.8.4 Tumbling and Autorotation....................................... 297 7.8.5 Lateral Dynamic Phenomenon.................................. 297 7.8.6 Flat Spin and Deep Spin............................................ 297 7.8.7 Wing Drop, Wing Rock and Nose Slice.................... 298 7.8.8 Fully Coupled Motions: The Falling Leaf................ 299 7.8.9 Regenerative Phenomenon........................................300 Chapter Highlights............................................................................ 301 Exercises............................................................................................ 301 References......................................................................................... 319 Chapter 8 Aircraft Flight Control...................................................................... 321 8.1 8.2 8.3

8.4

Automatic Flight Control Systems: An Introduction............. 321 Functions of a Flight Control System..................................... 323 Integrated Flight Control System........................................... 334 8.3.1 Guidance System: Interfacing to the Automatic Flight Control System............................................... 338 8.3.2 Flight Management System.......................................340 Flight Control System Design................................................ 341 8.4.1 Block Diagram Algebra............................................344 8.4.2 Return Difference Equation......................................348 8.4.3 Laplace Transform....................................................348 8.4.4 Stability of Uncontrolled and Controlled Systems.... 349 8.4.5 Routh’s Tabular Method............................................ 352 8.4.6 Frequency Response.................................................. 353 8.4.7 Bode Plots.................................................................. 355 8.4.8 Nyquist Plots............................................................. 355 8.4.9 Stability in the Frequency Domain........................... 356 8.4.10 Stability Margins: Gain and Phase Margins............. 356 8.4.11 Mapping Complex Functions and Nyquist Diagrams................................................................... 357 8.4.12 Time Domain: State Variable Representation........... 357 8.4.13 Solution of the State Equations and the Controllability Condition..........................................360 8.4.14 State-Space and Transfer Function Equivalence....... 362 8.4.15 Transformations of State Variables........................... 362 8.4.16 Design of a Full-State Variable Feedback Control Law............................................................... 363 8.4.17 Root Locus Method................................................... 365 8.4.18 Root Locus Principle................................................. 367 8.4.19 Root Locus Sketching Procedure.............................. 367

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8.4.20 Producing a Root Locus Using MATLAB®.............. 367 8.4.21 Application of the Root Locus Method: Unity Feedback with a PID Control Law............................ 368 8.5 Optimal Control of Flight Dynamics..................................... 371 8.5.1 Compensating Full-State Feedback: Observers and Compensators..................................................... 372 8.5.2 Observers for Controller Implementation................. 372 8.5.3 Observer Equations................................................... 373 8.5.4 Special Cases: Full- and First-Order Observers........ 374 8.5.5 Solving the Observer Equations................................ 374 8.5.6 Luenberger Observer................................................. 375 8.5.7 Optimisation Performance Criteria........................... 375 8.5.8 Good Handling Domains of Modal Response Parameters................................................................. 376 8.5.9 Cooper–Harper Rating Scale.................................... 377 8.6 Application to the Design of Stability Augmentation Systems and Autopilots.......................................................... 379 8.6.1 Design of a Pitch Attitude Autopilot Using PID Feedback and the Root Locus Method...................... 379 8.6.2 Example of Pitch Attitude Autopilot Design for the Lockheed F104 by the Root Locus Method........ 382 8.6.3 Example of Pitch Attitude Autopilot Design, Including a Stability Augmentation Inner Loop, by the Root Locus Method........................................ 382 8.6.4 Design of an Altitude Acquire-and-Hold Autopilot.................................................................... 383 8.6.5 Design of a Lateral Roll Attitude Autopilot.............. 386 8.6.6 Design of a Lateral Yaw Damper.............................. 387 8.6.7 Design of a Lateral Heading Autopilot..................... 388 8.6.8 Turn Coordination with Sideslip Suppression........... 388 8.6.9 Application of Optimal Control to Lateral Control Augmentation Design................................... 390 8.7 Performance Assessment of a Command or Control Augmentation System............................................................. 392 8.8 Linear Perturbation Dynamics Flight Control Law Design by Partial Dynamic Inversion.................................... 393 8.8.1 Design Example of a Longitudinal Autopilot Based on Partial Dynamic Inversion......................... 397 8.9 Design of Controllers for Multi-Input Systems...................... 399 8.9.1 Design Example of a Lateral Turn Coordination Using the Partial Inverse Dynamics Method............ 399 8.9.2 Design Example of the Simultaneously Operating Auto-Throttle and Pitch Attitude Autopilot..............400 8.9.3 Two-Input Lateral Attitude Control Autopilot..........402 8.10 Decoupling Control and Its Application: Longitudinal and Lateral Dynamics Decoupling Control................................................. 406

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8.11 Full Aircraft Six-DOF Flight Controller Design by Dynamic Inversion.................................................................408 8.11.1 Control Law Synthesis.............................................. 420 8.11.2 Example of Linear Control Law Synthesis by Partial Dynamic Inversion: Fully PropulsionControlled MD11 Aircraft......................................... 423 Chapter Highlights............................................................................ 425 Exercises............................................................................................ 426 Answers to Selected Exercises.......................................................... 437 References......................................................................................... 437 Chapter 9 Piloted Simulation and Pilot Modelling............................................ 439 9.1 9.2

Introduction............................................................................ 439 Piloted Flight Simulation....................................................... 439 9.2.1 Full Moving-Base Simulation: The Stewart Platform..................................................................... 442 9.2.2 Kinematics of Motion Systems................................. 443 9.2.3 Principles of Motion Control.....................................444 9.2.4 Motion Cueing Concepts...........................................444 9.3 Principles of Human Pilot Physiological Modelling..............448 9.3.1 Auricular and Ocular Sensors................................... 450 9.4 Human Physiological Control Mechanisms........................... 453 9.4.1 Crossover Model........................................................ 454 9.4.2 Neal–Smith Criterion................................................ 457 9.4.3 Pilot-Induced Oscillations......................................... 458 9.4.4 PIO Categories.......................................................... 459 9.4.5 PIOs Classified under Small Perturbation Modes.....460 9.4.6 Optimal Control Models........................................... 461 9.4.7 Generic Human Pilot Modelling............................... 461 9.4.8 Pilot–Vehicle Simulation...........................................465 9.5 Spatial Awareness...................................................................466 9.5.1 Visual Displays.......................................................... 467 9.5.2 Animation and Visual Cues......................................468 9.5.3 Visual Illusions..........................................................469 Chapter Highlights............................................................................ 472 Exercises............................................................................................ 472 References......................................................................................... 477

Chapter 10 Flight Dynamics of Elastic Aircraft.................................................. 479 10.1 10.2 10.3 10.4

Introduction............................................................................ 479 Flight Dynamics of Flexible Aircraft..................................... 479 Newton–Euler Equations of a Rigid Aircraft.........................480 Lagrangian Formulation......................................................... 486 10.4.1 Generalised Coordinates and Holonomic Dynamic Systems...................................................... 487

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10.4.2 10.4.3 10.4.4 10.4.5 10.4.6 10.4.7

Generalised Velocities............................................... 487 Virtual Displacements and Virtual Work................. 488 Principle of Virtual Work.......................................... 489 Euler–Lagrange Equations........................................ 489 Potential Energy and the Dissipation Function......... 493 Euler–Lagrange Equations of Motion in Quasi-Coordinates..................................................... 494 10.4.8 Transformation to Centre of Mass Coordinates........ 499 10.4.9 Application of the Lagrangian Method to a Rigid Aircraft............................................................ 501 10.5 Vibration of Elastic Structures in a Fluid Medium................506 10.5.1 Effects of Structural Flexibility in Aircraft Aeroelasticity............................................................. 510 10.5.2 Wing Divergence....................................................... 510 10.5.3 Control Reversal........................................................ 511 10.5.4 Wing Flutter.............................................................. 513 10.5.5 Aerofoil Flutter Analysis........................................... 513 10.6 Unsteady Aerodynamics of an Aerofoil................................. 521 10.7 Euler–Lagrange Formulation of Flexible Body Dynamics.... 524 10.8 Application to an Aircraft with a Flexible Wing Vibrating in Bending and Torsion.......................................... 526 10.8.1 Longitudinal Small Perturbation Equations with Flexibility.................................................................. 526 10.8.2 Lateral Small Perturbation Equations with Flexibility.................................................................. 530 10.9 Kinetic and Potential Energies of the Whole Elastic Aircraft....................................................................... 531 10.9.1 Kinetic Energy.......................................................... 532 10.9.2 Simplifying the General Expression......................... 533 10.9.3 Mean Axes................................................................ 534 10.9.4 Kinetic Energy in terms of Modal Amplitudes........ 534 10.9.5 Tisserand Frame........................................................ 534 10.10 Euler–Lagrange Matrix Equations of a Flexible Body in Quasi-Coordinates.............................................................. 535 10.11 Slender Elastic Aircraft.......................................................... 537 10.12 Aircraft with a Flexible Flat Body Component...................... 538 10.12.1 Elastic Large Aspect Ratio Flying Wing Model....... 539 10.12.2 Flexible Aircraft in Roll............................................ 539 10.13 Estimating the Aerodynamic Derivatives: Modified Strip Analysis......................................................................... 539 Chapter Highlights............................................................................540 Exercises............................................................................................540 Answers to Selected Exercises.......................................................... 548 References......................................................................................... 549

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Chapter 11 Dynamics and Control of Drones and Unmanned Aerial Vehicles.................................................................................. 551 11.1 Introduction............................................................................ 551 11.2 Dynamics of a Generic Drone................................................ 552 11.3 Rigid Body Kinematics.......................................................... 552 11.3.1 Defining the Body Frame....................................... 552 11.3.2 Defining the Body Angular Velocity Components............................................................ 555 11.4 Translational Dynamics.......................................................... 556 11.5 Attitude Dynamics................................................................. 557 11.6 Attitude Kinematics................................................................ 559 11.6.1 The Quaternion Representation of the Attitude...... 560 11.6.2 The Relations between Quaternion Rates and Angular Velocities.................................................. 561 11.7 Aerodynamic Forces............................................................... 562 11.8 Propulsion-Based Control......................................................564 11.9 Stability and Control..............................................................564 11.10 Automatic Flight Control....................................................... 565 11.11 Autonomous Flight Control.................................................... 565 11.12 The Quadrotor Drone............................................................. 566 11.12.1 Dynamics of the Quadrotor Drone......................... 566 11.12.2 Quadrotor Control Allocation................................. 569 11.12.3 Quadrotor Control Strategies.................................. 571 11.12.4 PID Control of a Quadrotor.................................... 571 11.13 Optimal Controller Synthesis for Drones............................... 577 11.14 Unconventional Multi-Rotor Drones...................................... 580 11.14.1 Quadrotors with Bi-Directional Motors................. 580 11.14.2 Quadcopters: Dynamics of the Quadcopter............ 580 11.14.3 Body Forces and Body Moments Acting on a Quadcopter.............................................................. 582 11.14.4 The Unsymmetrically Actuated Quadcopter.......... 583 11.14.5 The Pentacopter...................................................... 584 11.14.6 Equations of Motion of a Pentacopter.................... 584 11.14.7 The Hexa-Rotor and the Hexa-copter..................... 586 11.14.8 Dynamics of a Hexa-Rotor Drone.......................... 586 11.14.9 The Basic Hexa-Rotor Configuration: Derivation of the Body Forces and Moments......... 586 11.14.10 Alternate Tri-Axial Multi-Rotor Configurations.... 590 11.14.11 The Hexa-Rotor Configuration with Two Rotors Tilted: The Hexa-copter.......................................... 591 11.14.12 A Hexa-copter with Three Tilt-Controlled Rotors...................................................................... 593 11.14.13 Six DOFs Configuration: Derivation of the Body Forces and Moments..................................... 594

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11.14.14 Control of a Fully Actuated Hexa-rotor Drone: Decoupling.................................................. 596 11.14.15 The Octocopter and Over-Actuated Multi-Rotor Drones..................................................................... 599 11.14.16 Dynamics of an Octocopter Drone......................... 599 11.14.17 Nonlinear and Linear Dynamic Modelling of Multi-Rotor Drones.................................................603 11.14.18 H∞ Optimal Control of an Octocopter drone..........605 11.14.19 Typical Simulation Example...................................605 11.15 Drones and Unmanned Aerial Vehicles with Aerodynamic Lifting Surfaces...............................................606 Chapter Highlights............................................................................607 Exercises............................................................................................608 References.........................................................................................609 Index....................................................................................................................... 611

List of Acronyms AC ADF amc AR BDF CG CH CM CP DME DRONE DOF EFIS EIS EPR FBW FCU FDAU FMGS GPS HSI HUD IAS IFS ILS INS ISE KF LQG LQR NDF NP PD PID PIO psfc RMI SLAM SISO TCAS TR Tsfc

Aerodynamic centre Automatic direction finding Aerodynamic mean chord Aspect ratio Backward difference formula Centre of gravity Cooper–Harper (rating) Centre of mass Centre of pressure Distance measuring equipment Digital Remotely Operated Navigation Equipment Degrees of freedom Electronic flight information system Electronic information system Engine pressure ratio Fly by wire Flight control unit Flight data acquisition unit Flight management and guidance system Global positioning system Horizontal situation indicator Head-up displays Indicated airspeed In-flight simulation Instrument landing system Inertial navigation system Integral squared error Kalman filter Linear quadratic Gaussian Linear quadratic regulator Numerical differentiation formula Neutral point Proportion derivative Proportional, integral, derivative Pilot-induced oscillation Power-specific fuel consumption Radio magnetic indicator Simultaneous localisation and mapping Single input, single output Traffic collision avoidance system Trapezoidal rule Thrust-specific fuel consumption xix

xx

UAV VHF VOR VTOL

List of Acronyms

Unmanned aerial vehicle Very high frequency VHF omni-range or vestibulo–ocular reflex Vertical take-off and landing

Preface In the last decade, we have seen a phenomenal increase in air travel to phenomenal levels. A plethora of low-cost airlines have made it possible for the common man to travel between continents at relatively reasonable fares. This has also led to the design of newer energy-efficient aircraft incorporating the principles of feedback control. These aircraft have generally tended to be lighter and more flexible because of the use of composite structures and other smart materials. It therefore becomes important to consider the aircraft not as a rigid body, as has been done traditionally in the past, but as an inherently flexible body. Such considerations will require a revision of a number of traditional concepts, although many of them can be easily adapted to the flexible aircraft. This book addresses the core issues involved in the dynamic modelling, simulation, and control of a selection of aircraft. The principles of modelling and control could be applied to both traditional rigid aircraft and more modern flexible aircraft. A primary feature of this book is that it brings together a range of diverse topics relevant to the understanding of flight dynamics, its regulation and control, and the design of flight control systems and flight simulators. This book will help the reader understand the methods of modelling both rigid and flexible aircraft for controller design application as well as gain a basic understanding of the processes involved in the design of control systems and regulators. It will also serve as a useful guide to study the simulation of flight dynamics for implementing monitoring systems based on the estimation of internal system variables from measurements of observable system variables. The book brings together diverse topics in flight mechanics, aeroelasticity, and automatic controls. It would be useful to designers of hybrid flight control systems that involve advanced composite structure-based components in the wings, fuselage, and control surfaces. The distinctive feature of this book is that it introduces case studies of practical control laws for several modern aircraft and deals with the use of non-linear model-based techniques and their applications to flight control. Chapter 1 begins with an introduction and reviews the configuration of a typical aircraft and its components. Chapter 2 deals with the basic principles governing aerodynamic flows. Chapter 3 covers the mechanics of equilibrium flight and describes static equilibrium, trimmed steady level flight, the analysis of the static stability of an aircraft, static margins stick-fixed and stick-free, modelling of control-surface hinge moments, and the estimation of the elevator angle for trim. Basic concepts of stability based on disturbances to one parameter alone are discussed. The effects of a change in the angle of attack on the pitching moment and its application to stability assessment are discussed. Also considered are steady flight at an angle to the horizontal and the definition of flight path, incidence and pitch angles, and the heading, yaw, and sideslip angles. The assessment of manoeuvrability and the application of margins required for a steady pull-out from a dive are also introduced. Chapter 4 is dedicated to the development of the non-linear equations of motion of an aircraft, including simple two-dimensional dynamic models, and the development xxi

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of the aircraft’s equations of motion in three dimensions. The general Euler equations of rigid body and the definition and estimation of moments of inertia matrix are discussed. The definitions of motion-induced aerodynamic forces and moments and the need for various reference axes that are fixed in space, fixed to the body, and fixed in the wind as well as the definition of stability axes are clearly explained. The non-linear dynamics of aircraft motion in the stability axes is derived in terms of both body axis degrees of freedom and wind axis variables. The concept of nonlinear reduced-order modelling is introduced, and the short period approximation is discussed. Finally, the trimmed equations of motion as well as the non-linear perturbation equations of motion are derived. The concept of linearisation is also introduced, and the linear equations of aircraft motion are briefly discussed. In Chapter 5, the small perturbation equations of motion are described in detail, and the equations are expressed as two sets of decoupled equations representing the longitudinal and lateral dynamics. Chapter 6 introduces the methodology of linear stability analysis and provides a modal description of aircraft dynamics. The application of small perturbation equations in determining the control setting angles for executing typical manoeuvres is also discussed in this chapter. Chapter 7 covers the evaluation of aircraft dynamic response and the application of MATLAB®/Simulink® in determining the aircraft’s response to typical control inputs. A basic introduction to aircraft non-linear dynamic phenomenon is also presented in this chapter. Chapter 8 deals with aircraft flight control, the design of control laws, stability augmentation, autopilots, and the optimal design of feedback controllers. Chapter 9 describes flight simulators and the principles governing their design. The penultimate, The last chapter is dedicated to the flight dynamics of elastic aircraft, including the principles of aeroelasticity from an aircraft perspective. I thank my colleagues and students at the School of Engineering and Material Science, Queen Mary University of London, for their support in this endeavour. I thank my wife Sudha for her love, understanding, and patience. Her encouragement and support provided me the motivation to complete this project. I also thank our children Lullu, Satvi, and Abhinav for their understanding during the course of this project. Ranjan Vepa London, United Kingdom MATLAB® is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: [email protected] Web: www.mathworks.com

Author Dr. Ranjan Vepa earned his PhD in applied mechanics from Stanford University, Stanford, California, specialising in the area of aeroelasticity under the guidance of the late Professor. Holt Ashley. He currently serves as a Reader in Aerospace Engineering in the School of Engineering and Material Science, Queen Mary University of London, where he has also been the programme director of the Avionics Programme since 2001. Prior to joining Queen Mary, he was with the NASA Langley Research Center, where he was awarded a National Research Council Fellowship and conducted research in the area of unsteady aerodynamic modelling for active control applications. Subsequently, he was with the Structures Division of the National Aeronautical Laboratory, Bengaluru, India, and the Indian Institute of Technology, Chennai, India. Dr. Vepa’s research interests include the design of flight control systems and the aerodynamics of morphing wings and bodies with applications in smart structures, robotics, and biomedical engineering and energy systems, including wind turbines. He is particularly interested in the dynamics and in the robust adaptive estimation and the control of linear and non-linear aerospace, energy, and biological systems with uncertainties. The research in the area of the aerodynamics of morphing wings and bodies is dedicated to the study of aerodynamics and its control. This includes the use of smart structures and their applications to the control of aerospace vehicles, jet engines, robotics, and biomedical systems. Other applications of this work are to wind turbine and compressor control, maximum power point tracking, flow control over smart flaps, and the control of biodynamic systems. Dr. Vepa currently conducts research on biomimetic morphing and aerodynamic shape control and their applications, which include feedback control of aerofoil section shape in subsonic and transonic flow for UAV, airship and turbomachine applications and integration of computational aeroelasticity (CFD, computational fluid dynamics/CSD, computational structural dynamics) with deforming grids as well as their applications to active flow control. Of particular interest are the boundary layer instabilities in laminar flow arising due to various morphing-induced disturbances. Dr. Vepa has also been studying the optimal use and regulation of alternate power sources such as fuel cells in hybrid electric vehicle power trains, modelling of fuel cell degradation, and health monitoring of aircraft structures and systems. With regard to structural health monitoring and control, observer and Kalman filter–based crack detection filters are being designed and applied to crack detection and isolation in aeroelastic aircraft structures such as nacelles, casings, turbine rotors, and rotor blades. Feedback control of crack propagation and compliance compensation in cracked vibrating structures is also being investigated. Another issue is the modelling of damage in laminated composite plates, non-linear flutter analysis of their plates, and their interaction with unsteady aerodynamics. These research studies are contributing to the holistic design of vision-guided autonomous UAVs, which are expected to be extensively used in future.

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Dr. Vepa is the author of seven books titled: Electric Aircraft Dynamics: A Systems Engineering Approach (CRC Press, 2020), Dynamics and Control of Autonomous Space Vehicles and Robotics (Cambridge University Press, 2019), Nonlinear Control of Robots and Unmanned Aerial Vehicles: An Integrated Approach (CRC Press, 2016), Flight Dynamics Simulation and Control of Aircraft: Rigid and Flexible (CRC Press, 2014), Dynamic Modelling, Simulation and Control of Energy Generation (Springer, 2013), Dynamics of Smart Structures (John Wiley, 2010) and Biomimetic Robotics: Mechanisms and Control (Cambridge University Press, 2009). He is a member of the Royal Aeronautical Society, London; the Institution of Electrical and Electronic Engineers (IEEE), New York; a fellow of the Higher Education Academy; a member of the Royal Institute of Navigation, London; and a chartered engineer.

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Introduction to Flight Vehicles

1.1  INTRODUCTION While aerodynamics is the study of flows past and over bodies, the principles of flight are governed by the dynamics and aerodynamics of flight vehicles. The focus of this chapter is on the general principles of flight and primary features of aircraft. Further details may be found in Anderson [1] and Shevell [2]. As the aerodynamics of bodies is greatly influenced by their external geometry, the aerodynamics of flight vehicles is entirely determined by their external geometry. The external geometry is in turn completely influenced by the entire complement of components external to the vehicle. The basic architecture of a typical aeroplane, the simplest of flight vehicles, is well known to any cursory observer of aeroplanes. It can be considered to be the assemblage of a number of individual components. The principal external components are the fuselage, the left and right wings, the power plant pods or nacelles, the tail plane unit comprising of the horizontal and vertical stabilisers, the various control flaps and control surfaces and the landing gear. When the components are assembled or integrated together, a complete external picture of a typical aeroplane emerges. A typical planform or top-down view of an aeroplane is shown in Figure 1.1.

FIGURE 1.1  Typical planform view of an aeroplane. DOI: 10.1201/9781003266310-1

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1.2  COMPONENTS OF AN AEROPLANE The primary components of an aeroplane are the fuselage, the wing, the tail surfaces which are collectively referred to as the empennage, the power plant, the various control surfaces used to control the flight of the aeroplane and the landing gear.

1.2.1  Fuselage The fuselage is the main body of any aeroplane, housing the crew and passengers or the cargo or payload and the like.

1.2.2  Wings The wings are the main lifting element of the aeroplane. They are comprised of the wing leading and trailing edges, flaps and slats that are used to augment the lift on the wing, ailerons to enable the aeroplane to bank while turning and spoilers that are capable of reducing the wing lift during landing and act as speed brakes. The high-lift devices controlled and operated below the wing permit the wing to develop the necessary lift during take-off when a large passenger jet attains speeds of the order of 320 km/h after accelerating down a runway of length 3–4 km. The controls and drive mechanisms linking these devices are usually shrouded in canoe-shaped fairings attached to the underside of the wing. The wing essentially carries the entire aeroplane and all other associated systems. The wing is essentially a single aerodynamic element although it extends symmetrically on either side of the fuselage.

1.2.3  Tail Surfaces or Empennage The tail surfaces are the basic elements that stabilise and control the aeroplane. Normally, both the vertical and horizontal tail surfaces have a fixed forward portion and a hinged rearward portion. The forward portion of the horizontal tail surface is known as the stabiliser, while the rearward hinged portion on the same surface is known as the elevator. On many long-haul airliners, the horizontal stabiliser is an all-movable unit. On the vertical tail, the fixed forward portion is known as the fin, while the hinged rearward portion is known as the rudder. Both on the rudder and on the elevator are additional hinged surfaces known as the trim tabs which are used to adjust the forces on the pilot’s control column (which controls the movement of the elevator) and rudder pedals so that these are forcefree. Together, the entire horizontal and vertical tail surface assembly is known as the empennage.

1.2.4  Landing Gear To enable an aeroplane to operate from land, aeroplanes are provided with landing gear comprising of wheels with types mounted on axles. Brakes are integral elements while the axles are attached via supporting struts and shock absorbers to the

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fuselage. To minimise drag during take-off and in steady flight, cowlings and retractable mechanisms are provided. The latter permit the retraction of the entire landing gear to an enclosed housing within the fuselage once the aeroplane is airborne.

1.3  BASIC PRINCIPLES OF FLIGHT 1.3.1  Forces Acting on an Aeroplane Consider the equilibrium of an aeroplane on the ground. Its weight may be regarded as acting vertically downwards through the aeroplane’s centre of gravity (CG), and this is balanced by two sets of reactions acting vertically upwards, one at the points of contact of the main undercarriage and the ground surface and the other either at the nose wheel or at tail skid depending on the type of aeroplane. To maintain an aeroplane in vertical equilibrium during flight, the vertical reactions at the main undercarriage and nose wheels must be replaced by equivalent upward forces: the lift components acting on the main wing and tail plane surface. In the days of the lighter than air balloons, which were axially symmetric about the CG axis, the reaction was a single lift force due to the buoyancy. This force was due to the difference in the weight of the air displaced by the balloon and the gas contained within and acted in the vicinity of the CG. However, with the arrival of the airship, the forces were no longer acting in a single vertical line. Typically, a steady level flight is held in balance or equilibrium by a combination of forces (Figure 1.2a). The forces comprise the following:

1. The lift on the aeroplane with the principal contributions being due to the wing and horizontal tail; 2. The drag which consists of two main components the profile drag and the induced drag; 3. The thrust produced by the power plants; and 4. The weight of the aeroplane.

In addition to the equilibrium of forces, the forces on the tail plane contribute principally towards rotational moments acting on the aeroplane. All the rotational moments acting on the aeroplane must cancel each other to ensure that the aeroplane is in rotational equilibrium. Rotational equilibrium is essential so the aeroplane can maintain steady orientation during a long and sustained flight. Thus, the attitude of the aeroplane must remain steady during extended periods of flight. The principal phenomenon that is responsible for holding the aeroplane in flight is the wing lift which is caused as a result of the generation of a low-pressure or suction region over the top surface of the wing and high-pressure region below the lower surface of the wing (Figure 1.2b). The region of low pressure on the top surface of the wing is caused by the flow of air over the curved surface of the wing with a resultant increase in flow velocity and consequent decrease in pressure relative to the rest of the atmosphere. Similarly, the region of high pressure below the lower surface of the wing represents a region where the pressure is relatively greater than in the

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FIGURE 1.2  (a) Forces acting on aeroplane in steady, level, equilibrium flight and (b) pressure distribution on a wing: front and side view of a typical wing section.

surrounding air. The result of these two complementary effects on the two surfaces of the wing is the generation of lift. This generation is due to the fact that the two flows emerging from the upper and lower surfaces at the trailing edge of the wing result in a downwash or vortical flow. Thus, the wing experiences an upward and opposite reaction in the form of lift. The lift is directly proportional to the air density and also a function of the airspeed; the higher the airspeed, the greater the lift generated by the wings. An increase in the wing surface area increases the lift in direct proportion. The wing camber and angle of attack are the other parameters that cause the lift to increase.

1.3.2  Drag and Its Reduction As for the drag on the aeroplane, there are two distinct types of drag that act to retard the aeroplane when it is forward flight. The first is profile drag that is itself made up of two components, form drag and skin friction drag. The former is produced due to the finite shape of the aeroplane as the result of the streamlined flow around its body.

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Thus, the shape of the body is almost always optimally streamlined to reduce this component to minimum. The latter component is produced due to the viscous friction between the aeroplane’s skin and the airflow around the body. The airflow results in the formation of a thin boundary layer where the flow velocity reduces to zero as one gets closer to the skin of the aeroplane. This type of drag depends to a large extent on the thickness of the boundary layer that must be kept to a minimum to reduce the drag. These aforementioned two components that constitute the profile drag have one common feature: they both increase markedly as the speed of the aeroplane increases, and the increase is directly proportional to the square of the airspeed. The second type of drag experienced by an aeroplane is the induced drag. Due to the pressure difference between the top and bottom side of the wing surface, there is a spillover of air, particularly at the wing tips, from the bottom to the top. To a large extent, the induced drag is caused by a meeting of the airflow emerging from the upper and lower surfaces at the trailing edge, at a finite angle, resulting in the formation of vortices, set-up due to the air spilling over. The vortices accumulate at the wing tips to produce a rotating flow of air, rotating in the direction of the wing root and resulting in a wing tip vortex. These wing tip vortices are the principal contributors to the induced drag which is caused by the energy dissipated in rotating the air. Due to the wing tip vortices being washed away at a faster rate at higher airspeeds, there is a decrease in the induced drag with the increase in the speed. As a result of the different behaviours of the two types of drag as speed increases, there is a speed at which the drag is in fact a minimum. At this speed, the contributions to the total drag by the two types of drag are equal, and as a result, either an increase or a decrease in the airspeed causes the drag to increase. Aeroplanes are generally flown at a cruise speed that is just above the minimum drag speed as it is important to operate on the right side of the drag curve. Operating on the wrong side implies that a small reduction in the airspeed increases the drag substantially and unacceptably large increments of power are required to increase the aeroplane’s speed. Operating on the wrong side is not acceptable and unsafe especially when the power plant is already being operated near its maximum power output. Finally, the drag must be low and well below the thrust generated by the aeroplane’s power plants or propulsive units to ensure that the aeroplane may be accelerated fast enough as may be desired during various phases of the flight.

1.3.3  Aerodynamically Conforming Shapes: Streamlining There is a patent need to reduce the drag acting on an aeroplane. This is done by shaping the envelope of the various components in the flow or streamlining. By appropriately shaping the envelope so that all directions tangential to it are parallel to the directions of the flow adjacent to it, the drag could be considerably minimised. In most cases, this is because air is able to smoothly pass over the body generating any eddies or turbulence. The generation of a turbulent wake behind the body could substantially increase the drag. Streamlining is also necessary for the generation of lift. There are indeed three effects that contribute to wing lift: (1) the shape of the aerofoil or wing section is such that the velocity of the flow must necessarily be higher over the upper surface

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than below the lower surface; (2) the velocity of the flow field gives rise to a pressure differential or suction that is a principal contributor to the wing lift; and (3) there is the effect of the downward inclination of the streamlines behind the aerofoil section, known as downwash, as well as the slight upward inclination of the flow in the vicinity of the leading edge or front of the aerofoil, known as the upwash. Together, the upwash and the downwash are responsible for producing a curved streamlined flow with a resulting inertia force acting outwards. This is a significant contributor to the lift acting on the wing section.

1.3.4  Stability and Balance The weight distribution on an aeroplane also plays a critical role in ensuring a stable flight. By stability, we mean the ability of the aeroplane to return to its equilibrium orientation when disturbed by an external effect of any kind. To ensure stability, it is essential that the CG of the aeroplane is sufficiently forward. Thus, it is particularly important to ensure not only that the weight is laterally balanced but also that the aeroplane is not too tail heavy. Maintaining rotational balance is an important requirement in flight. Lift and weight generally do not act at the same point during a particular flight of an aeroplane. The centre of aerodynamic pressure can be expected to change continually depending on the selection of control surfaces deployed during the different phases of the flight. Moreover, the weight distribution around the aeroplane is also changing due to variations in payloads and fuel consumption. Fuel can account for up to 30%– 45% of an aeroplane’s weight, while in an airline, the total weight of the passengers and other payloads could weigh as much as 15%–20% of the maximum take-off weight. Thus, ensuring stability is a difficult proposition. The problem is overcome by making the entire horizontal tail plane movable so it could be deployed as a stabilising surface. The tail plane generates lift, and as a result of its long moment arm, it is adequate to restore the aeroplane to an equilibrium position when a disturbing force acts at the CG. The movable or variable position tail plane is used to rebalance the aeroplane and particularly to maintain equilibrium when there are changes in the aeroplane’s weight and CG location. Thus, when the CG is aft of the centre of pressure (CP), the aeroplane is tail heavy, and it is essential to stabilise the aeroplane. At this stage, the stabiliser is moved up to decrease the lift on the tail unit and hence rebalance the aeroplane. This process of balancing the aeroplane by movement of the stabiliser is known as trimming. On the smaller general aviation aeroplane, this function is performed by the trim tabs that are smaller movable control surfaces hinged to the rear of the elevator and rudder. Aeroplanes that are provided with trim tabs generally have fixed stabilisers. Some aeroplanes are provided with both an allmoving horizontal tail plane, for automatic trim, and a full set of trim tabs for manual trimming. To be able to trim the aeroplane, the pilot must have a feel of the out-ofbalance forces. A feel unit usually provides this feedback, and the pilot usually feels the pressure of out-of-balance forces on the control column. When the aeroplane is trimmed, the control column is relieved of the out-of-balance feedback and is free of any forces acting on it. Thus, the aeroplane may be flown in a stable condition with handsoff of the control column.

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1.4  F LYING CONTROL SURFACES: ELEVATOR, AILERONS AND RUDDER To understand the fundamentals of the dynamics of heavier than an aeroplane, it is essential to first understand not only the basic principles of flight but also its control. The aeroplane in level flight at constant speed can be considered to be flying in equilibrium. The weight of the craft is completely balanced by the lift generated by the wings of the aeroplane. The thrust imparted to it by the engines is completely balanced by the drag. The lift is generated by the flow of air over the surface of the wing that is designed to have a special cross section. When the aeroplane loses speed, there is also a loss of lift that must be compensated, if the aeroplane is to fly at constant altitude. The aeroplane compensates the loss of lift by increasing its angle of attack that results in an increased lift. However, there is a limiting angle (about 15°) beyond which any further increase in the angle of attack only causes the aeroplane to lose lift due to flow separation over the upper surface of the wing as illustrated in Figure 1.3. Consequently, the aeroplane stalls and any further increase in the angle of attack or reduction in speed results in a dramatic loss of lift. The speed at which this condition of stalling occurs is the stalling speed that is always the same for a particular aeroplane. The most dangerous moments in the flight of an aeroplane are during take-off and landing. At these stages in the flight, there is demand for maximum lift at low speeds. To generate additional lift during these low-speed stages of the flight, the aeroplane is provided with high-lift devices such as retractable flaps (the Fowler flaps) and movable slats in the leading-edge region which can effectively increase the curvature

FIGURE 1.3  Flow separation and the onset of stall. (a) Flat plate aerofoil at 0° incidence, (b) flat plate aerofoil at 15° incidence, (c) aerofoil at 13° incidence and (d) aerofoil at 18° incidence. Trailing edge separation initiated.

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of the wing section or aerofoil and thus generate the additional lift. After take-off, every effort is generally made to reduce the aeroplane’s drag, thereby increasing its flight speed. To do this, the landing gear is retracted and held within the belly of the aeroplane, so the shape of the aeroplane is apparently streamlined and the drag is minimised. The flight of the aeroplane is controlled by means of the controllers within the cockpit of the aeroplane: the control column, the throttle levers, the rudder pedals and the toe brakes. These controls allow a whole family of control surfaces to be controlled indirectly using intermediate, electro-hydraulically operated mechanisms, known as power control units. Figure 1.4 shows the complete complement of controls on a typical aeroplane. The control columns operate the elevator when moved fore and aft. When the elevator moves down, the additional lift generated on the tail plane forces the aeroplane to pitch nose downwards and vice versa. The elevators are hinged at the trailing edge of the horizontal stabiliser. The elevators are generally operated by the power control units, but on most aeroplanes, there is the option of manual reversion, so the pilot could, when necessary, take control and manually operate them. When operated by the power control units, there is need for some form of artificial feel. The artificial feel is provided by an actuator applying a force on the control column. The force is computed by the feel computer which receives its inputs from the pilot, the static pressure ports and the horizontal stabiliser setting.

FIGURE 1.4  The complete complement of controls on a typical aeroplane.

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The horizontal stabiliser’s function is to provide for longitudinal trim. This is accomplished by changing the incidence angle of the horizontal stabiliser. It may also be driven by an electro-hydraulic power control unit or manually by cables. On some aeroplanes, increase in the airspeed causes the CP to move aft and the aerodynamic centre forwards causing the aeroplane to tuck. In this state, the natural phugoid mode of the aeroplane is absent, and the aeroplane could come dangerously close to being unstable. To avoid this behaviour, the horizontal stabiliser is sometimes fitted with an automatic pitch/trim compensator. Horizontal stabilisers are generally set in motion by switches on the pilot’s control column. Trimming may be achieved either automatically or manually. When the control column is moved or rotated from left to right, the ailerons at the far end of each of the two wings rotate in a differential manner, thereby generating a rolling moment. Thus, the aeroplane banks, the angle of bank being directly proportional to the differential moment of the ailerons. Roll may not only be initiated by the ailerons but also be controlled by them. On some aeroplanes, there are two ailerons on each wing. The outboard pair is usually locked in with the wing in high-speed flight while they may be proportionally controlled at low speeds. The outboard pair is not used when the flaps are deployed. Like the case of the elevator, it is often possible to revert to manual control and artificial feel is also provided. The artificial feel unit is usually a spring-loaded roller cam mechanism which is responsible for providing a feedback force to the control column that is directly proportional to the roll actuator command input. The rudder pedals operate the rudder that generates the necessary turning moment to turn the aeroplane. The rudder generally provides for the control of yaw (nose right or nose left). Some aeroplanes are provided with dual rudders, each of which is split into two separately actuated sections. To protect the vertical tail from structural damage that may result from excessive rudder deflection, rudder travel is limited by incorporating signal limiters in the rudder control circuits. The rudder control system also incorporates, most often, a yaw damper which receives inputs from a yaw rate gyro and provides additional signals to the rudder power control unit so as to move the aeroplane in the direction opposing the yaw motion and in proportion to the yaw rate. The yaw damper is not usually operational in the manual reversionary mode. An artificial feel unit similar to the one fitted to the ailerons is also fitted to the rudder. The toe brakes apply braking to the wheel assemblies on the respective sides while allowing for differential braking to supplement the rudder on the ground. Various types of control tabs, balance tabs and differentially controlled balance panels are also used in aeroplane control. These devices are generally used to balance the forces or moments acting on the control column in the respective directions. This is achieved without adversely affecting the control forces and moments generated by the main control surface and thus maintaining the control column in a force-free condition. Thus, the tabs can mechanically fly the elevator, aileron or rudder while effectively relieving the pilot of having to provide a command input to the control column. The pilot may then fly the aeroplane in the particular trimmed condition in a hands-free mode. These controlling movements are illustrated in Figures 1.5–1.7.

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FIGURE 1.5  Operation of the elevator. (a) Elevator down results in aeroplane nose down and (b) elevator up results in aeroplane nose up.

FIGURE 1.6  Operation of the aileron: up aileron forces wing down and down aileron forces wing up, resulting in bank for turning left; aeroplane continues to turn left when ailerons are returned to the normal position.

FIGURE 1.7  Operation of the rudder: moving the rudder to the left turns the aeroplane to the left and vice versa.

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1.4.1  Flaps, High-Lift and Flow Control Devices There are a plethora of high-lift devices that may be used to improve the lift characteristics of the aeroplane’s primary lifting surfaces during take-off or other phases of the flight. Broadly, all flow control devices fall into five primary categories:

1. Short-chord/short-span passive devices. 2. Single multi-element/multi-surface variable camber or deployable systems. 3. Blown or suction systems. 4. Inflatable systems including leading-edge devices. 5. Active/passive vortex and circulation control systems.

High-lift devices are usually deployed to increase the lift force. However, there is also a substantial increase in the drag accompanying any increase in lift. During take-off, an increase in the lift is generally required to reduce the unstick speed and take-off run, while during landing, there is need to reduce the landing speed and to reduce the landing run. Thus, the increase in drag can be effectively optimised in reducing the take-off and landing runs. One of several short-chord/short-span passive devices is available to reduce the wing lift over sections of the wing surface to achieve flow control. Although there are several methods available to increase the wing lift, single control surfaces or multi-element/multi-surface variable camber or deployable systems are normally used on most aeroplanes. These generally offer almost negligible resistance when they are not deployed, and their deployment is completely controlled by the pilot. Wing leading-edge deflection, at high angles of attack, is essential to impede stall, thus enabling to attain higher angles of attack, thus generating greater lift. Effectively, the leading-edge deflection of the wing results in an increased curvature of the wing section. This is achieved by a combination of slats, slots and flaps (Figures 1.8 and 1.9).

FIGURE 1.8  Typical complement of trailing edge high-lift flaps. (a) Plain hinged flap, (b) slotted flap, (c) double-slotted flap, (d) Wragg or external aerofoil flap, (e) split flap and (f) fowler flap (which is moved down to the rear).

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FIGURE 1.9  Handley page leading-edge slat (which is pulled out into place by suction at high angles of attack).

1.4.2  Introducing Boundary Layers The very thin layer of air in which the velocity is gradually increasing from zero to that of the airstream is called the boundary layer. Viscous friction plays an important part in its evolution, and typically the boundary layer affects the streamline flow, which is outside it. The separation of the boundary layer from the surface of the wing can result in an extreme loss of lift. Boundary layer separation due to adverse pressure gradients on lifting surfaces due to high angles of attack or due to transonic shock effects is the primary cause for the flow separation followed by a loss of lift. Boundary layer separation also causes an increase in the drag. Thus, there is an increased demand for fuel and loss of performance. The unsteady flow associated with separation leads to a random loading on the wing that results in the so-called phenomenon of buffeting. There are several techniques used to control boundary layer separation, and these are as follows:

1. Vanes. 2. Flow control rails. 3. Boundary layer blowing. 4. Boundary layer suction. 5. Vortex mixing. 6. Passive control of boundary layer. 7. Control of wing camber and thickness. 8. Active control techniques.

Openings in the vicinity of the leading-edge wing surfaces allow the flow of air, with a higher energy, into the boundary layer of the upper surface to blow it off and inhibit the separation of airflow at that angle of attack. Thus, as a consequence of these openings or slots, separation now occurs at a much higher angle of attack. Thus, the result is an increase in the effective lift coefficient. The slots in the vicinity of the leading-edge wing surfaces are hydraulically opened only when the trailing edge laps are down and automatically closed when the flaps are up. Leading-edge segments that move on tracks and extend from the wing leading edge to form slots are essentially movable slots. They are known as slats and produce the same effect as fixed slots. Slats are also hydraulically operated, and the deployment and extension of slats is usually synchronised with the deployment and extension of trailing edge flaps. The coordinated movement of slots, slats and trailing edge flaps is designed to effectively increase the camber of the wing and thus improve the wing characteristics at low flight speeds. Leading-edge flaps, which can give the wing an additional droop when extended, may also be deployed to produce the same effect. The deployment of

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leading-edge flaps, known also as Krueger flaps, is also automatically synchronised with the deployment of trailing edge flaps by an electrically signalled and hydraulically operated power control unit. The deployment of trailing edge flaps is controlled by a flap handle that is located on the pilot’s control pedestal in the cockpit. Earlier forms of trailing edge flaps were usually split flaps although the use of plain flaps and extension flaps (Fowler flaps) is now widespread. In one form, trailing edge flaps are usually deployed in a twosection configuration, which are designated as the inboard and outboard sections. Each of the inboard and outboard sections is independently signalled electrically and can be programmed to operate, symmetrically, in one of several coordinated schedules. In many of the older aeroplanes, the coordination of the inboard and outboard sections and the symmetric operation of the left and right wing flaps are achieved by mechanical torque tubes and cabling. Wing lift may also be regulated by controlling the airflow over the wing. Typically, a narrow jet of air passing between the wing and trailing edge flaps blows off the boundary layer, thus providing for attached flow and consequently a higher lift coefficient. Theoretically, the most advantageous methods are the boundary layer blowing off and suction from the upper surface of the wing. Suction increases the rate flow, and consequently, there is an increase in the rarefaction close to the wing surface in the region ahead of the suction point. By contrast, the effect of blowing is an increase in the rarefaction close to the wing surface over the entire chord. With boundary layer blowing or suction, the wing drag decreases with increasing lift coefficient and consequently there is an increase in lift/drag ratio. A jet flap is another means of increasing the lift force. It is essentially established by blowing air through a special slot in the trailing edge of the wing, at angle to the extended chord line. The jet flap extends the wing virtually as well as its camber, and there is an increase in the total lift force acting on the wing. The magnitude of the pressure distribution in the vicinity of the trailing edge area is usually substantially greater than a wing without the jet flap. A hybrid boundary layer suction system coupled with a jet flap is considered to be a promising high-lift generating system. Inflatable wings are particularly suitable for compensating the aeroplane’s wing section for the in-flight ice accretion process. Ice accretion is particularly a problem in the vicinity of the aeroplane’s leading edge, and compensation is achieved by designing inflatable and deflatable wing leading edges. During the ice accretion period, an active controller is used to deflate the leading edge and thus compensate for the ice accretion. A circulation control system employs rearward tangential blowing over a rounded or near-rounded trailing edge, to reinforce the boundary layer and delay the separation. Separation is delayed due to the flow remaining attached to the trailing edge due to the Coanda effect. The location of the separation point may be controlled by varying the blowing rate, thus affecting the wing lift. Generally in the case of flaps with circulation control, there is substantial increase in wing lift than in the case of conventional mechanical flaps. A similar approach is adopted in the wings with vortex control jets. There are indeed several alternate methods of controlling and regulating wing lift. In the case of most high-speed jets, particularly those capable of flying faster than the speed of sound, wings are swept back to minimise drag. Yet it is well known that

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swept wings do not perform as well as straight wings at lower speeds. Thus, in the case of swing wing aircraft, the wings are movable and may be deployed as swept wings at high speeds with the ability to revert to a straight wing configuration at low speeds. There are also some vertical take-off and landing aircraft where the aircraft’s lift is controlled entirely by control jets. The jet’s nozzle can be mechanically swivelled and the jet’s exhaust directed accordingly, to alter the direction of the net thrust acting on the aeroplane. Fluidic jets, where the jet’s directional control is based on the socalled Coanda effect, have also been employed in some experimental programmes.

1.4.3  Spoilers Spoilers, so called because they are employed to spoil the lift on the wing by disrupting the streamlined airflow around it, are usually deployed at the instant of landing to place the full weight of an aircraft on the wheels and prevent it from bouncing back into the air after a heavy landing. They are also deployed automatically on an abandoned take-off following the selection of reverse thrust, again to place the full weight of the aircraft on the wheels and to improve braking performance. In-flight spoilers are deployed as speed brakes to slow the aircraft rapidly and to greatly increase the rate of descent (Figure 1.10). They are also employed occasionally for enhanced roll control. Deploying the spoilers on one side of the aircraft disrupts the lift on that side and aids the aircraft in rolling. Spoilers are normally actuated by electro-hydraulic power control units. On most civil aircraft, there are a number of spoilers and groups of these are actuated by one of several hydraulic channels to provide for redundancy and fault tolerance. During operation, they are designed to be adaptive; that is, the extension is generally much lower at higher speeds. Spoiler actuators are also designed to retract back to their unloaded position when hydraulic power to them is lost. In the case of flight spoilers, which are used to supplement the aileron in roll control, the spoiler inputs are generated by the aileron’s movement and moderated by a spoiler mixer mechanism or a spoiler control law. Ground spoiler actuators are normally activated only while the

FIGURE 1.10  Aircraft with spoilers deployed: spoilers function as lift dumpers or speed brakes.

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15

aircrafts are on the ground and are controlled so the entire weight of the aircraft acts on the landing gear just before touchdown.

1.5  P  ILOT’S CONTROLS: THE THROTTLE, THE CONTROL COLUMN AND YOKE, THE RUDDER PEDALS AND THE TOE BRAKES A primary complement of the pilot’s control in the cockpit are the throttle levers to control the fuel delivered to the power plant, the control column which may be pulled back or pushed forwards to rotate an aeroplane or to flare the aeroplane during landing, the yoke which when turned banks the aeroplane to one side or the other, the rudder pedal that is used to change the direction of the aeroplane’s flight path and toe brakes which allow for the differential braking of the wheels during landing.

1.6  MODES OF FLIGHT Speed and power are intimately connected with changes in attitude or the change in the direction in which the nose is pointing relative to the direction of flight. Vertical changes in the direction of flight as well as the changes in attitude affect the forces acting on an aeroplane. Flight at constant velocity is called steady, and we have already dealt with steady horizontal flight. The simplified force diagrams for steady flight other than horizontal are shown in Figure 1.11. These represent climbing, power gliding and gliding flight. In addition, there are also the cases of an aircraft, in steady spin, in a terminal velocity dive, climbing in a turn and gliding in a turn. We observe from Figure 1.11 that the direction of the airflow relative to the aeroplane is exactly opposite to the direction of motion of the aeroplane. The air itself is not moving, and it only has velocity relative to the aeroplane. The direction of airflow is important as it determines the directions of the lift and drag. Based on the figure, we may establish that the conditions for equilibrium flight may be obtained, as in the case of steady level flight, by resolving the forces in the directions of the lift and drag.

1.6.1  Static and In-Flight Stability Margins The problem of stability has already been discussed. Yet, the overall stability of an aeroplane is particularly important, and in large passenger aeroplane a good deal of stability is desirable. An important feature of these aeroplanes is the inherent stability in the three aeroplane attitudinal degrees of freedom of pitch, roll and yaw as well as the static stability in equilibrium flight. As already mentioned, the tail plane generates lift, and as a result of its long moment arm, it is adequate to restore the aeroplane to an equilibrium position when a disturbing force such as a gust of wind acts to displace the aeroplane from its equilibrium position. A measure of this characteristic is the distance of the aerodynamic centre, the location of the CP of all aerodynamic forces generated when the aeroplane pitches forwards or backwards from a position of equilibrium and the CG. This is known as the longitudinal static stability margin.

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Flight Dynamics, Simulation, and Control

FIGURE 1.11  Modes of flight. (a) Climbing flight, (b) power gliding and (c) gliding.

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Stability in roll is achieved due to the dihedral construction; that is, each half of the wing is positioned at a small positive angle (4°–10°) to the horizontal. Thus, when the aeroplane rolls to one side, there is an increased lift on the corresponding side of the wing resulting in a restoring moment and the aeroplane returns to a state of equilibrium. Stability in yawing motion is due to the tail fin. It plays the same role in yawing motion as the horizontal tail plane does in pitching motion. Similar to longitudinal static stability margins, one could define lateral static stability margins. The lateral or weathercock stability margin is essential to provide the aeroplane with directional stability. The aforementioned stability characteristics refer to the desirable static stability margins of an aeroplane. In addition, an aeroplane must possess certain dynamic stability characteristics; that is, although an aeroplane may return to state of equilibrium from a disturbed position, certain motion characteristics are essential during its return to equilibrium. These desirable motion features are observed when the aeroplane has acceptable dynamic stability margins that are equally important, if not more, than the static stability margins.

1.7  POWER PLANT Thus far, we have generated a thrust by drawing an arrow in the direction of thrust and indicated it by the letter T, but the production of thrust in reality is a different matter. A forward force can only be generated by pushing a quantity of air back, that is, by increasing the velocity of the relative airflow. The thrust produced is directly proportional to this increase in the relative air velocity. Thus, thrust is produced whenever energy is imparted to a stream of air. Without exception, all powered aircrafts are propelled by one or more thrust-producing thermal engines that convert heat energy released by fuel combustion into mechanical power. Thrust-producing power plants used on board an aircraft may be typical supercharged piston engines driving a propeller or one of a variety of jet engines. The former class of engines is used for the smaller purpose-built general aviation aeroplanes, while the latter class is used on most airliners. These power plants are mounted on the aeroplane in one of several ways such as on the wings inside specially built enclosures known as nacelles, on the tail plane, mounted on but external to the fuselage or integrated into the fuselage. As many as six of these power plant units may be used to propel a single aeroplane.

1.7.1  Propeller-Driven Aircraft These aircraft are primarily driven by typical supercharged piston engines driving a propeller. The propeller itself is constructed just like a wing of constant chord and a very high span to chord ratio, a uniform twist in the spanwise direction. It acts like a screw winding its way through the air, and the velocity of air relative to each part of the blade will be directed like a screw thread. The blade is designed such that the aerofoil sections along the span are inclined at the appropriate angle attack to the net airflow, and consequently, the lift components at each section will combine

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Flight Dynamics, Simulation, and Control

constructively to produce the thrust in the direction of motion of the aircraft, while the drag components combine to form a resisting torque. When this total resisting torque is less than torque of the engine, the engine speed will continue to increase. Consequently, there is an increase in the resisting torque, and when this torque balances the engine torque, the equilibrium engine speed is attained. The equilibrium engine speed and the corresponding thrust determine the conditions of equilibrium flight.

1.7.2  Jet Propulsion Jet propulsion is based on the production of thrust by means of the reaction of the force due to a rapid change in momentum of a jet of gas produced within the aircraft but directed rearwards. It is usually associated with gas turbine which is employed as a normal power unit to convert as much as possible the heat energy of the fuel into mechanical motion by causing a shaft to rotate. When this is coupled with a fan, it will draw a large mass of air through the aircraft where it expands and thus gain in kinetic energy. However, when the gas expands, not only is there a fall in the pressure accompanied by an increase in the kinetic energy, but there is also a fall in the temperature. To ensure that the combustion chamber, to which this charge of fuel–air mixture is delivered, functions efficiently, the temperature of the gas must be within certain limits and not fall too low. For this reason, the air delivered to the turbine is pre-compressed by a rotary compressor, fitted in front of the turbine on a common shaft. After burning in the combustion chamber where the gases acquire further energy, a jet of hot gas directed rearwards is produced, which in turn generates the desired thrust. Turbo-jet propulsion is particularly adaptable to aircraft, due to its high power to weight ratio, small size, the presence of a minimal number of moving parts, better performance at high speeds coupled with cheaper fuel cost and the ability to redirect pats of the jet for de-icing and flow (boundary layer) control. It has replaced the supercharged rotary petrol engine driving a propeller, as the main power plant on almost all large airliners. In fact, currently, research is already way so such propulsion systems could be employed for purposes of controlling an aircraft, in lieu of the usual control surfaces such as spoilers, flaps, elevators and ailerons.

1.8  AVIONICS, INSTRUMENTATION AND SYSTEMS Typically about 50% of a modern airliner’s cost is contributed by the entire complement of avionics on-board. Generally, this can be classified into the following three groups:

1. Stand-alone standard avionics equipment for communication, navigation and guidance, transponders, radar, audio, autopilots, displays, indicators, etc. 2. Cockpit instrumentation and supporting electronics. 3. Supporting electronics integrated with other subsystems such as power plants and FADEC avionics, landing gear and brakes, flight control s­ ystems, fuel control systems, hydraulics systems, electrical and power systems, lighting systems and cabin systems.

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Although a detailed description of each of these classes of systems or even their generic features is well beyond the scope of this introductory section, the subject of autonomous navigation will be briefly covered.

1.8.1  Autonomous Navigation A new breed of aircraft has been emerging during last decade. Although conceived, as digital remotely operated navigation equipment, from which the acronym DRONE arose and has now become a proper word in the English language, drones have evolved from remotely operated and controlled aircraft. A remotely operated and controlled drone, which involves manual control of an unmanned aerial vehicle (UAV), by a remote pilot, without sensory feedback for line-of-sight applications, has evolved to a fully autonomous platform, where the vehicle operates fully autonomously without human intervention, with only on-board sensory observations and feedback for adapting to operational and environmental changes. Two other classes of drones involve teleoperation when they are piloted by a remote operator using feedback from onboard sensors for motion control with no obstacle avoidance capabilities for beyondline-of-sight applications, and semi-autonomous operations when they are piloted by a human operator are needed for high-level mission planning for autonomous operation between waypoints with obstacle avoidance. The main hardware and software components in an UAV or drone are illustrated in Figures 1.12 and 1.13 (adapted from Elmokadem and Savkin [3]). The main planning and control tasks in a drone involve perception within the environment, for purposes of localisation and mapping, motion planning and obstacle avoidance and flight control. Localisation is the process of determining the vehicle’s position with respect to a reference frame. This can be achieved by referring to a map based on new information on landmarks. Localisation coupled with identification of landmarks also helps in adding to the map. Thus, localisation and mapping could be inter-related. The perception, localisation and mapping, motion planning and control naturally lead to autonomous navigation. Although there are several

FIGURE 1.12  Hardware components used in an UAV or drone.

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Flight Dynamics, Simulation, and Control

FIGURE 1.13  Software components used in an UAV or drone.

FIGURE 1.14  Feedback architecture of the integrated vehicle controller.

feasible navigation control structures for drones, all of them have a generic structure, leading to a feedback architecture, which is illustrated in Figure 1.14. Moreover, the concept of simultaneous localisation and mapping (SLAM) has emerged, and the main problem that SLAM addresses is the perception problem of a drone navigating an unknown environment. While navigating an unknown environment, a drone seeks to acquire a map of the region, and at the same time, it wishes to localise itself on the same map. SLAM addresses the issue of a limited and yet detailed enough model of the environment of the drone as well as its ability to localise (and position itself relative to the environment) itself. There are three main classes of SLAM: (1) Kalman filter (KF), extended KF, unscented KF-based; (2) based on particle filters;

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and (3) based on graph optimisation. The latter is equivalent to computing the state of minimal energy of the model, and the graph corresponds to a representation of the log of posterior conditional probability function. The need for three-dimensional SLAM led to the development of digital-camera-based visual SLAM technologies, which are now incorporated into most modern drones, capable of autonomous navigation.

1.9  GEOMETRY OF AEROFOILS AND WINGS The primary lifting surfaces in an aeroplane that is responsible to a large extent in maintaining the aeroplane in flight at a reasonable altitude are the two wings attached to the fuselage. It is quite natural to expect that the geometry of these wings, the planform shape and the cross sectional geometry in particular play a crucial role in the generation of the lift, which in turn is the primary force on the aeroplane that is responsible in keeping it aloft. The aerofoil itself is the envelope of the cross section of the wing. It essentially consists of a structural framework covered by a thin metallic or composite skin. While the structural framework gives the wing the required strength and stiffness, the skin is primarily responsible in shaping the aerofoil to match a prescribed aerofoil contour.

1.9.1  Aerofoil Geometry Aerofoil geometries have evolved over the years, and there are now a number of standardised aerofoil section geometries. Examples of typical symmetric and unsymmetric aerofoil sections are illustrated in Figure 1.15.

1.9.2  Chord Line A principal characteristic of any aerofoil section is the chord line or chord, and it defines the length of the aerofoil. It is a line drawn from the leading edge of the aerofoil near its nose to the trailing edge of the aerofoil (Figure 1.16a). Whether it falls totally within the aerofoil section or partially outside it, it is a primary reference for defining the various ordinates of the upper and lower surfaces of the aerofoil. It is normally designated by the lowercase letter ‘c’.

1.9.3  Camber The upper and lower surfaces of an aerofoil are known as the upper and lower cambers. The distance halfway between the upper and lower camber lines is known as the mean camber line. The mean camber line plays a significant role in the generation of lift and is a key parameter in determining the mean value of the section aerodynamic lift force under steady flow conditions. The maximum camber is the maximum distance of mean camber line from the chord line. Its magnitude and location along the chord are usually expressed as percentages of the chord. Typically, the magnitude is usually of the order 4% in the case of non-symmetric aerofoils and located at about 30% downstream from the leading edge (Figure 1.16b).

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FIGURE 1.15  Symmetric and unsymmetric aerofoil sections. (a) Two examples of symmetric aerofoil sections and (b) two examples of unsymmetric aerofoil sections.

FIGURE 1.16  (a) Location of chord line and the definition of aerofoil chord and (b) location of mean camber line.

1.9.4  Leading and Trailing Edges The leading edge is located at the forward tip of the aerofoil and the aerofoil chord. A circle drawn with its centre on the mean camber line and a radius so it passes through the forward most tip of the chord line is essential in locating the leading edge. The leading-edge radius and the coordinates of its centre are used to define the leadingedge circle. The trailing edge is defined in a far more simpler way and is the point where the upper and lower camber lines intersect. Although it is apparently represented as a knife edge, it is in fact a region characterised by a finite thickness depending on the thickness of the skin used to envelope the wing structure.

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Introduction to Flight Vehicles

1.9.5  Specifying Aerofoils The methodology for defining the shape of aerofoils has evolved over many years. Its early development was done exclusively at the U.S. National Advisory Committee on Aeronautics, (NACA) Langley Field Laboratory, by a number of contributors. Here, the NACA 4-digit, modified 4-digit, 5-digit, 6-digit and the 6A series of aerofoils were developed over a period spanning about 50 years. For example, the NACA 4-digit aerofoils are defined as follows:

NACA MXTT

where M is the maximum value of the mean line in hundredths of chord X is the chordwise position of the maximum camber in tenths of the chord TT is the maximum thickness ratio (t/c) in per cent chord The NACA 2410 refers to a 10% maximum thickness aerofoil, with maximum value of the camber of 0.02 at x/c = 0.4. In the case of the NACA 5-digit aerofoil (e.g. 23,015), the following applies: First digit: Twenty-thirds (20/3) times the design lift coefficient. It’s also safe to say that it represents the maximum height of the camber line expressed as a percentage of the aerofoil chord length. Second and third digits combined: The horizontal location of the maximum camber line height in 200th of the chord length. Also, if the third digit is 0, then the trailing camber line is a straight line. If it is equal to 1, then the trailing camber line is reflex, or bowed down. Last two digits combined: The maximum thickness of the aerofoil expressed as a percentage of the aerofoil chord length. It occurs at about 30% of the chord length with the NACA 5-digit series. The NACA 6-series aerofoils are designed for laminar flow unlike the NACA 4- or 5-digit aerofoils. In these aerofoils, The first digit: Tells us that the aerofoil is a six-series aerofoil. Second digit: The horizontal location of the minimum pressure coefficient (i.e. maximum suction from the accelerated air) in tenth of a chord length for the symmetrical uncambered shape. Third digit: Tells us the approximate design lift coefficient for that aerofoil in tenth. Last two digits combined: The maximum thickness of the aerofoil expressed as a percentage of the aerofoil chord length. Further details on the NACA series of aerofoils may be found in Abbott and von Doenhoff [4].

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1.9.6  Equations Defining Mean Camber Line Employing the thin aerofoil theory, we can show that a simple cubic camber line can be expressed in terms of the zero-lift angle of attack and the moment coefficient of the moment about the leading edge as

C  4x   x 3 zc = −2α L = 0 x  1 −   1 +  1 − mLE   1 −   α L=0   3c   c 10 

(1.1)

where αL = 0 = −CLi/2π is the zero-lift angle of attack (i.e. CL = 2π(α – αL = 0) = 0) CLi is the coefficient of the lift due to the aerodynamic pressure distribution around the aerofoil surface when the angle of attack is zero. CL is the coefficient of the lift due to the aerodynamic pressure distribution around the aerofoil surface (i.e. CL = 2π(α – αL = 0)). CmLE is the coefficient of the moment of the aerodynamic pressure distribution about the leading edge. In practice, it is customary to employ more than one curve to model the mean camber line. The shape of the mean camber lines of NACA 4-digit aerofoil sections can be expressed analytically as two parabolic arcs, tangent at the position of maximum mean-line ordinate. The equations defining the mean lines are taken as



zc m x x = − 2  2q −  , c q  c c

x ≤ q, (1.2a) c



2 zc m  2  x − q  , =− 1 − q − ( ) c   c (1 − q )2 

x ≥q c



(1.2b)

where m is the maximum ordinate of the mean line expressed as a fraction of the chord while q is the chordwise position of the maximum ordinate. For example, for the NACA 6400 aerofoil, m = 0.06 and q = 0.4, and for the NACA 4400 aerofoil, m = 0.04 and q = 0.4.

1.9.7  Aerofoil Thickness Distributions The distance between the upper and lower camber lines is the thickness, and it varies along the chord. A typical aerofoil thickness distribution (NACA 4-digit aerofoil) is given by

2 3 4  x x x x x  zt = 5t  a0 − a1 − a2   + a3   − a4    (1.3)  c  c  c  c c 

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Introduction to Flight Vehicles

TABLE 1.1 Coefficients Defining Thickness Distribution of NACA 4-digit Aerofoil a0 0.2969

a1

a2

a3

a4

0.1260

0.3516

0.2843

0.1015

where the coefficients a 0, a1, …, a4 are listed in Table 1.1. The maximum thickness is at x/c = 0.3, and the leading-edge radius and the included angle at the trailing edge are 2



t   δ TE = 2 tan −1 1.16925    .   c 

t rLE = c × 1.1019   ,  c

(1.4)

The equations defining the upper and lower surfaces are then given as

x u x zt = − sin θ, c c c

zu zc zt = − cos θ c c c



(1.5a)

x l x zt = + sin θ, c c c

zl zc zt = + cos θ, c c c



(1.5b)

and

θ = −α



(1.5c)

where α positive angle of slope at chordwise position x determined by differentiating the equation for the camber line. Hence,



tan α =

tan α =

2m dzc x = − 2  q −  , dx q  c

2m  dzc x q− , =− 2    dx c (1 − q )

x ≤ q, c x ≥ q, c

(1.6a)

(1.6b)

The shape of the mean camber lines of the NACA 16-series aerofoil section is given as and

zc C = − Li 4π c

x  x x x   1 − c  ln  1 − c  + c ln c 

(1.7a)

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Flight Dynamics, Simulation, and Control

dzc C Li = 4π dx



   − ln  1 −

x x + ln  .  c c

(1.7b)

1.9.8  Wing Geometry Apart from the geometry of an aerofoil section, there are a number of other characteristic features of wings that are also extremely important in the development of the aerodynamic forces and moments on an aeroplane. These include wing span, root chord, tip chord, mean geometric chord and mean aerodynamic chord, planform area and wing aspect ratio. Certain angles associated with the geometry of wings also play an important part in the aerodynamics of wings. These are the incidence, sweepback and dihedral angles. The principal characteristic of a typical section of an aerofoil is its chord. Considering a general wing planform, the horizontal distance between the wing tips is the wing span. The root of the wing is where the wing is attached to the fuselage and is normally different from the fuselage centreline. In many practical situations, the two are relatively located so close to each other that they are assumed to be at the same location, spanwise. The aspect ratio is defined as the ratio of the square of the span to the reference area (usually the area of the planform but sometimes could include the planform area of the horizontal tail plane or the horizontally projected area of the fuselage). The mean geometric chord is the ratio of the area of the wing planform to the span. Various integral properties of general wing planforms including the mean aerodynamic chord are defined in Table 1.2. Most wing planforms are trapezoidal-shaped as illustrated in Figure 1.17. The leading and trailing edges of a typical trapezoidal planform may be swept backwards or forwards. They play a significant role in determining the maximum lift on the aerofoil and hence the stall characteristics of the aeroplane. The principal geometrical relationships associated with trapezoidal symmetric planforms are tabulated in Table 1.3. The upward slope of the wing when viewed from the wing root is known as the dihedral or dihedral angle. The dihedral angle is essential as it is principally responsible in making the aeroplane sufficiently stable in roll. It is usually of the order of about 5°–10°. In the case of many modern planforms, which are also kinked, the dihedral angles corresponding to the inboard and outboard sections of the planform can be different. A typical example of a kinked planform is illustrated in Figure 1.18. The principal geometrical relationships associated with kinked-trapezoidal symmetric planforms are tabulated in Table 1.4.

CHAPTER HIGHLIGHTS Aircraft features: The primary aircraft components are fuselage, wings, empennage or tail plane, landing gear, power plant and avionics and instrumentation systems.

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Introduction to Flight Vehicles

TABLE 1.2 Integral Properties of General Symmetric Planforms Property

Relation s

Planform area, S

∫

S = 2 c ( y ) dy



0

s

Mean aerodynamic chord, c



2 2 c ( y ) dy S

∫

c=

0

s

Mean geometric chord, c



c=

2 c ( y ) dy b

∫ 0

 c ( y)  2 c ( y)  = + x LE ( y ) dy S  2  s

x position of centroid of area, xcen

xcen

Spanwise position of mean geometric chord



∫ 0

s

ycen =

2 c ( y ) ydy S

∫ 0

s

2 x LE ( y ) c ( y ) dy S

Leading-edge position of mean chord



Aspect ratio



AR =

Reference chord



Cref =

Taper ratio, λ, c0 is the chord at the centreline



x LEcen =

∫ 0

b2 Sref

Sref b cT λ= c0

c, chord; s, semi-span; b, span = 2s.

Control surfaces and other controls: throttle, elevators, trim tabs, horizontal stabiliser, aileron, rudder, vertical stabiliser, flaps (fowler flaps), spoilers, slats and other high lift devices. Each control has a specific function (e.g. speed, pitch attitude, stick force, bank and turn, added lift, lift dumping, stall delay, etc.)

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Flight Dynamics, Simulation, and Control

FIGURE 1.17  Typical trapezoidal planform.

TABLE 1.3 Properties of Trapezoidal Symmetric Planforms Property

Relation xLE (y) = xLE (0) + y tan ΛLE (y) xTE (y) = xTE (0) + y tan ΛTE (y) c(y) = c0 (1 – (1 – λ)η)

Leading-edge line Trailing edge line Local chord Sweepback at any element line, n, m, fractions of local chord

tan Λ n = tan Λ m −

4   1− λ    ( n − m )  AR  1 + λ  

tan Λn = (1 – n)tan ΛLE + n tan ΛLE

Sweepback at any element line in terms of leading-edge and trailing edge sweep Planform area

S = s × c0 × (1 + λ)

Average chord



Mean geometric chord



Centreline chord



Mean aerodynamic chord



Chord at fuselage junction, fuselage diameter = d



cT + c0 2 S c= b

cave =

 2  c0 =   1 − λ  c=

S AR

2c0  1 + λ + λ 2  3  1 + λ 

d  cR = c ( yd ) = c0  1 − (1 − λ )   b (Continued)

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Introduction to Flight Vehicles

TABLE 1.3 (Continued) Properties of Trapezoidal Symmetric Planforms Property

Relation

Aspect ratio s

y position of centroid of area, ycen =

2 c ( y ) ydy S

∫



AR =

s  4    c0  1 + λ 



ycen =

b  1 + 2λ  6  1 + λ 

0

Leading-edge location at above spanwise position, x LEcen



 1 + 2λ  x LEcen = x LE0 + c0  AR tan Λ LE  12 

Chord at y = ycen



c

x position of centroid of area, xcen



Spanwise position of mean aerodynamic chord



s

∫

c 2

2λ 2 − λ − 1

ymac =

(

)

3 λ2 − 1



ycen =



yp =

b  1 + 2λ  6  1 + λ 

0

s

Rolling moment arm y p = 2 c ( y ) y 2 dy Ss

∫ 0

c, chord; s, semi-span; b, span = 2s; η = y/s.

FIGURE 1.18  Kinked-trapezoidal planform.

s

xLE(ymac) = xLE(0) + ymactanΛLE(ymac)

Leading-edge position of mean aerodynamic chord Side-slip force moment arm ycen = 2 c ( y ) ydy S

xcen = x LEcen +

b  1 + 3λ  12  1 + λ 

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Flight Dynamics, Simulation, and Control

TABLE 1.4 Properties of Kinked-trapezoidal Symmetric Planforms Property

Relation

Tip chord

λ2c0 λ1c0

Local chord at kink Chord at any spanwise location



 η c ( y ) = c0  1 − (1 − λ1 )  , η1  



 1− η  c ( y ) = c0  λ 2 + ( λ1 − λ 2 ) , 1 − η1  

Mean geometric chord



Mean aerodynamic chord

c=

(k – 1)th moment arm, yk −1 =

1 c0 s k

s

∫c ( y) y

k −1

dy



c=

(

(

c0 λ1 + λ 2 + (1 − λ 2 ) η1 2

)

(

η1 ≤ η ≤ 1

) )

2 2 2 2c0 1 + λ1 + λ1 η1 + λ1 + λ 2 + λ1 λ 2 (1 − η1 ) 3 λ1 + λ 2 + (1 − λ 2 ) η1

(

)

 λ − λ 2  1 − η1k yk −1 =  λ 2 + 1                               1 − η1  k  −

0

0 ≤ η ≤ η1 ,

λ1 − λ 2 1 − η1k +1 1 + λ1 k + η1k k ( k + 1) 1 − η1 k + 1

c, chord; s, semi-span; b, span = 2s; η = y/s, ηj = ykink/s and centreline chord, c0.

EXERCISES

1.1  The mean aerodynamic chord of symmetric wing planform is defined by the integral 1



c=

2s 2  y   y  . c d  s  s S

∫ 0

 ypically, a wing of arbitrary planform shape is divided into J trapezoidal T panels, and the chord at any spanwise location is defined by the relations

 η −η c ( η) = c0  λ i +1 + ( λ i − λ i +1 ) i +1 , ηi +1 − ηi  

η=

y , s



ηi ≤ η ≤ ηi +1 , K    i = 0,2,…, J − 1,    η0 = 0,   ηJ = 1,   λ 0 = 1.



 btain equations for the planform area and the mean aerodynamic chord, O in terms of λ i , ηi , c0 and the semi-span s. 1.2 Obtain an equation for the centre chord length c0m for a straight tapered symmetric wing of an area of Sw, an aspect ratio AR and a taper ratio λ.

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Introduction to Flight Vehicles



1.3 Consider a trapezoidal planform. i. Show from the first principles that the sweep angle of any constant-chord fraction line can be related to that of the leading-edge sweep angle by tan Λ n = tan Λ 0 −



4n 1 − λ , AR 1 + λ

where 0 ≤ n ≤ 1 is the chord fraction (e.g. 0 for the leading edge, 1/4 for the quarter-chord line and 1 for the trailing edge). ii. Show that the location of any chord fraction point on the mean aerodynamic chord, relative to the wing apex, can be determined as





xn =

2 S

s

∫ ( nc

root

0

   1 + 2λ   1 + λ  + y tan Λ n ) c ( y ) dy = c  n +  AR tan Λ 0  .  8   1 + λ + λ 2   

1.4 Consider an elliptic wing planform. The root chord is given to be c0. The semi-span is given as s. Obtain equations for the mean aerodynamic chord and the planform area in terms of c0 and s. What is the aspect ratio of the planform? 1.5 Verify the formulas for the aspect ratio, the spanwise position of the centroid of area, leading-edge location and the chord at the spanwise position of the centroid of area and streamwise position of the centroid of area in Table 1.3. 1.6 Verify the formulas for spanwise position of the mean aerodynamic chord, leading-edge position of the mean aerodynamic chord, side-slip force moment arm and the rolling moment arm in Table 1.3. 1.7 Verify the formulas for tip chord, local chord at kink, chord at any spanwise location, mean geometric chord, mean aerodynamic chord and the (k – 1)th moment arm for kinked-trapezoidal symmetric planforms in Table 1.4.

ANSWERS TO SELECTED EXERCISES J −1



1.1  S = c0

∑(λ

i +1

+ λ i ) ( ηi +1 − ηi ) ,

i=0

          c =

2sc02 3S



1.2  c0 m =



1.4  c =

J −1

∑(λ

)

+ λ i λ i +1 + λ i2+1 ( ηi +1 − ηi ) .

i=0

2 1+ λ 1

2 i

Sw . AR

2s 2 8c c ( η) dη = 0 , S = 2s S 3π

∫ 0

1

∫c ( η) dη = sc

0

0

π 8s , AR = . πc0 2

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Flight Dynamics, Simulation, and Control

REFERENCES [1] Anderson, J., Introduction to Flight, 4th ed., McGraw-Hill, New York, 2000. [2] Shevell, R., Fundamentals of Flight, 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1989. [3] Abbott, I. H. and von Doenhoff, A. E., Theory of Wing Sections: Including a Summary of Data, Dover Publications Inc., New York, 1958. [4] Elmokadem, T. and Savkin, A. V., Towards fully autonomous UAVs: A survey, Sensors, 21, 6223, 2021. https://doi.org/10.3390/s21186223

2

Basic Principles Governing Aerodynamic Flows

2.1  INTRODUCTION Aerodynamic flows are encountered when one is dealing with any aspect of aeronautical engineering. Physical principles that govern aerodynamic flows are based on the conservation of mass, momentum and energy. Just as Euler’s equations of motion are derived from Newton’s laws of motion in classical mechanics, secondary laws governing the dynamics of rotational flows can be derived from the fundamental physical principles. The flows around aerofoils are the simplest examples of aerodynamic flows. In aerofoil theory, it is possible to idealise the flow by ignoring the influence of the compressibility of the fluid medium. When compressibility is included, one needs to consider three distinct cases: subsonic flow when the flow velocity is well below the speed of sound or pressure disturbances in the flow, transonic flow when the flow velocity is in the vicinity of the speed of sound and supersonic flow when the flow velocity is well above the speed of sound. Furthermore, the viscous forces, which result in friction at the boundaries, play a key role in aerodynamic flows. In this chapter, we review the basic principles governing aerodynamic flows, the influence of compressibility and viscosity, the definition of the standard properties of the atmosphere, the flow around an aerofoil, the generation of lift, drag and moment over an aerofoil and the basic properties of aerofoil aerodynamics.

2.2  CONTINUITY PRINCIPLE The continuity principle is statement of the principle of conservation of mass and states that the mass of a fluid flowing into a control volume is equal to the mass of the fluid flowing out of it.

2.2.1 Streamlines and Stream Tubes A streamline can be thought of as a continuous line in the fluid flow, along which the velocity of a fluid particle is tangent to it and the velocity components in a plane perpendicular to it are zero. At any point on the streamline, each particle will experience the same velocity and pressure as the preceding particles as they pass the point. Particles following a lead particle in a streamline will experience the same velocity and pressure as the lead particle as they pass its location in the streamline. These values of velocity and pressure may change from point to point along the streamline. A reduction in the velocity of streamline flow is indicated by wider spacing of DOI: 10.1201/9781003266310-2

33

34

Flight Dynamics, Simulation, and Control

the streamline, while increased velocity is indicated by a decrease in the spacing between the streamlines. In a steady flow, therefore, the shape of a streamline is invariant; that is, it does not change with time. Furthermore, the particles in a particular streamline maintain their position relative to the particles in another streamline. Thus, streamlines do not cross. If the relative positions of the streamlines are identical in all parallel cross sections of the flow, then the flow is said to be 2D. A set of streamlines not coincident with each other constitute a stream tube.

2.3  BERNOULLI’S PRINCIPLE Bernoulli’s principle is a statement of the principle of conservation of energy and states that along any point in a uniform irrotational flow without dissipation, the sum of the pressure energy, kinetic energy and gravitational potential energy is a constant. In aerodynamic flows, it is customary to refer to both kinetic and potential energies in terms of equivalent pressure energies. Thus, the pressure energy is distinguished from the other two energies by referring to the conventional pressure as static pressure (ps). The equivalent pressure that results in the same energy as the kinetic energy is the dynamic pressure and is obtained by dividing the expression for the kinetic energy by the volume of the flow. Thus, the expression for the dynamic pressure is q = ρV2/2, while the equivalent pressure corresponding to the potential energy is ph = ρgh.

2.4  LAMINAR FLOWS AND BOUNDARY LAYERS When a thin plate is placed in a uniform flow of air, the air encounters friction as it flows over the plate, and the flow next to the air is retarded and brought to rest adjacent to the surface of the plate. This retardation effect diminishes rapidly when the flow is farther away from the plate surface. When the flow is uniform, the retardation effect is restricted to a narrow layer close to the plate surface, and while the flow continues to be uniform beyond this layer, the flow is said to be laminar, and the layer is referred to as a boundary layer. The thickness of the boundary layer is governed by the viscosity of the flow and the friction of the surface. The viscous effects are only important within the layer and may be ignored outside it. The thickness of the boundary layer may be controlled by sucking it away. Further details on boundary layers may be found in Schetz [1].

2.5  TURBULENT FLOWS When disturbances are initiated in the flow, which cause the flow beyond the boundary layer to be non-uniform and disturbed, the flow is said to be turbulent. In aerodynamics, turbulent flows are undesirable as it results in energy loss due to the formation of eddies. Steady streamline flow is desirable in most phases of flight, and turbulent flow is best avoided. The transition of a laminar flow to a turbulent is usually a multistage process. In the first instance, the fully laminar flow region becomes partially turbulent and is characterised by a turbulent inner layer. The point on the surface of the thin plate where this happens is the transition point. Beyond the transition point, the boundary is slightly thicker. If and when the boundary layer separates from the plate, it causes the main airflow to break away and become turbulent.

Basic Principles Governing Aerodynamic Flows

35

The point where the boundary layer separates from the plate, if and when it does so, is known as the separation point. The occurrence of separation must generally be avoided in aircraft flight as it causes the aircraft to stall.

2.6  AERODYNAMICS OF AEROFOILS AND WINGS The relative flow of air past a wing results in the development of a pressure distribution over it. The characteristics of the aerodynamic pressure distribution over a wing are functions of several factors, which may be classified into four principal groups: 1. Flow effects such as compressibility and viscosity. 2. Aerofoil and wing shape and geometry parameters. 3. Size and scale effects. 4. Orientation of the body relative to the flow. An understanding of the flow past an aerofoil is essential for a holistic understanding of the aerodynamics of aerofoils and wings.

2.6.1 Flow around an Aerofoil The flow around an aerofoil has many similar characteristics to the flow around a thin plate. The two cases of streamlined flow around a symmetric and unsymmetric aerofoil are shown in Figure 2.1.

FIGURE 2.1  Flow around an aerofoil. (a) Flow around a symmetric aerofoil section and (b) flow around an unsymmetric aerofoil section.

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Flight Dynamics, Simulation, and Control

The higher velocity on the upper surface of the unsymmetric aerofoil causes the pressure to fall on it, resulting in a suction, which, in turn, is responsible in lifting the aerofoil.

2.6.2 Mach Number and Subsonic and Supersonic Flows Pressure fluctuation in a flow results in sympathetic fluctuation in density. In a fluid medium, only longitudinal waves manifesting themselves as compression or rarefaction waves can propagate through the medium. These disturbances in the flow medium are transmitted at the speed of sound propagation of that medium. Thus, it is customary to non-dimensionalise all flow speeds with the speed of sound. The ratio of the velocity of the free stream relative to the aircraft to the speed of sound is known as the Mach number and is denoted by the letter M. As the flight speed approaches the speed of sound, that is, the velocity of the free stream approaches the speed of sound, the Mach number approaches unity, and effects of compressibility are both pronounced and noticeable. Based on the Mach number of the flight, the types of flow fields one could encounter in flight could be broadly grouped into three types: subsonic, transonic and supersonic. In the subsonic case, M  1, the velocity of free stream past the aircraft is greater than the speed of sound at all points in the flow. This is the region of supersonic flow when any increases or decreases in speed must be accompanied by matching increases and decreases in the cross-sectional area of the flow. Any increases or decreases in speed are also accompanied by matching decreases or increases in pressure and density. Quite naturally, when M ≈ 1 at some points in the flow, the region of flow is in transition, from a subsonic to a supersonic, from a supersonic to subsonic or is a mixed region. It can therefore be expected that analysis of such flows is complex and requires special consideration. When M ≈ 1 at some points within a flow, it is said to be transonic. When the free-stream flow Mach number is well above unity and in the regions of 4 and 5 at all points in the flow, the flow may be considered to be hypersonic and the analysis of such flows is amenable to certain simplifications. When a body with an appropriate profile such as an aerofoil moves in a flow at subsonic speeds, the stream receives a signal of the body’s impending arrival well before the body arrives at a point in the flow. In the case of the body moving at supersonic speeds, the flow has absolutely no prior knowledge of the body’s arrival, and the body cuts through a completely undisturbed and unsuspecting flow. The situation can be described in terms of propagation of spherical disturbance waves in (1) an incompressible flow, (2) compressible flow at subsonic speeds, (3) transonic speeds and (4) supersonic speeds (see Figure 2.2). Considering Figure 2.2a, the case when the source of the disturbance is at rest relative to the flow field, the disturbance propagates with uniform velocity in all directions, and wavefronts propagate in the form of concentric circles (2D case). When the

Basic Principles Governing Aerodynamic Flows

37

FIGURE 2.2  Propagation of spherical disturbances from a point source in (a) an incompressible flow, (b) compressible flow at subsonic speeds, (c) transonic speeds and (d) supersonic speeds.

source of the disturbance is in motion with a speed well below the speed of sound, wavefronts continue to propagate with a speed equal to the difference between the speed of sound and the speed of the disturbance, in the direction of the motion of the disturbance. In the opposite direction, the disturbance propagates at the sum of the two speeds, and the wavefronts are no longer concentric circles. This situation is illustrated in Figure 2.2b. When the disturbance moves with the speed of sound, as in Figure 2.2c, the wavefront is stationary with respect to the source, and in the course of time, it evolves into a normal shock boundary, a surface of discontinuity across which there is discrete change in pressure and density. Finally, when the source moves faster than the speed of sound, it is always ahead of the wavefront, and the result is an oblique shock boundary, as shown in Figure 2.2d. The sine of the half angle of the cone, known as the Mach angle, is equal to the inverse of the Mach number. The surface of the cone forms a shock boundary. The shock boundary separates the free undisturbed flow and the region of the disturbed flow. At transonic and supersonic speeds, there is a substantial increase in the drag experienced by the body in the flow. The features of compressible flows are discussed by Shapiro [2].

38

Flight Dynamics, Simulation, and Control

2.7  PROPERTIES OF AIR IN THE ATMOSPHERE 2.7.1 Composition of the Atmosphere: The Troposphere, Stratosphere, Mesosphere, Ionosphere and Exosphere The envelope of air surrounding the Earth, which is essentially a cosmic boundary layer around the Earth, is known as the atmosphere. The depth of the atmosphere is very thin in comparison with the Earth’s radius. More than 50% of the mass of the atmosphere is within 6 km of the Earth’s surface, 75% of the mass of the atmosphere is within 10 km and 94% of the mass of the atmosphere is within 20 km. The atmosphere begins to decompose to an atomic state at an altitude of 120–150 km, and beyond 200 km, it is completely in an ionic state. It is a mixture of several gases, the primary constituents being nitrogen (78%) and oxygen (21%), while the remaining 1% is made up of argon, hydrogen, carbon dioxide and helium. Broadly, the atmosphere is divided into two regions: the lower atmosphere (up to 50 km) and the upper atmosphere. The lower atmosphere is further divided into two layers of varying thickness across the Earth’s surface: the troposphere (8 km in depth over the poles to 16 km over the equator) and the stratosphere. Likewise, the upper atmosphere is divided into three regions: the mesosphere (50–80 km), ionosphere (70–500 km) and exosphere (from about 450 km and beyond). The distinguishing feature between the upper and the lower atmosphere is the fact that while the lower atmosphere is practically a homogeneous mixture, the upper atmosphere is completely inhomogeneous both spatially and temporally. It is characterised by low air pressures and densities and by intense processes of dissociation and air ionisation, resulting in the splitting of molecules and in the formation of charged particles. Weather and thermal air currents originate in the troposphere, where condensation of water vapour and cloud formation are possible. There is a gradual linear fall in the temperature in the troposphere, followed by the region of constant temperature in the lower stratosphere (–56.5°C) and temperature versus altitude rise in upper regions of the stratosphere.

2.7.2 Air Density There is a rapid decrease in the atmospheric density and pressure with altitude. In the troposphere, there is considerable non-uniform turbulent activity. The unsteadiness of the troposphere both spatially and temporally gives rise to a number of difficulties, particularly in predicting flying characteristics of flight vehicles.

2.7.3 Temperature In the troposphere, the air temperature quickly decreases with altitude. In the stratosphere, it remains almost constant to roughly the 25- to 27-km level, above which it starts to rise intensely with altitude. It is approximately 0°C at an altitude of 50 km. In the mesosphere, the temperature falls again to −80°C at an altitude of 80 km.

Basic Principles Governing Aerodynamic Flows

39

2.7.4 Pressure At an altitude of 10 km, the air pressure is 3.8 times as low as at the ground level, while the air density is only three times as low. At an altitude of 25 km, the air pressure reduces to just 2.4% of its sea level value, while air density reduces to 3% of its corresponding sea level value. At an altitude of 220 km, the air pressure is only a billionth of the sea level value, while the air density drops to less than one billionth of its sea level value.

2.7.5 Effects of Pressure and Temperature The effects of pressure and temperature on air density can be stated in terms of the socalled universal gas law. The universal gas law, relating to a perfect gas, is as follows:

pv = NRT (2.1)

where p is the pressure in the gas Absolute T is the temperature of the gas R is the universal gas constant v is the volume corresponding to N moles of gas Thus, the density ρ is given by

ρ=

N (2.2) v

and the universal gas law may also be expressed as

p = ρRT . (2.3)

As the density is the ratio of the mass and volume, the pressure and temperature affect it indirectly, as a consequence of the universal gas law.

2.7.6 Viscosity When one layer of a fluid slides over another, there is a friction-like force and is termed as a viscous friction force. A standard measure of these forces is the coefficient of viscosity, μ, which is defined by considering a narrow layer of fluid flowing over a horizontal surface. The shear stress at the top of the layer τ is directly proportional to the rate of shear strain, that is, the rate of change of the flow velocity component parallel to the plane and in the direction of the flow with respect to the normal distance from the horizontal surface. The proportionality constant is the coefficient of dynamic viscosity, μ. Hence,

τ=µ

∂u . (2.4) ∂y

40

Flight Dynamics, Simulation, and Control

2.7.7 Bulk Modulus of Elasticity The bulk modulus of elasticity, although not a critical parameter in aerodynamics, is important as it indirectly influences the speed of sound. It is defined as the ratio of the stress to strain and is given as

K = − Lt

∆v→0

∆p ∆p dp = Lt = ρ (2.5) ∆ v → 0 ∆v /v ∆ρ/ρ dp

where Δp is the pressure difference Δv/v is the volume strain The square of the velocity of sound is given by

a2 =

K . (2.6) ρ

Assuming that the fluid is barotropic, the pressure–density relationship is unique for the entire flow, and considering the flow to be isentropic, the differentials are evaluated from the isentropic flow condition:

pv n = Constant. (2.7)

2.7.8 Temperature Variations with Altitude: The Lapse Rate The rate of decrease in temperature with altitude is known as the lapse rate and has a value of 6.5 K/km in the troposphere. The lapse rate is essentially different in different altitude bands in the troposphere and stratosphere. Based largely on mean values observed over extended periods of time, certain standard models for the variation of temperature with altitude have been developed. Amongst these standard models, the International Standard Atmosphere (ISA) developed by the International Civil Aviation Organisation is accepted worldwide as a typical model for the standard atmosphere.

2.8  INTERNATIONAL STANDARD ATMOSPHERE (FROM ESDU 77021, 1986) It has become a norm amongst aeronautical engineers worldwide to accept a common standard for measuring or considering the properties of air. Accordingly, an ISA has been established. It includes models of temperature, air density and air pressure variations with altitude. Sea level conditions are defined in the ISA as follows:

Pressure, p0 = 1.01325 × 105 N/m2

Density, ρ0 = 1.22505 kg/m3

Temperature, T0 = 15°C = 288.15 K Universal gas constant, R = 287.053 J/kg K

Speed of sound, a0 = 340.3 m/s g0 = 9.806 65 m/s2

41

Basic Principles Governing Aerodynamic Flows

The atmospheric bands for heights up to 150 km are treated separately in the following:

1. For altitudes in the range 0 ≤ h (in km) ≤ 11 km (troposphere),



Tropospheric lapse rate,  α = 6.5 ( Lk ,1 = 6.5 K/km ) , (2.8a)



Temperature variation with altitude  h, T1 = T0 − α × h, (2.8b)



α Temperature ratio, Tratio = 1 −   × h, (2.8c)  T0  Pressure ratio exponent, n =



g = 5.256, (2.8d) α   ×  R



Density ratio = Tratio ( n − 1) , (2.8e)



Pressure ratio = Tratio n, (2.8f) Ratio of the speed of sound,



Density at height, h = 1.22505 × Density ratio. (2.8h)





a1 = Tratio ( 0.5) , (2.8g) a0

2. For altitudes in the range 11 ≤ h (in km) ≤ 20 km (stratosphere), which is a constant temperature region,



Temperature, T2 = −56.5°C = 216.65 K, (2.9a)



α = 0.0  ( Stratospheric lapse rate ) , (2.9b) Over this entire altitude range, the ratio of the speed of sound is a20 = 0.8671, (2.9c) a0

At an altitude of 11 km,

p20 = 0.2234, p0



ρ 20 = 0.2971, (2.9d) ρ0

and

p2 ρ = 2 = exp  −0.15769 ( h − 11) . (2.9e) p20 ρ20

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Flight Dynamics, Simulation, and Control

3. At altitudes in the range 20 ≤ h (in km) ≤ 32 km, the temperature rises ­linearly from 216.65 to 228.65 K at 32 km, α = –1 (lapse rate): T3 = T2 − α × ( h − hb ) ,



4. For altitudes in the range 32 ≤ h (in km) ≤ 86 km, the temperature is given by the following equation: T4 = Tb − α × ( h − hb ) , (2.11)



where Tb and hb are given in Table 2.1. 5. For altitudes in the range 86 ≤ h (in km) ≤ 91 km, which is a constant temperature region, T = 186.8673 K. (2.12)



hb = 20 km. (2.10)

6. For altitudes in the range 91 ≤ h (in km) ≤ 110 km, 1

  h − 91  2  2 T = Tc − A 1 −    (2.13)   hc  





where Tc = 263.1905 K A = 76.3232 K hc = 19.9429 km 7. For altitudes in the range 110 ≤ h (in km) ≤ 120 km, T = T9 − Lk ,9 ( h − ha ) (2.14)





where T9 = 240 K Lk,9 = 12.0 K/km ha = 110.0 km 8. For altitudes in the range 120 ≤ h (in km) ≤ 150 km,  ( h − h10 )( r0 + h10 )  T = T∞ − ( T∞ − T10 ) exp  − λ  (2.15) ( r0 + h )  



TABLE 2.1 Table of lapse rates at various altitudes Altitude range(km) 32–47 47–51 51–71 71–86

hb(km)

Alpha

Tb(K)

32 47 51 71

–2.8 0 2.8 2

228.65 270.65 270.65 214.65

43

Basic Principles Governing Aerodynamic Flows

TABLE 2.2 Table of temperature and pressure model parameters at various altitudes Altitude range(km) 86–91 91–110 110–120 120–150

hb

LM, b

TM, b(K)

Equation

86 91 110 120

0 0 12 12

186.946 186.870 240 360

2.16 2.16 2.17 2.17

where T∞ = 1,000 K T10 = 360 K λ = 0.01875 h10 = 120 km r0 is the Earth’s radius, r0 = 6,356 km For altitudes above 86 km, ρ = p/RT and p is given by the following equations:

 g M ( h − hb )  p = pb exp  − 0 * 0  TM ,b   R



  TM ,b p = pb    TM ,b + LM ,b ( h − hb ) 

 g0 M 0 1   *  R LM ,b 

(2.16)



(2.17)

where pb is the pressure at the end of the previous height band g0 M0/R* = 34.16 hb is the height at the lower end of the height band TM, b and L M, b are given in Table 2.2

2.9  GENERATION OF LIFT AND DRAG To understand the process of the generation of lift and drag, consider a wing section in a 2D steady flow of a perfect ideal fluid. Consequently, the fluid is assumed to not possess any viscosity, and as a further consequence, there is no energy dissipation, no work done and no drag, and the wing cannot develop any circulation around it. Hence, there is also no lift! When considering the streamline of the flow around such an aerofoil, ideally, there are two stagnation points that separate the flow domain into two regions, one above and the other below the aerofoils. The streamline on the separating boundary ends on the aerofoil surface at a stagnation point, somewhere below the leading edge for a positive angle of attack. It re-emerges at a rear stagnation point somewhere on the upper surface of the aerofoil just before the trailing edge and

44

Flight Dynamics, Simulation, and Control

proceeds downstream along the flow field. When the developed pressure distribution is integrated over the surface, there is no lift and no drag. In the case of an aerofoil in real flow field, in addition to the aforementioned flow pattern, a circulation of the flow around the aerofoil develops. Initially, the flow pattern is as in the ideal case, as discussed earlier. Yet because of boundary layer effects, the flow on the lower surface is retarded more than the flow on the upper surface. The two flows meet somewhere on the upper surface very near the trailing edge and form a local shear flow pattern, which results in the formation of an eddy. Moreover, the process of the eddy formation is unstable, and consequently, the eddy is swept downstream. Since the eddy contains a vortex with a finite circulation, it follows from the fundamental physics of vortex flows that an equal and opposite circulation must develop around the wing. Towards the end of the 19th century, the German mathematician Wilhelm M. Kutta and the Russian mathematician Nicolai E. Joukowski independently put forward the theory of lift generation in physical terms while also expressing it mathematically. In particular, Kutta postulated that the strength of the circulation generated by the viscous effects was just enough to move the rear stagnation point to the trailing edge, thus maintaining a smooth flow in the vicinity of the trailing edge. He argued that excessive or less circulation would not necessarily result in a stable and physically meaningful flow and that the physics of the real viscous flow field required the rear stagnation point to remain at the trailing edge as the velocity of the flow on either side of the trailing edge must be identical as it leaves it. This would ensure that there is no jump in the velocity of pressure, thus requiring the flow to be smooth at the trailing edge. The condition of smooth flow at the trailing edge thus determines the strength of the circulation and is known as the Kutta condition. It allows one to superpose a circulation of a finite and known strength over and above the ideal flow field so as to generate the correct flow pattern on a wing in a real viscous flow with very small amounts of viscous dissipation. Thus, the real flow field tends to be an ideal flow in this case and corresponds to a situation where the Reynolds number, Re = ρUd/μ, is high. The Reynolds number is a non-­d imensional number that characterises the ratio of the inertia to viscous forces, where the numerator is the product of the density of the fluid in the free stream (ρ), the free-stream velocity of the fluid (U) and a characteristic distance (d), while the denominator is the coefficient of viscosity (μ). The characteristic distance is usually taken to be the aerodynamic mean chord length c. It should be mentioned in passing that if the Reynolds number of the flow is too high, the flow ceases to be laminar at certain points on aerofoil and experiences a transition to turbulent flow flowed by separation and loss of lift. Thus, it is essential that the Reynolds number of the flow is optimum if the flow is to remain attached, which is essential for the generation of lift. Considering Bernoulli’s principle of conservation of energy of an ideal flow around the aerofoil, we may relate the pressure p and velocity perturbation q on the upper and lower sides of the aerofoil by the following equation:

1 1 2 2 pu + ρ (U + qu ) = pl + ρ (U + ql ) . (2.18) 2 2

45

Basic Principles Governing Aerodynamic Flows

Hence, pl − pu = ρ ( qu − ql )  U + 



qu + ql  . (2.19) 2 

Thus, the lift is given by c

L=



c



∫ ( p − p ) dx = ρ ∫ ( q − q )  U + l

u

u

0

l

0

qu + ql  dx. (2.20) 2 

In the limit when U >>(qu + ql)/2,

L=



c

c

0

0

∫ ( pl − pu ) dx = ρU ∫ ( qu − ql ) dx = ρUΓ. (2.21)

This expression may also be written as 1

c



L=

∫ ( p − p ) dx = ρU c ∫ l

2

u

0

( qu − ql ) dx U

0

c

= ρ U 2 cΓˆ = ρ UΓ. (2.22)

This is the Kutta–Joukowski theorem that relates circulation, and therefore vorticity, around an aerofoil to the lift.

2.10  AERODYNAMIC FORCES AND MOMENTS In section 2.7, it was shown that for steady inviscid flows, the Kutta–Joukowski theorem relates circulation, and therefore vorticity, around an aerofoil to the lift L, by the following equation:

L = ρU 2 c

∫ uˆ ·d ˆl = ρU cΓˆ = ρUΓ (2.23) 2

Σ

where the circulation, Γ, is defined to be positive in the clockwise direction. Here, Γˆ is the lift per unit span non-dimensionalised with respect to the product of density of the fluid (ρ), the aerodynamic mean chord length, c, and the square of free-stream velocity of the fluid (U). This quantity(conventionally multiplied by two) is called the lift coefficient and is usually denoted in the literature by CL . For inviscid fluids undergoing steady(non-accelerated) flows,



d ˆ d Γ= dt dt

∫ uˆ ·d ˆl = 0 (2.24) Σ

46

Flight Dynamics, Simulation, and Control

and

Γˆ =

∫ uˆ ·d ˆl = Constant. (2.25) Σ

When an aerofoil starts from rest, the net circulation in the fluid before the start of the motion is zero. Thus, equation for Γˆ is simply a mathematical expression for Kelvin’s law, which states that the total circulation (and therefore the total vorticity) in an ideal fluid must remain zero at all times. In other words, if new vorticity (or circulation) is introduced in an inviscid fluid (e.g. through an application of the Kutta condition), then it must be accompanied by equal and opposite vorticity, which is usually distributed in the wake behind the trailing edge. However, as vorticity diffuses into the wake and is distributed in, it does not generate any forces within it. Thus, the only forces acting are those on the aerofoil itself. Yet this shed or diffused vorticity represents a loss in kinetic energy, and this contributes to the induced drag. The energy source is the energy in the circulation imposed on the aerofoil. So although one is dealing with an energy conservative potential flow problem and there is no energy dissipation overall, there is energy transport from a region near the aerofoil to the far field. It is this energy transported away from the aerofoil that manifests itself as the induced or vortex drag. Because the domain of the flow field is practically infinite, the fact that it is energy conservative within the domain is not very useful. In the aerofoil problem, one must necessarily consider a subdomain enclosing the body to evaluate the forces and moments acting on it. The shedding of vortices is considerably enhanced in a finite wing due to the formation of vortices at the tips of the wing. This process is illustrated in Figure 2.3a. Paired vortices are formed in the wake of the aircraft by an aerodynamic process that is directly related to the lift generated. The aerodynamic flow past the leading edge of each wing establishes a boundary layer that separated from the wing surface and rolls up into a spiral vortex sheet. At some distance behind the trailing edge of

FIGURE 2.3  (a) The mechanism of wing tip vortex formation and (b) definition of lift force, pitching moment and drag force.

Basic Principles Governing Aerodynamic Flows

47

the wing, the streamlines of the separated flow converge resulting in a dominant flow pattern consisting of a pair of vortices separated by a distance equal to the wingspan of the aircraft. The diameter of each tightly bound vortex core is only about 3% of the wing span. The vortices are remarkably stable and persist for long distances behind the aircraft. Wing tip vortices produced by the wings of smaller aircraft have a negligible effect in the wake, but wing tip vortices created by larger and heavier aircraft can be extremely dangerous to aircraft trailing behind, even at a distance of many miles. Wing tip vortices form a part of the entire wake roll-up vortex, and the turbulence generated in the wake of large aircraft can cause buffeting, instability, uncontrollable rolls and sudden loss of altitude in a trailing aircraft. This is due to the fact that the flow field between the vortex pair appears as an induced downwash, while beyond the vortex pair, it is an induced upwash, resulting in severe wind gradients in the vicinity of the vortices. There have been many incidents, especially at lower altitudes during landing approaches, when wake vortex effects have resulted in fatal accidents because of the inability of pilots to regain full control of their trailing aircraft after being buffeted violently by the powerful wake vortices. It is possible, in principle, to predict the time-dependent forces and moments acting on an aerofoil in an incompressible or in a compressible viscous flow with only the knowledge of the velocity or vorticity field in a finite and arbitrarily chosen region enclosing the body. Furthermore, the forces and moments can be conveniently expressed in terms of integrals of the moments of the vorticity and their time derivatives (see, e.g. Ashley and Landahl [3]). They are particularly useful in understanding the nature of the forces and moments acting on the aerofoil. For our part, it is much more convenient to define a set of non-dimensional aerodynamic coefficients, which may be used to represent the forces and moments acting on an aerofoil. Consider an isolated vortex of strength γ c located on the surface of a flat plate of chord c and infinite span, inclined to a uniform free stream with a velocity U at an angle α. The total circulation induced around the flat plate aerofoil is as follows: c



∫

Γ = γ c δ ( x − xγ ) dx = γ c. (2.26) 0

The total normal suction force acting vertically upwards per unit span is N = ρUcγ cos α , while the forward propulsive force per unit span is T = ρUcγ sin α . The lift force per unit span, normal to the direction of the free stream, is given by

L = N cos α + T  sin α = ρUcγ . (2.27)

The corresponding net drag force acting in the direction of the velocity per unit span is

D = N  sin α − T  cos α = 0. (2.28)

Assuming that the strength of the vortex is proportional to the velocity normal to the plate, U sin α, the lift force per unit span is

48

Flight Dynamics, Simulation, and Control

L = Kγ ρU 2 c sin α (2.29)



where it is assumed that γ = Kγ U sin α . One could also perform an exact analysis of the flow around a flat plate by ­mapping a circle in a uniform flow field on to a flat plate. This results in a nonuniform distribution of vortices, γ(x), along the surface of the flat plate and the total circulation induced is c

∫

Γ = γ ( x ) dx. (2.30)



0

While the complete analysis is beyond the scope of this section, the expression for the total lift force per unit span is L = πρU 2 c sin α . (2.31)



Thus, the simplified analysis presented earlier is valid provided Kγ = π, and the n­ on-dimensional lift force per unit span may be expressed as

CL =

L = 2Γˆ = 2π sin α . (2.32) 1 ρU 2 c 2

2.10.1 Aerodynamic Coefficients Assuming the wing to be unswept, of infinite span and of uniform chord, one may consider the forces and moment acting on a typical wing section. By convention, the lift, pitching moment and drag force are typically assumed to act at point along the chord of the wing section or aerofoil and in directions normal to and parallel to the airflow relative to the wing. These directions are illustrated in Figure 2.3b. The lift, pitching moment and drag are typically defined in terms of non-­dimensional lift, pitching moment and drag coefficients. The relationships between the non-dimensional wing section lift, wing section pitching moment and wing section drag coefficients, CL , Cm and CD, and the wing section lift force, L; wing section pitching moment, M; and the wing section drag force, D, are, respectively, given as follows:



L= M=

1 ρU 02 × S × C L (2.33a) 2

1 ρU 02 × S × Cref × Cm (2.33b) 2

and

D=

1 ρU 02 × S × C D (2.33c) 2

49

Basic Principles Governing Aerodynamic Flows

where ρ is the density of the free stream U0 is the velocity of the free stream relative to the aerofoil cref is the reference aerofoil chord, usually, c, the mean aerodynamic chord S is the reference area, usually, SW, the area of the planform of the wing An important feature of the lift coefficient, CL , is that it may be analytically determined under assumptions of an ideal flow field(incompressible and inviscid flow) assuming that the aerofoil is essentially a flat plate. The analytical expression under these assumptions of ideal flow is given as follows: C L = C L 0 + 2π × sin α . (2.34)



When the reference axis is also the zero-lift line, CL0 = 0, that is, CL = 0 when α = 0. The variation of the lift coefficient with respect to the angle of attack α has also been determined experimentally in a wind tunnel by several experimenters. In almost all these experiments, the relationship has been found to be linear for values of α less than a critical value, αs, αs ≈ 12°. For values of α  d m         



(7.55)

where Um is the gust magnitude dm is the gust length and x is the distance travelled The gust is assumed to be acting in one of the body directions, and the corresponding expression for the gust linear velocity in that body axis may be obtained. By the aforementioned process, the deterministic translational components of the gust are modelled by



 ∆ug   ∆vg   ∆wg

  ∆ug 0  1  = −  ∆vg 0 2    ∆wg 0

   (1 − cos ( ω t )) .  

(7.56)

The velocity and direction of the mean wind with respect to the ground are not always constant along the flight path. This variation of the mean wind, not including any random fluctuations or turbulence, along a flight path is known as wind shear. The influence of wind shear on aircraft motion is of particular importance when and wherever it is relatively very large and significant. Thus, it is particularly important during the landing and take-off phases of a flight. If one employs the International Civil Aviation Organization standard atmosphere as a model, the temperature profile

Numerical Simulation and Non-Linear Phenomenon

279

in the lower reaches of the atmosphere is characterised by a standard non-zero lapse rate. For this lapse rate, a typical idealised wind profile is given by the following expression:

Vw = Vw

h = 9.15 m

×

(h

0.2545

− 0.4097

1.3470

Vw = 2.86585 × Vw

h = 9.15 m

) ,    ( 0 < h < 300 m )

,    ( h ≥ 300 m ) ,

(7.57a) (7.57b)

where Vw|h = 9.15m is the wind speed at an altitude of 9.15 m. If the direction of the wind vector relative to the north is ψwind (which is zero when the wind is blowing from the north), the two horizontal plane components along the body axes of an aircraft with a heading angle equal to ψ are given by

uwind = Vw cos ( Ψ wind − π − Ψ ) ,  vwind = Vw sin ( Ψ wind − π − Ψ ).

(7.58)

Hence, the lateral gust velocity and gust rotation rate components, Δvg and Δrg, may be determined. Given a wind shear profile, due to atmospheric boundary effects, as a function of altitude and the measured wind speed at 20 ft (6 m) above the ground in the form



 z In    z0  ∆u = W20 ,    1, 000 > z > 3 ft  20  In    z0 

(7.59)

where Δu is the mean wind speed W20 is the measured wind speed at an altitude of 20 ft z is the altitude z0 is a constant equal to 0.15 ft for Category C flight phases and 2.0 ft for all other flight phases, Δug and Δqg may be obtained, when z = zCG, the height above ground of the aircraft’s centre of gravity (CG) (Category C flight phases are defined to be terminal flight phases, which include take-off, approach and landing.) A modified expression for the wind shear velocity profile in the vicinity of the Earth’s surface, taking into account the local Coriolis effect, is



 z 12ω s sin ϕ × z In   + 5.75  z0  U* ∆u = W20 ,    1, 000 > z > 3 ft.  20  12ω s sin ϕ × 20 In   + 5.75  z0  U*

(7.60)

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Flight Dynamics, Simulation, and Control

In Equation 7.60, ωs is the sidereal rate ϕ is the latitude U* is the friction velocity (proportional to the square root of the turbulent shear stress, which is assumed constant in the lower boundary layer) Thus, expressions for Δug and Δqg may be obtained, when z = zCG, the height above the ground of the aircraft’s CG. A microburst is a very severe type of wind shear where very large nose winds are immediately followed by large trail winds in a very short-time interval. Thus, an aircraft flying through a microburst could experience severe wind shear as illustrated in Figure 7.9. A microburst model developed by NASA in 1988, based on boundary layer stagnation flow, employs approximations for the velocities of the wind in the horizontal and vertical directions given by    u =



  r  2     z  λr  1     z   × 1 exp −  −     exp  −  *   − exp  −     , (7.61a) 2 2  ( r /R )    ε   z   R  

  r 2     z     z   w = −λ × exp  −     ε  exp  −    − 1 − z *  exp  −  *   − 1  ,    ε    z      R  

where λ is a scaling factor r is the radial distance

FIGURE 7.9  Flight of an aircraft through a microburst.

(7.61b)

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Numerical Simulation and Non-Linear Phenomenon

R is the radius of the downdraft shaft z is the altitude z* is the characteristic height out of the boundary layer ε is the characteristic height in the boundary layer The corresponding expressions for the gust angular velocities in a microburst about the body axes may then be obtained. Based on an earlier published model, Ivan, Zhao and Bryson [3,4] formulated a ring-vortex model to simulate a downburst’s flow field. The advantage of their revised model is that one could adopt the superposition principle and employ a multiple-ringvortex model to provide a better description of the flow field. The model employs six parameters per ring vortex: Γ the ring-vortex model circulation, R the radius of the vortex ring, Rc the radius of finite core and X, Y and Z the 3D coordinates of the ring centre. The induced velocities (u, w) at any point of interest, defined by the three coordinates (x, Y, h), are then computed through the following relations: x1 = x − X − R,    x 2 = x − X + R,    h p = h − H ,    hm = h + H ,    r1 p = x12 + h p2 ,  r2  r2 p = x 22 + h p2 ,    r1m = x12 + hm2 ,    r1m = x 22 + hm2 ,    ζ = 1 − exp  − 02  ,  Rc     r0 = min ( r1 p , r2 p )    rxp = ( x − X )2 + h p2 + R 2 ,     rxm = ( x − X )2 + hm2 + R 2 , 

 ( x − X )2  rhp =  2 2  4 + h + R  p

3/4

 ( x − X )2  ,    rhm =  2 2  4 + h + R 

3/4

.      

(7.62)

m

When r0< ε where ε is a small number, representing a point close to the ring filament, (u, w) = (0, 0). Otherwise,



u=

w=

1.182Γζ  R  h p h p  R − −  2π  rxp  r22p r12p  rxm

 hm hm    r 2 − r 2   , 2m 1m  

1.576Γζ  R  x1 x2  R  x1 x2   − 3/2   3/2 − 3/2  − .  3/2 2π  rhp  r1 p r2 p  rhm  r1m r2 m  

(7.63a)

(7.63b)

Zhao and Bryson [3] adopt a two-ring-vortex model for a typical downburst. Two sets of parameters, one representing a moderate downburst and the other a severe downburst, are listed in Table 7.1. A complete gust model corresponding to a downburst may be established from the aforementioned data.

282

Flight Dynamics, Simulation, and Control

TABLE 7.1 Downburst Parameters Parameter and Unit Γ1 (ft2/s) R1 (ft) H1 (ft) Rc1 (ft) Γ2 (ft2/s) R2 (ft) H2 (ft) Rc2 (ft)

Moderate Downburst

Severe Downburst

200,000 5,500 2,000 500 120,000 4,000 2,500 500

400,000 5,000 2,000 500 280,000 3,500 2,000 300

7.6  P  RINCIPLES OF RANDOM ATMOSPHERIC DISTURBANCE MODELLING 7.6.1  White Noise: Power Spectrum and Autocorrelation A filter is a SISO system that is a linear time invariant and can be represented by the relation Y(t) = L(X(t)) where L() is a linear operator. Considering X1(t) and X2(t) as two arbitrary inputs and a, b two arbitrary real constants, this system is said to be linear if

L ( aX1 ( t ) + bX 2 ( t )) = aL ( X1 ( t )) + bL ( X 2 ( t )) .

(7.64)

Moreover, the system is said to be time invariant if

L ( X ( t − t0 )) = Y ( t − t0 ) ,

(7.65)

where t0 is an arbitrary time shift. In probability theory, a stochastic process or sometimes simply a random process is a collection of one or more random variables which represent evolution of some random values, or system, over time. Associated with a stochastic process is a probability density function f(ζ, t) that maps a random variable or signal ζ over a domain. Let ζ be a continuous random variable. Then, the probability density function of ζ is a function f(ζ, t) such that

f (ζ, t ) dζ = p {ζ ≤ ζ ( t ) ≤ ζ + dζ} ,

(7.66)

where p{x (t)} is the probability function. For a stochastic process, first-order probability distribution and first-order probability density function f(ζ, t) are defined as the following:

F ( x, t ) = p { x (t ) ≤ x}

(7.67a)

Numerical Simulation and Non-Linear Phenomenon

F ( x,t ) =



∂F ( x, t ) . ∂x

283

(7.67b)

The ensemble average of a stochastic process for a stochastic variable x(t) is called an expected value and is defined as E ( x ( t )) =



∞

∫ xf ( x,t )dx.

(7.68)

−∞

Autocorrelation function is defined as ∞ ∞



Rx ( t1 , t2 ) =

∫ ∫ x x f ( x , x ; t ,t )dx dx 1 2

1

2

1

2

1

2

= E { x1 ( t ) x 2 ( t )} ,

(7.69)

−∞ −∞

where f is a stochastic variable and x(t1) = x1 and x(t2) = x2. A stochastic process x(t) is said to be strict-sense stationary, if it is statistically independent of distance from the origin. In other words, x(t) is statistically equal to x(t + c), where c is an arbitrary value. Also, a stochastic process x(t) is said to be widesense stationary (WSS) when the expected value (average) is constant – E{(x(t)} = η – and its autocorrelation relates only with the difference between t1 and t2 (τ = t1 – t2) and independent of t1 and t2 values:

{

}

E x ( t + τ ) x * ( t ) = R ( τ ) .

(7.70)

White noise is an idealisation, and as an excitation, it represents an ergodic random process that is a random noise with a power spectrum that is independent of frequency, and its value at any frequency is the same and defined as

Sw ( f ) = q =

N0 . 2

(7.71)

This noise is called white noise because the density spectrum of this process is widely distributed in the frequency domain like white light. The autocorrelation function is the inverse Fourier transform of power spectrum density (PSD). Therefore, the autocorrelation function of white noise can be represented as follows (see Figure 7.10a and b):

Rw ( τ ) = qδ ( τ ) =

N 0δ ( τ ) . 2

(7.72)

7.6.2  Linear Time-Invariant System with Stochastic Process Input For a linear operator L defining a linear system with input x(t) and output y(t), if the input is a stochastic process, then the output is also a stochastic process. To find the relation between output autocorrelation function and input autocorrelation function,

284

Flight Dynamics, Simulation, and Control

FIGURE 7.10  (a) Autocorrelation function of White noise and (b) its spectrum.

observe that the expectation operator applied to the output of the linear system is equal to the linear operator acting on the output of the expectation operator applied to the input. Thus, it follows that

{

}

E L [ x ( t )]   = L  E { x ( t )}  .

(7.73)

Since the linear operator L and the expectation operator E are both linear, we have

Rxy ( t1 , t2 ) = L2 [ Rxx ( t1 , t2 )] ,

(7.74)

where R xy is the cross-correlation of x(t) and y(t) Ln means that the system acts at tn, which is the independent variable, and tn is to be treated as a parameter Through this relation, we can develop a relationship between input and output spectrums. For x(t), a WSS process, the PSD is the Fourier transform of its autocorrelation function: ∞



S (ω ) =

∫R

xx

( τ ) e − jωτ dτ.

(7.75)

−∞

Since R(–τ) = R* (τ) and S(ω) is a real function of variable ω. Using the inverse Fourier transform, we have

1 Rxx ( τ ) = 2π

∞

∫ S (ω ) e

jωz

dω.

(7.76)

−∞

It means that R xx(τ) can be obtained from the spectrum S(ω). We can find infinite processes that have the same spectrum S(ω). There are two equivalent principles that are

285

Numerical Simulation and Non-Linear Phenomenon

commonly used to approximate a random process with a given spectrum by another with its own spectrum. Considering a random process as follows: x ( t ) = ae j(ωt −ϕ ) .



(7.77)

in which a is a real constant ω is a stochastic variable with a density of fω(ω) ϕ is an independent stochastic variable with uniform density on (0, 2π), and it can be proved that this process is a WSS process with zero mean and the autocorrelation function given by

Rx ( τ ) = a 2 E {e jωτ } = a 2



+∞

∫f

ω

( ω ) e jωτ dω.

(7.78)

−∞

Therefore, its spectrum can be found from the Fourier transform of the autocorrelation function which is S x ( ω ) = F { Rx ( τ )}



(7.79)

and 1 Rx ( τ ) = 2π



+∞

∫ S (ω ) e x

jωτ

dω .

(7.80)

−∞

Hence, it follows that S x ( ω ) = 2πa 2 fω ( ω ) .



(7.81)

This relates the PSD function of x(t), with the probability density function of ω. Substituting τ = 0 in Equation 7.81 yields +∞



Rx ( 0 ) = a

2

∫

−∞

1 fω ( ω ) dω = a = 2π 2

+∞

∫ S (ω ) dω. x

(7.82)

−∞

Thus, to find a stochastic process of its spectrum, we suppose that the probability density function is

fω =

Sx (ω ) . 2πa 2

(7.83)

286

Flight Dynamics, Simulation, and Control

Thus, a2 = R x(0) and R(0) is the signal’s power of process. In this way, the process with the autocorrelation function defined by Equation 7.78 would have the spectrum of S(ω). Another equivalent principle is based on the impulse response of a linear system. Consider a linear time-invariant system with an impulse response h(t). The response to any input x(t) is defined by the convolution integral t

∫

y ( t ) = h ( t − τ ) x ( τ ) dτ.



(7.84)

0

It is usually denoted by y(t) = h(t) * u(t). If x(t) is a WSS process, the input crosscorrelation is related to the input autocorrelation by ∞



∫ h (τ) R

Rxy ( t ) = h ( −t ) Rxx ( t ) = *

*

*

xx

( τ + t ) dτ,

(7.85)

−∞

where h* (t) is the complex conjugate of h(t). Similarly, the relationship between the output autocorrelation and input crosscorrelation would be ∞



Ryy ( t ) = h ( t ) Rxy ( t ) = *

∫ h (τ) R

xy

( t − τ ) dτ.



(7.86)

−∞

Eliminating the cross-correlations, the output and input autocorrelations are related by

Ryy ( t ) = Rxx ( t )* h ( t )* h* ( −t ) .

(7.87)

Taking Fourier transform from both sides of the earlier equation, we have

S yy ( f ) = S xx ( f ) H ( f ) H * ( f ) = S xx ( f ) H ( f ) , 2

(7.88)

If the input of system is white noise with q = 1, then

Rxx ( τ ) = qδ ( τ ) = δ ( τ ) ,    S xx ( t ) = 1



(7.89a)

and

S yy ( f ) = S xx ( f ) H ( f ) ⇒ H ( f ) = S yy ( f ) . 2

(7.89b)

So, to achieve a process with a certain spectrum, a system with the following TF can be defined as follows:

H ( f ) = S ( f ) ,   H ( f ) = 0.

(7.90)

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If the input of such a system is white noise, then the output of the system will have the spectrum S(f). Hence,

S yy ( f ) = H ( f ) S xx ( f ) = 2

(

S( f )

)

2

× 1 = S ( f ).

(7.91)

7.7  APPLICATION TO ATMOSPHERIC TURBULENCE MODELLING According to the standard references, turbulence is the fluctuations in the wind translational and rotational velocities about certain pre-defined means and is a stochastic process defined by velocity spectra. The von Karman and Dryden wind turbulence models employ corresponding spectral representations of turbulence fields. The Dryden model is particularly convenient to simulate, and it is possible to include realistic turbulence inputs by passing band-limited white noise through appropriate shaping filters. The Aerospace Blockset in MATLAB–Simulink not only provides for these models to be included in a typical simulation but also provides various spectral representation functions as defined in the Military Specification MIL-F8785C and Military Handbook MIL-HDBK-1797. It is not our intention to reproduce these functions here. To illustrate the use of one such spectral function, we consider a turbulence input in the vertical perturbation velocity field. The spectral representation, which is probably most appropriate for this when the wavelength of the gust is relatively larger than the length of the aircraft, L, (λ > 8L), is the corresponding von Karman spectrum, while the Dryden spectrum is a very good approximation of this over a wide bandwidth. The von Karman approximation to model the vertical Gust spectrum is



Φ ww ( ω ) = σ 2ww

Ls πU se

1 + 8 1.339 L ω 2  s )   3 (  ,  ω = ω , 2 11/6 U se 1 + (1.339 Ls ω )   

(7.92)

and the horizontal by

2 Φuu ( ω ) = σ uu

2 Ls 1 πU se 1 + (1.339 Ls ω )2 5/6  

(7.93)

where ω is the gust frequency σww and σuu are gust intensities Ls is the turbulence scale length Unfortunately, because of the fractional powers involved, the filters for simulating the gusts are physically realisable without further approximations. Hence, the Dryden approximations of the lateral and longitudinal spectra are commonly used. The vertical and horizontal Dryden spectra (MIL-F-8785C) may be approximated by

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Flight Dynamics, Simulation, and Control

Φ ww ( ω ) = σ 2ww

2 Ls 1 + 3 ( Ls ω )  1 2 2 Ls ,   Φuu ( ω ) = σ uu . (7.94) πU se 1 + ( Ls ω )2  2 πU se 1 + ( Ls ω )2     

The gust intensity is related to the area under the spectral density curve. Thus, ∞

∫

2 σ uu =  Φuu ( ω ) dω ,



(7.95)

0

and similar integral formulas apply in other directions. It is assumed that the gust can be derived from unit-intensity white noise which is an idealisation of random noise and is characterised by a flat power spectrum over a very large range of frequencies. Following the stochastic process theory, a typical power spectrum may be assumed to be the spectrum of the output of a noise-shaping filter with unit white noise input and a TF given by Guu ( s ) = σ uu



2 Ls 1 ,   s = iω. πU se 1 + Ls s

(7.96)

It then follows that

2 Φuu ( ω ) = Guu ( iω ) × Guu ( −iω ) × 1 = σ uu

2 Ls 1 . e πU s 1 + ( Ls ω )2   

(7.97)

The corresponding state vector equation of the shaping filter is Ls z ( t ) = −U se z ( t ) + K GUes w ( t ) ,   ∆ut = z ( t ) ,  K G = σ uu



2 Ls πU se

(7.98)

where w(t) is unit intensity white noise. An interesting feature of the horizontal and side-velocity components of the Dryden spectrum is that they can be simulated by passing white noise through a second-order filter. The velocity components may be defined by  ∆vt   ∆wt

 = 

 n1   n2

  ,   n = σ T 

LT /U 0  1 2π 

 x1 3 ( LT /U 0 )     x 2 

 e  ,  U 0 = U s , 

 (7.99a) d  x1 ( t )  dt  x 2 ( t )  

 = 

 0  2  − (U 0 /LT )

1 −2U 0 /LT

  x1 ( t )    x 2 ( t )

  +  

0

(U 0 /LT )

2

  wn ( t ) .  (7.99b)

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It can be shown that the spectrum of the output signal closely matches the Dryden spectrum. Thus,

Snn ( f ) = H ( f ) Sww ( f ) = S Dryden ( f ) . 2

(7.100)

Thus, the turbulence gust velocities in the three axes can be defined as   ∆u t    =  ∆vt   ∆w t  

 ∆ug   ∆vg   ∆wg



  .  

(7.101)

In general, the spectral approximations to the turbulence spectra according to the von Karman and Dryden approximations in all three translational and rotational axes are listed in Table 7.2.

TABLE 7.2 Table of Dryden and von Karman Spectra Models Spectral Comp. Φuu(ω)

Φvv(ω)

Dryden Model

von Karman Model

2σ Lu 1 πU 0 1 + ( Lu ω / U 0 )2

Φqq(ω)

5/6

m f σ 2ww Lw 1 + (8/3) (1.339m f Lw ω /U 0 ) 2 11/6 πU 0 1 + (1.339m f Lw ω /U 0 )

σ 2ww 0.8 ( Lw π /4 b ) U 0 Lw 1 + ( 4 bω /πU 0 )2

σ 2ww 0.8 ( Lw π /4 b ) U 0 Lw 1 + ( 4 bω /πU 0 )2

2

)

1/3

± ( ω /U 0 )

2

± ( ω /U 0 )

2

2

1 + ( 3bω /πU 0 ) Source: Gage [5].

)

m f σ 2ww Lw 1 + 3 ( m f Lw ω /U 0 ) 2 2 πU 0 1 + ( m f Lw ω /U 0 )

)

1 + ( 4 bω /πU 0 ) Φrr(ω)

(

m f σ 2vv Lv 1 + (8/3) (1.339m f Lv ω /U 0 ) 2 11/6 πU 0 1 + (1.339m f Lv ω /U 0 )

2

(

Φpp(ω)

2 uu

m f σ 2vv Lv 1 + 3 ( m f Lv ω /U 0 ) 2 2 πU 0 1 + ( m f Lv ω /U 0 )

(

Φww(ω)

2σ Lu 1 πU 0 1 + (1.339 L ω /U )2 0 u

2 uu

2

Φ ww ( ω )

Φ vv ( ω )

2

(

)

2

(

)

1/3

± ( ω /U 0 )

2

1 + ( 4 bω /πU 0 ) ± ( ω /U 0 )

2

Φ ww ( ω )

2

1 + ( 3bω /πU 0 )

2

Φ vv ω

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Flight Dynamics, Simulation, and Control

To state the continuous white-noise-shaping filters in a compact form, they are first related to a set of normalised filters, in terms of a normalised Laplace transform variable as Lu Lv Hu ( s ) ,  H v ( s ) = σ vv Hv ( s ), πU 0 πU 0

Hu ( s ) = σ uu

           H s ( s ) = σ ww



Lw Hw ( s ), πU 0

(7.102a)

See also the US military reference handbooks published in 1980 (MIL-F-8785C) and 1997 (MIL-HDBK-1797); mf = 1 for MIL-F-8785C and mf = 2 for MIL-HDBK-1797 and U 0 = U se. H p ( s ) = σ ww

Lw Lw Lv Hr ( s ) , H p ( s ) ,   H q ( s ) = σ ww H q ( s ) ,      Hr ( s ) = σ vv πU 0 πU 0 πU 0

 (7.102b) where s = s / U 0 . The continuous normalised white-noise-shaping filters are listed in Table 7.3. TABLE 7.3 Table of Continuous, Normalised Shaping Filters to Derive the Dryden and von Karman Spectral Approximations from White Noise Filter

Dryden Model

2 (1 + 0.25 Lu s /U 0 )

Hu ( s )

2 1 + Lu s

Hv ( s )

m f 1 + m f 3 Lv s

(

(1 + m Hw ( s )

(

Lv s )

1 + 1.357 Lu s + 0.1987 ( Lu s )

2

m f 1 + m f 3 Lw s

(1 + m Hp (s )

f

 0.128π 4   m 2 bL5   f w 

f

Lw s )

1/6

von Karman Model

2

) )

1 1 + 4 bs /π

1 + 2.7478 Lv s + 0.3398 ( Lv s )

2

1 + 2.9958 Lv s + 1.9754 ( Lv s ) + 0.1539 ( Lv s ) 2

1 + 2.7478 Lw s + 0.3398 ( Lw s )

3

2

1 + 2.9958 Lw s + 1.9754 ( Lw s ) + 0.1539 ( Lw s ) 2

 0.128π 4   bL5  w

1/6

1 1 + 4 bs /π

Hq ( s )

±s Hw (s) 1 + 4bs /π

±s Hw (s) 1 + 4bs /π

Hr ( s )

∓s Hv (s) 1 + 3bs /π

∓s Hv (s) 1 + 3bs /π

Source: Gage [5].

2

3

Numerical Simulation and Non-Linear Phenomenon

291

7.8  A  IRCRAFT NON-LINEAR DYNAMIC RESPONSE PHENOMENON In the past few years, the ability to operate an aircraft at high angles of attack and attitudes and rates well beyond conventional limits has become an important issue. The recent upsurge in the interest in a high angle-of-attack flight both in the case of military aircraft and commercial airlines has culminated in NASA earmarking a high-performance fighter, the F-18, to research and mitigate the practical constraints on such flights. Thus, there is a demand for increased agility and carefree manoeuvring throughout the envelope of the aircraft. This demand naturally leads to the need for a better understanding of the non-linear modes of aircraft motion. For a complete understanding of the modes, we need to understand not only the conditions and consequences of the dynamic balance between forces and torques acting on the aircraft but also the sources of energy available for the generation of aerodynamic forces, namely kinetic energy by virtue of its speed, potential energy by virtue of its altitude and chemical energy by virtue of the fuel expended in the propulsion unit. The term energy state is often employed to describe how much of each kind of energy is available to the aircraft (and its pilot) and how much is dissipated at each instant of time during a manoeuvre. One can then deliberately seek to manipulate and manage the energy state to establish a desired equilibrium state or manoeuvre. The net result of a manoeuvre is the generation of aerodynamic forces and moments, but the aircraft must also always operate between the limits on a number of state variables to ensure efficiency. Unfortunately, the pilot cannot exercise direct control of the energy transfer mechanisms but can only control some of the forces and moments by employing the throttle or deploying the control surfaces. The state variable limits then are of even greater significance. For example, to avoid flow separation over the wing surface, the angle of attack must typically be no more than 24° nose up, 10° nose down, the bank angle within 45° and the airspeed within appropriate limits for the conditions of flights. Exceeding the limits would result in flow separation. When the airflow around an aerofoil separates from the suction surface, the condition of aerodynamic stalling occurs. Entering a stalled state would lead to buffeting, lack of pitch or roll authority and result in the pilot being unable to arrest the descent rate. Some of the effects of flow separation and features of aerodynamic stall are discussed by Rom [6] and by Tobak et al. [7]. While pilots today are trained extensively in recovery techniques, the focus of this chapter is primarily on the non-linear modes of motion of an aircraft. Recovery from these modes either by the application of manual controls or by virtue of feedback control, although of fundamental importance, is not within the scope of this chapter. In developing the non-linear dynamics of an aircraft, it is customary to establish an appropriate set of reference axes followed by derivation of a complete set of nonlinear differential equations governing the motions of the aircraft. However, it must be said that very rarely in the literature is the process carried forwards to completion: enunciation of the solution of the equations followed by characterisation of the nonlinear modes of a flight. Although aircraft dynamics is seemingly the same no matter what the reference axes and aircraft responses are quite independent of the choice of these axes, certain equilibrium states are more readily found when a particular

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reference frame is selected. Moreover, the chosen frame is also most suitable for the study of the behaviour of small perturbations about the said equilibrium states as well as the stability of the equilibrium states themselves. A typical example is the socalled “falling leaf”, a fully coupled non-linear mode of motion where the analysis is not only simplified but also meaningful if the frame is chosen appropriately. In conducting simulations of non-linear aircraft dynamics, it is generally found that there is a need to have a complete understanding of the physics of the modes of motion as well as their principal features before establishing a matching set of reference axes and the associated governing EOMs, representing the balance of forces and torques. An album of aircraft non-linear motions is generally established in the first instance. In this section, based on such an approach, the principal non-linear modes of aircraft motion are summarised and then classified. The relationships between these modes which are able to capture a number of characteristic non-linear behaviours and the traditional linear modes based on linearised small perturbation equations are explained. The objective is a holistic view of an aircraft’s linear modes of motion and their non-linear counterparts.

7.8.1  Aircraft Dynamic Non-Linearities and Their Analysis The linear modes of motion corresponding to small perturbations from a state of equilibrium in a uniform uncontrolled flight with constant forward speed are well described in the literature. Non-linear models of aircraft dynamics involve kinematic, dynamic and aerodynamic non-linearities. There are tried and tested procedures for modelling kinematic and non-aerodynamic non-linearities. Aerodynamic non-linearities on the other hand pose a number of problems to the dynamicist. The aerodynamic loads associated with aerodynamically non-linear flows are characterised both by certain classical static non-linear behavioural models and dynamic non-linear hysteresis effects that need to be carefully modelled. Thus, the modelling of non-linear unsteady aerodynamic effects is of primary importance in the modal analysis of aircraft dynamics. It is recognised that models of non-linear aircraft dynamics are computational prototypes that are essential for understanding the behaviour of the aircraft in an aerodynamic environment. The basis for the modelling stems from the following facts:

1. Only certain aspects of the behaviour play an important role in the development of the pressure distribution and the forces and moments induced by it on the aircraft, which in turn influence the non-linear dynamics. The other aspects contribute to the build-up of secondary effects such as turbulence and separation. 2. From a practical point of view, it is probably unrealistic and inefficient to encapsulate the entire behaviour of non-linear aircraft dynamics in a single model. 3. While there is a need to understand the full range of qualitative behavioural models of aircraft dynamics, specific computational prototypes are designed with specific applications in mind; the empirical exemplars are

Numerical Simulation and Non-Linear Phenomenon

293

based on a need for models of moderate complexity of the aerodynamic forces and moments acting on the aircraft, enough to capture the qualitative behaviour of interest to the modeller. The evolution of non-linear motions from their linear counterparts, referred to as departure, is normally investigated by employing the methodology of bifurcation analysis. The term departure is also used to represent the boundary between controlled and uncontrolled states which is assumed to be synonymous with a boundary between linear and non-linear behaviour. A bifurcation analysis can be performed with reduced-order local models that capture the behaviour of the aircraft in the local neighbourhood of the state space. These are not tools for a global analysis. Application of bifurcation analysis has been carried out by several authors [8–15] who have not only demonstrated the capability of the technique but also established departure-prone regions in the flight envelope by determining all steady-state conditions attainable by the aircraft. This is done by a detailed inspection of the bifurcation diagrams or continuation methods. (Continuation methods [16], it may be recalled, are a class of methods complementary to bifurcation analysis, which seek solutions to non-linear equations in the form of parameterised curves and surfaces.) Bifurcation analysis cannot really provide an answer to the transient behaviours of the aircraft prior to departure or indeed to the exact behaviours near the points of departure. Thus, there is a need not only for the modelling of the complete non-linear dynamics of the aircraft for an understanding of the aircraft’s modes of non-linear behaviour. Although there are a number of classes of aerodynamic non-linearities that must be considered in the analysis of aircraft dynamics, they are primarily caused by shock waves in transonic flows, separated flows and vortex-induced flows (see, e.g. Lee et al. [17], Katz [18] and Ekaterinaris and Platzer [19]). When there is a flow over a wing, there is invariably a formation of a boundary layer. Initially, the boundary layers are very thin, viscous effects are confined to a rather small region, and the fluid friction may sometimes be neglected. Even if the boundary layers are thin to begin with, they can thicken rapidly with increasing angles of attack and the flow can separate. The flow over a wing surface can then produce a large region of separated flow downstream of the aircraft resulting in a wake where the flow is usually highly unsteady and large eddies or vortices are shed downstream. The large eddies are formed at a regular frequency and known to dissipate a lot of energy. Vortex-induced flows generally are associated with unsteady separation and a consequent local thickening of the boundary layer. These are initiated by the presence of a thick core vortex in the vicinity of the leading edge. There is considerable flow recirculation within the boundary layer leading to substantial energy dissipation and consequent loss of lift. At higher Mach numbers, 3D separated flows are further complicated by transitional turbulent boundary layers and shock waves, along with their mutual interactions. These interactions often lead to a dramatic fall in the lifting performance of the wing. When the flow, associated with the presence of shock waves in transonic flows, is inviscid, a separation does not occur. The unsteady forces generated by the motion of the shock wave have been shown to destabilise single-DOF aerofoil pitching motion

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and affect the bending–torsion flutter by lowering the flutter speed at the so-called transonic dip regime. It is known that the shock waves located on the upper and lower wing surfaces move periodically with large phase lags relative to the forced oscillatory aerofoil motion. These relative motions are the predominant factor in the anomalies observed when the Mach number approaches unity. At certain low frequencies relative to angular velocities of the wing that generate the same leadingedge velocity as its translational velocity, a peculiar shock-type behaviour is known to occur. During part of the oscillation cycle, the shock disappears only to reappear at a later time. The forces generated are non-linear and usually a number of harmonics are present. When viscous effects are present, non-linearities associated with flow separation can occur which are caused by the shock–boundary layer interaction. At a low speed, aerodynamic non-linearities associated with flow separation are found in the dynamic stall of helicopter blades. At a low angle of attack, the flow is attached, but leading edge separation is initiated as it approaches a certain critical angle. The leading edge vortex moves downstream along the aerofoil surface and, after traversing a certain distance from the leading edge, lifts off from the surface and finally convects away from the trailing edge. During the downstroke cycle, the flow reattaches on the upper surface starting from the trailing edge and moves upstream while separation occurs on the lower surface following events on the upper surface during the upstroke. In both these cases, the lift and moment coefficients exhibit a hysteresis type of behaviour. Yet, this behaviour is qualitatively quite different in each case (see, e.g. Prananta et al. [20], Wegener [21]). In the case of low-speed separated flows, three distinct types of hysteresis loops can be identified: one below CLmax, the second around CLmax and a third type of hysteresis loop well above CLmax. In the second case, there is clear evidence of aerodynamic bifurcations resulting from the instability of a substantial leading edge separation bubble in the flow. In transonic flows, the appearance of a shock wave can trigger flow separation. The shock wave motion causes a large pressure fluctuation on the wing surface which in turn triggers flow separation accompanied by hysteresis. In order to quantify the effect of shock wave motion on the unsteady aerodynamic characteristics, it is necessary to examine the variation of the amplitude and the phase angle of shock wave motion due to variations in the frequency. When a flow separation occurs, triggered by a shock wave, the lift coefficient starts to decrease and there is a noticeable dip in the CL–M plot. The drag coefficient increases sharply at these Mach numbers. This phenomenon is called shock stall. The hysteresis behaviour has some analogies with the aerofoil stall, past the angle of CLmax. Another type of distinct aerodynamic non-linearity arises from the formation of leading edge and wingtip vortices. The vortices oscillate with time and their strengths depend on the static angle of attack and the amplitude of aerofoil motion. This type of flow can readily be modelled using an unsteady distribution of vortices. The wing and wake are modelled as a vortex lattice. The position of the wing portion, called the bound vortex lattice, is specified, and there is a finite pressure jump across it. The position of the wake portion, referred to as the free-vortex lattice, is not specified but is force free and predicted as part of the solution. The aerodynamic loading is

Numerical Simulation and Non-Linear Phenomenon

295

determined by calculating the pressure jump across each individual element in the bound vortex. However, for very large amplitude motions when the roll-up vortices are formed alternatively on the upper and lower surfaces of the wing, the phase lag with respect to the wing oscillation can be very large leading to instability of the motion. Both separated and vortex-induced flows can cause rotary motion of various wings and aeroplane configurations leading at times to limit cycle roll oscillations, commonly referred to as wing rock. A study of this phenomenon, based on numerous experimental observations, demonstrates the role of the leading edge vortices in driving the motion (see, e.g. Katz [18]).

7.8.2  High-Angle-of-Attack Dynamics and Its Consequences When the longitudinal stability of a symmetric aircraft is examined beyond the wellknown linearisation about a steady mean state, leading to the phugoid and short period modes, the exact equations of longitudinal motion of a symmetric aircraft must be considered based on the balance of longitudinal and transverse forces (without side force) and balance of a pitching moment. Furthermore, the angle of attack relative to the angle of zero pitching moment and the flight path angle must be considered to be moderate. In the case of a small flight path angle and small angle of attack, the equations simplify to a pair of differential equations, containing a set of non-linear corrections. It may be shown, by a small perturbation method, that forced oscillations occur at the harmonics of the short-period frequency (at both double and triple frequency), and free oscillations can have a decaying or growing amplitude. Furthermore, a local bifurcation analysis indicates a Hopf type of bifurcation, indicating that the short-period linear mode evolves into a non-linear mode. The Hopf bifurcation occurs when a pair of complex conjugate eigenvalues crosses the imaginary axis of the complex s plane, and the result is one or more periodic responses that could be stable or unstable. The presence of both the stable higher harmonics and Hopf bifurcation indicates a relatively simple stability boundary and a relatively complex post-critical or post-bifurcation response. A global analysis and description of the modes are therefore essential. The most common bifurcation phenomenon in aircraft flight dynamics is the pitchfork and Hopf family of bifurcations [22–24]. In a pitchfork bifurcation, the number of equilibrium states goes from one to three as the bifurcation boundary is crossed, and the stability of the original equilibrium state changes. One or more of the bifurcation phenomena appear depending on the choice of the bifurcation parameter. Typical of these have been the steady components of the flight speed, angle of attack, sideslip angle, roll rate and elevator angle. While simple analytical models can only provide insight into the mechanics of these bifurcations, which result in a variety of undesirable motions, flight simulation can provide a complete understanding of these phenomena. In particular, we refer to the pitchfork bifurcation of the phugoid resulting in the aircraft tumbling or the Hopf bifurcation of short-period dynamics resulting in the stall/post-stall motions in the case of longitudinal dynamics. In the lateral case, we have the pitchfork and Hopf bifurcations of the spiral mode resulting in the so-called wing-rock-type roll oscillations at moderately high- and high-angle-of-attack flight regimes as well as longitudinal/lateral coupled post-stall

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modes resulting in stall–spin motions such as flat and deep spins. The problems of wing rock or stall-induced spin could be further accentuated if not properly controlled by the pilot resulting in pilot-induced oscillations [18,25]. The latter more complex coupled modes clearly indicate the existence of a deeper coupling between the flight dynamic and aerodynamic bifurcations on the one hand and pilot behavioural dynamics on the other. The influence of aerodynamic hysteresis on these phenomena can be significantly destabilising, and flight simulation can provide some additional insights into the behaviour when such features are present. These dynamic phenomena involve the coupling of flight dynamic and aerodynamic bifurcations, particularly saddle node and period-doubling-type bifurcations associated with vortical flows, which are extremely difficult to model in practice. Enunciation of the flight dynamical modes could not only provide valuable insights into these phenomena but also provide dynamicists and control engineers insights that would help them to design better controllers for an aircraft in a stall-induced spin or wing-rock-type roll mode.

7.8.3  Post-Stall Behaviour High-angle-of-attack flight, which may be construed as predominantly longitudinal, has a number of consequences that influence not just the longitudinal dynamics but the motion of the entire aircraft. The sequence of events that evolve into the phenomenon of stall after the initiation of flow separation is briefly reviewed here primarily because of its importance in triggering a number of lateral modes which in turn could lead to disastrous consequences. The initiation of flow separation with increasing angle of attack has already been discussed. As the aerofoil incidence increases further, a region of reverse flow appears in the vicinity of the trailing edge on the upper surface, which not only grows in size but propagates upstream towards the leading edge. As a consequence, there is a region of separation which is quite thin but all along the aerofoil’s upper surface. As a consequence of the formation of this separation layer, the stagnation point moves aft along the aerofoil’s lower surface and there is an increase in the leading edge suction peak, while the slope of the lift curve versus the angle of attack (the lift curve slope) and the moment coefficient remain almost constant. Thus, the lift increases almost linearly with the angle of attack. With a further increase in the angle of attack, the separation region eventually reaches the vicinity of the aerofoil leading edge. There is now a sudden halt in the increase of the leading edge suction peak accompanied by a reduction in the lift curve slope. However, a further increase in the angle of attack results in the formation of a flow vortex just aft of the leading edge, and this event triggers an increase in the lift curve slope. Furthermore, there is a subsequent motion downstream of the leading edge vortex, followed by the separation of this vortex from the main flow and a collapse of the leading edge suction. This event is the initiation of the moment stall and accompanied by a fall in the lift curve slope. With the periodic separation of the leading edge vortex, there is a periodic force acting on the wing, and the inherent asymmetry of the separation process over the two halves of the wing results in a rolling moment. Thus, the separation process not only generates oscillatory pitching moments but also couples the motion to the lateral modes.

Numerical Simulation and Non-Linear Phenomenon

297

The post-stall regime provides a number of mechanisms that favour the coupling of longitudinal and lateral modes. When one of the wings is moving down, there is a further increase in the angle of attack relative to the wing moving up. The wing moving down operates in the stalled region of the lift curve, and consequently, there is a reduction in lift. This tends to push the wing further down and hence assists the autorotation of the wing. The ensuing rolling motion with high angles of attack operating in the vicinity of the peak of the lift curve slope has a tendency to be regenerative; that is, it behaves like a system with positive feedback. The aircraft is then in a near autorotation state and this feature tends to accentuate the spin mode. Similarly, the increasing drag force with an increasing angle of attack results in a net yawing moment on the rolling wing. The result is a coupled spinning motion, with high angles of attack and near-constant roll and yaw rates.

7.8.4  Tumbling and Autorotation There is also the case of tumbling spins or autorotation where the aircraft traces a predominantly straight path with little spiralling. Tumbling about the pitch axis is particularly pronounced in certain aircraft due to the non-linear nature of the variation of the nose-down aerodynamic pitching moment with changes in the angle of attack. In many cases, the tuck mode, a degenerate version of the phugoid when the mode is non-oscillatory and convergent (or divergent when unstable), eventually results in tumbling, following a bifurcation. While, prior to stall, the nose-down aerodynamic pitching moment is either constant or increasing with the angle of attack, there is a sharp decrease in the post-stall region. When the aircraft is operating in the vicinity of the peak of this curve, any increase in the angle of attack tends to increase the angle of attack further, thus facilitating autorotation in pitch or tumbling. Thus, in the case of aircraft where moment stall is a possibility, the tumbling mode is also a feature of the non-linear motions.

7.8.5  Lateral Dynamic Phenomenon We have already discussed how high-angle-of-attack effects causing lift stall and tip stall as well as the asymmetry in the drag curve can cause lateral moments resulting in some characteristically large motions in the lateral directions. Yet, there are also a few cases when rolling and yawing moment stalls can cause some characteristic responses in the lateral modes. Tumbling or autorotation is a natural outcome of the moment stall about the yaw or roll axes. The mechanism is no different from the case of tumbling about the pitch axis. Generally, there are never three independent tumbling modes about three different axes. In practice, there is only one tumbling mode or autorotation about an axis that is steady but coupled and oriented quite arbitrarily in three dimensions.

7.8.6  Flat Spin and Deep Spin In certain instances, control surfaces such as the aileron and rudder are also capable of inducing a spin. Swept wings were employed to fly at high speeds by delaying the

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transonic drag rise. However, the introduction of the swept wing resulted in problems in the low-speed, high-α regime. This was due to the loss of control following the swept wing’s inclination to stall first at the tips. This not only caused a forward shift in the centre of the lift resulting in a tendency to pitch upwards but also limited the aileron’s authority. If one swept tip stalls more rapidly than the other, the resulting asymmetry could happily yaw and roll an aircraft into a departure leading to a spin. Chordwise stall fences to control the spanwise flow, leading edge slats and repositioning of the tail wing away from the wake of the main wing were some of the features introduced to avoid a tip stall in a high-speed flight. Spins are known to evolve through four phases: departure, post-stall gyration, incipient spin and developed spin. Following stall and departure, there is usually an uncontrolled motion about one or more axes which is known as a post-stall gyration. These are usually in the form of snap rotations and include a tumbling motion over a finite time. Post-stall gyrations are absent in some aircraft, which directly depart into an incipient spin. As a fully developed spin evolves, there is an intermediate stage when the inertial forces and moments are still relatively small. It is in this phase that the aircraft is said to be in an incipient spin state. The developed spin usually involves either steady rates of roll, yaw and sometimes pitch or oscillatory spin involving fluctuations in the pitch attitude and rates of roll and yaw. The steady spin is characterised by steady roll and yaw rates at large non-zero values accompanied by small or moderate fluctuations in the angle of attack (α) or sideslip angle (β) at minimal rates of change of α and β. In such a mode, the aircraft remains stalled throughout the motion. Oscillatory spin, with predominant pitch oscillation and modest lateral activity, is characteristic of a deep stall. It is characterised by inphase roll and yaw rates fluctuating about large non-zero means. In such a mode, the aircraft remains stalled throughout the motion. The fluctuations in α are moderate, while fluctuations in β could be large. The rates of change in α and β remain moderate during the motion. Spins are generally made up of yaw and roll motions with the spin axis being usually vertical. Flat spin modes consist mostly of yaw, while deep spins are steep but mostly roll. The aeroplane’s CG follows a helical path around a displaced spin axis. The motion can be described very simply by adopting the screw theory. (A screw is simply a combined rotation and translation about the same axis. Any combined rotation and translation could be expressed as a screw about some axis in three dimensions.)

7.8.7  Wing Drop, Wing Rock and Nose Slice Wing drop is a peculiar non-linear mode of motion usually caused by a typical nonsteady rolling moment stall combined with a sideslip. Wing drop is an unacceptable, uncommanded abrupt lateral roll-off that randomly occurs and involves rapid bank angle changes of up to 60°. The wing then settles into an equilibrium state at a finite bank angle. Wing rock is a self-excited roll-sideslip coupled oscillation caused by a loss of damping at high angles of attack. This often leads to a loss of control. Two particularly important cases are the highly swept wing and the slender-body wing rock

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(forebody-delta wing). The phenomenon of wing rock and the associated problems have been extensively reviewed in Orlik-Rükemann [26], Ericsson [27] and Katz [18]. When pitching to high angles of attack, the vertical tails become surrounded by turbulent, motionless air which tends to limit their directional control capability. Therefore, relatively small side forces on the nose, even at zero sideslip, can dominate directional stability, creating large yawing moments. These small side forces are a result of asymmetrical shedding of the forebody vortices. Small surface imperfections such as radome gaps, dents and sharp paint depth mismatches can affect the strength and path of one of the vortices. The resultant net side force can then increase and the flow becomes unstable. This condition of aircraft experiencing severe yawing moments at a high-angle-of-attack flight is called nose-slice departure. It is an unsteady phenomenon and can be catastrophic.

7.8.8  Fully Coupled Motions: The Falling Leaf When the air is motionless relative to the aircraft in a particular mode of motion, then that mode is relatively similar to an equivalent mode of the same aircraft in free fall. Although James Clerk Maxwell first considered the motion of an object falling freely through the air over 150 years ago, the equations remain unsolved even to this day for the most general case. As yet, there’s no satisfactory explanation for the well-observed phenomenon of flutter, and tumbling associated with such bodies cannot be characterised by a stable equilibrium and can at best be considered to be chaotic. Modern parachutes are designed specifically to avoid this problem. Freely falling bodies tend to serve as models for predicting the complex modal behaviour of aircraft. Three examples are a falling sphere, the falling circular lamina and the freely falling rolling dumbbell. The falling sphere being symmetric allows one to demonstrate the nature of the drag on translational motion, while the falling circular lamina demonstrates the coupling between the aerodynamic vortex generation and the influence of the resulting moments on the motion of the lamina. In this case, there are significant vortex-induced tumbling oscillations coupled with sideslip illustrating incipient adverse yaw-type coupling. The complex features of these motions can be observed by anyone by simply dropping a light flat circular coin in a long glass jar filled with water. The freely falling rolling dumbbell is a typical model of fighter aircraft and demonstrates the traditional roll–pitch inertial coupling; the dumbbell oscillates in a pitch as a consequence of the inertial coupling with the rolling motion due to the moments of the centrifugal forces about the pitch axis. The realm of aircraft flight dynamics stretches from local static stability of equilibrium flight to global dynamic behaviour. Coordinate systems then play a key role in the setting up of the governing dynamic equations. Traditionally, these have been defined in the context of small perturbations about uniform equilibrium flight with a constant velocity at a fixed attitude. They employ the well-known Euler angle sequences, which have a number of pitfalls associated with them. An alternate formulation may be based on quaternions. While these are elegant, they are not quite suitable for developing complementary aerodynamic models in the flight regimes of interest.

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Thus, it can be seen that by employing a variety of interrelated coordinate systems, a range of non-linear dynamic models could be established. The main contribution of these coordinate systems is that one can easily visualise how the instantaneous velocity vector relates to the instantaneous rotation vector, the angular rate vector of the aircraft, given that the aircraft is in a near-equilibrium flight, involving either a translational or rotational equilibrium state. Although there are a number of characteristic aircraft motions, the falling leaf and aircraft spin, including free-fall spin, flat spin and deep spin, are of primary importance. If one is able to simulate these with high fidelity, the simulation of all other modes of motion could also be effected, in principle. The falling leaf, in particular, is a motion that involves large variations in the angle of attack and sideslip coupled with significant rotation. This motion is produced by the interaction of dynamic and aerodynamic moments and involves strong coupling about all axes. Analysis of this motion can only be undertaken in terms of the custom coordinate systems. The analysis based on the custom coordinate systems can predict both the amplitude and the frequency of this mode. The falling leaf has also been described as an unstable post-stall gyration from which the aircraft cannot recover. The motion may be characterised by in-phase, periodic roll and yaw rates fluctuating about small or zero means. The rates of change in the aircraft’s attitudinal angles sometimes exceed ±90 deg/s. Unlike spin, the motion does not involve a continuous unidirectional heading angle change but oscillates in predominantly roll and yaw about zero or small means. In such a mode, the angle of attack may dip below the stall angle at some time during the motion. The fluctuations in α are large as are the fluctuations in β. The rates of change in α and β reveal large peaks during the motion. Simulations of the falling leaf reveal the main characteristics of the associated motions in terms of some key variables. In general, the motions can be divided into two general categories: slow- and fast-falling leaf motions. The primary differences between the slow and fast motions are the higher mean value of the angle of attack, the appearance of a yaw rate bias and a decrease in the period of the motion, that is, increase in the frequency of oscillation. Simulations also indicate that there is no distinct boundary delineating the fast- and slow-falling leaf. The differences are primarily due to the increases in the minimum angle of attack of the motion while maintaining the same angle-of-attack range, that is, approximately 60° between the minimum and maximum angles.

7.8.9  Regenerative Phenomenon Under the influence of small perturbations, an aircraft in a level flight is influenced by a certain regenerative phenomenon which results in the redistribution of energy and can therefore result in an unstable situation. The term regenerative is borrowed from the feedback amplifier theory and refers to feedback. Positive feedback can be destabilising, and there exist two distinct regenerative mechanisms associated with non-linear aerodynamics which have an adverse effect on the modes of motion: 1. Post-stall regenerative coupling: These are primarily due to steady or unsteady separation effects which can independently take place over each of the two wing/tail configurations, thus causing a steady or unsteady lateral

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imbalance. The lateral moments in turn can result in a loss of lift and a consequent rise or fall in the angle of attack. 2. Adverse roll–yaw coupling: This is the classic coupling that exists even in the linear case and results in the instability of the spiral mode. High-angleof-attack effects could substantially alter the nature of the coupling and are primarily the effects governing fully coupled motions. Both these mechanisms ultimately result in fully coupled motions that are difficult to distinguish. However, what set them apart are the mechanisms that are the primary cause of these effects. Amongst the control-coupled modes, the pilot-coupled modes seem to offer the maximum potential for regenerative-type instabilities, primarily due to the unpredictability of human behaviour. Pilot-induced oscillations are dealt with in a later chapter. Generally, it can be assumed that the aircraft’s automatic flight control systems are not only well defined but also optimally designed satisfying the most appropriate performance criteria. While it is quite difficult to identify a generic set of control-coupled modes associated with an automatically controlled flight, it can be safely assumed that there are no instabilities associated with them.

CHAPTER HIGHLIGHTS • Aircraft response Free response Response to transient disturbances Stick-free natural motions Fly-by-computer natural motions Forced response Response to control inputs Response to sustained disturbances: gusts, wind shear, microbursts and turbulence • Control response (step input) Pure rolling motion: roll subsidence mode Dutch roll motion Short period/phugoid motions • Forced response to disturbances Typical disturbances: gusts, wind shear, microbursts and turbulence

EXERCISES 7.1 Consider the longitudinal dynamics of the F15 fighter given in Exercise 6.1. If the pitch rate is measured by a rate gyro, the pitch rate measurement, qm, may be expressed by the state-space equations,

x = Ax + Bη,   qm = Cx + Dη.

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Denoting (sI – A) –1 as Φ, show that elevator to pitch rate TF may be expressed as,

Qm ( s ) = [ D + CΦB ] H (s)

where Qm (s) is the Laplace transform of qm(t) H (s) is the Laplace transform of η(t) Hence, or otherwise, determine, by elimination, the elevator to pitch rate measurement TF. 7.2 Consider the longitudinal dynamics of the F15 fighter given in Exercise 6.1. Express the characteristic equation as



∆ ( s ) =  s 2 + 2ζ pω ps + ω 2p   s 2 + 2ζsω s s + ω 2s  = 0. i. Obtain the damping ratio in the phugoid mode ζ p, the natural frequency in the phugoid mode ωp, the damping ratio in the short period ζs and the natural frequency in the short-period mode ωs. If the damping ratio in the phugoid mode ζ p is desired to be 0.7 and the damping ratio in the short period ζs is desired to be 0.75 while the corresponding natural frequencies remain the same, calculate the coefficients of the desired characteristic polynomial. ii. Express the, a. Elevator to forward velocity perturbation TF as 2 2 u ( s ) K u [ s + 1 / Tu ]  s + 2ζuω u s + ω u  = ; η( s ) ∆ (s)

b. Elevator to normal velocity perturbation TF as 2 2 w ( s ) K w [ s + 1 / Tw ]  s + 2ζ wω w s + ω w  = ; η( s ) ∆ (s)

c. Elevator to pitch rate TF as q ( s ) K q s [ s + 1 / Tθ ] ( s + 1) / Tq  = . η( s ) ∆ (s)

Obtain the poles and zeros of the TF. Hence, or otherwise, obtain the response of the relevant output for a step input to the elevator. 7.3 Consider the lateral dynamics of the F15 fighter given in Exercise 6.3. Express the characteristic equation as

∆ ( s ) = [ s + 1 / Ts ]  s + 1 / Tp   s 2 + 2ζd ω d s + ω d2  = 0.

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i. Obtain the time constant of the roll subsidence mode (Tp), the time constant of the spiral mode (Ts), the damping ratio and the natural frequency of the Dutch roll oscillations (ζd and ωd respectively). ii. Express the, a. Rudder to yaw rate TF as K r [ s + 1 / Tr ]  s 2 + 2ζr ω r s + ω r2  r (s) = ζ ( s ) [ s + 1 / Ts ]  s + 1 / Tp   s 2 + 2ζd ω d s + ω d2  and obtain Kr , Tr , ζr and ωr b. Aileron to yaw rate TF as 2  K ra [ s + 1 / Tra ]  s 2 + 2ζraω ra s + ω ra r (s) = 2 ξ ( s ) [ s + 1 / Ts ]  s + 1 / Tp   s + 2ζd ω d s + ω d2 

and obtain Kra , Tm, ζra and ωra c. Rudder to roll rate TF as K p  s + 1 / Tp1   s + 1 / Tp 2   s + 1 / Tp3  p(s) = . ζ ( s ) [ s + 1 / Ts ]  s + 1 / Tp   s 2 + 2ζd ω d s + ω d2  Obtain the corresponding zeros of the TF d. Aileron to roll angle TF as K ϕa Tϕa s + 1  s 2 + 2ζϕaω ϕa s + ω ϕ2 a  φ(s) = ξ ( s ) [ s + 1 / Ts ]  s + 1 / Tp   s 2 + 2ζd ω d s + ω 2d 

Obtain the poles and zeros of the TF. Hence, or otherwise, obtain the response of the relevant output for a step input to the rudder or aileron. 7.4 Consider the simplified longitudinal dynamics of an aircraft in a steadylevel flight. The equations are diagrammatically expressed as in Figure 7.11, in terms of integrators, gains, summing amplifiers and take-off points. In Figure 7.11, the quantity U is equilibrium flight velocity in stability axes, U se. Obtain the state equations corresponding to the diagram and verify that these correspond to the simplified aircraft longitudinal dynamics. 7.5 Consider the simplified lateral dynamics of an aircraft in a steady-level flight. The equations are diagrammatically expressed as in Figure 7.12, in terms of integrators, gains, summing amplifiers and take-off points. In the figure, the quantity U is U se. Obtain the state equations corresponding to the diagram and verify that these correspond to the simplified aircraft lateral dynamics.

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FIGURE 7.11  Aircraft longitudinal dynamics.

7.6 The simplified linearised longitudinal dynamics of the AFTI/F16 for small perturbations about a steady-level flight with the throttle in a trimmed setting may be expressed as



i. Obtain a reduced set of approximate equations governing the dynamics of the short-period modes of the aircraft.

       

∆us ∆α s ∆qs ∆θ s ∆h

 xu   z   ue   Us = m u     0     0  

U se x w

0

−g

zw

1

0

U se mw

mq

0

0 −U se

1 0

0 U se

0    0    0   0  0    

∆us ∆α s ∆qs ∆θs ∆h

 x ηζ    zη   e   Us  +  mη     0   0   

      ∆η     

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FIGURE 7.12  Aircraft lateral dynamics.



ii. Employing only simple blocks, such as a summer, integrator and gain or an attenuator and the appropriate sources and sinks, draw a neat Simulink block diagram representation of the approximate short-period dynamics of the fighter. Label your diagram so as to clearly indicate the relevant inputs and outputs to each block, signal sign changes and block TFs and simulate the short-period dynamics, assuming a unit step input to the elevator and monitor all the states on scopes. Comment on the responses you observe on the scope and justify your results. iii. Hence, or otherwise, employing the same generic blocks as in part (ii) as well as any blocks created in the earlier parts, draw a neat Simulink block diagram representation of the simplified, linearised, full, longitudinal dynamics of the fighter. Label your diagram so as to clearly indicate the relevant inputs and outputs to each block, signal sign changes and block TFs and simulate the longitudinal dynamics, assuming a unit step input to the elevator and monitor all the states on scopes. Comment on the responses you observe on the scope and justify your results. 7.7 The simplified linearised lateral dynamics of the AFTI/F16 for small perturbations about a steady-level flight may be expressed as

306

   ∆β s  ∆p s   ∆rs  ∆φs  s  ∆Ψ 



Flight Dynamics, Simulation, and Control

 yv     s   U e lv   s  =  U e nv   0    0    

yp Ues

yr U es

g Ues

lp

lr

0

np

nr

0

1 0

0 1

0 0

 0    ∆β s  0   ∆ps  0   ∆rs  0   ∆φs 0   ∆Ψ s  

 yξ   U es    lξ   nξ   0  0   

yζ U es lζ nζ 0 0

      ∆ξ      ∆ζ     

i. Employing only simple blocks, such as a summer, integrator and gain or an attenuator and the appropriate sources and sinks, draw a neat Simulink block diagram representation of the approximate dynamics of the roll subsidence mode of the fighter. Label your diagram so as to clearly indicate the relevant inputs and outputs to each block, signal sign changes and block TFs and simulate the roll subsidence dynamics, assuming a unit step input to the aileron, and monitor all the states on scopes. Comment on the responses you observe on the scope and justify your results. ii. In a typical fighter, the dynamics of the spiral mode is relatively slow and can be ignored. Hence, or otherwise, employing the same generic blocks as in part (i), as well as any blocks created in the earlier parts, draw a neat Simulink block diagram representation of the simplified, linearised coupled dynamics of the remaining faster lateral modes of the fighter. Label your diagram so as to clearly indicate the relevant inputs and outputs to each block, signal sign changes and block TFs and simulate the fast lateral dynamics, assuming a unit step input to the aileron, and monitor all the states on scopes. Comment on the responses you observe on the scope and justify your results. How do the responses differ if the step input is to the rudder? iii. Hence, or otherwise, employing the same generic blocks as in part (i) and (ii), as well as any blocks created in the earlier parts, draw a neat Simulink block diagram representation of the simplified, linearised, full, lateral dynamics of the fighter. Label your diagram so as to clearly indicate the relevant inputs and outputs to each block, signal sign changes and block TFs and simulate the full lateral dynamics, assuming a unit step input to the elevator, and monitor all the states on scopes. Comment on the responses you observe on the scope and justify your results. How do the responses differ if the step input is to the rudder? 7.8 Employing only simple blocks, such as a summation point, integrator of vector or scalar signals, scalar or vector product, scalar or matrix gain or attenuator, the appropriate bus multiplexers, bus demultiplexers, sources and sinks, a block diagram representation of the EOMs of a rigid body relating

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307

the body angular velocity vector to the vector of applied external torques is shown in the following. It is assumed that the reference frame is fixed to the body at its centre of mass (CM). In the diagram, I is the inertia matrix, Iin is the inverse of the inertia matrix, LMN is the vector of external torques, and pqr is the vector of angular velocities (Figure 7.13).



7.9

i. Draw a neat block diagram representation of the force equations of a rigid body in body-fixed reference frame fixed at the body’s CM and relate the applied forces to the translational velocities of the body. Include the external forces due to gravity and other external forces separately. ii. Draw a neat block diagram representation of the relationship between the Euler angle rates (attitude rates) and components of the angular velocity vector. iii. Modify your diagrams if the reference is fixed to an arbitrary point on the body. Include the forces and moments due to gravity as well as other external forces and moments separately. i. Starting with the relationship (Thomson [28]),

H ( iω ) = H ( iω ) exp ( iφ ( ω )) ,

show that,

H * ( iω ) = H ( iω ) exp ( −i 2φ ( ω )) . ii. Consider the convolution integral ∞



x (t ) =

∫ f (t − τ) h ( τ) dτ 0

FIGURE 7.13  Block diagram representation of Euler’s EOMs of a rigid body.

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and employing Fourier transforms, show that ∞



∫

X ( iω ) = F ( iω ) H ( iω )    and that,      x 2 = Φ F ( ω ) H ( iω ) dω 2

0

where 1 F ( iω ) F * ( iω ) . T →∞ 2πT

Φ F ( ω ) = lim



iii. The mean square response of the linear system to spectral noise excitation may be expressed as ∞

∫

y = Φ x ( f+ ) H ( f ) df . 2



2

0

For a single DOF vibrating system, the TF H(f) may be expressed as H(f)=



1 1 f × ,     r = . k fn 1 − r 2 + 2ζ  r −1

(

)

Hence, ∞



∞

∫

∫

y 2 = Φ x ( f+ ) H ( f ) df Φ x ( f+ ) 2

0

0

1 k2

  1   df , 2  1 − r 2 + 4ζ2 r 2 

(

)

and it follows that

y 2 = Φ x ( fn )

fn k2

  1 f π   dr = Φ x ( fn ) n2 2 k 4ζ  1 − r 2 + 4ζ2 r 2  0

∞

∫(

)

A simplified model of an automobile travelling over a rough road is made up of a body with a mass, m, supported by a spring with a spring constant, k, and a dash-pot with a damping constant, c. It is assumed that the tyres are rigid and in continuous contact with the surface and that the vehicle moves with constant speed V. The ground motion (input) to the vehicle due to the roughness of the road is assumed to be a single-point input. Assuming that the input PSD due to the road surface is S0, obtain the PSD of the body response in the vertical direction as well as the mean square value of the body’s vertical displacement. 7.10 A jet engine with a mass of 270 kg is tested on a stand which results in a natural frequency of 25 Hz. The spectral density of the jet force is bandlimited unit-intensity Gaussian white noise over a bandwidth of 1–100 Hz. The stand may be assumed to be lightly damped with a damping ratio of 0.1.

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Determine the probability of the vibration amplitude of the jet thrust exceeding 0.012 m. (Hint: Compute the mean square response and the variance and use the fact that the probability distribution is Gaussian). 7.11





i. Determine the TF for the shaping filter for deriving the vertical and horizontal Dryden spectra from unit-intensity white noise. Hence, or otherwise, determine the state equations for the noiseshaping filters in each of these cases. ii. The MIL-HDBK-1797 published in 1997 gives a better approximation for the vertical spectrum. Determine the TF for the shaping filter for deriving the newer vertical spectrum from unit-intensity white noise. Hence, or otherwise, determine the state equations for the noiseshaping filter in this case. iii. Compare the von Karman with each of the Dryden (MIL-F-8785C and MIL-HDBK-1797) spectra over a range of frequencies both in the vertical and horizontal cases. For the purposes of comparison, the intensity is chosen for a specified probability of occurrence at a specified altitude. For a 1% probability of occurrence at an altitude of 3,000 m, σ 2w is 8 m2/s2. When σw is 1 m/s, Ls is 800 m and U se is 200 m/s, KG = σ w



Ls = π ( Use )

4 π

which is a reasonable choice for the purposes of comparison. At high altitudes, it may be assumed that the turbulence intensities are the same in the vertical and horizontal directions. 7.12 The motion of a lifting vehicle about a steady flight path due to atmospheric turbulence may be represented as single-DOF vibration and represented by the governing equation ¨ F (t ) h ( t ) + 2ξω n h ( t ) + ω 2n h ( t ) = m



where h is the vertical displacement of the vehicle m is the mass of the vehicle F(t) is the net vertical force acting on it ωn is the natural frequency of vertical oscillations ζ is the damping ratio The spectral density of the aerodynamic forcing function, F(t) is given by,

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Sww (ω )  πω c   1 + U c  s

SFF (ω ) =



where c is the mean aerodynamic wing chord U se is the steady flight velocity Sww(ω) is the spectral density of the vertical component of the wind velocity, which is approximated by,







L Sww ( ω ) = σ se Us 2 w

1 + ( Lsω )2    ,   ω = ω . 2 2 U se 1 + ( Lsω )   

i. Find the PSD of the vehicle’s response in the vertical direction as well as the mean square value of the vehicle’s vertical displacement. ii. Assume that the PSD of the forcing function, F(t) is a constant, S 0, and that the corresponding mean square displacement of the mass is δ and show that the following relations hold: m=

c=

(

πS0

2δω ω − ω 2 n

(

2 n

2πS0 ω 2n − ω 2d δω

2 n

)

2 1/2 d

)

1/2

,   k = ω n2 m    and

(

= 16 m ω 2n − ω 2d

)

2

where ωd is the damped natural frequency of vertical oscillations c is the damping coefficient k is the stiffness constant 7.13



i. Consider the two DOF vibrating system [29] illustrated in Figure 7.14 and derive the EOMs. Assume the body with mass m1 is subjected to ideal white noise excitation representing an ergodic random process and derive general expressions for the cross-correlation function between the displacements of the two bodies by employing appropriate transformations to principal coordinates. Also, obtain the mean square response of the displacements of the two bodies. Assume that m1 = m, m2 = 2m, k1 = k2 = k and c1 = c2 = c = 0.02 km . ii. Rewrite the EOMs as first-order equations in state-space form and derive a generic equation for the cross-correlation between the states in the state vector. Again employ principal coordinates as far as possible and obtain the steady-state solution for the autocorrelation matrix.

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FIGURE 7.14  Two-DOF vibration model.

7.14 Consider the model of the longitudinal dynamics of the F-18 High Alpha (angle-of-attack) Research Vehicle (HARV) flying in straight-level equilibrium flight at sea level with a forward speed of U se = 250 ft/s and angle of attack of 30°. The controls are the stabilator angle, η, and the thrust vector angle, τ. Three disturbance inputs included are the horizontal gust ug, the vertical gust wg and the acceleration component of the vertical gust nwg. The perturbation velocities are in ft/s while the pitch rate is in rad/s and the pitch angle in radians. The body-axis dynamic equations take the form, x = Ax + Bu  with x =  ∆ub  and   u =  η τ ug 



wg

∆wb nwg  

∆qb

T

∆θb       

T

where,



 0.00950000  −0.13100000 A=  1.9900 E − 03  0 

0.02570 −0.23000 −0.003096697 0

−110.000 190.52559 −0.31000 1

−27.860037 −16.085000 0 0

  ,   

and  −1.23000  −13.1000 B =   −1.23000  0 

−0.0078000 −16.10000 −2.530000 0

−0.0095 0.13100 −0.0019 0

−0.0257 0.2300 0.003096697 0

0 0 0.001627078 0

  .   

In addition to the states, the following outputs are measurable: (1) the stability axis forward speed, (2) the angle of attack, (3) the flight path angle, (4) the altitude rate, (5) the body axis forward acceleration at the sensor

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location and (6) the body axis vertical acceleration at the sensor location. The C and D measurement matrices in the measurement equation y = Cx + Du

are given by



 0.8660254  −2.272727 E − 03  2.272727 E − 03 C=  0.500000  9.50000 E − 03  −0.1437300 

0.500000 3.936479 R − 03 −3.936479 R − 03 −0.8660254 0.0257000 −0.2092521

0 0 0 0 0 2.0770000

0 0 1 220 0 0

   ,    

and  0  0  0  D=  0  − 1.23000   −4.85900

0 0 0 0 −0.009500 0.851000

0 0 0 0 −0.0078000 −0.143730

0 0 0 0 −0.02570 0.2092521

0 0 0 0 0 −0.0109014

   .    

i. Assume that the vertical component of the gust is given by the Dryden model, employ the MATLAB matrix analysis toolbox and determine the response of the aircraft in each of the six outputs. ii. Assume in addition a constant (unit step) input for the horizontal gust equal to one-quarter of the maximum magnitude of the vertical component and repeat the calculation of the outputs. iii. Now include the effect of the acceleration component of the vertical gust and determine the maximum critical magnitude of this component that the aircraft can withstand. State the criteria for your assessment. iv. Assume that the aircraft experiences turbulence in the longitudinal and vertical directions and determine the output spectral density of the normal acceleration and pitch rate states, in each of the two cases. Employ a white noise-shaped input, with the shaping filter defined by a suitable approximation of the Dryden spectrum. (An approximation to the turbulence spectrum is given by the Dryden turbulence model, in US standard units. For the reference altitude in this study, it may be assumed that, σu = 10.8 ft/s, σw = 6.88 ft/s, Lu = 65,574 ft and Lw = 26,229 ft.) 7.15 Simulation case study: In this exercise, it is desired to simulate the complete dynamics of an airship. The diagrams generated in the preceding exercise must be extended to diagrammatically simulate the complete dynamics of

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an airship. The gravitational and buoyancy forces and moments, the aerodynamic forces and moments as well as the control forces and moments must be explicitly defined. The location of the centre of buoyancy is assumed to be defined by the vector rbu = [xbu 0 zbu]T and the location of the CG by the vector rcg = [xG 0 zG]T, both relative to the centre of volume. For static stability zbu > zG. A unit vector in the direction of the gravity vector by [k x k y kz]T  = TBI × [0 0 1]T, the vector of gravitational and buoyancy forces is, FBG = W – B, where W = mg[k x k y kz]T and B = B[k x k y kz]T, while the vector of gravitational and buoyancy moments at the CM is



 LG   MG  NG 

   = rcg × W − rbu × B =  

 k y ( Bzbu − mg  zG )   k x ( mg  zG − Bzbu ) + k z ( Bx bu − mg  xG )  k y ( mg  xG − Bx bu )  

  .   

The propulsive control forces due to the three thrusts, TL , TR, TS, acting at clockwise angles, θLε, θRε, θSε respectively about the body y-axis, are given by

FBT



 XT  =  YT  ZT 

  TL cos θ Le + TR cos θ Rε + TS cos θ Sε   0 =   −TL sin θ Le − TR sin θ Rε − TS sin θ Sε  

    

while the propulsive control moments in the body frame at the origin are given by,  LT   MT  NT 

  =  

 yL ( TL sinθ Lε − TRsinθ Rε )   TL ( z L cos θ Lε + x L sinθ Lε ) + TR ( z L cos θ Rε + x L sinθ Rε ) + TS ( z S cos θ Sε + x S sinθ Sε )  yL ( TL cosθ Lε − TR cosθ Rε )  where yL is the distance of the port thruster, from the body x-axis. The starboard thruster is assumed to be symmetrically placed with respect to the body xz plane. The body-axis coordinates, xS, yS, of the rear thruster, are assumed to be in the body axis xz plane. A typical set of non-zero parameters for a prototype airship are given in Table 7.4. It is common practice to describe the aerodynamic forces and moments of a flight vehicle in the flight path axes, that is, using the true speed U, angle of attack α and the sideslip angle β instead of the linear velocity components

    

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Flight Dynamics, Simulation, and Control

TABLE 7.4 Airship Parameters Dimensions Length, L (m) aλ a2/a1 xcv (m) Max. thickness, 2b (m) Hull volume (m3) Hull ref. area (m2) Span width (m) Fin area (m2) Inertia properties Mass (at altitude 200 m) (kg) Ixx (kg m2) Iyy (kg m2) Izz (kg m2) Ixx (kg m2)

CM Location (at Altitude 200 m) 15 5 2 6.875 3.7 107.42 22.5973 4.3 2.88 136.8 213

xc (m) 0 yc (m) 0 zc (m) 0.45 Port thruster location xL (m) 8.13 yL (m) 0.6 zL (m) 0.2 Stern thruster location xs (m) 12.5 0 ys (m) zs (m) 0.25 Max. thruster of each of the three identical thrusters: 120 N

3,310 3,211 88

Source: Modified from Kornienko [30].

u, v and w. These angles are determined using a block diagram of the type as shown in Figure 7.15. For the steady aerodynamic forces and moments as well as the virtual aerodynamic inertias, we adopt the non-dimensional expressions in Mueller

FIGURE 7.15  Simulink® diagram for computing the angles of attack and sideslip.

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Numerical Simulation and Non-Linear Phenomenon

FIGURE 7.16  Double ellipsoid geometry of airship hull. (From Mueller et al. [31].)

et al. [31]. For completeness, we present a summary of their results. Some of their notation has been altered to be consistent. For the purposes of computing the aerodynamic loads by the application of the slender body theory, the airship is modelled as an axisymmetric double ellipsoid as illustrated in Figure 7.16. The volume of the hull and the reference area in terms of the hull volume are respectively given by, VH =



2π ( a1 + a2 ) b 2 ,   Sref = VH2/3 . 3

For the double ellipsoid, where the origin of the body frame is located, the centre of volume is located along the x-axis at the point, xcv =



5a1 + 3a2 8

from the nose and directed to the tail of the hull. If we let p, q and r denote as angular velocities about the body x- y- and z- axes, respectively, ζT and ζB denote deflections of the top and bottom trailing edge flaps of the rudder and ξL and ξR denote deflections of the left and right trailing edge flaps of the elevator, the equations for the aerodynamic forces and moments are

FBA



 XA  =  YA  ZA 

  LA    ,   M BA  M A   NA  

  ,  

where,  XA   YA  ZA 

C X 1cos2 α cos2 β + CY 1 sin ( 2α ) sin ( α / 2 )                                                   1 2   sin 2 cos /2 sin 2 sin V C C C C β β + β + β + ζ + ζ = ρ ( ) ( ) ( ) ( ) T Y Y Y Y T B 1 2 3 4  2       CY 1 sin ( 2α ) cos ( α /2 ) + CY 2 sin ( 2α ) + C Z 3sin α  sin α + CY 4 ( ξ L + ξ R ) 

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Flight Dynamics, Simulation, and Control

and  LA   MA  NA 

  1 2  = 2 ρVT   C L1 ( ξ L − ξ R + ζ B − ζT ) + C L 2 sin β sin β                                                      C sin ( 2α ) cos ( α /2 ) + C sin ( 2α ) + C sin α  sin α + C ( ξ + ξ )  M2 M3 M4 L R  M1     −C M 1 sin ( 2β ) cos (β /2 ) − C M 2 sin ( 2β ) − C M 3 sin β  sin β − C M 4 (ζT + ζ B ) 

The 12 aerodynamic coefficients, CX1, CY1, CY2, …, CY4, CZ3, CL1, CL2, CM1, …, CM4 are defined in Tables 7.5 and 7.6 (see also Table 7.7). TABLE 7.5 Aerodynamic Coefficients and Related Parameters Coefficient CX1 CY1

Formula –[CDh0Sh + CDf0Sf + CDg0Sg] k21ηk Sh I1 ,  k21 =

βk αk − βk − 2 α k − 2

CY2

1  ∂C  −  L  S f ηf 2  ∂α  f

CY3

–[CDchJ1Sh + CDcfSf + CDcgSg]

CY4

1  ∂C  −  L  S f ηf 2  ∂ξ  f

CZ3

–[CDchJ1Sh + CDcfSf]

CL1

1  ∂C L  S f ηf lf 3 2  ∂ζ  f

CL2

–CDcgSglgz

CM1

k21ηk Sh I 2 L ,  k21 =

βk αk − , L = ( a1 + a2 ) βk − 2 α k − 2

CM2

1  ∂C  −  L  S f ηf lf1 2  ∂α  f

CM3

–[CDchJ2ShL + CDcfSflf2]

CM4

1  ∂C  −  L  S f ηf lf1 2  ∂ξ  f

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Numerical Simulation and Non-Linear Phenomenon

TABLE 7.6 Definition of Parameters in the Aerodynamic Coefficients Parameter

Definition or Formula

Value

CDh0 CDf0 CDg0 CDch CDcf CDcg

Hull zero-incidence drag coefficient Fin zero-incidence drag coefficient Gondola zero-incidence drag coefficient Hull cross-flow drag coefficient Fin cross-flow drag coefficient Gondola cross-flow drag coefficient

0.025 0.006 0.01 0.5 1.0 1.0

 ∂C L   ∂α  f

Derivative of fin lift coefficient with respect to the angle of attack at zero incidence

5.73

 ∂C L   ∂ξ 

Derivative of fin lift coefficient with respect to the fin-flap deflection angle

1.24

Sref, Sh Sf Sg lhf lh lf 1 lf2 lf 3 lgx lgz ηf ηk

Units

f

Hull reference area, VH2/3 Fin reference area Gondola reference area x distance from nose to fin leading edge x distance from origin to fin leading edge x distance from origin to aerodynamic centre of fins x distance from origin to geometric centre of fins y, z distances from origin to aerodynamic centre of fins x distance from origin to aerodynamic centre of gondola z distance from origin to aerodynamic centre of gondola Fin efficiency factor accounting for the effect of the hull on the fins

22.5973

m2

10.132 0.56

m2 m2

7.05 7.782 1.098 1.752 2.4 0.29

m m m m m

Hull efficiency factor accounting for the effect of the fins on the hull

1.19

The only unsteady effects included are the virtual mass and inertia effects. All other velocity-dependent unsteady aerodynamic loads are assumed to be negligible. The virtual mass and inertia matrices are based on that for a single axisymmetric ellipsoid with a semi-major and semiminor axis equal to a and b, respectively. The virtual mass and inertia matrices are



MVM

 αk  α  k −2  = −ρair VH  0    0  

0 βk βk − 2 0

    , 0   βk  βk − 2   0

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Flight Dynamics, Simulation, and Control

TABLE 7.7 Table of the Relevant Non-Dimensional Integrals in Tables 7.5 and 7.6 Parameter fh Sref

Definition or Formula

I1



0.5536



Sref = VH2/3 /πb 2 πb 2

2.1017

1 Sref



I2

1 Sref



J2

1 lh Sref

IVI =

a2

1 lh Sref



J1

Value

lhf − a1 a2

∫

− a1

(

∫

− a1

0.33

)

(

–0.69

− a1

2r ( x ) xdx = J1

)

1  a1 a2 fh  + π 2bSref  2

2r ( x ) dx =

a2

∫

dS ( x ) 1 1 − fh2 dx = dx Sref

dS ( x ) 1 x xdx = a1 − 3a1 fh2 − 2a2 fh3 − cv I1 dx 3 LSref L

a2

∫

a2

( a1 − xcv ) L

− a1

+

1 − fh2 +

(

1.31

2a2  sin 2 ( fh )  π

(

2 a22 − a12 − a22 1 − fh2 3πbLSref

)

3/ 2

)

0.53

ρair VH a 2 − b 2 × 5 a2 + b2

 0    0    0  

0

(b

2

2 b −a

2

(

2

−a

2

) (β

) + (b

2

0

0 k

− αk )

0

)

+ a 2 (β k − α k )

(b

2

2 b −a

2

(

2

)

− a 2 (β k − α k )

) + (b

2

)

+ a 2 (β k − α k )

where, a = 0.5(a1 + a2) and the constants αk and βk are defined in Table 7.8.

        

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Numerical Simulation and Non-Linear Phenomenon

TABLE 7.8 Parameters for Computing the Virtual Inertia and Mass Matrices Parameter b/a

Formula b/a

Value 0.2467 0.9691

e

1 − ( b /a )

f

 1+ e log   1 − e 

4.1543

g

(1 − e2)/e3

0.0669

αk

2g(0.5f − e)

0.1462

βk

1 fg − e2 2

0.9259

2

REFERENCES



1. Brian, L. S. and Frank, L. L., Aircraft Control and Simulation, 1st ed., Wiley Interscience, New York, 1992. 2. Peter, H. Z., Modeling and Simulation of Aerospace Vehicle Dynamics, 2nd revised ed., AIAA Education Series, American Institute of Aeronautics and Astronautics Inc., Reston, VA, 2007. 3. Zhao, Y. and Bryson, A. E. Jr., Optimal paths through downbursts, Journal of Guidance, Control and Dynamics, 13(5), 813–818, 1990. 4. Zhao, Y. and Bryson, A. E. Jr., Control of aircraft’s in downbursts, Journal of Guidance, Control and Dynamics, 13(5), 819–823, 1990. 5. Gage, S., Creating a unified graphical wind turbulence model from multiple specifications, AIAA 2003-5529, AIAA Modeling and Simulation Technologies Conference and Exhibit, Austin, TX, August 11–14, 2003. 6. Rom, J., High Angle-of-Attack Aerodynamics: Subsonic, Transonic, and Supersonic Flows, Springer Verlag, New York, 1992, pp. 166–170. 7. Tobak, M. and Chapman, G. T., Non-linear problems in flight dynamics involving aerodynamic bifurcations, AGARD Symposium on Unsteady Aerodynamics Fundamentals and Applications to Aircraft Dynamics, Göttingen, Germany, Paper No. 25, May 6–9, 1985. 8. Mehra, R. D., Kessel, W. C., and Carroll, J. V., Global stability and control analysis of aircraft at high angles-of-attack, Annual Technical Report 1, ONR-CR215-248-1, Scientific Systems Inc., Cambridge, MA, 1977. 9. Zagaynov, G. I. and Goman, M. G., Bifurcation analysis of critical aircraft flight regimes, In: Proceedings of the 17th Congress of International Council of the Aeronautical Sciences (ICAS), ICAS Paper 84-4.2.1, Joulouse, France, 1984. 10. Adams, W. M. Jr., Analytical prediction of airplane equilibrium spin characteristics. NASA Technical Note, NASA-TN-D-6926, Langley Research Centre, Hampton, VA, 1972. 11. Guicheteau, P., Bifurcation theory in flight dynamics: An application to a real combat aircraft. Proceedings of the 17th congress of International Council of the Aeronautical Sciences (ICAS), ICAS Paper 116 (90-5.10.4), Stockholm, Sweden, 1990. 12. Lowenberg, M., Stability and controllability evaluation of sustained flight manoeuvres. AIAA-96-3422, AIAA AFM Conference, San Diego, CA, 1996, pp. 490–499.

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13. Planeaux, J. B., Beck, J. A., and Baumann, D. D., Bifurcation analysis of a model fighter aircraft with control augmentation, American Institute of Aeronautics and Astronautics (AIAA) New York, AIAA Paper, 1990–2836, 1990. 14. Jahnke, C. C. and Culick, F. E. C., Application of bifurcation theory to high-angle-ofattack dynamics of the F-14, Journal of Aircraft, 31, 26–34, 1994. 15. Patel, Y. and Littleboy, D., Piloted simulation tools for aircraft departure analysis, Philosophical Transactions of the Royal Society of London A, 356(1745), 2203–2221, 1998. 16. Cummings, P. A., Continuation methods for qualitative analysis of aircraft dynamics, NASA/CR-2004–213035, NIA Report No. 2004-06, National Institute of Aerospace, Hampton, VI, July 2004. 17. Lee, B. H. K., Price, S. J., and Wong Y. S., Non-linear aeroelastic analysis of airfoils: Bifurcation and chaos, Progress in Aerospace Sciences, 35, 205–334, 1999. 18. Katz, J., Wing/vortex interactions and wing rock, Progress in Aerospace Sciences, 35, 727–750, 1999. 19. Ekaterinaris, J. A. and Platzer, M. F., Computational prediction of aerofoil dynamic stall, Progress in Aerospace Sciences, 33, 759–846, 1977. 20. Prasanta, B. B., Hounjet, M. H. L., and Zwaan, R. J., Thin layer Navier stokes solver and its application for aeroelastic analysis of an aerofoil in transonic flow, Proceedings of an International Forum on Aeroelasticity and Structural Dynamics, Paper No. 15, Royal Aeronautical Society, London, June 26–28, 1995. 21. Wegner, W., Prediction of unsteady aerodynamic forces for elastically oscillating wings using CFD methods, Proceedings of an International Forum on Aeroelasticity and Structural Dynamics, Paper No. 11, Royal Aeronautical Society, London, June 26–28, 1995. 22. Liebst, B. S., The dynamics, prediction and control of wing rock in high-performance aircraft, Philosophical Transactions of the Royal Society London A, 356, 2257–2276, 1998. 23. Macmillen, F. B. J. and Thompson, J. M. T., Bifurcation analysis in the flight dynamic process? A view from the aircraft industry, Philosophical Transactions of the Royal Society London A, 356, 2321–2333, 1998. 24. Gránásy, P., Thomasson, P. G., Sørensen, C. B., and Mosekilde, E., Nonlinear flight dynamics at high angles-of-attack, The Aeronautical Journal, 102(1016), 337–343, 1998. 25. Saad, A. A. and Liebst, B. S., Computational simulation of wing rock in three degrees of freedom for a generic fighter with chine-shaped forebody, The Aeronautical Journal, 107, 49–56, 2003. 26. Orlik-Rükemann, K. J., Aerodynamic aspects of aircraft dynamics at high angles of attack, Presented at AIAA AFM Conference, San Diego, CA, August 1982, AIAA-821363. Also in AIAA’s Journal of Aircraft, 20, September 1983. 27. Ericson, L. E., Wing rock analysis of slender delta wings, review and extension, Journal of Aircraft, 32(6), 1221–1226, 1995. 28. Thomson, W. T., Theory of Vibration with Applications, 4th ed., Chapman & Hall, Englewood Cliffs, NJ, 1993. 29. Rao, S. S., Mechanical Vibrations, 3rd ed., Addison-Wesley, Reading, MA, 1995. 31. Mueller, J. B., Paluszek, M. A., and Zhao, Y., Development of an aerodynamic model and control law design for a high altitude airship, AIAA Unmanned Unlimited Conference, No. AIAA-6479, AIAA, Chicago, IL, September, 2004. 30. Kornienko, A., System identification approach for determining flight dynamical characteristics of an airship from flight data, PhD dissertation, Institut für Flugmechanik und Flugregelung Universität, Stuttgart, Germany, 2006. http://elib.uni-stuttgart.de/opus/ volltexte/2006/2880/pdf/Dissertation_Kornienko.pdf. Accessed November 1, 2012.

8

Aircraft Flight Control

8.1  A  UTOMATIC FLIGHT CONTROL SYSTEMS: AN INTRODUCTION The design of an automatic flight control system including a suite of autopilots generally begins with selecting the appropriate desired functions of the autopilots. The next step involves choosing an appropriate control structure from a whole repertoire of such structures that are generally available. These may range from a simple proportional–integral–derivative (PID) controller to a much more complex observer/ estimator controller, adaptive or self-tuning controller, internal model controller or even a sophisticated non-linear control structure. The choice depends on the whole range of considerations driven by the functional requirements, safety, stability margins, robustness and handling qualities. The final stage involves choosing an appropriate architecture, given that the most modern controller for the large civil aircraft is implemented digitally. However, certain basic structures have emerged over the years and provide a baseline for the design of the automatic flight control systems. Early flight control systems were mechanically signalled (Figure 8.1), which are currently used as backup or standby systems for manual control in case of a failure of all the primary systems. Current methods of signalling and control are purely electrical in nature. The position of the control stick is measured by a typical position sensor or encoder, and the electrical output is a primary input to the flight control system (Figure 8.2). The output of the control system drives an integrated electro-hydrostatic actuator with a

FIGURE 8.1  Mechanically signalled control, used normally as backup.

DOI: 10.1201/9781003266310-8

321

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Flight Dynamics, Simulation, and Control

FIGURE 8.2  Electrically signalled (FBW) hydraulic control.

built-in reservoir for supplying the hydraulic fluid. Modern flight control systems are designed and built around electro-hydrostatic actuation systems of the types, which are schematically illustrated in Figures 8.3 and 8.4. A primary feature of this integrated system is that there are no servo valves or pipes conveying the fluid, and each actuator has its own built-in pump to drive it. In each electro-hydrostatic actuator, an electric motor drives a self-contained hydraulic system, comprising a pump and reservoir, which provides the motive force to power the control surface to the demanded position. The system is therefore potentially safer and more reliable as it is not plagued by problems associated with hydraulic leaks, etc. This is the basic principle behind fly-by-wire (FBW) actuation, and generally, when it is adopted, it is applied on an aircraft-wide basis. Thus, on most modern aircraft like the Boeing 787 Dreamliner and the Airbus A380, the ailerons, flaps or flaperons, in-board and out-board spoilers, horizontal stabiliser, elevator, rudder, wheel brakes, thrust reversers and a number of other electro-hydraulic and electro-hydrostatic servo-actuators use the FBW principle and are commanded electrically. An existing or off-the-shelf flight control computer is used to synthesise the commands and the throttle.

Aircraft Flight Control

323

FIGURE 8.3  Electro-hydrostatic system schematic incorporating a variable speed motor and a fixed displacement pump.

FIGURE 8.4  Electro-hydrostatic system schematic incorporating a fixed speed motor and a variable displacement pump.

8.2  FUNCTIONS OF A FLIGHT CONTROL SYSTEM The functions of a flight control system usually vary with the role and complexity of the aircraft. The foremost function of most flight control systems is to improve the stability of the aircraft while ensuring that other performance constraints are also met. The aircraft pitch motion is typically controlled by the elevator. The aircraft’s longitudinal motions, which include pitching motions, can be additionally controlled by the thrust or equivalently by the throttle. The aircraft’s coupled lateral motions in roll and yaw are controlled by the ailerons and rudder, respectively. The function of most control system designs is to regulate the small motions of the aircraft about a state of equilibrium characterised by steady-level flight. There are also a set of special autopilots that control the aircraft in an automatic landing mode or the automatic

324

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go-around mode. These small motions are the displacements and rotations about the three mutually perpendicular axes, with the origin fixed in the aircraft at its centre of gravity (CG) and coinciding with the aircraft’s geometric axes of symmetry. In Chapter 6, it was observed that the dynamics of any typical aircraft may be conveniently partitioned into two independent dynamical models, which are commonly referred to as longitudinal and lateral dynamics. The motion coordinates often used to describe the longitudinal dynamics are change in speed, angle of attack, pitch rate, pitch and altitude. The motion coordinates often used to describe the lateral dynamics are the cross-track displacement or sideslip, cross-track velocity, roll rate, yaw rate, roll angle and yaw angle. The aircraft’s motion response is best described in terms of its modes; the longitudinal modes are the phugoid and short-period modes; the lateral modes are the Dutch roll, spiral and roll subsidence. In a typical aircraft, considering the longitudinal modes, the forward speed normally changes very slowly relative to the pitching motions. Thus, the approximation of the forward speed being constant is often made, and this results in an approximate model for the short-period (fast) behaviour. On the other hand, because the pitching motions are relatively fast, the equations corresponding to these may be assumed to be instantaneously satisfied. This approximation results in a simplified model for the phugoid. Similar approximations may be derived in the lateral case on the basis that the roll subsidence takes place relatively faster than the other lateral motions and that roll rate integration takes place at a relatively slower rate than all other lateral motions. Corresponding to the longitudinal and lateral dynamics, aircraft flight control systems and autopilots may be broadly classified as longitudinal and lateral control systems and longitudinal and lateral autopilots, respectively. The functions of a flight control system may be broadly divided into two main areas: the inner loops, which are active throughout a particular flight regime, and the outer loops, which perform specific control functions such as holding the altitude of the aircraft, flying with a constant pitch attitude or bank angle and flying with constant velocity. They do this by replacing the pilot inputs with signals that are proportional to the appropriate demands that are injected into the inner loops. They may be operated only by switching on the appropriate autopilot mode. For purposes of designing autopilots, it is important to clearly understand the relationship between the dynamics of the aircraft and the kinematics of the flight path, as illustrated in Figure 8.5. The functions of the inner loops are to provide the pilot with an aircraft with good handling qualities over its operating flight envelope. The term handling qualities describes the response of the aircraft to pilot inputs in the presence of disturbing forces generated by atmospheric gusts and wind shear. The most commonly specified handling qualities are in terms of the damping and natural frequencies of the aircraft’s natural modes of motion. Others are usually specified in terms of the control sensitivity (stick force/‘g’) and the maximum sideslip or lateral acceleration allowable in turning flight. When longitudinal control is considered, it is noted that precision of pitch attitude control is compromised when the static stability margin is inadequate and when there are substantial trim changes due to thrust and flaps or by turbulence disturbances or by an easily excited phugoid mode. In the case of hands-free flight, the phugoid

Aircraft Flight Control

325

FIGURE 8.5  Flight path kinematics showing the inputs to the aircraft dynamics and the outputs, including yaw angle computation (a) longitudinal case and (b) lateral case.

can substantially alter the flight path and airspeed and degrade the tracking of the glide-slope command and may even require the pilot to intervene increasing the pilot’s workload. Even if precise attitude control is achieved, the aircraft’s response to a pitch attitude command is adversely affected at low speed and at speeds where

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induced drag exceeds profile drag when any increase in speed results in a decrease in the net drag. The sluggish initial flight path response to pitch attitude commands and the unsustainability of path corrections, in the long term, with attitude control, make path control with attitude inputs difficult to achieve. Although thrust control is a very powerful way of influencing the flight path, coupling of the flight path, airspeed and thrust response lags can make thrust control of flight path almost impossible. In the case of lateral/directional control, roll control capability can be substantially reduced by poor lateral control force characteristics, low control power and roll damping. Low directional stability, low Dutch roll damping and large unfavourable yawing moments from lateral controls reinforce each other in reducing the heading control capability. Stability augmentation systems where there is a direct connection between pilot input and control surface deflection with some additional stability augmentation in the form of inner loops are designed to improve the inherent stability characteristics of the aircraft. Command augmentation, on the other hand, implies that a flight computer is translating pilot inputs into a request and the flight computer then tries to satisfy that request, but there need not be a link between stick deflection and control surface deflection. Command augmentation provides the pilot with precise control of aircraft performance by measuring the aircraft response to a given command and by adding to or subtracting from the control surface deflection as required, to match the command to the ideal response. Thus, they are continuously compensating for the degradation of performance discussed earlier. What this means in practice is that even if the pilot is holding the stick at a particular position to command a certain pitch responses, the elevator is continuously moving to deliver the demanded pitch response. Thus, a control augmentation system for stabilisation and control of flight vehicles is a subsystem capable of seeking a specific response of the vehicle in proportion to a control input by altering the proportionality constant while simultaneously ensuring the stability of the vehicle. The system consists of both feedforward and feedback signal paths. The feedback paths provide augmentation of the vehicle’s inherent stability. The feedforward paths augment the controllability of certain specific modes of response, as commanded by the pilot. Probably the simplest of the longitudinal autopilots used in the control of current civil airliners is the normal acceleration demand autopilot (Figure 8.6). In this system, in addition to improving the damping of the less stable longitudinal modes, the aircraft is controlled so it responds to a typical manoeuvre demand such as a demand for a specific value of the normal acceleration or a certain level of ‘g’ as commanded by the pilot. In this system, the pilot moves the stick by a fixed amount to command a certain normal acceleration. The control system measures the aircraft’s ‘g’ value and continues to deflect the elevator till the error between this measured value of ‘g’ and the demanded value of ‘g’ is zero. This may be achieved by one of several control structures such as a PID controller. The actual gains in the control elements are selected from stability considerations in the short-period mode. Protection devices such as limiters are incorporated in the control loops to prevent the pilot from applying excessive ‘g’ or stalling the aircraft at low speeds. Further, as the aircraft’s performance depends on the atmospheric conditions, which are variable over the flight envelope, the gains of the various control elements and the parameters describing the features of the protection limits must be varied or scheduled according to the local

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FIGURE 8.6  Basic normal acceleration demand system including (optional) pitch rate feedback for stability augmentation and stall protection, qdynamic = ρU2/2.

dynamic pressure and the angle of attack. Generally, the principal loop gain reduces as the dynamic pressure increases. Another common example of an inner loop system is the yaw damper shown in Figure 8.7. Its task is to increase the damping in the Dutch roll mode, which is the principal mode associated with the yawing lateral dynamics of the aircraft. Thus, the yaw damper provides for lateral directional stability augmentation. The yaw damper senses the aircraft’s yaw rate and synthesises an appropriate control signal to the rudder servo, which in turn actuates the rudder to achieve the desired response.

FIGURE 8.7  Typical limited authority yaw damper for lateral directional stability augmentation with a manual/back-up loop.

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A variant of the normal acceleration demand system is the C*(t) demand system. A measure of the longitudinal flying qualities of an aircraft is C*(t), which is defined as

C * ( t ) = nzp ( t ) +

Ucθ g

(8.1)

where nzp is the normal acceleration of the load factor at the pilot’s seat Uc is a crossover velocity used to define C*(t). Thus, the C*(t) demand system is equivalent to the normal acceleration demand system, and it is essentially used to ensure that the aircraft handles in a predictable manner throughout the flight envelope. The importance of the inner loop system such as the normal acceleration demand system stems from the fact that it provides the basic ingredient for one of several autopilots. For example, it forms the heart of a height hold system (Figure 8.8), which is an essential autopilot in most civil airliners today. The flight management and guidance system demands a certain altitude, and the autopilot compares this with a blend of the measured altitude and altitude rates. The error signal is multiplied by an appropriate gain, which is then fed to the demand channel of the normal acceleration demand autopilot in place of the pilot’s commanded signal. Before the autopilot is engaged, the aircraft is trimmed to fly at the correct initial condition. Thereafter, the flight management and guidance system takes over and flies the aircraft until the pilot chooses to take over control of the aircraft again.

FIGURE 8.8  Height hold mode autopilot system including a normal acceleration demand inner loop system.

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Several other longitudinal autopilot modes are available in most commercial civil airliners either as independent autopilot modes or as subsidiary systems of more complex autopilots such as instrument landing system (ILS)-coupled autoland autopilot with automatic go-around facility. Typical examples of these are centred around the use of the elevator as the primary control input for controlling the aircraft and include the pitch attitude hold autopilot illustrated in Figure 8.9 and the Mach-hold autopilot illustrated in Figure 8.10. In the lateral mode, the basic autopilot is the one that allows the aircraft to be steered along a set direction. A change in the aircraft’s heading (the heading angle is the sum of the sideslip and yaw angles) is usually achieved by banking the aircraft and suppressing the accompanying sideslip response, which results in a coordinated turn. There are several approaches to implementing such a system. When the sideslip angle is measurable as is usually the case in military aircraft, the basic inner loop uses the roll rate, yaw rate and sideslip feedback. When the sideslip is not available for feedback, the inner loop utilises

FIGURE 8.9  Pitch attitude demand control system (in autopilot mode).

FIGURE 8.10  Mach-hold (elevator) control system (in autopilot mode), where a is the relevant speed of sound.

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cross feeds from the rudder and aileron channels to suppress the sideslip response, so the heading angle is essentially equivalent to the yaw. Two block diagrams of such systems that assume that the demand is for a coordinated turn—the first based on sideslip measurement and the second including cross feeds from the aileron to the rudder channel and sideslip suppression using yaw rate feedback—are shown in Figure 8.11. In the first case, only the inner loops are shown. The washout filter is included in the loop to eliminate the yaw rate feedback in a steady turn. Coupled with a very high-frequency omni range (VOR) heading acquisition loop, the autopilot forms the basis of a lateral autopilot for flying towards a way point. This autopilot is also a basic inner loop for a localiser acquisition system prior to the aircraft landing, which is also based on the same principle as the VOR heading acquisition system. Engine throttle control is the primary method of speed control and a typical autothrottle system for speed control, as is illustrated in Figure 8.12. Engine thrust-only control is often used as a backup method for both longitudinal and lateral flight

FIGURE 8.11  Typical lateral autopilots with and without sideslip feedback (a) includes sideslip feedback and a high-pass filter to eliminate (washout) the yaw rate feedback in a steady turn and inner loops only (b) with cross feeds connecting the aileron and rudder channels (when no sideslip suppression is included the yaw attitude gyro replaces the compass).

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FIGURE 8.12  Typical auto-throttle-type autopilot for aircraft speed control, showing additionally the role of a noise filter to attenuate sensor disturbances.

controls. These thrust-only-based longitudinal and lateral autopilot examples are illustrated in Figures 8.13 and 8.14. The differential thrust input is summed to the collective thrust input in the left engine and subtracted from the collective thrust input in the right engine. Figures 8.15 and 8.16 show typical ILS glide slope and localiser acquisition autopilots, which are built around the pitch attitude and heading acquisition autopilots discussed earlier. Often there is a push button switch available on the autopilot control panel, which permits switching between VOR and localiser acquisition. Figure 8.17 illustrates a typical autoland autopilot that is used in conjunction with a lateral heading hold autopilot to smoothly land the aircraft along a predefined landing trajectory referred to as the flare landing path.

FIGURE 8.13  Typical thrust-only longitudinal flight control system with inner loops omitted.

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FIGURE 8.14  Typical thrust-only lateral flight control system with inner loops omitted.

FIGURE 8.15  Longitudinal ILS capture autopilot for acquisition of glide slope.

The repertoire of autopilots illustrated earlier constitutes a minimal set that are installed in current civil airliners. Not shown are autopilots for lift dumping, air brakes and load alleviation, which make use of additional control surfaces such as spoilers, additional flaps and even flaperons, which are capable of functioning as both ailerons and flaps. Different sets of autopilots may be selected by appropriate switching on the autopilot control panel, which is sometimes known as the flight control unit (FCU). Typically, there is a button for selecting the autopilot mode (as opposed to manual

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FIGURE 8.16  VOR/lateral ILS capture autopilot for acquisition of VOR/localiser heading with the inner loops as in Figure 8.16.

FIGURE 8.17  Block diagram of autoland autopilot incorporating automatic flare control.

flying). Then, the pilot usually has the option of overriding the primary system and selecting the backup. He or she can also decide whether the manual inputs or autopilot inputs have priority using another override button. For the primary throttle control, he or she can choose between an indicated airspeed hold/acquisition autopilot and a Mach hold/acquisition autopilot. In the lateral mode, he or she can choose between a heading and a track hold/acquisition autopilot. In the longitudinal case, the elevator servos are driven by one of the outputs from the flight management guidance computer (height hold/acquisition) or a speed hold, vertical speed hold/acquisition or flight path angle hold/acquisition autopilots. Not all the autopilots may be available in the backup mode. He or she can choose between the normal autopilot functions and the approach and land mode by means of another selector button.

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The design of the FCU, which permits the selection of different sets of autopilots during different phases of the flight as well as provides for certain protection and safety mechanisms, is an extremely important phase in the total design of the aircraft’s automatic flight control system. The final phase in the design of an aircraft’s automatic flight control system is the selection of the control laws or parameter design and implementation of the system. Several examples of flight control laws design are presented later in this chapter. If the system is designed to be implemented digitally, there is a greater flexibility in the choice of the control laws. Another aspect is that both the hardware and software must be of equally high integrity. To understand how stringent these requirements are, it is worth noting that designers must be able to show that the probability of a catastrophic accident due to a control system failure should be of the order of 10 −10 per hour for civil airliners.

8.3  INTEGRATED FLIGHT CONTROL SYSTEM Historically, it is probably fair to say that the arrival of the Concorde supersonic airliner heralded the beginning of integrated avionics although the specification seemed primarily to deal with the navigation system. It was produced around the 1970s to meet the specifications written first in 1962. For example, the navigation system was required to maintain the aircraft within a corridor of ±20 nautical miles width, equivalent to a 2σ error. At a distance of 50 nautical miles from the destination, the position error was required to be less than five nautical miles. Thirty minutes before the estimated time of arrival, it was expected to be predicted with an error within 3 minutes and preferably within 1 minute. The Concorde avionics system is illustrated in Figure 8.18. The navigation equipment, initially installed in the Concorde, may be divided into two groups: conventional radio navigation equipment and other self-contained units. The former consisted of two VOR units, an ILS with two horizontal situation indicators to display this information in the cockpit, two automatic direction finding units with two radio magnetic indicators to display this information in the cockpit, a pair of radar altimeters and a pair of distance-measuring equipment units. The self-contained units consisted of three complete and independent inertial navigation systems, a pair of autopilots integrated with the flight directors and two air data computers. The next stage in the development of integrated avionics was the development of digital processing hardware and software that seems to have provided the impetus for further integration. This was achieved by replacing appropriate analogue filtering modules within the system by equivalent digital processing filters. This feature is illustrated by considering the case of the longitudinal, lateral and directional flight control systems in a typical modern fighter aircraft. The functional block diagrams of these systems are illustrated in Figures 8.19–8.21. A number of the functional blocks in Figures 8.19–8.21 represent analogue filtering modules. This becomes apparent when the modules are modelled mathematically and represented as I/O transfer functions. Thus, it is possible to replace the analogue signal processing modules by their digital equivalents. The principal analogue transfer functions as well as the model transfer functions of the sensors and actuators are well-known standard structures, discussed in the latter sections.

FIGURE 8.18  Concorde avionics system. (Reproduced from Hill [1]. With permission from the Canadian Aeronautics and Space Institute.)

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FIGURE 8.19  Simplified functional block diagram of a single-lane longitudinal flight control system.

FIGURE 8.20  Simplified functional block diagram of a single-lane lateral (roll channel) flight control system.

Fly-by-computer control systems are often confused with FBW control systems. In a typical FBW control system, illustrated in Figure 8.22, the motion of the rudder is sensed electrically, combined with the autopilot input, suitably amplified and then employed to drive the electro-hydraulic servo-actuators. To cater for the need to accommodate failures, a mechanical backup or reversion system is provided which could be engaged or disengaged by the use of a typical mechanical clutch. Fly-by-computer control systems are digitally integrated systems with all the computing functions performed by a set of distributed computer systems, illustrated in Figure 8.23 by a single block. The computers themselves are linked together typically by one of many custom digital data buses. The data bus is essentially a common medium for communication between the computers within the system. Data input to the computers, although shown in Figure 8.23 as modularly partitioned, is routed

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FIGURE 8.21  Simplified functional block diagram of a single-lane directional flight control system.

FIGURE 8.22  Basic FBW and fly-by-computer systems compared: principle of FBW rudder control with mechanical reversion.

via a flight data acquisition unit (FDAU) on some aircraft. The FDAU receives input signals from sensors and various aeroplane systems via input/ output (I/O) ports. In the case of one commercial airliner, the typical inputs are as listed in Table 8.1. The use of such a centralised FDAU has both its advantages and disadvantages. The principal advantage is that it facilitates integration, and the principal disadvantage stems from the need for redundancy. Control commands and other computed outputs from the flight control computers to the servo-actuators and other hardware systems including the redundant computer modules that are essentially the data sinks or data users are routed directly via the data buses, bypassing the FDAU. In Table 8.2 are listed a typical set of control functions (or control laws) programmed in software in a redundant set of flight control computers, which the pilot may select via selector switches located in the mode control section of the glare shield panel.

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FIGURE 8.23  Basic FBW and fly-by-computer systems compared: principle of fly-by-computer control.

8.3.1 Guidance System: Interfacing to the Automatic Flight Control System The FM computer provides altitude and airspeed commands to a longitudinal autopilot. With the throttle loop open, this autopilot functions as altitude hold autopilot. With the throttle loop closed, it is a vertical flight path hold/ acquire autopilot. A block diagram of a typical coupling system is shown in Figure 8.24. The inputs to the guidance system are switchable to either the flight management system (FMS)

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TABLE 8.1 Typical input signals to an FDAU Analogue and digital inputs Analogue inputs 1. Accelerometer 2. Aileron position transmitters 3. Brake pressure transmitters 4. Control column position sensors 5. Control wheel position sensors 6. Elevator position transmitters 7. Horizontal stabiliser position transmitters 8. Rudder pedal position sensor 9. Rudder position transmitters 10. Speed brake handle position transmitter Digital Inputs 1. Auto-throttle computer 2. Signal conditioner inputs 3. Clock–captain 4. Digital air data computer 5. Digital stall warning computer 6. Digital to analogue adapters 7. EFIS control panels 8. EFIS symbol generators 9. EIS primary 10. EIS secondary 11. Flight control computers 12. Flight management computer 13. Ground proximity warning computer 14. Inertial reference units

Discrete inputs Discrete inputs 1. Aeroplane ID shorting receptacle 2. Alternate brake select valve switch 3. Auto-throttle computer 4. Column switching module 5. Compartment overheat control module 6. EFIS transfer relay 7. Engine accessory unit 8. Engine low idle light inhibit relays 9. Fire detection module 10. Flight control computers 11. Flight control module 12. HF transceivers 13. IRS transfer relay 14. Landing gear logic shelf 15. Leading edge flaps/slats position indicator module 16. Light shield panel lights 17. Marker beacon receiver 18. Master caution annunciators 19. Stabiliser trim cut-out relay 20. Stall warning computers 21. Systems A and B elec pump low-pressure switches 22. Systems A and B eng pump low-pressure switches 23. Trailing edge flaps bypass valve 24. VHF transceivers

15. TCAS

or the pilot’s sidestick and throttle commands. The interfacing to the sidestick and throttle levers as well as the incorporation of the protection limits is not shown in Figure 8.24. In the example shown earlier, it is assumed that, in the longitudinal case, both elevator control and throttle control are concurrently available. Longitudinal and lateral flight control laws may be designed so that they operate in throttle or thrust-only mode with the appropriate roll/pitch priority logic, in control-surface-only mode with elevator only being effective in the longitudinal case and aileron and rudder only in the lateral case, or in a concurrent mode where both the throttle and the relevant control surfaces are effective. Examples of autopilots operating in each of the two non-concurrent modes in both the longitudinal and lateral cases have been discussed in the earlier sections.

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TABLE 8.2 Integrated flight control computing blocks or objects Category Longitudinal

Lateral

Auto-throttle Autoland

Modes Radar altitude trim/hold Pitch attitude trim/hold Mach trim/hold Auto/manual approach Track acquire Heading hold Heading acquire Bank attitude Auto/manual approach Calibrated airspeed hold Flare control Lateral ILS (acquire and hold) Longitudinal ILS

FIGURE 8.24  FM computer/longitudinal autopilot coupling subsystem (normal control law in cruise).

8.3.2 Flight Management System The collective name for the FM computer or the performance data computer system, the autopilots and flight director, auto-throttle and inertial reference systems (IRSs) when used as one integrated system is the FMS. The FMS can be defined as being

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capable of 4D area navigation (latitude, longitude, altitude and time) while optimising performance to achieve the most economical flight possible. Thus, the FMS is simply an integrated FMGCS. The heart of the FMS is thus the FMC which uses data from aircraft sensors and its own database to perform navigational and performance computations and control and guidance commands. The FMC eliminates many routine tasks and manual computations previously performed by pilots. Current FMSs can autonomously fly an aircraft from take-off through landing but may not provide robust responses to anomalous events. Thus, current FMSs are capable of autonomously controlling an aircraft from take-off through landing during nominal flight operations. A systems engineering approach may be adopted in the design of an FMS based on the partitioning of the design calculations. The design may therefore be broken up into the following subsystems: (1) performance system design, (2) FMS design, (3) guidance system design, (4) navigation interfaces and mixing filters system design, (5) flight control system design, (6) full authority digital engine control and interface to FMS, (8) display interface system design, (9) data communications interface system design and (10) integration and test. A complete discussion of a total systems engineering approach is well beyond the scope of this introduction and will be presented elsewhere.

8.4  FLIGHT CONTROL SYSTEM DESIGN The most common approach to flight control law design for a variety of flight conditions is gain scheduling, which requires the design of control laws for a large matrix of flight conditions. Each design can be a time-consuming process, based on one of many classical techniques either in the time domain or in the frequency domain, where there are a large number of choices both in structure and in parameters available to the designer. In designing a flight control system, it is essential to first select an appropriate structure for the closed-loop system. There are a number of options available for doing this and some of the more general structures are listed in the following: A. Proportional feedback control This is the simplest form of a feedback controller that is usually applied to a unity feedback system. The error between the demanded output and the actual output is multiplied by a simple gain. The resulting signal is then used as the control input for the system. B. Proportional derivative (PD) feedback control In this case, the simple proportional gain is replaced by a weighted linear combinational of the error and its first time derivative. This situation is equivalent to adding a zero to the closed-loop transfer function. In the case of the position controller, the use of a PD controller is equivalent to additional velocity feedback, and it allows the control system designer to select the closed-loop damping ratio and the closed-loop response frequency independently. This type of control is often adopted in the design of servo-motors.

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C. Integral control As a generalisation, proportional error feedback will adjust the natural frequency of a system, while feedback of rate of change of output or error will add damping to the system and increase stability. When the system under control does not contain an intrinsic integrator, it is not possible to obtain a situation of zero error in response to a step command with only these feedback terms. To obtain good system accuracy, an integral term proportional to the integral of the proportional term is added to the control input. The integrator is implemented as an electronic one using an operational amplifier or in terms of digital software in the system’s memory or a motor which moves to a new position and holds it when the input goes to zero. They all have the property that they can hold a fixed value of control command on the output even though the error is zero. This feature might be employed when, for example, a fixed rudder angle is required to hold an aircraft trimmed in a crosswind or with one engine failed. D. Compensators and control filters It is now apparent that the feedback path need not necessarily have a unity transfer function in many applications. Thus, a typical filter is introduced in the feedback path to process the output before it is subtracted from the input signal. The difference which can no longer be referred to as error may also be processed by another typical filter before it is amplified and provided as a control input to the system being controlled. The filters introduced in the feedback and forward paths are generally referred to as control filters or compensators are generally designed by the control system designer so that closed system meets the desired specifications. There are generally speaking a number of filters to choose from such as lag filters, lead filters, lead/lag and lag/lead filters and a variety of second- and higher order filters. E. Observer-based designs Observers are a special class of compensators or estimators which are designed so that the observer’s poles do not in any way influence the closedloop performance of the system, provided the system operates under the ideal design conditions. This is usually achieved by implementing the control filter in the forward path also as a feedback system. The forward path and the feedback path control filters are both chosen to have the same openloop characteristic equation. All the aforementioned control structures are basically linear controllers and may be implemented either analogue in the continuous time domain or in the discrete-time domain using digital hardware. F. Other controller structures There are a number of other structures possible for the controller and some of these are mentioned in the succeeding text. a. Non-linear controllers: In many practical situations, non-linear elements that do not satisfy the principle of superposition are deliberately used to enhance performance. Typical examples are control systems incorporating relays, servo-valves with artificial dither, servos with limiters,

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b.



c.



d.



e.



f.

343

etc. In these cases, linearisation techniques based on so-called describing functions are often used for analysis and design. Bang-bang (on-off) and variable structure systems: Bang-bang controllers are typical non-linear control systems incorporating a simple relay to switch the control input between two different levels. Such systems are also referred to as on-off controllers. Variable structure systems are generalisations of on-off controllers where the input is switched between several discrete levels based on a switching criterion. The switching criterion provides the logic based on which control input is varied from one to another discrete level. The switching criterion is essentially a set of rules based on whether or not the input signals to the controller belong or do not belong to certain predefined domains or sets. Fuzzy-logic-based controllers: Fuzzy logic controllers can be considered to be generalisations of variable structure controllers where the switching logic is based on the so-called concept of fuzzy subsets where membership to a set or domain is graded and the inputs to the controller can be assigned one of several discrete or continuous grades of membership to the predefined domains or sets. The rules account for the grade of membership in evaluating the control. The control which is usually a member of a fuzzy set is de-fuzzified before it is input to the system. Fuzzy logic controllers are particularly proving to be useful in controlling passive systems where the system description is vague and when stability is not a problem. Adaptive controllers: These are a class of non-linear controllers where the controller parameters are continuously modified in real time to adapt the system response characteristics to the requirements on the basis of an adaptation law. One approach to adaptive control is to use a parameter estimation method to identify the system parameters and use these to update and schedule the controller parameters in real time. This approach to adaptive control is known as self-tuning. Other approaches are based on a model of the plant to simulate a desirable output and using the error between the actual and desirable output to adjust the controller parameters. The latter class of adaptive controllers are usually referred to as model reference adaptive control. Self-tuning PID controllers have been used in the process control industry extensively. Expert controllers: Expert system-based controllers are a class of adaptive controllers where the adaptation law is based on a rule-based expert system. Here, the adaptation is achieved on the basis of logic and reasoning or an inference engine which is part of the expert system. Although both adaptive and expert controllers may be considered to be learning control systems, the learning is implicit and it is more appropriate to consider them as closed-loop controller schedulers. Fuzzy-logic-based expert controllers are being used extensively for tuning PID controller gains. Self-organising controllers: This is a class of adaptive controllers where the adaptive outer loop is not only a rule-based expert system but also

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the basic controller. In a sense, these controllers tend to have the capability to learn and the learning is explicit. For example, if the controller is a fuzzy-logic-based controller, adaptive outer loop is generally one which evaluates a performance index, and on this basis, a rule modifier modifies the switching rules in the fuzzy logic controller. The rule modifier is also a fuzzy-logic-based system which punishes rules that are used less and rewards rules that are used more often. Fuzzy logicbased self-organising controllers are being used to a limited extent in the process control industry. In the avionics industry, expert systems are being developed for use as low authority expert controllers and as control advisory systems. g. Neural-net-based controllers: These controllers are a recent innovation based on the concept of the Hopfield neuron. It is possible to solve optimisation and constraint resolution problems using a network of Hopfield neurons. The interesting feature of this approach is the ability of the network to learn during the process of the solution. For example, if one is minimising the weighted sum of two integrated performance measures, the weights which are initially unknown may be updated till an acceptable solution is obtained. It must be emphasised that the neural network methodology is only a solution technique and large optimisation problems difficult to solve by conventional means may be solved rapidly by the use of neural nets because of their parallel processing capability.

Control systems analysis and design, in general, can take place in one of two environments. These are referred to as the time domain and frequency domain. In timedomain analysis and design, computations are made on the physical variables that is ones which are directly observable and measurable. Such methods are computationally intensive requiring the use of computer-aided tools. Examples of measurable variables on which computations can be directly performed are voltage, current, position, velocity, temperature, flow rate, pressure, etc. In the frequency domain, however, the physical variables are subjected to some type of mathematical transformation before any analytical or design computations are initiated. Such transformations simplify the problem in some sense but make real-time calculations quite impossible. On the other hand, even though the variables are no longer physically observable, general I/O relations are more readily derived, and the properties of classes of systems can be explored and categorised. In this sense, therefore, frequency-domain techniques are still invaluable and indispensable.

8.4.1 Block Diagram Algebra The mathematical relationships expressed in the form of the governing equations of motion need to be manipulated if one is seeking fundamental system properties such as stability. However, the process of analysing stability is not only not straightforward but also involves a number of tedious mathematical operations. The process of this analysis may be greatly aided by representing the governing equations by

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block diagrams. Although we have already introduced these block diagrams earlier, no formal definitions of the entities that constitute the block diagrams or the rules governing the operations of the blocks were presented. The use of the block diagrams has the distinct advantage of indicating realistically the actual processes they represent. Further, by systematically applying the rules for combining the block diagrams, it is easy to form simpler block diagrams to be established. Based on these simpler block diagram representations, various control engineering components may be classified into groups represented by certain standard forms of block diagrams. A primary element of a block diagram is a single block with a directed input signal entering the block and a directed output signal leaving the block as illustrated in Figure 8.25. In the figure, the operator ‘D’ is the differential operator. It may be useful to consider the Laplace variable s which can be considered to be the differential operator ‘D’, so that s = D = d/dt. The integral operator can therefore be represented as

1 1 ≡ ≡ s D

t

∫ (     ) dt.

(8.2)

0

The relationship between the input signal and the output may be expressed as y(t) = G(D)u(t) where G(D) operates on the input signal to generate the output and is known as the block transfer function or simply as the transfer function. The box itself may then be interpreted as a symbol for multiplication. The other important element that may constitute a block diagram is the comparator. It is used to subtract the feedback signal from the reference input to generate the error output. The mathematical relationship may be expressed as e = yi – y. The block diagram of a typical comparator is illustrated in Figure 8.26. Thus, the circle may be interpreted as a symbol to indicate the summing operation, the arrows directed towards it indicating the inputs and the arrows pointing out representing the output. The sign at each input arrowhead indicates whether the quantity is to be added or subtracted.

FIGURE 8.25  Block diagram representation of a simple block with an input and an output.

FIGURE 8.26  Block diagram representation of a comparator.

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FIGURE 8.27  Block diagram representation of a non-unity feedback control system.

FIGURE 8.28  Single block representation of a non-unity feedback loop system.

The block diagram of a typical non-unity feedback system may be synthesised by combining the basic blocks, as illustrated earlier and shown in Figure 8.27. Considering the mathematical representations of each block, we obtain the following mathematical relations:

y = G ( D ) Ke,

e = yi − v

and

v = H ( D ) y.

(8.3)

Eliminating all variables except yi and y,

y = G ( D ) K ( yi − v ) = G ( D ) K ( yi − H ( D ) y ) .

(8.4)

Rearranging the equation, it follows that

(1 + G ( D ) KH ( D )) y = G ( D ) Kyi .



(8.5)

Hence, the ratio of the output to the input is G ( D) K y = . yi 1 + G ( D ) KH ( D )



(8.6)

The aforementioned equation represents an open- to closed-loop transformation and allows the feedback loop to be simplified and expressed in terms of a single block, as illustrated in Figure 8.28. The closed-loop transfer function then is

T ( D) =

KG ( D ) KG ( s )   and  T ( s ) = 1 + KG ( D ) H ( D ) 1 + KG ( s ) H ( s )′

(8.7)

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when the D operator is replaced by the Laplace transform variable, s. Several other reduction and simplification rules may be formulated. These are tabulated in Table 8.3. Thus, it is possible to reduce a number of other complex feedback loops to simpler block diagrams.

TABLE 8.3 Block diagram transformations: SISO systems

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8.4.2 Return Difference Equation The closed-loop characteristic equation is obtained from 1 + KG ( D ) H ( D ) = 0.



(8.8)

The right-hand side of the aforementioned equation can be seen as the ratio of the closed-loop to the open-loop characteristic polynomials. The equation given earlier is the return difference equation as it represents the difference between a notional input to the plant and the value returned at the end of the feedback loop, in the absence of any input, y(t). When the loop is closed, this difference is forced to be equal to zero. The function KG(D)H(D) is the open-loop transfer function, and the closedloop characteristic equation may be established by setting the open-loop transfer function equal to −1. Thus, not only are the roots of the closed-loop characteristic equation functions of the gain, K but we may also obtain the roots of the closed-loop characteristic equation from the open-loop transfer function. Given that the transfer functions G(D) and H(D) are defined as G(D) = N(D)/P(D) and H(D) = M(D)/L(D), the closed-loop transfer function can be expressed as GC ( D ) = =

KG ( D ) KN ( D ) /P ( D ) = 1 + KG( D ) H ( D ) 1 + KN ( D ) M ( D ) / P ( D ) L ( D ) KN ( D ) L ( D ) . P ( D ) L ( D ) + KN ( D ) M ( D )



(8.9)

                

The return difference equation is P ( D ) L ( D ) + KN ( D ) M ( D ) P ( D) L ( D) (8.10) Closed-loop characteristic equation =                  = 0. Open-loop characteristic polynomial

1 + KG ( D ) H ( D ) =

The closed-loop characteristic equation is

P ( D ) L ( D ) + KN ( D ) M ( D ) = 0.

(8.11)

8.4.3 Laplace Transform A large class of signals can be represented as a linear combination of complex exponentials, and complex exponentials are eigenfunctions of linear time-invariant systems. This leads to the development of the Laplace transform of a function of time, t. Any function of time is expressed as an infinite sum of complex exponentials

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(est, s = σ + iω). Laplace transforms can be applied to the analysis of stable or unstable systems (signals with infinite energy) and play a role in the analysis of system stability. The analysis of time-invariant linear systems can also be carried out by transform techniques. This is generally referred to as the ‘s’ plane or frequencydomain analysis. The most useful transformation for control systems analysis is the Laplace transform since the variable ‘s’ of this transformation is equivalent to the ‘D’ operator for all practical purposes. It is also related to the frequency of sinusoidal waveforms and therefore corresponds to an easily measured and interpreted quantity. Other transforms such as the ‘z’ transforms are used in the analysis of discrete systems. The Fourier transform is a generalisation of the Laplace transform and is also extremely useful in signal processing. The Laplace transform of a general signal x(t) is defined by the integral ∞



X (s) =

∫ x (t ) e

− st

dt.

(8.12)

−∞

The Laplace transform is thus a function in terms of the variable ‘s’, which is usually obtained from a standard table of transforms and inverse transforms, rather than from first principles.

8.4.4 Stability of Uncontrolled and Controlled Systems An issue of paramount importance is the design of globally stable multimode controllers in order to guarantee the safe performance of such systems. There are essentially two classes of methods of control law synthesis for non-linear plants. The first class of methods is based on stability analysis, while the second is based on optimisation. Each class of methods is associated with its own set of advantages and disadvantages. The analysis of the asymptotic systems was pioneered by A.M. Lyapunov in his thesis that was published almost 100 years ago. Lyapunov’s method is not only a basic tool in the analysis of system stability but also a very valuable tool for the synthesis of controllers for non-linear systems. Lyapunov divided the problem of stability analysis of non-linear systems into two groups. The first group of problems involved non-linear systems, which could be either solved exactly or reduced by some means to a linear system. These not only included exact methods but also, in some cases, approximated techniques, where the stability of the linearised system yields useful properties about the stability of equilibrium of the non-linear system. In the case of a linear system, it is possible to analyse the stability of the solution without having to derive elaborate general solutions to the problem. Methods of evaluating stability such as the Routh–Hurwitz method and Routh’s tabulation allow the control system designer to establish bounds on important system gains to guarantee stability. Consider the definition of stability. If the system is in a state of equilibrium, any disturbance of the finite magnitude applied to it will cause a free motion following the disturbance. If the free motion ultimately disappears, the motion is said to be stable. On the other hand, if the free motion gradually degenerates to a finite motion

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that eventually becomes larger limitlessly, the motion is said to be unstable. If the free motion neither disappears nor becomes large, the system is neutrally stable. To assess the stability of a system, one must consider the I/O transfer function. Given a transfer function, in the form

y (t ) Y ( D) = G ( D) = , u (t ) U ( D)

(8.13)

in the case of free motion, we can assume that u(t) = 0. Rationalising the numerator and the denominator, the equations for the free response, in the absence of any inputs, take the form

P ( D ) y (t ) = 0

(8.14)

where P(D) is generally a polynomial in the ‘D’ operator. To solve the equations, we assume



y ( t ) = Ae pt

(8.15)

Dy ( t ) = pAe pt , D 2 y ( t ) = p2 Ae pt , D 3 y ( t ) = p3 Ae pt , D 4 y ( t ) = p4 Ae pt ,…



(8.16)

Hence, it follows that But

P ( D ) y ( t ) = P ( p ) Ae pt = P ( p ) y ( t ) .

(8.17)

P ( D ) y ( t ) = P ( p ) y ( t ) = 0.

(8.18)

Since the response is not assumed to be trivial and hence is non-zero, y(t) ≠ 0 and P(p) = 0. The equation P(p) = 0 is known as the characteristic equation. The roots p1, p2, p3, …, pn of the characteristic equation will, in general, be complex quantities of the form p = q   +  ir, where i = −1 . The response y(t) is given by or by

y (t ) =

∑A e k

pkt



y ( t ) = A1 e p1t + A2 e p2t + A3 e p3t +  An e pn t

(8.19)

(8.20)

For the response to be stable, the real parts of the values of pk must be negative for all k.

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Each of the cases of quadratic, cubic and quartic characteristic equations may be independently considered as these can be assessed quite easily by inspection. Quadratic equation, p2 + a1p + a 0 = 0: The two roots have negative real parts if a1 and a 0 are both greater than zero. Cubic equation, p3 + a2p2 + a1p + a 0 = 0: A cubic has, in general, one real root and two complex conjugate roots p1 = q1 ,



p2 = q2 + ir2

p3 = q2 − ir2 .

and

(8.21)

The characteristic equation may be written as (p – q1)(p – q2 – ir2) (p – q2 + ir2) = 0 and a2 = − ( 2q2 + q1 ) ,



(

a1 = q22 + 2q2 q1 + r22 ,

)

a0 = − q1 q22 + r22 . (8.22)

It follows that when the system is stable, a2, a1 and a 0 are all greater than 0. Further,

(

)

a1 a2 − a0 = −2q2 2q22 + 2q2 q1 + r22 > 0.



(8.23)

Thus, a2 > 0,



a1 > 0,

a0 > 0,

a2 a1 − a0 > 0.



(8.24)

These four conditions guarantee stability. Quartic equation, p4 + a3p3 + a2p2 + a1p + a 0 = 0: The conditions of stability are a3 > 0,



a2 > 0,

a1 > 0,

a0 > 0,

(

a3 a2 a1 > a12 − a0 a32

)

(8.25)

Conditions for stability in the general case may be stated in a compact form. Given an algebraic characteristic equation of the form F ( s ) = a0 s n + a1 s n −1 +  + an −1 s + an = 0



(8.26)

where the coefficients ar are real, construct the following n determinants:

D1 = a1 , D2 =

Dn =

a1 a3

a0 a2

, D3 =

a1 a3 a5 

a0 a2 a4 

0 a1 a3 

a2 n −1

a2 n − 2

a2 n − 3

0 a0 a2 

a1 a3 a5

0 a1 

a0 a2 a4

a0 

0 a1 a3 … … … 

,

(8.27) 0 0

0 0

…

an

0 0

.

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A necessary/sufficient condition for the roots of the characteristic equation F(s) = 0 to have negative real parts is D1 > 0,



D2 > 0,

D3 > 0,

Dn > 0.



(8.28)

These conditions are verified by using Routh’s tabular method.

8.4.5 Routh’s Tabular Method Routh’s tabular method is designed for rapid assessment of stability, given the characteristic polynomial without having to find the roots of the characteristic polynomial. A tabular array of the form in Table 8.4 is produced. The first two rows are formed by writing down alternate coefficients of the characteristic equation. Each entry in the following rows is calculated from four of the previous entries according to the following equations:



b1 =

c1 =

a2 a1 − a0 a3 , a1

b2 =

a4 a1 − a0 a5 , a1

b3 =

a6 a1 − a0 a7 , a1

(8.29a)

a3 b1 − a1 b2 , b1

c2 =

a5 b1 − a1 b3 , b1

d1 =

b2 c1 − b1 c2 . c1

(8.29b)

Coefficients are calculated until only zeros are obtained, and the rows shortening until the p 0 row contains only one value. The conditions for stability are as follows:

1. Every change of sign in the first column of this table signifies the presence of a root with a positive real part. 2. For stability, therefore, all values in the first column of this array must be positive. 3. There are special cases and these require special treatment.

TABLE 8.4 Routh’s tabular array Pn Pn–1 Pn–2 Pn–3 Pn–4 … … P0

a0 a1 b1 c1 d1 .. .. ..

a2 a3 b2 c2 d2 .. ..

a4 a5 b3 c3 d3

a6 a7 b4 .. ..

.. .. .. .. ..

.. .. ..

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FIGURE 8.29  Frequency response of a control engineering subsystem.

8.4.6 Frequency Response An important technique of studying control engineering component is by consideration of their output to a typical sinusoidal input. Thus, consider a typical system with a sinusoidal input and a sinusoidal output, as illustrated in Figure 8.29. Consider the transfer function Y ( D) N ( D) = . (8.30) U ( D) P ( D)

Hence,

P ( D ) Y ( D ) = N ( D ) U ( D ) . (8.31)

If one assumes that

u ( t ) = exp ( iωt ) ,



d u ( t ) = iω × exp ( iωt ) dt



(8.32)

and DU ( D ) = iωU ( D ) , D 2U ( D ) = −ω 2U ( D ) , D 3U ( D ) = −iω 3U ( D ) ,...

(8.33)

Consequently,

Y ( iω ) N ( iω ) N ( iω ) = = exp ( −iφ ) = K exp ( −iφ ) . U ( iω ) P ( iω ) P ( iω )

(8.34)

The ratio of the magnitude of the output to the magnitude of the input, K, is the gain of the subsystem. Further, the output lags the input by a certain phase angle, ϕ. Since the same I/O relation must hold for a cosine or a sine input, one may express the output-to-input relation of the subsystem in the complex number domain as

vout KA exp ( iωt − φ ) vout = exp ( −iφ ) = K ( A, ω ) exp ( −iφ ) . = vin A exp ( iωt ) vin

(8.35)

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FIGURE 8.30  Typical subsystem frequency characteristics.

Ideally, one would require that K be independent of both A and ω and that the phase lag, ϕ, be as small as possible over a range of frequencies of interest to the user. The gain of a subsystem is most often specified in the decibel notation. It is defined as 2

v  v G = 10 log10  out  = 20 log10 out = 20 log10 K .  vin  vin



(8.36)

Thus, the frequency response of a typical subsystem is expressed as a plot of the gain versus the frequency on a logarithmic scale, as illustrated in Figure 8.30, when the system is operating within the domain where the response is linear (Figure 8.31). In the example shown in the succeeding text, it is a maximum of about 43 dB and over 40 dB in the frequency range of 0.2 ≤ f ≤ 20,000 Hz. The useful frequency range of the subsystem, the bandwidth, is defined as the range of frequencies, over which the gain is greater than the gain at the half power points. Since the power is proportional to the square of the voltage, the voltage gain at the half power points is defined as  1  v 2 G(1/ 2)P = 10 log10   out   = 20 log10  2  vin  



= 20 log10

vout 2 vin



(8.37)

vout − 20 log10 2. vin

Hence,

G(1− 2)P = 20 log10 K − 10 log10  2 ≈ 20 log10  K − 3 = ( G − 3)db .

(8.38)

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355

FIGURE 8.31  Non-linear subsystem characteristics and operation in the linear domain.

Thus, as the peak gain of this subsystem is 43 dB, the half power point is at 40 dB, and it follows that the bandwidth of the subsystem is 0.2 ≤ f ≤ 20,000 Hz.

8.4.7 Bode Plots In plotting the open-loop dB magnitude-phase information, it is often more convenient to first plot the dB magnitude and phase independently as functions of the frequency ω on a semi-logarithmic graph paper. The frequency ω is plotted on the log scale (horizontally), while the dB magnitude and phase are plotted vertically on linear scale. Although the two plots are independent, the same horizontal axis is used for the two plots. Thus, it is possible to read off for each frequency the corresponding dB magnitude and phase and vice versa. These plots are referred to as Bode plots and are valuable tool particularly for validation of a control system design.

8.4.8 Nyquist Plots The magnitude and phase of the phasor G(iω) may be conveniently plotted as the frequency ω is varied from ω = 0 to ∞ on the complex plane with the real axis representing the real part of G(iω) and the imaginary axis representing the imaginary

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part of G(iω). G(iω) is represented as a vector of magnitude M(ω) at an angle to the positive real axis equal to the phase angle ϕ(ω), which is considered to be positive counter-clockwise. Such a plot described by the vector G(iω) as ω varies from 0 to ∞ is referred to as a polar plot. The Nyquist plot is a special version of the polar plot which must be defined for variations of ω along a special contour.

8.4.9 Stability in the Frequency Domain To define the condition of stability in the frequency domain, one must return to the return difference equation. For a unity feedback system, 1 + G ( iω ) = 0. (8.39)



Since G(iω) is a complex quantity, the return difference equation must be satisfied both in terms of the magnitude and phase. Thus, since G(iω) = –1,

G ( iω ) = 1

and

arg ( G ( iω )) = arg ( −1) = 180°.

(8.40)

Since stability requires that there are no poles in the right half of the complex ‘s’ plane, for stability,

G ( iω ) = 1

and

arg ( G ( iω )) < arg ( −1) = 180°,

(8.41)

in the right half of the complex ‘s’ plane. Both requirements must be met simultaneously for the system to be stable. Thus, the assessment of stability is not straightforward in the frequency domain. The frequency at which

G ( iω ) = 1

and

arg ( G ( iω )) < arg ( −1) = 180°

(8.42a)

is known as the gain crossover frequency, while the frequency at which

G ( iω ) < 1

and

arg ( G ( iω )) = arg ( −1) = 180°

(8.42b)

is known as the phase crossover frequency. These frequencies are used to define two margins of stability in the frequency domain, rather than defining a simple criterion for the stability of the system. The stability margins are measures of relative stability. The system is considered to be adequately stable if both the margins are adequate.

8.4.10 Stability Margins: Gain and Phase Margins In view of the return difference equation, one cannot usually increase the loop gain indefinitely because the system will become oversensitive to noise or actually

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357

unstable. It is essential to design any control system with a safety margin between the design value of loop gain and the value which would make the system unstable. This is to allow for the effects of tolerance build up and wear on mechanical components during the operating life of the control system. The ratio of the additional loop gain that makes the system unstable to the actual loop gain is called the gain margin. A general practical design rule is that if the loop gain were doubled, the system should just about remain stable. The effect of wear and tolerance build-up can additionally manifest itself as extra destabilising phase lag due to delays in the system. The additional phase lag required to cause the system to become unstable at its nominal operating design point is called the phase margin. A suitable value for this safety margin is 45° or so. The gain margin is the amount by which the open-loop gain must be increased, at the frequency at which the phase shift is 180° (the phase crossover frequency, ωpc), in order to enclose the -1 point. The phase margin is the additional open-loop phase shift required, at the frequency where the gain is unity (the gain crossover frequency, ωgc), in order that the −1 point shall be enclosed. Stability margins are relatively easy to define on the Bode and Nyquist plots. From the Nyquist or polar plot of G(iω), the gain margin = 1 / G ( iω pc ) and the phase margin = Z(G(iωgc) – 180°).

8.4.11 Mapping Complex Functions and Nyquist Diagrams One may generalise the idea of the polar plot and construct mappings, that is plots of the transfer function G(s) on the complex ‘G(s)’ plane for values of ‘s’ corresponding to a curve or domain in the complex ‘s’ plane. The mapping of one particular closed contour is of fundamental importance in control engineering because it encloses the entire right half of the complex ‘s’ plane and therefore may be used to map the right half of the complex ‘s’ plane. This contour is known as the Nyquist D contour. The Nyquist D contour consists of the imaginary axis from −j∞ to +j∞ and a semicircle of radius R → ∞. In the case when G(s) has poles on the imaginary axis, they are excluded from the Nyquist contour by semi-circular indentations of infinitesimal radius around them. The plot of the complex function G(s) on the complex ‘G(s)’ plane for all complex values of the variable ‘s’ along the Nyquist D contour is known as the Nyquist diagram. It plays a key role in determining whether or not the zeros of any transfer function G(s) are enclosed by the D contour and therefore lie in the right half of the complex ‘s’ plane. The Nyquist stability criterion is thus based on the Nyquist diagram.

8.4.12 Time Domain: State Variable Representation The type of time-domain operations that most systems impose on physical variables either can be characterised by differential equations or can be suitably approximated by differential equations. Most dynamic system models can be described by linear and time-invariant models. In many situations, one is primarily interested in the

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steady-state behaviour of the system, that is when it has been in operation for a long time and all transients have subsided. In these cases, the asymptotic stability of the system is of primary concern. Generally speaking, information about the initial conditions can be neglected for stability analysis, and in these situations, the transfer function-based methods are relatively easy to use. A very powerful method of control systems analysis is the state variable method. State variables are internal variables and they can be used to reconstruct the I/O description of the system. A mass–damper–spring system can be expressed in statespace form by expressing the single higher order equation of motion as a set of coupled first-order differential equations. For example, in the mass–damper–spring system, one could define two state variables: the displacement of the mass and the velocity of the mass. If one assumes that only the displacement is measured, the velocity is completely internal to the system as it may not appear in the measurement unless it is actually also measured. The kinematic relation between the velocity and displacement is one of the state equations, while the equation of motion, expressed in terms of both the velocity and displacement such that the equation is a first-order differential equation in one or more variables, is the second state equation completing the state-space description of the dynamics of the system. Several alternate internal variables may be chosen to establish this relationship and the state variables are not unique. In the aforementioned example of the mass–damper–spring system, an alternate internal state variable could be the linear momentum. Thus, the state variable is simply an alternate way of representing the behaviour of the system which makes the internal dynamics transparent. A valid set of state variables is usually the minimum set of state variables of the system such that the knowledge of them at any initial time together with information about the inputs is sufficient to specify the states at any other time. The output is then synthesised from a linear or non-linear combination of linear convolutions of state variable vector and the inputs. Basically, the method requires that the variables used to formulate the system equations be chosen in such a way that the equations can be written compactly in terms of matrices. The state variable technique is simply a method of representing an nth-order differential equation as a set of n first-order differential equations which can be written in a standard form using matrix notation. Such a representation of system dynamics permits the formalisation of many properties as well as the analysis and design techniques pertaining to the system. Further, many of the methods associated with state variable representations can be generalised and applied to cases where transfer function-based techniques are not applicable. These cases include, amongst others, time-varying and non-linear systems, systems with multiple inputs and outputs and non-stationary random inputs. A classic example is the multi-degree-of-freedom (DOF) spring-mass system which is described as a set of coupled second-order differential equations:  ( t ) + Dd ( t ) + Kd ( t ) = u ( t ) Md

where

d(t) is a q × 1 vector of the displacement DOFs u(t) is a q × 1 vector of control forces generated by the actuators

(8.43)

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M is a q × q mass matrix associated with the flexible structure D is a q × q damping matrix associated with the structure K is the associated q × q stiffness matrix. The classical transfer function method is relatively easy to use when compared to the state variable technique especially if the control system designer has no access to a computer. On the other hand, most computer-aided design methods are based on the state variable representation as the state variable representation is particularly suited for that purpose. The dynamics of the spring-mass system may be expressed in statespace form as  M  0 



0 d  d   0  = M  dt  d   −K

M  d   0  u  + − D   d   I 

(8.44)

and reduced to 0 d  d    = dt  d   − M −1 K



 d   0  I +   u. −1 − M D   d   M −1 

If one assumes that certain linear combinations of the displacements, velocities and inputs are measured, the aforementioned equations may be expressed in state-space form as d x ( t ) = Ax + Bu, dt



y = Cx + Gu,

(8.45)

where  d  x= ,  d 

 0 A= −1 M K − 

 I  −1 − M D 

and

 0  B= . −1   M

(8.46)

The state-space description of a linear time-invariant system is given by the aforementioned state equation where x, u and y are the n × 1 state vector, the m × 1 input vector and the l × k output vector and A, B, C and G are n × n, n × m, k × n and m × m matrices. The first of Equation 8.45 represents the relationship between the inputs and the states, while the second represents the relationship of the states and the inputs to the outputs. Thus, in principle, the transfer function representation may be obtained by eliminating the states from the second of Equation 8.45 using the first. For a single input-single output (SISO) system, the number of inputs and number of outputs are equal to unity, that is m = k = 1.

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8.4.13 Solution of the State Equations and the Controllability Condition When state-space equations are small in number, then it is possible to solve them by conventional techniques without using a computer. The conventional method for solving them is the Laplace transform method. The Laplace transform may also be used to relate the transfer function of the system to the state-space equations. Thus, taking the Laplace transforms of Equation 8.45, we have

sX ( s ) − x ( 0 ) = A   X ( s ) + B   U ( s )

(8.47a)



Y ( s ) = C   X ( s ) + D  U ( s )

(8.47b)

where

X ( s ) = L ( x ( t )) ,

U ( s ) = L ( u ( t ))

and

Y ( s ) = L ( y ( t )). (8.48)

Solving Equation 8.47a for X(s), we have

X ( s ) = [ sI − A ]

−1

[ x ( 0 ) + B   U ( s )] ,

(8.49)

and substituting in the output equation, we have

Y ( s ) = C [ sI − A ] x ( 0 ) + C [ sI − A ] B + D    U ( s ) . −1

−1

(8.50)

It is possible to identify a key term in Equations 8.49 and 8.50:

R ( s ) = [ sI − A ] = L ( r ( t )) −1

where r(t) = exp(–At). As a consequence of the Cayley–Hamilton theorem which states that every matrix satisfies its own characteristic equation, the exponential of a matrix is a finite degree polynomial function of the matrix. Thus, it follows that the matrix R(s) can be expressed as the summation of powers of A multiplied by scalar functions of the Laplace transform variable, s. Thus, in principle, X(s) may be expressed as

X ( s ) =  B   AB   A 2 B   …  A n −1 B  f ( s )

where f(s) is a vector of functions of the Laplace transform variable, s.

(8.51)

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The inverse of a matrix is obtained by dividing the adjoint of the matrix by its determinant. It may be recalled from the theory of determinants that the adjoint of a matrix is defined as the transpose of the cofactor matrix. The matrix R(s) plays a fundamental role in the evolution of the system response and is referred to as the resolvent matrix. The polynomial given by

∆ ( s ) = det ( sI − A )

(8.52)

Is, in fact, the denominator polynomial of the transfer function and is therefore the left-hand side of the characteristic equation which is given by

∆ ( s ) = det ( sI − A ) = 0.

(8.53)

Thus, in order to find the response of the system to a particular input u(t) and for a given set of initial conditions, the best approach would be to find the resolvent matrix and then its inverse Laplace transform to find the state transition matrix. Once the state transition matrix is known, the solution can be predicted for any combination of inputs, for initial conditions and for any B and C matrices. It is needless to add that an efficient algorithm is essential to calculate the resolvent matrix. Normally, it is customary to use Cramer’s rule for evaluating the determinant of the matrix. This is generally not suitable for programming on a computer. An algorithm suitable for computer implementation is one that repeatedly uses the trace function. The trace of matrix is defined as the sum of all the diagonal elements of a matrix. The matrix

C o =  B  AB  A 2 B  …  A n −1 B 

(8.54)

plays a key role in control engineering and is known as the controllability matrix. It is quite clear from the equation defining the response that if one wants to generate a set of inputs in order to obtain a desired state vector as the state response, then the controllability matrix must be inverted to obtain a suitable solution for r(t). This is the basis for the controllability condition which requires that the controllability matrix be of full rank. The concept of observability is analogous to controllability. Loosely, a system is said to be observable if the initial state of the system can be determined from suitable measurements of the output. This is an important property that can be exploited in the development of filters for deriving information about the internal states of the system from a measurement of the output. From a practical point of view, it is desirable to know whether the measurement of the system output would provide all the information about the system states or if there are any system modes hidden from the observation. Observability guarantees that the reconstruction of state variables that cannot be measured directly, due to the limitations of the sensors used for measurement, is feasible.

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8.4.14 State-Space and Transfer Function Equivalence It may have already been noted by the reader that the state-space and transfer function representations are equivalent for linear time-invariant systems. Given a state-space representation, it is an easy to obtain the transfer function, and this is a unique relationship. However, given a transfer function, the internal dynamics of the system may be represented by an infinite number of state-space realisations as state-space realisations corresponding to a transfer function are not unique. In the latter case, it is important to construct a minimal realisation that is a realisation based on the minimum number of states. If we assume that the vector of initial conditions x(0) is set equal to zero, the I/O relationship in the Laplace transform domain is given by

−1 Y ( s ) = C [ sI − A ] B + G    U ( s ) .

(8.55)

Hence, the transfer function relating the output to the input is given by

H ( s ) = C [ sI − A ] B + G  . (8.56) −1

The transfer function is the ratio of the Laplace transform of the output to that of the input for zero initial conditions. The transfer function is of the form H (s) =



N (s)

( det [ sI − A ])

. (8.57)

The characteristic equation is

det [ sI − A ] = a0 s n + a1 s n −1 +  + an − 2 s 2 + an −1 s + an = 0. (8.58)

The roots of the characteristic polynomial are the poles of the transfer function, and they determine the nature of the response of the system to an input. The zeros of the transfer function are given by the roots of equation, N ( s ) = 0. The response of a linear system to a disturbance is given by the complimentary solution and is the weighted sum of exponentials of the form exp ( − pi t ), where pi are the poles of the transfer function. Thus, they must have negative real parts of sufficient magnitude to guarantee that the disturbance response decays to zero asymptotically as the time, t, tends to infinity.

8.4.15 Transformations of State Variables The fact that the state vector is a set of variables used to describe the internal dynamics of the system and implies that the state vector is essentially non-unique. Hence, it is always possible to transform the state variables from one set to another using a

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transformation, which may be linear or non-linear. Linear transformations preserve the linearity properties of the system. As an example, we consider the application of a transformation to Equation 8.45. Consider the transformation of state variables defined by x = Tz. (8.59)



Thus, Equation 8.45 may be expressed as

d Tz ( t ) = ATz + Bu, dt

y = CTz + Gu. (8.60)

Assuming T is independent of time and multiplying Equation 8.60 by its inverse,

d z ( t ) = T −1 ATz + T −1 Bu, dt

y = CTz + Gu. (8.61)

Thus, Equation 8.45 transforms another set of state-space equations, where A transforms to T–1ATz, B to T–1B and C to CT, while the matrix G is unchanged. It is worth noting that transformations such as the aforementioned are known as similarity transformations and generally preserve the key properties of the system in relation to stability and control. In particular, the characteristic equation of the system remains unchanged.

8.4.16 Design of a Full-State Variable Feedback Control Law If one wishes to transform a single input system described in one set of n states by

d x1 ( t ) = A1 x1 + B1 u, dt

y ( t ) = C1 x1 + Gu



(8.62)

to a system described in another set of n states

d x 2 ( t ) = A 2 x 2 + B 2 u, dt

y ( t ) = C 2 x 2 + Gu



(8.63)

and if A2 and B2 are prescribed, then one constructs the controllability matrices

C x1 =  B1 

A1 B1

A12 B1



A1n B1  , 

(8.64a)



C x 2 =  B2 

A2 B2

A 22 B2



A n2 B2  . 

(8.64b)

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It can be shown that the transformation relating the two sets of states satisfies

C x 2 = T −1 C x1 .

(8.65)

T = C x1 C −x12 .

(8.66)

Hence, it follows that

One may construct the observability matrices when A2 and C2 are prescribed, in a similar way replacing the B matrices in the controllability matrix expressions by the transpose of the corresponding C matrices. It can be shown that the transformation relating the two observability matrices satisfies a linear relationship, and the transformation relating the two sets of states can be obtained by solving a linear equation. The ability to transform the state-space representations from one set of internal states to another is extremely useful in the design of control systems to shape the dynamic response of the closed-loop system. In control systems design, one transforms the system to a certain canonical form, which facilitates the design of the control laws and the transforms back to the original variables to complete the design. For example, if the coefficient in Equation 8.58, a0 = 1 and n = 4, the control canonical form for a fourth-order system takes the form  0  0 A 2 = T −1 AT =   0  − a4 

1 0 0 − a3

0 1 0 − a2

  ,   

0 0 1 − a1

  B2 = T −1 B =    

0 0 0 1

     

(8.66)

where ai are the coefficients of characteristic Equation 8.58. Consider the corresponding state-space representation

d x 2 ( t ) = A 2 x 2 + B 2 u, dt

y ( t ) = C 2 x 2 + Gu.



(8.67)

 x2 , 

(8.68)

Given a control law of the form

u = v − K 2 x 2 = v −  k1

k2

k3

k4

the closed-loop system states are defined by

d x 2 ( t ) = {A 2 − B 2 K 2 } x 2 + B 2 v , dt

(8.69)

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where



 0  0 A2 − B2 K 2 =  0   − a4 − k1    B2 =    

1 0 0 − a3 − k2

0 1 0 − a2 − k3

  ,   

0 0 1 − a1 − k4

     

0 0 0 1

(8.70)

Given that the desired closed dynamics must be characterised by certain pole locations that correspond to a set of desired closed-loop characteristic polynomial coefficients, the desired closed-loop system may be expressed in control canonical form with



A d2 _ closed

 0  0 =  0  −ad 4 

1 0 0 − a3d

0 1 0 − a2d

0 0 1 − a1d

  ,   

  B2 =    

0 0 0 1

  .   

(8.71)

where aid are the desired coefficients of the closed-loop characteristic polynomial. Thus, the control gains satisfy the linear relations aid = ai + kn +1− i . Consequently, in the control canonical representation, the control gains are given by

K 2 = [ ki ] =  and+1− i − an +1− i  .

(8.72)

Transforming back to the original state-space representation using Equation 8.61,

K1 = K 2 T −1 . (8.73)

Thus, a full-state feedback control law may be easily constructed in the state-space domain.

8.4.17 Root Locus Method The root locus is a powerful approach for the design of a control law for stability and transient response. The root locus plot shows how changes in one of a system’s parameters (typically is loop gain) will change the closed-loop pole positions and thus change the system’s dynamic stability and closed-loop performance. It can

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also be used for determining the range of control loop gain for system stability and desired closed-loop performance. Consider a simple block diagram representation of a control system, as illustrated in Figure 8.27. The output of the closed-loop system is y(t) and is fed back. In the feedback path, we assume that it is modified by a filter with a transfer function H(D) or H(s). The closed-loop transfer function then is

T ( D) =

KG ( D ) 1 + KG ( D ) H ( D )

T (s) =

or

KG ( s ) , 1 + KG ( s ) H ( s )

(8.74)

when the ‘D’ operator is replaced by the Laplace transform variable, ‘s’. The root locus technique consists of plotting the closed-loop pole trajectories in the complex plane as the loop gain, K, varies. When the loop gain equals zero, the closed-loop poles coincide with the open-loop poles. On the other hand, when the loop gain is very high, the closed-loop poles coincide with the open-loop zeros. Thus, based on the pole and zero distributions of an open-loop system, the stability of the closed-loop system can be ascertained as a function of one scalar parameter, the loop gain, K. The plot can be employed to identify the loop gain value associated with a desired set of closed-loop poles. The case of a constant loop gain is the simplest one to consider. The proportional controller generally reduces the rise time, increases the overshoot and reduces the steady-state error. An integral controller also decreases the rise time and increases both the overshoot and the settling time but eliminates the steady-state error. On the other hand, the derivative controller reduces both the overshoot and the settling time but does not eliminate the steady-state error. However, the integral controller is destabilising when used on its own. For this reason, when it is essential to reduce the steady-state error, it is combined with the proportional controller, and the proportional–integral (PI) controller takes the form



K (s) = K

(s + T ) −1 I

s

(8.75)

where TI is the integral control time constant, which must be adjusted to obtain the desired rate of elimination of the steady-state error. In practice, it is essential to reduce the proportional gain, K, because the integral controller also reduces the rise time and increases the overshoot as the proportional controller does. When the steady-state error is not an issue, the proportional gain is combined with the derivative controller, and the PD controller takes the form

K ( s ) = K ( Td s + 1) .

(8.76)

The root locus method may be employed to design both these types of controllers when it is initially possible to make a suitable choice of either TI or Td.

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8.4.18 Root Locus Principle We will consider in this section how the roots of the characteristic equation can influence the stability and dynamic response of a closed-loop control system. The objective of feedback is to influence the characteristic equation in such a way that the roots of the closed-loop characteristic equation are in desired locations in the complex plane; that is the closed damping and natural frequencies are as selected by the control system designer. The closed-loop characteristic equation is obtained from

1 + KG ( D ) H ( D ) = 0.

(8.87)

Equation 8.77 is the return difference equation discussed in Section 8.4.2. Assuming that H(D) is known a priori, it is natural that we try and plot these roots as K varies from zero to infinity, so we may choose the gain, K, in an optimum way. The locus of the roots of the closed-loop characteristic equation as K varies from zero to infinity is the root locus plot.

8.4.19 Root Locus Sketching Procedure Most text books of classical control theory such as Ogata [2] discuss the rules of sketching the root loci. These rules are based on explicitly identifying where the loci are terminated, where they begin, the directions of the loci at these points, the directions of the asymptotes and the number of loci one can expect. However, as computational tools such as MATLAB are extensively available, root loci are no longer sketched but directly plotted from the roots of the characteristic polynomials, which are solved numerically.

8.4.20 Producing a Root Locus Using MATLAB® Considering a closed-loop third-order control system, the forward path transfer function and the feedback path transfer function are

H (s) =

1

( s + 3)

.

(8.89)

The characteristic equation needs to be written as

1+ K

( s + 1) = 0. s ( s + 2 )( s + 3)

(8.90)

The general form of the characteristic equation necessary for the application of the rlocus command in MATLAB® is

1+ K

p (s) =0 q (s)

(8.91)

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FIGURE 8.32  Root locus plot for the return difference equation defined by Equation 8.90.

where K is the parameter of interest to be varied from 0 ≤ K ≤ ∞. The steps involved in producing a root locus plot may be found by using the MATLAB® help command. A root locus plot corresponding to Equation 8.90 is shown in Figure 8.32. The MATLAB® function rlocfind can now be used to find the value of K corresponding to a pair of complex roots. rlocfind(p, q) %: This function needs to be used after the root locus has been plotted, and it needs to be in an active MATLAB® figure window in order to manipulate the cross-hair marker that locates the point of interest. The value of the parameter K and the value of the selected point will then be displayed.

8.4.21 Application of the Root Locus Method: Unity Feedback with a PID Control Law Consider a plant with a second-order transfer function and a unity feedback loop around it. The plant transfer function is assumed to be

G (s) =

(

K . s 2 + 2ω pζ ps + ω 2p

)

(8.92)

The controller transfer function is assumed to be obtained from a PID control law given by

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C (s) = KP +



K1 1   + K Ds = K P  1 + τ Ds   s τI s

(8.93)

where KI =



KP τI

K D = K Pτ D .

and



(8.94)

The PID control law may also be expressed as



C (s) = KP +

 s 2 + 2ζcω c s + ω c2  KI K s2 + K Ps + K I + K Ds = D = K Pτ D   (8.95) s s s 

where



1 τ DτI

ωc =

ζc =

and

1 = 2ω c τ D

τI . (8.96) 4τ D

The closed-loop transfer function is

Gc ( s ) =

C (s)G (s) = 1 + C (s)G (s)

 s 2 + 2ζc ω c s + ω c2  K KPτD   2 s   s + 2ω pζ p s + ω 2p

(

)

 s 2 + 2ζc ω c s + ω c2  K 1 + KPτD   s 2 + 2ω ζ s + ω 2 s  p p p

(

. (8.97)

)

Simplifying the closed-loop transfer function,



Gc ( s ) =

( s ( s + 2ω ζ s + ω ) + KK

KK P τ D s 2 + 2ζc ω c s + ω c2

2

2 p

p p

τ

P D

(s

2

)

+ 2ζc ω c s + ω c2

)



(8.98)

The steady-state response of the unit step command input to the closed-loop system may be found from the final value theorem and is

KK P τ Dω c2  G (s)  yss = lt y ( t ) = lt s  c  = lt Gc ( s ) = = 1. t→0 s→ 0  s  s→ 0 KK P τ Dω c2

(8.99)

Moreover, the closed-loop characteristic equation is

(

)

s 3 + ( 2ω pζ p + KK P τ D ) s 2 + ω 2p + 2 KK P τ Dζcω c s + KK P τ Dω c2 = 0. (8.100)

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The desired closed-loop characteristic polynomial is



(s

2

+ 2ζd ω d s + ω 2d

)(s + z

d

)

(8.101) = s 3 + ( 2ζd ω d + zd ) s 2 + ω 2d + 2ζd ω d zd s + ω d2 zd .

(

)

Thus, comparing coefficients, of the two polynomials in Equations 8.100 and 8.101, the controller parameters K D = τDKP, ωc and ζc may be chosen so that the closed poles lie at desired locations. However, when τI and τD can be selected a priori, only the gain KP needs to be selected, and this can be done by plotting the root locus of the return difference equation



1 + C ( s ) G ( s ) = 1 + KK P τ D

(s s (s

)

2

+ 2ζc ω c s + ω c2

2

+ 2ω pζc s + ω 2p

)

= 0. (8.102)

The method can be applied when the plant is of any order and not just second order. If the plant is of any order, the return difference equation is

1 + C ( s ) G ( s ) = 1 + K rl

(s

2

+ 2ζcω c s + ω c2 s

) G ( s ) = 0.

(8.103)

To select the controller constants τI and τD, one may employ the Ziegler–Nichols tuning rules. There are two sets of tuning rules based on two independent methods: the quarter decay ratio method and the ultimate sensitivity method. In practice, only one of the two methods can be applied. In the quarter decay ratio method, the application of the tuning rules should give a decay ratio, the ratio of the magnitudes of two consecutive peaks of an oscillation, equal to 0.25. To obtain the rules, consider a step input to the open-loop system. The lag L when the system begins to respond is noted. The slope R, which is the average slope of the system response during its rise to steady state, is noted. Based on the values of L and R, we may choose the controller parameters. For a purely proportional gain controller, KP = 1/RL. For a PI controller, KP = 0.9/RL and τI = L/0.3; for a PID controller, KP = 1.2/RL, τI = 2L and τD = 0.5L. These rules should then roughly give a decay ratio of 0.25. If not, the gain KP is adjusted by using the root locus method. Considering the ultimate sensitivity method, the gain KP is adjusted to value KP = KO such that the system is set in oscillation. If necessary, all parameters contributing to damping of the oscillations are set to zero. When the system is set in continuous oscillations, the root locus plot has to cross the imaginary axis at a point other than zero. The point on the imaginary axis gives the natural frequency of continuous oscillations ω O. The time period of the oscillations is noted and is denoted by TO = 2π/ωO. For a PI controller, τI = TO /1.2; for a PID controller, τI = TO /2 and τD = TO /8. The gain K P is set initially to K P = 0.5KO and then adjusted by using the root locus method.

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8.5  OPTIMAL CONTROL OF FLIGHT DYNAMICS The design of the control laws to optimise a performance criterion is done by minimising cost function

J LQR

∞

1 = 2

∫ 0

( x Qx + u RU ) dt = 12 T

T

∞

∫ (x

Q

+ u R ) dt.

(8.104)

0

∞

∫

The first term 1/2  x Q   dt represents the energy of the state response and provides 0

the energy of a combination of the states to assess the performance. The second term ∞

∫

in the cost function 1 / 2  u R   dt represents the energy contained in the control signal 0

that is fed back into the plant by the controller. The linear quadratic regulator (LQR) uses a linear controller that minimises the quadratic performance cost functional of the states and control inputs. Thus, the optimal gain matrix K is chosen such that for a given continuous time state-space model, the state feedback control law u = –Kx minimises the quadratic cost function as

J LQR

1 = 2

∞

∫ ( x Qx + u RU ) dt T

T

(8.105)

0

subject to the state vector satisfying the model’s dynamics equations x = Ax + Bu [2]. A first choice for the matrices Q and R in the expression for the cost function is given by Bryson’s rules: select Q and R as diagonal matrices with elements [3]

qii =

1 , Maximum expected value of xi2

(8.106a)



rii =

1 . Maximum expected value of ui2

(8.106b)

The solution to the optimal control problem may be conveniently expressed in terms of the constant Riccati matrix P, which is defined by d ( x T Px )



dt

= − x T ( Q + K T RK ) x.

(8.107)

Substituting the state feedback control input and using the aforementioned equation, ∞



J LQR

∞ 1 T 1 1 = x ( Q + K T RK ) x  dt = − ( x T Px ) = ( x T ( 0 ) Px ( 0 )) . (8.108) 0 2 2 2

∫ 0

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It then follows that JLQR has a minimum, which may be obtained by expanding Equation 8.107, and solving the resulting algebraic matrix Riccati equation,

AT P = PAT − PBR −1 BT P + Q = 0,

(8.109)

for P, by eigenvalue decomposition of the equations representing the optimal solution. The optimal gain is given by

K = R −1 BT P.

(8.110)

A crucial property of an LQR controller is that this closed loop is asymptotically stable as long as the system is at least controllable. Furthermore, LQR controllers are inherently robust with respect to process uncertainty. Moreover, the gain margin of an LQR controller is infinite for gain increase and −6 dB for gain decrease, and the phase margin is at least 60°. Thus, the LQR controllers have some very desirable robustness properties and can be tuned to meet other requirements and thus extremely well suited for the flight control system design.

8.5.1 Compensating Full-State Feedback: Observers and Compensators The entire theory of optimum full-state feedback appears to be too good to be true. Unfortunately, there is a serious problem when it comes to implementation. It was stated earlier that state variables are internal to the system and therefore not available for measurement. Thus, it may be only possible to measure fewer linear combinations of states than the total number of states present. In such a situation, it is not really possible to feed back all the states and there is no reason to believe that, in general, the measured linear combinations of states would correspond to the optimal fullstate control input. Thus, it is essential to compensate for the fact that all the states necessary for feedback are not measured. Such compensation can be provided by filtering the measurements and constructing additional outputs. There are several approaches for designing such compensating systems or compensators or filters. One approach is based on the concept of the observer. An observer is essentially a linear filter to which are fed as inputs both the outputs of and inputs to the system that is being compensated, that is the plant for which one is interested in designing a control system. The observer then reconstructs the desired states or linear combinations of states provided the state-space equations of the system are known. The outputs of the observer are then linearly combined with the system outputs with suitable gains to obtain the desired feedback inputs for controlling the system.

8.5.2 Observers for Controller Implementation When all the states of a system are not available for feedback, an electronic circuit known as an observer [4] is employed to generate additional feedbacks to supplement those available from the measurements. Rather than employing the states of the

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system for constructing the feedback control law, the actual available system outputs and the states of the observer are employed to construct a control law equivalent to the full-state control law. To compensate for the fact that all the states of the system which are assumed to be in the form

x = Ax + Bu,

y = Cx,



(8.111)

are not available for feedback, one may construct an observer which satisfies the equations z = Fz + Gy + Hu.



(8.112)

In the steady state, the observer states are related to the system model state vector by the linear transformation z = Sx.



(8.113)

where S is the constant matrix transformation. The full-state feedback control law is expressed in terms of the observer states, z, and the available outputs, y, as

u = u ′ − kx = u ′ − Ez − Ly.

(8.114)

Thus, only the states of the observer circuit and the measurements are employed for feedback.

8.5.3 Observer Equations To define the observer equations, the following steps must be followed:  i. Define the observer error vector as e = z – Sx and its derivative is e = z − Sx. Thus,



e = Fz + Gy + Hu − SAx − SBu = Fe + ( FS + GC − SA ) x + ( H − SB ) u.

(8.115)

To decouple the observer from the states and control, the observer must satisfy the general equations

SA − FS = GC,

H = SB, k = ES + LC.

(8.116)

The matrix F must have all its eigenvalues in the left half of the s-plane, for the observer error vector to be asymptotically stable.

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8.5.4 Special Cases: Full- and First-Order Observers In the special case when S = I, one has a full-order observer. In this case, the observer’s state equations are

z = ( A − GC ) z + Gy + Bu,

u = u ′ − kz.



(8.117)

One can compare and contrast the full-state-feedback- and full-order-observer-based closed-loop systems. Consider the special case when the rank of the matrix rank [C CA]T = n. In this case, a first-order observer will suffice; that is F and z are scalars. The first two observer equations in this case may be solved easily for an arbitrary choice of F = f, a scalar:

SA − FS = SA − Sf = GC,

S = ( A − If ) GC, −1

H = SB. (8.118)

The observer is a minimum-order observer designed to directly estimate the matrices E and L and hence obtain the control law in terms of the elements of the control gain vector, k.

8.5.5 Solving the Observer Equations Consider the second-order observer in the first instance. Assume that F is a 2 × 2 matrix and that its characteristic equation is

s2 s + 2ζn + 1 = 0. ωn ω 2n

λF (s) =

(8.119)

To solve the first of the observer Equation 8.132, the equation is pre- and post-multiplied by the matrices F and A, respectively. The process is repeated several times to show that i −1



SA − F S = i

i

∑F GCA j

i −1− j

.

(8.120)

j=0

However, as a consequence of the Cayley–Hamilton theorem, the matrix F satisfies its own characteristic polynomial, and hence,

λF (F) =

F2 F + 1 = 0. (8.121) 2 + 2ζ n ωn ωn

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It follows that

Sλ F ( A ) − λ F ( F ) S = Sλ F ( A ) =

2ζn FGC + GCA . GC + ωn ω n2

(8.122)

Hence, solving for S and rearranging as a product of matrices, the solution is



S =  G

 FG  C   

( 2ζn ω n I + A ) / ω 2n I / ω 2n

  λ −F1 ( A ) .  

(8.123)

Once S is determined, the observer’s state equation may be obtained and the feedback control law follows from the equation for the gain vector k. The method can be generalised to higher order observers.

8.5.6 Luenberger Observer When one seeks to seeks to only estimate the states that cannot be measured, only some of the elements of the state vector need to be estimated. In this case, the observer is the classical reduced-order Luenberger observer for estimating some elements of the state vector.

8.5.7 Optimisation Performance Criteria In the design of control systems, it becomes essential to define what constitutes a good response to a disturbance input. Thus, one can expect a certain type of response from a system following a disturbance, provided the control system has been designed to deliver the desired transient performance characteristics. Such characteristics in the time domain are as follows:

1. The maximum overshoot expressed as a percentage of the magnitude of the step input 2. The rise time is the time required for the system response to rise from 0% to 95% of the magnitude of the step input 3. The time constant refers to the time required for the system response to rise from 0% to 63.2% (= 100(1 – exp(−1))%) of the magnitude of the step input 4. The settling time is the time required for the system response to rise from to within 5% of the steady-state response 5. The number of oscillations that occurs over the settling time period 6. Integrated error performance measures can be expressed in terms of integrals of the error between the actual and desired outputs as ∞



I nm =

∫ e (t ) 0

n

t m dt ,

n = 1, 2,

m = 0,1

(8.124)

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Flight Dynamics, Simulation, and Control

In the frequency domain, certain characteristics may be defined representing a good response. These are 1. The peak magnification ratio of the frequency response amplitude which should not exceed certain limits for good transient behaviour 2. The bandwidth defined as the frequencies outside which the dynamic magnification ratio of the frequency response amplitude falls more than 3 dB from its peak value 3. The frequency at which the peak magnification occurs or the resonance frequency, if any 4. The drop rate or the rate at which the magnification falls beyond the bandwidth 5. The gain and phase margins Many of these performance criteria impose conflicting requirements, and these requirements must often be met in an optimal sense. In addition, in designing flight control systems, one must consider performance requirements imposed on the system to ensure the aircraft possesses good handling qualities, which are discussed in the next section.

8.5.8  Good Handling Domains of Modal Response Parameters Approximately 25 years ago, researchers at the U.S. NASA’s then Dryden Flight Research Center applied the military standard, MIL-F-8785B, criteria to data pertaining to two aircraft, YF-12 and XB-70. However, it is probably fair to state that pilots are the best judges as to how well an aircraft handles or performs. While it is quite difficult to quantify pilot opinion, a number of techniques have evolved that allow pilot opinion to be translated and quantified and expressed in quantitative terms. Based on these techniques, close relationships, useful in design, have been established between the small perturbation characteristics of aircraft dynamic modes and what the pilot regards as desirable handling or flying qualities. The flying quality requirements differ for different classes of aircraft and are defined for four classes of aircraft. Small light aeroplanes are class I, medium-weight aeroplanes are class II, big heavy aeroplanes are class III and high manoeuvrability aeroplanes are class IV. Class I aircraft represents general aviation aircraft, class II represents aircraft that do not belong to class I or III in terms of their weight, class III represents transport aircraft and class IV represents fighter aircraft. Thus, aircraft may be classed into four classes, as indicated in Table 8.5. The flying quality requirements are different for different phases of the flight. The phases of an aircraft’s flight are thus categorised, and three categories of flight are TABLE 8.5 Aircraft classes Class I II

Characteristic

Class

Characteristic

Low mass,  30,000 kg Highly manoeuvrable

Aircraft Flight Control

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defined: category A for rapid manoeuvring and precision tracking flight, category B for slow but gradual manoeuvring flight not requiring precision flight path control as in cruise flight and category C for slow but gradual manoeuvring flight requiring precision flight path control. Category A concerns non-terminal flight phases that require precise flight path control. Category B is about non-terminal flight phases that require less precise tracking and less accurate flight path control as in climb, descent and cruise. Finally, category C relates to terminal flight phases during take-off and landing. Aircraft designers must demonstrate compliance with the requirements. This is done by using flight tests where the flying quality of the aircraft is rated on a scale of 1–10 based on the Cooper–Harper (C–H) scale. In this scale, a rating of 1 means the aircraft has excellent handling qualities, and the pilot workload is low. On the other hand, a rating of 10 means that there are major deficiencies in the handling quality of the aircraft. A test is not performed for just the aircraft but for the combination of the aircraft and the control system. Flight requirements are generally specified for three levels of flying quality. Also, the levels of flying quality requirements are established depending on the level of pilot workload. Level 1 refers to flying qualities adequate for the mission flight phase, Level 2 also refers to flying qualities adequate for the mission flight phase but with an increase in the pilot’s workload and/or a degradation in mission effectiveness, and Level 3 refers to a degraded level of flying qualities while ensuring that the aircraft is still controllable but with inadequate mission effectiveness and high pilot workload. Level 1 means that the flying qualities are clearly adequate for the respective flight phases. Level 2 means that the flying qualities are adequate, but one can expect an increase in pilot’s workload and/or degradation in mission effectiveness. In Level 3, the flying qualities are degraded. However, the aeroplane can still be controlled, albeit with inadequate mission effectiveness and a high or limiting pilot workload. Aeroplanes are normally designed to satisfy Level 1 flying quality requirements with all systems in their normal operating state. The modal requirements may be easily stated for each of these classes of aircraft and categories of flight. These requirements are summarised in Table 8.6. There are, in fact, a number of other requirements such as the static stability margins and performance metrics for good handling and good behaviour.

8.5.9 Cooper–Harper Rating Scale In the military standard, pilot subjective flying quality ratings are quantified in terms of C–H ratings (enunciated first by Cooper and Harper [5]). These are based on certain definitions of pilot performance that are essential to quantify pilot opinion. Pilot opinion is dependent on their skills, backgrounds and experience, and the assessments of pilot compensation required in various instances can vary. Yet, a statistical evaluation of several skilled pilots provides a reasonably good indicator of pilot opinion. The C–H rating scale (and its predecessor the Cooper scale) is a numerical scale ranging from 1 to 10, with 1 being the best rating and 10 the worst. The C–H ratings are split into the three handling quality levels, where the correspondence between the C–H ratings and levels is listed in Table 8.7. In practice, the C–H ratings from 1 to 3 are referred to as Level 1, ratings from 4 to 6 as Level 2, and 7 to 9 as Level 3.

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TABLE 8.6 Modal damping and response time requirements based on good handling Mode

Class/category/level

Min. damping

Phugoid

Level 1 Level 2 Level 3

0.04–0.15 0 Negative (?)

Short period

A, Level 1 A, Level 2 A, B, Level 3 B, Level 1 B, Level 2 C, Level 1 C, Level 2 C, Level 3 I, IV, A/C, Level 1 I, IV, A/C, Level 1 II, III, A/C, Level 1 I, II, III, IV, B, Level 1 II, III, A/C, Level 2 I, II, III, IV, B, Level 2 Level 3 A/C, Level 1 A/C, Level 2 A/C, Level 3 B, Level 1 B, Level 2 B, Level 3 I, IV, A, Level 1 II, III, A, Level 1 B, Level 1 I, IV, C, Level 1 II, III, C, Level 1 Level 2 Level 3

Roll subsidence

Spiral

Dutch roll

0.35–1.3 0.25–2 0.1 0.3–2 0.2–2 0.5–1.3 0.35–2 0.25

0.19 0.19 0.08 0.08 0.08 0.02 0.0

TABLE 8.7 Three-level classification of C–H ratings Level 1 2 3

C–H rating C–H