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i i PRENTICE HALLINTERNATIONAL SERIES IN SYSTEMSAND - CONTROL ENGINEERING SERIF^ 'DITOR: M.J. GRIMBLEDONALD McLEAN
Automatic FlightControl Systems
Prentice Hall InternationalSeries in Systems and Control EngineeringM. J. Grimble, Series EditorBANKS,S. P., Control Systems Engineering: Modelling and Simulation, Control Theory and Microprocessor ImplementationBANKSS, . P., Mathematical Theories of Nonlinear SystemsBENNETTS,., Real-time Computer Control: An IntroductionCEGRELLT,. , Power Systems ControlCOOKP, . A., Nonlinear Dynamical SystemsLUNZEJ,., Robust Multivariable Feedback ControlPATTON,R., CLARKR, . N., FRANKP, . M. (editors), Fault Diagnosis in Dynamic SystemsSODERSTROTM., and STOICAP,., System IdentiJicationWARWICKK,., Control Systems: An Introduction
Automatic FlightControl Systems Donald McLean Westland Professor of Aeronautics University of Southampton, UK PRENTICE HALLNew York . London . Toronto - Sydney. Tokyo. Singapore
a First published 1990 by Prentice Hall International (UK) Ltd 66 Wood Lane End, Heme1 Hempstead Hertfordshire HP2 4RG A division of Simon & Schuster International Group @ 1990 Prentice Hall International (UK) Ltd All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission, in writing, from the publisher. For permission within the United States of America contact Prentice Hall Inc., Englewood Cliffs, NJ 07632. Typeset in 10112pt Times by Columns of Reading Printed and bound in Great Britain at the University Press, Cambridge Library o f Congress Cataloging-in-PublicationData McLean, Donald. 1 9 3 6 Automatic flight control systems I by Donald McLean. p. cm. Bibliography: p. Includes index. ISBN 0-13-054008-0: $60.00 1. Flight control. I. Title. TL589.4.M45 1990 629.132'6 - - dc20 89-22857 CIP British Library Cataloguing in Publication Data McLean, D. (Donald, 1936) Automatic flight control systems. 1. Aircraft. Automatic flight control systems I. Title 629.135'2 ISBN CL13-054008-0
ContentsPreface 1 Aircraft Flight Control 1.1 Introduction 1.2 Control Surfaces 1.3 Primary Flying Controls 1.4 Flight Control Systems 1.5 Brief History of Flight Control Systems 1.6 Outline of the Book 1.7 conclusions 1.8 Note 1.9 References 2 The Equationsof Motion of an Aircraft Introduction Axis (Coordinate) Systems The Equations of Motion of a Rigid Body Aircraft Complete Linearized Equations of Motion Equations of Motion in Stability Axis System Equations of Motion for Steady Manoeuvring Flight Conditions Additional Motion Variables State and Output Equations Obtaining a Transfer Function from State and Output Equations Important Stability Derivatives The Inclusion of the Equations of Motion of Thrust Effects Conclusions Exercises Notes References 3 Aircraft Stability and Dynamics 3.1 Introduction 3.2 Longitudinal Stability
vi Contents Static Stability Tranfer Functions Related to Longitudinal Motion Transfer Functions Obtained from Short Period Approximation Transfer Functions Obtained from Phugoid Approximation Lateral Stability Transfer Functions Related to Lateral Motion Three Degrees of Freedom Approximations Two Degrees of Freedom Approximations Single Degree of Freedom Approximation State Equation Formulation to Emphasize Lateral/ Directional Effects Conclusions Exercises Notes References 4 The Dynamic Effects of Structural Flexibility Upon the Motion of an Aircraft Introduction Bending Motion of the Wing Torsion of the Wing Coupled Motions The Dynamics of a Flexible Aircraft Mathematical Representation of the Dynamics of a Flexible Aircraft Lift Growth Effects Bending Moments Blade Flapping Motion Conclusion Exercises Notes References 5 DisturbancesAffecting Aircraft Motion Introduction Atmospheric Disturbances A Discrete Gust Function Power Spectral Density Functions Continuous Gust Representations State Variable Models Angular Gust Equations The Effects of Gusts on Aircraft Motion
Contents 5.9 Transient Analogue 5.10 Determination of the r.m.s. Value of Acceleration as a Result of Encountering Gusts 5.11 Wind Shear and Microbursts 5.12 Sensor Noise 5.13 Conclusions 5.14 Exercises 5.15 References6 Flying and Handling Qualities 6.1 Introduction 6.2 Some Definitions Required for Use with Flying Qualities' Specifications 6.3 Longitudinal Flying Qualities 6.4 LateraVDirectional Flying Qualities 6.5 The C* Criterion 6.6 Ride Discomfort Index 6.7 Helicopter Control and Flying Qualities 6.8 Conclusions 6.9 Exercises 6.10 References 7 Control System Design Methods I AFCS as a Control Problem Generalized AFCS Conventional Control Methods Parameter Optimization Conclusions Exercises Note References 8 Control System Design Methods I1 Introduction The Meaning of Optimal Control Controllability, Observability and Stabilizability Theory of the Linear Quadratic Problem Optimal Output Regulator Problem State Regulators with a Prescribed Degree of Stability Explicit Model-Following Optimal Command Control System Use of Integral Feedback in LQP
viii Contents 8.10 State Reconstruction 8.11 Conclusions 8.12 Exercises 8.13 Notes 8.14 References 9 Stability Augmentation Systems Introduction Actuator Dynamics Sensor Dynamics Longitudinal Control (Use of Elevator Only) Other Longitudinal Axis SASS Sensor Effects Scheduling Lateral Control Conclusions Exercises Notes Reference10 Attitude Control Systems Introduction Pitch Attitude Control Systems Roll Angle Control Systems Wing Leveller Co-ordinated Turn Systems Sideslip Suppression Systems Directional Stability During Ground Roll Conclusions Exercises Notes ReferencesI 1 Flight Path Control Systems Introduction Height Control Systems Speed Control Systems Mach Hold System Direction Control System Heading Control System VOR-Coupled Automatic Tracking System ILS Localizer-Coupled Control System ILS Glide-Path-Coupled Control System
Contents 11.10 Automatic Landing System 11.11 A Terrain-Following Control System 11.12 Conclusions 11.13 Exercises 11.14 Notes 11.15 References12 Active Control Systems Introduction ACT Control Funtions Some Benefits Expected from ACT Gust Alleviation Load Alleviation System for a Bomber Aircraft A Ride Control System for a Modern Fighter Aircraft Aircraft Positioning Control Systems Conclusions Exercises Note References13 Helicopter Flight Control Systems Introduction Equations of Motion Static Stability Dynamic Stability Stability Augmentation Systems Conclusions Exercises Notes References14 Digital Control Systems Introduction A Simple Discrete Control System A Data Hold Element The z-Transform Bilinear Transformations Discrete State Equation Stability of Digital Systems Optimal Discrete Control Use of Digital Computers in AFCSs Conclusions
Contents 14.11 Exercises 14.12 Notes 14.13 References15 Adaptive Flight Control Systems Introduction Model Reference Systems The MIT Scheme Example System A Lyapunov Scheme Parameter Adaptation Scheme Conclusions Notes ReferencesAppendicesA Actuators and Sensors A.1 Introduction A.2 Actuator Use in AFCSs A.3 Actuators A.4 Sensors A.5 Accelerometers A.6 Angle of Attack Sensor A.7 ReferencesB Stability Derivatives for Several Representative Modern Aircraft B.1 Nomenclature B.2 Aircraft DataC Mathematical Models of Human Pilots C.1 Introduction C.2 Classical Models C.3 References
PrefaceThis is an introductory textbook on automatic flight control systems (AFCSs) forundergraduate aeronautical engineers. It is hoped that the material and themanner of its presentation will increase the student's understanding of the basicproblems of controlling an aircraft's flight, and enhance his ability to assess thesolutions to the problems which are generally proposed. Not every method ortheory of control which can be used for designing a flight controller is dealt within this book; however, if a reader should find that some favourite technique orapproach has been omitted, the fault lies entirely with the author upon whosejudgement the selection depended. The method is not being impugned by itsomission. Before understanding how an aircraft may be controlled automatically inflight it is essential to know how any aircraft will respond dynamically to adeliberate movement of its control surfaces, or to an encounter with unexpectedand random disturbances of the air through which it is flying. A sound knowledgeof an aircraft's dynamic response is necessary for the succesful design of anyAFCS, but that knowledge is not sufficient. A knowledge of the quality of aircraftresponse, which can result in the aircraft's being considered by a pilot assatisfactory to fly, is also important. In this book the first six chapters are whollyconcerned with material relevant to such important matters. There are now so many methods of designing control systems that it wouldrequire another book to deal with them alone. Instead, Chapters 7 and 8 havebeen included to provide a reasonably self-contained account of the mostsignificant methods of designing linear control systems which find universal use inAFCSs. Emphasis has been placed upon what are spoken of as modern methodsof control (to distinguish them from the classical methods): it is most unlikely thattoday's students would not consider the use of a computer in arriving at therequired solution. Being firmly based upon time-domain methods, modern controltheory, particularly the use of state equations, is a natural and effective techniquefor use with computer aided engineering and harmonizes with the mathematicaldescription of the aircraft dynamics which are most completely, and conveniently,expressed in terms of a state and an output equation. The form involved leadsnaturally to the use of eigenvalues and eigenvectors which make consideration ofthe stability properties of the aircraft simple and straightforward. Since computersare to be used, the need for normalizing the dynamic equations can be dispensedwith and the differential equations can be solved to find the aircraft's motion inreal time. The slight cost to be borne for this convenience is that the stabilityderivatives of the aircraft which are used in the analysis are dimensional;
xii Prefacehowever, since the aircraft dynamics are in real time, the dynamics of the flightcontroller, the control surface actuators, and the motion sensors can also be dealtwith in real time, thereby avoiding the need for cumbersome and unnecessarytransformations. Since dimensional stability derivatives were to be used, theAmerican system of notation for the aircraft equations of motions was adopted:most papers and most data throughout the world now use this system. Chapters 9 to 11 relate to particular modes of an AFCS, being concernedwith stability augmentation systems, attitude and path coitrol systems. Aparticular AFCS may have some, or all, of these modes involved in its operation,some being active at all times in the flight, and others being switched in by thepilot only when required for a particular phase of flight. Although helicopter flightcontrol systems do not differ in principle from those used with fixed wing aircraft,they are fitted for different purposes. Furthermore, both the dynamics and themeans of controlling a helicopter's flight are radically different from fixed wingaircraft. Consequently, helicopter AFCSs are dealt with wholly in Chapter 13 toemphasize the distinctive stability and handling problems that their use isintended to overcome. Active control systems are dealt with in Chapter 12 and only a brieftreatment is given to indicate how structural motion can be controlledsimultaneously, for example, with controlling the aircraft's rigid body motion.Ride control and fuselage pointing are flight control modes dealt with in thischapter. In the thousands of commercial airliners, the tens of thousands of militaryaircraft, and the hundreds of thousands of general aviation aircraft which areflying throughout the world today, examples of the types of AFCS discussed inthis book can easily be found. But most modern AFCSs are digital, and toaccount for this trend Chapter 14 has been added to deal solely with digitalcontrol methods. The consequences for the dynamic response of the closed-loopsystem of implementing a continuous control law in a digital fashion isemphasized. Results complementary to those in Chapters 9 to 11, obtained usingwholly digital system analysis, are also shown. The final chapter deals briefly with the subject of adaptive flight controlsystems, and three appendices provide a summary of information relating toactuators, sensors, aircraft stability data, and human operators. In writing a textbook, ideas and techniques which have been used effectivelyand easily by the author over the years are discussed and presented, but theoriginal source is often forgotten. If others find their work used here butunacknowledged, please be assured that it was unintentional and has occurredmostly as a result of a middle-aged memory rather than malice, for I am consciousof having had many masters in this subject. At the risk of offending manymentors, I wish to acknowledge here only the special help of three people, for thelist of acknowledgements would be impossibly long otherwise. Two are Americanscholars: Professors Jack d'Azzo and Dino Houpis, of the United States Air ForceInstitute of Technology, in Dayton, Ohio. They are nonpareil as teachers ofcontrol and taught me in a too-short association the importance of the student and
Preface xiiihis needs. The other is my secretary, Liz Tedder, who now knows, to her lastingregret, more about automatic flight control systems than she ever wished to know. D. McLEAN Southampton
Aircraft Flight Control1.I INTRODUCTIONWhatever form a vehicle may take, its value to its user depends on how effectivelyit can be made to proceed in the time allowed on a precisely controllable pathbetween its point of departure and its intended destination. That is why, forinstance, kites and balloons find only limited application in modern warfare.When the motion of any type of vehicle is being studied it is possible to generalizeso that the vehicle can be regarded as being fully characterized by its velocityvector. The time integral of that vector is the path of the vehicle through space(McRuer et al., 1973). The velocity vector, which may be denoted as ;6, is affectedby the position, x, of the vehicle in space by whatever kind of control, u, can beused, by any disturbance, 6 , and by time, t. Thus, the motion of the vehiclecan be represented in the most general way by the vector differential equation:where f is some vector function. The means by which the path of any vehicle canbe controlled vary widely, depending chiefly on the physical constraints whichobtain. For example, everyone knows that a locomotive moves along the rails ofthe permanent way. It can be controlled only in its velocity; it cannot be steered,because its lateral direction is constrained by the contact of its wheel rims on therails. Automobiles move over the surface of the earth, but with both speed anddirection being controlled. Aircraft differ from locomotives and automobilesbecause they have six degrees of freedom: three associated with angular motionabout the aircraft's centre of gravity and three associated with the translation ofthe centre of gravity.1 Because of this greater freedom of motion, aircraft controlproblems are usually more complicated than those of other vehicles. Those qualities of an aircraft which tend to make it resist any change ofits velocity vector, either in its direction or its magnitude, or in both, are whatconstitutes its stability. The ease with which the velocity vector may be changed isrelated to the aircraft's quality of control. It is stability which makes possible themaintenance of a steady, unaccelerated flight path; aircraft manoeuvres areeffected by control. Of itself, the path of any aircraft is never stable; aircraft have only neutralstability in heading. Without control, aircraft tend to fly in a constant turn. Inorder to fly a straight and level course continuously-controlling corrections mustbe made, either through the agency of a human pilot, or by means of an automatic
2 Aircraft Flight Controlflight control system (AFCS). In aircraft, such AFCSs employ feedback control toachieve the following benefits:1. The speed of response is better than from the aircraft without closed loop control.2. The accuracy in following commands is better.3. The system is capable of suppressing, to some degree, unwanted effects which have arisen as a result of disturbances affecting the aircraft's flight.However, under certain conditions such feedback control systems have a tendencyto oscillate; the AFCS then has poor stability. Although the use of high values ofgain in the feedback loops can assist in the achievement of fast and accuratedynamic response, their use is invariably inimical to good stability. Hence,designers of AFCSs are obliged to strike an acceptable, but delicate, balancebetween the requirements for stability and for control. The early aeronautical experimenters hoped to make flying easier byproviding 'inherent' stability in their flying machines. What they tried to providewas a basic, self-restoring property of the airframe without the active use of anyfeedback. A number of them, such as Cayley, Langley and Lilienthal, discoveredhow to achieve longitudinal static stability with respect to the relative wind, e.g.by setting the incidence of the tailplane at some appropriate value. Thoseexperimenters also discovered how to use wing dihedral to achieve lateral staticstability. However, as aviation has developed, it has become increasingly evidentthat the motion of an aircraft designed to be inherently very stable, is particularlysusceptible to being affected by atmospheric turbulence. This characteristic is lessacceptable to pilots than poor static stability. It was the great achievement of the Wright brothers that they ignored theattainment of inherent stability in their aircraft, but concentrated instead onmaking it controllable in moderate weather conditions with average flying skill. Sofar in this introduction, the terms dynamic and static stability have been usedwithout definition, their imprecise sense being left to the reader to determinefrom the text. There is, however, only one dynamic property - stability - whichcan be established by any of the theories of stability appropriate to the differentialequations being considered. However, in aeronautical engineering, the two termsare still commonly used; they are given separate specifications for the flyingqualities to be attained by any particular aircraft. When the term static stability isused, what is meant is that if a disturbance to an aircraft causes the resultingforces and moments acting on the aircraft to tend initially to return the aircraft tothe kind of flight path for which its controls are set, the aircraft can be said to bestatically stable. Some modern aircraft are not capable of stable equilibrium -they are statically unstable. Essentially, the function of static stability is to recoverthe original speed of equilibrium flight. This does not mean that the initial flightpath is resumed, nor is the new direction of motion necessarily the same as theold. If, as a result of a disturbance, the resulting forces and moments do not tendinitially to restore the aircraft to its former equilibrium flight path, but leave it inits disturbed state, the aircraft is neutrally stable. If it tends initially to deviate
Control Surfaces 3further from its equilibrium flight path, it is statically unstable. When an aircraft isput in a state of equilibrium by the action'of the pilot adjusting the controls, it issaid to be trimmed. If, as a result of a disturbance, the aircraft tends to returneventually to its equilibrium flight path, and remains at that position, for sometime, the aircraft is said to be dynamically stable. Thus, dynamic stability governshow an aircraft recovers its equilibrium after a disturbance. It will be seen laterhow some aircraft may be statically stable, but are dynamically unstable, althoughaircraft which are statically unstable will be dynamically unstable.1.2 CONTROL SURFACESEvery aeronautical student knows that if a body is to be changed from its presentstate of motion then external forces, or moments, or both, must be applied to thebody, and the resulting acceleration vector can be determined by applyingNewton's Second Law of Motion. Every aircraft has control surfaces or othermeans which are used to generate the forces and moments required to producethe accelerations which cause the aircraft to be steered along its three-dimensionalflight path to its specified destination. A conventional aircraft is represented in Figure 1.1.It is shown with theusual control surfaces, namely elevator, ailerons, and rudder. Such conventionalaircraft have a fourth control, the change in thrust, which can be obtained fromthe engines. Many modern aircraft, particularly combat aircraft, have consider-ably more control surfaces, which produce additional control forces or moments.Some of these additional surfaces and motivators include horizontal and verticalcanards, spoilers, variable cambered wings, reaction jets, differentially operatinghorizontal tails and movable fins. One characteristic of flight control is that therequired motion often needs a number of control surfaces to be usedsimultaneously. It is shown later in this book that the use of a single controlsurface always produces other motion as well as the intended motion. When morethan one control surface is deployed simultaneously, there often results Figure 1.1 Conventional aircraft.
Aircraft Flight Control ading edge (LE) slats canard q4 'Vertical canard Figure 1.2 A proposed control configured vehicle.considerable coupling and interaction between motion variables. It is this physicalsituation which makes AFCS design both fascinating and difficult. When theseextra surfaces are added to the aircraft configuration to achieve particular flightcontrol functions, the aircraft is described as a 'control configured vehicle' (CCV).A sketch of a proposed CCV is illustrated in Figure 1.2 in which there are showna number of extra and unconventional control surfaces. When such extra controlsare provided it is not to be supposed that the pilot in the cockpit will have anequal number of extra levers, wheels, pedals, or whatever, to provide theappropriate commands. In a CCV such commands are obtained directly from anAFCS and the pilot has no direct control over the deployment of each individualsurface. The AFCS involved in this activity are said to be active control technologysystems. The surfaces are moved by actuators which are signalled electrically (fly-by-wire) or by means of fibre optic paths (fly-by-light). But, in a conventionalaircraft, the pilot has direct mechanical links to the surfaces, and how hecommands the deflections, or changes, he requires from the controls is by meansof what are called the primary flying controls.1.3 PRIMARY FLYING CONTROLSIn the UK, it is considered that what constitutes a flight control system is anarrangment of all those control elements which enable controlling forces andmoments to be applied to the aircraft. These elements are considered to belong tothree groups: pilot input elements, system output elements and interveninglinkages and elements. The primary flying controls are part of the flight control system and aredefined as the input elements moved directly by a human pilot to cause an
Primary Flying Controls 5operation of the control surfaces. The main primary flying controls are pitchcontrol, roll control and yaw control. The use of these flight controls affectsmotion principally about the transverse, the longitudinal, and the normal axesrespectively, although each may affect motion about the other axes. The use ofthrust control via the throttle levers is also effective, but its use is primarilygoverned by considerations of engine management. Figure 1.3 represents thecockpit layout of a typical, twin engined, general aviation aircraft. The yoke is theprimary flying control used for pitch and roll control. When the yoke is pulledtowards, or pushed away from, the pilot the elevator is moved correspondingly.When the yoke is rotated to the left or the right, the ailerons of the aircraft aremoved. Yaw control is effected by means of the pedals, which a pilot pushes leftor right with his feet to move the rudder. In the kind of aircraft with the kind ofcockpit illustrated here, the link between these primary flying controls and thecontrol surfaces is by means of cables and pulleys. This means that theaerodynamic forces acting on the control surfaces have to be countered directly bythe pilot. To maintain a control surface at a fixed position for any period of timemeans that the pilot must maintain the required counterforce, which can be verydifficult and fatiguing to sustain. Consequently, all aircraft have trim wheels (seeFigure 1.3) which the pilot adjusts until the command, which he has set initiallyon his primary flying control, is set on the control surface and the pilot is thenrelieved of the need to sustain the force. There are trim wheels for pitch, roll andyaw (which is sometimes referred to as 'nose trim'). Dual r.p.m. gauge Magnetic compass Dual exhaustDual manifold pressure gaug gas temperature Fuel pressure WR fuel quantity Avionics cluster ropellors blade pitch controlsInstrument landing system (I Pitch trim wheel Figure 1.3 Cockpit layout.
Aircraft Flight Control , . .,m~ievator I,. . Aileron Figure 1.4 Control surface deflection conventions. In large transport aircraft, or fast military aircraft, the aerodynamic forcesacting on the control surfaces are so large that it is impossible for any human pilotto supply or sustain the force required. Powered flying controls are then used.Usually the control surfaces are moved by means of mechanical linkages driven byelectrohydraulic actuators. A number of aircraft use electrical actuators, but thereare not many such types. The command signals to these electrohydraulic actuatorsare electrical voltages supplied from the controller of an AFCS, or directly from asuitable transducer on the primary flying control itself. By providing the pilot withpower assistance, so that the only force he needs to produce is a tiny force,sufficient to move the transducer, it has been found necessary to provide artificialfeel so that some force, representing what the aircraft is doing, is produced on theprimary flying control. Such forces are cues to a pilot and are essential to hisflying the aircraft successfully. The conventions adopted for the control surfacedeflections are shown in Figure 1.4. In the event of an electrical or hydraulic failure such a powered flyingcontrol system ceases to function, which would mean that the control surfacecould not be moved: the aircraft would therefore be out of control. To preventthis occurring, most civilian and military aircraft retain a direct, but parallel,mechanical connection from the primary flying control to the control surfacewhich can be used in an emergency. When this is done the control system is saidto have 'manual reversion'. Fly-by-wire (and fly-by-light) aircraft have essentiallythe same kind of flight control system, but are distinguished from conventionalaircraft by having no manual reversion. To meet the emergency situation, whenfailures occur in the system, fly-by-wire(FBW) aircraft have flight control systemswhich are triplicated, sometimes quadruplicated, to meet this stringent reliabilityrequirement. With FBW aircraft and CCVs it has been realized that there is no longera direct relationship between the pilot's command and the deflection, or even theuse, of a particular control surface. What the pilot of such aircraft is commandingfrom the AFCS is a particular manoeuvre. When this was understood, and when
Flight Control Systems Figure 1.5 Side arm controller.the increased complexity of flying was taken into account, it was found that theprovision of a yoke or a stick to introduce commands was unnecessary andinconvenient. Modern aircraft are being provided with side arm controllers (seeFigure 1.5) which provide signals corresponding to the forces applied by the pilot.Generally, these controllers do not move a great deal, but respond to appliedforce. By using such controllers a great deal of cockpit area is made available forthe growing number of avionics displays which modern aircraft require.1.4 FLIGHT CONTROL SYSTEMSIn addition to the control surfaces which are used for steering, every aircraftcontains motion sensors which provide measures of changes in motion variableswhich occur as the aircraft responds to the pilot's commands or as it encounterssome disturbance. The signals from these sensors can be used to provide the pilotwith a visual display, or they can be used as feedback signals for the AFCS. Thus,the general structure of an AFCS can be represented as the block schematic ofFigure 1.6. The purpose of the controller is to compare the commanded motionwith the measured motion and, if any discrepancy exists, to generate, inaccordance with the required control law, the command signals to the actuator toproduce the control surface deflections which will result in the correct controlforce or moment being applied. This, in turn, causes the aircraft to respondappropriately so that the measured motion and commanded motion are finally incorrespondence. How the required control law can be determined is one of theprincipal topics of this book. Whenever either the physical or abstract attributes of an aircraft, and itsmotion sensing and controlling elements, are considered in detail, their effects areso interrelated as almost to preclude discussion of any single aspect of the system,
Pilot's direct Aircraft Flight Controlcommand input Atmospheric controls disturbances actuators dynamics - Motion I -variables deflections - Flight sensors + controller (control law) Sensor noise Figure 1.6 General structure of an AFCS.without having to treat most of the other aspects at the same time. It is helpful,therefore, to define here, albeit somewhat broadly, the area of study upon whichthis book will concentrate.1. The development of forces and moments for the purpose of establishing an equilibrium state of motion (operating point) for an aircraft, and for the purpose of restoring a disturbed aircraft to its equilibrium state, and regulating within specific limits the departure of the aircraft's response from the operating point, are regarded here as constituting flight control.2 . Regulating the aircraft's response is frequently referred to as stabilization.3. Guidance is taken to mean the action of determining the course and speed to be followed by the aircraft, relative to some reference system. Flight control systems act as interfaces between the guidance systems and the aircraft being guided in that the flight control system receives, as inputs from the guidance systems, correction commands, and provides, as outputs, appropriate deflections of the necessary control surfaces to cause the required change in the motion of the aircraft (Draper, 1981). For this control action to be effective, the flight control system must ensure that the whole system has adequate stability. If an aircraft is to execute commands properly, in relation to earth coordinates, it must be provided with information about the aircraft's orientation so that right turn, left turn, up, down, roll left, roll right, for example, are related to the airborne geometrical reference. For about sixty years, it has been common practice to provide aircraft with
Flight Control Systems 9reference coordinates for control and stabilization by means of gyroscopicinstruments. The bank and climb indicator, for example, effectivelyprovides a horizontal reference plane, with an accuracy of a few degrees,and is as satisfactory today for the purposes of control as when it was firstintroduced. Similarly, the turn indicator, which shows the aircraft'sturning left or right, to about the same accuracy, is also a gyroscopicinstrument and the use of signals from both these devices, as feedbacksignals for an AFCS, is still effective and valid. However, the use ofconventional gyroscopic instruments in aircraft has fundamental limita-tions which lie in the inherent accuracy of indication, which is to within afew degrees only, and also in the inherent drift rates, of about ten degreesper hour. Such instruments are unsuitable for present-day navigation,which requires that the accumulated error in distance for each hour ofoperation, after an inertial fix, be not greater than 1.5 km. An angle ofone degree between local gravitational directions corresponds to adistance on the earth's surface of approximately 95 km. Consequently,special motion sensors, such as ring laser gyros, NMR gyros, strap-down,force-balance accelerometers, must be used in modern flight controlsystems. Because this book is concerned with control, rather thanguidance, it is more convenient to represent the motion of aircraft in asystem of coordinates which is fixed in the aircraft and moves with it. Bydoing this, the coordinate transformations generally required to obtainthe aircraft's motion in some other coordinate system, such as a systemfixed in the earth, can be avoided. When the origin of such a body-fixedsystem of coordinates is fixed at the centre of gravity of the aircraft,which is in an equilibrium (or trimmed) state of motion along a nominalflight path, then, when only small perturbations of the aircraft's motionabout this equilibrium state are considered, the corresponding equationsof motion can be linearized. Since many flight control problems are ofvery short duration (5-20 seconds), the coefficients of these equations ofmotion can be regarded as constant, so that transfer functions cansometimes be conveniently used to describe the dynamics of the aircraft.However, it must be remembered that a notable feature of an aircraft'sdynamic response is how it changes markedly with forward speed, height,and the aircraft's mass. Some of the most difficult problems of flightcontrol occurred with the introduction of jet propulsion, the consequentexpansion of the flight envelope of such aircraft, and the resultingchanges in configuration, most notable of which were the use of sweptwings, of very short span and greatly increased wing loading, and theconcentrated mass of the aircraft being distributed in a long and slenderfuselage. In aircraft of about 1956 these changes led to markeddeficiences in the damping of the classical modes of aircraft motion,namely the short period mode of the aircraft's longitudinal motion, andthe Dutch roll mode of its lateral motion. Other unknown, coupled
Aircraft Flight Control modes also appeared, such as fuel sloshing and roll instability; the use of thinner wings and more slender fuselages meant greater flexibility of the aircraft structure, and the modes associated with this structural flexibility coupled with the rigid-body modes of the aircraft's motion, caused further problems. One of the first solutions to these problems was the use of a stability augmentation system (SAS), which is simply a feedback control system designed to increase the relative damping of a particular mode of the motion of the aircraft. Such an increase in damping is achieved by augmenting one or more of the coefficients of the equations of motion by imposing on the aircraft appropriate forces or moments as a result of actuating the control surfaces in response to feedback signals derived from appropriate motion variables. After SAS, the following AFCS modes were developed: sideslip suppression SAS, pitch attitude hold, autothrottle (speed control system), much hold, height hold, and turn coordination systems. An integrated flight control system is a collection of such AFCS modes in a single comprehensive system, with particular modes being selected by the pilot to suit the task required for any particular phase of flight. In the past such functions were loosely referred to as an autopilot, but that name was a trademark registered by the German company Siemens in 1928. Today, AFCS not only augment the stability of an aircraft, but they can follow path and manoeuvre commands, thereby providing the means of automatic tracking and navigation; they can perform automatic take-off and landing; they can provide structural mode control, gust load alleviation, and active ride control.1.5 BRIEF HISTORY OF FLIGHT CONTROL SYSTEMSThe heavier-than-air machine designed and built by Hiram Maxim in 1891 wascolossal for its time: it was 34 m long and weighed 3 600 kg. Even now, the largestpropeller to be seen in the aviation collection of the Science Museum in London isone of the pair used by Maxim. It was obvious to Maxim, if to no-one else at thetime, that when his aircraft flew, its longitudinal stability would be inadequate, forhe installed in the machine a flight control system which used an actuator todeflect the elevator and employed a gyroscope to provide a feedback signal. Itwas identical, except in inconsequential detail, to a present-day pitch attitudecontrol system. Two of the minor details were the system's weight, over 130kg,and its power source, steam. The concept remains unique. Between 1910 and 1912 the American father-and-son team, the Sperrys,developed a two-axis stabilizer system in which the actuators were powered bycompressed air and the gyroscopes were also air-driven. The system couldmaintain both pitch and bank angles simultaneously and, from a photographic
Brief History 17record of a celebrated demonstration flight, in which Sperry Snr is seen in theopen cockpit, with his arms stretched up above his head, and a mechanic isstanding on the upper surface of the upper wing at the starboard wing tip,maintaining level flight automatically was easily within its capacity. During World War I, aircraft design improved sufficiently to provide, bythe sound choice of size, shape and location of the aerodynamic control surfaces,adequate stability for pilots' needs. Many aircraft were still unstable, but notdangerously so, or, to express that properly, the degree of damage was acceptablein terms of the loss rates of pilots and machines. In the 1920s, however, it was found that, although the early commercialairliners were quite easy to fly, it was difficult to hold heading in poor visibility.Frequently, in such conditions, a pilot and his co-pilot had to divide the flying taskbetween them. The pilot held the course by monitoring both the compass and theturn indicator and by using the rudder; the co-pilot held the speed and theattitude constant by monitoring both the airspeed and the pitch attitude indicatorand by controlling the airspeed via the engine throttles and the pitch attitude byusing the elevator. From the need to alleviate this workload grew the need tocontrol aircraft automatically. The most extensive period of development of early flight control systemstook place between 1922 and 1937: in Great Britain, at the Royal AircraftEstablishment (RAE) at Farnborough; in Germany, in the industrial firms ofAskania and Siemens; and in the USA, in Sperrys and at NACA (NationalAdvisory Committee for Aeronautics - now NASA). Like all other flight controlsystems up to 1922, the RAE'S Mk I system was two-axis, controlling pitchattitude and heading. It was a pneumatic system, but its superior performanceover its predecessors and competitors was due to the fact that it had beendesigned scientifically by applying the methods of dynamic stability analysis whichhad been developed in Great Britain by some very distinguished appliedmathematicians and aerodynamicists (see McRuer et al., 1973; Draper, 1981;Hopkin and Dunn, 1947; McRuer and Graham, 1981; Oppelt, 1976). Suchcomprehensive theoretical analysis, in association with extensive experimentalflight tests and trials carried out by the RAF, led to a clear understanding ofwhich particular motion variables were most effective for use as feedback signalsin flight control systems. In 1927, in Germany, the firm of Askania developed a pneumatic systemwhich controlled heading by means of the aircraft's rudder. It used an air-drivengyroscope, designed and manufactured by Sperrys of the USA. The first unit wasflight tested on the Graf Zeppelin-LZ127; the system merits mention only becauseof its registered trade name, Autopilot. However, the Germans soon decided thatas a drive medium, air, which is very compressible, gave inferior performancecompared to oil, which was considered to be very nearly incompressible. Thus, inits two-axis 'autopilot' of 1935, the Siemens company successfully used hydraulicactuators and thereby established the trend, still followed today, of usinghydraulic oil in preference to air, which in turn was used in preference to Maxim'ssteam. In 1950, the Bristol Aeroplane Company built a four-engined, turbo-prop
12 Aircraft Flight Controltransport aircraft which used electric actuators, but it was not copied by othermanufacturers. At present, NASA and the USAF are actively pursuing aprogramme of reasearch designed to lead to 'an all-electric airplane' by 1990. The reader should not infer from earlier statements that the RAE solvedevery flight control problem on the basis of having adequate theories. In 1934, theMk IV system, which was a three-axis pneumatic system, was designed forinstallation in the Hawker Hart, a biplane in service with the RAF. In flight, aconsiderable number of stability problems were experienced and these were neversolved. However, when the same system was subsequently fitted to the heavybombers then entering RAF service (the Hampdens, Whitleys and Wellingtons)all the stability problems vanished and no satisfactory reasons for thisimprovement were ever adduced. (McRuer and Graham (1981) suggest that theincreased inertia and the consequently slower response of the heavier aircraftwere the major improving factors.) In 1940, the RAE had developed a new AFCS, the Mk VII, which wasagain two-axis and pneumatic, but, in the longitudinal axis, used both airspeedand its rate of change as feedback signals, and, in the lateral axis, moved theailerons in response to a combination of roll and yaw angles. At cruising speed incalm weather the system was adjudged by pilots to give the best automatic controlyet devised. But, in some aircraft at low speeds, and in all aircraft in turbulence,the elevator motion caused such violent changes in the pitch attitude that theresultant vertical acceleration so affected the fuel supply that the engines stopped.It was only in 1943 that the problem was eventually solved by Neumark (seeNeumark, 1943) who conducted an analysis of the problem entirely by time-domain methods. He used a formulation of the aircraft dynamics that controlengineers now refer to as the state equation. German work did not keep pace with British efforts, since, until very latein World War 11, they concentrated on directional and lateral motion AFCSs,only providing a three-axis AFCS in 1944. The American developments had beenessentially derived from the Sperry Automatic Pilot used in the Curtiss 'Condors'operated by Eastern Airlines in 1931. Subsequently, electric, three-axis autopilotswere developed in the USA by firms such as Bendix, Honeywell and Sperry. TheMinneapolis Honeywell C l was developed from the Norden Stabilized Bomb-sight and was much used in World War I1 by both the American Air Forces andthe Royal Air Force. The development of automatic landing was due principally to the BlindLanding Experimental Unit of RAE, although in 1943 at the Flight DevelopmentEstablishment at Rechlin in Germany, at least one aircraft had been landedautomatically. The German efforts on flight control at this time were devoted tothe systems required for the V1 and V2 missiles. On 23 September 1947 anAmerican Douglas C-54 flew across the Atlantic completely under automaticcontrol, from take-off at Stephenville, in Newfoundland, Canada, to landing atBrize Norton, in England. A considerable effort has been given to developingAFCSs since that time to become the ultra-reliable integrated flight controlsystems which form the subject of this book. The interested reader is referred to
Outline of the Book 13Hopkin and Dunn (1947), McRuer and Graham (1981), Oppelt (1976) andHoward (1973) for further discussions of the history of flight control systems.1.6 OUTLINE OF THE BOOKChapters 2 and 3 deal with the dynamic nature and characteristics of aircraft and,in so doing, it is hoped to establish the significance and appropriateness of the axissystems commonly used, and to derive mathematical models upon which it isconvenient subsequently to base the designs of the AFCS. Chapters 4 and 5 have been included to provide the reader with a clearknowledge of those significant dynamic effects which greatly affect the nature ofan aircraft's flight, but over which a designer had no control. The complexity,which inevitably arises in providing a consistent account of the structuralflexibility effects in aircraft dynamics, has to be understood if the importantdevelopment of active control technology is to make sense. The principalobjective of Chapter 4 is to provide a reasonable and consistent development ofthe additional dynamical equations representing the structural flexibility effects,to show how to incorporate them into the mathematical model of the aircraft, andto provide the reader with an account of their physical significance. One of thechief reasons why aircraft require flight control systems is to achieve smooth flightin turbulent atmospheric conditions. An explanation of the important forms ofatmospheric turbulence is given in Chapter 5. How they can be representedmathematically, and how their effects can be properly introduced into the aircraftequations, are also covered there. Chapter 6 deals with the important subject offlying and handling qualities which are expressed mostly in terms of desirabledynamic properties which have been shown, from extensive flight and simulationexperiments, to be most suited to pilots' skills and passengers' comfort. Thesequalities are the chief source of the performance criteria by which AFCS designsare assessed. In a subject as extensive as AFCSs many methods of control systemdesign are tried, used and reported in the technical reports and journals. It isnecessary for any student to be competent in some of these methods, andreasonably familiar with the general nature of them all. Although it is notintended to provide a text book in control theory, Chapters 7 and 8 have beenincluded to provide students with a self-contained summary of the most com-monly applied methods, together with some indication of the relative advantagesand disadvantages of each from the viewpoint of a designer of AFCSs for aircraft. It is the objective of Chapters 9 to 11 to introduce students to the basicflight control modes which form the integrated flight control systems found inmost modern aircraft. The nature of the dynamic response and the effects uponthe performance of each subsystem of its inclusion as an inner loop in a largersystem are both dealt with. Chapter 12 provides students with a clear account of the type of AFCS
14 Aircraft Flight Controlwhich is now finding use in aircraft under current development, the so-calledcontrol configured vehicle. The flight control modes involved are specialist(except relaxed static stability, which can be handled by the methods outlined inChapter 8) and, since the assessment of the performance of these active controltechnology (ACT) systems is not based upon the criteria dealt with in Chapter 6,they have been gathered together and dealt with separately in this chapter. Rotary wing aircraft have quite distinctive methods of control and alsohave special dynamical problems. Although in forward flight, at all but the lowestspeeds, they can be treated in the same manner as fixed-wing aircraft, the controlproblems are, in general, so distinctive that they are dealt with separately inChapter 13, although the AFCSs employed in helicopters still involve stabilityaugmentation and attitude and path control. Chapter 14 demonstrates how the control laws developed earlier can betreated by digital control methods, so that digital AFCSs, which are commonlyfitted to modern aircraft, can be considered and also to provide an outline of theeffects upon the AFCS7sperformance in terms of the particular features of thedigital method used. Modern fighter and interdiction aircraft have flight envelopes which areso extensive that those changes which arise in the characteristic equation of theaircraft are too great to be handled by control laws devised on the basis of thecontrol methods dealt with earlier. For such situations, the use of adaptive controlis advocated. Chapter 15 presents some information about the theories which areused to develop such systems. Since the dynamic equations of these systems arenon-linear, special stability considerations apply and these are also dealt with.1.7 CONCLUSIONSIn considering the design of an AFCS an engineer will succeed only if he is ableboth to establish an adequate model representing the appropriate dynamicalbehaviour of the aircraft to be controlled and to recognize how an effectivecontrol system design can be realized. Consequently, the control engineer working with AFCSs must completelyunderstand the equations of the aircraft's motion, be familiar with their methodsof solution, understand the characteristic responses associated with them, knowwhat influence they have on the aircraft's flying qualities, appreciate howatmospheric disturbances can be characterized and know how such disturbancesaffect performance. Additionally, it is important to understand how primaryflying controls can be improved, or their worst effects reduced, so that the matchbetween a human pilot and the aircraft is optimized. In addition, the theory of control, with its attendant design techniques,must be thoroughly mastered so that it, and they, can be used to produce anAFCS based upon control surface actuators and motion sensors which areavailable, and whose dynamic behaviour is thoroughly known.
References 15 The alternative methods of carrying out the required computation toproduce the appropriate control laws have also to be completely understood, andthe engineer is expected to be sound in his appreciation of the limitations ofwhatever particular method was chosen to perform the control design. Detailed engineering considerations of installing and testing such AFCSs,particularly in regard to certification procedures for airworthiness requirements,and the special reliability considerations of the effect of subsystem failure uponthe integrity of the overall system, are special studies beyond this book. Theinfluence of these topics on the final form of the AFCS is profound and representsone of the most difficult aspects of flight control work. Any flight control engineerwill be obliged to master both subjects early in his professional career.1.8 NOTE1. Sometimes 'centre of mass' and 'centre of gravity' are used interchangeably. For any group of particles in a uniform gravitational field these centres coincide. For spacecraft, their separation is distinctive and this separation results in an appreciable moment due to gravity being exerted on the spacecraft. For aircraft flying in the atmosphere the centres are identically located.1.9 REFERENCESDRAPER, C.S. 1981. Control, navigation and guidance. ZEEE Control Systems Magazine. l(4): 4-17.HOPKIN, H.R. and R.W. DUNN. 1947. Theory and development of automatic pilots 1937-1947. RAE report. IAP 1459. August.HOWARD, R.W. 1973. Automatic flight controls in fixed wing aircraft - the first hundred years. Aero. J. 77(11): 553-62.McRUER, D.T., I.L. ASHKENAS and D.C. GRAHAM. 1973. Aircraft Dynamics and Automatic Control. Princeton University Press.McRUER, D.T. and&. GRAHAM. 1981. Eighty years of flight control: triumphs and pitfalls of the systems approach. J . Guid. and Cont. 4(4): 353-62.NEUMARK, S. 1943. The disturbed longitudinal motion of an uncontrolled aeroplane and of an aeroplane with automatic control. ARC R&M 2078. January.OPPELT, W. 1976. An historical review of Autopilot development, research and theory in Germany. J. Dyn. Sys., Meas. and Cont. 98(3): 215-23.
The Equations of Motion ofan Aircraft2.1 INTRODUCTIONIf the problems associated with designing an AFCS were solely concerned withlarge area navigation then an appropriate frame of reference, in which to expressthe equations of motion of an aircraft, would be inertial, with its centre in thefixed stars. But problems involving AFCSs are generally related to events whichdo not persist: the dynamic situation being considered rarely lasts for more than afew minutes. Consequently, a more convenient inertial reference frame is atropocentric coordinate system, i.e. one whose origin is regarded as being fixed atthe centre of the Earth: the Earth axis system. It is used primarily as a referencesystem to express gravitational effects, altitude, horizontal distance, and theorientation of the aircraft. A set of axes commonly used with the Earth axissystem is shown in Figure 2.1; the axis, XE, is chosen to point north, the axis, YE,then pointing east with the orthogonal triad being completed when the axis, ZE,points down. If the Earth axis system is used as a basic frame of reference, towhich any other axis frames employed in the study are referred, the aircraft itself XE (North) ZEFigure 2.1 Earth axis system.
Introduction Lift (positive upwards) All directions shown are positive U, V,R are the forward, side and yawing velocities L, M,Nare roll, pitch and yaw moments P, Q, R are the angular velocities, , Y are roll, pitch and yaw angles Thrust (positive forwards) Figure 2.2 Body axis system.must then have a suitable axis system. Several are available which all find use, toa greater or lesser extent, in AFCS work. The choice of axis system governs theform taken by the equations of motion. However, only body-fixed axis systems,i.e. only systems whose origins are located identically at an aircraft's centre ofgravity, are considered in this book. For such a system, the axis, XB, pointsforward out of the nose of the aircraft; the axis, YB, points out through thestarboard (right) wing, and the axis, ZB, points down (see Figure 2.2). Axes XB,YB and ZB emphasize that it is a body-fixed axis system which is being used.Forces, moments and velocities are also defined. By using a system of axes fixedin the aircraft the inertia terms, which appear in the equations of motion, may beconsidered to be constant. Furthermore, the aerodynamic forces and momentsdepend only upon the angles, a and P, which orient the total velocity vector,VT, in relation to the axis, XB. The angular orientation of the body axis systemwith respect to the Earth axis system depends strictly upon the orientationsequence. This sequence of rotations is customarily taken as follows (seeThelander, 1965):1. Rotate the Earth axes, XE, YE, and ZE, through some azimuthal angle, q ,about the axis, XE, to reach some intermediate axes XI, Y1 and Z1.2. Next, rotate these axes XI, Y1 and Z1 through some angle of elevation, O , about the axis Y1 to reach a second, intermediate set of axes, X2, Y2, and Z2.3. Finally, the axes X2, Y2 and Zz are rotated through an angle of bank, @, about the axis, X2, to reach the body axes XB, YB and ZB.Three other special axis systems are considered here, because they can be foundto have been used sufficiently often in AFCS studies. They are: the stability axis
Habilitative Systems Donald Dew
78 Equations of Motion of an Aircraftsystem; the principal axis system; and the wind axis system. In AFCS work, themost commonly used system is the stability axis system.2.2 AXIS (COORDINATE) SYSTEMS2.2.1 The Stability Axis SystemThe axis X, is chosen to coincide with the velocity vector, VT, at the start of themotion. Therefore, between the X-axis of the stability axis system and the X-axisof the body axis system, there is a trimmed angle of attack, a,. The equations ofmotion derived by using this axis system are a special subset of the set derived byusing the body axis system.2.2.2 The Principal Axis SystemThis set of body axes is specially chosen to coincide with the principal axes of theaircraft. The convenience of this system resides in the fact that in the equations ofmotion, all the product of inertia terms are zero, which greatly simplifies theequations.2.2.3 The Wind Axis SystemBecause this system is oriented with respect to the aircraft's flight path, time-varying terms which correspond to the moments and cross-products of inertiaappear in the equations of motion. Such terms considerably complicate theanalysis of aircraft motion and, consequently, wind axes are not used in this text.They have appeared frequently, however, in American papers on the subject.2.2.4 Sensor SignalsBecause an AFCS uses feedback signals from motion sensors, it is important toremember that such signals are relative to the axis system of the sensor and not tothe body-fked axis system of the aircraft. This simple fact can sometimes causethe performance obtained from an AFCS to be modified and, in certain flighttasks, may have to be taken into account. However, in straight and level flight atcruise it is insignificant.
Equations of Motion of a Rigid Body Aircraft2.3 THE EQUATIONS OF MOTION OF A RIGID BODY AIRCRAFT2.3.1 IntroductionThe treatment given here closely follows that of McRuer et al. (1953). It is assumed, first, that the aircraft is rigid-body; the distance betweenany points on the aircraft do not change in flight. Special methods to take intoaccount the flexible motion of the airframe are treated in Chapter 4. When theaircraft can be assumed to be a rigid body moving in space, its motion can beconsidered to have six degrees of freedom. By applying Newton's Second Law tothat rigid body the equations of motion can be established in terms of thetranslational and angular accelerations which occur as a consequence of someforces and moments being applied to the aircraft. In the introduction to this chapter it was stated that the form of theequations of motion depends upon the choice of axis system, and a few of theadvantages of using a body-fixed axis system were indicated there. In thedevelopment which follows, a body axis system is used with the change to thestability axis system being made at an appropriate point later in the text. In orderto be specific about the atmosphere in which the aircraft is moving, it is alsoassumed that the inertial frame of reference does not itself accelerate, in otherwords, the Earth is taken to be fixed in space.2.3.2 Translational Motion*~ewton's Second Law it can be deduced that:M = d {H} - dtwhere F represents the sum of all externally applied forces, M represents the sumof all applied torques, and H is the angular momentum. The sum of the external forces has three components: aerodynamic,gravitational and propulsive. In every aircraft some part of the propulsive (thrust)force is produced by expending some of the vehicle's mass. But it can easily beshown1 that if the mass, rn, of an aircraft is assumed to be constant, the thrust,which is a force equal to the relative velocity between the exhausted mass and theaircraft and the change of the aircraft's masslunit time, can be treated as anexternal force without impairing the accuracy of the equations of motion. If it isassumed, for the present, that there will be no change in the propulsive force,changes in the aircraft's state of motion from its equilibrium state can occur if andonly if there are changes in either the aerodynamic or gravitational forces (orboth). If it becomes necessary in a problem to include the changes of thrust (as it
20 Equations of Motion of an Aircraftwill be when dealing with airspeed control systems, for example) only a smallextension of the method being outlined here is required. Details in relation to thestability axis system are given in section 2.2. For the present, however, the thrustforce can be considered to be contained in the general applied force, F. When carrying out an analysis of an AFCS it is convenient to regard thesums of applied torque and force as consisting of an equilibrium and aperturbational component, namely: +d (2.4)M = Mo AM = -dt {H)The subscript 0 denotes the equilibrium component, A the component ofperturbation. Since the axis system being used as an inertial reference system isthe Earth axis system, eqs (2.3) and (2.4) can be re-expressed as:By definition, equilibrium flight must be unaccelerated flight along a straightpath; during this flight the linear velocity vector relative to fixed space isinvariant, and the angular velocity is zero. Thus, both Fo and Mo are zero. The rate of change of VT relative to the Earth axis system is given by:where w is the angular velocity of the aircraft with respect to the fixed axissystem. Wnen the vectors are expressed in coordinates in relation to the body-fixed axis system, both velocities may be written as the sum of their correspondingcomponents, with respect to XB, YB and ZB, as follows:VT = iU + jV + kW (2.8)o=iP+jQ+kR (2.9)and the cross-product, o x VT, is given by:
Material Flow And Conveyor Systems Donald Or
Equations of Motion of a Rigid Body Aircraft 21+ += i(QW - V R ) j(UR - P W ) k(PV - U Q ) (2.11)In a similar fashion, the components of the perturbation force can be expressed asHence, + + +AF = m { i ( ~ QW - V R ) j ( ~ UR - PW) + k(W + PV - U Q ) }From which it can be inferred that: (2.14) +AF, = m ( U QW - V R ) AFy = m ( ~ U+R + P W ) +AF, = m ( ~VP - U Q )Rather than continue the development using the cumbersome notation, AFi, todenote the ith component of the perturbational force, it is proposed to follow theAmerican custom and use the following notation:It must be remembered that now X, Y and Z denote forces. With thesesubstitutions in eqs (2.14)-(2.16),the equations of translational motion can beexpressed as: +A X = m ( ~QW - V R ) +AY = m ( ~UR - P W ) AZ = m ( ~ V+P - U Q )2.3.3 Rotational MotionFor a rigid body, angular momentum may be defined as: H =IwThe inertia matrix, I , is defined as:where Iiidenotes a moment of inertia, and IVa product of inertia j # i.
22 Equations of Motion of an AircraftTransforming from body axes to the Earth axis system (see Gaines and Hoffman,1972) allows eq. (2.23) to be re-expressed as:However, cl,xoboandwhere h, hy and h, are the components of H obtained from expanding eq. (2.21)thus:h, = IxxP - IxyQ - IxzR (2.28) (2.29)+h, = - I,P IyyQ - Zy,R (2.30)+h, = - I,P - I,Q I,RIn general, aircraft are symmetrical about the plane XZ, and consequently it isgenerally the case that:I, = Zyz = 0 (2.31)Therefore:h, = IxxP - Ix,R (2.32)hy = IyyQ (2.33) (2.34)h, = - I,P + I,Rand + +- I,(R P Q ) Q R (I, - I,) (2.35) AM, = I,P + +zX,(p2 - R2) PR(Z, - I,) (2.36) AM, = I,Q (2.37) AM, = I,R +- I,P PQ(Iyy - I,) + I,QRAgain, following American usage:AM, = A L AM, = AM AM, = AN (2.38)where L, M and N are moments about the rolling, pitching and yawing axesrespectively.
Equations of Motion of a Rigid Body Aircraft AL = IXxP- z,(R + P Q ) + (I, - Iyy)QR + +AM = I, Q I,(P' - R2) (Ixx- I,)PR ++AN = Z,R - IX,P P Q (Iyy- I,) Ix,QR2.3.4 Some Points Arising from the Derivation of the EquationsIt is worth emphasizing here that the form of equations arrived at, having used abody axis system, is not entirely convenient for flight simulation work (Fogartyand Howe, 1969). For example, suppose a fighter aircraft has a maximum velocityof 600 m s-' and a maximum angular velocity QB of 2.0 rad s-l. The term, UQ,in eq. (2.20) can have a value as large as 1200 m sK2, i.e. 120 g , whereas theterm, AZ, the normal acceleration due to the external forces (primarilyaerodynamic and gravitational) may have a maximum value in the range 10.0 to20.0 m s-' (i.e. 1-2 g). It can be seen, therefore, how a (dynamic) acceleration ofvery large value, perhaps fifty times greater than the physical accelerations, canoccur in the equations merely as a result of the high rate of rotation experiencedby the body axis system. Furthermore, it can be seen from inspection of eqs(2.18)-(2.20) how angular motion has been coupled into translational motion.Moreover, on the right-hand side of eqs (2.39)-(2.41) the third term is a non-linear, inertial coupling term. For large aircraft, such as transports, which cannotgenerate large angular rates, these terms are frequently neglected so that themoment equations become: +AL = ZP, - IX,(R P Q ) AM = IYyQ+ z ~ , ( P-~ R') AN = Z,R - IX,(P - QR )A number of other assumptions are frequently invoked in relation to theseequations:1. Sometimes, for a particular aircraft, the product of inertia, I, is sufficiently small to allow of its being neglected. This often happens when the body axes, XB, YB, and ZBhave been chosen to almost coincide with the principal axes.2. For aircraft whose maximum values of angular velocity are low, the terms PQ, QR, and p2- R2 can be neglected.3. Since R2 is frequently very much smaller than p2, it is often neglected.It is emphasized, however, that the neglect of such terms can only be practisedafter very careful consideration of both the aircraft's characteristics and the AFCSproblem being considered. Modern fighter aircraft, for example, may lose controlas a result of rolYpitch inertial coupling. In such aircraft, pitch-up is sensed whena roll manoeuvre is being carried out. When an AFCS is fitted, such a sensorsignal would cause an elevator deflection to be commanded to provide a
24 Equations of Motion of an Aircraftnose-down attitude until the elevator can be deflected no further and theaircraft cannot be controlled. Such a situation can happen whenever the term(Ixx - Z,)PR is large enough to cause an uncontrollable pitching movement.2.3.5 Contributions to the Equations of Motion of the Forces Due to GravityThe forces due to gravity are always present in an aircraft; however, by neglectingany consideration of gradients in the gravity field, which are important only inextra-atmospheric flight if all other external forces are essentially non-existent, itcan be properly assumed that gravity acts at the centre of gravity (c.g.) of theaircraft. Hence, since the centres of mass and gravity coincide in an aircraft, thereis no external moment produced by gravity about the c.g. Hence, for the bodyaxis system, gravity contributes only to the external force vector, F. The gravitational force acting upon an aircraft is most obviouslyexpressed in terms of the Earth axes. With respect to these axes the gravityvector, mg, is directed along the ZE axis. Figure 2.3 shows the alignment of thegravity vector with respect to the body-fixed axes. In Figure 2.3 O represents theangle between the gravity vector and the YBZBplane; the angle is positive whenthe nose of the aircraft goes up. @ represents the bank angle between the axis ZBand the projection of the gravity vector on the YBZBplane; the angle is positivewhen the right wing Is down. Direct resolution of the vector mg, into X, Y and Zcomponents produces:SY = mg cos [- O] sin @ = mg cos O sin @ (2.45)SZ = mg cos [- O] cos @ = mg cos O cos cDFigure 2.3 Orientation of gravity vector with body axis systems.
Equations of Motion of a Rigid Body Aircraft 25In general, the angles O and Q, are not simply the integrals of the angular velocityP and Q ; in effect, two new motion variables have been introduced and itis necessary to relate them and their derivatives to the angular velocities, P , Qand R. How this is done depends upon whether the gravitational vertical seenfrom the aircraft is fixed or whether it rotates relative to inertial space. Aircraftspeeds being very low compared to orbital velocities, the vertical may be regardedas fixed. In very high speed flight the vertical will be seen as rotating and thetreatment which is being presented here will then require some minoramendments. The manner in which the angular orientation and velocity of the body axissystem with respect to the gravity vector is expressed depends upon the angularvelocity of the body axes about the vector mg. This angular velocity is theazimuth rate, Zk; it is not normal to either 6 or 6 , but its projection in the YBZBplane is normal to both (see Figure 2.4). By resolution, it is seen that: +R = - 6 sin Q, Zk cos O cos Q,Also,6 = Q cos @ - R sin Q,Ip = R c o s @ + QsinQ,cos 0 cos 0Using substitution, it is easy to show that:Figure 2.4 Angular orientation and velocities of gravity vector, g, relativeto body axis.
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26 Equations of Motion of an Aircraft + +@ = P R tan 0 cos @ Q tan O sin @@, O and l-lr are referred to as the Euler angles.2.3.6 Axis TransformationsThe physical relationships established so far depend upon two frames ofreference: the Earth axis system and the body axis system. To orient thesesystems one to another requires the use of axis transformations. Any set of axescan be obtained from any other set by a sequence of three rotations. For eachrotation a transformation matrix is applied to the variables. The totaltransformation array is obtained simply by taking the product of the threematrices, multiplied in the order of the rotations. In aircraft dynamics, the mostcommon set of transformations is that between the Earth axis system whichincorporates the gravity vector, g, as one axis, and the body-fixed axes, XB,YBand ZB.The rotations follow the usual order: azimuth 'Y, pitch 0, and roll @. Thecorresponding matrices are:1cos 'Y sin 'Y O 1cos 0 0 - sin 01 sin 0 o cos 0 j The complete transformation matrix T is called the direction cosine arrayand is defined as:Before expressing the matrix T in full, a notational shorthand is proposedwhereby a term such as cos 6 is written as ct and a term such as sine is writtenas sE. Thus:
Equations of Motion of a Rigid Body Aircraft 27 It is worth noting that the order of rotation T-O-@ is that which resultsin the least complicated resolution of the gravity vector g into the body axissystem. It can easily be shown that: Another practical advantage is that the angles are those which aremeasured by a typically oriented vertical gyroscope. A two degree of freedom,gravity erected, vertical gyroscope, oriented such that the bearing axis of its outergimbal lies along OXB, measures on its inner and outer gimbals the Euler anglesO and @, respectively.2.3.7 Linearization of the Inertial and Gravitational TermsEquations (2.14)-(2.16) and (2.39)-(2.41) represent the inertial forces acting onthe aircraft. Equation (2.45) represents the contribution of the forces due togravity to those equations. All these forces are proportional to the mass af theaircraft. Consequently, these terms may be conveniently combined intocomponents to represent the accelerations which would be measured by sensorslocated on the aircraft in such a manner that the input axes of the sensors wouldbe coincident with the body axes XB, YB and ZB. The external forces acting onthe aircraft can be re-expressed as:where 6X, SY and SZ are the gravitational terms and AX, AY and AZrepresent the aerodynamic and thrust forces. For notational convenience, AL,AM and AN are now denoted by L , M and N. Thus the equations of motion ofthe rigid body, for its six degrees of freedom, may be expressed as: x 4 m a X = m [ ~ +Q W - R V + g s i n O ] cg ~ k m a , = m [ ~ R+ U - P W - g cosO sin@] C% Z4maZ =~[w+Pv-QU-~COSOCOS@] cg L = PIxx- z,(R + P Q ) + (I, - Iyy)QR + +M = QZ, I,(P* - R') (Ixx- I,)PR N = RZ, - I,P + PQ(Iyy- Ixx) + Ix,QR The auxiliary equations of eq. (2.46) must also be used since they relateT,O and @ to R, Q and P.
28 Equations of Motion of an Aircraft The equations which constitute eq. (2.56) are non-linear since theycontain terms which comprise the product of dependent variables, the squares ofdependent variables, and some of the terms are transcendental. Solutions of suchequations cannot be obtained analytically and would require the use of acomputer. Some simplification is possible, however, by considering the aircraft to'comprise two components: a mean motion which represents the equilibrium, ortrim, conditions, and a dynamic motion which accounts for the perturbationsabout the mean motion. In this form of analysis it is customary to assume that theperturbations are small. Thus, every motion variable is considered to have twocomponents. For example: U&iJo+u R A R ~ + ~ Q A Qo + q M A Mo + ml etc.The trim, or equilibrium, values are denoted by a subscript 0 and the smallperturbation values of a variable are denoted by the lower case letter.3 In trim there can be no translational or rotational acceleration. Hence,the equations which represent the trim conditions can be expressed as: Xo = m [QoW o - RoV o + g sin 00]Y o = m [UoRo- POWo- g cos O0 sin Oo]Zo = m[PoVo- QoUo - g cos O0 cos @o]LO = QoRo(Zzz - Zyy) - PoQoZxzMo += ( P ; - ~ ; ) z x z ( Z X X - Zzz)PoRoNO= Zxz QORO+ (Iyy- Zxx)PoQoSteady rolling, pitching and yawing motion can occur in the trimcondition; the equations which define Po, Q0 and Ro are given by eq. (2.46) butwith @, O and being subscripted by 0.The perturbed motion can be found either by substituting eq. (2.57) into(2.56), expanding the terms and then subtracting eq. (2.58) from the result, or bydifferentiating both sides of eq. (2.56). When perturbations from the meanconditions are small, the sines and cosines can be approximated to the anglesthemselves and the value unity, respectively. Moreover, the products and squaresof the perturbed quantities are negligible. Thus, the perturbed equations ofmotion for an aircraft can be written as:+dX = m [ u Woq + Qow - Vor - Rov + g cos OoO]+ +dY = m [ 3 Uor Rou - Wop - Pow - (g cos 0 0 cos @o)++ ( g sin sin @o)O]d Z = m [w + Vop + Pov - Uoq - Qou + (g cos 00sin @o)+ (2.59) + (g sin O0 cos cPo)0]
Equations of Motion of a Rigid Body Aircraft+where q o , O0 and Qo have been used to represent steady orientations, and Y,0 and the perturbations in the Euler angles. Equations (2.59) are now linear.Obviously, perturbation equations are required for the auxiliary set of equationsgiven as eq. (2.46), because the gravitional forces must be perturbed by any smallchange in the orientation of the body axis system with respect to the Earth axissystem. However, the full set of perturbed, auxiliary equations is rarely used sinceit is complicated. But the components of angular velocity which represent therotation of the body-fixed axes XB, YB and ZB relative to the Earth axes XE, YEand ZE are sometimes required. These are:p = C$ - 7k sin O0 - o ( $ ~cos BO) (2.60)4q = cos cPo - 0($0 sin Q sin 00) + $ sin Tocos O + +(.ire cos O0 cos Qo - 60sin Qo)+r = 7k cos B0 cos QO- +($0 cos OOsin Qo 7ko cos Qo)- 0 sin Qo - sin B0 cos Qo)Although these equations are linear, they are still too cumbersome for general useowing to the completely general trim conditions which have been allowed. Whatis commonly done in AFCS studies is to consider flight cases with simpler trimconditions, a case of great interest being, for example, when an aircraft has beentrimmed to fly straight in steady, symmetric flight, with its wings level. Steadyflight is motion with the rates of change of the components of linear and angularvelocity being zero. Possible steady flight conditions include level turns, steadysideslip and helical turns. Steady pitching flight must be regarded as merely a'quasi-steady' condition because u and w cannot both be zero for any appreciabletime if Q is not zero. Straight flight is motion with the components of angularvelocity being zero. Steady sideslips and dives and climbs without longitudinalacceleration are straight flight conditions. Symmetric flight is motion in which theplane of symmetry of the aircraft remains fixed in space throughout themanoeuvre taking place. Dives and climbs with wings level, and pull-ups withoutsideslipping, are examples of symmetric flight. Sideslip, rolls and turns are typicalasymmetric flight conditions. The significance of the specified trim conditions maybe judged when the following implications are understood:1. That straight flight implies $o = B0 = 0.2. That symmetric flight implies qo= Vo = 0.3. That flying with wings level implies Qo = 0.For this particular trimmed flight state, the aircraft will have particular values of
30 Equations of Motion of an AircraftUo, W o and OO. These may be zero, but for conventional aircraft the steadyforward speed, Uo,must be greater than the stall speed if flight is to be sustained.However, certain rotary wing and V/STOL aircraft can achieve a flying state inwhich Uo, W oand O0 may be zero; when Uo and W o are simultaneously zero theaircraft is said to be hovering. Hence, for straight, symmetric flight with wings level, the equationswhich represent translational motion in eq. (2.59) become: +z = m[w Pov - Uoq - Qou + g sin Oo0]The equations (2.59) which represent rotational motion are unaffected. Equation(2.60), however, becomes:r = cos O0From the same expression, for this trimmed flight state, it may be assumed that:Qo = Po = Ro = 0 (2.63)Therefore, it is possible to write eqs (2.59) and (2.61) in the new form: +x = m[u Woq - g cos OoO] ml = Iyyg n = Izz? - IxzpConsideration of eq. (2.64) indicates not only that the equations have beensimplified, but that the set can be separated into two distinct groups which aregiven below:z = m[w - Uoq + g sin OoO] (2.65)and +y = m[3 Uor - Wop - g cos OO+]
Complete Linearized Equations of Motion 37In eq. (2.65) the dependent variables are u, w, q and 0 and these are confined tothe plane XBZB.The set of equations is said to represent the longitudinal motion.The lateral/directional motion, consisting of sideslip, rolling and yawing motion isrepresented in eq. (2.66). Although it appears from this equation that the sideslipis not coupled to the rolling and yawing accelerations, the motion is, however,coupled (at least implicitly). In practice, a considerable amount of coupling canexist as a result of aerodynamic forces which are contained within the terms onthe left-hand side of the equations. It is noteworthy that this separation of lateral and longitudinal equationsis merely a separation of gravitational and inertial forces: this separation ispossible only because of the assumed trim conditions. But 'in flight', the sixdegrees of freedom model may be coupled strongly by those forces and momentswhich are associated with propulsion or with the aerodynamics.2.4 COMPLETE LINEARIZED EQUATIONS OF MOTION2.4.1 Expansion of Aerodynamic Force and Moment TermsTo expand the left-hand side of the equations of motion, a Taylor series is usedabout the trimmed flight condition. Thus, for example,Equation (2.67) supposes that the perturbed force z has a contribution from onlyone control surface, the elevator. However, if any other control surface on theaircraft being considered were involved, additional terms, accounting for theircontribution to z, would be used. For example, if changes of thrust (T), and thedeflection of flaps (F) and symmetrical spoilers (sp) were also used as controls forlongitudinal motion, additional terms, such asaz az az-ST &T, - SF and -ass, as, 3%would be added to eq. (2.67). Furthermore, some terms depending on othermotion variables, such as 0, are omitted because they are generally insignificant. For the moment only longitudinal motion is treated, and, for simplicity, itis assumed that only elevator deflection is involved in the control of the aircraft'slongitudinal motion. Thus, it is now possible to write eq. (2.65) as:
32 Equations of Motion of an Aircraft + $ = m [ w - Uoq + g sin OoO] ~SEUaM+ - t i +aM- W + -awM+ - q +dM- q + - aM aM aM &I3au au dw aw 89 aq asETo simplify the notation it is customary to make the following substitutions:x' = - -1- ax m axWhen this substitution is made the coefficients, such as M, Z, and X, arereferred to as the stability derivatives.2.4.2 Equations of Longitudinal MotionEquation (2.68) may now be rewritten in the following form: - g cos e,e + xSEs+ExQEw = Z,u + Z k u + Z,w + Z,+w + Z q q + Z4q + Uoq - g sin + Z,ESE + Z*E6E+q = Muu M& + M,w + M,+w + Mqq + M4q + MSESE+ M , $ ~ $For completeness, the second equation of (2.62) is usually added to eq. (2.70),i.e.0=q (2.70a)From studying the aerodynamic data of a large number of aircraft it becomesevident that not every stability derivative is significant and, frequently, a number
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Complete Linearized Equations of Motion 33can be neglected. However, it is essential to remember that such stabilityderivatives depend both upon the aircraft being considered and the. flightcondition which applies. Thus, before ignoring stability derivatives, it is importantto check the appropriate aerodynamic data. Without loss of generality it can beassumed that the following stability derivatives are often insignificant, and may beignored: X ~X, X,+, X*, ZC, Z,+, Mc, ZS, and Ms, .The stability derivative Z, is usually quite large but often ignored if the trimmedforward speed, Uo, is large. If the case being studied is hovering motion, then Z,ought not to be ignored. With these assumptions, the equations of perturbedlongitudinal motion, for straight, symmetric flight, with wings level, can beexpressed as:+w = Zuu + Zww Uoq - g sin OoO + Z6,SEq = M,u + M,w + M,+w+ Mqq + M6E8ENotice that each term in the first three equations of (2.71) is an acceleration term,but since the motion and control variables, u, w, q, 0 and SB, have such units asm s-l, and s-I the stability derivatives appearing in these equations aredimensional. It is possible to write similar equations using non-dimensionalstability derivatives, and this is frequently done in American literature and isalways done in the British system; but when it is done, the resulting equationsmust be written in terms of 'dimensionless' time. The responses obtainedfrom those equations are then expressed in units of time which differ from realtime. If the reader requires details of the use of non-dimensional stabilityderivatives, Babister (1961) should be consulted. It has been decided in this bookto use the form of equations given in (2.71) where dimensional stabilityderivatives must be used (these are the stability derivatives which are usuallyquoted in American works) but where time is real. Such a decision makes thedesign of AFCSs much easier and more direct for it allows direct simulation, andalso makes the interpretation of the aircraft responses in terms of flying qualitiesmore straightforward.2.4.3 Equations of Lateral MotionFrom eqs (2.64) and (2.62) the following set of equations applies to lateralmotion: +y = m [ Q Uor - Wop - g cos OO+] 1 = zxxp - ZXZ?
Equations of Motion of an Aircraft P = Ijr cos o0Expanding the left-hand side of the first three equations results in the following(subscripts A and R indicate aileron and rudder, respectively): = Zz2j.- zx2pAdopting the more convenient notation, namely:allows the eqs (2.73) to be written more simply as:For conventional aircraft, it can usually be assumed that the following stabilityderivatives are insignificant: Yc, Yp, Yj, Yr, Y1, YsA, Li, Li, Nc, Ni. .Note, however, that Yr may be significant if Uois small. When this assumption ismade the equations governing perturbed lateralldirectional motion of the aircraftare given by: 6 = YVv+ Uor- Wop- g cos Oo$+ YaRSR
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Equations of Motion in StabiEityAxis System p =$-*sin@0 i = Zfi. cos O02.5 EQUATIONS OF MOTION IN STABILITY AXIS SYSTEMThe aerodynamic forces which contribute to the x, y and z terms in eq. (2.65) arethe components of lift and drag resolved into the body-fixed axes. The angleswhich orient the forces of lift and drag relative to the body-fixed axes are: theangle of attack, a , and the angle of sideslip, P. The angles are defined in Figure2.5 where the subscript 'a' has been used to indicate that the velocity and itscomponents are relative in the sense of airframe to air mass. If the velocity of theair mass is constant relative to inertial space, then the subscript 'a' can bedropped. The velocity components along the body axes are:V, = VT sin PW, = VT cos p sin a 01Earlier it was shown that if symmetric flight was assumed, Vo would be zero.Therefore, if the axis system is oriented such that Wo is zero, then both a. and Poare zero. This orientation results in the XB axis, in the steady state, pointing intothe relative wind and the XB axis and the velocity vector being aligned such that:U' = VT (2.78)Such an orientation results in a stability axis system which, initially, is inclined tothe horizon at some flight path angle, yo, since:Figure 2.5 Orientation of relative wind with body axis system.