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Aircraft Dynamics and Automatic Control
By Duane McRuer, Irving Ashkenas, Dunstan Graham PRINCETON UNIVERSITY PRESS
Copyright © 1973 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08083-3
CHAPTER 1
INTRODUCTION AND ANTECEDENTS
We now know a method of mounting into the air, and, I think, are not likely to know more. The vehicles can serve no use till we can guide them; and they can gratify no curiosity till we mount with them to greater heights than we can reach without; till we rise above the tops of the highest mountains.
The economic or military value of any vehicle depends fundamentally on its ability to traverse a controllable path between its point of departure and its destination or "target." Abstractly, the vehicle is a velocity vector in space. It has a direction in which it is going and a speed with which it is going there. The time integral of the velocity vector is the path. Each type of vehicle, however, is made to move and carry in a certain medium and its motions may be subject to constraints. Means for the control of the path vary widely and depend on the constraints. Thus a train, for example, is constrained to move along a track; the control that is provided is merely a speed control. The train is not steered. An automobile or a ship, on the other hand, while constrained to move on the surface of the land or the sea, must be steered as well. Aircraft share with submarines and torpedos an unusual freedom from constraints, and the problems of the control of aircraft are of unusual complexity. We do indeed "know a method of mounting into the air," but the solution of the problems of control still requires both sensibility and diligence.
An aeronautical vehicle or weapon system contains spatial sensors, and guidance and control devices (possibly all subsumed in the human pilot) whose purpose it is to develop three-dimensional flight path commands appropriate to steering so as to reach a destination or target, and then to execute those commands by maintaining or modifying the forces on the vehicle so as to maintain or modify the velocity vector. This allows an intended purpose or "mission" to be accomplished.
Qualities of an aircraft that tend to make it resist changes in the direction or magnitude of its velocity vector are referred to as stability, while the ease and expedition with which the vector may be altered are referred to as the qualities of control. Stability makes a steady unaccelerated flight path possible; maneuvers are made with control. The path of an aircraft, however, is never stable of itself; whether through the intervention of the human pilot or by means of automatic control, stability is actually secured with the mechanism of feedback, a principle by which cause and effect systems are modified to secure certain desirable properties. Information about the effect (or output) is fed back (or returned) to the input and is used to modify the cause. Typical of feedback control is its speed of response and its accuracy in following commands and in suppressing the effects of disturbances. Also typical, however, is its tendency to "hunt" or oscillate. The particular advantages of feedback are enhanced by high gain, but this is inimical to dynamic stability, and high gain also increases the susceptibility of the system to spurious signals or "noise." Therefore, a designer intending to exploit the potential advantages of feedback is compelled to strike a fine balance between the desirable properties that might be secured and the pressing danger of disastrous performance.
The earliest aeronautical experimenters had hoped to achieve "inherent" stability (i.e., without feedback), and while many, such as Cayley, Penaud, Lilienthal, Chanute, and Langley, pursued this goal and discovered how to set the incidence of the tail plane so as to achieve longitudinal stability with respect to the relative wind, and to use wing dihedral so as to achieve "lateral stability," it gradually became clear that configurations with a large amount of such inherent stability were particularly, and distressingly, susceptible to being upset by gusts.
Speaking before the Western Society of Engineers in 1901, Wilbur Wright said: "Men already know how to construct wings or aeroplanes, which when driven through the air at sufficient speed, will not only sustain the weight of the wings themselves, but also that of the engine, and of the engineer as well. Men also know how to build engines and screws of sufficient lightness and power to drive these planes at sustaining speed. ... Inability to balance and steer still confronts students of the flying problem. ... When this one feature has been worked out, the age of flying machines will have arrived, for all other difficulties are of minor importance."
While this statement was somewhat optimistic with respect to the state of knowledge concerning airfoils and propellers, as the Wright brothers themselves soon discovered, it was correct in its essentials, and there is no doubt at all that suitable stability and control characteristics were the very last features of the first successful airplane to be developed. It is now generally agreed that the principal contribution of Wilbur and Orville Wright was their recognition that the frustrating search for inherent stability might well be abandoned if only the operator were provided with sufficiently powerful controls with which to balance and steer, i.e., that the human pilot, operating on feedback signals, could use the controls to stabilize a neutrally stable or an inherently unstable aircraft. Of course, the Wright brothers did not use this language, and indeed the recognition of the essential character of the airplane as an element in a feedback control loop came comparatively recently.
While the first automatic feedback control system for an airplane actually antedated the first successful flight by more than a decade, and the demonstration of completely automatic control of an airplane in full flight took place more than 50 years ago — in 1914, the means employed to secure satisfactory flying qualities of the aircraft themselves and to develop artificial stabilizers and automatic pilots were, at first, largely empirical arts. They seem to have made progress with a minimum amount of mathematics until after the end of the 1939–1945 war.
The modern view of the dynamics of aircraft and their control systems, in terms of the stability and response of the entire closed-loop (feedback) system, can be traced from its sources by way of three separate branches of technical knowledge, their confluence, and the recent advance and augmentation of the subject (see Fig. 1-1). During roughly the first 50 years of aviation's history, the study of the dynamics of aircraft and their control systems was of negligible interest to designers, who learned to get by with rules of thumb for proportioning the stabilizing and control surfaces and to develop automatic feedback controls by cut-and-try methods. This was in spite of the fact that a mathematical theory for the stability of the unattended motion and of the aircraft's response to control was developed at an early date. On the other hand, design trends since World War II, which have greatly extended the flight envelope of fixed-wing airplanes and introduced new types of vehicles such as helicopters, VTOL airplanes, ground effect machines, hydrofoil boats, winged missiles, and space launchers, have so enormously multiplied the number and type of problems that the techniques formerly employed in practice would have been totally inadequate. Very fortunately, wartime pressures produced two developments that fundamentally altered techniques for the design of automatic flight control systems. The first of these was the theory of servomechanisms; the second was the electronic computer. Analysis and simulation are today the twin pillars on which the entablature of aircraft flight control system design stands.
There has been an explosive growth in the practice of "experimenting" with mathematical models. It has been urged by both the expanding complexity of the problems and the increasing availability of appropriate methods and techniques. Further, the mathematical theory has served for the classification, interpretation, and extrapolation of the growing number of results of physical experiments.
It is to the development, exposition, and demonstration of methods of analysis and synthesis for aircraft automatic flight control systems that this monograph is addressed. It is not a text on design but is rather a guide to the consideration of the effects of vehicle and equipment features on the dynamic performance of the system. Where possible, the emphasis in treating the elements of the system is on the largest entities. Thus attention is directed to the response of the airplane to elevator motion, rather than to the change in airflow over the tail, and to the input/output characteristics of a rate gyro, rather than to detailed consideration of the torques acting on the gimbal. The vehicles considered are the ones that are heavier than the fluid in which they operate but which are acted on by significant fluid dynamical forces. This class includes at least the following types of vehicles:
Airplanes
Helicopters
Vertical takeoff and landing aircraft
Ground effect machines
Hydrofoil boats
Control, as somewhat distinct from guidance, is taken to be the subject of interest. For this reason it will ordinarily be possible to consider the motions in moving coordinate systems fixed in the vehicle and to avoid the coordinate axis transformations required to obtain the vehicle motion in, for example, a coordinate system fixed in the earth. When the origin of tae moving coordinate system is in an "equilibrium" state of motion along a nominal trajectory, the equations of motion of the vehicle can be linearized for small perturbations and the linearized equations will have constant coefficients. Then it is possible to use the convenient transfer function models for the dynamics of the vehicle, and all the analytical techniques for the study of linear feedback systems can be brought to bear on the problem.
Although there are a number of modern treatments of the stability and control of aircraft, all of which emphasize the same approach to the linearized dynamics that is to be adopted here, and there is also a very wide selection of both introductory texts and more advanced treatises on automatic feedback control, there has been a conspicuous lack of any significant treatment of these subjects in concert and therefore no proper introduction to the area between these fields. It is a fact that the methods of servomechanism analysis can be used as a powerful tool in the study of aircraft dynamics, and, additionally, that the characteristics of aircraft and their control systems provide a series of both subtle and complex problems that are likely to carry the student of feedback systems beyond what he may have learned in connection with the customary examples of remote position control, speed regulation, process control, and instrumentation. The discussion that follows will serve to bridge a gap between existing technical disciplines and to make more readily available some of the results contained in a scattered engineering report literature which is now familiar only to a small group of specialists.
The authors have adopted an eclectic view, taking from several fields what best appeals and suits but attempting, at the same time, to provide a unified treatment. Where a completely unified view is not feasible, the dominant theme is stated and the minor theme is contraposed.
It is the conviction of the authors that only the most thorough understanding of the dynamics of each element is a suitable basis for system synthesis. While digital and analog computers are now generally available to produce "solutions," even a sheaf of solutions may not clearly show the designer how to obtain the most satisfactory behavior and to avoid unpleasant surprises when the machinery is built. It is for this reason that the mathematical analysis of aircraft feedback control systems is emphasized throughout the treatment here. Of course, simulation and flight testing are valuable tools in the development of aircraft control systems, but, to an extent, a good theory is a summary of, and substitute for, experience, and the understanding which is conferred by analysis is a short-cut to the best results. It may seem, however, that a linearized theory is unrealistic because practical aircraft feedback control systems inevitably include nonlinear elements. The results that are achieved justify its use. Restrictions that are implicit in the use of linear theory are nowhere nearly as severe as might be imagined. In part, this is because linear approximations often have a substantial validity; in part it is so because feedback, in itself, tends to "linearize" the system.
Finally, it may or may not be true, as George Santayana said, that "those who cannot remember the past are condemned to repeat it," but there is enough truth there so that the history of the present subject can be studied with considerable profit. It is evident upon knowledgeable consideration that some costly mistakes might have been avoided with a better appreciation of the difficulties that confronted previous investigators of the problems of flight control.
1-1. Outline of the Volume: A Guide for the Reader
The subject of the feedback control of flight has a considerable scope and variety, and there is no canonical approach to its understanding. Its students will typically have acquired a considerable knowledge of the theory of linear feedback systems, and of the dynamic stability of aircraft and their response to control, as substantially independent subjects. The background of the typical reader will probably include some knowledge of operational or Laplace transform techniques for the solution of ordinary linear differential equations with constant coefficients, conventional servo analysis techniques such as the root locus and frequency response methods, response calculations with either deterministic or random inputs, and the describing function method for the treatment of common control system nonlinearities. While many of these matters are reviewed here before they are applied, the pace is brisk and the treatment is not intended as an introduction to the elements of the theory. The reader is further presumed to have some acquaintance with the dynamics of rigid bodies, although it is not, strictly speaking, necessary to have studied the dynamics of aircraft. Again, the latter subject is treated here ab initio but with a purpose not shared with the conventional texts cited in note 4.
Figure 1-2 is a graphical representation of the outline for this volume. The book begins, in this first chapter, with a definition of control appropriate to aeronautical vehicles and a distinction between control and guidance. This is followed by a brief summary of the advantages of feedback for control and an introduction to some of the earliest examples of feedback control. Historical sketches of the development of aircraft dynamic stability and control, practical automatic flight control systems, and feedback system analysis complete the introduction.
Chapter 2 comprises a review of those aspects of applied mathematics pertinent to the construction and use of linear mathematical models of aircraft and their control systems. The Laplace transform method and the transfer function model, which play such a prominent part later, are discussed in detail, and considerable emphasis is placed on graphical representations and graphical constructions. While the typical reader is assumed already to have a considerable familiarity with this material so that he should be able to move ahead rapidly, he is likely to find that certain matters such as time vectors and the steady-state response to polynomial inputs are treated here in a unique way that provides a background for subsequent developments.
The material of Chapter 3 is a condensed account of the particular topics in feedback system analysis on which the remainder of the monograph strongly depends. Here the reader will find not only a review of the root locus method and the conventional open-loop/closed-loop logarithmic frequency response methods but also their presentation as elements of a unified servoanalysis method that is a complete generalization of the semigraphical analytical techniques. The reader will also find here an exposition of multiloop analysis procedures particularly appropriate to the study of vehicular control systems and, finally, a discussion of sensitivity, including the connection between gain sensitivity and the modal response coefficients (time vectors or eigenvectors) of the system response. This chapter is one of the most unusual features of the volume because many of the techniques, and especially their highly organized connections, are not explained in the conventional textbooks on linear feedback system analysis.
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Excerpted from Aircraft Dynamics and Automatic Control by Duane McRuer, Irving Ashkenas, Dunstan Graham. Copyright © 1973 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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