j* STUDY* OF NONLINEAR SERVOMECHANISMS DISSERTATION Presented In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy la the Graduate School of The Ohio State University By Robert Lien Cosgriff, B.E.E., M.S The Ohio State University 1959 Approved, by* PREFACE It la the purpose or this dissertation to review the techniques that are available for the study of feed­ back control systems* It Is contended that all systems are nonlinear In one manner or another, and that If these devices are to be designed for optimum performance, the characteristics caused by system nonllnearlty must be recognized* Due to this contention, all the material beyond the first chapter (which reviews basic linear con­ cepts used In later chapters) Is connected with nonlinear systems* Most of the concepts and techniques that are considered are extensions of such linear concepts as gain, transfer function, frequency response, equivalent net­ works, phasor representation, and phasor (vector) diagrams, all of which are well known to most servo and electronic engineers. Some of the material such as phase plane techniques and Koohenburger 1 s technique la a review of the work of others in this field, which has been Included so that the development will be contIncus and not laoking Important information. The author believes that examples are as Important In illustrating a method or technique as In the formal A Q D 9 2 9 -11- deacriptlon of the method* For this reason, with every proposed method Is one or more examples of the use of the method In the analysis or design of typloal servo systems. Some of these examples have not been considered previously In detail. The accuracies of proposed solutions of some of the problems are indicated In most cases by comparisons of the analytic solutions and the analogue computer sol­ ution of the problem being considered. -111- NOTATXON During the past three years, the author has tried several notations for representing the Important variables of servo systems. After long and due consideration It was decided that the usual notation which used the Greek letters 9^ and 0Q as the Input and output variables of a servo system were not completely satisfactory for the study of nonlinear systems due to the number of subscripts that are cnecessary to designate the variations of these Input and output variables. Since this study of nonlinear servo systems grew out of the demand for servo function generators it has been very convenient to let the Input signal be represented by x and the servo output be designated by y, In which case the desired relationship between the input and output signals of the servo function generators becomes y z= f(x). The variations of the Input signal x and output y necessary for studying the response of a nonlinear system are given In the table which follows. A. Instantaneous x Input signal to the servo system y Output of the system B. Equilibrium condition (condition when all derivatives of x and y are zero) X Input at equilibrium Y Output at equilibrium -iv- G. Variation from equilibrium QC Input variation from equilibrium x - X -+ 3G q^°Output variation from equilibrium y — Y -+ V D • Phaaor Notation* X Input signal of the form a e*^w^ Y Output signal of the form b e^wfc * It should also be noted that the angular frequency in radians per second is represented by w so that the standard typewriter can be used In writing as many of the equations as possible* ACKNOWLEDGMENTS I wish to express my sincere appreciation Tor the Inspiration and guidance by my advisor, Professor C. E. Warren, which has led me to my present status* I am also very deeply indebted to the staff* of the Antenna Laboratory, particularly Bill Williams and Robert Lacky who have aided me In the construction of equipment, mak­ ing measurements and many other tasks, W* Ryan who did the photographic work and to A. Kail, Max Gorden and E. L. Huey who have done the most of the art work In this dissertation. I am particularly grateful to Professor Rumsey who saw fit to employ me during the last three years and who made arrangements with Wright Air Develop­ ment Center to sponsor the lnvestlgatlon reported In this dissertation* I wish to call attention to the fact that it would have been impossible for me to attend graduate school, if It had not been for my wife's encouragement and will­ ingness to accept the sacrlflcies during the first year of my pre-dootoral studies and her patience and thought­ fulness throughout this period* -vl- Table of Contenta Chapter I Fundamental Concepta page 1 A. Introduction 1 B- Differential Operatora 1 C* Block Diagrams 3 D. Frequency Respenaa 5 S. Characterlatic Modes of Linear Systems 7 Chapter II Detormlnation ef the Response of Analytic Systems about their Equilibrium States 12 A* Introduction 12 B. Theorem Concerning the Response of Analytic Systems in the Vicinity of an Equilibrium Point 14 C. Example of a Nonlinear System 19 D. Extension of Satisfactory Operating Range of Non­ linear System 26 E. First Order Linearization 31 F. Special Equilibrium Points 39 Chapter III Servomechanisms with Nonlinear Feedback 41 A* Introduction 41 B. Prototype Systems and their Variations 41 C# Multiple-Valued Function Generators 46 D. Accuracy Considerations 52 Chapter IV Phase Plane Analysis of Nonlinear Systems 56 A. Introduction 55 B• Definition ef Phase Plane and Phase Plane Trajectories 55 -Vi 1- C. Determination of the Reaponas Characteristics of Linear Systems by means of Phase Plane Trajectories 59 D* Phase Plane for Nonlinear Systems 63 E. Nonlinear Compensation for Servomeohaniams 67 Chapter V Large Amplitude Oscillation and Limit Cycles in Nonlinear Systems 83 A. Introduction 83 B. Gain of a Nonlinear System 83 C. Stability of Function Generators 85 D* Example of Logarithmic Recorder 89 E. Example of Square Root Generator 93 F. Example of Uncompensated Function Generator 96 G* General Method for Determination of Limit Cycles by Frequency Response Techniques 100 Chapter VI Frequency Response Characteristics of Nonlinear Systems 113 A- Introduction 113 3. Response of Open-Loop Frequency Response of Nonlinear Systems 114 C. Use of Linear Filters to Improve the Response of Nonlinear Systems 119 D* Indications of Limit Cycles and Subharmonlo Response 124 S. Response Hysteresis (Jump Phenomenon) 127 F. Reduction of Nonlinear Equations to Linear Equations with Time Varying Parameters 129 G • Network Representation 134 H. Maximum Sinusoidal Output 139 I* Response Hysteresis 142 J. Frequency Response of Nonsyrametrio Systems 147 -vin- Applications 1 5 2 Chapter VII Subharmonlo Response of Nonlinear Systems 158 A* Introduction 158 B m Analysis of a System with Subharmonic Response 162 C* Example 172 Chapter VIII Dimensional Analysis of Nonlinear Systems 172 A< Introduction 172 B. Change ef Variables 173 C. Dimensional Analysis and the Buckingham Theorem 176 D» Evaluation of Universal Functions for Servo Systems 178 Chapter IX System Performance Criteria 192 A- Introduction 192 B. Criterion fox* Recording Systems 196 C. Component Performance 201 Bibliography 2G6 Aut obiography 209 A ST'JDY OP NOhLINEAR SERVOMECHANISMS Chapter I Fundamental Concepts A. Introduction The material in this first chapter Is a review of the linear methods useful In the study of nonlinear systems. Each section of this chapter Illustrates a concept of nota­ tion which will be used in later chapters. B. Differential Operators I^eedback systems are characterized by an input signal represented by x , and a resultant or output signal y. The input x controls the output y. If the system is linear the relationship between the input and output can bo represent­ ed by a linear differential equation of thei form (1.1a) This equation Is generally written In the form (l.lb) where p la the differential operator d/dt; hence, pn is a shorthand notation for dn/dtn . The notation in (l.lb) may be further simplified to the form -1- -2- i ( p ) x — H(p)y, ( 1 . 1 c ) where C-(p) 3tands ror the operator and h(p) repre­ sents ^f^^p11* It snould be noted that these operators are not al eoraic quantities* Consider the meaning of the e/.oi’esai on (p ■+<*1 )(px) • ( 1 *2a } Tiiis means that p operates upon x and that (p - | - 0 ^ ) o p e r a t e s unon tne result ol the first operation* This can be ex­ pressed a 3 (P - ) (PX) = <*_^ g . £ * * V l | f . (1 • 2b ) d t o If the two operators are multiplied one obtains (p -pc^_p)x the resulting product has the same meaninr as the result riven in £l*2b). Multiplication, division, addition, and suo tract ion of operators can be handled ,ju3t as if the operat ors were alpeoraic as lonp as the c oef f ic ient so<f and ^ are constants* Care must be U3ed in handlin; equations in operational form, if the equations are nonlinear. For -xample, the operational expansion of p p ■+-y)y] (i.3a) ha3 a coefficient which is not constant, and the inner operation should be performed first, ^ivinp p(py-t- y2) - p2y -t py2- + 2 y If . (i.3 b) dt^ -3- The expression (p + y) (py)> ;iowev'T, Is equal to 2 (p + y)(py)* -t-y 3? • (1-4) d t fere we see that an Interchange of nonlinear operators results in a different differential equation* f . Hock diagrams The c’narac t er 1 s 11 c a of linear as well as nonlinear s/a t e-os can oe represented pi c t or la 11 y .