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 -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 . If x corresponds to a c 02': t roll in.; quantity wh ich produces a resultant y, trie system can j e represented by a clock diagram as snown In Tig . 1. The process cun oe losericed ,3 follows* fh e si nal x Is , or.orated ;.;v some G (p) y ^ H (p) source a;id Is fed Into the jox which operates upon t’d 3 f I . 1 Block dia ~ran signal producing n out put 3 I5 - roprose fatlon ci a system .,al y« If the system is linear the opera 11 o . per form el by the o ox can oe represented 07/ the operator G(o)/H(p). This means that for an input x, y can 3 e wr111en as y - l r ^ 7 *■ (1-5) wnare G(p) and H (p) are polynomials In p* Notice that the arrow heads indicate the direction of the signal flow t'nrough, the system* Often two or more inputs are summed and an operation is per forme 1 up :• this sum • This c; .n.n. ■; I "n la r- o ted in -4- olock form as shown it. I51!,”. £ and the equatlo:. represent ‘_ t:. - a y ate n i s ! j ( y ) c - . ~ > — y ■ (1 *6 ) T T p T l"*" 2 x 2 h i : . ; . 2 hi oa k lie, r a m w i t h t w o i n p u t 3 flock dia::ram3 may oe int ere onnec t ed in various ways 3 Two blocks int ereo n n e c ted b l o c k s in this figure represent the followin- operations; ) (31, + x9 ) — y, (1*7) P + P (1 .8) -( P + « 2 * notice that the input to the lower block of i'lg. 3 is v at id its output is X g . Equations (1.7) and (1.8) may be combined eliminating Xg, if desired, by substituting ‘■y/(p+-°<) for Xg in (1.7). - 5- -.loc k diai'.ra'ns -re actually m o d e l s of mu themu deal e q u a !. ons . Several models exist which, have the same equation s. for example, the diagram saown In fi^ . 3 la e.jol vuloMt mathemat i call y to that shown In hi; • 4. Slock diagrams w ’ll ne used In this paper lor the purposes of a 32. mple r e p r e s e ;tation of the problem and lor indicating method of solution# D* Frequency Heaponse One of the more Important types of calculations in servomechanism theory is the determination of the output y when the Input x Is of the form a sin wt• If the system ’.3 linear and staole the steady state output for this ex citation will oe of the form b ain(wth 0) . The quan titles b/a and © evaluated as functions of frequency are termed thi frequency response characteristics of the system* One can evaluate the quantities b and © by assuming the Input and output to be of the form just mentioned, but a more convenient method^" in. common use is to assume that the Input i wt Is tee complex quantity a e . The output signal y is ¥w t + ©) tnen of the form b e^ f • A special notation is used for these complex quaw§t— ties. Complex values of x and y are represented by complex x » "X » ae ^ ** ) , v , j(wt-t- 0) (1*9) complex y * Y -b e • 6- a x Pig. 4* A block diagram equivalent mathematically to * the blook diagram shown In Flg« 3* ,VL i.. t: i ; qua titles X and Y are used in place o:’ the euerul i uic 1 1 ons of time x arid y, the operators G ( p) and ■- ( p) sc one functions of jw. This can oe oo served rjy ca:’ryI . out the operation f(o) X = < £ V ^ - a ©>*= - aejwt . (l.lO) ndtn ^ 1 (doe o:,. (1.1) for the definition o:' G(o).) If G(jw) is ief* nod by -( jw) — )n (1.11) o n e has G ( p) X - G ( j w ) X . (1-12) notice that the algebraic quantity G(jw) can be obtained by suostituting jw for p. Equation (1.1c) can be written, If x is of the form ae^w ^, as G ( jw) a * J'wt:^= H(jw) be*1 G rlaracteristIc Modes of Linear Systems A linear system can have an output y when the input x is zero and still satisfy an equation of the form given by (1.1a). The solution of the system equation under tills -8- c v:d:t t ' >;n is the transient or complementary solution of the lineal’ i'ffere.l'.al equati no. If the output y Is repre sented by oe°J and the ’nput variable x Iszero, (1.1c) o o c o rr e s d(s) beSti 0 , CL.15) fotloe that the qua itltv a can be real , complex or irnagin- ary. Equation (1.15) has a solution other than b=0 , only if K (s)— 0. (1.16) This latter equation Is often called the characteristic equation and the values of s for which H(s) vanishes are called the roots of the characteristic equation. If (1.16) has m roots represented by s*^, 3g, • • • sm the general solution for y, when x Is zero, is the sum £2 snt y = b e - (1.17) sn t th Each term bn e of (1.17) Is said to be the n character istic mode of the system. It is desirable that these characteristic modes uecome small in magnitude in a short t■me since they are effectively error terms or unwanted terms In the system response. If the real part of each of the roots sn Is a large negative number, then each of the characteristic modes will become small in a short time. If one of the roots of (1.16) has a positive real part the characteristic mode corresponding to this root will - 9 - ':Vx t;h t* c: e arid a system wi t', one or more r o n t. a with I t!vft real parts is said to be unstable• dh-i tramfer functio.: olays an important role in the j ta oy ?f 1i :i ear systems, part 1 cularily when the s ;s t en is re ..roue. tod in block diayran form. If in Fi =•:. 1the in nut S 1j is ns sr.mod to o e ae ’ then trie output y will b e of the 3 * r1 31 • tj ro n s ’ • S ’rice p :e is equal to s‘JeJ , su stitution of the assumed or*ms of x and y into eo . (1.1c) results in the ■ y ;a i on G(s) aest= , h(3) be3t , (1.18) and tiie ratio of the output to the input then becomes y b e 3 ^ _ G ( s ) \ i - r y s r - Trnrr (1*19) fhe ratio f ( 3)/h ( s) is called the transfer fbmction of the n1ock• fire c r'!arac t er i s t i c modes of the feedback system shown In dip,, 3 may be determiried by the tech iques demonstrated an ->ve . r ir st, let the transfer function of the upper .lock hi . 3 be(1/ s )/H]_ (a) and that of the lower oe bg(s)/Hg(s). Let the output of the lower block consist of a single characteristic mode represented by Xg and the output of the upoer clock be y. Assuming that the input x-^ is zero, one has for y making use of the transfer func tion of the upper block - 1 0 - p ( S ) x 2 ~ ’/ - <1 .2 - 0 7.i i a .; In':!, la** maiiner the relationship between the input a i. . output o the lower block can be e a t ab 1 i she i t namely *2 C 3) _ y 'IgTsT ~ x2 * (1-21) It x0 Is e 11 mi nu t ed from the last two equa 11 on 3 there results ap <3) y = Hg (a) H ^ s ) y . (1.22) The characteristic equation of the system la l2 (s) Gita) - K 2 (s ) H i (a) — 0 * (1.23) Notice that the roots of (1.23) are determined by both the G and H functions when the blocks are Interconnected* Equation (1.22) could ; ave been written G2~.[p).. fJ£yP.l y = y, ( 1 .2 4 ) h JT r J Hg{ p) when y Is a natural mode. Here G0 (p) G-^ (p)/&-^( p)Hg (p) *s the loop operator or loop transfer function.4 There is a basic philosophy not apparent In (1.24) which Is nest descrloed veroally* In a feedback system wi tlx no Input ■K;ua 1 to y • f h a t i :i if art out pu fc y after oeinu ooerat t e d o n b ;; l:1 o loop t r a n s f uiil y is c onpos ed of ''u.idamental modes. - 12- Chapter IX Deurnlna'ion of the Response of1 Analytic Systems about their* Equilibrium States a . Introduction It is the purpose of this chapter to Introduce small signal theory for the purpose of studying the response of nonlinear systems- This method of analysis I3 very con venient for determining the response of a class of systems p which are termed analytic • The nonlinear terms in the equation expressing the relationship between the output varlaole y and the Input variable x of an analytic system can be expanded In a convergent Taylor's series In terms of the Input x and output y and their tine derivatives for small variations auout the equillbrlum condition. Consider for example a system whose output and In.>ut variables are related by the nonlinear relationship F(x, px, p x, • *;y, py, p2 y, • •) = 0 (2.1) If the rates of change of the variables x and y and all their nigher derivatives are zero and eq. (2.1) is satis fied for real values of the variables x and y, the system is said to be at an equilibrium state. The values of x and y at an equilibrium state are of importance in this chapter and for this reason the values of x and y at equilibrium are represented by the symbols X and Y respectively. The possible sets of values of X and Y can be found b y —13- ev&luating (2.1) wi th all the derivatives of x and y set equal to zero, namely by the evaluation of F(X,0,0, • •;Y,O,0, • .) — F(X,Y) = 0. (2.2) liocice that there may oe one or more values of Y for a , ■ I v q n 1 n pu t X • In determining the response of a nonlinear system In the vicinity of an oqulllcrlum state the nonlinear function as expressed in (2.1) is expanded In a Taylor's series aoout uhe equilibrium point (X,Y). Rather than write this series in terms of x - X and y - Y and their derivatives, two new variables 0C and s r are Introduced which are equal to x - X and y - Y respectively. After expanding (2.1) and substituting QG and into this expansion one obtains an equation of the form F(X,Y)+ (aQ -f a^-t* agp2, • ) cq d - c ^ +- cgp2 • • )?£ "*■ 2/"^d0't“ dlp r d2p2 * •)*** * - (bo + blP "*"b2 P2 * ■) 2T* (2.3) All the constants - the a 1?, b's and the others - are determined by the Taylor's expansion of F. For example an =• dF/d(pny) and dn — d^F/dyd(pnx ) both evaluated at the equilibrium point. Since the system Is nonlinear many of the constants may be functions of the quantities X and Y* -14- h. Theorem Concerning the Response of Analytic Systems In the Vicinity of an Soullibrium Point A useful inequality relating the magnitude of the nonlinear terms in eq. (2.3) will be developed. This ine quality can then be used in the proof of a theorem which states that in the neighborhood of the equilibrium point the response of an analytic system is determined y the linear terms of the Taylor series. First eq. (2.3) is divided Into linear terms and nonlinear terms as Indicated In eq. (2.4) the operators G(p) and H(p) are equivalent to eq. (2.3) respectively* The term F(X,Y) of (2.3) need not be Included In (2.4) since b;y (2.2- It is equal to zero. The nonlinear terms of (2.3) such as Three numbers will now be defined which will be used In the Inequality* L A number larger in value than the absolute value of any of the constants In eq. (2.4), that Is, the a*s, b 's, etc. k A number larger In value thaxi the absolute value of^G or 3T and of the derivatives of arid 3f for some period of time T. It is assumed that Initially the system Is displaced from the equilibrium coalition, j This number is equal to the total number of the variables and and the derivatives of and If s. occuring in eq. (2.3). p The terms such as and dQ from eq• (2.3) making up f are now divided into groups. The sum of the exponents of the variables «)& a n d ^ f and their derivatives occurim. In each of the terms of any given group are all the same and this exponent sum Is represented by q. For — jk example •& ^ and 0& d^T/dt belong to a group with q equal to five. The sum of all terms whose exponent sum Is q will be represented by Sq. The total num b e r of terms whose exponent sum is q Is less than for j and q equal to or greater than two."* The absolute value of each term In a * This Inequality can be demonstrated as follows: Con sider the ^ combinations that can be formed from j vari ables (2G , p3fc ,py , and the others) taken q at a time. If all of the varlaoles In any of these combinations are multiplied together one will have a term which ojcurs In the q®*1 group. Due to the permutations that occur In the combinations there are two or more combinations that correspond to terms In a group which contain two or more variables. For example the c omb Ina ti ons both correspond to the same term In the gi jup with q equal to three. This indicates that the number of terms in a group is less than the number of combinations possible. ;;roup is less tlian L kq si,ice L is larger than any coeffl- cienfc an i k is lar/rer than any variable, thereby making kq larger than any oroduct of variables in the i;roup with exponent sum q. Because of these two inequalities, one expresain. the maximum absolute value of each term in a ..roup and the ot er t ie total numoer of terms in the ;:roup one can write for the absolute value of SQ; (Sq| < L ( jk) q . (2.5) Since f is the sum of the Sq 's fox' q ranging from two to an upper limit which may be infinite, a relationship involving the absolute value of f may be written as f ol l o w s : |f| •=■ I t. s | ir £ |s I . (a.e) 1 q-2 q* q-2 q The righthand inequality hollows since the absolute value of a sum of terms is equal to or less than the sum of the absolute values of the terms. Equations (2.6) and (2.5) can be comoined , i vi.i the Inequality m e l ? (j n q • (2.v) q-2 Fortunately the sum indicated oy (2.7) can be summed in closed form and reduces to Theorem I t If the variables 3£» 9 and their derivatives occurlng In the nonlinear term f of eq. (2*4) also occur In the linear terms of eq. (2*4) and If their number Is finite, then the stability of the equilibrium state of the system Is determined by the linear part of eq. (2*4), neglecting the nonlinear portion f. Proof: Normalize eq. (2.4) by dividing by k giving (2.9) In the period T the upper bound of and their time derivatives is k by definition. If k approaches zero, the normalized variables 3&/k and /k as well as the time derivatives of these normalized variables still may remain finite over part of the period T. On the other hand the function f/k must approach zero In the limit for by (2.8), f/k can be made smaller than any small number £ by the proper oholoe of k. Since the term f/k becomes vanish ingly small In the limit as k approaches zere eq. (2.9) In this limit becomes equal to (2 .10 ) during the Interval T. This indicates that In the limit as k approaches zero the response of the system can be determined by the linear terms of eq. (2.3) alone. If the system Is stable the response can be determined for a period of T of any duration, but if the system Is unstable the response can be determined only over a short Interval of time alnoe the Variables <3o and will Increase In time forcing one to ohooae large values of k for long periods of T* Close and critical examination of physical phenomena leads to the conclusion that there are few if any linear systems* It also appears that there Is no system that is analytic in the sense defined in the previous discussion* In vacuum tubes, the current arriving at the plate cannot be expressed with complete exactness in a power series since the current is due to the transport of discrete charges on the electrons* In mechanical systems nonlinear effeots such as backlash and coulomb friction cannot be described by a power series due to the discontinuous forces that they cause* In the design of servomeohanlsms It Is generally the prooedure to reduce the effects of these discontinuous processes as much as possible, and then to assume, when making an analysis, that the effects that do occur are sufflatently small that they can be neglected* Ibis assumption may be valid for large variations about the equilibrium point, but It Is definitely not true for small variations* To Illustrate this point an instrument servo whose output position was proportional to the Input voltage was tested* First the input to the servo was changed suddenly by three units* The change of the Input and the resulting -19- change of the output are shown in Fig. 5a* Notloe that In time the output approached the new Input* After the recording in Fig* 5a was made the sensitivity of the recorder was Increased by a factor of 40* This time a very small step function was applied to the input of the servo* Both the Input and the resulting output of the servo for this small ohange of Input are shown In Fig* 5o* Notloe that In Fig* 5b there la an observable delay between the time that the step function is applied and the time when the output oeglns to move* This delay is due to back lash* Likewise the output position never reaches the position corresponding to the magnitude of the Input* This steady state error Is due to coulomb friction* Just as It Is convenient for analytical purposes to neglect the discontinuous effects in systems which are considered linear it also is convenient in the study of many nonlinear systems to negleot these same discontinuous effeots* This will be done in the remaining sections of this ohapter as well as In the following chapter* C. Example of a Nonlinear System Consider the nonlinear system shown in Fig* 6* The input voltage x is applied across a linear potentiometer* The wiper position of the potentiometer is controlled by the servo output y* Vhen the servo output position y is unity the potentiometer wiper Is at the ground point and -20- UT Fig* 5(a)* Response of servo output whan sarvo Input la a large step function. Effecta of backlash and coulomb friction not apparent. Fig* 5(b)* Response of sarvo output when servo Input la a small step function* Effects of backlash and coulomb friction are apparent* X -y / / o / J Pig* 6* Example of analytic nonlinear ayatem aa y moves towards It a zero position it moves the wiper of the potentiometer towards the high aide of the potentiom eter winding* The signal produced at the output terminal of the potentiometer oan be repreaented by (x)(l - y ) • The equations describing the operation of thla servo oan be written aa x(l - y) - C — e (2 .1 1 ) These two equations oan be oomblned giving (2 *1 2 ) The equilibrium condition aa Indicated in Section A of thla ohapter oan be determined oy substituting X and Y for x and y respectively and letting all time derivatives be z ero, thus X(1 - Y) ~ C . (2*13) - 2 2 - If -*• X and ~t~ T are substituted for x and y respec tively and the equilibrium condition Indioated In (2*13) Is dropped one has the equation but by the theorem in Section 3 product terms oan be neglected in studying the response near the equilibrium condition* For this reason for small values of 3d and one la assured that the response of the system, shown In Fig* 6, oan be determined by *** X = (pS P + X)ST • (8.15) Notloe that certain coefficients In (2*15) are dependent upon the equilibrium condition* The oharaoterlstlo modes of the linear equation given In (2*15) will now be studied to indicate the manner In whloh the response of the system will vary as the equi librium point of the system la changed* It was indicated in Chapter I that a natural mode la of the form be"fe* The characteristic equation of the system Is given by eq* (2*16) Is (a2 a -*■ X) e** = 0 . (2.16) The values of a whioh satlaify this equation and whloh are represented by s^ and Sg are 23- - 1 -+■ -IT - 4X "5“ (2.17) - 1 - TT - 4X s2 T" These roots, s^ and m%9 oan be plotted In the complex plane as shown In Fig* 7* Notloe that as X changes the position of these roots also change. For X 0, s^ Is equal to -1 and sg to zero. At X 1/4 the two roots are equal, and for X greater than 1/4 they become complex con jugates. The path* taken by the two roots as X varies from zero to greater than 1/4 are shown In Fig. 7. Plotting these roots In this manner Is similar to the method proposed by Evans9* If the system Is to have a rapid response with out exoesslve overshoot It is neeessary that these roots lie In the shaded portion of the complex plane Indicated In Fig. 8. If the roots lie to the right of the vertloal dotted line the characteristic modes of the system will not reduce in magnitude sufficiently rapidly. In other words the real parts of and Sg are too small to allow the system to ohange position rapidly* On the other hand. If the roots lie In regions 1 and 8 the system will be highly oscillatory* If the shaded region of Fig. 8 la superimposed on Fig. 7 It will be found that the response of the system Indicated In Pig* 6 will have a satisfactory response for only a limited range of X. For small values of X, s^ lies In the region to the right of the vertloal 2 4 - Locus Imaginary X = O Real X = -fr Fig. 7 Paths taken by the roots or the small signal characteristic equation of the nonlinear system shown in Fig. 6 as the equilibrium point changes -25- Region Imoginary Real Region Pig* 8* Shaded region lndloates acceptable poaltlon of roots of characteristic equation. - 2 6 - line In Fig. 8. For large values of X, e^ and Sg lie in regions 1 and 2* ri'ja this diaouaslon one may for* the conclusion that the roots of the linearised equation should be restriotod to a definite region if the system is to operate satisfactorily* D* Kxtenslon of the Satisfactory Operating Range of Nonlinear Systems In the laat section the quality of the response of a nonlinear ays ten near an equilibrium point mi determined by the position of the roots of the oharaeterlstlo equa tion of the system in the oomplex pla.e* If the response about some equilibrium points is undesirable It la neces sary that the linearised equation be modified so that the oharaoterlstlo roots will fall in an aooeptaole ran^;> of the ooaiplex plane, that Is, the roots given by (2*17) must fall in the shaded region of Fig. B. Also, If possible these roots should be Independent of the equilibrium con dition* A development of this type will now be oonsldered* The nonlinear system to be used In this discussion will consist of a linear section and a nonlinear section (see Fig* 9)* H«p§ the nonlinear seotlan performs an operation upon the input x and the output y and produces an output e^* The output e^ excites the linear section B chose output is y* - 2 7 - No n I j near Elem ent Fig* 9* Block diagram of nonlinear cloaed-loop system Let the transfer function of block B he represented by H(p), In whloh ease the relationship between e^ and y la y H(p) ex . (8.18) Notloe that H(p) Is a linear differential operator. The output e^ of the nonlinear devloe is a function of the variables x and y and their tine derivatives, that Is, ex = f(x,y; px,py; p2x,p2y; • •) ♦ (8.19) At equlllbrlun, x and y are replaced by X and Y aa before; and all the derivatives of x and y are aero. The equlllb rlun conditions oan be obtained by the solution ef the e quation Y = H(0) f(X,Y; 0,Oj * *) , (2.20) - 2 8 - whloh Is obtained from (2.18) and (2.19)•* Equation (2.19) oan be expanded in a Taylor series about the equilibrium point (X,Y) aa described In Section A of thla ohapter. Likewise and ^ will be substituted for x - X and y - Y respectively. The resulting linearized expression for ex which neglects all product terms la *1 - £ *n sp a& ■+ £. * a l P g f t (2.21) where the coefficients are given by -*L- I d (pnx ) I x - X, px * p2x = • • * 0 Y * Y, py ^ p2y - • * ~ O (2 .2 2 ) b n ~ - df d(pn y) i x - X, px ■» p2x » • • * O y = Y, py - p2y - • • * o It should be recalled that (2.21) Is a valid equation only for very small variations about the equilibrium * Thla relationship oan be demonstrated as follows} Let the transfer function H(p) of block 3 be of the -+ °1P ^ ft2P 8 * general form . . Not lag eq. (2.18) ptt(d0 -+* dxp -+ Ogp • •) and the value of e^ one may write Pn (d0 d.p -t- d2P2 * *)y - P*(dQ dip -r •flP2 * •)* • the variations of y about Y and f about YtX,Y) approach zero the derivatives of y and of f also approach zero. In the limit as these variations beoome zero the equilibrium condition can be determined by substituting zero for p. Generally m Is zero and a la one so that the equilibrium oonditlon Is given by c q f (X,Y) — O . - 2 9 - polnt at whloh the ootnstants are evaluated* A linear relationship between 3 6 a n d ‘If' la obtained If (2*21) La substituted Into (2*18), and the equlllorlum relationship given by (2*20) la removed* Thla gives - H If the Inverse of H(p) equal to l/H(p) la defined as Hip), the transfer function of the nonlinear ayatem for amall variations of 3* and ^ la given by y _ ^ ‘»pn . (2.84) 36 HlJ) -£bnp“ The stability of the equilibrium oondltlon oan now be Investigated by using tfyqulst's or Routh's orlterla* It should be noted that the a'a and b'a used In (2*24) are not constants In the same sense aa In linear systems, but are functions of the equilibrium variables X and Y* For thla reason the transient response of the system near the equilibrium point Is determined by the equilibrium state as was illustrated in Section B* If the a fa and b'a are constants. Independent of the equilibrium oondltlon, the system will have the same response about any equilibrium point* It will be shown that this is an impossibility If the equation relating X and Y Is nonlinear* If the relationship between X and Y Is nonlinear It la usually oonvenlent to foroe the b'a to - 30 - be constant and to lot a^ bo a function oonalstant with the relationship between X and Y. In thla oaae the num erator of (2.24) can be written aa a. £ (a p’Va . ) • If u ^ n o the ratloa *,*/•() ar# foro#d to b® oonatanta the system will have the aame reaponae about all equilibrium pointa exoept for the amplitude of the reaponae whloh la propor tional to and will change aa the equilibrium oondltlona changes. It will now be shown that the alope dY/dX la equal to the equilibrium transfer function given by (2*84) when p la set equal to zero. 3be do steady state values of X ■+■ cXa and Y * ^ If and ^ *r® infinitesimal In mag nitude. Thla la true alnoe very small values of^£ and gf whloh satlalfy (2*24) alao aatlalfy the original eqs. (2.18) and (2.18) by theorem 1, and by definition of equilibrium (aee section A) oonatant values of x and y whloh aatlalfy the original equations are equilibrium values. The 1nore- mental variations of X and Y, that la ateady atate do values of JG and If Oq and b^ are oonatanta not funotlona of X and Y, then dY/dX must be equal to the oonatant whloh results when p Is replaced b y zero In the transfer function given In (8.24) . If dY/dX la a constant, then the equilibrium relationship Is of the form Y D q -f- D^X, where both Dq and are constants; and therefore the equation relating Y and X must be linear* E. First Order Linearization The process of forcing a system to have a transfer function with oonstants a*a and b'a of a type that Insures a uniform response about any equilibrium point will be called first order linearization* This system lineariza tion can be accomplished by several methods* One method whloh can be used when f is a function of only x and y will be taken up now* Consider an amplifier whose gain Is con trolled by y as shown In Fig* 10* (Gain Is defined here as a transfer function whloh la Independent ef the differentIdL operator p*) The Input to this amplifier la e^ and Its out put e^ drives block B* Letting the gain of thla amplifier be denoted by G(y), the Input eg to block B oan be written as eg s: f U,y) G(y) • (2.25) The produot f(x,y) G(y) oan be expanded In a Taylor's series about the equilibrium values of X and Y giving -32 Nonlinear E lement Fig* 10* Blook diagram of a nonlinear ayatern with first order linearisation -33- •2 = f (X,Y) G(Y) -+- G(Y) (•0 3& -+ bQ^ f ) * ^lly*Yf(X'Y> * <8*26> Observing that the equilibrium condition for thla system is given by f(X,Y) G(Y) H(l) Y, (2.27) the small signal transfer funetlon of the system obtained from (2.26), (2.27) and the use of the Inverse transfer function H(p) of blook B becomes y o ( y ) * o ______ " h t J) - a(Y) bo - f(X,Y) B0 (2*20) where Is equal fee dG/dy evaluated at y - Y. If the response Is to be uniform, it Is neeessary that G(Y) bQ -+• f (X,Y) Bq be a constant Independent of X and Y. First order linearisation for this restricted ease is obtained by adjusting G(y) te satlsify G(Y) b Q -+ f(X,Y) Bq ^ -X (2.29) where K Is Independent of X and Y» Equation (2.28) then becomes X _ 0(Y) »o 'Y ~ —1 (2.30) 3 6 H?p) + K - 34 - Example: The blook diagram of a sarvo square root generator to be examined Is shown in Fig. 11. In this system the square of the output y Is fed back to the differential* For this reason the differential output e^ or f is equal to x - y2 * The transfer function of £ will 41 be assumed to be of the form p(p -+- d) • Two cases will be considered* In the first oase the gain of G will be assumed to be unity* Using values of aQ and bQ obtained by substituting f into (2*22), eq* (2*28) becomes ' ----- • (2.31) *** HTp) + 2Y The gain 0 for the second case will be assumed to be l/2y* This value of G satisifies (2*29) since H7p) Is of the form p(p ■+- d), and bp is equal to - 2Y* The value of the transfer function for this second case obtained from (2.28) becomes (2.32) 35 h TJ) + l The transient value of Is uniform in the second case while the response for the first case Indicated by (2.31) changes with Y* The responses for the two oases are shown * This assumption simplifies the equilibrium relation ship between X and Y to f(X,Y) — 0. See eqs* (2*27) and (2 *2 0 )* - 3 5 - Flg. 11. dlook diagram of aquara root ganarator - 3 6 - ln Pig• 12. The equilibrium conditions are shown aa straight lines while the transient responses are the small deviations from these straight lines* In our general ease f was considered to be a func11on not only of x and 7 but also of the time derivatives of x and y* At times it is possible to allow G to be a funotlon of x and y and their derivatives. In this general ease eg the Input to blook B oan be expanded In a Taylor series as In (2.26)* The equilibrium conditions oan be found In the manner Indicated by (2*27) and finally the small signal transfer function can be obtained. This transfer funotlon Is G(X,Y) £ a_pn ■+ f (X,Y) £ Anpn 2L ^ ------“------tt------(2.33) oc H7p) - G(X,Y)£ bnp» - f(X,Y) £ B^p11 where the constants an and bn are aa given by (2.22) and - do *n = dCp®*) x - X # px * p®x - • • •pnx * 0 y - Y, py ^ p2y - • * •pny' 0 (2.34) dCp^y) x - X, px = p2x • * •pnx - 0 y = Y, p y - p 2y • • •pny ^ O . If the sums a -+• A„ and b_ -r Bn are oonatant for all n } n n n n that la - 3 7 - UNCOMPENSATED COMPENSATED O O t o t Fig* 18* St«p funotloa rtsponi* of iqotre root generators \ - 3 8 - G(X,Y) An^ Kln (2.35) G(X,Y) + f(X,Y) Bn — ^2xi with K^n and Kgn independent of X and Y, then the response will be the sane about all equilibrium points. Slnoe the steady state ratio of r\f/Qd for constant ^ and ^ Is equal te dY/dX the ooefflolents Indicated by eq* (2*37) foroe the equilibrium relationship between X and Y to be a linear equation* (See the last paragraph of section D for the method of proof*) If the equilibrium relation ship between X and Y la nonlinear the transfer function (2*33) oan be written In the form (2.36) h7p> - 0(X,T) £ bnp“ - t (X,Y) £ The coefficients are now required te be B(X,T) »n -» f(X,Y) An J(X,Y) (2.37) Q(X»Y) ba + f(X,Y) Bn = . If the coefficients of (2*36) are as given by (2*37), the transfer function of ('Jf/J(X,yJ/3& will be Independent of X and Y* The reaponae of Jf/J(X,Y) for some Input 36 will alao be Independent ef X and Y* -3 9- F. Special Equilibrium Point a A possible plot of the equilibrium value of Y aa a funotlon of X la shown In Fig. 13. There are aeveral points whloh are of special Interest. These are located at points where the slope of the equilibrium curve dY/dX Is zero or infinite. Notice that at these points the transfer funotlon with the operator p set equal to zero must alao take on the values of zero and Infinity. For a system whloh is linearized these values of dX/dY require that Q(Y) aa given In (2.28) or J(X,Y) in (2.36) be zero or Infinite respectively at these speolal points. It la quite obvious that an amplifier with an infinite gain la impossible; likewise It would appear likely that the more general funotlon J(X,Y) can not be infinite* For this reason, difficulty la experienced when attempting to design a system which will operate satisfactory near an equilib rium point where dY/dX Is Infinite. It has been the custom to alter the equilibrium relationship In the region where dY/dX is very large in the manner shown by the dotted lines In Fig. 13, so that the slope will be finite* 40 - Flg* 13• Modification of equilibrium conditions near points where dY/dX le infinite -41 Chapter III Sorvomeohanlama with Honllnear Feedback A* Introduction The material in thla chapter haa been developed Tor the analysis of nonlinear recording systems and haa been used in the design of several of theae systems* In these recording aystems the feedback devlee Is purposely made nonlinear* This is done so that the servo output y approaches some funotlon, such as the logarithm or square root of the input x* Two typical nonlinear feedback systems were used as examples In Chapter II and are shown in block diagram form In Figs* 6 and 10* B. Prototype Systems and their Variations The baslo blook diagram of a nonlinear reoordlng device Is shown In Fig* 14* This configuration Is the prototype system* The prototype has two nonlinear blocks, N and G* Blook If produces an output f^(y) and Is called the nonlinear feedback path and blook G linearises the system* Generally the systems are designed so that e the Input and output variables x and y satisfy a relation tx (r ) = x (3*1) at equilibrium and that it be 1m error as little as possible when the Input signal x is varying* When (3*1) Is - 4 2 - — Hip) T ^ y ) N ZJ Pig* 14* Prototype function generator X „ Non linear ■ ••» H(p) Block y ...... + „ Pig* 1 6 * General representation of function generator - 4 3 - satlsfled the output y la the dealred function of x* Ihe equilibrium relationship indicated In (3*1) can be shewn to can be demanatrated by observing that f(x,y) aa defined In Section E of Chapter II, which la equivalent to x - f^(y) here, has an equilibrium value of zero when the transfer funo 11 on H(0 ) la zero a a Indicated by the solution of (2 .27)* If It la dealred to dealgn a aquare root generator, eq. (3.1) will be of the form x rs y2 * This Indicates that at equilibrium the serve output ahaft position la propor tional to the square root of the input x* If a recording pen la driven by the aervo output motor the pen position la proportional to the square root of the Input signal x* If the dealred function of x to be generated Is y = F(x) , (3.2) then the output function f^(y) generated by the nonlinear feedback path H must be such that y^FUj^y)) . (3.3) The latter equation la obtained by substituting (3.1) Into (3*2)• Thla means that f^(y) and F(x) are Inverse functions. The seoond nonlinear block G la neoeaaary If the ayatem la to have a uniform response for all equilibrium states aa waa pointed out In Chhpter II* The gain G(y) of -44- block G must be controlled by the servo shaft position y such that G(y) will satlslfy the linearization requirement given by eq. (2*29) • To evaluate G(y) observe that f(x,y) used in Chapter II Is equal to x - f^(y) and that f(X,Y) or X - f^(Y) is zero* For this reason G(Y) = -K/b^ therefore | x = X> P* p2x • • • =0 y y — Y, py p2y • • • 0 (3.4) since I s equal to df/dy evaluated at equilibrium con ditions by (2.22). As tho derivative df^(y)/dy evaluated at an equilibrium value of y will be used often In this chapter it will be denoted by f^(f). The gain G(y) will be allowed t# be equal to B^(d^(y)/dy) at all times slnoe this satlsifles (3.4). There are a large number of modifications that can be made In the prototype which will not alter the operation of the system. For example# consider Fig. 15. All non linear blocks whleh perform an operation upon x and y and produce the same input to bleok H, namely e^# will produoe the same output y, slnoe the operating equations* will be the same In all oases. * The term "operating equation* will be used often, and It Is defined as the equation shloh describes the relationships between the variables of the system* - 4 5 - Thaorea 2 : XT a flyfltda li modified In suoh a mjumf that the Inputs x and fx(y) to tkl# differential are multiplied by * 0 and the output from the differential la divided by s aa is Indicated by Pig. 16, then the operation of the system will be the same as that of the original system shown in Fig* 14* Proof: Consider the input to and the output from the bloolc Indicated by the dotted lines In Fig* 16* lhe output e'/i can be written aa e t x% - f^Cy)* f - = ------(3.6) * s or ■ z" = x , (3*6) jej xz z H (p) y y ) N Fig. 16* Modification of prototype system by multiplying input to differential by s and dividing output from differential by 9 -46- Equatlon (3.6) indicates that the Input to block G la the aame a a that of the prototype ah own In Pig* 14, thua proving the theorem* To give an Indication of the uae of thla theorem consider the prototype4 when f^(y) — e^ and G(y) ^ K e“y * If the value of z in the modified aysten la e~^ one haa the system shown In Fig* 17* Notice that the output from the differential In Fig* 17 la 20*7 - 1 and that the Input to G la (xe~y - l)®y or 2 - e^, which la the aame aa that of the prototype* The advantage of aueh a modification can be appreolated when one computes the gain between the differential and block H* Thla gain la the product of the gain of the two blooka* Thla la e^ Ke~y or almply K* For thla reason the system can he simplified to that ahown in Fig* 18* The gain of block G in thla figure is almply K* It la apparent that the complexity of the nonlinear system haa been reduced by modifying the prototype system to the form shown In Fig* 18* Unfortunately it la not poaalble to simplify all systems te the degree shewn by this example; however, a modification of thla type often will result In a system which haa elements which are easier te realize* C* Multiple-Valued Function Generators The material presented so far applies to function generators which have a single valued relationship between Input and output* At times one may wish to generate a - 4 7 - X Ke_yj±iO\ e* e'ey S' i v y O H (p) 1 ■yey=i i ey Fig* 17* Modification of prototypo system for db funotlon generator xe* H (p) Fig. 18* Simplification of block diagram shown la Fig. 17 -48- mult i pie-valued funotlon* The case to be considered In this section Is s funotlon which Is sxulti pie-valued In y* For example, y sin x 1s aucha function* An arbitrary curve of this type la shown In Fig* 19* The x,y plot can be divided into regions (see Fig* 19) such that In each region there Is a 1 to 1 relationship between Input and output* If y la oontlnuous the boundaries of the regions exist at points where dy/dx equals zero* It is obvious that the nonlinear feedbaok funotlon f^(y) Is multiple valued* The feedbaok blook N must produoe separate out puts of x^, Xg# Xg, and x^ for the same output shaft position y^* The plot of f^(y), corresponding to the plot In Fig* 19, Is shown In Fig* 20* Thla plot can be ob tained directly from Fig* 19 by a rotation of the coor dinates of thla figure* The nunfcer of feedback paths must be equal to the mutter of regions* The particular path In use Is determined by the region, which In turn Is determined by the magnitude of x* If f^(y) Is produced by potentiometers, a system such aa that shown in Fig* 21 could be used to generate a function of the type indicated In Fig* 19* The amplitude of x must control the position of switch S* It Is Interesting to note that aa the system changes from one region to the next the gain of blook 0 must change sign If first order linearisation la used slnoe the derivative df^(y)/dy changes sign* This la necessary If the system Is te be stable* Fig* 19* RegIona of a multipie-valued generator Region f( Region Region Region Fig* 80* Feedbaok funotlon for Multiple-valued generator H(p) Fig* 81 Method of generating Multiple-valued output D. Example or a Multiple-Valued Function Generator A simple example will nov be considered■ The desired output 7 la y “ 1*1 * ( 3 .7 ) The desired value of 7 as a funotlon of x Is shown in Fig. 22(a). A tabulation of the values of f^(y) and of the gain of blook G for the protot7 pe system Is given for the two regions in Fig. 22(b). My) Goin Region 1 y I Region 2 Region 1 Reg ion2 -y -1 Fig. 22(a). Desired output Fig. 22(b). Feedbaok 7 of funotlon generator as a funotlon and gain of the funotlon of the Input x compensating amplifier of protot7 pe funotlon genera* tor In the two regions shown in Fig. 2 2 (a) A modification and simplification of the pretotTpe system can be made by multiplying the Input to the differential b y - 1 and dividing the output by -1 in region 2. Thla requires that for x negative the Input to the differential due to x be *x. A tabulation of f^(y), gain of blook Q and the input to the differential due to x la given In Fig. 23(a). The blook diagram of the modified system la shown In Fig. 23(b). 51 Input to differentiol ^V) Go i n due to x Region 1 y 1 X Region 2 y 1 — X Fig* 23(a)* Feedbaok function* compensating amplifier gain and differential input signal caused by Input signal X of modified funotlon generator whose prototype oharacterlstlos are given in Fig* 22 I * ° * H(p) -X Fig* 23(b) • Blook diagram of modified function generator whose characteristics are given by table In Fig* 23(a) -52- Motloe that the ay at am la much simpler than tha pro tot ypa in that the gain of blook G la oonatant and single valued and only one feedbaok path la needed* The only complica tion la In providing an Input to tha differential whloh la equal to the absolute value of x* Thla can be done with the switch type arrangement as ahown in Fig* 23(b) or by the uae of rectifier tubea* The uae of theorem 2 has simplified a system* The Input to and output from a servo connected as ahown in Fig* 23(b) Is ahown In Fig. 24. D* Accuracy Considerations The question of acouraoy of funotlon generators of the type just considered will be dlaouased briefly* Generally the steady state aoouracy la determined by the static friction of the motor* Xn fact, the aoouracy can be specified In terms of the torque just necessary to cause the motor to start rotating. It will be assumed that the torque developed by the motor Is proportional to the input to blook H* (See Fig* 14») Let the value of this Input whloh Is just necessary to cause the motor to rotate be represented by E* As the gain of blook G Is equal to l^f^(Y) by eq* (3*4) and the discussion whloh follows eq* (3.4), the differential output e equal to Ef^(Y)/K will just cause the motor to rotate* Let the system in put x remain constant and let the motor output position y -53- OUTPUT 5 S E C INPUT Pig* 24* Reaponae of aa abaolute-value aorro generator -54- be rotated by an external force away from lta equilibrium value corresponding to the given Input until the differ ential output e la equal to E f^(Y)/K- Let the deviation of y from the equilibrium value Y be represented by jy® • If e Is expanded about the equilibrium values of x and y there results the equation • s X - fx(Y) - (3.8) for small. Observing that X - f^(Y) Is zero by eq. (3.1), one oan determine the variation In jy® which can take plaoe before the motor will develop sufficient torque to start rotating. This oan he obtained by equating eq. (3.8) to the value of e necessary to oause the motor to rotate and then solving for , that is setting - fi Notice that the solution f or will always be - E/K* This means that the error in y due to friction of the motor will not be dependent upon the nonlinear feedback funotlon if gain compensation Is used. -55- Chapter IV 5 , 6 Phase Plane Analysis of Nonlinear Systems A. Introduction In the preceding discussion the quasi-linear characteristics of nonlinear systems have been considered. The variations of the dependent variables of the system from the equilibrium condition have been assumed to be sufficiently small so that the relationship between these variables can be represented by a set of linear diff erential equations. It is now necessary to Introduce new concepts and terminology so that large deviations of the variables from their equilibrium state can be investigated. Since many of the concepts and much of the terminology used in the study of nonlinear systems are the result of phase space studies, this subject is briefly Introduced In this chapter. Likewise some of the proposed methods for improving the response of servo systems by means of nonlinear compensation are studied briefly since their design is largely based upon phase plane techni ques. B. Definition of Phase Plane and Phase Plane Trajectories To Introduce the phase plane terminology the follow ing linear second order differential equation Is conslder- ed- (p2 + 2 0 ^p + l)y o* (i^-.l) - 56- (The system Input variable x Is assumed to be zero.) Equation (1^.1) Is equivalent to the following two first order equations (4-2) M. + 2o(v + y » 0, These two equations can be combined* eliminating the independent variable tt Notice that If (J+.l) were a third order equation, a set of three first order equations would be necessary to represent the behavior of the system. Elimination of the variable t from the three first order equations would result in a set of two equations descriptive of the be havior of the system. When eq. C^4.*3) is solved for a given set of initial conditions one has a curve in the y,v plane. For example, if 0< — 0 the solution of (^.3) is T2^y2-k2, where k is a constant determined by the initial values of v and y. The solution of (if.l) with o( equal to zero Is y = a sln(t -*- ©) v “ — a cos(t *+■©). -57- If v and y are squared and added one has v^-t-y^=A^, the same as the solution of (4 *3 )* The plane whose cartesian coordinates are y and v or y and py Is known as a phase plane and the trajectory obtained by plotting the solution of eq. (I4..3 ) on this phase plane is known as the phase plane trajectory or the equation or of* the system. If the original equation, eq. (^•1 ), had been a third order equation, the phase trajectory would have to be plotted in a three dimensional space. Due to difficulties in representing third and higher dimen sional plots, the use of the phase space trajectories is limited in practice to plane trajectories except in special cases. Due to this limitation in representing the tra jectories, only the trajectories obtained from the solution of second order differential equations can be represented conveniently. Several types of trajectories exist for linear systems. The equilibrium points of systems, often called singular points, are given special names descriptive of the type of trajectories associated with the singular point. The equation of motion, a phase plane plot and the name of the equilibrium point are given for each of several types of singular points in Fig. 2 £. * 5 8 - fhnoo nmm* FIo> Typo of Singularity Vortu point p y+scfpy + y * o Saddl* point PS7 + B « p y + 7 * 0 O < •< < 1 Stablo fooal point p y + 8« p y^Bapy+y * o Stnbln nodnl point o( > 1 P y+**«(P7 4-7 * O tfnotablo nodnl point oC < - 1 lg. 25. Equation *f motion and piimao plan* plot Tor arious typaa or singular point* -59- C. Determination or the Response Characteristics or Linear Systems by means or Phase Plane Trajectories. It is the purpose or this section to show that the phase plane trajectory conveys the same information as that given by the solution or eq. (ij..l) ror y. The locus of points in the phase plane at which the slopes of the phase trajectories are equal to a constant, that is the locus or points for which the slope dv/dy is constant, is called an Isocline. The equation Tor an isocline is obtained by setting dv/dy in the phase plane equation, for example eq. plane shown in Fig. 2 6 are the locus or points ror which dv/dy is constant. The short parallel lines Intersecting each or these isoclines have a slope equal to the value or k used in drawing the isoclines. Arter these guide lines have been drawn on the phase plane, it is possible, with some care and imagination, to sketch a trajectory on the phase plane. The trajectory is constructed rrom the initial condition point on the phase plane passing across each isocline with the slope ror which the isocline was determined. This technique ror the determination or phase trajectories la known as the method of isoclines.^ The trajectory shown in Fig. 26 was drawn using this method. 60 I— I— I L ilt Pig* 26. fhaee plane trajectory drawn by uae of laoallnee - 61- Certaln Important characteristics or the system response may be obtained from the phase trajectories of the system. The first of these Is the amount of overshoot In the system output y resulting from the application of a step function input.® The overshoot can be determined from the phase trajectory as Illustrated in Pig. 26. In this figure, Pi Indicates the initial equilibrium point or original error in y after the application of an input step function. The final value of y, in this case, is found at the origin. The value of 7 at P2 Is the amount by which the output overshoots its final value of zero. The percent age overshoot is thereby given by the ratio of 100 where y2 and yi are the values Indicated by points Pg and Pj_ In Pig. 26. The second characteristic or quality of a servo system that can be determined by Its step function response is the time td required for the output of the system y to first reach Its equilibrium value. On the phase plane plot it is the time taken for the system to move from an initial state given by point in the phase plane to the state indicated by point P^, the point at which y first reaches its equilibrium value. If the velocity v or py Indicated by the ordinate in the phase plane is consistently large, the time td in general becomes short. A large area included between the phase plane axes and the trajectory from F-^to P^ indicates a rapid response cF a small value of td. - 6 2 - The period td required by the system to move from state Pj_ to state of Pig. 26 is given by td= r3 i dy* (4*5) = / 3 i ” ■ F 1 It is possible to simplify eq. (4.5) by making a change of variable. If u =x. l/v then the phase plot in the yu-plane Indicates directly both the overshoot and t^« since td In this case is given by P3 *d or the area under the trajectory from P^ to P^ in this new plane.^ When this change of variables Is made du — - 1 dv zz. - u^dv, (4*7) ” “ Z v and djj — _ dv u 2 (4 *8 ) dt — dt Fquation (4 *2 ) may be written in terms of u and y as follows - -h 2 ofu -f-y u2 = 0 d t • S - u ' These equations can be combined giving .g& » 2 c(u2+y u^. (4 .1 0 ) -63- Here again the method or isoclines can be used to sketch the phase plane path in the yu-plane. A curve corresponding to the trajectory in the yv-plane in Fig. 26 has been sketched in the yu-plane (see Fig. 27). In summary the following points should be observed in regard to the typical phase plane trajectories as shown in Fig. 25: 1 . The phase plane trajectories can be constructed only for a second order differential equation. 2. The trajectories do not cross. 3 . The trajectories approach the origin for **< positive and leave the origin for<\ negative. 4* The trajectories are closed curves around the origin when the solution is simple harmonic motIon• Section D. Phase Plane for Nonlinear Systems The most important use of the phase plane approach is for systems which are characterized by second order non linear differential equations. One of the more important characteristics of nonlinear systems, namely the limit cycle will be discussed in this section. Consider the equation (4.11) If again v is substituted for dv. eq. (^.11) is transformed dt to the set of two first order equations. (4 .12) -64- y Fig. 27* Zsr«r«« pIm i m pi >!• A i l laAloatlag H a t «f r s a p t K M t - 6 5 - These equations oan be combined, giving fi= - . (4.13) Generally, the exact solution of (4*13) for y cannot be ob tained, but the phase plane trajectories can be sketched by the method of isoclines as in the case of a linear system* At times this is the moat convenient method for determining something about the quality of a servosystem whose equation of motion is a second order nonlinear differential equation* The phase plane trajectories near the singular point of a nonlinear system approach those of a linear system since the equations of motion for small variations of the dependent variables approaah those of a linear system, as was indicated In Chapter IX* Per larger deviations about equilibrium the phase plane trajectories take on forma not encountered In linear systems* The moat Important of these trajectories for large deviations la the limit cycle* The physical meaning of a limit cycle oan best be described by means of an example* Consider an electronic system such as a vacuum tube oaolllator, which has an unstable fooal point (see Pig* 25)* Initially the system output y will be of ta| 4* the form er* sin wt* Im time as y becomes larger and lar ger due to the term ert, nanllnearltles of the tube such aa grid ourrent and tube cutoff change the effeotlve gain of the tube and llsilt the amplitude of the oseilla tion to some well defined value* - 6 6 - If an oscillation larger than the amplitude limit just discussed is initiated the amplitude or the oscillation will decrease in time, finally reaching the same limit conditions as before* The general character of this process is as indicated in Fig. 28. Notice that all trajectories approach one closed trajectory. This closed trajectory, corresponds to a steady state oscillation and is termed a stable limit cycle since all trajectories in the neigh borhood of this limit cycle approach the limit cycle. A second type of limit cycle is shown In Fig. 2 9 . This limit cycle is termed unstable because every trajectory in the neighborhood of the limit cycle moves away from this limit cycle. +rt* ir. Fig. 8 8 . Stable limit oyole Fig* 89. Unstable limit oyele -67- Xhe limit c; cle tyje of rts^OnLc cun occur in .. urvomechunisus ii' the systems ^r-. improperly assigned, m fact .ouny . o culled unstable cervo -yecens are actually . ybtecs v.-ith etasle 1 imit cycles. In the iollov.ing chapter i hhapti r V) methods ba^. td u; on the steady ^tasu fre ueiicy i l U: uiot- ..ill he use a to investigate the presence of li...it cycles in uhe system response. s. nonlinear Uompensaoion for aervomec^cinisms During the past few years several writers have proposed the use of nonlinear elements in an otherwise linear sysaem to increase the opted of response of the original linear system.-*-® ~ ^ The improvement in the rtsionse of the resulting nonlinear system in most cases is aemonstruted by use of the phase plane methods discussed in section 0. One group of methods for nonlinear compen sation which cause the servomechanism to have an equation of motion of the type d2y f Cx-y)dy_ + y A x (JV.14) O ^ r i t - . v.ill be discussed in this section. The damping coeffi cient f(x-y) in all of these systems is a function of the error (x - y ). - 68- To inaiCutt the reasoning th. t i- uatd in the x'or- ..ialation of the a o . i i n e a r systems whoso c uatiori ci motion it at 0iven ov eq. (4 •1 4 ), conaiaer the linear ay stem obtained by substituting the constant 2 OC Tor f(x-y) in eq. (4 *1 4 ). If* oC is ttjual to l/l, and x i^ a etep AncLiOn, the uutput y '..ill be oi‘ lor..: indicated bv ii.u daiiip-;.e sinusoidal curve in Big. 3b. If Oi is equal to 2 the output y will ue as ..i.otn by the second solid curve in this figure. Both of the responses shown in hia. 30 are considered to be undesirable. In the case ■..here 0 < is small the system is very oscillatory and forOC I^r. e the system is sluggish or slow. If CK is replaced bp the variable term f(x-y) and f(x-y) is initally small, it is found that y approaches equilibrium as si .own by hic first ..art of the damped sinusoidal curve. But if f(x-y) becomes large ( 4 or greater) for small values of \ x - y\ , the system will decelerate ra] idly and trie out- ; .it y ’will deviate from the original damped sinusoidal iurm in the .aanner shown by the uotttd line in Big. 3 0 . several forms for f(x-v) wnich h.-.ve been uuggestea re given in (4 *1 5 )* bervo systems w.ose s> stem equation is given by (4.14) using the first three of tne functions ^iven in (4-15) are discussed in reference 12. The fourth type is discussed in 13* Apparently an error in the fourth function was made by the author uecause it is apparent that tne compensation function should be independent of the sign 69- Large Domping Smal I Damping Fig* SO Step function reapani* «f aarvo output oi x - y. l’he _ ifth type is probably the ^unction e.esired in 4 4-,-'c e the fourth 1 # f(x-\ ) ___ K 1 + s u - yi -oix-yl 2 . fU-y) — h e it ✓3 .• f(x-y) (4.15) 1 + 1 x - y +- C ( px - i y )l 4 * f(x-y) = K - b(x - y) 5 . f (x-y) h - C |x - y| . tiie of these uaiiupiiijj, tGnas is however, it can result in system instability under certain jonaitions. ihe phase plane trajectories oi the equation 2 . -+* (2 - \y\ t y = 0 (h .lo ) dt <_uv illustrated in Fig. j>l* equation (4-lc>) determines tie free resy.-onse of a system ’.:ith type 5 damping, in this case f(x-y) is given by 5 of (4.11) with h equal .0 cv.o, 0 equal to one and x is assumed to be sero. .jit t,rposeu on the ph^se ^ lane being considered is a tra jectory of a linear system, th^t is a system with i’(x-y) cqu_l to 2 o< a constant. Notice that trajectory "n" of the nonlinear system starting at the same initial :oint "a" as the linear system attains a higher velocity than uo .0 the linear system. This means that the nonlinear system starting ~t conditions given by “a" will reach the -72- c.rc , usition, y = a , u e L r e the linear system. (d-e uccv^ii b . ) Lihewisc, the overshoot uf the nonlinear „p wteia Iru.ic^tcd by p.oint ’’b11 is leas tl an the overshoot the coxap.arison Iint..r system indicated bv po int "c". . e'O .rt however, two bad characteristics of the non linear system. The first is th~t tlie system has an un stable li.uit cycle, and the second is the- fact th- t if the ..w-imum value of y is smaller than one, the damping term - |yl ) is too large for satisfactory response* A second type of nonlinear compensation, called ; ia.se i-lune switching, arises i.i.en f(x-y) is altered attending on the op.orating1 ^tate of the system. For trample in the region of operation where 1-rge velocities ..rt important f(n-y) is giv^n a small or even negative constant value. In the reeion of operation where a r-pid deceleration is required, f(n-y) is -lade to be large and positive. This example is illustrated by means of .he .na.se plane plot shown in Fig. >2. uegions where a slow- irr, down process occurs are shown shaded. These arc the -irst uiid taird quacrants. In the first ana third quad rant- a large damping term Is used whereas a small damping term is used in the second and f o u r t h quadrants to a l l o w rapid changes in these two regions. Phase plane switching is v ery thoroughly discussed in reference 1 4 . 73 l o w ._-— ^ X ^ V ^ hlsh damping / y y y s / y / y x damping / l o w da mping --- 'damping IP Fig* 38* Phaaa plana Indicating high and law damping region® or phaaa plana awl tolling ayatam -74- Two examples of nonlinear compensation typical of proposed systems have been briefly considered* Although the arguments for these methods of compensation seem plau sible , it is the belief of the author that any linear system can be Improved more simply by linear means than by means of nonlinear damping* To Illustrate this point the responses of two nonlinear systems have been compared to the responses of slmlllar linear systems by means of an analogue computer* The following test proceedure was used* Both the nonlinear system and the linear system whose responses were being compared were simulated by means of the oomputer and both were excited simultane ously with the same Input x* This input was a quasl- random signal generated manually* The absolute error |e) and the output y of each of the systems was raoorded as well as the Input x . To evaluate the performance of the two systems, the percentage of time the absolute error lei exceeded a positive number n was determined for both systems for several values of n* The percentage of time that |e| :>n was represented by the function P(n) • For example If |e|> 1 12 percent of the time then P(l) ~ 12* These functions are called error distributions* The first test was made on two systems whose oper ating equations are respectively -75- o p y -v- (2 - jx-yl )py +- y - x (4.17) P2y -r 1*41 py -+ 2 j — 2x . It la the author*a contention that Increasing the gain of the error amplifier In the linear aervo to a value of two will do more good than the nonlinear damping factor* The gain of the error amplifier la,Indicated by the coeffi cients of x and y In the linear equation of (4*17) • A sample of the teat record for these aysterna la shown In Fig. 33* The top graphlo record la the Input x* The out puts of the linear and the nonlinear systems are shown In the second reoord, and the absolute error ef the linear and nonlinear systems are shown in the botton record* Data were taken from a long run of this type and were used to determine error distributions P(n) of the linear and nonlinear systems* These two distributions are shown In Pig* 34* Notice that the error distribution of the lin ear system la consistently smaller and thereby better than that of the nonlinear system* A second test was made comparing the responses of a phase-plane switching system and a linear system* The operating equations of these two systems were p2 y + f p y y « x (4*18) p2y + py + y = x* Iflpul. X l-m Ut t \ Lfaa4J$ai^LL o)\Q.ut pul y □ a I fo o /a ia Erro r: j£ WWWAUUH Fig* 33. Qraphio reoord oompalrlng responses of llneer end nonlinear damping servos Pig* 34* Srror distribution* of IIbmup mad nonlinear damping nonlinear mad IIbmup of distribution* Srror Pig* 34* ays tbo axolt ton* by ad ■P(e) 0 4 0 9 0 8 20 0 6 0 7 0 5 0 3 0.2 0.4 m m m m ipt signal input 0.6 77 Linear i ar ea lin n o N 0.8 1.0 1.2 - 7 8 - In eq* (4.18) the damping factor f ef the nonlinear system has one of two values as shown In Fig* 35* Fig* 35* Phase plane Indicating amount of damping used in nonlinear system During the period where rapid velooltles are desired f Is zero, and during periods where slowing dawn Is desired f is two* This method of phase plane switching Is better than that shown In Fig* 38 since the damping is increased before the error ohanges sign* Fig. 36 indicates the quaal-randem input signal used to test the two systems* The outputs of the two teat systems are shown In the second record ef Fig* 36, and the errors of the linear and nonlinear system are shown In the bottom record* tfotlee that for the sudden dis placement the "phase-plane servo* Is better than the m m W f f n n[ln#.£ 79- Ut»ar. i ii 11 i n £ r r o n Howriieajr5l«la&m MS n Fig* 36* Graphic record oompalrlng linear and phase plane switching servo responses - 80 - linear servo, but that otherwise the two are very similar. The short section of the graphic record shown In Fig. 36 was part of a longer run used to determine the error dl^> trlbutlons shown In Fig. 37* Both examples, comparing the response of nonlinear systems to that of similar linear systems, oear out the fact that the linear systems were as good or better than the two nonlinear systems tested for the random input x that was used. In fact If the original system can be represented by a linear second order differential equation then In principle a perfect servo with zero response time t^ can be made by allowing the gain of the error amplifier to approach Infinity. Actually servomechanisms can only be approximated by second order linear differential equa tions. Generally if the gain of the error amplifier of any servo system is Increased sufficiently the system breaks Into oscillation. This oscillation may be oaused by nonllnearlty such as backlash or because the system can be represented aoourately enly by third or higher order differential equation. Due to these observations it would appear that one oan not compare analytloally the response of a servo designed using linear techniques and the response of one designed using nonlinear compensation if the operation of the system without the nonlinear com pensation Is described by a linear seoond order differen tial equation. i. 7 3 Fig. switching switching 0 4 0 7 zo 30 Pie) 60 50 O.i • Brrar distribution llnaar and pbus* plana and pbus* llnaar distribution Brrar • systaaa axoltad systaaaaxoltad = hs Pae wthn Distribution Switching Plane Phase = x = ier ytm Distribution System Linear = • j b 81- tha mmmm Ipt signal Input ______ -92- It la difficult to propose a method for evaluating the performance of a nonlinear system* The process of testing a nonlinear servo by observing the output response when the input Is a step function, when it Is obvious that typical system Inputs are not step functions, Is open to question* Test procedures for nonlinear systems will be discussed In more detail In Chapter IX* -83- Chapter V Large Amplitude Oaolllatlons and Limit Cycles in Nonlinear Systems A* Introduction In this chapter nonlinear systems operating with large variations about the equilibrium point will be con sidered* This type or operation will be investigated by frequency response methods, rather than the phase plane techniques discussed in Chapter IV* The frequency response approach has the advantage over the phase plane methods In that systems represented by third and higher order differ ential equations ean be studied* The frequency response 15 method used by Kochenburger will be used to study the stability oharaoteristics of the function generators dis cussed in Chapter III* The concept of nonlinear gain will be introduced and will be used to investigate the presence and properties of limit oyoles* This method Is somewhat more general and more accurate than the method used by Kochenburger'*'® and oan be used to determine the fundamen tal frequency, and wave shape of the output y when the system la operating on a stable limit cycle* 5* Qain of a Nonlinear System The output response y of a stable nonlinear system when the input to the system is sinusoidal differs from the output of a stable linear system with the same input in that -84- the steady state output of the latter will consist of a single component whose frequency la equal to that of the input signal. Die output of the nonlinear system will usually contain harmonic components In addition to the fundamental component. If the Input x Is of the form x — sin wt , (5.1) then the output y of the nonlinear system will he of the form y = b0 + *3. aln^w t -t b2 Sln(2wt+ * * (5.2) If the system shown in Pig. 38 is linear then, of course, all h fs except b^ are zero* Fig. 38. Block diagram of a system The gain of the linear system Is written as G = — =----- . (5.3) ■l The gain is the ratio ef output to Input when both are represented in phasor form. If the system la nonlinear (5.3) has a similar meaning. Sinoe the nonlinear system has several frequency components, one may form the concept of a gain for eaoh component by means of the phaaor repre- i.U sentation* A general gain for the n harmonic component is defined as G (5*4) Examples of the use of the nonlinear gain G^ ^ will be con sidered In the next section* C. Stability of Function Generators The function generators described In Chapter III have a prototype block diagram as shown in Fig* 39* All nonlinearities are assumed to exist in the feedback path and the compensating amplifier G^* In Chapter II a method was described for obtaining a uniform response of the system for small variations about equilibrium* The methods that were presented there are not valid, when the variations about equilibrium are large* It is necessary,"when this condition prevails, to determine whether or not there are limit cycles before the absolute stability of the system can be determined* In references 15 and 15, It is indicated that when steady state oscillations or stable limit cycles ooour in a practical servo system, the system output y will gener ally be approximately sinusoidal* The system output Is nearly sinusoidal since harmonies oontalned in the signal -86- cx e I,- G(P) — y 'f(y) I pie* 3 9 Bl*ek Dlagraa af Funotioa Oaneratoa* -87- which excites the output block G(p) are attenuated much more than the fundamental component for the usual types of transfer functions G(p) found in practical servo systems* In the following discussion of the stability of funotlon generators it will be assumed that they do have a limit cycle and that the system output y la of the fora y — bQ b^ cos wt, (5-6) when the input signal x Is a constant Sq. Wie output of, the compensating amplifier G^ Is equal to the difference (x - fx = («0 — f^(y)) (f^(y) )"1* (5-6) Notice that the gain of the compensating amplifier G^ is set equal to (df^/dy) ^ or (f^(y))"’*' as Indicated by eq. (3.4). Since the Input x is equal to the constant Sq the output e^ of G^ can be represented as a funotlon of y and aQ, namely e^(y,ao)« The value of e^ la then the same for y at times t^ and -t^ since y by (5.5) Is the same at these two times* The signal e oan therefore be repre- 1 sented by the cosine series ®i - o *+* o- oos wt •+* o„ cos 2wt * * (5.7) A O * * The gain G^ ^ relating y and e^, that Is the overall gain of the nonlinear feedback block and block G^ oan be - 88 - expressod as * ( 6 -8) Now that the relationship between fundamental In put and output of the two nonlinear aectlona has been specified by G _ the same will be done for the linear x , x block G(p) shown In Fig- 39* If a steady atate oscilla tion of the form indicated by (5«5) la to exist In the out put , the linear transfer function G(p) must convert e^ into this assumed fora of y given by (5-5)- The transfer funo tlon oan be simplified to the algebralo form G(jw) when the Input e^ la repreaented In phaaor form- For thla reason the Input phaaor o^e^wt times G(jw) is equal to the output phaaor b e^wt. Canoelllng the terms e^*** one obtains G( Jw) o1 = b^ (5-9) and from (5-8) °i,i bi= °i • (5*10) Jwt In the future the e- terms will not be written alnoe they oan always be cancelled from both aides of the equa tion- If eqa- (5-9) and (5-10) are to be satisfied It la necessary that G, , G(Jw) = I. (5.11) - 89 - This last aquation assumes a simpler form if G(jw) Is written In the Inverse form G(jw). Then — G(Jw). (5.12) Generally, G^ x Is real for function generators considered while gT^w) la complex. The possible frequencies of the oscillations oan be found If the frequencies for which la real are known. Hie amplitude of oscillation la given by the value of b^ whloh oauaea ^ to be equal to G(^w)• For example, oT^w) oan be plotted In the complex plane as w varies from aero towards Infinity, and G^ x a real negative number oan be plotted as a funotlon of b^ In the complex plane (aee Fig. 40). Notice that the two curves cross when b^ and w have values of 4 and 3 respectively. This Indicates that a steady state oscillation In the out put of the form 4 ooa 3t oan exist. The gain °x,l will be evaluated for two types of funotlon generators In the following sections. D* Example of Logarlthmlo Recorder In Section B ef Chapter 1 1 1 It was indicated that the feedback path ef a logarlthstlo system must generate the funotlon f ^ y ) = .y, (6-13) which Is fed Into the differential as indicated In Fig. 39. The output ef blook G^ oan be shown te be of the form -90- 60 Locus Pig* 40* Vonllnaar Oaln and Invara* Tranafar Funotlon «f Syataa with. tft&atabla 1*1 alt Oyola by substituting the valut of f^(y) given by (5*13) Into (5 . 6 ) • One oan substitute the assumed form of y as given by (5.5) into (5.14) and then expend e^ in a power series In terms of b^oos wt• This power series Is b 2 b 3 e1 =«Qe"toO(i - b^oos wt —icoa^wt - ^ o o i 3wt+- ••)-! (5.15) It will be necessary to obtain the magnitudes of the do and the fundamental frequency terms of (5*15) • To help In detm* mining the amplitudes of the do and fundamental terms, the magnitudes of the zero and the fundamental frequency components of oosnwt represented by d and g_ respectively are: S' l*3-5* • • (n - 1) * _ ( g yi'.gv ■. ^ ------for n *ven d — ) (5.16) n / n for n odd S The do component ef (5.15) represented by oan be written using (5.16) and(5.15) -92 The amplitude of the fundamental oomponent of e^ represented t>y °x la •l = -0«‘b° In the following dlsousaion the transfer function of the linear block G(p) will be assumed to be of the form g 1/p (dQ dj^p-f-dgp * *)• If there Is a dc Input to this linear blook the steady state value of y will contain a term of the form CQt/dg* However, since a term of this type cannot exist In a physical system it Is necessary -bn that Oq be zero* For this reason the value of aQO can be found as a function of b^ by setting In (5*18) equal -bn to zero* Substituting the value of a^e so obtained into eq* (5*19) one has for o o, __ 5* ^ (5.20) b T b* ^ 21 '2' 41 V 2 *4 I and the gain 0^ ^ by (5*8) then becomes 1 / 3\ .bl ° i . i = . STT V TP B T (5*21) b? ( L i t 2T v s r r r a * This gain G la plotted in Pig* 41 aa a function of the 1.1 sinusoidal output amplitude b-^* Notice that thla gain la a maximum when b^ la aero* This expression for the gain 0^ can be used with the Inverse transfer function g T^ w ) of the linear block to determine the possible existence of limit cycles aa shown In Fig* 40* E. Example of Square-Root Function Generator The value for e^ In terms of the input aQ and the system output y can be determined aa explained In the previous example except that the feedback function f^ (5*22) When the assumed form of y given by eq. (5*5) Is aubstl tuted Into (5*28) e^ becomes and eq* (5*23) can be expanded Into the series 94- Pig* 41* Q«1b of> AoallaMT bltoka of logorltholo genorot or - 95 - whexi b1/bQ la leas than one* Making uae of (5*16) and (5-17) one may write Tor the constant and fundamental components of e^ (5.25) The value for Oq given by (5*25) can be written In the closed form (5.26) and O- in the form (5.27) (The justification for setting cQ equal to zero la given la the laat example*) Xt la desirable to eliminate from this last expression so that the gain can be determined In terms of a^ and b^. This oan be done by solving (5*26) 2 for bQ* The following expression oan be written after eliminating b^ from (5*27) • -96- °1 = (5.28) The nonlinear gain then becomes -2 -i 4 * G - 1 (5.29) 1,1 -1 -t-V H- 4 where ^ Is b^/aQ . This gain la plotted In Fig. 42. This expression Tor the gain G^ ^ oan be used to determine the stability or the system If the transfer function of the linear block Is known as previously explained with reference to Fig. 40* F. Example of Unoompensated Function Generator The two examples Just considered do not in general have limit oycles aa their gain G does not vary greatly 1,1 with the magnitude of the sinusoidal output• To Illus trate one system that does have an unstable limit cycle consider the case where the output of the nonlinear feed back path la a function of the form f^y) - y-i- y3 . (5.30) 1.2 ' .. .. ______------— <— ■ ^ ____I*----- ___ ^ ------1.0 I-- ■ Maximum Value Of — G. . = 1:5 T"-r Fig* 42* Gain i of nonlinear aec- tlona of aquare root generator(Isbf/ag) 0.8 -G I.' i to 0.6 I 0.4 0.2 0 0.2 0.4 ^ 0.6 0.8 1.0 1.2 -98- The transfer function of the linear block will be assumed to be of the form °(p) — ~S ^5------* (5.31) P + V -f 4P Since the system Is assumed to be uncompensated the gain of block G^ of Pig* 39 will be unity. The value of e^ Is simply -fx (y) assuming that the Input signal x la zero* •1 = -(y+ y3) (5.32) The output y will be assumed to be b-^ cos wt with the result that the fundamental frequency term of e^ Is (-b-^ - 3b®/4) cos wt* the gain of the section between the output y and the input to the linear block of Fig* 39 Is therefore Gx ^ = -(1 + | b®) , (5-33) Hie Inverse transfer function with Jw substituted for p oan be written as G(^w) — - Jw3 - w2 J4w. (5*34) A plot of this Inverse transfer function and the nonlinear gain G. . as given by eqs- (5*34) and (5*33) respectively Is shown In Fig* 43* Notice that the twe curves cross for w = 2 and for b^ sx 2* It has been feund using an analogue computer that an unstable limit cycle with a peak amplitude -99- 0(jw) and ^ lool of ayatora with limit cycle of 7 equal to 2,02 exista for thla system* The deviation between the calculated value and the measured amplitude la due to the harmonica neglected in the analysis* It should be observed that It would be difficult to use phase plane teohnlques In this example slnoe the differential equation describing the system response la of third order* G* General Method for Determination of Limit Cycles by Frequency Response Teohnlques In this section the output y of a system operating In a limit cycle condition will be approximated by fun damental and harmonic components rather than the fundamen tal alone as was the case In the preoeedlng sections of this ohapter* The use of harmonlo components not only gives a better approximation of the wave form of y but also gives a better approximation for the fundamental frequency than does the use of the fundamental oomponent by itself* Before proceeding with the discussion of limit cycles some of the properties of nonlinear differential aquations will be Investigated* First, nonlinear differ ential equations will be divided Into two classifications* Equations In the first classification will be oalled symmetric differential equations* They are defined as followsc A symmetric differential equation oan be expanded o 2 In a power series about xapx^p x •* y-spy-rp y • • = 0, and has the property that the sum of the exponents of x and - 1 0 1 - and their derivatives Tor eaoh term In the power aeries la odd* For example (5*35) is symmetrlc-the sum or the exponents Is one for all terms except the nonlinear term whioh la three* Other nonlinear differential equations whioh have terms whose variables have exponent sums which are both even and odd are oalled mixed equations* For example, p2y -t 7 py t y = * (5*36) is a mixed equation slnoe the second term has a sum of exponents totaling two while the exponent sum for the remaining terms is unity* ▲ seoond order equation will he used as an example in the following discussion; however, this fact should not be considered as a restriction of the method* A general second order equation with sero excitation term is of the form c>(py,y) p2y +B(py,y) py + A (py,y) y = 0 • (5.37) This oan be written In the form (5*38) C B -102- It Is assumed that eaoh of the ooefflolents» B*/Cf and a '/C** oan 150 expanded In a series In terms of j and py. If (5*38) oan he expanded It la convenient to write the expansion In the form P2y -+- -+- ^ y 2 3t 3+ * * * '/^oo Aoy ■+ /io *2 + ' • • /oipt -v /^.iypy t- • * py = 0 /§2 (py*2-+ /^2y(py)2 ■+ (5.39) Many of the terms given In (5.59) do not ooour If the equation being expanded Is symmetric since terms of the form oc Q# o^gy2, / ^ 1y2(py)2 do not occur. The general form of a symmetric equation la given by eq. (6.40). p2y y + oc"3 y3 -»■ • - foo •+ • • Ziiypy /six3 py-*- • * p y ^ . o (5.40) 8 I ft»(py) If a symmetric equation has a limit oyole whioh encloses the singular point x s 0, y • 0, than the output y when the system Is operating In a limit oyole state oan be - 103 - expanded in a Fourier aeries containing no even harmonics or dc terms, that Is, y la of the form y = cos wt -+- b3 oos(3wt +■ O jj) + bg coa(6wt-h©g) • •• (5.41) It can he demonatrated that y la of the form of eq* (5.41) as follows: Assume that a solution for y, namely y^, Is not symmetrical about the ?ero axis - this means that terms of the form \>q, bg oos(2wt 9g) and other even harmonics are Included In y^ In addition to those given In (5.41). It is obvious that If y^ is a solution of (5.40), -y-^ is also a solution. If y - y^ and v -=.py^ as well as y — -y-^ and py *P7x ar* plotted aa trajectories in a phase plane these two trajectories about the same singular point cross. This la Impossible since two trajectories oan cross only at a singular point. This contradiction indicates that the limit oyole solution for y must be aymmetrlo for In this case the trajectories are the same* It la convenient when studying nonlinear systems to develop a model In the form of a block diagram. To Indi cate a form for this block diagram eq. (5.39) is written in the form )y — f(py,y), (5.42) where f(py,y) contains all the nonlinear terms of (5.39) A blook diagram satisfying eq. (5.42) is shown In Fig. 44. -104- f(py,y) Linear Block Nonltneor Block Fig. 44* Blook Diagram Representation of Nonlinear System Xt Is seen that the nonlinear bleok generates f(py,y) whioh excites the linear blook and this blook in turn generates y. The transfer function of the linear block, G(p), is equal te G(p) — (5.43) p 2 + f i > o p + The oomplex gain of the linear blook oan be written as O(Jw) (5.44) " w & / o O w -#-0^1 The method for determining limit oyoles will be indicated by a solution of the limit oyole for a system which has an equation ef motion of the form (y8 - Dpy -t- y — o . (6.45) Note that this 1s a symmetric equation so that a limit oyole of the form y = b_ sin wt (5.46) - 105- can be used as a first approximation for the system output when operating on the limit oyole* (If the equation of motion had been unsytnmetrlo It vould have been necessary to assume y In the form b^ -+• sin wt as a first order solution due to the unsymmetrical nature of the system*) One can write f(y*py) for eq* (5*45) with y as given In (5*46) as wb® j—1- (cos wt - cos 3wt) • (5*47) The fundamental component of f(y,py) can be represented In 3 . the complex* phasor form as -Jetoj/4, and y In the phasor form la b^* The nonlinear gain ^ as expressed by (5*4) becomes G = - jw (5.48) •LiA 4 and a limit cycle exists when the product G G(jw) is 1*1 unity by (5*11)• That Is* when b2 <-J* _ i ) i---- \ - l (5.49) 4 I -w8 -Jw -+ 1 I Equation (5*49) determines specific values of b^ and w which can be found very easily by cross multiplying and equating real and imaginary parts* as follows: -106- jw (5.50) - w2 1 ^ 0 . The solutions art w»l and b^ s ± 2. If a better approximation to the amplitude and frequenoy of the system response when operating on the limit oyole Is desired, y oan be assumed to be of the form y — b^ aim wt -+• b 3 sln(3wt -+ 0 ^) (5.51) This Is called a third order solution since harmonica up to and lnoludlng the third harmonic are included In this expression. In this case one oan divide both the third harmonic and fundamental terms in the output of the nonlinear blook Into two parts. One part Is due to the fundamental term of the Input, and the second Is due to the third har monic term of the Input* In Section B of this chapter the concept of nonlinear gain was introduced and now a further extension la made. The output of the nonlinear section of Fig. 44 is represented by e with e^ being the fundamental component and e the third harmonlo component. TJslng the 3 nonlinear gain concept one may represent e^ and e^ in the complex form? (5.52) Y 3,3 3 * - 1 0 7 - Here G^ x la the fundamental oomponent of e represented In complex form due to the fundamental output term 7X, while G Y la the fundamental oomponent of e. In complex 3|1 5 1 form due to the third harmonlo oomponent Y_ • Similar o terminology la used In the second equation of (5*52). The Inter-relationship existing between and 7^ aa well as EL and 7 . due to the linear bleolc of Fig* 44 ° 3 with gain G(Jw), la ^ G( Jw) - 7 X (5.53) Ej G(J3w):= 7 3 . The two seta of equations (5*52) and (5*53) oan be combined to give <0 1 . 1 - > * 1 + ° 3 . ! Y 3 * ° (5.54) °1.3 71 + (G3.3 ' °Ti3w) > 73 = ° The solution of this last set of equations for b^, b-j, w a:id Q will result in y of the form ^iven in (5*51)* The o above equations appear simple but the algebraic solution for b^, b^, eto* on the limit oyole may be quite complicated. In the preoeedlng example, eq* (5*45), It oan be shown that the fundamental and third harmonlo terms of f(y#py)» namely e^ and e^, are of the form -100- el f e i + ooa wt - ^ * ® J ] e -= [- — coa 3wtt- ^jbj + J-bjbgj ooa(3«t+ 3 (5.55) These two equations oan be written In the complex form as follows: / bf b.bfi? b?b bfb, \ , _ - / 1 . 1 3 _ 1 3 JO3 ) i -i»lT~ +■ S i— • / (5.56) e 5 = -J Slnoe T 1 S ^ and (5.57) s' 3 “ 3 The expressions Tor and Eg In (5*56) oan be written, using (5.57), aa follows: / b^ b?\ b? E1 — " Jw \ * “ + 3T7 Yl^ Jw T~ 73 # b 2 (5.58) *3 = >^*1 - ^ 3,2 + 3^ The gain terms O^ 3 # ° 3 1 ***** G 3 3 aro found from (5.58) to bo The fundamental sat of aquations to bo solved as lndloated by (5.54), oan than be written as The method used here for the solution of eq* (5*60) for b^, bg, w and 8 5 la to divide both equations by Yj_, giving First and b^ are assumed to be zero* Solutions for b^ and w are obtained from the first equation of (5*51)• These values of and w are then substituted into the -110- aecond aquation, a a aural ng only b to be zero, giving a o solution for which In turn gives an approximate value for bj as a value Tor b^ has been obtained. This value or b^ as well aa ¥ / ¥ are substituted Into the first equation of 3 l (5*61) giving new values for b^ and w, as well as a new value for b^ In terms of b^* These values are then used to obtain a new value for ?3/¥^* This oyollo process Is continued until the solution converges* In order to verify the results of the analysis proposed In this section a solution for the steady state oscillation of the system whose system equation la given by (5.45) has been found by means of an analogue computer* The upper graphloal record shown In Fig* 45 Is the computer solution of (5*45) showing y as a funot 1 on of time* A sin usoidal signal y^ shown as the second trace of the record ing, was adjusted in both amplitude and frequency until the sum of the computer solution y and the signal y^ contained no visible fundamental oomponent• Since y -f. y^ has no fundamental oomponent -y^ Is equal to the fundamental com ponent of y* The sum y -v y^, which la largely third har monic, Is recorded aa the third trace on Fig* 45* One cycle of the calculated third harmonlo la plotted for com parison with the measured harmonlo terms In the system re sponse* The characteristic parameters of the third order solution, the fundamental solution and the computer solu tion are summarized in Table I which follews* Pig. 45. Graphical raoord of oonputar solution of aq* (5*45) -112- The phase of the third harmonlo o f the ooaputer solution with respect to the fundamental could not he determined accurately. Table I First Order Third Order Computer Solution Solution Solution Fundamental Amplitude b^ 2 2.017 2.07 Fundamental Frequenoy w 1 0.9439 0.95 Third Harmonic Amplitude b_ 0.254 0.25 Third Harmonic Phase 9^ 294° -113- Chapter VI Frequency Response Characterlatics of Nonlinear Systems A* Introduction In this chapter a study will be made of the steady state response of nonlinear feedback systems excited by large periodlo input signals* In the first sections of this chapter a method is developed for representing open-loop response data in graphical form* This graphical represen tation of measured or calculated open-loop response data la used to lndioate oertaln properties of the olosed-loop re sponse of the nonlinear system* In the last sections of this chapter analytical methods for determining the response characteristics of systems are considered* The term "frequency response” must be defined for a nonlinear system* If the system input x varies sinusoidally with time, the output signal y and the error (x - y) con tain terms of fundamental and harmonlo frequencies* On the other hand if y la sinusoldal, then the input x and error (x - y) will have fundamental and harmonic terms* In the general case of a nonlinear system only one variable at one time oan be represented by a single frequency, sinusoidal component* The remaining variables generally will contain fundamental and harmonlo frequency components* ▲ frequency response will be defined as the evaluation ef the steady state response when any one variable, such as the Input x. -114- output 7 or error (x - y), consists of a single, sinusoidal, fundamental frequency component* B. Representation of Open-Loop Frequency Response of Nonlinear Systems It is the purpose of this section to Introduce a graphical representation for open-loop frequency response data which oan be used to determine the olosed-loop re sponse of the system* This representation can be used for open-loop response data obtained either experimentally er analytically* It might appear that methods for the deter mination of the olosed-loop response of systems would be more appropriate than for the corresponding open-loop con dition; however, this is not the case* Open-loop, fre quency response data oan be obtained experimentally even when the clesed-loop system Is unstable • This last advan tage of the open-loop response is particularly Important in testing large servo systems, since many large servos are capable of dlstroylng part of the system If the olosed-loop servo system Is unstable* A blook diagram of a typical servomechanism Is shown in Fig;. 46* The Input x to the servo controls the output y* The error is defined, as usual, by e =r x - y , (6 *1 ) 115- Fig* 46. Blook diagram of a servo system The open-loop frequency response characteristics oan be de termined by opening the loop at "a* and applying the signal e r Oq + a in (wt t 9) f (6*2) directly to the input ef blook 0 ? Xn this oase the output generally will be ef the form y= bQ ■+■ b^ sin wt -*- bg sln(2wt -+ ^ g ) -r b 3 ain(3wt + • ** (6*3) It la necessary when measuring the open-loop response of a ay a tern whose approximate epen-leop transfer function (of blook G in Fig* 46 for example) is of the form l/(p) (d0 -t- d^p-f* * *) to allow the input e to have a small constant value Oq; otherwise, slnee there Is no restoring force proportional to the displacement of the output from the neutral or sere position, the output y tends to creep away from the neutral position, that la, the output Inoludes a term of the form qt In addition to the terms Indicated * Here and In the remainder of this section the error will be considered to consist of a small constant term and a single frequency component* in eq. (6-3) • This term qt will distort the output 7 and will make It difficult to measure the desired frequency com ponenta of eq* (6*3)* The fundamental frequency, complex gain of blook G of Fig* 46 whose input Is given by (6*2) and whose output la given by (6*3) oan be written as 0(jw) = *“ 3® . (6.4) In the study of linear systems there Is but one value of G(jw) for a given value of w; however, when the blook G of Fig• 46 la nonlinear, the gain G(jw) la a function not only of w but also of the fundamental error amplitude c^* It is convenient In the study of linear systems to plot the complex gain G(jw) or Its Inverse Oil,) ; however, for 17 nonlinear systems a more useful graphical representation * will be Introduced* Observe that the Input x oan be ex pressed as o ■f' y; and using (6*2) and (6*3), one oan write an expression for x, for the corresponding olosed-loop conditlon x r e + y — oQ + c^ sIn(wt -+©) b Q -+ b^ sin wt -+ b 2 sin (2 wt *+ 2) * * — *q-t- a^ a In (w t -*-<*< ) -+ ag sln(2 wt -+ o( g ) • • (6 .6 ) The fundamental Input oomponent ef x oan be written In the -117- c omplex f ePB X( Jw) ~ 0 X bj • (6 *6 ) Notice that If Oj_ e^®-|- b^ or X( Jw) la plottad on the oom- plex plane rather than e^®/b^} the amplitude and phase of the fundamental input oomponent of x with respect to the fundamental output Is Indicated directly* In making an X( jw) plot It is desirable to make each point in this o ompl ex plane convey as much useful information as possible* The position of the point X( Jw) Indioates only the fundamental amplitude and the phase of the Input x with respect to the fundamental output component of y* It Is desirable that each point also Indicate the fundamental output amplitude and fundamental frequenoy w* For this reason values of eJ are determined for a set of values of b^ at a given set of fundamental frequencies w and tabulated as shown in Table lit After data has been obtained from the open-loop re sponse of a nonlinear system In the form lndloated b y T a b l e d each point la then recorded In the X( Jw) plane* For example the information given by the lewer right hand square In a To illustrate the meaning of this tabulation consider the lower right hand number In the matrix of values given in Table II* This complex number Indioates that when b, = 0*25 -indicated by the matrix row- and v&3.0 - indicated oy the matrix column - the error signal S(J3*0) is equal te 3.21 |J249*4<> cnd the fundamental Input signal Is X(J3.0) — 0.25 + 3.21 - 118- Tab !• IX \ w 0 0.25 0.5 1 . 0 1.5 2 . 0 3.0 ' b^ N 1.63 1.63 1.70 1.87 2.45 3 . 8 6 .93 2 1 . 6 Ix q q J 5-73' Z171- Aao° /204° /225° / 245* 1.5 1.27 1 .33 1.48 2 . 0 2 3.28 6 . 2 19.8 /l80 * A 7 2 * Afeg" A s o * /S05.a fZ2&" >fe46' 1.16 0.58 0.63 0.75 1 .16 2.17 3 .78 15.0 /ISO* tL67* A ^ Q T yfelO - £ 3 0 ° ■ b & L 0 0.5 .047 .086 0.14 0.30 0.77 1.83 6.43 /180 ** /137* /180“ /218a ^535* /3400 1 0.25 .0059 • 032 .060 0.13 0.37 0.9 0 3.21 1 /iao* A 15* ZJL39* A g o : A>19- /236 * /3Afi* Table II Is plotted as given by (6.7). After all the data In a table have bean plotted In the complex plane, two sets of loci are sketched* The first set la made by connecting all points which correspond to a conn on b-^ or all points which come from a common row in a table corresponding to Table II. The second set is made by connecting points of a common fre quency or points Indicated in one column of Table II. A set of values similar to those given in Table II have been obtained for a servo system whose output unit was a two-phase 400 cycle Induction motor* Haase data were obtained by measuring the open-loop frequency response of the servo system with an Input of the form given by (6 *2 )* The open-loop data of this servo are plotted as Indicated by eq. (6 .6 ) In Fig* 47* The following Information la available from Fig* 47* Consider point Fg which has a radius ef 1*33* For a funda mental Input of amplitude 1*33 and of frequency one ops, the output is of the form 1*3 sin 6*28t and the Input leads the output b y an angle ef 56# * Hie angular frequency Is 6*28 radians/sec•, since Fg Ilea am the one ops locus. The out put amplitude la estimated as 1*3, since Fg lies between the 1.25 and the 1*50 amplitude or b^ loci* C. Use of Linear Filters to Improve the Response of Nonlinear Systems Considerable material Is available concerning methods for Improving the response ef linear systems by the Intro duction ef linear filters in the ferward or error amplified?*3* 0 I Pig. 47. X(jv) plat af an aotual atrvo ayataa -121- Technlquea of this typo oan be used with some modifications and reatr lot Iona In the oaao of nonlinear ay a tenia. For ex ample consider the system ahown In Fig* 43. Thla ayatern la similar to that ah own In Fig* 46 except that a linear filter F la introduced Into the forward or error aectlon of the loop* If the open-loop frequency raaponse of the ayaten without the filter F la known and If the gain of filter F la given, then the open-loop frequency response ef the ayaten with filter F oan be determined aa followsi First It la assumed that the values of the error S( jw) fer the original system of the form shown In Fig* 46 are known and tabulated as In Table II* Thla means that values ef Fig* 48) which oauae y to take on the desired values ef at a glvsn set of frequenolea w #u*e equal to the set ef values of S(jw) of the erlglnsLL system* Knowing the complex gain of the linear block F, one oan tabulate a new set of values of jw) which will cause the output y to have the same set ef values of b^ for the given aet ef Frequenolea w, by dividing the known aet ef values ef X( Jw) by the gain of block F* As an example consider the oase where a filter with gain 1 /( 1 +-jw) is introduced as shown in Fig* 48 ef the aystem whose orig inal open-loop response Is aa tabulated In Table II* The new table oan be obtained by dividing each of the values of 3g(jw) by 1/(1 + jw) obtaining the new value ef S^jw). Fer oxample the value ef E^(jw) fer the lower right hand comer of the new table la found by multiplying the value ef 3( jw) 122- Fig. 48. Blook dltgru of serve syatm with filter 1m error aapllfler aaetlaa in thla position In Table XI by (14- jw) or S^(jw) Is equal to (14- J3)(3.21) OJ249*40. Tbs b^ + e^® or X( Jw) plot can bs modified when a linear filter is added as In Pig* 48* To Indicate this process consider the oase where F, which is assumed to have a gain of two, Is Inserted In a system whose original res ponse is shown b y the solid line In Fig* 49. N e w /> b,*l Locus 0 . 5 1.0 - b, -*4 Pig* 49* Method ef oenst root lag X(Jw) plet ef systeai with aapllfler ef gals 8 (see Fig* 48) using X( Jw) plet ef system without tapllflsr The value of e or e^® of the original systen la given by the line drawn from a point dlsplaoed to the right of the origin by a distance b^ (Is this oase unity) to the given point on the b ^ s 1 leous * In the new system an error Input equal to half the former value will produee the same output* For thla reason eaoh point in the X( jw) plet la found by di viding the original Z(Jw) by two and drawing the new b ~ 1 loous -124- through thaae points aa shown by the dot tad. lino* D. Iadioationa of Limit Cyolea and Subharmonlo Raaponae In thla aaction tha question of tha stability of non linear systems will be discussed in terms of tha apan loop reaponaa data plottad In tha !2(Jw) plana* To Indloata tha type of atabllity that will ba diaouaaed, obaerva tha origin In Fig* 50* Thla point oorreaponda to a fundamental Input of zero amplitude, yat tha two lool, b^=. 1*155 and w = l , whioh interaaot thla point Indloata that a fundamental out put of tha form 1*155 ooa t may axlat* One might expeot that tha Interaeotlon of a finite b^ laoua and a finite fraquenoy looua at tha origin of tha X(jw) plana would Indloata tha fundamental fraquenoy and amplitude of tha output y whan tha ayatem la In a H a l t oyola state* However, by tha definition of a In (6 *8 ), x must contain harmonic terma If y ha a any harmonlo taraa other than the fundamental. For thla reason one muat In general aaaume that x ha a harmonlo tarma oven whan tha X( Jw) plat Indloatea that x doea not have a fundamental component* Thla moans that tha moat that oan ba said far a ayatem with lool crossing on tha origin la that tha ayatem has a aubharmonlo reaponaa* Thla sub harmonlo reaponaa la different from the usual sub ha m on - lo response alnoe x by definition reneve a the harmonlo tarma that generally oause aubharmonlo reaponaa from the Input te block G of Fig* 46* Thla type ef reaponaa la very closely related to limit oyola phenomenon and usually Indloatea the presence ef a limit oyola* 125 Pig* 50* xTl*) pi it of »y«t« with Halt oyolt - 1 2 6 - If the system Is analytic one oan determine the stab1 1 - 21 Ity of the equilibrium point by using Nyqulst's criterion* This can be done by observing the enoirolaments of the origin of the X(jw) plane by a b^ locus.* This faot oan be estab lished rigorously by observing that Theorem I Indloatea that as the variations about equilibrium ef an analitloal system approach zero the system becomes linear. Fer this reason as b^-w-O, 3£( Jw) la equal to b^ times the inverse transfer funotlon of the olosed loop system* An extension of the b^~ 0*25 looua for the negative and complex frequency range (see reference 21) Is indicated by the crosses In Fig* 50* This locus, b^=: 0*25 and its extension indicated by the crosses, encircles the origin ef the X(jw) plane twloe In dicating that the system has an unstable equilibrium point* It should be observed that a second pair of loci, the b^n 1*656 and the w -=.0 lool, cross at the origin* Thla seoond orossing corresponds to a second equilibrium or singu lar point where the Input and the output are constants since w ~ 0 * The points that fall on the origin yield information concerning the stability ef the system* If possible all b^ lool should circle oounteroloolcmlae to the left of the origin and continue circling in this direction splrallng outward as w varies from sere towards plus Infinity* In general systems * Netloe that stability fer the oloaed-loop oase Is deter mined by the encirclements ef the origin by a X(Jw) locus, namely, a b^ locus* -127- wi th. trajectories of th.1 a type do not have limit oyoles ox* aubharmonlo response* If tha lool do not follow daalrad paths one oan use tha technique lndloatad In Section C of thla dmpt^r to change tha lool to a more desirable position* B. Response Hysteresis (Jump Phenomenon) Whan measuring the olosad-loop response of a servo using a two-phase motor as the output device, It has been observed that the output amplitude response depends upon the past history of the frequency and amplitude variation of the input signal x* This phenomenon Is partloularly notioeable If the input signal frequency is held constant and the input amplitude la varied* This characteristic of nonlinear systems can be described with the aid of Pig* 51* Pi Pig* 51. Fundamental output b1 ef nonlinear nerve system with response hysteresis phenomenon oocuring when the slnu«* aoldal input has amplitude a^ -128- If the fundament* 1 Input amplitude a^ la Increased, the fun damental output amplitude b^ increases until a^ reaohea am * If a^ beoomea larger than am, b^ Jumps to the lower ourve* If a^ la initially larger than am and then la decreased it will be found that the jump between the two ourvea doea not occur until a^ la leas than Thla oharaoterlatlo la go often called jump phenomenon beoauae of the Jump in the steady atate response* However* a definite and often a very gradual transition occurs between the two states • For thla reason thla phenomenon la oalled reaponaa hyatereala by the author* The presence of this phenomenon oan be determined from the X(jw) plot (see Fig* 47)* Consider sue input with a frequency of one oyole/seo whose amplitude la slowly In creased until the atate of the ayatem la determined by point P^* There is no point near P^ on the one oyole/aeo locus corresponding to a radius greater than 1*33* the radius of P^* exoept la the neighborhood of point P2 * For thla reason the flrat Jump ef the hysteresis cycle oooura when the Input amplitude becomes larger than 1*33* Aa the input amplitude la decreased from above 1*33 a second tran sit i on occura when the input amplitude becomes less than the radlua to point Pg aa there la no point near P^ with a radius less than P g * The operation of the system is then described by the point P^ whloh has the same radlua aa P^* completing the hysteresis cycle* The typical jump polnta -129- and ? 3 will ooour wherever tha constant fraquenoy locus la tangent to a oirole of constant Input magnitude* For the servo whose response is indicated by Fig* 47 the most prom inent characteristic of the response hysteresis is the large change of phase that results when the system passes through the critical points* F* Reduction of Nonlinear Equations to Linear Equations with Time Varying Parameters It is easier in many oases to solve an equation with time varying parameters than it is to solve a nonlinear equation* In determining the frequency response of a non linear system* the sinusoidal variable often can be ohoosen such that the nonlinear terms of the system equation are transformed to linear terms with time varying coefficients* Tills method* similar to the one proposed In referenoe 23* will now be illustrated by an example* Consider the following equations P2y -f-[l -H (x - y)2 ] py =>10(x - y). (6*7) The term (x - y) py is nonlinear and may be converted to a linear term by assuming (x - y) Is a sinusoidal function of theIndependent variable t* If thla equation Is written with v substituted for py* and e for (x - y) * one has pv + ( 1 -f- e2 )v = 1 0 e* (6 *8 ) I f the error e l a assumed to be of the form e s= o ooa wt. (6.9) eq. (6 .8 ) l a converted to a linear equation with one time varying coefficient. It l a convenient, in deriving a solution for v, to represent (6 .8 ) In terms of a block diagram. This diagram la broken into a fixed and a time varying block (aee Fig. 52)* The gain G(Jw) of the fixed or time Invariant component of the block diagram l a G( jw) _ JL, (6 .10) > t 1 The velocity v will be assumed to have the general form v = d^oos(wt -+ -+ dj C o a O w t -+■ ^ 3 ) • • •« (6 .1 1 ) The term -e2v can be broken down Into sinusoidal components. The fundamental term of -e**v l a and the third harmonic term la <3wt4- /^3 ) d5ooa (3wt +■ g) 2 Higher odd harmonic terms are of the form coa(nwt + -131- 10 e Time Varying Fig. 52. Block dltgran of syatMi with time vtrying ptnntfeeri -132- Now that the form of the various frequency components at the input to the fixed block has been found, a relation ship between the input and assumed output for eaoh frequency component oan be found by means of the transfer funatlam or complex gain* It is easier to handle these equations if phasor notation is used* For example oos(nwt -f~ 3) oan be represented by je^® in an equation involving n*1*1 harmonic terms* Since the fundamental Input to the Invariant section multiplied by the fundamental gain G(jw) is equal to the fun damental output of the Invariant section, the relationship between fundamental input and output ef the fixed blook oan be written as (6 .1 2 ) This expression can be simplified by multiplying by (jw -t l)/j which re stilts In the equation (6.13) Terms of the form d_ e ^ “ oan be written as D while the — 1 /S n complex conjugate, djf ' n Is written as In this man ner eq. (6*13) beoomos 133- q 2 10 ox - 5 D* -*-§ d 3 ) = (Jw + 1). (6.14) The third harmonlo equation oan be written aa °? D n D fi - +D3 + 2 } = °3 (J3w “+ 1} ' (6.15) and the general odd harmonlo equation beoomea D o 5» - gi- ( -§=£ -t- D„ -*■ “ g- " ) = Dn (Jnw h - 1) . (6.16) These equations oan be expressed in the form of a matrix. Notice that thla matrix haa Infinite dimensions due to the infinite number of harmonlo a which are present In the non linear system, operating under the assumed conditions. 1 * 1 4* 10c]_ _ — =■ D 0 0 0 a a 0 4 1 zi z t d i 0 0 0 • e 0 z t Z 3 z t D 2 0 0 0 • • o D t 5 t 3 • e • • e • • o o • e *• a e • a e e a * J (6.17) 2 O? In eq. (6.17) Zfc la equal to 0^/4 while (jwn 1 -+ ) la represented by Z • n -134 G. Network Representation The determination of an exact solution for each com ponent of v by use of the matrix equation given by (6.17) Is difficult if not Impossible. One oan, however, draw an equivalent electric network with currents Dg, * • Dn which satlsify (6.17); and one can determine the solution for these network currents as accurately as required fer finite values of n by use ef familiar network Impedanoe con cepts. The linear network shown in Fig. 53 has current variables Dg, * *Dn which satlsify (6.17). An exact so lution oan be obtained for If the equivalent Impedanoe between terminals aa' oan be evaluated. Let this Impedanoe looking to the right at terminals a a* be Za « J1 1 a i3 J5 I l a Fig. 53. Equivalent network for determining response of a system with time varying parameters The solution for 1a terns of Za is given by 10 °1 ’ ^ DX = D1( 1+ + Za). (6.18) To obtain a solution both and Z^ must be separated into real and Imaginary part s• Dl = ° 1 R + JDir D* = C 6 >10) d xr - JDi x z — a z n + JZ.I Separating (6.18) Into a real and Imaginary part one has 2 10 cx - ZaR^ “ D1I*W + 2al^ (6 .2 0 ) 0 - DlR iw -h ZaI) DXI (1 *+■ Z ^) Notice that these equations In turn break down Into a set of antisysusetrio network equations. These equations oan b< combined to give 10 °i= dir [1 + r z «LR~t~ <~~ ~ Z * I ) 2 ] . <«•*!> * 1 ■+■ Z aR where -136- The difficulty in solving for liss in evaluating Z • If the network were similar section by section, trans- a mission line equations could be used for evaluating thla impedance* Thla oan be done only for the oase of zero fre quency* One possible method, which la not necessarily the best* la to assume that some component Dn is aero* (This corresponds physically to opening the n^*1 loop and mathe matically to crossing out all rows and columns below and to the right of the (n-l)/2 row and column, respectively.) If e Is of the form 2 cos t, and it la assumed that does not appreciably affeot the value D^, then a solution oan be found whloh is of the form v = 5.10 coa(t - 31.8°) -t- 1*160 oos(3t 101*) -t- 0*109 ooa(5t +• 231.»°) (6.23) The values of the 7 and higher harmonica can be found based upon the assumption that the solution for the first harmonic term la accurate* A plet of (6*23) la compared with the analog computer solution of eq* (6.8) In Fig* 54. Notice that the computer solution and network solution are very almlllar In fora, aa ahown In Fig 54, although only the first, third and fifth hamonloa are used In the calculated ourve * This particular problem once It la In network form oan be solved aa accurately and almost aa easily using net work Procedures as ualng the analogue computer* Greater accuracy can be sbtalned using network procedures, If time -137- ACTUAL RESPONSE CALCULATED RESPONSE Pig* M* «f itiwl VMpMH MXmUltd r«apon«« af ar«t« vilk U m i vtryiaf i i m u m Ii m -138- ig taken to evaluate the impedances accurately. The output displacement y oan he round hy integrating (6.23) with respect to t giving y -= 5.10 sin(t - 31.8°) -+ 0.387 sln(3t-M01°) +- 0.0399 sln(5t+ 231.9°) (6.24) This output occurs when the input x Is x — y -t- e - 2 cos t-+ y ^ 4.38 sin(t - 9°) t-0.387 sin(3t 4 101°) + 0.0399 sin(5t+- 231.9°) (6.25) A few pertinent points will he indloated before leav ing this subject. This method of analysis oan he oarrled out only if the equation being solved oan he reduced to one in irtilch the nonlinear terms are products of one variable or products of the one variable times the first power of the second variable. For example, the term (^|-t2e)SH- + °an be reduoed to linear form when e is as given by (6.9 )• Terms of the form •2PT+ T2 * cannot be reduoed to linear form slnoe squared terms ef both e and y exist. It la Interesting to observe that the impedanoe of the series arm in the clroult shown In Fig. 53 Is equal to the Inverse gain o7)»w) of the fixed blook ef Fig. 52. The value of n corresponds to the harmonlo Dm whloh flews -139- throu&h the arm* For example If G(^w) was of tha form Jw + 1 +■ l/jw , than aaoh series arm would consist ef an R-L-C series network* H. Maximum Sinusoidal Output Often In determining the frequency response of a de vice the output Is assumed to be sinusoidal, that Is, y Is of the form b eos wt. After making this assumption con cerning the output y, the corresponding Input to the device must be determined* Generally if the system equations are an acourate representation ef the system, one oan find a value of the output amplitude whloh causes the Input to take on a form which Is not physloally possible, aa for ex ample, when the solution for the Input signal to the device becomes Infinite, complex: or Imaginary* It la believed that a specification ef this maximum sinusoidal output Indi cates more about the quality ef a system and Its ability to respond satlsfaotorlly than does the more usual static limi tations or restrictions* Applications of the oonoept ef maximum sinusoidal output are considered in Chapter IX* To Illustrate the concept of maximum sinusoidal out put consider eq* (6*8)* If v la assumed to be sinusoidal and of the form d cos wt, eq* (6*8) oan be written as -wd sin wt 1*(lt t®) d cos wt = 10 •• (6*26) -140- The value of e which latlsflti (6*£6) oan be shown to be where cos 0 =* 1 / ^ 1 -+ w2 • The signal e oan be plotted dir ectly or If d la sufficiently small the binomial expansion can be used to obtain a series expansion for eq. (6.27)* The lmportanee of thla development lies in the faot that by specifying the form of the output signal the maximum output for that particular specified form oan be found b y observing the value of d which oauses e to ohange from a real value to a complex value* To Illustrate this procedure, eq. (6.27) Is rewritten giving 0.02 d If the amplitude of the sinusoidal component under the radl- p oal becomes larger than the fixed value (1 - 0*02d ) then during certain periods e becomes complex due to the negative value of the stun ef terms under the radical. The error signal e will be real only as long as # The orltloal value ef the amplltui d P o is -141- __ 7.07 _ t d-"V'ic * (6'30) The magnitude of the amplitude of y, namely b, la equal to d/w. For thla reason one cannot expeot to have pure ainu- aoldal motion la the output of the servo greater than dc/w< Thla critical value of b, namely, bQ , haa a value for the example under oonalderatlon ©r b - --- 7‘07 . (6.31) o w ^ H - A -t- wZ For example. If w = 1, la equal t© 4*55* nie amplitude bQ la called the maximum sinusoidal output. The equation, (6.7), used aa am ©xampl© in thla auc tion la of practical importanc© lm aervomeohanlama, alnoe amay two-phaae aervo motera have operating equations which osm be approximated by an equation very almlllar to (6*7). It haa been shown that th© ©quatlan of mctlon of a two-phaa© indue- tion motor with a larg© rotor roalatano© is of the form J p2®+ [f + |(l*+ I*)] P®= Ktx Zr w 0 (6.32) where J la the rotor in©rtla, f th© ooofflolont of vlaooua friction between th© rot©r u d stator, Th© current Ix la a fixed sinusoidal ourrent flowing in an© meter phas©. Th© control current I flowing lm the aeoond phaa© la 90° out y of phaa© ©l©otrloally with th© fixed current Ix • -142- The variable 0 Is the angular rotor position, Wq la the aynohr enus speed of the motor and K la a constant depend ing upon the motor design* Hie value of W q K oan be deter mined by dividing the stalled torque of the motor by the product of the two currents, I * and I y . producing this torque* One oan show that the amplitude of the maximum sinusoidal output bQ for two-phase motors with an equation of motion given by <6*32) Is (6.33) The value of b given In (6*33) oan be used to approximate o the maximum sinusoidal output of nest two-phase servo motors* (The results of (6*33) will net be In serious error provided the two-phase motor being considered will not run as a single phase motor*) X* Response Hysteresis In Seotlon B of this ohapter the phenomenon of re sponse hysteresis was briefly considered* In this seotlon this type of response will be studied by analytlo methods and by means of analogue computer solutions • In Seotlon H It was shown that devices such as serve motors have a definite maximum sinusoidal output* This maxi mum output Is closely related with the phenomenon ef response hysteresis* In all the systems which the author -143- ha a studied* the cause of response hysteresis oan be attributed to the Imposition of an input to the servo larger than the servo motor is capable of following* This response hysteresis or break down occurring in servo systems is simi lar in many respects to limitations observed in human con trolled systems such as the steering of an automobile on a winding road* To illustrate the actual transient phenomena oocuring in a response hysteresis cycle* eq. (6.7) has been solved using an analogue computer with an Input x equal to a sin wt• Sections of the graphlo record of the computer solution are shown In Fig* 55(a)* (b) and (c)• In each section the upper record Is the Input x* the second is the output y and the lower record Is the error (x-y)• In Fig* 55(a) the input x is sufficiently small that the output y follows the input with a rather small error* Fig* 55(b) Indicates the output y when the input x is of the correct magnitude to cause the first transition in the hysteresis cycle* Notice that the error before the transition is much smaller than the error after the transition* Notloe also that after the first transition* an increase in the Input amplitude causes the peak-to-peak amplitude of the output y to decrease* Hie third graphic record shows the system Input* output and error during the second transition* Notloe that the value of a at the second, transition is smaller than the value of a at the first trans it! on and that the change In error and output y at this transi- m m m U W u W W v m /£Bw°" WFFR W W W l in VYU a (o) First transition t t t t f f u PUJ l m i \ \T\ 11 i k i \WWW w \w \ w[ w \w \ n l PUT I 1 1 1 f ffigg? W W lu l m (n - f f f f f * **ror f j-j-f w w w w (a) Snail Input signal (o) Seoond Transit ion 56* Recording ot output y and error (n-y) of alnulated servo with sinusoidal Input x tion I» not as noticeable as for the first transition* The transition in steady state response from one stable state to another is not abrupt and the transition period may be equivalent to several oyoles of the Input signal* It is possible to determine analytically the approxi mate magnitude of the sinusoidal input at whloh the transi tions or response Jumps take plaoe* This oan be done moat conveniently for the example solved b y the ooraputer, eq* (6.7), by determining analytically as in Seotlon F the open-loop response assuming that the error la sinusoidal* The analytical open-loop response data can then be plotted In the jw) plane as Indloated In Seotlon B and the points at whloh the jumps ocour oan be found as In Seotlon D* This process for determining the jump points has been oarrled out for the system represented by eq* (6*7) and the T{Jw) plot is Indloated by the dotted line In Fig. 56* In determining the analytical data the error rather than the Input x Is assumed sinusoidal, thereby causing the Input x to oontaln harmonio components In addition to the fundamental* Since these analytical data were based on this type of assumption, the value of the oaloulated fundamental amplitude at whloh the Jumps take plaoe differs from the computer data* A second set of open-loop response data were deter mined analytically In order to verify the computer solution* These data were obtained by assuming that the error e and output y were of the form -146 Pig * 56. X( Jw) plot of ayatom with roaponae hyatoreaia -147- o - oos wt o^ ooa(3vt -+ 0g) (6*34) y -=. bx ooa(wt - o3 co«(3wt + 0^) , The values of b^, /^L* ° 3 6X1(1 ®3 ware determined using the concept of nonlinear gains used in Seotlon G of Chapter V* Notloe that In this solution the signal x whloh la the sum of e and y will not contain a third harmonic component* The X(Jw) plot of this last set of analytical open-loop data Is shown by the solid line in Fig* 56* This solution whloh assumed a third harmonic term In e was dlffloult to compute; however, the data so obtained checked the computer solution quite well* To Illustrate the quality of this agreement the peak output amplitude of y as determined by the two analytic solutions mentioned In this seotlon as well as the computer solution were plotted as a function of the fundamental Input amplitude In Fig* 57* The upper ourve of the analytic solu tion which included the third harmonic term was very nearly the same as that of the computer (see Pig* 57). In order to show this analytic solution it was purposely plotted slightly above the oomputer solution* J* Frequency Response of Nonaymnetrlo Systems The analytical problems considered In the preceding sections all dealt with the response of the symmetric equation, eq* (6*8)* The steady state solution for y con sisted of the sum of fundamental and ether odd harmonic com ponents for an error ef the form o oos wt* Experimental ?4 Measured Response. Theoritical Response Considering Fundamental And Third Harmonics. Theoritical Response Considering On y Fundamental . __ 0 2 3 4 7 8 9 Input Amplitud 5 ( , Sinusoidal) 6 Fig. Ef. Calculated and ceaeured peak output of ayatern with response hysteresis -149- •vldenoe indloates that any stable devloe with a symmetric system equation will have a steady state output solution composed of the sum of edd harmonic components when the in put consists of a single frequency component* Many prac tical systems suoh as the function generators considered In Chapters III and V are nonsymmetrlo systems, and for this reason the frequency response of nonsymmetrlo systems will be considered briefly in this seotlon* The output of an unsymmetrlo system, unlike the symmetric syste^ will generally have do, fundamental, even and odd harmonic output components when the Input consists of a single frequenoy component* In the study of symmetric servomechanisms, it is often possible to obtain a good approximation of system performance by neglecting all fre quency components except the fundamental since the servo output motor will attenuate the third and higher odd har monic components exciting the motor* In nonsymmetrlo sys tems the output will generally have a do term and second harmonic components in addition to the fundamental and third harmonic components* It should be observed that the second harmonic component being closer to the fundamental in fre quenoy will not be attenuated as much by the motor as the third harmonic • For this reason the seoond harmonic Is apt to be more pronounoed In the output than the third harmonic* Likewise the do signal In the output causes the average value ef the output te be shifted from the sero position* It Is often possible to neglect the seoond and higher -150- harmonlo components of the output of a nonlinear dovloe, but the do oemponent oannot bo neglected If reliable result* are to bo obtained. A simple example will now be considered to Illustrate the method of analysis whloh oan be used to determine the steady state response of a nonlinear nonsymmetrlo system* Consider the equation (p® + p8 + 2p + 1) y -+ y2 -at x. (6.35) The nonsymmetrlo term In eq. (6.35) la y2 . a frequency re sponse characteristic oan be obtained for this nonlinear sys tem by letting the error a be defined as e = x - y2 , (6.36) and assiimlng that o la of the form: e r: o^ sin ft ^ oQ • (8.37) One oan determine the frequency response of this system con sidering that a contains only a fundamental component. If this Is done, the Input x will oontaln a do term In addition to the fundamental and seoond harmonlo terms. In order te Isolate the effects of the do term and the time varying terms in the Input, the error la chosen suoh that the Input will not have a do term. Beoauae of the oholoe of e as given by eq. (6.37), eq. (6.35) beoones (p® + p2 + 2p +1) y — Oq o^ sin wt • (8.38) Conventional linear methods oan now be used for finding y. The steady state solution of (6*38) will be of the form y =r oQ - V b sln(wt - 8) . (6.30) The value of b la given by b = 0 1 ft- w 3 ■ + ■ 2 w )8 + ( 1 - t ^ ) S J s and 6 oan be determined from tan 9 — 1 - w2 Substituting the value of y given by (6*30) into (6.36) and observing the form of e given by (6*37), one has for xt x - o1 sin wt oQ -t- e2 -+ 2oQ b sin(wt-8) 8 -h 2 “ £l - ooa(2wt - 2 9)j (6*40) As the purpose ef oq is to eliminate the oonatant term of z f one oan set the oonatant terms of (6*40) equal te zero, thus Oq +■ Oq + g— -s. 0 • (6*41) The values ef whloh aatlalfy eq* (6*41) are -152- Aa Oq la zero when the Input end b ere zero, the oholoe for oQ la (6*43) When the value of Oq given by (6*43) la aubatltuted Into eq • (6*40) one hea for x ooa(2wt -26)* (6.44) It oan be aeen at onoe that a physical input exists only ao long as b2 1/2; for if 2 b2 becomes larger than one, x be- comes complex* One oan state that 0.707 la the maximum available sinusoidal output when x does not have a do or oonatant term. It should be noted that the limit Imposed upon b la determined by the value of b which onuses the con stant or do term of the error oQ to change from a real value to a oomplex value* K. Applications In this chapter methods for determining the open-loep frequency response of nonlinear systems have been considered* In the actual design of a nonlinear serve system, it is suggested that the open-loop frequenoy response data ef the system, determined either experimentally er analytically, be plotted In the T(Jw) plane as Indloated la Seotlon B* The stability oan then be determined as Indloated In Seotlon D. -153- If the system la unstable or if the b^ lool approach the origin too closely, it la advlaabl# that llnaar filters ba insertad In the error amplifier. These flltara should oauaa the Mlnlnum radii of the b^ loci to fall In a aoro appropriate region than the b^ leoi ef the system without the flltara* In the design of llnaar aervomeohanlaaa there la a rule-of-thumb oonoernlng the olosed-loop frequency response* Let the oloaed-loep ayatem Input ba a aln wt and the output be b aln(wt + 0) • This rule-of-thuirib states that the maxl- 25 mum value of the ratio b/a should be equal to 1*3* The frequency at whloh b/a la maximum la called the resonant fre quency and It la generally made aa large aa possible* A criterion similar to this may ba stated for nonlinear ayatema* First observe that the radius from the origin to the b^ looua la equal to the fundamental olosed-loop Input amplitude a^ which produces the fundamental output amplitude of b^* Likewise observe that the minimum radius to a b^ looua corraaponds to the olosed-loop Input at the resonant fre quency* For this reason the rule-of-thumb criterion becomest The minimum radius to a b^ looua shall be 1/1*3 times the value of b^. It Is Impossible to make this criterion hold for all values of b^ and should therefore be used as a guide rather than a hard and fast rule* Consider the X( jw) plane shown In Fig* 59* Notloe that point Indicates that a unit amplitude output la possible when the fundamental Input amplitude la 0*1 at the Pig* 59* Possible T{jw) plot of servo frequenoy response - 155- frequency of two radlana/aeo* A filter must ba uaad In tha error amplifier whloh will shift points on tha w - 2 looua away from tha origin* If poaalbla this shift should ba to tha right so as to Inoraasa tha width of tha frequenoy pass band* Using Fig* 49 as a guide, it oan ba observed that a filter whose gain G(jw) or g(w)a^ la near unity in magni tude and whose phase angle la leading by 20 to 45 degrees near w equal to two will have the desired effect* For this reason the lead filter shown in Fig* 60 Is proposed as a com pensating network* Notloe that the gain G(Jw) of this filter - v w v •SM&if Fig. 60* Lead filter proposed far Uprsviaf thn response ef the' system whose TL( Jw) Is sh ew La Fig* 19 Is equal to (6*45) Te Illustrate the method ef drawing the lool for the new system with the lead filter shown In Fig* 60 consider points P^ and Pg shown in Fig* 59* Hie phasor from the ori gin to point *8 represents the fundamental component of the system output in phasor form when the error signal In phasor - 156- forra la equivalent to the phasor from point Pg to point P^, namely 1.08 1 174° • (The frequency is two radlana/sec.) The same output will be produced by the system with the com pensating filter if the output of the filter is 1.08 | 174°. This means that the input to the compensating filter or the error signal of the new system oan be found by dividing 1.08 | 174° by the complex gain of the filter or (1.08 [ 174*)/ . This input la 0.857 \ 155.5°. Since the fundamental Input represented in phasor form, that is X( j2)# la equal to the sum of the fundamental output and the error In complex form, the new point P^, for the com pensated system la given by 1 -t* 0.867 \ 155.6° or by the sum of the output phasor from the origin to Pg shown In Fig. 61 plus the phasor 0.857 |155.6° - the phasor from point Pg to point P^ In the same figure. The complete X(jw) plot shown lm Fig. 50 ha a been transformed In this manner Just considered for point P^ to the one shown In Fig. 61. Notice that the b^ loci In Fig. 61 do not approach the origin of the X(Jw) plot as closely as shown In Fig. 59 and that the frequency at the nearest approach of the b^ loci to the origin, that is the resonant frequenoy, has been increased from about 2 radlana/sec. to 3 radlana/aeo. for the system with the filter. This X(jw) plot indloates an Improvement in the response of the system; however, the response la still not entirely satisfactory since the minimum radius to the b^— 1 locus 1st still only 0.18. Pig* 61* X(jw) plot of servo frequenoy response after using filter shown In Fig* 60 -158- Chapter VII 26 Subharmonlo Response of Nonlinear Systems A. Introduction If a nonlinear system Is excited by a sinusoidal Input signal, the output, In general, will consist of a com ponent of fundamental frequency and various harmonic fre quency components* In some oases, other output components whose frequencies are rational fractions of the fundamental input frequency appear* Components of this type are oalled subharmonlos* In this chapter subharmonlo response Is briefly investigated using frequency response characteristics* A slrrple example will be considered to illustrate the method of analysis* Fundamentally the subharmonlo response Is the result of an unstable condition existing In the system* Consider the oase where a system such as Is shown in Fig* 62 has an unstable singular point and a stable limit oyole* Ttie steady state or limit cycle output when the Input signal x Is zero is of the form y = b^ oos wQt ■+• bg oos(2wot + 0g) * • (7.1) If an Input signal x equal to a2 oos(2wQt -I- oC ) is applied to the system, the output y will still be of the same form as that given by (7*1) and the system is then said to have a subharmonlo response * Generally, It Is possible In oases -159- fty) N o n l in « a r S e c t i o n Fig* 6t. B l M k dlagraa m t « »«alln«ar mjmt -1 6 0 - of this type to let the Input x be of the form a2 coa(2(w0t -%■ S and the resulting output then takes on the form y = b^ cos (wo + S ) t + b2 oos(2(w0 “l- & ) t-V- Bg) • •, (7 .2 ) providing S is sufficiently small. This shift of the fre quency Indicated by (7.2) la known as frequency entralnnent.27 In other oases, the system with subharmonlo response may have a stable singular point. But when the Input la finite and of the form a ooa(mwt + 0), where w Is near a natural or resonate frequency ef the system and n la a small Interger or rational fraction greater than one, the system may develop a large subharmonlo output component with a fundamental com ponent of the form b oos wt. Both of the systems Just discussed, the one with an unstable and the latter with a stable singular point, have subharmonlo response. It is again pointed out that subharmonlo response la an unstable phenomenon* The presence of the input signal causes the effeotive gain of the nonlinear sections to be altered suoh that a oharaoterlstlo mode of finite-dmplltude, as discussed In Seotlon B ef Chapter I, oan exist and whose fundamental frequency Is some rational fraction of the exci tation frequency. This characteristic er subharmonlo mode oan be Initiated by an Initial condition such as noise as Is the oase of the oommon electronic oscillator. -161- In the analysis of a nonlinear system with subhar monic response. It Is convenient to represent the system by means of a two seotlon block diagram - a linear seotlon and an nonlinear seotlon - as shown by Fig* 62. The system will be assumed to be excited by an Input signal x of the form x = a sln(nwt -+* The assumed steady state output signal y^, which does not contain the subharmonlo term, will be of the form r= b0 + t>n ain n w t -+* bgn alm(2wnt ■+- ®2n^ * ** (7*4) It will also be assumed that Initially a small subharmonlo component A(t) oan exist In the output. The total output signal y 1s then of the form y^ -+* A • The following system equation can be written for the block diagram given In Fig* 63: 3(p) (x f (y) ) = y . (7.6) As y1 Is a solution of (7*6) when A Is zero, one also oan write G(p) (^(y^^) + x) = ^ • (7.7) Subtracting eq. (7.7) from (7*6) and substituting y^4*-A for y, one ebtains the equation 0(p) - f < n > > = (7.8) In this study, th. aubhanonlo ooaipon.nt ^ will bs aasvua.d to be small Ijs the vicinity ef time t — 0 • During the period of time when 4 Is small equation (7*8) oan be approximated by* A = o T p ) 4 ]y = yi , (7.9) where gTp) i» the inverse of G(p) * Notloe that eq. (7.9) Is a linear equation with time varying coefficients. For this reason linear techniques oan be used for determining ^ during the period when <4 Is small in amplitude compared to B. Analysis of a System with Subharmonlo Response The relationship between the input and output of the nonlinear seotlon shown in Fig. 62 will now be assumed to be of the form f(y) = - y2* (7.10) The input signal x is set equal to xsag sln(2wt 4 o( )# (7.11) whloh preduoes the steady state output y^ (subharmonlo terms not considered) yj it b^ -V bg sin 2wt b^ sin(4wt + 9^) • •• (7.12) * Notloe that if f is a function of x # px, eto. then eq. (7.9) is of the fern [rf + ycpr)p ■+ p2 + • • ] A ^ - 163- The subharmonlo oomponent AS , having a term of tha form i i t sin (at -VO) must than satlslfy tha oondltlan given by •q. (7.9) whan f(y) Is as given by (7.10), namely - 2yx^ ' °^P) ^ • (7.13) I f the subharmonlo la to build up to finite size it 1s neoessary that ^ contain a faotor or faotors whloh Inorease with time. Consider tha oaae where A contains a do term and all harmonios of sin wt up to the n**1 harmonlo • nils means that there are (n +1) harmonlo aquations to be solved and n « a there are 2n -f 1 unknowns - bQ , b^, • • bm , the amplitudes M S H of aaoh of the harmonlo terms, and 8^, ®2'* * th# phases of aaoh of the harmonlo terms. Beoause of the form of (7.13), m n If bQ, bi ' ' ar# a possible amplitudes ef the subharmonlo * • frequenoy components, then kbQ, kb^, * * are also a possible amplitudes of the frequenoy components. This means that any one term say oan be ohosen arbitrarily* The n ^ l harmonlo equations oan be written as n -+* 1 complex equations which upon equating real and Imaginary parts reduce to 2n-v 1 equations. (The equation relating the do terms will not con tain an imaginary part when written In complex form.) If these 2m ■+ 1 equations are Independent, only in special oases m . s m w will It be possible to obtain a solution for tj/DQ, * * • t * and e1# ©2* •to# If however b0 is not a constant but la equal to bQ e^ the new unknown J Introduced makes a solu tion possible (see last seotlon of Chapter I). The equation for ^3 , based upon the preceding reasoning, la - 164- = e ^ * ^ 1 "*■ ^ 2 sin(2 wt -+ 8 2 ) * • J (7.14) during tha period when tha maximum value of ^ Is vanishingly snail with respeot to the maximum value of y^. Generally tha subharmonlo frequenoy w must be olose te tha resonate frequenoy of tha llnaar block (see Fig. 62). For this reason all terms of A other than the fundamental subharmonlo fre quenoy term will be attenuated to a muoh greater extent thas the subharmonlo term of frequenoy w. Often all terms exoept the tenable* * sln(wt + 9^) oan be negleoted as Is now done. The subharmonlo term Zk will be approximated by the single term jy - e^* bx sln(wt+8), (7.15) Likewise, the same arguments oan be used to show that the term bg sin 2wt and the do term bQ are generally the largest components ef the steady state solution y^. Therefore, y^ will be approximated by Tl — bo **" **2 *** 2wt* (7.16) (The primes are dropped from 9 and b^ In (7.15) as they no longer oanfllot with the corresponding terms In eq. (7.16).) Substituting (7.15) and (7.16) Into eq. (7.13) and equating fundamental components one obtains - 2 e^ * * oos(wt - 8) SZ olJ)l»i *(3 bi sin(wt-Y9). (7.17) -165- It la convenient to convert (7.17) to the complex form -(8 bxb0 e*^ + Jw^t'+J9+ b^g e*? + ) - G7p> b x .t#+J9 (7.18) Notloe that the term b^ e^ equivalent to the notation e*^ used In Seotlon S of Chapter I la the Investi gation of oharaoterlstlo modes. It was indloated la the same seotlon of Chapter I that the operators auoh as g 7p ) become algebralo multipliers when operating upon quanltles of the form e ^ Jw)* if -+ Jw) la substituted fer p. For this reason (7.17) becomes after substituting (‘p -4 jw) for p and canceling the term e^? ^ JwJt-t J© on both sides of the equations -(8 ‘o t J ‘s •'***) * 0T« + » • (7.19) Equation (7.19) Is very similar to the oharaoterlstlo equa tion considered In Chapter I* However, here values of © and ^ must be determined rather than ^ and w as was the oase in Chapter I* One ef the simpler methods of solving (7.19) for £ and © Is to plot 07^-frjw) on a complex plane allowing^ to vary, and also to plot on the same plane -(2 bQ -v jbg e"®^9 ) allowing © to vary. Solutions for ^ and © are the values of these parameters at the intersection of the two plots. —166 — C . Example Consider the oase where G(p) la of the form G(p) “ (p2-*- 0.02p -tlf1, (7.20) lb oan be shown that the values ef b^ and b2, the do and sinusoidal terms of y^, when the subharmonlo does not exist are equal to - 0.016 and 0.1651 respectively, for an ln- put signal x equal to 0.5 sln(2t ■+■ ) • Hieae values are found by substituting the assumed value oi G(p) and x given above and f(y) given by (7.10) Into eq. (7.7) and obtaining a solution fer y1 (see reference 28). Plots ef the oomplex quanltlea 0(^ -+ J) er ^ 8+0.02^ +■ J(2 ^ + 0.02) and (0.0272 - J 0.1651 e“J2Q) are shown In *lg. 63. The solutions for ^ and 9 Indloated by the Intersection of the two ourves in Fig. 63 are 0.071 and 86.7° respectively. If the frequency of the Input signal x la 1.1, rather than 1.0 as previously assumed, the value of bQ and b^ would be approximately the same; however, gT3^ +J1.1) then becomes J 2 *+* 0.02^- 0.21 -f* J (0 .022 * 2.2 J ) . The loous of oT^y-t-Jl*l) lies to the right of the olrole In Fig. 63 and does not Intersect the olrole. This Indicates that a subharmonlo oan not build up. The range of values of w for whloh a subharmonlo term of the fora given by eq. ("XU) oan build up oan be very easily determined from a plot ef the type shown In Fig. 63 9 by assuming that values of bQ and b2 remain nearly oonatant for small changes ef the frequenoy w. d w m m I • 1 £ I ' ' - iliTi: TTrF^i ;..rc;.„.;^?aii ...... et: ;HI«*3 c-M^i ^ -Hm Ul -168- The output y of the aye tea oonalsta of the sum of the steady state response y^ and the subharmonic term A and oan be written aa y + ^ - 0*0136 -f b x e°*071t sln(t + 86.7°) -^0.1651 sin 2t (7.21) during the period when b^ #0*071t amall* The aystem equation, eq. (7.6), with f(y) aa given by (7.10) and G(p) aa given by (7.20) was aet up on an analogue computer and the Initial conditions were very carefully aet ao that lnltally the amplitude of the aubharmonlc was very small. After twenty or thirty seconds of operation the eubharmonlo component of the response had grown until It became quite apparent. This component continued to grow In amplitude until It dominated the fundamental output component and finally overloaded the computer. In order to verify the values of the"constants b^ and 9 found for the aubharmonlc term it 1s necessary to Investigate the aubharmonlc term during the period when Its amplitude is muoh smaller than the ateady state terms of y. It la therefore necessary to eliminate the ateady state terms from the oomputer output so that the aubharmonlc term may be observed alone. This was done by passing the output signal y through a filter whloh had a transfer function of of the form (p^ + O.lp + l) 1 • The output y and the out put of this filter are shown in the top and bottom recordings -169- of Fig* 64* Not* that the aubharmonlc torn la not apparent In the output y while the filtered output consists primarily of the aubharmonlc term* The maximum and minimum values of the filtered output shown In the graph were measured and these values after being oorreoted for the dc term In the filtered output were used to determine the rate of build up of the subharmonlo, that Is, ^ was determined from the envelope of the filtered signal* The value of ^ so deter mined was approximately 0*073* This measured value of ^ Is In fair agreement with the calculated value of 0*071; although, the aubharmonlc was of sizable magnitude relative to the fundamental when this measurement was made* Likewise, the distortion of the filter would cause this measurement ^ to be in error. Conclusions The material In this chapter dealt primarily with a method for determining wheather or not the aubharmonlc component will build up In amplitude* Xm carrying out this analysis, the form of both the aubharmonlc A (see ©q. (7*14) ) and the steady state response y^ (see eq* (7*15)) have bean simplified* It Is believed that a solution for the sub harmonic term A which Includes the harmonic terms negleoted In eq* (7*14) oould be obtained by use of the net work procedures developed In Chapter VI, for the period of time that the aubharmonlc & la very smII In amplitude* -170- 5 sec LTJiri rt 11 i n 4 / f i l t e r o u t p u t L— ------ i Fig* 64* CMjmtcr •ImOmtlta «f system «ltk n M M i m l e reap«»M - upper vtoerllag lmdloetea computer aeltit&Mi, lower tree# Imdloetes irtilnrwitc oempomemt osmputer ^ j*v output (ebtmimed top fllterlac eemputer output} -171- The small signal theory used In determining the pos sibility of a subharmonic term A Indloatea that IT the sub- harmonio A builds up, It will continue to build up without an upper limit In amplitude* If the actual upper limit of the amplitude of the subharmonlo term la to be Investigated, the original equation, eq* (7*8), must be used rather than the small signal approximation, eq* (7*9). It would appear likely that the techniques, used in Chapters V and VI to determine the steady state response of nonlinear systems, oould be used to Investigate the steady state aubharmonlc term A * -172- Chapter VIII Dimenslonal Analysis of Nonlinear Systems A- Introduction After studying the routine operational nethoda for determining the reaponae of linear systems, one la awed by the amount of labor Involved In obtaining even an approxi mate solution of a nonlinear ayaten* The difficulty in determining the aolution of nonlinear ayatem equations doea not lie in the complexity of the systems, for exact aolutiona cannot be found even for the equatioca of many of the aimpler aysterna• It Is convenient In determining the solution of a nonlinear system equation to review the solutions or methods of aolution of other nonlinear equations which are similar In form to that of the system equation being Investigated* If at all possible, the variables of the aystem being inves tigated should be ao chosen that the problem reduoes to one which haa bean previously studied* In this manner, the behavior of the system can be determined without duplicating previous work* In this chapter two phases of this problem are con sidered • The first section illustrates the manner in which a relatively oexplicated nonlinear system equation ota be reduced to the form of a simpler nonlinear equation dis cussed in Chapter VI* The latter part of this chapter deals with the Buckingham7T Theorem* This theorem Indicates the -173- functional form of universal curves which describe the characteristics of nonlinear systems* B• Change of Variables To lndloate how the problem may be simplified by means of a change of variables, consider the equation p®r +• (1*1" 0*1 u2)pr +-(1+ u2) r — u (8.1) This equation, eq* (8.1), will be converted to a form suoh that the network solutions given in Section G of Chapter VI can be used* That la, eq* (8*1) must be converted by a change of variables to the form (8 .2 ) in which case the equivalent block diagram la aa shown In Fig* 65* IOe 6 Time Vorying Fig* ee. Block diagram of system with time varying parameters Equation (8*1) can ba reduoad to tha form glvan by aq. (8*2) by first setting u s 10 a • (8.3) In thla ease, aq. (8.1) becomea p3r +■ (1 -1-10 e^)pr Hr (1-blOOa^) r =s 10 e (8.4) Now tbe coefficient of a^ In aq. (8.4) la aat aqual to v; that la, 10 pr +* lOOr — v. (8.5) Thla means that r la equivalent ta r ~ i~orP ^ toy ; <8*6> and eq. (8*4) radueaa to f a f p + i o ) ▼ + ^ = io ., (8 .7 ) the desired form given by aq* (8.5)* Xn Chapter VI It was damonatratad that tha equivalent circuit of tha above equation la aa shown In Fig* 56* The currenta D^, Dg, • • indicated In Fig* 66 are equivalent to the hamonle component a of v written In phasor form (see Section F, Chapter VI)* Likewise the Impedance Z(njw) Is aqual te tha complex, Inverse gain olnjw) or (P3 -+ P -+ D / ( 1 0 ) (p -1- 10) . Z(jw) / O E Fig. 66* Circuit ftr dtttwlnlng hamottla ecMpMMta in tk« output of • noaliiaar qratM M i -176- C. Dimensional Analysis and the Buoklngham TP Theoram One of the properties of equations relating physical quantities Is that eaoh additive term of the equation must be of the same dimension* For example, the simple equalon r =; f Is pure nonsense if r Is the distance to the moon and f Is the number ef fish in the Atlantic Ocean* This dimensional property can be a powerful tool if properly handled* A good discussion of this teohnlque is given In Reference 28* A simple example of the Buckingham If Theorem la that of dsbsrmtaing the veloolty v at any time t of a body of mass falling In a region of gravity g* (The body Is assumed to be at rest at time t -s. 0*) Baoh of the terms - m, g, v and t - have a dimension made up ef the three basic dimen sions - mass or M, time or T, and distance or L* For ex ample the dimension of v Is L/*T or length over time* Baoh parameter and Its dimension which enters thla problem is tabulated below* Term Dimension m M g L T“2 t - T v I T“X Only one dlmenalonless constant, oai he formed, namely -177- t (8.8) Tha Buckingham 'if' theorem states that there is a functional relationship among a aet of Independent ^ Ta* or (8-9) Since there la only one dlmenslonleaa ratio In the oase of the falling masa, one then concludea that (8 *10) Thla result la a well known faot, aa v « gt« HotIce that the "Ye theorem lndloatea that the veloolty la Independent of the aaaa m* Functional relationships, developed in the manner just considered, are of great Importance in many fields• Often exaot mathematical evaluation ef functions may he nearly Impossible* By use of the 'Tf~theorem a functional relationship between certain aeta of the problem variables Is Indicated. The functions can be evaluated experimentally resulting In a set of universal curves relating the dimen- alonlesa products of the problem variables* This process has been used with great success In the field ef fluid flow* it la believed that this YT theorem will also be found to be of value In the analysis of nonlinear servo systems* Several examples will be considered In the next section* -178- D * Evaluation of Universal Functions for Ssrvo Systems Example 1. Serve with backlash. When the output and input displacements of a gear train are measured. It Is found that the two displacements are not simply related by the gear ratio of the gear train* In fact the Input shaft can be held stationary and the out put shaft can be rotated through a small angle / g without exerting any force upon the Input shaft • This angle V % la called the backlash angle referred to the output* For this reason, if the input displacement represented by y^ is fixed the output displacement of the gear train y8 oan be of the form T2=-«Tl (8.11) & where g is the gear ratio ef the gear train, and oan take on any value between -1 and 1* To Illustrate this backlash phenomenon oonslder the oase where g is unity and the input y^ varies sinusoidally, as shewn In Fig* 67* The extreme output shaft positions of ya are shown by the middle plot of Fig. 67* The equations of motion of a 2-gear train aa shown in Fig* 68 oan be written as (JlP® *iP) Tx -F T18 - Ti (8 .12) t12 - (a) Dead Zone In Which Lies (b) ^ ✓Assumed ^ N ^ v / F o r m Of (C) '* y^r~ Motor Positionosition N. V / / Fig. 67. («) Sinusoidal not of position y^ In whloh otae y2 oan 11s in dead sens shown In (b ) . (o) motor shsrt position yx showing assumed load position yg 180 y. Pig. 68. Goar train shoving notation uaod In thla soot Ion -181- Here and are the Inertia and ooefflolent of vlaooua frlotion of the input shaft and the gear attached to this shaft* Jg and are the respective values for the output gear and load* For periods when T^g Is positive y2 la less than yj_ by the angle //2 and for T^g negative y^ is larger than y2 by ^/2, that is syi - g— for T12 positive (8.13a) -+• — for T12 negative Also when T^g is zero yg oan be any place in the dead zone, or i i Tf 172 “ S^ll ^ 2 ” t o T Tl2 zero (8.13b) In this first example ef the use ef the 77"theorem, relationships for determining the amplitude of nsolllatlan and the frequency of osoillation ef a system with backlash In the gear train are determined* The equation of motion is assumed to be of the form (Jp8 kp)yi =. 8(x - yg) * (8.14) It is assumed that the Inertia m and the ooefflolent of friction k ef the motor rotor, shaft, and first gear are the only effective inertia and frlotion terms and that these -182- values are referred to the gear train output by meana of the gear ratio g ao that yg and yj_ are related by Tg = fl^ (8.15) except for the periods when yg la In the dead zone region where yg la stationary. In the bottom plot of Fig. 67 the form of the motor position y^ Is shown (steady state oscil lation la oaused by the baoklash) as well as the assumed form of the output y^. Note how the baoklash causes y2 to lag behind the motor position y^. It is assumed that the peak-to-peak amplitude of the oscillation B Is dependent upon J, k and G (see eq. (8.14)) as well as the dead space if • For thla reason eaoh of the terms just mentioned together with their dimensions are tabulated In the table which follows. Term Dimension Gain G ML2T“2 Inertia J ML2 Coefficient of k m l 2t -i viscous friction Dead spaoe L Peak-to-peak amplitude B* L of oscillation (Here the dead spaoe ^ JJ, is the displacement of a point on a unit radius shaft and BJt is the peak-to-peak displacement of a unit radius motor shaft.) -183- g The dl men a Ionian# produota GJ/k and B/y oan bo formed from the terms in the table Just considered* For thla reason the ratio of the amplitude of oscillation to the backlash or dead spaoe/ is given by (8*16) The frequency of the oscillation V In cycles per second oan be determined by replacing the amplitude of oscillation In the table on the preoedlng table by the frequency W which has a dimension T“l* In this case the dlmenslonleas products Gj/k^ and JW/k result, Indicating that (8.17) The funotlons 7^ and 7g Indicated by eqa• (8*16) and (8*17) have been evaluated using Koohenburger* a method, (see refer ence 16), the analogue computer and an exact mathematical treatment* The values of F^ and Fg evaluated by these three methods are shown in Figs* 69 and 70 respectively* The cause of the error in the computer solution is not known although other results might have been observed if the author had waited a longer period for the oscillation to die down in amplitude• 10 x Computer Results 8 A Kochenburger Technique 6 — Theoretical 4 3 B 7 2 1.5 I 2 2.5 3 4 5 8 7 8 9 10 2 0 30 40 6 J -186- .0 0.8 0.6 0 . 4 0 . 3 0.2 WJ . I 0 . 0 8 Computer Results Kochenburger Technique 0 . 0 6 Theoretical 0 . 0 4 1.5 2 3 4 6 8 IO 20 3 0 4 0 G J Pig. 70. normalized fraquanoy of oaolllation, VJ/k aa a function of GJ/fc^ for ayatom with baoklaah. -186- Example 2. System with third order differentlei equation and baoklash Consider the oasa where the equation of motion of a system with baoklash is (p3 ■+ hp2 + bp) 7 ^ 1 o y i 0 . (8*18) Here y^ and yg have the same meaning as la example 1. The dimensions of the terms and the frequency w of oscillation are as In the following table: * L a T”1 b T “2 o --3 T"1 . 3 . o Three Independent dlsienslonless products, namely o/a , b/a and w/a can be obtained* This Indicates that a funotion of the form exists ? = P I ^=. • (s.19) O b ) This funotion oan be evaluated but in general the parameters o/a3 and b/a2 are ohosen la an aotual serve system so that ■* no oscillation is present* This means that they are ohosen so that w/a - 0* If w is sere (8*19) oan be written as (8*20) -187- Equatlon (8*20) holda juat at the point of transition from w = 0 to w - S where & la lnorlmental In magnitude* Thla funotion haa been evaluated ualng the analogue computer and la ahown in Fig* 71* Example 3. Maximum alnuaoldal output of devloe with non linear damping and ooulomb frlotion In chapter VI a ayatom equation with a damping term of the form (k -t e2) py was considered (see eq* (6*8)). In the following discussion the maximum alnuaoldal output (see Section H of Chapter VI) will be determined for a aystom equation which Inaludes nonlinear damping term, (k+e®) py, as well aa a coulomb frlotion term* The syatem equation to be considered la pv + (J ) vi ^ = | e . (8*21) Hero F la the torque oauaed by ooulomb frlotion and the sign is the same aa the sign of the output velocity v* It la proposed to determine the functional relationship giving the maximum alnuaoldal amplitude d for a given angular frequenoy w, when v la ef the form v == d aln wt • (8*22) The parameters and their dimensions In (8*21) and (8*22) are tabulated In the table Whloh follows* Notice that the Input signal e has the dimension ef the voltage 2 and that the velocity v Is assumed to have the dimension of L T . Slnoe 0.6 Unstable With 0.4 And Without Backlash 0.2 0.06 0.04 0.02 0.01 Stable Region With 0.006 And Without Backlash 0.004 0.002 0.001 0.01 0.02 0.04 0 . 0 6 0.1 0.2 0 . 4 0.6 b a2 Fig. 71. Regions of stability for serve aystee whose operating equation Is given by (8.18) -189- —2 the dimension of pv Is L T and that of e Is E, the dimen* si on of G/J la equal to L T~®* This Is obtained by equat ing the dlmenslen of the terms GS/J and pv* 'Rio dimension of the other parameters oan be found In the same manner* k/J T" 1 o P/J L T" 1 G/J L T* 2 -1 w T d It T” 1 Three Independent dimenslonless products or ^*s oan be determined from the preceding table; they are 1 = *k w - fLfcd2 (8*23) 2 o2 V r » -- • These three rr. a are then related by the functional form - f( ^ 3 ) • (8*24) The function f( ^ 3 ^ hafl bean •valuated for a few fixed values ef Jw/k and Is shown In Pig* 72* ^6 7* o* a m 1 u m nix i x a m a g n i n i m r e t e d for* e v r u o l a s r e v i n U 72* ^^■6* utput o system wt, se q (S.21 eq. (see g n i p m a d r a e n i l n o n with, m e t s y s of t u p t ou a k d OJ CM 5 0 . 0 0.15 0.2 0.1 TT J W - 190 - 10 7T, = 0. 4 20 4 0 6 0 IOO 200 4 0 0 6 0 0 IO ■ oldLal L)) -191- As an example of the use of this universal ourve, consider the oaae where the variables In (8.21) are k/J - 1, P/J ~ 0.1, <*C/J - 1 , G/J = 1, w --3. The values of and are then TP^ - 3 and ^ 100. In this case, the value of T^g, indicated by point P In Fig. 72, Is 0.108 so that d^ becomes equal to (8.25) or d - 0.329• This means that the maximum amplitude of the sinusoidal output velocity v with a frequency of 3 radians per sec. Is of the form v = 0.329 sin 3t It can be shown that if if^ le sufficiently large ^ 2 reduces to (8.27) Equation (8.27) can be used as an approximation If TP satisfies the Inequality 1 -*TT® Equation (8.27) can be verified by letting P In (8.21) be zero and solving for d as In Section H of Chapter VI. -192- Chapter IX System Porformanoa Criteria A- Introduction There have been many methods proposed for evaluating the performance of automatic controls* Many of these methods are hased upon frequency response techniques or step function transient response and are quite simple but In general they are satisfactory only for linear systems* Every device, partioularily servo output devices such as electric or hydraullo motors, have maximum ratings* These ratings may be dictated by any of several factors* Basically the maximum ratings can be divided into the two categories — 1. Saturation effects of the machine, 2* Safe operating oaa - dltions which will Insure that the machine will not be physi cally damaged* If there la danger of damaging a machine the servo system must be designed so that the machine rat ing will not be exoeeded* On the other hand if the full capa bilities of a servo are te be realized, the servo system must be operated, at least part of the time, In Its satitrated or limit state. This Implies that the ability of a servo to respond Is dependent, at least In the extreme oases, upon the nonllnearlties of the machine* For this reason, simple linear transient response and frequency response techniques are not sufficient to Indicate quality of performance during the nonlinear operating periods* -193- Before the performance of a devloe can be evaluated one muat first define performance, preferably, so that It can be evaluated numerically* Such a definition Is called a performance criterion* Unfortunately, there is not a standard criterion for all servomechanisms. The criterion for an ant1-alroraft position servo is usually quite dif ferent from that of a servo controlling an industrial process• In determining a criterion the purpose of the problem must be clearly stated* The purpose of an anti-aircraft gun la to destroy an aircraft In as short a time as possible. For this reason the criterion must be based upon the average life of an aircraft in range of the antl-airoraft gun* One criterion for testing auoh a servo can be stated as the percent time the error e between the gun position and the desired position is less than some small deviation ee, when the input is a. typloal operational input♦ The criterion for a servo controlling the air flow In a combustion chamber Is much different from that Just con sidered* Here the error corresponds to a loss of energy which has a value In dollars and oenti. If too much air flows into the combustion chamber energy is lost heating the excess air* If too little air is introduced the fuel will not be completely consumed* For this reason the error e between the desired air flow and the actual air flow de termines the heat energy that Is wasted* The amount of -194 wasted energy per unit of time can be represented as f(e) and the criterion of goodness becomes the amount of heat wasted for a typical operating condition* This means the evaluation of d t . There are two general criteria that have been pro- posed and used for a wide variety of systems* 29 These are (9.1) The first of these is often oalled the absolute error cri terion and the second Is slmillar to that oalled the RMS p q error criterion* * 30, 31 These Integrals are often evaluated when the Input to the system Is a unit step function* It Is the author's opinion that these criteria have a reasonable meaning only when the input is a typioal op erating input. To Illustrate this point consider the simple servo equation (p2 -t 2 5 p •+ l)y - x (9.2) If the above criteria are valid one shoud be able to find one optimum value for & for each criterion* The table which follows indicates three possible optimum values of & , for each figure of merit Q as defined by (9*1). In other words, the optimum damping as evaluated b~r the criteria In (9*1) Is dependent upon both the criterion and the Input signal* » -195 Typo of Input Unit ramp 0.01- £-0.0 + Func11 on Unit step 0.66* £ * 0.5 function Unit S ^ l.o £ “ 00 impulso Table of optimum valuoa for 5 in eq. (9.2) for various typos of Input functions. Observing that the criteria given by (9.1) give a different optimum value (ranging from zero to Infinity) for £ for each Input, one must conclude that these criteria are not complete until the form of the input is specified. It has 31 32 been pointed out by several authors * that good results can be obtained by the RMS error criterion for linear systems if the spectral density ** of the Input to the system la known. This is not a contradiction of the pre vious statement since a limited knowledge of the input signal la Implied when its spectral density la known. It * This value for $ was obtained from an analogue computer s tudy. The spectral density of an input x(t) whose Laplace transform la X(a) Is given by X(jw)X(Jw) where X(j*) is the complex conjugate of X(jw). ~ - 1 9 6 - should be apparent that the two criteria indicated by (9.1) are complete only when they are evaluated for an input to the system which is typical of the operating condition. Since any natural input to a servo will not be as abrupt as a step function, one would expect that the opti mum value for§ as evaluated by the RMS error criterion is somewhat less than 0.5. Several tests were made exciting two different systems (nrlth system equations of the form given by (9.2)) with the same Input. One of these systems had a damping factor S of 0.5 and the second of 0.3. When the input to each system was random and relatively smooth, the RMS error of the two systems was about the same. It was found that either system could be made to appear better than the other by allowing the input to be of a special form. B. Criterion for Recording Systems The criterion used for determining the performance d* graphical recording systems Is quite different from the criteria for most servo systems. The difference lies in the fact that the graphical record is used after the input signal is recorded rather than instantaneously as the record is recorded. If one knows the operating equations of the recording system one can take a graphical record and from it reconstruct the Input. Although this reconstructlon of the Input or correction process is possible In theory It la not practical. It has oeen observed that a large part of the error -197- between the recorded output and the actual Input signal can be eliminated by assuming that the servo output la a repro duction or the servo input delayed in time* That la if the input la x(t), the output la assumed to be y(t) = x(t - T) (9.3) This phenomenon was observed in graphical records of antenna patterns* To illustrate this characteristic an antenna pattern represented aa x '©) waa reoorded twice aa a function of time* Clhe flrat pattern was recorded in 4 minutes so that every degree of the angle 9 corresponded to 2/3 of a second in time* This period of time waa suffi ciently long that the rate of change of the input signal was small and the difference between the recorded pattern and the actual pattern was small* The pattern was then recorded a second time with a total recording time of 40 seconds so that each degree of 9 corresponded to 1/9 of a seoond* These two graphical records are drawn in polar form In Pig* 73 aa a function of the antenna position 9* Notice that the two patterns are similar In shape, but displaced by a small angle due to the tine delay of the servo* The 4 minute pattern is assumed to be correct aa It has a much 'smaller error than does the 40 second pattern* The error in the 40 second pattern (baaed upon the 4 minute pattern as being oorreot) as well as the 4 minute pattern r*ig. 73. Antenna pattern recorded in 40 a n d 2 4 0 seoonds 4 Min. Pattorn UnoM fto* Absolut Error Abooluto Error -20 O 10 2 0 3 0 4 0 Oofrooo 74. Mrror of pattern recorded in 40 Mooodi twfer end after shifting of the tine axis are shown in Fig. 73. The polar coordinates of the 40 second pattern were then rotated (corresponding to a shift in time) until the two patterns fell on top of each other. The error of the 40 second pattern before and after this shift of the time are shown in Fig. 74. This shift of the time axis reduced the RMS error of the 40 second pattern by a factor of 5. The value of the time delay can be found by shifting a recording of a known input until the error is minimized by observation as waa done in the example just considered, or more refined methods based upon the RMS error criterion can be used. For example the Integral T (9.4) can be used. Hie value of T which makes I (T ) a minimum can be defined as the optimum time delay of the system providing x(t), the desired input, is typical of all Inputs to the system. The minimum value of I ( T ) represented by I is also a figure of merit for a system. The best recording system can be defined as the system which has a minimum 1^. Equation (9.4) can also be written in the form -2 90- n ■ rr ‘h e 2 T 3 v-.. In'. ..• .• the * n if T-.-1X3 -VI i* ••••••rded litT'Lit :r. votica that th»• v - lue. of Y 1 is ■J". -t"' ‘ re d y t.hr* vs 2 ue i_ f *P • 'no!’ ocm'-~s r,ci” rs * ■ ■1 r e. 1. t i ' 'H 1 : t' « e r :: r n -"1 v t ° a . ;i jt>;u p . v t:si f, 5. ve vf-1 lie , If f,’ o 'nn ct i- =n r f^r x(t) re nr>*to esuT u. to iT t’ t \ a lue s " V : • 11 e f-un.l . 7or e--'*-le it ■ n 1 ; •.-» rel-teT ' .y (a2 -*- 2 $ n 4 l)y - x (9.6) il:- optimum v :. 1 ur1. of y* tor 9 . the form s' r ut if- deuen- . er.t 'i''c n v:. ’’or or: oil v lues of \ ,¥ - 2 S . If :c is •* u ol tc !:t, f is si so eoual to 2 £ ; ho\;?v r , if Is a us it tv- function th- valu'- f Y1 v ■■ - v■ -1 eer 1 -ur' 2 -S’ ■■*t’ sen 0 -nd 1 .25. The use of t:..i m r.hif t i.n • procec urc con 1- - used to due .reuse the error o C the rrr:' i e 1 reco:e r os '.ell nr. to tr.rycr.se the s v>eec! ot ’ r Ich the ’'-ttrn ch; rn cteri r,t i cs con •» '*« cre-d , Tor exe-rle th e srt^rma pattern used 3 on ::.or ••>la In this section can he cnmnlet ely re corded in a 1' •* r 1 n i of t ne :inute vitli a .'eon sar red error oft/-r shift— * n • * the 1 1 r © axis avout emicl to the venr. r* ~ ir: r e d r. rror of the uns: if ted -rtt' rn recorded in pour ruinates. * cross correlation of x and y is > i ven by T v 0(x,y) ? Lim _J f *(t) y(t + r7)dt T‘ - TU T-T~.*- / rp * i - 2 0 1 - If time shifting procedure is used to reduce the error of graphical records, then the servo recording system can be designed on the basis of introducing a uniform time delay- This means that the design parameters are chosen such that the system has a flat frequency response and a phase shift proportional to frequency over the pass hand of the servo system- Neither the frequency response con dition nor the phase salft condition can oe met exactly but they can be chosen such that oetter reproduction will be obtained than that of a servo designed using conventional servo design procedures- 33 C • Component Performance At times one Is led to believe that there exists a linear component for every need In the servomechanism field. This situation Is unfortunately not true and one must use a great deal of Ingenuity In the choice of major components and the design of coupling units between these components- Undoubtedly the most difficult task In the design of a servo system is the choice of motor and gear train to drive the mechanical load- If one could depend upon the linear theory so commonly used for all systems, the design beoomes absurd, for any motor could be used to drive any load and the frequency pass band could be set to any desired value by an adjustment of the amplifier gain and tachometer feedback- - 2 0 2 - It la generally quite easy to choose the class of motors which can he used to drive a given load out the choice of a particular motor and gear train is difficult at the present time mainly be cause we depend to such a large extent upon linear theory* If the motor Is to be used to its best advantage It Is necessary that the machine at least part of the time be operated in a saturated state* It is the opinion of the author that the static curves of motors and other data such as maximum stalled torque, maximum unloaded speed, reversing time indicate motor performance in a manner that is difficult to use* One possible method of making the material more readily available to the designer Is a set of universal curves Indicating either peak sinusoidal amplitude or velocity for a motor evaluated with several typical loads* This method Is suggested since a sinusoidal output is more representative of actual performance than Is the static condition given by the speed torque curves* A typical example of this technique Is discussed In Chapters VI and VIII. To illustrate this procedure, consider the problem of choosing a 2 -phase Induction motor and gear train for p driving a load which has an inertia of 1 0 0 gram-om and a coulomb friction torque of 70,000 dyne-cm. It is desired that the frequency response of the closed loop servo remain flat up to 30 radians/sec. and that the -203- maximum sinusoidal displacement or the load be as large as possible up to this frequency- The manufacturers data for two motors, one of which Is to be chosen to drive this load are listed below* The problem is to choose the motor and a gear ratio to go with this motor which, when connected to the load, will cause the load sinusoidal displacement to be as large as possible at a frequency of 3D radians/sec* Motor Specifications Motor 1 Motor 2 Motor and motor shaft inertia 7 *4 gram cm2 23 *2 gram cm.2 ciynohronus speed 3600 RPM 1200 RPM Current/phase for 40 ma 185 ma max* stalled torque Using the terminology in Chapter VI for two-phaae motors one can show that the parameters used In evaluating V* '7/^2 and F i g • 72 are as Indicated in the table which follows* In this table g is the gear ratio between the motor and the load* (Notice that the effective inertia of the motor Is the motor inertia plus the referred load inertia or iDQ/g®, and the effective coulosto friction torque F is the load coulomb friction divided by the gear ratio g*) Motor 1 Motor 2 K 73.9 69.5 F 70,000/s 70,000/g ^ 46,150 2030 W 30 30 O 1,390,000 939,000 J 7.4 100 23.3 + ~ ~ 1 0 0 The values In the proceeding table have been deter mined Tor a few values of g and these values substituted Into the equations for 'Tf*^ and ^ 3 * By ua® of the universal curves given in Pig. 72 values of the maximum sinusoidal velocity of the motors oan be determined. The value of the gear ratio which will cause the load driven by motor "I" to have a maximum sinusoidal output Is about 3. For g “ 3 the maximum sinusoidal motor velocity is of the form 172 cos 30t. This means that the maximum sinusoidal load displacement is of the form 172 3 ( ~ 30t. (9*7) For motor n2n the optimum value of g is approximately 1; and in this case the maximum sinusoidal displacement of -205- the load is of the form4 8.3 sin 3 0 t . (9.8) Motor n2n la thus better than motor **1". The author la not fully satisfied with the proce dure of determining 2-phase motor performance based upon maximum sinusoidal displacement by means of the universal curves given In Fig. 72. It is believed that these universal curves can be modified so that the Input signal applied to the device will not exceed some critical value. Likewise It Is possible to express ^*2 and ^ 3 In terms of the two-phase motor constants. It would appear that universal curves of the type Just indicated could also be found for a number of servomechanisms components such as amplidynes, shunt dc motors, and various types of hydraulic motors. 4 The approximation for discussed on the last page of Chapter VIII was used for evaluating the performance of motor n2 m• - 2 0 6 - Blbllography X. Brown, G. 3. and Campbell, D. P., Principlea of Servomechanisms. John Wiley and Sons, Inc., New York, if.' ¥., 1348. pp. 94-98. 2. Andronow, A. A. and Chaikin, C. E., Theory of 0soIllations. Prinoeton University Press^ Princeton, N • J., (Engl1ah Language Edition), 1949* p. 61. 3. Evans, W. R., '•Control System Synthesis by Root Locus Method". AXES Transactions, Vol. 69, Part I, 1950. pp. 66-69. 4. Bacon, Jack, "A Logarithmic Recorder of Unique Design". M. S* Thesis, Ohio State University, 1951. 5. Andronow, A. A. and Chaikin, C. E., Op. Clt. pp. 6-11. 6. Minoraky, N., Non-Linear Mechanics. Edwards Brothers, Inc . , Ann Arb or, Michigan, 1947". Chapt er I . 7. Ibid. p. 20. 8. Brown, 0. S. and Campbell, D. P., Op. Clt. p- 16. 9. McDonald, D, "Multiple Mode Operation ef Servomechanisms". Review of Scientlflo Instruments, Vol. 23, No. 1, Jan. 1952. pp. 22-30. 10. Mathews, K. C. and Boe, R. C., "The Application of Nonlinear Techniques to Servomechanisma". Paper presented at the National Electronics Conference, 1952. (Paper will appear In Vol. VIII of Proceedings ef the National Electronics Conference.) 11. McDonald, D., "Nonlinear Techniques for Improving Servo Performance". Proceedings of the National Electronics Conference, Vol. VI, 1950. pp. 400-421. 12. Lanthrop, R. C., "A Topological and Analogue Computer Study of Certain Servomeohanlams Employing Electronic Components". Doctoral Thesis, University of Wisconsin, 1951. 13. Lewis, J. B., "The Use of Nonlinear Feedbaok to Improve the Transient Response of a Servomechanism". Applications and Industry, No. 4, January, 1953, (Published by AIEE)• pp. 449-453. -2 07 14* Burns, K. N., "The Transient Response of a Single Point Non-Linear Servomechanism"• Paper presented at the National Electronics Conference, 1952. (Paper will appear in Vol. VIII of Proceedings of the National Electronics Conference•) 15• Kochenburger, R. J., "A Frequency Response Method of Analyzing and Synthesizing Contactor Servomechanisms"• A1EE Transactions, Vol. 69, Part I, 1950. pp. 270-284. 16. Johnson, E. C., "Sinusoidal Analysis of Feedback- Control Systems Containing Nonlinear Elements". AIEE Transactions, Vol.,71, Part II, 1952. pp. 169-131. 17* James, H. M«, Nichols, B. B*, Phillips, R. S., Theory of Servomechanisms. McGraw-Hill Book Co*. New York. N. Y*. TBT47.” pp. 155-153. 18. Brown, G. S*, Campbell, D. P., Op. Clt. pp. 146-154. 19. Ibid* pp* 132—236. 20. MacColl, L. A., Fundamental Theory of Servomechanisms. D. Van Nostrand Co., New York, lT. Y ., 1945• Chapter Vll. 21* Bode, H. W., Network Analysis and Feedbaok Amplifier Design* D. Van Nostrand Co., toew York, N. Y#, 1945* pp". T 5 1-157 . 22* Scott, W. E., "An Introduction to the Analysis of Non-Linear Closed Cycle Control Systems". An article In the book Automatic and Manual Control. Academic Press Inc., New York, N. Y., 1952. pp. 249-261. 23. Lyon, W. V., Applications of the Method of Symmetrical Components. McGraw-Hill Book Co.. New York. if. Y.. 1937. pp. 435-452. 24. Cosgrlff, R. L., "Integral Controller for Use in Carrier-Type Servomechanisms". AIEE Transactions, Vol. 65, 1950. pp. 1379-1383. 25. Brown, G. S., Campbell, D. P., Op. Clt. p. 188. 26. Minorsky, N., Op. Clt. Chapter XVII. 27. Ibid. Chapter XVIII. 28. Doherty, R. E., Keller, E. G., Mathematics ef Modem Wiley and Sons, Inc. New York, N. Y., pp. 131-162 - 208- 29• Flckeiaen, F. C., Stout, T. M., "Analogue Methods for Optimum Servomechanism Design". AIEE Transactions, Vol. 71, Fart II, 1952. pp. 244-250. 30. Hall, A. C., The Analysis and Synthesis of Linear Servomeohanlsms. Technology Press^ Cambridge, Mass., 1943. p •” 23 . 31. James, H. M., et. a l ., L o c • Clt. Chapter 7. 32. Wiener, N*, Extrapolation, Interpolation and Smoothing of Stationary Time Ser lea. John Wiley and 3ona, Tnc., New York, N . Y ., 1950. 33. "The Specification and Measurement of Performance In Servo Systems". Proceedings of the Institution of Elec trical Engineers, Vol. 99, Part II, No* 71, Oct., 1952. pp. 494-496. 34. Cherry, C., Pulses and Transients in Commumloatlon Circuits. Chapmand and Hall Ltd., London, 'England, 1949• pp. 145-148. -209- AUTOBIOGRAFHY I, Robert Lien Cosgriff, was b o m In Sweet Grass County near Big Timber, Montana. I received my primary and secondary school education in the Sweet Grass County public schools* My undergraduate training was obtained at Montana State College and The Ohio State University. A large part of the training obtained at the ^ l o State University was obtained while In the Army Specialized Training Program* In June of 1947 I received the degree Bachelor of Electrical Engineering from the °hio State University* During the following three years, I was employed as a research engineer at the Curtiss Wright Corp* and at the same time attended twilight school at the Ohio State University and obtained the degree ^astar of Science in Deoember of 1949* In 1950 I was employed by the Research Foundation as a Research Assistant, In 1951 was given the title of Research Associate and became a part-time Instructor in the Department of Electrical Engineering in 1952* While holding the latter positions at the Ohio State University, I completed the requirements for the degree Doctor of f*hilesophy*