(10+It+20)d2600 D+,DtSta i ity or Varying-Element dt2S| . . | n | * 1 ~~~~~~~~~~~~~~~~~~212t+D2t2)d o +(Go+Gjt)0oervomechanisms with Polynomial dt

= (Go+GJt)0j (2 )

Coefhcients For this system the a, b, and c coeffi-cients of equation 1 would be:

a2= Io b2 = Il C2 = 12M. J. KIRBY R. M. GIULIANELLI a,=Do+11 b1=D,+212 c,=D2MEMBER AIEE NONMEMBER AIEE ao=Go bo=Gi co=0

Equations for practical systems havingtIME-varving elements in servomecha- output 00, to input 01, and time, t, is of the more elements will contain a great manyI nisms are found in practice, for form: additional terms involving the rates of

example, in systems which must operate dnoo dn-1 0 change of the elements.during their warm-up periods, when the fZ(t) +f,,I(t) dt + +f(t)00 The definition of stability used through-gain or the damping, or both, are chang- dt dt out this paper is based on the fact thating materially. More frequentlyencoun- F(t) (1) equation I is linear, and its solution, 00,tered, however, are cases in which a sys- where can be separated into a transient responsetem element varies with velocity or some = or complimentary function, and a par-other dependent variable, but the pro- fi ) (aj+bjt+cjt,+ +KjtK) ticular integral or steady state response.gram of the process is known so that the The right side, F(t) involves 0i, pos- The transient is the solution to equationvariation can be adequately represented sibly its derivatives and products of these 3, which is equation 1 with the input, 01,as a function of time. An example is a by t. equal to zero:speed control for a reel on which strip A previous paper' showed that when the dn dn-1steel is being rolled or unrolled. The coefficients in equation 1 contain only fn(t) -00o +fn-(t)d7 00 + +f0(t)00 = 0inertia changes with the amount of steel (a+bt) terms, the stability can be deter- (3)on the reel, but if either the shaft speed of mined by forming a polynomial from thethe reel or the linear speed of the strip is "b" terms, and finding its roots. The In accordance with Nyquist's widelyknown, the inertia can be expressed as a present criterion extends this, and shows accepted definition,2 a servomechanism isfunction of time. that when the coefficients in equation 1 defined as being stable if, when excitedWhen a servomechanism must be an- contain higher powers of t, stability is de- by any input or initial energy storage, the

alyzed as a varying-element system, it is termined by the roots of a polynomial transient part of the output dies out com-always desirable to represent all varia- made up of the coefficients of the highest pletely as time approaches infinity. Iftions by straight lines if possible. How- power. any part of the transient persists orever, the rates of variation of some ele- As an illustration of how an equation builds up indefinitely as time becomes in-ments may change radically, or the of the form of equation 1 is obtained, con- finite, the system is unstable.product of the values of two elements may sider an elementary system in which an A restriction on the method is that theintroduce higher powers of time into the error actuated amplifier and motor drive highest power of t appearing in the co-

coefficients of the system equation. a load consisting of an inertia and viscous efficient of the highest derivative term,This paper presents a criterion for de- damping; assume the elements have been d'/dtn0o in equation 3, must be equal to or

termining the eventual stability of a found to vary thus greater than the highest power of t ap-servomechanism in such a case, when its pearing in the other derivative terms.elements vary as higher powers of time so inertia=IO+Iit+12t2that the differential equation relating damping=+ D,+D,t+D,P The Stability Criterion

- ~~~~~~~~~~~Thedifferential equlation relating out- It can be shown3 that the transient re-Paper 51-268, recommended by the AIEE Feedback ptto input combines two equations stat- sponse of a servomechanism described byControl Systems Committee and approved by the puAIEE Technical Program Committee for presenta- ing that motor torque is the product of equation 1 approaches a sum of terms oftion at the AIEE Summer General Meeting, error times gain, and is balanced by a load exponential form

script submitted March 14, 1951; made available torque which is the sumn of a viscousfor printing May 28, 1951. j =n .

M. J. KIaBY and R. M. G1ULIANELLI are with the dapn oqepu h iert f ooZAL X(j+i) (4)Sperry Gyroscope Company, Great Neck, N. Y. change of angular momentum: = t

1951, VOLUME 70 Kirby, Giulianelli-Varying-Element Servomechanisms 1447

Nomenclature Examples

ao, a,.. al.. an= constant part of coefficienits A =coefficient in equation 4 Plots of P(s) and time responses arein equation 1 D = damping of servomechanism, equation 2 given for two second order systems, one

bo, b by . bn = coefficients of t in equation 1 G = gain of servomechanism, equation 2 stable and one unstable. The time re-Co, Cl. . cl. cn = coefficients of t2 in equation 1 I= inertia of servomotor and load, equa-

s

and so on tion 2 spouses were outaineu by means of anP(s) = polynomial function of (s), equation 5 electronic analogue computer; the num-

j= V-i; also used as a subscript denoting v(s) =function of s in contour integral, ber of computing elements availablea general or typical value equation 13 limited the order of the equations and de-

k, K = when used as a coefficient or an ex- a= real part of y termined the form of the input. Theponent, is associated with the highest y=root of P(s) termmed beform ine in Thepower of t in the coefficients of equa- X =exponent of t in equation 4 stability could be determined in each casetion 1 Oi= input to servomechanism simply by solving the quadratic equation,

n =order of equation 1 00=output of servomechanism P(s) =0, but polar plots are included tos = arbitrary complex variable w = imaginary part of y show the characteristic shapes for stablet = tine +(s) =analytic function of s, equation 17

4(s) =analytic ~~~and unstable cases.

as t-> o. The A's are constants depend- Nyquist plot may be employed. The EXAMPLE 1ing upon initial conditions and system method used will depend upon the in-inputs; the X's depend on the servo sys- dividual and the problem at hand. Which- The system equation is:tem parameters, and the y's are the zeros ever method is used, the process is essen- d2 dOoof the polynomial P(s) formed from the tially that of finding the signs of the real (1+t+t2)do,+( 2+t±12) d+coefficients of the highest power of t in parts of the roots of the characteristic (1+t+t2)o6 = (1+t+t2)oi (6)equation 1, thus: equation of a system.

P(s)- Kns+Kn_-is"-+ . +Ko (5) The steps of the preceding section do From this, P(s) isnot cover the case in which P(s) has one or P(s) 52+s+ 1 (7)

The s is an arbitrary complex variable. more roots with zero real parts. TheThis asymptotic foim of 0, is the basis polar plot of P(s) indicates this condition Figure 1, the plot of P(s), indicates that

of the stability criterion. The e7' in each by passing through the origin for a finite, the system is stable; this is verified by theterm can be written as EtEI, where a imaginary value of s. If the jth root of response to an inputand co are the real and imaginary parts, P(s) has a zero real part the jth term in 1respectively of 'y. If a is not zero, the the eventual fonr oftthe transient, t2 (8)eat eventually overpowers all the other 7 telements in each term. If an a is nega- Aj + which is shown in Figure 2.tive, that term in the transient responseapproaches zero as t- co, regardless of A will be an oscillation which builds up or EXAMPLE 2or X. If an a is positive, the term becomes decays as 110(i+1), since eli' is simply a The system equation is:infinite if A is not zero. The process of sine wave, equal to I"it when yj is a puredetermining stability is therefore reduced imaginary. If any other root of P(s) has (1+t-0.2t2+t3) -0o+(0.2+t+O.1t2-4t3)to finding the wy's which are the roots of a positive real part, the servomechanism dtlP(s). is unstable. If one or more roots of P(s) d-o+(l+t+O.lt2+t3)00The specific steps are: have zero real parts and all other roots dt

1. Express the time varying elements as have negative real parts, stability is de- =(1+t+O.lt2+t3)oi (9)polynomial functions of time, and write the termined by the l/tX portions of these P(s) is made up of the coefficients of thedifferential equation relating output 0, to roots. When the coefficients of equation t3 termsinput oi and time t in the form of equation 1. 1 are of the form (a+bt), a straightfor-2. Form the polynomial P(s) as in equa- ward procedure' is available for evaluat- P(s) = s2 -4s+1 (10)tion 5. ing the X's. When higher powers of t are3. Find the sigIls of real parts (a's) of the present, however, the X's are so much Therplo, Figure 3, tocthe inptaroots of P(s) by any convenient method. more difficult to determine that this is not The response, Figure 3, to the inputIf all the a's have negative signs, the system worthwhile. In a practical situation the 1is stable. If any a has a positive sign the wo- n a p lsi=tuto the ___I__t___lt2 (11)system is unstable. If one or more a's build-up or decay of l/t is muchare zero, a special case arises which now is slower than that of an exponential term;discussed. this is more significant than the eventual shows this.

stability or instability. If the system orPractical Considerations process is to operate only a short time, Conclusions

such a transient might be toleratedA number of methods are available for whether stable or unstable. If the process The criterion may appear to be of some-

finding the roots of P(s). In the simpler continues for a long period this type of what limited use because it predictscases, inspection and the use of rules transient usually cannot be allowed to eventual stability or instability, that is,based on the signs of the terms in P(s), the persist. the build-up or decay of the final phasepresence of all terms or the absence of Rather than spend time trying to de- of the transient, while many servomecha-some, will suffice. Tables are available termine the eventual stability by evaluat- nisms with varying elements operatefor finding the roots of cubic and quartic ing X therefore, it is better to recognize only for finite periods of time. Further-polynomials. In more complicated cases that such a situation exists and to rem- more, the criterion tells little about thethe Routh-Hurwitz criterion can be used edy it by adding damping or by other size of the early part of the transient re-to advantage, or a polar plot similar to a physical changes. sponse.

1448 Kirby, Giulianelli- Varying-Element Servomechanisms AIEE TRANSACTIONS

In practice, however, a transient doesnot necessarily require a long time toreach its "eventual" or final phase. As EXAMPLEsoon as sufficient time has elapsed for thehighest power (tk) terms in the coefficientsof equation 1 to overpower the otherterms, the final phase is reached. Manyclosed-loop systems operate long enoughfor their transients to progress well intothis phase, even though the systems 1.5operate on programs of finite duration. /=1. \The two examples in an earlier section w/0.8illustrate this. In the remaining cases in W=0.6which the program stops short of the final 0o.4phase of the transient response, the even- -_, 3 - I 2 3 4tual stability or instability is still im-portant. For example, a designer wouldprovide extra safety devices to stop aprocess if it were known that the systemwould enter a destructive oscillation if theprocess were to continue beyond itsnormal time.The criterion states that the eventual

stability depends upon the eventual ratesof variation of the system elements as in-dicated by the coefficients of the highestpower of time (tk) in the polynomial co-efficients. When k=1 the criterion be-comes identical to that given previously'for systems described by equations whose EXAMPLE 2coefficients vary as (a+bt). When k = 0,P(s) reduces to the characteristic equationfor a constant coefficient system, which isthe basis for the common stability criteriafor such systems.The significance of the tk terms suggests

that stability might be determineddirectly by dividing each term in equation1 by tk, then letting t approach infinity.This procedure does not save any labor,since it arrives at the same polynomial, Z0.0

P(s), which is written down immediately -4 -3 -2 -I 2 3 4

when the present criterion is used. Fur-thermore, this amounts to saying that the W=0.2

differential equation 1 approaches a con-stant coefficient equation as t approaches W -.4infinity, therefore, its solution approachesa sum of exponential functions. There isa tendency to extend this type of intuitive } -.6analysis to predict properties other thanabsolute stability, and even numericalvalues of the response. When such pro- ZW.8cedures cannot be rigorously justified,they may involve considerable errors.

Honwever, sivnce P(s) 'hasq some of the* Figure 1. Plots of P(s) for Examples I and 2

properties of a characteristic equation,related to the final phase of the transient,it yields useful information as to whether t2 except one, which varies as (a+bt) (a+bt) coefficient varies as t2. If this isthe system response is satisfactory, and if only. The P(s) for this systemn will have impossible, the eventual response maynot what can be done to remedy it. This a missing power of s, indicating a root on sometimes be postponed or attenuated bywill become apparent as the criterion is the imaginary axis, and the eventual re- increasing the constant (a) or first orderapplied to specific servosystems; only sponse will decay or build-up slowly as a variation (bt) term of this coefficient.two examples will be given here. 1/1X+1 term. This may be an undesirable Likewise, consider a system for which

Consider a servomechanism for which condition; it can be remedied ideally only all the coefficients in equation 1 vary asall the coefficients in equation 1 vary as by changing the system so that the the same power of t, say t3, but one of the

1951, VOLUME 70 Kirby, Giulianelli- Varying-Element:Servomechanisms 1449

O 0 Z t

0 1 2 3 4 5 6 0 2 3 4 5 6TIME IN SECONDS TIME IN SECONDS

Figure 2. Response of system of Example 1 to input of form oi = 1/- Figure 3. Response of system of Example 2 to input of form ai = If/(I +t+t2) (I +t+0.1 t2+t3)

terms in P(s) is opposite in sign to all into this equation; the result is this would require the solution of the differ-others. The system will eventually be r r2b2 2t2 ential equation under the integral sign. Aonsthler The system afo(a1+b t+cIt2)S1 knowledge of some of the properties of v(s)unstable and can be made absolutely

n±+(a1+blt+cit2)s eItv(s)ds 0 (14) is sufficient to determine the form of 0 as

stable only by making all the signs in ±(ao+bot+±ct2) t .

P(s) the same. (This is necessary, but . . The y's are the singular points of themay not be sufficient if P(s) is higher than Differentiation under the integral sign is differential equation for v(s). In the neigh-the second order.) However, the in- equation 9 is next integrated by parts giv- borhood of each -y in the s-plane, say 'YK,

stability may be postponed by increasing ing:the coefficients of the lower powers of t v(s)=(s -K)XKfOK(S) (17)in equation 1 so that a longer period of I(b2s2+c2ts2-c2ts+b1s+ccts -ci+time must elapse before the tK terms pre- b t)st ( - (C2S+ where 4 (s) is analytic. Each of these solu-tomim atem eapseb renthenters)ivts) - tions can be put in equation 13 for 0, and

dr dv(s) integrated around the contour which en-final phase. For example, if the system is cis+Co)E d + circles the appropriate -y; this gives n solu-second order and the damping varies, as ds c tions to equation 12. Each such integrall+t+t2-t3 while the t3 coefficients in the (a2s2-2b2s+2c2+ais -b1+ can be divided into three parts, two alonginetiaandgai*tems recoe it ine the aO)f v(s)+(-b2S2+4c2s- the line parallel to the real axis and the thirdnertlia and gain terms are positivethe f t( dv(s)+ encircling -y. By expandingl(s) in a power

-t3 will overpower the positive termsin l b1stl2c1-bo)e81 ds ds=0 series and by further steps it can be shownthe damping before, say t=3. However, Jc d2(s) that as tao, the first two ititegrals ap-if the damping term is changed to 1+t+ (c2s2+ cIs+co)&El dS2 proach zero and the third aproaches:4t2-t3, the -t3 term will not predomi- t15 prAcer t d onate until later, say, t= 6.When the polvnomial coefficients it the Two groups of terms, as in equation 15, t(XK+1)

I m result regardless of the highest power of t ortime equation imust be determined from of the order of the time differential equation The intermediate steps omitted in thisexperimental or other numerical data, the corresponding to equation 12. The first summary are discussed by E. T. Ince.3coefficients of tK must be obtained ac- group is the expression in brackets, to becurately. Methods of representing em- evaluated around the contour c; productspirical data by polynomials are given by of powers of t and of s with v(s) auid Et References

entering this expression. The second group,Sokolnikoff4 and Scarborough,5 among under the integral sign, contains est multi-others. plied by a differential equation for v(s) with 1. STABILITY OP SERVOMECHANISMS WITH LIN-

coeficentwhch re owes o s,t des otEARLY VARYING ELIEMENTS, M. J. Kirby. AIREcoefficients which are powers of s, 1 does not Transactions, volume 69, 1950, part II, pages 1662-

enter the differential equation. 68.

Appendix The order of this differential equation for 2. RIEGENERATION THEORY, H. Nyquist. Bellnv(s) is always equal to the highest power of t System Technical Journal (New York, N. Y.),in the coefficients of equation 12, and the volume 11, January 1932.

The solution of many differential equa- coefficient of the highest derivative of v(s) is 3. ORDINARY DIFFE:RENTIAL EQUATIONS (bOOk),tions relating an unknown such as o0 to the polynomial P(s) formed from the co- E. T. Ince. Dover Publishing Company, Newtime can be materially aided by introducing efficients of the highest powers of t in equa- York, N. Y., chapter 18, 1946.a new complex variable (here called s) and tion 12. The highest power of s in P(s) is 4. HIGHER MATHEMATICS FOR ENGINEERS ANDexpressing 0 as an iiitegral: ff(s,t)ds of a always sn, where n is the order of equation PHYSICISTS, I. S. and E. S. Sokolnikoff. McGraw-function of s around a contour c in the s 12. For the present example, equation 12 Hill Book Company, New York, N. Y., 1941.plane. Time, t, enters the integral only as a 5. NUMERICAL MATHEMATICAL ANALYSIS, J. B.parameter. P(s) = c5s2+c1s+cO (16) Scarborough. Johns Hopkins Press, Baltimore,The object in the present case is to show

first that when the solution 0 is assumed to In any case, when the contour c in the s-be I'. eStv(s)ds, the function v(s) and the con- plane is chosen so that the quantity in thetour, c, can be found so that equation 3 is brackets is identically zero when evaluatedsatisfied, anid, second, that this solution around c, and when in addition v(s) satisfies xapproaches an exponential form as t-a.. the differential equation under the integral s PLANE

As neampeonsderth tim dife- sign, then equation 15 is satisfied by the XIential equationl: assumed form of 60o 'I

A satisfactory contour begins at - on ai)ed2 ~~~do0 line parallel to the real axis and passing - > REAL AXIS + a)

(csi+b2t+cit2) --00+(ai+bit±cit2)-± through one of the y's which are the roots ofdt2 dt P(s); the contour follows this line until it

(ao+bot+cOt2)0o =(aobot+cst2)Ot (12) nears the Sy, encircles it, and returns alongthe line to -a, Figure 4. A group of n W

The first step is to substitute the assumed such coultours can be found, one for each oftransient solution the n roots of P(s).

In order to determine stability, it is not Figure 4. Contour in S plane encircling zy's800= .f&1Vv(s)ds (13) necessary to determine v(s) completely; used for determining v(s)

1450 Kirby, Giulianeli- Varying-Element Servomechanisms AIEE TRANSACTIONS

Discussion knowledge of stability at large values of this somewhat round about method resultstime. in a superior understanding of the problemThe method employed here is called by and thus in a better control system.

W. G. Heffron (General Electric Company, some "use of the theory of small oscilla-Schenectady, N. Y.): The practical results tions" and by others "use of incrementalized REFERENCEof Messrs. Kirby and Giulianelli's method or linearized equations." The problem dis-can be reached by another method, quite cussed here is, technically, not a nonlinear 1. A New Differential Analyzer, H. P. Kuehni,similar in actual theory, but a bit more easily one, but many types of nonlinear problems H. A. Peterson. AJEE Transactions, volume 63,visualized. By substituting particular mnay be studied profitably in this manner. ' p 7

values of time in the coefficients and then Of course, there are some problems where thesubjecting the numerical results to the nonlinearities are too complicated to make M. J. Kirby and R. M. Giulianelli: Mr.Routh-Hurwitz test, the stability of the the results of a linearized study profitable: Heffron's comments are quite pertinent.system can be predicted, with the same such cases are studied with an analogue Linearization as he defines it is widely usedresults for large values of time as their computer' which we have available. in the study of varying-element systems,method gives. In addition, this method will As the authors point out, engineering particularly when an analogue computer isindicate at what time the system will change judgment must be used carefully to deter- available to check the results. Experiencefrom stable to unstable, if such a change does mine whether the results of a linearized with particular types of systems builds upoccur. For the simple Example 1 of the study may be extenided to predict the tran- judgment as to the degree of accuracy whichpaper it can easily be seen that the system sient performance of the system. Never- can be expected from the results of a linear-is stable at times greater than 1 second and theless, in many instances, linearization of ized analysis in a particular case. Theis unstable from time equal zero until 1 the equations has the important advantage criterion in the present paper was developedsecond. In the second example, this method of showing more clearly the functional rela- not to change this procedure but to justifyindicates stability from zero time up to tionships that control the system's per- part of it. Absolute stability is perhaps the0.59 second, and instability thereafter. It formance, pointing out the variations in most important single property of a system.would seem that knowledge of the periods inherent gain and time constants during the The present derivation provides a rigorousof instability also is essential to the de- operating cycle, and thus, delineating the basis for approximating the latter part ofsigner, in addition to the ever-important requirements for a control. Wre feel that the transient response.

1951, VOLUME 70 Kirby, Giulianelli- Varying-Element Servomechanisms 1451