Mikhailov Stability Criterion For Time-Delayed Systems - NASA

88Mikhailov Stability Criterionfor Time-Delayed SystemsFOP, EFERE-1NCE4"L. KeithBarkerJANUARY1979L:,;,."i.tL.'.'L!'E.:.;i CE '7[ER

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NASA TechnicalMemorandum78803Mikhailov Stability Criterionfor Time-Delayed SystemsL. Keith BarkerLangley Research CenterHampton, VirginiaNational Aeronauticsand Space AdministrationScientific and TechnicalInformation Office1979

SUMMARYThe validand invalidapplicationofthe Mikhailovcriterionto linear,time-invariantsystems with time delays isdiscussed. The Mikhailovcriterionis a graphicalprocedurewhich was initiallydevelopedto examine the stabilityof linear,time-invariantsystemswith no time delays. For these systems,there are two equivalentformulationsof thecriterion.Attemptingto applythe second formulationwhen thereare time delaysin thesystems can leadto erroneous results,as shown by an example. However, thefirstformulationremains validfor time-delayedsystems ofthe retardedtype,with the unde:.'standingthatthe Mikhailovcurve need not necessarilyalways rotateinthe counterclockwisedirectionfor a stablesystem. At present,applicationof the Mikhailovcriterionto neutralsystems has notbeen justified.INTRODUCTIONThe Mikhailov criterionis used to examine the stability of dynamicalsystemsare describedby linear ordinarydifferentialequations with constant coefficients.thatKashiwagiwhenthere(ref.1) indicatesare constanttimethat thisdelayscriterionin the system.can be used with only limitedsuccessYet recently,(ref.Chen and Tsay2) haveshown that a modified Nyquist criterion,which is similar to the Mikhailov criterion,doeshave valid applicationto time-delayedsystems.Why is there a discrepancybetween theearlierand recent works?The purpose of this paper is to remove the confusion relatedto the valid applicationof the MikhailovcriterionSYMBOLSA,Aj,Bjreal constantsiimaginaryjintegerindexkrealnumberunit,to time-delayedsystems.

L(s)characteristicpolynomial for system with no time delays, or characteristicquasi-polynomial for systems with time delaysLl(S),L2(s )parts of L(s)shown in equations (42) and (43)Mnumber of roots of L(s)Norderscomplexsjjth root ofX(w)realY(w)imaginaryof systemwith positive real partswith zero timedelaysvariableL(s) 0part ofL(iw)partangle in figureofL(iw)2damping parameterO(w),Oj(w)change in argumentsfrom zero, jtotal change in arguments of L(iw)increasingly from zero to infinityrealpartaasymptote7, jconstantwimaginaryofof L(iw)andiw - sj, respectively,andasiw - sj, respectively,w increasesasw variessof realreal-timepartofpart of largemodulusrootsofL(s)delayssANALYSISBy referringMikhailov criterionto the complex geometryinvolved,for a system with no time delays.Popov (ref.Kashiwagi3) gives insight into theused Popov as a basic

reference.Hence, in order to appreciatethe earlierthoughts on the Mikhailov criterion,it is necessaryto first examine the criterionfor no time delays as viewed by Popov.Let the characteristicL(s)--A0sN with realcoefficients,MikhailovCriterionpolynomialof a linearAI sN-I .or in factoredL(s) A0(s-(No Delays)systembe expressed AN IS A Nformas(A0 0)(I)asSll(S- s2) . (s- SN)(2)where the complex rootsappear in complex conjugatepairs. Withcan be writtenass iw, equation(1)L(iw) X(w) iY(w)(3)X(w) A N - AN 2W2 AN 4 o4 - . . .(4)whereY(w) AN lWand equation- AN 3 w3 . . .J(2) becomesNow considerthe plot ofL(iw)in the complexplane aswis increasedfrom0 to 0% as depicted hypotheticallyin figure 1. This is the so-calledMikhailov curve.The argument of L(iw) is simply the sum of the argumentsof the linear factors ofL(iw)in equation(5), that is,NargL(iw) arg (iw-sj)j l(6)

Let O(w) and 0j(w) denote the changes in the argumentsrespectively, as w increases from zero. Then,of L(iw)andiw-sj,N Z ej( )(7)j l(The argument of the real number A0 0 is zero.) Furthermore,if 0 and j denotethe total change in the arguments of L(iw) and iw - sj, respectively,as w variesfrom 0 to % thenN 0jj lto0(8)Consider the followingin equation (8).two specialcasesto gain insightinto the contributionsNegative real root.- The change in the argumentof the factoriw - Sl, whena negative real number, is examined by using figure 2(a). It is geometricallyclearas w varies from zero to infinityofjs 1 isthat7/1 0,)-*lim oo 61(w ) Complexfactors(iwnegative realconjugateroots.-(9)Considers2) and (iw- s3) , whens 2 andparts, as indicated in figure 2(b).w 0, the argument of the complex vectorto infinity, the change in the argumentis02Similarly,the angular limw-02(w) 2 for the vector3 limw-changein the argumentsof the twos 3 are complex conjugate roots withGeometrically,it can be seen that foriw - s 2is-7.Aswincreasesfromzero(10)iw - s3,7r03(w) 2 - 7(ii)

Hence,2 3 n 77It is indicated(12)by equations(9) and (12) that each root ofL(s)with a negativerealpart contributesto the argument of L(iw) in equation (8). By similar geometry,can be seen that any root of L(s) with a positive real part will result in an argumentit7/change in L(iw) of - as w goes from 0 to . Popov (ref. 3) also discussestheeffects of a zero root, purely imaginaryroots, and an infinite root, but these cases arenot presentedhere.SupposeN - Mrootstherearewith negative (N- M) nIn orderMrootsrealparts;L(s)in equation(1) with positivethen it is not difficultto inferrealparts,geometricallyandthatM 77 (N - 2M) 2forthe systemsrealparts.have negativeofunder(13)considerationIn this respect,to be stable,Popov (ref.all roots3) writes:"ForofL(s)stabilitymustof annth-orderlinearthe characteristicsystem it is necessaryand sufficient that the Mikhailov curve plotted forequation of the given system pass throughn quadrantsin successioncounterclockwise,L(iw)completescircling the origin of coordinates."a rotation by the angleThus,sincen Nin this paper,77 N(14)which is equation (13) with M 0 (no rootsformulationof the Mikhailov criterion.The rootsSl,s2, ands3ofL(s)with positiverealparts).are shown in figureThisis the first2 and have negativerealparts.The associatedvectorsiw - Sl, ice - s2, and iw - s 3 always rotate counterclockwiseas w increases;or equivalently,01, 62, and 03 always increase.Therefore, from equation (7), if all roots of L(s) have negative real parts, it is not possiblefor the vectorvectorL(iw)L(iw) to ever rotate by any amount in the clockwisecontinuallyrotates in the counterclockwisedirectiondirection;that is, theas w increasesn (eq. (14)). This leads to the second formulationfrom 0 to , and O(w) approachesN of the Mikhailov criterion,which states that a necessaryand sufficient condition for alinear, time-invariantsystem with no time delays to be stable (characteristicroots withnegative real parts) is that the real part,X(w), and the imaginaryin equation (3) alternatelyvanish a finite number of times.part,Y(w), ofL(iw)5

TranscendentalPreviousworks(refs.Characteristic3 and 4, for example)terms of a polynomialcharacteristicequation.interpretationto systems with time delays (ref.Suppose the system has time delaysexample, the culty1).the Mikhailovoccurscriterionin tryingso that the characteristicquasi-polynomialinto apply thisfunctionis, forL(s) s 2 2 se -Ts 1where- is a constant time delay.equation (15) has an infinite numberby the time delay.(15)Unlike equationof roots because(1) which has exactlyN roots,of the exponentialterm introducedFor illustration,let 6.2 and 0.2 in equation (15). In this specific case,all the roots of equation (15) have negative real parts, making the system stable (ref. 1).The Mikhailovcurvecorrespondingto equation(15) is shown in figurethat the curve moves in both the clockwise and counterclockwisethe roots have negative real parts.This could never happen if3.Noticedirectionseven though allL(s) were a polynominalwith negative real parts.To approximatethe curve in figure 3 very closely with a polynomial would require that the approximatingpolynomialhave an unstable root (positivereal part) in order that the approximatingcurve ever move in a clockwise direction overany portionof the solution.Clearly,from figure3,X(w)andY(w)do not alternatelyvanish;and, conse-quently (by counterexample),the second formulationof the Mikhailov criteriondefinitelydoes not apply in general.However, as noted in the section entitled "MikhailovCriterionfor Delayed Systems,"with proper interpretation,the first formulationof the Mikhailovcriterioncan be applied.MikhailovCriterionfor DelayedSystemsRecently,Chen and Tsay (ref. 2) derived a modified Nyquist stability criterionwhich,in the context of this paper, is similar to the first formulationof the Mikhailov criterion.The criterion6was shown to hold for characteristic(1)L(s)has no purely(2)L*(s) L(s*), where(3)L(s)acts likeimaginaryAs -k*meansasfunctionsL(s)for which:roots.complexconjugate.s - oo in the righthalf of the s-plane;that is,

lim L(s) A 0s-oo s-kwhereA(a - 0)is a nonzero constant and(16)k is an integer.Chen and Tsay (ref. 2) show that, asrotates by the amountw increasesfrom 0 to % the vectorL(iw)(17) (-k - 2M)whereMis the numberEquationof roots(17) indicateswith positivea stablesystemrealparts.if and only ifM 0; that is(18) -kIt is importantwise rotations.to note that thereis no requirementWith respect to the characteristiction (16), and equation (18) becomesthatpolynomialresultin equationonly from(1),counterclock-k -Nin equa- N -2(19)which is the same as equationit is clear that in equationHowever,this is not explicitRetardedthe formsystems.-in the derivationConsiderNZA N 0AN sN.To show this,tion (20) relativeandtoa system(ref.view of the Mikhailov criterion,only of counterclockwiserotations.2) of equationwith a characteristic(18).quasi-polynomialofN-IZj Owhere(14). Now, from Popov's(19) will, indeed, consistj O7j 0considerANsN.(ref.5).The dominantthe magnitudeThus,termof the rationote that forj ig N,in equation(20) asof any remainings-termoo isin equa-

limAsL I,I1-- lim- 0(21)Isl-ooANsN1 Isl- Isl -jsince N- j 0.limAlso,-- N eIsl-ooANSwhereis the realpart ofL I, lime- 0(22)Isl-ooIslN-js.Hence,lim L(s) AN 0s-oo sN(a 0)(23)for the retardedsystem, and itfollowsfrom equation(16)that k -N. Therefore,fromequation(18),the system is stableifand onlyif N 2For example,equationsuppose(15).(24)the retardedThen, equationsystem(24) indicateshas the characteristicthatquasi-polynomialL(iw), which is shown in figurein3, shouldrotate by the angle2( ) for stability.The real part of L(iw) in figure 3 isincreasingmuch faster than its imaginary part; therefore,the change in the argument ofL(iw) as w varies from 0 to ooappears to be approaching. This conjectureis verified algebraicallyas follows.Withs iw, equation(15) becomesL(iw) (1 - w 2 2 w sin wT) (2 w cos wT)iThe realpartofL(iw)in equation(25)(25) isX(w) 1 - w 2 2 w sin wT(26)Notice in equation (26) that ifw2 1 2 w sin w -(27)

thenXCw) 0.Certainly,equation (27) will be true ifw2 1 2 w(28)The right-hand side of equation (28) dominates the left side as 0) approaches 0. However, as 0) increases, the left side will eventually dominate the right, and equation (28)will be valid. Equation (28) is valid for all 0) tom, where 0)m is the largest positivereal root of the polynomial equation0)2 2 w - 1 0(29)and is given by0)m - / 2 1For the specificand (30) become,being considered, 0.2and- 6.2.Thus,equations(26)respectively,X(0)) 1 - 0)2 0.4w sin 6.20)(31) om 1.2(32)The realincreasingexample(30)and imaginaryalong the Mikhailovthe correspondingX(w)sponding values of X(w)Mikhailov curve in figurepartscurveofL(iw)in figureare parametric3.Afterequations0) increasesintow, with0)0) w m 1.2,value is X(0)m) 0.056.Thereafter,0) 0)m and correwill remain negative.In other words, if continued, the3 will remain in the left half plane (second and third quadrants).Since the curve in figure 3 does not encircle the origin,L(i0)) as w varies from 0 to 0olies in the intervalthe changein the argumentof- 0 3 .22Since the argument(33)ofL(i0))in figure3 is zeroatw 0,O(w) arg(i0)) .Hence,cos 8(0)) 1 - w 2 2 0) sin 0)7(2 0)cos 0)'r)2 (1-0)2 2 0)sinwT) 2(34)