absolute stability of nonlinear systems of automatic control

8/20/2019 Absolute Stability of Nonlinear Systems of Automatic Control

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Absolute Stability of Nonlinear Systemsof Automatic Control

V.M.POPOV

A CENTRALspecification imposedonacontrolmechanismis invariably its stability. Maxwell's paper On Governors [10],settled the problem of stability of systems described by third

order linear differential equations in terms of the coefficientsof the associated characteristic polynomial. Subsequent to thispaper, the problem of stability of systems described by lineardifferential equations in terms of the coefficients of the characteristic polynomial was solved by Routh and Hurwitz. It is worthnoting that the mathematical problem of determining when theroots of a polynomial lie in the left half of the complex planehad actually been effectively solved by Hermite [5] a decadebefore Maxwell posed the stability problem as a question incontrol.

In his thesis in 1895, Lyapunov put forward a method, nowcalled 'the second method of Lyapunov', for determining thestability of an equilibrium of a system described by nonlineardifferential equations. While Lyapunov was primarily interestedin applications to mechanics, his method and stability conceptsbecame very influential and widely applied in control. His ideaswere especially popular in the Eastern European control literature, since it was customary in these countries to use differentialequations as models for control mechanisms.

By contrast, in the Western literature, stability analysis in control had been centered on frequency-domain methods, propelledby the beauty, practicality, and generality (for linear systems) ofthe Nyquist criterion [11].

Each of these approaches has its advantages. Frequencydomain methods use very compact model specifications andare able to address robustness through gain and phase margins. However, the generalization to nonlinear systems of frequency-domain methods proved awkward. Differential equationmodels by and large proved a much better way of addressingnonlinearities.

Through the problem of determining conditions for the stability of systems with one nonlinear element, control witnesseda symbiosis between stability theory of Lyapunov methods andfrequency-domain techniques. A key result that was the catalystin this development was Popov's stability criterion, the subjectof the paper that follows.

The history of the problem is as follows (see [2] for moredetails). In 1944, Lur'e and Postnikov [7] formulated the problem of stability in the large of a finite dimensional system

with one nonlinear element I : JR

JR satisfying a I (a )

°for all a E JR. In subsequent publications, Lur'e obtained, usingLyapunov methods, a set of resolving equations that allowedto conclude stability. These ideas were further developed in [9],[16], [8], but the resulting conditions remained difficult to verifyand to interpret.

This problem of stability of a system with one nonlinear element led also to the Aizerman conjecture [1]. This states thatif there exist kl , k2 such that kl < f ( a ) < k2 for all a E JR, andaif the linear system obtained by replacing the nonlinear ele-ment by a linear one with gain k, is asymptotically stable forall k E [k1, k2 ], then the nonlinear system is also asymptoticallystable in the large. Aizerman's conjecture is valid for secondorder systems, but was proven to be false for third and higherorder systems [12] (see also [14], [15]).

By introducing the frequency response of a system, Popovobtained a sufficient condition for the stability of such systems.This criterion can be formulated as follows. Consider the feedback system described by (see Figure 3 in Popov's article):

d x = Ax + bu; y = cT x; U = - I (y)dt

with A E JRn x n , b, C E JRn , and I : JR JRa continuous nonlinearity with 1(0) = O. Denote by G the transfer function of thelinear part, i.e., G(s) := cT(Is - A)-lb. Assume furthermorethat A is a stability matrix [meaning that t E IR exp(At) EjRnxn is bounded on [0, (0)]. The matrix A need not have all itseigenvalues in the open left half of the complex plane. In fact,Popov assumes that A has all its eigenvalues in the open lefthalf of the complex plane, except, possibly, one eigenvalue atthe origin. The assumption that A need not be Hurwitz in factcomplicates his analysis considerably.

Popov proves that the equilibrium trajectory x = 0 is globallyasymptotically stable if the following conditions are satisfied:

1. a I (a ) > 0 for all 0 =1=a E JR,

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2. thereexists q :::: 0 suchthatRe(1 + q j co)G ( j ())) :::: 0forall()) E JR.

This result is known as the Popov criterion. We have statedthe criterion here for the case that the graph of f is containedin the first and third quadrant, but it is easily generalized tononlinearities in an arbitraryconic sector. The aboveconditionsonthefrequency responsecanreadilybeverifiedusinggraphicalmethods.

There are twomain techniquesfor proving this result. One isbasedon Lyapunov theory, and the other on studyingan associated integralequation.The relevantLyapunov equationconsistsof a quadraticplus an integralterm,

cT x

V(x) = x T Px +1 (1f«(1) de,with P = p T E R n x n a positivedefinitematrix that can be derivedfrom what is nowknownas the Positive-RealLemma [alsoknown as the KYP (Kalman-Yakubovich-Popov) lemma]. Asmentioned in the paper, Popov had developed the Lyapunov

proof in earlier articles. Actually, the above type of Lyapunovfunctionconsistingof a quadraticform in the state plus an integralof thenonlinearity, wasalreadyusedin [7].TheequivalencebetweenLur'e resolving equations, Popov's criterion, and theexistence of a Lyapunov function of the quadratic form plusintegraltype wasproven in [17](see also [6]).

In the present paper, Popov presents a new proof that involvesan integralequation,and in his analysis,he makesuseoftruncation operators. Truncation operators, and the associatedextended Ep -spaces, are elegant andeffectivetoolsfor stabilityanalysisfor feedbacksystems,usedalso for exampleby Zames[18] and Sandberg[13] (seealso [15]).

The ideas that led to and emerged from the Popov criterionbecameof central importancein the field. It was a first instanceof theuse of 'multipliers' forfeedbackstability. It demonstratedthe very fruitful interplay between time-domain and frequencydomainmethods. It paved the way for importantstability principles, such as the smallgain and the passiveoperatortheorem.Themethodof proof stimulatedthevery successfuluse of functional analysisideas in stabilityanalysis.

Popov's workimmediatelyinfluenced manyother leadingresearchersinthe field; forexample,Brockett[3], [4],Kalman[6],

Yakubovich [17], and Zames [18]. It was one of key elementsin the flowering of control theory in the sixties.

REFERENCES

[1] M.A. AIZERMAN, On the effectof nonlinearfunctionsof severalvariableson the stability of automatic control systems, Autom. i Telemekh., 8: 1,1947.

[2] M.A. AIZERMANAND ER. OANTMACHER,Absolute Stability of RegulatorSystems, Holden-Day, 1964 (translationof the 1963 Russian edition).

[3] R.W. BROCKETI ANDJ.L. WILLEMS, Frequency domainstabilitycriteria,PartsI andII, IEEE Trans.Aut. Contr., AC-I0:255-261 & 407-413, 1965.

[4] R.W. BROCKETI, The statusof stabilitytheory for deterministicsystems,IEEE Trans. Aut. Contr., AC-ll:596-606, 1966.

[5] C. HERMITE, On thenumberofrootsof an algebraicequationcontainedbetweentwo limits, extractof a letterfromMr.Ch HermitetoMr.Borchardtof Berlin, J. Reine Angew. Math., 52:39-51, 1856. (Translationby P CParks, Int. J. Cont., 26(2):183-196, 1977.)

[6] R.E. KALMAN, Liapunovfunctions for theproblem of Lur'e in automaticcontrol, Proc. Nat.Acad. Sci. U.S., 49:2, 1963.

[7] A.I. LUR'E AND V.N. POSTNIKOV, On the theory of stability of controlsystems, Prikl. Mat. i Mekh., 8:3, 1944.

[8] J.P. LASALLE AND S. LEFSCHETZ,Stability by Liapunov s Direct Methodwith Applications, Academic Press (New York), 1961.

[9] 1.0. MALKIN, On thetheory of stabilityof control systems, Prikl. Mat. iMekh., 15:1, 1951.

[10] J.C.MAXWELL, On governors, Proc. Royal Soc. London, 16:270-283,1868.

[11] H. NYQillST, Regeneration theory, Bell System Tech. J., 11:126-147,1932.

[12] V.A. PLISS, On the Aizermanproblem for a system of three differentialequations, Dokl. Acad. Nauk. SSSR, 121:3, 1958.

[13] I.W. SANDBERG, A frequency-domain conditionfor the stabilityof feedback systems containing a single time-varyingnonlinear element, BellSystem Tech. J., 43:1601-1608, 1974.

[14] J.C. WILLEMS, Perturbation theoryfor the analysis of instabilityin nonlinear feedback systems, Proc. 4th Allerton Conf. on Circuit and SystemTheory,pp.836-848, 1966.

[15] J.C. WILLEMS,The Analysis of Feedback Systems, The MIT Press (Cambridge, MA), 1971.

[16] V.A. YAKUBOVICH, On nonlinear differentialequations for control systemswith a singleregulator, Vest.LGU, 2:7,1960.

[17] V.A. YAKUBOVICH, The solution of certain matrix inequalities in automatic control theory, Dokl. Akad. Nauk SSSR, 143(6):1304-1307, 1962.(English Translation: Soviet Mathematics, 1962, pp 620-623.)

[18] O. ZAMES, On the input-outputstabilityof time-varying nonlinear feedbacksystems.PartI: Conditionsderivedusingconceptsof loopgain,conicity, and positivity; Part II: Conditions involvingcircles in the frequencyplane and sectornonlinearities, IEEE Trans.Aut. Contr., AC-ll:228-238& 465-476, 1966.

J.C.W.

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ABSOLUTE STABILITY OF NONLINEAR SYSTEMS OF

AUTOMATIC CONTROL

V. M. Popov (Bucharest)

Tr&DIlated from Avromatlka i Telemekbanika. Vol. 22, ~ 8,pp. 961-9'19, August, 1961Original article submitted January 17, 1961

Th e problem of absolute stabUity of an indirect cODuol system with a lingle DODlineartty i l lnvestlsated by uslDg a metbocl which differs from the second metbocl of Lyapuoov. The maiD condltioa ofthe obtafDecl criterion of absolute stability Is expressed In tenns of the trlJllfer functiob of the syltemlinear paR. It is also shown that by fonning the standard Lyapunov function --a quadratic form plusthe lDtegral of the nonlinearity It is DOt pouible In the case coosldered here to obtain a wider I tabil i tydomain than th e one obtained from the presented criterion. Graphical criteria of absolute contfDuityare also g iven by means of th e phase'-amplitude characteristic or by what is known as tile modifiedpbue-amplltude characteristic· of the system linear part.

In the praent paper the ablOlute stability is investigated of nonlinear systems of indirect CORbOl. Th e existing literature in this field (see [1] for example) deals exclusively with a dlmct application of LyapullOY sQIetbod.In this paper the solutionis obtained by a different method. and this enables on e to se t new leIulD.

It is assumed that the reader is Dol familiar with me author' , previous publlcatloDl. Therefore. not only themost generalwultl are given here but also a very simple example sball be considered.

By using a new method the author has also lDvestlgated the absolute .tabUity of other types of systems of differeatla1equations (for example.of the system of ·direct control- as well as of other classea of noallDear functioDi(for example-of functions whose graph is contaiD8d within a lector). In al l these cases the absolute stability of thesystem with several nonllneanties Is also studied (the case of systems with many CODtrolllng deVices).

In his most recent papen DOW in press, the autbor bas studied the stabi l ity in certain critical C I S e I and also th eltabUlty of systems of differential equations with an aftereffect .

1 . S t a t e m e n t o f th e Pro b le m

Systems of indirect control are considered which can be described by the following system of differentialequations:

(I ==1,2, . . • , 11 . (1.1)

(1.2)

(1.3)

where .Z k .• bE. cl .and Y are constant, and qJ (0 ) Is a function of clau A ·. that IS,a continuous function .adsfylnsthe condition

qJ (0) = 0 (1.4)and also the inequality

-It is assumed tbatthe quanti )' o . introduced in [6] alwaysvanbhes. It should just be mentioned that if a . ,-0. thetrivialsolution of the system (1.1)-(1 ..3)cannot be asymptotically stable.

Reprinted with permission from Automation and Remote Control, V.M. Popov, Absolute Stability of Nonlinear Systems ofAutomatic Control, Vol. 22, February 1962,pp. 857-875.

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qJ (0 ) a> 0 for a_ O

Th e system (1.1)-(1.3) admits the triviallO)utioD

Xl = C =0 =0,whose atabU ity il under investigation.

It fs assumed that the trlvlal101utlon of the linear system with c o ~ t tcoefficients

(1.5)

(1.6)

(l = i t 2, • • . ,A).(1.7)

where a Zk are th e lame as in (1.1).11 asymptotically stable, or (wh ich is equivalent) t ha t a ll real pans of th e eigen'values of th e matrix Calk) are nesadve. 11le conditioDt ale beIDa sought which would be satisfied by th e quantitiesa I k, b I, cl . and y in order that the triviallOlution of the .Ystem(1.1)-(1.3) be uymptotically a b l e . whatever thefunction tP (0 ) of th e clua A (In other words, a condition of uymptotic absolute stabiUty of the trivial solution). Aswe know to aChieve thil.lt Js neceauy that.

o.

We shall therefore consider In the sequel the inequality ( l .8) to be satisfied.

(1.8)

2. I n tr o d uc t or y D e f i n it i o n s. F o r m u l a t i o n o f C r i t e r i o n o f A b s o l u t e S t a b i l i t y

Consider th e functions ... Zm (I ) (Z I 1, 2, • • . , n; m = 1. 2, • • • , n), def ined when t c: O. and being the solut ionof the equations

together with the Initial conditions

( , == f. 2 • . . . • )m ==t, 2•• . • • I I (2.1)

. , m (0) = 6Z m l =1 . 2 , . . . . n; m =1. 2, • • • , 0) , (2.2)where 6Z m • 0 when I - m, aDd6 I m

=1 when I

=m,

The functions • I m (I) form th e fundamenwlJltem of solutions for the system (1 . 7).

Let Xl (I), E (t) be the solution of the equations (1.1)-(1.3)e. which satisfies the initial conditions Xl (OJIt X l' ' ~ (0) = I . and le t [0 ( t ) ] be a funct ion of t obtained by substituting th e function a (t ) = C1Z, t ) - T ; (t)in • (0 ) [see (1.3)].

As a solut ion of th e system (1.1) we obtain

ft ,

%, (t) = 111 ,,(1) z. . + ~ ~ I I (t - t) b .cp (a (Cn tit.- - I 1 0

It follows from (1.3) that

f t ,

e (I) = 1J II , (I) tna - ~ I- t) cp(cs(endt -rs (t). I 0

(2.3)

(2.4)

-II y < O. the UlvlallOlution is unstable w h e n . (0 ) =: ha. b>O.lf Y =: 0, the condi tion e (t) = 0 is not sa ti sf ied foral1101uti0DI of the system.• • Th e existence of solutions is a consequence of th e assumptions made Jn section 1 . But their uniqueness I. notusumed in the sequel. A IOlutJoll ca n always be extended in view of the conditions forstabilityas formulated below.

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where

tI

I&-(I) == }J e l ~ l .(t).' ' '1

ft •

V(t) ==- ~ ~ C ~ I . t ) b M .1 - 1 - - 1

(2.5)

(2.6)

Ia agreement with our aaumptioDi the tdvialsolutloD of the system (1.7) Is asymptotically stable. and thereforetwo positive constants I 0,

(2.10)

(2.11)

takes place. then the trivialsolutioD of the system (1.1)-(1.3) is asymptotically absolutely stable provided the assurnptions made in Sectlonl remain valid.

• ReX (or ImX) denotes the real (or,respec:tively. imaginary) pan of a complex quantity X.

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3. Proof of tbe Theorem In Sec t i on 2

COIDlder a.atn die .olation xl (t), E (t). a (t) of th e .ystem 1 ~ 1 - 1 . 3as well 81 the function. (a (I ». correapoacliftS to II. For any po.ltlve quantity T we define auxiliary functions

r (l) =={' (a (t») for 0 < t < T,cp 0 for t > T, (3.1)

• • (t) - - Y (I - t) ,.,. (t ) tit - q dv ~ t ) ' r (C)dt-ql (O) +rl 9.,.(t). (3.2)• 0

where S It the quantity occurring in the- theorem. and v (I) is given by the fonnul. (2.6); it follows that [see (2.2)]

(0) ==- c,b,.

'-1(3.3)

It can eal ly be seen dlat the function XT (t) is bounded for all 0 :s t T. When t > T., the Inequality (seeAppendix 1)

t. .place In which Ke II independent of t. ThiJ guarantees the existence of th e Fourier ualllfof\1lco

Lr Um) == r - J• ).,., (t) dl..' ·here exull also th e If.nsfann

00

r; (/f.) == rJ·'CPr (I) dt:o

(3.4)

(3.5)

(3.6)

In view of (2.6). (2.1),and (2.?) the Fourier transform of the function dv (1)1 dt exists and ca n be written as [se.(2.9)]

00

~ e-J - f d., ::l dl _ /J{ ( j . ) - (0).o

Tberefore.bytaldDstbeFourier transform of (3.2) [see (3.5), (3.61and (3.1)] we obtain

Lr U.) ==- ( t + jwq) N Uw) + 9r] F r , Uw).Th e follow lng function of T is inuoduced:

co

p (7') = ~ Ar to , r (I) dl.o

202

(3.'1)

(3.8)

(3.9)


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The Paneval fonnula can be applied in (1.9) because the function T (t) I. continuous for 0 s t s Ta n d Itvanishes for t > T. and the function AT (t ) Is allO continuous for 0 :$ t :$ T; and for t > T the inequal ity (3.4) takespla ce. We DOW obtain

CD

peT) == Lr (I.) F r u.)tI.,- 0 0

where PT (jell) 11 the conjusate complex of F-rtl(i + I - f ) N U-) + fTl fl..

--By camp.daB their real parts we obtain-(1') - - IFrU.)IIBe«i +1.,)NU.)+fTl ll••--

Observing that [see (2.10) and (2.11)]

Be [(t + 1_,) N UfIJ) + 911 -= B8 (1 +1-,) NUe) + ~ ==88 (1 + i-,) , U-).? o.

gives the inequality

p (T ) o.

By substltutlnl (3.1) into (3.9) we obtain

,-p (T) .... ~ 1 r(i) cp(a(t» dt < O.

o

(3.10)

(3.11)

(3.12)

(3.18)

(3.14)

(3.15)

(3.16)

By substituting the expression for T (t ) into (3.2) and by applying the formula (2.4) as well as the fannula

, :,') _ t 1 P t ; < ~SOIllO- ~d { ~ ~t> cp(a (t» tIC- Iv (0) + 11 cp (a (I», (3.1 'I). . . . . 1 0

eaily following from (2.4). we obtain within the IntelYal 0 :S t $ T

'-2' (t) == a (t) + f : t ~ t + 1 ~ (I) - (1 111(t) +, dJtr;,C') ) •m - l

By substituting this expression into (3.16) obtain

203

(3.18)


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r 2' r ., (a (I » a (I) fit + 9 ., (a (I» :: ') dt + T . , (a (I)) (t) dt• 0 •

t d- ~ [z.. . ~ (p.(I) + 9 ~ , ) ) . ,(a (I)) fit -c O.

- - , t(3.19)

Each lean of the expreaioD (8.19) .ball be considered in tum. Por the fint term we have the inequal ity [see (1and (1.5)]

where

T

- ~ ., (a (I» a (t) dJ -c O.•

The second term can be lewritten with the aid of the ident ity

T~ cp (a (I)) d ~ ~ )tIt·::::: F (a (7'» - F (a (0»,o

•.F. (a) .... ~ ., (a) tiD.

G

(3.20)

(3.21)

In the c u e when., (0 ) belongs to the class A [lee (1.4) and (1.5)], the condition

P (0 ) 2: 0 3 2 2 ~

i l fulfilled. with the equality only taking place when a = O. As q is nonnegative, one has. of course, the inequality

- qF (0 (1'» o.

Th e third term can be rewritten 81 follows [lee (1.2)]:r T ., (a I» (t) dt = t l l , , ~, ( I ) dt = +,-(7') - ~ .• D

A. Y > 0 [see (l.8)] we have

1 r ~ t I( T ) , O.

(3.23;

(3.25:

AI l u u me remaining tem1S are concemed we note (see Appendix 3] that there exists a positive quantity IUch that the inequality:

(3.26

takes place.

Uling th e identities (3.21) and (3 .24) u well 81 th e inequalities (3.26), on e i l able to rewrite the ioequal ity(3.19) as

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T

.., (a (t» a (t) dt + IF (a (7') + -} 1£1 (7')o

- K : : ~ ~...J1 I > : ~;(t>l , 'IF (a (0» + 1 ~ . (3.21)We shallsbow that the trivtalsolution of the system (1.1)-(1.3) II.table. To this end we combine the inequali

ties (3.21), (8.20).and (3.23)

T,I (7') - K . ma x (Is..ol) sup I ' (t ) I , 9F(a (0» + T ~. . . . . ,I , ••.• n o


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By applying (3.30). (3.84),and (3.35) we finally obtain

IE(I) I0 such that wben I x I t 1< 6 and J 10 I < 6 . th erlgbtbmd lidea of tbe Inequalities (3.36) and (3.3'1) are leu than ~ n dtherefore the inequalities J x·, (t ) 1


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andallO

lim , a (I» == O.,...because (0 ) is • contlnuoUl faDction.

Prom (3.43) an d from the equatlo . . (2.3) one obtains [see Appendix 6]

lim s, (t) ==0.. -.00

Th e relation (l.3) implies .[see (8.42) and 'S.44)]

lim e(I) - l i m . . . . . ~:r:,(t) - at) == O., . . . . 1-.00 T '_1

Tb e theorem has thus been plOved.

(3.43)

(3.44)

(3.45)

4. C o m p a r i n g th e R esu lt s w it h th e Resul ts O b t a i n a b l e with th e Aid o f Ly a p u n o v F u n ct iO D ' o f Known Kind

It is of interest to compare the criteria (2.11) of absolute stability with th e ones which ca n be obtained by constructing a LyapUDOv function of the kind described u a -quadratic form plus the integral of the nonlinearity [6-8].Wesball show that the latter are included in me criteria of the theorem In section 2. that Is, i f for the systemunder iDvestigatioD there extla a LyapuDOv function of the above Jdnd, then a Donnegative quantity q also exlsb suchthat the inequality (2.11) takes place. -

By differentiating the equation (1.3) we obtain a system equivalent to the (1.1)-(1.3):

(I : t. 2, • • • • II),

: - ~ c, a,k t + 6,4p(3) )-r4p(CS}. -I 1'. . ,

(4.1)

I t is shown In Appendix 7 that the mOlt g ~ r lform of a negative definite Lyapunov function which 11 of thekind des cri bed u a -quadratic fonn plus th e integral of the nonlinearity and which ca n be formed for the system (4.1)takes the form

tI ' 2 •

V == ~ ~ :r:,r ,,:r:.. - I t ( a - ~ c,x,) - 2Jl~4p (a) tlJs.1 - 1 - - 1 1 -1 I

with the parameters fZ m,a, 8 satisfying the conditions:

r, m==r , (J ::>Ot P :> 0. (I + P> o.Th e derivative of this function which, in accordaace with the s.ystem (1.4)e

f t+::-W(z,. 0) ==~ ~ :r:,r,.. IJrntXa + b.. cp 0»)1- 1 '-I ·1t==1ft n f t

+ «....,(0) ( a - ~ c,:r:,) - p4p (0) c, 1I,.:r:.+ h, 4p (a)) -fCP (a») l=-1 k=-1

mUit be positive definite for all functions ( 0 ) of the class A. and in particular when tl ft• We note that iii (0 - c, .,)=- - , . (0).

I- I

200

(4.2)

(4.3)

(4.4)


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rp (0 ) = h o, h>O. (4.5)

By putting (4.5) in (4.4) we obtain a quadrat ic fonn of real var iables Xl and 0 which must be posit ive definite.This implies that for al l complex values of Xl and a not vanishing simultaneously the inequality

rJft . ~ ;;r,,,, 40nk % +b ,ha) + ar M ( a - C, ,)

1- 1 t n - I I t - I I- I

tL

~ P h a ~c, ~ all 'Zk + b,ha) - rha )}>0.'=-1 k -I

(4.6)

i l valid, the quantities xl and 'lJ being conjugate to Xl and a It That this is so ca n be seen by putting in (4.6) xl = ul+ iV, • CI= 1£ + jU t with ul • vI ..... and 11real; the left-hand side of (4.6) assumes the form 1 (W. (U l .IJ) + W,('1 • II». where W. il the quadratic form (4.4) with tp (0 ) =he , ThUl, if uz. vI t Il.and v do no t vanish simultaneous)we have the Jnequality (4.8).

Th e Inequality (4.6) i l sat isfied in the particular case of

ta = hs ==M. (jCIJ) (l = 1, 2, ... . 0,11),

where Ml (jcu) satilfy th e synem of limultaneous equation

n

(4.7)

/mM, (jm) = tz,ItM. Um) -t- hi.=-1

(I = t, 2 .. . . . . ,ra).(4.8)

Th e system (4.8) hal a unique solution for any leal was according to our assumptions (see section 1) the matrix(a l k) has no purel y imaginary eigenvalues.

By taking Fourier uansfonns of the system (2.1) including (2.2l we obtain

.f t

where

jfJIF { ~ I m(l)} == o,r.F ('I 'hI (t)} + 6'mt-J ,

( ' = 1. 2 0, I I )m ==i ,2 , II ' (4.9)

DO

F ( . (I)) = ~ e-J-'1I' . (t) dl.G

By comparing (4.9) and (4.8) we obtain

ft ft

M, (j0) -

F ( . . . (t» h. = ~

ri

'1jI,,,(t) b ,dt.- - 1 t'ft=al0Substituting (4.7) into (4.6) and using th e relations (4.8) we obtain

{

ft

ft . MJ(jm) r mjwM. Uco) + «r{+- c,M,U.») -1 . I- I

- IS ~ c,j.M,U-) - 1 )} > o.I- I

208

(4.10)

(4.11)

{4.12)


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Suttee (4.11), (2.6). (2.9)]

C» CIO

cIM, (jeo)==1} ritl'ci'l'IM (I) bllltit == - (t) tit - - N U.). (4.13)1- 1 1-1. . . . . 1 . 0

In viewof r l m = rm I . we have-

ft • •

Be M, U.) r'lII j.-MIII U.) ==Be {+ j . :E :E )I,(j.) I I . U.)1. . 1- - t 1-1 1

+ M, U.) M . U.) )r,,,,}==o. (4.14)The inequality (4.12) can therefore be rewritten u

or [lee (2.10)]

+ He (Clr + j.P) G (j .) > o,

(4.15)

(4.16)

The above consideration remain valid for any real w. Thus. it b u been shown that for a LyapUDOvfunction of theconsidered kind to exist it is necessary that th e inequality (4.16) be valid for al l h>O and for any w.

If a .. O. [see (4.3)] and thus if a y >0 [see (1.8)]. from (4.16) must follow th e iDequality

(4.1 ')

Indeed. should th e inequality (4.17) be invalid for some w = Wo. then a positive quantity h exfla luch that theinequality (4.16) wUl not lake place for w =wo. The necessazy condition (4.17) il identical with the inequality (2,11)if q =Slay> O.

If a . O. we obtain from (4.3) aDd (4.16) the inequalitiel

P >0,RejmG U.) > e.

(4.18)

(4.19)

We mall nowconsider certain properties.af the fUDCtion N(jw) [see (2.9)]. Th e fUDCtion N(jw)is a caadnuousfUDCtloo of w 10 view of th e ioequality (2.8). By tbe R1emanD-Lebesque Lemma

li m N (j .) = O.'-I-.eo

Therefore. there exists a posit ive number PI such that for any CIl

HeN (jw)>- Pt-

By me Rlemann-Lebesque Lemma we also obtaia* • [see (3.3)]

lim IwN UCD)==v(0) ==- :E(,b,.

I • 100+0)

(4.20)

(4.21)

(4.22)

·W e note that 1m (M.(jfl) M ,(jUJ) + M, (j-l A I . (;(1)) EO .• • From th e relations(2.6) ,(2.1) and (2. ' ) It is ealyto find that the Integral

209

r,dvtt>l- r III must eonverge,o


14/21

It follows from (4.21). (4.22) and (2.10) that

ReG(j_) == ReNU.» -P l ,

lim i-e U-)=r

11m JJ i U.) +T==

-

c,b, + T>O.I • 1-..00 , - 1....oJ -I

(4.23)

The laner II • necawy COIlcl1don aDd ca n be obtalDed from(4.6) and (4.18) by putt1Dg Xl • O. 0 =l Ib . aI: O. It follows from (4.19>. (4.24) and from Ibe CODtimdty of die function G(Jf4 dlat a posit ive qua.lty p. can befound such that for aoy real w the Inequality

He JflG (Jw» p.. (4.25)

takes place.

By multiplying the inequality (4.25) by 21\/ P.and by adding the result to the inequality (4.23), we obtain

Re( t + 2 ;1if)))G(jCIJ)>- PI + 2Pl >O.I .

(4.26)

1be laner implies Ibat th e inequality (2.11) II valid (a t a IUODS inequality) when q =2Pl/P. >0.ThUilt ba l been proved that dle 1Dequallty (1.II)·suffices for dle trtY1alsolutioD of the .y.em delClibed In

section 1 to be abeolately asymptotically .able and also to be a Ilecellary cond1dOD for the exiltenec of a LyapUllovflDlCtiOD of the conddaed type.

Remarks. (1) In order to construct a Lyapuaov fuactioa of die -quadradc form- type (that 11 of the IdDd al In(4,2) with 8 • 0) It i t Deceauy that th e laeqaality (4.1 '1) bevalid with 8 • 0, Ibat i l that

(4.27)

for any real w.

The coDdldoo (4.2'1) IIlufftclent far the ahIOlute uymptotlc liability of the trivial lolution of the tyltem uDderinvatlsatloa. • • 1 In ddI c a e Ihe inequality (2.11) occun far q • O.

(2) Molt of the LyapUDOY func1iol1llO far CODltnlCted In [8] are of the Idnd al fa (4.2). with a = O. That Iswhy i t 11 nec- . ry that th e i_quality (4.19) be l i d ~die latter I, .110auffJcleat for the ablolut. Itabillty of theIystem trivial IOlutlon.

Tb e condition (4 .19) ca n allO be written for any real positive w al

1m G (-jw)< O.

11111 foliowl from the re la t ion [see (2.9) and (2.10)]

Im G (,61) .. - 1 m G ( - ; . )

and from tile inequality [see (2.10). (3.'1). and(1.8)].

(Re i. e (;1) )..-0 - T>O.

(4.28)

(4.29)

(4.30)

(3) It would be interesting to find th e seneralsolul1OD of th e followi . . mverse problem: U the coaditiOll (2.11)is satisfied il it always possible to coastrUCt a LyapUDov funetloD t i th e kind al in (4.2) ' Pot lome relatively Simplecases th e anawer i I In the aftlrmative.

5. Va r i o u s A n a l y t i c aD d G r a p h i c Porms o f t h e ( 2 .11 ) C r lt er io D

The function (1 + j fdq)G(jw) CAn be written al

210


15/21

(1 + . )G U·) P (iCd) (6.1)lfM1 fa) ==Q ( I . ) •( Appeaclix 1) with P (Jw) and 0 (jw) being polynomiall of jW. Th e coad1t1oD(2.11) can DOW be written a l

or

R(z) :> 0, s ==m , (5.3)vb R (x) 11 a polJDOmfal of the variable -=The condition ~ 1 1 11 reduced. therefore. to a polyaomfal of. . belngaoaaepdve for x O. ' be IOlud . ca D be abtamed by . . . . ItaDdaal alsebme metbocJs.

AD ubltrary DODDe.at1ve parameter ..9.appean III th e criterian, whlch caD be lelected in a IUltable manner inevery apec1flc c u e . · The algebraic methodl for obtaining the optimal valoea of 1 • • quite Jtraflbtfcrward.

'lbe (0110w_ F.phlcal criteria of absolute alymptotic subBlty are of Ipecl.llDterest fa practical .ppUc.tlons.

TbelOCU1 o f . e poflltl (u. v) III th e plane of (u. v) such that

II (co) == Re G (/OJ), 17(0) ==mImG(jC'd) (6.4)

. a l1 be cal led the -modified pbue-ampUtude characteristic- (M.P. A.C.). Directly from (1.11) we obtalD che Inquality

u(m) + qD m) :> 0, q > 0,which meaulbat the M. P.A.C. 11 In a half-plane.

A srapb1cal cdtmOD (PfS. 1) . If there JI • lIraJsbt Ua e dtuated _ III the fInI: . . . . the dWd qaadraDII d Ib e(a . v) pi or I t die orcUDate axl l - · aDd Ia add1doa lIauch that d1eM.P.A.C.ls-oathedgbt·of tbb atralptlfne.then the trivial solution of the lDYeatipted system 11 absolutely asymptotically stable. One may add that theM.P.A.c. II ·o n th e right- of tbliltraight llDe if allY p o ~of the M.P.A.C. Is either on the straight line or it II Inthe bali-pia . . bounded by this sttalsbt llDe and coatalnlng tile polDt . (+-,0). • • •

FiS. 1.

of.) or JiIJl-J

Fig. 2.

• We note that in die cri teria (4.27) or (4.28) which described the results obtainable by the usual kind of LyapUDOvfuactlOO (lee Remarks (1 ) and (2). there Is DO arbitrary parameter.• • Sucb • straight llDe obviously puses through the origin. Its equation is of the form u + qv =0 with q 2:: O.• • • Or. otherwise. that the inequality (S.S) takes place.

211


16/21

Fig. 3. Fig. 4.

Th e oldinary phase-amplitude characteristic ca n also be made use of in order to obtain simplified graphicalcriteria of absolute stability.

The following graphical criterion is obtained from the condition (4.2'7).

Simplified graPhical criterion No.1. If all the points of the ordinary (or modified) pbase -amplitude characteristic are situated on tbe right of the ordinate axis. the trival solution of the system under investigation is asymptoti

cally absolutely stable. (Fig. 2)It is necessary that this graphical criterion be latisfied in order that a Lyapunov funcnon of th e quadratic form

kind may exist r.ee Remark (1)].

Prom the sufficient condition (4.28) of stability another simplified graphical criterion is obtained.

Simplified Sf.plca criterion No. 2. l f . when w>O. all the points of the ordinary (or modified) P.A.C. are situatedin the third or the rounh quadrant or on the negative ordinate semi-axis. then the trivial selutton of the system underinvestigation is asymptotically absolutely stable (Fig. 3).

It 11 necessary that this criterion be fulfilled in o lder tha t a LyapuDov function {5.3)with a :::0 may exist [seeRemark (2)].

We should like to mention that no simple method exists to express the general graphical criterion (Fig. 1) bymeans of the ordinary P.A.C. knowing the ordinary P.A.C.,one is able to obtain the modified P.A.C. by mUltiplying

the ordinate ofeach-

point byth e corresponding

valueof

the variable.

6 . C o n c l u d i n g Remarks

The maiority of the argumenfJ developed in die preceding sections can be awlled with practically no alterationsto more general cases meDtioaed in me introductlon,and results of similar nature ar e obtained.

The fact that in the cri teria omy the ansfer function of Ibe system 'lInear pan appean. apart flOm the simpleassumptions of section 1. seems to constitute the main characteriltlc of th e achieved results. The latter need not beevaluated with the aid of the fODDula (2.9) but can be obtained by more direct methods which have been deVelopedfo r linear systems of automatic control. The graphical criteria of absolute stability developed above are allO applicablewhen DOthlng but iu linearity and independellCe are known about th e linear block of the system. Its phase-amplitudecharacteristic being determined eXperimentally.

The author wishes to exprea his thanks to the eeneenve of research-worken in th e field of ordinary differentialequations of the Mathematical IDstitute of the Academy of Sciences of the Romanian National Republic, in particular to Professor A. Khalan,for the interest they have shown aDd for their valuable r,·,narks.

APP ENDIX 1

Th e investigated system can always be represented in the form of a block diagram as in Fig. 4 where the l inearblock II denoted by L. the laner described by the system of equations

212


17/21

11 :=- CIS, + Tl-

1. . 1

(1.1)

and N representing tbe nonlinear block. Th e input and the output quantities of the linear and nonlinear bloclcl ar erelated by the eCluadons

fJ =- - cr; • =cp (er).

coLet z ' (t) be a known functfoa such that the integral ~ I at (. ) I

u ~ ~ n

(1..2)

exists. and le t L {z· et)} be its Laplace

co

L (. . (t») = ~ .-· 'ae{t) tit. fte _;> e.o

(1'.3)

Let further x' ,Ct), I (t). JJ' (t ) be th e 101utionof '(1.1) when z :: zO (t ) aDdwhen the initial conditiolll a re a ll

nul. It follows from the auumptioas in sec tion 1 that L{xl el)} exists at least for Re I 2 O. and L {C ( t ) } and L {'J'( t ) } exist for Re I > O.

Taking Laplace transforms of (1.1) we obtain

. L (s f (t ) =- ~ . ~(ao: (I») + baL flD (t)). Rs 8; > 0 (I = f t 2, . • . • n).~

(1.4)

(1.5)

0.6)

1).

,L ( , ' (I» ==L (sO (t)). Re 8>e, L ('l'( » == - e,L ~ ('»+ rL ~ Oo». Re , >o.

/ - 1

Th e system of simultaneous equations (1.4) has a unique solution wben Re s > 0 (see the assumptions in section

Now taking Laplace transforms of th e system (2.1) with the in it ial conditions (2.2):

where

,L (1),,,, (,»- t J l ~< ,__ (I)} + aim' Be 8 ;> 0 ( = t. 2, • • •• ) .

t::l:l m = = t , 2 • • • • ,1 1

By comparing (1.4) and (1.7) we obtain

•L (sf (I») = ~ L(.l' emb ,)L (zO (t)). Re - ;;> 0 (I = 1. 2 • • . . • II).

m:- l

Using (I.S).(I.6),and (J.80)gives

nN (.) = - ~ e,L .,m(t)} b , II 8 8 ' > 0

l==l 71'=-1

(1.7)

(1.8)

(1.9)

(I.10)

(1.11)

~ function G(s) defined by the equations (1.10)-(1.11) is the transfer function of th e system linear block. Tbefunction G (I> Is obtained in the fonnof a rational function of.. : The above considerations ar e only valid when Re

213


18/21

s > 0. we shall say nevertheless that G (s) II a traosfer funCtiOD if it is a rat ional Cunction def ined in the whole s..pia,and obtained as the analytic continuation of the G(I) function.

It should be mentioned that th e function L {.zm ( t)} exlsu when Re I =0 and that when I =jw (w real). thenL { t ' m (t)} =P {tLm (t)} [see (4.10)]. By comparlDg (1.11). with (2.6) and (2.9) we see that when I = jw ttte Cunctic(1.1.1) II equal to the function of (2.9). Tberefore.the function (2.10) 11 equal to the above defined function G (s) Wi1•=jw .

APP ENDIX 2

Therefore,

By uslog (3.1) and (3.2) we obtain for t > TT

AT e,)==- ~ ( e,- t ) + 9 dv e ~ ; -C)).CoeC»dC.o

ButJee (2.6). (2.1) and (2. ' l)j

i ft ft

1 (')+9 ~ : I ~ ~10

11(1 +9 ~ ~ J / l h I ) ~'''elKle-K

. ' . i- I t n - I i - I

T

, AT (') , ~ ~ ~ ~(0 e» 1 Iv (t - e + 9 d V e ~ ; -t)Itito

f t f t ft T

~ ~ b . ( O ( c ) H ~1011(1 + -s ~ I / l l r m l ) ~ f I K 1 S . - K , C - t ~ .1 -1 i- I --I i __1 0

Thia gives the inequality (3.4) ,whereft n ft ~

X. == B1Ip P (0 «»I ~ 1eel(1+ 9 ~ 1Clml)~ ' ' 'd ~ :(,K.r - t ) .o


19/21

By lubldtutlng (111.4) and (DI.S) into (DI.S) .nd b, using the inequality

T CD t~ 4I- oftit 0, lim f ==+ 0 0 . Let th e solutloD in question [see (3.41), (3.36), (S.3'1), and (1.3)] be

Th e sequence tk can be selected such that

c1 6 t - ' k - t > M. t t l > 2M , ·

c\Wben I ' - ' t l < 2M

1wehave

t

M . > I (') / = 13 (It) + ~ d ~ ~dt I>Ia ( (t) 1 - JlI l / , - ' t I> -} 6.' t

We abo· bave [see (1.4) and (1.5)] t .+- ...T NfT) Ii IN .

~ • (G«»cs(.) d. > tp (cs(.»cs (I ) tI .o t.==l t at - i i i ;

where N (T ) is an lDdex such mat 'N(T) + ?M ,Tan de + _6_ > T 1 N(T)+1 2M ·

215

(V.I)

(V.2)

(V.S)

(V.4)

(V..6)


20/21

.,.~ . G t » ~ l d t~ N ( 7 1 ) ~ . . m~ M. 2

Obviously1i.1I J\i (11;· 00.

., •• 0

Let i n f 1 ,, (0 )1 =m. Then we obtain from (V.&) [see (1.6)]

~


21/21

«(7)= a'cs,tJ = 1,

(VI1.5)

(V11.6)

wbere .JI one of the lntegen 1. 2. • • • • n: t t m (t ) the functions Introduced in section 2, and E an arbitrary (positiveor negltive) quantity. The function (VII.5) I. of coone of the class A. By integrating (2.1) we obtain. In view of. ' m (t ) [see a110 (2.2)],

n co

- 'm(0) =-a'm = I. ~ +k m (I) dl.k = l 0

Consequently [see V I I . 4 ~

t J l k s ~- e ~ m ·

k l:l

It is easily seen that for th e values of (VII.4)-(VII.6) the function i dV /dtcan be wri tt en as

; : - = ~ ~ z.t4) a + 0 (8 ).b = l k==l

in which 0 (E·) denotes terms with the property J 0 (E I ) J< KE J . with K being a constant.

Prom (VI1.9) using (Vlt8) we obtain

(V11.7)

(VII.B)

(YO.9)

(VI1.10)

If rm ;IIIO.then for a sufficiently small E the expfelSioD (Vl1.I0) has the sign of - fm £ ; rile latter, however. isarbitrary because the sign of € is also arbiuary. It is therefole necestary that Em = 0 be true, As was taken asarbitrary we see that the equalities (VII.3) mUlt be true,

The function (4.2) Is negative when Xl = O.a ='L, tp (a ) =ho , h > O. Tliis gives the inequali ty

C l 1 . a

absolute stability of nonlinear systems of automatic control

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