Absolute Stability of Regulator Systems

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•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •

ABSOLUTE STABILITY OF REGULATOR SYSTEMS

M. A. Aizerman and F. R. Gantmacher

HOLDEN-DAY SERIES IN INFORMATION SYSTEMS

ABSOLUTE STABILITY

Norman Abramson, Stanford University

OF REGULATOR SYSTEMS

Gene F. Franklin, Stanford University

Ernest S. Kuh, University of Califoruia, Berkeley Consulting Editors

Translated by E. Polak

>

HOLDEN-DAY, INC.

San Frantisr-u, !.mulon, Amsterdam 1964

TABLE OF CONTENTS

I.

Definition of the Problem

1

~

II.

Ill.

I. Equations of motion ~ 2. Formulation of the problem ~ 3. The his tory of the problem

6 10

Solution of the Absolute Stability Problem by the Direct Method. The Resolving Equations of Lur'e

15

~I.

The principal case of(!)

§ 2.

1\ particular case of l!)

A Frequency Method for Solving the Problem of Absolute Stability (The V. M. Popov Method) ~ I. § 2.

§ 3. ~ ·1. !i.

*

§6. ~ 7. § S.

IV.

Frequency response func tions Pre liminary investigation. Absolute s tabi lity of linear sy&tems T he V. 1\•1. Popov theorem The pnx1f of the V. 1\1. Popov 1heorcm Condi tions for stability-in-the-limit J\ new fo rmu lat io n of the Popov critt>rion The case k =oo Examplc.>s of the use of the Popov criterion for solvinJt I he problem of absolute s tabilit y fo r syslt>ms with unspecified parameters

I

•• •• •• •• i •• :j ••

15 3-1 45 4!1 49 !)I 58 67 79 82

&'I

The Connec tion Between the Popov Criterion ond the Existence of o Liopunov Function. Further Refinement of the Frequency Criteria .for the Absolute Stability of the Particular Cases of ( ! )

89

§ I. Introduction §2. The principal c-ase of(!)

!39 91

I •• •• •• •• •• •

•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •

TRANSLATOR 'S PREFACE

/ ...

Originally published as

/l b~lllutnaya

UslIUil' ~tabi l it r fur sy~•ll'IIIS irf('OU!'ibJt• to the sy'>tl' lll (!)

J,lfi

§ti.

Appendix.

The Connection Between the Method of Lur'e

Resolving Equations and the Popov Frequency Method

1-19

**l.I.

J.J9 152

~:I.

l11l rnduct iun The fundame~ttnl theorems fl ruof or the t heorccn'l

Bibliography Index

ABSOLUTE STABILI TY OF REGULATOR SYSTEMS

132

cr iterion V.

102

156

DEFINITION OF THE PROBLEM

§ 1.

Equations of Motion

In t his book we consider d y namic processes which can be described by a system o f differe ntial equations differing from a linear homogeneous set with constant coefficients only by an additively-en ter ing non linear fun cti on whose argume nt is any one or a linear com bination o f the state variables:

.

! a,;x1 + b,y j ~•

(i

= 1,2, ···,11),

y = \•(u),

. =!

u

l

(!)

rtXt.

I

In these equat ions. a,1 , b,, a nd Ct a re real constant coefficients o f thform the original system ( ! l into a system ( ! ) surh that the las t column of its c haracteristic determinant ( 2) consists (>f the l'lt'll1cnts 0, 0, · · ·. 0, - l. . t hat is , so that in the dcri ved ~ys tt•cn ( ! ) a ll the cnclflcil'nts a,. = 0 (i I. 2, · · ·, n). Let us tknoty z1, zr, .. · , z.. We can now I' Xpress tlw d trivt:d

Occasionall y, instead of the system ( !! ), one conside rs the system o f differential equations

l~t~m~ (! ! ) and t!!!J will not he trans formed into the ft,rms ( !! ) o r (!!!). If. In additio n. the zero root is not a repeated one, then (!)can be transformed inw (!!!!). . . .4 sysl cm ( ! ) surh /fwl all /he roofs of its c/l(~ra~lerislu: eqz~at~011 ( 2 ), lir i 11 1he lc/1 hal/ i.· plane u•i/1 be mlled a Prmct pal caSl' OJ ( · ), or briejfl·. a fHiwipal case. . . A s~·.~lcm ( ! ) such /flat some of the ronts ,f its rhnrnclensllC equa· lion ( 2) arc ary anti problem. lie considered an equation of "direct control" (Lur'e terminulogy) as a particular case IJf equations of ''indirect ccullrt·l'· !Lur'c tl'l'nlinolog)·). As in the above-mt'lllioned papers by A.L Lur'e and \ '.:\. Yakubl>\ itch, l.G. 1\ltem for which the lar~cst !>eCtor lk,. k: l. inside which absolute !'.lability is ensured. is onl y a portion o f the · ll urwitl scttnr." 11 This. theu ~1 \'f a negative answe r tn thtHHI in I Ill. a ud only tlw prublt'lll o f ideut ifying Lhe cla!-.S or ~y!ill'lllS for whirh thl' answer is a llirmati vt.> remained unsol\'ed. Siug ling out such a l'!a!>~ c)f systcrns \vould 1"\!clun·. in tlwir ras(', the problc:cn or ab*'olult' ~ •abi l ity tu a linN1r probl case for llw systCill ( I I, it is always possible to n•ustruct n positive definite q uadra tic form L (x ) -

"I ..:. , 1 .t , .r ;

( I ,;

l ; ,) •

L(x):::.,\"'TLX

?) (_

ij

such that its dcrivati\·c, computed by taking the t•qua t i on~ ( I J into cou~iclt"raliou .

Lex) = !

i'JL(:c) dx ,

. ax.

tit

.r •

12 lJ l,,x; It a,"""d .

::) L(.lft.,:r,

llll

D =

••

b,y' (!)

j

!\laking use of the Liapunov diJ'CCt method Ior the s tudy or g lobal

...

(tt, _, ·~!

•

i

~- ~ '4til

~· i

-~ -

•• •• •• •• •• •

•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• ••

18

II.

RfSOl VING EQUATIONS OF LUR"£

§ 1. TH£ PRINCIPAl CAS£ OF (I)

stabilit>'· we shall seek a continuously differentiable function I' of the 11 variables x,, x1 • • • •• x. with the following properties: 1°. The function Vis positive definite over the whole n·dimensional space. that is, V > 0 in the whole space with the exception of the origin x, x, x. 0, where it is zero.' 2°. The derivative with respect to time li, computed by taking equations ( I ) into consideration

= = · ·· = = ·

av

V = I -II • iJx, •

·a,.x. + b. ~(o) ) .

is a negative definite function defined on the whole space pi < 0). 3°. lim V = co, that is, the function V increases without bound when th~-Point with co-rdinates x,. x., · · ·. x. goes to infinity (this is the Barbashin·Krasovski condition, see (29)) . The system of different ia l equations (!) has a zero solution x, = x, = ... x. = 0. The existence of a Liapunov function V with properties 1°. 2°, 3° is a sufficient condi tion for the zero solution to be globally asymptotically stablt>. that is, for it to be asymptotirally stable a11d for its region of attral'tion to be the tdwle stale space . Let us now take a Liapunov function of the form

In the case considered CP ~ 0) it is always true that V and since lim L(x) = oo, it follows that lim V = oo also.

...

= L(x) + p~:1p(o)da ,

where /.(:c) =

I l 1,x,x, (/ 01

[ 1 ,}

and

p is

(8)

a fixed rl'al numl>cr.

'·I

As in the linear case, we define the coefficients of the quadratic form L(x) by means of the Liapunov operator

!! I r;(x) I

= L ,

that is. by using the equations (5). Ll'l at first {I ~ 0. Then. no matter how we chou. 0,

.·-

Therefore, when p ~ 0, conditions 1° and 3° are satisfied automatical· ly, and we have only to sat isfy condition 2°. lf on the other hand fJ < 0, then it does not follow immediately from the form (8) of the function V and the inequality L{x) > 0 that V is positive definite (cond ition 1°), nor does condition 3°. However, as w ill be shown at the conclus ion of this chapter, when {1 < 0, conditions l 0 and 3° follow from condition 2. As a result, for {1 ~ 0 or for fJ < 0 we need only to satisfy condition 2°, that is, we must make sure that the derivative V is negative definite. We now proceed to do this. Let us compute the derivative V taking equations (!) into account:

·

V

=

I'

19

= t av ox;

dx · d/

=

t (2 f l uXt + {I~(II)C, II f

fl;jX ;

+ b,(/. f.

r,J.r,

j

(o{! - p t b,c.]-

From (1 2) it is clear that f• will be negative dctinitOSHI\:c tfefJuite. and 2) making certain that Stx'. •)' "> 0 not ti, is tl • 1 1 d •J as a form in ~ lilt Cl.>t'll cnt \"ariables. whereas in faU t' • • hy the rdalio11 aurl \ an· t' ounen mrthod lor k - oo . T hi s method was aclaptt•d lur Lhc finite k rnli""~· in th: I) Fur this pur!)( , g•-~ .lc r clat. how to ~tisfy the as!>ume" 0. then it i'\ not C\'tn pussiblc

I(•

n11 ~ method . ' Lcfschet r., S.. lli,tfer-r.ntial P-lt-101.'111 ~>

we ea,ily ubtain the ident u r:

"_' n,J" ,

,

.

I

-I!•J

'•I

..

''•"J·

T his t y pe nf incqua lit y wa ~ li r:;l 1• bl ai 11 ed l1y :,, Ll'fticlu: tz 1821 for syste ms nf "i ndirect " C011l rl' l. See a lso 1581, 1681. " It is assumed tha t i n~tcad of the .,, tht• f'Xpressit~ns Cl>l ) ;:. n· bei n~: used here. I ) T he cocflil'ients n;. goi and {s are a"~umed values .

to be

~:i ven ancl ll.t have fixed

24

~I.

II. ltfSOlVIHG EQUATIONS OF LUR"E

concentrically compressed. 11 The second apfJroach. Let us recall that we arc considering the case w hen the following condition is satisfied

r

= 1..k -

fi'J.b;c;> O.

(17)

J

Let us construct the square of a linear form in x,, x,, · · ·, x., y in the expression ( 13) for the form S(x, y), so that the remaining terms do not contain y: (18)

Since, by assumption, the condition r > 0 is satisfied, for the form to be positive definite it is necessary and sufficient that the quadratic form below be positive definite:

( 18)

Q(x)

=

I Qux.x; i. 1

=

I

~d

KiiX;.'C;

_l(I r

a;.'C,)

1

THE PRINCIPAL CASE Of ( I)

= '~·. we obtain from (20) a system of 1/2n(n + I) q uadratic ttons tn ~he same number of the unknown variables I equa· . •1 • dIf th . ISf system of quad rat ·rc equatrons has a real solution then the

1•.'

qfua ralht~ hor.m L(~). thus defined is positive definite. orm w I C 1s posrttve definite is G(x)

=

'i.g;JX;XJ

'·i

7I

thus \:~~~al~ show that tbe system of I J2n(n + 1) quadratic equations . .'"e .can be ~educed to a system consist ing only of II quadra· t rc equatrons m 11 var1ables a, . :otr thfis purpoS: let us return to the identity (19). This identity A 1. consts s o quadrat 1c forms on l 't .· It y. PP ymg the Liapunov operator to 1 s rr g 1 • and left-hand s ides, we get:

J

2[Q(x)j =

~ I G(rJI

_!_"(("t" . r)') r ... 7 u,. i ·

bi linunrl y since they cont ain produ('t s a;tt;. Should we asslllnc Rll arbitrary JXISitive definite form Q(x), that is. -.hould we as~ume thl' ccwffkicnrs q;,, tht:n. sinct' q,, q,, aud

~[G(x)J

a."d al-.11 llun-

l\\(1

\!I(~ only. In that c:~~e. the h yi\cl'nrface f/lo,b. d - I k is i\n t>llitlti

= L(.r) = i.J 5' l,,x,x,,

mher Cluadratic forms f I. I or w HC l we inl roduce the nuta· l'IQ(x)J

•• H ,, 0. that is. if the l.iapunov function is a quadratic furrn (wi thout an intt•~:ral''· then the form 1/l(b. (") ~ 1!, 1 ",., ' " 0. and therefore cvn~i,lb of !)(lSi·

{i/; 1r 1

(_?I)

This equation contains the quadratic form (20)

a , a; .

Earlier (see (5) and (II)) il was shown that both the rocfficients Jf, 1 and the coefficients o, depend linearly on the coefficients /, 1 of the quadmtic form L(x) w hich e nter~ inttl the Liapunov funrtion I' being considered here. The cocflicit•n ts q,; depend on the co CJWttead o f the (•quatiuns (26) we ob tai n the "limit" system t•f quadratic cqua·

or, equaling

t ion ~

Subs tituting lhrse ,·alucs of tlw / ,, ir1to l'qua1io11 (29), wtahtlit )- which ran be establh.. lwd hv the tc'ioiVItll{ limit rqnations (:~~) . 1\t tlw l in"' th is book was writtl'll, the relationship :1111011 g t lw se fo u r r(a)(q - (\"(q)/k )), in the expression for the derivative V, and by ensuring that the remaining part is also negative (for any x; and y). We need not have isolated th is ter m (that is, used the S -method); we could bave trans fo rmed the whole expression ( 10) for \i in exactly the same manner as we transformed the form S(x. y), that is, separa ted the square of t he linear fo rm con· taining y from the remaining form Q(x). We then would have obtained "resolving'' prelimit and limit systems o f quadratic equations. dif· ferent from the systems (33) and (3-t) by the absence of the terms c,/2 and the te rms contai ning 1/k. llowevcr. such a sys tem o f prelim it equations. clearl y. would not have any real solution. ln fact. if it did have real solutions. the nega· t ivc dcfmiteness of li as a function of 11 + 1 variables would be ensured . As was s tated at t he ber,tinning o f t his section (see p. 20) this is im· possible. Concernin g limit equations constructed in s uc h a manne r. even if lhcy did have r('al solu tions . absolute s tability would s till not be ensured s ince the prk for res()lvi ug the problem uf abs olu te s tabilit y

system cannot have any real solutio ns. . Thus, for e~ample, for 11 = I. this system o f equations reduces to a smgle quadrattc equation .~

v - fir u -

d:r. til

p

=o

1/-Tr" - ~ J.r + ~: = o . This limit equation has two identical roots

u

=-

l

v - fJr,

and the prelirnit equation has t11e roots ~~ :;: - l

v - fh

I

V2).p'

which . . are a. lways com plex s ince />> 0• •and .1 < 0· Consequen t 1y. t hc l11111t obtained without the • · · dequat1ons fi · . S·mcthocl do not ensure t hc ncga· ttvr c mttrnc~s of the function 1'. ~~~ ~ondud ing we s hould like to make a few re marks about tht' app hcauon o f the Liapunov mC'lhod tn tlw principal c ;t!'f ,. 1ht l>r,rv· rtit•s I 0 • 2" t • • ~ fur• . ·an d •'!

=I

l;.z,z•. C(z) = - 2

' t

l,tZ.-1

p,E -1

= 2fGCz)l,

= L(z) +~(IT - I T

•·(z) t 2

where

~ ~t,y)( 7d;jZ, + s,y)

(.II)

llc te, let

r

z1

iu \\ hu-h 1 he r, arc arbitrary, usf.unwd r:-tl m•rnbt'r!.. .:;inn• rlw •• be 0 • clt'\t't minant of this systern, J ,. is uol 7.. from

the f:rcl !hat t' 0.

(55)

And thus. for finite k (and a l ~o for k >J provicl 0. If on the other hand. k =en with Y. > 0, t hen the condition V < 0 is equ ivale nt to I he requ irement that the fo rm $(:. yl bcrome pns itin' s llltlomajil'llfly I rom f· 0 t hai I' , 0. W e s h:lll ~ how that thi s abo take~ pl:tt'l' w h en ~ < 0. Let us con· Sider the linear 0. We s hall now show that for any fun ction IO(o) contained in the sector (0. k) V-> oo when the point (z 1 , q) tends to infinity. This follows immediately from equat ion (43) with p ~ 0 and x > 0, since L(z) > 0. For p < 0, we must have x > 0 and k finite (see inequality (51)). Tbe inequality V ~ V 10(o) = k, is then always satisfied for the function V defined by (43) with an arbitrary characteristic contained in the s ecto r (0, k]. But II r(ol = k 0 and x = 0 still remains unresolved. In th is case the fun c lion

\' = L( z)

1-

{I ~: l{'(t~)dtt

~~'{'(tJ)tf,t

-

~:~'f'(a)d 0.

(60)

This can be recas t into the Lefschetz condition 11

..~; g~} a;a; < r,

(60')

wher~ g~;·~ = G.;ltl: Ll.is .the determinant whose e lements are the g;;. that 1~, 1t 1s the d1scnmman t of the quadratic form G(x), and G;1 is the mmur co rres ponding to g., in t h is determinant. 3 ' Subst~tuting i~1to (60') for a, and r the expressions from (45) and (53) and . t~trodu;rn~ the corresponding notation, we can now write the. conditiOn (60 ), JUSt as for t he principal case of (!), in the form of an mcquality: f/J(s, t)
0 is atJ arbitrarily small numhcr). it is sttfficietu that there exist a finite real mtmber q such that /or all c.1 > 0 the following inequality is satisfied8

r(,

Re (l

+ iqw) W(im) + ~ > 0

(P)

In (lrdcr that /he system (!) be absolutely stable in the sertor (0, kJ for lite principal case, or in the sector [c, k) for the particular cases, it is sufficient that there exist in the W*(w) Plane a straight line, passing through the paint 011 the ret~l axis with abscissa - 1/k, such that the modified frequency response W* lies strictly to the rigltt of it, and additionally, that for the parli· X cular cases the conditions for stability·in·thc·limit be salisfied. As an example, Figure 7 s hows locii of W•(w) for the principal case o r(!) whic h satisf y the conditions of this theorem (the Popov lines are drawn heavily), and Figure 8 shows locii or IV *(m) for which a Popov I i II C does not exis t with the given value of k, even though the fre· qucncy res ponse W*(w) does not interse~ L the forbidden zone on

(a)

and, additionally for the particular cases, that the conditions for stability· in·the·limit be satisfied. Parallel with the above given analytic formulati on of the Popov theorem, we s hall also give a purely geometric formula tion . us ing tht: modified freque ncy response IV*(w) instead or the fre(Juem:y respnnsc IV(iw). The refore, let W *(m) = X r i Y. The n Re( l + iQ(u) contained in the sector (0, k - t,] fo r the princ ipal case a nd in the sector 1~, k - £,] for the par ticular cases (here c a nd e1 are arbitrarily small positive numbers) . Indeed, once t he Popov line w hich has points in common with W*(id cs o f this scnur: thr c haraneristic rl•'l - 0 brcumes the t· htabk fur 1 (•I) IN . llltlced. I>}' assumption, tiH· nwdi l1•·cl fn·qiif (!) lltr' JIH)dili!•d fn·l pll'l l \ y II'~J KHI:-.1' H ' ( n) 1nav ho

follows from the inequality (P) and from the identity (9). This inequality differs from the inequality (P) only by the sign of q a nd the c hange of ~V(im) to lV(iw). Hence, for the derived system, the conditions of the Popov theorem for the principal case of (!) with q '> 0 arc satisfied. Since the theorem may be assumed to be proved for that rase, the derived system is absolutely stable. But the absolute s tability of tl1e ori~inal system follows from the absolute stabilit y o f tlw derived sp;tcm, s ince these two systems, in fact. are the same. diffcrin~ only in uutal io n. Consequently, the ori~inal system is absu· lutely ~tnblc l'\'CII though the number q in the basic Popov iucquality is negati\'C. Le t us now c:-.:uninc the case q =0. The t·ondition (P) fo r q 0 immediately rcclul'l'!> to the t•rmditirm !P) for fJ > 0. In fact, if the nwdified frCfiUl'ncy ll':.J)(•nM' 11'•(..,) lil"> to the right nf the vertical linr passin~ t hrou~h 1hl' point (- I k. 0). then it also must lie tu Lhe ri~ht of c;onw ... trai!.!IH lirw pu:-..-.in~ through the same poinl and havi ng a !>Uflkicnll~· largt• posili\'l' -;lo pe." Hct·auSt'ltiou applies only tu the 1·a;,c (/ > 0 .

I.

dl

and [ G[f(l)ldt

= c < oo .

Let us assume that the assertion of the lemma is not correct. Then, it foll ows from the definition of a limit that there exists an unbounded increasing sequence of numbers It > 0 such that (k - I , 2,·· ·).

Here we may assume that, for all k. Ik~ I

-

'·

~

111

>0.

since if this condition is not satisfied for the original sequence, it is always possible, by om itting some te rms, to construc t a new seq uence which docs satisfy this inequalitr . ll follows from the fact that the d('rivnth·c d/U)tdl is bounded II d/(1) dl l ~ b), that fur an y I ? 0 1/(1) - /(lk)l ~ blt - I,J .

Then. taking intn account that f;(.r)

k

But fur /,

m ·'J. ·

I ~

1

t

hooc;c

111

0.

\1'- 0, the inequality Re ( I - iquMV(io>)

59

THE PROOF OF THE V. M. POPOV THEOREM

.J(/, )

1.'111

-

c;ufllciently mall, then

t

.f(/.1 ~

0.

n!!!.. 2

Thc•n. from the

lhe Proof of the V. M. Popov Theorem

Auxi lia r y Lemmas

If l illie ronlinuous runl'lion {(!)and itsdr•riYaliYl' d/11) ell we· htllll lcl t>CI ( \ •1' I (), :~ ) tiH' Ctll tlillllllliS [une li(m {;(.r) .l) for auy l.illlllltl

I , C), (,'(1))

1),

~ hould

:mel:~)

[(:jf(l)lt/1

n, thl'll

!il~.l

Jlf,IIIC.

f4

t WI

(; l' ('' 't!'

''" ·

'

wherr

rniu

r

/(1) _ (l,

likl' w r.:mind tlw rcadl'r that the lilllil !JOint II'"' I lies '"' th•· iomoL;ina • I' nx i ~ a nd ltl' lll'l' o·annot be 011 nur vertical line. ~inct- in our c·a!tC k 1 ,. • I· ntlhcrmr.rr. a~ }J;J., alrt·:td)· lwcon pni11tccl nul •see p. ~9o. for 1hc• pri11dUil l t·a~c •tf '1 tlw Cll llll' IIICitliiJccl frt' \\ ,.

Jr·· ~

I.

,

llcun·,

u ~ing:

.,

inequalit y (12), wt· ••llt a in

[ c:I;U )Idt whid1

0.

(,'( 1) f ..

r t•lll radic ts

thl' assumptiunc; of tlw

co not slol 0. Let x ;(t) (i = 1,· · ·, n) be an arbitrary solution of the system (!) with an arbitrary fixed function ~(11) contained in the sector (0, kl, where

1 2;r

= -

r : F,(iw}F3 (iw}dw .

f~ !H(iw) F,(iw) + F,(it,))J F 3(iw) dt" . J_co

is a fixed functi on of time and, in this sense, xk (t) (k solution of the linear nonhomogeneous system dx; dt

i

j

l

V-r (t) . -_

{ y(f)

where T is an arbitrnry fixed

0

=

2;-

f '.,,

.

.

J__ 1/ HeH(lw)F3(tw) +

-

F. (it•>) ,. 2vHef!(illJ} dw -

-

1 8:r

1'..

I [ ' "' I f~(iw)i 2

Brr

J "' ReH(iw)dw .

" Wf!' s hall denote this property as follows: f T,

(19)

-

number, and let the function,

w

)_ F , (tlil) . + 2 !/ Fl~2eu,.,> ][-/ . -.. /!(im) v Re H(tw)f 3 (t~>~)

= p [a(l)l •

y(/)

2

+

(t) .

Here

Since lhe left·hand side of this equatio n is real, the imaginary pan of the right-hand side integral must be equal to zero. Consequently, 1 f "'/,(t)/3 (t)d/ = -2 f j"' Re[H(iw} I F 3 (it,>) l'+ f',(iw)F,(iw)jdw Jo 1r ) ... ro

=

Ct Xt

k~l

Substituting t he expression f or F,(im) from the ide ntity (13). we obtain,

f., Jof,(l)f:,U)dt

.

=I

a(/.)

I c. Xt-r (I)

k

,

I

be defined by the solut ion of 1he linear unnhomnge ncous syste m, dx;r

-;it =

,

.t.. flijXj-r

+

b

;y'l'(t)

(k = 1, · · · , 11)

with the same initial conditions as for the func tion 11(/): (j = 1, .. ·, n).

(20)

•••

•• •• •• •• •• •• •I ••• •• •• •• •• •

•• -

••

•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •

62

we get:

Then it is obvious that o,.(t)

= o(t)

0~t

fo r

~

F ,(i01)

T .

A s a resul t of a property o f linear nonhomogeneous s ys te ms o r(l)

= i1r(t) + p(t) ,

= [o+ iqw) W (iro) + ~

ar(t)

+ f)(t) .

(22)

+ Qti r(/) -

F ,(iw)

= H (iw) Yr(iw) + F s(iw) .

Let us note tha t all the cond itions o f Lemma II are satisfied m t his identity , s ince fro m t he m a in a ssumption of th e t heorem be ing proved (see Remark 3 on p. 54), we may ded uce that the follow ing inequa lity is satisfied

In addition. f,(t) =-~ 0, /s(t) =~ 0 , and Lemma II,

k

Vr( T)

· = arU) + qiJr(l) -

-· -k- ·I p(/)

+ q{l(l) ·

(23)

-

ft(/) -

.

- 0 .

y,-(1)

- [ [.(1) _vr(l) dl ut.. a. In term'> of the new nutation. rquatinn (231 may I'K· wriu [u- ¥ ] dt + q

M =

rrl

~a) du ~ c•.

(25)

~~ ~ 0 is small. This linear system is

=

dx,

I~,c ( I

. ) II' .(/HI) . ).. kI ~ "• - >0 I Ufltl

.

In this last inequality. it is possible to change k to k -t 2s (c ..., 0 a suffiticntly small number) after havin~ det' inl'qu.1loties ca n bl' mado: :~rhitrarily s mall t.tnif~~rml y in ,,., bv d1oosing a sufficoently s mall 0. In

C:~n,

. .I

I a.,x, -t Eb,IIOW$ t hat for ~ II

I W(i,..\1· K . for

dt

+ ~ ':: ii > 0 •

follows the inequality ...

=

Conditions for Stability-in-the-Limit

For c > 0 small, the functions W1(iw) and W (iw) are not very dif · ferent. and hence from the inequality Re (I

67

CONDITIONS FOR STAI ILOY·IN·THE·liMIT

1

•

I i,,,w (i,) I < K, .

0 ~111:111. the inequality K · l / 2e is sa ti~fio:d.

Then u The stabi lity of the systern 133\ for q>1(,,) - 0 also followo; because the CO·

l f)J

t>fficirnts

a::•of

I his

s ystem differ lilllr frorn tht.• nocflicients

ll ii

fo r , . 0 s mall.

l

Il

l I

6S

Ill.

§ S.

THE V. M. POPOV METHOD

Theorem: For a particular case of (!) to be stable·i1z·the·limit. it is necessary and sufficient that each root on the imaginary axis, iw, . (wt is a rea/number) of its characteristic equaliOil have a multiplicity v o not greater than 1wo. and that the jollowi11g conditions be satisfied: a) for Wo = 0, llo = 1:

-··

lim Jm W (iw)

.. for w• 1= o,

=-

oo or Im W*(O) < 0;

lit = l : when m traverses the poi11l

II

-:

r,

"'

I

c,w - ....

(40')

...

--------------------------------------------- - 1 Ill.

72

THE V. M. POPOV METHOD

~ S.

c,.

Condition c acquires the form: r. > 0, and, in the series c,, c~ , - c1 , •••• the first no nzero te rm must be positive. Case D: w 0 0, v0 = 2. In this case it is easy to see from t he equations (40) that the coefficients Ck dk + ie. (k = 0, I, 2,· · ·) in the expans ion (38) must satis fy t he conditions: do >" 0, eo= 0, and t hat t he 1 first nonzero term in the series d,, -er1 , - da, - e(, d,, - e, , - d, ,· · · mus t be positive. Remark 4. The case D above (w0 ~ 0 , 110 = 2) of the theorem on stability-in-the-limit cannot occur for a frequency response W(ie01) which satisfies the Popov condition (P) (even if it is weakened to :2:. 0), since the locus of W*(w) cannot lie to the right of the Popov line if wo :t= 0 and v0 2.u Remark 5. We s hall now establis h whe the r the branches of W*(w) goi ng off t o infinity (fo r w-> w.) ha,•e asymptotes and, if such asymptotes exist. where they are located. ln Case A ( (110 = 0, v 0 = 1) there a re no branches goi ng off to in· fin ity, and case D has al ready been eliminated (see Remark 2). Hence \\ e need consider on ly cases U and C of t he theorem. Ca~ R. (w0 > 0, lJo = l ).u For the real and imaginary parts of t he modified frequency response, that is, for the quantities X = U and )"= wl' , mak ing use of (39), we obtain the following expansions about the point (II = (l••:t•

*

=

=

X

= _r_,_ + d, + ... vl -

w0

(d~ 1- I'~ > 0)

\.

- -

d•"'• --I

(II -

Wo

Como

( II )

I

l•

1- ·--.

d,....

"I' to i nhuit c~imals

1

-

tI 1

-+ (d• -

the functions W (i 0 k

+ iqm)

Re ( l

.~ ), f •

and conversely.

~~~ ~rticula~, if llze slrrngllzcned Popou condition (/' l or (zw), that IS, the striclnl!ss of tlzr inequalil\• p . • is satisfied for (JJ = oo • then it is also satisfied for the :. ( ) .'s pre~rvcd even quency response w (iw) The . . . • Simplest parucular fre· 1lml 1 pomt W (ro ) f h I • . QliCOCy response JV; (w) then lieS 1(1 tl • ,• h I f 0 t C ffiUJ)

( P ,)

-:~ 1 ~~

· '· . lint

•

I . Now, 111 a I. • Ill I II'\ I' CISl'. ti)U, • ar•< that tht• Prop' '" l'ondition \'l'rl '

•

the condition I

R(' '"' IV,(it.o) ' 0 .

whih• the sln:ngthcned Popov ('llllrlit i•rll ' ~··lt'l :1: 0, the magnitudes and the orientation of the angles between arc originating at p. are preserved in the conformal mapping. For the root locus of /J0 (t) to lie to the left of the imaginary axis in the />-plane, it is necessary and sufficient that the above- mentioned segmen t of the negative real axis of the z- plane lie to the left of the locus of W(iw). When this condition is transfered lo the locus of W(iw), we get condition b of the tbeorem.u For w. = 0, condition b becomes condition a, since, in this case, the I wo infinite branches of t he frequ ency response W(i111) , for , < 0 and M > 0, a re mirror images of one another about Lhe real axis. Now, from the s ix cases, J.ll, 52, 54, 58, 72, 7tl (see Fig. 11), there remain onl y two cases, 54 and 72, in which

= - oo. IV(/1 = ~i"(p.) :: 0,

lim Im W(i(lt)

• - ., o

0*

Case 2. vo ~ 2. fn this case but ~V"(/J ) 0, 0) and the double root p, of the equation (50') (fur ' 0) separates into two dist inct roots for£ > 0. We obtain twn branches in the root locus

Rez

z-ptane

p - plane

p-pfone FIG. 15.

of /l( t), /11(>. ) a nd /J2(f) (sec Fig. I5). FIG. 1-4.

30 It i s c l e~ r· l' rtlm what ha R S w0 and for c.1 < u10 ), which is 2:r, must contain twice the segment of the negative real axis. This is possible only when the branch o f the locus o f IV(iw) (for 1" > (tlo) is tangent to th e negative real axis from al>o ve while the other branch (for 111 < lllo) is tange nt lCI the nega ti ve rea l axis from below (see Fig. 15). ln a ll Olhe r cases, e ither one or both branches, p ,(e) a nd p.(d. of I he root locus lie to the ri g ht of the ima~inary axis. Proccccling fro m J.ir (il,.) to lmW the f rcqucncy response IV (i,l), we lind that the location of the branches o f this runc tion is as s hown in Figure 16, which cxReW prt•::.St'S ~-teometrically condition d. For t••• , 0 this condi t iun be comes nmd it ion c . Case ."J. ~~~ > 2. In t his case. FIG. 16. hran rhcs o f the roo t torus t·m:,natc from the point Po ;,•• o f t hl' p110 plane. Si nce tlwy all map c·onfor mally into a segment uf the ncgati\'e rea l ax is of the z pla ne and, si nce in the case under cnnsidcrat iou. a ll t lw angles become mu lt iplied by vo in the conformal mapping. it follnws that the direct inn nf t he br:ln 0). ft is no t difficult to s hown th:~t the function G(p) will be a positive (or, respectively, strict ly positive) rea l fun ctio n o f the argument p, if and onl y if, the fo llowing conditions are satisfied : I. For real values o f p t he func t ion C(p) takes em rc·al values cmly. 2. T he func tion C(p) has no poles lCI th e ri gh t of the imagina r y axis. 3. On the imaginary axis t ht• function G(p} can ha\C unlr :-.implc poles w ith positive res idues. ·1. The inequality Re C(il••> '· 0. (or, > 0, respectively) holds fo r all values of 111 (wit h the 0.

Assuming m = 0 in the inequality Re j.,, W(im) > 0, we immediately o btain the condition I, > 0.

*

lf, H'(p) for p 0 is equal to 10 • / 0 > 0. Thus we have obtained all the conditions for stability-in-the-limit for the case q = oo .

=

Repeating t he above- g iven reasoning in reverse o rder, we can also prcwe the converse: name ly. that the strict pos it iveness o f the function (l + qp) W(p) + 1/k follows from the Popov condition and from the s tabi lity· in·t he·limi t of the pa r t icular case of (!}. From our presentat ion we deduce t he following ne w formu latio ns of the above proved Popov theore m (see f78J).

or

f/ the f unction W(p) has an index fl, where p. is the number chamt:tuisfit roof s (lhdr multiplicity must he included) mt the imaginar.v

axis of the system (! ), and if /or some finite real values of k rmd q th~ /unction G(p) = (1 + Q/J) W(/J) + 1/k is' a strictly positiz1e function of the complex valued argunumt p, t/um the system (!) is absolutelv stable iu /he sector [0, kl .for the principal case and in the seclor [£, /; ) ((1)£/h ~ arbif rarilv small) for llze partimlar cases.

•• •• •• •• •• •• ••·t

.;.j ,!.

:I•.. ~~

' •

. ..

•• •• •• •• ••

•• •• ••

•• •• •• •• •• •• •• •• •• •• •• •• •• •

82

Ill.

THE V. 1/1, POPOV METHOD

S8.

The Case k = ex>

§ 7.

So far we have proved the Popov theorem for finite k only. We

shall now show how this theorem is extended to include the case k = • Let us consider the principal case o f (!). First, observe that in 09 the proof o f the Popov theorem for k finite, we s ubsti tuted t ile in· equality (Pa) for the inequalit y (P ) (see Remar k 3 on p 54). This was possible because.the limit point w •(oo) did not coincide with t he point (- 1/k, 0) since by rotating the Popov line, it was possible to make · sure t hat the limit point W •(oo) did not lie on the Popov line. All these arguments remain also valid for the case k = oo, provided w•( oo) If the limit point does coincide with the orig in ( w•( oo) = 0) • then. s ince fork = oo, the Popov line always passes through the o rigin, it is not possible to substit ute the inequality (Pa) for the inequality

* o.

(P).

The refore, for the proo f to re main valid for the case k = , we mu:>t im pose the additiona l require ment that the inequa lity W " (oo ) 0 be satisfied, that is, (see ~ I) that in the rational function lV(p) :.... K(p)/D(J>) the difference between the degree of the de nominat or and the de){ree of the numerator be equal to I (11- m = I ). If this condition is satisfied. then. since ll''(oo ) =- iB. (see§ 1), B, ~ 0. But then the necessary condition for absulute stability in the sector (0. oo ), that is, stability of the linear srstcm nbtaincd from(!) for ~'(11} 1111, 0 .- , ... ro , requires that. for any ~urh h the equation

*

/)(fJ)

+ hK(p} -

/>" HA.-1 1!80 ) p• + .. ·- 0,

He ( I 1 iqo•) ll'(i"') ._,. 0 .

0 rmd all "'

>

0, nnd, in addilinn,"

lim iw W (i o11 Pc · · n examples of th 1· · >IX"' erne non to problems f h' . . . e app tCatton o f the V. f-1 Tl o t IS ktnd wdl be . . te application of the Po v . . g tven here. ~Cncral form (b\· S)' tnboJs a d po Crtlerton to problems Sl)(•cifit'(J .Ill .., 1 . · n not by 1 . " i\nt ':" 0 t I1e cond.rtron 1 rty·m-tle· 1

c I l (see pp. 79 80) . We uow lind the frequency response W(iw) : '' ( . ) _ (U' m (1-1) is s table f111 the lini.':tr dwracterist ics ~· (o) - lw contained in the sector (0, ~). where

c h

Y = whe re u

l

"'·

I rum

W

+

--

U(U

'

Ca!'r 2.

I• '

r•.

I f I 11 t I"." t':t!;i.' llC nrct .•on ric• .;...;..._--:b) ,.!.. .rs a I ways 11

I r

+ c') '

nc~a-

1

b r - - fDr 11

1iv(· :uHI :1!--'iumes 1he value

('

-

0.

l l· • ~c IV*( oo .

X

In order to determine if (!>-1) is stable·in·the-limit. we compute the n·sidue 1

is always posi·

b)

u 1 c'

.

Thus we find ourselves confronted with a particular case of the system (!). T he characteristic equation has the roots

= - i;

b)

-

+ e'

b < c•. In this case the fraction r(c• -

1:

tive and tends to zero for u

p•- b

2:

• 11

1

Let us examine three cases separately.

/> 1 - b.

lV(p)

= _ e + c (e

X-qY = -e u+b

1

. 0 impossible for any k and corresponding values of u = w•.u And so, for q = - c

= K (p)y ,

= p• t· ep' + p + c,

K(p)

u ; r• ·

I

=

=

d · ie

+ qu ..!UJ!_

11-

which depends on two positive parameters: b > 0, and e > 0 . Note that o x , in (54). T o find the fwlCtion W(p), we let cp(u) y, and then, by elim inating x1 and x1 , we obtain the resulting equation

Conseq uen tl y

EXAMPLES OF USE OF POPOV CRITER ION

" The rq uality q - - r follows (rom the cquatinn (43) sincf'

tl

, b I l ·.::~ 1·

e

r b 1 I 2~

:llld

""'

I .

.:1\'. \

I.

~1 .I

~ t..

./

•••

111. THE V. M. ,O,OV METHOD

88

is satisfied only for

1 b - ca ->c; - - k c

b

IV

c

t h at ·IS, for any k which satisfies t he inequalit y

k < .E_ = g. b Thus the Popov theorem guarantees absolute stability i~ the sector lc, c/b _ t l where e > 0 is arbitr~rily s mall. T he system IS s table fo r linear c haracteristics contained 10 the sector {0, c/b). Co~ 3 : b = c:. In this case, the entire modified frequency res ponse lies on the straight line

X -q Y +c

= 0,

THE CONNECTION BETWEEN THE POPOV CRITERION AND THE EXISTENCE OF A LIAPUNOV FUNCTION. FURTHER REFINEMENT OF THE FREQUENCY CRITERIA FOR THE ABSOLUTE STABILITY OF THE PARTICULAR CASES OF (!)

a nd the Popov cond ition is satisfied for a ll k < l fc. The li~ear c~arac· teristics for which the system (54) is stable are also contamcd 10 the

sector (0, 1/c). The location of the modified frequency response and o f its asymp· y

'

b X

(a)

tote ( for t he Cases 1, 2. 3) a rc s hown rcspccti\·ely in Figures 17 a. 17 b. and 17 c. The al>o\'C results aJ·e idcnti· cal wi1 h 1he ones obtained b) Pliss (501 by a different met hod for the equations (5·1), and scn·c m; a 11 example wh ic h dis provt•s the h y pothes is that the sector or absolute s tability coincides wit h the llurwitz sector (see Chapter I, p. 12).

'

y ~

~

~

''

'

-c

''

X

'

',

- l

'

{b)

y

' - -b '' c

X

§ 1.

Introduction

In Chapters II and 111. we cunsidercd two different approaches to the problem to absolute stability. In Chapter II , the me thod o f Lur'e resolving equations was u~ for attacking this problem for the principal and the s implest particular cases of (! ). The Liapunov functi on used was of the type: "a quadratic form plus an integral of the nonlineari ty." It wi ll be reca lled that. in lhc parameter space , the region o f s tability defined by lhis method depe nded on the c hoice of the quadra1ic form Q(.rJ > O a nd that t his region was dt•fincd by means o f a corres ponding sySl't:' nl or quadratic resolvi og equa tions. Assumi ng that ()(x) 0, IV(' obtained a "limit'' ~ysw rn o f rc~ulving equations. The qut•s t ion of whether the region of stabi lity defined by means 0 was not answered.' In this chapter we l'hall examine· in detai l not onlr the principal and simplest panicuJar cast'5 o f (! ), bul a lso all the othc1 pankular caM's whuSt· ll'n 0.

The second variant of the S·method (used for r the nonnegativeness of the form S. that is

S(x. y)

> 0 is a fixed positive number, and

93

a; = 0,

J

( 4)

since a nonnegative quadratic form which does not contain the square of a particular variable does not conta in that variable. Hence, S(x, y) becomes reduced to CCx). Making use of the second requirement, we obtain

In more compact notation, the exprcssiou (4) may be written as follows

= G(.r) -

S(x, y)

I a; x; y + rl ,

2

G(x)

( 5)

i

a; = 0,

where G(x)

a,

-2

r

GCx)

f ( ~ 1, x t a,.x.),

(6)

1 1) (

~ 1,1 b, + ~ ~a,; c.,+ ;

= .;:_k -

(7)

c;,

fl I b1 c, .

( 8)

J

In Chapter 11, we described a procedure for constructing a Liapunov . . function called the S·method. Two variants of t he S·method were described: the fi rst vanant tS applicable for the case r >O. and the second for the case r =-. 0.' . . The first variant of the S·method requires that the denvaltve V be represen ted in the form (3), where S(x, y) must be ~)OSiti ve definite when considered as a quadratic form of the 11 + 1 vanables x, and y , that is , • In Chapter II . S l, we assumed thrtt r = I. t Let us recall that r :. r/k - f1 L. bJCJ 0 a lways, if a) k = oo and •

=

=

> 0.

The conditions

2;.J 0101

=

clearl y, define the second variant of the S·method (for r 0). When we say that a Liapunov function of the for m (2) can be constructed by means of the S·method, we mean that the function can be constructed either by means of the first or the second variant of the S·method, de;x:nding on whether r >O or r - 0. The connection between the Popov condition and the existence of a Liapunov function of rhe form (2), constructed by means of the S· method. is established in the following theorem. Theorem: Tire Popo,, emu/ilion CP> ;s a llf'Ccssary and suj]icieul condition for tire existence of a Liapunov fu nction, of the form (2), which um bf' collslruciCll by means of the S-method. Before proceeding with the proof of the the01·cm. we sha ll prove the fo llow ing preliminary lemma. Lemma: If the posili11e deji11ile quadratic form K(.r) is represented as thP. sum of products o( linear forms /,(x) and 1/h(.r)'~

0,

which indi cates that n - m> I, or b) k oo rtnd it is assumed thO by a suitable cboice or the scaling factor r.

(10)

> 0.

K(x)

=

:L. l,(x)m.(x) , >

(II)

then for m1y complex values x,,x: , ···,x. , not all zero, the follou•iltg 10

The number of term s in the s um M the ril( ht hand s ide o f (I I) is a rbitrary.

••

•• •• •• •• •• •I , .•• ; t: • -•• •• •• •• •• ~

•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •

94

IV.

§ 2.

THE POPOV CRITERION ANO THE EXISTENCE Of A LIAPUNOV FUNCT ION

incquulity holds K(x, .f) - Rc

Let ~s apply the lemma to the quad ratic form Sex. yl making use It is then possiblr to asser t that, for any x1 and y • not a ll zero, t he following i ncqua li t r holds

t~f equatton (.J ).

(12)

! l.(x)m,(i) > 0 . >

Proof

vl the

Lemma: The following idc n titr holds w he n the variables Sex. y; i. j)

= u1 + i11i arc complex: K(:c, i) = Rc ! l,(u + iv)m.(u- iv) = Rc ! (l,(tt) + if.(v) ) [m,(u)- im,(1•))

x;

•

-= !

[f,(u)m,(tt)

.,

+ l,(v) m,(v)] =

K(u)

+

95

THE PRINCIPAL CASE OF II)

= - Re [ !

p

+

( 13)

(2!I I" X'

-1 {k,y)

!!4 a,4 x4

1 h,y)

T((1- ~)J ]>0.

( 14)

Le t us now choose values x, and v. not a ll zero. such that the following system o f linear equation!> is ~tisfiedu

/\(11) .

Here K(u) ;:- 0, K(v) ~ 0, and at least one of these quantities is positive, since. by assumption, the 11 variables x,, and hence also the 2tt variables , u;. are not a ll zero. Hence the inequality (12) follows immediately

(15)

For these values o f x. and y (see Chapter Ill , § I)

11 1

from the identity ( 1 ~). Ncmarll 1. The ex pression K(x, .r) in (12) is defined uniquely by the quadratic form /((x) and is independent of the manner o f reprli'SCnt· ing the quadratic form /((x) in the form (II). This follows immediately

.

= L a,. x, + b, .v .

ir11x,

a -

Substituting the values or

-

(15' )

W(im)y .

x,, nnd .v lhu!i ddincd into I he \:X pression

(I ll) for .)'(x, y ; i, _ii) , we obtain

Z.) '1 /.• .1

from Lite identity (13). In p:utinalar. if /((;c) -

..

I

,

(II)) SH

- Re [ 2 ;., 1:1, x,

x, x• .

1 (:-

1 i(1w)•7."

~· I

thrn"

But th(• ex pressinn

,~, :,r,x,.x" . For n·al ,nlut•:. of tht• \'arialllc:.

X1

•

2 1.1 x,. i,

the expre~sion K(x, x)

Uc(:\ltliCS

Rl'matk :!. t\ ~tucrali zation of the lemma abo f,>lluw:. from the i 11l-ntit' (13). If /\(x) i-. a positi,·c quadratic form. Ko) · 0 for rl'nl values .uf x 1 • then. for the t·mnplcx ,·ariables Xj = u, + i1•1. J\1.\ ,

llt- n · the l'qualit)

I

I

0 .

·~·u hotelr I .

11,,1 I 1,

r...

1 •

We now wake usc of ano1lw1 IIH.! OI'Plll uf Yakubn v itJlUIIcli n~:t degree:. (I( the denominator :1nd nucncr;,wr uf the rationa l fu111 1iuu lllt p j K I'IJJilJ< p)

+'

100

IV.

THE PO,OV CRITERION ANO THE EXISTENCE OF A LIAPUNOV FUNCTION

S2.

conditions d dt [L(x)).

< 0, I 1;11 b11 + n = o.

•

(26)

for tbe principal case of (!), it is necessary and sufficient that the following inequality hold for all 0 Re I (a;, - i00;,)'-"b,r;

> 0,

(27)

j./1

where il 111 is the Kronecker delta, and that

I OJ,bpTJ * ,,,

(28)

0.

The inequality (27) is identical with t he ineq uality (25) for r = 0, and the inequality (25), as has been shown above, coincides with the Popov condition (P). T o show the existence of a quadratic form L(x) , and thus of a Lia punov fun ction of the form (2), we must still show that the COil · dition (28) can be satisfied in the case under consideration : 11 - m = 2. But this is a lways true. Indeed, suppose that for q = {3/r, the follow · ing equality holds true

I a;,b,r; /•~

= 0.

Then, substituting here the expressions for

n.

I•P

•

101

of the function W (iw) into a series of powers of 1/w." For 11 - m= 2 this coefficient is not zero. Hence we may consider in the case k = oo, n - m = 2, that thl' inequality (30) is satisfied." But then, as a resuJt of the second theorem of Yakubovitch (see p. 99), the Popov condition is sufficient for the existence of a Liapunov function (2) determinable by the S·metbod (in the given case, by the second variant). In the third case (n - m > 2 and k oo) the system(!) is not stable even with linear characteristics IW, for h > 0 large: and obviously, it cannot be absolutely stable.'' Hence there docs not exist a Liapunov fu nction for this system (!); nor can the Popov condition be satisfied, s ince this condition guarantees stability for linear characterist ics.

=

" Indeed, lV(p) is defined by the system of linear equlltlons

px, =

L 2. the lc.cus of W(im) mu st inCN~ct·c Cht• r ea l axib.

=

•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •

·-----•

••· •• •• •• ••

•• •• •• •• •• •• •• •• •• •• •

102

IV.

There fore, this third case is completely irrelevant to tlte given theorem, and thus the theorem is proved in full. We observe a t this time that the above t heorem also extends the Popov criterion to the case k = oo, rt - m > I. At the conc lus ion of Chapter Ill, the Popov theorem with k = oo was proved only for 11 - m = 1. Thus, for any k (ji11ile or infinite), the Popov cmzdition (P) is sufficient fo r the absolute stability of 1/ze principal case of the system (!).

.

The above proof of the theorem allows us to arri ve at addi tiona l conclusions about the properties of Liapunov functions of the form (2) determinable by the S· method. These conclusions depend on the direction of the Popov line: 1) If the Liapunov function in ques tion is a quadratic form only (without an integral, that is, of the form (2) for p = 0), then for such a form to be determinable by the S·method, it is necessary and sufficient that the Popov condition be :>atisfied for q = 0 (the Popov line is " vertical"!). 2) Aualogously. for the S·mcthod to yield a Liapunov function or t ill' form (2) wi th f' > 0 (or, respecti vely, {3 < 0). it is ueces:->ary and sufficient that t he Popov condit ion be satisfied for q > 0 (or, respcct.ively, fur q < 0). In conclusion, we s hould like to draw attention to the fac t that . as a resul l of the a bove proved thcm·em, the Po!)('v C()ndition applies to a ll thoSI\ and onl y to ihose, ~ystem s for wh k h t he S·mcthod gives a l.iapunov functiun ~>f th(' form (2). llowe\·er, for t he pri ncipal case of (I). then· exist Lia puuov fu nctions of the form (2) which cannot be dl'ttrminc-d b y tlw S.methocl . Whe th ilin~. that 1s •

fil'

·I

~

Nott• that tht' expression (12) fur I' is linear

111

(49)

in the form S(z.

y)

is always non·

:::: 0.

(50)

Thus, in order tu obtain all tht.' l.iapunov function s of the form quadratic for m plus an intl.'gral") with a ncgati,·c dcrivati\'C, it is necessary to impose m linear rela tions (33) on the crJefficients f, a nd It', and to require that the inequal ities given below be satisfied: (:!:~) (''a

· - 2(11'p-

! s;f;) :::: 0, v - Pt> + ; -

(,9 + 2;'' )p - ; ~ s.j,

· 0.

T lw quan ti ties w, f,. {J. anci I, , . ~'lt isfying the abn\'e inequalit ics. as W(a) contained in the sector (0, k). Then, for either k fin ite or k = oo and :- = 0, t he quadratic form S(z. yl · 0 is nondegenerate with respen to the variable y. If, how· ever, k - ru and " > 0. the n, for an y form S(z, y) ~ 0, the deriva tive f· is nondegenerate with respect to the va riable a. In t he next section we shall s how how the Popov condition ({') follows from either the positive definiteness of the for m S(z. y) or from it:. nundegcneracy wi th respect to y. On the other hand , the weal..cncd Popov condition CP ) (with the s ig n :;?:. ) al ways follnws from the mere po..,itiveness of the form Stz. y).

§ .4 .

Form Plus on Integral" for Particular Cases of { ! ) with a Zero Characteristic Root. 111 ordt>r to obtain the Po1X1V ineq ua li ty for the particular l'ases of (!) in a manner similar tn the onc used for the prindpal c-;1se. we must apply lhc lt•mma proved in §2 to tht' quadratic form S(z. y). after it hns l>t•C'n suitably t ransfunned.

~

fJy) ( ~ CJ Z; - py)

-! ;(fdv;/,

1- wes)z;y +! ( wp - f s,f,)l·

Making use of the relations (33) and the notation (44), we obta in the fo llo wing representation of the form S(z, y) as a sum of products of liuenr forms

Scz.y)

= -2~( il,.z•)( - 2(

~ _f, Zj +

fd,,z,+s,y )

+/i)') (

~ 0. Then, as a result of t he le mma for arbitrary complex values of z, a nd y. not all zero. the fo llowing incqualit y holds Sfz. y: z. j) ...,. 0 .

(:,{))

where

z..vl = Re [- 2~(tt..t zt)( fti"z; .-s...v) 2 ( ~ /; z; + ~ ~.v) ( t ~'• z. - r•.1') ~

S

+ iqw)W(it:~~) +

0

(- oo

< w < + oo).

(P)

We have obtained the Popov inequality from the a ssumption that the form S(z. y) is positive definite. llowe,·er, the entire derivat ion also remains valid when the form S(z, y) is only positive (S(z, y) ~ 0), provided it is nondegenerate with res pect t o y(S(z, y) > 0 for y 0). Indeed, in this event, for complex z1 a nd y

*

But

S(z, y; z, ji)

>0

for y

1!

(65)

0.

But, in deriving the Popov inequality, we assume in the inequality is a real quantity.

(65), values of z1 and y w hich satis fy tht: re lations (58) with y J:. 0. Observe that for both S(z, y) > 0 and S(z, y) ~ 0, in the event of nondegenerncy of S(z, y) with rt.'spect to y, the rocflicicnt v of l in

Hence

Re

l-

2im! l,., z,z.]

•••

=

0. It then lakes on

the fo r m: SL: y. (\s a

,. /. :;, (tt•t• -

Ch:Ht~ing

In

-l

2:, s- .t -.v - -

*$~/,)

y -

w (~

- t•Yl

(n2)

[iu•. dl'(it••l- ; ].v.

:- He [l. inl\t~·ad elf 1 ill' sign , Wf' \\!Ht ld get the s ig n >- , that is , we would obtain llw wt•ak••uc·d Popov l'ondil iun

S ub!'tituting this c; xpressio11 inlo (61), we ohtain:

Rc [ 2iw1•1 I ll'li•·•) 1" 1- (r

(66)

Thus, in lhe cases cctn~idercd. wc ubt:lin not only the Popov condition (P), but e,·en the s t ren}{thened t'und ition

0 small) .

(78) •·

'·~

lie nee

I

..

lim zJ(t ) -= 0 .

,

But a no nzero solution like (77) cannot decay for 1- •oo. zJ(I)

Hence

=0 ,

which was req u ired t o be proved .

Th11s, wizen llzc CQndilions for slabilily·ill·llzC'·Iimil arc salisjicd."' llbsolulc stability for I he parlicular rases of (!) with a zero clwracteristic root is ensured hy the existence u/ a Liaf,mov f unclion ll'lzich has a IIC/lalil•e derivative V s 0 willz a m inimal degeneracy (prot•ided c01ulitions I" and 3° arc satisfied). On rondition 1°. Not e t hat it follows from t he negativeness of t he deriva tive V. which has a m inimal degeneracy. that Lhe funct ion V

(76)

of t he form (32) is posit ive defi n ite. Indeed. t he Popov condit ion (P. ) follows from t he 11ondegeueracy of the der ivati ve V ~ 0 wit h respect to u for fi n ite k: and. for k = c-..• • t lte P11pov condit ion (P. ) similarly fol lows.•• For ~ysl erns (!!) wh irh a rc stable ·in-the·limil , the condi tion (P. ) g uara ntees, as a resul t o f

Wh 1• 11 he int rudt~co•d thi~ c 0 .

(80)

0, fJ > -

On rmulilion 3".

~

0, fi >-: 0,

IIJ

+2 f

/ z,o + (w + ~ {Jk)ot > 0. 1

(81)

= .. (crl

=

L(x) + 2 I fJzJO

+ Wot > 0.

and hence condition 3° is satisfied again. This reasoning ceases to be valid for {J > 0. A more carefu l a nalysis of the structure of the function V, whicJt will be carried out in the next section, will show that for {J > 0 and w > 0, the quadratic form

+ 2 Ij / 1z1o + wot,

(82)

will always be positive definite. Then this quadratic form, and hence also the fun ction V (since fJ > 0), w ill tend to infinity when the point (z~r a) goes to infinity, that is, in this case, too, condition 3° will be satisfied. Now let fJ > 0 and w = 0. The quadratic form (82) is always greater than or equal to zero. This follows from the fact that the function Vas defined by (32) is positi ve for ~(u) = t/1, where e > 0 is a small positive number. But, for w = 0, the quadratic form {82) docs not contain or. But then, ooJy if a ll the coefficients of the terms contain· ing products ziu are zero, will it not assume both pusiti\·e and ne.l{a· ti ve \'alues, that is, only if

/, = 0.

k2w .

(80')

In panirular, it follnws frnm this that w and fJ cannot both be zero at t lw same timr. If k , , then the inequality (80) can be satisfied f(lr ;'Ill h 0, provided IU

v..

L(z)

I£ k is a finite number. the n this is possible only when /II "C:::

L{z)

Hence it follows that V-> oo when the point (z,, a) goes to in· tinily. If {J = 0, then

I •* • 2'' o ,

where 0 < ,~ ~ k. We have thus shown tha t for the particular cases of (!!) considered in this section the condition 1° is a consequence of the condition 2°, even in its weaker furm. Note also that the positi\'eness of the coefficient of o• in 1he c1uadra tic form (i 9) follows from the condition L0 • that is, from the inf'quality I' .> 0. lienee. for 0 < h ~ k,

117

note that for fJ < 0, the number k is finite (see (801' )), and t hat for any characteristic p(u) in the sector (0, kJ

=

r..Jo

MINIMALlY DEGENERATE DUIVATIVE

1- fi > 0 .

(80" )

(The 11arbashin· Krasovskii cond itio n).

Lcl us

v.

3 1 In ch-ret, l( at so rne point V assumes the \';llue < 0. then a long the solution c ur ve stllrt•nlt fr om thi~ poi nt, we ha"e V :::- Vo. Hence the con1'''" " lim V(t) 0

·-

can not he ~ti sCied , si nt·e this condition is a l'Onsequence o f lim :r:(t)

,_ 0.

T hus, in the case under consideration (8.1)

and condition 3" is satisfied. provided the rlt~u·:tcterisl ics y>(o ) s:tl is fy the add i I iona I restrictions f1 ·~ 0 be satisfied, a nd that in the expression

n A. P. Tuzov (381 and V. A. Plis.o; (37] have s hown for the ca~ when I hi' charac ter istic eq uation (3 1) o f the system (!!) has one zero rovl and the remainin~ roots are to the left of the imaginary axi s, tha t. for m = 2. necessary and su fficien t conditions for nbsolute s tability are the Lur'e inequality /'2 '• 0. whic h, for this case, is the condition fo r stability-in -the-limit, and the rrquirement that .

{or both

L(z) -

fi must satisfy the conditionsu

2(11'f' -

/C)

=

Hence the coefficients of the quadratic fo rms L(z)

+ 11101 + Pf q~-{0.

l4

The relations (87) and (88) also remain valid for w

that I I ( . ): -'L,J, Zj = 0,- 'Lf, %1 0• 10 1

10

J

= 0, provided we ;lssume

--- -----------120

IV.

THE POPOV CRITERION AND THE EXISTENCE OF A LIAPUNOV FUNCTION

§ 6.

the quadratic form (89) (for finite k, we assume that h = k; when k = oo, we assume that r = 0). It is then necessary for Q(z) ~ 0, since for Q(l) < 0, assuming that z; = zi, y = 1/v( :? a;zJ), we obtain V > 0, which

,

is impossible. Now let Q(z) = 0 for z; = zJ. Then, for these values Zj, with y = 1/vO: a1 z~). we have V = 0. But then from the condition

of minimal dege~eracy of V, it follows that z~ is a point in the. null space. Since for z1 in the nul l space and 11 = 0 (that is, y = 0), V = 0 always, it always follows that Q(z) = 0 for these values of the z;. Thus, the quadratic form Q(z) becomes zero in the entire null space and is not zero anywhere outside this subspace. It follows from the fact that V = 0 for y = 0 and arbitrary z; in the null space, that (see (89)), at any point of this subspace, the linear form I a;z; becomes zero. Thus, the entire null space must lie in ;

the hyperplane k

The above f~rmulated conditions also are sufficient for the case , > 0, but they are no longer necessary. 35 The quadratic forms H(z) and G(z) are related as follows : 3 ~

d dt

-[H(z)j,, = - G(z),

(91)

H(z) = ~~~[G(z)j .

(91')

.

or

+ 21

/i

{3e;- r Zw •

= 0,

choice of the form H(z). For this purpose, we first introduce a change of variables transforming the system (!!) into the form 38

dz; dt dz; dt

.,

..

I

I

k~ l

k= .. J+ l

d;kZk

+ s;y

d;kZk+s;y

=

(j (j

=

1, .. ·, m,),

m,

+ 1,

· · ·, m) ,

(93') (93")

(93'" ) The characteristic equation

d .. -A

d,,. 1 0

d.,,,

(94')

· · · d,. 1,., - i

has roots with negative real parts, and the equation

*

j

(92)

Having made t he foregoing preliminary remarks, we next analyze the requirements imposed on the quadrati O. t he n!'cessary and s utlicienl conditi ons are ag& J.ct us recall that d!clt(H(z)) 11 d enotes the derivati,·e of the form Hms o f defi ning the quadratic forms H,(z) and H a(z). Let us examine the firs t problem. In t his case the c ha racteristic equation is Hurwitz and. in essence. we have to repeat the rt'asoning presented in § 2 of this c hapte r for the principal case of (!) • l nt n ld uci ng into (101') instead of C,(z) the quadratic fo rm Q ,(z), ' ' T his assert ion is verified easi ly in the case when the m:~trix with f'lements

G,(Z) which had no variables in co~n:on." But tht>n, it follows f rum

root~o

( 102")

t

+ ~\! ,cr; z;l

i~ also a positi ve definite form in the va riab~es z,. · · ·, z.. , · . Above the form C(z} was decomposed mlo two forms (J,(ZI a nd

It

(102' )

I

d z1 dl

posit ive definite quadratic form in the variables

C(z)

k

or

and Q(z)

= ..r, d;t 'lk •

d,k can be reduced to the Jurdan dia~:onal fo r m d,t - d, ,;,t (•iok is the K ronecke r della: i. k J. 2, · · · . ml. In that case, f rom th e relations (99). we obta in simple furmulae relating the coeffident'i t>f the fflrmS /liz) and GCz): g;k -

-(d , Hlt)h,t

(t,

k - I, · · ·.

m) •

It is ionmecliately clear from these relati ons th:lt. for i· nto. k ' mo. the equalities hik 0 foll ow from the equaliti es g, 4 - 0. since cl; ~dk J 0 (here d, a ud d~r are rMts of th e cha racter istic C!C'Iu:.tit)ll ~ (9~') and (94") rc~ pcctivcl y , a ncl hen ce Re t/ 1 0 also, that is, the quadratic form H,(z}, which we have determined, is positive definite. We now proceed to determine the quadratic form H.(z}, in the variables z., .. , · · ·, z. , from the conditions (97) and (101"). The characteris tic equation (94") has simple roots located on the

r; = · .l E· 2 '

I, = iw,, / 1 = - iw,, Ia = iCJJt, I, = - iw. ,

=

=

If a m - m, is odd, then there is one zero root 1. 0. By a trans formation of the variables z., .. , · · ·, z. (usually a complex trans· formation), it is possible to cast the system of differential equations

W,(p)

dZ; dt -- I, Z.,

= z,, Z, = Z

+ SJY

(J.

...

(/

I

..

(1 13)

py ,

·l

···:

3,

· · · . a) ,

f!_

p

_I )

I

S 1 E; P

1

It h ~l!i no nzero rP al n.etlirienls J/ ,:. 11 .••. .. ·, (//•• ). The tall!- nf lhif> form is a ..- 111 111,. In order lo c ha nge over to real valuerl qua nti ties, it •s sufficient to put

/., = x, + ix:. IS

Here, we make

Zt

=x, -

ix,,

z, -

use of the rt:"lations

x. + ix, , Z,

s, S:

j,

> 0, then a lso H(z) > 0.

J

/3Cr1•1• . 2!5.1"

= H, z. we

a nd s ince H 1(z)

S:Sr

(

L(z) + 2 I f, zi " -1 wa= - u•

7dn -

flu ....,

satisfying the conditions (97) a nd ( 101" ). At the same time, we have found the quadratic form H(z), in equation (87), which ensures the existence of a Lia punov function o f the type (87) with a minimally degenerate derivative li ~ 0. But then V > 0 (see ~ 5), and, consequently, as is immediately seen from equa· tion (87), H(z) ~ 0. Therefore H 1 (z) ~ 0 also. But, from the above proof, ll~( z) has rank m - m, , that is, it is not degenerate. Therefore

..

1- Pdof•J, - 0, :1nd l!,2 has the real value

/1,._

It is possible to return from the variables x, , · · ·, x. to the variables z.. ,.,. · · ·, z.. by means of a real coefficient transformation. We have thus shown that there exists a quadratic form H1 (z) of the variables z.. ,.. ,, · · ·, z., with real coefficients and o f rank'' m - m,,

Consequently, the quadratic for m

c.

dome ·

-

+ 2Ha,(.t; + xi) + · ·· + (H•• x:) .

I· fit/ow, 21Sd 2

But accord i11g to equation (43) of Chapter Il l, the Popov coefficient q is defined by the equation q

(fo r a odd, = x.). The form //2 then becomes a quadratic form in real variables:

Hz = 2Ha(x: + xi)

We shal l now show that the values of the coefficients defined by these equations are pairwise equal (H" Hu, Hs. = H.,, · · ·) and tha t they are real numbers. For this purpose, we write the equations (121) as fo llows' 8 H12

129

SUFFICIENT CONDITIONS

x, - :.,·. ,

Is, : =

s~ 1 •

I

I

,.

.=

'' - -/l' j I I liz,)+

J/(z)

(126)

of the 11oriab/es ZJ ami u 111flich enters info the sunr for the fu nction V w ill alway:; b ).

=

The res ults which we re e s tablis hed in § 3 throug h §6 o f this chapter for pa rticu lar cases o f (!) wil h a zero charactel-'i s t it· root c-an be stated in the form o f the foll owing theorem.

Theorem . I. If for any particular casr of 1/ze $,\'Stem (!!) which has a zero tlwraclu istic root . tlzc conditions fo r stabilily·in-tlu·-timil (S{W Chapter Ill. ~ 5} rmd the strrngfht!IICd fopw condition Re (I

+ iqm)

!H im)

+

!>

0

(-

ro

arc satisfied Czl'ith q real all(/ ,linitr or with

~

,,,

~

+

ro )

(P )

e dejiuile f unction V of t he ty jJe " t1 quadmtic fo rm fllus an integral" (l(iiJen by (32)) which has a JUmncxati1•c. minimally degenemte dcrit•atit•f? V, for all clwracleristics ~~( ., ) iu the serlor (0, !d. wh.:re I~ is either a finite number or k = o~. 2. For ll finite, and also fork '= oo, -r = 0 (see (32) and ('1-1)), stability· in·thc-/imil and the emu/ilion

0, $tability·in·thiJ·Iimil ami the tceak('}led Popo,, cmz· ditiv11

Rc [( I

arc nerrssary rontfilions.

+ iq,,r)

IV(im)J ' 0

SUFFICIENT CONDITIONS

131

3. For all the particular cases of (!} considered, except /or the simPlest particular case when equation (31 ) is Hurwitz, the Popov con· slant q is uniquely defined by the system / rCQuency response W(itu). If the characteristic equatiou ol the system (!) has a double zero root, that i$, equation (31 ) has a simple zero root , then q = oo . 4. The conditions of item 1 above, stability-in-the·limil and the stnmgthened Popov condition . are sufficient conditions fo r the absolute stability in the sector (0, k), k finite or k = oo , ·/or all particular cases · of the system (! ) with a simple zero characteristic roof. These conditions will be srtj}icient, also, in the case of a double zero characteristic root, provided that the additional conditions below are imposed on the character· istics IJ'(a) contahwd i11 the sector (0, k] :

{

...

)o

If/(ility· in·thc-l imit and the streng the ned Popov condition are satisfied for some q, t he n it is a lways possible to find a Liapunov functi on of the t y pe · •a quad ratic form plus an integ ra l" wi t h P and ~ such tha t q = l{i ~~ is sat isfied. For the s implest pa r t ic ular case of ( 1), if the bas ic P opo v condi · t ion (P) is satisfied for some g iven q, it is always possible to change llows from t he above theorem .

Theorem. In order that 1/tcre exist /or lhr simplr.st particular casr• of (1) a Liapmzou / unr:tion. of lhr l _vt>e "a qu.adrnl£l' form /Jlus an in · tegml," with a nc).fative de_linile tl••riunlll'l! such /hal if ensures ttbso/ulc stability in lite sector (0, kl f or ll jinilc, il is nect!ssary am/ :;ufjirieuf 10 In tlacst• cases , the limit poin t W(oo ) d(lt~~ not coincide w ith the poin t ( - l fk , 0). In the next section. we shall s how that the rest rict.ion p ,z.. 0 for k _ ,,, may be dropped, pnw ided that the Li a pu nov fuu ct i11n used has a wea kl y degene rate

derivat h•c and that il ensures absolute s tabi l irr .

,·

132

IV.

S 7.

THE POPOV CRITERION AND THE EXI STENCE OF A LIAP UNOV FUNCTION

that the frequency response W(im) satisfy the Popov condition (P) and the condition (129). • Wlum k - CS of (!) considen::d (ln111~ H' · t 1'vn t'. :'lthown on pp. 11 3 116, it follows from the tinct from the mcth(l(l of resolving equatio ns, the Popu\' method red uces the problem to a n effective graphical construe! inn wh it·h is simple and con venie nt in e ngineering computatiuns. There arc ma ny reasons to suppose' (as was mcn1 ioned in Chapter Il l t hat the l.ur'e resolvin).! limit·e 1he rt"gions g ivt'n by the resolving prclirnit equations with an arbi1rary choice of the quadn:al ic form. If this conjecture turns cmt to be correct, it will be possible to cons ider th e Pop()v method as a n e ffec t ivc: wa y o f demons trating that it is pos~ ib l e to daoosc I he varinblc parameter in 1he Lu r'e resolving equations so that the t'quations w ill ha v(• real so l ution s.~ •

~e

the Appendix .

• ~e the Appendix .

144

V.

The question t hen arises as to whether the P opov cr ite r ion includes the entire region of absolute stability in the parameter space, that is, whether it is not only a sufficient but also a necessary condition for absolute stability. T his quest ion immediately breaks dow n into two questions: 1. Do absolutely stable principal cases of t he system (!) exist for whic h absolute stabili t y can be deter mined by means of Liapu nov functions o f the form (3), but not by means of the S-method? 2. Do any (that is, principal or par ticular) cases o f absolutely stable systems (!) exist for which it is not possible to construct a Liapunov function, of the form (3), capable of establishing tbe fact of absolute stability? At the present t ime these problems are sti ll unso lved, a nd, therefo re , t he question of whet her and in w hat manner the Popov criter ia can be modified so that they will g ive not only sufficient b ut a lso necessary conditions for abso lute stability is a lso unsolved.

§ 5.

S5.

REVIEW O F THE STATE OF THE PRO BLEM

The Pro b le m o f Absolute Stability in the Hurwitz Se ctor

In Chapter (, the following problem, formulated in 161. was mentioued briefly: s uppose that the principal case of Lhe system (!) is stablt- for any linear characteristic contained in the sector 10. kl (that is. for y he~. if 0 5; h .; k). Does it therefore fo llow that the system (!) is absolu te ly stable for any nonlinear characteristics contained in the sector 10. kl? If the answer to this question were affi r mative. the problem o f absolute stabilit y simply would not exist, since absolute stability could then be determined by means of the usual linear criteria o f s tability l llurwit z. ~Jikhaitov. Nyquist, etc.). However, this Question has been answered negatively 1501. The problem therefore is to find amtmg the systems (!) a subC'Iass o f systems for which the above question can l>c answered in thP affirm:u i\·e. In Chapter Ill. *2, it was shown that the stability o f the syste111 1!1 i~ t·n;~ thecuem 7

iu 180).

S6.

SURVEY OF SYSTEM S IRREDUCIBlE TO (I)

147

systems of differential equations o f th~ ty~ (!), but wti_h a delayed argument (that is, descr ibed by differential d1fference equattons). Such systems o f equations describe processes in automatic control systems with a delay element . The absolute .stability of systems with nonsingle valued c haracteristics (that is, characteristics with h ysteresis loops) was s tudied by Popov [78) and Yakubovitch [1021. Gelik (85) studied conditions for the absolute s t ability of systems with discontinuous characteristics. In this case, equations of the type (!) are insufficient to describe the motion of the system, and the equations must be supplemented to take care of the so-called "sliding" motions. F or this p ur pose, both discontinuities at the point o = 0 (for k = oo) and discontinuities in intermediate points where o :f! 0 are important. y.z. Tzypkin (881, [105] constructed an analog of the Popov criterion with q =0 for discrete automatic control systems which are descri bed by d iffere ntia l difference equations. . . . V. M. Popov [98] introduced the concept of hypers tabd1ty. wh1ch he defined in such a way that it was possible to give both m:cessary and sufficient conditions for hyperstability . Even from this brief survey , it is c lear that the concept o f ''absolute stability'' is being applied to quite differen t systems. and t hat the s phere of applicability uf the theory o f abst•lute stability is ~onstantly becoming wider. In the near future, we may expect new and Important results in th is region .

APPENDIX

THE CONNECTION BETWEEN THE METHOD OF LUR'E RESOLVING EQUATIONS AND THE POPOV FREQUENCY METHOD

§ 1.

Introduction

Some time after thi ~ manuscript was submitted to the publisher. some new facts re lated to t he va rious problems descr ibed in this book came to light. Firs t of a ll, t he authors became acquainted with the s till unpublished paper A. Yaku bovitch. In essence. he pro\-ecl that the Lur'e resol vin~ limit -equations define a region of s tabilitv, wb ic h is the union of all the regions obta inable by use o f a Liapunov function o f the type "a quadratic form plus an intt>gral"" (fo r the prin r i pal case o f (!) by means o f th e S- methOrivat ive I These resu lts. togctlu•r with a uum l>l'r o f consequcrW

0

whe re the coefficient t' ::::: 0. We obtain the Lu r'e resolvinK limit ·equ;nious by repre!>cnting the t he derivative I ' o f the Liapunov funClion in the form

I' = - S(x, y) -

(6')

-;-~?(!J) (a -

71)',

(J)

Both for th e pri ur ipu l case o f (!). and t he si mplt•s t particu lar G l!-.t' : F'u r the !.)'Sit:lll I! 111 1. we used al numbe rs. . Sim:e in the condition ( fl l we find only tht' ratio If

. ac; the c. 01.-fltt·ient in the usual form o f writing the l'o J)I >\' nlllditumsl. it is possihle. without loss of generalit y, to cons ider that :- al>sumcs or~l r two ''alues: J or 0. A finite 11 corrCSJXHlds to th e value:I, \\hrle IJ "' corresponds to the value r 0. ,~- (If

Theorem I. If the system (!) is l'omPfclcly raufmffablt• aml ~f if :-~ fwinripa/ cas11 or a !'impll'sf tmrliculnr rasr, then thr s:rslem oJ l.u r _r: I C'Wift,ing limil -equaliom (5), 1t.:if11 r :;:; 0, has a rral s~J/1~fwn ."•· · · ·, u. 1} nnd only if till' H'eakened H J/)(11' ronditiou ( P_) is .wtllsjwtf ,lur thl' snnu ,.,i/ "' '·~ c1/ r and p.

153

On the basis o f this theorem. the Lu r'e resolving limit·equations ha Ye rea l solutions whenever the weakened Popov condition (P_ ) is satisfied with any finite q. The ana logous equations, obtained without the S method, have real solutions only if the weakened Popo,· condition (P ) is satisfied with q = oo , that is . when the frequency response lies either above or below the real axis, and, possibly, has points in common with it.

2. A l though Theorem I establ is hes only conditions for the exist ence o f real solutions for the equations (5), in the proof o f this theorem we s hall give an effective method for constructing all such solutions w ith the aid o f the ident it y gi ven below, which will be pro ved later.•

' l t is possible to prove 198) that a system is nondegenerate i f and on ly if it is both completely controllable and co mpletely observable.

§ 2.

THE FUNDAMENTAL THEOREMS

Re (:- + i fk••) WCho)

+ ~

= I Vr + W,.(iw) 1'.

This enables us to use the resolvi ng equations not only to deter· mine abso lu te s tabi lit y, but a lso to cous truct the correspond ing l.iapunov function.

Theorem II . Let the system (!) or (!!!!) be uondegenrralr•. and J..1 Ihe / tmclion rlin ttdt with the rc~o h·ill ~-t limit t·quatioll'- t:ll) of Chaplt·t I I , \\hidt "LrlH.:trd both w ith and wi t luHtl the S method. • It is easy to sec that. hor llw rorinlival cocli rinn ct• ' lh:H ,. 0 1

•• •• •• •• •• •• •• •• •• ••... •• •• •• •• •• ••

•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •

154

~ 2.

lUR'E RESOLVING EQUATIONS AND POPOV FREQUENCY MET HOD

rurvc degeneratrs into the origin. Theorem Ill. If the conditions of Theorem ll are S(Jtisfit'd for thr Liapunou / wzclion V,, then V, e1zsures the absolute stability of the system (!) in tile sector [0, k), and the absolute stability of the system (!!!!) in the m :tor (0, k) /or all q = {3/k ~ 0. If f3 < 0. then the additional conditum br/ol(l m/ISI also be. satisfied'

r • [ka- \l)(o)J do = r·· J, J, [ka -

\P(a)Jdo = oo .

(I I)

Remark. Absolute stability can be ensured (even if a Liapunov fu nction V, of the form (9). (9') does not exist) if. in the conditions of Theorem Ill, the requireme nt of nondegeneracy is c hanged to the weaker re· qui rement of complete controllability . As dist inc t from the first two theorems, Theorem Ill no longer establis hes the connection between the Lur'e resolving equations a nd the Popov method. Instead, it e nlarges the possibilities of the Popov me thod. Indeed, as s tated in Chapter V, the Popov condit io n (P_) e nsures absolute s tabili ty for t he s implest particular case of (!) with k ro. However, it is s till not clear if it is possible to s ubstitute the weakened Popov condition (P ) for the basic Popov condit ion (P) for the re maining cases of (!). Theorem Ill establis hes the possibility of s uch a s ubstitution, for k finite or k = oo , for the principal a nd fo r the s implest particular cases of (!), when the additional weak res tric tions indkau:d in tht• texts of the Theorems I I ancl Ill are satisfied. T hio.; ~~~h~titution allmvs the Popov line and the modified frcquenry rc~ponst> II' •fm) to ha \'C points in common. Until nnw, in this appendix, we considered resolving li mit·equa· I ions only of the type (51. Let us now recall that. in Chapter II. the l imit ·equations were obtained from the prclimit equations having the form l

ru, -

I m,,(u,. · · ·.u.)/1,

1-

J

f

I n,,r, ; I p,,b, 1

1

~ r,

}

( 12}

-

f'or thC' principal 0, that is, for the S-method, the above·given theorems prov ide ans wers to this question. However, for< = 0, that is, when the S- metbod is not used. the situation is different. For < 0, the existence of real solutions for the equations (3) indicates that the condition (P_) is satisfied for q = oo. However, if the modified frequency response W• (w) touches tbe real axis to the left of the point (- I lk. 0). then the princ ipal case of (!) will not be s table even fo r linear characteristics contained in the secto r (0, k), although the conditio n (P. ) for q = oo may be satisfied. Hence the existence o f real solutions, for the limit-equations (5) obtained wi thout usi ng the S-method, does not guarantee absolute s tability fo r the principal case of (! ) for k - ('Q either. The ex istence of real solutions for the preli mit equations ( 12) implies that the bas ic (not the weake ned) Popov inl s in common with the real ax is. But, fo r the principa l ('ase of (! ), thee' s ta rting point W(O) always lies on the n•al a xis. Thus. fo r th(' principal ) "" c'(A - pEf' b

~

.

In both cases for 1PI large, the following expansion holds:

w,.(PJ

p y

2

Ab -= u'(A - pE) 1b -.. - u ·(b -1 Ab 1- fT

+ · · ·) ,

from which it is easy to see that fliP system is completely controllable (/hat i.~, W ..(p) 0 implies that u - 0) if and only if /he vectors

=

b. /lb ... . . A" I b

t

(14)

are li nearly inclependeut. Analogous ly it ma>' be s lwwn that lhe system is comp!clcly obsen•· a/111! if and only if lhl' t•eclors

c. A'r.· · ·, rt '• 'r.

( 15)

are linearly iude!)(•nde nl. Therefore, (sec [981). the system is nondeglmeralr• if and nnly if both illt't"l'dors (1:1) and thr t•rctor:; ( 15), r r spcrficPiy. arc lillearly iJI(/e{!endmt. W e first notP a few or the pwpcrties of completely ('(lntro llahle systt·tns whit'h will be net•cled later on. It [QIIows frot n the equatiun

11',.1 {! ) -

l l ' (l\ -

.

/> I~)

I

1\ ! /1) ) - IJ(f>)

I

( I 6)

that t he. r·oe llkil'nl s of tlw polynomial f((fi) l' 0. Let F , == 0 at some point. Taking this poin t as an initial state, let us construct the solu t ion cu r ve pa~sin.g through it. Along th is curve (r tabilit)' o f the suhuiun o f a partinllar nun linr7 44.

v. r.

Zubov.

Metllntl~

of

A. Ill.

), ia ]JIIIHlV

und their 1\pplil'atiOII.(;r,

"011 th., !-t:Jilll it~ n f nunlln t':lr rcmtrol " ' " lt'nh," (loki.

Akad. .Va11k SSSR. IIi • .J. I!>Si.

1954 1\:ull.l' hin. :-I 'II 1\ra"· "'''l.ti. "On tlr;; exi~t enc... ••f a l.i;tpuno' func tion 111 tht· '·'"'' uf ~:luhal a-.unpwtit· ~taloilil\, " l'rik /. .1/nll'm. i .llrkh .. '\\' Ill , 3. 111.'\ I. :Ill. ;\ i\1 I l'IU\ \ . I' 1111\akln. "On tht' ' lability nr Clrn trnl :.y~ll·m~ \\ilh IWII n:..:ul.ttof:.. l'rikl . Mut.·m. i .llrkh .. XVIII. 2. 19a~. :I I 1>llt·t't' lll'f "'' , lu/ "111 ( 'ot~ll''" I, MPsc-o w Ak ad Nauk ~SSH . l\l5!j. \ \I I,.,,, SltJirllit, ••1 .Volllillr~tr ( ',.ltl rul SJI.•Ie-m ~. :\losc•>w. (;{"tl'khi7.

·~

l' M11it ,,,

y, '"''"'·

11011, 111 ,;, '~""· t•d • t!lh:~)..., h .. ~ 1 .11• ,,r tlw prol. l•·m "' ~1. 1hili1> in th•· lht•' ~·: " ' ~ut ..mat ..: .

- - - . "N•'t't•,.;;•n· ancl ' Ufllt ir ul cnndil i"' '' r.. r 1ht • ,.J,.h t1 • l "•l•t Iii 1 llf! uf tIll' :-.111111 it•oil C1111di1run'·" l .,, 1•

~l :•hilit y," AI'Lt~m . i 'l'dt•mP.kh .. XI X. 1. 195!!.

'"'"li'"'"

r ,., ..,t rut "' , ,,. 10, rl, •.,.. , j ht•d 5 1. E. N. Rnzo:m·as.~f>l', "On the ~t:thitit) nf by differcnlialrqu:uiun' nf hfah .tnd ·· i~ l h "'dt•l' ," 111·111111 1 'l'r•l•·mrkh. XI X. 2, l!J5l!.

•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• ••

i

•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• ••

166

IIILIOGRAPHY

IIILIOGRAPHY

55. V. B. Shirokorad. ··on the

exist~nce

o f a cycle o utside the conditions of absolute stabilit y fo r a three dimensional system," A ~-tom. i T elemeklt. ..

XIX. 10, 1958. 56. V. A. Yakubovitch.

"On the boundedness and the global stability o f the solutions o f certain nonlinear differential equations," Dokl. Akad. Na"k SSSR,

121. 6. 1958. 57. V. 1\1. Popov.

.

"Criterii analitice s implificate pentru stabilitatea sistemelor liniare de reglare automa ta," Auto.matica si Electronic.a., 2. 3, 1958.

1959 58. A. K. Bedel'bae,·. "On some simplified stability criteria for non linear conrrol systems," At'Lom. i Telemeklt. .• XX. 6, 1959. 59. V. I. Zubov . Mathematical Methoos for Analyzing Automatic Control Systems. Leningrad, Sudpromgiz. 1959. 60. 0 . I. Komornit skaya . "On the stability of nonlinear control systems." J>rinkl . Matem. i Meklr ., XX II I. 3. 1959. 61. N. N. 1\rassov~kii. Some Pn1blems in the Theo·r y of :)tability c1f 1\llotion, Moscow. Fizmatgiz. 1959. 62. L. 1. Kuprianova. "On the st~l>il it y of a nonlinear control sy!>tcm with an inertialt>~S l'lcment." 111'tom . i 'f~ll' m ekh .• XX. 2. 1959. 63. E. N. ({ozen vasse r. V. A. Plis~. "A discussion o r the paper b y \'u. $. Sobo lt>v. ·On the absolurc st:tbility o f certain control systems ', " Adomt. i Ttle · n~tkh .• XX. ~. 1959. 6 :ota bilitate pentru sisterndc nliniarc de r. I. l!JCi2. 89. V. /\. Yakubovit c h. " Tht• svlttliou v f ('f'rt. J. "· Z~ pkin. " I lit tabilitat nichtlinearer impulseregclsysteme," Ht fi'IIIII!}SIUitllik. I, 1963.

1950 I. P. V. Bromberg. "'On the problem o f stability of " class of nonlinear systems."' Prikl. Matem . i !tlekh .. IX. 5, 1950.

1953 I. P.

Stubility al tasc, 4. 7, 15. ·19. :.o. f>2. 5:1.

Resolving equations, 11, 28, 93. 150-

152, 156 Romiti, A., 169 Rozenvasser. E. N., 11, 12, 32, 165-168 Rumiantzev, V. V., 146, 165 S·method, 21, 93, 139, 140 first variant, 27. 94 s~cond variant, 27, 96 Sampled-data systems, 14, 170 Sector , 6 Hurwitz, l2, 88, 1-14 Shimanov, S. N., 164 Shirokorad. V. B., 166 S implest particular case, 7, 3~1, 35, 83, 132, 151 S liding motions, 147 Stability , absolute , 7 in a sector , 9 pro blem. 10. 138 Stab il ity, global asympt otic, 6 Stability-in·the ·limit. 35. 5 1. !i7-79. 141, H, I :>2. L!l7 completely obS