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15.part1 On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems Part I Conditions








O n the Input-Output Stability of Time-Varying Nonlinear Feedback Systems Part I: Conditions Derived Using Concepts of Loop Gain, Conicity, and Positivity

Abstract-The object of this paper is to outline a stability theory terns we might ask: ll'hat are the kinds feedback that of for input-output problems using functional methods. More particuare stabilizing? Il-hat kinds lead to a stable s_vstem? larly, the aim is to derive open loop conditions for the boundedness Can some of the effects of feedback on stability be deand continuity of feedback systems, without,at thebeginning, placing scribed without assunling a very specific system restrictions on linearity or time invariance. It will be recalled that, in the special case of a linear time invari- representation? ant feedback system, stability can be assessed using Nyquist's criPart I of this paper is devoted to thes)-stem of Fig. 1, terion; roughly speaking, stability dependson the amounts by which which consists of two elements in a feedback 10op.l This signals are amplified and delayed in flowingaround the loop. An simpleconfiguration is a modelfor manycontrollers, attempt is made here to show that similar considerations govern the amplifiers, and modulators; its rangeof application will behavior of feedback systems in general-that stability of nonlinear be extended to include multi-element and distributed time-varying feedback systems can often be assessed from certain gross features of input-output behavior, which are related toamplifisystems, by allowing the system variables to be multication and delay. dimensional or infinite-dimensional. This paper is dividedintotwoparts: P r I containsgeneral at theorems, free of restrictions on linearity or time invariance; Part II, which wappear in a later issue, contains applications to a loop l i with one nonlinear element. There are three main results in P r I, at which follow the introduction of concepts of gain, conicity, positivity, -+ , , X and strong positivity:
THEOREM 1:I the open loop gain is less than one, then the closed f

. feedback loop with two elements. I

loop is bounded.

Fig. 1.

two, suitably proportioned, conic relations, then the closed loop is bounded.
If the open loop can be factored into

The traditional approach to stabilit\- involves


pullov's method; here it is proposed to take a different

THEOREM 3: If the open loop can be factored into two positive relations, one of which is strongly positive and has k i t e gain, then the closed loop is bounded.

Results analogous to Theorems 1-3, but with boundedness placed by continuity, are also obtained.



course. and to stress the relation between input-output beha\Tior and stability. AAninput-oz~tfiztf system is one in which a function of time, called the output, is required to track another function time, called the input; more of general[>-the output might be required to track sonle function of the input. In order to behave properly an input-output s?-stem must usualll- have two properties: 1) Bounded inputs must produce bounded outputsi.e., the system must be nonexplosive. 2 ) Outputs must not be criticall!- sensitive to small changes i n inputs-changes such as those caused by noise.
1 The system of Fig. 1 has a sil7gle input x , multiplied b>-constants Q Z , and added i n at t z o points. This arrangement has been chosen because it is symmetrical and thus convenient for analysis: it also remains invariant under someof the transformations thatwill be needed. Of course. a single input loop can be obtained by setting u l or an t o zero. The terms z-1and v: are fixed bins functions, which will be used to account for the effects of initial conditions. The variables e l , e?, y ~ and y2are outputs. ,

EEDBACK, broadly speaking, affects a system in
one of two opposing wa>-s:depending on circumstances i t is either degenerative or regenerativeeither stabilizing or destabilizing. In trying to gain some perspective on the qualitative behavior of feedback SJ-shIanuscript received December 29, 1964: revised October 1, 1963: Februar5- 2, 1966. This work was carried out at the lI.I.T. Electronic Systems Laboratory in part under support extended by S.4SX under Contract SsG-196 with the Center for Space Research. Parts of this paper were presented a t the 1964 Sational Electronics Conference, Chicago, Ill., and at the 1964 International Conference on 1Iicrowaves. Circuit Theor>-, and Information Theory, Tokyo, Japan. The author is with the Department of ElectricalEngineering, Jlassachusetts Institute of Technology, Cambridge, XIass.

a 1 and



These two propertieswill form the basisof the definition of these results are paralleled in the work of Brockett of stability presented in this paper. It is desired to find and Willems [4], who use Lyapunov basedmethods. conditions on the elements H and H 2 (in Fig. 1) which Several others have obtained similar or related results I functional methods: Sandberg extended [Sa] the will ensure that the overall loop will remain stable after by H I and H? are interconnected. I t is customary to refer nonlinear distortion theory mentioned above; later [Sb] to HI and H prior to interconnection as the "open-loop'' he obtained a stability theorem similar to Theorem1 of P this paper. Kudrewicz [6] has obtained circle conditions elements?andtotheinterconnectedstructureasthe by fixed point methods. Contraction methods for incre"closed loop." The problem to be considered here can therefore be described as seeking open-loop conditions for mentallypositiveoperatorshavebeendevelopedby closed-loop stability. Zarantonello [7], Kolodner [8], Minty 191, and Although the problem a t hand is posed as a feedback Browder [lo]. A stability condition linear for timeproblem, it can equally well be interpreted as problem varying systems has been described by Bongiorno [I 11. a in networks;it will befound,forexample,thatthe 2. FOKML-LATIOS H E PROBLEM OF T equations of the system of Fig. 1 have the same form as those of the circuit of Fig. 2, which consists of two eleThere are several preliminaries to settle, namely, to ments in series with a voltage source, and in parallel specify a system model, t o define stability, and to write with a current source.2 feedback equations. What is a suitable mathematical model of a feedback element? A "black box" point of view towards defining a model will be taken. That is to say, only input-output behavior, which is a purely external property, will be considered ; details of internal structure which underlie this behavior will be omitted. Accordingly, throughout Part I, a feedback element will be represented by an abstract relation, whichcanbe interpreted as a mapping from a space of input functions into a space of output functions. More concrete representations,involvingconvolutionintegrals,characteristic graphs, etc., will be considered in Part 11. Some of the elementary notionsof functional analysis w i l l beused,thoughlimitations of spaceprevent an introduction to this ~ubject.~ Xmong the concepts which will be needed and used freely are those of an abstract relation, a normed linear space, an inner product space, and the L, spaces. The practice of omitting the quantifier "for all" shall be utilized. For example, the statement








v = Z2i2. 2

i, = yv,.

Fig. 2.

X circuit equivalent t o the loop of Fig. 1.

1.1 Historical ;Vote

The problem of Lyapunov stability has a substantial history with which the names of Lur'e, Malkin, Yakubowitch, Iialman, and many others, are associated. On theotherhand,functionalmethodsforstabilityreceived less attention until relatively recently, although someversions of the well-known Popov [ I ] theorem might be considered as fitting into this category. The present paper has its origin in studies [2a, b] of nonlinear distortion in bandlimited feedback loops, in which contraction methods were used t o prove the existence stability and of aninversionscheme.The author'sapplication of contractionmethodstomore general stability problems was inspired in part by conversations Sarendra with during 1962-1963; using Lyapunov's method, Narendra and Goldwyn [3] later obtained a result similar to the circle condition of Part I1 of this paper. The kel- resultsof this paper, in a son~ewhat different formulation, were first presented in 1964 [2dl e]. Many

-$ <

E -y)!'

is to be read:
"for all x

E X, -x? 5 x?."

COSVENTION: A n y expression co.rztaining a condition of the type ''xEX,"free of quantijers, lzolds for all x E X .

2.1 The Extended ;Yormed Linear Space X,

In order to specify what meant b): a system, a suitis able space of input and output functions will first be defined.4 Since unstable systems will be involved, this space contain must functions which "explode," i.e., which grow without bound as time increases [for example, the exponential exp( t ) ] . Such functions are not contained in the spaces commonly used in analysis, for example, in the L, spaces.Thereforeit is necessary t o
of output

? I t is assumed that the source voltage 'J and the source current i . good reference is Kolmogorovand Fomin1121. I are inputs, with = a l x + w l and i = a ? x + w ? ; the currents and voltages 1 The space of inputfunctions will equal the space in the two elements are outputs. functions.






construct a special space,which will be called X,. X,, will The point of assumptions (2)-(3) on X can now be apcontain both "well-behaved" and "exploding" functions, preciated; these assumptions make it possible to determine whether or not an element X , has a finite norm, xE which will be distinguished from each other by assigning by observing whether or not limt-a :/xf!! exists. For exfinitenorms to the former and infinitenorms t o t h e ample : latter. X , will be an extension, or enlargement, of a n associated normed linear space in thefollowing sense. X EXAMPLE 1: Let Lp [0, x ) be the normed linear space Each finite-time truncation of every function in X , will consisting of those real-valued functions x on [0, 5 ) for lie in X ; in other words, the restriction of x E X , t o a which theintegral S,"x2(t)dt exists, and let thisintefinite timeinterval,sayto [0, t ] , will have a finite gral equal !lxIl2.Let X = L 2 [ 0 , x ) , and let L s ~ = X , t h a t ; norm-but this norm may grow without limit asf+ 50. is, Lpr is the extension of L2[0, x ) . Let x be the function First a time interval T and a range of input or output on [0, 5 ) given by x ( t ) =exp ( t ) . Is finite, t h a t is, values T will be fixed. T is x in X ? No, because llxtl; growswithoutlimit as

type [to,

a > or

T i s a given subinterval of the reals, of the ( - ~i , a). v is a given linear space.

t--+ , or in other CT. words,



[For example, the in analysis of multielement (or distributed) networks, V multidimensional is (or infinite-dimensional) .] Second, the notion a truncated of function is introduced.

2.2 Input-Output Relations
The mathematical model of an input-output system will be a relation on X,:
DEFINITION: A relation H on X,is a n y subset of the product space X , X X , . If ( x , y ) is any pairbelonging to H then y will be said to be H-related to x ; y willalso be said to be an image of x under H.j

Let x be any function mapping T itzto V, that is, x : T+ V ; let t be any point in T ; then the symbol x t denotesthe truncated function, x f :T+V, zchich assumes the values x() = x ( T ) f o r ~ < and xt(7) = O elset. t where.

In other words, relation is a set pairs of functions in a of X,. I t will be convenient to refer to the first element (A truncatedfunction is showninFig. 3.) Kext,the in any pairas an input, and the second element as an to space X is defined. output, even though the reverse interpretation is sometimes more appropriate. A relation can also be thought V of as a mapping, which maps some (not necessarily all) inputsintooutputs. In general,arelationismultivalued; i.e., a single input can be mapped into many outputs. The conceptof state, which is essential to Lyapunov's method, will not be used here. This does not mean that initialconditionscannotbeconsidered.Onewa~7 of T accounting for various initial conditions is to represent a system by multi-valued relation, inwhich each input a Fig. 3. A truncatedfunction. is paired with many outputs, one output per initial condition.Anotherpossibilityis tointroduce a separate DEFINITIOE;: X is a space consisting of functions of the relation for each initial condition. type x : T 4 V; the following assumptions are made conNote that the restrictions placed on X, tend to limit, a cerning X : priori, the class of allowable systems. In particular, the (1) X is a normed linear space; the norm of x ~ is requirement that truncated outputs have finite norms X means, roughly speaking, that only systems having infidenoted by llxil. nite "escape times," i.e., systems which do not blow up ( 2 ) If x E X then x t E X f o r all t E T . in finite time, shall be considered. ( 3 ) If x: T+V, and i x t E X f o r all t E T , then: f Some additional nomenclature follows: ( a ) [ ! x t ! ! i a nondecreasing function of t E T. s (b) If limt+- l/xtll exists, then x E X and the Zimit DEFINITION: I f H is a relation on X,, then the domain of equals I IXI I . H denoted D o ( H ) , and the range of H denoted R a ( H ) ,are

Do(H) = ( x 1 x E X , , and there exists y E X , such that The extension of X,denoted b y X,, is the (x,Y)EHl space consisting of those functions whose truncations lie i n X , t h a t i s , ~ , = { x lx : ~ - - , ~ ' , a n d x ~ ~ X f o r a Z Z t R a~ H .) = {yl y E X , , and there exists x E X , such that ~ (j ( x , Y)EHJ (NOTE: X, is a linear space.) A n extended norm, denoted IIxll,, is assigned to each x E X , as follows: !!x\le=IIxII if %EX,and IIxII,= oc ifxgx. 5 I n general x can have many images.

(For example, it can be verified that assunlptions (1)( 3 ) are satisfied b y t h eL, spaces.) Next, X , is defined.

the sets,



23 1

NOTATION: If H i s a relation on X,, and if x i s a given element of X , , then the symbol Hx denotes a n image of x under H.5

The idea here is to use a special symbol for an element instead of indicating that the element belongs to a certain set. For example, the statement, "there exists Hx having property P" is shorthand "there for exists y E R a ( H ) , such that y is a n image of x,and y has property P.'16 Observing that H z is, accordingto the definitions used here, a function on T , the following symbol for the value of H x at time t is adopted:
NOTATION: The symbol Hx(t) denotes fhe vahe assumed by the function at time t E T. Hz

DEFINITION: A relation H on X,is continuous i H f has .following the property: Given a n y x E X (thatis, ]Ix\le< m), and any A>O, there exists 6>0 such that, for nZZyEX, i f IIx-yll < 6 then IIHx-Hyll <A. DEFINITION: A relation H on X , i s input-output stable if H i s bounded and continuous.

2.5 Feedback Equations
Although negative feedback loops will be of interest, the positive feedback configuration of Fig. 1 has been chosenbecause i t is symmetrical.' The equations describingthissystem, t o beknownasthe FEEDBACK EQUATIOXS, are :


= x11

+ ulx + y 2

Occasionally a special type operator, will be used:

of relation, called


e2 = w2 f u2z y 2 = H2e2

+ y1

(la, (1 b)

DEFINTION: A n operator H i s a relation on X , which satisfiestwoconditions: 1) D o ( H ) =X,. 2) H i s singlevalued; that is, i x,y , and z are elements of X,, and if y f and z are images of x under H , then y = z.





in which it is assumed that:

2.3 The Class 6l

DEFINITION: (3 i s the class of those relations H on X , having the property that the zero elemenf, denofed0, lies in D o ( H ) , a n d H= 0. o The assumption that maps zero into zero simplifies (The biases are used t o compensate for nonzero zeroH of many derivations;if this conditionis not met at the out-input responses and, inparticular,fortheeffects initial conditions.) T h e closed-looprelations El, E2, F1, set, it can be obtained by adding a compensating bias to and Fz, are now defined as follows. the feedback equations. If H a n d K are relations in( i l l and c is a real constant, DEFINITION: E1 i s therelationthatrelatesel to x or, then the sum ( H + K ) , the product cH, and the co-mposi- more precisely, El= { ( x , el) I ( x , e l ) E X , x X , , and there tion product K H of K following H, are defined in the exist e2, yl, Hlel, and H2ez, all in X,, such that (1) and y?, usual way17 and are relations in 6l. T h e inverse of H in ( 2 ) are satisfied. ] Similarly E2 relates e2 to x ; F I relates a, denoted by H-l, always exists. T h e identity operator y l to x; F 2 relates y 2 to x. on X,is denoted by I . All the prerequisites arenow assembled for defining the problem of interest which is: Find conditions on H1 and 2.4 Irzput-Output Stability The term "stable" has been used in a variety of ways, H z which ensure that El, Ez, Fl, and F2 are bounded or t o indicate that a system is somehow well behaved. A stable. In general it will be enough to be concerned with system'shall be called stable if it iswell behaved in two E1 and E2 only, and to neglect F1 and F2, since every respects: (1) I t is bounded, i.e., not explosive. (2) i t is F z x is related to some E l x by the equation F2x= E l x -alx -w1, so t h a t F Z is bounded (or stable) whenever continuous, i.e., not critically sensitive to noise. El is, and similarly for F1 vs. E2. DEFINITION : A subset Y of X , is boundedi f there exists I t should be noted that by posing the feedback prob1-1 > O such that, for all y E Y, (lyl!,<A. A relation H on lem interms of relations (rather than in terms of X,is boundeds i f the image underH of every bounded sub- operators) all questions of existence and uniqueness of set of X , i s a bounded subset of X,. solutions are avoided. For the resuIts to be practically significant, it must usually be known from some other 6 I n keeping with the usual convention used here, any statement sourceg that solutions exist and are unique (and have containing H z free o quantifiers holdsfov all z in Ra(N). example, f For and for all H z in infinite "escape times"). " H x > l (xEX,)" means t h a t "for all x in X,,

H1 and H are relations in (3 Z a1 and a 2 are real constants w1 and w2 are fixed biases in X x in X , is an input el and e2 in X,are (error) outputs y l and y z in X , are outputs.

Ra(H), Hx>l." In particular,Do(H+K) = Do(H)nDo(K). Note that @is nota linear space; for example, if Do(H) #Do@) then Do[(H+K) --K] #Do(H). This definition implies that inputs of finite norm produce outi t implies thatthesort of puts of finitenorm.klorethanthat, situation is avoided in which a boundedsequence of inputs,say IIxnll< 1 where n= 1, 2, . . , produces a sequence of outputs having norms that are finite but increasing without limit, say llHxnll =n.



Existence and stability can frequently be deduced from entirely separate assumptions. For example, existence can often be deduced, by- iteration methods, solely from the fact that (loosely speaking) the open loop delays signals; stability can not. (The connection between existence and generalized delay is discussed in G. Zames, "Realizability conditions for nonlinear feedback systems," IEEE Trans. on Circuit Theory, vol. CF-11, pp. 186-194, June 1964.)



3. SMALL LOOPGAINCOKDITIOKS T o secure a foothold on this problem a simple situationissoughtinwhichitseemslikely,onintuitive grounds, that the feedback system will be stable. Such a situation occurs when the open loop attenuates all signals. This intuitive idea be formalized in Theorem ; will 1 in later sections, a more comprehensive theory will be derived from Theorem 1. T o express this idea, a measure of attenuation, i.e., a notion of gain, is needed.
3.1 Gains

I;Hx - Hylje 5 g(H) . / , X -

[x,y E D o ( H ) ] .


In Feedback the Equations (1)-(2), the product g(H1) .g(H2) will be called the open-ioopgain-product, and similarly, HI) . g(H2) will be called theincremental

open-loop gain-product.
3.2 A Stability Theorem
Consider the Feedback Equations


THEOREM 1:lo a ) If g(H$ eg(H2)<1, then theclosed loop relations E1 and EB are bounded. b) If g(H1). g(H,) 1, < then El and El are input-output stable.

Gain will be measured in terms of the ratio of the norm of a truncated output to the norm of the related, truncated input.

T h e gain

sf a relafioz H in a, denoted by

g W ) , is

Theorem 1 is inspired by the well known Contraction Principle.'l PROOF OF THEOREM 1: (a) Since eqs. (1)-(2) are symmetrical in the subscripts 1 and 2, i t is enough t o consider El. This proof will consist of showing that there are positive constantsa , b, and c, with the property that any pair (x, belonging t o E1 [and so being a solution el) of eqs. (1)-(2)], satisfies the inequality

where the supremum is taken over all x in D o ( H ) , all H X i n R a ( H ) ,and all t in T for which xt#O.
In other words, the supremum is taken over possible all input-output pairs, and over possible truncations. The all reason for using truncated (rather than whole) functions is that the norms of truncated functions are known to be finite a priori. I t can be verified that gains haveall the properties of norms. In addition, if H a n d K belong to then g(KH) <g(K)g(H). Gainsalsosatisfythe following inequalities:

IlellI I allml)



+ c\lx\I.


I t will follom t h a t if x is confined t o a bounded region, say \!x:! 5.4,then el \vi11 also be confined to a bounded region, in this case jlel:l ~ u . ~ ~ z 1+bilw21! + c A . Thus El 11\ will be bounded. PROOF OF ISEQUALITY (10): If (x, el) belongsto El then, after truncating eqs. (la) and (lb), and using the triangle inequality to bound their norms, the following inequalities are obtained :

) ( . ! I HI 4g(H)-1)xt11 ; ! : H . v 5~g(H).11x11. ! ~


E D m ) ;1 E T I [X E Do(H)]

(4 (5)

+I Il l w t ] l + I
5 liwtll

a1 * I I x t 1 1
a 2

I + jlyztll I . I I x t ] l + llyltll
, I

(t E T ) (loa)
( f E T ) (lob)

where (4)is implied b y (3), and ( 5 ) is derived from (4) by taking the limit as x . t-+ If g ( H ) < then (5) implies that H isbounded.In fact, conditions for boundedness will be derived using the notion of gain and inequalities such as(5). In a similar way, conditions for continuity will be derived using the notion of incremental gain, which is defined as follows :
DEFINITION: T h e incremental gain of noted by g ( H ) ,i s

Furthermore,applyingInequality (4) toeqs. (2), the following is obtained, for each t in T :

5 g(HJ .ile?tl/


(1 la>
(1 1b)


I g(H,) .i!eltj.

Letting g(H1)La and g ( H J andapplying (lla) t o (loa) and (1 lb) to (lob), the following inequalities are obtained :


H in a, de-

I l)=-ltll

+ I u1












Q.Z ~ -iI.rtll t


+ aljeltll
~ ~


E TI. (12b)

applying (12b) t o ~ l e ~ tin (12a), and rearranging, ll

where the supremumi s t a k e n over all x and y in D o ( H ) ,all H x and H y in R a ( H ) , and all t .in T f o r which x t # y t .
Incremental gains have all the properties norms, and of satisfy the inequalities

(1 - aa)i,eltll


1 ~ ~ 1 1 ~ 1 1 +~]






C 8 ' (12 I ) ; I x t i l


E T).


g(KH) I .t(H) 1; (H-t-)- ( H ~ ) t l ( t



10 -4 variation of Theorem 1 wasoriginallypresentedin[2d]. 4 related continuity theorem was used in [2c]. An independent, re-

_< g(H) *IIxt - ytjl [x,yEDo(H); / E T ] (8)

lated result is Sandberg's [Sb]. 11 If X is a complete space, if all relations are in fact operators, and if the hypothesis of Theorem l b holds, then the Contraction Principleimpliesexistence and uniquenes of solutions-a matter that has been disregarded here.




Since (1- ayP) > 0 (as ap < 1, by hypothesis), Inequality (13) can be divided by (1-a@) ; after dividing and taking the limit of both sides as t - m , the Inequality (10) remains. Q.E.D. (b) Let ( x ! , el') and (x", el") be any two pairs belonging t o El. Proceeding as in Part (a) an inequality of the form l\el"-el'/' Ici;x"-x'II is obtained, which implies t h a t El is continuous. Moreover, since the hgpothesis of Part (b) implies the hypothesis of Part (a), El is bounded too. Therefore E1 is input-output stable. EXAMPLE 2: I n eqs. (1)-(2) (and in Fig. 1) let one of the two relations, say HI,be the identity on Lz,. ( L z ,is defined in Example 1.) Let the other relation,H z on LZe, be given by the equation Hg(t) kN[x(t)], where k>O = is a constant, and L is a function whose graph is shown V in Fig. 4. For what values of k are the closed loop relations ( a ) bounded? (b) stable? (a) First the gain is calculated.
LV[x(f)] dt

unaffected; however, H is changed into a new relation 1 HI', as in effect --I appears in feedbackaround HI. Under what conditions does this transformation give a gain product less than one? It will appear that a sufficient condition is that the input-output relationsof the open loop elements be confined certain"conic" regions to in the product space, x X,. X

Fig. 5.

X transformation.


l a x 2 ( t ) dt}

RESTRICTIOK: I n the remainder of this paper, assume that X i s a n inner-product space, that (x, y) denotes the inner product otz X, atzd that (x,x > =I;x;I2.

This restriction is made with the intention of working mainlyintheextended L2[0, m ) norm,12 with (x, y j

2 real

wherethefirstsupisover [ x E D o ( H ) ;H x E R a ( H ) ; t E T , x t # O ] . T h a t is, g ( H ) is k times the supremum of theabsoluteslopesoflinesdrawnfromtheoriginto points on t h e graph of N. Here g ( H ) = K , so Theorem 1 implies boundedness for k < 1. This example is trivial in at least one respect, namely, in thatH has no memory; examples with memory will be given in Part 11. (b) g(H) can be worked out to be k times the supremum of the absolute Lipsclzitzian slopes of :IT, t h a t is, g(H)= k sups, N ( x ) -:V(y)/x-yI = 2k. T h e closed loop i s therefore stablef o r k < 1/2.
N (x)

1.1 Definit.ions of Conic and Posifize Relations
DEFIKITION: A relation H in 6i i s interiorconic i f there are real constants r>_O afad c f o r which the inequality

IJ( H x- ~~ I J )CX

I rJj.~tlJ E Do(H); t E T ] (14) {X

i s satisfied. H is exterior conic if the inequality sign in (14) i s reversed. H i s conic if i t i sexterior conic or interior conic. The constant c will be called the center parameter of H , a n d r will be called the radius parameter.
Thetruncatedoutput(Hx), of a conic relation lies either inside or outside a sphere in X, with center proportional to the truncated input x t and radius proportional to IIxtll. The region thus determined in X , X X , will be called a "cone," a term suggested by the folloming special case: EXAMPLE 3: Let H be a relation on Lps (see Example 1); let Hx(t) be a function of x ( t ) , say Hx(t) =N[x(t)], where N has a graph in the plane; then, as shown in Fig. 6, the graph lies inside or outside a conic sector of the plane, witha center line of slope c and boundaries of slopes c-r and c+r. More generally, for H t o be conic [without Hx(t) necessarily being a function of x ( t ) ,t h a t is, if H has memory], i t is enough for the point [x(t), Hn.(t)] t o be confined t o a sector of the plane. In this case, it will be said t h a t H is instantaneously confined to a sector of the plane. Inequality (14) can be expressed in the form ! ! ( H x ) t - c x t l l ? - r l ! x , ! ; 2 1 0If norms . are expressed in
l2 However, in engineering applicationsit is often more interesting to prove stability in the L, norm. The present theory has been extended in that direction in the author's[2f]. The idea is [Zf] is t o transform L? functions into L , functions by means of exponential weighting factors.


I ..a;'-denotes slope

/ 4:



Fig. 4 . Graph of the relation in Example 2.

The usefulness of Theorem 1 is limited by the condition that the open-loop gain-product be less t h a n o n e a condition seldom met in practice. However,a reduced gain product can often be obtained by transforming the feedback equations. For example, if G Iis added to and subtracted from Hz, shown in Fig. 5, then ez remains as






I f

a>O then H-' is i n s i d e f l / b ,


lb: I a<O then H-l i s outside { l / a , f l/b CASE 2 : If a=O t h e n H - l - ( l / b ) I ) ( is

1. 1.

(v) Properties (ii), (iii), and(iv)remainvalidwith theterms"inside f } " and "outside { interckanged throughout. (vi) g ( H ) s m a x ( l a l , I b l ) . H e n c e i f H i s i n { - r , r ] th,en g(H) Ir.


Interior of sector is shaded.

Fig. 6 . A conic sector i n the plane.

T h e proofs are in AppendixA. One consequence o these f properties is that it is relatively easy to estimate conic bounds for simple interconnections, where it might be more difficult, say, to find Lyapunov functions.

terms of inner products then, after factoring, there is obtained the equivalent inequality

4.3 A Theorem on Boundedness
Consider the feedback system of Fig. 1, and suppose t h a t Hz is confined t o a sector f a , b I t is desirable to find a condition on H I which will ensure the boundedness of the closed loop. h condition will be found, which places H1 inside or outside sector dependingon a and b , a and which requires either H or Hz to be bounded away 1 from edge the of itssectorbyanarbitrarilysmall amount, A or 6. THEOREM 2a: [ I n eqs. (1)-(2)] Let H and H be Conic 1 z relations. Let A and 6 be constants, of which one is strictly positize and one is zero. Suppose that

- axt, ( H x- bxt) I 0 [ x E Do(H); t E T ] (15) )~
\\-here a = c - r and b = c + r . I t will often be desirable t o manipulateinequalitiessuchas (15), and a notation inspired by Fig. 6 is introduced:
NOTATIOS: A conic relation H is said to be inside the sector { a , b } , i a s b and i Inequality (15) holds. H is f f outside the sector { a , b } i a s b and i ( 1 5 ) holds with f f the inequality sign rezersed.


T h e following relationship will frequently be used: If H is interior (exterior) conic with center c and radius r then H is inside (outside) the sector { c - r , c + r }. Conversely, if H is inside (outside) the sector { a, b } , then H is interior (exterior) conic, wit5 center (b+a)/2 and radius ( b - a)/2. DEFINITION: A relation H in ( is positive13 i f R

(I) - H z is insidethe sector { a + A , b - A ) where b > O , and, (11) H1 satisfies one of the following conditions. CASE l a : If a> 0 then H I is outside


[ E Do(H); t E T I . . x



A positiverelationcan be regarded as degenerately conic,withasectorfrom 0 t o E . [Compare (15) and (16).] For example, the relationH on L Z e positive if i t is is instantaneously confined (see Example 3) to the first and third quadrants of the plane.
4.2 Some Properties of Conic Relations Some simple properties will be listed. I t will be assumed, in these properties, that H and H are conic 1 relations; that H is inside sector the { a , b ] , with b>O; t h a t H isinside { a , , b l ) with b1>0; and that 1 k 2 0 is a constant. (i> I is inside { I , I (ii) K is inside { k a , k b } ; - H i s inside { -b, - a } . H (iii) SUM RVLE: (H+H1) is inside { a + a l , b + b l } . (iv) INVERSE RULE
13 Short for "positive semidefinite." "nondissipative" have also been used.

l b : If a < O then H1is inside


2: I a = 0 then f


+ (+- 6)

is posit.ive; .in addition, i A=O then g(H1) f

< =.
Tlzen El and E2 are bounded. T h e proof of Theorem 2a is in Appendix B. Kote that the minus sign in front H z reflects an interest in negaof tive feedback. EXAMPLE 4: If H1 and Hz are relations on LZe instantaneouslyconfined to sectors of the plane (as in Example 3), then the closed loop will be bounded if the sectors are related asin Fig. 7. (Uore realistic examples will be discussed in Part 11.)






THEOREM 2b: Let HI and Hz be incrementallyconic relations. Let A and 6 be constants, of which one is strictly positive and one i s zero. Suppose that,

CASE 2. a = O

(I) - H P isincrementallyimidethe sector {a+A, b - A } , where b>0, and, (11) H1 satisfies one of the following conditions:

l a : If a>O then H1is incrementally outside


l b : If a <O thetz


is incrementally inside


---6 a


If a=O then



Fig. 7.

H2and H I

is incrementally positive; in addition, i f A = 0 then g(H1) c13 . < Then El and Ez are input-output stable. T h e proof is similar t o t h a t of Theorem la, and is omitted.

4.4 Incrementally Conic and Positive Relations

N e s t , i t is desired t o find a stability result similar to the preceding theorem on boundedness. T o t h i s e n d t h e 5 . CONDITIONS V O L ~ I N G IS POSITIVE REIATIONS recent steps are repeated with all definitions replaced A special case of Theorem 2, of interest in the theory by their "incremental" counterparts. of passivenetworks, is obtainedby,ineffect,letting DEFINITION: A relation H in & i s incrementally intex rior (exterior) conic <f there are real comtants r>O ami c a = 0 and b - + . Both relations then become positive; also, one of the two relations becomes strongly positive, for which the inequality 1.e. : (Hx - H y ) t - C ( % - u)tll 5 (x - y)tl( DEFINITION: A relation H in & i s strongly (incrementally) positive i f , for some u>O, the relation (H--crI) is [x,y E W H ) ;t E TI (17) (incrementally) positive. i s satisfied (with inequality sign reversed). A n incremenThe theorem, whose proof is in Appendix C , is: tally conic relation H i s incrementally inside (outside) THEoREbf 3:'s (a) [In eqs. (1)-(a)] If H1 and -Hz are the sector { a , b } , if a 4 b and i f the inequality positive, and if - H Z is strongly positive and has finite ( ( H X - ~ 3 ' - ~ ( x - Y)~, - H Y ) - b ( r ) Q (HX ~ I gain, then E l and E? are bounded. (b) If H1 and -H2 o are incrementally positive, andif -Hz is strongly incre[x, y E W H ) ; E TI (18) mentally positive and has finite incremental gain, then t i s satisfied (with ineqzlality sign reversed). A relation H E1 and Ez are input-output stable. For example, if H 2 on LZe instantaneously confined is in i s incrementally positive'* if t o a sector of the plane, then, under the provisions of ( ( X - y ) t , (Hx - H>l)t)2 0 [ X , y E Do(H); t E TI. (19) Theorem 3, the sector of H z lies in the first and third quadrants, and is bounded away from both axes. E x A m w z 5 : Consider therelation H on L2e, with H x ( t ) = L V [ x ( t ) ]where :I7is a function having a graph , in the plane. If M is incrementally inside f a , b then B 5.1 Positivity and Passivity in :Yetzmrks satisfies the Lipschitz conditions, -y) 4 ;V(x) - X(y) a(x A passive element is one that always absorbs energy. 5 b ( x - y ) . T h u s llr' lies in a sector of the plane, as in the I s a. networkcomisting of passiveelementsnecessarily nonincrementalcase (see Fig. 6), and inadditionhas stable? An attempt will be made to answer this question upper and lower bounds to its slope. for the special case of the circuit of Fig. 2. Incrementally conic relations have properties similar First, an elaboration is given on what is meant by a t o those of conic relations (see Section 4.2).



The terms "monotone" and "incrementally passive" have also been used.

15 Aq variation of this resultwasoriginallypresented in [2dl. a restriction of related results, with Kolodner [8] has obtained linearity on one of the elements.




positiveconstant."Infact,the conicsectorsdefined a Kyquist hereresemble thedisk-shapedregionson j,Di(t)-,l(f)dt, the condition for passivity is that this chart. However, Theorem 2 differs from Kyquist's Criand integral be non-negative. Xow, let 2 be a relation map- terion in two important respects: ( 1 ) Unlike Kyquist's ping i into o; by analogy with the linear theory, it is Criterion, Theorem 2 is not necessary, which is hardly Z natural to think of Z as an impedance relation; suppose surprising, since bounds on i Y 1 and H are assumed in Z is defined on L2e, where the energl; integral equals the place of a more detailed characterization. (2) Kyquist's innerproduct {i, thenpassivity of the element is criterion assesses stability from observation of only the equivalent to positivitl-of Z. Similarl>-? Y on L Z e an eigenfunctions exp (jwt), where Theorem 2 involves all if is admittance relation, which maps c into i, then passivity inputs in X,. There is also a resemblance between the use of the is equivalent to positivity of Y. Kow consider the circuit of Fig. 2 . This circuit con- notions of gain and inner productas discussed here, and sists of an element characterized by an impedance rela- the use of attenuation and phaseshift in the linear thefurther discussion of this topic is postponedto tion 2 .a n element characterized by an admittance rela-ory. 2 a tion Y1, voltage source E', and a current source i. T h e Part 11, where linear systems will be examined in some detail. equations of this circuit are, One of thebroaderimplications of the theory developed here concerns the use of functional analysis for the study of poorly defined systems. I t seems possible, from only coarse information about a system, and perhaps even without knowing details internal structure, of to make useful assessments of qualitative behavior. I t is observed that these equations have the same form APPESDIS as the Feedback Equations, provided that the sources i and E' are constrained by the equations v=alx+wl, d . Proofs of Properties (i-vi) and i=a,s+w,. (By letting al=O the familiar ''parallel Properties (i, ii). These two properties are immedicircuit" obtained. is Similarly, letting by a 2 = 0 the ately implied by the inequalities "series circuit" is obtained.) Thus there is a correspondence between the feedback system and the network con( ( 1 x ) t - 1.xt,(I.x)t - 1.mt> = 0 sidered here. Corresponding to the closed loop relation ( ( c H x )- cast, (cH.Y)~ CbSt) ~ E1 there is a voltage transfer relation mapping z into = c'((H.Y)~ ax1, (H.t-),- bxt) 5 0 "1. Similarly,correspondingto E? there is a current transfer relation mapping i into i z . If Theorem 3 is now in which c is a (positive or negative) real constant. applied to eqs. (20)-(21) i t ma>- be concluded t h a t : If Property (iii). I t is enough t o showthat ( H + H l ) both elements are passfile, and ti, in addition, the relation has center +(b+bl+a+al) and radius +(b+bl-a-al); of one of the elements is strongly positisle and has -tiTzite the following inequalities establish this: gain, then the network tra-nsfer relations are bozlnded.

passive element. Consider an element having

a current

i and a voltage z!; the absorbed energy is the integral



The main resulthere is Theorem 2. This theorem provides sufficient conditions for continuit>- and boundedness of the closed loop, without restricting the open loop t o be linear time or invariant. Theorem 2 includes Theorems 1 and 3 as special cases. However, all three whereeq. (,4lb) follows from eq. (Ala) since H has theorems are equivalent, in the sense that each can be center $(b+a.) andradius +(b-a), andsince H I has derived from a n y of the others by suitable transforma- center +(bl+aJ and radius t ( b l - a l ) . a tion. Property (iv). There are resemblances between Theorem 2 and Kyquist's Criterion. For example, consider the followCASES l a AND l b : H e r e a#O and b>O, and ing, easill; derived,limitingform of Theorem 2 : "If Hz = k I then a sufficient condition for boundedness of the closed loop is t h a t H l be bounded away from the critical value - ( l / k ) I , i n the sense that

for all x in

X,and t in T , where 6 is an arbitrarily small




where H-'x=y and x = Hy.Since, by hypothesis, H is inside { a , b ] and b > 0, the sign of the last expression is opposite t o t h a tof a. Thus the Inverse Rule is obtained. CASE 2: Here a =O. The property is implied by the inequality,

(111) (Using Fig. 5 as a guide,) define two new ments of X,,
yz' =

ele(A6a) (A6b)


el' = el

+ cyl.

+ cez

Property (v). Simply reverse all the inequality signs. Property (vi).
II(Hx>,ll I II(H4, - +(b a)xt;I li+(b u)xt)i (TriangleIneq.)





I t shall now be shown that there are elements H1'eI' and Hz'ez' in X,that satisfy eqs. (A3)-(A4) simultaneously withtheelements defined in (1)-(111). Takingeqs. (A3)-(A4) one at a time: Equation (A3a). Substitutingeq.(la) for el i n eq. (A6b), and eq. (lb) for yl,
e; =

- $(b


- a)ll~tll




1 b l ).ljxt,l

++ I b +u I

(Ma) (42b)

- cwz)


(a1 - cu2)x

+ (yp + c e ? ) .



where eq ( X b ) follows from eq (A2a) since, from the hypothesis, H has center +(b+a) and radius +(b-a). I t follows that g ( H )S m a x ( l a \ , l b l ) . Q.E.D.

B. Proof of Th.eorem 2a
T h e proof is divided into three parts: (1) The transformation of Fig. 5 is carried out, giving a new relation Ez'; is shown to contain Ez. (2) T h e new gain prodEz' uct is shown to be less than one. (3) Ez' shown to be is 1 bounded, by Theorem 1; the boundedness of Ezand E follows. Let c=+(b+a) and r = $ ( b - a ) .

If wl'=wl-cwz and a ~ ' = a ~ - c a then, with the aid of ~, eq. (-464, eq. (-47) reduces to eq. (-43a). Equation (A3b) : This is merely eq. (lb), repeated. Equation (A4a): Recalling that H2'=H2+cI, i t follows, from eqs. (A6a) and (2a), that there is an Hzrez in X,for which eq. (A4a) holds. Equation (A4b): If eq.(A6b)issubstitutedfor el in eq. (Zb), i t is found that there exists Hl(el'-cyl) in X,such that y1= Hl(e1' -cyl). Therefore, (after inversion) (after rearrangement) (after inversion)



el' - cyl el'

+ cl)yl


+ cI)-le;.

B.1 Transfornzation of Eqs. (1)-(2)
T h e proof will be worked backwards from the end; the equations of the transformed system of Fig. 5 are,
el' =

T h a t is, there exists Hl'el' in X, for which eq. (r14b) holds.Sinceeqs. (A3)-(A4) are all satisfied, (x, e 2 ) is in ES'. Since ( x , en) is an arbitrary element of E z , Ez' contains Ez.

B.2 Boundedness of E,'
I t will be shown thatg(H1') -g(Hz') 1. < The Case A > 0, 6 = 0 : g(Hz') will be bounded first. Since Hzis in { -b+A, - a - A } by hypothesis, ( H z + c I ) is in { -b+A+c, -a-A+c) by the Sum Rule of Section 4.2. Observing that (H,+cl) = H 2 ' , t h a t (-b+c) = - r , and that (-a+c) = Y , i t is concluded that Hz' is in { - r + 4 , r-A). Therefore g ( H s ' ) _ < r - 4 . Kext, g(H1') will be bounded. In Case l a , where a>O and HI is outside

w; + u { x + ypl

= wz

+ + y1

(A3a) (A3b) (A4a) ( i Ab)

y; = He'ez

yl = Hl'e;

Hpl = (Hz c l )

H '= I


+ +



(A5 b)

(It may be observed that these equations have the same the Inverse Rule of Section 4.2 implies that H1-l is outform as eqs. (1)-[2), but H1 is replaced b y HI'and H2 side { - b , -a ] ; the same result is obtained in Casesl b is replaced by Hz'.) Ezr be the closed-loop relation Let and 2. In all cases, therefore, the Sum Rule implies that t h a t consistsofallpairs (x, ez) satisfyingeqs. (A3)(H~-'+cl) isoutside { - r , r ) . By theInverseRule (A4). I t shall now be shown t h a t Ez' contains E z , t h a t again, (H1-l+cI)-l is i n is, that any solution of eqs. (1)-(2) is also a solution of 1 eqs. (A3)-(A4); thusboundedness of E,' will imply boundedness of E z . In greater detail Therefore g(H1') 5 l/r. (I) let (x, ea) be any given element of Ez. Finally, (11) Let el, yl, yz, Hlel, and H2ez be fixed elements of X,that satisfy eqs. (1)-(2) simultaneously with x and e2.


- -1 ,
u '

- 1


{- 7 , f).



The Case A = 0, 6 >0: It shall be shown that this - (xt,( ~ 2 % ) ~ )2 uJI ztj!z (A91 I' I is a special case of the case A > 0, 6 = 0. In other words, (A10) II(H%&ll2 I I ( 1 q l 2 X2 i t will be shown that there are real constants b*, and a*, A* for which the conditions of the case A > 0 , 6 = 0 are for any x in X, and for any t in T. Hence, for any Y > 0, fulfilled, but witha repIaced by a*, b by b*, and A by A*. Y.# 5 (x' - 2 y U r?)ll.v,liz. (AH) Consider Case la, in which a > 0. (Cases l b a n d have 2 similar proofs, n-hichwill be omitted.) It must be shou-nEquation ( A l l ) was obtained by expanding the square on its I.h.s., and applying eqs. (A9) and (A10). Cont h a t : (1) -H2 is in {a*+A, b*-A}. (2) H1 is outside stants h, r , and A, are selected so t h a t X>a, r =XZ/u, 1 and A = r [I - 4 - (U/X)~]. Now it can be verified that, 1 {-a.J;}. for this choice of constants, the term ( X Z - ~ Y U + ~ ~ ) in ; .X Without loss of generality it can be assumed that 6 is eq. (All) equals ( Y - A ) ~also, 0 < A < r since (/ ) <1); therefore eq. (All) implies t h a t H is conic with center z smaller than either l / a or l / b . Choose a* and b* in the --I andradius r-A. Q.E.D. ranges a b -ACKNOKLEDGMEKT < a.* < a and b < b* < ___ 1 - b6 l+d The author thanks Dr. P. Falb for carefully reading the draft this paper, and for making a number of of valuSince - H z is in { a , b ] by hypothesis, and since a* <a able suggestions concerning its arrangement and conand b*> b by construction, there must be a A*>O such cerning the mathematical formulation. t h a t H z satisfies condition (1). Since H1 is outside

]I(H~.Y)~ +



by hypothesis, and since by construction


condition (2) is satisfied. Hence this is, indeed, a special case of the previous one.
B.3 Conclmion of the Proof

Sinceg(H,')-g(Hz')<1, Ez' is bounded by Theorem 1, and so is Ez, which is contained in E2'. i\Ioreover, the boundedness of Ez implies the boundedness of E l ; for, if (x,el) is in E1 and (x, e2) is in EB,then

Thus, if 1 x :<const. and IIe211 <const., then ljelll I c o n s t . 11 (Inequality (AS) wasobtainedbyapplyingtheTriangleInequalityandInequality (4) toeq.(la),and taking the limit as t+=. It may be noted that g(H3) < m , since -HZis in { a , b by hypothesis.) Q.E.D.


C. Proof of Theorem 2
This shall be reduced to a special case of Theorem 2 [Case 2 with 6=0]. In particular, it shallbe shown t h a t there are constantsb > 0 and A > 0 for which (I) - H z is inside { A , b - A l , and, (11) the relation [ H l + ( l , ! b ) l ]is positive. [ H l + ( l / b ) I ] is clearly positive for any b > 0, since byhypothesis H 2 is positive;thesecondcondition is thereforesatisfied. T o provethefirstconditionit is enough t o show t h a t Hzis conic with center --I and radius r - A , where r = b/2. This is shown as follows: Thehypothesisimpliesthat,forsomeconstant u>O and for any constant h > g ( H z ) , the following inequalities are true

[l] V . 31. Popov, "Absolute stability of nonlinear systems of automaticcontrol," Azdomationand RenzoteControl, pp. 857-875, March 1962. (Russian original in August 1961.) [2] (a) G;, Zames, "Conservation of bandwidth in nonlinear operatlons, M.I.T. Res. Lab, of Electronics, Cambridge, Mass., Quarterly Progress Rept. 55, pp. 98-109, October 15, 1959. (b) , "Yonlinearoperatorsforsystemanalysis,"M.I.T. Res. Lab. of Electronics, Tech. Rept. 370, September 1960. "Functional analysisappliedtononlinear feedback ; , (c) vol. CT-10, pp. 392systems, IEEE Trans. 0% Circuit Tl~eouy, 404, September 1963. idi , "On the stability of nonlinear, time-vaqing feedback systems," Proc. A7EC, vol. 20, pp. 725-730, October 1964. (e) , "Contracting transformations--l t h e o n of stability and iteration for nonlinear, time-vaning systems,'! Suntnzuries, 1964 Internat'l Conf. olt &furuzzases, Circuit Tlteory, and Information T h e o ~ y pp. 121-122. , if) , "Sonlineartimevarl-ingfeedback systems-Conditions forLOC -boundedness derived using conic operators on exponentially weighted spaces," Proc. 1965 dllertott Conf., pp. 460-4i 1. [3] K. S. Xarendra andR. AI. Goldwyn, " X geometrical criterion for thestability of certain nonlinearnonautonomous s:-stems," IEEE Trans. on Circuit Theory (Correspondence), vol. CT-11, pp. 406-108, September 1964. [4]R. I' BrockettandJ. \V. U'illems, "Frequencydomain sta\. bility criteria," pts. I and 11, 1965 Proc. J i t dulometic Cotztrol on Conf., pp. 735-747. [j](a) I. I[:. Sandberg, "On the properties of somesystems that distort signals," Bell Sys. Tech. J., vol. 42, p. 2033, September 1963, and vol. 43, pp. 91-112, January 1964. (b) -,"On the L?-boundedness of solutions of nonlinear functionalequations," Bell. Sys. Tech. J., vol. 43, pt. 11, pp. 1581-1599, July 1964. [6] J. Kudrewicz, "Stab!lity of nonlinear feedback systems," Automatzka i Telenteckm&a, vol. 25, no. 8, 1964 (and other papers). [i] E. H. Zarantoldlo, "Solving functional equations by contracuve averagmg, U. S. A%rmy hIath.Res.Ctr.,hIadison, Ii?s. Tech. Summary Rept. 160, 1960. [8] I . I. Kolodner, "Contractive methods for the Hammerstein equation in Hilbert spaces," Cniversity of S e w Mexico, Albuquerque, Tech. Rept. 35, July 1963. [9] G. J . l.finty, "On nonlinear integral equations of the Hammerstem type," survey appearing in ,Vononlirzear Integral Epzratiotzs, P. A I . Anselone, Ed. XIadison, [Vis.: UniversityPress, 1964, pp. 99-154. [IO] E'. E. Browder, "The solvability of nonlinear functional equations." Duke X a t h . J . , vol. 30, pp. 557-566, 1963. [ l l ] J. J. Bongiorno, Jr., ''An extension of the Nyquist-Barkhausen stability criterion to linear lumped-parameter systems with time var)-ing elements," IEEE Trans. on rlzrtomatic Control(Corresponden.ce), vol. AC-8, pp. 166-170, April 1963. [ 121 -4. h-. Kolmogorov and S . V. Fomin, Functional d nalysis, vols. I and 11. Xew York: Graylock Press, 195i.



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