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708

which contradicts the requirement that ek be boun ded. It is not difficultto show that there exist no i cop, an d P that ensur e stability for a l lallowable parameter variations. Q.E.D.

REFERENCES

J. A. Roluik and I. M. Horowitz, "Feedback control system synthesis for plants1. M. Horowiu, Synrhesis of Feedbuck Sysrems. New York: Academic, 1963.with arge parameter variations,- I E E E T r m . Auromr . Conrr. (ShortPapers), vol.AC-l4,-pp. 714-718, Dec. 1969.D. D. Siljak, Nonlinear Systems. New York: Wiley,1969.D. M. Salmon, "Minimax controllerdesign," I E E E T rm . Auromor. Conrr., vol.AC-13. pp. 369-376, Aug. 1968.H. P.Horisberger,"Controlof linear systems with argeparametervariations."E. W. Cheney and A. A. Goldstein, "Newton's method of convex progamming andPh.D. dissertation. McGill Univ., Montreal, P.Q., Canada. Aug. 1973.Chebyscheff approximation," Numer. " a h . . pp. 253-259. 1959.E. Pol&, CompurnrionalMerhods in Oprimizarion. New York Academic, 1971.T. L. Johnson and M . Athans "On the design of optimal constrained dynamiccompensators for linear constant systems,'' I E E E T rm . Auromr . Conrr. (ShortPapers), vol. AC-15, pp. 658-660, Dec. 1970.H. P.Horisberger and P. R . Bilanger, "Solution of the optimal constant outputNotes and Corresp.), vol. AC-19, pp. 434435. Aug. 1974.feedbackproblembyconjugategradients," I E E E T r m . Auromor . Conrr. (Tech.J . M . Danskm. The %o y of Max-Min. New York: Springer, 1967.

The Stabilityof Nonlinear DissipativeSystemsDAVID HILL, M E M B E R , IEEE,AVDPETER MOYLAN, MEMBER,EEE

Abstract--This short paper presents aechniqueoreneratingLyapunov functions fo r a broad class of nonlinear systems represented bystat e eqnations. The system, fo r which a L yapunov function is required, isassumed t o have a property called dissipativeness. R oughly speaking, thismeans hat the system absorbsmore energy from the external world than itsupplies. Differenttypes of dissipativenesscan be considered dependingonhow on e chooses to defin e "power input." Dissipativeness is shorn to becharacterized by the existence of acomputable unctionwhich can beinterpretedas the "stored energy"of l b e system. Under certain conditions,this energy function is a Lyapunov function which establishes stability, andin some cases asymptotic stability, of the isolated system.

I. INTRODUCTIONIn the study of a ph ysical system, such as an electrical network or amechanicalmachine, heconcept of stored energy is often useful indedu cing the behavior of the system. In many contr ol problems, how-ever, one is dealing with an abstract mathem atical model where it maybe difficult or even impossible to find some property of the model whichcorresponds to physical energy. In this short pap er we show, for a certainclass of nonlinear systems, that an ' 'energy" approach can still be usefulin stability analysis, despite the fact that the "energy" might not haveany physical meaning. In effect, OUT echnique is a method for generat-ing Lyap unov functions.The theorydescribed herehas its origins in workbyMoylan andAnderson [ 1]-[3] a nd Willems [4] on the prop erties of passive systems.

Manuscript received August 19, 1975; revised May 18, 1976. Paper recommended byG.L. Blankenship, Chairman of the IEEE S C S Stability. Nonlinear, and DistributedSystems Committee. This work was supported by the Australian Research Grants Com-mlttee.tle, Newcastle. N.S.W., Australia.The authors

are with the Department of Electrical Engineering. University of Ne wa s-

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, OCTOBER 1976F or our purposes, a passive system can be defined a s a system whichalways dissipates energy, provided the energy input to the system is suchthat he produ ct of system input and outpu t represents input power.(This makes sense phy sically if the system is a n electrical network [SI; nother cases, there mightbe no simplephysical interpr etation.) Severaluseful properties of passive systems have been noted in [2] for the linearcase and in [l], [3] for nonlinear systems.Th e notion of passivity was extended by Willems [4] to allow a mo regeneral definition of input power. A "dissipative system" is defined in [4]to be one for which a supply rate (input power) and storage function(stored energy) can be found, with the property that (in a sense mademore precise in [4]) energy is alw ays dissipated. As one might expect,there are som e useful con nectio ns between dissipativeness (in Willems'sense)' and Lya pun ov stability.Thepresent hortpaperexten ds he passivity esults of [I] usingconc epts very similar to those of Willems [4]. Our app roach differs fromthat of Willemsprimarily because we treat bs ipa tiv en ess as an in-put-ou tput prope rty; hat is,we do not postu late he existence of anintern al storage func tion. However, we ded uce the existence of such astorage function (in general nonunique), so it could be argued that thisdifference is unimportant. A morenoticeabledifferencebetween OUTresults and those of [4] is that, by sacrificing som e generality, we ob tainresults which are considerably mo re explicit. The centra l result of thisshort paper is an algebraic criterion, in terms of fu nction s of the systemstate, for the input-o utput property of dissipativeness. This result is thenused to d erive stability criteria.Th e structure of the short paper is as follows. In Section I1 we derivethe algebraic criterion or dissipativeness,which leads to compu tablestorage unctions.These unctionshave the properties of Lyapunovfunctions and in Section I11 a re use d to g et s t a b ~ t yesults. In SectionIV, the special cas e of pa ssive systems is consid ered.

11. ALGEBRAIC CRITERIONOR DISSIPATIVENESSTh e systems to be studied are described by the equation s

i = f ( x ) + G ( x ) uy = h ( x ) + J ( x ) u . (1)

The values of x, u, and y lie in R n , R m , and RP, respectively. Theadmissable contr ols re taken tobe locally squar e integrable. T h efunc t ions f :R" - tR" , G : R " + R n X m ,h : R " - t R P of the state vector xsatisfy f(0)=0, (0 )=0, and the following assumption.Assumption I : The functions appearing in ( I ) have sufficient smo oth-ness to make the system well define d; hat is, for any x ( t J E R" an dadmissible u ( .), here exists a unique solution on [ to .00) uch that y ( ) islocally square integrable.Th eabove ormulation includes abro ad class of systems. In fact,Balakrishnan [ 7] has claimed that, und er quite unrestrictive conditions,any time-invariant finite dimensional system can be represented by ( I )with an appro priate choice of the state vector. However, it is not know nwhether the transformation in [7] wi preserve our Assumption 1 andfinite-dimensionality of the state space. Rath er than explore these diffi-culties, we will adopt the attitudeha theepresentation ( I ) issufficiently general to warr ant study in its own right.Motivated by Wdlems [4], we associate with ( I ) a sup& rate

w ( u , y ) = y ' 9 + 2 y ' S u + u ' R u (2)where Q E R P X p ,S E R P X m , nd R ERmX" are constant matrices withQ and R symmetric. The supply rate is an abstraction of the concept ofinput power. In physicalsystems, inpu t power is associatedwith theconc ept of sto red energy. For moregeneral abstrac t systems,such asgiven by ( I ) , physical reasoning fails us , but we can define a possible

on differential equations (see, for example, [6D. Despite the coincidence in terminology,'The word "dissipative" is used in an entirely different sense in some of the iteraturethere IS no connection between the two Qpes of dissipativeness defined in [4] and [6]. Ourdefinition is essential ly equivalent to that of Willems [4].

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PAPERS 709nam e stored energy.Consider the unc tionand t l > to , an dan y x(ro) , straightforward use of (5) gives

(1 ) and x(O)= xW This is called the available storage in [4]; ite interpreted as the maximum amount of energy which may befrom the system (1). No te that @= ( x )> 0 fo r all x .he seque l, we shall impose the following assumptions on the system(2).2: Th e state spa ce of the system (1 ) is reachable from the

x , and I , , there exists a to< t , and anble contro l u( .) such that the state can be riven from x( ta )=O to3: The available storage +, (x ) , when it exists, is a differen-function of x .4: For any y # 0 there exists some u such that the supply(2) satisfies w ( u , y ) t2, we have

~ o ~ l w ( t ) d r0 (4 )x ( t o ) =O an d w ( t ) = w [ u ( t ) ,y ( t ) ]evaluated along the trajectory of

por tant special cases of dissipative systems are the following.w = uygain systems w = k%u -yy, k being a fixed scalar.

al, but useful, observation is that if (1) is dissipative with respectwi, i = 1, . . I , then it is also dissipative with respect to

{q} f nonnegative coefficients.dissipativeness, as just defined, is an input-output prop-e system. The following theorem, which is the central result ofer, shows tha t dissipativeness can also be characterized in1). A restricted version of

[ I ] .I : A necessary and sufficient condition for (1) to be dissipa-respect to supply rate (2) is that there exist r e a l functions: R - + R q ,and W : + R q X m (for some integer q ) satisfying

@ ( x )> 0, d o ) = 0V @ ( ~ ) f ( x ) = h ( x ) Q h ( ~ ) - I ( x ) l ( ~ ) ( 5 )

$ G ( x ) V @ ( x )= S ( x ) h ( x ) - W ( x ) I ( x )i ( x ) =W ( x )W ( x )

x, whered ( x )= R + J ( ~ ) s +J ( x )+J J ( ~ ) Q Jx )

s ( x ) = Q J ( x ) + S .To prove sufficiency we suppo se that +(a), I ( - ) , and E(.)

( 5 ) is satisfied. Then for any admissible u ( any to

+ O t 1 [ I ( x ) + ( x ) u l [ I ( x )+W(x)uldt . ( 6 )Setting x( ta )= 0, we have condition (4).For necessity, we proceed to show that +,(-) given by (3) is a solutionof (5) for some appropriate functions ( . ) an d W( . ) .For any state x. at r =O , there exists by Assumption 2 a time r - < Oand an admissible control u ( . ) defined on [t-l,O] such that x(t-,)=Oand x ( 0 ) = x W From (4), then,

The righ t-hand side of this inequality dep ends only on x@ whereas u ( . )can e hosen rbitrarily on [O,T]. Hence, there exists aunction .C :R + R of x such that(7)

whenever x ( 0 ) = x W From (9,e have @,(x )< co for all x . Also, dis-sipativeness implies tha t $1~(0 )0.Now it is shown in [4]ha t @a satisfies

for all t , > to and all admissible u ( + ) ,where x ( t o ) = x o and x ( t l ) = x l .Assumption 3 then gives

along any trajectory of (1). To turn this inequality into an equality, weintroduce a functiond :R X R m + R via

= - V @ = ( x ) [ f ( ~ ) + G ( x ) u l + ~ [ ~ , h ( x ) + J ( x ) u l .9 )From (8), d ( x ,u )> 0 for all x an d u . In addition, it is clear from (9)that d ( x , u ) is quadra tic n u. Com bining these wo observations, itfollows that d ( x , u ) may be factored as

d ( x , u ) = [ I ( x ) +( x ) u l [ I ( x ) + W ( x ) u l (10)for some functions I : R n + R q , W : R n + R q X m , nd some integer q.(No tice, however, that he choice of q, I , an d W is far rom beingunique.)Substituting (10) into (9) gives- V @ , ( x ) f ( ~ ) - V @ , ( x ) G ( x ) u + h ( x ) Q h ( x ) + 2 h ( x ) S ( x ) u + u d ( x ) u

= I ( x ) I ( x ) + 2 I ( x ) W ( x ) u + u W ( x ) W ( x ) ufor all x and u . Equ ating coefficients of like powers of u , we obtain (5)with @=9 nEquation (6) can be interpreted as expressing an energy balance forsystem (l),and shows that he unctions @(e) are torage unctionssatisfying Willems definition [4]. The p roof of Th eorem 1 uses a dif-ferent approa ch to that adopted in [I ] for the special case of passivity.The re the necessity part of th e proof relied uponHamilton-Jacobitheory. For a mo re complete discussion of dissipativeness alon g the linesof this short paper, the report [8] can be consulted. In particular, it isshown in [8] that healgebraicequations (5) possess maximum andminimum solutions which correspond to the equired suppb and aoailabksrorage, defined in [4]. (Equation (3) provides the minimum solution.This observation will be useful ina ater section.) -We can now give characterizations of dissipativeness, with respect to

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710 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, OaOBER 1976particular supply rates, by substituting the appropriate Q, , S into (5).Example I : Fo r finite gain, we set Q= - , S=0, an d R = k21 wherek is a scalar. This gives ( 5 ) as

V + ( x ) f ( x )= - ( x ) h ( x )- ( x ) [ x )f G ( x ) V @ ( x ) = J ( x ) h ( x ) - W ( X ) ~ ( X )

k Z I - J ( x ) J ( x ) = W ( x ) W ( x )which,when pecialized to linear systems, s a generalization of th eBounded Real Lemma [5].A differential version of (6)s given by the following result.Corollaly: If system (1) is dissipative with respect to supply ra te (2),then there exists a real func tion +(.) satisfying + ( x ) > 0, +(O)= 0, suchthat I

for the system (1).Proof: This is simply a restatement of (9), with d ( x , u ) defined as in(10). A

111. STASILITY F DISSIPATIVEYSTEMSWillem s [4] suggests the usefulne ss of th e theory of dissipativ e systemsin he investigatio n of system tabilityvia Lyapun ov methods. Th eLya pun ov func tions are genertllized energy functions, corresponding to

I$(.) in Theorem 1. This leads us to consider conditions for which +(.) ispositive definite, in the sense that + ( x )>0 for all x #O .Definition 2: Th e system (1 ) is zero-state detecta ble if, for any trajec-tory such that u ( t ) = O , y ( t ) = O implies x ( t ) =O .Lemma I: If th e system (1) is dissipative with respect to supply r ate(2) and zero-state detectable, then all solutio ns +(.) of ( 5 ) are positivedefinite.Proof: The minimum solution of ( 5 ) , given by (3), is positive defi-nite if .the re exists a con trol such t hat w(t)< 0 on [ r , , w), with strictinequality on asubset of positive meashe. Now from Assumption 4,there certainly exists u such that w(b,y) 0 .

One way to interpret these definitions is as follows. A YS P system is apassive system for which a small am oun t of positive feedback does no tdestroy the passivity property; a USP system can be similarly interpretedin terms of feedforward. (The usual definition [lo], [111of strict p assivitycorresponds to our definition of USP.) A VS P system is, of course, onewhich s both USP an d YSP. Another example of the possibility ofcombining supply rates in the manner mentioned in Section I1 is pro-vided by obsewing that a USP and finite gain system is VSP.We now summarize stability results for passive systems which followimmediately from Theorem 2.Theorem 3: For systems of the form (l), passive and USP systems arestable, while YS P an d VSP systems are asymptotically stable.Th e following example serves to illustrate the ideas of tlus short paper.Example 2: We consider the equationi + f ( x ) i + g ( x ) = u (12)

wheref( . ) , g ( . ) are functions of the scalar variable x an d u is a forcingterm. Setting u = 0 gives the Lienard equation [9].Letting F ( x ) = J G f ( u ) d u, a set of first-order eq uation s equivalent t o(12) isx, = - F ( x , )+x 2x2= - g ( x , ) + u

where x , = x .We combine (13) with the system output equation defined byy=au,-PF(x,), a > B >O

an d consider the passivity of this ystem (of cour se we could findcond itions for other forms of dissipativeness). It is convenient to defineG ( x ) = J G g ( u ) d u .Th e calculations are straightforward and involve substitutio n into (5)

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of conditions for theexistence of a solution. We on lytwo special casesor a = 1/2, B = 0, passivity follows if

G ( x )> 0

g ( x ) F ( x ) 0.is+ , ( x ) = t [ x + F ( x ) I 2 + G ( x ) .

a = p = $, passivity follows ifG ( x ) > O

f x ) 0.

+ 2 ( x ) = $x2+G ( x ) .+2 ar estandard Lyapu nov unctions used for thef th e stability of th eLienard equation: choice of +2 as ais motivated by its interpretation as the sum of thetic and pote ntial energies of (12), and + I = + 2 + i F ( x ) + i F ( ~ ) 2sd the modified energy fun ction [9]. We have now shown that theyas stored energy functions (depending on howdefines the system outpu t) arising from a study of the passivity of

V. CONCLUSIONhemain result presented here is Theo rem 2,which relates theility of abroad class of non linear systems to the input-outp ut

y of dissipativeness. We can char acter ize dissipativeness by theof a computable function +(.) of the state. This fun ction isf the system and unde r certain condi-is a Lyapunov function.ur approa ch can also be applied profitably to intercon nected sys-

can, for example, deriveuno v versions of t he stability criteria of Za mes [ l ], using afunction which is the sum of the storage functions for thef these and oth er results are currently inon and will be reported separately.

REFERENCES

P. J. Moylan, Implications of passivity n a class of nonlinear systems, I E E ETrans. Automur . Con tr., vol. AC-19, pp. 373-381, Aug. 1974.B. D. Anderson, A system theory criterion for positive real matrices, S I A M 1.Contr, vol. 5, pp. 171-182. May 1967.P. J. Moylan and B. D. Anderson, Nonlinear regulator theory and an nverseoptimal control problem, I E E E Truns. Automn r. Conrr., vol. AC-18, pp. 460-465,J. C. Willems, Dissipative dynamical systems Part I : General theory; Part 11:Dec. 1973.

vol. 45, no. , pp. 321-393, 1972.Linear systemswith quadratic supply rates, Arch. RafioMi Mechanics and Ana!vsis,

Cliffs, NJ: Prentice-Hall, 1973.B.D. Anderson and S. Vongpanitlerd, Nerwork Analysis and Synrhesis. EnglewoodJ. P. LaSalle, Dissipative systems, n OrdiMly Differential Epuntiom 1971 N R L -MRC Conference, L.Weiss, Ed. New York: Academic, 1972.A. V. Balakrishnan, On the controllability of a nonlinear System Proc. Nur.A d ci. U.S.A., vol. 55, pp. 4 6 5 4 6 8 , 1966.for nonlinear dynamical systems, Dep. Elec. h g . , U n i v . Newcastle, Newcastle,D. J. Hill and P. J. Moylan, Cyclo-dissipativeness,dissipativenessand lorslessnessN ew South Wales,Australia, Tech. Rep. EE-7526, Nov. 1975.J. LaSaIle and S. Lefschea Smbi/ity by Lyapnms DirectMefhod. New Ya k :Academic, 1961.J. C.WiUems,The generation of Lyapunov functions for input-output stablesystems, S I A M 1.Contr., vol. 9, pp. 105-134, Feb. 1971.G. a m q On the input-output stabilityof ime-varying nonlinear feedbacksystems; Part I: Conditions derived using concepts of loop gain, conicity andpositivity, I E E E Trans. Auto mat. Conrr., vol. AGI I, pp. 228-238, Apr. 1966.

711Identification of Linear Systems with Time-DelayOperating in a Closed Loop in the Presence of Noise

E. GABAY AND S.J. MERHAV, MEMBER, IEEEAbstruct-Xbe subject of this short paper is on -lin e i d e n t i f i i n of theparameter vector defining a linear dynamical system which operates in aclased loop in the presence of noise, and incorporates a time delay. The

method is based on the equation error. Other known subsystems n theclosed-loop system increase the dimension of the closed-loop parametervector which ends to degrade the estimationconvergenmprocess. Bymeans of composite state variablq introduced in this short paper, thisincrease is prevented and the open-loop parameters are directly iden tifiedfrom dosed-loop input-output data.The pure time delay in the closed loopcauses a representation problem inthe equation error formulation.Tbis isovercome by composite delayed state variables. The value of the timedelay is determined by means of ex- parameters provided by a higherordermodelanda imple on-line search procednre. The method isillustrated by simulated examples.

I. IhTRODUCnON ANm STATEMENT OF THE PROBLEMThe subsystem G(s) of unknow n structure and order to be identifiedincludes a pure time delay and is part of a closed-loop system inco rpo-rating other know n dynam ical ubsystems (Fig. 1). The system is excitedby a given stationary random input, and Gaussian uncorrelated additivenoise is assumed to be present in the loop. In terms of L aplace trans-forms, G ( s ) s given by

me- b p i

nG(s)=e-- N ( s b) = i = O

D (s.a) 1 + c. ajs/J =

where rn, n denote he highest num eratoran ddenominator powers,respectively, and rn < n. G(s) ope rates n a closed loop (Fig. 1) withother dynamical elements G,(s), G2(s)and G,(s) having known parame-ters. Tracking or regulating tasks in man ual contro l are special cases ofthe system in Fig. 1. The following assumptions regarding the input,system, and noise are made.Assumption I: The input~ ( t )s a samp le of a given random stationarymean square boun ded ergodic process. Its spectral distribution guaran-tees a persistent excitation of all the mo des of G(s).Assumption 2: G ( s ) and he closed-loop system denoted by T ( s ) sstable and time invariant.Assumption 3: The noise nl ( t ) (Fig. 1 ) is a zero-mean stationaryergodic Gaussian process uncorrelated with x ( t ) .It is required to provide unbiased estimates of Q and b from ~ ( t )nd asuitably cho sen closed-loop system ou tput. Since the sensitivity of theclosed-loop system output output of G2(s) ] to variations n a, b isreduced by the loop gain, the system error,denoted by z ( t ) (Fig. 1)

which retains this sensitivity, is ch osen as the approp riate system ou tput.The corresponding closed-loop transfer function relating x(t) to r ( t ) sT(s)=G,(s)[l+Gl(s)G2(s)Gj(s)G(s)1-. (2 )

In-t ime domain, ( t ) is given by~ z ( t ) = T ( p ) x ( t ) - f n ( t ) = y ( ~ ) + n ( t ) (3 )

wherep d/dt, i=O, 1,2;.., andn ( t )= - G2 (zJ)G, ( p ) G ( p ) T ( p ) n ~ ( t ) . (4)

y ( t ) s the system error in the absence of noise. The known subsystems

Manuscript received August 12, 1975; revised April I , 1976. Paper recommended by E.The authors are with he Depamnent of Aeronautical Engineering, Technion-Israel

Tse, Past Chairman of the IEEE S C S StochasticControl Committee.Institute of Technology, Haifa, Israel.