A. Jameson and E. Kreindler, âInverse problem of linear optimal control,â J. SZAM Contr. vol. 11, pp. 1-19, Feb. 1973. J. C. Willems,,, âDissipative dinarnica1.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.
AC-19,
NO.
4,
AUQUST
[31 L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze, and E. Mischenko, TheMathemticalTheory of Optimal Processes. Reading, Mass.: Addison-Wesley, 1962. [41 W. Rudin, Principles of Mathematical Analysis, 2nd ed. New York: McGraw-Hill, 1964. [51 M. Athans and P. L. Falb, Optimal Control-An Introduction to the Theory and its Applications. New York: McGraw-Hill, 1966. C. A. Desoer, Notes for a Second Cwrse in LinearSystems. Princeton, N.J.: Van Nostrand-Reinhold 1970. C. E. Gray, “Optimal control of systems wlth combined pulsewidth and pulse-frequency modulation,” ESL Inc., Sunnyvale, Calif., Rep. IR-76, Feb. 24, 1971. R. M. Haviraand J . B. Lewis, “Computation of quantized controls using differential dynamic programming,” IEEE Trans. Automat. Contr., vol. AC-17, pp. 191-196, Apr. 1972. extrema1 controls for a 191 M. Athans, “On the uniqueness of class of minimum fuel problems, IEEE Trans.Automat. Contr., vol. AC-11, pp. 660-668, Oct. 1966. L. W. Neustadt, -“The existence of optimal controlsin the absence of convexity conditions,” J . Math. Anal. Appl., vol. 7, pp. 110-117, 1963. J. B. Lewis’and J. T. Tou, “Optimum sampled-data systems with quantized control signals,” IEEE Trans. Appl. I d . , vol. 82, pp. 229-233, July 1963.
h ,!e
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Charles E. Gray (S’67-M’71) was born in Helsinki, Finland on July 13,1938. He received the degree of IngCnieur Civil MCcanicien Electricien from the University of Brussels, Brussels, Belgium, in 1961, the S.M. degree in electrical engineering from the MassachusettsInstitute of Technology, Cambridge, in 1963 and the Ph.D. degree in electrical engineering and computer sciences from the University of California, Berkeley, in 1971. From 1963 to 1966, he was with the M.I.T. Instrumentation Laboratory,working on Project Apollo. From 1966 to 1970, he was a Research Assistant in the Departmentof Electrical Engineering and Computer Sciences of the University of California, Berkeley. During this time he also held summer positions at the M.I.T. Instrumentation Laboratory and a t E.S.L. Inc., Sunnyvale, Calif. From 1971 to 1973, he was with E.S.L. Inc., where he worked on advanced comumnicat>ionssyst,ems. He is now a consultant. His currentinterests include optimal control,computationalmethods and methods for testing digital circuits. Dr. Gray is a member of Sigma Xi and of the Societe Royale Belge des Ingdnieurs e t Industriels.
Implications of Passivity in a Class of Nonlinear Systems
Abstract-For a broad class of nonlinear systems, a connection is establishedbetweenthe input-output property of passivity and a set of constraints on the state equations of the system. These conof thestoredenergyand straintsarethen-interpretedinterms dissipation of apassive system. Applications are given in two problems of optimal control theory, and a generalized form of the circle criterion is also derived.
I. INTRODUCTION
I
Nelectricalnetworktheory, the distinctionbetween active and passive networks has long been considered an important one. The distinction is simple; active networks maycontaininternalenergy sources, whereas passive networks-at least in principle-do not. Because of the obviousphysicalimplications of classifying networks in this way, a large body of literature has grown up on the subject of passive networks [1],[2].
For more general c,ontrol systems, t,he physical meaning andimplications of passivity are less clear. The usual approach-and one that will be adopted in this paper-is t o Proceed by analogy with the networks case and define an“inputenergy”forthesystem.Passivity is then defined in termsof a nonnegativity condition on this input energy. One ofmain motivations the for studying passivity in the control theory context has been its connection with stability [3],[4]. An especially importantresultinthis area is the Kalman-Yakubovich-Popov lemma [5]-[7] or Positive Real Lemma,which was used in solving the welIknown stability problem of Lur’e [8]. I n essence, this lemma states that a transfer function is positive real if and only if a certain set of matrix equations has a nonnegative definite solution. The connection between positive real functions and passive systems is, of course, well known 191. Generalizations of the Dositive real lemma are now available which extend the results to positive rea.1 matrices [51, [lo], and in Particular Anderson 1111 has obtained a solution to the multivariable version of the problem of Lur’e. Applications of the positive real lemma L
Manuscript received July 16, 1973. Paper recommended byR. A. Skoog, Chairman of the IEEE S-CS Stability, Nonlinear, and Distributed Systems Committee. This work W ~ supported S in part by the Australian Research Grants Committee. The author is withtheDepartment of ElectricalEngineering, University of Newcastle, N. S.W., Australia.
A
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IEEE TRANSACTIONSONAUTOMATIC
also include spectral fact.orization [E],network synthesis [21], andthe solut,ion of an inverseproblem of linear optimal cont.rol 1131. The main limit,ation of the posit,ive real concept is t,hat i t applies only to 1inea.r systems. For nonlinear systems, the most promising approach to studying the properties of passive syst,ems appears to be t,hat of Willems [14]. The starting point in Willems’ theory is the definition of a “dissipat,ivc” syst.em in t.erms of an incquality involving. in effect, the stored energy of t.he system. Passivity, as defined in this paper, then appears as a special case of dissipativeness. The results of this paper lie somewhere bctw-m those for linear systcmsand those for the rxtrcmely general class of systems studied by Willems. The principal restriction imposed is that the state equations for t.he systems consideredshouldinvolve the controlvector only linearly. (Thisconditionis less rest,rict,ivet,han it nught at. first appear t,o be.Balakrishnan 1221 has shown that,, under very mild restridons,there existsa choice of st,at,e vector for almost any controllable finitedimensional syst.em such that the cont,rol appears linearly in the &ate equations.) By making t.his restriction, it, turns out t.0 be possible to obt,ainresults thatare considerablymore explicit than have hitherto been available for nonlinear systems. In fact, t.he central result could be interprcted as a nonlinear version of the positive real lemma. This result, given as Theorem 1 in Sect.ion 11, is t,he cornerstone of the subsequent development.; as with the posit,ivc real lemma, it. shows that a system is passive if and only if there exists a cert.ain scalar function of t.hc &ate which is nonnegat.ive. The remainder of Scction 11 is devoted to shoxing that,t.his funct,ion hasthe properties of storedenergy,thusarrivingby a. different routeat. some of the results of [14]. I n Sect.ions 111-V the results of Section I1 are applied t o a class of inverse opt.imal cont.ro1 problems, a class of singularcontrolproblems, and anonlinearextension of the Popov stability problem [S],[lj]. X11of t,hese results are believed to benew, alt,hough a restricted versionof the results of Sect,ion 111 has beenreport,edin [16]. The algorithm of Section IV is part.icularly interest.ing, inthat i t achieves a partial solution of a singular optimal cont,rol problem by met-hods which are equally applicable to nonsingular problems. It appears t.hat t.hese ideas could be extendedconsiderably, ahhough the detailsstillremain to beworked out. 11.
The systems to be st.udied in t.his paper are described by the equa.t.ions
+ G(x)u h.(x) + J G ) u
k = f(x) y =
(1)
1974
where f( and h( .) are real vect.or functions of the state vector a, wit.h f(0) = 0, h.(O) = 0, and G ( - ) and J ( are real matrix funct.ions of x. I n order toguaranteethe existence of solut,ions t.o the equations t,o be introduced below, it will b r assumed that f(.), G( .), h ( .), and J ( .) all possess cont,inuousderivat.ives of allorders,although weaker conditions would probably suffice. The input, vector 11 and theout.put. vectory ha.ve tlhe same dimension, so that J ( s ) is a square matrix. Thesyst.em is assumed t o be complet,cly controllable, in the sense t,hat.for any finite stat,cs x. and 11, t.herc exists a finite time tl and a squareintegrable cont,rol u(t) defined on [O,t,] such that t.he state can be driven from x(0) = 10 t,o x(t1) = 2 1 . I n addit.ion, a form of local cont,rollabilit>-is assumed; for any x 0 and any x1 in a suitably small open neighborhood of ro,t,here exists a choice of u(.) and tl as above 1vit.h t,he additional pr0pert.y that a)
0
ilf’
u’(t)y(t) i l l 1
5
P(ll2-1
)
- aoll)
for some continuous p ( . ) such that p ( 0 ) = 0. The dynamicalsystem (1) will be called pa.ssine if, whenever x(to) = 0 (for any a), JOT
2u’(t)y(t) clt 2. 0
for all T 2 io and all square-integrable u(t). (The mu1t.iplier 2 is simpl:: a scaling fact,or Tvhich it will be convenient to use in t.he following equations.) The quantit.y on t.he left-hand side of (‘2) will sometimes be called the input energy. An alternat,ive formulation of condit,ion (‘2) is given in the following theorem. Theorem 1: A necessary and sufficient condit,ion for system (1) to be passive is that there exist real functions +(.), Z(.), and W ( . ) ,mith +(x) continuous and with
$(x) 2 0,
for all x
and
do) = 0 such that V’$5(z)f(x)
=
--Z‘(x)Z(.r)
+G‘(z)V~(X) = h ( ~ ) W’(Z)Z(X)
J(x)
h i ALGEBRAIC CONDITIOX FOR PASSIVITY
CONTROL, AUGUST
+ J’(x)
=
rn‘(x)W(x).
(3)
1\Ioreover, if J is a constant mat.rix,t.hen W may be taken t.o be const,ant.. Proof: 1) Suficimcy: Suppose 4 ( . ) , I ( - ) , and W ( - ) are given sucht,hat (3) are satisfied. Then for any squareintegrable u.(t), and any fo and T 2 to, and any a(&),
375
MOYLW: PASSIVITY IN N O N L I N E m SYSTEMS
lT
T
Bu’(t)y(t)dt
{2u‘h(z)
= to
=
lT
+ u’[J(x)+ J’(z)]u)dt
lT +
[u’G’(z)V+(z)
+ 2u’W’(z)l(x)
2u’(t)y(t) dt
2 -C(z,)
(5)
whenever z(0) = xo. Kow define +(x) - a s
u’W’(z)W(z)u]dt
=
lT
+(ao> = -1im inf
+ G(x)uI
2u’(t)y(t) dt
T-+a u ( . ) S O T
{V’+(x>[f(z)
(6)
subject to (1) andtheboundarycondition x(0) = 20. Exist.ence of +(x) follows2 from the inequality (5); moreover, it is clear t,hat,
+ 2u’W’(z)Z(z) + u‘W‘(z)W(z)u- V’+(z)f(z) ) dt
+(O)
=
0
and
+ u’W’(z)W(z)u + Z’(x)Z(z) =
1
+(x) 2 0,
clt
for all x.
+
+[z(T)I - +[z(ta)I
+ L T I Z ( x )+ W(z)uI’[Z(z) + W(x)uIdt.
Let us suppose t.emporarily that [ J ( x ) J’(z)]is nonsingular for all x. Then, performing the optimization indicat-ed in (6) above, it is easily shown3 t.hat+(x) satisfies the (Hamilton-Jacobi) equation
+
+
V’+(z)f(~) [h(z)- +G’(z)V+(x)I’(J J’)-’[h(z) Setting z(to) = 0, the result, follows. - +G‘(z)V+(z)] = 0. 2 ) Necessity: As before, assume that the syst.em (1) is started int.he init,ial state z(to) = 0. From the definition If W ( z )in (4)is chosen t.0 be the posit.ive definite square of y ( t ) , it follows that, root of ( J J’), t,hen a functionZ(x) may be defined as
+
lT
2u’(t)y(t) dt =
LT
{2u’h(z)
+ y’[J(x) + J’(x)]zc) dt.
7(~)
=
(W’)-’[h(z) - +G’(z)V+(z)]
from which (3) follows. If the system is passive, this integralshould be nonnegat,ive If ( J J‘) issingular, t.he above procedure must be for any T 2 to and any u ( - ) .This implies that, [ J ( z ) slight,ly modified. I n t.his case, a new matrix J , ( z ) is J’(x)] is a nonnegative definite matrix for all z, since defined as otherwiseonecould findl a u(t) suchthat.theintegral + I J b ) = J(x) became negative for some T . Consequently, the symmetric matrix [ J ( z ) J‘(x)].a.dmit,sof a (nonunique) ract,oriza- for some small positive constant, E . Functions +.(x), Ze(x), tion and W e ( x )can t,hen be definedint,heobviousmanner. From (4), (5), and (6), it isthenstraight,forward to J ( z ) J ’ ( x ) = W‘(z)W(z). (4) establish the existence of the limit,s Nom for any state zo, there exist,s by controllability a W ( z ) = lim W , ( x ) time to < 0 a.nd a control u ( .) defined on the interval r+O [to,O] such t.hat x(0) = 2 0 . By passivit,y +(x) = lim 4 h )
+
+
+
+
+
lo
Bu’(t)y(t) dt
+
ST
c-0
2u‘(t)y(t) dt
2 0.
0
and Z(z)
That is
=
lim ZAz). e-0
Sincetheselimits a.re a.pproachedcontinuously, it is simple to establish that (3) areinfact satisfied. vvv The significance of Theorem 1 is that conditions for The right-hand side of this inequality dependsonly on xo, passivity-essentially a property of theinput-output whereas u(t) can be chosen arbit,rarily for t 2 0. There relationship of a systemhave been stated in terns of therefore exists a funct.ion C(zo) of zo alone such that functions of the st.ate vector; ina. sense, (3) are conditions on t.he internal struct,ureof the system. It is m7ort.h noting SOT
2u’(t)y(t) dl
2 -
s,”
2u’(t)y(t) dt.
1 The construction is roughly as follows. First, h d a control driving to a point where (J + J’)is no longer nonnegative definite; then follow this wit.h a “pulse” in u. large enough t o overcome any posit.ive contribution to the integral.
X
2.3 The details are not entirely trivial. However, the argument follows the same lines as, say, [ 161.
that (3) would remain unchanged if +(x) where changed by any constantamount.That,is,theconstraint +(O) = 0 could hare been dropped,andthe condition 4(x) 2 0 replaced by +(x) 2 4(0). To avoid trivia, hon-ever, only t,hose solutionsfor which 4(0) = 0 will be considered in the sequel. If the physical significance of t,he vect.ors u(t) and y(t) are such t.hat>the int,egral in (2) represents input energy, one can go even further. Recall t.hatr JUT Zu’(t)y(t)
+
dt =
~ [ x ( T) ]+[.(to)]
[[(x) + W ( . Z > Z l ] ’ [ l ( X )
J T
+ W(X)21] clt
(7)
c
=
y,(t)
Final energy - 1nit.ial energy
+ Dissipated energy.
Zi[X(t)]
=
+
wi[I(f)]2l(t).
Then (7) implies that
41[x(T) 1 - 41[2(to) 1 =
+
JOT
4 2 W )1 -
Yl’(t)Yl(t) dt
h [ 4 f 0 )
I
+
T
y?‘(t)y?(t>dt
(S)
for any u(.) and any boundary condit,ions z ( t 0 ) ~ x ( T ) . Let +l(s) be &(x). Then from t,he definition of +Jz), there exists a sequence of controls { u ( j ) ( . )] such that,, with
x(0)
which may be int,erpret,edas the “conservation of energy” equation Input energy
Let ( ~ j ( . ) . r j ( . ) ? m r j ( . )il ,= 1,3,be t,wo sets of solutions of ( 3 ) , and for simplicity define “dissipation outputs”
= xg,
S, T
linl lim T-2
j+a
~ u ( j ) ’ ( t ) y ( j ) ( dt t ) = -+.l(xo)
where y(j) (t) is the corresponding solut.ion of (1). Thus,
The nonnegative quantity +(x) then appears as the stored energy of the syst.em, while the second integral corresponds t o dissipated energy. -4s expected for passive systems, the \\-here .r(j’(t) and yl(J’(t) are defined sindarly. Equation (8) can now be rearranged to give dissipatedenergy is aln-ays nonnegat.ive,although it is pat,h-dependent. The change in stored energy while moving from state r(to)t o x(T)is, of course? independent of the path taken. Since no physical Significance has yet. been attached to t,he state vect.or 5 , the stored energy is not, in gcneral, uniquely specified by the state equations (1). Typically, (3) n-ill have a large number of solutions. However, it is possible t o place bounds on these solutions, as shown in the The second limit. is zero, from above, xhereas the first, is following theorem. Theorem. 8: Equat3ions (3) possess a maximum and a nonnegative.Therefore, minimum solut.ion, in the sense that, t.here exist solutions &(so) 2 +1(x0), forall so, &(x) and &(x) such that That is, +a(.) I +(X) 5 d h ) , for all 5 +(x) 2 &(x), for all x where +(x) is any solution of (3). Moreover, where +( . ) is any solution of ( 3 ) . This completes t,heproof that &(x) is the minimumsolution. That &(x) is t.he nlaxirnunl solut.ion may be shown in a similar manner. vvv subject, to (1) and x(0) = zo; a.nd In the terminology of Willems [4], + a ( x ) is the aznilabte energy of the system (1)-by definition? it is the maximum +r(.l-o) = lim inf “ u ’ ( t ) y ( t ) dt h--- u : . ) amount. o f energy t,hat can be extracted at the terminals when starting from t.he init,ial state s. Similarly, + T ( x ) subject. t o (1). x(to) = O ? and ~ ( 0 =) ;ro. may be called the required energy, since it is the nlinimunl Prooj: From (7), every solution set {~(.),~?IV(~)~ energy required to excite the system t o a state I from the of ( 3 ) satisfies equilibrium (zero-energy) state. These two functions will be identical if the system is reversible, [Ir]. JOT 2 ? i ’ ( f ) y ( f d) t = $ [ x ( T ) ]- + [ . ~ ( t g ) ] Theorem 3 shows that. all solutions 4(.r) of (3) lie in a bounded set.. It is also 1vort.hnotling that this set, is convex. [l(.c) m ( s ) z l ] ’ [ r ( x ) W ( X ) 1 1 ] dt (7) to Lenrma 1: If +1(x) and 4?(.r)are anytn-o solutions of (3) , where r ( f )and y ( t ) are solutions of (1).The left side of this then
JCo
+ J’
+
+
equation depends on z l ( . ) and bhe boundary conditions on set.
x ( t ) , hut, is independent of t.he particularsolution {4(.>,u.>?w(.> 1.
377
MOYL.4N: PMSIVITP IN NONLINEAR SYSTEMS
Proof: Corresponding tothe solutions +1(z) and +&), there are vectors Zl(z) and Z&), and mat,rices W1(x) and W 2 ( x ) appearing , in (3). Kow define,
Then &(x) isreadily seen to be asolution of (3), with Z(x) = i(z)and W ( x ) = ? $ T ( x ) . vvv Actually, the convexity of the set of energy functions hasalready been proved b y Willems [14]. The novel point, of Lemma 1 is t,hat itis possible t o explicitly exhibit, the associa.t,ed dissipation terms i(x) and %(x). I n a later section of this paper, the functions +(x) w i l l be used as Lyapunov functions;it is therefore of interest,to not,e conditions under which +(x) will be posit.ive definite (in the sense that +(x) = 0 implies z = 0). Definition: The system (1)will be called obsermble if, for any traject.orS7 suchthat, u(t) 0, y ( t ) 0 implies z(t) = 0. (ITot,e that this is a. weaker condition tha.n is required by more st.andard definitions of observabilitp. 0bservabilit.y in the abovesense does not imply t,he abilit,yt o deduce the initial state from output measurements alone, although it does imply t,he ability to det.ect the presence of a nonzero stat.e.) Lenznza 2: If tlhe system (1) is passive a.nd observable, then all solut,ions +(x) of (3) are posit.ive definite. Proof: Suppose that there ekists some zo sucht>ha.t +(zo) = 0, and let z ( t ) be t,he solut,ion of R ( t ) = f[z(t)],
z(0) = zo.
proved. This generally requires stronger conditions than ha.ve been noted a.bove. The preceding discussion has shown t.hat. thereis a natural ordering of t.he solutions of (3), corresponding t.o an ordering in the stored energy +(x). As an altmnat,ive, one could define a.n ordering of solutionsint,erms of dissipated energy. This approach is illustratedbelow. Dejnifion.: With {&i(z),li(x),Wi]any solut,ion set of (3), let,
represent the input.-t.0-dissipat,ion out>put,relationship of the system 3i- = f ( z )
Then rl has less delay tha.n rs, written r1 < rs, if, for any time tl 2 fa,and any common input u( - )~ L ~ [ t ~ , t ~ ] , PI,
-+[&)I dt
loT
yl’(t)yl(t) clt
=
V‘+(z)f(z)
=
-Z’(x)L(z).
(9)
lo T
=
yz’(t)yz(t) dt
(12)
for any such t.hat z ( T ) = z(t0). Proof: The proof i s obvious from (8). vvv The mea.ning of the description “less de1a.y” should now be clear. For any common input to the txo systems, the dissipat,ion responses have the same mean square value (see (12)) but (11) shows t.hat the main part of the response occurs earlier for the first systemt,hanfor the second. I n particular,t.he syst>cm n-hose stored energy function is &(x) is a nzinimwn delay syst.em. At. t.hc ot,her extreme, the system whose storedenergyis &(x) isa rnmi.mum delay system. For linearsystems, the concept. of “minimumdelay” act,ually turns out t.o correspond wit.h the classical notion of “minimumphase.” This special case has been previously st,udied by rlnderson [l’i],in the cont.ext of spectral factorization of power spect,rum matrices. Reference [ 1 7 ] also establishes t,hat there is a sense in which the term !Ln~axinmm phase” could be applied t>omaximumdelay systems, as defined here. .(e)
Therefore,
4 [ 4 01
P fl
ivhenever z(f+)= 0, and if equality does not hold for all u(t) and all tl. The notmationrl 5 rs will be used in case (11) holds, with the possibi1it.y t,hat equalityalways holds. The relat.ion 5 represents a partial ordering, based on the amount of energy dissipated along any given trajectory. Theorem 3: Let. rl and rs be any two members of the family (10). Then rl 5 r r if and only if +1(x) 5 & ( x ) for all x. AIoreover, regardless of any orderingbetween rl and r2,
Then, from (3),
cl
+ G(z)u
i +[201,
for t 2 0.
It follows that +[z(t)] is ident,ically zero along t.his traject,or>-,so that Z[z(t)] is also zero. Since +(x) is a continuous nonnegat,ivefunction of x, it also follows that B4(z) is zero along t>he sametrajectory. From the second of equations (3), t,hen, k [ z ( t ) ]= 0 along the same trajectory. Unless zo = 0, this contra.dicts the assumedobservabi1it.y of (1). It.must, therefore be t.rue that. +(x) is positive definite. VVV Corollary: If the system (1) is passive and observable, then the free syst.em 3i- = f(x)
is (Lyapunov) stable. Proof: This follows directly from (9). vvv Note, however, that, a.symptotic stability ha.s not been
111.
APPLICATIOS T O OPTIK4L
A t.ypica1 probleminoptimalcontrol following:
CONTROL theoryisthe
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IEEE TRANSACTIONS O N AlJTOblATIC COXTROL, A U G U S T
1974
Given the system equations 3i. = f(z)
+ GU
(13)
+
+
+
S O T [u k(z)l’[u L(z>]at. of t.he vector z possessing where f( .) is a real function continuous derivatives of all orders, f(0) = 0, and G is a QVV constant matrix, find a control u(t) that will minimize t4he Sett.ing n ( x ) = +(x), the result follows. I n view of Theorems 1 and 2, t.hese resultsmay be performance index summarized as follows. Theorem 4: A necessary and sufficient condition for t,he V I = lim ( n [ x ( T ) ] S O T [E’(z)l(z) u‘.u] d l } (14) T- m existence of a solut,ion to the inverse problem is that the where n ( z ) 2 0 for all x. Wit.hout. substantial loss of system 3i. = f ( ~ ) +Gk(x) GU generality, it mag beassumed that E(0) = 0 and n(0) = 0. The optimal control will in general be of the form y = k(x)
+
+
+
u* = - k ( z )
(18
and the minimum value of the performance index will be some function +(xo) of the initial stat.ezo. However, it may n-ell be that, thereexist many different funct,ions I(r),and correspondingly different.+(x), for which the same control law (15) isopt,imal. I f the system (13) and control Ian(1.5) arc given, we shall say that t,he pair {+(-),Z(.) ] is a solution to the inowse problem if the performance index (14)-for some nonnegative choice of n(z)-is minimized by (ls),the minimum value being +(xo). Necessary and sufficient,condit.ions for t.he exist,ence of a solut.ion to the inverse problem have been given in [16]. For linear systems with linear stat.e feedback, one can in factobtain extremely det.ailed results, the best-known results being those of Kalman [ l S ] . However, these references are primarily concerned wit,h obtainingone solution t.0 the inverse problem. A method of generating all solut.ions is indicated in t.he follou-ing lemma. L e m m a 3: Suppose that the system equations (13) and control law (15) are given. Then apair { + ( . ) q l ( . ) isa solution to the inverseproblem if and only if $(.r) and Z(z) satisfy the equations
be passive. If t.his condit,ibn is sdsfied, then there exist tn-o solut,ions { 4 a ( * ) : I a ( - ) and {4,(.).2,(.) not necessarily dist,inct,, such that all other solut.ions { + ( . ) , I ( - ) satisfy the const,raint
1,
1,
I +(E) I
+a(.)
+T(.T),
1
for all E .
117. SIXGULAR OPTIMIZATION PROBLEMS a further applicat,ion of the results of Section 11, consider the following problem: 11-it.hx ( t ) governed by i =
f(x)
+ G(x)u,
~ ( 0 =) x.
(16)
find u(.) such that the performance index PT
I.’,
=
lim T-c
J0
[l(.u)
+ W(x)u]’[l(s)+ W ( E ) . Udt]
(17)
is minimized. I n addition to the stateequat.ion (16), t.here may bo boundary conditions on x and possibly some constraints on u. If W ’ ( z ) W ( s )is invertible for all x , the solut.ion of the aboveproblemisrelativelystraightforward. If, on the V’+(X)V(E)- +Gk(z)]= -rf(x)l(~) other hand, W’(z)W(.z)is singular, most, st,andardoptimizationmethods fail to find an opt.imalsolution. Al+G’V+(z) = k(x) though it is st.raight,forward t.0 establish t>hat.the optimal +(O) = 0 cont.ro1is bang-bang (that is! t,hc control attainsits limits, or isinfinite if there are no a. priori bounds)formost and values of E , *thereis t,he possibility of singular regions $(x) 2 0, for all x. where the optimal control is determined by criteria indeProof: 1) Suppose {+(.),Z(.) } is known to be a solution pendent of the bounds on u. The determination of t.he space is of the inverseproblem. Then k ( r ) is given [16]from location of the singular regions inthestate usually t.he most, difficult, step in solving t.he above probstandard Hamilton-Jacobi theory as lem. k ( z ) = +G‘V+(x) For the purposes of this section, it sill beassumed t,hat which shows that the second equat.ion is satisfied. -41~0, there are no a priori boundson IC, and that boundary [ l G ] , +(x) must. be a nonnegative solutionof the Hamilton- conditions for E have not yet, been specified. That is, we wish to derive the optimal controls for all possible comJacobi equation (in its limiting form) binations of boundaryconditions. A sin.gularregion will V ’ $ ( ~ ) f ( r) ~V’+(E)GG’V$(X) rf(s)Z(.r) = 0 then be defined as any region in (x.u) spacp in which the optimalcontrol is finite. (This differs fromt,heusual from which t,he first equation follows. 2) Suppose that + ( E ) and I(x) are knon-n to satisfy the definition, in t.hat t,he case where W ’ ( z ) W ( x )is invertible actual conaboveequations.Thenit is easily shown that, for any hasnot been excluded.) Dependingont,he straints on 11 a.nd on the boundaryconditions, some of square-int,egrable u(- ),
1
+
379
SYSTEMS MOYLAN: P?LsSIvITY IN NONLINEAR
these regions ma.y not in fact enter into the optimalsolution,butthese considerations a.re most ea.sily handled after the singularregions have been located. An obviouscandidate asa singular regionis the region
Z(x)
+G’(x)V+j(z)
=
h(x) - Wi’(x)Zt(~)
W,’(x)W,(x)
=
J(x)
3i. = f(x),
x(0)
= x0
u
+ W,(x)u = 0
l&)
db)
and (16) and (17) modified a.ppropriately. This does not change the form of (16) or (17).] With this assumption, define5 a.function +(x) via
(20)
=
+(x>
- 4ib)
the following set of equat,ions may be derived: V’&r)f(x)
=
zi’(z)l&) - Z’(z)Z(z)
+G’(x)V&(x) = Wi’(x)Zi(~)- W ’ ( ~ ) l ( x )
(18)
- k(x)
,
is a singular region. Proof: By defining a new function
W,’(x)W,(x) = W’(x)W(x).
isasymptoticallystable. [Unst,able systemscan also be handled if it ca.n be shorn that tohereexists a stabilizing feedback control law4 for the origind system (16). Supposing that t.his control is of the form - k ( z ) ,a new control can be defined via u 1=
(19)
Then theregion
+ W(x)u = 0.
However, trajectoriesin this regionmay not ha.ve the property that the boundaryconditions a.re met. Moreover, the region may be a trivial one, consisting only of the origin. To proceed further, it is obviously necessaryto find all the singular regions, a.nd then to exclude those which do not meet the side constraints. In the sequel, it. will be assumed that the free system
+ J’(x).
Then, with z(t) given by (16), d
- d[r(t)l
a.
=
+ V’d(x)G(x)u
V’d(x)j(~)
for any u(t). With the aid of the above equations, this reduces to d -
at
d [ x ( t ) ]= [li(x)
+ W,(x)ul’[lt(x) + W,(z)uI [ W + rn(x>uI’[Z(x)+ W(x)uI. -
Therefore, for any times 4 and tl 2 to,
[where x(t) is governedby (18)], a.nd then define functions h(x) and J ( x ) as
+ W’(x)Z(x)
h(x)
=
+G’(x)V+(x)
J(x)
=
+W’(x)W(x).
LtL
+ w(x)ul’[z(x)+ W(x>uldt d[x(tdI - d [ x ( t J ]+ J“ + Wi(x)u]’[li(x)+ W ~ ( X )dt-U ] [Z(x)
=
[It(.)
to
Equation (20) is therefore a sufficient condition for optiThe significance of these functionsis s h o r n in the following malit,y, provided that the boundary condit,ions are such lemma. that $[x(tl)] is also minimized. It follows that (20) deLemma 4: The system singular scribes a region. vvv 3i. = f(x) G(x)u Note, incidentally, that t.he above theorem holds whether or not W’(x)W(x)is invertible.If this matrix is in Y = h(x) J ( 4 u factinvertible,thenthe “singular region”covers the is passive. entire st,ate space, and the solution for u reduces to that Proof: From thedefinition of +(x), i t follows that found for nonsingular problems. I n all other cases, (20) can besatisfied only in a subsetof the st.atespace. It is felt 0’4(x)f(x) = -Z’(x)Z(x) that a major cont,ributionof t.his sectmion is t.hat Theorem5 and that covers singular and nonsingular problems equally well, by considering nonsinguhr problems in x space as “singular” +(x) 2 0, for all x. problems in (r,u)space. Equations (3) are therefore trivially satisfied, and Theorem 1 may applied. vvv V . TOLERANCE OF SECTOR NOXLINEARITIES A number of singular regions may now be derived as follows. In this section we invest,igat.ethe stabilityof the-system Theorem 5: Wit.h h(x) and J ( x ) defined as above, let {+i(x),li(x),W,(x) be any solution set of the equations W = f [ W 1 G$ {k[x(O1) (21) V’+,(x>f(x) = -&’(4&(x) where f ( - ) and k( are known linear or nonlinear func-
+
+
be
1
+
e)
4 Controllability appears to be a sufficient condition for such a cont.ro1law to exist [16]. 5 Boundedness of +(. ) needs t.o be explicitly assumed.
tions, G is a lm0n.n matrix, and $( .>is an unknown nonlinearity.However, #(.) is partlyconstrainedbythe conditions
380
IEEE TRAYSACTIONS
u’[#(u) -
-
+
T U ]2 [#(a)- T u ] ’ M [ # ( u ) T U ] eu’u, forall u f 0 (22)
ox
ACTONATTICCOSTROL, AUGUST
1974
VI. CONCLUSIONS
The central point of Ohis paper is of course, Theorem 1. Here, it is shonm that passivity of a certain class of systems is equivalent to the existence of a state function #(O) = 0. +(x) satisfyinga set of well-defined equations. The sigIn the above inequality, T and Jif are knon-n constant nificance of t.his t,heorem is that the input-output property matrices, nith M nonnegabive and symmetric, and e is a of passivity may be replaced by aset. of const,raints on the internal structure of a system. The interpretat,ion of +(x) small positive scalar constant. In the case of independent nonlinearities, i.e., t.he jt.h as a.n energy function is in a sense a secondary issue, but component of # ( u ) depends only on t.hej t h component of it allows some useful insight,s into themeaning of Theorem 1. u, we may t.ake T and Jf to be diagonal. The inequality As with the more general storage functions of Willems then reduces to a set of inequalit.ies of t,he form [4],[14],it turns out t.hat, t,he stored energyis nonunique; this implies, among other things, that when one is interested in computinginternalenergy t.he state equations provide an incomplete description of a system. Consider That is, the graph of # % ( u )as a function of ut is known t.0 for example t.he w r e of linear systems. It is natural to call lie in the sect,or bounded by st,raight lines of slope f i i and tii ( l / m i i ) .I n general, t,hen, we call + ( u ) a sector non.- two systems equivalent [20] if t.heir state vectors x(t) and P ( 1 ) a.re such that .?(t) = Tz(t) for all t , for some invertible linearity. matrix T . Supposing that the stored energies for the t.mo The main result isas follows. sptems are given by x‘Px and .?’$.?, then it, is not necesTheorem 6: If the system sarily true that, P = T’PT. This conclusion has a natural 1 = f(z) - G T k ( z ) GU and fairly obvious physical int.erpretation in the case of electrical networks, but t.he implications for more general y = k(x) Mu controlsystems arenot ent,irely clear. It would be of is passive and observable, then t.he system (21) is a s y m p considerable interest to develop a theory of control systot,ically stable. tems which t,ook account of t,he differences between Proof: The condit.ions of the theorem inlply the ex- different systems having the same state equat.ions. istence of {+(+),Z(-),W},with +(-) positive definite, such The applications list.ed in Sections 111-V are int,ended that only as a brief survey; it is clear t,hat. one could obtain stronger or more general results for the problems menV‘+(X)~(X -) V’+(z)GTk(z) = - Z’(X)Z(X) tioned in thesesections. For example, the Lyapunov +G’V+(z) = k ( x ) - W’Z(z) function of Section T‘ could be augmented by an integral term involving the unknon-n non1inearit.y (as x a s done in AT 111‘ = W‘W. [ 11I), with a corresponding increase in generality of t,he With z(t) governed by (21), i t follows easily t,hat system of Theorem 6. Another area where the theory shows promise is in the s>-nt,hesisof nonlinear elect,rical = v‘+(x)f(x) - v’+(z)G$[k(x)] networks; the concepts of stored and dissipated energy are dt readily translat,ed int,o physical terms, and the only real problem is todetermine hon- t,he net,n-ork e1ement.s should = -).(I! - W + [ k ( x ) ] W T k ( z )1’ {Z(X) be interconnected. Yet, anot.her application is in problems - W # [ k ( x ) ] W T k ( x )} involving the stabi1it.y of int,erconnected systcms-t.he - 2 {k’(4 [# I w c ) 1 - T W x ) I rvork of Zames [3] is especially relevant here. The precise - [ # [ k ( ~ )-] T k ( ~ ) ] ‘ d I [ + [ k (-~ )T] k ( ~ ) ] i .details in theselast. two cases still remainto be worked out.
and
+
+
+
+
-’
+
+
The derivative is clearly nonnegative, so t,hat @(x) is a Lyapunov function establishing stability for (21). Moreover, the above derivative can never be identically zero along a. nontrivial trajectory, for by the inequality (22) this mould imply that k(x) 0 along the same trajectory; this possibility is ruled outby observability. It then follows [19]t,hat lim f-+
=
+ [z(f)] = 0
which is sufficient to proveasymptoticstability.
VVV
REFERESCES Sew York: [2] L. Weinberg, Setwnrk Am.lysis and Synthesis. Kew York: lIcGraw-Hfll, 1962. [3] G. Z a m q On the input-output stability of timevarying nonlinearfeedback systenLs-Part I: Conditionsderived using concepts of loop gain, conicit.y, and positivity,” I E E E Trans. d ~ t o n ~ aCi~. n t r . V , O ~ .AC-11, pp. 22s-233, APT. 1966. -, “On the input-output stabilitv of timevarying nonlinear feedback system-Part 11: Conditions involving circles in the frequency plane and sector nonlinearities,” ibid., pp. 463-476, July 1966.
IEEE AUTOMATIC CONTROL, TRANSACTIONS ON
VOL.
AC-19,
NO.
4,
AUGUST
[4] J. C. Willems, “The generation of Lyapunov functions for input-output stable systems,” J. S I A M Contr., vol. 9, pp. 105134, Feb. 1971. [Ti] V. M. Popov, “Hyperstabilityandoptimality of automat’ic systems with several control functions,” Rev. Roum. Sci. Tech., Ser. Electrotech. Energ., vol. 9, no. 4,. pp. 629-690, 1964. [6] V.A. Yakubovich, “Absolute s t a b i k y of nonlinear control in critical cases-Part I,” Automat.RemoteContr., vol. 24, pp. 273-282,1963. -_ “Absolute stability of nonlinear control in critical casesPart’II,” ibid., pp. 6554338, 1963. [7] It. E. Kalman, “On a new characterization of linear passive systems,” in Proc. 1st Allerton C’onf. Circuit and System Theory, Univ. of Illinois, Urbana, Nov. 1963, pp. 456-470. [8] --, “Lyapunov functions for the problem of Lur’e in automatic control,” Proc. Nut. Acad. Sci. (U.S.A.), vol. 49, pp. 201205, Feb. 1963. [9] 1). C. Youla, L. J. Castriota, and H. J. Carlin, “Bounded real scattering matrices and the foundations of linear pwsive network theory,” I R E Trans.CircuitTheory (Sequential Transducer Issue), vol. CT-6, pp. 102-124, Mar. 1959. [lo] B. D. 0. Anderson, “A systemtheory criterion for positive real matrices,” J . SZAM Contr., vol. 5 , pp. 171-182, May 1967. 1111 --. “Stabilitv of controlsystems with multiple nonlinearities,” J . Frankiin Inst., vol. Sl,Sept. 1966. __ , “An algebraic solution t o t h e spectralfactorization problem,” I E E E Trans. Automat. Contr., vol. AC-12, pp. 410414, Aig. 1967. A. Jameson andE. Kreindler, “Inverse problem of linear optimal control,” J . SZAM Contr. vol. 11, pp. 1-19, Feb. 1973. J. C. Willems,,, “Dissipative dinarnica1 systems-Part I: General Theory, Arch. Ration. Mech. Anal., vol. 45, no. ,5, pp. 321-351,1972. --, “Dissipative dynamical systems-Part 11: Linear systems with quadratic supply rates,” ibid., pp. 352-393. V. N. Popov, “Absolute stability of nonlinear systems of automatic control,” Automat. Remote Contr., vol. 22, pp. 8.57875, March 1962.
1974
381
I161 P. J. Moylan and B. D. Anderson, “Nonlinear regulator theory and an inverse optimal control problem,’’ ZEEE Trans. Automat. Contr., vol. AC-18, pp. 460464, Oct. 1973. [I71 B. I>. Anderson, “Algebraic properties of minimal degree spectral factors,” Automatica, vol. 9, pp. 491-500, 1973. [I81 It. E. Kalman, “When is alinearcontrol system optimal?”, Trans. A S M E (J. Basic Eng.), vol. 86, Series D, pp. 51-60, Mar. 1964. [19] J. La Salle and S. Lefschetz, Stability byLyapunov’sDirect Method. New York: Academic, 1961. [20] It. E. Kalman, “Mathematical descript,ion of linear dynamical ’ systems,” J . SZAM Contr., vol. 1, no. 2, pp. 1.52-192, 1963. (211 B. I). Anderson and S. Vongpanitlerd, Network Analysis and Synthesis. Englewood Cliffs,N. J.: Prentice-Ilall, 1973. [22] A. V. Balakrkhnan, “On the controllability of a nonlinear system,” Proc. Nut. Acad. Sci. (U.S.A.), vol. 55, pp. 465468, Mar. 1966.
Peter J. Moylan (“73) was born in Hamilton, Vic., Australia,on February 24, 1948. He received the B.E. degree in electrical engineering fromthe University of Melbourne,Melbourne,Australia, andthe M.E. a.nd Ph.1). degrees from the University of Newcastle, Newcastle, Australia. Since 1972 he has been a Lecturer in Electrical Engineering at the University of Newcastle. His current research interests are in the fields of control of nonlinearsystems, pattern recognition, and electrical networks.
Autonomous Periodic Motion in Nonlinear Feedback Systems E R I K J. L. NOLDUS
Abstract-Sufficient conditions are presented for the existence of periodic motions in aclass of autonomous, time-invariant,nonlinear feedback systems. The conditions can be interpreted as circle typecriteria, stated in terms of thefrequencyresponse androot locus diagramsfor the system’s linear part,and in terms of the characteristics of the nonlinearity.
I. INTRODUCTION
T
HE present paper is concerned with establishing the existence of nontrivial periodic solutions,for the
Manuscript received April 18, 1973; revised August. 16, 1973. Paper recommended by R.. A. Skoog, Chairman of the IEISE S-CS Stability, Nonlinear, and l)ist,ributed Systems Committee. The author is with the Automatic Control Laboratory, University of Ghent, Ghent, Belgium.
diff erontial equation of an autonomous, t’imc-invariant, nonlincar fwdback system. While a general thcory of nonlinear oscillations isavailableforsecond-ordersystems, whcre analyt,ical-t’opological methods may be applied very nicely, thr problem remains a challenging one for systcms of order higher than t.wo. For its solution,only a few basic principlescanbcfoundin the textbooks.Truly, some result’s of this kind, for specific cases of third-order syst,cms,havebeen known forarelatively long time,one much citcd rxamplc by Rauch [ l ] going back as far as 19.50. Other examples can bc found in the Russian literature [ 2 ] . It srems difficult however t,o dcrivcasimple crit.rxrion that applies to the casc of a general system of order 7 1 , containing an arbitrary nonlinear amplifier.
