- Email: [email protected]
Stability of nonlinear time-varying feedback systems
Automatfea, Vol. 4, pp. 309-322. Pergamon Press, 1968. Printed in Great Britain
STABILITY OF N O N L I N E A R TIME-VARYING FEEDBACK SYSTEMS Yo-SUNG CHO Honeywell Inc., Waltham, Mass. and KUM~AT~ S. NARENDRA Yale University, New Haven, Conn. Summary--In this paper, the stability of nonlinear time-varying feedback systems is studied using a "passive operator" technique. The feedback system is assumed to consist of a linear time-invariant operator G(s) in the forward path and a nonlinear time-varying gain function f ( • )K(t) in the feedback path. The stability condition indicates that the bound on the time derivative [dK(t)/(dt)] depends both on the nonlinearity f ( • ) and the multiplier Z(s) chosen to make G(s)Z(s) positive real. It is also shown that the main result in this paper can be specialized to yield many of the results obtained so far for nonlinear time-invariant systems and linear time-varying systems. 1. I N T R O D U C T I O N IN R~CENT years m a n y new results have been o b t a i n e d in the stability t h e o r y o f f e e d b a c k systems [1-8]. T h e f e e d b a c k system considered generally consists o f a linear t i m e - i n v a r i a n t o p e r a t o r in the f o r w a r d p a t h a n d a n o n l i n e a r o r tim~-varying gain f u n c t i o n in the f e e d b a c k p a t h . T h e results o b t a i n e d a r e generally expressed as sufficienct c o n d i t i o n s o n t h e transfer function G(s) o f the linear p a r t o f the system. I f U(s)Z(s) is positive real for a multiplier Z(s) b e l o n g i n g to some class Zc the f e e d b a c k system is s h o w n to b e stable f o r n o n l i n e a r gains f ( • ) b e l o n g i n g to a c o r r e s p o n d i n g class f t . I n this p a p e r , using a passive o p e r a t o r t ~ h n i q u e ,
Ill v. M. PoPOV: Absolute stability of nonlinear systems of automatic control. Automation and Remote Control 857-875, March 0962). [2] K. S. NAR~r~RA and R. M. GOL~WY~: A geometrical crite~'ion for the stability of certain nonlinear nonautonomous systems. IEEE Trans. Ciremt Theory CToll, 406-408 0954). [3] (a) I. W. SANDnERG: Some results on the theory of physical systems governed by nonlinear functional equations. Bell Syst. Tech. J'. XLIV, 871-898 (1965). (b) I. W. SA~a)BER6: A frequency domain condition for the stability of f~xiback systems containing a single time-varying nonlinear element. Bell Syst. Tech. J. XL~I, 1601-1608 0964). (c) I. W. SA~r~B~R6: On generalizations and extensions of the,Popov criterion. IEEE Trans. Circuit Theory 117-118, March (1966). [4] (a) G. Z~a~.s: Nonlinear time-varying feedback systems-conditions for L®-boundedn~ss derived using conic operators on exponentially weighte~d spaces. Prec. 1965 Allerton Conf., pp. 460-471. (b) G. ZA~,w~s: On the input-output stability of time-varying nonlinear feedback systems, Pts. I and II. IEEE Trans. Automatic Control AC-11, 228-238; 465--476 (1966). [5] R. W. B g o c ~ r r and L. FogYs: On the stability of systems containing a time-varying gain. Prec. 1964 Allerton Conf., pp. 413-430. [6] C. T. L ~ and C. A. D~so~g: Stability of single-loop nonlinear feedback systems. El~t. Res. Lab., Univ. of Cal., Berkeley, Cal., Rpt. No. ERL 66-13, May 0966). [7] Y. S. C~o and K. S. N ~ a ~ a ) ~ : An off-axis circle ~'iterion for the stability of fe~lback systems with a monotonic nonlinearity. Proc. 1967 Allerton Conf, pp. 238-248. [8] (a) K. S. N ~ r , ro~A and Y. S. C8o: Stability of linear time-varying feedback systems. Dunham Lab., Yale Univ., New Haven, Conn., Tech. Rpt. CT-8, Feb. (1967). (b) K. S. NARE~qDRAand Y. S. C8o: Stability of feedback systems containing a single odd monotonic nonlinearity. IEEE Trans. Automatic Control AC-12, August (1967).
309
310
Yo-Suy(; CHO a n d KUMPATI S. NARENDRA
sufficient conditions are derived for the stability of nonlinear and time-varying systems which can be specialized to yield many of the results obtained earlier, e.g. Popov criterion [1], circie criteria for systems with linear time-varying [3b, 5, 8a] and nonlinear time-varying [2, 3b, 4~t1 gains.
2.
MATHEMATICAL
PRELIMINARIES
We define an Lz~ space which is an extension o f t h e L z space of square integrable function s of real variable t in [0, m] as follows. Lz~ contains both functions with Lz finite norms and functions with infinite norms. Each finite time truncation of every function x(t) belonging to Lz~ will lie in L~. Let x, denote the truncated function which assumes the values x,(t) = x(t) ( t < x and reT where T is the subinterval of the reals [0, oo]), and x M ) = 0 elsewhere. For xeL2, let ]lx[[2 denote the norm of x. Definition: The extension of L2, denoted by L2~, is the space consisting of those functions whose truncations lie in L2. An extended norm denoted by [[x[l:~ is assigned to each x(t) as follows.
Ilxitze=
xZdt =llxiJ2 oo
iixiize=
x¢c .
if
x,~L 2
and
Assuming G and F i n Fig. la map L2e into itself, one may formulate feedback equations for the given system as follows: If Ge(t) indicates the operation of G on an input function e(t), then
Ge(t) = y(t) FGe(t)+e(t)=x(t)
l)
The operators G and F considered in this paper belong to the classes described in 2.1 and 2.2. The problem of interest may now be stated as follows: Assuming that a solution exists for equation (1) and lies in Lze, find sufficient conditions on G and F under which x(t)eL2 and e(t), y(t)eLze imply e(t), y(t)eL,_ and lim ]y(t)[~O. t ~
2.1 Classes of nonlinear t#ne-varying gains F Let f ( . ) represent a memoryless nonlinear time-invariant function and K(t) be a differentiable linear time-varying function. The operators F considered here belong to the class of memoryless nonlinear timevarying functions which can be expressed as F ( . , t ) = f ( . )K(t) and may be defined as follows: (a) F(a, t)=O for all t>O i f a - - O
Stability of nonlinear time-varying feedback systems
311
(o] Loop
-l y
-
I
. . . .
l
I
~
.~ Loop 2 (b)
~
, 1~-~ I~
I z ~¢1 FIG. l(a) Non-linear time-varying feedback system; (b) proof of theorem 2.1 ; (c) proof of theorem 3.1.
Fp: If F satisfies the condition (a) and if O< F(~r' t)<~oo for all t>_O and for all real numbers tr ~0, then FeFp. tr F~,k: If K N and K, are positive constants such that 0 < f ( a . ) < K r ¢ and O
Kz=KN • K,, and if F satisfies condition (a) and lies in the sector 0 < F(tr' t)
tr
F,.: If FeFp and (F(cr 1, t ) - F ( t r 2, t)} • (0"1 - t r 2 ) > 0 for all t ~ 0 and for all real numbers ~r1 and a 2, then FeF,.. Frnk'. If Fe.Fra~Ft,k, then
for ~r1 q:a2.
FeF,.k. If Fe,Fmk , it should be noted that 0_< F(trl, t ) -
F(tr2, t) < oO
~rt--tr2
312
Yo-SuNG Cr~o a n d KUMPATI S. NARENDRA
Fo,.: IfF~F,. and F(a, t)= - F ( - g ,
t) for all t_>0, then FeFo,..
Fo.,~: If F~Fo,.c~F.,,, then FeFo.,~. In all cases, given a nonlinear function f ( . ), the positive real constant Fn,~nis defined as Fmi~=min f(w)w where w is a real number. w f'~f(v)dv ~rO
If the nonlinear function lies in the first and third quadrants, F,.i. lies in the range 0 < Fm~~ < m. If the nonlinear function is a monotonic increasing function, F=~. lies in the range 1 < F ~ . < oc. 2.2 Linear time-invariant operator G In the given feedback system, Fig. l(a), the operator G is linear time-invarianl and is assumed to satisfy the following condition. t
Ge(t) =
f O(t-
~)e(~)d~ + gi(t)
(3)
do
where g(t)tLx taL2 is a unit impulse response of G and gi(t)eLz is the result of initial conditions. It can be shown [4b] that in all the problems considered in this paper the initial condilions can be assumed to be zero without any loss of generality. This is accomplished by suitably modifying the input to the system. If x(t) represents the input to the system (1) with initial conditions and x*(t) is defined as
x*(t)
~(t-¢)x(p)dp +~(t)
(4)
0
then the system represented by the equation
f
' g ( t - p)F[y(p)]dp + y(t) = x*(t) 0
with zero initial conditions has the same response as the system (1). If x(t) and gi(t)eL2 then x*(t)eL 2 and hence the initial conditions can always be assumed to be zero. 2.3 Passive operators If x, yeL2, an inner product in L2 is defined by setting
=fo~ x(t)y(t)dt. 2o Since every truncated function x~(t), y~(t)eL:, if x(t), y(t)eL2e, we can define the following notation;
~ =
x~(t)y~(t)dt=
y~(t)> 0
where reT.
x(Oy(t)dt 0
Stability of nonlinear time-varyingfc~Ibaek systems
313
(i) Passive operator: Let an operator S map L~, into itself. If < x , Sx>~>O for all zeT and for all xeL~,, then the operator S is said to be passive. (ii) &passiveoperator:Ifthereexistsapositiverealnumber6suchthat < x, Sx > ,> 6 < x, x > ~, the operator S is 6-passive. (iii) 6-M passive operator: if < x , S x > , > 6 < S x , Sx>,, S is a 6-M passive operator. (iv) - e
passive operator: If there exists a positive real number e such that
,>_ - e < x , x>,, the operator is - e Passive. 2.4. Basic theorem The results derived in sections 3 and 4 are direct consequences of the following basic stability theorem and Lemma 2.1.
Theorem 2.1: In the feedback system of Fig. l(a)if the operators G and F map L~, into itself and there exist two positive constants e and 6, 6 > ~ such that F is 6-M passive or 6 passive and G is - e passive or - e - M passive then the system is stable in the sense (i) x(t)t~L2 implies y(t)*L2 (ii) lira ly(t)l-~O. I-*t~
only the proof of (i) is given here. Similar results are also found in [3a, 4b, 6]. The proof of (ii) may be found in [3a, 8b]. Proof: Case 1: F is 6-M passive and G is - 8 passive. Referring to Fig. 1, the system equation is
x = [~ + ~ ] e .
~s)
Multiplying both sides of equation (5) by 26,
2fix=[26I + 26FG]e = [261+ G]e + [ 2 6 F - I]G[26I + G]- t[2al + G]e.
(6)
Defining xl = 26xeL2 and ei = [26I+ G]eeLue if eeL2,, Equation (6) becomes xl = e~ + [ 2 6 F - I ] G [ 2 6 I + G]- Vet.
(7)
Defining F l = 2 6 F - I and G~=G[2M+G] -t, it can be shown that ~>- ~ < el, e ~> ~(where ~ < 1 is a positive real constant), and < F~e2, Fie2 >, < < e2 e2 > , if the hypotheses of Theorem 2.1 are satisfied. Since F1Gt is a contraction operator in L2, space therefore elsL2 if x~eL2 from equation (7). Hence yeJ_.2 and eeL>
Case 2: G is i S - M passive and F is - ~ passive. Operating on both sides of the system equation with G, the equation x t = e l +GFel is obtained where xt = Gx and el = Ge. This has the same form as equation (5) of Case 1. Case 3: F is t~-passive and G is - 8M passive.
314
Yo-SUNG CHO and KUMPATI S. NARENDRA
x=e+FGe = [I + 6G]e + I F - 61]G[I + 6Gj - I[1+ fiGJe
=e 1+ F~GIe ~ where
e1=[l+6G]e,
F,=[F-61],
G, = G [ I + f G ] -~
18)
Then, if the conditions of Theorem 2.1 are satisfied,
,>(6-e)
G~e> ~
, > 0 .
and
Hence, equation (8) has the form of equation (5) of Case 1.
Case 4: G is f-passive and F is - e M passive. The proof follows along the lines indicated in Cases 2 and 3. 2.4.
Lernmas
Let linear time-invariant operators the forms:
(i) (ii)
Zu, Z~z, Z~Lc be Laplace transformable and have
Z~(s)=s Z~z(s)= I~I s + ~where ~o = 0 and 0 < 8 0 < ~ <81
i=oS+8~
and (iii)
Z~zc(S)= ~ ~,+bos+ Vl es+c'~i+ ~ g y s ~S"~, 2, j=l
i~ =l
' S+~i
j=l
"~- L
,=, ' S2"~kS~-I"k
~ s z + ~s + p~
where all the coefficients are non-negative and satisfy 0
~ ~>0, ~= ~
tb -- ~' -- ----> - - ' 2~ 2~
~-->/~'
p~>v~
and
l~ p~-vk v~(~-/~k)>O (k-- t~k 2 (Pk-- V~) (the form ofZ~Lc(S) appeared in [8b] and is used to prove the stability of systems containing an odd monotonic function). Then ZN, ZRL and Z~Lc are passive operators. Let Z~,, Z,., Zorn be defined by
(i) Z;l(s)=ZN(s+8)=s+[~ (ii)
Z7, l(s)=Z~z(s+8)
(iii) Z~'m~(S) =
Z~t.c(S+ 8)
where fl is a positive real constant. Then, the following Lemma 2.1 may be stated.
Stability of nonlinear time-varying feedback systems
315
Lemma 2.1. If a nonlinear time-varying function F is preceded by a linear operator Z and there exists a positive constant/3 such that R(t) < flF.,i. K(t) then (i) FZ v is 6 - M passive if FeF v (ii) FZ,, is 6 - M passive if FeF,, (iii) FZo,. is 6 - M
passive if FeFo,,
Proof: Case (i) Let x and y be in L2e and defined by the relation y(t) = K(t)f[Zx]. Then, it must be shown that < x , y > , > 6 < y , y>~. I f w and p are defined by the equations *~,+flw=y (or Z~y=w) and p=Kf(w)
(9)
the inner product < y , p>~ can be shown to be greater than or equal to zero. If fl=fll +~ where e is an arbitrarily small positive number
, = <~+/~w, gf(w)>
+~
,f
K (~)~dt+~
o
r
do
Kf(~)~d¢
If w(O) is chosen to be zero the first term in the r.h.s, of equation (10) is greater than zero. If
~ .
< ~ l F m i n
the second and third terms together are greater than or equal to zero. The last term is greater than or equal to
Hence
< Y, P>.>~-2, and the operator K(t)f{z,y} is 6 passive.
Case (ii) In this case it is required to show that the operator K(t)f{Z,.y} is 6-passive, or,~b
p>,
(11)
316
Y o - S u N G C n o a n d KUMPATI S. NARENDRA
Let ~= ~ (s + rh)
where 0 < r h < ? ~ < r h < 7 2 < . . . respect to s + fl,
and tl~>_ft. Expanding Z[,t(s) in partial fractions with ~
. s+fl
x
Z~ t(s)=~o+fio(S+fl)+ L O i - - ~ Z(~(s)+Z(~)(s)+ ~ Zi(s) i=1
S .@ ~i ~-
where all coefficients are non-negative real numbers. y(t)=Z2, Xw(t), the l.h.s, of inequality (11) becomes
(12)
i=I
Defining w(t) by the relationship N
Kf(w)>~+
Kf(w)>~+ ~
Kf(w)>~.
(13)
i=1
passive. By Case (i), the second term of (13) is
The first term ot (13)is obviously 6 - M also~6- M passive if
/~ -- < flFmi. . K The third term can be shown to be positive as follows: Defining u(t) by ti+y~u=w. Z i w = 6 i ( ( I .@flu). Then,
Kf(w)>~=fi<(fi+flu),
Kf(fi+y~u)>~.
(14)
Subtracting
(15) from (14), one obtains a non-negative quantity by the monotonic property (2) of F. But the expression (15) is greater than zero if
R - - ,< f l F m i n
K
by case (i). Hence, the expression (14) is greater than zero and FZ,, is 6 - M passive.
Case (iii). By employing the same methods as were used in the previous cases and Lemmas 2.3-2.5 of [8b], passivity of FZo,, can be proved. Howevtr, the proof is omitted here. 3.
EXTENSION
It is well k n o w n that if
OF P O P O V ' S C R I T E R I O N TO TIME-VARYING SYSTEMS
NONLINEAR
Stability of nonlinear time-varyingfeedback systems
3! 7
is positive real, the feedback system in Fig. l(a)is asymptotically stable for all nonlinear time-varying gains F(17,t) in the range
(16)
o
by the circle criterion [2, 3b]. Further, by Popov's criterion if a gain
Kp is found
so that
G(s,(~+1)+~ is positive real, then Kp>_Ko and the nonlinear autonomous system is stable for all timeinvariant nonlinear functions f ( • ) in the sector
0 < S(a) < Kr.
(17)
17
In this section stability criteria are derived for feedback systems with nonlinear time-varying gains. The results can be specialized to yield the criteria of (16) or (17). In the general case, the criterion derived, involves a trade-off between the rate of variation of K(t) and the range of the nonlinear gain function f ( • ).
Main theorem 3.1. In the feedback system of Fig. (1) if operator and the function K(t) satisfies the inequality
FeF,, G(s)Z~l(s) is a
passive
/~(t!
K(t) then the system is stable in the sense (i)
x(t)eL2 implies y(t)eLz
(ii) lim
[y(t) I- 0.
['-* O~
Proof. FZp is a f i - M passive operator by lemma 2.1. Hence by theorems 2.1 and 2.2 the system is asymptotically stable. Theorem 3.2. In
the feedback system of Fig. 1 if
F~Fpk, is a passive operator and the function
[G(s)Z'~'(s) +--~ K(t) satisfies the inequality
/'~'(t?< ]TF.i.rl K(t)1 tO(t)
then the system is stable.
L --~-~ J
(18)
318
Yo-SuNG CHO and KUMPATIS. NARENDRA
Proq£ It
is sufficient to show that FZp is a ~ - M passive operator or
y > ~ ];
FZ,y>~(r~T)
Defining w(i) by the relation ~ ( t ) + flw(t)=y(t) we have w(/)=Z,y(t) and Hence
~19~
FZ,y(I)=.~w(t)K(t)]
./~wK)>~
y>~=
+~/~ f~d~.
K,J
do
do t"
The first term is greater than or equal to zero: the second and the third terms together are greater than or equal to zero if condition (18) is satisfied. The last term yields
~[ K'(t)f(wlwdt~ ~o K,"
K,
K,
K2Jo
{f(w)K(t)}~dt
which is non-negative. Hence inequality (19) is satisfied.
Comments on theorems 3.1 and 3.2.
The condition
~(t) < ~Fm,~['
K(t~
"~ -
K(t)~ K, /
indicates that the rate of variation KU) as indicated by ~ / K depends on the nonlinear function f ( • ) that is considered. For example, if f ( • ) is a purely linear time invariant gain [i.e. F(w, t ) = wK(t)], the stability condition of Theorem 3.2 b~omes
a result which is known for linear time-varying systems [3c,'5, 8a]. If the entire class of nonlinear functions in the sector 0 < J'te~ > K~
is considerered, then F ~ 0 (i.e. ~ 0 ) or F( •, l) has to be a linear time-invariant nonlinear function, as in Popov's criterion [1]. However. if the s p ~ i ~ c nonlinear function exists such that F ~ has a value 7 [e.g. iff(w)=w a, ~ 4 ] , then •
<~(1-~
.
Stability of nonlinear time-varying feedback systems
310
For the particular case, when K2-,Kc, ~-, oo [i.e. G(s) + (I/K2) becomes positive real] so that the sytem is asymptotically stable for all nonlinear gains 0 < F ( a, I)
which is the range predicted by the circle criterion [2, 3b]. FOR THE CLASSES Fm AND F0,~
4. STABILITY CRITERION
If the class of monotonic increasing nonlinear functions is considered, it is well known that the range of stability may be increased in many situations beyond the Popov sector to 0 < f ( a ) < KM O"
where KM > Kp. A sufficient condition for stability in this sector is that
G(s) ÷ ~ J be P.R. [4b], where Z(s) has the form ofZaL(s ) or Z~(s). The extension of this criterion to nonlinear time-varying systems is considered in this section.
Theorem 4.1. (a) If in the feedback system of Fig. I FeF~,and there exist a multiplier Z~t(s) such that
G(s)Z~l(s) is P.R. and
/~(t)< #Fmln
K(t) then the system is stable in the sense that
(i) x(t)eL 2 implies y(t)eL 2 (ii) lim
ly(t)[--,0.
t~06
(b) If FeFo,, and a multiplier Zff~(s) exists such that G(s)Z~(s) is P.R. and
~!t!<~F.~. K(t) then the system is asymptotically stable.
Proof. (a) Since G(s)' Z~,t(s) and F[Z,~(. ), t] are passive and 6 - M passive by the hypothesis and case (ii) of Lemma 2.2 respectively, the system is asymptotically stable by Theorems 2.1 and 2.2. The same arguments can be used to prove case (b).
32O
Yo-SuNG CHO a n d KUMPATI S. NARENDRA
Theorem 4.2 (a) If in the feedback system of Fig. 1 FeF,,~and there exists a multiplier Z~, l(s) such that IG(s)Z~,~(s)+~lisP.R.
120)
and
"'-~<[]Fmi,(1-K(t)~ ,
(1
g~/'
then the system is stable. (b) If F~Fo,,~ and there exists a multiplier
Zom(s) such
t21)
that
[G(s) "Zo-~(s)+-~2] is P.R.
(21)
where a = ao + bob and
~<,F~i.(l- ~(t)) ~
k
~,J
then the system is stable.
Praa~ Case (a):
By theorems 2.1 and 2.2 it is su~¢ient to show
< FZmy , y > ~ < F Z ~ y ,
that
FZ~y>~
(23)
.
Defining w(t) by the relation w(t)=Z~y(t) an inequality similar to (13) may be obtained. The 1.h.s. of inequality (23) may be expressed as:
~o ~+ , + ~ ~. i=1
The first term is greater than or equal to
~°,. Kz
If the inequality (21) is satisfied the second and third terms are greater than or equal to zero. In pa~icular the second term is greater than or equal to
~ < r(t)f(w), ~(t)f(w)>~. Ka H e n ~ inequality (23) is satisfied.
Case (b):
The proof follows along the same lines as in (a) and is omitted here.
Stabmty ox nommear time-varying feedback systems 5. I N T E R P R E T A T I O N
OF
321
RESULTS
Theorems 3.1, 4.1, and 4.2 give sufficient conditions for the stability o f feedback systems with nonlinear time-varying feedback gains having the general f o r m f ( . )K(t). These conditions have the same f o r m as well established frequency domain conditions for nonlinear a u t o n o m o u s sytems and are expressed in terms of the existence o f a multiplier Z(s) which makes G(s)Z(s) (or G(s)Z(s)+[1/Kx] in the finite sector case) positive real. The results indicate that the same multiplier used in the nonlinear time-invariant case can also be used for the nonlinear time-varying case to prove stability provided (I~/K) is b o u n d e d in some fashion. The b o u n d on (R/K) is seen to depend on two factors: fl and Fmin where fl is the distance o f the nearest singularity of Z(s) f r o m the imaginary axis (for RL and RC multipliers) and Fmin -~ 'fw(~)w I
"
n
Hence fl depends on the particular multiplier chosen and F=i.(w) on the particular nonlinear function included in the feedback path. The latter can be zero when f ( • ) includes the entire class o f nonlinearities in the sector
o
but is finite and lies in the interval 0 < F m i n < ~ for a specific nonlinear function. F o r m o n o t o n i c nonlinear functions Fml, satisfies l < F m i n < ~ , indicating that the stability criterion derived for nonlinear time-invariant systems is also valid for time-varying cases for some slowly varying function K(0. R6sum6--Dans cet article la stabilit6 de syst6mes non lin6aires/t r6action variables dans le temps est 6tudi6e en utilisant une technique d' "op6rateur passif". Le syst6me A r6action est suppos6 consister d'un op6rateur lin~aire non-variable dans le temps G(s) dans la chaine d'action et d'un gain non-lin6aire variable dans le temps f(.)K(t) darts la chaine de r~action. La condition de stabilit6 indique que la limitation de la deriv6e par rapport au temps [dK(t)/(dt)] depend A la fois de ia non-lin6arit6 f ( ' ) et du multiplicateur Z(s) choisi pour rendre G(s)Z(s) positif r6el, I1 est 6galement montr6 que le r6sultat g6n6ral dam cet article peut 6tre sl~ialis6 pour fournir plusieurs des r6sultats obtenus jusq'ici pour les syst6mes nonlin6aires non variables darts le temps et pour les syst6mes lin6aires variables dans le temps. Zusammenfassung--ln der Arbeit wird die Stabilit~it nichtlinearer zeitvariabler rtickgeftihrter Systeme mit Hilfe einer Methode vines "passiven Operators" untersucht. Dabei wird angenommen, dab das riickgekoppelte System aus einem linearen zeitinvarianten Operator G(s) im Vorw~irtszweig und einem nichtlinearen zeitvariablen Verstarker f(.)K(t) im Riickfiihrzweig besteht. Die Stabilitiitsbedingungen zeigen, dab der Grenzwert der zeitlichen Ableitung dK(t)/dt sowohl yon der Nichtlinearit,~t J(.) als auch yon dem Multiplikator Z(s) abhiingt, der so gvw/ihlt ist, dab G(s)Z(s) positiv reell wird. Es wird auch gezeigt, dab die haupts'achlichcn Ergcbnisse for nichtlineare zeitinvariante Systeme und lineare zeitvariable Systeme erhalten werden kSnnen. Ko~cne~T--B 3Tol~ CTaT~c, yCTOI~SOCT~ HeTIHHe~HbIX CHCTeM C o6paTHOR CBH3blO 3aBHCHMblX OT B ~ M e H H
H3y~IaeTcR C HOMOIIIhIO TVXHHKH "rlaCCHBHOro o n v p a T o p a " .
CHCTCMa C o6paTHOI~CBS3b~ pacCMaTpHBaCTCaKaK COCTO~R~a~H3 JIHHe~IHOrO oncpaTopa H e 3 a n e c r t M o r o OT BI:~MVHH G(s) B 31~He n p a M o R c e s 3 r t H ~3 HVJ1HHegHOFO KO:~b~HHHeHTa
322
5o-St;~;
C ~ o a n d KUMPAT! S. NARENDRA
y c t t a e ~ n ~ 3 a a ~ c n M o r o OT apCMent~f(t)K(') a 3aerie o 6 p a T 8 o i ~ C a a 3 n . Y c ~ o a u e yCTO~q~BOC7~ yra3~er ~ r o o r p a n n q e n n e npOH3BO~HOB nO a p e M e n a [dK(t)]/(dt) OflHOBpeMeHHo 3 a B a c n T OT a e ~ n n e a H o c T n f ( . ) a OT u n O ~ n T e a f l Z(s) a ~ 6 p a n n o r o T a r ~ r 0 6 ~ G(s)Z(s)CTa~ a o a o ~ n Te~bHblM ~e~CTBHTe~bHblM. T a K ~ ¢ n o K a 3 a u o ~To 0 6 ~ p e 3 y h b T a T 3TO~ CTaTbH MO~eY 6blTb c H e H ~ 3 ~ p o B a H qTO6bl ~ a T b MHOF~e f13 p e 3 y ~ b T a T O a H O ~ e H H b l X ~O C~X llOp ~ f l He~HHe~HblX CHCTeM He3aB~CMMblX OT BpeMeH~ H ~ ~MHe~HblX CHCTeM "laB~CltMbl~ O~ apeMeHn.
STABILITY OF N O N L I N E A R TIME-VARYING FEEDBACK SYSTEMS Yo-SUNG CHO Honeywell Inc., Waltham, Mass. and KUM~AT~ S. NARENDRA Yale University, New Haven, Conn. Summary--In this paper, the stability of nonlinear time-varying feedback systems is studied using a "passive operator" technique. The feedback system is assumed to consist of a linear time-invariant operator G(s) in the forward path and a nonlinear time-varying gain function f ( • )K(t) in the feedback path. The stability condition indicates that the bound on the time derivative [dK(t)/(dt)] depends both on the nonlinearity f ( • ) and the multiplier Z(s) chosen to make G(s)Z(s) positive real. It is also shown that the main result in this paper can be specialized to yield many of the results obtained so far for nonlinear time-invariant systems and linear time-varying systems. 1. I N T R O D U C T I O N IN R~CENT years m a n y new results have been o b t a i n e d in the stability t h e o r y o f f e e d b a c k systems [1-8]. T h e f e e d b a c k system considered generally consists o f a linear t i m e - i n v a r i a n t o p e r a t o r in the f o r w a r d p a t h a n d a n o n l i n e a r o r tim~-varying gain f u n c t i o n in the f e e d b a c k p a t h . T h e results o b t a i n e d a r e generally expressed as sufficienct c o n d i t i o n s o n t h e transfer function G(s) o f the linear p a r t o f the system. I f U(s)Z(s) is positive real for a multiplier Z(s) b e l o n g i n g to some class Zc the f e e d b a c k system is s h o w n to b e stable f o r n o n l i n e a r gains f ( • ) b e l o n g i n g to a c o r r e s p o n d i n g class f t . I n this p a p e r , using a passive o p e r a t o r t ~ h n i q u e ,
Ill v. M. PoPOV: Absolute stability of nonlinear systems of automatic control. Automation and Remote Control 857-875, March 0962). [2] K. S. NAR~r~RA and R. M. GOL~WY~: A geometrical crite~'ion for the stability of certain nonlinear nonautonomous systems. IEEE Trans. Ciremt Theory CToll, 406-408 0954). [3] (a) I. W. SANDnERG: Some results on the theory of physical systems governed by nonlinear functional equations. Bell Syst. Tech. J'. XLIV, 871-898 (1965). (b) I. W. SA~a)BER6: A frequency domain condition for the stability of f~xiback systems containing a single time-varying nonlinear element. Bell Syst. Tech. J. XL~I, 1601-1608 0964). (c) I. W. SA~r~B~R6: On generalizations and extensions of the,Popov criterion. IEEE Trans. Circuit Theory 117-118, March (1966). [4] (a) G. Z~a~.s: Nonlinear time-varying feedback systems-conditions for L®-boundedn~ss derived using conic operators on exponentially weighte~d spaces. Prec. 1965 Allerton Conf., pp. 460-471. (b) G. ZA~,w~s: On the input-output stability of time-varying nonlinear feedback systems, Pts. I and II. IEEE Trans. Automatic Control AC-11, 228-238; 465--476 (1966). [5] R. W. B g o c ~ r r and L. FogYs: On the stability of systems containing a time-varying gain. Prec. 1964 Allerton Conf., pp. 413-430. [6] C. T. L ~ and C. A. D~so~g: Stability of single-loop nonlinear feedback systems. El~t. Res. Lab., Univ. of Cal., Berkeley, Cal., Rpt. No. ERL 66-13, May 0966). [7] Y. S. C~o and K. S. N ~ a ~ a ) ~ : An off-axis circle ~'iterion for the stability of fe~lback systems with a monotonic nonlinearity. Proc. 1967 Allerton Conf, pp. 238-248. [8] (a) K. S. N ~ r , ro~A and Y. S. C8o: Stability of linear time-varying feedback systems. Dunham Lab., Yale Univ., New Haven, Conn., Tech. Rpt. CT-8, Feb. (1967). (b) K. S. NARE~qDRAand Y. S. C8o: Stability of feedback systems containing a single odd monotonic nonlinearity. IEEE Trans. Automatic Control AC-12, August (1967).
309
310
Yo-Suy(; CHO a n d KUMPATI S. NARENDRA
sufficient conditions are derived for the stability of nonlinear and time-varying systems which can be specialized to yield many of the results obtained earlier, e.g. Popov criterion [1], circie criteria for systems with linear time-varying [3b, 5, 8a] and nonlinear time-varying [2, 3b, 4~t1 gains.
2.
MATHEMATICAL
PRELIMINARIES
We define an Lz~ space which is an extension o f t h e L z space of square integrable function s of real variable t in [0, m] as follows. Lz~ contains both functions with Lz finite norms and functions with infinite norms. Each finite time truncation of every function x(t) belonging to Lz~ will lie in L~. Let x, denote the truncated function which assumes the values x,(t) = x(t) ( t < x and reT where T is the subinterval of the reals [0, oo]), and x M ) = 0 elsewhere. For xeL2, let ]lx[[2 denote the norm of x. Definition: The extension of L2, denoted by L2~, is the space consisting of those functions whose truncations lie in L2. An extended norm denoted by [[x[l:~ is assigned to each x(t) as follows.
Ilxitze=
xZdt =llxiJ2 oo
iixiize=
x¢c .
if
x,~L 2
and
Assuming G and F i n Fig. la map L2e into itself, one may formulate feedback equations for the given system as follows: If Ge(t) indicates the operation of G on an input function e(t), then
Ge(t) = y(t) FGe(t)+e(t)=x(t)
l)
The operators G and F considered in this paper belong to the classes described in 2.1 and 2.2. The problem of interest may now be stated as follows: Assuming that a solution exists for equation (1) and lies in Lze, find sufficient conditions on G and F under which x(t)eL2 and e(t), y(t)eLze imply e(t), y(t)eL,_ and lim ]y(t)[~O. t ~
2.1 Classes of nonlinear t#ne-varying gains F Let f ( . ) represent a memoryless nonlinear time-invariant function and K(t) be a differentiable linear time-varying function. The operators F considered here belong to the class of memoryless nonlinear timevarying functions which can be expressed as F ( . , t ) = f ( . )K(t) and may be defined as follows: (a) F(a, t)=O for all t>O i f a - - O
Stability of nonlinear time-varying feedback systems
311
(o] Loop
-l y
-
I
. . . .
l
I
~
.~ Loop 2 (b)
~
, 1~-~ I~
I z ~¢1 FIG. l(a) Non-linear time-varying feedback system; (b) proof of theorem 2.1 ; (c) proof of theorem 3.1.
Fp: If F satisfies the condition (a) and if O< F(~r' t)<~oo for all t>_O and for all real numbers tr ~0, then FeFp. tr F~,k: If K N and K, are positive constants such that 0 < f ( a . ) < K r ¢ and O
Kz=KN • K,, and if F satisfies condition (a) and lies in the sector 0 < F(tr' t)
tr
F,.: If FeFp and (F(cr 1, t ) - F ( t r 2, t)} • (0"1 - t r 2 ) > 0 for all t ~ 0 and for all real numbers ~r1 and a 2, then FeF,.. Frnk'. If Fe.Fra~Ft,k, then
for ~r1 q:a2.
FeF,.k. If Fe,Fmk , it should be noted that 0_< F(trl, t ) -
F(tr2, t) < oO
~rt--tr2
312
Yo-SuNG Cr~o a n d KUMPATI S. NARENDRA
Fo,.: IfF~F,. and F(a, t)= - F ( - g ,
t) for all t_>0, then FeFo,..
Fo.,~: If F~Fo,.c~F.,,, then FeFo.,~. In all cases, given a nonlinear function f ( . ), the positive real constant Fn,~nis defined as Fmi~=min f(w)w where w is a real number. w f'~f(v)dv ~rO
If the nonlinear function lies in the first and third quadrants, F,.i. lies in the range 0 < Fm~~ < m. If the nonlinear function is a monotonic increasing function, F=~. lies in the range 1 < F ~ . < oc. 2.2 Linear time-invariant operator G In the given feedback system, Fig. l(a), the operator G is linear time-invarianl and is assumed to satisfy the following condition. t
Ge(t) =
f O(t-
~)e(~)d~ + gi(t)
(3)
do
where g(t)tLx taL2 is a unit impulse response of G and gi(t)eLz is the result of initial conditions. It can be shown [4b] that in all the problems considered in this paper the initial condilions can be assumed to be zero without any loss of generality. This is accomplished by suitably modifying the input to the system. If x(t) represents the input to the system (1) with initial conditions and x*(t) is defined as
x*(t)
~(t-¢)x(p)dp +~(t)
(4)
0
then the system represented by the equation
f
' g ( t - p)F[y(p)]dp + y(t) = x*(t) 0
with zero initial conditions has the same response as the system (1). If x(t) and gi(t)eL2 then x*(t)eL 2 and hence the initial conditions can always be assumed to be zero. 2.3 Passive operators If x, yeL2, an inner product in L2 is defined by setting
x~(t)y~(t)dt=
y~(t)> 0
where reT.
x(Oy(t)dt 0
Stability of nonlinear time-varyingfc~Ibaek systems
313
(i) Passive operator: Let an operator S map L~, into itself. If < x , Sx>~>O for all zeT and for all xeL~,, then the operator S is said to be passive. (ii) &passiveoperator:Ifthereexistsapositiverealnumber6suchthat < x, Sx > ,> 6 < x, x > ~, the operator S is 6-passive. (iii) 6-M passive operator: if < x , S x > , > 6 < S x , Sx>,, S is a 6-M passive operator. (iv) - e
passive operator: If there exists a positive real number e such that
Theorem 2.1: In the feedback system of Fig. l(a)if the operators G and F map L~, into itself and there exist two positive constants e and 6, 6 > ~ such that F is 6-M passive or 6 passive and G is - e passive or - e - M passive then the system is stable in the sense (i) x(t)t~L2 implies y(t)*L2 (ii) lira ly(t)l-~O. I-*t~
only the proof of (i) is given here. Similar results are also found in [3a, 4b, 6]. The proof of (ii) may be found in [3a, 8b]. Proof: Case 1: F is 6-M passive and G is - 8 passive. Referring to Fig. 1, the system equation is
x = [~ + ~ ] e .
~s)
Multiplying both sides of equation (5) by 26,
2fix=[26I + 26FG]e = [261+ G]e + [ 2 6 F - I]G[26I + G]- t[2al + G]e.
(6)
Defining xl = 26xeL2 and ei = [26I+ G]eeLue if eeL2,, Equation (6) becomes xl = e~ + [ 2 6 F - I ] G [ 2 6 I + G]- Vet.
(7)
Defining F l = 2 6 F - I and G~=G[2M+G] -t, it can be shown that
Case 2: G is i S - M passive and F is - ~ passive. Operating on both sides of the system equation with G, the equation x t = e l +GFel is obtained where xt = Gx and el = Ge. This has the same form as equation (5) of Case 1. Case 3: F is t~-passive and G is - 8M passive.
314
Yo-SUNG CHO and KUMPATI S. NARENDRA
x=e+FGe = [I + 6G]e + I F - 61]G[I + 6Gj - I[1+ fiGJe
=e 1+ F~GIe ~ where
e1=[l+6G]e,
F,=[F-61],
G, = G [ I + f G ] -~
18)
Then, if the conditions of Theorem 2.1 are satisfied,
G~e> ~
and
Hence, equation (8) has the form of equation (5) of Case 1.
Case 4: G is f-passive and F is - e M passive. The proof follows along the lines indicated in Cases 2 and 3. 2.4.
Lernmas
Let linear time-invariant operators the forms:
(i) (ii)
Zu, Z~z, Z~Lc be Laplace transformable and have
Z~(s)=s Z~z(s)= I~I s + ~where ~o = 0 and 0 < 8 0 < ~ <81
i=oS+8~
and (iii)
Z~zc(S)= ~ ~,+bos+ Vl es+c'~i+ ~ g y s ~S"~, 2, j=l
i~ =l
' S+~i
j=l
"~- L
,=, ' S2"~kS~-I"k
~ s z + ~s + p~
where all the coefficients are non-negative and satisfy 0
~ ~>0, ~= ~
tb -- ~' -- ----> - - ' 2~ 2~
~-->/~'
p~>v~
and
l~ p~-vk v~(~-/~k)>O (k-- t~k 2 (Pk-- V~) (the form ofZ~Lc(S) appeared in [8b] and is used to prove the stability of systems containing an odd monotonic function). Then ZN, ZRL and Z~Lc are passive operators. Let Z~,, Z,., Zorn be defined by
(i) Z;l(s)=ZN(s+8)=s+[~ (ii)
Z7, l(s)=Z~z(s+8)
(iii) Z~'m~(S) =
Z~t.c(S+ 8)
where fl is a positive real constant. Then, the following Lemma 2.1 may be stated.
Stability of nonlinear time-varying feedback systems
315
Lemma 2.1. If a nonlinear time-varying function F is preceded by a linear operator Z and there exists a positive constant/3 such that R(t) < flF.,i. K(t) then (i) FZ v is 6 - M passive if FeF v (ii) FZ,, is 6 - M passive if FeF,, (iii) FZo,. is 6 - M
passive if FeFo,,
Proof: Case (i) Let x and y be in L2e and defined by the relation y(t) = K(t)f[Zx]. Then, it must be shown that < x , y > , > 6 < y , y>~. I f w and p are defined by the equations *~,+flw=y (or Z~y=w) and p=Kf(w)
(9)
the inner product < y , p>~ can be shown to be greater than or equal to zero. If fl=fll +~ where e is an arbitrarily small positive number
+~
,f
K (~)~dt+~
o
r
do
Kf(~)~d¢
If w(O) is chosen to be zero the first term in the r.h.s, of equation (10) is greater than zero. If
~ .
< ~ l F m i n
the second and third terms together are greater than or equal to zero. The last term is greater than or equal to
Hence
< Y, P>.>~-2
Case (ii) In this case it is required to show that the operator K(t)f{Z,.y} is 6-passive, or
p>,
(11)
316
Y o - S u N G C n o a n d KUMPATI S. NARENDRA
Let ~= ~ (s + rh)
where 0 < r h < ? ~ < r h < 7 2 < . . . respect to s + fl,
and tl~>_ft. Expanding Z[,t(s) in partial fractions with ~
. s+fl
x
Z~ t(s)=~o+fio(S+fl)+ L O i - - ~ Z(~(s)+Z(~)(s)+ ~ Zi(s) i=1
S .@ ~i ~-
where all coefficients are non-negative real numbers. y(t)=Z2, Xw(t), the l.h.s, of inequality (11) becomes
(12)
i=I
Defining w(t) by the relationship N
Kf(w)>~+
Kf(w)>~+ ~
Kf(w)>~.
(13)
i=1
passive. By Case (i), the second term of (13) is
The first term ot (13)is obviously 6 - M also~6- M passive if
/~ -- < flFmi. . K The third term can be shown to be positive as follows: Defining u(t) by ti+y~u=w. Z i w = 6 i ( ( I .@flu). Then,
Kf(w)>~=fi<(fi+flu),
Kf(fi+y~u)>~.
(14)
Subtracting
(15) from (14), one obtains a non-negative quantity by the monotonic property (2) of F. But the expression (15) is greater than zero if
R - - ,< f l F m i n
K
by case (i). Hence, the expression (14) is greater than zero and FZ,, is 6 - M passive.
Case (iii). By employing the same methods as were used in the previous cases and Lemmas 2.3-2.5 of [8b], passivity of FZo,, can be proved. Howevtr, the proof is omitted here. 3.
EXTENSION
It is well k n o w n that if
OF P O P O V ' S C R I T E R I O N TO TIME-VARYING SYSTEMS
NONLINEAR
Stability of nonlinear time-varyingfeedback systems
3! 7
is positive real, the feedback system in Fig. l(a)is asymptotically stable for all nonlinear time-varying gains F(17,t) in the range
(16)
o
by the circle criterion [2, 3b]. Further, by Popov's criterion if a gain
Kp is found
so that
G(s,(~+1)+~ is positive real, then Kp>_Ko and the nonlinear autonomous system is stable for all timeinvariant nonlinear functions f ( • ) in the sector
0 < S(a) < Kr.
(17)
17
In this section stability criteria are derived for feedback systems with nonlinear time-varying gains. The results can be specialized to yield the criteria of (16) or (17). In the general case, the criterion derived, involves a trade-off between the rate of variation of K(t) and the range of the nonlinear gain function f ( • ).
Main theorem 3.1. In the feedback system of Fig. (1) if operator and the function K(t) satisfies the inequality
FeF,, G(s)Z~l(s) is a
passive
/~(t!
K(t) then the system is stable in the sense (i)
x(t)eL2 implies y(t)eLz
(ii) lim
[y(t) I- 0.
['-* O~
Proof. FZp is a f i - M passive operator by lemma 2.1. Hence by theorems 2.1 and 2.2 the system is asymptotically stable. Theorem 3.2. In
the feedback system of Fig. 1 if
F~Fpk, is a passive operator and the function
[G(s)Z'~'(s) +--~ K(t) satisfies the inequality
/'~'(t?< ]TF.i.rl K(t)1 tO(t)
then the system is stable.
L --~-~ J
(18)
318
Yo-SuNG CHO and KUMPATIS. NARENDRA
Proq£ It
is sufficient to show that FZp is a ~ - M passive operator or
y > ~ ];
FZ,y>~(r~T)
Defining w(i) by the relation ~ ( t ) + flw(t)=y(t) we have w(/)=Z,y(t) and Hence
~19~
FZ,y(I)=.~w(t)K(t)]
./~wK)>~
y>~=
+~/~ f~d~.
K,J
do
do t"
The first term is greater than or equal to zero: the second and the third terms together are greater than or equal to zero if condition (18) is satisfied. The last term yields
~[ K'(t)f(wlwdt~ ~o K,"
K,
K,
K2Jo
{f(w)K(t)}~dt
which is non-negative. Hence inequality (19) is satisfied.
Comments on theorems 3.1 and 3.2.
The condition
~(t) < ~Fm,~['
K(t~
"~ -
K(t)~ K, /
indicates that the rate of variation KU) as indicated by ~ / K depends on the nonlinear function f ( • ) that is considered. For example, if f ( • ) is a purely linear time invariant gain [i.e. F(w, t ) = wK(t)], the stability condition of Theorem 3.2 b~omes
a result which is known for linear time-varying systems [3c,'5, 8a]. If the entire class of nonlinear functions in the sector 0 < J'te~ > K~
is considerered, then F ~ 0 (i.e. ~ 0 ) or F( •, l) has to be a linear time-invariant nonlinear function, as in Popov's criterion [1]. However. if the s p ~ i ~ c nonlinear function exists such that F ~ has a value 7 [e.g. iff(w)=w a, ~ 4 ] , then •
<~(1-~
.
Stability of nonlinear time-varying feedback systems
310
For the particular case, when K2-,Kc, ~-, oo [i.e. G(s) + (I/K2) becomes positive real] so that the sytem is asymptotically stable for all nonlinear gains 0 < F ( a, I)
which is the range predicted by the circle criterion [2, 3b]. FOR THE CLASSES Fm AND F0,~
4. STABILITY CRITERION
If the class of monotonic increasing nonlinear functions is considered, it is well known that the range of stability may be increased in many situations beyond the Popov sector to 0 < f ( a ) < KM O"
where KM > Kp. A sufficient condition for stability in this sector is that
G(s) ÷ ~ J be P.R. [4b], where Z(s) has the form ofZaL(s ) or Z~(s). The extension of this criterion to nonlinear time-varying systems is considered in this section.
Theorem 4.1. (a) If in the feedback system of Fig. I FeF~,and there exist a multiplier Z~t(s) such that
G(s)Z~l(s) is P.R. and
/~(t)< #Fmln
K(t) then the system is stable in the sense that
(i) x(t)eL 2 implies y(t)eL 2 (ii) lim
ly(t)[--,0.
t~06
(b) If FeFo,, and a multiplier Zff~(s) exists such that G(s)Z~(s) is P.R. and
~!t!<~F.~. K(t) then the system is asymptotically stable.
Proof. (a) Since G(s)' Z~,t(s) and F[Z,~(. ), t] are passive and 6 - M passive by the hypothesis and case (ii) of Lemma 2.2 respectively, the system is asymptotically stable by Theorems 2.1 and 2.2. The same arguments can be used to prove case (b).
32O
Yo-SuNG CHO a n d KUMPATI S. NARENDRA
Theorem 4.2 (a) If in the feedback system of Fig. 1 FeF,,~and there exists a multiplier Z~, l(s) such that IG(s)Z~,~(s)+~lisP.R.
120)
and
"'-~<[]Fmi,(1-K(t)~ ,
(1
g~/'
then the system is stable. (b) If F~Fo,,~ and there exists a multiplier
Zom(s) such
t21)
that
[G(s) "Zo-~(s)+-~2] is P.R.
(21)
where a = ao + bob and
~<,F~i.(l- ~(t)) ~
k
~,J
then the system is stable.
Praa~ Case (a):
By theorems 2.1 and 2.2 it is su~¢ient to show
< FZmy , y > ~ < F Z ~ y ,
that
FZ~y>~
(23)
.
Defining w(t) by the relation w(t)=Z~y(t) an inequality similar to (13) may be obtained. The 1.h.s. of inequality (23) may be expressed as:
~o
The first term is greater than or equal to
~°
If the inequality (21) is satisfied the second and third terms are greater than or equal to zero. In pa~icular the second term is greater than or equal to
~ < r(t)f(w), ~(t)f(w)>~. Ka H e n ~ inequality (23) is satisfied.
Case (b):
The proof follows along the same lines as in (a) and is omitted here.
Stabmty ox nommear time-varying feedback systems 5. I N T E R P R E T A T I O N
OF
321
RESULTS
Theorems 3.1, 4.1, and 4.2 give sufficient conditions for the stability o f feedback systems with nonlinear time-varying feedback gains having the general f o r m f ( . )K(t). These conditions have the same f o r m as well established frequency domain conditions for nonlinear a u t o n o m o u s sytems and are expressed in terms of the existence o f a multiplier Z(s) which makes G(s)Z(s) (or G(s)Z(s)+[1/Kx] in the finite sector case) positive real. The results indicate that the same multiplier used in the nonlinear time-invariant case can also be used for the nonlinear time-varying case to prove stability provided (I~/K) is b o u n d e d in some fashion. The b o u n d on (R/K) is seen to depend on two factors: fl and Fmin where fl is the distance o f the nearest singularity of Z(s) f r o m the imaginary axis (for RL and RC multipliers) and Fmin -~ 'fw(~)w I
"
n
Hence fl depends on the particular multiplier chosen and F=i.(w) on the particular nonlinear function included in the feedback path. The latter can be zero when f ( • ) includes the entire class o f nonlinearities in the sector
o
but is finite and lies in the interval 0 < F m i n < ~ for a specific nonlinear function. F o r m o n o t o n i c nonlinear functions Fml, satisfies l < F m i n < ~ , indicating that the stability criterion derived for nonlinear time-invariant systems is also valid for time-varying cases for some slowly varying function K(0. R6sum6--Dans cet article la stabilit6 de syst6mes non lin6aires/t r6action variables dans le temps est 6tudi6e en utilisant une technique d' "op6rateur passif". Le syst6me A r6action est suppos6 consister d'un op6rateur lin~aire non-variable dans le temps G(s) dans la chaine d'action et d'un gain non-lin6aire variable dans le temps f(.)K(t) darts la chaine de r~action. La condition de stabilit6 indique que la limitation de la deriv6e par rapport au temps [dK(t)/(dt)] depend A la fois de ia non-lin6arit6 f ( ' ) et du multiplicateur Z(s) choisi pour rendre G(s)Z(s) positif r6el, I1 est 6galement montr6 que le r6sultat g6n6ral dam cet article peut 6tre sl~ialis6 pour fournir plusieurs des r6sultats obtenus jusq'ici pour les syst6mes nonlin6aires non variables darts le temps et pour les syst6mes lin6aires variables dans le temps. Zusammenfassung--ln der Arbeit wird die Stabilit~it nichtlinearer zeitvariabler rtickgeftihrter Systeme mit Hilfe einer Methode vines "passiven Operators" untersucht. Dabei wird angenommen, dab das riickgekoppelte System aus einem linearen zeitinvarianten Operator G(s) im Vorw~irtszweig und einem nichtlinearen zeitvariablen Verstarker f(.)K(t) im Riickfiihrzweig besteht. Die Stabilitiitsbedingungen zeigen, dab der Grenzwert der zeitlichen Ableitung dK(t)/dt sowohl yon der Nichtlinearit,~t J(.) als auch yon dem Multiplikator Z(s) abhiingt, der so gvw/ihlt ist, dab G(s)Z(s) positiv reell wird. Es wird auch gezeigt, dab die haupts'achlichcn Ergcbnisse for nichtlineare zeitinvariante Systeme und lineare zeitvariable Systeme erhalten werden kSnnen. Ko~cne~T--B 3Tol~ CTaT~c, yCTOI~SOCT~ HeTIHHe~HbIX CHCTeM C o6paTHOR CBH3blO 3aBHCHMblX OT B ~ M e H H
H3y~IaeTcR C HOMOIIIhIO TVXHHKH "rlaCCHBHOro o n v p a T o p a " .
CHCTCMa C o6paTHOI~CBS3b~ pacCMaTpHBaCTCaKaK COCTO~R~a~H3 JIHHe~IHOrO oncpaTopa H e 3 a n e c r t M o r o OT BI:~MVHH G(s) B 31~He n p a M o R c e s 3 r t H ~3 HVJ1HHegHOFO KO:~b~HHHeHTa
322
5o-St;~;
C ~ o a n d KUMPAT! S. NARENDRA
y c t t a e ~ n ~ 3 a a ~ c n M o r o OT apCMent~f(t)K(') a 3aerie o 6 p a T 8 o i ~ C a a 3 n . Y c ~ o a u e yCTO~q~BOC7~ yra3~er ~ r o o r p a n n q e n n e npOH3BO~HOB nO a p e M e n a [dK(t)]/(dt) OflHOBpeMeHHo 3 a B a c n T OT a e ~ n n e a H o c T n f ( . ) a OT u n O ~ n T e a f l Z(s) a ~ 6 p a n n o r o T a r ~ r 0 6 ~ G(s)Z(s)CTa~ a o a o ~ n Te~bHblM ~e~CTBHTe~bHblM. T a K ~ ¢ n o K a 3 a u o ~To 0 6 ~ p e 3 y h b T a T 3TO~ CTaTbH MO~eY 6blTb c H e H ~ 3 ~ p o B a H qTO6bl ~ a T b MHOF~e f13 p e 3 y ~ b T a T O a H O ~ e H H b l X ~O C~X llOp ~ f l He~HHe~HblX CHCTeM He3aB~CMMblX OT BpeMeH~ H ~ ~MHe~HblX CHCTeM "laB~CltMbl~ O~ apeMeHn.