32 1
TECHNICALNOTES AND CORRESPONDENCE
equation shows the system to be stable all
for positive valuesof T, K ,
and thefeedback parameters.
This generalized error coefficient approach can be used to analyze
the effects of any t,ype of n0nunit.y feedback, whether inherent, or
deliberate, upon t.he performance of a linear system. The simple examples of this paper indicate t.he need for caution in using nonunit,y
feedback since the effects can be significant and are not
always
apparent. The examples further illustrat.e why the outer (position)
loop of so many control systems is indeed unity and suggest the
posible use of the generalized error coefficients in the synthesis of
feedback transfer funct.ions that will improve the accuracy as well
as the transientresponse of a system. Finally, stat.e variablefeedback
can be handled by t.he reduct.ion of the block diagram or signal flow
graph to obt.ain the H , of Melsa and Schultz [;I, which can then be
substitutedfor H intothe expressions for the generalized error
coefficients.
< (1
for all
then
T
+ 6,
and any z(.) in LZ (-
s;m
-a
i)
m, m).
e"'z(t)X(z(t))dt
(2)
If, in addition, - V ( - ) is odd,
REFERENCES
(11 B. C. Kuo, Automdie Confrol Systems. EnglewoodCliffs,N.J.:PrenticeHall, 1 9 6 i pp. 252-263.
[2] 0. I. Elgird. Control SystemsTheory.
New York: IvIcGraw-Hill, 1966, pp.
215-227.
[3] G. J. Murphy, BasicAutomatic
Control Theory. Princet,on,K.J.:Van
Nostrand, 1966, pp. 159-161, 437-443.
[4] G. S. Brownand D . P. Campbell, Principles of Servomechanism. New
York: Wiley, 1948, pp. 227-230.
[5] J. L. Mel? and D . G. Schultz. Linear Control Systems. New York:
McGraw-HI11, 1969, pp. 10&104, 383-393.
pf
m
:f,,
e
t ) N ( ~ ( t dt
))
-
eO'P(z(t)) dt
ep(t-r)z(t -
-
dt
T)N(z(~))
e.fP(z(t - .)e-".)
dt.
(4)
Consider the
last
integral of inequalit,y (4).By a change of the variable of integration to tl = t - 7, we get
On the Positivity of Certain Nonlinear
Time-Varying Operators
Y. V. VENKATESH
Abstract-Sufficient conditions are given for the positivity of a
composition of two positive operators, one of which is nonlinear and
time varying.
We consider the following question, which is of some interest in
stability. Given two positive operators, one of which is nonlinear and
time varying, under what conditions is t,heir composition positive?
This correspondence gives the main lemma (Lemma 2), which
forms the pivot,al result in the author's report [l] establishing sufficient. conditions (more general t.han those of [2]) for the stability
of nonlinear timevarying feedback systems.
Let N (. ) be a real-valued funct.ion on (- m , m ) with the following properties: 1) X ( 0 ) = 0; 2) N is monotone nondecreasing, Le.,
(T - s ) ( N ( r )- -V(s))
0; 3) there is aconstant C > 0 such that
lN(r)l _< C(TIfor all T . N will be called odd if N ( - v ) = - N ( v ) for
all real v. Let Lz(- m , m ) be the space of real-valued square integrable functions on (- m , m ).
The derivat,ion of conditions for the composit.ion of operators to
be positive is based on t.he following area inequality (1).
Lemma 1: If lV(.) is monotone nondecreasing, t,hen
z N ( z ) - yAT(z) 2 P ( z ) - P ( y )
(1)
for all z and y, where P ( z ) = f i A r ( s )ds.
Proof: See the proof of Lemma 7 in Zames and Falb [3].
Define
sa
= sup
[P(z)/N(z)z]
2
6i =
inf ( P ( z ) / N ( z ) z ] .
Manuscript
received
October
1972.
24,
The author is with the Department of Electrical Engineering, Indian Institute
India.
Bangalore,
of Science,
Thus, (3) holds.
Inequalities (2) and (3) can be used t o derive positivity conditions
where X(. ) is as defined above; and k
absolutely
continuous
on
[0, m ).
(e
) is a real-valued function,
322
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, .TUNE
I. INTRODUCTIOS
Let Lte be the space of real-valued functions x(. ) on t.he interval
03, m ), for which for every finite t 3 0, z ~ ( Td7) < a.
Let.2 denote the class of operators 2:Lz, + Lb sat.isfying an equation of t,he type
1;
(-
1973
The Shckelberg solut.ion [ l ] for a two-player nonzero-sum differential game where, because of a bias in the prior information sets,
one player assumes the role of a leader and t.he other player assumes
t.he role of a follower has been investigated in [2] and [3]. In this note
m
a
t.he Stackelberg concept, isrridened t o include games where the
number of players are more than t.m-0. The players are assumed to be
divided into two groups, a group of leaders and a group of followers.
z ( ~ ) z (t T ) dT
r ( t ) (6) The group of leaders announce its st.rategies before the group of
followers, and everyplayer may or may notbe cooperating wit.hin his
where z;, zj’ (for all i, j ) are real constants;
l(lzil
izi‘l) < m ; own group. Naturally, in order t o be able t o define a Stackelberg
st.rategy for this game, the leaders as a group are assumed t o know
z ( . ) is a real-valued function on (- m , 03 ), and j?
dr < m;
the rationale according to which the followers are playing, and t,he
0 < 7 1 , 71’ < T Z , ~ 2 <
’
. . . . It is shown in [3] t.hat any convolution of followers are assumed t o be rat.iona1in the sense that. t.heg will play
form (6) is a bounded linear transformation of L?(- =, m ) into itself according t o this rationale. Several interesting games can be f o m and that the set
of all such convolutions can be vieaed as a con1mut.a- d a t e d using the above model such as, for example, oligopoly markets
tive Banach algebra with an identity.
where there are several large firm- as price leaders and several other
Finally, let X be a given operatorin C. X is called posit.ive if
smaller firms as price followers.
JFz(t)m(t)dt
0 for all z in the domain of X and all T
0.
In this note,preliminary results as a basis foramore
general
The lemma given below is proved (under somewhat, less general theory of a tao-group Stackelberg game are obtained. Because of its
conditions) in [I]; its proof is based on inequalities (2) and (3).
analytical simplicity, only the case where Nash strategies are the
Lemma 3: The inequality
rationale used withineach group is t.reated. Other cas- where
minimax, Pareto, and Stackelberg st.rategies are used within each
J T a(t)3?Yh(t)dt 3 0,
doZ; t t T 2 0
( i ) group can be similarly studied; however, this will not be done in this
note.
+
+
~I-.(T)I
>
>
where doz denotes t.he domain of Z, holds for all monotonely nondecreasing 37. if t,he following conditions are satisfied:
1) zi,zj’ 6 0 for i,j = 0,1,2,. . 2) Z ( T ) 6 0 almost everywhere;
a ;
where y , Y arenonnegativeconstants; 4) with f = nmx -( and =
max Y for which 3) is verified, k ( t ) e - 8 is nonincreasing, and k(t)ecfis
nondecreasing.
The inequality (7) holds for all monotonically nondecreasing o d d X
if conditions 3) and 4 ) are satisfied.
11. DEFIA-ITION
AA-D PROPERTIES
Let. S and ’11denote the number of players in groups 1 and 2,
respectively, and let, Ci, i = 1, . *,a\r, and Vi, i = 1,. . . , X , be their
U i and T’ = T
;
:
Vi be the
sets of admissible controls. Let, C = ::T
sets of admissible cont,rols for groups 1 and 2, respectively. Let
J ~ ~ ( u.I. , u. . Y ; v I , - ..,s.u)
Jji(u;v) (where u = (ul,. .,ux)EG and
v = ( ~ ] , . . . , ~ . w ) E T ’ ) f o r j =l ; i = 1 , . - . , I \ : a n d j = 2 ; i = 1 , . - . , M ,
be t.he cost functions for the X
M players, i.e., J i i : li X V + R
for all j and i where j refers to t.he group, and i refers to the player
in this group.
Dqfinition 1: If there exist mappings Ti:T’ + Gi, i = l,.. -,N,
such that. for any ~ € 1 ’
+
ACKNOR-LEDGMENT
The author wishes to thank Dr. 31. A . L. Thathachar for some
useful discussions.
REFERENCES
[ l ] Y . V. Vykatesh. “Improved stability conditions for nonlinear time varying
of Science.Uan~alore.India.
systems.Dep.Elec.Eng.,IndianInstitute
Rep. EE 124. June 1972.
“?ioncausal multipliers for nonlinear system stability,” I E E E T r a n s .
(21
Aufomrcl. Confr.. vol. XC-1.5. pp. 19.5-204. hpr. 1YiO.
[3] G . Zames and P. Falh. “Stability conditions for sl-stems with monotone and
slope restricted nonlinearities.” SI.4.V J . Cor.fr..vol. 6, pp. 89-108. 1968.
-.
A Stackelberg Solution for Games with Many Players
M . SIhIXAN A N D
J. B. CRUZ, J R .
Abstract-The concept of Stackelbergsolution
is widened to
include games with many leaders and many followers. Necessary
conditions for the existence of an open-loop Stackelberg solution in
differential gameswhereeach
player is usingaNash
strategy
within his group are also derived.
2 2 , 1973.This
Jianuscript. receix-ed September 21, 1972;revisedJanuary
work was supportedinpartby
the T.S. Air ForceunderGrant.iFOSR-6&
l579p. in part by the XSF underGrant GK-36276, and in part by the Joint
Serrlces Electronics Prozram under Contract D.iAB-07-72-C-02.59.
The authors arewith the Coordinated Science Laboratory and the Department
of Electrical Engineering, University of Illinois, Urbana. Ill. 61801.
where Tv
= (7‘10,.
.. , T x v )and if there exists a v,,EV such that
J?i(TVs2;v,*)
where v s J i )
= (oxm.
5
Jzi(TV,,c’);V*,c’));
i
=
1,. *
.,M
. ., ~ ~ i - l ) ~ . , v i , ~ ~.i.+,V.W~:),
~ ) . ~ then the strategies
(u,,,~,,)EC
X V ahere
uls: = T v ~ are
, , called Stackelberg strategies
with group 9 as -’ash leaders and group 1 as Sashfollowers.
I n other words, this Stackelberg strategy is the optimal (in the
sense of Nash) strategy for the leaders when the followers react by
playingaccording t o a K s h optimalsolution. It protects every
player in t,he leader group fromattempts by any other one player
in
the leader groupt o deviate from his Stackelberg strategy, causing the
followers t o deviate also, in order t o furt,her reduce their costs. In
this sense this St.ackelberg strat,egy is safer for theleaders than their
corresponding Kash strategies, which in effect, safeguards everyplayer from attempts by anyone other player only to furtherreduce
his cost.
Following t,he same terminology as in [3], the Nash rationalreaction set of group 1 when group 2 is the leader group is defined by
D1
=
((u,v)EC: X V :u = TU,
YvEV).
A similar definition can be made for the Nash rationalreaction set
D2 of group 2 when group 1 is the leader group. It follows from the
definitions of DI and D,that t.he Nash solution for the given game
with 9 X players isobtained byt.aking the intersect,ion of D1 and
+
Dz.
When :
V = 1 the game reduces to a many-leader onefollower