IEEE TR.4KShCTIONS OX AVTOBUTIC CONTROL, VOL.
AC-19,
XO.
3,
JUNE
D. G. Luenberger, ‘fXat,henlaticalprogramming and control
theory: Trends of interplay,”Presented
a t t,he 7thIntern.
Symp. Math. Programming, The Hague, Ketherlands,Sept.
1970.
__ , “Introduction t,o Linear and Nonlinear Programming.”
Reading, M a s . : Addison-Wesley, 1973.
-,
“Convergence rate of a penalty function scheme,” J .
Optimiz. Theory AWL, vol. 7, no. 1, pp. 39-51, 1971.
E. Polak, Compututional Methods in Optimization. Kew York:
Academic, 1971;‘
B. T. Pofyak,The c0njugat.e gradient met.hod in extrema1
problem, USSR Comput. N a f h . Math. Phys., vol. 9, no. 4,
1969 (English Translation).
W . I. Zangwill, Xmlinear Programming: A U n i f i d Approach.
Englewood Cliffs, X. J.: Prentice-Hall, 1969.
I). K. Faddeev and V. N. Faddeeva, Comptutional Methods of
Linear A7gebra. San Francisco, Calif.: Freeman, 1963.
G. E. Forsvt,he. “On the asvrmtotic directions of the sdimensional optiG1um gradient mgthbd,” A’um.er. Nath.., vol. 11, pp.
57-76. 1Qm.
G. P. McCornlick, “8nt.i-zig-zagging by bending,” Management Sci., V O ~ . 15, pp. 315-319, 1969.
A. A. Goldstein, “Convex programming in Hilbert space,” Bull.
Amer. Math. Soc., vol. 70, no. 5, pp. 709-710, 1964.
E. S. Le7,itin and B. T. Polyak, “Const.rained minimization
methods, Zh. Vychisl. X a t . M a t . Fiz., vol. 6, no. 5 , pp. 787823, 1966.
--7
-II-
1974
217
Dimitri P. Bertsekas mas born in Athens,
Greece, in 1942. He received the Mechanical
and Electrical Engineering Diploma from the
National TechnicalUniversity
of Athens,
Athens, Greece, in 1965, the N.S.E.E. degree
’, ,
from George WashingtonUniversity, Washington, D.C., in 1969, and the Ph.D. degree
in system science from theMassachusetts
Institute of Technology, Cambridge, in 1971.
From 1966 to 196’7, he performed research
atthe
National Technical
Vniversity
of
Athens, and from 1967 to 1969, he was with t.he U. S. Army Research
Laboratories, Fort. Belvoir, Va. In the summer of 1971, he worked
at Systems Control, Inc., Palo Alto, Calif. From September 1971 to
December 1973, he was in the faculty of the Engineering-Economic
SystemsDepart.ment
of St.anford University %.here hetaught
courses inoptimization by vectorspacemethods,nonlinear
programming, optinlizat,ion under uncertainty,and convex analysis.
In January 1974, he joined t,he faculty of t.he Electrical Engineering
Department, Universit.y of Illinois, Urbana, where hecurrently
holds the rank of Assist.ant.Professor. His research int.erests include
the areas of estimation and control of uncertain syst.ems, dynamic
programming, optimixat.ion theory, and nonlinear programming
algorithms.
,I
L-Stability of Nonstationary Feedback Systems:
Frequency-Domain Criteria
MALUR K. SUKDARESHAN
Abstract-A frequency-domain positivity condition is derived for
develop &
linear time-varying operators in L andisusedto
stability criteria for linear and nonlinear feedback systems. These
criteria permit the useof a very general classof operators inL with
nonstationary kernels, a s multipliers. More specific resultsare
a timeobtained by using a first-order differential operator with
varying coefficient a s multiplier. Finally, by employing periodic
multipliers, improved stability criteria are derived for the nonlinear
damped Mathieu equation with a forcing function.
I. IKTHODUCTIOS
T
HE employment of an operator theoretic framework
stabi1it.y studies hasresulted in the development of the
posit,ivit.y theorem (due t o Zames [l])as a versatile tool
for the input-ou6put sta.bility analysisof feedback syst.ems.
Many useful criteria for t,he L-stability of t.ime-invsriant
Imdinear syst,ems [2] and syst.ems lyith isolatedtimeva.rying gains[3],[4],have
been developed in the last
Manwcript received October 30, 1972; revised March 2.3, 1973.
Paper recommended by J. C. ‘A7illem, Past Chairman, and R. A.
Skoog, Chairman of t h e I E E E S C SStability, Konlinear, and Distributed Systems Conunittee.
The authors are with the Department of Electrical Engineering,
Indian Institute of Science, Bangalore, India.
AND
&I. A.
L. THATHACHAR
few years, following t,his approach. However, the derivat,ion of sindar resu1t.s in the more general situa,tion of
systemscontaininglineartime-varyingoperators
which
do not, admit, a separation of time-va.ria.tions, has not received much at,tention. Recalling that t.he application of
t,he positivity t,heorem requires t.he open-loop to be fact.ored into a composition of t*wopositive operators, the unpopularit,; of this problem ma>- be atkribut.ed t.0 t,he
difficult,; of obt.aining positivity conditions for a.rbitrary
t,ime-varying operators.Recently, by using aninternal
(state-space)description
of t,heoperat,or, &’illems [51,
for
Estrada and Desoer [ 6 ] ,and the authors[7] have obtained
posit,ivit,y condit,ions in thc time-domain (and from t,hese
L?-stability criteria [’i] for feedback system5 containing a
time-varyinglinear
part). Theseposit>ivityconditions
require the solut,ion of certain associat.ed Riccat,i equations
and are not easy t o check, except in a few simplc cases.
This paperpresentssimplerfrequency-domain
conditions for the positivity and L2-stability of t.ime-varying
systems, derived with the imposition of certain additional
c0nstraint.s of differentiabilit,g, null initial condit.ions, etc.
The nwthod draws inspirat.ion from a recent paper due to
BIodgett, and Young [SI, which gives an absolutc sta.bility
218
IEEE T W S A C T I O N S ON AUTOMATIC CONTROL, JUNE
1974
criterion for a zero-input feedback system with a. time- K , and a being positive const.ants and 11- IIB denoting the
varying linear part anda Popov-type nonlinea.rit,y,using a Euclidean norm.
mult.iplier (1 qs), y > 0. It should however be mentioned
4) H has zero initial condit,ions, i.e., ~ ~ ( =0 0.)
t,hat, although [SI cont,ains ma.ng valuable ideas, the main
Because of propert.- 3 above, it is simple to observe t,hat,
stability t.heorem appears to be incorrect. A corrected
H
E O I P A implies H E O E with the kernel restricted by
version of the st.abilitg criterion of [SI may be obtained as
Ih(f,T)1 6 k
‘ esp(--a(f - T)), which in turn implies that
a special case of the criteria derived in this paper, which
H:L2
+ L.
permit. the use of more general time-varying multipliers.
An operat.or H E O E is said t o have “finite ga.in” if
+
11. PROBLEM
FORMULATION
Notations and D e j h i t i m s
A cert.ain familiarity with the not,ions of L,-spaces will
be assumed. Let R, Ri, and R“ denote, respect.ively, the
real numbers, t,henonnega.tive real numbers, and .n-dimensional Euclidean space.
The concept of the extended L2-space (L?,)is defined by,
LPe
=
{x(.):x*(.) E Lp,
T
‘+
T E R+)
(2.1)
where x*(.) is the t.runcation of x(.), x,(t) = z(t), ++ t E
[O,T], and is zero otherwise.
An operator H in L, (LBe)is defined a$ n single-valued
mapping of L,(L?,) into it.self. H is said to be a, “causal”
operat,or in L2 (Lze)if ( H z ( . ) ) , = (Hz,(.)), ++ x(.) E L2
(L.28)and ++ T E R+.
Let. OB den0t.e t.he class of linear causal operators H in
L2, withanexternal(input-output))
deecript-ion, i.e., if
H E OB, t,hen there exists a map (t.he kernel of H ) h ( . 7 - ) :
R+ X R+ R such that,
-
?/B(t)
=
HuH(t)
=
1-
h(t,T)UH(T) dT
v
uH(’)
the normsindicat,edbeing
t.he
Lz-norms.
Xotethat
H E O I P A => r(H) <
An operator H E ee is said to be“positive(e)” [strongly
posit,ive(e)] if t.he inequality,
EL
and V T E R+ holds Xvith e = O[E > 01. If in a,ddition
H E Or, then y H ( . ) given by (2.3) may beusedin theplace
of
in (2.5).
If H is a causal operator in Lf,then
H positive(e) (=) ( u H ( . ) , H u H ( . )3) 0,
‘4
E L2. (2.6)
System D e s c r i p f i m
The system under consideration
ha3 t.he configuration
as in Fig. 1 and is described by the input,-output relations,
e
el(t)
-
(2.2)
.
Z~N(.)
=
ul(t) -
1c2(t)
+
e?(t) = u&)
zc1(t)
(2.7)
where 2tH(.):R+
-+ R is t.he input to H , yH(.):R+
R
rrl(t) = Gel(t) and m ( t ) = Fez(t)
is t,he output of H and h ( t , ~ =
) 0 V T > t.
Let OI den0t.e the class of linear causal operators H in with the following assumptions:
Lze withaninternal
(state-space) description, i.e., if
) L2e.
H E Or, thenthere exists aquadruple { A H ( - ) , b B ( . ) , Assumptim 1: t c 1 ( . ) , u 2 ( . ) E b,and e l ( - ) , e B ( .E
Asrumptiott.
2:
G
E
oE
n
eI.
c H ( . ) ,and d R ( . ) f and a positive integer 71, where A H ( . ) :
Assumptiott. 3: F is a time-invariant. Popor-type nonR + +- R” X R”, b,(.):R+ -+ R”,c H ( . ) : R + + R” and
linear
mcmoryless operator in I,, defined b>-:
d H ( .) :R
R, such t.hatH is described by t h r dynamical
equat,ions,
+
-
+ bx(t)
CR’(t) XH(t) +
*x(O = A x ( Q zH(0
Y&)
=
~ . H ( O
dH(t) UH(t)
(2.3)
Let us denot>c the class of operators F satisfying
5p.
(2s)by
where cH‘(t) d e n o h the t.ranspose of c H ( t ) , x H ( - ) : R + - R“
is the stateof H and u H ( . ) , yx( - ) are defined as earlier.
The 31aitl Problem.
Let. O I p A be t.he subset of aI consist,ing of those operators
Find conditions on C a,nd F which ensure that the sysH which satisfy the following requirements:
tem drscribed by(2.7) is L,-stablr, i.e., Z L ~ (-),?le(.) E L? =>
1) The internal dcscriplion of H is in the phase-variable e l ( . ) , e d . ) E h .
canonical form.
111. SOLCTIOS
OF THE 1 \ I ~ r sPROBLEM
2) The elements of A,(t) and cH(t) are ?)-times differentiable n i t h respcct to f, 71 being t,he dimension of the
In this section, it is proposed to provide a solution to
state-vector of H .
t,hr a,bove problem byapplying the po3itivitytheorem
3) H is exponentiallyasymptoticallystablewith
zero [1],[2] after transforming the system [9] n-ith t,he introinput,, i.e., I’xH(t)I’g K llzH(to)
;IE esp(-a(t - to)) v t 3 to, duction of “multiplicrs” (as shown in Fig. 2 ) . Recall that
<
219
SUNDIRESHAN AND THATHACHAR: NONSTATIONARY FEEDBACK SYSTEMS
5 a = i ~ * ( ‘ -2~ )0,
i = l
++
w
ER
(3.4)
,
are satisfied.
If, in addition t,o t.he above hypotheses, &(t)
Y t E R+,then H is st,rongly positive(e).
Fig. 1. The feedback system under consideration.
3 E >0
Proof:
Po&ivity(e) of H: Since H E OIPA, xHi(t)= 0 at
i = 1,2,-. *,nandlim xHi(t) = 0 ++ i = 1 , 2 , - . . , n l
t = 0 ‘4
t-
m
Now, since H E O I P A =) H is a causal operator in L,
it, is sufficient,to prove, in view of (2.6), t.hat
3
v
0,
(ZLH(.),YH(.))
UH(.)
E L?.
(3.5)
Sow, Left-hand side of (3.5)
t,he application of the theorem requires a. factorization of
t,he loop into a composition of two operators, one of which
isstrongly positive(e) witjh finitegain and t.he other is
positive(e).
=
= (UH(.),CIf’(.)
ZH(.))
+ M - ) ud-)),from (2.3)
+
?&I(.)).
(3.6)
(%I(.)rdH(.)
Second term on t.he right-ha.nd side (RHS) of (3.6)
A F.requency-Do?~~.ailz
Positivity Cmzdition for Time-Varying
Linear Operators
h criterion for the positivit.y(e) of linear time-varying
operat.ors H E O I p A 15-ilI now beenunciat,ed. This is a
sufficient. condit,ion and requires the establishment of the
nonnegativit,y of a.n evendegree polynomial whose coefficients reault. from the minimizat,ion of certain combinat.ions of aHi(t) and c H j ( t )and their derivat.ives.
Lenzma.: Let, { a H i }i, = 1,2,. . . , ? I . be an .n-tuple of const.ant.sassociated wit,han operator H E O I p a and defined by,
(u.d.),cH’(.) ax(.)
=
lrn
dt
dH(U
t )H Z ( t )
> 0,
a
?AH(.)
E Lz
(3.7)
since cl,(t) 3 0, Y t E R’.
Further, since H E 01,,, subst,ituting for u H ( . ) from
(2.3), we ha,ve, first tern1 on the R.HS of (3.6)
(3.1)
where, -aHi(t), i = 1,2; . .,?Aare the elements of the nt,h
row of A H ( t ) ,c H j ( t ) , i = 1,2, . . .,n.are t.he element,s of
cH(t) a.nd Pt,iare the coefficients defined by,
S(j)
+ k S ( j - 1)
1
- l)!
+ kh.(j - 2 ) (k(x:--j 2j)!.j!
h(k - 2 j )
(3.2)
+
f ( t >y i ( t ) y i + k ( t ) dt
=
(the sL1perscript.s d h i n the brackets in (3.1) den0t.e the
order of the derivative with respect to time; terms with
negative superscripts should be discarded).
Then H is positiw(e) if 1)
cl,(I)
3
0,
Y t E R+
and 2) the frequency-domain inequality,
(3.3)
(3.8)
We will next simplify t,he first and third terms on the
RHS of ( 3 3 ) bg: repeated integration by parts. With this
motive, let us now state a simple result.
Proposition: If f ( t ) is a real-valued funct.ion different.iable k-times aud if yi(t), i = 1,2; . . arc a family of timefunctions sat.isfying 1) j i ( t ) = yi+](t), i = 1,2,. . . and 2 )
yi(tl) = gi(f.2) = 0, Y i = 1,2,. . . and some t l , t2 E R+,hhen
j’r
where
aHl,(t.)~~i(~)]~~i(t).~H~(t)
clt.
J
t?
ti
k
pp,j(f(t))‘”-’i’yi+j2(f) dt,
(3.9)
j = O
‘This last assertion is a result. of the following arguments. If
h i ( t , ~ are
)
the kernels relating the components of the state-vector to
the input,, i.e., x ~ i ( t =
)
~ ~ ( L , T ) U H ( T ) dr, ’ ~ ~ i ( f ) !
[hi(t,T)’
Il(H(r)i d s
ICs; e - a ( t - r ) lu~(s)I
d7 since HEBrpn; noting that t.he
last term is a convolution oft.mo Icfunctions, a st.raight.forward
application of t,he Riemann-Lehesgue theorem gives the intended
result.
<
< st
where Br ,jare t,he coefficients defined as in (3.3) (the superscriptwithin the bracketsin (3.9) denoting, as earlier,
the order of the time-derivative).
The proof of this resultissimplyestablishedby
repeated integration by parts and is omitt.ed as it is readily
available in [SI.
i\Tow, simplifying t.he first, and third terms on the RHS
of (3.5)from an a.pplicat.ionof the above result2, we have,
RHS of (3.S)
ing condit.ions are satisfied:
A Few Remarks
Rem.a.rk 1: I n compa.risonwith tjhe esisting results
for the
stability of similarsystems,itnlaybe
observed that
Theorem 1 pernut,s t>heuse of a. very gcncral class of timeaHI:(t)CHj(t)](L+j--2i)XHi?(1) dt
varying mult,ipliers, much like the results of [ i ] However,
.
2
aH+xHi2(t)df, aHibeing defined as in (3.1)
it should be not.ed t,hat, while ['i] uses a minimal realizai = l
tion t.0 givestabilit.y criteria. in the time-domain which
require
t.he solution of ccrt,ain matrix Riccati equat>ions,
(3.10)
t.hcpresentresults
startwith
a canonical realization
sat,isfying
additional
differentiabilit.!- requirements, t.0
by an application of Parscval's t.heorem, where X H i ( j o )is
>-ield simplerfrequcncy-domain
conditions. The not,cthe Fourier-transform of .rHi(f).
worthy feat.ureof the present results isthe ease in checking
Xow, eincv r H j ( t=
) anci-l)(t),
V i = l!?; . . $ u , XHi(ju).
the stabi1it.y conditions, which has rcsulted from the im= j~S,,,-~,(j6~)
nnd by induction, X H , ( j w ) = ( j ~ ) ~ - l
position of certain addit.ionn1constraints on the system.
X H i ( j u ) ,V i = 1.2,. . .,v.
Remark 2: A stabi1it.y criterion for nonlinear system,
Hence, on substitution, we have
similar to the Theorem 1, may be formulat.ed by using a
time-varyingmultiplier
:If which isdrcomposible int.0
X =E
Z with jZ'I < 1 (the norm used is the opcrat,or
norm) andrestricting the nonlinea.rity t o be odd and
3 0, from (3.4). (3.11) monot.onically nondecreasing. The statement. of the rcsult
will br omit,ted, i n view of t.hc popularity (see Zames xnd
Thus, combining (3.7) and (3.11), (3.5) results.
Stro,l!/po~jtil'ity(e)o f H : It is sinlplc to observe that! Fnlb [2] or Willems and Brockett [lo]) of such methods.
H is stronglypositirc(e) if ( H - € E ) is positivc(e) for I t should. however. bc nwntioned that the conditions inlsome e > 0. E being the identit.! operator in I+. Further, posed by this criterion on the linear part of the system
H E @Ip.-l => ( H - EA') E @ I P A with the. st:Lte-equation would be simplrr and nlorc explicit than the continuousunchanged,
but
the
output
cquation
nlodificd into, time version of the results of [lo].
Remark 3: The difficult step in thr application of the
B H ( . ) = cH'(-).cH(.)
[(I,(.)
- e ] u H ( . ) . H t ~ l c c !t h c .
stability
criterin of the present type lics in ensuring that
positivity(e) of ( H - € E )ma?- b(3 cstablisllcd by working
the
composite
oprrator L E Oyp2,(in fact?thi!: is the only
as bcforc, [reprating thr stvps from (X.;) to (:<.11)] and
difficult
step;
oncc
such n rcprcsentntiotl is obtnincd, the
noting that d H ( f ) 3 E, +f t E R'.
Q.E.D.
determination of theconstants { cyLi\ and {
although
at first. glance appears t.o be difficult, is quite simple
Frequellcy-Donlaitl Stability Criteria
The positivity condition obtnincd in the previous srct.ion irrespective of the dimensions of (; and U ) . Even in this
mnkwthe proof of the following stnbilit?-criterion for stcg, thr difficult part is only to ensure R phase-variable
linrar system :I straightforward application ofthcl posi- rcalizntion of I,, sincc. withsuch a representation being
ensured, tllc other requirements result in a simplc manner
tivity tllcorem.
from
Tlreorem 1: If t h t w exists an operator AI E o ~ such
~ . ~ the imposition of suitablr restrictions on the describdifficulty isgreatlyreducedby
the
that J/ is invertible in I,?. L = J I ( ; E O I p . L nnd the follow- ingelements.This
availabi1it.y of
11-rstablished algorithms [ 111. [12] m-hich
ensure for any uniformly controllable and differentiable
realizat,ion of L , thc c.xistencc of n uniquenonsingular
2 Sote that all the requirements of the proposition are fulfilled
A
a H i ( f )and c ~ i ( tare
)
differentiablen-time. and
since H f ~ I P implies
u L x u L matrix, which transforms the realization of L
rff;(t)= 0 at f = 0 and f = a.Further, r,r;(f)
being the state-vector
to the phnsc-variable form.
components of H , i,;(t) = ~ H , . i + l ) ( f ) , V i = 1,2,. . .,(n - 1).
+
2
0
+
+
~~--c.
221
SUNDARESHAN A N D THATHACHAR: NONSTATIONARY FEEDBACK SYSTEAfS
being the coefficients defined as in (3.2), n. being the
dimension of G and cGO(t)= 0.
Then the syst,em described by (2.7), mith the additiona.1
The Case of the Time-carying Popoc ll4ultipdiw
assumption uz(.) = uzf(.)
3 hf(.)
E L2, is &-stable for
Alt.hough it is evident that,t,he stability criteria. given inall F E SP.
The proof of t.his thcorenl is omitt.ed due t.o space rethe previous sect.ion, which permit, the use of a very
general class of t,ime-varying operat,ors in
Lz as ndtipliers, quirements. The general patt,ernof development, however,
are more general t,hanthe existing results,the advantage of folloms very closely the proof of the lemma in Section 11,
using a t,ime-varying mult,iplier is yet. to be decisively dem- in establishing the st.rong positivity(e) of MG. 4 complete
onst.rated. Inthis
sect.ion, we propose t.o do t>his by proof may be found in [ 131.
using asimple
first.-order differentialoperator
asthe
nlultiplierandderiveastabilitycriterion
which will be Discussion of the Result
compared d h the existing results that. employ similar but.
1) AhhoughTheorem 2 contains a. frequency-domain
time-invaria,nt
multipliers.
As an a.pplicat,ion of the
resultsderived, L2-st.a.bilityconditions for the nonlinear inequality similar to t,hose appearing in Theorem 1, the
da.mped Jlathieu equation arc obtained in a subsequent. conditions are explicitJy on the e1ement.s describing G'
section and compared qith the results of t,he earlier in- (note in comparison that, Theorem 1 imposes condit.ions
on the composite operat.or L = JIG) and hence, are far
vestigat.ors.
simplcr t.0 check. The complexity of the expression for t,he
The multipliers that will be considered in this srction
are operators E OE having a decomposition M = Q D coefficients { a i l nrednot beaseriousdrawback,since
where Q is atime-varying gain defined by Qz(t)= q ( t ) x ( t ) , the evaluat,ion of these is very simple even for systems of
'd: x(.) E Lz,, q(t) being a nonnegativefunction on R+ large dimensions.
3) The advantage of using a time-varying multiplier is
diffrrentiable almost everywhere, and
D, t,he differential
opera.tor in Lf,, D x ( t ) = i ( t ) , ++ x(.) E L2,.Let us denote illustrated in t,he following cxa.mplr.
Excmple: Consider the syst.em Tvith t.hr linrar part G
t,he class of such operators by a?.
It is sinlple to observe
.
that X E -mp=) M - I exists as an operator in Lfe. One gowrned by t.he nona,utonomousdifferrntialequation,
particular point needs to be emphasized when the use of
multipliers M E
is cont,emplated.Since M = Q
D
is not bounded operat,or in L2?
the familiar introduction
of &I into the loop (see Fig. 2 ) will change the nature of
theinputsto
t.hcsystem(notet.hat
M u ? E L?). This
where @(t)is a gain?the bounds on whosr rate of variation
difKculty may however be ovwcome, following Zanxs [I], arc. to be determined for the sta,bility of t.he feedba.ck loop
by restricting u.l(.) t.0 be a fixed funct,ion tizf( .) in & such comprising of a nonlinear operator F E 5 pin cascade wit.h
(i.
that GI(.)
E LI.
Theorem 9: If G' E O I p A and t>here exists an operat,or
Rcpresent.ing G in the phase-variable canonical form, we
ME%, satisfying the following conditions:
have the describing q ~ a d r u p l e , ~
117. SIMPLIFIED
CRITERIAUSING FIRST-ORDER
&,j
MULTIPLIERS
+
+
r,
Using Brockett,'s result! [14], it maybeobserved
that.
G' E o~~~~
if 0 < e' < @(t) 6 11.5.
For a more rest.ricted form of the syst,em (i.e., with
dG(t) 0 and no inputs into the feedback loop), Blodget,t
(4.3)
+ kGj(t) + CG(j-I)(t) - cGn(t)aGj(t))]'"+'-"'
The considerat,ion of t,he particular form of $ ( u ) in (4.1) is
merely t.o facilitate a simple phasevariable realization of G.
222
IEEE TRANSACTIOSS
ON AUTOMATIC CONTROL, JUNE
1974
I
€
1
Fig. 3. Stability regions for t.he hiat.hieu equation from (4.9).
+
G E f l I p A is 0 < ( b E cos p t ) 6 11.5; this will be satisfied
if b and E arerestricted by 0 < ( b f e ) 6 11.5. [Sote
howevcr that this bound does not make US(> of the fact.
that the time varying coefficient is periodic with period
5 ~ : p ;more relaxed bounds, for larger values of p , may be
obtained by using 1J-illrms' [15] result for periodic systems. ]
In the follon-ing, two diffcrcnt choices of q ( t ) are made
and corrwponding stability regions are obtained in order
to dcmonstratc tllnt diffcrcnt choice::of q(t) often Irad to
improved results.
Clroice X o . 1: Let
so as to satisfy (4.G). Substituting this in ( 4 . i ) and simplifying, we have
Sot(. that (4.G) is satisfied. Substituting in (4.7) and simplifying, we 1 1 : ~ ~
223
S U N D A R E S W N A N D T H L T H A C W R : XONSTATIONARY FEEDBACK SYSTEMS
Fig. 4.
Stability regions for the Mathieu equation from (4.11).
+
( e / p ) sin p t ) gives a larger
that the stability region obt,a.ined presentlyis close to p = 1, the choice of q ( t ) = (1
Parks' region in the range -2 < E < +S. Further, it may region in the rm1ge.b 6 0.5 than the choice of q(t) =
be observed from Fig. 4 that the condit,ion (4.11) gives (1 - ( E , / ? ) cos kt), while the reverse is true for b > 0.5.
t,hesame regions asParks*intheparameter
ra.nges However, since the stability results obtained through the
b 6 0.5 for p = 1 and b
1.3 for p = 2: whereas for use of these periodic mukipliers are only sufficient. conditions, the bounds on the parameters of t,he 11at.hieu equap = 5, the region presently obtained is hrger in therange
tion (4.5) which ensure st.ability of t.he feedback sgst.enl
b 4.5.
2 ) I n comparison with the L,-stabilit,\T crit.erion of Wu ma.y correspond to either of these two regions. I n other
and Desoer [17] (for t.he problem present.ly considered, words, a union of t.hc twostability regions is itself a
i.e., damping coefficient. = 2, t.he condition from [17] is stabilitJ7 region for tjhe syst,em. Continuing on this theme,
b > I E ~ 0.75), it. may be seen that. (4.9) gives a la.rger one ma.y conclude that the present, met.hod of analysis
region in t.he pa.rameter ranges, b
2.75 for p = 1 and involving t.he use of periodic multipliers, has opened a new
b 6 2.25 for p = 2 , while (4.11) gives a larger region in avenue for obtaining enlarged stability regions t,hrough a
t,he ranges, b 6 1.5 when p = 1, b 6 2.25 when U, = 2 and judicious choice of different periodic functions q(1) in the
b 6 4.75 when p = 5. Further, it may be noted that in
mult,iplier.
t.he latter case, the improvement, in $he stabilit,J: region
V. CONCLUSIONS
increases nit.h increased p. Thus one of the st,rong points
of t.he present criteria is that information on the value of
The problem of developing L d a b i l i t y criteriafor
linear and nonlinear feedback systems containing a timethe parameter p is exploit,ed to obtain enlarged stability
regions.
varying linear operatorin Lz, has been trea,t.ed. A frepositivity
condition
for
t.ime-varying
3) It should however be noted that the above compari- quency-domain
son is notactuallyfair
to the presentcriteria;for,
the 1inea.r operat,orsisderived andis used to generate LZresults of Parks [16] a,nd Wuand
Desoer [17] were stability criteria, which permit t.he use of a very general
class of mult,iplicrs withnonstationarykernels.
originally derived for the Mathieu equation without any
For the
feedback nonlinearity. If the latter results are t,o remain case of feedba.ck systems containing a Popov-type nonvalid in the nonlinear case, the nonlinearity should be linearity, more explicit. conditions
in
the frcquencyconfined t.0 the sector [ O , E ] whereas t.he present critcria
domain, involving a firsbordertime-varyingdifferentmid
permit it to bein the infinit,e sect.or. Thus in spiteof con- multiplier are derived and a.re shown t.o improve upon the
sidering a more general syst.em, the present. results give existing results. The use of periodic multipliers for the Lzcertain st.ability rrgions not contained in t,hc earlier ones.stability of nonlinear damped3Iat.hieuequationwitha
A Ge)lera.l Remark: A comparison of the stabi1it.y regions forcing function,issuggested and is shown to result in
regions not. contained in those given by the existobtained in t.his section through the choice of t.wo diffrrcnt stnbilit>~
periodic functions q ( f ) , shown in Figs. 3 and 4, reveals an ing critcria even for the case of the linear equa,tion. The
int.erest,ing point.. Note that for the part,icular case of positivity
criterion
obtained
in
this
paper,
although
motivated from a desire t o use it in t.he stability analysis,
is of independent interest and has applications in various
4 Parks' result is derived by the use of the Circle crit.erion and
other area5 of IIathrmatical System Theory.
hence does not, give different conditions for different. values of p .
<
<
+
<
224
IEEE TRANSACTIOXS ON AUTOMATIC CONTROL, JUNE
REFEREXCES
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31. K. Sundareshan and 31. .I. L. Thathachar, “Time-domain
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J . C. Willen1s, The Analysis of Feedback S y s t e m . Cambridge,
Mass.: 31. 1. T. Press, 1970.
J. C. Willems and R. W. Brockett, “Some nex rearrangement
inequalities havingapplication in stabilitv analysis,“ IEEE
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[121 B. Rdnlasn-ami and K. Ramar, “On the transformation of
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31. K. Sundareshan and 11. A . L. Thathachar. “Sew results
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~ 0 1 7,
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IC. Sundareshan wa5 born in ltobertsonpet, Kolar Gold Fields, India, on June 16,
1946. He received the B.E. degree i n electrical engineering from the Bangalore University,Bangalore,
India in 1966 andthe
3I.E. and Ph.1). degrees in control s;?-stems
engineering from the Indian Institute of Science, Bangalore, India in 1969 and 1Yi3,
respectively.
Since January 1973, he has been with the
School of Automation, IndianInstitute of
Science aud is engaged i n the design and development of systems for
flight path control. Hi:: current research interests lie i n the areas of
linesr and nonlinear 2;>-stemtheor>-, optimal performance of flight
control sh-stemsand computational methods.
Malur
M. A. L. Thathachar w s born in 31ysore
City,India in 1939. He received the B.E.
degree in electrical engineering from the
University of Llysore, 313-sore, India, in 1959
andthe 3I.E. and Ph.11. degrees from the
Indian Institute of Science, Bangalore i n 1061
and 1968, respectively.
He has been on the faculty of the Indian
Institute of Technology, 31adra (during
1961-64) and of theIndianlnstitute
of
Science. Bangalore
(since10611, where heis
currently -k+ociate Profesor. He spent a Sabbatical year in 19T3-54
at Tnle Illriversit>-,S e a Haven, Conn., and at Sir George \I-illiams
Vniversity, Jlontreal,Canada. His research interestsare in the
areas of Stability Theory and Learning Systems.