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,.
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504
IEEE TRANSACTIONS
ON
-4TjTOMATIC CONTROL, VOL.
AC-17,KO. 4,
AUGUST
1972
[SI C. D. Johnson, "A unified canonical form for cont.rollable and
uncontrollable linear dynamical systems," in Proc. 1969 Joint
Automatic Control Conf., 1969, pp. 189-199.
L,-Stability of Linear Time-Varying SystemsConditions Involving Noncausal Multipliers
MALUR, K. SUT\TDARESHAK AND M. A. L. THATHACHAR
Absfracf-New criteria in the multiplier form are presented for
the input-output stability in the &-space of a linear system with a
time-varying element k ( t ) in a feedback loop. These are suf6cient
conditions for the system stability and involve conditions on the
shifted imaginary-axis behavior of the multipliers. Thecriteria
permit the use of noncausal multipliers, and it is shown that this
necessitates dk/dt to be bounded from above a s well a s from below.
The method of derivation draws on the theory of positivity of compositions of operators and time-varying gains, and the results are
shown to be more general than the
existing criteria.
I. INTRODUCTIOK
HE STABILITY properties of the system containing a linear time-invariant convolution operator
G and a time-varying ga.in k(t) in cascade, in a feedback
loop, have drawn a
good deal of attention in the past,
and theresult most popularly known is the circle criterion
[l], [2], [3]. Though t,he application of the circle criterion
requiresasimple
geometrical construction,it.is highly
conserva.tive in requiring the 1inea.rtra.nsfer function to be
positive real. Improved results were obtained by Brocltett,
and Forys[4]by employing an upper bound onthe rateof
time variation clk/dt and using an RC multiplier. These
results were later generalized by Gruber and Willems [SI,
whose criterion assured stabilityif there exists a multiplier
Z(s) such that 1) dk/clt 6 2ak(L) and 2) Z(s)G(s) and
Z(s - a) are strict<ly positive
real. However, the multiplier
T
Manuscript received September 29,1971; revised Februaly 24,
1972. Paper recommended by J. C. Willems, Chairman of the IEEE
S-CS Stability, Nonlinear, and Distributed Systems Committee.
The authors are with the Department. of Electrical Engineering,
Indian Institute of Science, Bangalore 12, India.
employed here is causal, and hence is not. the most general
one.
Recently,Venkatesh
[6] derived some useful crit.eria
involving noncausal multipliers for the absolute stability
of nonlinear t.ime-varying systems andobtained result,s
for linear syst.ems by a limit. process. But the anticausal
terms employed in [6] are of a particular form containing
poles and zeros on t.he positive real
axis only. In thepresent
paper, more genera.1ant.icausa1 multipliers are introduced
andcriteria
for the L2-input, Lm_-out.put stabilityare
derived by employing a lower as well as an upper bound
on dkldt. The setting of the problem is the one used by
Zames [3] and t.he method of solut.ion emplovs the positive
operator theory.
11. FORMULATION
O F THE
PROBLEM
A . Notutions and Definitions
It is assumed that the reader has a cert,ain familiarity
with the not,ions of normed spaces,linear spaces, inner
products, Banach spaces, and Banach algebra,. (Reference
[7] isgood for details on t,hese.)
Let R and R+ denote, respectively, the real and nonnegative real numbers. Ll is the space of all measurable
real-valued functions x( on R+ that have afinite Lebesgue
integral; the L1-norm is defined by lix111 = So" (x(l)I dt.
Lp is the space of allmeasurablereal-valuedfunctions
x(-) on B+ which have finite energy. The inner product
on Lz is defined by (x(.), y(.)) = $," x ( t ) y ( t ) dt, and the
&norm is related to the inner product by I I x l l p = [(x(-),
x( J3. LBe is the extension of L.defined by
e )
0
)
)
.
SUNDARESHAN AND THATHACHAR:
L e
=
{x(*)IxT(.)
LTSTABILITY
OF LINEAR
TIME-VARYING
E LZ ++ T E E+]
(2.1)
where zT(- ) is the truncation of x( -), xT(t) = z(t), t E
[0, T I , and zero otherwise.
An operator H on Lz is a single-valued mapping of L.2
into itself. The gain and p0sitivit.y of H are defined as in
Zames and Falb [9].
1) H is said to be “causal” (nonanticipative) if
Y X(*) E Lz,
(Hz(.))T = (Hz:T(.)):T,
++ T
505
SYSTEMS
R+. (2.2)
Fig. 1. The feedback system under consideration.
wl(t)
=
Gel(t) and wZ(t) = Kez(t)
(2.9)
with the following assumpt,ions.
Assumption 1: 6‘ is a 1inea.r time-invariant convolution
operator inL2 defined by
2) The “adjoint”operator
of H , denot,ed H*, isa
mapping of Lz into itself such that
++ 4 . 1 E Lz, (2.10)
(4.1, HY(-))
= (H*X(.), ?I(->),
++ X(.), Y(*) E La. (2.3) where { rt] isasequence
H is saidto be self-adjoint if H = H*.
3 ) H issaid to beanticausal(anticipative)
if H * is
causal. H is said to be noncausal if it is a sum of causal
and anticausal operators.’ (For an informat.ive discussion
on the causality of operators, sec Saeks [SI.)
S o t c that,, if H is a convolut.ion operat.or defined by
Hz(t) = h(t) @ x(t)
( @ denotesconvolution),
=
J-:mh(C;)z(t- C;)dC;
H causal implies h(t)
in R+, {si]is an l1-sequence
lgi(is finite), and g(-) E L1.
(i.e.,
Let G ( j w ) denote the frequency function of G defined by
x:==,
m
W w ) =
,Esi exp ( - j w 7 J
r=l
Assumption 2.- X is the class of all operatorsK in L p such
that.
= 0,
K E X =) K x ( ~ =
) k(t)~(t),
t < 0, and H a.nticausa1implies h(t) = 0, t > 0.
4) H is said t o h a w “finite gain” if
0
<
6
k(t)
6 iT < 03,
dk (t)
c X 3 K E X’=)
X’
-
dt
(2.4)
6
v X( .) E Lz
v t E I?+; (2.12)
++ t E R+
Bbk(t),
and some /3 E R f ;
(2.13)
5 ) H is said to be “positive” if
(X(*),H z ( * ) )>,
0,
b! X( * ) E
(2.5)
v
and is said to be strongly positive if,for some 6 > 0,
(X(-),HZ(.))
>,
6(~(.), X(.)),
++ X(.) E Lt.
(2.6)
6) Let H now beassumed to be acausaloperat,or
having LBea.s its domain and range. Then, H is said to be
positive ( e ) if
(ZT(.)r
(HZ(.))T)
>, 0,
++ X(.> E L2e,
++ T E R + .
(2.7)
However, if in addition it is knou-n that H is a causal
operator inLz, it. can ea.sily be shown that
H ispositive (=) H ispositive (e).
(2.8)
B. System Description
The system under consideration has the configuration
in Fig. 1 and has the input-outputrelat.ion defined by
el(t)
=
ul(t) - z t 2 ( t )
e&)
=
4 0
+ w(t)
1 This characterization of a noncausal operator is not associated
with any loss of generality, since every linear operator in LScan be
written tls ths sum of a causal and ananticausal operator [8].
Let
=
t E R+ and some
CY
E R + . (2.14)
x, n X@.
C . The &lain Problem and the Method of Solution
The problem of interest may now be stat.ed. Given the
system described by (2.9) wit.11 the related Assumptions
1 and 2, find the conditions on G which are sufficient t o
ensure t,hat, the system is Lz-input, Lz-output stable, i.e.,
given that the inputpair ul(-),
E & and the errors
el(
ez( .) E L,,, find the conditions which emure that
el(-), e2(-)E Lz.
The solution to thisproblem will be sought by thenow
well-established principle [ 3 ] of factoring the openloop
int,otwopositiveoperat,ors,one
of which isstrongly
positive and has finite ga.in. To render flexibility to this
approach,a“multiplier”
M that satisfies the following
conditions: 1) M is a 1inea.rconvolution operatorin Lp; 2) M
is invertible in L,; and 3) - y ( M ) < 03, is artificially introduced into the systenl as in Fig. 2. It may be recognized
[ 8 ] , [9] that. the operators M satisfyingconditions 1-3
form a commutative Banach algebra(€3 with an identityE.
Also, a = aCU aaC
where ac(@ac)
is the Banach algebra
of causal(anticausal)operators
&I satisfyingconditions
% ( a )
e),
506
IEEE TEANSBCTIONS ON AUTOMATIC CONTROL, AUGUST
1972
being an operator in Lz,
QP will be an operator L2.
in
Proof: It is required to prove tha,t
(x(-), Q P x ( . ) ) 2 0,
Fig. 2. M o a e d system with t.he multiplier M .
left-hand side (LHS)
1-3, with E being a.n element of both Bc and @uc.2 This
characterization permitsthe stabilityproblem t,o be treated
within theframework of this algebra.
I n what follon7s, conditions mill be established which
ensure that.: 1) M E has a suitable factorization M =
&I_M+, M - E BUcand ill+E Bc, M - and ill+being
invertible
such
t,hat
M--l
E BQcand
E aC;2)
M G is strongly positive and has finite gain; and3) KM-'
is posit.ive.
These are sufficient 191 to prove that el( .) and e,( - ) E
Lz. It needs to beemphasizedhere
thatthe funct,ion
spaces in [9] are defined over the entirerea.1line R, whereas
they are definedover R+ here. The proof of the basic
theorem in [9] holds in toto, even in the present
case,
since, although the functionsare defined over R+, the
convolution is definable over (- m ,
a).This follows
from the fact that, ..if dl is a general noncausal operator
belonging t.o a, it can bedecomposed into 31, M,,
M c E BCand Mu,
E &, and further,
=
c
+
M4t) =
=
=
ln
m(t - M E ) dE
lt
mAt - M E ) d l
s-+:
m(r)z(t -
+
sm
t
m d
7) d r .
111. MATEEMATICAL P F ~ E L ~ I N A R I E S
Jm
z(t)Pz(t)dt -
0
lm
5
Pt
on int.egration by parts. Term
I on the righbhand side
mHS)
0 since q( is nonnegative and P is positive
( e ) . Term I1 on the RHS = (dpldt)
zt(u)
Pzr(u) du dt and is nonnegative since (dqldt)
0 and P
is positive ( e ) . Hence (3.1) follows.
Lemma 2: If P is a causal operator in Lz and Q is a selfadjoint operator in Lze, then QPQ is posit.ive (e) if P is
positive.
Proof: If P is causal and positive, P is posit.ive (e)
from (2.8) and the lemma requiresto show that
>
a )
<
v x(.)
{ z d . ) , (QPQz(.))T)>, 0,
EL
e ,
++ T E R+
LHS
(3.3)
(xT( .), QPQxT(
= (&xT(.), PQzT(
since Q is self-adjoint
= {?IT(*),PYT(.)),
Y(.)
=
since P is causal
a)),
a)),
- E)z(E)dE
(3.1)
z(t)q(t)Px(t)dt
= q(a)
-
+
V x(*) E Lz
=
&x(.) E
L2e
(PY(.))T)
=
(YT(.),
3
0 since P is positive (e).
Lemma 3 (Shifting Lemma): If P is a linear convolution
opera.tor in L2and Q, is atime-va.rying gain definedby
Q,z(O
= q,(Mt)
and qJt)
=
exp (PO,
then
In this section,conditions will be established for the
positivity of combinations of alineartime-invariant
M * > , Q p W - ) ) = (z(.>, PpshQ,z(.)),
convolutionoperator P defined by: 1) Px(t) = p ( t ) @
V 4 . 1 E Lz (3.4)
z(t); and 2) P(ju) = S[p(
5( -) denot,ing the Fourier where PPshis a linear convolution operator related to P
transform and a timevarying gainQ having t,he following
bY
properties: 1) Q x ( t ) = q(t)z(t); 2) Q has an inverse Q-'
S[p'"h(.)] = P"""(jw) = P(ju - p)
(3.5)
defined by Q - k ( t ) = q-l(t)z(t);and 3) if Q is an operator
in L,&-I is an operat.or in Lze and vice versa. As an if the inner products in (3.4) exist.
example, theoperator Q such that q(t) = exp (-ut),
Proof:
u > 0, satisfies the preceding properties 1-3. It must be
noted that Q and Q-I defined as above are self-adjoint
in the respective spaces of definition.
Lemma 1: If P is a causa.1 operator in Lee and Q is an
operator in Lz such that: 1) P is positive (e) ; 2) QP is an
operat,or in L z ; and 3) the derivative of q ( + ) exists almost
everywhere, then QP is positiveif g(. ) is nonnegative and
monot.one nonincreasing.
Remark: Condition 2) needs some exphation. If P
is an operator in Leand Q in Lp,QP in general will be an
operat.orin LZ,. Horever, in cert.ain special sitaations,
for example, when P is of the form P = Q-IPI with PI
a)],
This corresponds to Saeks' concept of "weak causdity" 181.
making a choice of r in a f i d e portion of the complex
pla.ne possible. Hence the integral on the RHS of (4.11)
is well defined and M E ( R c =} log 114 E 63,. (It may be
(x( PP"x( .)) = (x( (P*)-pshx(.)),
recognkedt.hat, the RHS of (4.1) iscomparable t.0 the
Cauchy
integral formula in the complex variable t,heory.)
V x(*) E La (3.6)
Having t.hus settled t.he question of the existence of log
if the indicated inner products exist,.
M as a bounded operator in L2e, it. is an easy exercise t o
Proof: The proof involves simple operator manipula- show that &I:L2+-Lt =) log M: L2 L,2,i.e., log Jl E @.
tions and repeated applicationsof Lemma 3, and is hence
The rest of the proof is simple and follom as inZames
omitted.
1
1
,
a.nd Falb [9].It nmy further bechecked that M A T + = 2
since (R is a commutative algebra..
IV. FACTORIZATION
OF OPERATORS
Remurk: The a,bove lemmaillustrates
the relation
The importance of t,hefactorization of an operator between the factorizability and the posit,ivity of operators
dl E (R into a suitable composition of elements of (R is inaBanach a.lgebra a.nd, in the a,uthors'opinion, is a
apparent fromSection II-C. I n this section,conditions
considerableimprovement,overt,heresults
of [9] and
forsucha fact.orizat,ion will be enunciated. Let. ,e(@,@) [lo]. It. may be recalled t,hat t,hese references prove the
be the spa.ce of all cont.inuous linear ma.ps of (R into desired factorization under such restrictive conditions on
Let. P+ be a M a s AI = aE 2, 2 E @,a E R with l!Zl < lal.
itself and let E2 be the ident,ity of .e(@,@).
projection on (R and let P- = E2 - P+.Let (R+ and @V . SOLUTION
OF THE L-STABILITY
PROBLEM
denote,respectively, the ranges of P+ and P-. Then,
conforming &h the earlier notation, it is easy t o see that
This sect.ion containsa
few t.heorems t,ha.tprovide
Bc is t,he subspace spanned by @+ and E and Bac is the sufficient conditionsfor the Lz-stability of t,hesystem
subspace spanned by (R- and E ( E being the ident,it.y of under consideration. Theorem 3 is the major result. of t.he
paper and involves noncausal nmlt.ipliers. However,
The following lemma is a considerable generalization of Theorems 1 and 2, which, respectively, consider the
simi1a.r lemmas of Zames a.nd Falb [9] and Willems and introduction of causal and anticausal multipliers, provide
Brocketk [lo].
the necessary mot.ivation for the technique employed to
Lemma 5: With the notation introduced a.bove, if M take into account. the effect of the timewrying gain k ( t )
isanyarbitrarystrongly
positiveoperator in @,there in theotherwise time-invariant feedbackloop.
exist elements M + = exp [P+ log MIand ilI- = exp
Theorem 1:If t.here is anoperator dl1 in @, such that
[P- log ill] such that: 1) 1W+ E BC and M- E (Rat; 2 )
Re Jfl(ja)G(ju)
6 > 0,
++ w E R
(5.1)
AT = M - M + ; 3) M + and dl- a.re invertible with M+-l E
Bc and M - - l E (Rat.
R.e &Il($
- 0) 0,
Y w ER
(5.2)
Proof: The crucial part of t.he proof is to shorn- t.he
for some nonnegativeconsta.nt 0, then t.hesystem deexistence of log M as an element of (R.
scribed by (2.9), with the relat,ed Assumptions l and 2, is
Let L2c denote the space of allmeasurable complex&stable for all K E X'.
valued functions on R+ that have finite energy. Clearly
Proof:
La c LPc.Now, 1VI can be regarded as an operator in LzC
1) The factorizat.ion condit,ion is satisfied t.rivially
and theBanach algebra of these operators can be denoted
since
by W . Let. a ( M ) denote t,he spect,rum of M . ill strongly
R,=) M l = EMl, E E aaC
and M I E (R,.
posit,ive =) a(M) c the part of the complex plane { ,$: M1 E (
R e ,$ 2 E > 0).Hence it is possible to t.ake a simply con2 ) M1G is strongly positive by (5.1) and an applinected doma.in D inthe complex plane, excluding the cation of Parseval'stheorem. Also MIG has finitegain
negative real line,3 suchthat, a ( M ) c D. Let I' be a simple since g( - ) E Ll and M I E =) y(ilTl) < a.
closed curve (in t.he positive sense) in D enclosing all the
3) Positivity of K M - ' . To prove
spect,rumpoints.Since
f(E) = log 5 isa holomorphic
(X(.),K & l l - l ~ { . ) ) 0,
!V X( .) E L? (5.3)
function inD, the logarithm of M exists and is given by the
Dunford-Ta.ylor integral [7, pp. 566-5761 as
LHS = (IIfly( K y (
v( .) = I l r l l - l ~ ( E LI
= (KMly(
y( -)},
since K isself-adjoint
Lemma
lemma,
e),
4: If P and P hare defined as in t.he previous
e
)
,
--f
+
W
f
>
>
>
e),
,
)
e
e )
e),
=
(K&2,-'Qa,#1~(*), Y(.>),
QzaY(0 = Y(t> exp (2Pt)
where a([; 44) = ( [ E - M)-l is the resolvent, of M ,
which exists as aboundedoperatorfor
a.11 E E p ( l k 0 ,
2 0,
V $4.) E L2
(5.4)
t.he resolvent. set. Further, the fact that M is a bounded
if QaaJll is posit,ive (e). [By Lemma 1, ident,ifying P with
operat.or ensuresthe spect.ral radius of $1 to be finite, t,hus
QSsMl and Q with KQ2'-l,
is a.n operator in Lz,,
and, since K E Xa, k(t) exp (-2Pt) ismonotonenonD is required t o exclude the negative real l i e so as to make increasing. Hence,allt.heconditions
of Lemma 1 are
provision for a branch cut for defining the logarithm of complex
satisfied. ]
quant,ities.
508
Nom-,
(xd.1,
=
(xT(-), Q ~ M ~ B ~ ~ Q , ~ ) )
[by Lemma 3 where M P h is a convolution operator such
that M , 8 " ( j w ) = Ml(jw - B)]
> 0,
++ x(-) E L2,and '4 T E R + ,
(5.5)
if MlBsh is posit,ive (by Lemma 2).
Af? is indeed positive because of (5.2) and Parseval's
(5.5), (5.4), and (5.3) resultand,in
theorem.Hence
view of the observations made in Section11-C, the syst.em
is Lpstable.
Th.eorem2: If t,here isan operator M2 in
such that,
Re M2Cjw)G(jw)
> S > 0,
+ a) 2 0,
Re
++ w E R
v
w
ER
(5.12)
++
w
ER
(5.13)
(5.14)
M2 E
@a=)
M2
=
'4 w E R
(5.7)
e )
(x(.),KM2-4c(*)) 3 0,
=
(M2y(*),Kg(*)), Y(.)
=
(!I(*>,Mz*K!I(.))
=
(h( .), K-'(Mz")-'h( .)>,
'4 x(*) E L2
= Jf2-54.)
(5.8)
E Lz
if ( ~ 2 * ) " ' ~is positive, appealing to Theorem 1. (Since
K E X, =) K-' E X" and Mz E @a=) M2* E aC,
the
steps from (5.3)-(5.5) of Theorem 1 ma,ybe repeated,
giving t.he claimed result.)
Now, positivity of (M2*)ashfollows from (5.7) since
+ a) > 0,
'4'wER
>
(=> (x(-),M2-""h~(*))0,
(=> (x(.>,(M2*)ash4.))2 0,
'4 X(*) E Lz
v x(.) E L2
by Lemma 4.
Hence, (5.9) and (5.8) hold and, from the observations
made inSect.ion 11-C, the system is &-stable.
Theorem 3: If there is an operator M in such that
M
=
MI
+ M2;
E = E l + h > O
for some nonnega,tive constants a a.nd and an arbit,rarily
small positive constant e, then the system described by
(2.9), with the related Assumptions 1 and 2, is &-stable
for all K E XaB.
Proof:
1) Factorization Condition. From (5.10),
Re M ( j w )
=
Re J f l ( j w )
2
€'+E2
= ~ > 0 ,
+ Re 1M2(jw)
YwER
(5.15)
because of (5.12)-(5.14). Hence,by
a.n application of
Parseval's theorem, it c a n be shonm that
Hence, ilf isstrongly positive and byLemma
5 the
factorization condition is satisfied.
2) M G is strongly positive by (5.11) and an applica.tion of Parseval's t.heorem. Further, M G has finite gain '
sinceg(.) E L 1 a n d M E @ = ) y ( M ) < 03.
3) Positivity of KM-I:
M2 E a,,, E E aC.
M2E1
2) M2G isstronglypositivefrom
(5.6) andan
application of Parseval's theorem. Also M2G hm finite
gain since g( E L1 and M 2 E aaC
=> 7(..M2) < a.
3) Posit,ivity of KM2-'. To prove
Re MAjw
2 0,
2 0,
(5.6)
for some nonnegative constant
a,then the system
described
by (2.9), nith t,herelatedAssumptions 1 a.nd 2, is L-r
&able for all K E X,.
Proof:
1) M 2 &ides the factorization condit.ion t.rivially
since
LHS
2
Re M 2 ( j w + a) 2
(xT( .), QzaM1xT(.)) since MI E aC=) QzaMlis causal
= @A*),
Q,&$f14'))
=
COXTROL, AUGUST
Re M I & - P )
(Q2&14.>>~)
1972
IEEE TR-U~SACTIOKSON AUTOMATIC
E
M2
E
@
"
(5'10)
Term I on RHS is nonnegat.ive by (5.12) according t o
Theorem 1, and
Term
I1 is
nonnegative
b r (5.13)
a.ccording to Theorem 2. Hence, KAf-l is positive.
Thus, since all t,he conditions of the basic lemma in
Zames and Falb [9] are fulfilled, the system is L2-stable.
A Few Remarks
Remark 1: Theorem 1 gives the sa.me conditions as
Gruberand Willems (derived in 151 for the absolute
stability problem), the most general resultinvolving
causal mult,ipliers. Theorem 3, n-hich permits ant,icausal
terms in the multiplier, provides more relaxation on the
phaseexcursions of G ( j w ) outsidet,he *90" band,and
hence, is a considerable improvement over [5]. Also, the
present met,hod of derivation is simpler and more elegant
than [5], which is derived in t,he framen-ork of Lyapunov
theory.
Remark 2: Recently Venka.t.esh [6] has given a method
of introducing noncausa.1 mukipliers for the absolute
stability problemusing thePopovapproach.Butthe
anticausalfunctionsemployedin
[6] areonly
of the
part.icular form
The results derived here permit. more general
multipliers
(see the numerical example below) and are derived for a
&rower definition of st,abilitv.However. t.he a.ssertion in
SUNDARESX4N AND THATHACHAR:
L 2 - ~ ~O F. LINEAR
% ~ TIME-VXRYIKG
~ ~ ~ ~
bility multiplier necessitate a lower bound on the rate of
time variat.ion dk/dt is justified here.
Rema& 3: The &ability criteria. given in the preceding
t,heorems involve checking a frequency-domain inequality
of the form Re M ( j w ) G ( j u )
6 > 0 for a given G(s).
The applicability of the criteria is great.ly simplified by
constructing the stability multiplier M(s) from a knowledge of arg G‘(jw) by the methods enumerated in Sundareshan and Thathachar1111 and bychecking whether the
multiplier so constructed satisfies the other conditions of
the theorems.
Remark 4: More general multipliers of the form
>
=
M,(jW)
M(jW)
+ qjw,
q
>0
can be used in (5.11) if an additional condition
Iim ] w ~ ‘ ( j o ) \=
o
to--) m
is imposed on G (see Zames and Falb [12]). This is necessary since a causal multiplier of the form M,,(s) = M,(s)
qs will not. have a finite gain, and hence will not be an
element of aC.
+
Observe that n/I,,(s) contains complex poIes and hence is
not, permittedby Venkatesh[6]. Also, byusingcausal
multipliers only, it is unlikely that a result better than
(dkldt) 6 3.5 k ( t ) can be obtained.
VII. COWLUSION
Crit.eriafor the input-out.put stabilityin Ls-spaces,
more general t,han the already existing results, have been
derived forlinear t.ime-varying systems.Thesecriteria
permit noncausal multipliers to be used, and hence relax
the conditions on t,he 1inea.r part t.o a great extent. A byproduct of the method of derivation has been t.he relation
between the factorizabilit,y andpositivity propert,ies of
operators in a Banach algebra. However, t.he resultsof the
present findings can be improved by furt,her investiga,tion
along two lines:1) derivation of similar criteria (conditions
onthe
shiftedmultipliers)forsystemswithatimeinvariant, nonlinearity in cascade with K in Fig. 1; and 2)
employing a global bound on clk/dt as in Freedma,n and
Zames [13] insteadof the local bounds used here.
ACKNOWLEDGMENT
VI. EXAMPLE
Consider a system with
G(s) =
(s2
+ 10.6)(s2+ 200.1s + 20).
(s’ + 2s + 10)(s2+ s + 16)
4- 4.22s
The systcmisstable
forallconstant,gains
Choosing a multiplier
M(s)
=
( 9 - 2s
+ 1O)(s + 4)
(6.1)
in [O,
> 6 > 0,
1
(6.2)
y u E R.
NOW,
M(s)
=
(s
(s
~
+ 3)
+ 4)
+
(82
(1 - 2s)
- 2s
10)
+
whcre
is the causal part and
Muc(s)=
- 2s
+ 10)
is t.he ant.icausa1part.
> - > 0, ++ w € R
(3 - ~ / 3 and
) Re M u c ( j w + a) 2 0,
Re Al,(jw - P)
if
6
v w E R if a
< 0.5.
Hence, t.he system is st.able for all time-varying gains
satisfying
-k(t)
(0
6 dk
-6
dt
(6 - ~ ) k ( t ) .
for the Stability of certain nonlinear nonautonomous syst.ems,”
IEEE Trans. Circuit Theory (Corresp.), vol. CT-11, pp. 406408, Sept.. 1964.
I. W. Sandberg, “A frequency-domain condition for t.he
stabi1it.y of s y s t e m containinga
single t.ime-varying nonlinearelement,” Bell Sysf. Tech. J., voI.43, pp. 1601-1638,
1964.
G. Zames, On the input-output,st.ability
of t,ime-varying
nonlinear feedback systems-Part I: Conditions derived using
concepts of loop gain, conicity, and positivity,” IEEE Trans.
Automat. Cmtr., vol. Ac-11, pp. 228-238, Apr. 1966. .
-,
“On the input-outfjut stability of time-varymg nonlinear
feedback system-Part, 11: Condit.ions involving circles in the
frequency plane andsector nonlinearities,” IEEE Trans. Automat. Contr., vol. AC-11, pp. 465-476, July 1966.
141 R. W. Brockett and L. J. Forys, “On the stability of systems
containing a time-varying gain,” in PTOC.2ndAllerton Cmj.
Circuit and System Theory, 1964, pp. 413430.
M . Gruber and J. L. Willems, “On a generalization of the
circle criterion,’, in Proc. 4th Allerton Conf. Circuit and System
Theory, 1966, pp. 827-848;
Y. V. Venkat.esh, “Koncausal multipliers for nonlinear syst.em
vol. AC-15. DW.
stabilit.v.” IEEE Trans.Automat.contr..
~.~
~.~
(1 - 2s)
(s*
REFERENCES
a).
onc can verify that
Re M ( j w ) G ( j u )
The authorswish to thankDr. R. Vital Raofor valuable
discussions snd the revietvers for their suggestions which
improved the value of the paper.
K. S. Narendra and R. &I. Goldwyn, “A geometrical criterion
4- 3.22)(s2 - 4.22s + 10.6)
(S
509
SYSTEMS
(6.3)
N. DdOYd- and J . T. Schwartz, Linear Operators, part 1.
New York: Interscience, 1966.
R. Saeks, “Causalit.y in Hilbert space,” SIAM Rev., vol. 12,
no. 3, pp. 357-383,19’70.
G . Zames and P. L. Falb,“Stability cqndit.ions forsystems
with monotone and sloDe-restricted nonlinearities,” SIAi%f J .
Contr., vol. 6, no. 1, pp. 89-108, 1968.
t 101 J. C. Willems and R. ‘A7. Brockett, “Some new rearrangement,
inequalitieshaving applicat.ion in stabi1it.y analysis,” IEEE
Trans. Automat. Contr., vol. AC-13, pp. 539-549, Oct,. 1968.
3.1. K. Sundareshan and M . A. L. Thathachar, “Construct.ion
of stability multipliers for nonlinear autonomous syst.em,”
Dep. Elec. Eng.,Indian Instituteof Science, Bangalore, Mysore,
India, Rep. EE-16, 1971.
G. Zames and P. L. Falb, “On the stability of systems with
monotone a.nd odd monotonenonlinearities,” IEEE Trans.
Automat. Contr. (Corresp.), vol. AC-12, pp. 221-223, Apr. 1967.
L 131 M. Freedman and G. Zames, “Logarithmic variation criteria
for the stability of systems with time-varying gains,” S I A X J .
Contr., vol. 6, no. 3, pp. 487-507, 1968.
510
IEEE TRANSACTTONSON AUTOMATIC
Malur K. Sundareshan was born in &be+
sonpet,Kolar Gold Fields, India, on June
16, 1946. H e received theB.E. degree in
electrical engineering fromBangalore University,
Bangalore,
Mysore, India,
and
electronia and
the X E . degreeinapplied
servomechanisms from t,he Indian 1nst.itute
of Science, Bangalore, Mysore, India, in
1966 and 1969, respectively. He is currently
working toward the Ph.D. degree in control
1nstit.uk
of
engineering at t.he Indian
Science.
His research interests are in &ability t.heory and comput.ational
met,hods.
~~
CONTROL,
VOL. AC-17,
NO.
4,
AUGUST
1972
M. A. L. Thathachar WPS born in Mysore
City, Mysore, India, in 1939. He received the
B.E. degree in e l e c t r i d engineering from
Mysore University, Mysore, India, and the
M.E. and Ph.D.degreesfrom
t.he Indian
1nst.itute of Science, Bangalore, Mysore,
India, in 19.59, 1961, and 1968, respectively.
He was a member of the st.aff of t,he Indian
1nst.itut.e of Technology, Madras,Madras,
during 1961-1964. Since 1964 he hasbeen
withtheDepartment.
of ElectricalEngineering, Indian Inst,itute of Science, where he is currently Assistant
Professor. His research interests are in st,ability theory and learning systems.
On the Existence of a Trap State for
OSO-Type Satellites
Abstract-The possibility of theexistence of “trap states” in
orbiting-solar-observatory-type satellites which use a single-degree
of-freedom mass-spring-dashpot nutation damper is investigated.
It is shown that the spacecraft exhibits a trap state which corresponds toa transverse angular rate of on,where wn denotes thenatural
frequency of the specific damping system used. In particular, it
is indicated that for wfo < wn, where COLD is the initial transverse
angular rate of the spacecraft, a trap state occurs when the initial
displacement zo of the damper mass with respect to the spacecraft
principal axis exceeds a certain critical value; while for o f O> on,
the trap stateoccurs for any nonzero value of 20.
form,asinthe
case of the orbiting-solar-observatory
(OS0)-type satellit,es considered here.
I n general, a directsolution to t.he problemmaybe
provided by using the form of damping mechanism suggested recently by this author [ 5 ] , [6]. Fortunately, it is
observedt.hat.a sa.tel1it.eusingsuchadamper
does not
exhibitanytra.pstat.e
[7]. An alternative approach nil1
probably be t o use any arbitrary form of damper which
may result in trap st.ates, but is considered suitable for
other reasons. Then, by aconvenient, design, t.he trap
states are located in a position well removed from t,he
I. IKTRODUCTION
operating zone on the system state space. Clearly, this
CCURRENCE of “trapst.ates”in
spin-stabilized latterapproach,though
dif6cult t o pursue in general,
spacecraft,s is well-known
a
phenomenon. Physically, may be unavoidable in many practical situations.
the phenomenon arises because of t,he nonlinear effects of
I n t.his paper, the possibilit,y of the exist.ence of trap
the energy-dissipa,ting mechanismsnormallypresent
in states nil1 be investigated for OSO-type satellit.es which
the spacecraft. Several reports [1]-[4] are now available use asingle-degree-of-freedom mass-spring-dashpot damper
in which the problem has been treatedby considering for attitude stabilization of the spacecra,ft. It nil1 be
spacecrafts with specific models of energy dissipators. Of shomn that such satellites do exhibit a trap state whose
these, apparent.Iy t.he earliest paper that is devoted to a location on t,he qst,em state space is det.ermined by the
dual-spin configvration of the satellite is by Cloutier [3]. parameters of the particulardampingmechanism used.
An interesting feature of t,he trap mode investigated by Finally, computer resu1t.s will be presented t o corroborate
Cloutier is, however, that it does not a.ppear when the t.he theoretical obsermtions.
energy dissipator is loca,ted on a completely despun plat11. THEEQUATIONS
OF MOTION FOR
OSO-TYPE
SATELLITES
;Manuscript received June 18, 1971; revised November 15, 1971.
0
Paper recommended by E. I. Axelband, Chairman of the IEEE
S-CS Applicat.ions, SystemsEvaluations,ComponentsCommittee.
This paper mas presented a t t h e 1971 Joint Automat.ic Cont.ro1
Conference, Universit.y of Washington, St.. Louis, $10. This work
was supported bv K M A and performed while the aut,horrras a
Nat.iona1 h e a r t h Council Post-Doctoral Resident Research Associate a t t h eNASA Goddard Space Flight Center, Greenbelt, Md.
The aut.hor is with theCommunications Research Centre, Ot,tawa,
Ont., Canada.
The orbiting solar observat,ory is a
dual-spin satellite
which, in actual design, uses a cantilevered-mass nutat.ion
damper on the despun portion for attitude st.abilization
of the spacecraft [SI. By considering the damper t o be
composed ofa. mass, spring, and a. dashpot as illustrated
in Fig. 1 and by noting that t.he inertial spin rate of an