L,-Stability of Linear Time-Varying Systems ... - ePrints@IISc

. ,. . . .

. .

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 Systems- Conditions 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. The criteria permit the use of noncausal multipliers, and it is shown that this necessitates d k / d t to be bounded from above a s well a s from below.

The method of derivation draws on the theory of positivity of com- positions of operators and time-varying gains, and the results are shown to be more general than the existing criteria.

T

I. INTRODUCTIOK

HE STABILITY properties of the system con- taining 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 the result most popularly known is the circle criterion

[l], [2], [3]. Though t,he application of the circle criterion requires a simple geometrical construction, it. is highly conserva.tive in requiring the 1inea.r tra.nsfer function to be positive real. Improved results were obtained by Brocltett, and Forys [4] by employing an upper bound on the rate of time variation clk/dt and using an RC multiplier. These results were later generalized by Gruber and Willems [SI, whose criterion assured stability if 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 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 and obtained 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 the present paper, more genera.1 ant.icausa1 multipliers are introduced and criteria for the L2-input, Lm_-out.put stability are derived by employing a lower as well as an upper bound on d k l d t . 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 non- negative real numbers. Ll is the space of all measurable real-valued functions x( ^{e )} on R+ that have a finite Lebesgue integral; the L1-norm is defined by ^lix111 =

So"

(x(l)I dt.

Lp is the space of all measurable real-valued functions 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( ⁰⁾⁾ J3. LBe is the extension of L. defined by

SUNDARESHAN AND THATHACHAR: LTSTABILITY OF LINEAR TIME-VARYING SYSTEMS 505

L e = { x ( * ) I x T ( . )

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 (Hz(.))T ⁼ (Hz:T(.)):T, Y X(*)

E

Lz,

++

T R+. (2.2) 2) The “adjoint” operator of H , denot,ed H*, is a mapping of Lz into itself such that

(4.1, HY(-))

= ( H * X ( . ) , ?I(->),

++

^X(.), Y(*)

E

La. (2.3) H is said to be self-adjoint if H ⁼ H*.

3 ) H is said to be anticausal (anticipative) if H * is causal. H is said t o 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 H z ( t ) = h ( t ) @ x ( t ) =

J-:m

^{h(C;)z(t - C;) dC;}

( @ denotes convolution), H causal implies h(t) = 0, t

<

0, and H a.nticausa1 implies h(t) = 0 , t

>

0.

4) H is said t o h a w “finite gain” if

(2.4) 5 ) H is said to be “positive” if

(X(*), H z ( * ) )

>,

0 , b!X( * ) E (2.5) 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 be assumed t o be a causal operat,or having LBe a.s its domain and range. Then, H is said t o be positive ( e ) if

(ZT(.)r ( H Z ( . ) ) T )

>,

0 ,

++

X(.>

E

L2e,

++ T E

R + . (2.7) However, if in addition it is knou-n that H is a causal operator in Lz, it. can ea.sily be shown that

H is positive (=) H is positive ( e ) . (2.8) B. System Description

The system under consideration has the configuration in Fig. 1 and has the input-output relat.ion defined by

el(t) = ul(t) - z t 2 ( t ) e&) =

4 0 +

^{w ( t )}

with any loss of generality, since every linear operator in LS can be

1 This characterization of a noncausal operator is not associated written ^tls ths sum of a causal and an anticausal operator [8].

Fig. 1. The feedback system under consideration.

wl(t) = Gel(t) and wZ(t) ⁼ Kez(t) (2.9) with the following assumpt,ions.

operator in L2 defined by

Assumption 1: 6‘ is a 1inea.r time-invariant convolution

++ 4 . 1 E

Lz, (2.10) where

{

rt] is a sequence in R+,

{si]

is a n l1-sequence (i.e.,

x:==,

^lgi( is finite), and g ( - ) E L1.

Let G ( j w ) denote the frequency function of G defined by

m

W w ) =

,E si

^exp ( - j w 7 J

r = l

Assumption 2.- X is the class of all operators K in L p such that.

K

E

X =) K x ( ~ ) ⁼ k ( t ) ~ ( t ) ,

v

^{X( .)}

E

Lz 0

< 6

k ( t )

6 i T <

^{0 3 ,}

v

^t

E

^{I?+; (2.12)}

X’

c

X

3

K E X’=) - ^{dk (t)}

6

Bbk(t),

++

^{t E R+}

dt

and some /3 E R f ; (2.13)

v

^t

E

^{R+ and some CY E R + . (2.14)}

Let =

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 input pair ul( -), ^{% ( a )}

E

^& and the errors el( ^e), ez( .) E L,,, find the conditions which emure that el(-), e2(-) E Lz.

The solution to this problem will be sought by the now well-established principle [ 3 ] of factoring the open loop int,o two positive operat,ors, one of which is strongly 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.r convolution operator in Lp; 2) M is invertible in L,; and 3) - y ( M )

<

^03, is artificially intro- duced into the systenl as in Fig. 2. It may be recognized

[ 8 ] , [9] that. the operators M satisfying conditions 1-3 form a commutative Banach algebra (€3 with an identity E.

Also,

a

= aC U aaC where ac(@ac) is the Banach algebra of causal (anticausal) operators &I satisfying conditions

506 IEEE TEANSBCTIONS ON AUTOMATIC CONTROL, AUGUST 1972

Fig. 2. M o a e d system with t.he multiplier M .

1-3, with E being a.n element of both Bc and ^@uc.2 This characterization permits the stability problem t,o be treated within the framework 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

BUc and ill+ E Bc, M - and ill+ being invertible such t,hat M - - l

E

BQc and E aC; 2) M G is strongly positive and has finite gain; and 3) KM-' is posit.ive.

These are sufficient 191 to prove that el( .) and e,( - ) E Lz. It needs to be emphasized here that the funct,ion spaces in [9] are defined over the entire rea.1 line R , whereas they are defined over R+ here. The proof of the basic theorem in [9] holds in toto, even in the present case, since, although the functions are 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 be decomposed into 31,

+

^M,,

M c E BC and

Mu,

E

&,

and further, M 4 t ) =

ln

^{m(t -}

M E )

^dE

=

lt

^{mAt -}

M E )

^{d l}

+ sm t m d - E)z(E) dE

=

s-+:

^{m(r)z(t - 7) d r .}

111. MATEEMATICAL P F ~ E L ~ I N A R I E S In this section, conditions will be established for the positivity of combinations of a linear time-invariant

convolution operator P defined by: 1) Px(t) = p ( t ) @ z(t) ; and 2) P ( j u ) = S[p( a ) ] , 5( -) denot,ing t he Fourier transform and a timevarying gain Q having t,he following properties: 1) Q x ( t ) = q(t)z(t); 2) Q has an inverse Q-' 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

example, the operator Q such that q(t) = exp (-ut), 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 tha t: 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 positive if g(. ) is nonnegative and monot.one nonincreasing.

Remark: Condition 2) needs some e x p h a t i o n . If P is an operator in

Le

and Q in Lp, QP in general will be an operat.or in LZ,. Horever, in cert.ain special sitaations, for example, when P is of the form P ⁼ Q-IPI with P I

This corresponds to Saeks' concept of "weak causdity" 181.

being an operator in

Lz,

QP will be an operator in L2.

Proof: It is required to prove tha,t

(x(-), Q P x ( . ) )

2

0, V x(*)

E

Lz (3.1) left-hand side (LHS) =

c

z(t)q(t)Px(t) dt

= q ( a ) Jm ₀ z(t)Pz(t) dt -

lm 5

Pt

on int.egration by parts. Term I on the righbhand side mHS)

>

^{0 since q( a )} 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 self- adjoint 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 requires to show th a t

{ z d . ) , (QPQz(.))T)

>,

0,

v

^{x(.) E L e ,}

++

T E R+ (3.3) LHS ⁼ (xT(

.),

QPQxT( ^{a ) ) ,} since P is causal

= (&xT(

.),

PQzT( ^{a ) ) ,} since Q is self-adjoint

= {?IT(*), PYT(.)),

Y(.)

= & x ( . )

E

^{L 2 e}

= (YT(.), (PY(.))T)

3

0 since P is positive ( e ) .

Lemma 3 (Shifting Lemma): If P is a linear convolution opera.tor in L2 and Q, is a time-va.rying gain defined by

Q,z(O = q , ( M t ) and qJt) = exp ( P O , then

M * > , Q p W - ) ) = (z(.>, PpshQ,z(.)),

V

4 . 1 E

Lz (3.4) where PPsh is a linear convolution operator related to P bY

S[p'"h(.)] = P"""(jw) = P(ju

-

^p) (3.5) if the inner products in (3.4) exist.

Proof:

Lemma

4:

If P and P h are defined as in t.he previous lemma,

(x( ^e), PP"x( .)) = (x( ^e), (P*) -pshx( .)),

V x(*) E La (3.6) if the indicated inner products exist,.

Proof: The proof involves simple operator manipula- tions and repeated applications of Lemma 3, and is hence omitted.

IV. FACTORIZATION ^OF OPERATORS

The importance of t,he factorization of an operator dl E (R into a suitable composition of elements of (R is apparent from Section II-C. I n this section, conditions for such a fact.orizat,ion will be enunciated. Let. ,e(@,@) be the spa.ce of all cont.inuous linear ma.ps of ^(R into itself and let E2 be the ident,ity of .e(@,@). Let. P+ be a projection on ^(R and let P- = E2 ^- P+. Let (R+ and @-

denote, respectively, the ranges of P+ and P-. Then, conforming &h the earlier notation, it is easy t o see that Bc is t,he subspace spanned by @+ and E and Bac is the subspace spanned by ^(R- and E ( E being the ident,it.y of

The following lemma is a considerable generalization of simi1a.r lemmas of Zames a.nd Falb [9] and Willems and Brocketk [lo].

Lemma 5: With the notation introduced a.bove, if M is any arbitrary strongly positive operator in @, there exist elements ^{M +} = exp [P+ log MI and ilI- = exp [P- log ill] such that: 1)

1W+ E

BC and M- E (Rat; 2 )

AT

= M - M + ; 3) M + and dl- a.re invertible with M+-l E Bc and M - - l E (Rat.

Proof: The crucial part of t.he proof is to shorn- t.he existence of log M as a n element of

(R.

Let L2c denote the space of all measurable complex- valued functions on R+ that have finite energy. Clearly La c LPc. Now, 1VI can be regarded as an operator in LzC and the Banach algebra of these operators can be denoted by W . Let. a ( M ) denote t,he spect,rum of M . ill strongly posit,ive =) a(M)

c

the part of the complex plane

{

,$:

R e ,$

2

^E

>

^0). Hence it is possible t o t.ake a simply con- nected doma.in D in the complex plane, excluding the negative real line,3 such that, a ( M ) c D. Let I' be a simple closed curve (in t.he positive sense) in D enclosing all the spect,rum points. Since

f(E)

⁼ log 5 is a holomorphic function in D, the logarithm of M exists and is given by the Dunford-Ta.ylor integral [7, pp. 566-5761 as

W f

where

a([;

44) = ( [ E - M ) - l is the resolvent, of M , which exists as a bounded operator for a.11 E E p ( l k 0 , t.he resolvent. set. Further, the fact that

M

is a bounded operat.or ensures the spect.ral radius of $1 t o be finite, t,hus

provision D is required for a branch t o exclude the negative real l i e so cut for defining the logarithm of complex as to make quant,ities.

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 recognked t.hat, the RHS of (4.1) is comparable t.0 the Cauchy integral formula in the complex variable t,heory.) Having t.hus settled t.he question of the existence of log M as a bounded operator in L2e, it. is a n easy exercise t o show that &I: L2 +- Lt =) log M: L2 ^--f L,2, i.e., log Jl E @.

The rest of the proof is simple and follom as in Zames a.nd Falb [9]. It nmy further be checked that M A T + ⁼ 211, since (R is a commutative algebra..

Remurk: The a,bove lemma illustrates the relation between the factorizability and the posit,ivity of operators in a Banach a.lgebra a.nd, in the a,uthors' opinion, is a considerable improvement, over t,he results of [9] and [lo]. It. may be recalled t,hat t,hese references prove the desired factorization under such restrictive conditions on M a s AI = aE

+

^{2, 2 E @, a E R with}

l!Zll <

^lal.

V . SOLUTION ^{OF THE} L-STABILITY PROBLEM This sect.ion contains a few t.heorems t,ha.t provide sufficient conditions for the Lz-stability of t,he system under consideration. Theorem 3 is the major result. of t.he paper and involves noncausal nmlt.ipliers. However, Theorems 1 and 2, which, respectively, consider the introduction of causal and anticausal multipliers, provide the necessary mot.ivation for the technique employed to take into account. the effect of the timewrying gain k ( t ) in the otherwise time-invariant feedback loop.

Theorem 1 : If t.here is an operator ^dl1 in @, such that R e Jfl(ja)G(ju)

>

⁶

>

^0,

++

^{w E R (5.1)}

R.e &Il($ - 0)

>

^{0, Y w E R (5.2)}

for some nonnegative consta.nt 0, then t.he system de- scribed by (2.9), with the relat,ed Assumptions l and 2, is

&stable for all K E X'.

Proof:

1) The factorizat.ion condit,ion is satisfied t.rivially since

M1 E

( R ,

=) M l = EMl, E

E

aaC and M I E

(R,.

2 ) M1G is strongly positive by (5.1) and an appli- cation of Parseval's theorem. Also M I G has finite gain since g( - ) E Ll and M I E =) y(ilTl)

<

^a.

3) Positivity of K M - ' . To prove

(X(.), K & l l - l ~ { . ) )

>

^{0, !V X( .) E L? (5.3)}

LHS = ( I I f l y ( ^{e ) ,} K y ( ^e)),

v(

.) = I l r l l - l ~ ( e )

E

LI

= (KMly( ^{e ) ,} y( -)}, since K is self-adjoint

= (K&2,-'Qa,#1~(*), Y(.>),

QzaY(0 ⁼ Y(t> exp (2Pt)

2

0, V $4.) E L2 (5.4)

if QaaJll is posit,ive ( e ) . [By Lemma 1, ident,ifying P with QSsMl and Q with KQ2'-l, is a.n operator in Lz,, and, since K E Xa, k ( t ) exp (-2Pt) is monotone non- increasing. Hence, all t.he conditions of Lemma 1 are satisfied. ]

508 ^IEEE TR-U~SACTIOKS ON AUTOMATIC COXTROL, AUGUST 1972

Nom-,

(xd.1, ( Q 2 & 1 4 . > > ~ )

= (xT( .), QzaM1xT(. )) since MI E aC =) QzaMl is causal

= @ A * ) , Q , & $ f 1 4 ' ) )

= (xT(-), Q ~ M ~ B ~ ~ Q , ~ ) )

[by Lemma 3 where M P h is a convolution operator such that M , 8 " ( j w ) = M l ( j w - 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 theorem. Hence (5.5), (5.4), and (5.3) result and, in view of the observations made in Section 11-C, the syst.em is Lpstable.

Th.eorem 2: If t,here is an operator M2 in such that, Re M 2 C j w ) G ( j w )

>

^S

>

0,

++

^w E R (5.6)

Re

+

^a)

2

^{0, '4 w E R (5.7)}

for some nonnegative constant a, then the system described by (2.9), ni t h t,he related Assumptions 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

M2

E

@a =) M2 ⁼ M2E1 M2 E

a,,,

E

E

aC.

2) M2G is strongly positive from (5.6) and an application of Parseval's theorem. Also M2G hm finite gain since g( ^{e )} E L1 and M 2

E

aaC => 7(..M2)

<

^a.

3) Posit,ivity of KM2-'. To prove

(x(.), KM2-4c(*))

3

0, '4 x(*) E L2 (5.8) LHS = (M2y(*),

Kg(*)),

Y(.) = J f 2 - 5 4 . )

E Lz

= (!I(*>, Mz*K!I(.))

= (h( .), K-'(Mz")-'h(

.)>,

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,y be repeated, giving t.he claimed result.)

Now, positivity of (M2*)ash follows from (5.7) since Re MAjw

+

^a)

>

^0, ' 4 ' w E R

(=> (x(-), M 2 - " " h ~ ( * ) )

>

^{0 , '4 X(*) E Lz}

(=> (x(.>, (M2*)ash4.))

2

0,

v

^{x(.) E L2}

by Lemma 4.

made in Sect.ion 11-C, the system is &-stable.

Hence, (5.9) and (5.8) hold and, from the observations Theorem 3: If there is an operator M in such that

Re M I & -

P ) 2 2

0 ,

v

w

E

R _(5.12) R e M 2 ( j w

+

^a)

2 2

0,

++

^w E R (5.13)

E = E l + h > O (5.14)

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 X a B .

Proof:

1) Factorization Condition. From (5.10), Re M ( j w ) = Re J f l ( j w )

+

^{Re 1M2(jw)}

2

€ ' + E 2

= ~ > 0 , Y w E R (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, i l f is strongly positive and by Lemma 5 the factorization condition is satisfied.

2) M G is strongly positive by (5.11) and an appli- ca.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 K M - I :

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 Gruber and Willems (derived in 151 for the absolute stability problem), the most general result involving causal mult,ipliers. Theorem 3, n-hich permits ant,icausal terms in the multiplier, provides more relaxation on the phase excursions of G ( j w ) outside t,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 problem using the Popov approach. But the

anticausal functions employed in [6] are only of the part.icular form

The results derived here permit. more general multipliers

-

M = MI

+

^M2; (see the numerical example below) and are derived for a

E M 2 E ^@" (5'10) &rower definition of st,abilitv. However. t.he a.ssertion in

SUNDARESX4N A N D THATHACHAR: L 2 - ~ ~ . % ~ ~ ~ ~ ~ O F LINEAR TIME-VXRYIKG SYSTEMS 509 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 knowl- edge of arg G‘(jw) by the methods enumerated in Sund- areshan and Thathachar 1111 and by checking whether the multiplier so constructed satisfies the other conditions of the theorems.

Remark

4:

More general multipliers of the form

M , ( j W ) = M ( j W )

+

^{q j w , q}

>

⁰

can be used in (5.11) if an additional condition Iim ] w ~ ‘ ( j o ) \ =

o

is imposed on G (see Zames and Falb [12]). This is neces- sary 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.

to--) m

VI. EXAMPLE Consider a system with

G(s) = (s2 4- 4.22s

+

^10.6)(s2

+

^200.1s

+

²⁰⁾

(s’

+

^2s

+

^10)(s2

+

^s

+

¹⁶⁾

.

^(6.1)

The systcm is stable for all constant, gains in [O, ^a).

Choosing a multiplier

M(s) = ^(S

4-

3.22)(s2 - 4.22s

+

^10.6)

( 9 - 2s

+

^1O)(s

+

^{4) 1 (6.2)}

onc can verify that

R e M ( j w ) G ( j u )

>

⁶

>

^{0 , y u}

E

R . NOW,

(s

+

^{3) (1 - 2s)}

M ( s ) = ^~

(s

+

^{4) + (82 - 2s}

+

¹⁰⁾

whcre

is the causal part and

M u c ( s ) ^{= (1 - 2s)} (s* - 2s

+

¹⁰⁾

is t.he ant.icausa1 part.

R e A l , ( j w - P)

> - >

^{0, ++ w € R}

if

6

(3 - ~ / 3 ) and Re M u c ( j w

+

^a)

2

^{0 ,}

v

^w

E R

if ^a

<

^0.5.

Hence, t.he system is st.able for all time-varying gains satisfying

dk

(0

- k ( t )

6

^-

6

(6 - ~ ) k ( t ) . (6.3) dt

Observe that n/I,,(s) contains complex poIes and hence is not, permitted by Venkatesh [6]. Also, by using causal multipliers only, it is unlikely that a result better than ( d k l d t )

6

3.5 k ( t ) can be obtained.

VII. COWLUSION

Crit.eria for the input-out.put stability in Ls-spaces, more general t,han the already existing results, have been derived for linear t.ime-varying systems. These criteria permit noncausal multipliers to be used, and hence relax the conditions on t,he 1inea.r part t.o a great extent. A by- product of the method of derivation has been t.he relation between the factorizabilit,y and positivity propert,ies of operators in a Banach algebra. However, t.he results of the present findings can be improved by furt,her investiga,tion along two lines: 1) derivation of similar criteria (conditions on the shifted multipliers) for systems with a time-

invariant, nonlinearity in cascade with K in Fig. 1 ; and 2) employing a global bound on clk/dt as in Freedma,n and Zames [13] instead of the local bounds used here.

ACKNOWLEDGMENT

The authors wish to thank Dr. R. Vital Rao for valuable discussions snd the revietvers for their suggestions which improved the value of the paper.

141

t 101

L 131

REFERENCES

K. S. Narendra and R. &I. Goldwyn, “A geometrical criterion for the Stability of certain nonlinear nonautonomous syst.ems,”

IEEE Trans. Circuit Theory (Corresp.), vol. CT-11, pp. 406- 408, Sept.. 1964.

I. W. Sandberg, “A frequency-domain condition for t.he stabi1it.y of s y s t e m containing a single t.ime-varying non- linear element,” 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 Automat. Cmtr., vol. Ac-11, pp. 228-238, Apr. 1966.

.

concepts of loop gain, conicity, and positivity,” IEEE Trans.

-, “On the input-outfjut stability of time-varymg nonlinear feedback system-Part, 11: Condit.ions involving circles in the frequency plane and sector nonlinearities,” IEEE Trans. Auto- mat. Contr., vol. AC-11, pp. 465-476, July 1966.

R. W. Brockett and L. J. Forys, “On the stability of systems containing a time-varying gain,” in PTOC. 2nd Allerton C m j . 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 stabilit.v.” IEEE Trans. Automat. contr.. vol. AC-15. DW.

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 for systems with monotone and sloDe-restricted nonlinearities,” SIAi%f J .

~.~ ~.~

Contr., vol. 6, no. 1, pp. 89-108, 1968.

J. C. Willems and R. ^‘A7. Brockett, “Some new rearrangement, inequalities having 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 Institute of Science, Bangalore, Mysore, India, Rep. EE-16, 1971.

monotone a.nd odd monotone nonlinearities,” IEEE Trans.

G. Zames and P. L. Falb, “On the stability of systems with Automat. Contr. (Corresp.), vol. AC-12, pp. 221-223, Apr. 1967.

for the stability of systems with time-varying gains,” S I A X J . M. Freedman and G. Zames, “Logarithmic variation criteria Contr., vol. 6, no. 3, pp. 487-507, 1968.

510 ^IEEE TRANSACTTONS ON AUTOMATIC CONTROL, VOL. AC-17, NO. 4, AUGUST 1972

Science.

Malur K. Sundareshan was born in &be+

sonpet, Kolar Gold Fields, India, on June 16, 1946. H e received the B.E. degree in electrical engineering from Bangalore Uni- versity, Bangalore, Mysore, India, and the X E . degree in applied electronia and 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 engineering at t.he Indian 1nstit.uk of

~~

His research interests are in &ability t.heory and comput.ational met,hods.

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. degrees from 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 has been with the Department. of Electrical Engi- neering, Indian Inst,itute of Science, where he is currently Assistant Professor. His research interests are in st,ability theory and learn- ing systems.

On the Existence of a Trap State for OSO-Type Satellites

Abstract-The possibility of the existence 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 corre- sponds to a transverse angular rate of on, where wn denotes the natural 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 state occurs for any nonzero value of 20.

I. IKTRODUCTION

0

^CCURRENCE of “trap st.ates” in spin-stabilized spacecraft,s is a well-known phenomenon. Physically, the phenomenon arises because of t,he nonlinear effects of the energy-dissipa,ting mechanisms normally present in the spacecraft. Several reports [1]-[4] are now available in which the problem has been treated by considering spacecrafts with specific models of energy dissipators. Of these, apparent.Iy t.he earliest paper that is devoted t o a dual-spin configvration of the satellite is by Cloutier [3].

An interesting feature of t,he trap mode investigated by Cloutier is, however, that it does not a.ppear when the energy dissipator is loca,ted on a completely despun plat-

;Manuscript received June 18, 1971; revised November 15, 1971.

Paper recommended by E. I. Axelband, Chairman of the IEEE This paper mas presented a t t h e 1971 Joint Automat.ic Cont.ro1 S-CS Applicat.ions, Systems Evaluations, Components Committee.

was supported bv Conference, Universit.y of Washington, St.. Louis, K M A and performed while the aut,hor rras $10. This work a Nat.iona1 h e a r t h Council Post-Doctoral Resident Research Asso- ciate a t t h e NASA Goddard Space Flight Center, Greenbelt, Md.

Ont., Canada. The aut.hor is with the Communications Research Centre, Ot,tawa,

form, as in the case of the orbiting-solar-observatory (OS0)-type satellit,es considered here.

I n general, a direct solution t o t.he problem may be provided by using the form of damping mechanism sug- gested recently by this author [ 5 ] , [6]. Fortunately, it is observed t.hat. a sa.tel1it.e using such a damper does not exhibit any tra.p stat.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 a convenient, design, t.he trap states are located in a position well removed from t,he operating zone on the system state space. Clearly, this latter approach, though dif6cult t o pursue in general, may be unavoidable in many practical situations.

I n t.his paper, the possibilit,y of the exist.ence of trap states nil1 be investigated for OSO-type satellit.es which use a single-degree-of-freedom mass-spring-dashpot damper for attitude stabilization of the spacecra,ft. It nil1 be shomn that such satellites do exhibit a trap state whose location on t,he qst,em state space is det.ermined by the parameters of the particular damping mechanism used.

Finally, computer resu1t.s will be presented t o corroborate t.he theoretical obsermtions.

11. THE EQUATIONS ^{OF MOTION FOR} OSO-TYPE SATELLITES

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 of a. mass, spring, and a. dashpot as illustrated in Fig. 1 and by noting that t.he inertial spin rate of an