427
SHORT PAPERS
- “On
the input-output stahi1it.y of time-yaryingnonlinearfeedback
systems-Part 11: Conditions Involving clrcles In the frerluencs plane and
sector nonlinearities,” I E E E Trans. Automaf. Contr., 1.01. AC-11. pp. 465476, July 1966.
I. Is7. Sandherg, “On theLrhoundedness of solutions of nonlinear functional equations.” Bell S y s t . Tech. J . , vol. 43, part 2, pp. 1581-1599, July
1964.
- “Someresults on thetheory of physicalsystemsgoverned by nonlinear functional equations,” Bejl Sgst. Tech. J., vol. 44, pp. 871-898, MayJune 1965.
E. I. Jury an:, B. IV. Lee ”The absolute st.abilitv of svstems with many
nonlinearities Azltomat. iemote Contr.. vol. 26, pp. 9451966, .June 1966.
13. D. 0.r\ndkrson. “Stabihty of control systems with multiple nonlinearities,” J . Franklin Inat., vol. 282. pp. 15.5160. Sept. 1966.
V. A. Yacubovich “Frequency conditions for the absolute shhility of control systems with ‘several nonlinear or linear nonstationary blocks.” Automat. Remote Contr.. vol. 28. pp. &30, June 1967.
F. X. I3ailey, “Theapplication of Lyapunov’ssecondmethodtointerconnected systems ” S I d X J . Contr.. ser. .‘ vol. 3. pp. 143-462. 1966.
R. F. Estrada On the arahility of multiloop feedback systems.” I E E E
Trans. Automitic Control. r o ~ ~. ~ - 1 pp.
7 .781-791, Dec. 1972.
F . R. Gantmacher. M a t r i x Theory. New York: Chelsea, 1959.
VIJI. CONCLUSIONS
The stabilityof multjloop systems is investigated here in a fashion
which directlyincorporates the ‘Lstnwt.ure” of the system in the
analysis. This can be accomplished because system stability can
often be interpreted in terms of margins within which subloops are
stable. The method of analysis is straight.forward, and a general
analysis procedure is given.
The theory is applied to t.wo specific multiloop systems. A system
composed of two single-loop systems, each having a linear and a
nonlinear element, in cascade with anouter feedback loop is examined. Conditions are derived which as the out.er feedback loop
gain approaches zero reduce to the familiar circle criterion for each
of the single-loop systems. &o a system is analyzed which consists
of an interconnect.ion of three lineartime invariant elements, a
linear t h e varying element,, a piecewise linear nonlinearity, and a
hysterisis nonlinearity.
Average Variation L2-Stability Criteria for Time-Varying
Feedback Systems-A Unified Approach
APPENDIX
C o h l P L E T I O N OF THE P R O O F O F THEOREM
1
In order to show that thehypotheses of Theorem 1 are sufficient to
I - [(bijIg(Hj)]has an inverse whose
guaranteethatthematrix
elements are all nonnegative, the following preliminary results (see,
e.g., [ l o ] pp. 66 and 71 of vol. 11) are employed.
Theorem A : 9 mat,rix A havingall elements nonnegativehas
always a nonnegative eigenvalue r such that the moduli of all the
eigenvalues of A do not exceed r. To this “maximal” eigenvalue r
there corresponds an eigenvect.or y such that y 2 0 and y # 0 (refer
t o Sect.ion I1 for notation). Moreover, the adjoint matrix B(X) =
(AI - A)-llXZ - 81 has all elements nonnegative for X 2 r.
Theorem B: If a mat.rix G has all off diagonal elements negative
or zero and the successive principalminors are posit.ive, then all
principal minors are positive.
As in Theorem 1, assume that the successive principal minors of
I - [Ih,,Ig(Hj)] are all positive. Since the last, successive principal
minor is the det.erminant of t.his matrix, the matrix is nonsingular and
has an inverse. From Theorem A, it is clear that. mat.rix [Ibi,lg(Hj)]
has
a
“maximal” eigenvalue r. Further,
the
matrix
(I [lbijlg(Hj)])-1!I- [lbiilg(Hj)]lhas all elements nonnegat.ive if r 5 1 .
But [ I - [/bij1g(H;)]Iis t.he last successive principal minor of I [Ibijlg(Hj)]and is positive. Hence, if r 5 1 then Z - [!bijlg(Hj)]has
an inverse with all nonnegat,ive e1ement.s.
Iiow it only need be shown that r 5 1 . Since r is an eigenvalue,
it is found that
31.K. SUNDARESH-U’T A N D 1.1.A . L. THATHACHAR.
Abstract-The
problem of developing L-stability criteriafor
feedback systems with a single time-varying gain, which impose
on the gain, is treated. A unified
averagevariationconstraints
approach is presented which facilitates the development of such
averagevariationcriteria
for both linear and nonlinearsystems.
The stability criteria derived here are shown to be more general
than the existing results.
I. IKTRODUCTIOS
Following the work of Freedman and Za.mes [ l ] in developing Lnstability criteria for linear systems containing a time-varying gain
k ( t ) in an otherwise t.ime-invariant. feedback loop, by imposing
averaging constraints on the 1ogarit.hmicvariat.ion of k ( t ) , there has
been a considerable act.ivity in obtaining similar results for linear
and nonlinearsystems.
3Iajor resultsin
t.his direction are t.he
absolute stabilit,y criteria of Venkatesh [ 2 ] for linear aystenw and of
[3] for nonlinear systems. Venkatesh’s criKarendraandTaylor
terion imposesaconstraint on k ( t ) more general t.han [ l ] in considering only the positive lobes of d log k ( t ) / d t for averaging. Hoaever, both [ 2 ] and [ 3 ] make use of Corduneanu’s ext.ension of t.he
Lyapunov method for the derivat,ion of results and hence, do not
o = 1r1 - [ I b i j l g ( ~ j ) l I= - [ I b i j l g ( ~ j ) l ( r - 1111.
facilit,ate a straightforward extension t.o the L-stability problem.
This means ( 1 - r ) is an eigenvalue of t.he matrix I - [Ibi,lg(Hj)]. I n a recent paper, the authors[4] have presented Ln-stability criteria
Now the charact,erist,ic equat.ion for an n X n mat.rix B can be for linear systems which impose instantaneous bounds on l / k ( t )
writ.ten as
dk(t)ldt, by employing the t.heory of positivity of compositions of
causal operatorsandtime-vaving
gains. The mainaim
of the
present paper is t o emphasize that [ 4 ] contains t.he core of a proIB - X I 1 = (-X)”
Sp(-X)n-k
= 0
cedure which inconjunction with asuit.ablefactorization of k ( t )
E=l
yields anextension to averagevariation stability criteria. This is
where each S k denotes the sum of all principal minors of order k of
demonstrated here by deriving within the framework developed in
the matrix B. Let.ting B = I - [lbijlg(Hj)]and X = 1 - r results in
[4],stability crit.eria which are moregeneral than [ 2 ] forlinear
n
systems and [ 3 ] for non1inea.rsystems.
(r - 1)”
~ k ( r 1)f-k = o
+
1
+
2
+ L=l
where each
Sk
represents the sum of all principal minors of order k of
I - [ l b i j / g ( H j ) ] .But from Theorem B it is clear t.hat. all principal
minors of I - [Ibij!g(Hj)]are positive. Hence, each Sk > 0. Thus,
the above charact.erist.ic equationcannot
Hence r, 5 1.
be satisfied for
7
>
1.
REFEREWES
[l]
G.Zames, “On the inpuboutput stability of time-varying nonlinear feed-
back systems-Part I : Conditions derived using the concepts of loop gain,
conicity.andpositivity,” I B E E Trans. Automat. Contr., vol. AC-11, pp.
228-238, Apr. 1966.
A.
:vOtUtiQ,lS
urd Dejinitiuns
While t.he notations employed in [ 4 ] nil1 be folloxed, a few of the
import,ant ones are briefly recapit,ulat.ed. Let,, R, R’, and J + denote;
respectively, the reak, the nonnegative reds, and the nonnegat.ive
integers. An operat.or H in L? (L?,) is a single-valued mapping of Ln
Manuscriptreceived May 21973:revisedOctober12,
19i3.Paper recommended by R. A . S l t o o ~ .Chairman of t h e I E E E S C SStability, Nonlinear, and
Distributed Systems Committee.
The authors are with the Department of Electrical Engineering. Indian Institute of Science, Bangalore, India.
428
IEEE TR.LYSACTIONS
CONTROL,
O N AUTOMATIC
(,he)
into itself. If H is an operator in L e , then H is said t o be "positive(e)" [st.rongly positive(e)] if bhe inequalit.y ( x T ( .), ( H z ( .))T)
2 B ( z T ( . 1, TT(.)) holds wit.h 6 = 0[6 > 01, V x(. ) E I?,and V T E'
R+, TT(.) being the truncation of z( .) defined by, z T ( t ) = ~ ( t V)
t E [0,T]and zero otherwise.
Letdenotethe
Banachalgebrawith
an ident.ity E, oflinear
time-invariant causal convolution operators in
that, are defined by
where { r i ) ,i E J is a sequence inR +,{ h i ) ,i E J is an Zl-sequence
in R and h ( . ) E L. H ( j w ) =
h i esp(-jwr,)
f?: h ( t )
esp(-jot) dt. An operator H E ( B c issaid to be "regular in (Be''
if H-3 E Cat.
X is the clasof time-varying operators K in 4,defined by K z ( t ) =
k(t)z(t)V T( . ) E I,,, 0 < inf k ( t ) 6 k ( t ) sup k(t) < m V t E R+.
Let. I denote the identity of K . X @ c X 3 K E X@=) dkct)ldt 6
28k(t) for some 8 E K + V t E R'. &cE 3 K E X, =)dk(t):'dt
3 --2ak(t) for some 0: E R' V t E R+.
With this notation, it is simple t o obtain the following result.
Propositiolnl: If K E X@(Xs)has a decomposition K = e I
I?,
0 < e < inf k ( t ) , then there exists a scalar @'
> @ such that I? E
J;@'(xa'jand the difference (6' - 8 ) ran be made arbitrarily small.
ciEJ+
+
+
+
<
+
B. System Descriptio11
The system (Fig. 1) is described by the input-output relations,
e l ( . ) = ul(.) - w ( . ) , e?(.) = 7 4 . )
.al(.),
w ( . ) = Gel(.),and
w2(.
) = Ke,( . ) where G E 63,and K E 3;.
+
C . The JIai1t Problem
Given that u 1 ( . ) , 7 1 1 ( . ) E L . 2 and e l ( . ) , e,(-) E I,?,, find conditions
on G and K which ensure that the system is &stable, i.e., el( .),
e?(.) E L .
111. PRELIMIKARIRESULTS
The following lenunas give conditions for thepositivity(e) of
and time-varying gains.
compositions of causal operators in I, (I2?<)
The proofs of these may be found in [4].
Lemma 1: Let P be a causal operator in I/>,and Q be an operator
in L?, defined by & x ( t ) = q(t)r(t)V T(.) E L?,, q ( . ) : R ++ R is
absolutely continuous on R+ and is bounded for all t E R+.Then
QP is positive(e) if 1 ) P is positive(e) and 3) q ( . ) is nonnegative and
monotone nonincreasing.
Lemma 8: Let. P and Q be defined as inlemma 1 . Then PQ is
positive(e) if 1 ) P is posit.ive(e) and2) q ( . ) is nonnegat.ive and
monotone nondecreasing.
Lemma 3: Let H E C a C sat,isfy Re H ( j w - 8 ) 2 0 V w E K and
some 8 E R-. Then, KH is positive(e) for all K E X@ and H K is
positive for all h- E KO.
If H further satisfies He H ( j w ) 2 e > 0 V w E K , then KH is
strongly positive(e) for all K E X@" and HK is strongly positive(ej
for all K E Xy,where 8" < 8, the difference (0 - a")being arbitrarily small.
;Tote: The first part of Lemma 3, i.e., positivity(e) of KH and
Hi? is proved in [4](refer to t.he steps from ( 5 . 3 ) to ( 5 . 5 ) in [ 4 ] ) .
The proof of the strong posit,ivity(e) also follows similarly, after
making use of the proposition in Section 11-A.
IV. AIAIN RKSULTS
The lemmas enunciated in the previous section will be crucially
used in the derivation of the stabi1it.y criteria for the system under
consideration.
Stability Criteria for Liltear System
Fig. 1.
AUGUST
The linear feedback system under consideration.
for some T > 0 and some scalars a$ E R +,'d t
denoting only the posit.ive lobes of the argument.
%XaB
1974
E R +, with [ .]
+
c x 3 K E anx,@=)
for some T > 0 and some scalal-s a,@ E R+, V t E X + wit.h [ . ] denoting only t.he negative lobe5 of the argument.
The following lemmas depict certainfactorization properties of
the operators belonging to these classes.
Lennmma 3 : If h-E @,Xa$, t.hen K admits a factorization h-= K l K ,
such that K I E X,q and K , E X,.
Proof: The proof follow along the same lines a s the proof of a
similar factorizationlemma in Freedmanand Zanles [ I ] , by deadditive decomposition of
fining I(t) = log k(t) andobtainingan
I(t). This, hoaever, is contained in Ranlarajan and Thathachar.
Lemma 5: If K E @nX,p, then K admits afactorization K =
KlK, such that K ZE X, and K , E X@.
Proof: The proof can be obtainedfroma
slight, variat.ion of
that of Lemma 4, by defining / ( t ) = -log k ( t ) .
The following theorem gives a criterion for the Lstability of t.he
system under consideration.
a
,and constants ( ~ $ 1
Theorem 1: If there esists an operator X E C
E R+ such that,
'
1) M is regular i n bc
2) Re
M ( j w ) GGw)
3) I<c ,IrCjw
-
2
(4.3)
6
>0V
8') G ( j w - 8 ' )
ER
w
2
0 V
(4.4)
w
ER
(4.5)
and
4)
Re X C j w -
a)
20V
w
E R,
(4.6)
then the system (Fig. 1) is &stable for all K E @,X,@ U &Jia@,
where @ < p', the difference (8' - 8 ) being arbitrarily small.
Proof: The proofwill be outlinedfor the case K E a p X a g
only, since the other follows fromsimilararguments.
Since K E
=) K = K&,, Kl E 6 8 and K , E P , make the system
transformations as shoKn in Fig. 2. An application of lemma 3 ensures the st.rong positivity(e) of X G K l from (4.4)and (4.5) and the
positivity(e) of K J - ' from (4.6). The finiteness of gain of XGKI
(for the definition of the gain of an operator, see [ 4 ] ) is also ensured
since X , G E aCand I C ! E 3;. Hence, invoking a basic result due t o
Zanles [6], Lrstability of the system ensues.
Stability Criteria f o r :Yonliltear Systems
In this section, we will consider a syst.enl having a different configuration (Fig. 3) from the one described earlier in Section I1 B,
in having a nonlinear operator F in the feedback path. Let '5, denote
the rlas of nonlinear operat.ors F in I,, defined by, F z ( . ) = f ( r ( .) ) V
) E L e , f(0) = 0, zf(r) 2 0 V r E R and [xl - x ? ] [ ~ ( I I-)
f(+z)] 2 0 V rI,Z? E R. Let .&.II c 3.tf 3 F E &.v =)f(. ) is odd.
Let. us introduce the follom-ing subclas of 3;.
D e h e the following subclases of X :
QpX& c x 3 K E
@,X,@
=)
1'
- ~ ( CU @)
dr
< 26
(4.7)
(4.1)
for some scalars 8 , E~ R
+
++ t E R +.
429
SHORT PAPERS
G
M-1
w;
Fig. 2.
in
(0
)
e;(*)
u;(+MU~
Transformations of the system for Theorem 1.
Fig. 4.
Transformat.ions of the system for Theorem 2.
time-varying systems with monotone nonlinearities. It. is of interest
to observe in this context that, in spite of the ample atkention this
problem has received ever since the appearance in the literature of
O'Shea's [9]results for t.ime-invariant syst.ems, it remained unsolved
even for the simpler and less general case of the imposition of instantaneous bounds on the logarithmic variation of the gain.
[n] have also obtained
C) Recently, Ramarajan and Thathachar
Fig. 3. The nonlinear feedback system under considerat.ion.
results similar t o Theorem 1 by folloning the approach of Freedman
and Zames [l]. However, the present. met.hod of derivation is
It may be shown, following Freedman [7], t.hatthis averaging different and has followed from an ext.ension of the techniques decondition is sufficient to ensure a useful fact.orization of K.
veloped in [4] for the inst.ant.aneous bounds case, thus providing a
Lemma 6: If K E @X@,then K admits a factorization K = K I K , unified approach for thederivcztion of aoerage variation criferia with
such t.hat K CE X6 and K , E Xo, 3 i o being the class [Xa]6=0.
various types of aLleraging constraints. In this context., it is interesting
This lemma' lends itself useful in proving the following stability to notet.hat.the earlier invest.igators had toresort to entirelydifferent
criterion.
methods for t.he derivation of stability criteria wit.h bounds of either
Theorem 2: If there exists an operat,or X E aCand a constant type on t.he gain (for exa.nlple compare the results of Karendra and
p' E R + such t,hat,,
Taylor in [3] and [8]; while the criteria in [SI which impose instanregular
1) M is
aC,
(4.8) taneous bounds were derived using the Popov approach, generalisation of these results with averaging bounds in [3] could be obt.ained
only by employing Corduneanu's method).
2)
= E
z, E ac3 11z11=
:zi;
iz(t); dt < 1
D) It may be noted that thehypotheses of Theorem 1 differ
isJ+
from Freedman andZames [ l ] only in t.he c0nstraint.s on k ( t ) (which
(4'9) are more general). Hence, it is possible t,o state this result in purely
3) Re M ( j o ) G ( j w ) 3 6 > 0,
'd w E R,
(4.10) geometrical t.enns, involving certain "shifted" Nyquist diagrams
and dispensing wit.h t.he elrplicit w e of multipliers, following Freedand
man [TI.
x
+ z
4) Re M ( j w - p')G(jw
+
- 8') 3
++ w
0,
E R,
(4.11)
V. CONCLUSIONS
then
the
system
(Fig. 3) is &stable for all F E
and all K E U X ~ ,
New criteria for the
of
containing a single
where fl < B', the difference (0' - p ) being a r b i h r i l y small.
time-varying gain k ( t ) in an otherwise time-invariant negative feedcOrollury: If in addition to the hypotheses Of t'he theorem,
back loop, are developed bJ, inlposing averaging constraints on the
E '+ and z ( t )
E R C t then the system gain, A unified approach for thederivation of such average variation
is L s t a b l e for all F E 5.w and K E u X ~ .
criteria forboth linear and nonlinear systems is delineat.ed. The
Since
E
=)
= KzK"9K 2 E xB.and K u E
stabilitycriteriapresented here permitavery
general c l a s of causal
make thesystemtransformations as shown in Fig. 4. It is now
operators in L2 to be used as multipliers and are more general than
sufficientt.0 prove that. K2'N-l is positive(e) and MGKz is st.rongly the existing results. However, t,he results of the present findings
posit,ive(e) wit.h finit.e gain. The former follows from (4.9) and
can be improved by further investigation in developing similar
Lemma 2,
the latter folloTVsfrom (4.10), (4.11)7 and h n m a 3. average variation crite& which
the
of nonThe proof of the corollary mas' also be obtained from similar argucausal multipliers, analogous to the results of [4]for the instantaments.
neous bounds case.
'"
"
A Few Remarks
A) In comparison
criteria
the
with
of the earlier investigat,ors,
it is easy to realize that Theorem 1 is more general than t.he results
of ~~~~d~~~and Zames Ill a,nd Venkat,esh [21,while Theorem is
genera,l than Narendra and ~~~l~~ 131. It, should especially be
noted that the technique employed in [2] and [3] has been to me
~
~ extension of
~ the L
d ~met,hod,~~which requires
,~
the multiplier ;Jf(s)to be a rational funct,ion of L ' s ) J and the sJ7st,em
t o be described by a differential equation. Hence the present results
aTe improve men^ upon
a,nd [31in the follon,ing specis: 1)
general multipliers employed, 2 ) a broader c l a s of systems considered, 3 ) less restrictive averaging conditions on t.he gain, 4)a stronger
stabilit,y concept
and 3 ) elegance of the met,hod of deriva-
REFERENCES
[I] M .
I. Freedmanand
if?;;;,
;;;;,stems
G . Zames."Logarithmicvariation
criteria for the
withtime-varying s i n s . " SIA.II J Contr., vel. 5. PP.
[z] T . V.Venkatesh."Glohalvariation
criteria for the stahi1it.y of linear t i m e
varying syst.ems." S I A M J . Contr;: vol. 9. pp. 431-440. 19i1.
[B] J. H. Taylorand K. S. Narendra, The Corduneanu-Popovapproach t.0 the
~ stability
~
~of nonlinear
~ ~timevarying
~~systems,'' ~S I A~M J . A p p l . Math..
~ vel. 18,
pp. 267-281, 1970.
141 &I. K . Sundareshanand M . .\. I,. Thathachar. "L?stal>ilit.y of linear timevarying
systems-Conditions
involving
noncausal
multipliers,"
IEEE
Trans.Automat. C o d r . . vol. AC-17,pp. 5 0 S 2 1 0 , hua. 1972.
[ti] 8. Ramarajan
and
M . .I\. L. Thathachar.
L1-stal)ilits
of
time-=w'ing
sysrems with glohal enndidone on the rime-varying gain." I n l . J . SWt. S e i . ,
[sl
~ ~ ~ ~ ~ ~ ~ ~ ~of time-varying
~ ~ l ~ nonlinear
~ ~ l ieedl,ack
~ s ~
g19';,3hz;
wstems-PartI:Conditionsderivedueineconceptsof
loop gain.conicity,
and
positivity,"
I E B E Trans. - 4 ~ O m Q l .COntr.. v.01. A G 1 1 . PP. 2 2 s 2 3 8 ,
,\pr.1966.
- "On the inpuc-output staldity oftime-varyingnonlinearieedhack
tion.
syste'ms-Part 11: Conditions involving circles in the frequency plane and
B) An additionalnovel feature of Theorem 2 is the use of
sector nonlinearities." ibid., vol. .\C-11, pp. 4 6 5 4 7 5 , July 1966.
OK&ea t,vpe mult,ipliers wit,h integral constraints on t,he kernel, for
[i] &x. I.Freedman,"Lrstahi1it.yoftimevaryingsystems-Constmction
of
multiplierswith prescrihed phasecharacteristics," SI.4.V J . Contr.. V O ~ . 6.
DD. 559-5i8 1968.
(81 K. $. Narenhra and J. 15. Taylor. "Lyapunov functions for nonlinear time1 I t shouldbenotedthatFreedman
[7] hasobtainedaslightlydifferent
varying systems:" Inform. Contr., vol. 12. pp. 378-393, 1968.
= [ 3 & 9 ] ~ - ~Hoaever,
).
[9]R.
P. O'Shea, h combinedfrequency-timedomainstability
criterion for
factorioat.ion. K = K z K uK~I 5 Xoand K&XB (where
6 alsofolloas from (4.7>, the proof heinsautonommlscontinuolls
systems." I R E 6 Trans. A u t o m i . Contr.. vol..\C-ll.
thefactorizationgiveninlemma
171. after defining Z(t) = -log L(1).
pp. 177184,July 1966.
similarthat
given
to in
r
a
i