IEEE TFiAXSACTIOSS O X AUTOJIATIC COhTROL,
VOL. AC-15, KO.
2,
APEIL
1970
195
Noncausal Multipliers forNonlinear
System Stability
YEDATORE V. VENKATESH
Abstract-Using
the Popov approach, new absolute stability
conditions in multiplier form are derived for a single-loop system
with a time-invariant stable linear element G in the forward path
) the feedback path.
and a nonlinear time-varying gain k ( f ) @ ( -in
Theclasses of nonlinearitiesconsidered arethe monotonic, odd
monotonic, and power law. The stability multiplier contains causal
and noncausal terms; for absolute stability, the latter give rise to a
lower bound (which is believed to be new)on d k / d f and theformer,
as in earlier investigations, to an upper bound on d k / d f .Asymptotic
stability conditions
linear
a for
system
are
realized as a limiting
case of theabsolute stability conditions derivedfor the power
law nonlinearity.
-
ff(t)
G(s)
Linear time invariant
k(t)$(r)
stable
Nonlinear rnernoryless
I
Fig. 1. Nonlinear timevarying feedback system.
I. IKTRODUCTION
where A isan n. X 12. (constant,)stablematrix;
b,c,x
(state) are 77. x 1 vectors; u the output of the system is
ascalar. The time-varyinggain k ( t ) is assumed to be
cont,inuous a.nd bounded; for convenience, it is allowed to
belong to t.he infinite sector or 0 < E I k ( t ) < m . The
transfer function of the syst.em is G(s) = c’(s1 - A)-%.
+(a) is a memoryleas continuousnonlinearitysatisfying
in general t,he conditions
+(O)
=
0, u+(u)
> 0,
for all
# 0.
This class of functions is denoted by P, i.e., +(u) C P.
For simplicit,y in the proofs of the theorems it. is assumed
t.hat there exist positive constants 121 (however small) and
h? such t.ha.t~
hlu’ 5 u+(u) < h d .
If + (u) sat<isfiesadditional conditions, then it belongs to
other classes. For example,
1) +(a)
PL,l,the class of monot,onicallg nondecreasing
functions
c
[ + ( a ) - + ( u 2 ) ] (ul -
u2)
2. 0, for a.11ul and u2 (2)
Manuscript. received June 26, 1969.
The ant.hor is wit.h t.he Department of Electrical Engineering,
Indian 1natit.ut.eof Science, Bangalore 12, India.
for ul I 2. I u2 I and m
~
o
2. 1.
Or, equivalentJy, (2) and
where
(4)
As mo + x , c -+ 1 so that +(u) C P,>~O
and (4) reduces
t o (3) ; as m o --+ 1, c + 0 giving a linear feedback; c =
0.3536 corresponds to mo = 3. (See Thathachar et al. [lo].)
Assuming that the closed-loop system is asympt,otica.lly
stable for all constant linear feedback with k ( t ) = K C
(0,w ) , the problem is to find condit,ions to guarantee the
absolute stability of the null solution of (1) when k ( t ) C
( 0 , )~ and +(u) belongs to any one of the preceding
clmses.
1) A causal (or nonant,icipative) system is one whose
response to an impulse applied a.t t = 0 is nonzero for
196
IEEE TRANSACTIOXS O R A U T O U T I C COXTROL, APRIL
t 2 0 and zero for t < 0 ; the conlplex-frequency function
of such a syst,em is said to be causal.
It is to be noted t,hat passive systems (or systems nit.h
a positive real impedance funct,ion) are necessarily causal.
2 ) A noncallsal (or anticipative) system is one whose
response to an impulseapplied a t t = 0 is nonzero for
t 5 0 and zero for t > 0; the complex-frequency function
of such a syst.em is said to be noncausal.
The a.bsolute stability of the null solution (SS) of ( 5 )
implies the absolute stability of the iSS of (1). It is obvious t,hata knowledge of t,he behayior of u ( t ) enables
one to deduce the beha.vior of x ( t ) from (Sa); for example, if u ( t ) is bounded, so is x ( t ) , that is, all 6he components of x ( t ) are bounded; if u ( t ) tends to zero, so do
t,he component,s of x ( t ).
The method used by Popov [l] is to obtain an integral
inequality of the form
Historical Credits
It appearsfromasurvey
of literature that Popov’s
approach [l] has notbeen applied to thepresent problem.
Using Zames’ positive operatortheory
[2], Cho and
Sarendra [3], a,nd in a Lyapunov framework, Narendra
a,nd Taylor [4] derived abso1ut.e stability condit.ions in
terms of positive real (or causal) mukipliers and a local
bound on clh-/dt (n-hich depends upon 4 ( u ) and t.he multiplier employed). But they do not consider a nonca,usal
multiplier and hence theirresultsare
less general than
those presented in Section 111.
Major contents of this paper are thefollowing.
1) A new method is presented for establishing the nonnegativeness of the integral
[~ ( t > 4 ( u ) 1 Z u (dtt ) l
nith Z t.he operator representation of the multiplier chosen.
This met,hod easily accommodates noncausal operators in
contrast wit.h Zames’ [ 3 ] and Narendra a.nd Taylor’s [4]
approaches.
2) Theorems 3 and 4 arepresented (in Section 111),
which contain in part a. special noncausal mult.iplier. The
resulting additional lower bound on dk/dt appears to be
the first of its kind.
The paper isdivided into two main parts: t,he first part
introduces the Popov approa.ch and deals 1i-it.h causal
multipliers; Theorem 1, containing an R L R C multiplier,
is proved in det,ail to provide motivat.ion for the second
part, which deals with carnal and noncausal multipliers.
The proofs of the mainresults(Theorems 3 and 4) do
not differ very much from the proof of Theorem 1; therefore, only the necessary cha.ngm are indicated.
OF THE M A I X
11. SOLUTIOX
PROBLEM-CAUSAL
MULTIPLIER
4 1 UI)
+ I’b(l
x ( t ) = exp (At)x(O)
u(t) =
[
exp [ A ( t - T ) ] ~ ~ ( T ) @ ( u ( T dT
) )
c’exp (At)x(O)
-
c’exp [ A ( t - T ) ] ~ ~ ( T ) # ( u ( T d) T) .
0
I)
dt
< d(l Q(0) I)
(6)
where a ( T ) , b (F), and d ( r ) are continuousfunct,ions zero in
the origin, the first twofunctions being monotonically
increasing. From inequality (G) , the absolute stability of
the S S of (1) can be proved [l].
I n order to realize the integral inequalit). of the desired
form (6), a quadratic functional p ( T ) is considered in u
for which one seeks, firstly, a lon-er bound of the form
4 1 I)
+ (b(l
u
I)
dt
+ c h ( l u(0) I)
(7)
(where dl (I u(0) i ) is a. function of the initial condition a ( 0 ) ) and, secondly, an upperbound of taheform
&(I u(0) ;). The Fouriertransformationis used to obtainthis upperboundaftera
choice (following Popov
[ I ] ) of a convenient funct,ional p ( T ) .
Let g ( t ) = c‘ exp ( S t )b be the Laplace inverse of G(s),
and f ( t ) = c’ exp ( A t ) x( 0 ) . The asumpt.ion of a stable
matrix A implies the existence of constants 1.1, 1-0 > 0,
such that
I g ( t ) I 5 r1 esp
(--rot), for all t 2 0.
Also, for the same reason, t.here exists a, constant
such that
If(t) [ I
1‘9 exp (-rot),
t
r2
>0
L 0.
Consequently, t,he Fourier transforms G ( j w ) and F ( j w )
of g ( t ) and f ( t ) , respectively, exist.
Notation :
@(u)
=
[4(u)
du
> 0,
for u # 0
6, = @ . ( u ) / 4 ( u > u
dm,
=
max @ ( u ) / + ( u ) u
LT
A. Introduction to Popov Approach
The integral equation representation of t h e system is
1970
#.(.A
=
k:(t)4(o).
8 ~ (-t 2’) denotes a unit impulse function occurring a t
t = T . Z is the operator represent.ation of an impedance
function z ( s ) . The small Greek let.t,ers a, 8, y, p, Y , X, e,
I , 6, and are const,ants. The subscript i when applied to
different 1ett.ers need not have the same range of values,
for example,
ml
mi ma
C T i ; C C Ti’i=l
i=l
197
TiEXtCATESH: XOSCACSAL XULTIPLIERS FOR A-OXLISEAR SYSTFAI STABILITY
B. Xatlmnatical Preliminaries
so that
Lemma 1
The integral
l
T
Io
=
$ ( f f , t ) [ a f f (+t )P ( d f f / d t ) ] d t ,
a
is positive (except for aconstantterm
of the form
d l ( I a(0) 1 ) appearing in (7) ) if, for some €0 > 0,
I (a/,8)k(t) - ,e,
(dkicZt)G,,
ff1 = ff
> 0, p 2. 0
for all t.
(TI - a2 =
from which
(8)
uZ = - p i ( v i - 1)
Ppooj:
Io
=
[
ak(t)r$(a)(Tdt
+
and
, 8 k ( t ) 4 ( f f()d f f / d t )at.
dffiidt =
l
l
exp
[-pi(t
-
T)]c(T)
exp [ - p i ( t - T ) ] c ( T ) dr
-pia?
+ Pi(l -
Vi)ff(t)
T
I,
=
(12)
[ a k ( t ) - ,8sC(dk/dt)]4(ff)(Tdt
+ B k ! T ) @ ( f f ( T )-) B k ( O ) @ ( f l ( O ) )
n-hich is positive(exceptfor
(8) is sat.isfied.
Consider
mi
Z I ( ~ )=
C
i=l
+
Y~(S
viPi)/(s
+
cZr
or
Integrate the second term by parts t.o get
l
+
~ ( t ) ( v i - 1)pi
if
Q.E.D.
(11) the expression
Adding toandsubtractingfrom
+ ( a ~((TI
) - m) , one has
the last constant term)
T
I I=
~
k(t)[9(ad - 4(a2)](al - a?) dt
+ J k ( t ) 4 ( r r , ) ( Q - n ) dt.
pi),
(13)
0
y i , p i > 0: v i 2. 0, for all i.
Lemnn 2
For @ ( u ) C P-11: the integral
I1 =
(9)
1
The first integral of (13) is nonnegative in virtue of ( 2 ) ;
the second integral on using (12) becomes
[ k ( t ) 4 ( f f 2 ) (ff1 -
a?) clt
$(ff,t) [Zlff( t ) 1dt
is nonnegative (except for a constant
term of the form
(lo)
T
1
-
k ( t )~ ( c z[clgs/dt
)
+
~ipic~?
dt.
]
(14)
When k ( t ) is a constant,, int.egra.1(14) (and hence 11)is
nonnega,tive if 0 5 vi < 1 (for all i), t.hereby verifying
earlier results [j], [C]. However, when k ( t ) is time-varying, use Lemma 1 to get (10) for the n0nnegat.ivit.y of I,.
Q.E.D.
Consider
m2
22(s) =
c
i=l
Yi’(S
+
Yi’Pi’)/(S
yi’,p;’
+ Pi’),
> 0,
vi’
2 0, for all i. (15)
The proof of the folloming lemma. is based on (3) and is
similar to the proof of Lemma 2.
Lemma 3
For t$ (c) C PMO,the integral
$ (.,t) [ZZr ( t ) ] dt
198
IEEE TRANSACTIOXS ON AUTOJLITIC
COXTROL, APRIL
1970
Proof: Let
and
(dk/dt)6,, 5 min (2 - v ( ) p ( k : ( t ) .
(16)
0’
Corollary: For 4(a) C PMO,the integral
In =
[#
(a$) [ ( Z l
=
+ Zd ~ ( t1 )dt
UT
is nonnegative (except for a constant term of the form
d l ( / a(0) I) in (7)) if (10) a.nd (16) a.re simultaneously
satisfied.
The follon-ing two lemmas are extensions of the results
found in Popov [l]; the proofs are omitted here.
t>T
0,
05t5T
= ~ ( t ) ,
=
t>T
0,
Lemma 4
If I f ( t )
I 5 rpexp (-rot), nith r2, TO > 0, then there
exists a constant R1 independent of T such that
1[
+ + Zz)f(t) M(.,t)
+
{af(t> H d f l d t )
(21
< R1
-
SUP
I~
dt
( I. 0 (17)
OStST
k n m a 5 (Popov-Barbalat)
Let y ( t ) map [O,m) into the real line, and be differentiable. If y and d y / d t are bounded on[0, =), J ( y ( t ) ) = 0,
for y(t) = 0 continuous, a,nd J ( y ( t ) ) > 0, for y(t) # 0,
0 < eo 2 ~ ( t )for
, all t, and
+ 1-
+ z2)(aT - f T i1dt.
# T ~ ( ~ l
0
Let the Fourier transforms
of UT,+T be, respectivel.,
(20)
&,QT.
By virtue of thetrunca.tionandtheassumptionon
then
lim y ( t )
G,
Parseval’s theorem is applicable to (20) and on application gives
!
=
0.
f*oc
C . Absolu.te Stability
Basedon
the preceding preliminaries, theabsolute
st.ability conditions for ( 5 ) are derivedassuming that
4 ( u ) C P , ~ oThe
. stability multiplier, being then an RCRL impedance, contains as special cases the multipliers
for $(u) C P and 4 (u) C P u . Therefore, when 4 ( u ) C P
or P,+f,Theorem 1 holds after casting out the inadmissible
terms from the multiplier. The functions z1(s) and zz(s)
are as defined in (9) and (15), respectively.
0 according t.0 hypothesis a )
Since I QT lz is real, p l ( T ) I
of Theorem 1, implying thereby that
[
[CYu(t)
+ B(da/’dt)]#(u:t)clt
Theorem 1
Thesystem governed by(5)
is absolutely stablefor
4(u) C PHOif there exist constant CY$ > 0; ~ ~ , - y i ’ , p ~2, p0;
~’
0 I vi < 1, 1 < vi’ 5 2, for all i;and a mult,iplier
Consider the left-hand side of the inequality ( 2 2 ) : from
+ Ps + z1(s) +
z(s) = a
Lemma. 1, sat.isfaction of ( 8 ) gua.rant,ees the positiveness
of the first integral. Let
zs(s)
such t,hat
a) Re 2 ( j w ) G ( j w ) 2 0, for all real
eo > 0, as small as desired,
b) ( d k / d t ,),,6
5 min
( ( ~ / B , v i p (i ,2
w,
and, for some
- v i ‘ ) p i ’ )12 ( t ) - €0.
i
(18)
&(t)
- B(cZk/dt)6,,
=
e l ( t ) 2 EO
> 0,
for all t .
The second int,egral of (X!)is nonnegat.ire (from Corollary of Lemma 3) if (10) and(16) a,re simultaneously
satisfied. AS for t,he right-hand side of ( 2 ) , Lemma 4
gives its upper bound in the inequality (17).
199
VEWKATESH : KONCAUSAL XULTIPLIERS FOR NONLlhTAR SYSTEM STABlLlTP
Based on t,he preceding results, t.he following theorem
generalizes Theorem 1. Its proof, being analogous to the
proof of Theorem 1, is omitted.
Consequently, (22) becomes
[el(f)+(u)ucZt
+Bk(T)@(u(T))
+ J ” J . ( U , t ) [ ( Z+1 z d U ( t > l clt
0
+
<
- ~ k ( O ) @ ( u ( O ) ) R1 SUP
I~
( t I.)
(23)
0 5 t<T
By assumption,there exists apositiveconstant hl such
t,hat, +(a) u 2 hlu2, from which @ (u) 2 hlu2/’3. Let 81 =
fill& ( T )1 2 ; R, = Pk ( 0 ) ( u (0) ) . Then t,he crucial inequalit,y (23) takes the form
+
[ a ( t ) + ( o ) r d t + 81u2(T)
c
SUP ; u ( t )
I
(24)
0 5 t<T
where constants Ro,R1are independent of T . Boundedness
of u ( t ) and asymptotic stabilit): follow from (24) asin
Popov’s proof [I].
Q.E.D.
Corollary 1: 4 ( U ) C P , z1 (s) and z2(s) are inadmissible.
Hypothesis b) reads
5 ( a / P ) k ( t ) - En.
S0t.e that,for a general +(u) C E‘: 0 < 6,,
5 X.
Corollwy 2 : $ ( u ) C P,,I,z2(s) is inadmissible. Hyp0t.hesis b) accordingly rea.ds
(dk/dt)6m,,
< min
(a/’@,v+i)k( t ) -
C P J ~0, < 6max 5
Preliminaries: I n view of t.he inequality (2) characterizing + ( a ) Pprno,Lemma 2 holdshere,but.Lemma
3
needs modification, as it is based on (3) instead of (4).
The proof is sinlilar to that of Lemma 3.
c
Lemma G
C Ppmo,the integral
13
=
[
$ ( u , t )[ & a ( t )
is nonnegative (except for
cll(l u ( 0 ) I) in (7)) if
1 < vi’
( d k l d,t, ),6
5
5
+ ljc?),
mill
(1
+ e!
d k j d t 2 2 min ( a j f l , v i p i , p i ‘ ) k ( t ) - €0.
i
Rema,rk: It is to be observed tha.t t,he multiplier of
Corolla,ry 2 is not a general positive real function (which
is in fact expected in view of Gruber and Willems’ result
[SI). The asymptotic stability conditions for the linear
system obtained by a. limit process are consequently not
the best possible.
A. Prelimimries
1) Consider the impulse response funct,ion
m3
-
=
S,
ti exp ( { i t ) ,
Xi =
zla(s) =
pi’k ( t ) .
{i
C Pprn0,the integral
$ ( u , t ) L ( z l + z Z ) u ( t ) l rlt
isnonnegative(except for a const,a.nt term of the form
cll(l u ( 0 j I) in (7)) if (10) and (25) Rre simultaneously
obeyed.
-
0;
fi,
C vi({i0i
C vi[l
i
(25)
ti,{;
> 0,
for all i
= Xi/{i
- s)/(Ti
- 8)
i
=
Vi‘?)
5 0,
0,
t>o
which describes a nonca,usal system. The Fourier trans- f i / ({i - ju).
form exists and is equal to
2) Let
2 0, for all i,
for all i
-
t
i=l
1dt
T
113
+ c2 - v i ’ c 2 ) p i ’ )
Corollary 1: If +(a) C P,w, replace c by 1 toget
Theorem 1.
Corollary 2 : If +(u) = u, c = 0, and 6,,
= 1/2, Theorem 2 then gives conditions for the asymptotic stability
of the linear system. I n this case, vi’ > 1, for all i, and
hypothesis b) reads
Define
a
Corollarg : When 4 (U )
and, for some
i
a constant term of the form
(1
w,
111. SOLUTION
OF THE M A I N PROBLEM-CAUSAL
AND
S o ~ c a n s MULTIPLIERS
a~
1.
D. P o w w L a w Nonlinearity (Class PpmO)
For +(u)
a) Re z ( j w ) G ( j w ) 2 0, for all real
t o > 0, as small as desired,
b) (clk/dt)d,,, _< min (ai/& v L p i , ( 1
€0.
i
Note t,hat, for +(u)
+
+
+ +
. k ( t ) - EO.
II,(.,t)[(Zl+ Z d a ( i ) l d t
5 Ro + R1
(dkjdf)6,,,
Theorem 2
The systemgoverned by (5) is absolutely stablefor
4 ( u ) C P p , if there exist const.ants a , 6 > 0; yi, yi’, p f ?
pi’ 2 0 ; 0 5 vi < 1, 1 < vi‘ 5 (1
l/2), for a.11 i, and
a multiplier z (s) = a 8s z1( s ) z2 (s) such t.hat
Ip
=
-
ti/(Ti
-
~)l-
(‘-26)
200
IEEE TRANSACTIOSS O N AUTOXATIC CO?JTROL, APRIL
Let
1950
define
1)
- ti/
)
exp [{i(t - T ) ] u ( T ) dr dt
t
so t,hat Ila =
When +(u)
a.nd
x i qJlia.
(28)
C P:tI, (2) is satisfied. I n (88) ,let u1 = u ( t ) ,
=
[
[cyk(t) - P(dk/dt)6,]4(u)udt
+ B k ( T ) @ ( u ( T )-) ,8k(O)@(u(O))+
Let it. be assumed that. (for some E,,
lla.
(36)
> 0)
from which
and
+ [ ' k ( t ) + ( u J (ul -
u2)
dt.
(31)
0
If +(u) C Pw, the first integral of (31) is nonnegat.ive in
virt,ue of ( 2 ) . The second integral of (31), on using (30)
a.nd integrating by parts, gives
.-T
The first term of (38) is positive if fi > Ciq;/ti.As regards the second term of (38), let T be so chosen t-hat
u ( t ) a.ttains its extremum value uext a t t = T (i.e., &her
its posit,ive ma.ximum urnas+ or its negativemaximum
umns-). R.ecalling that
uz(T)
=
t;
11)
exp [ l i (-~T)]u(T) c1r
one has
Since 0 < 6,,
5 1 for +(u) C P,u and k ( t ) C (O,.o),
the right-hand side integral of (32) is nonnegative if
2 -ri&k(t),
(dk/dt)6,,,
for all t.
(33)
Consequently, if +(u) C P.u and (33) is satisfied, Iliais
nonnegative except for t.he t.erm - k ( T )@(u2(?"))/& in
(32). This in turn implies that theinequality
(clk/clt) 6,
2
-
min [i&k ( t ), forall t
(34)
be
noted
that. u?(T) 2
implies a?(T ) >
urnax-if 7 < 1, and j a?(T ) I 5 qu,,,,+ implies I u?(T)
I<
urnax+
if < 1. Then, for 0 < /Ii < 1 or
> ti. one concludes from (39) and (40) t.hat u2(T ) > urnas-when uz(T)
is negative, and u2 (T)< ulnnS+when u?( T ) is posit.ive.
Consequently, the integral
It isto
I
JU"'
guarant.ees the nonnegativity of Ilagiven in (27), save
for the tern1
C - ~ & ( T ) @ ( ~ ~ ( T ) ) / t i - (35)
+(a) flu
m(T)
is nonnegative by virt,ue of the fact that @(u) C P . The
folloning is a summary of the preceding findings.
i
3) It is now int,ended to make Ila nonnegative by adding a posit'ive term to (35). To this end! add CY PS
(vit.h ~ r , p> 0) to ZI"(S). Let
+
Zd(S) =
CY
+ Ps +
ZIU(S)
Result I
Theintegral (36) (,\-ith
> 0 , and T so chosen that
u attains ita extremum value a t t = T ) is positive (except
for
a
conshnt
term of the form dl (I u(0) 1) in (7) ) if
201
% E S K i T E B H :NOSCAUSAL MCLTIPLIERS FOR X O N L I S E A R SYSTEM STABILITY
> 0)
(for some eo
-
Lemma 7
If I f(t) j 5 r2 esp (-rot) with 1‘2, 1’0 > 0, then there
exists a const.ant R1’ independent of T such that
I ( a / p ) k ( t ) - eo
nlinj-if?ik(t)5 (cZk/clt)6,,,,,
i
p>
vi/li(l
e i ) , 0 < ei < 1, for all i.
-
+ Z ~ u ) J ( t ) l dI
t ! R1‘ sup I
I[k(t)4(u)I(-&‘
i
OgtST
u(t)l.
4) Consider the noncausal impulse response function
ml
(43)
ti’ exp (ti?), t 5 0, ti‘,li’ > 0, for all i
Proof: The proof is similar to t.hat.found in Popov 111.
i=l
0,
t > 0.
The Fourier transformexists and is given by
B. X a i n Results
C ti’/ (-ti’ - j u ) .
z
Let vi‘ 2 0, for all i; xi‘
=
li’+ ti’,Bit
vi/(ejyit- s ) / ( l i ’
z.?=( s ) =
and
=
- S)
.L
+
- s)].
(41)
The proof of the following result is similar
Result 1 and is hence omitted.
to that of
=
Vi’[l
&’/({i’
Having set,tled t,he preliminaries, a major result (Theorem 3) of t,his section can be sta,ted. This t,heorem, believed to be new, shows that the price to be paid for the
introduction of a. noncausal functionint,o t.he sta,bility
multiplier is a lower bound on dk/dt. Theorem 3 includes
Theorem 1 as a special case. When 4(u) C P.11, the theorem still holds after casting out inadmissible terms from
t,he mult,iplier.
a’
Result 2
When +(u) C P , ~ o the
,
integral (with O C ,>~ 0, and T
so chosen that. u at.tains its mkximum value urnaxa t t = T )
I0.a
[s(t)4(u)[Qg
=
+ B(clu/dt) + Z,%(t)] clt
is posit-ive (exceptforaconst.ant
d l ( ] u ( 0 ) i ) in (7)) if (for some eo
-
term of theform
> 0)
nlin ( 2 - e j ‘ ) l i f k ( t )5 (dk/dt)6,,,,
5 ~ k ( t ) /p eo
i
>
,8
vi’/li’(&‘
- I),
1 < ei’
< 8,
for all i.
i
A combination of Lemmas 1-3 and Results 1 and 8 yields
the following.
Result 3
When + ( u ) C P~uo,the int.egra1 (mit.h a$ > 0, and T
so chosen tha.t u ( t ) att,a.insi6s maximum value a t t = T )
1
T
lo=
-
k(t)C(O)
[QU
+ B (du/clt) +
(21
+ + ZP + ZP)u ( t )] czt
2 2
is positive(except
for a constant tern1 of the form
d l ( ] u(0) I) in ( 7 ) ) if (forsomeeo > 0 )
-
min (li&,(2 - e,’){,‘)k(t)
a,) Re x ( j u ) G ( ju) 1 0, for all real a, and for some
€0 > 0, as small as desired,
b)
nlin ( l i e i , ( 2 - Oi’)li’)k(t)
i
5 ( d L / d t )6,,
5 nlin
Proof: It is similar t.0 the proof of Theorem 1. The
point,s of departure are 1) the negative terns due to 21a
a.nd Z 2 a are to be dominated by the positive contribution
of ps (seepreliminaryresults) ; 2) Lemma 7 is t o be
taken into account.
;Is in the proof of Theorem 1, hypothesis a ) leads t o
the inequalit,y
/da(t)4(u)[a.
i
( a / p , v i p j , ( 2 - v i ’ ) p i ’ ) k ( t ) - eo.
i
+ B(du/dt) + (21+ 22 + Zla + Z,Q)CT(t)]clt
I (dklclt)6,ax
5 min
( a / B , v ; p i 7 ( 2 - vi’)pi’)k(t) - eo
(42)
1
B
> C [ q i / ( l - ei)lil + C [ q i ’ / ( e i ’
1
+ (2, + + + Z ? a ) f ( t )czt]
- 1)~i’],
2 2
21=
1
I
R1
0 2 vi < 1, 1 < Vi’ 2 2,
0
< Bi < 1,
1 < Bi’
< 2,
for all i.
SUP
I .(t) I
o< t$T
the constant R1 being independent of T .
202
IEEE TRAXSACTIONS ON AUTONATIC CONTROL, APRIL
is positive(except
for aconstantterm
dl(l u(O) I) in (’7)) if (for some €0 > 0 )
Concerning Result 2, let
u , ( ~=
)
Lrn
19iO
of the form
- min (1 + e? - c?Oi‘){i’k(t) 5 (dk;dt)6,nxx
exp [{{(T - r > ] u ( . > c~r.
i
For t.he definition of up(T ) , see (29). Hypothesis b) on
a.pplication to the left-hand side of (44) gives, as in the
proof of Theorem 1, for an E.? ( t ) 2 eo > 0, for all t, and a
const.ant Ro independent of T ,
[
c.(t)+(u)o dt
+ L - ( T ){ P @ ( ~ ( T ) -) c v i a ( u p ( T ) ) / t i
i
-
c
>
fl
A combination of the Corollaryto Lenlnla Gt Result. 1,
and Result 4 gives the following.
+
When ( u ) C Pprnoand G # 0, t,he integral (xvith a,fi > 0
and T chosen so as to allo\v u ( f ) to reach it,s maximum
value at. t = T )
vi’@(Q(n)/&’l
+ (nonnegat,ive terms due to Z1,Z?,Zla,Z?a)
I,a
Since the inequality (45) has been stated forevery
positive T , it holds in particular for T so chosen as to
permit u ( f )t.o attain its maximum value a.t t = T . Therefore, u ( f )in @ ( u (T ) ) of the left-hand side of (45) can be
replaced by umnr without affecting the inequality.This
enables one t.ouse Result 3, which guarantees that the
expression inside the braces of (45) is positive and greater
than or equal to Bok ( T )@ ( u ( T )) for some P o > 0. Consequent.ly,
e2(t)+(u)u
r/t
vi‘lc?(Oi’I
Result 5
i
[
+ Pak(T)@(u(T))+ (nonnegative t e r m )
=
Ii(t)+(U)
.[au
+ B(da/dt) + (Z, + 2, + z1a + Z.”)U(t)] dt
is positive (except for aconstant,
cll (I u ( 0 ) I) in (’7) ) if (for some eo
- nlin (Oiri,(l+ c?
I Ro + E1 SUP I ~
I
( t )
5 nlin
C. Potter Law Norzli.nearity
The present, aim is to include zIa(s) and zza (s) in the
multiplier of Theorem 2. By doing so, Theorem 3 is
generalized.
Because the property ( 2 ) of +( u ) is conmon for both
+ ( U ) C Ppq)l0a.nd +( U ) C P M ,Result 1 concerning zla ( s )
holdsfor + ( u ) C Ppmo. As regards z j a ( s ) ,an analysis
similar to the one leading to Result 2 gives the following.
+ e? - C2ui’)pi’)k(t) -
(a/P,vipj,(l
€0
i
B
>
vJ(1 - 8i)j-i
+ C vi’/c2(Oi‘- l)fi’,
d
I vj < 1,
0 < Oi < 1,
0
(46)
c
- c?Oi’)ri’)k(f)
I (dklrlt)6n,,r
O<tjT
from which stability and asymptotic stability
of u ( t ) ensue
as in Popov’s proof [I].
Corollary: If ( u ) P M ,the terms x2 (s) and zga (s) are
inadmissible in t,he multiplier. Hypothesis b) accordingly
reads
t.erm of the form
> 0)
1
2
+
I ( d b ) k ( t ) - €0
1 < Bit < 1 + I/?, for all i.
+ 1;s;
1 < ei’ < 1 + I//?,
1 < Vi’
5
1
for all i.
A limiting case of R.esult 5 is t,he following.
Result 6
When + ( u ) = u, t.he int.egrnl (with a$ > 0 and T
chosen so as t o allon- ~ ( t attain
)
i h maximum value at
t = T)
IL
=
[k(t)C(U)
8
[a.
+ P(dU/dt) + (2, + + Z1= + Z,.)
2 2
1 dt
u(t)
is positive(except
for a. constant t,erm of the form
d,(J u(O) I) in (7)) if (for some €0 > 0)
- 2 nlin ( e i [ i , ( i r ) k ( t5
) dk/dt
I
5 1 min
( a / B , V i p i . p i ’ ) k ( f ) - €0
i
Resulf 4
When + ( u ) C Ppnloand e # 0, the integral (witha$ > 0
and T chosen so as to make ~ ( t attain
)
its maximum a t
f = T)
Iopa
=
[k(t)+(.)Cau
+ P(dU/dt) + ( Z , % ( t ) ) ] clt
P
> C vi/(l - &){i
0 5 v i < 1,
+C
qi’(Oi’
- I ) / <t. ’?
i
z
vi‘
> 1,
0
< ei < 1, eif > 1;
for all i.
Based on the preceding results, the following theorem
generalizes Theorem 3. Its proof resembles the proof of
Theorem 3 and is hence not given.
Theorem 4
The system governed by (5) is a.bsolutely stablefor
4 ( u ) C Ppmo, c # 0, if there exist constants a,B > 0;
qi,7]i’,yi,yif:pi,plil2 0 ; 0 5 vi < 1, 1 < V i f 5 (1
1/c2),
0 < ei < 1, 1 < eir < (1 1/c2), for all i,
+
+
p
>
0i/(1 -
Bi)Ti
+ C [oi’/s(eif
- 1)ri’l
4
i
E. Examples
a.nd a multiplier
z(s) =
an inconsequential bound on dk/dt ; na.mely, li ( t ) is either
a constant or a. nonincreasing funct,ion of time. However,
if the time-var>+ng gain k ( t ) is periodic, one can arrive
a t useful conditions for absolute stability of the system
interms of a special multiplier;a det.ailed correlationboundanalysis is not essent.ia1. Thiscont.ribution uill
appear elsex-here.
01
+ ps + z1(s) +
Zi(S)
+
z1a(s)
+
1) Let
G(S) =
Z.lQ(S)
such t,hat
a) Re z ( j w ) G ( jw) 2. 0, for all real w, and, for some
eo > 0, as small as desired,
b) - nlin ( o i l i , (1 c2 - 20i’){i’)k ( t )
+
< min
(a/p,vjpj,(l
+ c? - r % i ’ ) p i ’ ) k ( t ) -
EO.
2
+
Corollary 1 : When c = 1 (or (cr)C P;\fo), Theorem 3
is obtained.
CoroZZa,ry 2: (Based on Result 6) When ( u ) = u, the
linear system governed by (5) is asymptot,ically stable if
there exist constar1t.s a$ > 0; vi,vi‘,yi,yi’,pi,p;’ 2. 0; 0 5
v i < 1, vi’ > 1, 0 < Bi < 1, ei‘ > 1, for all i,
+
B > C [vi/(1
- e;){i]
+ Xi C v i W
-
1)/t4’1
I
and a mult.iplier
z(sj
+ ps +
=
+
~ ~ ( 8 )z 2 ( s )
+ zlQ(s)+ ~.P(s)
such that
b) -2 n i n (6’i{iJi’)k(t)
1. d k / d f .
I
< 2 n i n (01/,8,vipz?pi’)k(t) - E O .
-
1
D.1Zenza1.X-s
1) The present, method of introducing a noncausal
functioninto t.he sta.bility nlult.iplier is different from
O’Shea’s [7] correlation method developed explicitly for
a time-invariant system. (See Zames and Falb [9] for a
rigorous treatment, of the correlation met,hod).
2 ) The multiplier introduced b>-O’Shea [7] is very
general, but the accompanying time domain rest,riction on
t,he multiplier is not always easy to verify. The stabilit,y
multiplier of the paper contains an RC-RL reflected impedance, and there is no explicit, time domain rest,rict.ion
on it.. It. is possible to est,end t,he class of multipliers (for
+ ( u ) C P.lloJ’pmO) by considering biquadratic funct.ions.
3 ) It has been found that the correlation technique? on
extension to the time-varying feedback problem, leads to
+(GI
0 3 ,
cp J f .
c
x(s)
=
=
+ s + [l - 2/(6 - s)]
2 + s + (4 - s ) / ( 6 - s ) .
2
E = 3-, { = 6, Q! = 2, p = 1. Note t,hat { > E , @ > lit.
Also 0 < 6,,
5 1. Corollary 1 of Theorem 3 is applicable.
It can be verified tha.t Ree ( j w ) G ( j w ) 2 0: for all real w .
The system is, therefore, absolutely stable for + ( u ) C Px
if, for some > 0, as small as desired
--4k(t) 5 (dk/dt)6,,
When the system is linear, 6,,
bound on dk/clt becomes
1. 2 k ( t ) - €0.
=
l/2 and the preceding
-8k ( t ) 5 dkjdt 5 4k (1)
-
EO.
Observe that for the 1inea.r system, using positive real
multipliers alone, it is unlikely that a result better t,han
dk/dt 5 2 k ( t ) - eo (for asymptotic stability) can be established.
2) Let
G(s) = s3/(s5
a) Re z ( j w ) G ( j w ) 2. 0 for all real w , and, for some
> 0, as small as desired,
+
With constant. linear feedback, the system is asymptoti[ O , a ). Choose
cally stable for all gains
i
1. (dklclt)6,,,,
S/(S
+ 5s4 + 4s3 + 3s’ + 2s + 5 ) ,
+(u)
C P.wo.
Wit,h linear constant feedback, the system is asympt0t.ically stable for all gains C [O,= ) . Choose
Z(S)
=
2s
+ 4 + [l + 1/(6 - s ) ] .
Here E’ = 1, 1’ = 6, = 4,,B = 2, {’ - f’ > 0; ,8> l i t ’ ,
0 < 6,,
5 1. Theorem 3 is applica,ble. It is easy to verify
that Re e ( j w ) G ( ju) 2 0, for all real w.
Therefore, the nonlinear time-va.rying feedback system
is absolutely stable for 4 ( u ) C P.WOif, for some eo > 0; as
small as desired,
-5k ( t )
I ( d k / d t )6,,
When the system is linear, 6,
straint on dkjdt becomes
- 1Ok ( t ) 1. dk/dt
5 2k ( t ) - eo.
=
I/?. The preceding con-
5 4k (1)
-
€0.
Observe that in the linear time-varying ca.se, because of
the third-order zero in G(s), it is not possible t.o find a
positive real z ( s ) such that e (s)G (s) is st.rictly positive
real. Consequently Gruber and Willems’ [SI criterion for
the linear time-varying system is not applicable here.
204
APRIL
IEEE TRkNSACTIOXS
CONTROL,
ON AUTOMATIC
IV. COSCLUSIOW
S e n - absolute st.ability criteria are derived for a nonlinear time-varying feedback system illustrated in Fig. 1.
The classes of the nonlinearities considered are PAW,P.,Io,
and Pp7l‘o (nit,h n10 2 1). The criteria are expressed in
t,erms of a multipliercontaining causal and noncausal
functions. X significant out,come of the presence of noncausal functions in the mult,iplier is that absolute st,ability
oan be established in c a m where a. purely causal mu1t.iplier is ineffective because thephme angle of G(s) is
outside the f 9 0 ” band in many intervals
along the j w
asis; in return, dR/dt is to be bounded from beloxv also.
Such a lower bound on dk/clt appears to be t.he first of
its kind. As in earlier investigations, the causal part of
&/dl.
the mult,iplier gives rise to an upperboundon
These bounds on dk/dt a.re dependent on the form of t.he
nonlinearity and the multiplier chosen.
The search for a z(s) to satisfyhypothesis a) of the
theorems is ra.ther cumbersome. It would be very useful
in practice t.0 have a direct method by which a. candidate
for z ( s ) is obtained directJy from t.he phase angle charmt.eristic of G (s) . This is an area deserving st,udy.
ACILUOWLEDGMENT
The aut,hor is grateful to Prof.H. X. Ramachandra Rao
for his interestandstimulation,
a.nd to Dr. A i . A . L.
Thathachar for useful discussions and advice. Thanks axe
est,endedto the reviewers whose suggestions have improved the value of this paper.
REFERENCES
[l] 1.. 31. Popov, “Absolate stability of nonlinear systems of
automatic cont.rol,” -4uto. andRemoteConirol,
pp. 857-875,
March 1962.
121 G. Zamw, ”On the stability
of nonlinear time varying feedback syetems,“ Proc. 196d XEC, vol. 20, pp. 725-730.
1970
-,
“Nonlinear time vanringfeedback systems-conditions
derived using conic operators on esponentially weighted spaces,” Proc. 3rd Ann. Allerbn. Cot~f.
C.ircuifand Sysfem Theory, 1965, pp. 4
6
H
X
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[3] 1’.S. Cho and K. S. Narepdra, “Stability of nonlinear time
varying feedback systems, Bzctomatica., vol. 4, pp. 309-31i,
1968.
[-I] K. S. Sarendra and J. H. Taylor, “Ly~punovfunctions for
nonlinear time varying feedback systems, Inform. and Control:
vol.13, pp. 378-387, 1968.
[5] K. S. Narendra and C. P. Neuman,”Stabilit,y of a class of
differential equat.ions wjt.h single monotone nonlinearity,”
SIdlll J . Control, vol. 4, pp. 295-308, 1966.
161 R. W
. Brockettand J. L. Willems, “Frequency domain
stabi1it.y crit.eria, pt,. I,” IEEE T r a m . duto,natic Control,
vol. AC-10, pp. 255-261, July 1965.
“Frequencydomainstabilit.y
criteria, pt,. 11,” IEEE
Trans. Automatic Coonfrol, vol. AC-10, pp. M i 4 1 3 , December
1965.
[ i ]R. P. O’Shea, “An improved frequency time domain stabiky
criterion for aut,onomous cont.inuous systems,” IEEE Trans.
dulonmiic Conlrol, vol. -4C-12, pp. $25-731, December 1967.
[8] 31. Gruberand J. L.?Tillem,
“On a generalization of t.he
Circuit and
circle criterion,” PTOC.4th Ann. rillertonConf.
Systern Theory, 1966, pp. 827-8.28.
[9] G. Zames and P. L. Falb, “Stability conditions for systems
with monotone and slope-restricted nonlinearities,” SIdJl J .
Co-ntrol, vol. 6, no. 1, pp. 89-108, 1968.
[lo] 31. A. L. Thathachar, M. D. Srinat.h, and H. K. Hamapriyan,
“On a modified Lur’e problem,” I E E E Trans. dutowmtic
Control, vol. AC-12, pp. i31-i40, December 1%;.
for L,-boundedness
!
Yedatore V. Venkatesh was born
in hlysore, South India.He
was
cducated a t the rational 1nstit.ut.e
of Engineering, hlysore, and received the h1.E. arid Ph.D. degrees
in electrical engineering from the
Indian Irxtitute of Science, Bangalore, in 1965 and lY70, respect.ivcly.
He is presently with the Department. of Electrical Engineering
at the Indian Institute of Science.
His areas of interest. include optimal
control and automntn theory.