TECHNICAL AOTES AND
CORRESPONDENCE
139
I n order to apply the theorem to (13) let [note assumption (2)]
so t.hat, (13) becomes
1 4
I
Iml
+
lh(t,.)Jmdx(.)I
&.
(14)
Inequality (14) is of t,he same form as (3). Therefore, the theorem
can be used to get a bound on s(t). A more customary problem is to
find a sector for +(z,t) of the following form:
-m 5 +(x,t) 5 +a,
0
5
c = constant
Iv.
I +(x,t) I
c2z,
7),
CI
+ 1)
=
A s ( n ) - d$(u,n)
u ( n ) = 2b‘x(n)
pole shifting [2, pp. 382-3851 modifies the sector in (15) to
CIX
s(n
(15)
so that Ir(t)l does not exceed a given bound.
If in (1)
h(t,7) = h(t -
The stabilityof nonlinear time-invariant (NLTI) discrete systems
has been considered earlier by Szego [ l ] using the Lyapunov technique, and similar results were also independently obtained by Jury
and Lee [2] and by Tsypkin [a] via Popov’s met.hod. Chen [4] has
considered linear time-varying (LTV) discrete systems and
derived a
stabilit,y criterion. Inthis correspondence nonlineartime-varying
(NLTV) discrete systems with a separable time-varying nonlinearity
are considered, and a frequency domain stability crit,erion is derived
using Lyapunov’s met.hod. It is also shownt.hat Szego’s [ l ] and
Chen’s [4] criteria can be obtained as special cases from the criterion
derived here.
Consider the syst.em given by
where the time-varying non1inearit.y $(u,n)is assumed to be separable and can be m i t t e n as $(u,n) = k(n)+(u), A is a constant square
matrix, d and b are const.ant vectors, and the prime denote5 matrix
t.ranspose. The open-loop pulse t.ransfer function i7 given by
# c2.
CObTMENTS AND CONCLUSIONS
A simple comparison theoremhas been discussed and various
applications have been shorn. Wit.h little expenditure, the theorem
can be extended to a nonlinear integral equation of the following
type :
W ( Z )= 2b’(d - A ) - ’ d .
< k(n) I L < m
A k ( n - 1) = k ( n ) - k ( n 0
< k*(n) 5
Ak*(n - 1) = k*(n)
1
- k * ( n - 1).
(4)
Also let, us assume t,hat thenonlinea.rity +( u) sat.isfiest,he f o l l o ~ g
two conditions: namely,
or
O
2 1,
1).
Define k * ( n ) = k ( n ) / L so that (3) becomes
0
a
(2)
Let
Ft
assuming, for instance, that either
(1)
f ( t ) anarbitrary continuousfunction.
The scope of the theorem is adequate for the requirements of various stability investigations. Admittedly, if i t is desired to generalize
the theorem regardless of any application, one can proceed to a more
general comparison theorem as is suggestedby Nohel[5],for example.
< u+(u) I Ku2,
u # 0,
+(O)
=
0, K
<
(5)
and
Define Fmin,a measure [5] of the nonlinearity, as
REFEREACES
F. G. Tricomi Iniegral Equations. NeK York: Interscience, 1967, pp. 10-15.
J. C. Hsu anh A. U. Meier, Modern Conhol Principles and Applications.
New York: IvIcGraw-Hi11,,1968.
G . Sansone and R . Contg. .?Tonlinear DifferentialEquations. NewPork:
Macmillan, 1964, pp. 11. 12.
A. Halany, Diferenlial Equations. New Pork: Academic, 1966, pp. 7,
8.
J. A. Nohel. “Some problems in nonlinear integml
C q U R t i O U S , ” Bull. Amer.
Math. Soc., vol. 68, pp. 323-329, July 1962.
It can easily be seen that for linear cases F,i,
nonlinearities Fmin2 1.
Under condition (6) we have [11,[61
Absolute Stability of NLTV Discrete Systems
S. RAMARAJAN
AND
+ 9 [ u ( n + 1) - u(n)12
S. N. RAO
Absiracf-A frequency domainstabilitycriterionfornonlinear
time-varying (NLTV) discrete systems with a separable time-varying nonlineariwi s derived using the Lyapunov approach. It is shown
that Szegii’s [I] criterion for nonlinear time-invariant discrete systems and Chen’s [4] criterion for linear time-varying discrete systems can be obtaineda s special cases of the criterion derived here.
= 2 and for monotonic
where p is a real number. If B is restricted to be a positive real number, then ( 8 ) is true [3] for - m < d+(u)/du < p. On the other hand,
if p is rest.ricted t.o be a negative real number, then (8) holds for
-p < d4/du < m .
The system (1) can also be writ.ten as
+
~ ( n 1)
=
AZ(TL)
- ak*(n)+[u(n)]
(9)
.(n) = 26‘4n)
Manuscript received April 14, 1971; revised July 20, 1971.
S. Rarnaraian is with the Deuartment
of Electrical Engineering, Indian Instit.ute of ScienceBangalore12-India.
S. N . RROis wiih the Deprhment of Mechanical Engineering, Indian Institute
of Science, Bangalore 12, India.
(8)
where a
=
dL and thenew open-loop pulse t,ransfer functionbecomes
W * ( Z )= 2b’(zZ - A)-’u
=
LW(Z).
(10)
140
1972
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, FEBRUARY
The following theorem gives the stabilitycriterion for system (1).
Theorem: Consider the system (1)where A is stable and the linear
part is completely controllable and completely observable. Let the
linearized system be stable for dl h e a r gains in the range ( 0 , K L J .
Then
the equilibrium r(n) = s(n
1) = 0 is asymptotically absolutely stable if there exkt constants6, e, and X such that
+ lal~k*(n)u(n)+[u(n)l- Isl7k*"(n)u(nk[u(n)I.
+
+ Re W ( z )] + ;K- + Re {[e d= ( z - l ) ]W ( z ) }
- @ I(z
2
- l)W(Z)(*2
Y
0,
121
=
The first term in the right-hand side of the preceding equation is
nonpositive if
Ak*(n - 1 ) ~ g fln
1;
5 Fmi, Sk*(n)
Hence, using (5)we get
- IsI.r k*e(n)+e[u(n)]. (16)
.$ = a'b 7 p(a'b)2;
K
3)
&Ak(n
-
1)
5 6Fmb k ( n )
Using (13), (14),and (16) in (12), we obtain
where H is a symmetric positive-deKite matrix and0 is a real constant. The V difference of (11) along the solutions of system ( 9 )is
+
+
AV(n) = ~ ' ( n l ) H ~ ( n 1 ) - d(n)Hz(n)
+ ,911k*(n) l ( n + l ) + ( p )
c(n)+(~)
Substituting for u(n)and u(n f 1 ) and adding and subtracting
dp
- k*(n - 1)
[
dp]
a+[u(n)lk*(n) u b ) - +['(n)lk*(n)l
K
= s'(n)[A'HA - H ] z ( ~-) 2+[o(n)]k*(n)z'(n)A'Ha
+ +2[u(n)lk*2(n)afHa
dn
where a
have
+1)
l'(R)+(~)
+(p)dp
- k*(n -
1)
+(PI
+(PIdp I ~k*(n)+[dn)l
[dn
+ 1) - dn)l
PIS1
-r ['(n
-
(14)
By adding t.o and subtracting from the first term in the right-hand
(13),the
side of quantity
1bl6k*(n)u(n)6[U(n)]+ l a l T k * ' ( n ) U ( n ) + [ U ( n ) l ]
where 6
> 0 and 0 5
T/&
- Bc]
d ~ ]
2 1, we get(aftersimplication)are
- ( 2 p b ' ~sgn B
c
E
Pu(n+l)
+
- l&b
c(n)
Using (4) and (S), the second term in the right-hand side of (13) can
be mitten as
Bk*(n) Ju(,,
+
AV(n) I d ( n ) [ A ' H A - H
w']z(~)
- 2k*(n)+[u(n)]z'(n)[A'Ha- ab
The last term in (12) can be written as
[k%)
2 0, from the right-hand side of the preceding inequality we
= a'b
- 1)(A - 1)'b
- p(a'b)2 sgn p.
(18)
Now let us consider a slightly modiiied version of the lemma due to
Szego [ I ] , 171.
Lemma: Necessary and sufficient conditions for the existence of a
real scalar y, a real vector q, and areal matrix H = H' > 0 such that
A'HA
-H
+ W' + qq'
A'Ha - ab - Is18b
2p.$
= 0
- PC = yq
+ a/K + la17/K - a'Aa
= y2
>0
(19)
TECHNICAL
141
AND CORRESPONDENCE
(20b)
1
+ 6 + Re W(z) + Re { [B f ( z - l)]W(z)}
1)
Under the conditions of (20), (17) becomes
AV(n) 5
+ -/+[dn)lk*(n)l’.
- [p’z(n)
(21)
Following the arguments of Szego [ l ] , we can show that 4V(n) = 0
if and only if z ( n 1) = s(n)= 0 (Le., at. t.he equilibrium point) and
A V ( n ) < 0 for s(n) # 0. Also, since the right-hand side of (21) is independent of 8, by an argument similar to that of Szego [l],we see
that V is posit.ive definite, even when p is negative.
Thus t.he system (9) and, hence, the system (1) are asymptotically
stable if t.he conditions of (20) are satisfied for some CY 2 0 along
with (15). Substitu6ing for 4, c, and r in (20j, we get
+
2,
+ 6X
P +2t>o,
3)
=kAk(n - 1)
e
920,
5 26k(n)
05x51;
6>0,
[
1 - __
X k 3
One way of eliminating condit.ion 2 is to allow K -+ 0 and L -+ m
such t.hat.K L = I; (a constant) so t.hat condition 2 is obviouslysatisfied, since the first term becomes hrge as K
0 and E -+ (a’b), which
is finite. So the criterion reduces to (with k(n) = K k ( n ) so t,hat
0 < k(n) 5 k ) t.he following:
--f
(22b)
NONreplacing W*(z) by LW(z) and dividing throughoutby
and denohing CY/]^] = 9 2 0, (22) becomes
IpI
=kAk(n - 1)
2)
9
11
5 26k(.n)
2 0,
6
> 0,
0
5X5
1, 0
< k(n) 5
k.
(26b)
Chen’s criterion [4] for LTV cases can be obtained by putting
= 1, 6 = l/q, and using only the positive sign in (26).
e = 0, X
REFERERCES
2)
9
e
2 0,
[ I ] G. P. Szego. “On the absolutestability of sampled-datasyst.ems,” Proc.
.Vat. Acad. Sci., vol. 5 0 , ~ p . ~ p 5 8 - 5 6 01963:
,
[ a ] E. I . Juryand B. ‘X, Lee, On the stablhty of acertainclass of nonlinear
samnled-dat.a srstems.” I E E E Trans. A u t o m d . contr.. vol. AC-9.. nD. 51-61,
Jan.<1964.
[ 3 ] P a . Z. Tsypkin,“Absolute stabi1it.y of nonlinearautomaticsampled-det.a
systems,” Automat. Remote Conlr. (USSR), vol. 25, pp. 1030-1036, ,l964,.
[1] C. T. Chen, “On the stability of sampled-data feedback systems m t h timevarying pain,” I B E E Trans. Autamat. Conir. (Corresp.), vol. AG11,pp.
624425,July 1966.
151 K. 5. Narendra and J. H. Taylor, “Liapunov functions for nonlinear t i m e
varying systems,”Inform. Contr.. vol. 1: pp. 3i8-393, 1968.
of a
[61 J. B. Pearson. Jr. and J. E. Gibson, On theasympt.oticstability
class of saturating’sampled-data systems,” I E E E Trans. A p p l . Ind., vol. 83,
pp. 81-86. Mar. 1964.
[ i ]G. P. SzegS and R.,,E. Kalman.“A1xolut.estability
of a system of finit.0
difference equations, C.R. Acad. Sei., vol. 256, pp. 388-390, 1963.
+ -K + 2E sgn p > 0;
7
__
and correspondingly, (15) becomes
Ak(n - 1) sgn p 2 Fmh6k(n)
’-
Putting r/6 = X, (23) and ( 2 4 ) can be written as
1)
6
[I x
-
LK
+ Re W ( z ) ] + Ee
+ Re [e +
-
I(z
-
(2
-
1) sgn PlW(z))
1)W(Z)12
a
2 0,
2
e
IZI =
1;
Nonlinear
RIIN-YEN WU
6X
(25b)
3)
Stability Criteria for a Class of Multiplicative
Time-Varying
Systems
Ak(n - 1) ~ g pn 5 6Fmin R(n)
(254
which are
but
of the theorem \<<t,hsgn
eit,her positive or negative), and hence the proof.
Konlinear
Time-Invariant
= fl
(i.e.,
Case
Szego’s [l] criterion for the NLTI case can easily be obtained by
putting 6 = 0, L = 1, K = 12, and 8 = ( y @ ) / [ pin
i (25).
+
Linear
Time-Varying
Case
For the linear case, Fmin= 2. Taking + ( u ) = Ku, Ne see that
k(n),$(g)u = k(n)Ku2 5 KLu2 or
s k u 2 . Now we get the following condit,ions:
1.1 = K . Let KI, = k, so that
Absfracf-Sufficient conditions for the stabilityof a class of multiplicative time-varging a s well a s time-invariant nonlinearsystems
are presented.
The stabbilit.y problem of feedback systemswithmultiplicative
nonlinearities has been of current interest, and several results are
available in the literature [ l ] , [2]. R.ecently Zames’ passivity theorem
[3] has been applied
obtain
to
sufficient conditions
in
the frequency
domain for the stability of a class of multiplicative time-invariant
nonlinear systems [ l ] . In this correspondence we present sufficient
conditions for the st.ability for the salne system shown in [I, fig. I),
with
but
a broader class ofsubsystems
linear
and nonlinearities.
Manuscript received July21,1971’ revised September 7, 1971.
The author is \nth the Departmhnt of Electrical Engineering, Unirersit.y of
Colorado, Boulder, Colo. 80302.