A note on exponential stability of partially slowly time-varying
nonlinear systems
Joan Peuteman
Dirk Aeyels
SYSTeMS
Universiteit Gent
Technologiepark-Zwijnaarde, 9
9052 GENT (Zwijnaarde)
BELGIUM
e-mail: Joan.Peuteman@rug.ac.be
e-mail: Dirk.Aeyels@rug.ac.be
Preliminary version
Abstract
1 Introduction
The classical Liapunov approach to exponential stability of an equilibrium point of a dynamical system x_ = f (x; t) requires the existence of a positive
denite decrescent Liapunov function V (x; t) with
vminxT x V (x; t) vmax xT x (vmin; vmax 2 R+0 )
such that for some 2 R+0 : V_ (x; t) ? kxk2. In
general it may be hard to construct a Liapunov function which satises this condition. A weaker condition, still guaranteeing exponential stability of the
equilibrium point has been proved in [1].
The main theorem of [1] proves that the origin of a system is exponentially stable under
the condition that for a positive denite decrescent V (x; t) with vmin xT x V (x; t) vmax xT x
(vmin ; vmax 2 R+0 ), 9T > 0 and a sequence of times tk
such that V (x(tk+1 ; tk ; x(tk )); tk+1 ) ? V (x(tk ); tk ) ? kx(tk )k2 with 2 R+0 and tk+1 ? tk T 8k and
tk ! +1 as k ! +1 and tk ! ?1 as k ! ?1.
This theorem allows to establish, by means of a Liapunov approach, that the system x_ = f (x; t), with
suciently large, is exponentially stable when its
Consider a system x_ = f (x; t; t ) with a time-
varying vectoreld which contains a regular and
a slow time scale ( large). Assume there exists
( ) such that kx (t; t0 ; x0 )k K ( )kx0 ke( )(t?t0)
where x (t; t0 ; x0 ) is the solution of the system x_ =
f (x; t; ) with initial state x0 at t0 . We show that
for suciently large, x_ = f (x; t; t ) is exponentially stable when the average of ( ) is negative.
This result can be used to extend the circle criterion i.e. to obtain a sucient condition for exponential stability of a feedback interconnection of a
slowly time-varying linear system and a sector nonlinearity. An example is included which shows that
the technique can be used to obtain an exponential
stability result for a pendulum with a nonlinear partially slowly time-varying friction attaining positive
and negative values.
Keywords: Nonlinear dynamics, exponential stability, circle criterion, pendulum.
1
averaged system is exponentially stable. This averaging result allows to study exponential stability of
fast time-varying nonlinear systems. In [5], the averaging approach is generalized. This generalization,
which we call `partial averaging', allows to study exponential stability of a system x_ = f (x; t; t) with suciently large (see also [2], pp. 194-195).
In the present paper, the main theorem of [1] is used
to investigate exponential stability of slowly timevarying nonlinear systems. The investigation of exponential stability of slowly time-varying systems is
already discussed in [3, 6, 7]. Rugh [6] and Solo [7]
only deal with linear systems. Rugh [6] considers
only systems whose eigenvalues are pointwise in the
open left half-plane. Solo [7] generalizes this result
and considers linear systems with eigenvalues which
are allowed to `wander' into the right half-plane as
long as `on average' they are strictly in the left halfplane. Khalil ([3], pp. 239-250) discusses exponential stability of slowly time-varying nonlinear systems with the restriction that the `frozen' systems,
i.e. the time-invariant system where t is treated as a
`frozen' parameter, are all exponentially stable. An
extension of the results of Solo [7] and Khalil ([3],
pp. 239-250) is the study of a nonlinear slowly timevarying system x_ = f (x; t ) with suciently large
where the `frozen' systems are exponentially stable
or exponentially unstable.
The main theorem of the present paper does not only
allow to study exponential stability of slowly timevarying systems x_ = f (x; t ). The main theorem also
studies exponential stability of partially slowly timevarying systems x_ = f (x; t; t ).
Let ( ) and K ( ) be chosen such that 8t0
8x0 : kx (t; t0 ; x0 )k K ( )kx0 ke( )(t?t0) where
x (t; t0 ; x0 ) is the solution of the system x_ =
f (x; t; ) with initial state x0 at t0 . We prove that
for suciently large x_ = f (x; t; t ) is exponentially
stable if the average of ( ) is negative.
It is important that ( ) is allowed to take positive
and negative values.
This result can be used to extend the circle criterion
i.e. to obtain a sucient condition for exponential
stability of a feedback interconnection of a slowly
time-varying linear system and a sector nonlinearity.
The classical circle criterion only gives a sucient
condition for exponential stability of a feedback interconnection of a time-invariant linear system and
a sector nonlinearity.
An example is included which demonstrates that the
technique can be used to obtain an exponential stability result for a pendulum with a nonlinear partially slowly time-varying friction attaining positive
and negative values.
2 Exponential stability
We recall the main theorem for exponential stability
of [1] since the proof of Theorem 2 relies on it.
Consider
x_ (t) = f (x; t)
(1)
with f : W R ! Rn , W open, W Rn , f (x; t) is
measurable for each xed x, 0 2 W and f (0; t) = 0
8 t 2 R. Furthermore we assume that conditions are
imposed on (1) such that existence and uniqueness
of its solutions is secured. These conditions are
imposed on all the dierential equations mentioned
in the present paper and of these conditions, we
single out the local Lipschitz condition: f is locally
Lipschitz on W , i.e. for 8x 2 W , 9 a neighborhood
N (x) W , such that the restriction f jN (x) is
Lipschitz with Lipschitz function lx(t). We assume
that lx(t) is bounded 8 t 2 R.
Theorem 1. Consider a V : U R ! R, with U W an open neighborhood of 0. We assume that the
following conditions are satised:
Condition 1: V (x; t) is positive denite and decrescent (V (0; t) = 0 8t) and there exist positive numbers vmin and vmax such that 8x 2 U
vmin xT x V (x; t) vmax xT x.
Condition 2: There exists an increasing sequence of times tk (k 2 Z) with tk ! 1 as
k ! 1 and tk ! ?1 as k ! ?1, 9 nite
T > 0 : tk+1 ? tk T (8k 2 Z), 9 > 0 and
an open set U 0 U which contains the origin
2
( ) > 0 and 8r0 : 0 < r0 Kr , 8t0,
8t 2 [t0 ; t0 + 1 ln( Krr 0 )]: 8x0 2 B r0 (0):
such that the Liapunov function V (x(t); t) has
the property that 8 k 2 Z and 8 x(tk ) 2 U 0 nf0g
V (x(tk+1 ; tk ; x(tk ));tk+1 ) ? V (x(tk ); tk )
(2)
? kx(tk )k2 < 0
where x(tk+1 ; tk ; x(tk )) is the solution of (1) at
tk+1 with initial condition x(tk ) at tk .
then the equilibrium point x = 0 of (1) is exponen-
kx (t; t0 ; x0 )k K ( )kx0 ke( )(t?t0)
(4)
Here x (t; t0 ; x0 ) is the solution at t of the
system x_ = f (x; t; ) with initial condition
x0 at t0 .
{ K ( ) is bounded i.e. 9Kmax 1 such that
8 : K ( ) Kmax.
Condition 2: 9 Tm > 0, 9 TM > 0 and
a sequence of times k with the property that
8k 2 Z : Tm k+1 ? k TM . The sequence
k has the additional properties that
{ For some > 0 and 8k
tially stable.
Proof. For a proof, the reader is referred to [1].
Remark 1. When x_ = f (x; t) is globally Lipschitz
and the conditions of Theorem 1 hold globally then
the equilibrium point x = 0 is globally exponentially
stable.
3 Partially slowly time-varying
systems
Z k+1
k
( )d ?(k+1 ? k )
(5)
{ f (x; t; ), for each xed x 2 B r (0) and
each xed t, has nd (k) ? 1 discontinuities
k; 1 ; :::; k;n
d (k)?1 in ]k ; k+1 [ and 9nd 2 N
such that 8k: nd(k) nd .
In the present section, we will investigate exponential
stability of partially slowly time-varying nonlinear
systems i.e. systems of the form x_ = f (x; t; t ) with
suciently large.
Theorem 2. Consider the system x_ = f (x; t; t )
which satises the assumptions made in Section 2:
f is locally Lipschitz on W , i.e. for 8x 2 W , 9 a
neighborhood N (x) W , such that the restriction
f jN (x) is Lipschitz with Lipschitz function lx (t; t )
which is assumed to be bounded on R with a bound
independent of .
If 9r > 0 (B r (0) W ) such that
f (x; t; ) is allowed to have discontinuities
at k .
{ We require that 80 9r (0) such that j1 ?
2 j r (0 ) with the additional condition
that 9k 2 Z and 9i 2 f0; :::; nd(k) ? 1g such
; [ and 2 2 [ ; [
that 1 2 [k;i
k;i+1
k;i k;i+1
implies that 8x 2 B r (0), 8t: jf (x; t; 1 ) ?
f (x; t; 2 )j 0 kxk.
{ We require that 80 9(0) such that j1 ?
2 j (0 ) with the additional condition
Condition 1:
that 9k 2 Z and 9i 2 f0; :::; nd(k) ? 1g such
; [ and 2 2 [ ; [
that 1 2 [k;i
k;i+1
k;i k;i+1
{ 8 , 9K ( ) 1 and 9( ) such that either
implies that j(1 ) ? (2 )j 0 .
( ) 0 and 8t0 , 8t t0 : 8x0 2
then there exists 1 > 0 such that 8 1 the equiB r0 (0):
librium point x = 0 of x_ = f (x; t; t ) is exponentially
(
)(
t
?
t
)
0
kx (t; t0 ; x0 )k K ( )kx0 ke
stable.
(3)
Proof. The proof relies on Theorem 1 where
with r0 = Kr .
V (x; t) = kxk2. For suciently large, Condition
3
2 of Theorem 1 is satised with tk = k .
The entire proof of the theorem will be submitted
for publication elsewhere.
Remark 2. Partially slowly time-varying systems
x_ = f (x; t; t ) with suciently large are a generalization of slowly time varying systems x_ = f (x; t ).
Remark 3. When x_ = f (x; t; t ) is globally Lipschitz
and when the conditions of Theorem 2 hold globally
than the system is globally exponentially stable.
When there exists a positive denite symmetric P , a
n-dimensional L and a constant 0 2 R such that
PA + AT P = ?LT L + 0 P
p
PB = C T k ? 2LT
then
V_ (x; t) 0 xT Px
(14)
p
p
T
? (Lx ? 2k(x; t)Cx) (Lx ? 2k(x; t)Cx)
0 xT Px
(15)
4 Systems of the Lure type
q
max (P ) that
This implies that with K =
min (P )
0
kx(t; t0 ; x0 )k K kx0 ke(t?t0 ) 2 where x(t; t0 ; x0 ) denotes the solution of (9) at t with initial condition
x0 at t0 .
This result is a generalization of the classical circle
criterion in the sense that 0 is allowed to be positive or negative whereas the classical circle criterion
is only interested in negative 0 .
By the Kalman-Yakubovich-Popov lemma ([3], pp.
468-469), (12) and 0 (13) are satised if and only if
1 + kC (sI ? A + 2 I )?1 B is positive real. Positive
0
realness of 1 + kC (sI ? A + 2 I )?1 B is by denition
equivalent with
4.1 Lure systems with time-invariant
linear part
Consider a SISO system which is built up as a feedback interconnection of a linear time-invariant system, with (A; B ) controllable and (A; C ) observable
and a nonlinear time-varying element. When the system is unforced, the behavior is represented by
x_ (t) = Ax(t) + Bu(t)
(6)
y(t) = Cx(t)
(7)
u(t) = ?k(x(t); t)y(t)
(8)
where for some k > 0: 0 k(x; t) k 8x 2 Rn 8t
and u; y 2 R. Taking (6), (7) and (8) into consider-
0
Re(1 + kC ((j! + 2 )I ? A)?1 B ) 0 8! 2 R
ation one obtains that
x_ (t) = Ax(t) ? Bk(x(t); t)Cx(t)
(12)
(13)
(16)
(9)
Therefore, the ow of the system
(9) satises
0
(
t
?
t
)
0
kx(t; t0 ; x0 )k K kx0 ke 0 2 = K kx0 ke(t?t0)
when (16) is satised ( = 2 ).
Consider V (x; t) = V (x) = xT Px such that along
the ow of (9)
V_ (x; t) = xT (PA + AT P )x ? 2xT PBk(x; t)Cx
(10)
Notice that k(x; t)y(k(x; t)y ? ky) 0 such that
?2k(x; t)Cx(k(x; t)Cx?k Cx) 0. One obtains that
V_ (x; t) is less or equal than
xT (PA + AT P )x + 2xT (C T k ? PB )k(x; t)Cx
? 2xT C T k2 (x; t)Cx
(11)
4.2 Lure system with time-varying
linear part
Consider a SISO system which is built up as a feedback interconnection of a linear time-varying system,
with A(t), B (t) and C (t) time-variant, and a nonlinear time-varying element. We assume that A(t),
B (t) and C (t) are bounded, periodic and piecewise
continuous with each period T a nite number of
discontinuities. Here (A(t); B (t)) is controllable 8t
4
and (A(t); C (t)) is observable 8t. When the system
is unforced, the behavior is represented by
the discontinuities (if any) of A(t), B (t) and C (t)
then Condition 2 of Theorem 2 is satised with a
sequence k := kT (k 2 Z). Theorem 2 implies that
91 > 0 such that 8 1 , the partially slowly
time-varying system
x_ (t) = A(t)x(t) + B (t)u(t)
(17)
y(t) = C (t)x(t)
(18)
u(t) = ?k(x(t); t)y(t)
(19)
x_ (t) = A( t )x(t) ? B ( t )k(x(t); t)C ( t )x(t) (26)
where for some k > 0: 0 k(x; t) k 8x 2 Rn is exponentially stable.
8t and u; y 2 R. Taking (17), (18) and (19) into Remark 4. In Section 5, we show by means of an
consideration one obtains that
example how piecewise continuity of ( ) and the
x_ (t) = A(t)x(t) ? B (t)k(x(t); t)C (t)x(t) (20) existence of a Kmax may be established.
Consider now for every the system
x_ (t) = A( )x(t) ? B ( )k(x(t); t)C ( )x(t)
with 0 ( ) 2 R such that
0
1 + kC ( )((s + ( ) )I ? A( ))?1 B ( )
2
5 Pendulum with nonlinear
partially slowly time-varying
friction which attains positive and negative values
(21)
(22)
Consider the pendulum described by
is positive real.
x_ (t) = M (t; t ; x)x
(27)
Recall that by the Kalman-Yakubovich-Popov
lemma, positive realness of (22) is equivalent with with
the existence of an 0 ( ) 2 R, a symmetric positive
t
0
1
denite matrix P ( ) and a n-dimensional L( ) such
(28)
M (t; ; x) = ?1 ?f (t; t ; x)
that
t
t
P ( )A( ) + AT ( )P ( ) = ?LT ( )L( ) + 0 ( )P ( ) and f (t; ; x) = 1+b( 2)k(x; t). Here, for some k > 0,
k(x; t) k 8x 2 R 8t.
(23) 0Notice
that the system (27) can be written in the
p
P ( )B ( ) = C T ( )k ? 2LT ( )
form (26) with
(24)
A = A( t ) = ?01 ?11
(29)
By Section 4.1, this positive realness implies
that the ow of0 (21) satises kx (t; t0 ; x0 )k and B T ( t ) = [0 b( t )] and C = C ( t ) = [0 1].
K ( )kx0 ke(t?t0) 2( ) with
We consider the case where k = 1 and we assume
that b( ) is a continuous function with period T and
s
that 8 : b( ) 2 [?1:5; ?0:5]. Notice that 8 > 0,
(
P
(
))
max
(25) 8t: (A; B ( t )) is controllable and (A; C ( t )) is obK ( ) = (P ( ))
min
servable.
When there exists a Kmax 1 such that 8 : Kmax In order to apply the results of Section 5.2, we investigate the positive realness of (22) i.e.
K ( ) then Condition 1 Rof Theorem 2R is satised.
0
When 9 > 0 such that 0T ( )d = 0T 2( ) d ?
0 ( ) )I ? 0 1 ]?1 0
0
0
1
1
+
[(
s
+
b( )
?1 ?1
2
?T and ( ) = 2( ) is bounded and piecewise con(30)
tinuous with discontinuities which synchronize with
5
6 Acknowledgement
Recall that (30) is positive real when 8! 2 R its real
part is nonnegative. After some calculations, one
obtains that this is satised when
This paper presents research results of the Belgian
Programme on Interuniversity Poles of Attraction
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p
0 ( ) ?2 (b( ) + 2)(b( ) ? 3) + 2jb( )j 3 ? b( )
(b( ) ? 2)(b( ) + 6)
(31)
Since b( ) 2 [?1:5; ?0:5], one obtains by taking 0 ( )
equal
to the right hand side of inequality (31) that
0 ( ) 2 [?0:246; +0:262] and attains positive and
2
negative values.
This implies that for 0 ( ), (30) is positive real such
that x_ (t) = M (t; ; x)x has solutions satisfying
0
kx (t; t0 ; x0 )k K ( )kx0 ke(t?t0) 2( )
References
[1] D. Aeyels and J. Peuteman, On exponential
stability of non-linear time-varying dierential
equations, submitted for publication.
[2] J.K. Hale, Ordinary dierential equations,
Robert E. Krieger Publishing Company, (1980),
pp. 190-194.
[3] H.K. Khalil, Nonlinear systems, Printice Hall,
(1996).
[4] P. Lancaster and L. Rodman, Algebraic Riccati
Equations, Oxford Science Publications, (1995).
[5] J. Peuteman and D. Aeyels, Exponential stability of non-linear time-variant dierential equations and partial averaging, in preparation.
[6] W.J. Rugh, Linear system theory, Prentice Hall
New Jersey, (1993).
[7] V. Solo, On the stability of Slowly Time-Varying
Linear Systems, Mathematics of Control, Signals, and Systems, (1994) pp. 331-350.
(32)
We assumed that b( ) is periodic and continuous
which implies by (31) (and 0our choice of 0 ( )) continuity and periodicity of 2( ) = ( ).
By the Kalman-Yakubovich-Popov lemma, positive
realness of (30) is equivalent with (23) and (24) when
A( ), B ( ) and C ( ) take the values indicated in
the present section. By eliminating L( ) in (23) and
(24), one obtains a Riccati equation with solution
P ( ). Relying on the continuity properties of A( ),
B ( ), C ( ) and 0 ( ), it is possible to establish continuity of P ( ) ([4], pp. 260-266) since (?A( ) +
1
T
2 B ( )C ( )+ ( )I; B ( )B ( )) is c-stabilizable 8 .
This implies the existence of a max > 0 and a
min > 0 such that 8 : max max (qP ( )) and
min min (P ( )). Therefore, Kmax = max
min 1
such that 8 : K ( ) Kmax.
One obtains by Theorem 2 that 91 such that 8 1 , (27) is exponenially stable when
T
Z
0
0 ( )d < 0
(33)
or equivalently
Z
0
T (b( ) + 2)(b( ) ? 3) + 2jb( )jp3 ? b( )
(b( ) ? 2)(b( ) + 6)
d > 0
(34)
6