- Email: [email protected]
Converse Lyapunov Theorems for Non-Uniform in Time Global Asymptotic Stability and Stabilization by Means of Time-Varying Feedback
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CONVERSE LYAPUNOV THEOREMS FOR NON-UNIFORM IN TIME GLOBAL ASYMPTOTIC STABILITY AND STABILIZATION BY MEANS OF TIME-VARYING FEEDBACK I. Karafyllis and J. Tsinias Department of Mathematics, National Technical University ofAthens, Zografou Campus 15780, Athens, Greece Email: [email protected]
Abstract: Two converse Lyapunov Theorems are presented concerning non-uniform in time global asymptotic stability. A necessary and sufficient condition for stabilization of affine in the control systems by means ofa time-varying feedback is established. Copyright (Q 2001lFAC Keywords: Lyapunov function, Feedback stabilization, Time-varying systems
I. INTRODUCTION
x
= f(t,x) + g(t,x)u
(3)
We consider time-varying systems of the form: x=f(t,x,d) XE
9\n ,d E D, t;? 0
where f(·) and g(.) are both CO and locally Lipschitz with respect to x, with f(-'O) =O. The corresponding feedback can be explicitly expressed in terms of f and g.
(1 )
where Dc 9\m is a non-empty compact set and f: 9\+ x9\n x D ~ 9\n is a CO map with f(t,O,d) = 0,
The present work constitutes a continuation of Karafyllis and Tsinias (2001); Tsinias and Karafyllis (1999) and Tsinias (2000) dealing with non-uniform in time stability and its applicability to the feedback stabilization problem.
for all (t,d)E 9\+ x D, which is locally Lipschitz with respect to x, uniformly in d ; equivalently, there exists a positive CO function L: 9\+ x9\+ ~ 9\+ such that for each fixed s;? 0 the mappings L(· ,s) and L(s,') are non-decreasing and the following holds: If(t,x,d) - f(t,y,d)1 ~ L(t,lxl +
Notations: Throughout this paper we use the following notations. We denote by MD the set of all measurable
functions from 9\+ to D. By K+ is denoted the class of
IYI) Ix - YI
V(t,x,y,d) E 9\+ x9\n x9\n x D
positive non-decreasing C~ functions ~ : 9\ + ~ 9\ + and (2)
by
Our goal is to provide Lyapunov characterizations for the concepts of non-unifonn in time robust global asymptotic stability (Theorem 3.1) and robust global exponential stability (Theorem 4.1) for the case of systems (1). A necessary and sufficient condition for non-uniform in time-global asymptotic stabilization by means of a timevarying feedback is established (Theorem 5.1) for affine in the control systems
E
the
class
-
of non-negative
CO
functions
<2':9\+ ~9\+, for which it holds: J<2'(t)dt<+oo
and
°
lim <2'( t) =0 . For the definitions 0 f classes K, K ~ ,KL
t ..... _
see Khalil (1996) and Lin et al (1996). We denote by Il the class of Cl positive functions <2': 9\ ~ 9\+ that satisfy
769
iiP(t, to , Xo ;d)i ~ K(m(t»)exp{m(to) - m(t)]xoi
t a ~rp(t)~tP for certain constants a,/l and t >0 away from zero. 2. NOTIONS OF RGAS AND RGES
Vt~to,xoE9\n,dEMD
(6)
3. A CONVERSE LY APUNOV THEOREM FOR ROBUST GLOBAL ASYMPTOTIC STABILITY
Consider the system (1) and denote by iP(t) =iP(t, to, xo; d) its solution at time t that corresponds to some input dE MD' initiated from Xo at time to .
The proof of the following converse Lyapunov theorem is based on a similar procedure with this employed in Lin et al (1996) for the case of RUG AS.
Definition 2.1 We say that 0 E 9\n is Robustly Globally Asymptotically Stable (RGAS) for (1) if:
Theorem 3.1 For the system (1) OE 9\n is RGAS, if and
I. For every to~, dEMD and Xo E 9\n the corresponding
functions a,
solution iP(·) exists for all t ~ to .
definite function p: 9\ + ~ 9\ +such that:
only if. there exist a C- function V: 9\+ x9\n ~ 9\+,
IL For every
such that:
£
> 0, T
ixo i~ 0,
~
a2 E K_,
0, there exists a 0:= 0(£, T) > 0
a(i~) ~ V(t,x) ~ P(t) a2 q~)
to E [0, T] ~
at
ax
V(t,x,d)E 9\+ x9\n x D
~
0 and R ~ 0, there exists a 'i:=r"(c,T,R)~O,suchthat: ixoi~R,toE[O,T] ~
We give an outline of the proof. Details will be provided in future work.
As in the case of uniform in time RUGAS (see [3]) we have:
if and
(8)
iiP(t,to,xo; d)i ~ £ , Vt ~ to + 'l", Vd EM D
Proposition 2.2 OE 9\n is RGAS for (1),
(7)
~(t,x) + ~(t,x)f(t,x,d) ~ -p(V(t, x»)
iiP(t,t o ,xo;d)i ~ £ ,Vt ~to, Vd EM D ilL For every c > 0, T
PE K+ and a C- positive
only
Proof Suppose first that OE 9\n is RGAS. Then by Proposition 2.2 there exist functions al ,a2 E K_ and
if
/l E K+ such that for all t ~ to and dE M D we have:
there exist functions a l ,a2 E K_, /lE K+ such that for
al (iiP(t,to,xo;d)D ~ exp( -2(t -to»/l(tO)a2
every dE MD' to ~ 0 and Xo E 9\n it holds:
qxoi)
V(t,x,d)E9\+x9\n x M D , Vt~to
(9)
a l ~(t,to,xo;d~)~ exp(-t + to)/l(to)a2(ixoi) , Vt ~ to (4)
Without loss of generality we may assume that al is
locally Lipschitz on (0,+00) . Combining (2) and (9) it
A proof of Proposition 2.2 will be given in a forthcoming paper.
follows that for every (to,x,y,d)E 9\+ x9\n x9\n xM D
The notion of RGAS above is an extension of the well known Sontag's robust uniform GAS (RUGAS) for timeinvariant systems, namely, when the solution iP(·)
I~(t,to,x;d) - ~(t,to,y;d)l:s; exp( F(s,lxl + IYI)ds)lx -YI (10)
and time t
~
to: I
satisfies iiP(t~~/l(ixoi,t-to) for certain /l of class KL (see for instance [2,3]). To justify this, we may recall Proposition 7, in Sontag (1998), which asserts that for any /lE KL there exist functions GJ,~EK;,. with
t
~(t,to,x;d)i ~ exp(- JL(s,i~)ds)i~
/l(s,t)~all(exp(--t)a2(S»). It turns out that RUGAS is characterized by the inequality al~iP(t~)~exp(-t+to)a2qxoi), which obviously is a special case of (4).
(12)
where L(t,s):= L(t,2a- 1(/l(y(t)s»). We define:
V(t,X):= sup{al <\fP<'l",t,x;d~)exp(.. -t); .. ~ t ,d EM D}(13)
Definition 2.3 We say that OE 9\n is Robustly Globally Exponentially Stable (RGES) for (1), if there exists a pair offunctions K, m of class n and a constant a > 0 such that: ta-I ~ m(t), Vt ~ 0 (5)
By (9) and (13) we have:
770
This shows that V: 9t+ x9t n ~ 9t+ is continuous at x = 0 with V(t,O) = 0 for all t ~ O. We next establish that V(-)
and this implies that for all dE D :
.
is locally Lipschitz on 9t+ x (9t n \ {O}). By using (9) and ( 14) and defining
T1 (t,x):=log( 11
2p(t)a2(lxl)
11)
2
T2(t,I~)~
(15)
2
2
\f(t,X)E 9t+ x9t
n
(23)
Notice that for each fixed s > 0, 11 (.,s) (i = 1,2 ) are nondecreasing. It turns out from (15) and (16) that
aW aw -(t,x)+-a (t,x)f(t,x,d) at x
IV(t,y) - V(t,x)1
sup
(22)
4
~al (Ixl) ~ W(t,x) ~ ~ !3(t)a2 (I~),
(16)
sup T1(t,IJi) lY-zl~~.rJ 2
{o})
extended version of Theorem B.l in Lin et aI (1996), which asserts the existence of a function n W:9t+ x9t ~9t+ of class ca(9t+ x~ }nC-(9t+ x(9tn \{O})), that satisfies:
V(t,x) =sup{al(I{I>(T,t,x;d~)exp(T-t); t ~T~t+;, dE MD}
V;~1](t,lxl), x;tO
\
In order to build a smooth Lyapunov function, define 1 1 p(t,x) :=-V(t,x) and v(t,x):= -V(t,x), and use an
al(-I~) 2
it follows:
:>;
av
a.e. (t,X)E9t+ x(9t n
{2,8(t)a2(~~)1 1
T( 1..1)-1 2t'4·-0
al(x)
av
V(t,x,d) :=-(t,x) +-(t,x)f(t,x,d) ~ -V(t,X) at ax
~
1 2
--W(t,x)
V(t,x,d)E9t+ x(9t n \{O})xD
{exp( T - t)lal (I~("r,t, y;d)l) - a1 q~(T,t, x;d)I)1 }
dEM D• ISTSl+T2(I~zl)
(24)
Then applying Lemma 17 in Teel and Praly (2000) as well as Corollary lOin Sontag (1998), we can build a C-
\flY - xl ~ ~ Ixl ' x;t 0
(17)
function V: 9t+ x9t n ~ 9t+ that satisfies the desired (7) and (8) for certain 121
Exploiting (10), (12), (17) and the fact that al is locally Lipschitz on (0,+:>0), it follows that for every compact
,122 E K_,
PE K+ and p(s):= s .
Conversely, if (7) and (8) hold, it follows by virtue of Lemma 4.4 in Lin et al (1996) that there exists a KL function (j such that:
interval J c 9t+ a constant M I > 0 can be found such that: (18)
which implies that OE 9t n is RGAS with respect to (1). The proof is complete.
(19)
This establishes that
V(t,·)
4. A CONVERSE LY APUNOV THEOREM FOR ROBUST GLOBAL EXPONENTIAL STABILITY
is locally Lipschitz on
9t n \ {O} for each t ~ 0 . Similarly we can establish that for x
;t
The following result is presented in Tsinias (2001).
0 it holds:
Theorem 4.1 For the system (I) assume that there exists a function L(·) E n such that: for some CO function G2 :9t+x(O,+:>o)~(O,+:>o), such
If(t,x,d~~L(t)I~, \fXE9t n ;t~O;dED
(26)
that for each fixed s > 0 the mapping G2 ( ., s) is nondecreasing. From (18), (19) and (20) we deduce that V(·)
Then OE 9t n is RGES, if and only if there exist afunction
is locally Lipschitz on 9t + x (9t n \ {O}) and consequently
V, of class Cl (9t+x9t n )nc-(9t+ x(9t n \ {O})) , a pair of
almost everywhere differentiable in 9t+
X
9t n
.
functions K, rp of class
We next
tl ~
and a constant q > 0 such
that:
evaluate the time derivative U( .) of V(·) along the solutions of (1). Notice first that, for definition (13) gives:
n
t q - 1 ~ ,pet)
t and dE MD'
(27)
1~2 ~ Vet, x) ~ K(rp(t»lxI2, \f(t, X)E 9t+ X 9t n (28)
771
dV dV . :;-(t,x) + -(t,x)f(t,x,d) S; -rp(t) V(t, x) ut dx
W(t,x) S; fP(t) =>
av
dV ! -(t,x) + -(t,x)f(t,x) s; -- p(v(t,x»)+ 2,(t) (36) dt dx 2
Vx *,0, de D, t away from zero (29) 5. ST ABILIZATION OF AFFINE CONTROL SYSTEMS BY SMOOTH TIME-VARYING FEEDBACK
Let 0: 9t ~ 9\ + be any C~ non-decreasing function which satisfies: O(s)=O, for sS;O; O(s)
The following theorem is an extension of the "ArtsteinSontag" theorem (Sontag, 1989; Tsinias, 1989). Theorem 5.1 For the system (3) the following statements are equivalent:
k(l,x):= -
(i) There exists a CO jUnction k: 9t+
X
9t n ~ 9t with
k(t,O) =0 for all t ~ O. being locally Lipschitz in x. in such a way that Oe 9t n is GAS with respect to the closedloop system (3) with u = k(t,x) (ii) There exists a
c~
function
~ 9t+
(I, X)g(l, x)
1l_~W~;;) )+(~:
where (.) is any positive
C~
)
(37)
1 (I, x) (I.X)g(t,X»)
function that satisfies:
dV dV -(t,x) + -(t,x)f(t,x) + p(V(t,x»)S; (t,x) dt dx
(30) V: 9t+ x9t n
~:
V(t,x)e 9t+ x9t n
•
(38)
jUnctions a 1,a 2 e K~. fJe K+. fPe E and a C~ positive
Notice that k has the same regularity properties with
definite jUnction p: 9t + ~ 9t + such that: a 1(I~) S; V(t,x) S; fJ(t)a 2
Qxl)
(31)
dV -(t,x)g(t,x) =O~ dx
Indeed,
dV dV -(t,x) + -(t, x)f(t, x) S; -p(V(t,x»)+ fP(t) (32) dt dx
vl
(t,x)e9\+x9t n . It then W(t,x)~fP(t)
and
denominator in (37) are nonzero. We next estimate the derivative V(.) of VC-) along the trajectories of the solutions of the closed-loop system (3) with (30), (37). We find:
~(t,x) + d V (t,x)V(t,x) + g(t,x)k(t,x»)s; -p(V(t,x») dx
.
av
av
av
V(t, x) := -(t, x) + -(t, x)f(t, x) + - (t, x)g(t, x)k(t,x) at dx dX (39) 1 :::; -- p(V(t,x»)+ ~(t) 2
(33)
Then (32) (with fP(t) == Oe E ) is a direct consequence of (33).
Indeed, for those t,x for which W(t,x) S; fP(t) we have by taking into account (36) and (37) that:
(i) ~ (ii) Without loss of generality we may assume that fP C~
for certain
. dV thus by Virtue of (35) both h(t,x)g(t,x) and the
Since Oe 9t n is assumed to be GAS for this system, Theorem 3.1 guarantees the existence of a C~ function V: 9\+ X 9t n ~ 9t+ in such a way that (5.2) holds and
is
for all t and x and suppose that
follows from definition of 0(·) that
V(t,x)e 9t+ x9t n
l~O(W(t'X») fP(t)
l=jW(t,X») fP(t)
Proof (i)~ (ii) We apply the converse Lyapunov Theorem 3. I for the closed-loop system (3) with (30).
dt
f
and g, and is well defined for all (t,x), since the denominator in (37) is everywhere strictly positive.
with fP(t) > 0 for all t ~ O. Define:
~(t,x) + d V (t,x)f(t,x) S; -~ p(V(t,x»)+ 2fP(t)
dV dV 1 Wet, x):= -(t,x) +-(t,x)f(t, x) +- P(V(t,x»)- 9(t) (34) dt dX 2
dt
dV
= 0 => W(t,x) S; 0
2
dV
-(t, x)g(t, x)k(t, x) S; 0 dx
It turns out from (32) and (34) that: ~(t,x)g(t,x)
dx
which implies (39). On the other hand for those t,x for which W(t,x) ~ fP(t), it follows from (35) and (37) that:
(35)
772
oV (t,x)g(t, x) *" 0; a-;
Ix(t)I' "'P
oV ox
U
L(r, t"
Ix,l)dr) Ix,1
(45)
-(t,x)g(t,x)k(t,x) = -(t,x) Combining (43) and (45) we get for any T ~ 0 :
which by virtue of(38) implies V(t, x) ~ -p(V(t,x»)~
sup Ix(t)I~A(t-to,T,lxol)
-.!.2 p(V(t,x») + 2~(t)
A(t,T,s) :=mir{exriL(T+t,T,s)t}s, all (o(M+ t~})a2(S) , t))}(47)
This establishes (39). In order to establish global asymptotic stability for the closed-loop system (3) with (30), we need the following lemma, which constitutes generalization of Lemma 4.4 in Lin et al (1996).
The desired GAS of zero is a consequence of (46), (47).
Corollary 5.3 Consider the system
i = f(x) + g(x)u
Lemma 5.2 Let y: 9\+ ~ 9\+ be an absolutely continuous function that satisfies the following differential inequality:
y ~ -p(y) + i(t),
y(t o ) = yo ~ 0
xE9\n,uE9\
et
is of class E . Then there exists a KL function
a: (9\+
feedback Then there exists a Cl time-varying feedback
y~ 9\+ such that: 0
l
law: u = k(t,x), with k(·,O) = 0 , such that OE 9\n is GAS with respect to the closed-loop system (48) with u = k(t,x) .
Vt ""
(41)
REFERENCES
By using (39) and Lemma 5.2 with
i
P =.!. p
and 2 = 2~ E E , it follows that there exists a KL function
a: (9\+
Karafyllis I. and 1. Tsinias (2001). "Global Stabilization and Asymptotic Tracking for a Class of Nonlinear Systems by Means of Time-Varying Feedback It, submitted. Khalil, H.K. (1996). Nonlinear Systems. Second Edition. Prentice-Hall. Lin Y. , E.D. Sontag and Y. Wang (1996). "A Smooth Converse Lyapunov Theorem for Robust Stability", SIAM J. Control and Optimization, 34, 124-160. Sontag, E. (1989). "A "universal" Construction of Artstein's Theorem on Nonlinear Stabilization", Systems and Control Letters, 13, 117-123. Sontag, E.D. (1998). "Comments on Integral Variants of ISS" , Systems and Control Letters, 34, 93-100. Teel, A.R. and L. Praly (2000). "A Smooth Lyapunov Function From A Class- KL Estimate Involving Two Positi ve Semi definite Functions", COCV-ESAIM. Tsinias J. (1989). "Sufficient Lyapunovlike Conditions for Stabilization", Math. Control. Signals Systems, 2, 343357. Tsinias 1. and I. Karafyllis (1999). "ISS Property for TimeVarying Systems and Application to Partial-Static Feedback Stabilization and Asymptotic Tracking", IEEE Trans. Automat. Contr., 44, 2179-2185. Tsinias 1. (2000). "Backstepping Design for Time-Varying Nonlinear Systems with Unknown Parameters", Systems and Control Letters, 39, 219-227. Tsinias J. (2001). "A Converse Lyapunov Theorem for Non-Unifonn in Time, Global Exponential Robust Stability", submitted.
y~ 9\+ such that:
where M := 2
J~(t)dt
for any t ~ to
, and thus by virtue of (31)
and initial
(t 0' Xo ). This establishes
attractivity. In order to establish that OE 9\n is GAS, let L:(9\+)3 ~9\+ be a CO function such that for each t~O
and
s~O
the mappings L(t, s,), L(t,·,s) and
L( · ,t,s) are non-decreasing and in such a way that: JV(t ,x) + g(t,x)k(t, x~
L(t,to,r)~su1
I.xj
_I
(48)
(1 ~ I ~ 00) with f(O) = 0 and where f and g are suppose that the origin of (48) can be globally asymptotically stabilized by means of a CO static
(40)
where p :9\+ ~ 9\+ is a positive definite Cl function and
i
(46)
loE[O,T)
}
; O
(44)
Consequently:
773
Copyright (I IFAC Nonlinear Control Systems, SI. Petersburg, Russia, 200 I
c:
0
C>
Publications www.elsevier.comllocate/ifac
CONVERSE LYAPUNOV THEOREMS FOR NON-UNIFORM IN TIME GLOBAL ASYMPTOTIC STABILITY AND STABILIZATION BY MEANS OF TIME-VARYING FEEDBACK I. Karafyllis and J. Tsinias Department of Mathematics, National Technical University ofAthens, Zografou Campus 15780, Athens, Greece Email: [email protected]
Abstract: Two converse Lyapunov Theorems are presented concerning non-uniform in time global asymptotic stability. A necessary and sufficient condition for stabilization of affine in the control systems by means ofa time-varying feedback is established. Copyright (Q 2001lFAC Keywords: Lyapunov function, Feedback stabilization, Time-varying systems
I. INTRODUCTION
x
= f(t,x) + g(t,x)u
(3)
We consider time-varying systems of the form: x=f(t,x,d) XE
9\n ,d E D, t;? 0
where f(·) and g(.) are both CO and locally Lipschitz with respect to x, with f(-'O) =O. The corresponding feedback can be explicitly expressed in terms of f and g.
(1 )
where Dc 9\m is a non-empty compact set and f: 9\+ x9\n x D ~ 9\n is a CO map with f(t,O,d) = 0,
The present work constitutes a continuation of Karafyllis and Tsinias (2001); Tsinias and Karafyllis (1999) and Tsinias (2000) dealing with non-uniform in time stability and its applicability to the feedback stabilization problem.
for all (t,d)E 9\+ x D, which is locally Lipschitz with respect to x, uniformly in d ; equivalently, there exists a positive CO function L: 9\+ x9\+ ~ 9\+ such that for each fixed s;? 0 the mappings L(· ,s) and L(s,') are non-decreasing and the following holds: If(t,x,d) - f(t,y,d)1 ~ L(t,lxl +
Notations: Throughout this paper we use the following notations. We denote by MD the set of all measurable
functions from 9\+ to D. By K+ is denoted the class of
IYI) Ix - YI
V(t,x,y,d) E 9\+ x9\n x9\n x D
positive non-decreasing C~ functions ~ : 9\ + ~ 9\ + and (2)
by
Our goal is to provide Lyapunov characterizations for the concepts of non-unifonn in time robust global asymptotic stability (Theorem 3.1) and robust global exponential stability (Theorem 4.1) for the case of systems (1). A necessary and sufficient condition for non-uniform in time-global asymptotic stabilization by means of a timevarying feedback is established (Theorem 5.1) for affine in the control systems
E
the
class
-
of non-negative
CO
functions
<2':9\+ ~9\+, for which it holds: J<2'(t)dt<+oo
and
°
lim <2'( t) =0 . For the definitions 0 f classes K, K ~ ,KL
t ..... _
see Khalil (1996) and Lin et al (1996). We denote by Il the class of Cl positive functions <2': 9\ ~ 9\+ that satisfy
769
iiP(t, to , Xo ;d)i ~ K(m(t»)exp{m(to) - m(t)]xoi
t a ~rp(t)~tP for certain constants a,/l and t >0 away from zero. 2. NOTIONS OF RGAS AND RGES
Vt~to,xoE9\n,dEMD
(6)
3. A CONVERSE LY APUNOV THEOREM FOR ROBUST GLOBAL ASYMPTOTIC STABILITY
Consider the system (1) and denote by iP(t) =iP(t, to, xo; d) its solution at time t that corresponds to some input dE MD' initiated from Xo at time to .
The proof of the following converse Lyapunov theorem is based on a similar procedure with this employed in Lin et al (1996) for the case of RUG AS.
Definition 2.1 We say that 0 E 9\n is Robustly Globally Asymptotically Stable (RGAS) for (1) if:
Theorem 3.1 For the system (1) OE 9\n is RGAS, if and
I. For every to~, dEMD and Xo E 9\n the corresponding
functions a,
solution iP(·) exists for all t ~ to .
definite function p: 9\ + ~ 9\ +such that:
only if. there exist a C- function V: 9\+ x9\n ~ 9\+,
IL For every
such that:
£
> 0, T
ixo i~ 0,
~
a2 E K_,
0, there exists a 0:= 0(£, T) > 0
a(i~) ~ V(t,x) ~ P(t) a2 q~)
to E [0, T] ~
at
ax
V(t,x,d)E 9\+ x9\n x D
~
0 and R ~ 0, there exists a 'i:=r"(c,T,R)~O,suchthat: ixoi~R,toE[O,T] ~
We give an outline of the proof. Details will be provided in future work.
As in the case of uniform in time RUGAS (see [3]) we have:
if and
(8)
iiP(t,to,xo; d)i ~ £ , Vt ~ to + 'l", Vd EM D
Proposition 2.2 OE 9\n is RGAS for (1),
(7)
~(t,x) + ~(t,x)f(t,x,d) ~ -p(V(t, x»)
iiP(t,t o ,xo;d)i ~ £ ,Vt ~to, Vd EM D ilL For every c > 0, T
PE K+ and a C- positive
only
Proof Suppose first that OE 9\n is RGAS. Then by Proposition 2.2 there exist functions al ,a2 E K_ and
if
/l E K+ such that for all t ~ to and dE M D we have:
there exist functions a l ,a2 E K_, /lE K+ such that for
al (iiP(t,to,xo;d)D ~ exp( -2(t -to»/l(tO)a2
every dE MD' to ~ 0 and Xo E 9\n it holds:
qxoi)
V(t,x,d)E9\+x9\n x M D , Vt~to
(9)
a l ~(t,to,xo;d~)~ exp(-t + to)/l(to)a2(ixoi) , Vt ~ to (4)
Without loss of generality we may assume that al is
locally Lipschitz on (0,+00) . Combining (2) and (9) it
A proof of Proposition 2.2 will be given in a forthcoming paper.
follows that for every (to,x,y,d)E 9\+ x9\n x9\n xM D
The notion of RGAS above is an extension of the well known Sontag's robust uniform GAS (RUGAS) for timeinvariant systems, namely, when the solution iP(·)
I~(t,to,x;d) - ~(t,to,y;d)l:s; exp( F(s,lxl + IYI)ds)lx -YI (10)
and time t
~
to: I
satisfies iiP(t~~/l(ixoi,t-to) for certain /l of class KL (see for instance [2,3]). To justify this, we may recall Proposition 7, in Sontag (1998), which asserts that for any /lE KL there exist functions GJ,~EK;,. with
t
~(t,to,x;d)i ~ exp(- JL(s,i~)ds)i~
/l(s,t)~all(exp(--t)a2(S»). It turns out that RUGAS is characterized by the inequality al~iP(t~)~exp(-t+to)a2qxoi), which obviously is a special case of (4).
(12)
where L(t,s):= L(t,2a- 1(/l(y(t)s»). We define:
V(t,X):= sup{al <\fP<'l",t,x;d~)exp(.. -t); .. ~ t ,d EM D}(13)
Definition 2.3 We say that OE 9\n is Robustly Globally Exponentially Stable (RGES) for (1), if there exists a pair offunctions K, m of class n and a constant a > 0 such that: ta-I ~ m(t), Vt ~ 0 (5)
By (9) and (13) we have:
770
This shows that V: 9t+ x9t n ~ 9t+ is continuous at x = 0 with V(t,O) = 0 for all t ~ O. We next establish that V(-)
and this implies that for all dE D :
.
is locally Lipschitz on 9t+ x (9t n \ {O}). By using (9) and ( 14) and defining
T1 (t,x):=log( 11
2p(t)a2(lxl)
11)
2
T2(t,I~)~
(15)
2
2
\f(t,X)E 9t+ x9t
n
(23)
Notice that for each fixed s > 0, 11 (.,s) (i = 1,2 ) are nondecreasing. It turns out from (15) and (16) that
aW aw -(t,x)+-a (t,x)f(t,x,d) at x
IV(t,y) - V(t,x)1
sup
(22)
4
~al (Ixl) ~ W(t,x) ~ ~ !3(t)a2 (I~),
(16)
sup T1(t,IJi) lY-zl~~.rJ 2
{o})
extended version of Theorem B.l in Lin et aI (1996), which asserts the existence of a function n W:9t+ x9t ~9t+ of class ca(9t+ x~ }nC-(9t+ x(9tn \{O})), that satisfies:
V(t,x) =sup{al(I{I>(T,t,x;d~)exp(T-t); t ~T~t+;, dE MD}
V;~1](t,lxl), x;tO
\
In order to build a smooth Lyapunov function, define 1 1 p(t,x) :=-V(t,x) and v(t,x):= -V(t,x), and use an
al(-I~) 2
it follows:
:>;
av
a.e. (t,X)E9t+ x(9t n
{2,8(t)a2(~~)1 1
T( 1..1)-1 2t'4·-0
al(x)
av
V(t,x,d) :=-(t,x) +-(t,x)f(t,x,d) ~ -V(t,X) at ax
~
1 2
--W(t,x)
V(t,x,d)E9t+ x(9t n \{O})xD
{exp( T - t)lal (I~("r,t, y;d)l) - a1 q~(T,t, x;d)I)1 }
dEM D• ISTSl+T2(I~zl)
(24)
Then applying Lemma 17 in Teel and Praly (2000) as well as Corollary lOin Sontag (1998), we can build a C-
\flY - xl ~ ~ Ixl ' x;t 0
(17)
function V: 9t+ x9t n ~ 9t+ that satisfies the desired (7) and (8) for certain 121
Exploiting (10), (12), (17) and the fact that al is locally Lipschitz on (0,+:>0), it follows that for every compact
,122 E K_,
PE K+ and p(s):= s .
Conversely, if (7) and (8) hold, it follows by virtue of Lemma 4.4 in Lin et al (1996) that there exists a KL function (j such that:
interval J c 9t+ a constant M I > 0 can be found such that: (18)
which implies that OE 9t n is RGAS with respect to (1). The proof is complete.
(19)
This establishes that
V(t,·)
4. A CONVERSE LY APUNOV THEOREM FOR ROBUST GLOBAL EXPONENTIAL STABILITY
is locally Lipschitz on
9t n \ {O} for each t ~ 0 . Similarly we can establish that for x
;t
The following result is presented in Tsinias (2001).
0 it holds:
Theorem 4.1 For the system (I) assume that there exists a function L(·) E n such that: for some CO function G2 :9t+x(O,+:>o)~(O,+:>o), such
If(t,x,d~~L(t)I~, \fXE9t n ;t~O;dED
(26)
that for each fixed s > 0 the mapping G2 ( ., s) is nondecreasing. From (18), (19) and (20) we deduce that V(·)
Then OE 9t n is RGES, if and only if there exist afunction
is locally Lipschitz on 9t + x (9t n \ {O}) and consequently
V, of class Cl (9t+x9t n )nc-(9t+ x(9t n \ {O})) , a pair of
almost everywhere differentiable in 9t+
X
9t n
.
functions K, rp of class
We next
tl ~
and a constant q > 0 such
that:
evaluate the time derivative U( .) of V(·) along the solutions of (1). Notice first that, for definition (13) gives:
n
t q - 1 ~ ,pet)
t and dE MD'
(27)
1~2 ~ Vet, x) ~ K(rp(t»lxI2, \f(t, X)E 9t+ X 9t n (28)
771
dV dV . :;-(t,x) + -(t,x)f(t,x,d) S; -rp(t) V(t, x) ut dx
W(t,x) S; fP(t) =>
av
dV ! -(t,x) + -(t,x)f(t,x) s; -- p(v(t,x»)+ 2,(t) (36) dt dx 2
Vx *,0, de D, t away from zero (29) 5. ST ABILIZATION OF AFFINE CONTROL SYSTEMS BY SMOOTH TIME-VARYING FEEDBACK
Let 0: 9t ~ 9\ + be any C~ non-decreasing function which satisfies: O(s)=O, for sS;O; O(s)
The following theorem is an extension of the "ArtsteinSontag" theorem (Sontag, 1989; Tsinias, 1989). Theorem 5.1 For the system (3) the following statements are equivalent:
k(l,x):= -
(i) There exists a CO jUnction k: 9t+
X
9t n ~ 9t with
k(t,O) =0 for all t ~ O. being locally Lipschitz in x. in such a way that Oe 9t n is GAS with respect to the closedloop system (3) with u = k(t,x) (ii) There exists a
c~
function
~ 9t+
(I, X)g(l, x)
1l_~W~;;) )+(~:
where (.) is any positive
C~
)
(37)
1 (I, x) (I.X)g(t,X»)
function that satisfies:
dV dV -(t,x) + -(t,x)f(t,x) + p(V(t,x»)S; (t,x) dt dx
(30) V: 9t+ x9t n
~:
V(t,x)e 9t+ x9t n
•
(38)
jUnctions a 1,a 2 e K~. fJe K+. fPe E and a C~ positive
Notice that k has the same regularity properties with
definite jUnction p: 9t + ~ 9t + such that: a 1(I~) S; V(t,x) S; fJ(t)a 2
Qxl)
(31)
dV -(t,x)g(t,x) =O~ dx
Indeed,
dV dV -(t,x) + -(t, x)f(t, x) S; -p(V(t,x»)+ fP(t) (32) dt dx
vl
(t,x)e9\+x9t n . It then W(t,x)~fP(t)
and
denominator in (37) are nonzero. We next estimate the derivative V(.) of VC-) along the trajectories of the solutions of the closed-loop system (3) with (30), (37). We find:
~(t,x) + d V (t,x)V(t,x) + g(t,x)k(t,x»)s; -p(V(t,x») dx
.
av
av
av
V(t, x) := -(t, x) + -(t, x)f(t, x) + - (t, x)g(t, x)k(t,x) at dx dX (39) 1 :::; -- p(V(t,x»)+ ~(t) 2
(33)
Then (32) (with fP(t) == Oe E ) is a direct consequence of (33).
Indeed, for those t,x for which W(t,x) S; fP(t) we have by taking into account (36) and (37) that:
(i) ~ (ii) Without loss of generality we may assume that fP C~
for certain
. dV thus by Virtue of (35) both h(t,x)g(t,x) and the
Since Oe 9t n is assumed to be GAS for this system, Theorem 3.1 guarantees the existence of a C~ function V: 9\+ X 9t n ~ 9t+ in such a way that (5.2) holds and
is
for all t and x and suppose that
follows from definition of 0(·) that
V(t,x)e 9t+ x9t n
l~O(W(t'X») fP(t)
l=jW(t,X») fP(t)
Proof (i)~ (ii) We apply the converse Lyapunov Theorem 3. I for the closed-loop system (3) with (30).
dt
f
and g, and is well defined for all (t,x), since the denominator in (37) is everywhere strictly positive.
with fP(t) > 0 for all t ~ O. Define:
~(t,x) + d V (t,x)f(t,x) S; -~ p(V(t,x»)+ 2fP(t)
dV dV 1 Wet, x):= -(t,x) +-(t,x)f(t, x) +- P(V(t,x»)- 9(t) (34) dt dX 2
dt
dV
= 0 => W(t,x) S; 0
2
dV
-(t, x)g(t, x)k(t, x) S; 0 dx
It turns out from (32) and (34) that: ~(t,x)g(t,x)
dx
which implies (39). On the other hand for those t,x for which W(t,x) ~ fP(t), it follows from (35) and (37) that:
(35)
772
oV (t,x)g(t, x) *" 0; a-;
Ix(t)I' "'P
oV ox
U
L(r, t"
Ix,l)dr) Ix,1
(45)
-(t,x)g(t,x)k(t,x) = -(t,x) Combining (43) and (45) we get for any T ~ 0 :
which by virtue of(38) implies V(t, x) ~ -p(V(t,x»)~
sup Ix(t)I~A(t-to,T,lxol)
-.!.2 p(V(t,x») + 2~(t)
A(t,T,s) :=mir{exriL(T+t,T,s)t}s, all (o(M+ t~})a2(S) , t))}(47)
This establishes (39). In order to establish global asymptotic stability for the closed-loop system (3) with (30), we need the following lemma, which constitutes generalization of Lemma 4.4 in Lin et al (1996).
The desired GAS of zero is a consequence of (46), (47).
Corollary 5.3 Consider the system
i = f(x) + g(x)u
Lemma 5.2 Let y: 9\+ ~ 9\+ be an absolutely continuous function that satisfies the following differential inequality:
y ~ -p(y) + i(t),
y(t o ) = yo ~ 0
xE9\n,uE9\
et
is of class E . Then there exists a KL function
a: (9\+
feedback Then there exists a Cl time-varying feedback
y~ 9\+ such that: 0
l
law: u = k(t,x), with k(·,O) = 0 , such that OE 9\n is GAS with respect to the closed-loop system (48) with u = k(t,x) .
Vt ""
(41)
REFERENCES
By using (39) and Lemma 5.2 with
i
P =.!. p
and 2 = 2~ E E , it follows that there exists a KL function
a: (9\+
Karafyllis I. and 1. Tsinias (2001). "Global Stabilization and Asymptotic Tracking for a Class of Nonlinear Systems by Means of Time-Varying Feedback It, submitted. Khalil, H.K. (1996). Nonlinear Systems. Second Edition. Prentice-Hall. Lin Y. , E.D. Sontag and Y. Wang (1996). "A Smooth Converse Lyapunov Theorem for Robust Stability", SIAM J. Control and Optimization, 34, 124-160. Sontag, E. (1989). "A "universal" Construction of Artstein's Theorem on Nonlinear Stabilization", Systems and Control Letters, 13, 117-123. Sontag, E.D. (1998). "Comments on Integral Variants of ISS" , Systems and Control Letters, 34, 93-100. Teel, A.R. and L. Praly (2000). "A Smooth Lyapunov Function From A Class- KL Estimate Involving Two Positi ve Semi definite Functions", COCV-ESAIM. Tsinias J. (1989). "Sufficient Lyapunovlike Conditions for Stabilization", Math. Control. Signals Systems, 2, 343357. Tsinias 1. and I. Karafyllis (1999). "ISS Property for TimeVarying Systems and Application to Partial-Static Feedback Stabilization and Asymptotic Tracking", IEEE Trans. Automat. Contr., 44, 2179-2185. Tsinias 1. (2000). "Backstepping Design for Time-Varying Nonlinear Systems with Unknown Parameters", Systems and Control Letters, 39, 219-227. Tsinias J. (2001). "A Converse Lyapunov Theorem for Non-Unifonn in Time, Global Exponential Robust Stability", submitted.
y~ 9\+ such that:
where M := 2
J~(t)dt
for any t ~ to
, and thus by virtue of (31)
and initial
(t 0' Xo ). This establishes
attractivity. In order to establish that OE 9\n is GAS, let L:(9\+)3 ~9\+ be a CO function such that for each t~O
and
s~O
the mappings L(t, s,), L(t,·,s) and
L( · ,t,s) are non-decreasing and in such a way that: JV(t ,x) + g(t,x)k(t, x~
L(t,to,r)~su1
I.xj
_I
(48)
(1 ~ I ~ 00) with f(O) = 0 and where f and g are suppose that the origin of (48) can be globally asymptotically stabilized by means of a CO static
(40)
where p :9\+ ~ 9\+ is a positive definite Cl function and
i
(46)
loE[O,T)
}
; O
(44)
Consequently:
773