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Input-to-State Stability with Respect to Noncompact Sets1
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2b-03 1
131h Triennial World Congrc~<; . S3J1 Fr:mci!'>co. USA
INPUT-TO-STATE STABILITY WITH RESPECT TO NONCOMPACT SETS I Yuandan Lin
Depa,·tme,.t 01 Mathematics, Florida Atlantic Un;ve"sity, Roc. Raton, FL 334.11, USA
Abstract. Our objed is to study the input-w-state stability with respect to closed, not necessarily compad, sets. A previous result on feedback st.abilization with respect to equiHbria is generalized to the case of noncompact sets. A Lyapunov characterization for t he input-t.o-state stability is also established for time-varying systems. Keywords. Input.-to-state stability, Lyapunov method, Nonlinear systemf->
1. INTRODUCTION
Onc of the most. importaut issues in t he study of tontrol systems is that of understanding t.he dependence of state trajectories 011 the magnitude of inputs. In the late 80 's, a precise definit.ion for input/output stability (ISS) was iutroduced in (Sont.ag, 1989). Since then, the notion of ISS has bcx:n proved a fundamental c.onccpt [or t he study of system stability, and has been subsequently employed by many authors in different area.'!, see for inst.ance (Tsinias: 1989; .Jiang et al. ) 1994; Krst.ic et al., 1995). The ISS property was originally stateu for stability with respect to an equilibrium state of interest. Various results on characterizat.ions of the ISS property with respect to equilibria 01 (:ompact set~ were established in (Son tag, 1990; Sontag and Wang, 1995). The main object of this work is to invf:',stigate ! hE' IS5 property with respect to dosed, not necessarily compact, sets. We are motivated by potentia l applications to a wide variety of areas. Examples of such applicatioll!i indude problems of robust control and OUtpu t stauilization (w here t he interesting set is t.he zero set of tlte observation map). This motivat.ed t.he st.udy of the "set" v~rsion of the rss property, originally in (Sontag and Liu , 1993), and developed with applicat.ions to t.he st.udy of parameterized famiJies of systems in (Lin et (ll. , 1995). The first purpose of thb wor k is
1
Lo
('xl,cnd from the e'lui-
Supp orted in part by NSf Grant DM S-9403921
librium to the set ISS case the result.s given in (Sontag, 1990) regarding feedback equivalence. We show that a system is stabilizable in the state-space sense, with respect to a given dosed invariant set , if and only if it is feedback equivalent to a system which is set-ISS with respect to t.he same set. The proof is analogous to that. giVfm for the case of eq uilibria, but several subtle details, especially when the completelless property of systems is involved, have to be dealt with. (A preliminary version of these results was given in Chapt.er 3 of (Lin , 1992).) The second purpose of this work is to develop the Lyapunav characterization for the ISS property for timevarying systems. To this purpose, we first treat a t.imevarying system as an augmented time invariant systf.m. The ISS property for t he original system is then COnverted to the set ISS probleIn for the augmented system. vs ing the same technique as in (Sontag and Wang, 1995), we show that a time-varying system is ISS with respect to a equilibrium state if and only if it admits an Iss-Lyapunov function .
2. INP t:T TO STATE STABILITY WITH RESPECT TO SETS Consider the following systen I
:i:(t) = I(x(t), u(t») , (E) where, for each t ~ 0, x(t) .e lRn, u(t ) E lRm, and I : lRfl x lRm -+ W is smooth. Controls, denoted by u, j are measurable, locally eS!lIp.nt.ially bounded fUIlctions from
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tR>o to am. For each { E I!i:n and each 'U E L ~ , we denote by x(t, ~, u) the trajectory of the system E with ~ and the input u. This solution initi1ll state x(O) is uniquely defined 011 some maximal interval [0, TJ;", u), with T(, u :S +00 . If T(, .. = 00 for all E an d all u, the system E i:-; fo rward r:omplete.
=
Wc use H to indicate Euclidean norm . Consider any nonempty ~ ubs~t A ~ IR'~; t hen for each ~ E Rn , the point~to-set distance from ~ t o A is denoted by
2.1 Chamcterization via DectJY Estimate In this section l we provide a characterization of the ISS property in terms of comparison functions , which generalizes to arbitrary sets A th e similar condition given earlier ill (Sontag and Wang, 1995).
Proposition 3. Assume tha.t. t.he system E is forward complete. Then E is ISS wit.h respect to A if and only if t here are a K:L-fllnct.ion P a nd a K:-function "Y so that (3)
In particular, 1~llO) .= I~I. A noneIllpty subset A is 0invariant for :E if eve))' solution with zero input start.ing from A is defined for a ll t > 0 and stavs in A when the control is u == 0; i.e.! for Ule corresponding "zero-input" or "undisturhed" system :i:
= 10(:<) = f( x, Il) ,
it holds that x(t , E,O ) E A for each
~
(Eo )
E A and all t 2: O.
Recall that a funetic,n "( : [0,00) -+ [0,00) is of class if it is continuOU9, strictly increasing, and satisfies "((0) = 0. A fUll ction "( is of dass K= if it is of class K a.nd it is unbound ed . A function {3 ; !Iho x D h o -7 iho is of class KL if, for e",:h t 2: 0 , the function fJ(·, tn, of cl ..., K , and , for ,,~ch s 2: 0, the fun ction fl(s , .) is decreasing and teuds 1,0 zero a t infinity. K,
Definition 1. Let A ~ lltn be a IlOIlempty subset. The function 'Y is called an a~~ymptotic yain fot' a given system L if, for each control u , ( 1) uniformly on subsets of the form {~ E litn, I{I A :S r} , 2: O. The system I: is said to have );:>asgmptolic gain with 1'Cspect to A if there is some asymptotic gain r E Kfor it.
holds for each t 2: 0,
U
E
L~,
and
EE ]Rn.
It is clear that if (3) holds for a forward complete system E, then E is ISS, Any fUllction "( for which an estimate (3) holds (for some fJ) will be called here an ISSgain for t.he syst.em (with respect to A); such a, is in part.icular an asymptotic gaiE. To prove the other implication, one can use exactly the same argument as in t he proof of (Sontag and Wa ng, 1995, Lemma 2.7). The only "xception is t.hat the a rgurnenl, in (Sontag and Wang, 1995, Remark 2.8) requires A to be compad. To drop thE' compactness assumpt ion , we est.a blish the following result to replace (Sontag and Wang, 1995, Remark 2, 8). Remark 4- In Definition 2, Property 2 is equivalent t.o t.h e following:
2' . There exisr.s a Koo-funclion such that, if I~IA :S J(E) , Ix(t , C u)IA :S
E
8 and
a K-function
+ ' I(lIull), 'It
~
l'
0, lIu. (4)
T
Thus, "'( is an asyrnpw tic gain if the t;ystem is forward complete and for every pair of reals r ,6" > 0, there is some T = T(r ," ) 2: so that, for all " , alll~IA :S r, and all t 2: T ,
°
Definition 2. The system E is said to he input~to-!~tate stable (ISS) with respect to a noncmpty subset A <,;; lltn if the following properties hold: (1) there exists a K ,~ -fllfi( : ti{)n 6(·) so that
Ix(t, ~, uJ IA :S
C,
1112:0,
(2)
whenever I~ I A < 6(0) and 11,,11 :; 6(0); and (2) E has K-asymptotic gain with respect to A. Observe that if Eis lSS , then in part.icular the autonomous system Eo is globally asympt.otica.lly stahle with respect t.O A (in t he sense in (Lin et nl. , 1996) ).
Clearly Property 2' implies Property 2. To show that Property 2 implies Property 21 , first observe t hat since the function 6(·) in Property 1 of Definition 2 is of class }Coo, it follows that for any l' 2: 0, there exists some M > 0 such that
Now for "(l
(~ach
(r) =
r 2: 0)
w(;!
let
sup{lx(t ,~,u) IA
. t 2: 0, I~IA :S
t·,
lIull :S r}.
°
Not.e t.hen 71 is a nondecreasiog function, and it follows from (2) that "(.(r) -+ as r -+ O. Let 1 be a Koofunction t hat majorizes the function 'rl. Then it hoJds that
To complete the proof of Re!llark 4 , one chooses J(r) = 1,I(r ). 0
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2.2 ISS-Lyapu1lov Pvnetinn .• An ISs-Lyapunov junction with respect to a set A for system 1: is a smoot ~l fUIlction V: TI{n -+ lR>o which satisfies the following conditions: -
• V is proper a.uG. positi ve definit.p- with respect to t he set A, t.hat is, therp. exist il l , (}2 E }Coo such t hat for a U I;. E DI" , (5)
• there exist a continuous, positive definite function <>3 a.Illi a Koo -function X snch that t here holds the following implication:
11;.1 ...
~
x(lvi) => V V(Of (~, 11)
(for
ca~h
state
~
E
{R1l
vlaim on page 441 of (Sontag, 1989) (see also (Sontag and Wang, 1995, Lemma 2.U», one can show t.he following: Lemma 7. Assume that syst.em E is forward complete.
If system E admits an lss-Lyapunov function, then it is [SS with the funct.ion X in (6) aB an [ss-gain function. 2.3 Stabilizability and Feedba ck Equivalence Definition 8. The syst em E is smoothly slubi/izable with respect to a dosed set A if there exists a. smooth map k : lR,n --+ ]Rm so t.h a t. t.h e dosed-loop system
S - <>3 (It;. I... )
(6)
amI control value v E IRm).
Remark 5. A slightly different way of dd1.uillg ISSLyapuIlov fUIlctions is to require the fUIlction 0'3 in (6) to be a K,:x>-function t.hat is smooth on (0,00). These t.wo variants are equivalent: if E admi t.s an rs s-Lyapll110V function V with (6), t.hen one can a lways get an ISSLyaplIllO\< functio n HI by modifying l' so that t he following implicat.ion
= J(x,
k(x) is globally asymptot.icaJly sl,a!>le with respect to A , t hat is, system Ec is forward complete, and there exists a KL-function {i so that Ix( t,O I... S {i(l~ I A' t) for all t ~ 0, for every solution x(t , ,;) of Eo. :i;
Definition 9. Two systems :i; = f(x, u) and :i; = .q(x, u), wit h t he same state-space a n ;ind the same control-value space lRTTl, are feedba ck t;:quivalent if g(~,
u) = f(~, k(O
+ r({),,) ,
\I ~ E IItn
,
\I " E IRm ,
where k : Rn --+ wn is smooth, and the m )( 1ft ma trix function r is smooth, and nonsingular for each ~ E Rn. holds for some Koo-function n:1 that is smooth on (0, oo)! where X is t.he same fun ction as in (6). For details of const.ru cting such a fun ction itV , see Remark 4.1 of (Lin et al. , 1996).
Remark 6. In (Sontag and Wang, 1995) , it was shown th at property (6) is also equivalent. to the existence of (} 4 E Kao and (J E K. :such t hat
This equivalence relation is in general not true when A is not. compact. As a.n example . eonsid(;!r the following system in 1R2 :
;', = _·x, + xlq(U -
jl-21),
--7 lIt is a ~IIlooth fuudion sat.isfying q(r) = S 0 and q(7-) > 0 for all r > O. Let. A be the set {(~l' 6) : 6 = O} . Consider the fun ction V (O = ~V2. Clearly V satisfies (5) (as I ~I ... = 1 ~,Il , a nd
where q: R
ofor all r
I(lA ~
Ivl => vF(O J(t;., 11) S -{i·
But it is easy to see that V fails to satisfy property (8). Note that an lss-Lya.punov function V for E is automatically also a Lyapunov fUIlction for th e 7.cro-inpnt system Eo with respect t.o A . Following the proof of the
It is clear that the above not.iun indeed defines an equivalfmcf. relat.ion. Definition 10. The system E is smoothly [SS stabilizable with ,....pecl to a closed set A if it is feedback equivalent t o a sys tem whkh is ISS with respect to A.
Assume that the ~ystem E is :,moothly stabilizable with resp ect. to a closed set A. The objective is to find a system , feedback equivaleIlt to the given one, which is ISS with respect 10 A. We fi rst establish the following result.
Lemma 11. Assume that the autonomous system 'Eo is forward complet.e. Then , there exists a smooth function IPa : [Rn -+ IR>o so that for any smooth function satisfying 0 S 'P(O S 'Po(~) for al l ~ E ~", the syst.em :i; f (x, ,p(xlu) (Er)
=
is also forward complete. We fi rst. need t he following tr ivial consequence of cont.inuity on initial conditions and controls.
Rema,7'k 12. Assume tha t Eo is forward complete. Then for any compact subset K ~ IRn, there exist some compact neighborhood L of K and some 6 > 0 so t.hat for any ~ E K and any lIull < ,I, t.he solut.ion x(t, ~,u) is define.j ou [0, 1J a nd
1956
x(t,
~ ,u) EL,
'I t E [0, I], 'I~ E K ,
'111,,11 < 8.(9)
Indeed, let K be any I~ompact subset of RU. By the com-
plet.eness assumption of Eo, one knows that x(t, €, 0) is defined for all t ~ 0, and. in particular, it is defined on {x(l , ~, 0) E E K , t E [0, In , and [0 , IJ. Let Lo let L be any compact neighborhood of La . It t hen follows from t.he eontinuity of the map (~ , u) >-t x(·, { , u) C([O, 1]) t.hat t.here exists some o( > 0 so that for each 1 < 0(, the solu'/ so t.hat IT/ - El < o( a nd each 11111 t.ion x(l, 'I, '1.1) of 1:: is defined OIl [0 , 1 and x(t , ~, n) E L, 'It E [0, 1], 'I 1'/- ~I < 0(, '1 11" 11 < 0(. By t.he
PROOF. Assume that system E is smoothly stabilizable with respect to A by u k(x). Then by the converse Lyapunov theorem given in (Lin et al. , 1996), there exists a smooth Lyapunov function for the closed-loop system so that the following holns:
=
=
• V is positive definite with respect to A i • t here exists a Koo-fullction (l' so that
c:
<.:ompactne..ss of <5
f( ,
onc knows t.hat there
ex ist~
VV(Of (~, k«())
S
-a
Sinc:e f , l/ aud k are smooth, there exists a smooth function G(E, v) such that
VVWf({, k(O
some
> 0 so that (9) hold,.
+ v)
= VV(Of(E, k({»)
+ G(~ , ti)".
For imtance, G can be defined by
,
Wc now ret urn to prove Lemma 11. Proof of Lemma 11. \Vest.art. with the compact set Kt = El , where for allY int.eger k > 0, we use Bk to denote t he set {E: IEI ~ k ) . Let L, and 0, be as in Lemma 12. Let I<2 = Lt U B'2' Uy inductioll, one knows that there p.xist an sequence of compact. suhset.s
G({ , v) = / ..~ F({, Av)dA , (11.1.
n
where
F(~,
'lE E ](.,
'11 E [O,IJ,
'I llull < Ok.
Now let !fo : II~n '~I~> o be a smooth function so that 'Po(O < o,-/k fol' all ~ E 1(. \ Kk _' (w here onc takes Ko as t he empt.y set) . In t he followin~ we show that the (:onclusion of Lemma 11 holds for sueh a choice of 'Po.
V
the claim is raise, t.hen thf'fP. exists some T > 0 so t hat l:c(l)l -+ 00 as t. -+ T. For the input u and the initial state C there is some M > 0 so t hat lu(t)1 ~ M for almost. a ll t E [0, T + 1J and £ E f( M. Pick lip I, > T - 1/2 so t hat x(t,) E K T \ [{r- ' fol' some I > M. Then 1'P(x(t,)u(t) 1 < 0" and hmce, x(t + t,,{) x(l, x(t,)) is defined fo r all 0 ~ t ~ 1, contradicting to t.h e assumption t. hat x(t ) is not defin ed fO f t > T. 0 Suppo~ e
Corollan) 1S. If syst"", 1:: is slIIoothly st.ahilizable by = k (x), then there exists a smooth function CPu : an --t lR>u so that for every smooth funct ion 'P satis(ying 0 ::; 'P (~) ~ 'Po(O fOl a ll ( . the s)'stem T = f(x , k(x) +
H
T he foUmving is th p. main
l'e~mlt
in tltis section.
Thcor-c m 14. The sys t.em 1:: is smoothly stabili.able with rp$pect. t o A if and only if it is feedb a,ck equivalent to
V
h(~) ~
1
~ lvi ,
for all { E II!.", all vEII!."'. (For the details on t.he existence of such functions, s.,e (Lin, 1992, Sec. 3.4.1).)
"'ith
~uch
a choice of VV(Of
h~
we
I ~ave
((k(~) + h;~))
=vV({)f«( , k«()) + G ({, h~{)) ,,~{)
=
~ (J.
\
G ( {, h({)) h({)
Let'P be any function satisfying 0 ~ A{) ~ 'Po (E) for all ~ E IRn. Pir.k lip a ny { E IRn and a ll." input 11. Let x(t ) denote the corrc_'ponding trajectory of:i: f( x, 'P(x)u) defined on some interval.
Claim: x(t) i, d cnned for all t
= vV( Of(~, '<0 + It).
Now we choose a smoot h fUlJctio n h({) wit.h for all Esuch th at
wit h Ek c;;: Kk and positive numhers 81, 02 • ... such t hat
x(t,E,u) E ](1+1,
u)
~
-a(I{I A ) + 11!1
for all Eand v. We now deHne a function p : 1R~0 -t 1R~ 0 ~:
-
o( s) p(S,I) = '--2-
+r .
-¥
Note then that p(s,O) = < 0 fat' all s > O. By Lemma 3.1 in (Sontag, 1990), onc knows t.hat there exist a smooth function 9 : IIh o ~ IEt>o and a K oo -fullction Cl'4 so that for ea.eh pair (8~ r) '::: IR;o for which (l4(r) < s, necessarily p(s, 9(5)r ) < O. Wc would like t.o define t h,' matrix function f (O in terms of h(O and g(I~IA) ' However, the function I ~IA is in general not smooth. Since g(.) itself is smooth and the fUllctio n E >-> I{I A is globally Lipsdlitz 011 II!', it follows that g(lEI A ) is locally Lipschitz. It is a standard fact tbat there is some smooth function 9 : !Rn -+ 1Ft sa.tisfying
an Iss-syst.em .
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for all { E !Rn (see for instance the a.pproximation theorem given in the appendix of (Lin et al., 1996)). We now define the smooth m(~trix function r 0 uy: for{EJRn
.
As in the ti.me invariant case, ISS implies that t.he system is UGAS if the inpllt. is identically zero, t.hat is, when u ~ 0, one has
Ix(t)1 :0 !9(I~I, t - to),
(ll)
Vt;::: O.
(15)
(S.., (Lin et aL 1996) for mc.re detailed discussions on
Then it holds t.hat
IJGA S.)
whenever I{I A ;::: x(lvl), where x(r) = ,,;-' (r), and consequently, V is an Iss·· Lyapunov function for the system
We can study the ISS property of the time-varying system E, by reducing to the study of the ISS property of the following time· invaria nt system:
± = I( x, z,u),
i
= 1.
(16)
(12)
To conclude that the system ( 12) is ISS, one needs to modify the procedure of constructing the function r 0 SO that the system i, forward complete. Let '1'0 be the function given in the ~tatement of Lemma 11 . Then pick up any smooth function 9 : IRn --+ IR>o so t.hat
Y(O :0
min {
Then one defines
r
~i.~~~,
Because of the nniql1cne.r.;s property of the systems Et and (16), for every solution of the systems, it holds that
'PoW},
(x(t
to, u), t + to) =
k(x) + [(x),,)
(13)
Now we assume that t.he sY:item Et is ISS! Le., there exist (J E K.L and l' E K. sucb that (14) holds. Let A = {O } x JR. Consider any solu tion ('I'(t, €, to , u), '/J(t, to» of the system (16). By (17) , olle has
I('I'( I, ~, to, It), !/J(t , to») lA = 1'I'(t, e ta, "«)1 = Ix(t + to, €. to , "t;)1 :S !9(1{I, (t +to) - to) + l' (1I["t;l~ll) = !9(1(( to) lA' t) + ~ <11"11)
3. ISS FOR. TIME-VARYING SYSTEMS Consider the following time-varying
::: = j(:c , t,
sy~tems:
u),
(E, )
where f : IRIl x III x l~m -4 (Rn is locally Lipschitz. and frO, t, 0) = 0 for all t. For each ~ E JRn and each measurable locally essentially bounded t<, we use x(t, ~, to , u) t.o denote the solution of Et with t.he init.ial condition x(to) = ~. III this section, we consider thf~ with respeet t.o th r. origi n.
ISS
property of system Et
Definition 15. "Vc say that the time-varying system El is ISS if t.here exi!:it a K:L-function f3 .md a. K:'fnnction ; ~u(;h that., for each nu: ~asurable locally C8.'I)entially bounded u and each ~ E IRn , the soLution :t: (t) = x (t , X, to, u) satisfies ,~(I€I, t - t n )
+ (11,," 11)
(17)
=
is forward complete, and V is an lss-Lyapunov function for the system. By L"mma 7, Olle sees that system (13) is ISS. 0
Ic(l)1 :0
~,
for all t,to,:I;O and u, where uio is defined by ut(t) = u(t + to). We also define ut; b.v letting ut; (t) ,,(t - to).
Rn.
It is then dear that the system
= f(x,
+ to,
('I'( t , x" , to , u~), ",(t , t.o)),
by ~ E
j;
Let ('I'(t, (, A, u), ",(t , A)) d""ote the solution of (16) with the initial condi tion '1'(0) = ~ , ",(0) = A and the control ". Note that "'(t, A) =, A + t is independent of { and u.
Thus, t,he system (16) is ISS with respect to the set
A.
Using all al1alogous argumenl., one can show that, conversely, if (16) is ISS with '.'peet to ,,4, the n the syst.em E t is ISS.
Ohserve that if system (16) admits an function V with respect to A, then the tomatically forward cO ll1pl eV~_ Now , let Lyapunov fUIlctiuII for syste Hl (16) with then there exist (\1 , a2 , 0:3 E /Coo a.nd X E
for all
(~,
lss-Lyapunov system is auV be a ll ISSr Csp f!ct to .A, K.'X) such that
.\) E lI!:" x 111., nnd
(14)
D, V(E, A)f({,.\, 11)
for all t 2: to. where Ufo is the function defined by "I,(t) = 0 if t < to , a nd « 'O (t) = u (t) if t ;::: to. 0
+ D, V(~, A) :S -a,(I(E, A)I A)
1958
(19)
whenever I(~, A)I." 2: a nd (19) become
x(lvl). A,
::;
"1 (I~I)
V(~, .\) ::;
A)I..
I(~,
= I~I,
(18)
Not.e then that (23) and (24) a re the same as:
Ql(1W :5
a,(IW
for all for all (e.\) E II!.n x lil, and
whenever I~I 2:
(E, A) E Rn
whenever I~ I 2:
x( lvi).
We say that a smoot h function V is an IsS-LyapuIlov function for the time-varying system Et if there are "1,02,0, E 1(= a nd X E !C oo such that (20)-(21) hold. T he above discllssion shows that the existence of an ISSLyapunov function for E" guarant('Cs that. the system is
X
A) :5 Q2(1W
(25)
II!., and
D( ll (~, .\)/((, A, v)
+ D, qC.\):5 -03( 1( 1) (21)
D{V(~ , ).)f(~ , )" ·v )
V(~,
(20)
+ D, V(~,
x(lvi), where t(,·)
).) :5
-",(lW, (26)
= p-'(r ).
Hence, we proved the followillg: Proposition 16. The time-varying system I:: t i[i ISS if and only if it. admits an Iss-LyapllIlov function.
ISS .
Assume now that. system (16) is lSS. Note also that the [SS property guarantws that the system is forward complete. Following exa('. t Iy the same steps as in the proof of Lemma 2.2 in (Sontag and '\Tang, 1995) (essentially, onc only needs to repi ar.e the Euclidean norm in the state space by the norm I-lA)' one shows that there exists a smooth K:x.o-fnnct ion p such that the syst em
i: = f i x, z, d(I.)fI
i
= 1,
(22 )
is UGAS with fe!:lpect (.0 .A in the sense of (Lin et al. , 1996), where 'lit ) [0, (0) -+ [0, lJ'" st.and, for timevarying parameters. That is, there exists some ,Bo E K.L such that for ever." d with IIdll :5 1, it holds that 1(t,~, )., d)IA
:5
d(l(~, A)IA ' f.),
V t 2: 0,
where (t , ~ , A, d) is the solution of (22) with the initial value ..\.1 and t.he time-varying pa.ramct er d.
«(
Using the converse Lyapunov theorem in (Lin et al., 1996, Theorem 2.8), one knows that syst.em (22) admits a Lyapunov function V wit.h respect t.o -4, that is, a smooth function l/ satisfying the following: • There exist
0 1 , Ct 2
"1 (I(~ , ).) 1.. )
E K:
:5 V (e .\) :5 "2 (1 (~ , ).) 1.. ) (23)
for all (C .\) E m:" x IR; • There exis t 0 :) E !COt.:: such that. D( V(~, ). ) f(~, A,
:5 for all
~,
all ), a nd all
Idl :0:
.\)I A)
,Jiang, Zhong-Ping, Andrew Teel and Lament Praly (1994) . Small-gain theorem for ISS systems and applications. Malhemn.tiui of Control, Signals, and Systems 7, 95- 120.
Krstic, Miroslav, Ioannis Kan ellakopouios and Petar V. Kokotovic (1995) . NonUIlcar and Adaptive Control D esign. John Wiley & Sr,ns . New York. Lin, Yuandan (1992). Lya punov FUnction Techniques for Stabilization. PhD t.hesis. Rutgers, The State University of New Jerscy. New Brunswick: New Jersey. Lin, Yuandan, Eduardo D. Son tag and Yna n Wang (1995). Input to state stabilizability for para meteri~ed families of systems. International Journal of Robust and Nonlinear Control 5(3), 187- 205. Lin, YuandaIl, Eduardo D. Sontag and Yuan Wang (1996). A smooth convel-se Lyapunov theorem for robu, t stability. SIA M }"umal on Control and Optimization. Sontag, Eduardo D. (1989). Smooth stabilization implies coprime factorization. IEEE TranMlctions on katomatic Control AC-34, 435-443 . Sontag, Eduardo D. (1990) . FUrther facts about input t.o sta.te stabiUzation . IEEE Tran.'iuctions on Automatic Contml AC-35, 473-476. Sontag, E
dp(IW) + D, V(~, ).)
- ",,(I(~,
4. REFERENCES
(24)
1.
A potential problem here is that one does not know if system (22) is cornplr:t.r:. However. one cau apply Theorem 2.9 in (Un et at., 1996) to get the exist.ence of t he Lyapunov function . Though A fail::! to be compact, Theorem 2.9 st.ill a pplies to (22) with t.he set A. To ,ee t,his~ note that the key step in the proof of Theorem 2.9 is to estahlish Lemma 7.2 in (Lin d ut , 1996). Onc ca.n prove the same. rp-suIt for system (22) with A by following exactly t.he sru ne st.eps.
Sontag, Eduardo D. and Yuandan Lin (1993) . Stabilization with respect to noncompact sets: LyapullOV characterizations and effect of bounded inputs. In : Nonliuf'!ar Control S!;stems Design 1992/ IFAG S1/1"1109ia Se"es (Miche! Fliess, Ed.) . pp. 43-49. Pel'gamon Press. Oxford . T sinias, John (1989). Suffir.icllt. Lyapunovlike conditions for stabilization. Math t::matics of Control, Signals, and Systems 2, 343 ..357.
1959
2b-03 1
131h Triennial World Congrc~<; . S3J1 Fr:mci!'>co. USA
INPUT-TO-STATE STABILITY WITH RESPECT TO NONCOMPACT SETS I Yuandan Lin
Depa,·tme,.t 01 Mathematics, Florida Atlantic Un;ve"sity, Roc. Raton, FL 334.11, USA
Abstract. Our objed is to study the input-w-state stability with respect to closed, not necessarily compad, sets. A previous result on feedback st.abilization with respect to equiHbria is generalized to the case of noncompact sets. A Lyapunov characterization for t he input-t.o-state stability is also established for time-varying systems. Keywords. Input.-to-state stability, Lyapunov method, Nonlinear systemf->
1. INTRODUCTION
Onc of the most. importaut issues in t he study of tontrol systems is that of understanding t.he dependence of state trajectories 011 the magnitude of inputs. In the late 80 's, a precise definit.ion for input/output stability (ISS) was iutroduced in (Sont.ag, 1989). Since then, the notion of ISS has bcx:n proved a fundamental c.onccpt [or t he study of system stability, and has been subsequently employed by many authors in different area.'!, see for inst.ance (Tsinias: 1989; .Jiang et al. ) 1994; Krst.ic et al., 1995). The ISS property was originally stateu for stability with respect to an equilibrium state of interest. Various results on characterizat.ions of the ISS property with respect to equilibria 01 (:ompact set~ were established in (Son tag, 1990; Sontag and Wang, 1995). The main object of this work is to invf:',stigate ! hE' IS5 property with respect to dosed, not necessarily compact, sets. We are motivated by potentia l applications to a wide variety of areas. Examples of such applicatioll!i indude problems of robust control and OUtpu t stauilization (w here t he interesting set is t.he zero set of tlte observation map). This motivat.ed t.he st.udy of the "set" v~rsion of the rss property, originally in (Sontag and Liu , 1993), and developed with applicat.ions to t.he st.udy of parameterized famiJies of systems in (Lin et (ll. , 1995). The first purpose of thb wor k is
1
Lo
('xl,cnd from the e'lui-
Supp orted in part by NSf Grant DM S-9403921
librium to the set ISS case the result.s given in (Sontag, 1990) regarding feedback equivalence. We show that a system is stabilizable in the state-space sense, with respect to a given dosed invariant set , if and only if it is feedback equivalent to a system which is set-ISS with respect to t.he same set. The proof is analogous to that. giVfm for the case of eq uilibria, but several subtle details, especially when the completelless property of systems is involved, have to be dealt with. (A preliminary version of these results was given in Chapt.er 3 of (Lin , 1992).) The second purpose of this work is to develop the Lyapunav characterization for the ISS property for timevarying systems. To this purpose, we first treat a t.imevarying system as an augmented time invariant systf.m. The ISS property for t he original system is then COnverted to the set ISS probleIn for the augmented system. vs ing the same technique as in (Sontag and Wang, 1995), we show that a time-varying system is ISS with respect to a equilibrium state if and only if it admits an Iss-Lyapunov function .
2. INP t:T TO STATE STABILITY WITH RESPECT TO SETS Consider the following systen I
:i:(t) = I(x(t), u(t») , (E) where, for each t ~ 0, x(t) .e lRn, u(t ) E lRm, and I : lRfl x lRm -+ W is smooth. Controls, denoted by u, j are measurable, locally eS!lIp.nt.ially bounded fUIlctions from
1954
tR>o to am. For each { E I!i:n and each 'U E L ~ , we denote by x(t, ~, u) the trajectory of the system E with ~ and the input u. This solution initi1ll state x(O) is uniquely defined 011 some maximal interval [0, TJ;", u), with T(, u :S +00 . If T(, .. = 00 for all E an d all u, the system E i:-; fo rward r:omplete.
=
Wc use H to indicate Euclidean norm . Consider any nonempty ~ ubs~t A ~ IR'~; t hen for each ~ E Rn , the point~to-set distance from ~ t o A is denoted by
2.1 Chamcterization via DectJY Estimate In this section l we provide a characterization of the ISS property in terms of comparison functions , which generalizes to arbitrary sets A th e similar condition given earlier ill (Sontag and Wang, 1995).
Proposition 3. Assume tha.t. t.he system E is forward complete. Then E is ISS wit.h respect to A if and only if t here are a K:L-fllnct.ion P a nd a K:-function "Y so that (3)
In particular, 1~llO) .= I~I. A noneIllpty subset A is 0invariant for :E if eve))' solution with zero input start.ing from A is defined for a ll t > 0 and stavs in A when the control is u == 0; i.e.! for Ule corresponding "zero-input" or "undisturhed" system :i:
= 10(:<) = f( x, Il) ,
it holds that x(t , E,O ) E A for each
~
(Eo )
E A and all t 2: O.
Recall that a funetic,n "( : [0,00) -+ [0,00) is of class if it is continuOU9, strictly increasing, and satisfies "((0) = 0. A fUll ction "( is of dass K= if it is of class K a.nd it is unbound ed . A function {3 ; !Iho x D h o -7 iho is of class KL if, for e",:h t 2: 0 , the function fJ(·, tn, of cl ..., K , and , for ,,~ch s 2: 0, the fun ction fl(s , .) is decreasing and teuds 1,0 zero a t infinity. K,
Definition 1. Let A ~ lltn be a IlOIlempty subset. The function 'Y is called an a~~ymptotic yain fot' a given system L if, for each control u , ( 1) uniformly on subsets of the form {~ E litn, I{I A :S r} , 2: O. The system I: is said to have );:>asgmptolic gain with 1'Cspect to A if there is some asymptotic gain r E Kfor it.
holds for each t 2: 0,
U
E
L~,
and
EE ]Rn.
It is clear that if (3) holds for a forward complete system E, then E is ISS, Any fUllction "( for which an estimate (3) holds (for some fJ) will be called here an ISSgain for t.he syst.em (with respect to A); such a, is in part.icular an asymptotic gaiE. To prove the other implication, one can use exactly the same argument as in t he proof of (Sontag and Wa ng, 1995, Lemma 2.7). The only "xception is t.hat the a rgurnenl, in (Sontag and Wang, 1995, Remark 2.8) requires A to be compad. To drop thE' compactness assumpt ion , we est.a blish the following result to replace (Sontag and Wang, 1995, Remark 2, 8). Remark 4- In Definition 2, Property 2 is equivalent t.o t.h e following:
2' . There exisr.s a Koo-funclion such that, if I~IA :S J(E) , Ix(t , C u)IA :S
E
8 and
a K-function
+ ' I(lIull), 'It
~
l'
0, lIu. (4)
T
Thus, "'( is an asyrnpw tic gain if the t;ystem is forward complete and for every pair of reals r ,6" > 0, there is some T = T(r ," ) 2: so that, for all " , alll~IA :S r, and all t 2: T ,
°
Definition 2. The system E is said to he input~to-!~tate stable (ISS) with respect to a noncmpty subset A <,;; lltn if the following properties hold: (1) there exists a K ,~ -fllfi( : ti{)n 6(·) so that
Ix(t, ~, uJ IA :S
C,
1112:0,
(2)
whenever I~ I A < 6(0) and 11,,11 :; 6(0); and (2) E has K-asymptotic gain with respect to A. Observe that if Eis lSS , then in part.icular the autonomous system Eo is globally asympt.otica.lly stahle with respect t.O A (in t he sense in (Lin et nl. , 1996) ).
Clearly Property 2' implies Property 2. To show that Property 2 implies Property 21 , first observe t hat since the function 6(·) in Property 1 of Definition 2 is of class }Coo, it follows that for any l' 2: 0, there exists some M > 0 such that
Now for "(l
(~ach
(r) =
r 2: 0)
w(;!
let
sup{lx(t ,~,u) IA
. t 2: 0, I~IA :S
t·,
lIull :S r}.
°
Not.e t.hen 71 is a nondecreasiog function, and it follows from (2) that "(.(r) -+ as r -+ O. Let 1 be a Koofunction t hat majorizes the function 'rl. Then it hoJds that
To complete the proof of Re!llark 4 , one chooses J(r) = 1,I(r ). 0
1955
2.2 ISS-Lyapu1lov Pvnetinn .• An ISs-Lyapunov junction with respect to a set A for system 1: is a smoot ~l fUIlction V: TI{n -+ lR>o which satisfies the following conditions: -
• V is proper a.uG. positi ve definit.p- with respect to t he set A, t.hat is, therp. exist il l , (}2 E }Coo such t hat for a U I;. E DI" , (5)
• there exist a continuous, positive definite function <>3 a.Illi a Koo -function X snch that t here holds the following implication:
11;.1 ...
~
x(lvi) => V V(Of (~, 11)
(for
ca~h
state
~
E
{R1l
vlaim on page 441 of (Sontag, 1989) (see also (Sontag and Wang, 1995, Lemma 2.U», one can show t.he following: Lemma 7. Assume that syst.em E is forward complete.
If system E admits an lss-Lyapunov function, then it is [SS with the funct.ion X in (6) aB an [ss-gain function. 2.3 Stabilizability and Feedba ck Equivalence Definition 8. The syst em E is smoothly slubi/izable with respect to a dosed set A if there exists a. smooth map k : lR,n --+ ]Rm so t.h a t. t.h e dosed-loop system
S - <>3 (It;. I... )
(6)
amI control value v E IRm).
Remark 5. A slightly different way of dd1.uillg ISSLyapuIlov fUIlctions is to require the fUIlction 0'3 in (6) to be a K,:x>-function t.hat is smooth on (0,00). These t.wo variants are equivalent: if E admi t.s an rs s-Lyapll110V function V with (6), t.hen one can a lways get an ISSLyaplIllO\< functio n HI by modifying l' so that t he following implicat.ion
= J(x,
k(x) is globally asymptot.icaJly sl,a!>le with respect to A , t hat is, system Ec is forward complete, and there exists a KL-function {i so that Ix( t,O I... S {i(l~ I A' t) for all t ~ 0, for every solution x(t , ,;) of Eo. :i;
Definition 9. Two systems :i; = f(x, u) and :i; = .q(x, u), wit h t he same state-space a n ;ind the same control-value space lRTTl, are feedba ck t;:quivalent if g(~,
u) = f(~, k(O
+ r({),,) ,
\I ~ E IItn
,
\I " E IRm ,
where k : Rn --+ wn is smooth, and the m )( 1ft ma trix function r is smooth, and nonsingular for each ~ E Rn. holds for some Koo-function n:1 that is smooth on (0, oo)! where X is t.he same fun ction as in (6). For details of const.ru cting such a fun ction itV , see Remark 4.1 of (Lin et al. , 1996).
Remark 6. In (Sontag and Wang, 1995) , it was shown th at property (6) is also equivalent. to the existence of (} 4 E Kao and (J E K. :such t hat
This equivalence relation is in general not true when A is not. compact. As a.n example . eonsid(;!r the following system in 1R2 :
;', = _·x, + xlq(U -
jl-21),
--7 lIt is a ~IIlooth fuudion sat.isfying q(r) = S 0 and q(7-) > 0 for all r > O. Let. A be the set {(~l' 6) : 6 = O} . Consider the fun ction V (O = ~V2. Clearly V satisfies (5) (as I ~I ... = 1 ~,Il , a nd
where q: R
ofor all r
I(lA ~
Ivl => vF(O J(t;., 11) S -{i·
But it is easy to see that V fails to satisfy property (8). Note that an lss-Lya.punov function V for E is automatically also a Lyapunov fUIlction for th e 7.cro-inpnt system Eo with respect t.o A . Following the proof of the
It is clear that the above not.iun indeed defines an equivalfmcf. relat.ion. Definition 10. The system E is smoothly [SS stabilizable with ,....pecl to a closed set A if it is feedback equivalent t o a sys tem whkh is ISS with respect to A.
Assume that the ~ystem E is :,moothly stabilizable with resp ect. to a closed set A. The objective is to find a system , feedback equivaleIlt to the given one, which is ISS with respect 10 A. We fi rst establish the following result.
Lemma 11. Assume that the autonomous system 'Eo is forward complet.e. Then , there exists a smooth function IPa : [Rn -+ IR>o so that for any smooth function satisfying 0 S 'P(O S 'Po(~) for al l ~ E ~", the syst.em :i; f (x, ,p(xlu) (Er)
=
is also forward complete. We fi rst. need t he following tr ivial consequence of cont.inuity on initial conditions and controls.
Rema,7'k 12. Assume tha t Eo is forward complete. Then for any compact subset K ~ IRn, there exist some compact neighborhood L of K and some 6 > 0 so t.hat for any ~ E K and any lIull < ,I, t.he solut.ion x(t, ~,u) is define.j ou [0, 1J a nd
1956
x(t,
~ ,u) EL,
'I t E [0, I], 'I~ E K ,
'111,,11 < 8.(9)
Indeed, let K be any I~ompact subset of RU. By the com-
plet.eness assumption of Eo, one knows that x(t, €, 0) is defined for all t ~ 0, and. in particular, it is defined on {x(l , ~, 0) E E K , t E [0, In , and [0 , IJ. Let Lo let L be any compact neighborhood of La . It t hen follows from t.he eontinuity of the map (~ , u) >-t x(·, { , u) C([O, 1]) t.hat t.here exists some o( > 0 so that for each 1 < 0(, the solu'/ so t.hat IT/ - El < o( a nd each 11111 t.ion x(l, 'I, '1.1) of 1:: is defined OIl [0 , 1 and x(t , ~, n) E L, 'It E [0, 1], 'I 1'/- ~I < 0(, '1 11" 11 < 0(. By t.he
PROOF. Assume that system E is smoothly stabilizable with respect to A by u k(x). Then by the converse Lyapunov theorem given in (Lin et al. , 1996), there exists a smooth Lyapunov function for the closed-loop system so that the following holns:
=
=
• V is positive definite with respect to A i • t here exists a Koo-fullction (l' so that
c:
<.:ompactne..ss of <5
f( ,
onc knows t.hat there
ex ist~
VV(Of (~, k«())
S
-a
Sinc:e f , l/ aud k are smooth, there exists a smooth function G(E, v) such that
VVWf({, k(O
some
> 0 so that (9) hold,.
+ v)
= VV(Of(E, k({»)
+ G(~ , ti)".
For imtance, G can be defined by
,
Wc now ret urn to prove Lemma 11. Proof of Lemma 11. \Vest.art. with the compact set Kt = El , where for allY int.eger k > 0, we use Bk to denote t he set {E: IEI ~ k ) . Let L, and 0, be as in Lemma 12. Let I<2 = Lt U B'2' Uy inductioll, one knows that there p.xist an sequence of compact. suhset.s
G({ , v) = / ..~ F({, Av)dA , (11.1.
n
where
F(~,
'lE E ](.,
'11 E [O,IJ,
'I llull < Ok.
Now let !fo : II~n '~I~> o be a smooth function so that 'Po(O < o,-/k fol' all ~ E 1(. \ Kk _' (w here onc takes Ko as t he empt.y set) . In t he followin~ we show that the (:onclusion of Lemma 11 holds for sueh a choice of 'Po.
V
the claim is raise, t.hen thf'fP. exists some T > 0 so t hat l:c(l)l -+ 00 as t. -+ T. For the input u and the initial state C there is some M > 0 so t hat lu(t)1 ~ M for almost. a ll t E [0, T + 1J and £ E f( M. Pick lip I, > T - 1/2 so t hat x(t,) E K T \ [{r- ' fol' some I > M. Then 1'P(x(t,)u(t) 1 < 0" and hmce, x(t + t,,{) x(l, x(t,)) is defined fo r all 0 ~ t ~ 1, contradicting to t.h e assumption t. hat x(t ) is not defin ed fO f t > T. 0 Suppo~ e
Corollan) 1S. If syst"", 1:: is slIIoothly st.ahilizable by = k (x), then there exists a smooth function CPu : an --t lR>u so that for every smooth funct ion 'P satis(ying 0 ::; 'P (~) ~ 'Po(O fOl a ll ( . the s)'stem T = f(x , k(x) +
H
T he foUmving is th p. main
l'e~mlt
in tltis section.
Thcor-c m 14. The sys t.em 1:: is smoothly stabili.able with rp$pect. t o A if and only if it is feedb a,ck equivalent to
V
h(~) ~
1
~ lvi ,
for all { E II!.", all vEII!."'. (For the details on t.he existence of such functions, s.,e (Lin, 1992, Sec. 3.4.1).)
"'ith
~uch
a choice of VV(Of
h~
we
I ~ave
((k(~) + h;~))
=vV({)f«( , k«()) + G ({, h~{)) ,,~{)
=
~ (J.
\
G ( {, h({)) h({)
Let'P be any function satisfying 0 ~ A{) ~ 'Po (E) for all ~ E IRn. Pir.k lip a ny { E IRn and a ll." input 11. Let x(t ) denote the corrc_'ponding trajectory of:i: f( x, 'P(x)u) defined on some interval.
Claim: x(t) i, d cnned for all t
= vV( Of(~, '<0 + It).
Now we choose a smoot h fUlJctio n h({) wit.h for all Esuch th at
wit h Ek c;;: Kk and positive numhers 81, 02 • ... such t hat
x(t,E,u) E ](1+1,
u)
~
-a(I{I A ) + 11!1
for all Eand v. We now deHne a function p : 1R~0 -t 1R~ 0 ~:
-
o( s) p(S,I) = '--2-
+r .
-¥
Note then that p(s,O) = < 0 fat' all s > O. By Lemma 3.1 in (Sontag, 1990), onc knows t.hat there exist a smooth function 9 : IIh o ~ IEt>o and a K oo -fullction Cl'4 so that for ea.eh pair (8~ r) '::: IR;o for which (l4(r) < s, necessarily p(s, 9(5)r ) < O. Wc would like t.o define t h,' matrix function f (O in terms of h(O and g(I~IA) ' However, the function I ~IA is in general not smooth. Since g(.) itself is smooth and the fUllctio n E >-> I{I A is globally Lipsdlitz 011 II!', it follows that g(lEI A ) is locally Lipschitz. It is a standard fact tbat there is some smooth function 9 : !Rn -+ 1Ft sa.tisfying
an Iss-syst.em .
1957
for all { E !Rn (see for instance the a.pproximation theorem given in the appendix of (Lin et al., 1996)). We now define the smooth m(~trix function r 0 uy: for{EJRn
.
As in the ti.me invariant case, ISS implies that t.he system is UGAS if the inpllt. is identically zero, t.hat is, when u ~ 0, one has
Ix(t)1 :0 !9(I~I, t - to),
(ll)
Vt;::: O.
(15)
(S.., (Lin et aL 1996) for mc.re detailed discussions on
Then it holds t.hat
IJGA S.)
whenever I{I A ;::: x(lvl), where x(r) = ,,;-' (r), and consequently, V is an Iss·· Lyapunov function for the system
We can study the ISS property of the time-varying system E, by reducing to the study of the ISS property of the following time· invaria nt system:
± = I( x, z,u),
i
= 1.
(16)
(12)
To conclude that the system ( 12) is ISS, one needs to modify the procedure of constructing the function r 0 SO that the system i, forward complete. Let '1'0 be the function given in the ~tatement of Lemma 11 . Then pick up any smooth function 9 : IRn --+ IR>o so t.hat
Y(O :0
min {
Then one defines
r
~i.~~~,
Because of the nniql1cne.r.;s property of the systems Et and (16), for every solution of the systems, it holds that
'PoW},
(x(t
to, u), t + to) =
k(x) + [(x),,)
(13)
Now we assume that t.he sY:item Et is ISS! Le., there exist (J E K.L and l' E K. sucb that (14) holds. Let A = {O } x JR. Consider any solu tion ('I'(t, €, to , u), '/J(t, to» of the system (16). By (17) , olle has
I('I'( I, ~, to, It), !/J(t , to») lA = 1'I'(t, e ta, "«)1 = Ix(t + to, €. to , "t;)1 :S !9(1{I, (t +to) - to) + l' (1I["t;l~ll) = !9(1(( to) lA' t) + ~ <11"11)
3. ISS FOR. TIME-VARYING SYSTEMS Consider the following time-varying
::: = j(:c , t,
sy~tems:
u),
(E, )
where f : IRIl x III x l~m -4 (Rn is locally Lipschitz. and frO, t, 0) = 0 for all t. For each ~ E JRn and each measurable locally essentially bounded t<, we use x(t, ~, to , u) t.o denote the solution of Et with t.he init.ial condition x(to) = ~. III this section, we consider thf~ with respeet t.o th r. origi n.
ISS
property of system Et
Definition 15. "Vc say that the time-varying system El is ISS if t.here exi!:it a K:L-function f3 .md a. K:'fnnction ; ~u(;h that., for each nu: ~asurable locally C8.'I)entially bounded u and each ~ E IRn , the soLution :t: (t) = x (t , X, to, u) satisfies ,~(I€I, t - t n )
+ (11,," 11)
(17)
=
is forward complete, and V is an lss-Lyapunov function for the system. By L"mma 7, Olle sees that system (13) is ISS. 0
Ic(l)1 :0
~,
for all t,to,:I;O and u, where uio is defined by ut(t) = u(t + to). We also define ut; b.v letting ut; (t) ,,(t - to).
Rn.
It is then dear that the system
= f(x,
+ to,
('I'( t , x" , to , u~), ",(t , t.o)),
by ~ E
j;
Let ('I'(t, (, A, u), ",(t , A)) d""ote the solution of (16) with the initial condi tion '1'(0) = ~ , ",(0) = A and the control ". Note that "'(t, A) =, A + t is independent of { and u.
Thus, t,he system (16) is ISS with respect to the set
A.
Using all al1alogous argumenl., one can show that, conversely, if (16) is ISS with '.'peet to ,,4, the n the syst.em E t is ISS.
Ohserve that if system (16) admits an function V with respect to A, then the tomatically forward cO ll1pl eV~_ Now , let Lyapunov fUIlctiuII for syste Hl (16) with then there exist (\1 , a2 , 0:3 E /Coo a.nd X E
for all
(~,
lss-Lyapunov system is auV be a ll ISSr Csp f!ct to .A, K.'X) such that
.\) E lI!:" x 111., nnd
(14)
D, V(E, A)f({,.\, 11)
for all t 2: to. where Ufo is the function defined by "I,(t) = 0 if t < to , a nd « 'O (t) = u (t) if t ;::: to. 0
+ D, V(~, A) :S -a,(I(E, A)I A)
1958
(19)
whenever I(~, A)I." 2: a nd (19) become
x(lvl). A,
::;
"1 (I~I)
V(~, .\) ::;
A)I..
I(~,
= I~I,
(18)
Not.e then that (23) and (24) a re the same as:
Ql(1W :5
a,(IW
for all for all (e.\) E II!.n x lil, and
whenever I~I 2:
(E, A) E Rn
whenever I~ I 2:
x( lvi).
We say that a smoot h function V is an IsS-LyapuIlov function for the time-varying system Et if there are "1,02,0, E 1(= a nd X E !C oo such that (20)-(21) hold. T he above discllssion shows that the existence of an ISSLyapunov function for E" guarant('Cs that. the system is
X
A) :5 Q2(1W
(25)
II!., and
D( ll (~, .\)/((, A, v)
+ D, qC.\):5 -03( 1( 1) (21)
D{V(~ , ).)f(~ , )" ·v )
V(~,
(20)
+ D, V(~,
x(lvi), where t(,·)
).) :5
-",(lW, (26)
= p-'(r ).
Hence, we proved the followillg: Proposition 16. The time-varying system I:: t i[i ISS if and only if it. admits an Iss-LyapllIlov function.
ISS .
Assume now that. system (16) is lSS. Note also that the [SS property guarantws that the system is forward complete. Following exa('. t Iy the same steps as in the proof of Lemma 2.2 in (Sontag and '\Tang, 1995) (essentially, onc only needs to repi ar.e the Euclidean norm in the state space by the norm I-lA)' one shows that there exists a smooth K:x.o-fnnct ion p such that the syst em
i: = f i x, z, d(I.)fI
i
= 1,
(22 )
is UGAS with fe!:lpect (.0 .A in the sense of (Lin et al. , 1996), where 'lit ) [0, (0) -+ [0, lJ'" st.and, for timevarying parameters. That is, there exists some ,Bo E K.L such that for ever." d with IIdll :5 1, it holds that 1(t,~, )., d)IA
:5
d(l(~, A)IA ' f.),
V t 2: 0,
where (t , ~ , A, d) is the solution of (22) with the initial value ..\.1 and t.he time-varying pa.ramct er d.
«(
Using the converse Lyapunov theorem in (Lin et al., 1996, Theorem 2.8), one knows that syst.em (22) admits a Lyapunov function V wit.h respect t.o -4, that is, a smooth function l/ satisfying the following: • There exist
0 1 , Ct 2
"1 (I(~ , ).) 1.. )
E K:
:5 V (e .\) :5 "2 (1 (~ , ).) 1.. ) (23)
for all (C .\) E m:" x IR; • There exis t 0 :) E !COt.:: such that. D( V(~, ). ) f(~, A,
:5 for all
~,
all ), a nd all
Idl :0:
.\)I A)
,Jiang, Zhong-Ping, Andrew Teel and Lament Praly (1994) . Small-gain theorem for ISS systems and applications. Malhemn.tiui of Control, Signals, and Systems 7, 95- 120.
Krstic, Miroslav, Ioannis Kan ellakopouios and Petar V. Kokotovic (1995) . NonUIlcar and Adaptive Control D esign. John Wiley & Sr,ns . New York. Lin, Yuandan (1992). Lya punov FUnction Techniques for Stabilization. PhD t.hesis. Rutgers, The State University of New Jerscy. New Brunswick: New Jersey. Lin, Yuandan, Eduardo D. Son tag and Yna n Wang (1995). Input to state stabilizability for para meteri~ed families of systems. International Journal of Robust and Nonlinear Control 5(3), 187- 205. Lin, YuandaIl, Eduardo D. Sontag and Yuan Wang (1996). A smooth convel-se Lyapunov theorem for robu, t stability. SIA M }"umal on Control and Optimization. Sontag, Eduardo D. (1989). Smooth stabilization implies coprime factorization. IEEE TranMlctions on katomatic Control AC-34, 435-443 . Sontag, Eduardo D. (1990) . FUrther facts about input t.o sta.te stabiUzation . IEEE Tran.'iuctions on Automatic Contml AC-35, 473-476. Sontag, E
dp(IW) + D, V(~, ).)
- ",,(I(~,
4. REFERENCES
(24)
1.
A potential problem here is that one does not know if system (22) is cornplr:t.r:. However. one cau apply Theorem 2.9 in (Un et at., 1996) to get the exist.ence of t he Lyapunov function . Though A fail::! to be compact, Theorem 2.9 st.ill a pplies to (22) with t.he set A. To ,ee t,his~ note that the key step in the proof of Theorem 2.9 is to estahlish Lemma 7.2 in (Lin d ut , 1996). Onc ca.n prove the same. rp-suIt for system (22) with A by following exactly t.he sru ne st.eps.
Sontag, Eduardo D. and Yuandan Lin (1993) . Stabilization with respect to noncompact sets: LyapullOV characterizations and effect of bounded inputs. In : Nonliuf'!ar Control S!;stems Design 1992/ IFAG S1/1"1109ia Se"es (Miche! Fliess, Ed.) . pp. 43-49. Pel'gamon Press. Oxford . T sinias, John (1989). Suffir.icllt. Lyapunovlike conditions for stabilization. Math t::matics of Control, Signals, and Systems 2, 343 ..357.
1959