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Feedback Stabilization of Perturbed Nonlinear Discrete-Time Systems
Copyright © 1996 IFAC (3th Triennial World Congress, San Francisco, USA
2b-14 5
FEEDBACK STABILIZATION OF PERTURBED NONLINEAR DISCRETE-TIME SYSTEMS I Cesar Cruz' and Henk Nijrneijer" • CICESE, Mexico 2 ,.. University of l'wente, The Netherlands 3
Abstract. A robu:-it feedback IStabilization problem of nonlincar discrde-time syst.f!m~ i~
studied, A sufficient condition for feooback ~tabilization of a class of perturbed nonlinear systermi (which are not necessarily feedback linearizable) i~ obtained_ It is l:ihown that. if the Jacobian linearization of the llominalsYl:itcm io stabilizable l then the pert.urbed non linear system ilS asymptotically stabilizablc. A linear feedback that stabilizes the nominal system also locally stabilizes the perturbed system. A simulation example illustrate the obtained results.
Keywords. :\"onlinear discrete-time systems 1 feedback stabilization pert.urbed j
systems,
robllstnes~_
I. INTRODVCTlO:\
The stabilization problem plays a central role in dynamical control systems, A matter of practical importance is t.o determine the extent of aHymptotic stability of an asymptotically stable equilibrium point, Le., how large a perturbation from the equilibrium can be allowed and we can still be sure t hat the traject.ory will return to that equilibrium. In the majority of the applications 1 we encounter perturbed system:;; which are the product of modf:~lillg errors) uncertaintie~ and disturbances. Hence a property desirable of cOlltrollers is t.heir robustness, i.e. 1 their capacity to work suitably even when t.hif'! controlled. system differ::; from lhe system designed. .For continuom; time nonlinear systems several papers have been published where the stability of the closcdloop system i8 analyzed when uncertainties are present (e.g .. Khalil, 1992; Liao, et al., 1992: Marino and Tomei, H193). It. ha.., been proved that. a controller may tolerate uncertainties if the clo.'ieu-loop nominal system has an RSymptoti(:ally stable equilibrium, and t.he perturbation sati'ifiili some olher conditiollf:i. In the discrete-time case, in (Byrnes, et aI., 199:3) it is shown t.hat. any nOIlI Research performed during a visit in Alltumn'94 of Cesar Cruz at the University of 'l'wente, supported by the Faculty of Applied Mathematics. 2 Department of Electronics and Telecommunications, p.a. Box 434944, San Dif'gu, CA 92143, USA. PhUIlp.: +52.(617).4450lx2,~30; fax: +rI2.(617).4.'iISa; e-mail: ccruzOciCCSC.UlX 3 Department of Applied Mathematic!'<, p.a. Box 217, 7500 AE Enschede, The Netherland!'J. Phone: +31.53.893442; fax: 31.53.340733; e-mail: [email protected]
linear system with Lyapunov-stable unforced dynamics can always be globally stabilized by smooth state feedback if suitable cont.rollability-like rank conditions are satisfied. In particular, in (Cruz and Alvarez, 1994) the authors have di-.;cm:;soo the stability of a class of nonlinear fef!dback linearizablf' :
2424
2. PROBLEM STATEMENT
ConlSider the class of discret.e-time lIonlinear systems d epending on a real-valued parameter vector c E n q
1:(k I
I) ~.
(1 )
f(x (k) . lI(k ),£) ,
where x :..:; (X t , ___ ,x n ) and Tt -- (u! , ... , u m ) are smooth 10(:a1 coordinat.es for the state spa(X~ Ai and input i:ipace U respe(~t ive ly, \Ve 3s;;urne that f i:-; a analytic vcct.orfield ill J: , 'It and 1.;:, anrl that f(ll, 0, E)
(2)
0, '1£.
c
Problem 1. (Robust feedback stabilization of nonlinear control systems): Con.id,,· the .• y.• tem (1 ) .• atisfying (2). Under which mnditions does there exist" smoolll. static staJ.e feedba ck
/(x ,u) c' f(x , u,a). the system (1) corresponding to the value e ::: 0 is called the unpertur-bed or nominal sy&tern, while it will be called lite pC7'turbed syst em whe n e .f O. T he verturbed sys t.em could result from modcling errors , l:lg iug of (Jh'y~i(:81 cumponents, uncertainties and d~t.urUaIl(: es which exit-I. ill any TCH listic problem .
Let us int. rodu<.:c the notation B(O , R) to indicate the set of v"d oTS 1: satisfying 11 x - (I 11 < R ; that. is, it. is the sct. of all point.~ who...e Ellclidt'..an dist ance fro m 0 ie,: less t han R .
u(k) -- (» (x(k)) ,
where () , M loop system
+ 1)
(3)
J(J:(k)),
(3) is stable if there ., an Ro >
a for which the following
is true: For e very R < Ro. there is an r, 0 < r < R, such t.ha t if x(O ) i, in,ide R(O, r ), t.hen x (k ) i, inside R(O, R) for a ll k 2: a, (a) is aBymptotically .table if it stable aud B(O . r ) (;811 he chOljcn such that IIx (O) 1I
=>
lim x (k) =
k -- .x,
a.
Defin;tion f. (Luenuerger, 1979) A function v defined un Ii region n of the stat e space of (3) and containing t he equilibrium point. ;1: . - 0 i~ a Lyapunov function for the disc rete· tirnp. r.;ystem (8) if it sati.<;fies t.h p. foll owing three requiremenb;: (1) 1.1 is (:o ntinuous. (2) 1! (3:') has R unique minimum at O. (3) The function L'!.v(x(k)) . 11 (f(x(k))) - v(x (k)) sa!.isne. L'!.1'{x(k)) SO, for all J: E n. Theo rem 1. (Asymptotic stability) (Luenberger,1979) H t here ex',ts 8 L.Yapunov function v(x) in B(a , Ro) with Ce nt.er the o rigill , s uch that the function .6.v{x) is strict ly negative at every point (except O) ~ t hen the origin or t.he non linear syste m (:i) is asymptotically stable.
Rm, w;U, 0'(0) - 0 , .•".oh that the closed-
x(k
+ I)
-, f(r.(k) ,cr(x (k)),e:)
(r.)
has 0 a.s a locally asympto tically stable equilibrium for all c in a neighborhood of l: ;;;;;; 0 ? Note that 0' is selected ind~pende nt. of c. Let us write the ~y8tcm (5) in the form
x(k
+ I) =
/( x (k) , n (x(k)))
+ g(x(k) , o(x (k)) , E) , (6)
where g(x , n(x) ,E) = f( x , <>(x) , €) - /(x , n(x ») is the perturbation function . \Ve make the following a....sllmption :
(A) : The linearizat;on of Ihe nominal system x(k + 1) -- j(x(k ), tI(k))
Dejinition 1. (Luenberger, 1979) Let x = 0 be an equilibrium point of tlw nonlinear syst.em x(k
~
around the origin} is
(7)
stabili ~ able.
If We define Ao ~ ~(a,O) and Ro = ~(a, a), thell the Assumption (A) guarantees t he existe nce of a matrix FE R.mxn such t.hat all eigt mvalue~ of (.1\0 + BoF) have modulu:o> st.ric:tiy less than one .
Proposition £. Consider t he nomina) syst e m (7). A..." umc Assumption (A) holds and 1<1. F he such that (An I RoF) is asymptotically st.a ble, t hen the c1ose
+ I)
= j(x( k ), Fx(k) ).
has 0 as a locally asymptot ically
Proof.
stabl~
(8)
e;quilibriurn.
Suppo,e Assumption (A) hoIck It follows that
for any positive df!finitc ITwtrix Q = qT , t,he solution p .;.;; pT of the discrct.c-timt, ~ Lyapunov matrix equation
(Ao + BoF)Tp(Ao 1 RoF) - P ~ -Q , is a posit.ive definite matrix see, e.g. : (Vidyli.'Sagar, 1993) . \Ve will ~how that v(~: ) =c x T Px is also M Lyapunov funet.ion for the c1",cd-Ioop system (8) in a lIeighborhood of /(a, a) ~ o. To thl'i end, we evaluat.e t.1I(x(k)) = IAox(k)
Our goal in this not.c is to study the following problem.
+ Ro,,(k) + j
(x(k ), u(k))
-A"x(k) - Bou(k)rr .
2425
.PIA"x(k) I R"vik:1 = I(A" I B"F) x( k) I
,PI(Ao
-
I,
+ j(x(k), v(k)) (x( k) , Fx(klll
T
Suppose v(x) is we ll-defined in B(O, ~). Let R be arbitrary with 0 < R < ~. Let R, < R be selected so that if x E B(O , R,) the n i(x , Fx) E H(O . ~) (Le. , if the state lies illtiide 8(0, RI) , it will not jump o ut of B(O , R,,) in one step). The continuity of t.he funct.ion I gllarant fCi'j the existence of RI'
.
+ BoF)x(k) + I, (x(k) , Fx(k»)J
-3:I'(kjPx(k) where
It ...
Fx. I,- (x. Fx)
6
fix . F.r) - (Ao
.cc
+ BoF ) x,
Let m ,.,. minRI <; lI r U:=; nu V(3:), the exist.ence oft-he minimum m is guaranteed from t.he continuity of v(x) aud the compactnpJ;.'; of the region defined uy R, :Sl\ x I\ :S Ra . Moreover m > v(O) :;ince ll has a unique minimum at O. From t.he conti nuity tJ[ v(x), it. is po.':iSible to select an r, 0 < r < R, such that for x E B(O, r) it holds v(x) < m . This is be(:allSf! near the origin v mu...,'t take value.s cIOtiC to viOl = o. If x(O) E H(O ,r), '.he n v(x(O» < m . Sillce t. v(x) < 0 t he value of" decrea,es wit h time. Therefore, the corresponding trajectory can never go out~idf! 8(0 , RI), and consequently it can ne ver go outside R(O, R), i.c ..
t.v( 3:(k» .o -xT(k)Qx(k)
t2x T (k) (Ao I
with
+ BuF}7' pi,
(x(k) , Fx(k»)
iT (x(!:) . F.r(k») pi, (x(k). F.r(k)) ,
i, (x , Fx) satisfies
Ili, (x(!:), Fx(k))11 <", Ilx(k}11 + c~ IIFx(k)11 :S c,II:"(k)11 + c,c3 1Ix(k)ll, so
\I I(x(!:) , Fx(k»
\I
- Aox(k) - BoFx(k) II x(k) II
Ilx(O)1\ < r ""
I\x(/')I\ < R, < R ,
'I k ~ O.
a.
< c, + C2 03,
Finally, it i< suffic ient to show that v(x (k» ~ 0 k~ Since t.v(x) < 0 for all:c except at origin, t hen vex) must act.ually deCrCA!;f! rnoU4)tonousiy. Thus v(x(k» COIIverges t.o some limiting value m. The only que.c;t.ion i~ whether it is possible fur .ft > viOl = O. This i, nut poosible since v(x(k» eOnVi!rges to tnl it must be t rue that t.v(x(k) converges to zero. But t.v(x) < 0 except at origin. Thus, since L\v(x(k» is continuous x(k) m1L~·t converge to origin and v(x(k)) must (:onverge to v(O) . By Theorem 1, we (,;onclude that the origin is (locally) asymptot.ically stable equilibrium for the nomiutl) sy:-;tem (8). 00.
for all x satisfying I\x(k)1\ < r
t.v(x(k)):S
- IIQllllx(k)II' t 2I1xT(k) IIII(Ao + BoFfll· ·IIPIIIII,(x(k). FT.(k»11 + lifT (x(k). FX(kll lIIIPl\lli, (x(k) , Fx(k))ll ·
Now Jet
- xT(k)Qx(k) :S -).,»>,,,(Q)
II(Au
I
RoFl"'I l ~ cs. I\PI\ -
Ilx(k)1I
2
).,m •• W)
=
,
-c,llx(k)11 ,
= C6,
C, =
2cscn.
Hel1l.:e
t.v(x(k))
< -c, l\x(k) I\' + c, (c, + C,C3) I\x(k) 1\' +C6 (c, + c,c3)' l\x(k) I\' =
I- c, + e7 (c , +c. (c, t
t C,C3:1
c,c3)'lllx(k)II' .
choosing
The following theorem shows how t he origin of (1) will remain asymptotically sta ble fOT !;mall valuC2:0 of parameter g 'I 0; that is the robust stabilizatioll Problem 1 is solvahle for small e. Theorem 3. Consider t he class of discrete-time nonlinear perturbed systems ( 1) . Assume that Assumption (A) hold,. Then the perturhed system (5) has 00,<; a locall y a.~ymptotic:Rll'y stable ('lCluilibrium for c sufficie:ntly small.
Proof. Assume t hat Assumpt.ion (A) holds. We will show that v(x) ;;... x T Px is also a Lyapunov function for the closoo-loop perturbed s~'stem
x(k ensures that t.v (x(k))
< 0, for all IIx(k)1\ < r.
From standard Lyapunov Theorems see, e.g. } (Luenberger, 1979) , we have t hat all traj ector ies starting inside 8(0, r) will ('on verge to the origi n.
+ 1) =
I(x(k) , FX( k),e:)
(with u - Fx , where F such that (Ao + BuF) is asymptotically siable) in a neighborhood of (O,O,c), i.e .,
x(k
2426
+ 1) '- (A. + B,F)x(k) + ft(x(k) , PX(k},E),
IIp(x(kllll S (C8 f.(3:,1-'X.<:} ~ fix , FT, c) - (A.
+ B. F)x,
(9)
+ C9) IIx(k)II 2 + (c6 1 CIO ) IIx(k)II Ill,
(x(k) , Fx(k) ,E)11
+c. IIf, (3(k) , Fx(k)")112 < C" IIx(k)II 2 +C'2 II,,(k) 11 [Cl
with
8f
A, - ,,(O,O,e), vX
+ R.F 0
A,
(Ao
{)f B, - Ou (0, 0, e),
+ A (e»
1 (80
+ B(e»F.
+ B,Fj'"P(A + B.F) -
1>v(x(k» - xT(k)l(A,
Plx (k)
+2x'" (k) (A, +- H,F)" P j,(x(k) , Fx(k), c)
+Jl" (x(k), Px(k),c) Pf, (x(k), Fx(k),e) , i(x,t,,) +- g(x,Px,e) where Let f(x , F x,e ) g(x, FX ,e) ~ f(x , Fx ,e) - I(x,Fx), thu8, we write (9) in the form I, (x , Fx . c) ~ J (x, Fx) - (110 I BoP) x - (.4 (e) + H (c) F) x.
~. C"
for all IIx(k)II
+ C3 (C2 + 1'2"")IIlx(k)II , 2 +C6 [Cl -1- (" + C3 (C2 + c21)1 II x(k)II 2 2 IIx(k)1I +- c" (k, + k21) IIx(k)II 2 2 ~c, (k, + kn) II x(k) " ,
< c"
I C,
where k, = c, '1 '" I C,C3, k, = C2e",
IIp(x(kllll < (k3 +- k41
+ k,1') IIx(k)II' ,
\I IIx(k) 11
< T"
wit.h
1>1'(x(k)) S -x" (k)Qx(k)
+ (k3 + k,"") + k.1') IIx(k) 11' ,
I g (x, Fx ,.) Choose
Now IIJ(x(k)' Fx(k» - (Ao we
choo~e 1:\11
+ BoF)x(k) 11 < (c, + C2C3) IIx(k)II ,
arbitrary
,0
> U and assume that
IIg(x(k), FJ:(k),£) - (A(e) I B(e)FJx(k)11 < c, IIx(k)11 I cno IIFx(k)II (10)
S (c, holds for some "'Y
:5
')'0-
I
c2c:n'<>!lIx(k)11
So
11 f, (x(k). Fx(k) , e} I I H ' , + C3(C, + C2~O) , 11 x (k) 11 < c, , for all x satisfying
IIx(k) II < 1\OW W~
,
'2 = mm
(
'"
(
c. ~ C2
.,) + C2"YO
).
have t hat
1>1'(x(k»
= -x"l"(k)Qx(k) + p(x(k»,
then 1>v(x(k)) S -x T (k)Q3:(k), for all IIx(k)1I < T3, if g(x, Fx, e) - (A(e) + B(e) F),""sti:;fics (10) with"") S "")1. In this point, we repeat the :-isme tlrguments of the pre-
vious proof, i.e., we show t,hat all trajectories starting inside B(O, r3) converge to the origin provided (10) iH sati"ified with 1 ~ ')'1 . Remark 1. If one consiuer/') I he closed-loop syst.em
x(k
+ 1) ~ I
(3'(k) , Fx(k),e)
and linearize6 it at. x ~ 0 for
€
+ 1)
=
(11)
0, we obtain the
- (A" 'f BoF) x(k). Clearly th" linf".arization of (11) about x .,-:- 0 for small g will have it~<;
linearization x(k
eigenvalues clooe t.o the eigfmvalues of (Ao + HoP). Since all the eigenvalues of (AD + lioF) have modulus strictly
less than one the same hold,~ true for the linearizat.ioll of (11) for sufficiently small ,.
wh ere
+ BaF)'" prAte) + B(e)F) x +xT (A(E) + B(e)P)'" l' (A(e) + B(e) F) x
3, EXA'vIPLE
p(x) = 2xT (Aa
12xT (Aol BoF)T Ph(x , Fx , e)
+2XT (A( e)
Consider the following Ilonlinear system wliich is not feedback linearizahlc
+ B(e)F)'" PJ.{x , Fx ,.)
x,(H I) =x3(k)(a, 1 x,(k» ,
I fnx , Fx , e:)P/, (x. FX, E),
x2(k
+ 1) = x, (k) + (a, + xz(k)) u(k) ,
x3(k + 1) ~x2(k) (a3 ;. x,(k» - x3(k)u(k),
and 1'H:ttisfiet;
2427
t.he or igin is an equilibrium point of the nominall'iYl'it.em, i.e., j(O , O) __ O. T he linearization a round U is z(k
+
E,
..
E,
,.
I) - Aoz(k) f- Bou(k) ,
.
• •
with
•
.
.,
~
.
,
":, (k
+ I)
+ x , (k))·-
t l) =
A,z(k)
••
~.
"-.- '
• ( • .bl
."f; E,
[ (0.006
+ l.00(je,)(1
+ e,) [< 1 - (0.07 + 0,07"2), (12)
+ 1.00602 )(1
I E,))2 ,( 13)
]n Fig. 1.a we t;how the guarant.eed stabilit.Y region given by the conditions (12) and (1 3).
B ,u(k) ,
Remark 2. If onc would change t.h e original F , then aL':'O thL,", region in t he (El,e2)-plane may change; other F's could lead to more or less c' )J1servat ivp. h oun<.lli, i.e., the size or the region iu l ite (e"1~ E2)-space can be change, sce Fig.1.b, we choose F - I- I,O ,O[ such that the p oh have b een fixed gt t he origin, in this C&ie, t he values of E l , c2 sat,is fy t he conditions :
with
t.h ~ closerl~loop
. ..
,
[-(0.07+ 0.07£, )[< 1- «0.006
x3(k)u(k) ,
The linearization around 0 is
*
•
11 . _1
has all it.~ root~ wit h modulus strictly less t han one if and only if El and £2 :-;atisf.\· t he inequali t ie8
+ x,(k)) + E, xo(k), + (az + x.(k)) u(k ) + E,u(k) ,
x3(k I 1) ~.. x,( k) (a3
" ."
_:'I n
Fig. I. Guaranteed stability regions given by: (La) t he ineq ualities (12) ann (13), and (1. b) the inequalities (14)
- xo(k) (a,
x.(k I I) =, x ,(k)
and
.
.
.,.
" where [R", A"R", A~B"J ha., full ran k (n = 3) for lhe nominal parameter values a] :-: ; u" '''' a:'1 = 1, so (AO l Bo) is .tabilizahle, If we ch.,.,.., u ~ F z, wilh F z ~ -1.006z, + O.07Z3i then t.he closed-loop poJ('f) have been fixed a t 0.1, 0.2 a nd - 0.3. Let us ~llPp~ C that a·1 and a2 are perturbed by £ 1 = D.a] ano £2 = Aa:,? r{'~.,pectively. Then t.he pert,llrherl :'iy~tem is givt'n by
•• '" '" ,
perturbed :;ystem is given by
z(k + I) - (A , + R, F) z(k) = (Ao t BoF):(k) t (A (E) + B(e)F)z(k), where
I (1 + e,Je, 1< 1,
J - [(I
,
+ "de, I > O.
(14)
The following numerical re<.; ults are obtained with t he first choice for F .
In Fig .2, we show t he response of the closed-loop IiAo
A(E)
+
1 ] BoF - 0,006 0 0.07 , [ o I 0
+ B(E)F =
0
0
e, ]
0 0 - 1.006e, OO.07e, [ 000
'
By T heorem 3 the perturbed ~yste m has a asymptotic.ally stable closed-loop linf'..ari7.at ion for El. £2 sufficiently s mall. 111 the present setting (Af:'" R~F) will be a stahle matrix, if the following hold... : from the SchurCohu Criterion (LaSalle, 1986) ~ the characteristic polynomial
.\', - (0.07
+ 0.07C2)' + (O.OO(j
I L006o,)(1
nearizctl I)y~t.em under t.he values of Cl = 10 and € :2 ,,:: 0.05, wlwn the states are ~tarting at Zl = Z2 ...." 0.4 a nd Z3 = 0.5. Fig.3.a exhibit s t:he behavior of t,he nonlin ear syste m controlled by the linear feedback under the :;aUle perturbat.ionl';, but. with initial condit ions: XI X2 = 0.01 and X3 = 0.1. From Figs.2 and 3.a, we can see t.he obvioUb fact that. t,h(' stability region of the nOHlinear syste m constitutel'l f\ !-imall region in t.he stability region or t he linearized syst.em. Finally~ F ig.3.b shows the corm:-;ponding respons~ of lhe closed-loop linearized :;ystem exactly wit. h the same r:olldit ioIlS of the experiment shown in Fig.:l.a to illustrat.r. t hat in this case, both r ~ ponses are very ~ im i l a r .
+ e,) ,
2428
4,
-::~
"
o
5
'0
'5
-: :: ~
"
o
!i
10
15
20
25
38
- ~--
25
---- =1
3D
35
.0
Time (k:I
L
,3 D,5
-0,: 0f ~ 10 5
15
25
20
I
o
5
15
'0
'"
25
]n this note we have presented all study of the robU/'j l. feedback stabilization proble m for a class o f perturbed non linear dlli<:rete·time. \V.: have I)hown that. the COIltroller ils ruLm;l if the li nearir.at.ion oft·he nominal system is st.abilizaLle. We haVf~ remarked that t.hi."i robustnes.o;; property depends of the choice of the feedbac k applied. that is, the si:u~ of the ... tability reg ion may be cha nge. \Ve have generali:r.ed the rel'ults given in (Crnz and Alvarez, 1994) for the cla....."i 01' lIlultivaria ble systemH that are not necesHarily feedback lin carizablc.
.0 35 nm'(I<1
3D
u,:~ -'0
I
•0
Timeltl
20
35
30
3S
I
.0
REFERENCES Byrncs, C,I. , W, Lin and H K. Ghosh (1993), Stabilizat.ion of di.-;cret.e-time nnnlinr.ar system:; by smoot.h state feedback , System.' & Control',elters, 21 , pp,-
Ome (ij
Fig. 2 . .H.csponse of the linearized ~i.)'stem under t he perturbs!,ions s, ~ 10 and 02 ~ 0,05; wilh . , (0) = "2(0) -- 0 ,4 and "3 (0) - 0,5_
CO~CLCSJONS
25,,)-263,
Cruz, C. and J . Alvarez (1994) _ Stability Analysis uf Li ncarizing Controllers f(,r a C lass of Perturbed Nonlinear Discrete-time SY:"itcms, Proe. 1994 American Control Conference , Balt.imore, .YlD , CSA, pp,5.58,562.
Xl_:fF~_3_ o
211
.110 Time
x2
~
0
11
20
40
fkl
o,;ftF'_____ -0.1
":F------~
liD
,2
20
60
Time IkI
o:~r-~----11 -<1.1
!;=o---=,,;:---'."o--'.. ,;! Time IkI
zJ
';F....--o-11 -G.I 0
20
40
'0 Time M
_:Ew------1 _:F-----11 u
u
0204060 13.a)
Time (kJ
02040&0 Time fleJ
!l_bl
Khalil, 1l.K. (1992), No nlin. ar systems, Macmillan I'll bli~hing Company. LaSalle, J,P, (19S6),The Stability and Control of Discrete Processes, Applied 1\1athematical Sciences, Vol. 62, Springf:r- Verlag. Liao, T., L, Fll and Ch, H
Luenberger, I),G. (1979), Introduction to Dynamic SystentS, Theory, Models, & Applications, Johll Wiley & Sons, Marillo ~ R . and P . Tomei (lH93) . Robust st.abi li:r.ation of feedback linearizable t. un e· var ying uncertain 11011linear systems, Automniica, Vo1.29~ No. i , pp.181189. Vidyasagar, M, (199:]). Non/inea,' systems analy,';... Se c.ond F...d., Prentice Hall.
Fig. 3. (3.a) RespOIl8e of the nonlineRr system under the perr.urbations c, - 10 and "2 " 0.05; with x,(O) ~ xAO) ,~ om and X3(0). (3,b) RCBponse of the lincarized ~y.stf!m ~xact. ly under the same conditions.
Remark ,'1. T1H.~ approximal,ioIl givf'n in (Cruz ann AIvarez 1 1994) is for nonlinear feedback linearizable systeJTIN only. it is clear that such a strategy is not possible to apply to the dA...';S of non linear s'ystem ao;; in this ex· ampl~ which it; not feedback lineari:t.ablc tiystem.
2429
2b-14 5
FEEDBACK STABILIZATION OF PERTURBED NONLINEAR DISCRETE-TIME SYSTEMS I Cesar Cruz' and Henk Nijrneijer" • CICESE, Mexico 2 ,.. University of l'wente, The Netherlands 3
Abstract. A robu:-it feedback IStabilization problem of nonlincar discrde-time syst.f!m~ i~
studied, A sufficient condition for feooback ~tabilization of a class of perturbed nonlinear systermi (which are not necessarily feedback linearizable) i~ obtained_ It is l:ihown that. if the Jacobian linearization of the llominalsYl:itcm io stabilizable l then the pert.urbed non linear system ilS asymptotically stabilizablc. A linear feedback that stabilizes the nominal system also locally stabilizes the perturbed system. A simulation example illustrate the obtained results.
Keywords. :\"onlinear discrete-time systems 1 feedback stabilization pert.urbed j
systems,
robllstnes~_
I. INTRODVCTlO:\
The stabilization problem plays a central role in dynamical control systems, A matter of practical importance is t.o determine the extent of aHymptotic stability of an asymptotically stable equilibrium point, Le., how large a perturbation from the equilibrium can be allowed and we can still be sure t hat the traject.ory will return to that equilibrium. In the majority of the applications 1 we encounter perturbed system:;; which are the product of modf:~lillg errors) uncertaintie~ and disturbances. Hence a property desirable of cOlltrollers is t.heir robustness, i.e. 1 their capacity to work suitably even when t.hif'! controlled. system differ::; from lhe system designed. .For continuom; time nonlinear systems several papers have been published where the stability of the closcdloop system i8 analyzed when uncertainties are present (e.g .. Khalil, 1992; Liao, et al., 1992: Marino and Tomei, H193). It. ha.., been proved that. a controller may tolerate uncertainties if the clo.'ieu-loop nominal system has an RSymptoti(:ally stable equilibrium, and t.he perturbation sati'ifiili some olher conditiollf:i. In the discrete-time case, in (Byrnes, et aI., 199:3) it is shown t.hat. any nOIlI Research performed during a visit in Alltumn'94 of Cesar Cruz at the University of 'l'wente, supported by the Faculty of Applied Mathematics. 2 Department of Electronics and Telecommunications, p.a. Box 434944, San Dif'gu, CA 92143, USA. PhUIlp.: +52.(617).4450lx2,~30; fax: +rI2.(617).4.'iISa; e-mail: ccruzOciCCSC.UlX 3 Department of Applied Mathematic!'<, p.a. Box 217, 7500 AE Enschede, The Netherland!'J. Phone: +31.53.893442; fax: 31.53.340733; e-mail: [email protected]
linear system with Lyapunov-stable unforced dynamics can always be globally stabilized by smooth state feedback if suitable cont.rollability-like rank conditions are satisfied. In particular, in (Cruz and Alvarez, 1994) the authors have di-.;cm:;soo the stability of a class of nonlinear fef!dback linearizablf' :
2424
2. PROBLEM STATEMENT
ConlSider the class of discret.e-time lIonlinear systems d epending on a real-valued parameter vector c E n q
1:(k I
I) ~.
(1 )
f(x (k) . lI(k ),£) ,
where x :..:; (X t , ___ ,x n ) and Tt -- (u! , ... , u m ) are smooth 10(:a1 coordinat.es for the state spa(X~ Ai and input i:ipace U respe(~t ive ly, \Ve 3s;;urne that f i:-; a analytic vcct.orfield ill J: , 'It and 1.;:, anrl that f(ll, 0, E)
(2)
0, '1£.
c
Problem 1. (Robust feedback stabilization of nonlinear control systems): Con.id,,· the .• y.• tem (1 ) .• atisfying (2). Under which mnditions does there exist" smoolll. static staJ.e feedba ck
/(x ,u) c' f(x , u,a). the system (1) corresponding to the value e ::: 0 is called the unpertur-bed or nominal sy&tern, while it will be called lite pC7'turbed syst em whe n e .f O. T he verturbed sys t.em could result from modcling errors , l:lg iug of (Jh'y~i(:81 cumponents, uncertainties and d~t.urUaIl(: es which exit-I. ill any TCH listic problem .
Let us int. rodu<.:c the notation B(O , R) to indicate the set of v"d oTS 1: satisfying 11 x - (I 11 < R ; that. is, it. is the sct. of all point.~ who...e Ellclidt'..an dist ance fro m 0 ie,: less t han R .
u(k) -- (» (x(k)) ,
where () , M loop system
+ 1)
(3)
J(J:(k)),
(3) is stable if there ., an Ro >
a for which the following
is true: For e very R < Ro. there is an r, 0 < r < R, such t.ha t if x(O ) i, in,ide R(O, r ), t.hen x (k ) i, inside R(O, R) for a ll k 2: a, (a) is aBymptotically .table if it stable aud B(O . r ) (;811 he chOljcn such that IIx (O) 1I
=>
lim x (k) =
k -- .x,
a.
Defin;tion f. (Luenuerger, 1979) A function v defined un Ii region n of the stat e space of (3) and containing t he equilibrium point. ;1: . - 0 i~ a Lyapunov function for the disc rete· tirnp. r.;ystem (8) if it sati.<;fies t.h p. foll owing three requiremenb;: (1) 1.1 is (:o ntinuous. (2) 1! (3:') has R unique minimum at O. (3) The function L'!.v(x(k)) . 11 (f(x(k))) - v(x (k)) sa!.isne. L'!.1'{x(k)) SO, for all J: E n. Theo rem 1. (Asymptotic stability) (Luenberger,1979) H t here ex',ts 8 L.Yapunov function v(x) in B(a , Ro) with Ce nt.er the o rigill , s uch that the function .6.v{x) is strict ly negative at every point (except O) ~ t hen the origin or t.he non linear syste m (:i) is asymptotically stable.
Rm, w;U, 0'(0) - 0 , .•".oh that the closed-
x(k
+ I)
-, f(r.(k) ,cr(x (k)),e:)
(r.)
has 0 a.s a locally asympto tically stable equilibrium for all c in a neighborhood of l: ;;;;;; 0 ? Note that 0' is selected ind~pende nt. of c. Let us write the ~y8tcm (5) in the form
x(k
+ I) =
/( x (k) , n (x(k)))
+ g(x(k) , o(x (k)) , E) , (6)
where g(x , n(x) ,E) = f( x , <>(x) , €) - /(x , n(x ») is the perturbation function . \Ve make the following a....sllmption :
(A) : The linearizat;on of Ihe nominal system x(k + 1) -- j(x(k ), tI(k))
Dejinition 1. (Luenberger, 1979) Let x = 0 be an equilibrium point of tlw nonlinear syst.em x(k
~
around the origin} is
(7)
stabili ~ able.
If We define Ao ~ ~(a,O) and Ro = ~(a, a), thell the Assumption (A) guarantees t he existe nce of a matrix FE R.mxn such t.hat all eigt mvalue~ of (.1\0 + BoF) have modulu:o> st.ric:tiy less than one .
Proposition £. Consider t he nomina) syst e m (7). A..." umc Assumption (A) holds and 1<1. F he such that (An I RoF) is asymptotically st.a ble, t hen the c1ose
+ I)
= j(x( k ), Fx(k) ).
has 0 as a locally asymptot ically
Proof.
stabl~
(8)
e;quilibriurn.
Suppo,e Assumption (A) hoIck It follows that
for any positive df!finitc ITwtrix Q = qT , t,he solution p .;.;; pT of the discrct.c-timt, ~ Lyapunov matrix equation
(Ao + BoF)Tp(Ao 1 RoF) - P ~ -Q , is a posit.ive definite matrix see, e.g. : (Vidyli.'Sagar, 1993) . \Ve will ~how that v(~: ) =c x T Px is also M Lyapunov funet.ion for the c1",cd-Ioop system (8) in a lIeighborhood of /(a, a) ~ o. To thl'i end, we evaluat.e t.1I(x(k)) = IAox(k)
Our goal in this not.c is to study the following problem.
+ Ro,,(k) + j
(x(k ), u(k))
-A"x(k) - Bou(k)rr .
2425
.PIA"x(k) I R"vik:1 = I(A" I B"F) x( k) I
,PI(Ao
-
I,
+ j(x(k), v(k)) (x( k) , Fx(klll
T
Suppose v(x) is we ll-defined in B(O, ~). Let R be arbitrary with 0 < R < ~. Let R, < R be selected so that if x E B(O , R,) the n i(x , Fx) E H(O . ~) (Le. , if the state lies illtiide 8(0, RI) , it will not jump o ut of B(O , R,,) in one step). The continuity of t.he funct.ion I gllarant fCi'j the existence of RI'
.
+ BoF)x(k) + I, (x(k) , Fx(k»)J
-3:I'(kjPx(k) where
It ...
Fx. I,- (x. Fx)
6
fix . F.r) - (Ao
.cc
+ BoF ) x,
Let m ,.,. minRI <; lI r U:=; nu V(3:), the exist.ence oft-he minimum m is guaranteed from t.he continuity of v(x) aud the compactnpJ;.'; of the region defined uy R, :Sl\ x I\ :S Ra . Moreover m > v(O) :;ince ll has a unique minimum at O. From t.he conti nuity tJ[ v(x), it. is po.':iSible to select an r, 0 < r < R, such that for x E B(O, r) it holds v(x) < m . This is be(:allSf! near the origin v mu...,'t take value.s cIOtiC to viOl = o. If x(O) E H(O ,r), '.he n v(x(O» < m . Sillce t. v(x) < 0 t he value of" decrea,es wit h time. Therefore, the corresponding trajectory can never go out~idf! 8(0 , RI), and consequently it can ne ver go outside R(O, R), i.c ..
t.v( 3:(k» .o -xT(k)Qx(k)
t2x T (k) (Ao I
with
+ BuF}7' pi,
(x(k) , Fx(k»)
iT (x(!:) . F.r(k») pi, (x(k). F.r(k)) ,
i, (x , Fx) satisfies
Ili, (x(!:), Fx(k))11 <", Ilx(k}11 + c~ IIFx(k)11 :S c,II:"(k)11 + c,c3 1Ix(k)ll, so
\I I(x(!:) , Fx(k»
\I
- Aox(k) - BoFx(k) II x(k) II
Ilx(O)1\ < r ""
I\x(/')I\ < R, < R ,
'I k ~ O.
a.
< c, + C2 03,
Finally, it i< suffic ient to show that v(x (k» ~ 0 k~ Since t.v(x) < 0 for all:c except at origin, t hen vex) must act.ually deCrCA!;f! rnoU4)tonousiy. Thus v(x(k» COIIverges t.o some limiting value m. The only que.c;t.ion i~ whether it is possible fur .ft > viOl = O. This i, nut poosible since v(x(k» eOnVi!rges to tnl it must be t rue that t.v(x(k) converges to zero. But t.v(x) < 0 except at origin. Thus, since L\v(x(k» is continuous x(k) m1L~·t converge to origin and v(x(k)) must (:onverge to v(O) . By Theorem 1, we (,;onclude that the origin is (locally) asymptot.ically stable equilibrium for the nomiutl) sy:-;tem (8). 00.
for all x satisfying I\x(k)1\ < r
t.v(x(k)):S
- IIQllllx(k)II' t 2I1xT(k) IIII(Ao + BoFfll· ·IIPIIIII,(x(k). FT.(k»11 + lifT (x(k). FX(kll lIIIPl\lli, (x(k) , Fx(k))ll ·
Now Jet
- xT(k)Qx(k) :S -).,»>,,,(Q)
II(Au
I
RoFl"'I l ~ cs. I\PI\ -
Ilx(k)1I
2
).,m •• W)
=
,
-c,llx(k)11 ,
= C6,
C, =
2cscn.
Hel1l.:e
t.v(x(k))
< -c, l\x(k) I\' + c, (c, + C,C3) I\x(k) 1\' +C6 (c, + c,c3)' l\x(k) I\' =
I- c, + e7 (c , +c. (c, t
t C,C3:1
c,c3)'lllx(k)II' .
choosing
The following theorem shows how t he origin of (1) will remain asymptotically sta ble fOT !;mall valuC2:0 of parameter g 'I 0; that is the robust stabilizatioll Problem 1 is solvahle for small e. Theorem 3. Consider t he class of discrete-time nonlinear perturbed systems ( 1) . Assume that Assumption (A) hold,. Then the perturhed system (5) has 00,<; a locall y a.~ymptotic:Rll'y stable ('lCluilibrium for c sufficie:ntly small.
Proof. Assume t hat Assumpt.ion (A) holds. We will show that v(x) ;;... x T Px is also a Lyapunov function for the closoo-loop perturbed s~'stem
x(k ensures that t.v (x(k))
< 0, for all IIx(k)1\ < r.
From standard Lyapunov Theorems see, e.g. } (Luenberger, 1979) , we have t hat all traj ector ies starting inside 8(0, r) will ('on verge to the origi n.
+ 1) =
I(x(k) , FX( k),e:)
(with u - Fx , where F such that (Ao + BuF) is asymptotically siable) in a neighborhood of (O,O,c), i.e .,
x(k
2426
+ 1) '- (A. + B,F)x(k) + ft(x(k) , PX(k},E),
IIp(x(kllll S (C8 f.(3:,1-'X.<:} ~ fix , FT, c) - (A.
+ B. F)x,
(9)
+ C9) IIx(k)II 2 + (c6 1 CIO ) IIx(k)II Ill,
(x(k) , Fx(k) ,E)11
+c. IIf, (3(k) , Fx(k)")112 < C" IIx(k)II 2 +C'2 II,,(k) 11 [Cl
with
8f
A, - ,,(O,O,e), vX
+ R.F 0
A,
(Ao
{)f B, - Ou (0, 0, e),
+ A (e»
1 (80
+ B(e»F.
+ B,Fj'"P(A + B.F) -
1>v(x(k» - xT(k)l(A,
Plx (k)
+2x'" (k) (A, +- H,F)" P j,(x(k) , Fx(k), c)
+Jl" (x(k), Px(k),c) Pf, (x(k), Fx(k),e) , i(x,t,,) +- g(x,Px,e) where Let f(x , F x,e ) g(x, FX ,e) ~ f(x , Fx ,e) - I(x,Fx), thu8, we write (9) in the form I, (x , Fx . c) ~ J (x, Fx) - (110 I BoP) x - (.4 (e) + H (c) F) x.
~. C"
for all IIx(k)II
+ C3 (C2 + 1'2"")IIlx(k)II , 2 +C6 [Cl -1- (" + C3 (C2 + c21)1 II x(k)II 2 2 IIx(k)1I +- c" (k, + k21) IIx(k)II 2 2 ~c, (k, + kn) II x(k) " ,
< c"
I C,
where k, = c, '1 '" I C,C3, k, = C2e",
IIp(x(kllll < (k3 +- k41
+ k,1') IIx(k)II' ,
\I IIx(k) 11
< T"
wit.h
1>1'(x(k)) S -x" (k)Qx(k)
+ (k3 + k,"") + k.1') IIx(k) 11' ,
I g (x, Fx ,.) Choose
Now IIJ(x(k)' Fx(k» - (Ao we
choo~e 1:\11
+ BoF)x(k) 11 < (c, + C2C3) IIx(k)II ,
arbitrary
,0
> U and assume that
IIg(x(k), FJ:(k),£) - (A(e) I B(e)FJx(k)11 < c, IIx(k)11 I cno IIFx(k)II (10)
S (c, holds for some "'Y
:5
')'0-
I
c2c:n'<>!lIx(k)11
So
11 f, (x(k). Fx(k) , e} I I H ' , + C3(C, + C2~O) , 11 x (k) 11 < c, , for all x satisfying
IIx(k) II < 1\OW W~
,
'2 = mm
(
'"
(
c. ~ C2
.,) + C2"YO
).
have t hat
1>1'(x(k»
= -x"l"(k)Qx(k) + p(x(k»,
then 1>v(x(k)) S -x T (k)Q3:(k), for all IIx(k)1I < T3, if g(x, Fx, e) - (A(e) + B(e) F),""sti:;fics (10) with"") S "")1. In this point, we repeat the :-isme tlrguments of the pre-
vious proof, i.e., we show t,hat all trajectories starting inside B(O, r3) converge to the origin provided (10) iH sati"ified with 1 ~ ')'1 . Remark 1. If one consiuer/') I he closed-loop syst.em
x(k
+ 1) ~ I
(3'(k) , Fx(k),e)
and linearize6 it at. x ~ 0 for
€
+ 1)
=
(11)
0, we obtain the
- (A" 'f BoF) x(k). Clearly th" linf".arization of (11) about x .,-:- 0 for small g will have it~<;
linearization x(k
eigenvalues clooe t.o the eigfmvalues of (Ao + HoP). Since all the eigenvalues of (AD + lioF) have modulus strictly
less than one the same hold,~ true for the linearizat.ioll of (11) for sufficiently small ,.
wh ere
+ BaF)'" prAte) + B(e)F) x +xT (A(E) + B(e)P)'" l' (A(e) + B(e) F) x
3, EXA'vIPLE
p(x) = 2xT (Aa
12xT (Aol BoF)T Ph(x , Fx , e)
+2XT (A( e)
Consider the following Ilonlinear system wliich is not feedback linearizahlc
+ B(e)F)'" PJ.{x , Fx ,.)
x,(H I) =x3(k)(a, 1 x,(k» ,
I fnx , Fx , e:)P/, (x. FX, E),
x2(k
+ 1) = x, (k) + (a, + xz(k)) u(k) ,
x3(k + 1) ~x2(k) (a3 ;. x,(k» - x3(k)u(k),
and 1'H:ttisfiet;
2427
t.he or igin is an equilibrium point of the nominall'iYl'it.em, i.e., j(O , O) __ O. T he linearization a round U is z(k
+
E,
..
E,
,.
I) - Aoz(k) f- Bou(k) ,
.
• •
with
•
.
.,
~
.
,
":, (k
+ I)
+ x , (k))·-
t l) =
A,z(k)
••
~.
"-.- '
• ( • .bl
."f; E,
[ (0.006
+ l.00(je,)(1
+ e,) [< 1 - (0.07 + 0,07"2), (12)
+ 1.00602 )(1
I E,))2 ,( 13)
]n Fig. 1.a we t;how the guarant.eed stabilit.Y region given by the conditions (12) and (1 3).
B ,u(k) ,
Remark 2. If onc would change t.h e original F , then aL':'O thL,", region in t he (El,e2)-plane may change; other F's could lead to more or less c' )J1servat ivp. h oun<.lli, i.e., the size or the region iu l ite (e"1~ E2)-space can be change, sce Fig.1.b, we choose F - I- I,O ,O[ such that the p oh have b een fixed gt t he origin, in this C&ie, t he values of E l , c2 sat,is fy t he conditions :
with
t.h ~ closerl~loop
. ..
,
[-(0.07+ 0.07£, )[< 1- «0.006
x3(k)u(k) ,
The linearization around 0 is
*
•
11 . _1
has all it.~ root~ wit h modulus strictly less t han one if and only if El and £2 :-;atisf.\· t he inequali t ie8
+ x,(k)) + E, xo(k), + (az + x.(k)) u(k ) + E,u(k) ,
x3(k I 1) ~.. x,( k) (a3
" ."
_:'I n
Fig. I. Guaranteed stability regions given by: (La) t he ineq ualities (12) ann (13), and (1. b) the inequalities (14)
- xo(k) (a,
x.(k I I) =, x ,(k)
and
.
.
.,.
" where [R", A"R", A~B"J ha., full ran k (n = 3) for lhe nominal parameter values a] :-: ; u" '''' a:'1 = 1, so (AO l Bo) is .tabilizahle, If we ch.,.,.., u ~ F z, wilh F z ~ -1.006z, + O.07Z3i then t.he closed-loop poJ('f) have been fixed a t 0.1, 0.2 a nd - 0.3. Let us ~llPp~ C that a·1 and a2 are perturbed by £ 1 = D.a] ano £2 = Aa:,? r{'~.,pectively. Then t.he pert,llrherl :'iy~tem is givt'n by
•• '" '" ,
perturbed :;ystem is given by
z(k + I) - (A , + R, F) z(k) = (Ao t BoF):(k) t (A (E) + B(e)F)z(k), where
I (1 + e,Je, 1< 1,
J - [(I
,
+ "de, I > O.
(14)
The following numerical re<.; ults are obtained with t he first choice for F .
In Fig .2, we show t he response of the closed-loop IiAo
A(E)
+
1 ] BoF - 0,006 0 0.07 , [ o I 0
+ B(E)F =
0
0
e, ]
0 0 - 1.006e, OO.07e, [ 000
'
By T heorem 3 the perturbed ~yste m has a asymptotic.ally stable closed-loop linf'..ari7.at ion for El. £2 sufficiently s mall. 111 the present setting (Af:'" R~F) will be a stahle matrix, if the following hold... : from the SchurCohu Criterion (LaSalle, 1986) ~ the characteristic polynomial
.\', - (0.07
+ 0.07C2)' + (O.OO(j
I L006o,)(1
nearizctl I)y~t.em under t.he values of Cl = 10 and € :2 ,,:: 0.05, wlwn the states are ~tarting at Zl = Z2 ...." 0.4 a nd Z3 = 0.5. Fig.3.a exhibit s t:he behavior of t,he nonlin ear syste m controlled by the linear feedback under the :;aUle perturbat.ionl';, but. with initial condit ions: XI X2 = 0.01 and X3 = 0.1. From Figs.2 and 3.a, we can see t.he obvioUb fact that. t,h(' stability region of the nOHlinear syste m constitutel'l f\ !-imall region in t.he stability region or t he linearized syst.em. Finally~ F ig.3.b shows the corm:-;ponding respons~ of lhe closed-loop linearized :;ystem exactly wit. h the same r:olldit ioIlS of the experiment shown in Fig.:l.a to illustrat.r. t hat in this case, both r ~ ponses are very ~ im i l a r .
+ e,) ,
2428
4,
-::~
"
o
5
'0
'5
-: :: ~
"
o
!i
10
15
20
25
38
- ~--
25
---- =1
3D
35
.0
Time (k:I
L
,3 D,5
-0,: 0f ~ 10 5
15
25
20
I
o
5
15
'0
'"
25
]n this note we have presented all study of the robU/'j l. feedback stabilization proble m for a class o f perturbed non linear dlli<:rete·time. \V.: have I)hown that. the COIltroller ils ruLm;l if the li nearir.at.ion oft·he nominal system is st.abilizaLle. We haVf~ remarked that t.hi."i robustnes.o;; property depends of the choice of the feedbac k applied. that is, the si:u~ of the ... tability reg ion may be cha nge. \Ve have generali:r.ed the rel'ults given in (Crnz and Alvarez, 1994) for the cla....."i 01' lIlultivaria ble systemH that are not necesHarily feedback lin carizablc.
.0 35 nm'(I<1
3D
u,:~ -'0
I
•0
Timeltl
20
35
30
3S
I
.0
REFERENCES Byrncs, C,I. , W, Lin and H K. Ghosh (1993), Stabilizat.ion of di.-;cret.e-time nnnlinr.ar system:; by smoot.h state feedback , System.' & Control',elters, 21 , pp,-
Ome (ij
Fig. 2 . .H.csponse of the linearized ~i.)'stem under t he perturbs!,ions s, ~ 10 and 02 ~ 0,05; wilh . , (0) = "2(0) -- 0 ,4 and "3 (0) - 0,5_
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Cruz, C. and J . Alvarez (1994) _ Stability Analysis uf Li ncarizing Controllers f(,r a C lass of Perturbed Nonlinear Discrete-time SY:"itcms, Proe. 1994 American Control Conference , Balt.imore, .YlD , CSA, pp,5.58,562.
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Khalil, 1l.K. (1992), No nlin. ar systems, Macmillan I'll bli~hing Company. LaSalle, J,P, (19S6),The Stability and Control of Discrete Processes, Applied 1\1athematical Sciences, Vol. 62, Springf:r- Verlag. Liao, T., L, Fll and Ch, H
Luenberger, I),G. (1979), Introduction to Dynamic SystentS, Theory, Models, & Applications, Johll Wiley & Sons, Marillo ~ R . and P . Tomei (lH93) . Robust st.abi li:r.ation of feedback linearizable t. un e· var ying uncertain 11011linear systems, Automniica, Vo1.29~ No. i , pp.181189. Vidyasagar, M, (199:]). Non/inea,' systems analy,';... Se c.ond F...d., Prentice Hall.
Fig. 3. (3.a) RespOIl8e of the nonlineRr system under the perr.urbations c, - 10 and "2 " 0.05; with x,(O) ~ xAO) ,~ om and X3(0). (3,b) RCBponse of the lincarized ~y.stf!m ~xact. ly under the same conditions.
Remark ,'1. T1H.~ approximal,ioIl givf'n in (Cruz ann AIvarez 1 1994) is for nonlinear feedback linearizable systeJTIN only. it is clear that such a strategy is not possible to apply to the dA...';S of non linear s'ystem ao;; in this ex· ampl~ which it; not feedback lineari:t.ablc tiystem.
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