Robust control for nonlinear similar composite systems with uncertain parameters

Y.-H.Wang and S.-Y.Zhang

Abstract: The paper describcs tlic similarity between two nonlinear control systcms, sliowii in R large number of physical systcrns of intcrcst in inany engineering applications. By utilising similar information, robust cotitrollcrs h r large-scale nonlinear similar compositc systcnis with tinccrtain parnmctcrs arc proposed. The results obtaincd show that thc similar structure of coinposite systcrns possesses a holographic propcrty and can simplify the analysis and design of nonlinear systems.

1 Introduction

The fundamental structure fentiire displayed in Illally rcal large-scale systems is that, iii corrcsponding subsystems, thew arc almost the same qualities of nature or appcarancc (e.g., in the stabilising problcin For ;1 power system in R

plant with identical units [I] , the voltage control o f the fccding nodes in an electric power system consisting of connected synchronous inachines [Z]). Syinnictric and similar composite systcrns [3. ,151 also possess this property.

Analysing systems by utilising similar itiformation caii simpliry the analysis arid design of thc systcins [1-15]. Control design far similar composite systems has not yet been thoroughly explored, especially robust control design for nonlinear similar composite systems with w"a in ty . One reason inay be that a standardiscd mathematical description has not yct hccn presented for universal- systems.

Bascd on carlicr studies, this paper is concerned with similarity between two control systcms, with spccial emphasis on the difkrcntial geometric approach. Robust stabilkation far a class of nonliriear similar cornpasiic systems with nricertaintics in parainctcrs, both in subsys- tems and intcrcoiincctcd tcrms is lhen discussed. Based 011

siniilar structure information, robust stabil isatioii o f similar coinposite systenis depends mainly on external subsystcin and similar paraineters. Thus control dcsigo is wcakcned.

2 Preliminaries and similarity description

Consider two control systems as follows:

0 IEE, 2000

I I X ProcceilirrgJ online im. 20000 10R ,901: IO. I04~~/ip~cta:2OOilO108 Paper reccivcd 15111 Scptcmbcr 1998 and in revised form 8th June 1999 Ttic nuthorn arc with thc Dcpartinciit of Aitumatio Coiaml, Nnrlhea4crri University, Slien Yung 110006, P. R. Chian E-inail: s ~ z h n ~ ~ g ~ ~ n ~ i l . n e ~ r . e d u . c n

BO

whcreRE i 7 ~ R'", X E uc X" arc state vcctors, respectively; ni 1 ti' 0 and U are open sets in RI'' and RI', respectively; a,

respectively; f ( f, 0, #), f(x, U, b) arc smooth inappings, respectively; t ~ l = ( a , + w,).

From [ 16, 171, we givc now thc conceptions o l regular cmbcdding arid kcdhack, respectively. Dejnitinn I : Lct us R"', U C R" (m > 0) be open sets, respcctivcly. h rcgular cinbcdrling is a one-to-one mapping 'p: U + U which is hoineomorphism of U oiito its imagc with m n k q = n at every point of U Defwition 2: A regular feedback o f system as eqri. 2 is a feedback U = a(x, I ) + /j(x, k)v such that ~ ( x , t ) is a smooth mapping and P(x, f) an invertible smooth function matrix, of red dimensions.

Then the similarity between two control systems i s described RS follows; Dehition 3: We can say that the control systcm (cqn. 2 ) i s similar to the systctn (eyi. I ) if thcrt: cxist thc regular fccdback g= Z( $, t ) + J ( f , i)v, U = ~ ( x , f ) + P(x, t)v and regular embedding (or differential homeoniorphism) rp:U+ cp(U) c x -> X, such that thc tangent mapping rp satisfies:

i r cR J arc thc-inputs of the system (eqn. 1) and (eqn. 21,

where (q, ~ l , j, a, /I> is a group o f similar p"nctcrr; between the control systcm (eqns. 1 and 2), a(x, i) is a d- tiilnctlsioii sninoth mapping, and PIX, t ) is a il-dimension smooth invertible function matrix. Remark 1: Consider the following systems:

According to Definition 3 , system (eqn. 5) is similar to system (eqn. 4) ineaiiings that thcre cxist thc rcgular feedback a = i( f , r ) f /?(f, t)v, U = a(x, t ) + P(x, t)v and

Kfl Proc.-Cu:unrroi T h m j Appf., W. 167, No. 1. knvary 2000

the regular embedding (or the differential homcomorph- ism) cp:U + cp(U), x + ,t such that:

2 (f(., 0 + gix, ,)a(& t ) )

dJcg(x, OB(& 0 = b'(m, M Y ( X ) , 4

8X

841

=.m4 t ) -!-i(V(.T), f)G(&), t> (6)

(7)

j = r l k $ & j (8)

(9)

Furthermore, for thr: following linear systems;

.i = AX -+ Bu

According lo Definition 3, if thcrc cxist the linear feedback ~ i = J % t / J v , u = K x + p v aiid the m x n inatrix E runkF= n, such that:

F(A+13K)=( ; i+ .BK)F (10)

1QIup = Bp (1 I > then the linear system (cqn. 9) is similar to thc linear system (eqn. 8), cspccially, if ni = n , when th is i s equal to the concept o f a similar system i n lincar theory [IS]. Therefore, the concept o f il similar system described by Definition 3 expands tlic corresponding one in lincar theory.

3 Stabilisation of similar composite systems

Cotisidcr thc foliowing nonlinear composite system composed of cxtcriial system Eo and N subsystems El, I I . , ,EN with uncertain panmctcrs:

if =,/$(xi, I ) + & ( X i , 0, I tl 4- &,(+ t)Ui t ai(.& 0 2 , t ) (12) i = o , I , ..., N

where x - E Vi 2 X"', no = mnx(ni); input i f i E P I ; E (a, -+ GO);

gj(*), (I),(*) are smooth mappings of proper dimcnsions; the uncertain paramctcrs 01 E I l l , O 2 E LIZ, whcrc Cl,, R2 are the compact sets in R"', rcspcctively; J ; (0, t ) = X i (0, 0 1, t> = 'I+ (0, 02, t ) = 0; Ui is an open co-ordinate neigh- bourhood ut xi = 0 in R"'.

System jCi=.fi (x,, t) t gi(xi, t) U,, (0 5 i 5 N), is called thc ith nominal subsystem of the conipositc system (eqn. 12), especially, jm =h(xo, t ) +gD(.uu, t ) is callcd the external nominal subsystein. DeJinitiort 4: Wc say that the cainpositr: system (eqn. 12) is a similar composite system or composite system with similar structure, if cach nominal subsystems of cqn. 12 is similar to thc cxtcmal nominal subsystcin according to eqns 6 and 7,

Tf system (eqn. 12) is n similar coniposite system, and (vi, an, Po, air /Ii) is a group of similar parameters hclween the ith nominal subsystem and thc cxtemal nominal subsystem, for convenience ((pi, ao, Po, xi, pi) are callcd the similar parameters of thc composite system (eqn. 12), where is a identity mapping, vi (O)=O, i = 0,1,2,. . . , N , i.c. we obtain:

+> x = (XIJ,. . . ,xJT; U = U, x U, x * x U,; A{*), Ri(*),

Remurk 2: Invoking the nominal subsystcms of composite system (eqn. 12), the class of similar compositc systems describes B large number of physical systems of intcrcst in many engineering applications, including the fui-mer class dispIayed in [1-15].

I n this paper, a robust controller (KC) of compositc systcin (cqn. 12) is involved, the sense of which is shown in the rollowing definition. Ilefinitiori 5; A robust controlier (Re) of com ositc

U:(., I ) , . . . , uT(x, t)): satisfying that the resulting dosed- loop system can be locally uniformly asymptotically stable in the Lyapnnov sense Ibr VOl € C l , , U 2 f Q 2 . Remark 3: In this papcr, wc propose a discontinuous robust controller (DRC) for composite system (eqn. 12). Dy DRC, we mean that each component of U, (x, t) in ~efihit ian 5 may be a piecewise-continuous funclion. This type ofDRC i s synthesised in order to overcome the 'worst CAW' uncertainty (the uticertainty that inight result in the rnost positive time derivative o f Lyapunov function chosen)

Assimplion I : l'lie cxtcmai nominal subsystem of composite system (eqn. 12) is locally uniformly asympto- tically stabiliscd at TO = 0 via feedback 110 = $0 (xu, t), and tlic corresponding closed-loop system as follows:

system (eqn. 12) is a control law I I = u(x, t ) = (uo(x, P t),

[ 19-23],

Assuriiption 2: Terms with uncertuin pararnetcrs satisfy the fallowing conditions:

Then the I3RC o f thc composite system (eqn. 12) is proposed E I S follows:

t i , = +U: + i = 0, I , 2, + . . , N (15)

where

81

RePncirk 4: [t is notcd tlial. the DRC (eqn. 15) depends mainly on the external subsystcm and the group of similar paranieters (i.c., by ulilising fully similar inforniation, thc DRC (cqn. 15) h r large-scale nonlinear similar cflinpositc systcins (eqn. 12) can easily be rectmstructcd). To a certain extent, the role of DRC (eyn. 15) is analogous with the hologrnphic role i n biology.

Theorem I : Coiisidcr t l ic similar composite system (cqn. 12) satisfying assumptioil I and Assumption 2 . Then there cxists DRC (cqn. 15) so that the similar conipositc systcnl (cqn. 12) can be locally uniformly asymplotically robust stabiliscd at x = 0.

Wotc that, in order to apply '['lieorcm 1 arid cnnslrtict the DRC (eqn. 15>, finding a group of similar paranieters of composite system (cqn. 12) apparcii~ly involves the solu- tion of a systein of partial differential equntioris (eqri. 13). Gcncraliy, it is difficult to obtain a group of sinlilar parainctcrs nf composite system (eqn. 12) by soiving the partial differential equations (cqii. 13). However, under special conditions, it is possible to obtain a group o f similar paraincters of composite systciii (cqn. 12) by using the exact linearisation mcthod in [24]. We now show an algorithm by which similar paranieters are obtniiicd fnr a class of nonlinear time-invariant single- input systems as follows:

i, =A(q) +RR,(X,, U,) +g,(x,)u, + mi(x, 0,) (16) i = 0 , 1 , . . . , N

where i t i€ X; the scnscs olthc other corresponding signs of eqn. 16 arc analogous with eqn. 12.

According to [24], the following conccptions arc intro- duced first. Consider tlic fullowing noillincar he- invar - iant system with singlc-input:

"i. = , f ( x > + g(x))u (17)

where the state X E U s R", z i E R, f (x ) and ~ ( n ) are analo- gous with cqn. 5.

DeJinltion 6: Cansidcr systcin (eqn. 17). If there exist a function A(x) and a number r > 0 such that:

Ihc systcm (eqn. 17) is called to have auxiliary-rclativc degree r in U, the function i(x> is called auxiliary-oatput nl' system (eqn. 17).

I'rom [24], considcr the system (equ. 17), if one can find R numbcr r z 2 such that: (i) thc distribulion D =span {g , d f g , . . , , adj-2.q] is nonsingular involutive in U. Assunic that mnk D = s. (ii) [adj- ' g , m;g] $ D, Tar certain k sutli that

Thcn the auxiliary-output A(.) C R I ~ be obtaincd by using tllc following steps:

0 ~ k ~ r - Z .

RZ

Step I : Lct XI X 2 , . , ,,&. . I dcnnlc (lic vector fields g, ad'g, . , . ,ad;-' g, rcspcctively. Choosc X,, X,.+ 1 , . . . ,X,, to bc a coinplcincnlaiy set of the vector fields with property that X,, X,, . . . , X,. ~~ ,, X, Xr+ . . . ,X,, are litiearly indc- pendent in U. Step 2: Construct the following nisppitig:

where $2 denotes the flow of the vector field Xi ( i= 1,

Xtep 3: Calculate the inverse t'/ - I (x),

2, . . . , n ) .

'I'tien cadi zk (x) (s + I 5 k 5 n) can be chosen as the auxiliary-output i(x) of cqn. 17

Still rrom [24], it is known that, if the external tiotninal subsystem of composite system {eqn. 16) has an aiixiliary- relntive degree n 5 no and auxiliary-output lo (xo) in U , , by the following co-ordinatcs transforination:

(19) 1'

4 0 : x g t+ ZO = (zn,, 202, . . . , ~ o , ~ ~ )

ZOI = W o ) z02 = L & h l )

. . .

Zone =I 4 0 n f l ( d

where + I (x~) , . . . , (bo,,(, (xo) arc pao '- II iirnclions such that thc inapping (cqn. 19) has a nmsitigular jacobim matrix i n [lo, Thc cxtcrnal nominal subsystem of compo- site system (eqn. LO) can be transformed as follows;

501 = 202

z02 = 303

. . .

. . . = q,,(io? V U ) +1jn,(50* d t l o

wticre ao(zo> = .L~,~A~((/>"- '(z0)), h,(z,) = L,&: ' 10

( 4 - 'Izo)); To = (zo1, . . . ,ZflS) vo = (ZOn i. 1 r , . * ,zmJ +

Meanwhile, if the ith ( i # O ) noinitial subsystcm of composite system (cqn. 16) also possesscs aiixiliary-rcla- tive dcgrcc n, = ri slid auxiliary-output l i ( x i ) , the ith nowinal subsystcin can be exactly linearised into the following form:

T T

ii, = Z,Q, , , . , Z j + , = zjn,

i,, = Ljyi{fb;'(zi)) + LgiL;-!il(Q(z,))ui (21)

the corresponding co-ordinatcs transformation is as follows:

T f j l :xi I+ Zl = (z;,, ' ' ' ,Z jn , ) ,

Z i k = $ j n ! ( x , ) , k = 1,2, . . . , R (22)

i E L Pru~c,-Crm!rol Tlreory Appt , , HJL 147, No. i, Jiinuary 2UOO

Consequently, from eqns. 13 and 16-22, tlic following algorithm is obvious. Algurifhna I : Consider the composite systein (cqn. 16) with nl = n2 = . . =TI ';z 2. If the ilh noininal subsystcin ( i =0 , 1 , 2,. , . ,N) has the sanic auxiliary-relativc degree n and auxiliary-nulput Ai (xi), respcctivcly, a group o f similar paramctcrs of composite systcin (eqn. 16) can bc obtained Ram thc .following steps: Step A: Under tlic previous condition (i) a i d ( i i ) for cadi nominal subsystem of cotiipnsite system (eqii. 1 b), carry on the Stcpl-Step?. The c h o o ~ certain aoxiliary-output i,o

Step B: For the external iiomiiiel subsystcin, construct thc co-ordinates trnnsforination (eqn. I9>. Mcanwhile, presciii out the system (eqn. 20). ,%2[J C: For each noininal subsysteni, construct thc co- ordinates transforination (eqn. 22). Mcanwhile, prcscni out the correspnnding system (eqti. 2 I ) . Step D: For the ith noniinal subsystcin, construct tlic rollowing mapping:

Fo : zo I+ q, is n identity mapping;

I (-U . I I , in, @hi), rcspectively.

Fj : zj 13 20 , i # 0 (23) ZOl = Zil

Zo, = h:'0(Zi)

where the i to - n hnctions hi!' '(zi), . . . ,h:'O(z,) arc chosen such that:

Q4h)

= (q, . . . ~ zin>l ti0 = (h: + '(z ,) , . . . , hf'"{zj))7: ~f tllcrc exist A:' '(zJ,. , , , h;'"(z,) satisfying eqn.24 tIien carry out the following stcp E, else Algorithm 1 has no effectivcness. Bep E: If Step D i s effective, thc siniilar paramctcrs of composite systciii (eqn,16) arc as follows:

Renrark 5: Algorithm 1 is a spccial algorithm fbr similar paramctcrs of tlie composite system (eqn. 12). In Algu- rithm I, solving partial differential equations (cqn. 24) is thc key Link and simpler than cqn. 13 becausc or the reduced nuinbcr of equations atid simple struct~ire, cspe- cially if gk vu) and p k (Tn, 17,~) in eqn. 20 have the propcrty yk (CO, O)=O, p k (L, O)=% k = = n + I, I 1 , ,no, lhen lay + (zi) = . ' =A',!" (zi) = 0 arc snlution of cqn. 24, tt

ifl6 Pinc.-Cmiirrol Theory Appl. , K)!. 147, No. 1. Jnnrimy 2OMi

i s noted that, if tlie nominal subsystems of the campmite system (eqn. 12) arc linear, an algorithm of similar paiu- meters is shown in [ I 51. [rowever, a universal algorithm of siinilar parameters to the composite system (eqn. 12) is still possible

4 Example

Considcr the following nonlinear composite system with uncertain parameters:

whcse

where 01 E { U1 10 i 0 I i 21; O2 E ,U2J - 3 i U,, 5 3, - 1 5 0 2 2 5 I ) .

11 can be verified that the external nominal siibsystein of eqn. 25 has auxiliary-relative degree 2 and auxiliary-output ~ o ( . Y o ) = ? c o ~ hy carrying out the prevhus Stcp I-Step3 with & = ( I , 0, 0)"and X3=(0, 0, 1)' in Step 1. Thcrcrore, similar parameters c p ~ ; s o -+ xo is a identity mapping, a,,

For the first nominal subsystem of cqn. 25, i t is verified that the auxiliary-relative dcgrcc is equal to 2 and anxili- my-output L1(xl)=xll by chnosiiig X 2 = ( l , 0 ) T in Stcp 1 .

(,To) h o i $- 3x02, f io (Xo) = 1 .

R3

Thus, al(xI) = - (xl +xl?) e*”, fll(xl) = 6‘”. Now, eqns. 24 in Step 11 have thc following simplc form:

It i s easily sccn that h;(zl) =211212 is a solution of eqn. 26. Therefore, the corresponding mapping F1 in eqn. 23 is obtained as follows:

201 =Zllr202 =Z12r203 = z l l z I Z (27)

9, ; z l l =x1jIzI2 = X I 2 (28)

The co-ordinates transformation eqn. 22 is now as follows:

Consequently, rrom eqns. 26-28, we can get similar para- meters qI:xol=xll, xnZ=xl2,

By the same calculation, for the second nominal subsys- tem of (eqn. 25), the auxiliary-relative degree r = 2 uiid auxiliary-output &(Q) = x 2 1 can bc obtaiiicd by choosing X2=(l,0) in Stcp 1 . Similar pacainctcrs ~ I Z : X O I =XZI,

From the above calculation, we verify that composite system (eqn. 2 5 ) is a similar composite system. The corresponding similar parameters arc {vi, En, f i n , cli, pi, i =o , l,Z}.

To conslriict the DRC (eqn. 15), lirsl of‘ all, we have to construct the Lyapunov function Vo(xo) and redback un = satisfying Assumption 1. Since the linenr controller is both simple and practical in application, assume t,bo(xn) =mol + bxoz fcx03. Then the correspond- ing closed-loop system of the external nomitiul subsystem o f (2.5) cnn be represented as follows:

2 ~ 0 2 = X Z ~ . ~ 0 3 = ~ 2 1 ~ 2 2 , 2 2 ( ~ 2 ) = - ~ ” x Z Z , / I ~ ( J C , > = ~ ~ ~ .

rg =/la0 + B (29)

where

0 1 0

A = a - 2 b - 3 ( 0 0

B = (0 0 ( r a - 2)x& + x i , + (b - 2)x,,x(),, + CXo,XJ By observing eqn. 29, first to construct quadratic Lyapunov function of the following linear system:

i o = Ax0 (30)

then the Lyupunov function of eqn. 29 may be obtained by adding certain propcr function. Choose ta = I, b = 2, c = 0. Then A is a Hurwitz stable matrix. By solving the rollow- ing Lyapunov equation for Q = 2 i ( I denotes identity matrix),

A”P + PA = -Q (31)

we obtain the positive symmetric inatrix

P = 1 2 0 (1 : Thcrcfore, thc Lyapunov function of eqo. 30 is us follows: -

Y&O) = 3 4 1 + %Nxo2 + 2x202 + 43 (32)

84

Notc tlint 3x,& + 2xolxnz + 2x& is still positive for xnl and xop, so wc can assuinc that thc Lyapunov function of eqn. 29 possesses the rollowing Form:

( 3 3 ) whcrc e = e((xol, x02) such that e(0,O) = 0. By inspection, in order to ciisurc that y(lcD)l(,,, is negative definite, we obtain e=xf11xn2. That is, by the Lyapunov function V&n) = 3xh i - 2 ~ 0 1 ~ 0 2 + 2 d + (xu3 - x01xo2)‘ and the fccdback uo =xDI + hOz, thc external noinitial subsystem ol‘ lhe conipositc systcm (cqn. 25) is locally uniformly asymptotically stabilised at xo = 0. Conscqucntly, the corresponding DRC of conipositc system (cqn. 25) i s as follows:

J J ~ ( + ~ ~ ) = 3X& -i- 2101xo2 -I- 2& + (xo3 - e)’

= X’OI + 2x01 ~ (41 + 942 + 42 + ki,) x sign(h,,, + 4xOz) (340)

x sign[(2X1 I + 4xl,)e-”~l] (346)

x sign[(2x2, + 4xzz)eC22] (344

21, = 42x1, S X 1 2 +x:2)(+ - (x2 I 1 + XI2X21122 + 943)

la, = -(.T~~ +xZz +.&)ev2z - (e2siti2x21 + (xO1 + x ~ ~ ) ’ )

According to paramclcrs U t = I , O2 =(- 1.5 0.5)”and the initial states x,=(1 - 2 3)’; xI =(1.5 I)”; x2 = (2 - 1.5); corresponding sirnulation figures are given as ill Figs. 1-3.

5r

[U

0 Y

c

-’ f

2 .0 r

-1.5 0 2 1, 6 8 10

The abovc siinulation o f thc composite system (cqn. 25) show that the similar structure of composite system can simplify the tliialysis antl design of a nonliiieilr compositc control system.

5 Acknowledgment

This work is supported by the National Sciences Founda- tion (Projcct 69774005) and National Key Projccr of P. R. China. Tlic atilhars acknowledgc the referees and cditors for their proper comments.

6 References

I ARAUJO, C.S., and CASTRO, J.C.: 'Application of powcr system stabilizers iii a plant with idcntical units', K E Pwc. C, Geire?: Tmism Distrih., 1991, 138, pp, 11-18 UAKULL, L., xivl L U M R d.: 'Ucccntralixd dcsign offccdbnck cuiitrol far large-scale systcins', Kybcnretiku, 19x8, 24. pp. 1-1 0 RAKULE, L., and ROTIF,LLAR 1.: 'Dcccntralizetl contml dcsign for unccimin syinmelric compositc systems', Procccdings IFAC/lFORS/ IMACS syniposiutn o n Large Scab Sy.sisms: Theory nnd A p k c d i o n , 19'12, Rcijiiig, pi'. 79-84

4 CASTKO, 1.C.: 'Modal h " w k for system with idcntical uiiitu', Prcprirts of 1 llh IFAC world congrcss, 1990, Tallinn 2, pp. 257-261

5 IJJNZE, 1.: 'Dynamics of stroi~gly couplcd symmetric cornpositc systems', In/. I Conhid, 1986, 44, lip. 1651-I 670

6 LUNZF,, J.: 'Stahility analysis oi' largc-scale systems composed of strongly conncctcd similar subsystcin', Auforiinrim, 1989, ZS, pp. 561-570

7 XIAOPING, I-,: 'Otitpirt m~iilntion of s(rong1y coupled symniclric composi(l: systcm', Atr~omaricrr, 11192, 28, pp. 1037-1041

8 (IIAN, L.J., and ZIIANG S.Y.: 'Synthcsis of quadratic stability control for similar composite systems', Autonrdco, 1998, 24, (4), pp. 4 9 3 4 9 8

9 SUNDARESIIAN, M.K., nnd ELUANNA R.M.: 'Qiialitativc analysis and dccmtrtiliml controllcr synthesis Tor a class of large-scnlc systems with symmcrdcally i~~tt?i-c~~nncctcd subsystcins', , 4 ~ o n w 1 i c o , 1991, 27, pp. 383-388

I O RRKULE, L., slid 1WT)ELLAR 1.: 'Dcccntnilized control design o f uiicciTnin nominally lincar symmetric compositc systems', IRE Proc., Curtrrnl Theory Appl., 1996. 143, pp. 530-536

1 1 BAKULU, L., and TIE LA SEN, 11.: 'Decenlmlizcd stabilization or irnceitnin syminelric composite system', Preprints IIACflFORSIIMACS sytnposium un hip! S i d e Systems: T~icory curd Appht ior i s , 1998, i'alras 1, pp. 2655270

12 IIOVD, M., and SKOGBSTAD, S. : 'Control of syinnielrimly Itilcrcon- ncctcd plants', Artrupncriicxt, i 994,30, pp. 957-973

13 IIUANG, S.D., niid ZHANCi, S.Y.: 'Output rcgulntioii nntl mbust output replatinn for a class of lnrgc-scalc systcnis with symmclric circulanl sniictiirc', Preprints of IFAC/tl'ORS/IMACS symposiiim an Lnae Scale

14 YAN, X.G., WANC;, J.C., Lu X.Y., nnrl ZHANG, S.Y.: 'Decetitmlizcd output lccdback robust mbilizarion for a class of nnnlincar intercon- ticctcd systeins wilh similarity', 7inns., 199& AC43,pp. 294 ,299

15 WANG, Y.H., U, Y.S,, WAN(.; X.D., and 21 IANG S.Y.; 'Thc sliirctiire properties of control systcm with similarity', Procectlings of thc 14Ih WAC world congress, 1999, Ucijing, China

111 DOOTHRY W.M.: An intmrluction lo diffcrcntitlble manifolds and Rieniannian gcoinctiy (Acatlcinic Prcss, INC., New York, 198G) 2nd cdn.

2

3

*3b?ll,Y. The0rJJ Ofid ~ p p / f c d O m , 1998, PdtrtlS 2, pp. 920-925

IEE Pmc -Corrlrni Theory Appi., Yo/. 147. NI). I , J ~ t ~ ~ r i r r y 2000

17 MARINO, R.: 'On thc largcst feedback lincariznblc subsystcm', Qst.

18 KOSENBROCK, H.H.: State-space and niulliunrinble theory (Wilcy, Conirnl Lei{,, 1986, 1, pi]. 345-351

New York. 1970) 19 SONGTAG, E.D.: 'Nonlinear rcgiilntion: thc yicce-wise lincar

approach', {EEE P ~ n s . , 198 I , AC-26, pp. 346-.37 1 20 CHEN, Y.H.; 'Dcsign uF robust contmllcr for unccrlaiii dynsmiwl

systems', IEEE k n s . , 1988, AC-33, (5), pp. 4871195 21 STEFEN, H., and WANISLAW, H.Z.: 'Robust control syiitlicsis for

uiiccrtainlnonlinear dynamical systcins', Aittoornoiica, 1992;30, (2), pp. 289-298

22 RAG, S.K., SPARGEON S.K., and EDWARDS C.: 'Output feetlhack sliding modo dcsigii for lincar uncertain systems', [/?E pmc,, Control Thtory dppl., 1997, 144, (3), pp 209-216

23 YAN, X.G., antl nA1 G.2.: 'L)cccntralizcd output feedbnck whist control for nonlinear hrgo-acalc systcins', Autumatxn, 1998, 34, (1 I), pp, 1466-1472

24 ISIDORI, A.: Nonlinear control systcins (Spriiiger-Verlag Londnn Ltd, 1995) 3rd cdn.

7 Appendix

where ci (p) : R., + R belong to class K.

(cqn. 12) and the controllcr (cqn. 15) is as follows: Now, the closed-loop system composed of the system

Coiisidcr the Lyapunov hmcrioii candidate V(r, t ) = YZo V,, (q j (x i ) , t). Froin eqn. 35, we obtain:

Thcn, the derivative of V(x, r ) along the closed-loop system (eqn. 36) i s as follows:

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85

liivoking Assurnplion 1 , 2, controller (eqn. ISra), equalities (eqn. 13) and the inequalities (eqn. 37), we obtain:

From Assumption 2 , controller (eqns. 15b) and LSc, we obtain:

That is, by using the DRC (eqn. 5 ) , the similar composite system (eqn. 12) can be locally uniformly asymptotically stabilised at x = 0. This completes the proof of Theorem 1 .

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