Structure and stabilization of delayed neutral nonliear interconnected control systems

STRUCTURE AND STABILIZATION OF DELAYED NEUTRAL NON...

14th World Congress ofIFAC

L-5a-03-4

Copyright © 1999 IFAC 14th Triennial World Congre~s~ Beijjng~ P.R. China

STRUCTURE AND STABILIZATION OF DELAYED NEUTRAL NONLIEAR INTERCONNECTED CONTROL SYSTEMS ,.. Zhang Xinzheng Guangdang Univ. of Technology, GUaJ.lgzhou 510090, P. R. Cll ill a. Elllail: [email protected]

Liu Yongqing

Department of Automation, South ChinE. University of Technology, Guangzhou 51 0641~ P,R.Cl1illla

Abstract: Some new conceptR of the structul'e are established for the lllultigroup rnulti-delays neutral nOlllinear interconnected control systern. A positive definite quadratic form V -function via choosing the syullnetric positive definite solution n1.atrix of Riccati Inatrix differential equation is ulade up. On the base of the equivalence 111ethod of Lyapunoy"s function found, A sufficient criteria is obtained for the interconnected stabilization of the control systenl without delays and perturbation paral11.eters iU1.ply the interconnected stahihza.tion of the multi-group n1ulti-delay~ and perturbation paranleters neutral IlonlinearinterCOlll1ected control system. At the san]e tinle~ some cstiluate formulae of the bounded for both till1e-delays and perturbation paranleters are given. Copyright @ 1999 IFAC Keywords: Int.erconnec.ted stabilization, equivalence met.hod of Lyapunov's function, nentral control systern. 111ulti-delays

control systenl with the time-delay control vector fUllctions.

1. QUESTIONS AND METHODIC FC)RMULATION Tht~ concept of the structure and intercolluected Btability f01' large scale continuous dynalnic systen! had been fornlulated by D.D.Siljak in 1978, and he. had systenlat.ically analyzed thenl for COlltinuous la.rge scale systelu. In 1994~ Liu Yongqing a.nd Zhang Xinzheng had studied the structure and interc.onnected stahilization problclllS of the lllulti-group 111ulti-delays and perturbation paranleteI't; retarded linear constant interconnected

"'The projec:t Wtl,S supported by the 8cie:n~e fl.lll.dof the chinese nature science(69874005). This work is supported by Natural S(·.ience Foundation of Guallgdong Province and ShlUlrlong Prnvince(970235)

In this paper. via constructing positive definite quadratic fornl V-function fronl the positive definite ~Yl1l111.etric ~olution of the Riccati n1atl'ix algebraic equation, regarding the optinlal negative feedback vector function of the lineal' constant control systeln without tirne delays as the suboptilnal negative feedbaek vector function of the llJ.ulti-grollp luulti-dclays neutral nonlinear constant interconnected control syst~In with the tinle-delay control vector functions, by using the Lyapullov function equivalence ll1ethod proposed by Lin Yongqing~ we give out the structure and

5927

Copyright 1999 IFAC

ISBN: 0 08 043248 4

STRUCTURE AND STABILIZATION OF DELAYED NEUTRAL NON...

14th World Congress of IFAC

interconnected stabilization of the lllulti.. group Innlti-delays and perturbation parameters neutral llonlinear constant interconnec ted control sys tenl with the tilne delay control vector fUIlctions~ we alRo give ou t, SOHle Hmitations of tilne delay and perturbation structure paranleters.

2. STRUCTURE AND INTERCONNECTED

STABILIZATION Let us consider a ll-"Iulti-group illulti-delays and perturbation paranletCl'S neutral nonlillcar consta.nt interconnected cont.rol systenls \vith the tirne-delay control vector functions

where I is an identity ruatl"ix: A == (aij)n xn ~V:o. ----0).

(I-I: A 3

-1-

(aij+

)

1=1

J.V 1 -(r) (1')

L

aUjelij+

r::=l

B

/V-2 _(tl'")

E

.!l=1

a2ije2ij)nxn~

I: A~))-l(b;]

(I -

(b i ] )nxm

==

(8)

+

l=l 1\r

~ ~(dJ i...J bli j

(d)·

e b f )n

X

n~ , A

(r)

1

(r))

== (a hj

(t )

n xn

== (1-

(/=1 I'l

~ -(1) - 1 -::-< r ) LJ As) ( aiij)n x n

1,'"·, N 1 ),

(r

1==1

(" ) )11. x 1l ( a 2ij

(I .

::=

1.···, N 2 ),

~ A~(3l).) -

L..... l=l

-

A~)

1 (-( If) )

a 2ij

(a~~~)"xn ==

;::::

(a1~j)11 Xn (1 =- 1~··· ~ N3)~ --;.

(2.1 - a) :c( t)

== rP( t ) ~

r ===

nlax{

x(t)

==

;p(t ) ,

71~ 72 r3~ T4},

1. . ... ..IV2 J, r T3

;=:

2.:C=

-u( t) == .ljJ( t )1

lJ~~

{T

1<1.)<1'1

luax.

{T~;];

Z==

l~

w

••

r ::;

II?~X'5 n {T~:};

l~J,J

J: j:

d

-(d) l:';i.s;~~~~JSm {.Tiif ;

lS;g:;p,l~J::;n

~

71

-

==

t ~ D~ s ==

I == 1.··· ~ N 3 } ~

1

~I

L

A3

1~1

into

il <

+ h=l ~

d e1ays

-( z) ' CZgj

r ~~;

-,(hJ ~hl C19je6gj

> 0

""\'-7

~ A~I)-l

(I -

1=1

C == (cgj )pxn

~zJ ~(.: J)

+ .;:==:1 2:: CZgj (s ) T 1ij

(8

x 11;

e 7!Jj pXn,

> 0

( I == 1 ~ ... , N 3)

(S T

== 1 ~ ••.

~:) > 0

~

N2)

Cv:=

1 N 5 ) and r~~} > 0 (z == 1~··· ~ N7~ ~,J == 1~ ~ n~ f ::::: 1 r • • • ~ -rrL g :::: 1 ~ .. - ,p) are COllstants or functions of variable t. Hi (.) :::= (I1

~

4

••

~

N}

5 ; T4

=

I: A~)-l

~ N7}~ 1J(i) is a given

M

H;(.),

g (.):;::

M g (.).

~::: 1

continuous derivable initial vector function. 4,(t) is a given continuous initial vector function. E is the N x N interconnected Inatrix~ E = (ei.i) Vi7hic.h is generated by fundarnental intercOllUf:cted lllat.rix E == (ei)) (denoted by E E E, i.e.. where e ij is either 0 or 1. H (. ) is a gi vel1 U11Cel'tain function of x(t)~ y(t - T)~ x(t - r), 'u(t.),u(t TL El, E 2 , E3~ E 41 E 5 ; M g (.) is a given uncertain function of x(t)~ x(t-rL E6~ E 7 . If iV s ....(l)

(c g )

}\le

n

1. then (2.1) (2.1-a) can be rewritten

(Xl{t}~·· .. ~ xn{t))T ~ ·u(t)

And vectors x(t) (tLl(t)~···,1Lrn(t))T, Yp(t))T.

Definition 1 1,

o

and

For delays 0 (.[

(I) T2ij

Ti;j

==

(Y1(t), ... ,

yet)

2

0

(s (t')

'2:: :=: 1,··-,N3L J3ij 2=: (v == 1~··· ~N5), 7~;J ~ 0 (2 == 1~ ... ,N7:i~j ==

... ~

N) 2,

1 ~ . ~ . ~ n;

f

=

1,

4

4

•

~ 'nt; y

== 1 ~ ... ,p;) and

n1U-

tual connection Hlatrices E~<» E ~~O) (u == r~ s, t. d. ,tJ_h~ z: /3 =:: 1,2, 3, 4~ 5~ 6, 7), it the trivial ~olution of the elosed-loop systern of the lllUlti-

5928

Copyright 1999 IFAC

ISBN: 0 08 043248 4

STRUCTURE AND STABILIZATION OF DELAYED NEUTRAL NON...

14th World Congress of IFAC

group rI!111tj-d(~lays ano perturbation paraIlleters Hell tral llOlllillear constant interconnected control sys tenlS with the tiule-delay control vectoT functions (2.1) is aSYlnptotically stable, then we call the system (2.1) is interconnected stabilization.

V(x(i))(2.6) == x T (t)(K T B T P + Q)x(t.) == ~XT (t)(P BR- 1 B T P + Q)x(t).

In the lllulti-group lllulti-delays and perturbation para111eters neutral llolllinear constant interconnected control systems with the time-delay control vector functions (2.1 )~when T :== 0, if we dOll ~t consider the perturbation structure interconnected term and the perturbation interconnected ternl~ then (2.1) (2.1-a) heconles linear COllstallt control systeul without delay and perturbation structure pararneter

Due to PBR-lnT P+Q :::::: PBR- 1 ET P+C T C is a symnletric positive definite matrix there are the maxinlUJ11 and the minimuI11 characteristic values 0:1 > 0, 0:2 > O~ respectively~ such that

Q1X T

x(t)

== Ax(t) + Bu(t) y(t)

=

where A

(aij )nxn~

= Cx(t.).

(2.3 - a)

c

:S

x T (t)(P BR- 1 ET p+ eT C)x(t) 02 X T (t)x(t).

(t)x{t)

(2.3)

B:::= (bit )n:xrn~

(2.9)

:s

(2.10)

FrOIll (2.8) and (2.9) we obtain

===

(c9 J )pXH. ' .

A~sulne

that the lllatrix pair (A ; B) is controllable. and the matrix pair (A : C) is ob8crvablc~ then there exist the positive definite function V(x(t)) == x T (t)Px(t) of quadratic fornl~ such that LeIllIIlR 1

V(x(t))

V( x(t) ){2.6)

S

-al x

T

(t)x(t)

< O.

(2.11 )

Thjs SllOWS that the zero solution of cloRed-loop system (2.6) is asyn1.ptotically stable. the proof of L(~n1l11a 1 is COlllplcted.

< O.

Let

that is~ che zero solution of the nonlinear constant <:losed-Ioop systern (2.3) is aSYll1ptotica.lly stable. Where P is the only one symmetric positive definite solution of Riccati nlatrix differential nonlinear equation

Proof Calculating the derivative VC x(t») along the trajectory of closed-loop system of (2.3)

xCt) ==

(A ~ BK)x(t)

(2.6 )

~T ( ;r ( 1. )) l 2,6 J =.: [x T ( t ) P x ( t) + X T (t) P x (t ) + x T (t )PX(t)](2.6) X T (t)Px{t)+ x T (t)A T Px(t) - x T (t)K T B T Px(t)

=

+x T(t)PAx(t) - x T (t)P BKx(t) ( 2.7)

By

(2.5)~

we have

i) + AT P +

FA - PBll-lnT p

= -Q.

(2.]2)

(2.8)

Putt.ing (2.8) into (2.7), and paying attention to K == R- 1 B T P~ we obtain

In the definition region of x( t) and 'u(t L H (.) is continuous with respect to all variables, and satisfying

5929

Copyright 1999 IFAC

ISBN: 0 08 043248 4

STRUCTURE AND STABILIZATION OF DELAYED NEUTRAL NON...

14th World Congress of IFAC

Fl'Olll optin1al negative feed hack vector funct ion U(t) == -Kx(t) of (2.3) and (2.12) we can obtain n

!uj(t)1 ~

E Ik fj (t)lIxj(i)1

n

::; k 1

L:

IXj(t)\,

j=l

j~l

l~···,m..

f::::

(2.15 ) 3 (Liu Yongqing, et aL, 1988) Assunle that the conditions of Lemma 2 aTe satisfied 1 then we have the estilnate formulae as the following: LelllIDft

(2.13) Reg-al'diug' optilual negative feedback vector function U(t) == ~Kx(t) of (2.3) as the Buboptinlal negative f~edback vector func.tion of (2.1) with the lllultigroup 111ultidelays, we obtain a closedloop systelIl

:-ttf (t

~ T ~;))

n

l1 f

-

( t)!.

:S

T

k1 { . ~

{[ R 1 (

1+

t,J=l

+ (NI + 1)(0:1 + RI) + (N4 + + RI )k 1 lnJl x i (t~/) I + (al + _rv'2 ~u ($') RI) L IXi(t J - T 1ji )/ + (a 1 + 1T~kl)

1)

(b!

-=I

5=1 ...V 3

Rl )h l

R 1 )k 1

2:

l== 1 ~V::..

(1)'

m

IXi(t J - TZ)i)j rH

L L

lit

1.'l~1:(tJ -

-

+ (h l + (v)-

T 3wi

)I},

11:=lw=1

(2.18) (2.14)

LeIllll).a 2 (Li u Yongqing~ et al. ~ 1992) ASSU111e that. ·nl\r31L~ < 1. and x(t) is a solution of (2.14), and tl:.e(t)ll < 15, Ili(t)!1 < b, lIi~(t)" is bounded whell t ~ T :::; t S to· If llx(t)jf ::; j\l (to - 'T ~ t S tt. to < tl), then

Ilx(t)l1

where hI == -n[(d1 NZ)(fll RI) (N 4 nN3(a~ + R 1 )]. h~

+

tn

+

L bi f ( t ) k J j ( t ) I;

Ilx(t)il::; h~N~ + R 1 (1 + tn.k 1 ) + (Ni + + N 5 )mk 1 (b 1 + R1)]!(1 -

~ hlN~

=

i~ j

hi.

==

21

=

1"" ~ 1~ )

where t Cv) 7"3ij ::;

ifJ

tf

(8), Jlij ::; t j

::;

:s t,

t -

(l) T 2ij

::;

((

tj

::;

B == {X~ X E Rn, V(X) ~ 4V(Xl(t),··· ~ xn,(i))J. (2.19) LeIDIll84 (Liu Yongqing~ et al., 1992) If Pj = (Xl (tj - 'T"ji) , ••• ~ X n (tj - Tjn)) E B, then

SUp{faij(t)-

N can be a

t-

. t. DefinIng a set B as t h e £ollowIng: •

(2.20)

t~to

•

t,

where /3 11

:::::

4{32!{11 is a positive constant.

1=1

constant or function of variable t.

Owing to the two sum-sign items at the head

Proof The proof of of Lenlllla 2 i~ sinlilar as that of corresponding theorem 'in (Liu Y()ngqing~ et al.. 1992) ~ we onut it here.

of (2.2) i~ the right side of linear constant control systeln without tin1.e delay and perturba.tion structure paranletel' (2.3 L while six SUlll-sign iterllS at the back of (2.2) can be regard as the

5930

Copyright 1999 IFAC

ISBN: 0 08 043248 4

STRUCTURE AND STABILIZATION OF DELAYED NEUTRAL NON...

14th World Congress of IFAC

totically stabilizatioll when

perturbed tcrrn of the lllulti-group rllulti-clclays neutral control system (2.2) with the t.ime-delay control vector fuuc.tions. vVhen there arell ~t pertUI'batioll~ al = h 1 == T == 0, the right of (2.3) equal::; zero. Therefore, when perturbation strucis) (lJ (t)) ture parameter al ~ b 1 an cl cl e l ays T lij ~ '2ij' T 3ij arc very slnall~ regarding the six sU111-sign itenlS at tlle back of (2.2) as the perturbation terlll of (2.3). Thus~ taking optinlal negat.ive feedback Vf~C tor fUllctiol1 II (t) :.= - K :r( t) of (2.3) as the negative feedback vector function of control systerll (2.1) or (2.2)~ and taking V-function of (2.3) as the symnletric positive definite Lyapunov\) Vfunction of the quadratic forn1. of control systenl (2.1) 01' (2.2) ~ we can obtain the following Theol'en1 1: ~(r

.IV a

Theorelll 1

Suppose

11

L: As

J IJ

< 1, the

(Cl

+ E2 + £3::=

1~[1

>

O~£2

>

O~[3

>

0),

(2.25) Proof: The proof of TheoreIIr 2 is corTlpletely ilar as that of Theorem 1~ we onnt it here. (1) _

(1) _

-u)_

SiUl-

=

ReIIla.rk 1 When T 2 ij == 0, T 2ij == 0, u 3ij == 0 (I 1~· .. ~ ~f\l:3L (2.1) (2.1-a) becorne to the nonlinear

c.onstant interconnected control systelu with the multigroup luultidelays which has tinlc-delays in its control vector functions in the following:

COll-

':;;;1

ditions of Lenuna 2 are satisfied~ (A : B) is COlltl011able~ (A : C) is obscrvable~ then fol' for all (0)

Ea

~a)

E Et,

;-:uld 63

(2.26)

, there are constants 6. 1 > 0, L.l 2 > 0,

> 0 such t.hat the interconnected asynlp-

toticaIly stabilization for the nlulti-grollp lTlultidelays neutral nonlinear constant closed-loop control systenl with the tirne-delay control vector functions (2.2) or (2.1) when

o ~ D 1 < cl~l. 0 ::S RI < £26. 2 , o -:S T < [3.6. 3 , ([1 + C2 + E3 = 1, El £3

> 0, C2 > 0,

> 0),

(2.21 ) V\rhere

~1

==

QI12Pl'n 2 (N 1

(2.26 - a) From Theorem 2; we can obtain the sinlilar result in (Zhang Xinzheng, et aI., 1993), but, the variable range of pa1:anleter point (D, r) for corollary of Theorem 2 in this paper is 4.5 tinles of Theorenl 2 in [2L and we have solved interconnected stabilizatioIl pl"oblerll of the control systelTI included the derivative X(t-7!~).) of the state vari, .ut.j

+ N 2 + k 1 7n(2N4 + N 5 + 1))J-1 (2.22)

~2

:::::

Ctl[2Pln2(N3h1

+ 1)]-1

(2.23)

~3 == 0:1 {P1 n3 ( 1 + tj'll )[(D 1 (N 2 + kl1nn.J.V~) (R 1 (1 + mk 1 ) + (N1 + Nz + l)(al + R 1 ) +1\T3 h 1 (a'1 + RI) + (N 4 + 1)(b1 + R 1 )k 1 ?71 +(b2 + R 1 )k t mN5 ) + N3h~(a~ + R1)J}-1.

(l)

a bI e x (t - ,.2ij ) .

(2.24 )

Proof

The proof of Theoreru 1 ornit it here.

TlleoreIll 2

Suppose

}vs ~ll) JI

L

AJ

.

U<

1~ the C011-

1=1

Remark 2 For the lllulti-group lnulti-delays and perturbation paranleters neutral nonlinear tin1.evarying control systeul with the tinle-delay COlltrol vector functions

ditions of Lemma. 2 are 8atisfied~ and the trivial solution of the linear constant closed-loop systelll (2.6) of (2.3) is aSYluptotically ~table~ then th~r~ exist constants 6. 1 > 0, Ll 2 > 0, and (a)

-:;::::;(0)

6.. 3 > O. such that for \:j E 3 E E.:} (/3 == 1 ~ 2 ~ 3 ~ 4 ~ 5. 6, 7 ~ 0: :::: T ~ S ~ l, d ~ l' ~ "h, z), t he trivial solution of the closed-loop systeIIl of the lllulti-group rnulti-delays nentral nonlinear constant control systenl with the tirne-delay control vector fUllctions (2.2") or (2.1) is the interconnected asynlp-

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

STRUCTURE AND STABILIZATION OF DELAYED NEUTRAL NON...

jV'l

+- L:

Tn -Id)

L

14th World Congress of IFAC

(d)

b 1i f ( t )e 4i j (t ) H f ( t )

d::=;lf:=.l

~~

-m -( 1~',

+ L: I: (1::=1

I

(

t' ).

f=: 1

+Hi(t,

X(t)~

(2'

.

b2if~t)e5'ij(t)"Uf(t-

x(t - T)\ x(t -

i

T 3if )

r))~

(1::::

1~

_... n) (2.27)

(2.27 - a) here Hi and M; are the nonlinear tcrrns. Wc cau discuss the problems of the structure and intercounccted uniforluly stabilization for the systenl by using the linearal';ablc illcthod and the frcquence equivilel1ce nlethod~ we ol11it it here.

REFERENCES Siljak ~ D.O ~ Large-S cale Dynaluic S y s t erus: S ta bili ty and Strl1cture. North-Holland~ New York, 1978. Zhang Xilt~hellg. Liu ¥ongqillg, Structnre and illter~ {:onuecterl stabilization of linea! constant interconnected control SystCIUS. with Inultigroups aIld IIlUltidelays (2L Advances in Modelling & ~L\nalY8i5. C~ A1\1SE Pr(·s~. 39( 3) .1993. 45-51. Lin YOll~qillg. and Svn~ Z]lOllgkull, TIH:~ory and Applications of LaI:ge-scale Dynamic Systems~ Vol.l ~ The Press of South China Univer~ity of Technology~ GuaIL~zhou~ 1988.

Si Ligeng, Stability of linea.r delay systems of neutral with variable coefficients, Acta Mathclllutica Sinica, 26(1983)(1)4194-198.

Qin Yua.nxul1, Liu Yongqing~ and Wang 1ia.n, ()n the equivalence problem of differentia.l equation and differential difference equation in the theory of ~tability. Acta Mathen1atica Sinica. 9( 1 959)(3)~ 333-359.

Liu Yongqiug~ and Xu WeidiIlg, Theory and Applications of Large-scale Dynalnic. Systen1A. Vo1.2~ The Press of South China lTniversity of Technology~ 1989~

152-179.

Liu Yongqing~ Gao Cunchen~ On stabilization for the multi...groups and the multi-delays neutral lineal" control systelTlS (4) ~ Proceedings of International Couferencc on Modelling, Siululation and ControL AMSE MSC~92. Hefei. China. Oct.6-8, 1992~ USTC Prees, Vol.l( 1993), 564-

575. Gao Cunchen, Zheng Fenyong~ Expansion of the delays stilbilil;:ation range for linear tilne-varying control systerns with time delays. See that T'he Proceedings of "93 Chinese Control and Decision Conference, Northeast U niversi ty Pr~R~. 1993. 100-104.

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ISBN: 0 08 043248 4