Variable structure control of linear delayed large-scale systems*

VARIABLE STRUCTURE CONTROL OF LINEAR DELAYED LARGE...

14th World Congress of IFAC

E-2c-21-4

Copyright © 1999 IFAC 14th Triennial World Congress, Belling, P.R. China

VARIABLE STRUCTURE CONTROL OF LINEAR DELAYED LARGE-SCALE SYSTEMS * Zhang Xinzheng

Guangdong Univ. of Technology~ Guangzhou 510090, P. R. China. Emai1: [email protected] Liu Yongqing Depa.r~ment of A.utomation~

South Cbina University of Technology, Guangzhou 510641, P.. R.Chima

Abstract: This paper presents the design of decontralized variable structure controllers in the linear large scale control systems with time-delays, includ-

ing some design methods of control systems under certain given conditions as well as local decentralized controllers and the multilayer decentralized variable structure controllers by means of the concept and method of approaching rates. Copyright © 1999 IFAC Keywords: Variable structure control; time-delays large scale interconnected system; local decentralized controller; multilayer decentralized controller

1. INTRODUCTION

and interconnected stabilization problem of the linear continuous interconnected control systen1S with delays, its mathematical model is always differential equation with time delay variables. The problems of the interconnected control system are

In engineering and social practice, propagation of any signal and information is always to exist time delays, Due to the existence of some perturbation factor ~ some loops temporary interruption or get on again~ this kinks of control system with delays will take place some change in structure. Therefore, there are problems of the structure and variable structure control (VSC) not only in the retarded interconnected control systelTIB but also

naturally in infinitive dimension space, so the structure and (VSC) for interconnected linear COllstant interconnected control system with the timedelays becomes more difficult and more complicated. In this paper, based on Lyapunov stability

theory, the concept and method of approaching

in the large scale interconnected control systems

with

delays~

lation to x( t

rates the problem is solved.

namely, the control system bear re~

r)of the state variable x(t). We

The variable structure control (YSC) concepts and theorical founda.tion have been generalized to the control design of systems with delays[1-4]~ Outstanding advantage of delay VSC is tthat the sliding mode has invariance (or completely robustness) for the parameter perturbation and the external disrurbance of the system, which is the

are again in need of studying multilevel hierarchy multi-delays interconnected control systenl in practice~ now, that is appearing from structure "'The project was supported by the science fWldof the chinese nature science(69874005). This work is supported by N a.tural Science Foundation of Guangdong Province and Shandong Province(970235)

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VARIABLE STRUCTURE CONTROL OF LINEAR DELAYED LARGE...

14th World Congress of IFAC

problem that robust control needs to solve; its IIlajor shortcoming is that delay is produced by round-trip switching with finite frequency in sliding process, therefore oscillation is caused by delays. This leads to study vsc (of linear large scale control system with time-delays) but as far as the authors kn(jw~ there have not been any other papers dealing with structure and VSC of interconnected control systems with delays.

~:; (t)

=

(0)

e{3ij

=

A iXi

~r )

(s)

E El ,E2

-;:;{ ")

E E2

•

Al} Each state Xi can be locally accessible4 A(A) d 0 (r) (6) A 2 ) .A i 'Ai(r) j' ij an B i , E 1ij , E 2ij , (i == 1,· .. ,N) are all constant matrix; A3) BP is a matrix with the column rank filled:; A4) (Ai, Bf) is a pair of controllable matrix; A5) E(T) A~~) E Im(Bf?) E($l A~~) E Im(B9) 1 '1J t ~ 2 IJ t '

pre~

We are to design a. series of decentralized VSC laws Ui(t) (i == 1,44. ,N) so as to make the equilibrium state be global asymptotically stable for th'e

closed-loop time-delay linear large scale system.

+

34 DESIGN OF THE SWITCHING FUNCTION

--:;{r) () + ~ L..J ,Aij Xj t +

B i0 Ui

Suppose that the switching functions have the fol-

r=]

L:

~=1

(a)

(t - 'Hj) ~ G 1i

lowing linear forms

(1)

Xi(t) = rPi(t), ui(i) = ?/Ji(t), -7 :s;t ~ 0, i::::::: 1,4' . , N Xi

1

by

N 2 ---:'<8) A ij Xj

where

(0)

or e/3ij =

For i-th (i == 1~·· . , N) subsystems of Eq. (1), suppose that the following conditions hold.

Considering a linear large scale interconnected control systems with time-delays l:' c.ompcsed of N subsystems l:i ~ i EN:::: {1~ 2, .. · ~ N}~ each of

:

0

(r )

2. PROBLEM FORMULATION

. Xi

==

denote that El

sented in this paper can be extended to other cases of the multi-delay non lineal' large scale control systems with tinle-delay input control loop.

""'" ~i

=

r, s, d; {3 ~ 1,2,3). and if ~~] == 1 then either

comrnunication and improve the motion proper~ ties of controlled systems, it is necessary to design locally decentralized VS controllers of the large scale control systems with time-delay in order to

d~scribed

0,

responding fundamental mutual connection ma. -r.i1") Y:;{E 3:) ·f -:J n) 0 (a) 0 ( trlces E l ' 2 ' 1 ef3ij ,e/3ij ,a ~

lenls of control law, decrease the expenditure of

which is

1, :=

=

ditions. In practice, to simplify the design prob-

The approach

O~

Definition 2 We call that matrices E~r)(t) (ei:](t)nl)(nl~ E~8){t) = (e~~l(t)nlxnl' are mutual connection matrices to be produced by cor-

Today, the technology of decentralized VSC design of the multi-delay large scale interconnected control system has been generalized [5], especially structure of the time-delay linear large scale interconnected control systlenls under some given COIl-

satisfy the different needs.

1,

e1:J (t) =

inertia in the switching equipment, the control

E Rni and

Ui

S.(tJ = CiXi(i); i = 1~ .... , N

The switching surfaces are given by

E Hmi are respectively

the state vector and the input control vector of i-thsubsystems; x (xr,···,.x~)T E Rn and u == (u'[, .. ", uX,. ) T ER'Tn are the state vector and the input control vector of the combination large scale control systems; 4Ji(t) E C, 1/Ji (t) E C (i == 1~··· ~ N) are initial vector function and initial control vector function of i-th subsystems;

(2)

=

= A~;) E~;J, ~;) A~;) E~:J, are all ni x nj coefficient matrix; B? ~ are ni x m.; control rnatrix~ rn, mi~ nO, m~ M and N are all positive integers,

=

A}?

and n

==

nl

+ ... +n 1v,

m

= ml +

4

••

+ mN.

where Si E Rmi; Cs E .RJTl:iXni; is to be determined. First of all, choose proper transform matrix C i ( i = 1,· ~ ~ ,N) so that the equation of the sliding mode is asymptotically stable.. In view of A2)-A4), we can always carry out a. series of local state transformations.

we can choose

Definition 1 We call the fundamental mutual connection matrix for corresponding systems (1) .

-r.;( T

to the matrlX El (t)

(4~~(t) )nl xn l'

If

:::J"r)

)

=:

-;;;( $ )

(e1ij(t)nlxnl' E 2 (t)

11

so that T.(l) ]

(5)

y = 1j:c = [ T,(2)

= where T~l) E R(ni-mdxni , T~2) E " J ing ~

RTUiXni

,

Defin~

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VARIABLE STRUCTURE CONTROL OF LINEAR DELAYED LARGE...

T. -nA r)T-l i

/TT-nA $)T .Li

..

1 -

SJ

"'11

==

A is

[

A'" IS 2.1

ij n

12 ]

-

[

. . 11 Air . . 12]

A iT

A~l

tr

A is

A'" tS" 2. 2

_.

fTlBO _ .J.; ij -

A~2

[

tr

expected poles) and Ci == (efl) JC!2)). The deter-

,

]

O. B.

'

1iB~:) = [ ~;;)

];

'

TiA.1i-l=[~~~ ~~:]

f,

14th World Congress of IFAC

mination of Cfl) and Cj2) makes the equation of the switching surface and the switching function of the sliding mode be asymptotically stable.

4. DASIGN OF VSC LAW OF TIME-DELAYS LINEAR LARGE SCALE INTERCONNECTED CONTROL SYSTEM

(i=l,···,N)

(6) In the local state linear transformations (4), the control system (1) is changed into the following

form

By using the concept and method of approach~ ing rate established by W.B.Gao in [7], we provide two decentralized variable structure control

schemes of Eq. (1) linear large, scale interconnected control system with time-delays.

4.1 Design of the Local Decentralized VSC Law

For interconnected item suppose that the following inequalities of norm hold. Xi ( t)

== T z~ i ( t ), t ::; 0,

t) = Tr4Ji (t ), (i = 1, ~ .. , N)

Uj (

- T ::;

Vrn(t) ERm" Yi == ~l)). T (y~2))T)T a.nd the nmtrices A~l A~2 ((y :t ' '" I ~. t' ~' A;l and Bi all have corresponding dimensions. The equation of the switching surface beconles where yJl)(t) E Rni-m i

IISi(t)lI~ IIB;l ~ [A;;yPJ(t) + A;;y12 )(t)]

(9)

r=l

'ilS

,

c (2)yf2)(t) j

== 0·,

i

== 1, ... ,N (10)

.

1 _

where Cl~ -

(1)1 (2) (Ci C i ).

S

ZJ

IS

S

11

8=1 /y.

::;; E

,A;2,

C1(1)y~1)(t) +

.

+~ A~ly(l)(t - T.(~») + A~2y(2)(t - T~~)] L.. j=l

Pij

IISj(t)ll ~ 1IIlYj (t)1 L

(i == 1)···, N) (13)

Choose the VSC law in order tha·t the attainable condition ST S < 0 is satisfied. That is, the state orbits from any point of the stat~ space can all get to the switching surface in finite time, and

can eventually slip to the origin on the switching

Lemma 1[6] If (Ai, Bj) is a pair of controllable matrices, then (At l , A;2) is a.lso a pair of controllable matrices.

surface. Therefore, the global asymptotic stability of the close-loop system is achieved. For this purpose, we can easily obtain the following formula.

Si(t)l(l) :;: 1)yJl)(t) + C~2)y!2)(t) Cfl)(Afly~l)(t) + A;2y~2)(t)+ Bil(Atly~l)(t) + A12 y j2)(t) + BiUi )+

cl

di-

Obviously, C j(2) can be taken as Tni X Tni mension invertible matrix. To make the calculation simple and convenient) we take

N

B;l ~ [Afrly~l)(t) + A~;y!2)(t)]

(11 )

lows~

r=l

N

+

From differential equations (7),(10) and (11), the sliding Illode equation of i-th subsystem is as fo1-

=

~ A?ly(l)(t - T~~) L.J t$:J ' I)

+ A~2y(2)(t ~s

8

r~~»] IJ

3=1 i~

1,···,N (14) where y == (Yi,···, y_~ ) T. Let the locally decentralized variable control law of the i-th subsystem (i 1~· .. }N) be the following form.

=

From Lemma 1~ we know that (Afl, A;2) is a pair of controllable matrices. Because of invertibility of Bi' we can properly choose the matrix 1) which makes the sliding n10de (12) of i-th subsystem be asymptotically stable~ approach to its equilibrium at any needed tate or have arbitrary previously

Ui

cf

(t) = - [C~l) Ar 1 .:...-[CJl) Af2 -

+ Bi 1 A~l ]y;l)Ct)

+ B;l At 2]y;2)(t) 2 [gi + hiJlIYi(t)11 ] II~~~:~II

(15)

where (i ::::: 1,···, N), gi and all proper positytve

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VARIABLE STRUCTURE CONTROL OF LINEAR DELAYED LARGE...

constants~

where gi, hi are all proper positive numbers, thus

We have

]'y"

L:

ST S ==-

i=l

A};yP\t)] +

I:

=: -

I[yz

i=l 1'1

,

2: [Ar;y~l)(t)+

S~T Si = -gi"Si(t) 11 2

r=l

EAl:yPJ(t - T.j))+

8=1

- 2:

SiT {B;l

i=l

A;;y~2)(t - ri~~))] 1',[

the approaching condition

J.v 1

1'.r

ST Si

14th World Congress of IFAC

[gi

+ hi IIYi (t)

j

(20)

+ hi IIYi(t)11 2 ] 1I~~f:iI1}

2

1 111 Si (t)

.

being satisfied.

::;

11-

5. DESIGN OF VS CLOSED-LOOP SYSTEMS

WITHOUT MATCHING CONDITIONS

I: PiilIYillllSi(t)111/21ISj(t)111/2! = -FTQF i=l

( 16)

If the matcing condition A5) is not satisfied but the other assumed conditions in Section 2 hold~

-:f2],

VSC closed-loop system is to design the switching

F (F1T ~ FIT IlylDT , Fi 1 2 (1IS1(t) 11 / ~ . . . ,IIS.rv(t) 11 1 / 2 ) T, Ilyll

where

(1IY1(t)II,··· ,IIYN(t)I/)T, Q

=[

hiS! (t)sgn(Sj(t)) ~ 0

-

-;/2

G == diag (gl,···, 9}v)'} H ==diag (h t ,···, h N (Pij )"vx ~v·

then the main design work of the decentralized

),

P

function and the switching surface equations, the design of decentralized VSC law being the same as Section 4.

=

When the matching condition A5) in Section 4 does not hold, the interconnected terms of the i-th subsystem have the following form

Therefore, we can obtain t·he following theorern. TheoreIIl 1 The sliding Inode equation' (12) of i-th subsystem of Eq.(l) is the asymptotically stable if and only if all chosen 9i > 0 and hi > 0 (i == 1, ... , N) are great enough so that Q is a diagonal dominant matrix.

N

2:

M

A1ssXs(t)

JV

+E E

$=l,j;i:l

(21)

€PlisXa(t - r)

. j=1 s=1

Letting

4.2 Design of Multilayer Decentralized VSC Law From Section 4.1, we known that the local decentralized VSC law is not always realized in all

The siiding made equation of i-th 8ubsystem~ with the same from of switching surface equation, be-

cases~ So we suppose that the decentralized VSC law ui(i) (i = 1~··· ,N) of the i-th subsystem has the following multilayer forul

comes

(17) where fU~ and

our

are respectively the local control

component and the global control component of the i-th subsystem, u~ only uses the state information up to the i-th subsystem, while uf can use

Furthermore, letting Y r

the information of all subsystems. Therefore (17) is called the multi-order decentralized control law .

diag(Ci r ),

Fronl formula (14), if we let

u~(t) == -{Ci(l)[A~l'Y}t)(t) + AI 2y;2)(t)] +Bi- 1 [A;l Y 1 )(t) + A;2y~2)(t)} -9iSi(t) - hjsgn(Si(t))

1

((Yl(r»

T , ••• ,

Y~;)

T) T,

er

,et»),

r = 1,2,; B=diag (B l , · · · , B N ), on the switching surface. we have the following formula •• •

Y z == -C'2 1 C 1 Yl

(23)

(18) Taking C z as

(24) and

and introducing certain transformation, we obtain

l~r(t)

:::: -B i- 1

+~ L.J

A?l y (l)(t $8 S

lE [Ar/y~l)(t) + At;y~2)(t) 7V

the vector form of sliding mode equation

,'=1

-

T(~)) + A~2y(2)(t - 'T.(~»)] ~~ IJ

S

y1(1)(t)

==

(AI - A 2BC1)Yl(t)+ M

IJ

8=1

ASY1(t)

(19)

+ E

P j Y l(t - Tj)

(25)

j=l

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Copyright 1999 IFAC

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14th World Congress of IFAC

VARIABLE STRUCTURE CONTROL OF LINEAR DELAYED LARGE...

Gao C

where

C~

Liu Y Q. Stabilization and synthesis of two

order linear constant neutral variable structure COlltrol systeIlls. Kongzhi Lilun Yn Yingyong, To Appear ~ 1996 (in Chinese)

Gao C C, Lin Y Q. On the design of decentralized variable structure control of neutral type nonlinear large scale control systeIn with delays~ Proceedings

of IEEE'96 International Conference on Systems, Man and Cybernation t 1996, Beijing. Matthew8 G P., Decentralized variable structure con-

"N

2: (tP~f$ ~ q.~f~B1C~1)),

Pj = diag(

a=l,s:;t:i N

E

s=-l,s;t:i

... ,

trol for class of multiinpu t multioutput nonlinear systems, Proc 24th Annual IEEE Conf on Decision and Control, Fort Lauderriale~ Foride, 1985:1719~

(q>t}v~ - ~~~~BNCiJ))),

1724

Gao W B, The basic theory of the variable structure control. China Press of Sciences and Tech-

We can deternline the stability of the sliding n10de eq uation (25) in t he light of the er iteria of small-delay or delay-independent stability 01' asymptotic stability. Now that (AI, A z ) is a pair of controllable matrices, we can always proper choose the matrix C to determine the boundary

nology, Bejjing, China) 1990(in Chinese):22-24 Gao Cunchen, Liu Yongqing. Variable structure control ofroulti-delays neutral nonlinear large scale control systems-with time-delay input con.,. trol, Journal of South China University of Te-

chnology, 1996,24(5 ):102-108

of the delay I so as to make the sliding mode ha..;; various stabilities.

The method presented in this paper can be easily expanded to the other multi-delay nonlinear

large scale control systems with multi-delay input control

vectors~

6. CONCLUSION

This paper dealt with the decentralized VSC problems linear large scale interconnected control systems (1) with time-delays under different conditions, and offered the design methods of local and luultilayer decentralized VS controllers of these systeIlls by rnaking clever use of the property of diagonal dOIIlinant matrices as well as the concept and method of approaching rate established by W.B.Gao in [7], th,us ensuring the global asymptotic stability of these large scale VS closeloop control syst em8~

REFERENCES Jafarove

E~M~Analysis and

sional VSS with delays

synthesis of multidiInen-

in sliding modes~ Preprints

of l1-th IFAC World Congress on Automatic Con-

trol in the Service of mankind, Tallinn, U.S.S.R, 1990,6:46-49

Yuan F .S. and Liu Y.Q. On the variable struceure con-

trol of N-dimension invariant systems with single time-delay and multi input, Advances in Modelling Simulation~ 1992,31(3):57-64

Y.M~ Zhou QJ. Variable strncenre control of control systems with delay, Acte Automatica Sinica~

Rn

1991 (in Chinese) ~ 1 7(5) :587-591

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