Decentralized Variable-Structure Control for Uncertain Large-Scale Systems

Copyright © IFAC Youth Automation. Beijing. PRC. 1995

DECEl'\TRALIZED VAllUAB:LE-STRUCTURE CONTROL FOR UNCERTAIN LAHGE-SCALE SYSTEMS

Qing Wang

HualoDl: XU

Changhua Hu

Xinhai Chen

College of Mranautics, NurUnr:estl!rn Polyteclmical UnWersity Xi' an, Shaan:.ci Pravmce, 710072, P. R. Chino.

Abstract: A deomtralized variable-structure robust .:onttol for large-scale systems, which are subjected to bounded system parameter variations, uncertain interconnections and external disturbances, is proposed. The uncertainties under consideratic,n are no'; only one-order uncertainties but also higher-order uncertainties. lr. comparision with the decentralized variable-structure control proposed in previous papers, besides the higer-order uncertainties Ilre considered, the number of switching coefficients in local controller is redllced by times, which makl: the con':roller implementation easy. Simulation results are given to demons:rate the effectiveness of the proposed control approach. Key words: Dec~ntralized control; Variabb-structu::e control; Uncertainty; large-scale system; robust control; interconnection

1. INTRODUcrION

many switching coefficients in the local controller, which make the controller implementation difficult. Moreover, the interconnections under consideration in these approaches are only limited to linear interconnections, :'0 while the interconnections are higher-roder interconnections, these approaches are no longer suitable. But in practical large-scale systems, higher-order interconnections exist widely.

In recent years, much attention has been pad to the probem of design decentralized robust controllers for uncertain la;ge-scale systems. Because variable-:itructure syste. llS possess the advantage of insensitivity to system I>arameter variations and to external disturbances, Ihey provide powerful aIr proaches to deal with the complex control pro:>lem of large-:;cale systems, which are subjected to interconnections, system parameter variations and external disturbances. Decentralized varulblestructure control for large-scale systems has been investigated by Al-Abbass et al. (1985), Khurana et al. (1 H8 6), Ozguner et al. (1987), Xiaoh:le et al. (199)) and Chiang (1993) in the recent past. The control approaches proposed by Al-Abbass et al. Cl 98.5), Khurana ~ al. (1986), and Ozguner et al. (1987) are in hi:rarchical control struC!ture indeed. :By using the global reaching condition, Xiaohao et al. (1990) and Chiang (1993) dl:sign the really decentralized variable-structure controller for linear large-scale system with interconnections. However, the approaches mentioned above all have the di~. advantage of needing too

This paper proposes a decentralized variable-structure robust control approach for uncentain largescale systems. The uncertainties under consideration are not only one-order uncertainties but also higher-order uncertainties. By using the global reaching condition, the decentralized feedback form control law is designed. The control law can be expressed by linear control part with non linear control part. The designed decentralized controller can guarantee the system global stability under the effects of system parameter variations, external disturbances, linear interconnections, and even higher-order interconnections as well. In addition, in comparision with the decentralized variablestructure control proposed by Chiang (1993), the

253

significant advantages of the control approach are that it greatly reduces the numble of switching coefficients in the local controller and that the value of control input is decreased to some extent, which make the control approach applicable. The designed control law is in very simple form and easy to implement.

11 11

Et(t.x,) 11 ~U 11 x, 11 +tH Hij(t,xJ) 11 ~~ 11 XJ 11 +t8

( 4) The pair (A" Bt) are completely controllable Then the ith subsystem can be expressed as ",(t) =A,x,(t)+BtlIJ (t)+B,~(t) ei(t)=D,(t)x, (t)+Et(t.x,(t»+ +F,Ct)u, (t) +G,(t)ft(t)

So. the sliding of the composite manifold is

Based on the unit vector control technique in variable-structure control. the following decentralized variable-structure robust control law is proposed

N

i=1,2,···,N

(3)

where St E RIXal is a constant matrix and it is chosen according to the desired system performance.

Xi (t) = [AI+ ~AI (t)x, (t)+[Bt+;illt(t) Ju,(t)

j*'

J*'

(4)

Considering the following large-scale systems with one-order uncertainties, which can be expressed by N interconnected subsystems as

~ gij (t, Xj (t»

~Hij(t.xj(t»

The equation of the sliding submanifold for the ith subsystem can be written as

2. DECENTRALIZED CONTROL FOR UNCERTAIN LARGE-SCALE SYSTEMS

+ d i (t, Xi (t) ) +

(2)

N

The remaider of the paper is organized as follows. In the next section, the decentralized control law for large-scale systems with one-order uncertainties is proposed. In section 3, the decentralized control law for large-scale systems, which are subjected to higher-order uncertainties is developed. Numerical simulations are given in section 4, and conclusions are summarized in section 5.

+ C, (t) f, (t) (1)

U,=UU+UNi

where XI (t) E Rni, Ui (t) E RI, f, (t) E R U are state, control and external disturbace of subsystem respectively. Ai and Bt are nominal system matrices with appropriate dimension. ~,(t). ~ (t). and Ci (t) represent variations of system parameters d, ( t , Xi) is subsystem uncertainty. and g'j (t • Xj) are uncertain interconnections.

(6)

Ci= 1 ,2.··· .N)

(7)

(8) N

To complete the discription of the large-scale system with one-order uncertanties in eq. (1), the following assumptions are introduced.

cP=(1-

11

F,

cl = Cl -

11

F,

11

G,

11

11 ) - I ( 11

D,

11

11

)-I(

11

f,1I +

~t8)+
J-I (
+ 11 F, 11

11

K,

11

N

+U+y~~\)

Assumptians 1

J*'

Cl)

(y~max 1~Bt 1/min 1s,B,I)

~i (t), ~ (t),

d, (t.XI)' C(t). ftCt) and g'J (t, Xi) are bounded on their arguments.

where Uu expresses linear control part. which makes the behavior of the subsystem restrict to the switching subsurface; UN' expresses nonlinear control part. which suppresses the effects of the system parameter variations. uncertain interconnections. subsystem uncertanties and external disturbances. and drives the subsystem trajectories toward the switching subsurface.

(2) Matching conditions: There exist appropriate dimension matrices D, Ct) , E;(t.Xi)' F,(t), G,(t) and Hij(t,xJ)' so that ~, (t)=BiDi (t)

d i(t ,x,) =BiE, (t,x,) .m Ct) =BtF, (t) C(t)=BiGi(t) gij(t ,xi) =B,H,j(t ,Xi) and max 11 Fi(t) 11 <1

Throrem 1: Consider the uncertain large-scale system (1). Suppose that the assumptions 1 (1)'-'" (4) are valid. If the decentralized variable-structure robust control law (6)'-'" (8) are adopted. then the mo-

(3) E; (t , x,) and Hij (t. Aj) satisfy the following inequalities

254

tion of the composite manifold (5) is asymptotically stable.

can be expressed as

Proof: In order to prove the proposition, the following lyapunov function is adopted (Xiaohao et al. ,

3. DECENTRALIZED CONTROL FOR

1990; Chiang, 1993)

LARGE-SCALE SYSTEMS WITH HIGHER-ORDER UNCERTAINTES

N

V(t)= L;dl(t)ufCt) 1=1

Considering the following large-scale systems with higher-order uncertainties, which can be expressed by N interconnected subsystems as

>

where d l (t) 0 satisfy dl (t) I UI (t) 1= p (p is a positive constant). By calculation, the following equation can be obtained

XI (t) = [AI + ~ (t) ]XI (t) + [BI+~ (t) ]UI (t) +gl (t,x(t) )+(;(t)fl (t) (10) i=I,2,···,N

N

V (t) =

L; d l (t)UI (t)OI (t) 1-1

(9)

where XI(t) ERnl, UI (t) ERI, and fl(t) E R U are state, control and external disturbance of subsys-

N

N

( L; d l (t) UI (t) 01 (t) < 0 is I-I

called global reaching

tem respectively. n = L;nl, and X (t) E RD are I-I system state. AI, and ~ are nominal system matrices with appropriate dimension • .lAI (t),
condition. ) N

L; d l (t)UI (t)OI (t) 1=1

N

=

L; d l (t)UI (t)SI,(1 (t) I-I

In order to discribe the large-scale system with higher-order uncertainties in eq. (10), the following assumptions are introduced.

N

= L;dl(t)UI(t)SIBI(uNI+el) I-I N

~- L;dl(t) IUI(t)SIBII (~o+clll xIII) 1=1 N

+ L;dl(t) I UI(t)SIBd [( 1=1

+M)

11 0 1 11

XI

11

11

C et), Mt) and gl (t, x) are bounded on their arguments.

(1) .lA1 (t), .lE(t),

N

+ L; M 11 Xj 11 ,""I .

+

+cl

L; ~~+ j=1

11 xIII) +

N

11

FI

11 ( 11

KI

11

11

XI

11

+CP (2) Matching conditions: There exist appropriate dimension matrices 0 1( t) , ~(t,x), FI(t), and GI(t), so that .lA1 (t) =BIDI (t) gl (t,x) =B;EI (t ,x)

N

11 GI 11 11 fill) ] N

L; dl (t) IUI (t)SIB.! L; M 11 Xj 11 1=1 J;oOI N

~(t)=BIFI(t)

N

~ L;dl(t) IUI(t)SIBII (y L;~D J;oOI

11 xIII

CI(t) =BIGI(t) and max 11 FI(t) 11 <1

I~I

(y~max ISI~ I/min ISI~ I)

Then using the expressions cP and inequality is obtained

(3) ~(t,x) are bounded by a pth-order polynomial in states, which can be expressed as

cl, the following

P

N.

V(t)~- L;dl(t) IUI(t)SI~ la; <0 (a; >0)

11

I-I

E(t,x)

11

=

N

~ ~~ k-O j-l

11

Xj

11

k

(11)

(4) The pair (AI, BI) are completely controllable.

So the motion of the composite manifeld ( 5) is asymptotically stable.

Then the ith subsystem can be written as

In application, boundary layer modification for nonlinear control part can be introduced to improve

XI (t) =Alxl (t) + ~UI (t) + Blel (t) el(t) = 0 1(t)XI (t) +E (t;x (t» + FI (t)UI (t) +G I(t)fl (t)

the variable-structure system performance, which

255

{12)

(13)

N

~- ~d,(t) 10'1 (t)&Bd (~er 11 XIII

The equations of the sliding submanifold and the motion of composite manifold are defined as the same as eq. (4) and eq. (5).

I-I

~d,(t) 10'1 (t)&Bd[

+

1-1

11

0 1 11

11 xIII

N

+ t-O ~~ ~ j-I

Then the following decentralized variable-structure robust control law for large-scale system with higher-order uncertainties is proposed

11

t+

Xj 11

11

F, 11

P

• ( 11

KI

+ 11 Gill

(14)

..

(15)

o

P

11

11

11

N

XI

11

+

~ er

11

t-O

XI

t)

11

XI

N

~d,(t) 10'1(t)&~ I t~ ~~ 11 Xj 11 t I-I - I j,,1 P

N

~~ d l (t) I 0'1 (t) &Bi I ~ (y ~ ~) I-I t-I j,,1

0'1=0

11

fill ] P

N

UNI= { -sgn(O'ISIBI)( ~er 11 xIII t)

11 t

(y;,;?;max ISI~ I/min ISI~ I)

(16)

er

and by using the expressions of (k = 0,1 , •.• P) the following inequality is obtained

N

cP=(1- 11 FI 11- 1 ) ( 11 Gill 11 fill +

t)

t-O

N

P

o

P

~tD+
N

~ dl(t)O'I (t)OI (t)

cl =O-IIF,II-I)(IID,II+IIF,IIIIK,1I

I-I

N

N

+t\+y~t.D i,,1

~- ~d,(t) 1-1

I0'1 (t)SIBt 10.: <0

(0.:>0)

N

c~=(l-II F,II-I)(~+y~~D j,,1

(2~k~P)

Thus, the motion of the composite manifold (5) is asymptotically stable. In application, boundary layer modification for nonlinear control part can be introduced to improve the control system performance, which can be expressed as

where ULl expresses linear control part and UNI expresses nonlinear control part. y;,;?;max IS,B, I/min IS,B,I and
Theorem 2: Consider the large-scale system with higher-order uncertainties shown in eq. (l0). Suppose that the assumptions 2 (1)"'" ( 4) are valid. If the decentralized variable-structure robust control law (14) --- ( 16) are adopted, then the motion of the composite manifold (5) is asymptotically stable.

4. SIMULATION RESULTS

Proof: In order to prove the proposition, the same lyapunov function as in the proof of theorem 1 is adopted, so

In this section, two examples is given to demonstrate the performance of the proposed decentralized variable-structure robust control approach for large-scale systems. &le 1

N

V(t)=

~dl(t)O'I(t)ol(t) 1=1

The plant is given as following

N

=

~dl(t)O'I(t)SIXI(t)

3~bJxI +[ ~Jul 0: 4Jx2+[~. 05J

1=1 N

=

~dl(t)O'I(t)S!BI(uN!+el) 1=1

N

P

~- ~d,(t) I0'1 (t)S,B, 1(~cr 11 1=1

XI 11 t)

t-O

N

+

~dl(t) I01 (t)S,B, I 11 el 11 1=1

256

&le 2

where a= O. 2sin (2t). b= O. 3cos(3t)

The plant is given as following

XI=[_~+a 3!bJxI+[~JUI

The sliding submanifolds are chosen as

+[ The decentralized control law (6) - (8) is applied. All the initial condition of the states are O. 5. Fig. 1 shows the dynamiC responses of subsystem states Xll and X12. Fig. 2 and Fig. 3 show the local control signal UI and Ut respectively.

O~ 5 O~ 4Jx2+[~¥I+x~J +[~. 05J

X2=[

O~ 2 O.12Jx2+[~Ju2

+[O~3 O~4Jxl+[~.lJ where a= O. 2sin(2t). b= O. 3sin(3t)

QS~------------------~

The sliding submanifolds are chosen as

The decentralized control law (14) - (16) is applied. All the initial condition of the states are O. 5. L....__-'-__..o-.__.....__.......__..J

~

o

2

4

8

8

10

Fig. 4 shows the dynamic responses of subsystem states Xll and XI2. Fig. 5 and Fig. 6 show the local control signal UI and U2 respectively.

t(s) Fig. 1

The dynamic responses of states XII

&. XI2

QSr-------------------~

4r---~--------------~ 3 2

·1 ~

I-..--_____""'-_""--_.....__.J

o

8

8

10

t(s)

t(s) Fig. 2

2

Fig. 4

The control input u I

The dynamic responses of states Xll 4r-------------------~

2

.1

0

·1 -2

-2

-3

0

2

8

4

8

10

t(s)

t(s) Fig. 3

Fig. 5

The control input U2

257

The control input UI

&. XI2

scale systems with higher-order uncertain interconnections. Second, in comparision with the decentralized variable-structure robust control proposed by Chiang (1993), the number of switching coefficients in the designed local controller is greatly reduced, and the value of control input is decreased, which make the designed controller implementation easy. The two distinct advantages enable the proposed approach very applicable to complex uncertain large-scale systems •

8

.. 2

0

·2

... 0

2

.

8

8

REFERENCE

10

t(S)

Fig. 6

The control input

Al-Abbass, F. and Ozguner, U. (1985), Decentralized Model Reference Adaptive System Using a Variable Structure. Proa?Jedings 25th Canf. Decision and Control. pp. 1473-1478. Chiang, C. -C. (1993), Decentralized VariableStructure Adaptive Controller Synthesis of Large-Scale Systems Subjected to Bounded Disturbances. Ird J. System Sci. Vol. 24, pp. 1101-1111. Khurana, H. , Ahson, S. I. , and Lamba, S. S. (1986). On stabilization of Large-Scale System Using Variable Structure System Theory, IEEE TrailS on AC. Vol. 31, pp. 176-178. Ozguner, U., Yurkovich, S., and Al-Abbass, F. (1987). Decentralized Variable Structure Control of a Two-arm Robotic System. Proc. IEEE Conf. ROOotic &. AuIornDIitm. pp. 12481254. Xiaohao, Xu., Wu Yaohua and Huang Wenhu (1990), Variable-Structure Approach of Decentralized Model-Reference Adaptive System. lEE proa?Jedings. Vol. 137, pt. 0, pp. 302-306.

Uz

From these simulation results, it is demonstrated that the proposed decentralized variable-structure robust control law enables the states of large-scale interconnected systems stable quickly in the presence of system parameter variations, bounded disturbances, and even under the effects of higher-order interconnections and uncertaties.

5. CONCLUSIONS The proposed decentralized variable-structure robust control for large-scale systems can guarantee the system stability under the effects of system parameter variations, subsystem uncertainties, uncertain interconnections and external disturbances. The control approaches have two distinct advantages. First. while designing decentralized robust controller for large-scale systems, not only one-order uncertainties but also higher-order uncertainties are taken into consideration, which make the proposed approach still applicable to complex large-

258