Robust Stability of Nonlinear Large-Scale Systems with Uncertain Parameters

5a-025

Copyright © 1996 IFAC 13th Triennial World Congress. San Francisco, USA

ROBUST STABILITY OF NONLINEAR LARGE-SCALE SYSTEMS WITH UNCERTAIN PARAMETERS

Xinggang Van, Xingya Lii, Yuanwei Jing and Siying Zhang

P.D.Bole 195, Dep,. 0/ Aulomal;c Oontrol, NortAeutern Urti.er,;t, Shen,."I, LiGOn;n" 110006, P.R.Ohina

Abstract: In this paper, noulinear large-scale composite systems with uncertain parameters are investigated. By applying differential geometric theory, the problem of robust stability of nonlinear composite systems is transformed to the corresponding problem of a class of noulinear composite systems in which each isolated subsystem is linear. Then, some criteria of robust stability for the noulinear large-scale composite systems with uncertain parameters are presented. Finally, an example is introduced to illustrate the result of this paper and the method of finding the parametric robust space of large-scale composite systems. Keywords: large-scale systems, asymptotic stability, Lyapunov function, differential geometric method, singular value.

1. INTRODUCTION

It's well-known that composite systems widely exist in the practical world. Recently, the study of composite systems has been receiving much attention, see (Qu and Dawson, 1994; Yang and Zhang, 1993; Jiang et al., 1995). The problem of robust stability is also very important in practical engineering. So it's of great meanings to study the robust stability of composite systems.

The main task of studying robust stability of systems is to seek for the maximum robust bounds or robust region. Its general trend is to find some criteria which are unconsenative on one hand, and easy to be verified on the other hand. Recently, many results about robust stability of linear system (e.g., Chou, 1990; Zelent6vsky, 1994) and nonlinear systems (e.g., Lewkowicz and Raphael Sivan, 1993) have been achieved. The corresponding results about nonlinear large-scale systems, see (Barbieri, 1993), are very little because of the noulinearity and interconnection. In this paper, robust stability of nonlinear large-scale

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composite systems with uncertain parameters is to be studied. This paper is organized as follows. In section 2, systems are described and problem is formulated. In section 3, the preliminaries such as some definitions, notations and some fundamental results are given. In section 4, some criteria of robust stability for a class of nonlinear composite systems with uncertain parameters are presented. An illustrative example is introduced to show the method of finding the parametric robust space of large-scale composite systems in section 5. Section 6 concludes the paper.

2. SYSTEM DESCRIPTION AND PROBLEM STATEMENT

Consider an uncertain composite systems described by the following equations

Allumption 9 There exist ni ni-dimensional linear independent commutative vector fields

defined in 0 n Rn; such that [Ji (Zi), X;I = 0 for all q= 1,2,···,n;, andi= 1,2,···,N. Remark £.1 Assumption 3 guarantees that the isolated subsystems of systems (1) are linearizable by diffeomorphism. Now, the purpose of this paper is to solve the following problem: For systems (1), under Assumption 1-3, find a parametric perturbed region, i.e. parametric robust space (P RS) as unconservative as possible, in which parameters are perturbed, systems (1) are asymptotically stable in domain 0. This problem is called local robust stability of composite systems (1). If 0 = Rn, then, the coresponding problem is called global.

K

Zj=!j(Xj)+LEjjHjj(X)+9j(X),

i=I,2,···,N (1)

j=1

where x =

col( xl, x" ... , XN)

E

3. PRELIMINARIES

Rn and Xi are N

nj-dimensional state vectors of subsystem,

Lnj

=

j=1

n, j;( Zj) E Rn; are no-dimensional smooth vector fields defined in Rn;, Hij(Z),9i(Z) E Rn; are analytic interconnecting items defined in Rn, Ei; E R are uncertain parameters.

Let ~ (zo) denote the integral curve started from the initial state Zo after z times, Pf denote the matrix Q if QTQ = P, DM(A) denote the maximum singular value of matrix A, 11 Z 11 denote the 2-norm defined as follows n

For simplicity, the equilibrium points of uncertain systems in this paper all point to isolated invariant equilibrium points under perturbations if any special illustrations are not given. Definition £.1 The vector fields X and Y are said to be commutative if [X, YI = 0, where [X, YI denotes the Lie bracket of vector fields X and Y. The investigation in this paper is based on the following assumptions. Allumption 1 0 C Rn is a given domain which is a neighborhood of Xo = (ZIO, X20,' .. ,ZNO), and

= Hij(XO) = 9j(Zo} = = 1, 2, ... , N, i = 1, 2, ... ,K.

1

IIzlI=(LZ~P',

Z=(ZI,X2,"',Zn)'

(3)

j=1

Other norms such as matrix norm are all derived from it. Definition 9.1 (Huang, 1988) Suppose A(z) is m x n 1 function matrix defined in 0. Oii(Z) = (>';(Z»I (i = 1,2"",n) are called singular values of matrix A(z) at Z E 0, if >.;(z) (i = 1,2,··· ,n) are eigenvalues of AT(z)A(z) at each fixed point Z E 0. Lemma 9.1 (Huang, 1988) Suppose A(x) is a m x n matrix defined in 0. Then, 11 A(z) 11= DM(A(z» at each fixed point Z E 0.

(2)

It's necessary to give the following result which plays a great role in solving the problem stated in section 2.

Allumption £ The isolated subsystems of systems (1) are asymptotically stable at Zo in domain 0.

Lemma 9.1 For system (1), under assumption 1-3, there exist a diffeomorphism T : z 1-+ z defined in T-I(O)

fo(XiO)

for all i

0

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such that in new coordinates z, composite systems (1) have the following form K

Zi = Aizi+ L Eij+ij(Z) + Gi(Z),

i = 1,2,,'" N (4)

where

and

j=1

where Z = col(zI, Z2,' .. , ZN) E Rn , all Ai (i = 1,2, ... , N) are Hurwitz stable and

(+Ij(z), +f;(z), ... , +~j(z))T = [8T~:('")(H~(x),Hl(x)"" ,H.tj(x»T]",=T(Z) (j=I,2,···,N)

(Gf(Z), Gi(z), ... , G~(Z»T = [8T;;:(s) (gHz), gi(z),.· . , g~(z))Tl"'=T(z)

are n-dimensional analytic vector fields defined in T-I(O).

Proof: From assumption 3, there exist ni linear independent commutative vector fields xi, X~, ... ,X~; with [X~,X;) = 0 for all q, r = 1,2"", ni, and i = 1,2"", N. Now construct ni n-dimensional vector fields of the form

---

iT , 0"",0 )T Xqi = ( 0, ... ,0 ,(Aq) '---

where q = 1,2"," ni, i

= 1,2"", N.

(5)

OT,;-I (Xi) Gi(Z) = [ { ) gi(Z)]z=T(z) Zi

It's obvious that systems (4) are the version of systems (1) in new coordinates z. So from the character of diffeomorphism and Assumption 2, it is observed that Ai are Hurwitz stable for all i = 1,2"" ,N, and Z = 0 is an isolated equilibrium point of systems (4). Remark 9.1 From Lemma 4.1, it follows that under Assumption 1-3, the study of robust stability for systems (I) can be transformed to the corresponding problem of systems (4). It's necessary to emphasize that our linearization method for isolated subsystems of systems (1) is different from approximate linearization because the linearizable region here may be very large and sometimes may be the whole state space. Remark 9.2 Pherhaps the equations similar to (4) can be got easily from system (1) by other method, for N

example, Xi = Mz i+ LEiiH,}(z)+9i(Z)+(Ji(Zi}-Mxi). i=1

N

By Lni

= n, n

i=1

n-dimensional linear independent commutative vector fields defined in 0 are obtained as the following

But such identical deformation is of no value for solving the problem because the properties of !.-(Zj) are not been used sufficiently and the interconnection becomes more complicated.

(6)

Then, construct a map T : Z T(ZI'~"" ,ZN)

1-+

4. ROBUST STABILITY

x as follows

=';;10 +;'20 ... 0 ~:(zo)

(1)

x.X·. = ,x.x\ x. x~; h were '*' *,' 0 ,x.x~ *,. . = T. .(ZIi ... Z ~i ) ~ ~ 4 0'" 0 , * ,~i'" for all i = 1, 2, ... , N. From the structure of vector fields (6), [Ji(Zi)'X~] = 0 for all q = 1,2, .. ·,n; and i = 1,2" .. , N, it follows that

[/(z},X~) = 0

Next, we are to investigate the robust stability of systems (4) in the domain T-I(O). For simplicity, let E = T-I(O). Now, consider systems (4). For any given group of positive definite matrices Pi (i = 1,2· .. , N), the Lyapunov equations

A!Q· + Q·A· = -P.., I' "

q = 1,2, ... ,n;, i = 1,2" ., , N. (8)

i=I,2 .. ·,N.

,

(10)

have unique corresponding group of positive definite solution matrices Qi because Ai are Hurwitz stable matrices. Now, by Assumption 1 and Cheng 1988, it is concluded that T is a diifeomorphism satisfied with T-I(ZO) = 0, and in new coordinates z, systems (1) are characterized by the following equations

+ij(Z)ZT

K

Zi

= Aizi + L

i=1

Theorem 4.1 Suppose Z = 0 EEC Rn is an isolated equilibrium point of systeJllB (4). Let

Eij+ij(Z)

+ Gi(Z), i =

R;;(z} = \

1,2,···,N (9)

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TzT'

zi "" 0

0,

Zi

=0

(11)

Li(Z)

=

I

systems (4) is

Gi(Z)i[

11 Zi 11' ,

0,

Zi

#: 0

Zj

=0

(12)

and W = (Wij(Z» is a function matrix defined in E, where Wij(Z) are defined as follows .) 'f K 1

1

N

=, Wij

() Z

=

{I - 2eij, i = j

ii) if K > N, Wij(Z) =

Hi) if K < N, Wij(Z) =

-2eij,

i

#: j

1 - 2eij, i = j and i $ N -2eij, i #: j and i $ N 0, N < i$ K

1

1 - 2eij, i = j and j $ K -2eij, i #: j and j $ K 0, K
1

N

I

$ - Ld,.[ 11 j=1

Plz,. 11' -

l '

1

N

1

211 P/ Zj 11 DM«Pj-J)TQi(L£jjR;j(z) + Li(Z))Pi-')] _1 T 1_1 where eij = OM«Pi ') Qi(£ij~i(z) + KLi(z»Pi '), and Pi, Qi for all i = 1,2,···, N are determined by (10) . If there exists a diagonal matrix D = diag{dI,~,···,dN} (di > 0 lorall i = 1,2,···,N) such that WT D + DW is a positive definite function matrix in E, then, Z = 0 is an asymptotically stable equilibrium point of systems (4) in domain E.

Proof: We mainly give the proof of the case K = N. Other two cases can be easily got by it.

i) K = N.

N

L

d;zTQiZi

$ - L {d; 11 Pi' %i 11' (1 - 2 L eij)} j,=1

i=1

= _!yT(WTD + DW)Y 2

1

1

1

(16)

where Y = (11 Pt %1 11,11 Pi Z2 11"",11 PNZN II)T. Therefore, Y = 0 if and only if Z = 0 because Pi for all i = 1,2,"', N are positive definite. Then, V(z) 1(4) is a negative definite function in E from the positive definiteness of WTD + DW. Hence, Z = 0 is an asymptotic stable equilibrium point of system (4) in domain E.

(13)

Construct new composite systems as follows by adding K - N subsystems to systems (4). It follows that

i=1

Zj = Ajzj + Lf=1 £ii~jj(z) + Gj(z), i

where di > 0 (i = 1,2,· .. , N) are weighted coefficients. It is obvious that V(z) is a positive definite function the positive definitiveness of Qi and di > 0 for all i = 1,2,···,N.

It follows from (11) and (12) that

G,·(Z) = Li(Z)Zi ,

N

I

ii) K > N

Construct a Lyapunov function

V(z) =

j=1

N

i,j=I,2,···,N

(14)

i = 1,2,···,N

(15)

ZN+l

= -ZN+I

= 1,2,· .. , N

(17) where COI(ZN+I,ZN+"""ZK) E RK-N. By applying the conclusion of the case K = N to systems (17), the corresponding result in case K > N is got at once.

iii) K < N systems (4) can be described by N

and the derivative of Lyapunov function V (z) along the

Zj = Ajzj+

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L £ii~jj(%)+Gj(z),

j=1

i = 1,2, ... , N (18)

°

where ~ij(Z) = if K < i:5 N. Then, applying the conclusion in case i) to systems (18), we deduce that the corresponding result of the case K < N is true. Hence the result. From the proof of theorem 4.1, it is observed that the conservativeness of the P RS can be reduced by choosing appropriate Pi and di (i = 1,2,··· ,N), and the following result can be obtained easily.

select R;j(z) and Li(Z). Appropriate R;j(z) and Li(Z) are able to make the computation simple. It is necessary to emphasize that PRS of systems (4) can be got directly by PRS of systems (1). Therefore, the PRS of systems (1) can be found by theorem 4.1 and corollary 4.1. Further, PRS of systems (1) can be obtained by the positive definitiveness of matrix WT(z)D + DW(z).

Corollary ~.1 Suppose Z = 0 EEc Rn is an isolated equilibrium point of systems (4), Rij(z), Li(Z) is defined by (11) and (12), E is a neighborhood of

5. ILLUSTRATIVE EXAMPLE

Z = 0, 8UPZEED~«Pi-t)TQi(EijR;j~Z)+!fI Li(Z»Pi-~) = Ai;' W = (Wij) IS a constant matnx an Wij defined as follows

Consider the following uncertain nonlinear composite systems

.. _ { 1- 2Aij, i = i - , w' l 2\ .~ .

.) 'f K - N

I

I

-

Aij,

1 - 2Aij, -2Ai;, 0, 1 - 2Aij, -2Aij,

ii) if K > N,Wij =

1

iii) if K < N,Wij =

1

0,

'r 1 i =j

and i:5 N i =F i and i:5 N N < i:5 K i = j and i:5 K i =F j and j:5 K K < i:5 N

where Pi, Qi are determined by equations (10). If Z = 0 is an asymptotic stable equilibrium point of systems (4) in domain E. WT + W is a positive definite matrix, then,

Remark ~.1 The key differences between theorem 4.1 and corollary 4.1 are that W is a function matrix in theorem 4.1 and a constant matrix in corollary 4.1, and the latter is easier to be veryfied than the former but it is more conservative than the former. Remark

~.e

.

In corollary 4.1, Aij can be replaced I

by the upper bounds of DM«Pi-·)TQi(EijRij(Z) +

~Li(Z»Pi-!)' but the results obtained in such Ai; for all i = 1,2,···, N, i = 1,2"", K will be more conservative than corollary 4.1. Remark ,j..9 It is observed that (11) and (12) can be replaced by (14) and (15) respectively, from the proof of theorem 4.1. A method of choosing R;j(z) and Li(Z) with i = 1,2,···,N; j = I,2,···,K is shown by (11) and (12). It is emphasized that there are many Rij(Z) and Li(Z)(i = 1,2"", Nj i = 1,2,···, K) satisfying (14) and (15), and there are also many methods to

1

1

2

. 2

ZI = --ZI + -Elz,e

-I:'

I

srnz.

1

+E2Z1 srn Z. + 2zlz. . 1 ( ZIZ2+Z, ) +EI (1-Z2Cosz.+-e 1 _% 'Z2srnz. 2' ) z,=-2 4 1 2 +e2(z,COS z a + ZI Z2sinZ.) + 2Z'(ZI + ZIZ.)

2:8 = -ZI + e.z. sin(zze- ZI ) + ZIZ. 2:. = -Z. + EaX; cos Xl + Z3

(19) Study the local robust stability of systems (19) at eqUilibrium point Z = 0 in domain

n=

{(Z., z" Za, x.)1 i

1Xi 1< I, 1Zj 1< ~, = 1,2; i = 3, 4}

Let

Then, according to the structure of X},X2 ,X"X. construct 4 4-dimensionallinear independent commutative vector fields XI, X" XI and X. as follows XI X, X. X.

= = = =

(O,eZ',O,O)T, (l,z"O,O)T, (0,0,1, O)T, (0,0,0, I)T

Let

Then, in new coordinates z defined by T-I, systems

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studied. One of the items is perturbed interconnecting item and the other is unperturbed interconnecting item. It is emphasized that the method of this paper is different from the method of approximate linearization, because the domain which the investigation concerned with may be very large. Furthermore, the method of finding P RS of nonlinear large-scale composite systems is presented.

(19) are described by .

1

1

1

ZI = -izl +elizlcOSZf +e2Z1 COSZI + iZIZ3

(21) Now, the robust stability of system (19) in 0 is transformed to the corresponding problem of system (21) in domain

Barbieri, E. (1993). Stability Analysis of a Class of Interconnected Systems, Tram. ASME J. Dyn. Syst. measurement and control, 115, 546-551.

E = {(ZI,z2,Za,Zf)1 1zIe Z2 1< 1, 1z21< 1, 1

1

1ZI 1< 2' 1Z4 1< 2} Let PI = I, Pt = 21. Then Ql directly, it follows that

~ll ~2J

1

Computing

1

21 e) 1+8' 1 ::;1 el 1+2'

Chou, J.H. (1990). Stability Robustness of Linear State-Space Models with Structured perturbations. Syst. & contr. Lett. 15, 201-210.

1 1]

-2 1e, 1-- 1el 1-2 2 3 - 2 1e2 1- - 1ea 1-2 2 2 Therefore, by corollary 4.1, the set

=

21 e) 1

2

1

1

is a PRS of systems (19) in domain O.

Remark 5.1 Pay attention to Remark 4.6. In this example, R;j(z) and L;(z) satisfying (14) and (15) with i,j = 1,2 are chosen as follows

L1(z)

liCOBZ4

= [ ••• 0 o

R22 (z)

= [~

,0

1 0.

r-;·mz,

], R 12 (z)

= [COoBZa

R21(z) = [

-Z2

0 COBZ,

Linear Algebra in SYBtemB Huang Lin (1988). and Control Theory (in Chinese), Science Press, Beijing. Jiang Bin, Xiaoping Liu and Siying Zhang (1995). Robust StabiIization for a Class of Composite Nonlinear Large-Scale Systems with Unmatched Uncertainties, Proc. 0/ ACC., Seattle, Washington.

{(el,e2,e.)1 I e.l< 0.15, 31 el 1+12 1 e2 1 +HI E, 1 -I)]' < 2.25}

RIl(z) =

Cheng Daizhan (1988). The Geometric Theory 0/ Nonlinear Systems (in Chinese), Science Press, Beijing.

::;

T+~- -

w[

= Q2 = I.

REFERENOES

0 J Binz4 '

BinZl]

o '

~f.

6. CONCLUSION

Lewkowicz Izchak and Raphal Sivan (1993). Stability Robustness of Almost Linear State Equations, IEEE Tram. Aut. Contr., S8, 262-266. Qu Zhihua and Dawson Darren M. (1994). Robust Control of Cascaded and Individually Feedback Linearizable Nonlinear Systems, Automatica, SO, 1051-1064. Yang Guanghong and Siying Zhang (1993). LargeScale Systems Containing Multi-Partite Symmetrically Composite Subsystems: analysis and synthesis of decentralized controllers, Proc. 12th IFAC. World Congress, Sydney. ZelenMvsky, A.L.(1994). Nonquadratic Lyapunov Functions for Robust Stability Analysis of Linear Uncertain Systems, IEEE. Trans. Aut. Control, S9,

In this paper, nonlinear large-scale composite systems including two kinds of interconnecting items have been

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