Some Results on Nonlinear Decentralized Systems1

Copyright © IFAC Large Scale Systems, Beijing, PRC, 1992

SOME RESULTS ON NONLINEAR DECENTRALIZED SYSTEMSl Feng Gao, Zheng-zhi Han and Zhong-jun Zhang Department of Automatic Control, Shanghai iiaotong University, Shanghai 200030, PRC

This

Abstract

decentralized decoupling

paper

studies

systems,

(DDD)

Some

and

two

design

problems

for

decentralized

criteria

decentralized

lineal'zation

(DL)

for

nonlinear disturbance

are

given

via

differential geometry method. nonlinesr

Keywords

systems,

decentralized

control,

disturbance

decoupling, linearization

1 INTRODUCTION

nonlinear systems with decentralized imformation

problem of disturbance decoupling is of

THE

structure.

fundamental importance in the designing theory

This

of control systems. This problem has been widely

about

paper DDD

will

and

present

DL

for

some

nonlinear

conclusions systems.

In

investigated for various types of systems. The

Section

problem has been solved for linear and nonlineal'

including notations, definitions and lemmas. are

systems by using the notions of (A,B) invariant

introduced. Section III studies the

subspace

DDD . Two theorems are established respectively

(Wonham

distribution 1975, the

(Isidori

problem

research

of

large

scale

decoupling systems

I

)

by

systems,

condition

decentralized

(A,B

invariant

respectively.

1989),

disturbance established

(f,g)

has been introduced

necessary

and

and

1979)

using

invariant

a

of

the

necessity

and

problem of

sufficiency.

Section

IV

problem of DL. we also establish

two theorems, one gives the necessary and the other the sufficient conditions. The problem for

linear

further

was

section.

stations

notion

subspace

the

consider the

meterial .. :

sufficient

for

two

for

prelimenary

Dome

decentralized

(DDD)

with

Since to the

11,

investigation

are

given

in

the

last

of structural

(Hamano

Furuta

&

2.

PRELIMINARIES

1975) . Many research papers were then presented

THE paper takes the following notations:

to investigate how to extend the results to the

M or N: smooth manifold

systems with more than

V(M): the set of smooth vector fields on M

two stations (e.g. Cury

et al 1982, Hu & Zheng 1987, Leite 1985). Some general

criteria

were

obtained.

Recently,

research for DDD was further extended nonlinear been

systems.

obtained

A

for

sufficient

C"'(M,N):

the

maps M to N

to the

condition

Sp

{gl'

has

spanned g €V(M)

a

class

of

nonlinear

by

using

the

notion

the set of sDlooth mappings which

by

the

g ):

gl' gl'

r

gl'

distribution

where

... ,

r

decentralized structural Zhang

systems

(f,gl)

invariant

1990). This

investigation

for

distribution

paper will continue nonlinear

(Han

of

h.:

&

reduced

map

such a

k: the set

fordesigning

of

technique, the

is

one

of

nonlinear

the

useful

systems.

By

designing approaches

method such

a

m

developed

XI =f II (XI )+f ll (x 2 )+

for

problem of linearization

centralized

systems

(Hunt

et

is

I

linearization

(DL)

for

nonlinear system

I

1. I

l

x2=f ll (xl )+f2l (X2 )+L gll(x l )u 21 +Pl(x l )"'l

al

(1)

,. I

1982), it is natural to consider the problem of

decentralized

the

L gli (X 1 )U ll +PI (XI )"'1 m

the

if

with disturbance described by

for linear systems can be used to the non linear After

h.=ah/ax

(1, 2, ... , k)

Consider the decentralized

Linearization

solved

h,

Ker h.: the kernal of h.

decentralized

systems.

systems.

of

coordinate system x is specified

YI =h l (Xl)

those Y2=h 1 (X 2 )

The project is supported by National Natural Science Foundation of China

where all

299

for

belong

f H' glk (k€!!!,) and PI i=I,2 x €M l " to V(M ), for i,joSf. with i"j flJ 1

which hi

and

are

the

output

(i=1,2)

are

the

functions,

(L"T(x»a(x) = -L"T(X)

the T Ut =[uu UIl,,"u lm ] TT

disturbances. for

and

Denote

that

where I is the identity matrix and 0 the zero

and

matrix.

",=[",T",T ]T

i=1,2,

(6)

controls,

(L T(x»{3(x) = [I 0]

t

'"l

T

simplcity) as follows:

to C (M J ,M I ) are interconnections ,

hl

Uti 'U 11 '···'U 1m

'"l'

(J)

belon~

1

l

"

g

Lemma

I

y=[Yl Yl] .

(Isidori 1989) System (5) is completely

linearizable

The feed backs are said to be decentralized

at

Xo

by

state

feed back

iff

the

following conditions satisfy simultaneously:

if

(2)

for i=1, 2

(1). there exist N integers n N

1

Definition there

The

exist

DDD

problem

decentralized

is

solvable

feed backs

that the disturbance'" do not

L ni- n

the properties that

with (31 (XI) is invertible for i=1 or 2. (2)

affec~

~

n

l

~

... ~ nN with

and, ni- m such that

I-I

if

ad c g 1

such

are

outp~t

the

1

y, i.e., for two disturbances", and "', with "'''''',

at xo' (2).

the output of system (1) satisfies that

System

is

(5)

, .. aclfg n

,

l

independent

vectors

input-output linearizable

y(t,x lO ,x lO ,u1 ,ul,,,,)=y(t,xl0,xlO,Ul ,ul,W) (3) for all inputs u 1 ' u l and initial states x 10 and

at xo'

XlO ·

solution of S-I equation (6) such that the set

(3). there exisl a(x) and

If (3) holds only in some neighbourhood of

(gl'

T

ad;:

commutable,

gl'

1:

Remsrk m

xl=f ll (x 1 )+f 1l (X l )+

I-I m

linear

ll

... , ad (4)

z=T(x)

in

g

nN

g:

feedback

y l=hl(x l ) The

problem

of

DL

feed backs

tranformation under

is

f(xo)=O,

If

and

the

the

(2)

and

a

3

T

z=T(x)

(X=(X1 'Xl) ) coordinate system

new

describe

the

disturbance

the

centralized

for

-

gives

two

theorems

for

the

1 Let xlOe:MI and xloe:Ml' and let VI and

Vl be two neighborhood of x lO and x lO ' respectively. Suppose there exist kl -dimensional

of

distribution t.l

systems,

t.l

such

and

that

kl-dimensional following

the

distribution

conditions

are

satisfied: the

nonlinear

affine

system

(1) for i=1 or 2, t. I is involutive on V I; (2)

(5)

I-I

(3)

where xE:M, f, gl (i=1,2, ... ,m)"V(M), he:C (M,N). Suppose that t.

distribution

at

Xo e:M,

or

for

i=1

or

[fll,t.I]ct. 1

is an

let

i=1

2,

t. I

2,

ia

the

is nonsingular at x

Cl)

(Isidori 1989)

for

(f ,gl) l

invariant

distribution on VI;

y=h(x)

involutive

by

DDD

section

Theorem

and

ml x=f(x)+ Lgt(X)U I

1

then

transformation

with, we present sufficient conditions for ODD.

described by

Lemma

h(xo)=O,

problem of DDD of nonlinear systems. To start

-

solvability decoupling

RESULTS FOR THIS

respecti vely. Consider

and

y=Cz

of DDD and DL. To end this section we recall two

linearization

the

z=Az+Bv

In this paper, we only study local problems

of

maps

gn '

coordinate

locally

nonlinear system (4) becomes a linear system.

which

T

System (5) becomes

there exist decentralized

problems

then fields

into a/azl, a/az l , ... , a/az n

solvable (DL) if in some neighbourhood of xo'

lemmas

2,

vector

coordinate

I

N I -

Remark

Yt=h 1 (x l )

that

-

the

T(xo)=O.

I-I

such

be

Lell\ma

independent C

L glt(xl)u ll

xl=fll(Xl)+fll(xl)+

Let

transformation

I

L gll(X I )U l

coordinate

where

f=f+ga and g=g /3.

is described as follows:

g

..... ,

-

The decentralized system without disturbance

Definition

...,

gl'

rt-1 ....

XO=(Xl0,xlO) then we say that the problem of DDD is locally solvable.

/3 (x) which are the

let

G=Sp

f~1

and

and

quotient

distribution

;

1o [fll,t.l]ct. 1;

~l

be

last

the

(gi,ie:!!ll, if G/t. is nonsingular at xo' then t. is

components and the last (nl-k ) components of 1

an

fll and f ll , respectively, then

(f,g)-invariant

distribution

at

Xo

iff t.

weakly (f,g) -invariant. Let T(x)

be Silveman-Isidori matrix f

is

P l e:t. 1 c ker hunker

-

1989)constructed

by

and

Silveman-Isidori

equation

gl (S-L

ie:!!l,

(Isidori

1

P1e:t.1Cker h1.nker f ll •

consider

equation

f~lO

then the problem of ODD for decentralized systam

for

300

Before verifying Theorem 1. we give a remark.

;!:

Remark

In

Theorem

1.

there

are

the facts

five

nl-kl

Z =(ZT .ZT )T.

Z11 ElRkl •

Zll EIR

new coordinate

system.

so long

where

(1) is locally solvable.

l

11

that

II

[fn.A,]CA,.

[tl"AllCA l

.

Under

with

and

the

A,cker

conditions for the solvability of the problem of DDD.

but except Condition

(4)

the others are

•

similar to those of Lemma 1. Proof:

",

Since

is

there

(f,.g, I-invariant.

exist a smooth vector field a, (x, )EV(M,) and a smooth matrix fJ, (x,) such that

where

[f, +g,a,.A,]cA,

(7)

[(g,i3,) ,.A,]C",

(8)

g, =[gll.g'l.· ... g'm

,J.

coordinate

'th column of (g,/l,)' Because there

A,

is

such

involutive

and

decentralized

transformations

(i=1.2).

under

distribution. transformation

coordinate

that

feedbacks.

System 1 becomes an

a

exists

Z, =z, (x).

the (i=I.2)

the

is

(g,i3, ),

by

therefore.

the

new

system A, has the form of

"

coordinate

,

A, =Sp{8 /8z,.8 /8z l ...... 8 /8Zk Let 1=f +g a

1'1

g.11 =(g111 i3 ).

and

From (7)

(8).

we

get a,l/8z'=0 and agl /8z'=0

"

~l

y, =h, (zll)

for '=1.2 ..... k,

'J'

~l

Yl=h l (zn)

where f, and g'J are the last n-k, components of "

and g'J' respectively. Thus.

f,

Evidently.

and g'J' have

the

system

above

is

disturbance decoupled at (x'o.xlO ). Usually. it is difficult to find

the following forms

locally

•

,

out A

and

Al which satisfy the conditions of Theorem

1.

The following theorem presents a more convenient method to construct A, and Al' of course. the conditions are stronger than previous ones.

where

T T T T z, =(Zl1,Zll) •

zllEIR

k'

,z'lEIR

n-ki

•

By

the

Theorem

£

weakly

(f, .g, I-invariant

in

same reason. A, E ker ha implies that h, can be

ker

For i=1 or 2. let A, be the maximal

hUnfJU

in

distribution some and

thatA,

represented distribution suppose

On the other

Gl/A l

G,/A,are

thatAl

are

and

at

[fl,.Al]CA l

and

the problem of DDD for Simlarly. smooth

and

matrix

coordinate

a

Proof: Since A, Lemma

and

XlO • respectively. If p,EA, PlEA l • then

System

is locally

(1)

gl=(gl1. g ll ..... g lm), ~

(glfJ l ),

(glfJ l ). and f l • gl' Plo represented as the following forms

1.

A,

is weakly is

(f,.g, I-invariant.

(f,.g, I-invariant.

The

by

same

conclusion holds for Al'

[f l +glal·Al]cA l [(gli3 l ),·A l ]CA l

column of

x,o'

distribution

solvable at (X'O ~o) .

transformation Zl =Zl (xl)' such that

where

of

quotient

at

quotient

nonsingular

[fu.A,]C",.

the

nonsingular the

included

neighborhood

By the method used in the proof of Theorem 1. is

the

there

ith

exist

decentralized

feedbacks. and

hl

can

be

transformations. System form:

301

(1)

is

z, =z, (x,)

(i= 1.2).

transformed

into

u,(x,)=

coordinate such the

that

following

the properties of Remark I, then,

Yl=h 1 (zl1l where

z ERkl,

1,

i=l,2, and with

Z

11

Z,'

1.

E!I?nl-kl

8-,

and

with

dim

!:.,=k

[fll,DI1=O,

for

Proof: By the feed backs

" are partitioned conformably

and the decentralized coordinate transformations

is locally disturbance decoupled at (X'O x10 ) . • Remark i: 1:., and I:.~ which meet the

zl=T I (XI)'

T

algorithm

TT

(Isidori

T

1989).

Let

TT

h,=(fll,h l ) , h1=(fll,h l ) , for every calculate co- distributions Ok as follows: 0o=sp(dh

l1

zl=Tl(x1 )

System (4) becomes

requirements of Theorem 2 can be obtained by a recursive

k,

hl1 is the ith component of hi)

,

m' L (G~n

'm ,)

gl =(gl1. g l l ... •• g

{\) for k~O I-I where G~ is the orthogonal complement of G, at

where

x

equations.

Syotem

completely

linearizable.

0k+1=Ok+Lf,(G~n 0k+

tO

k

k

then I:.,=Q~ •• !:.lcan be constructed by the same

(T,.lfn(T

THIS

DL the

DL

Y, =fl1 (x,)

xl=fu(xl)+

yl =fll ( x2 1

that

the

linearity

(T,.lf,"(T;'(Zl»

-I

(13)

L gli(xl)u li 1-,

It implies that D=O . Therefore

-,

(T,.)fn(T l (zl»=CZ 2 ,

This

[(T,.)fll.%z'k 1=0.

k=1,2, ... ,n

(ZI1· .. •• z ,n

(10)

equation 1

leads

where

,

to Z

I

Because

X'O

and

System

(9)

and

1',.0, =[d/dzlkl. we get

• Theorem 3 gives the necessary condition,; for decentralized

X are the equilium 10 and System (10),

suppose

that

f2' (xtO)=O

then,

at

by

2

Lemma

1',

integars N, N' and integers N

[f

iE~'

1-1

l1

(4)

N'

L r i =n,' L r~=n1

with the properties that

i

Theorem

(4).

The

input-state

for

If

and

.D.l=0.

System

then

is

(9)

the

if

the

by

of

System

decentralized coordinate

d Bcen tralized

and

are

[fl l ,D,I=O.

equation

state

linearizable

feed backs

(10)

System and

linearizable.

completely

exist

and r;

system

linearization.

XlO '

there

iE~

of

oufficient

also

are

and

and

linearization

following theorem verifies that these conditions

fll(x.)=O. If system (9) and system (10) can be lineariza b le

,=

)

coordinate transformation z=T{x)

Suppose that

respectively,

are

(10)

prove

only depends on Z2'

by

for every z, EM"

(9)

of z,=T, (XI) and zl=T1(xll.

completely

we

0=(1',.) ' fn(T l (O»=C
is said to be decentralized if it takes the form

of

System

Now

Because T, is a homeomorphism and fll (x JO )=0.

ml

respectively.

and

-1

m' ,- ,

x,=f .. (x,)+L gl1 (x,)u l i

points

linear

Tl(xlO)=O. (13) leuds to that

(4). Define two isolated systems as follows:

A

are

(12)

(zJ) I=Cz J +Dz, where D and C are constant matrices.

that {x,o,xlo)EM, xMl is an equilibrium of System

Definition;!

(9)

(T,. )fn (T

problem for

the nonlinear decentralized System (4). Suppose

{

and

has the form of

section deals with

{

-, (Zl)}

l Otherwise.

way. RESUL TS FOR

and

1

=0.

+1

(11)

Because

.... glm I.

. If there exist an integer k· >0 such that O.

4.

(1=1.21

u,(x,)=a , {x,)+IJ,(X,)V,

Then, it is easy to see thst System (1)

i-1

transformations.

r,=m , rl=m such that the following two sets l l

Proof: Since System completely

(9)

linearizable

System the

(10)

are

decentralized coordinate

dece n tralized

and

feed backs

and by

transformations. Equation (10) and Equation (11) are

valid.

We

now

prove

that

-I

(T,. )f," (T, (Z,»

-1

X

and (TA. )f , (T, (ZI I) are linear functions. 2 we Because

10'

respectively. Theorem;!

i.e., Consider System

fl1 (xIO)=fll(xlOI=O

and

(4), suppose

that

f l , (xtOl=flA{xlol=O.

If

contain linear

D~,

(10)

System

linearizable. such

Moreover, that

transformations

the

completely

are if

there

exist

decentralized

Zl =T I (XI)

and

D,

variable

and

function

a linear function of Z,'

coordinate

Zl =T1 (Xl)

Z2'

z,'

thus.

5

satisfy

302

CONCLUSION

Hence,

fll

is

only

not a

{T,. )f ll can be represented Cz • with a constant matrix C. By l the same reason. we can verify that (TA.)f l , is

the problem of DL is solvable, then System (9) and

the

get

does

IN

this paper, the preliminary solutions for

the problems of ODD and DL have been obtained. There are a lot of remainder problems which are worth

beresearched.

to

problem

of

ODD,

the

For

example,

stability

of

for

the

the

systems

which are disturbance decoupled, the problem of disturbance decoupling for decentralized systems with n subsystems; for the problem of DL, our results are established only for state equation, what

are

Iinearization

the

conditions

of

decentralized

for

completely

sY6tems.

These

problems are of importance for applications. Reference

p.

J.E.R.,

Curl'

decentralized

in

problem

multivariable

linear

Noog:

C.H.

Cuel' chet decoupling

Disturbance

systems,

1nt. J Control, Vol.35 (6) pp.957-962, 1982 Hamano

F.

variable

in

Localization

Furuta:

K.

&

disturbance

decentralized

systems,

Int.

J

linear

Control,

of

multi-

Vol.22

(4)

pp.551-562, 1975 Han 2.Z. & Z.J. Zhang: Disturbance rejection for nonlinear

decentralized

systems,

in

"Selected

papers of Professor Zhang Zhong-jun"· Sidit!(l Press 1990 Hu

S.C.

Y.F.

&

Zheng:

Remurks

on

the

decentralized disturbance decoupling problem, Int •. J Control, Vol.46 (1) pp.97-109, 1987 Hunt

L.R.,

R.

Su

&

G.

1'leyer:

Nulti-input

nonlinear systems, Diff. Geometry Control Conf. Birkhauser, Boston, Cambridge, Mass, 1982

A. :

!sidori

Nonlinear

systems:

control

a

introduction, 2nd ed. Springer-Verlag, 1989 Leite

V.

1'1.

decentralized

P.:

Disturbance

linear

systems

decoupling by

in

nondynamic

feedback of state measurement, Int. J Control, Vol.42 (1). pp.913-937, 1985 Won ham

W.M.:

Linear

multivariable

geometric approach, 2nd

ed.

control:

a

Springer-Verlag,

1979

303