Approximation property of partition of unity and its ... - Springer Link

Journal of Control Theory and Applications 3 (2004) 267- 275

Approximation property of partition of unity and its applications Yinhe W A N G 1 , Zhiyuan LI2 , Siying Z H A N G 3 ( 1. Science College, Shantou University, Shantou Guangdong 515063, China; 2. Department of Mathematics,Imaer Mongolia NonnaI University, Huhhot Inner Mongolia 010022, China; 3. Deparmaent of Automatic Control, Northeastern University, Shenyang Liaoning 110006, China ) A b s t r a c t : The linear combination of certain partition of unity, subordinate to certain open covering of a compact set, is proved to be capable of approximating to a continuous function at arbitrarily precision. By using proper open covering and partition of unity, the robust nonlinear controllers and adaptive laws are designed for a class of nonlinear systems with uncertainties. The states and parameters of the closed-loop systems can be stabilized in the meaning of UUB ( unifomaly ultimately bounded) via the robust nonlinear controllers and adaptive laws. Finally, an example shows the validity of method in this paper.

Keywords: Partition of unity; Approximatingtheorem; Uncertain system; UUB

1

Introduction Control design for nonlinear systems has been a very

closed-loop systems and parameters to be UUB (uniformly ultimately bounded).

active field of research during the past two decades. The

The partition of unity is interrelated with the open covering of compact set considered. It is utilized to

introduction of differential geometric techniques led to a

approximate the uncertainty of system controlled. In the

great success in the development of controllers for nonlinear systems [ 1 - 14]. For example, the invariant distribution

past years, the uncertainties of systems controlled have also

was utilized to describe the internal structure of nonlinear systems. With the aid of this concept, it is shown that a

are represented by finite linear combinations of basis functions such as rational and spline functions and so on.

nonlinear system locally exhibits decompositions into the reachable/unreachable parts and or observable/unobservable

general by a finite linear combination of basis functions, it is

parts [ 1,15 - 21 ]. The local coordinates transformations were utilized to explain how a nonlinear system can be transformed into a linear and controllable one or another one lineafilized partly with zero dynamics, which led to the well-known exact linearization method [ 1 , 4 , 5 , 8 , 1 0 , 2 1 23]. By using differential forms and their properties, the construction of a canonical representation for nonlinear systems, similar to the Morse canonical form in the linear case, has been obtained [ 24 ]. The other canonical forms for nonlinear systems have been also discussed by using differential geometry method [ 2 5 , 2 6 ] . These examples show that the concepts and techniques of differential geometry have proved to be an effective means of analysis and design of nonlinear systems. In this paper, we adopt the partition of unity concept to design the adaptive robust controllers for a class of uncertain nonlinear systems. The adaptive controllers can ensure that all the states of the

been investigated in [ 2 7 - 32]. The unknown uncertainties

Since the unknown uncertainties cannot be represented in important to design the robust controllers with the errors of approximating the unknown uncerto_inties. Our goal in this paper is to provide a design UUB control method based on partition of unity for M I M O uncertain nonlinear systems. The paper is organized as follows : In Section 2, the basic concepts are introduced. In Section 3, a universal approximation theorem is proved, which is advantageous to understand the approximation property of partition of unity. In Section 4, we present the design UUB control method by using the universal approximation theorem for a class of M I M O uncertain systems. Finally, an example simulation shows the validity of this paper in Section 5.

2

Preliminaries Definition 11333

Assume t Ui }, i = 1 , 2 , " ' , N , be

an open coveting of the compact region U o f R n. A C ~

Received 13 February 2004; Revised 3 August 2004. This work was supported by the Natura/ Science Foundation of Guangdong Province (032035) and the Nature Science foundation of Inner Mongolia (200208020201).

Y. WANGet al./ Journal of Control Theoryand Applications 3 (21304) 267 - 275

268

partition of unity subordinate to the open covering { Ui } , i = 1 , 2 , " ' , N , is a collection o f C ~~functions { c~i} defined N

on the open set U = ~

Ui with the following properties:

i=i

1) ai >I 0 on U; 2) For each i, there is a Ui such that the support of ai s u p p ( a i ) C Ui; = 1 for e v e r y x E U.

i=1

{x E U I ai(x) r 0} (the closure

Where s u p p ( a i ) s

o f the set {x ~ U t a i ( x ) r 0 } ) . T h e o r e m 1 [ 33 ~ Associated to each open covering

real continuous functions on a compact set U. If 1 ) Z is an algebra, i . e . , the set Z is closed under an addition, multiplication, and scalar multiplication; 2) Z separates points on U , i. e. , for every x , y E U , x # y , there exists f ~ Z such that f ( x ) # f ( y ) ; 3) Z vanishes at no point of U , i. e . , for each x E U there exists f ~ Z such that

{ Uz} o f U C R'~, there is a C = partition of unity {az}

(C(~),d,,). Proof o f Theorem 1 : First, we prove that ( F , d ~ ) is an algebra. L e t f l ,f2 E F , so that we can write them as Nt N2

ft(x)

subordinate to l Ui}. N

the open set U = ~ , Ui with the properties (1) - (3) o f i=1

Definition 1, then {a i } is called a C ~ partition o f unity to

the

open

coveting

{Ui}.

Obviously,

~,22iflj(x), j=l

(1) where { ~, t and tg} are C = partition of unity subordinate to the open coveting { Ui } and { gj} of the compact region C R ", respectively. ~li,a2j,i 2 , ' " , N2, are real numbers.

1,2,'",Nl;j

=

= 1,

W e therefore obtain that

Theorem 1 is also tree for a C 1 partition of unity. Definition 2

= ~,2,,a~(x),f2(x)= i=1

Especially, if a collection of C 1 functions { ai } defined on

subordinate

Let Z be a set o f

f ( x ) # 0, then the uniform closure of Z consists of all real continuous functions on U ; i . e . , ( Z , d = ) is dense in

N

3) ~ a i

Theorem[34]

Stone-Weierstrass

Let F be the set of real continuous

functions defined on a compact region [) of R" with the

f,(x)

+fz(x)

N2

N,

= ~2A,i~ -i-= l

Oti(X) + E 2 A 2 j g ( x "/~

j=l

) 2

F if and only if there exist an open

(2)

coveting { Ui, i = 1 , 2 , ' " , N } of the compact region U , a

From Definition 1, it is easily verified that the collection of

{ai} subordinate to the open

C ~ functions {0.5ai,O.5flj,i = 1 , 2 , " ' , N I ; j = 1,2, " " , N21 is the partition of unity subordinate to the open covering {U i, Vi,i = 1 , 2 , ' " , N 1 ; j = 1,2,'",N21. So

condition: f ~

C ~ partition of unity

coveting t Ui} and N real numbers 21, a2, " " , 2N such that N

f(x)

= ~2,ai(x).

fl+fz~

i=!

The set F in Definition 2 is a metric space with the sup-metric d ~ ( f l , f 2 )

=

suplf~(x)

-f2(x)l.

F.

Similarly, for arbitrary real number e, Nj

The

efl(x) = ~-~(c2,i)ai(x).

xEU

following theorem shows that the metric space F is dense in C ( U ) , where C ( U ) denotes the set of all real continuous functions defined on the compact region U with the above metric d |

Obviously, cf l E F. Finally, we obtain that N1

fl(x)fz(x)=

For convenience sake, the set F is called as an expansion

Approximation in ESPU- U

property

of e x p a n s i o n s

i=1

suplg(x)

there

exists f

-f(x)l

j=l

= ~, ~,2,i22jai(x)flj(x).

E

F

(ESPU-U)

such that

< e.

xEU

W e use the following Stone-Weierstrass theorem to prove Theorem 2.

(4)

i=l j=l

Invoking N1

N2

i=I

j=l

N1

= E

T h e o r e m 2 For any given real continuous fimction g defined on the compact region U C ~n and arbitrary s > 0,

N2

[~2,iaz(x)][~azjpy(x)~ N~ N2

set of partition o f unity based on U. F is abbreviated to ESPU-U.

3

(3)

i=1

N2

i(x)JEP, p i ( x ) J = 1, i=1

j=l

it is easily verified that the collection of C • functions {ai

g,i

= 1,2,---,N1;j

= 1,2,...,N2}

is the partition o f

unity subordinate to the open coveting { Ui ['1 Vj, i = 1, 2,"',N1;j = 1,2,'",N2} of>. Therefore, ( F , d | ) is an algebra.

Hence, f ~ f 2

E

F.

Next, we prove that F separates points on U. Suppose

Y. WANGet ol./Journal of Control Theoryand Applications 3 (2004) 267 - 275 that arbitrarily given xo, Yo E U with x0 # Y0. It is seen that there exist r - ball Sr(XO) = {p E U I [t p - xo

< r} C R ~ a n d s - b a l l & ( y o )

= {pC UI

ltp-

II

y0ii

~ S,(yo) = 0 - B e c a u s e U

< st C ~ s u c h t h a t S r ( x 0 )

269

aN+1(XN+2) = 0, hence, f ( x o ) # f ( Y o ) . Therefore, F separates points on U. Finally, we prove that F vanishes at no point o f U. W e prove this by constructing a required f ~

F ; i. e . , we

is compact region, we can choose a hmited number o f open

specify f E F such that f ( x 0 ) # 0 for arbitrarily given x0

balls ISi, i =

E U. Let the hmited number o f balls { Si, i = 1 , 2 , " ' ,

1,2,"',N}

so as to construct an open

coveting o f U. Obviously, { Si , Sr ( Xo) , S, ( yo) , i = 1,

N I be an open covering o f the compact region U. Obvi-

2 , . . . , N t is also an open covering o f U. For convenience

ously, there exists a ball St E

sake, we denote Sr(xo) and S,(yo) by SN+I and SN+2,

that x0 E St. Similar to the above (5) - (7) ,we can ob-

respectively. N o w , we prove that F separates points on

tain a C ~ partition of unity {ai, i = 1 , 2 , ' " ,

by constructing a C = partition o f unity {ak, k = 1 , 2 , ' " ,

nate to { Si, i = 1 , 2 , - " , N } . Without loss o f generality,

{ Si, i = 1 , 2 , ' " ,

N} such

N} subordi-

N , N + 1 , N + 2t subordinate to l S k , k = 1 , 2 , " ' , N , N

suppose that s u p p ( a z ) C S1. C h o o s e f = at. It is easily

+ 1 , N + 21, where the support o f a k s u p p ( a k ) C Sk,

verified t h a t f ( x 0 )

1 ~< k ~< N + 2 .

That is, we s p e c i f y f E

F such that

f ( x o ) # f ( Y o ) . Without loss o f generality, suppose that the radius and center o f ball Sk are denoted as rk and xk, re-

=

r , rN+ 2 =

XN+I

=

X0,XN+2

This completes the proof o f Theorem 2. Remark

1

1 ) Theorem 2 shows that the expansions

o f E S P U - U are "universal approximators". As apphcations, in order to specify the e x p a n s i o n f ( x )

=

= ~X~ai(x)

of

i=l

s;

E S P U - U , w e can fix the partition o f unity {ai, i = 1 , 2 , Yo-

9" , N} at the very beginning o f design procedure, so that

Consider the following function:

It(t) =

# O.

N

spectively, i. e . , Sk = Sr~ ( x k ) , and rN+l

= at(Xo) > 0, i . e . , f ( x o )

the only free design parameters are 2 i ; in this case, f ( x ) is

0,

t ~< 0,

e-{,

t > O.

(5)

hnear in the parameters. W e can use least-square method or adaptive techniques.2) The partition o f unity {ai, i = 1,

W e can prove by direct computation that all o f h ( t )

2 , - - - , N} can be chosen by using the method similar to

derivatives exist and are zero at t = 0, and it is analytic for

(5) - (7) o f the proof o f Theorem 2. Another method is

other values o f t. Hence, h ( t ) is C = .

introduced as follows. Consider the following function:

Let ~(t)

h(rk -

g k ( x ) = h(rk -

II x 11 ) II x II ) + h ( II x II - 0. S r k ) '

k = 1,2,...,N,N+

1,N+2.

(6)

=

e t , t # 0, tO, t = 0.

(8)

W e can prove by direct computation that all o f q~(t)

Since h ( t ) is C = and the denominator o f gk ( x ) is never

derivatives exist and are zero at t = 0. Hence, ~9( t ) is

zero, g k ( x ) is also a C = function. It is easily seen that

C ~ . Let ai and b i denote real numbers such that ai < bi,

gk(x)

II x

= Ofor

II I> r k a n d g k ( x )

> 0 f o r II x II
0 for xi E ( ai, bi), gi ( x l ) = gk(~ ~+~

-

xk)

0 f o r xi q~ ( a i , b i ) .

,

P, g (x -

g(x)

j=l

k = 1,2,'",N,N

Let x = (Xl

= ~ [ g i ( x i ) , hence, g ( x ) i=1

+ 1 , N + 2.

It is easily verified that {ak, k = 1 , 2 , ' " ,

x2

"'"

xn) y and

> Ofor x E ~ [ ( a i , i=1

(7)

N , N + 1, N +

21 is a C | partition o f unity subordinate to { Sk, k = 1 , 2 ,

1,N+21. L e t f = aN+l. Invoking SN+I (7 SN+2 = &(Xo) f'] &(Yo) = O,f(xo) O~N+I(XN+I) > 0 a n d f ( y o ) =

bi),g(x)

= 0 for x ~ I I ( a i , b i ) .

This shows that if

i=1

the open covering { Uk ; k = 1 , 2 , " ' ,

N} o f the compact

9" , N , N +

=

( a/k, bi ), k = 1 , 2 , "", N ,

set U has the form: Uk = i=1

then we can prove that the partition o f unity subordinate to

Y. WANGet al./ Journal of Control Theoryand Applications 3 (2004) 267 - 275

270

this open covering is the following collection of C = func-

inequality

(13)

shows

that

rl(x)