()
( The fundamentals of the theoretical robotics and the fuzzy methods (review)
Preprint, Inst. Appl. Math., the Russian Academy of Science)

..
(A.A.Petrin)

. ..

, 2005
( 00-01-00403, 00-15-96135, 02-01-00750, 02-07-90-425, 02-61-00-671)

, , . , .

Abstract

General definitions, concepts, properties of fuzzy relations and operators for fuzzy transformations were treated. The measure of fuzziness and fuzzy complement which are expressed in distance-based measures are discussed.

 

 

3

1. ,

 

4

2.

10

3.

15

4.

18

5.

22

6.

23

 


 

 

, . , , . [1], , , .

, , n . .1.


.1

, [2]. . . .1, 1 2 1,2. 3 1,2 , 1,2,3 , . 1,2 , , - 1,2,3 - . , n ().

, , . , , . , . : ; ; . , , , . , ( ) ( ) . , , .

 

1. ,

 

E, , x - E. E [3, 4]

= {(xï mA (x))} ,

~ ~

 

mA (x) x A.

~

mA(x) x ÎE


~


.


. [3] [0,1]. . , ( ) , , . , : , , , , , . , , , . , ; , a , , 1/a , ..

: , , , , , , .

. - , -

A B - E ; ,

~ ~ ~

B,

~

"x Î E :mA(x) £ mB(x) A Ì B .

~ ~ ~ ~

.

E = {x1, x2 , x3 , x4 }, M = [0, 1].

A = {(x1ê0,4), (x2ê0,2), (x3ê0), (x4ê1)}.

~

B = {(x1ê0,3), (x2ê0), (x3ê0), (x4ê0)}.

~

B Ì A , 0,3 < 0,4, 0 < 0,2, 0 = 0, 0 < 1.

~ ~

. - , ,

A B - . , A B

~            ~ ~ ~

, " x Î E : mA(x) = mB(x).

~     ~

A = B .

~ ~

 

x E, mA(x) = mB(x) ,

~ ~

, A B A ¹ B .

~ ~ ~ ~

.    - ,    = [0, 1] -      , A B - . ,

~ ~

A B ,

~ ~

" x Î E : mB(x) = 1 - mA(x) .

~ ~     ¾


: B = `A `A = B . , (`A ) = A .

~ ~ ~ ~                                           ~ ~

.

E = { x 1 x2 x3 x4 x5 x6 }, M = [0, 1].

A = {(x1ê0,13), (x2ê0,61), (x3ê0), (x4ê0), (x5ê1), (x6ê0,03)},

~

B = {(x1ê0,87), (x2ê0,39), (x3ê1), (x4ê1), (x5ê0), (x6ê0,97)}.

~

`A = B .

~ ~

. = [0, 1] - , - ;

~ ~

Ç ,

~ ~

:

~ ~

" x Î E : mAÇB(x) = min (mA(x), mB(x)).

~ ~ ~ ~

.

E = {x1 x2 x3 x4 x5 }, M = [0, 1].

A = {(x1ê0,2), (x2ê0,7), (x3ê1), (x4ê0), (x5ê0,5)}.

~

B = {(x1ê0,5), (x2ê0,3), (x3ê1), (x4ê0,1), (x5ê0,5)}.

~

A Ç B = {(x1ê0,2), (x2ê0,3), (x3 ê1), (x4ê0), (x5ê0,5)}.

~ ~

, ,

" x Î E : x Î A x Î B Þ x Î A Ç B .

m A ~ mB ~ mAÇB~    ~


~ ~ ~ ~


. - = [ 0, 1] - , - ;

   ~ ~

È ,

                                                                                   ~ ~

, .

~ ~

" x Î E : mAÈB(x) = max (mA(x), mB(x)).

~ ~ ~ ~

,

A È B = {(x1ê0,5), (x2ê0,7), (x3ê1), (x4ê0,1), (x5ê0,5)}.

~ ~

. :

A Å B = ( A Ç`B) È (`A Ç B) .

~ ~ ~ ~ ~ ~

, :

A = {(x1ê0,2), (x2ê0,7), (x3ê1), (x4ê0), (x5ê0,5)},

~

B = {(x1ê0,5), (x2ê0,3), (x3ê1), (x4ê0,1), (x5ê0,5)}.

~

`A = {(x1ê0,8), (x2ê0,3), (x3ê0), (x4ê1), (x5ê0,5)}.

  ~

`B = {(x1ê0,5), (x2ê0,7), (x3ê0), (x4ê0,9), (x5ê0,5)}.

~

A Ç`B = {(x1ê0,2), (x2ê0,7), (x3ê0), (x4ê0), (x5ê0,5)}.

~ ~

`A Ç B = {(x1ê0,5), (x2ê0,3), (x3ê0), (x4ê0,1), (x5ê0,5)}.

~ ~

A Å B = {(x1ê0,5), (x2ê0,7), (x3ê0), (x4ê0,1), (x5ê0,5)}.

~ ~

. A - B = A Ç`B .

                                                          ~ ~ ~ ~

. , :

 

A Ç`B = {(x1ê0,2), (x2ê0,7), (x3ê0), (x4ê0), (x5ê0,5)}.

~ ~

, , , . a - . , a - , aI. aÎ[0, 1]; a - A Aa = { xïmA (x) ³ a}.

~

.

x1 x2 x3 x4 x5 x6 x7

A = 0,8 0,1 1 0,3 0,6 0,2 0,5 .

~

x1 x2 x3 x4 x5 x6 x7

A0,3 = 1 0 1 1 1 0 1 ,

 

x1 x2 x3 x4 x5 x6 x7

A0,55 = 1 0 1 0 1 0 0 .

 

 

. A

                     ~

aI :

 

A = max [ a1 × Aa1 , a2 × Aa2 , , an × Aan ],

~ ai

0 < ai £ 1 , i = 1,2, , n.

.

ì 1, mA (x) ³ ai ,

mAaI (x) = í ~

î 0, mA (x) < ai .

~

, A

m(x) = max [ ai × Aai ] =

max [ai × Aai ] ai

ai = max [ ai ] =

ai £ mA(x)

  ~

= mA (x) .

~

.

x1 x2 x3 x4 x5 x1 x2 x3 x4 x5 x1 x2 x3 x4 x5

0,2 0 0,5 1 0,7 = max ( (0,2) × 1 0 1 1 1 , (0,5) × 0 0 1 1 1 ,

 

x1 x2 x3 x4 x5 x1 x2 x3 x4 x5

(0,7) × 0 0 0 1 1 , (1) × 0 0 0 1 0 ) .

 

, ( ). [ a, 1], 0 < a £ 1,

   ì 1, mA(x) Î [ a, 1] ,

mAa (x) =   í ~

î 0, mA(x) Ï [a, 1] .

    ~

, a - 0 < a £ 1 .

, . A1 Ì Ì A2 Ì Ì Ì Ì An a1 A1 , a2 A2 , , an An , , a1 > a2 > > an , A .

 

 

X, , X, , , , , , , [5,7] X .

. :

1)          ,

2)          ,

3)          ,

4)          ,

5)          ,

6)          ,

7)          , È` = Ç` = .

. :

n = (k1, k2 , , kn)

n¢ = (k¢1, k¢2 , , k¢n) ,

ki k¢i , i = 1, 2, , n K. ³. , n¢ n, n¢ ³ n ( ), , k¢1 ³ k1 , k¢2 ³ k2 , , k¢n ³ kn . > Ð , , n¢ n. , n¢> n, k¢1 ³ k1 , k¢2 ³ k2, , k¢1 ³ kn k¢i ki , .

, , , , . , , .

 

2.

 

, . S , A = {a0, , an-1} [10]. a0 a1, , a1 - , S, 0 - , S , .. ms(a1) = 1, ms(a0) = 0. S. , . Xm n xi = {xi1, ,xim}, d(xi, xj) . : ai aj , dij = 0, ai, aj ai, ak, dij > dik ; ai, aj ai, ak, dij = dik.

[11].

1. S , .

. , a0 a1, .. S , , ai 1 , S.

2. ai aj S ai aj 1, S.

, ai aj S , .. c êd1i - d1j ê = êms (ai) ê, - .

ai a0, a1, :

 

c êd10 d1j ê = ms (aj), cd1j = 1 - ms (aj), .. d11 = 0.

 

,

 

d10 - d1j d1j

ms (aj) = ¾¾¾¾ = 1 - ¾¾ .

d1j d10

 

, , S, Xm .

, . d X, Y E, [10]: " X, Y, Z Î E:

1)                        d(X,Y).

2)                        d (X, Y) ³ 0 - ,

3)                        d (X, Y) = d (Y, X) - ,

4)                        d (X, Z) £ d ( X, Y) * d (Y, Z) - .

* - , d(X, X) = 0.

 

, n . , , 1 - 4, .

, , f(A) . f f(A) = f(`A).

, `. , , n. n . , (i) i- , . (1 (i)) i , `. , `.

D n , D(A,`A) `. [5]:

 

n

Dp(A,`A) = [ å çAC(xi) A(xi)çp]1/p

I=1

 

p = 1,2,3, .

 

,

 

p = 1: ,

n

D1[A,`A] = å êA(xi) - `A(xi) ç.

i=1

 

, `() = 1 (),

n

D1 [A,`A] = å ç2A(xi) 1 ê.

i=1

 

p = 2: ,

 

D2 [A,`A] = ( å (A(xi) - `A(xi))2)1/2

 

`() = 1 (),

 

D2 [A,`A] = ( å (2A(xi) 1)2)1/2.

 

p = ¥: Sup

 

Dx [A,`A] = Sup çA(xi) - `A(xi) ê

i=1,2,,n

 

Dx [A,`A] = å ê2A(xi) - 1ê

i=1,2,,n

, , ` . .

 

1 [5]. ` ,

 

Dp = Dp[C,`C] = n1/p

 

,

D1 = n ,

D2 = n1/2,

D¥ = 1.

 

. , (), `() 1 0 , ,

ç() - `() ç = 1

n

Dp(C,`C) = ( å 1)1/p = n1/p.


i=1

 

2 [5]. ,

Dp(A,`A) £ Dp(C,`C).

 

. ç(i) - `C(xi)ç = 1 i

ê2(i) 1 ç £ 1 i

 

, `. , fp(A) ,

 

Dp Dp(A,`A) Dp(A,`A)

fp(A) = ¾¾¾¾¾¾ = 1 - ¾¾¾¾ .

Dp Dp

 

,

 

n

f1(A) = 1 1/n( å ê2A(xi) 1 ê)

i=1

 

n

f2(A) = 1 1/n1/2 ( å ç2A(xi) 1 ç2)1/2

i=1

 

f¥(A) = 1 - Sup ç 2A(xi) 1 ç.

i= 1,2,3,

 

, De Luca Termini [9]:

1) , (1) = 1 0 ç2(i) 1 ç = 1 i = 1, , n Dp(A,`A) = Dp

 

 

Dp

fp(A) = 1 - ¾ = 0.

Dp

 

2)    fp(A) = 1 (Dp(A,`A)) / (Dp) max, Dp(A,`A) min. Dp (A,`A) min, ç2A(xi) 1 ç= 0 i, ç2(i) 1 ç= 0, (i)=1/2. , fp(A) max, (i)=1/2 Î.

3)      *() ³ () () ³1/2,

*() £ () () £1/2.

ç2*(i) 1 ç³ ç2A(xi) 1 ç i

 

, , Dp(A*,`A*) ³ Dp(A,`A) fp(A*) £ fp(A).

, (a, b) , , . , = [a, b] Î R, ` ,

b

Dp(A,`A) = [ ò çA(x) - `A(x) çp dx]1/p (p=1,2,, ¥).

`() = 1 (),

b

Dp(A,`A) = [ ò ê2A(x) 1 êp dx]1/p .

p = 1,

b

D1(A,`A) = ò ç2A(x) 1 ç dx,

a

p = 2,

b

D2(A,`A) = [ ò (2A(x) 1)2dx]1/2.

a

E ,

 

ç2() 1 ç = 1

b

Dp(C,`C) = [ ò dx]1/p = (b a)1/p.

a

,

D1 = b a

D2 = (b a)1/2

Dp = (b a)1/p.

, ,

Dp ³ Dp(A,`A).

, :

1 b

fp(A) = 1 - ¾¾¾ [ ò ê2A(x) 1 êp dx]1/p.

(b a)1/p a

 

3.

 

. [3] , F(X) F = min G = max :

1) F (mA, mB) = F (mB, mA) ; G (mA, mB) = G (mB, mA) ( ) ;

2) F (mA, F (mB, mC)) = F (F (mA , mB), mC) ;

G (mA, G (mB, mC)) = G (G (mA, mB), mC) ; () ;

3) F (mA, mB) £ F (mC, mD) ; G (mA, mB) £ G (mC, mD), mA £ mC, mB £ mD

() ;

4) F (mA, mA) < F (mB, mB) ; G(mA, mA) < G (mB, mB) , mA < mB ;

5) F (1, 1) = 1 ; G (0, 0) = 0 ;

6) F (mA, mB) £ min (mA, mB) ; G (mA, mB) ³ max (mA, mB) ;

7)     F G - ;

8)     F (mA, G (mB, mC)) = G (F (mA, mB), F (mA, mC)) ;

G (mA, F (mB, mC)) = F (G (mA, mB), G (mA, mC)) ().

 

 

. `m (x) = 1 - m(x) "x Î X ().

. U , . U

mA : U [0,1],

m(u) u .

c:[0,1] [0,1],

mA(u) c(mA) ():

(1) (0) = 1 (1) =0,

.. ; (2) 1 < 2, c(1) ³ c(2), .. .

:

(3) c ;

(4) , .. c(()) = Î [0,1].

, , ,

ì 1 £

c() = í

î 0 >,

Î [0,1].

 

C P (U) U,

C: P(U) P(U),

, = (U), C(A) = B, (m(u)) = mB(u) u ÎU. . , [7] .

 

 

1. c . , c() = , .

1 [9]. .

. . c() = 0, Î[0,1]. 1 , , c() = d, d , .

, 1 2 c() = d, 1<2. (1) 1 = d (2) 2 = d; ,

c(1) 1 = (2) 2. (1)

, (1) ³ c(2) 1<2,

c(1) 1 >(2) 2.

(1). , .

2 [9]. , c , ( 1).

£c(), £

³(), ³.

. <, = >, . ()³(), () = () () £(), , . () = () ³ , () = () £ , , , () >, () = () <, . , £ ()³ ³ () £. .

3 [9]. c , c .

. , c() = 0. c() = , Î[-1,1] , (1), (0) 0 =1 (1) -1= -1. , , Î[-1,1] , c() = . 1.

 

2. Î[0,1], d Î[0,1] ,

c(d) - dx = x c(x) (2)

. 1, d ( ). , . 3, d Î [0,1], c .

4 [9]. c ,

d = .

. = , () = 0 1. 2 d = ec.

5 [9]. Î[0,1], dx = c(x), (()) = .

. d = c(x). (()) () = (); , (()) = . (()) = . (d) - dx = c(c(x)) c(x); , d = c(x).

5 , , . , , .

, , 15, .

 

 

4.

 

P(U) U.

f:P(U) [0,1],

U f(A). f(A) , f . , () P(U), , .

3. , ÎP(U)

 

çmA(u) c(mA(u))ç³çmB(u) c(mB(u))ç

u Î U, c . , , ; Ð .

, f(A) > f(B), Ð. () f:

(f1) f(A) = 0, ;

(f2) Ð, f(A)£f(B);

(f3) , f(A) ,

m(u) = ec u ÎU.

(f1), , .

(f2) . .

çm (u) c(mA(u)) ç³ çmB(u) c(mB(u))

m(u) £ mB(u) m(u) c(mA(u)) ³c(mB(u)) - mB(u), mB(u) £c(mB(u)),

()
mA(u) ³ mB(u) mB(u) c(mA(u)) £ c(mB(u)) - mB(u), mB(u) ³ c(mB(u)).

 

(?) :

. Ð ,

m(u) £ mB(u) mA(u) ³ mB(d u), mB(u) £ ec

(b)

mA(u) ³ mB(u) mA(u) £ mB(du), mB(u) ³ ec

 

u ÎU ( 2 3).

1.     . Ð ,

mA(u) £mB (u) mA(u) £ c(mB(u)), mB(u) ³ ec

u ÎU ( 5).

2.     () = 1 . Ð ,

mB(u) £ ½

(c)

mA(u) ³ mB(u) mA(u) £ 1 - mB(u), mB(u) ³ ½,

u ÎU.

, [7-9], , [3]. : ,

 

mA(u) £ mB(u), mB(u) £ ½

()

mA(u) ³ mB(u), mB(u) ³ ½.

 

() = 1 , :

 

mA(u) £ mB(u), mB(u) £ c(mB(u))

()

mA(u) ³ mB(u), mB(u) ³ c(mB(u)) .

 

()

 

mA(u) £ mB(u), mB(u) £ ec

()

mA(u) ³ mB(u), mB(u) ³ ec.

() = 1 , (), ().

(), (), () (), (), (), , , . (), Ð . , (f2) :

(f2)¢ () , f(A) £ f(B).

(f3) f(A) - f(A) , mA(u) = ½ u ÎU, () = 1 . , , .

-, , U U ={u1, u2, ,un}. ÎP(U)

d,: U [0,1]

,

d,(ui) = çmA(ui) c(mA(ui))ç; (3)

fdc: P(U) [0,1]n

,

fdc(A) = (dc,A(u1), dc,A(u2), , dc,A(un));

( f , , U). ,

d: [0,1] [0,1]

, d() = ç ()ç

Ñ = d()ç Î[0,1];

, Ñ Í [0,1]. d .

6 [9]. fdc(P(U)) = Ñcn , U .

. A Î P(U). mA(ui) Î[0,1] ui ÎU, dc,A(ui) ÎÑc ui ÎU d,A , , fdc(A) ÎÑcn. , fdc(P(U)) Í Ñcn.

(dc(x1), dc(x2), ..., dc(xn)) Î Ñcn. Î P(U) , fdc(A) = (dc(x1), dc(x2), ,dc(xn)), .. mA(ui) = xi ui ÎU. , Ñn Í fdc(P(U)).

() d,(ui) () U.

4. , U ,

gc:Ñcn[0,1]

, :

(g1) gc(a1,a2,, an) ³ (a1,a2,,an) Î Ñcn;

(g2) gc ;

(g3) gc(a1,a2,,an) i = 1 i;

(g4) 0ÎÑ , gc(a1,a2,,an) ai = 0 i.

gc , .

gc dc,A(ui), ui ÎU, d,A , gc (). Dc,g(A, C(A)) .

Dc,g(A,C(A)) = gc(fdc(A)).

, U . : (3) :

d,(u) = çmA(u) c(mA))ç, uÎU;

fdc

dc:P(U) [0,1]U

dc(A) = dc,A, d,A .

7. dc(P(U)) = ÑcU U .

. 6.

7 , a ÎÑcU d,Îdc(P(U)).

, U .

5. , , U ,

Gc:ÑcU[0,1]

, :

(g1)¢ gc(a)³0 aÎÑU;

(g2)¢ gc ,

gc(a) ³gc(a¢) a(u) ³a¢(u) uÎU;

(g3)¢ gc(a) , a(u) = 1 u ÎU;

(g4)¢ 0ÎÑ, g(a) , a(u) = 0 uÎU.

gc , . , 5 , 4, .., . , , , 5 . gc d,A, gc , (). () , . U = {u1,u2,,un},

Dc,p[A,C(A)] = [ å dpc,A(ui) ]1/p ,

= 1,2, , , (.. = 1, = 2, = ¥). U = [a, b],

Dc,p[A, C(A)] = [ òab dpc,A(u)du ]1/p.

Dc,g [A,C(A)], A ÎP(U),

0 £ Dc,g [A,C(A)] £ Dc,g [N,C(N)],

N P(U).

^

Dc,g Dc,g

 

^ Dc,g [A,C(A)]

Dc,g [A,C(A)] = ¾¾¾¾¾¾ .

Dc,g [N,C(N)]

 

  ^

0 £ Dc,g [A,C(A)] £ 1.

 

 

5.

 

. , , , , : ( ) , , ( ) . , , , .

, ( ) ( ) . , , .

, , .

, .

Ù

fc,g(A) = 1 - Dc[A,C(A)], Dc[A,C(A)] ().

Ù

Dc[A,C(A)] : (1) (2) g, êmA(u) c(mA(u))ê U.

 

 

6.

 

1.     .. . // .: 42, - . . . .. , 2003, 22.

2.     Loo Ren C., Kay Michael G. Multisensor Integration and Fusion in Intelligent Systems. //IEEE Trans. On Systems, Man, und Cybernetics, 1989, v.19, 5, p.901-931.

3.     .. : . . // .: , 1976, 167.

4.     . // : . . .: , 1982, 417.

5.     Yager R.R. On the Measure of Fuzziness and Negation. Part 1: Membership in the Unit Interval. // International Journal of General Systems,1979, v.5, p.221-229.

6.     Yager R.R. On the measure of fuzziness and negation. Part II: lattices. // Information and Control,1980, v.44, 3, p.236-260.

7.     Yager R.R. Robot planning with fuzzy sets. // Robotica, 1983, v.1, 1, p.41-50.

8.     DeLuca A. and Termini S. On convergence of entropy measure of a fuzzy set. // Kybernetes, 1977, v.6, p.219-222.

9.     Higashi M., and Klir G.J. On Measure of Fuzziness and Fuzzy Complement. // International Journal of General Systems, 1982, v.8, p.169-180.

10. . . .. . // .,1986, 312.

11. .., .. . // .- .: , 1989, 304.