Polynomial Fuzzy Systems: Stability and Control - Fuzzy Modeling and Control: Theory and Applications - page 93

T n ( ) − ψ 2

ψ 1 − ψ 2

x

μ( x ) := (5.9)

x n ,wehave f

Hence, as f ( x ) = f n ( x ) + T n ( x ) ( x ) = f n ( x ) + (μ( x )ψ 1 + ( 1 −

x n so the polynomial p 1 in ( 5.5 ) is given by p 1 ( ) + ψ 1 x n , and μ( x ))ψ 2 ) x ) = f n ( x

) + ψ 2 x n . p 2 ( x ) = f n ( x

Note As in TS modeling, the representation ( 5.5 )is exact , i.e., there is no approx-
imation involved and there is no uncertainty in the membership functions, defined
in ( 5.9 ). As a conclusion, using the Taylor-based modeling any smooth nonlinear
system can be exactly expressed as a fuzzy polynomial one in a compact domain

.

Example 5.1 Let us model sin ( x ) , x ∈[− 1 , 1 ] , considering its Taylor series around

x = 0:

x 3
3

x 5
5

x 7
7

sin ( x ) = x − + − + ... (5.10)

! ! !

Let us define f 1 = 0, and T 1 ( x ) = ( sin ( x ) − 0 )/ x from the previously discussed

notation. The maximum and minimum of sin ( x )/ x in [ − 1 , 1] are 1 and 0 . 8415, so

we may express:

sin x = μ 1 ( 0 + 1 · x ) + μ 2 ( 0 + 0 . 8415 x ) (5.11)

which is coincident with the standard sector-nonlinearity TS model. Now, using the
cubic term in the Taylor series, the fuzzy model needs considering f 3 = 0 + x ,

x 3 ,so T 3 ( T 3 (

) = ( ( ) − )/ ) − . − .

x sin x x x has maximum 0 1585 and minimum 0 1667

[− , ]

in the interval 1 1 . Hence,

1585 x 3 1667 x 3

sin x = μ 1 ( 0 + x − 0 . ) + μ 2 ( 0 + x − 0 . ) (5.12)

x 3 x 3 x 5 ,

If we proceed to 5th order with f 5 = 0 + x − / 6, T 5 ( x ) = ( sin ( x ) − x + / 6 )/

then in [− 1 , 1 ] , T 5 ( x ) has maximum 0.0083336, minimum 0.0081376, so we get:

x 3 0083336 x 5 0081376 x 5

sin x = x − / 6 + (μ 1 · 0 . + μ 2 · 0 . ) (5.13)

where, for instance, μ 1 would have the usual interpolation expression:

μ 1 ( x ) = ( T 5 ( x ) − 0 . 0081376 )/( 0 . 0083336 − 0 . 0081376 ) (5.14)

If the bounding polynomials are plotted, it can be seen that they are closer as their
degree increases, see Sala and Ariño ( 2009 ) for details. Both 3rd- and 5th-order
bounds are very accurate (maximum error 0.0082 and 0.0002, respectively) in the
chosen interval.