Modeling, Identification and Control of Irrigation Station with Sprinkling: Takagi-Sugeno Approach - Complex System Modelling and Control Through Intelligent Soft Computations - page 471

3.3 Determination of Consequent System Parameters

The identi cation of consequent parameters is necessary to determine the equiva-

lent TS model such system, we cation methods

such as the method of ordinary least square (Bertrand and Moonen 2012 ) (LMS
used for linear system) Method of recursive least square (RLS) (Duan et al. 2011 ),
weighted least square (WLS) (Li et al. 2009 ), recursive least square weighted
(RWLS) (Soltani and Chaari 2013 ) (this method is used for the noisy nonlinear
systems).
In our case we used in the identi

find in the literature many identi

cation algorithm method of recursive least
square (RLS) (Duan et al. 2011 ; Chakchouk et al. 2014 ).
We know the form of T-S model fi i ¼

a i x

þ d i , then the vector of consequent

parameters written as follow:

T

a i h i ¼ ; d i ð 13 Þ

the increased regression matrix is de ned by:

; ð Þ

X e ¼½ X 1 14

then we de ned the gains matrix with the follow equation:

1

X T N PN ðÞ ¼ ðÞ XN ðÞ ð 15 Þ

P(N) can be written as follows:

P 1 N x T N ðÞ ¼k ðÞþ k N ðÞ l N ðÞ N xN Þ ðÞ ð 16 Þ

If we applied the matrix inverse theorem then:

" #

x T N 1

k

PN ð 1 Þ ðÞ xN ðÞ PN ð 1 Þ

PN ðÞ ¼ PN ð 1 Þ ð 17 Þ

1

l ðÞ þ

x T N ðÞ PN ð 1 Þ xN ðÞ

Then we de ned the gain G(N) with the following equation:

" #

ð Þ

x T N

ðÞ

PN 1

G ðÞ ¼ ð 18 Þ

l ðÞ þ

1

x T N ðÞ PN ð 1 Þ xN ðÞ

Then the regression matrix and the parameters consistent vector is as follow:

h

x T N

h ðÞ ¼ N I GN ðÞ ðÞ ð N 1 Þ

yN

x T N

þ l ðÞ N PN ð 1 Þ xN ðÞ GN ðÞ ðÞ PN ð 1 Þ xN ðÞ ðÞ ð 19 Þ