Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 663

The matrix A ð:Þ contains two nonlinear continuous terms:

n 1 ð x ð t ÞÞ ¼ a bx 2 ð 105 Þ

n 2 ð x ð t ÞÞ ¼ cx 2 ð 106 Þ

where n j , ð j ¼ 1 ; 2 Þ are two premise variables depend on state x 2 ð t Þ . Each premise

variable is bounded in a compact state space

n 1 ð x ð t ÞÞ 2 ½ n min1 ; n max1 ð 107 Þ

n 2 ð ð ÞÞ 2 ½ n min2 ; n max2 ð Þ

x t 108

Using the polytopic transformation (Chadli and Borne 2012 ; Tanaka et al. 1998 ),
the nonlinear continuous terms can be written as:

F 1 ð F 1 ð

n 1 ð ð ÞÞ ¼ ð ÞÞ n max1 þ ð ÞÞ n min1 ð Þ

x t x t x t 109

F 2 ð F 2 ð n 2 ð x ð t ÞÞ ¼ x ð t ÞÞ n max2 þ x ð t ÞÞ n min2 ð 110 Þ

where the functions F 1 , F 1 , F 2 , and F 2

are respectively given by:

ÞÞ n min1
n max1 n min1
F 1 ð x ð t ÞÞ ¼ n max1 n 1 ð

ÞÞ ¼ n 1 ð

x ð t

F 1 ð x ð t

ÞÞ
n max1 n min1
F 2 ð x ð t ÞÞ ¼ n 2 ð

x ð t

ÞÞ n min2
n max2 n min2
F 2 ð x ð t ÞÞ ¼ n max2 n 2 ð

x ð t

ÞÞ
n max2 n min2

ð

x t

The decomposition which will be carried out with the combination of nonlinear
terms bounds leads to four local models 2 2

. Then, the fuzzy bilinear
model is obtained by an interpolation of these local models with four nonlinear
activation functions. Let consider the studied case where the decision variable is
unmeasurable, the weighting functions depend on the estimated state

ð r ¼ ¼ 4 Þ

^

x 2 ð

Þ

. Hence,
the fuzzy bilinear representation of the predator-prey system studied subject to
unknown input d

t

ð t Þ and unmeasurable decision variable x 2 ð t Þ can be written as:

<

ðÞ ¼ P

4

xt _ h i ð^ x 2 ð t ÞÞ ð A i x ð t Þþ B i u ð t Þþ N i x ð t Þ u ð t Þþ F i d ð t Þþdð t Þ Þ

ð 111 Þ

:

i¼1

y ð t Þ ¼ Cx ð t Þ