Robust Fault Tolerant Control Framework Using Uncertain Takagi-Sugeno Fuzzy Models - Fuzzy Modeling and Control: Theory and Applications - page 114

Fig. 6.1 Uncertain Takagi-Sugeno system

A i = A Ni + A i (6.6)

B i = B Ni + B i (6.7)

where A Ni and B Ni denote the nominal matrices while B i represent the

respective uncertain part, for the i th subsystem, that are described in a polytopic way
as follows:

A i and

M

A i =

1 η ij A ij

(6.8)

j =

M

B i =

1 η ij B ij

(6.9)

j =

with j = 1 η ij =

η ij ≥

1 and 0. Hence, following ( 6.2 ), the state of the TS system is

inferred as:

N

1 ρ i (ϑ(τ)) A Ni + A i x

(τ ) + B Ni + B i u (τ ) σ. x (τ ) = (6.10)

i =

with defined as in ( 6.4 ).

Given an LMI region

ρ i (ϑ(τ))

D = { z ∈ C : f D ( z )< 0 } with the following characteristic

function (Chilali and Gahinet 1996 ):

T

f

D (

z ) = α + z β +¯ z β = [ α kl + β kl z + β lk ¯ z ] 1 k , l m (6.11)

m is a
matrix, the problem to be solved consists in finding a Takagi-Sugeno state-feedback
controller, as follows:

m × m m ×

where α = [ α kl ] ∈ R is a symmetric matrix and β =[ β kl ]∈R