By Ahmad Taher Azar

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Thus, we shall also focus on taking advantage of the FWLF approach to derive less conservative conditions for the solvability of the stabilization problem (for lack of space, the PLF approach will be not discussed in this chapter). B. Main issues Most stabilization conditions based on the FWLF are formulated in terms of parameterized linear matrix inequalities (PLMIs), which causes the following primary practical difficulty: the PLMI-based condition involves an infinite number of LMI-based conditions and thus the task of establishing a controller is intractable numerically.

Then, the robust stability with H∞ disturbance attenuation problem of fuzzy system is converted as the robust stability and robust stability with H∞ disturbance attenuation problems of equivalent uncertain system. Since λi ( x( k )) satisfies (13), we have λr ( x( k )) = 1 − [ λ1 ( x( k )) + λ2 ( x( k )) + " + λr −1 ( x( k ))]. It follows from this relationship that r ∑ λi ( x( k ))( Ai + ΔAi ) = Ar + H 1r F( k )E1r + λ1 ( x( k ))[ A1r + H 1r F1r ( k )E1r ] i =1 + " + λ r − 1 ( x( k ))[ Ar − 1 ,r + H r − 1 ,r Fr − 1 ,r ( k )Er − 1 ,r ] ~ ~ = Ar + H 1F ( k ) A where Air = Ai − Ar , i = 1 ," , r − 1 H 1 = [ I I " I H 11 H 12 " H 1r ], F( k ) = diag [λ1 ( k )I " λr − 1 ( k )I λ1 (t )F1 ( k ) " λr (t )Fr ( k )], ~ T T A = [ A1Tr AT2 r " ArT− 1,r E11 E12 " E1Tr ]T .

Thus, we consider local sector nonlinearity. This is reasonable as variables of physical systems are always bounded. 1- (b) shows the local sector nonlinearity, where two lines become the local sectors under x(t ) ∈ [t1 , t2 ] . The T-S fuzzy model exactly represents the nonlinear system in the “local” region, that is, x(t ) ∈ [t1 , t2 ] , which is described as follows: 2 ∇x(t ) = f ( x(t )) = ∑θi (t )ai x(t ), (16) i =1 where θ 1 (t ) = f ( x(t )) − a2 x(t ) w2 = , w1 + w2 a1x(t ) − a2 x(t ) (17) θ 2 (t ) = a x(t ) − f ( x(t )) w1 = 1 .