By Ahmad Taher Azar
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Extra resources for Fuzzy Systems
For the following auxiliary systems, we can show that the following theorems hold. Consider the following systems: x( k + 1) = Ar x( k ) + [ γ −1B1r ~( k ) + B u( k ), ε −1H 1 0 0]w 2r ⎡ γ −1D11r ⎡C1r ⎤ ⎢ −1 ~ ~⎥ ⎢ εA εγ B ~ z ( k ) = ⎢ ~ ⎥ x( k ) + ⎢⎢ −1 ~ 1 ⎢εC 2 ⎥ εγ D21 ⎢ −1 ~ ⎢ ~ ⎥ ⎣⎢ εC1 ⎦⎥ ⎣⎢εγ D11 y(k ) = C 2 r x( k ) + [ γ −1D21r ⎡D12r ⎤ 0 0 ε −1 H 2 ⎤ ⎥ ~ ⎥ ⎢ εB 0 0 0 ⎥~ ⎢ ~ 2 ⎥u( k ), w ( k ) + ⎢εD22 ⎥ 0 0 0 ⎥ ⎥ ⎢ ~ ⎥ 0 0 0 ⎦⎥ ⎣⎢εD12 ⎦⎥ (20) ~( k ) + D u( k ) 0 ε −1 H 3 0]w 22 r and x( k + 1) = Ar x( k ) + [ ε −1 H 1 0 ]w( k ) + B2 r u( k ), ~⎤ ~ ⎤ ⎡ εA ⎡ εB z(k ) = ⎢ ~ ⎥ x( k ) + ⎢ ~ 2 ⎥u( k ), ⎣⎢εD22 ⎦⎥ ⎣⎢εC2 ⎦⎥ (21) y(k ) = C 2 r x( k ) + [0 ε −1 H 3 ]w( k ) + D22r u( k ), where ε > 0 is a scaling parameter.
However, note that it is sometimes difficult to find global sectors for general nonlinear systems. 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 .
4 indicates that a controller that achieves a unitary H∞ disturbance attenuation γ . This leads to the fact that the same controller can be used to achieve the stability and the prescribed H∞ disturbance attenuation level of the fuzzy system (6). We also note that research on an H∞ output feedback controller design has been extensively investigated, and a design method of H∞ controllers has already been given. Therefore, the existing results on stability with H∞ disturbance attenuation can be applied to solve the robust H∞ output feedback stabilization problem for fuzzy systems with the immeasurable premise variables.