By Knopp K.

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The content of this book can be divided into three parts. The first part will be focused on analysis and synthesis of T-S fuzzy time-delay systems. Some sufficient conditions are derived for the stability and some optimal performances by developing new techniques for the considered TS fuzzy time-delay systems. The developed methodologies include the fuzzy LKF approach, the slack matrix approach, the delay-partitioning approach, the small gain theorem based input-output approach, and the reciprocally convex approach, etc.

Al further improved Park’s Inequality, and presented a new bounding inequality, named Moon’s Inequality. 18. [151] (Moon’s Inequality) Assume that a(α) ∈ Rna , b(α) ∈ Rnb and N ∈ Rna ×nb are given for α ∈ Ω. Then, for any matrices X ∈ Rna ×na , Y ∈ Rna ×nb and Z ∈ Rnb ×nb , the following inequality holds: −2 a(α) b(α) aT (α)N b(α)dα ≤ Ω Ω T X Y −N Z a(α) dα, b(α) where X Y Z ≥ 0. Based on the above bounding inequality, and choose the LKF as t V (t) = xT (t)P x(t) + xT (s)Qx(s)ds t−d 0 t −d t+θ x˙ T (s)Z x(s)dsdθ, ˙ + where P > 0, Q > 0 and Z > 0, a new less conservative stability condition was presented in [151].

2 J(m+2)i − ε D(m+2)i − D(m+2)i Proof. 15a) that Π22t < 0. Since Pt > 0, we have Gt + GTt > 0, which ensure that G−1 exists.

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