By Hongyi Li, Ligang Wu, Hak-Keung Lam, Yabin Gao
This ebook develops a collection of reference equipment in a position to modeling uncertainties latest in club features, and reading and synthesizing the period type-2 fuzzy structures with wanted performances. It additionally presents quite a few simulation effects for numerous examples, which fill convinced gaps during this zone of analysis and should function benchmark ideas for the readers.
Interval type-2 T-S fuzzy versions offer a handy and versatile process for research and synthesis of advanced nonlinear structures with uncertainties.
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Extra resources for Analysis and Synthesis for Interval Type-2 Fuzzy-Model-Based Systems
5) where η j (x(t)) denotes the lower grades of membership and η j (x(t)) denotes the upper grades of membership, μM (gs (x(t))) stands for the LMF and μMjs (gs (x(t))) js stands for the UMF. μMjs (gs (x(t))) ≥ μM (gs (x(t))) ≥ 0 and η j (x(t)) ≥ η j (x(t)) ≥ 0 js for all j. 6) j=1 where ηj (x(t)) = ν j (x(t))η j (x(t)) + ν j (x(t))η j (x(t)) r l=1 ν l (x(t))η l (x(t)) + ν l (x(t))η l (x(t)) ≥ 0, ∀j, with r ηj (x(t)) = 1, j=1 0 ≤ ν j (x(t)) ≤ 1, ∀j, 0 ≤ ν j (x(t)) ≤ 1, ∀j, ν j (x(t)) + ν j (x(t)) = 1, ∀j, in which ν j (x(t)) and ν j (x(t)) are predefined functions, and ηj (x(t)) stands for the grades of membership of the embedded membership functions.
16), it can be obtained that C˜ iT Φ C˜ i ≤ G. For any t ≥ 0, the following inequalities hold: t t J(s)ds − zT (t)Φz(t) ≥ 0 r r J(s)ds − 0 θi ηj i=1 j=1 × (Ci x(t) + D2i w(t))T Φ (Ci x(t) + D2i w(t)) t = r 0 θi ηj g T (t)C˜ iT Φ C˜ i g(t) i=1 j=1 t ≥ r J(s)ds − r r J(s)ds − 0 θi ηj x T (t)Gx(t) ≥ ρ. 1. 23) that V˙ (t) ≤ zT (t)Ψ1 z(t) − c |ξ(t)|2 . 1, we have V˙ (t) ≤ −c |ξ(t)|2 . 7) with w(t) = 0 is asymptotically stable. This completes the proof. 3). 3, the following theorem is obtained directly.
1, the LMFs and UMFs are defined as follows: q 2 2 h i jl (x(t)) = n ... 9) n ... in kl are constant scalars to be determined; 0 ≤ vris kl (xr (t)) ≤ 1 and vr 1kl (xr (t)) + vr 2kl (xr (t)) = 1 for r, s = 1, 2, . . , n; l = 1, 2, . . , τ + 1; q ir = 1, 2; x(t) ∈ Φk ; otherwise, vris k (xr (t)) = 0. As a result, we have k=1 i21 =1 2 2 n i 2 =1 . . i n =1 r =1 vrir kl (xr (t)) = 1 for all l, which is used in the stability analysis. 11) l=1 with p c h˜ i j (x(t)) = 1. 12) i=1 j=1 In addition, 0 ≤ γ i jl (x(t)) ≤ γ i jl (x(t)) ≤ 1 are two functions, which are not necessary to be known, exhibiting the property that γ i jl (x(t)) + γ i jl (x(t)) = 1 for all i, j, l; ξi jl (x(t)) = 1 if the membership function h i jl (x(t)) is within the subFOU l, otherwise, ξi jl (x(t)) = 0.