# Analysis and Synthesis for Interval Type-2 Fuzzy-Model-Based by Hongyi Li, Ligang Wu, Hak-Keung Lam, Yabin Gao

By Hongyi Li, Ligang Wu, Hak-Keung Lam, Yabin Gao

This publication develops a collection of reference equipment able to modeling uncertainties latest in club services, and examining and synthesizing the period type-2 fuzzy structures with wanted performances. It additionally offers a number of simulation effects for numerous examples, which fill definite gaps during this region of study and should function benchmark strategies for the readers.

Interval type-2 T-S fuzzy types offer a handy and versatile technique for research and synthesis of complicated nonlinear structures with uncertainties.

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**Sample text**

7 with l = 1 is employed to determine the system stability and synthesize the feedback gains. 8562 . The IT2 fuzzy controller is employed to stabilize the inverted pendulum with m p = 3 kg and Mc = 8 kg. The state responses of the system with different initial sates are shown in Fig. 2, which shows that the inverted pendulum can be stabilized subject to different values of m p and Mc , and different initial conditions. For comparison purposes, considering the simulation result in [97], it can be seen that the IT2 fuzzy controller can also stabilize the inverted pendulum.

C, satisfy h i j (ξ(k)) ≥ 0, and p c h i j (ξ(k)) = 1. i=1 j=1 Proof Based on the fact that 2M T Q N ≤ inf Q>0 M T Q M + N T Q N , it is easily obtained that ⎡ p c 2⎣ ⎤T c h κι (ξ(k)) Nκι κ=1 ι=1 i=1 j=1 p p h i j (ξ(k)) Mi j ⎦ Q c p c ≤ T h i j (ξ(k)) h κι (ξ(k)) MiTj Q Mi j + Nκι Q Nκι i=1 j=1 κ=1 ι=1 p p c = c h i j (ξ(k)) MiTj Q Mi j + κ=1 ι=1 i=1 j=1 p T h κι (ξ(k)) Nκι Q Nκι c = h i j (ξ(k)) MiTj Q Mi j + NiTj Q Ni j . i=1 j=1 This completes the proof. 8 ([211]) Given any matrices X , Y and Z > 0 with appropriate dimensions, then the inequality X T Y + Y T X ≤ X T Z X + Y T Z −1 Y holds.

19) 44 3 Output-Feedback Control of Interval Type-2 Fuzzy-Model-Based Systems where ⎡ Ω¯ 11ij ⎣ Ωij = ∗ ∗ Ω¯ 12ij Ω¯ 22ij ∗ ⎤ Ω¯ 13ij Ω¯ 23ij ⎦ , −I −Q Q −G C˜ iT Φ˜ T , , Θ2ij = ∗ G − 2I ∗ −I C˜ i = Ci Q, Ω¯ 13ij = C˜ iT Ψ˜ 1T , Ω¯ 11ij = He(Ai Q + Bi Mj ), T T ˜T Ω¯ 12ij = D1i − C˜ iT Ψ2 , Ω¯ 22ij = −He(D2i Ψ1 . Ψ2 ) − Ψ3 , Ω¯ 23ij = D2i Θ1ij = Then the IT2 fuzzy state-feedback controller gain matrices are given as Kj = Mj Q−1 . 1 can be chosen as ρ = −V (x(0)). 21) V (x(t)) = x T (t)Px(t), where P = PT > 0.