# Foundations of complex systems: nonlinear dynamics, by Gregoire Nicolis

By Gregoire Nicolis

Complexity is rising as a post-Newtonian paradigm for imminent a wide physique of phenomena of shock on the crossroads of actual, engineering, environmental, lifestyles and human sciences from a unifying standpoint. This e-book outlines the rules of contemporary complexity examine because it arose from the cross-fertilization of rules and instruments from nonlinear technology, statistical physics and numerical simulation. it truly is proven how those advancements result in an realizing, either qualitative and quantitative, of the complicated platforms encountered in nature and in daily adventure and, conversely, how ordinary complexity acts as a resource of notion for growth on the primary point.

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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.