By Tengfei Liu, Zhong-Ping Jiang, David J. Hill

Major development has been made on nonlinear regulate structures some time past 20 years. although, a number of the current nonlinear keep watch over tools can't be effortlessly used to deal with verbal exchange and networking concerns with out nontrivial changes.

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**Extra info for Nonlinear Control of Dynamic Networks**

**Sample text**

4 presents the equivalence between AS and local ISS. 42) is AS at the origin. Proof. 2] on the sufficiency of GAS for local ISS. The necessity part is obvious. We prove the sufficiency part. 18) hold. 44) Nonlinear Control of Dynamic Networks 12 where α, α ∈ K∞ and α is a continuous and positive definite function. 45) 2 for all |ǫ| ≤ δ. 45) holds for all |ǫ| ≤ χ0 (|x|). 46) for all 0 ≤ s ≤ max{|x| : x ∈ Ω0 }. It can be directly proved that if x ∈ Ω0 and χ(|u|) ≤ |x|, then 1 ∇V (x)f (x, u) ≤ − α(V (x)).

3) ρ ◦ γ < Id. , 19 Nonlinear Control of Dynamic Networks 20 As shown later, the robust stability property of ISS can be considered as a special case of the ISS small-gain theorem. 4) for GAS, which is a reduced version of the original proof of the ISS small-gain theorem given by Jiang, Teel, and Praly [130]. The proof is carried out in two steps. 1. 1]. Suppose that the solution x(t) of the interconnected system is right maximally defined on [0, T ) with 0 < T ≤ ∞. 5) for all 0 ≤ t < T . 4) satisfied, γ ◦ ρ( x x [0,T ) [0,T ) ) [0,T ) )}.

54) whenever V (p) ≥ χ ˆu2 (|v2 |), where α ˆ 2 = α2 and χ ˆu2 = χu2 . 55) t=0 for almost all p and all v, with ϕ(t) = [ϕT1 (t), ϕT2 (t)]T being the solution of the initial-value problem ϕ(t) ˙ = f (ϕ(t), v), ϕ(0) = p. 56) In this case, assume p = (p1 , p2 ) = (0, 0) and V1 (p1 ) ≥ χu1 (|v1 |), V2 (p2 ) ≥ χu2 (|v2 |). 60) where α ˆ 1 and α ˆ 2 are continuous and positive definite functions. Note that in this case p1 = 0 and p2 = 0. 62) for all x ∈ X1 × X2 . Note also that there exists δ > 0 such that ϕ(t) ∈ X1 × X2 for all 0 ≤ t < δ.