Notions and Preliminaries Sample Clauses

Notions and Preliminaries. → ◦ {| | ∈ ∞ } ≤ ∞∞ → ∈ | | Basic Notations The notations used in this paper are standard. We use Z+ to denote the set of all nonnegative integers. For any ω RN , ωT is its transpose and ω its Euclidean norm. For any function x : R+ RN , we denote x = sup x(t) , t [0, ) . For two functions γ1, γ2 : R+ R+, use γ1 γ2 to represent the composition of the two functions. We use Id to denote the identity function defined on R+. Comparison Functions For ease of presentation, two classes of functions are introduced. A function β : R+ × Copyright lies with the authors 5721 R+ → R+ is called a class I+L function if β ∈ KL, The objective of this paper is to design ϕi and φi such ∈ β(s, 0) = s for s R+, and for any specified T > 0, there exist continuous, positive definite and nondecreasing functions α1, α2 which depend on T and less than the that a robust output agreement objective is achieved, i.e., there exist γω, γω ∈ K such that ω lim |ηi(t) − ηj(t)|≤ γ ( ω ∞), i, j = 1, . . . , N (5) identity function, such that for all s ∈ R+, β(s, t) ≥ α1(s) t→∞ ω for t ∈ [0, T ] and β(s, t) ≤ α2(s) for t ∈ [T, ∞). A function lim |ζi(t)|≤ γ ( ω ∞), i = 1, . . . , N (6) β : R × R → R is called a class IL function if there exist t→∞ β′, β′′ ∈ I+ L such that for t ≥ 0, β(s, t) = β′(s, t) for where ω = [ω11 , . . . , ω1N , . . . , ωN1 , . . . , ωNN ]T . s ≥ 0, and β(s, t) = −β′′(−s, t) for s < 0. The definitions of class X, X∞ and KL functions can be found in ?. time interval [t1, t2] as G([t1, t2]) = (N , t∈[t1 ,t2 ] E(t)). Digraph We begin by introducing some basic concepts from graph theory; see e.g., ?. A digraph G = (N , E) is called quasi-strongly connected (QSC) if there exists a c ∈ N , referred as the center of G, such that there is a directed path from c to i for each i ∈ N . For a switching digraph G(t) = (N , E(t)), we denote the unionSdigraph over