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HOMOGENEOUS OBSERVER DESIGN 1837Consequently, by Lemma 2.13, there exists a positive real number c ∗ such that, for allc > c ∗ and all (ˆX n ,E 1 ) ≠ (0, 0), the Lyapunov function U, defined in (5.12), satisfies∂U()∂ ˆX(ˆX n ,E 1 ) S n ˆXn + B n φ n (ˆX n )+K 1 (e 1 )n+ ∂U∂E 1(ˆX n ,E 1 )(E 1 )(S n E 1 + K 1 (e 1 )) < 0and the same holds for the homogeneous approximations in the 0-limit and in the∞-limit; hence the claim.Proof of Corollary 5.2. We write the dynamics of the system (5.3) in the coordinatesˆX n and E 1 and in the time τ given in (5.8). This yields(5.13)with⎧⎪⎨⎪⎩ddτ ̂X n = S n ̂Xn + B n φ n (ˆX n )) + K 1 (e 1 ),ddτ E 1 = S n E 1 + K 1 (e 1 )+D(L)(δ1D(L) =L ,..., δ )nL n .We denote the solution of this system, starting from (̂X n (0),E 1 (0)) in R 2n at time τ,by (̂X τ,n (τ),E τ,1 (τ)). We have(5.14) x i (t) = L i−1 (ˆX τ,i (Lt) − e τ,i (Lt)) .The right-hand side of (5.13) is a vector field which is homogeneous in the bi-limitwith weights (r 0 ,r 0 ), (r ∞ ,r ∞ ) for (̂X n ,E 1 ) and (r 0 , r ∞ ) for D(L), where r 0,i = r 0,i +d 0and r ∞,i = r ∞,i + d ∞ for each i in {1,...,n}.The time function τ ↦→ δ( τ L) is considered as an input, and when D(L) = 0,Theorem 5.1 implies global asymptotic stability of the origin of the system (5.13)and of its homogeneous approximations. To complete the proof we show that thereexists L ∗ such that the “input” D(L) satisfies the small-gain condition (2.11) of Corollary2.22 for all L>L ∗ . Using (5.8) and (5.14), assumption (5.4) becomes, for all0 ≤ σ ≤ τ < LT and all i in {1,...,n},∣ δi( τL)∣ ∣L i{ (1 ∣∣∣z ( σ)∣ )∣∣≤ maxL i β τ − σδ , ,L L{i∑L −i ∣sup σ≤κ≤τ ∣L (j−1) (ˆX τ,j(κ) − e τ,j (κ)) ∣c 0j=11−d 0 (n−i−1)1−d 0 (n−j)i∑∣(5.15) + c ∞ ∣L (j−1) (ˆX τ,j(κ) − e τ,j (κ)) ∣Note that when 1 ≤ j ≤ i ≤ n, the function s ↦→ 1−(n−i−1) s1−(n−j) sis strictly increas-n−i) in (n+1−j , ij−1 ). As d 0 ≤ d ∞ < 1n−1, we have for all1ing, mapping (−1,1 ≤ j ≤ i ≤ n,n−1j=11−d∞(n−i−1)1−d∞(n−j)}}1 − d 0 (n − i − 1)1 − d 0 (n − j)≤ 1 − d ∞(n − i − 1)1 − d ∞ (n − j)