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HOMOGENEOUS OBSERVER DESIGN 1847wherec qi 1+cpi s pi+qi1+c qi s pi+qi ≤ max{cqi ,c pi } ,Hence, by continuity at 0, we havec qi s qi1+c qi s qi+pi ≤ c p i q iq i +p i ,c qi c pi s pi1+s pi ≤ c pi+qi .(G.7) µ i (cs) ≤ C i (c) µ i (s) ∀ (c, s) ∈ R 2 + .Consider the positive real numbers c 1 , c 2 , c δ , and c V previously introduced, and selectc G in R + satisfying( )c V(G.8) c G < min1≤i≤m C−1 i.2 mc 1 c 2 c δTo show that such a selection for c G is appropriate, observe that by (G.6) and (G.7)and µ i acting on both sides of the inequality (2.11), we get for each i in {1,...,m}and all 0 ≤ s ≤ t ≤ T ,µ i (|δ i (t)|) ≤ max{µ i ◦ β δ (|z(s)|,t− s) ,C i (c G ) c 2{ (sup Hs≤κ≤tConsequentlym∑{µ i (|δ i (t)|) ≤ max m max {µ i ◦ β δ (|z(s)|,t− s)} ,1≤i≤mi=1{ ((G.9) (m max 1≤i≤m C i (c G ) c 2 ) sup s≤κ≤t HSince (G.8) yields2c 1 c δc V|x(κ)| d0+d V 0r 0|x(κ)| d0+d V 0r 0m max1≤i≤m C i(c G ) c 2 < 1 ,)} }, |x(κ)| d∞+d V∞r ∞.)} }, |x(κ)| d∞+d V∞r ∞.the existence of the function β x follows from (2.10), (G.5), (G.9), and the (proof ofthe) small-gain theorem [11].Appendix H. Proof of Corollary 2.24. First, observe that the continuity off 0 , at least, on R n \{0} implies|d 0 | = −d 0 ≤ min1≤i≤n r 0,i ≤ max1≤i≤n r 0,i < d V0 .Then, let V be the function given in Theorem 2.20 and, since d 0 < 0 < d ∞ , the functionφ(x) =V (x)d V0 +d 0d V0+ V (x)d V∞ +d∞d V∞is homogeneous in the bi-limit with weightsr 0 and r ∞ , degrees d V0 + d 0 and d V∞ + d ∞ , and approximating functions V (x)d V∞ +d∞d V∞d V0 +d 0d V0and V (x) . Moreover, the function ζ(x) = − ∂V∂x(x) f(x) is homogeneous inthe bi-limit with the same weights and degrees as φ. Furthermore, since the functionζ and its homogeneous approximations are positive definite, Corollary 2.15 yields astrictly positive real number c such that(H.1)(d∂VV0 +d 0∂x (x) f(x) ≤ −c dV (x) V0d V∞)+d∞d+ V (x) V∞∀ x ∈ R n .Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.