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HOMOGENEOUS OBSERVER DESIGN 1843or in other words,(D.2)∂V ∞∂x (x) f(x) < 0 ∀ x : |x| r ∞≥ λ ∞ .This establishes global asymptotic stability of the compact setwhere v ∞ is given byC ∞ = {x : V ∞ (x) ≤ v ∞ } ,v ∞ = max|x| r∞ = λ ∞{V ∞ (x)} .Appendix E. Proof of Theorem 2.20. The proof is divided into three steps.First, we define three Lyapunov functions V 0 , V m , and V ∞ . Then we build anotherLyapunov function V from these three. Finally, we show that its derivative alongthe trajectories of the system (2.7) and its homogeneous approximations are negativedefinite.1. As established in the proof of Proposition 2.18, there exist a positive realnumber λ ∞ and a C 1 positive definite, proper, and homogeneous functionV ∞ : R n → R + , with weight r ∞ and degree d V∞ satisfying (D.2). Similarly,there exist a number λ 0 > 0 and a C 1 positive definite, proper, and homogeneousfunction V 0 : R n → R + , with weight r 0 and degree d V0 , satisfying(E.1)∂V 0∂x (x) f(x) < 0 ∀ x : 0 < |x| r 0≤ λ 0 .Finally, the global asymptotic stability of the origin of the system ẋ = f(x)implies the existence of a C 1 , positive definite, and proper function V m :R n → R + satisfying(E.2)∂V m(x) f(x) < 0 ∀ x ≠0.∂x2. Now we build a function V from the functions V m , V ∞ , and V 0 . For this, wefollow a technique used by Mazenc in [17] (see also [15]). Let v ∞ and v 0 betwo strictly positive real numbers such that v 0
1844 V. ANDRIEU, L. PRALY, AND A. ASTOLFILetV (x) = ω ∞ ϕ ∞ (V m (x))V ∞ (x)+ [1 − ϕ ∞ (V m (x))] ϕ 0 (V m (x)) V m (x)+ω 0 [1 − ϕ 0 (V m (x))] V 0 (x) ,where ϕ 0 and ϕ ∞ are C 1 nondecreasing functions satisfying(E.3)(E.4)ϕ 0 (s) = 0 ∀ s ≤ 1 2 v 0, ϕ 0 (s) = 1 ∀ s ≥ v 0 ,ϕ ∞ (s) = 0 ∀ s ≤ v ∞ , ϕ ∞ (s) = 1 ∀ s ≥ 2v ∞ .Then V is C 1 , positive definite, and proper. Moreover, by construction,⎧ω 0 V 0 (x) ∀ x : V m (x) ≤ 1 2 v 0 ,⎪⎨V (x) =⎪⎩ϕ 0 (V m (x)) V m (x)+ω 0 [1 − ϕ 0 (V m (x))] V 0 (x)∀ x :12 v 0 ≤ V m (x) ≤ v 0 ,V m (x) ∀ x : v 0 ≤ V m (x) ≤ v ∞ ,ω ∞ ϕ ∞ (V m (x))V ∞ (x) + [1 − ϕ ∞ (V m (x))] V m (x)∀ x : v ∞ ≤ V m (x) ≤ 2 v ∞ ,ω ∞ V ∞ (x) ∀ x : V m (x) ≥ 2 v ∞ .Thus for each i in {1,...,n},(E.5)and(E.6)∂V∂x i(x) = ω ∞∂V ∞∂x i(x)∀ x : V m (x) > 2 v ∞∂V∂x i(x) = ω 0∂V 0∂x i(x) ∀ x : V m (x) < 1 2 v 0 .Since ∂V∞∂x iand ∂V0∂x iare homogeneous in the standard sense, this proves that∂Vfor each i in {1,...,n},∂x iis homogeneous in the bi-limit, with weights r 0and r ∞ and degrees d V0 − r 0,i and d V∞ − r ∞,i .3. It remains to show that the Lie derivative of V along f is negative definite.To this end note that, for all x such that 1 2 v 0 ≤ V m (x) ≤ v 0 ,∂V∂x (x)f(x) = ϕ′ 0(V m (x)) [V m (x) − ω 0 V 0 (x)] ∂V m∂x (x)f(x)+ ω 0 [1 − ϕ 0 (V m (x))] ∂V 0∂x (x)f(x)+ϕ 0(V m (x)) ∂V m∂x (x)f(x)and, for all x such that v ∞ ≤ V m (x) ≤ 2 v ∞ ,∂V∂x (x)f(x) = ϕ′ ∞(V m (x)) [ω ∞ V ∞ (x) − V m (x)] ∂V m∂x (x)f(x)+ ω ∞ ϕ ∞ (V m (x)) ∂V ∞∂x (x)f(x) + [1 − ϕ ∞(V m (x))] ∂V m∂x (x)f(x) .By (D.2), (E.1), (E.2), (E.3), and (E.4), these inequalities imply∂V(x) f(x) < 0 ∀ x ≠ 0 ,∂xwhich proves the claim.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
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