HOMOGENEOUS OBSERVER DESIGN 1843or in other words,(D.2)∂V ∞∂x (x) f(x) < 0 ∀ x : |x| r ∞≥ λ ∞ .Th<strong>is</strong> establ<strong>is</strong>hes global asymptotic stability <strong>of</strong> the compact setwhere v ∞ <strong>is</strong> given <strong>by</strong>C ∞ = {x : V ∞ (x) ≤ v ∞ } ,v ∞ = max|x| r∞ = λ ∞{V ∞ (x)} .Appendix E. Pro<strong>of</strong> <strong>of</strong> Theorem 2.20. The pro<strong>of</strong> <strong>is</strong> 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 <strong>of</strong> the system (2.7) and its homogeneous approximations are negativedefinite.1. As establ<strong>is</strong>hed in the pro<strong>of</strong> <strong>of</strong> Proposition 2.18, there ex<strong>is</strong>t a positive realnumber λ ∞ and a C 1 positive definite, proper, and homogeneous functionV ∞ : R n → R + , with weight r ∞ and degree d V∞ sat<strong>is</strong>fying (D.2). Similarly,there ex<strong>is</strong>t 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 , sat<strong>is</strong>fying(E.1)∂V 0∂x (x) f(x) < 0 ∀ x : 0 < |x| r 0≤ λ 0 .Finally, the global asymptotic stability <strong>of</strong> the origin <strong>of</strong> the system ẋ = f(x)implies the ex<strong>is</strong>tence <strong>of</strong> a C 1 , positive definite, and proper function V m :R n → R + sat<strong>is</strong>fying(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 <strong>th<strong>is</strong></strong>, wefollow a technique used <strong>by</strong> 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 sat<strong>is</strong>fying(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 <strong>is</strong> C 1 , positive definite, and proper. Moreover, <strong>by</strong> 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, <strong>th<strong>is</strong></strong> proves that∂Vfor each i in {1,...,n},∂x i<strong>is</strong> 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 <strong>of</strong> V along f <strong>is</strong> negative definite.To <strong>th<strong>is</strong></strong> 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.<strong>Copyright</strong> © <strong>by</strong> <strong>SIAM</strong>. <strong>Unauthorized</strong> <strong>reproduction</strong> <strong>of</strong> <strong>th<strong>is</strong></strong> <strong>article</strong> <strong>is</strong> prohibited.
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