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HOMOGENEOUS OBSERVER DESIGN 1839From Corollary 2.22, the result holds for all L ∗ small enough to satisfy c 1 L ∗ε ≤c G .6. Conclusion. We have presented two new tools that can be useful in nonlinearcontrol design. The first one is introduced to formalize the notion of homogeneousapproximation valid both at the origin and at infinity. With this formalism we havegiven several novel results concerning asymptotic stability, robustness analysis, andalso finite time convergence (uniformly in the initial conditions). The second one isa new recursive design for an observer for a chain of integrators. The combination ofthese two tools allows us to obtain a new result on stabilization by output feedbackfor systems whose dominant homogeneous in the bi-limit part is a chain of integrators.Appendix A. Proof of Proposition 2.10. We give the proof only in the 0-limit case since the ∞-limit case is similar. Let C be an arbitrary compact subset ofR n \{0} and ɛ any strictly positive real number. By the definition of homogeneity inthe 0-limit, there exists λ 1 > 0 such that we have∣φ(λ r φ,0⋄ x)λ d φ,0− φ 0 (x)∣ ≤ 1 ∀ x ∈ C, ∀ λ ∈ (0,λ 1] .Hence, as φ 0 is a continuous function on R n , for all λ in (0,λ 1 ], the function x ↦→φ(λ r 0 ⋄ x)λ d φ,0takes its values in a compact set C φ = φ 0 (C) +B 1 , where B 1 is theunity ball.Now, as ζ 0 is continuous on the compact subset C φ , it is uniformly continuous;i.e., there exists ν> 0 such that|z 1 − z 2 | < ν =⇒ |ζ 0 (z 1 ) − ζ 0 (z 2 )| < ɛ .Also there exists µ ɛ > 0 satisfyingζ(µ r ζ,0z)∣ µ d − ζζ,0 0 (z)∣ ≤ ɛ ∀ z ∈ C φ , ∀ µ ∈ (0,µ ɛ ] ,or equivalently, since d φ,0 > 0,ζ(λ d φ,0z)(− ζ∣0 (z)∣ ≤ ɛ ∀ z ∈ C φ , ∀ λ ∈ 0,µd φ,0 d ζ,0rλ ζ,0r ζ,0d φ,0ɛSimilarly, there exists λ ν such thatφ(λ r φ,0⋄ x)∣ λ d − φφ,0 0 (x)∣ ≤ ν ∀ x ∈ C, ∀ λ ∈ (0,λ ν] .It follows that∣ ζ(φ(λ r φ,0⋄ x))∣∣∣∣ d∣ φ,0 d ζ,0− ζ 0 (φ 0 (x))rλ ζ,0∣ ≤ ζ(φ(λ r (φ,0⋄ x)) φ(λr φ,0) ∣ ⋄ x)∣∣∣d φ,0 d ζ,0− ζ 0rλ ζ,0λ d φ,0( +φ(λr∣ ζ φ,0)⋄ x)0λ d − ζφ,00 (φ 0 (x))∣{≤ 2 ɛ ∀ x ∈ C, ∀ λ ∈ min λ 1 ,λ ν ,µ].r ζ,0d φ,0ɛ}.This establishes homogeneity in the 0-limit of the function ζ ◦ φ.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
1840 V. ANDRIEU, L. PRALY, AND A. ASTOLFIAppendix B. Proof of Proposition 2.11. We give the proof only in the 0-limit case since the ∞-limit case is similar. The function φ being a bijection, we canassume without loss of generality that it is a strictly increasing function (otherwise wetake −φ). This, together with homogeneity in the 0-limit, implies that ϕ 0 is strictlypositive. Moreover, for each δ> 0, there exists t 0 (δ) > 0 such that∣ φ(t) ∣∣∣∣ − ϕ 0 ≤ δ ∀ t ∈ (0,t 0 (δ)] .t d0By letting λ = φ(t), this givesϕ 0 − δ ≤λφ −1 (λ) d0 ≤ ϕ 0 + δ ∀ λ ∈ (0,φ(t 0 (δ))] , ∀ δ> 0 .Since for δ < ϕ 0 the term on the left is strictly positive, these inequalities give( 1) 1d 0ϕ 0 + δ≤ φ−1 (λ)λ 1d 01Then since the function δ ↦→ (ϕ ) 1d 00−δexists δ 1 (ɛ 1 ) > 0 satisfying( 1ϕ 0) 1d 0− ɛ 1 ≤This yields(φ −1 (λ)∣λ 1d 0≤1ϕ 0 + δ 1 (ɛ 1 )( ) 11d 0∀ λ ∈ (0,φ −1 (t 0 (δ))], ∀ δ ∈ (0,ϕ 0 ) .ϕ 0 − δ) 1d 0is continuous at zero, for every ɛ 1 > 0 there≤(1ϕ 0 − δ 1 (ɛ 1 )) 1d 0( ) ∣11d 0 ∣∣∣∣− ≤ ɛ 1 ∀ λ ∈ (0,λ − (ɛ 1 )] ,ϕ 0with λ − (ɛ 1 )=φ(t 0 (δ 1 (ɛ 1 ))). With a similar argument, we getφ −1 ( ) ∣1(−λ) 1d 0 ∣∣∣∣ + ≤ ɛ 1 ∀ λ ∈ (0,λ + (ε 1 )]∣ d 0 ϕ 0λ 1for some λ + > 0. Let λ 0 = min{λ − ,λ + }.Now, for x ≠ 0 and λ> 0, we have∣φ −1 (λx)∣λ 1d 0( ) ∣1xd 0 ∣∣∣∣− = |x| 1d 0ϕ 0∣ ∣∣∣∣φ −1 (λx)(xλ) 1d 0≤( ) ∣11d 0 ∣∣∣∣− .ϕ 0Therefore, for any compact set C of R\{0} and any ɛ> 0, by letting ɛ 1 =we have|x| 1d 0 ɛ 1 ≤ ɛ, 0 < |λx| ≤ λ 0 (ɛ 1 ) ∀ λ ∈and thereforeφ −1 (λx)∣λ 1d 0( ) ∣1xd 0 ∣∣∣∣− ≤ ɛϕ 0∀ λ ∈(0,( 1ϕ 0) 1d 0+ ɛ 1 .ɛmax x∈C |x| 1d 0,]λ 0 (ɛ 1 ), ∀ x ∈ C,max x∈C |x|(]λ 0 (ɛ 1 )0,, ∀ x ∈ C.max x∈C |x|This establishes homogeneity in the 0-limit of the function φ −1 .Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
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