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HOMOGENEOUS OBSERVER DESIGN 1819then the function Φ i (x) = ∫ x iφ(x0 1 ,...,x i−1 , s, x i+1 ,...,x n ) ds is homogeneous inthe 0- (resp., ∞-) limit with associated triple (r 0 ,d 0 + r 0,i , Φ i,0 ) (resp., (r ∞ ,d ∞ +r ∞,i , Φ i,∞ )), with Φ i,0 (x) = ∫ x iφ0 0 (x 1 ,...,x i−1 , s, x i+1 ,...,x n ) ds (resp., Φ i,∞ (x) =∫ xiφ0 ∞ (x 1 ,...,x i−1 , s, x i+1 ,...,x n ) ds).By exploiting the definition of homogeneity in the bi-limit, it is possible to establishresults which are straightforward extensions of well-known results based on thestandard notion of homogeneity. These results are given as corollaries of the followingkey technical lemma, the proof of which is given in Appendix C.Lemma 2.13 (key technical lemma). Let η : R n → R and γ : R n → R + be twofunctions homogeneous in the bi-limit, with weights r 0 and r ∞ , degrees d 0 and d ∞ ,and approximating functions, η 0 , η ∞ and γ 0 , γ ∞ such that the following hold:{ x ∈ R n \{0} : γ(x) =0} ⊆{ x ∈ R n : η(x) < 0 } ,{ x ∈ R n \{0} : γ 0 (x) =0} ⊆{ x ∈ R n : η 0 (x) < 0 } ,{ x ∈ R n \{0} : γ ∞ (x) =0} ⊆{ x ∈ R n : η ∞ (x) < 0 } .Then there exists a real number c ∗ such that, for all c ≥ c ∗ and for all x in R n \{0},(2.4)η(x) − cγ(x) < 0 , η 0 (x) − cγ 0 (x) < 0 , η ∞ (x) − cγ ∞ (x) < 0 .Example 2.14. To illustrate the importance of this lemma, consider, for (x 1 ,x 2 )in R 2 , the functionsη(x 1 ,x 2 )=x 1 x 2 −|x 1 | r 1 +r 2r 1 , γ(x 1 ,x 2 )=|x 2 | r 1 +r 2r 2 ,with r 1 > 0 and r 2 > 0. They are homogeneous in the standard sense, and thereforein the bi-limit, with the same weight r =(r 1 ,r 2 ) and the same degree d = r 1 + r 2 .Furthermore, the function γ takes positive values, and for all (x 1 ,x 2 ) in {(x 1 ,x 2 ) ∈R 2 \{0} : γ(x 1 ,x 2 )=0} we haveη(x 1 ,x 2 ) = −|x 1 | r 1 +r 2r 1 < 0 .Thus Lemma 2.13 yields the existence of a positive real number c ∗ such that for allc ≥ c∗, we have(2.5) x 1 x 2 −|x 1 | r 1 +r 2r 1 − c |x 2 | r 1 +r 2r 2 < 0 ∀ (x 1 ,x 2 ) ∈ R 2 \{0} .This is a generalization of the procedure known as the completion of the squares inwhich, however, the constant c ∗ 1 is not specified.Corollary 2.15. Let φ : R n → R and ζ : R n → R + be two homogeneous in thebi-limit functions with the same weights r 0 and r ∞ , degrees d φ,0 , d φ,∞ and d ζ,0 , d ζ,∞ ,and approximating functions η 0 , φ ∞ and ζ 0 , ζ ∞ . If the degrees satisfy d φ,0 ≥ d ζ,0 andd φ,∞ ≤ d ζ,∞ , and the functions ζ, ζ 0 and ζ ∞ are positive definite, then there exists apositive real number c satisfyingProof. Consider the two functionsφ(x) ≤ cζ(x) ∀ x ∈ R n .η(x) := φ(x)+ζ(x), γ(x) := ζ(x) .Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
1820 V. ANDRIEU, L. PRALY, AND A. ASTOLFIBy property P2 (or P3) 2 in section 2.2, they are homogeneous in the bi-limit withdegrees d ζ,0 and d ζ,∞ . The function γ and its homogeneous approximations beingpositive definite, all assumptions of Lemma 2.13 are satisfied. Therefore there existsa positive real number c such thatcγ(x) > η(x) > φ(x) ∀ x ∈ R n \{0} .Finally, by continuity of the functions φ and ζ at zero, we can obtain the claim.2.3. Stability and homogeneous approximation. A very basic property ofasymptotic stability is its robustness. This fact was already known to Lyapunov, whoproposed his second method, in which (local) asymptotic stability of an equilibrium isestablished by looking at the first order approximation of the system. The case of localhomogeneous approximations of higher degree has been investigated by Massera [16],Hermes [9], Rosier [29], and Kawski [12].Proposition 2.16 (see [29]). Consider a homogeneous in the 0-limit vector fieldf : R n → R n with associated triple (r 0 , d 0 ,f 0 ). If the origin of the systemẋ = f 0 (x)is locally asymptotically stable, then the origin ofẋ = f(x)is locally asymptotically stable.Consequently, a natural strategy to ensure local asymptotic stability of an equilibriumof a system is to design a stabilizing homogeneous control law for the homogeneousapproximation in the 0-limit (see [9, 12, 5], for instance).Example 2.17. Consider the system (1.1), with q = 1 and p>q, and the linearcontrol lawu = −(c 0 + 1) x 2 − x 1 .The closed-loop vector field is homogeneous in the 0-limit with degree d 0 = 0,weight (1, 1) (i.e., we are in the linear case), and associated vector field f 0 (x 1 ,x 2 )=(x 2 , −x 1 − x 2 ) T . Selecting the Lyapunov function of degree 2,yieldsV 0 (x 1 ,x 2 ) = 1 2 |x 1| 2 + 1 2 |x 2 + x 1 | 2 ,∂V 0∂x (x) f 0(x) = −|x 1 | 2 −|x 2 + x 1 | 2 .It follows, from Lyapunov’s second method, that the control law locally asymptoticallystabilizes the equilibrium of the system. Furthermore, local asymptotic stability ispreserved in the presence of any perturbation which does not change the approximatinghomogeneous function, i.e., in the presence of perturbations which are dominatedby the linear part (see P2 in section 2.2).2 If φ 0 (x) +ζ 0 (x) = 0 (resp., φ ∞(x) +ζ ∞(x) = 0), the proof can be completed by replacing ζwith 2ζ.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
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