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HOMOGENEOUS OBSERVER DESIGN 1817R + is the degree, and φ ∞ : R n → R is the approximating function, if φ iscontinuous, φ ∞ is continuous and not identically zero, and, for each compactset C in R n \{0} and each ε> 0, there exists λ ∞ such thatmaxφ(λ r∞ ⋄ x)x ∈ C∣ λ d∞− φ ∞ (x)∣ ≤ ε ∀ λ ≥ λ ∞ .• A vector field f = ∑ ni=1 f i ∂∂x iis said to be homogeneous in the ∞-limit withassociated triple (r ∞ , d ∞ ,f ∞ ), where r ∞ in (R + \{0}) n is the weight, d ∞ inR is the degree, and f ∞ = ∑ ni=1 f ∞,i ∂∂x iis the approximating vector field, if,for each i in {1,...,n}, d ∞ + r ∞,i ≥ 0 and the function f i is homogeneousin the ∞-limit with associated triple (r ∞ , d ∞ + r ∞,i ,f ∞,i ).Example 2.4. The function δ 2 : R → R given in the illustrative system (1.1) ishomogeneous in the ∞-limit with associated triple (r ∞ ,d ∞ ,δ 2,∞ ) = (1, p, c ∞ x p 2 ).Furthermore, when p d 0r 0,i∀ i ∈ {1,...,n} .Example 2.9. We recall (1.6) and consider two homogeneous and positive definitefunctions φ 0 : R n → R + and φ ∞ : R n → R + with weights (r 0 ,r ∞ ) in (R + \{0}) 2n anddegrees (d 0 ,d ∞ ) in (R + \{0}) 2 . The function x ↦→ H(φ 0 (x),φ ∞ (x)) is positive definiteand homogeneous in the bi-limit with associated triples (r 0 ,d 0 ,φ 0 ) and (r ∞ ,d ∞ ,φ ∞ ).This way of constructing a homogeneous in the bi-limit function from two positivedefinite homogenous functions is extensively used in this paper.2.2. Properties of homogeneous approximations. To begin, we note thatthe weight and degree of a homogeneous in the 0- (resp., ∞-) limit function are1 This is proved by noting that, for all x in R n and all µ in R + \{0},φ 0 (µ r 0 ⋄ x)µ d 0= 1µ d lim φ (λ r 0 ⋄ (µ r 0 ⋄ x)) φ ((λµ) r 0 ⋄ x)0 λ→0 λ d = lim0λ→0 (λµ) d = φ 0 (x) ,0and similarly for the homogeneous in the ∞-limit function.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.