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As the function q −1iHOMOGENEOUS OBSERVER DESIGN 1827is continuous, strictly increasing and onto, the functionϑd W0 −r 0,ir 0,iid− q −1W0 −r 0,iri (e i+1 ) 0,id W∞ −r ∞,ir+ ϑ∞,ii − q −1d W∞ −r ∞,iri (e i+1 ) ∞,ihas a unique zero at q i (ϑ i )=e i+1 and has the same sign as q i (ϑ i ) − e i+1 . It followsthatOn the other hand, for all E i ≠ 0,T 2 (E i+1 ,ϑ i ) ≥ 0 ∀ (E i+1 ,ϑ i ) ∈ R n−i ,T 2 (E i+1 ,ϑ i )=0 ⇒ q i (ϑ i ) = e i+1 .T 1 (E i+1 ,q −1i (e i+1 )) = ∂W i+1∂E i+1(E i+1 )(S n−i E i+1 + K i+1 (e i+1 )) < 0 .Hence (3.9) yields{(Ei+1 ,ϑ i ) ∈ R n−i+1 \{0} : T 2 (E i+1 ,ϑ i ) = 0 }⊆{(Ei+1 ,ϑ i ) ∈ R n−i+1 : T 1 (E i+1 ,ϑ i ) < 0 } .By following the same argument, it can be shown that this property holds also forthe homogeneous approximations, i.e.,{(Ei+1 ,ϑ i ) ∈ R n−i+1 \{0} : T 2,0 (E i+1 ,ϑ i ) = 0 }{⊆ (Ei+1 ,ϑ i ) ∈ R n−i+1 : T 1,0 (E i+1 ,ϑ i ) < 0 } ,{(Ei+1 ,ϑ i ) ∈ R n−i+1 \{0} : T 2,∞ (E i+1 ,ϑ i ) = 0 }⊆{(Ei+1 ,ϑ i ) ∈ R n−i+1 : T 1,∞ (E i+1 ,ϑ i ) < 0 } .Therefore, by Lemma 2.13, there exists l ∗ such that, for all l ≥ l ∗ and all (E i+1 ,ϑ i ) ≠0,T 1 (E i+1 ,ϑ i ) − lT 2 (E i+1 ,ϑ i ) < 0 ,T 1,0 (E i+1 ,ϑ i ) − lT 2,0 (E i+1 ,ϑ i ) < 0 ,T 1,∞ (E i+1 ,ϑ i ) − lT 2,∞ (E i+1 ,ϑ i ) < 0 .This implies that the origin is a globally asymptotically stable equilibrium of thesystems (3.6), which concludes the proof.To construct the function K 1 , which defines the observer (3.3), it is sufficient toiterate the construction proposed in Theorem 3.1 starting from{ 11+dK n (e n ) = −0(l n e n ) 1+d0 , |l n e n |≤ 1 ,11+d ∞(l n e n ) 1+d∞ + 11+d 0− 11+d ∞, |l n e n |≥ 1 ,where l n is any strictly positive real number. Indeed, K n is a homogeneous in thebi-limit vector field with approximating functions K n,0 (e n )= 11+d 0(l n e n ) 1+d0 andK n,∞ (e n )= 11+d ∞(l n e n ) 1+d∞ . This selection implies that the origin is a globallyasymptotically stable equilibrium for the systems ė n = K n (e n ), ė n = K n,0 (e n ),and ė n = K n,∞ (e n ).Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
1828 V. ANDRIEU, L. PRALY, AND A. ASTOLFIConsequently the assumptions of Theorem 3.1 are satisfied for i + 1 = n. We canapply it recursively up to i = 1, obtaining the vector field K 1 .As a result of this procedure we obtain a homogeneous in the bi-limit observer,which globally asymptotically observes the state of the system (3.1), and also thestate for its homogeneous approximations around the origin and at infinity. In otherwords, the origin is a globally asymptotically stable equilibrium of the systems(3.12) Ė 1 = S n E 1 + K 1 (e 1 ) , Ė 1 = S n E 1 + K 1,0 (e 1 ) , Ė 1 = S n E 1 + K 1,∞ (e 1 ) .Remark 3.2. Note that when 0 ≤ d 0 ≤ d ∞ , we have 1 ≤ r0,i+d0r 0,ii =1,...,n and we can replace the function q i in (3.10) by the simpler function≤ r∞,i+d∞r ∞,iforr 0,i +d 0rq i (s) = s 0,ir ∞,i +d∞+ sr ∞,i,which has been used already in [1].Example 3.3. Consider a chain of integrators of dimension two, with the followingweights and degrees:()(r 0 , d 0 ) = (2 − q, 1), q− 1 , (r ∞ , d ∞ ) =()(2 − p, 1), p− 1 .When q ≥ p (i.e., d 0 ≤ d ∞ ), by following the above recursive observer design weobtain two positive real numbers l 1 and l 2 such that the systemwith(3.13) q 2 (s) =˙ˆX 1 = ˆX 2 − q 1 (l 1 e 1 ) , ˙ˆX 2 = u − q 2 (l 2 q 1 (l 1 e 1 )) ,e 1 = ˆX 1 − y,{ 1q sq , |s| ≤ 1,1p sp + 1 q − 1 p , |s| ≥ 1, q 1 (s) ={(2 − q) s12−q , |s| ≤ 1,(2 − p)s 12−p + p − q, |s| ≥ 1,is a global observer for the system Ẋ 1 = X 2, Ẋ 2 = u, y = X 1. Furthermore, its homogeneousapproximations around the origin and at infinity are also global observersfor the same system.4. Recursive design of a homogeneous in the bi-limit state feedback.It is well known that the system (3.1) can be rendered homogeneous by using astabilizing homogeneous state feedback which can be designed by backstepping (see[21, 25, 19, 26, 33, 10], for instance). We show in this section that this property canbe extended to the case of homogeneity in the bi-limit. More precisely, we show thatthere exists a homogeneous in the bi-limit function φ n such that the system (3.1) withu = φ n (X n ) is homogeneous in the bi-limit, with weights r 0 and r ∞ and degrees d 0and d ∞ . Furthermore, its origin and the origin of the approximating systems in the0-limit and in the ∞-limit are globally asymptotically stable equilibria.To design the state feedback we follow the approach of Praly and Mazenc [25]. Tothis end, consider the auxiliary system with state X i =(X 1,...,X i) in R i ,1≤ i
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