Copyright © by SIAM. Unauthorized reproduction of this article is ...

SIAM J. CONTROL OPTIM.Vol. 47, No. 4, pp. 1814–1850c○ 2008 Society for Industrial and Applied MathematicsHOMOGENEOUS APPROXIMATION, RECURSIVE OBSERVERDESIGN, AND OUTPUT FEEDBACK ∗VINCENT ANDRIEU † , LAURENT PRALY ‡ , AND ALESSANDRO ASTOLFI §Abstract. We introduce two new tools that can be useful in nonlinear observer and outputfeedback design. The first one is a simple extension of the notion of homogeneous approximationto make it valid both at the origin and at infinity (homogeneity in the bi-limit). Exploiting thisextension, we give several results concerning stability and robustness for a homogeneous in thebi-limit vector field. The second tool is a new recursive observer design procedure for a chain ofintegrator. Combining these two tools, we propose a new global asymptotic stabilization result byoutput feedback for feedback and feedforward systems.Key words. homogeneous approximation, output feedback and observerAMS subject classifications. 93B51, 93B52, 93D05, 93D15, 34D20DOI. 10.1137/0606758611. Introduction. The problems of designing globally convergent observers andglobally asymptotically stabilizing output feedback control laws for nonlinear systemshave been addressed by many authors following different routes. Many of these approachesexploit domination ideas and robustness of stability and/or convergence. Inview of possibly clarifying and developing further these techniques we introduce twonew tools. The first one is a simple extension of the technique of homogeneous approximationto make it valid both at the origin and at infinity. The second tool is anew recursive observer design procedure for a chain of integrator. Combining thesetwo tools, we propose a new global asymptotic stabilization result by output feedbackfor feedback and feedforward systems.To place our contribution in perspective, we consider the following system, forwhich we want to design a global asymptotic stabilizing output feedback:(1.1) ẋ 1 = x 2 , ẋ 2 = u + δ 2 (x 1 ,x 2 ), y = x 1 ,where (see notation (1.4))(1.2) δ 2 (x 1 ,x 2 ) = c 0 x q 2 + c ∞ x p 2 , (c 0,c ∞ ) ∈ R 2 , p > q > 0 .In the domination’s approach, the nonlinear function δ 2 is not treated per se inthe design but considered as a perturbation. In this framework the output feedbackcontroller is designed on the linear system(1.3) ẋ 1 = x 2 , ẋ 2 = u, y = x 1 ,∗ Received by the editors November 24, 2006; accepted for publication (in revised form) February17, 2008; published electronically June 25, 2008. The work of the first and third authors was partlysupported by the Leverhulme Trust.http://www.siam.org/journals/sicon/47-4/67586.html† LAAS-CNRS, University of Toulouse, 31077 Toulouse, France (vincent.andrieu@gmail.com).This author’s work was done while at Electrical and Electronic Engineering Department, ImperialCollege, London.‡ Centre d’Automatique et Systèmes, École des Mines de Paris, 35 Rue Saint Honoré, 77305Fontainebleau, France (Laurent.Praly@ensmp.fr).§ Electrical and Electronic Engineering Department, Imperial College London, London, SW7 2AZ,UK (a.astolfi@ic.ac.uk), and Dipartimento di Informatica Sistemi e Produzione, University of RomeTor Vergata, Via del Politecnico 1, 00133 Rome, Italy.1814Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

HOMOGENEOUS OBSERVER DESIGN 1815and will be suitable for the nonlinear system (1.1), provided the global asymptoticstability obtained for the origin of the closed-loop system is robust to the nonlineardisturbance δ 2 . For instance, the design given in [13, 27] provides a linear outputfeedback controller which is suitable for the nonlinear system (1.1) when q = 1 andc ∞ = 0. This result has been extended recently in [26] employing a homogeneousoutput feedback controller which allows us to deal with p ≥ 1 and c 0 = 0.Homogeneity in the bi-limit and the novel recursive observer design proposed inthis paper allow us to deal with the case in which c 0 ≠ 0 and c ∞ ≠ 0. In this case,the function δ 2 is such that1. when |x 2 | is small and q = 1, δ 2 (x 2 ) can be approximated by c 0 x 2 and thenonlinearity can be approximated by a linear function;2. when |x 2 | is large, δ 2 (x 2 ) can be approximated by c ∞ x p 2 , and hence we havea polynomial growth which can be handled by a weighted homogeneous controlleras in [26].To deal with both linear and polynomial terms we introduce a generalization ofweighted homogeneity which highlights the fact that a function becomes homogeneousas the state tends to the origin or to infinity but with different weights anddegrees.The paper is organized as follows. Section 2 is devoted to general propertiesrelated to homogeneity. After giving the definition of homogeneous approximationwe introduce homogeneous in the bi-limit functions and vector fields (section 2.1)and list some of their properties (section 2.2). Various results concerning stabilityand robustness for homogeneous in the bi-limit vector fields are given in section 2.3.In section 3 we introduce a novel recursive observer design method for a chain ofintegrator. Section 4 is devoted to the homogeneous in the bi-limit state feedback.Finally, in section 5, using the previous tools, we establish new results on stabilizationby output feedback.Notation.• R + denotes the set [0, +∞).• For any nonnegative real number r the function w ↦→ w r is defined as(1.4) w r = sign(w) |w| r ∀ w ∈ R .According to this definition,(1.5)dw rdw = r|w|r−1 ,w 2 = w|w| , (w 1 >w 2 and r>0) ⇒ w r 1 >w r 2 .• The function H : R 2 + → R + is defined as(1.6) H(a, b) = a [1 + b] .1+a• Given r =(r 1 ,...,r n ) T in R n + and λ in R + , λ r ⋄ x =(λ r1 x 1 , . . . , λ rn x n ) T isthe dilation of a vector x in R n with weight r. Note thatλ r 1 ⋄ (λ r 2 ⋄ x) = (λ 1 λ 2 ) r ⋄ x.• Given r =(r 1 ,...,r n ) T in (R + \{0}) n , |x| r = |x 1 | 1r 1 + ··· + |x n | 1rnhomogeneous norm with weight r and degree 1. Note that( ) r |λ r ⋄ x| r = λ |x| r ,1∣ ⋄ x|x| r∣ = 1 .ris theCopyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1816 V. ANDRIEU, L. PRALY, AND A. ASTOLFI• Given r in (R + \{0}) n , S r = {x ∈ R n ||x| r = 1} is the unity homogeneoussphere. Note that each x in R n can be decomposed in polar coordinates; i.e.,there exist λ in R + and θ in S r satisfying(1.7) x = λ r ⋄ θ with{ λ = |x|r(,) r1θ =|x| r⋄ x.2. Homogeneous approximation.2.1. Definitions. The use of homogeneous approximations has a long historyin the study of stability of an equilibrium. It can be traced back to the Lyapunovfirst order approximation theorem and has been pursued by many authors; see, forexample, Massera [16], Hahn [8], Hermes [9], and Rosier [29]. Similarly, this techniquehas been used to investigate the behavior of the solutions of dynamical systems atinfinity; see, for instance, Lefschetz in [14, Chapter IX.5] and Orsi, Praly, and Mareelsin [20]. In this section, we recall the definitions of homogeneous approximation at theorigin and at infinity and restate and/or complete some related results.Definition 2.1 (homogeneity in the 0-limit).• A function φ : R n → R is said to be homogeneous in the 0-limit with associatedtriple (r 0 ,d 0 ,φ 0 ), where r 0 in (R + \{0}) n is the weight, d 0 in R + thedegree, and φ 0 : R n → R the approximating function, if φ is continuous, φ 0is continuous and not identically zero, and, for each compact set C in R n \{0}and each ε> 0, there exists λ 0 such thatmaxφ(λ r0 ⋄ x)x ∈ C∣ λ d0− φ 0 (x)∣ ≤ ε ∀ λ ∈ (0,λ 0] .• A vector field f = ∑ ni=1 f i ∂∂x iis said to be homogeneous in the 0-limit withassociated triple (r 0 , d 0 ,f 0 ), where r 0 in (R + \{0}) n is the weight, d 0 in Ris the degree, and f 0 = ∑ ni=1 f 0,i ∂∂x iis the approximating vector field, if, foreach i in {1,...,n}, d 0 + r 0,i ≥ 0 and the function f i is homogeneous in the0-limit with associated triple (r 0 , d 0 + r 0,i ,f 0,i ).This notion of local approximation of a function or of a vector field can be foundin [9, 29, 2, 10].Example 2.2. The function δ 2 : R → R introduced in the illustrative system (1.1)is homogeneous in the 0-limit with associated triple (r 0 ,d 0 ,δ 2,0 ) = (1, q, c 0 x q 2 ). Furthermore,if q

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.

1818 V. ANDRIEU, L. PRALY, AND A. ASTOLFInot uniquely defined. Indeed, if φ is homogeneous in the 0- (resp., ∞-) limit withassociated triple (r 0 ,d 0 ,φ 0 ) (resp., (r ∞ ,d ∞ ,φ ∞ )), then it is also homogeneous in the0- (resp., ∞-) limit with associated triple (kr 0 , k d 0 ,φ 0 ) (resp., (kr ∞ , k d ∞ ,φ ∞ )) forall k>0. (Simply change λ into λ k .)It is straightforward to show that if φ and ζ are two functions homogeneous inthe 0- (resp., ∞-) limit, with weights r φ,0 and r ζ,0 (resp., r φ,∞ and r ζ,∞ ), degrees d φ,0and d ζ,0 (resp., d φ,∞ and d ζ,∞ ), and approximating functions φ 0 and ζ 0 (resp., φ ∞and ζ ∞ ), then the following hold:P1: If there exists k in R + such that kr φ,0 = r ζ,0 (resp., kr φ,∞ = r ζ,∞ ), thenthe function x ↦→ φ(x) ζ(x) is homogeneous in the 0- (resp., ∞-) limit withweight r ζ,0 , degree kd φ,0 +d ζ,0 (resp., r ζ,∞ , kd φ,∞ +d ζ,∞ ) and approximatingfunction x ↦→ φ 0 (x) ζ 0 (x) (resp., x ↦→ φ ∞ (x) ζ ∞ (x)).dP2: If, for each j in {1,...,n},φ,0r φ,0,j< d ζ,0dr ζ,0,j(resp.,φ,∞r φ,∞,j> d ζ,∞r ζ,∞,j), thenthe function x ↦→ φ(x) +ζ(x) is homogeneous in the 0- (resp., ∞-) limitwith degree d φ,0 and weight r φ,0 (resp., d φ,∞ and r φ,∞ ) and approximatingfunction x ↦→ φ 0 (x) (resp., x ↦→ φ ∞ (x)). In this case we say that the functionφ dominates the function ζ in the 0-limit (resp., in the ∞-limit).P3: If the function φ 0 + ζ 0 (resp., φ ∞ + ζ ∞ ) is not identically zero and, for eachd φ,0r φ,0,j= d ζ,0r ζ,0,jdj in {1,...,n},(resp.,φ,∞r φ,∞,j= d ζ,∞r ζ,∞,j), then the functionx ↦→ φ(x) +ζ(x) is homogeneous in the 0- (resp., ∞-) limit with degreed φ,0 and weight r φ,0 (resp., d φ,∞ and r φ,∞ ) and approximating function x ↦→φ 0 (x) +ζ 0 (x) (resp., x ↦→ φ ∞ (x)+ζ ∞ (x)).Some properties of the composition or inverse of functions are given in the followingtwo propositions, the proofs of which are given in Appendices A and B.Proposition 2.10 (composition function). If φ : R n → R and ζ : R → R arehomogeneous in the 0- (resp., ∞-) limit functions, with weights r φ,0 and r ζ,0 (resp.,r φ,∞ and r ζ,∞ ), degrees d φ,0 > 0 and d ζ,0 ≥ 0 (resp., d φ,∞ > 0 and d ζ,∞ ≥ 0),and approximating functions φ 0 and ζ 0 (resp., φ ∞ and ζ ∞ ), then ζ ◦ φ is homogeneousin the 0- (resp., ∞-) limit with weight r φ,0 (resp., r φ,∞ ), degree d ζ,0 d φ,0r ζ,0(resp.,d ζ,∞ d φ,∞r ζ,∞), and approximating function ζ 0 ◦ φ 0 (resp., ζ ∞ ◦ φ ∞ ).Proposition 2.11 (inverse function). Let φ : R → R be a bijective homogeneousin the 0- (resp., ∞-) limit function with associated triple ( )1,d 0 ,ϕ 0 x d0 with ϕ0 ≠0and d 0 > 0 (resp., ( )1,d ∞ ,ϕ ∞ x d∞ with ϕ∞ ≠0and d ∞ > 0). Then the inversefunction φ −1 : R → R is a homogeneous in the 0- (resp., ∞-) limit function withassociated triple (1, 1 d 0, ( x ϕ 0) 1d 01) (resp., (1,d ∞, ( xϕ ∞) 1d∞ )).Despite the existence of well-known results concerning the derivative of a homogeneousfunction, it is not possible to say anything, in general, when dealing withhomogeneity in the limit. For example, the functionφ(x) =x 3 + x 2 sin(x 2 )+x 3 sin(1/x)+x 2 , x ∈ R ,is homogeneous in the bi-limit with associated triples(1, 2,x2 ) (, 1, 3,x3 ) .However, its derivative is homogeneous in neither the 0-limit nor the ∞-limit. Neverthelessthe following result holds, the proof of which is elementary.Proposition 2.12 (integral function). If the function φ : R n → R is homogeneousin the 0- (resp., ∞-) limit with associated triple (r 0 ,d 0 ,φ 0 ) (resp., (r ∞ ,d ∞ ,φ ∞ )),Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

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.

HOMOGENEOUS OBSERVER DESIGN 1821In the context of homogeneity in the ∞-limit, we have the following result.Proposition 2.18. Consider a homogeneous in the ∞-limit vector field f :R n → R n with associated triple (r ∞ , d ∞ ,f ∞ ). If the origin of the systemẋ = f ∞ (x)is globally asymptotically stable, then there exists an invariant compact subset of R n ,denoted C ∞ , which is globally asymptotically stable 3 for the systemẋ = f(x) .The proof of the proposition is given in Appendix D.As in the case of homogeneity in the 0-limit, this property can be used to designa feedback, ensuring boundedness of solutions.Example 2.19. Consider the system (1.1) with 0 < q < p < 2 and the controllaw(2.6) u = − 1(2 − p x p−12−p1 x 2 − x p2−p1 − c ∞ x p 2 −x 2 + x 12−p1) p.This control law is such that the closed-loop vector field is homogeneous in the ∞-limitwith degree d ∞ = p − 1, weight (2 − p, 1), and associated vector field f ∞ (x 1 ,x 2 )=(x 2 , − 12−p x p−12−p1 x 2 − x p2−p1 −(x 2 + x 12−p1 ) p ) T . For the homogeneous Lyapunov functionof degree 2,V ∞ (x 1 ,x 2 ) = 2 − p2 1|x 1 |2−p + 22 ∣ x 2 + x 12−p1 ∣2,we get∂V ∣ ∣∣∣ ∞∂x (x) f ∞(x) = −|x 1 | p+12−p − x 2 + x 112−p∣p+1.It follows that the control law (2.6) guarantees boundedness of the solutions of theclosed-loop system. Furthermore, boundedness of solutions is preserved in the presenceof any perturbation which does not change the approximating homogeneousfunction in the ∞-limit, i.e., in the presence of perturbations which are negligiblewith respect to the dominant homogeneous part (see P2 in section 2.2).The key step in the proof of Propositions 2.16 and 2.18 is the converse Lyapunovtheorem given by Rosier in [29]. This result can also be extended to the case ofhomogeneity in the bi-limit.Theorem 2.20 (homogeneous in the bi-limit Lyapunov functions). Considera homogeneous in the bi-limit vector field f : R n → R n , with associated triples(r ∞ , d ∞ ,f ∞ ) and (r 0 , d 0 ,f 0 ) such that the origins of the systems(2.7) ẋ = f(x), ẋ = f ∞ (x), ẋ = f 0 (x)are globally asymptotically stable equilibria. Let d V∞ and d V0 be real numbers suchthat d V∞ > max 1≤i≤n r ∞,i and d V0 > max 1≤i≤n r 0,i . Then there exists a C 1 , positivedefinite, and proper function V : R n → R + such that, for each i in {1,...,n},3 See [34] for the definition of global asymptotical stability for invariant compact sets.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1822 V. ANDRIEU, L. PRALY, AND A. ASTOLFI∂Vthe function x ↦→∂x iis homogeneous in the bi-limit with associated triples (r 0 ,d V0 − r 0,i , ∂V0∂x i) and (r ∞ ,d V∞ − r ∞,i , ∂V∞∂x i), and the functions x ↦→ ∂V∂x(x) f(x), x ↦→∂V 0∂x (x) f 0(x), and x ↦→ ∂V∞∂x (x) f ∞(x) are negative definite.The proof is given in Appendix E. A direct consequence of this result is an inputto-statestability (ISS) property with respect to disturbances (see [31]). To illustratethis property, consider the system with exogenous disturbance δ =(δ 1 ,...,δ m ) inR m ,(2.8) ẋ = f(x, δ) ,with f : R n ×R m a continuous vector field homogeneous in the bi-limit with associatedtriples (d 0 , (r 0 , r 0 ),f 0 ) and (d ∞ , (r ∞ , r ∞ ),f ∞ ), where r 0 and r ∞ in (R + \{0}) m arethe weights associated with the disturbance δ.Corollary 2.21 (ISS property). If the origins of the systemsẋ = f(x, 0), ẋ = f 0 (x, 0), ẋ = f ∞ (x, 0)are globally asymptotically stable equilibria, then under the hypotheses of Theorem2.20 the function V given by Theorem 2.20 satisfies, 4 for all δ =(δ 1 ,...,δ m ) in R mand x in R n ,(d∂VV0 +d 0d∂x (x) f(x, δ) ≤ −c V∞)+d∞dV H V (x) V0d, V(x) V∞∑ m (d V0 +d 0d V∞)+d∞r(2.9) + c δ H |δ j | 0,j r, |δ j | ∞,jwhere c V and c δ are positive real numbers.In other words, system (2.8) with δ as input satisfies an ISS property. The proofof this corollary is given in Appendix F.Finally, we have also the following small-gain result for homogeneous in the bilimitvector fields.Corollary 2.22 (small-gain). Under the hypotheses of Corollary 2.21, thereexists a real number c G > 0 such that, for each class K function γ z and KL functionβ δ , there exists a class KL function β x such that, for each function t ∈ [0,T) ↦→(x(t),δ(t),z(t)), T ≤ +∞, with xC 1 and δ and z continuous, which satisfy (2.8) on[0,T) and, for all 0 ≤ s ≤ t ≤ T ,(2.10)(2.11)we have( )|z(t)| ≤ max{β δ |z(s)|,t− s , sups≤κ≤tj=1}γ z (|x(κ)|)( ) { (|δ i (t)| ≤ max{β δ |z(s)|,t− s ,c G sup H |x(κ)|r 0,i)} }r 0, |x(κ)| r∞,ir ∞,s≤κ≤t(2.12) |x(t)| ≤ β x (|(x(s),z(s))|,t− s), 0 ≤ s ≤ t ≤ T.,,4 The function H is defined in (1.6).Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

HOMOGENEOUS OBSERVER DESIGN 1823The proof is given in Appendix G.Example 2.23. An interesting case, which can be dealt with by Corollary 2.22, iswhen the δ i ’s are outputs of auxiliary systems with state z i in R ni , i.e.,(2.13) δ i (t) := δ i (z i (t),x(t)), ż i = g i (z i ,x) .It can be checked that the bounds (2.11) and (2.10) are satisfied by all the solutionsof (2.8) and (2.13) if there exist positive definite and radially unbounded functionsZ i : R ni → R + ; class K functions ω 1 , ω 2 , and ω 3 ; and a positive real number ɛ in(0, 1) such that for all x in R n , for all i in {1,...,m}, and z i in R ni , we have∂Z i|δ i (z i ,x)| ≤ ω 1 (x)+ω 2 (Z i (z i )), (z i ) g i (z i ,x) ≤ −Z i (z i )+ω 3 (|x|) ,∂z iω 1 (x)+ω 2 ([1 + ɛ] ω 3 (|x|)) ≤ c G H ( )|x| r0,ir 0, |x| r∞,ir ∞ .Another important result exploiting Theorem 2.20 deals with finite time convergenceof solutions toward a globally asymptotically stable equilibrium (see [4]). It iswell known that when the origin of the homogeneous approximation in the 0-limitis globally asymptotically stable with a strictly negative degree, then solutions convergeto the origin in finite time (see [3]). We extend this result by showing that if,furthermore, the origin of the homogeneous approximation in the ∞-limit is globallyasymptotically stable with strictly positive degree, then the convergence time doesn’tdepend on the initial condition. This is expressed by the following corollary.Corollary 2.24 (uniform and finite time convergence). Under the hypothesesof Theorem 2.20, if we have d ∞ > 0 > d 0 , then all solutions of the system ẋ = f(x)converge in finite time to the origin, uniformly in the initial condition.The proof is given in Appendix H.3. Recursive observer design for a chain of integrators. The notion ofhomogeneity in the bi-limit is instrumental in introducing a new observer designmethod. Throughout this section we consider a chain of integrators, with state X n =(X 1,...,X n) in R n , namely,(3.1) Ẋ 1 = X 2 , . . . , Ẋ n = u, or in compact form, Ẋ n = S n X n + B n u,where S n is the shift matrix of order n, i.e., S n X n = (X 2,...,X n, 0) T and B n =(0,...,0, 1) T 1. By selecting arbitrary vector field degrees d 0 and d ∞ in (−1,n−1), wesee that, to possibly obtain homogeneity in the bi-limit of the associated vector field,we must choose the weights r 0 =(r 0,1 ,...,r 0,n ) and r ∞ =(r ∞,1 ,...,r ∞,n ) as(3.2)r 0,n = 1 , r 0,i = r 0,i+1 − d 0 = 1 − d 0 (n − i) ,r ∞,n = 1 , r ∞,i = r ∞,i+1 − d ∞ = 1 − d ∞ (n − i) .The goal of this section is to introduce a global homogeneous in the bi-limit observerfor the system (3.1). This design follows a recursive method, which constitutes one ofthe main contributions of this paper.The idea of designing an observer recursively starting from X n and going backwardstowards X 1 is not new. It can be found, for instance, in [28, 26, 23, 30, 35] andin [7, Lemma 6.2.1]. Nevertheless, the procedure we propose is new and extends theresults in [23, Lemmas 1 and 2] to the homogeneous in the bi-limit case.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1824 V. ANDRIEU, L. PRALY, AND A. ASTOLFIAlso, as opposed to what is proposed in [28, 26], 5 this observer is an exact observer(with any input u) for a chain of integrators. The observer is given by the system 6(3.3)˙ˆX n = S n ˆXn + B n u + K 1 (ˆX 1 − X 1) ,with state ˆX n = (ˆX 1,...,ˆX n), and where K 1 : R n → R n is a homogeneous in thebi-limit vector field with weights r 0 , r ∞ and degrees d 0 , d ∞ . The output injectionvector field K 1 has to be selected such that the origin is a globally asymptoticallystable equilibrium for the system(3.4) Ė 1 = S n E 1 + K 1 (e 1 ), E 1 = (e 1 ,...,e n ) T ,and also for its homogeneous approximations. The construction of K 1 is performedvia a recursive procedure whose induction argument is as follows.Consider the system on R n−i given by(3.5) Ė i+1 = S n−i E i+1 + K i+1 (e i+1 ), E i+1 =(e i+1 ,...,e n ) T ,with S n−i the shift matrix of order n − i, i.e., S n−i E i+1 =(e i+2 ,...,e n , 0) T , andK i+1 : R n−i → R n−i a homogeneous in the bi-limit vector field, whose associatedtriples are ((r 0,i+1 ,...,r 0,n ), d 0 ,K i+1,0 ) and ((r ∞,i+1 ,...,r ∞,n ), d ∞ ,K i+1,∞ ).Theorem 3.1 (homogeneous in the bi-limit observer design). Consider the system(3.5) and its homogeneous approximation at infinity and around the origin,Ė i+1 = S n−i E i+1 + K i+1,0 (e i+1 ), Ė i+1 = S n−i E i+1 + K i+1,∞ (e i+1 ) .Suppose the origin is a globally asymptotically stable equilibrium for these systems.Then there exists a homogeneous in the bi-limit vector field K i : R n−i+1 → R n−i+1 ,with associated triples ((r 0,i ,...,r 0,n ), d 0 ,K i,0 ) and ((r ∞,i ,...,r ∞,n ), d ∞ ,K i,∞ ), suchthat the origin is a globally asymptotically stable equilibrium for the systems(3.6)Ė i = S n−i+1 E i + K i (e i ) ,Ė i = S n−i+1 E i + K i,0 (e i ), E i =(e i ,...,e n ) T ,Ė i = S n−i+1 E i + K i,∞ (e i ) .Proof. We prove this result in two steps. First, we define a homogeneous in thebi-limit Lyapunov function. Then we construct the vector field K i , depending on aparameter l which, if sufficiently large, renders negative definite the derivative of thisLyapunov function along the solutions of the system.Step 1. Definition of the Lyapunov function. Let d W0 and d W∞ be positive realnumbers satisfying(3.7) d W0 > 2 max 1≤j≤ n r 0,j + d 0 , d W∞ > 2 max 1≤j≤ n r ∞,j + d ∞ ,and(3.8)d W∞r ∞,i≥ d W 0r 0,i.5 Note the term x i in (3.15) of [28], for instance.6 To simplify the presentation, we use the compact notation K 1 (ˆX 1−X 1) for K 1 (ˆX 1−X 1, 0,...,0).Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

HOMOGENEOUS OBSERVER DESIGN 1825The selection (3.2) implies r 0,j + d 0 > 0 and r ∞,j + d ∞ > 0 for each j in {1,...,n}.Hence,d W0> max1≤j≤ n r 0,j, d W∞ > max1≤j≤ n r ∞,j ,and we can invoke Theorem 2.20 for the system (3.4) and its homogeneous approximationsgiven in (3.5). This implies that there exists a C 1 , positive definite, and properfunction W i+1 : R n−i → R + such that, for each j in {i+1,...,n}, the function ∂Wi+1∂e jis homogeneous in the bi-limit with associated triples((r 0,i+1 ,...,r 0,n ),d W0 − r 0,j , ∂W )i+1,0and∂e j((r ∞,i+1 ,...,r ∞,n ),d W∞ − r ∞,j , ∂W )i+1,∞.∂e jMoreover, for all E i+1 ∈ R n−i \{0}, we have(3.9)(3.10) q i (s) =∂W i+1∂E i+1(E i+1 )(S n−i E i+1 + K i+1 (e i+1 )) < 0 ,∂W i+1,0(E i+1 )(S n−i E i+1 + K i+1,0 (e i+1 )) < 0 ,∂E i+1∂W i+1,∞(E i+1 )(S n−i E i+1 + K i+1,∞ (e i+1 )) < 0 .∂E i+1Consider the function q i : R → R defined as⎧⎨⎩r 0,i +d 0r 0,ir 0,i+d 0sr ∞,ir ∞,i+d ∞sr 0,i, |s| ≤ 1 ,r ∞,i +d∞r ∞,i+ r0,ir 0,i+d 0− r∞,ir ∞,i+d ∞, |s| ≥ 1 .Since we have 0

1826 V. ANDRIEU, L. PRALY, AND A. ASTOLFILet W i: R n−i+1 → R + be defined byW i (E i+1 ,s) = W i+1 (E i+1 )+∫ s−∫ sq −1i (e i+1)q −1i (e i+1)(d W0 −r 0,irh 0,i)d W∞ −r ∞,i+ hr ∞,idh(dq −1W0 −r 0,iri (e i+1 ) 0,i+ q −1d W∞ −r ∞,i)ri (e i+1 ) ∞,idh .This function is C 1 , and by (3.8), Proposition 2.12 yields that it is homogeneous inthe bi-limit with weights (r 0,i+1 ,...,r 0,n ) and (r ∞,i+1 ,...,r ∞,n ) for E i+1 , r 0,i andr ∞,i for s, and degrees d W0 and d W∞ . Furthermore, for each j in {i +1,...,n}, thefunctions ∂Wi∂e j(E i+1 ,s) are also homogeneous in the bi-limit with the same weightsand with degrees d W0 − r 0,j and d W∞ − r ∞,j .Step 2. Construction of the vector field K i . Given a positive real number l, wedefine the vector field K i : R n−i → R n−i as()−qK i (e i )=i (le i ).K i+1 (q i (le i ))By Proposition 2.10 and the properties we have established for q i , K i is a homogeneousin the bi-limit vector field. We show now that selecting l large enough yields theasymptotic stability properties. To begin with, note that for all E i =(E i+1 ,e i ) inR n−i ,∂W i (E i+1 , le i )∂E i(E i )(S n−i+1 E i + K i (e i )) ≤ T 1 (E i+1 , le i ) − lT 2 (E i+1 , le i ) ,with the functions T 1 and T 2 defined asT 1 (E i+1 ,ϑ i )=T 2 (E i+1 ,ϑ i )=∂W i(E i+1 ,ϑ i )(S n−i E i+1 + K i+1 (q i (ϑ i ))) ,∂E i+1(dϑ − q −1W0 −r 0,iri (e i+1 ) 0,id W0 −r 0,ir 0,ii)d W∞ −r ∞,ir+ ϑ∞,ii − q −1d W∞ −r ∞,iri (e i+1 ) ∞,i×(q i (ϑ i ) − e i+1 ) .These functions are homogeneous in the bi-limit with weights (r ∞,i ,...,r ∞,n ) and(r 0,i ,...,r 0,n ), degrees d 0 + d W0 and d ∞ + d W∞ , and continuous approximating functionsT 1,0 (E i+1 ,ϑ i )= ∂W i,0∂E i+1(E i+1 ,ϑ i )(S n−i E i+1 + K i+1,0 (q i,0 (ϑ i ))) ,T 1,∞ (E i+1 ,ϑ i )= ∂W i,∞(E i+1 ,ϑ i )(S n−i E i+1 + K i+1,∞ (q i,∞ (ϑ i ))) ,∂E i+1()dT 2,0 (E i+1 ,ϑ i )= ϑ − q −1i,0 (e W0 −r 0,iri+1) 0,i(q i,0 (ϑ i ) − e i+1 ) ,d W0 −r 0,ir 0,iiandT 2,∞ (E i+1 ,ϑ i )=(ϑd W∞ −r ∞,ir ∞,ii)d W∞ −r ∞,i− q −1i,∞ (e i+1)r ∞,i(q i,∞ (ϑ i ) − e i+1 ) .Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

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

HOMOGENEOUS OBSERVER DESIGN 1829bi-limit stabilizing control law for the origin of the system (4.1), then there is one forthe origin of the system with state X i+1 =(X 1,...,X i+1) in R i+1 defined by(4.2) Ẋ 1 = X 2,...,Ẋ i+1 = u, i.e., Ẋ i+1 = S i+1 X i+1 + B i+1 u.1Let d 0 and d ∞ be in (−1,n−1) and consider the weights and degrees defined in (3.2).Theorem 4.1 (homogeneous in the bi-limit backstepping). Suppose there existsa homogeneous in the bi-limit function φ i : R i → R with associated triples (r 0 , d 0 +r 0,i ,φ i,0 ) and (r ∞ , d ∞ + r ∞,i ,φ i,∞ ) such that the following hold:1. There exists α i ≥ 1 such that the function ψ i (X i ) = φ i (X i ) αi is C 1 andfor each j in {1,...,i} the function ∂ψi is homogeneous in the bi-limit with∂X jweights (r 0,1 ,...,r 0,i ), (r ∞,1 ,...,r ∞,i ), degrees α i (r 0,i + d 0 ) − r 0,j , α i (r ∞,i +d ∞ ) − r ∞,j , and approximating functions ∂ψi0, ∂ψi∞.∂X j ∂X j2. The origin is a globally asymptotically stable equilibrium of the systems(4.3)Ẋ i = S i X i + B i φ i (X i ) , Ẋ i = S i X i + B i φ i,0 (X i ) , Ẋ i = S i X i + B i φ i,∞ (X i ) .Then there exists a homogeneous in the bi-limit function φ i+1 : R i+1 → R withassociated triples (r 0 , d 0 + r 0,i+1 ,φ i+1,0 ) and (r ∞ , d ∞ + r ∞,i+1 ,φ i+1,∞ ) such that thesame properties hold, i.e.,1. there exists a real number α i+1 > 1 such that the function ψ i+1 (X i+1 ) =φ i+1 (X i+1 ) αi+1 is C 1 and for each j in {1,...,i+1} the function ∂ψi+1is∂X jhomogeneous in the bi-limit with weights (r 0,1 ,...,r 0,i+1 ), (r ∞,1 ,...,r ∞,i+1 ),degrees α i+1 (r 0,i+1 + d 0 ) − r 0,j , α i+1 (r ∞,i+1 + d ∞ ) − r ∞,j , and approximatingfunctions ∂ψi+1,0∂X j, ∂ψi+1,∞;∂X j2. the origin is a globally asymptotically stable equilibrium of the systems(4.4)X i+1 = S i+1 X i+1 + B i+1 φ i+1 (X i+1 ) ,X i+1 = S i+1 X i+1 + B i+1 φ i+1,0 (X i+1 ) ,X i+1 = S i+1 X i+1 + B i+1 φ i+1,∞ (X i+1 ) .Proof. We prove this result in three steps. First, we construct a homogeneous inthe bi-limit Lyapunov function; then we define a control law parametrized by a realnumber k. Finally, we show that there exists k such that the time derivative, alongthe trajectories of systems (4.4), of the Lyapunov function and of its approximatingfunctions is negative definite.Step 1. Construction of the Lyapunov function. Let d V0 and d V∞ be positive realnumbers satisfying(4.5) d V0 > maxj∈{1,...,n} {r 0,j}, d V∞ > maxj∈{1,...,n} {r ∞,j} ,and(4.6)d V∞r ∞,i+1≥ d V 0r 0,i+1> 1+α i .With this selection, Theorem 2.20 gives the existence of a C 1 , proper, and positivedefinite function V i : R i → R + such that, for each j in {1,...,n}, the function ∂Vi∂X jCopyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1830 V. ANDRIEU, L. PRALY, AND A. ASTOLFIis homogeneous in the bi-limit with weights (r 0,1 ,...,r 0,i ), (r ∞,1 ,...,r ∞,i ), degreesd V0 − r 0,j , d V∞ − r ∞,j , and approximating functions ∂Vi,0. Moreover, we havefor all X i ∈ R i \{0},∂X j, ∂Vi,∞∂X j(4.7)∂V i∂X i(X i )[S i X i + B i φ i (X i )] < 0 ,∂V i,0(X i )[S i X i + B i φ i,0 (X i )] < 0 ,∂X i∂V i,∞(X i )[S i X i + B i φ i,∞ (X i )] < 0 .∂X iFollowing [21], consider the Lyapunov function V i+1 : R i+1 → R + defined by∫ X i+1(d V0 −r 0,i+1d V0 −r 0,i+1)rV i+1 (X i+1 ) = V i (X i )+ h 0,i+1 r− φ i (X i ) 0,i+1dhφ i(X i)∫ X i+1(d V∞ −r ∞,i+1d V∞ −r ∞,i+1)r+ h ∞,i+1 r− φ i (X i ) ∞,i+1dh .φ i(X i)This function is positive definite and proper. Furthermore, as d V∞ and d V0 satisfy(4.6), we haved V∞ − r ∞,i+1r ∞,i+1≥ d V 0− r 0,i+1r 0,i+1>α i ≥ 1 .Since the function ψ i (X i )=φ i (X i ) αi is C 1 , this inequality yields that the functionV i+1 is C 1 . Finally, for each j in {1,...,n}, the function ∂Vi+1is homogeneous in∂X jthe bi-limit with associated triples((r 0,1 ,...,r 0,i+1 ),d V0 − r 0,j , ∂V ) (i+1,0, (r ∞,1 ,...,r ∞,i+1 ),d V∞ − r ∞,j , ∂V )i+1,∞.ψ i+1∂X jStep 2. Definition of the control law. Recall (1.6) and consider the function: R i+1 → R defined by∫ αXii+1 −φi(Xi)α i (ψ i+1 (X i+1 )=−kH |s| αi+1 d 0 +r 0,i+10α i r 0,i+1−1, |s|α i+1d∞+r ∞,i+1where k in R + is a design parameter and α i+1 is selected as{ }αi r 0,i+1 α i r ∞,i+1α i+1 ≥ max,, 1d 0 + r 0,i+1 d ∞ + r ∞,i+1α i r ∞,i+1.∂X j)−1ds ,ψ i+1 takes values with the same sign as X i+1 − φ i (X i ), is C 1 , and, by Proposition2.12, is homogeneous in the bi-limit. Furthermore, by Proposition 2.10, for each jin {1,...,i +1}, the function ∂ψi+1is homogeneous in the bi-limit, with weights∂X j(r 0,1 ,...,r 0,i+1 ), (r ∞,1 ,...,r ∞,i+1 ), degrees α i+1 (r 0,i+1 + d 0 ) − r 0,j , α i+1 (r ∞,i+1 +d ∞ ) − r ∞,j , and approximating functions ∂ψi+1,0 ∂ψ,i+1,∞. With this at hand, we∂X j ∂X jchoose the control law φ i+1 asφ i+1 (X i+1 ) = ψ i+1 (X i+1 ) 1α i+1 .Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

(4.8)Step 3. Selection of k. Note thatHOMOGENEOUS OBSERVER DESIGN 1831∂V i+1∂X i+1(X i+1 )[S i+1 X i+1 + B i+1 φ i+1 (X i+1 )] = T 1 (X i+1 ) − kT 2 (X i+1 ) ,with the functions T 1 and T 2 defined asT 1 (X i+1 )= ∂V i+1(X i+1 )[S i X i + B i X i+1)]∂X i(T 2 (X i+1 )=Xd V0 −r 0,i+1r 0,i+1i+1 − φ i (X i )d V0 −r 0,i+1r 0,i+1+ Xd V∞ −r ∞,i+1r ∞,i+1i+1 − φ i (X i )d V∞ −r ∞,i+1r ∞,i+1)φ i+1 (X i+1 ) .By the definition of homogeneity in the bi-limit and Proposition 2.10, these functionsare homogeneous in the bi-limit with weights (r 0,1 ,...,r 0,i+1 ) and (r ∞,1 ,...,r ∞,i+1 )and degrees d V0 + d 0 and d V∞ + d ∞ . Moreover, since φ i+1 (X i+1 ) has the same sign asX i+1 − φ i (X i ), T 2 (X i+1 ) is nonnegative for all X i+1 in R i+1 and, as φ i+1 (X i+1 )=0only if X i+1 − φ i (X i ) = 0, we getT 2 (X i+1 ) = 0 =⇒ X i+1 = φ i (X i ) ,X i+1 = φ i (X i ) =⇒ T 1 (X i+1 )= ∂V i∂X i(X i )[S i X i + B i φ i (X i )] .Consequently, equations (4.7) yield{Xi+1 ∈ R i+1 \{0} : T 2 (X i+1 )=0 } ⊆ { X i+1 ∈ R i+1 : T 1 (X i+1 ) < 0 } .The same implication holds for the homogeneous approximations of the two functionsat infinity and around the origin, i.e.,{Xi+1 ∈ R i+1 \{0} : T 2,0 (X i+1 )=0 } ⊆ { X i+1 ∈ R i+1 : T 1,0 (X i+1 ) < 0 } ,{Xi+1 ∈ R i+1 \{0} : T 2,∞ (X i+1 )=0 } ⊆ { X i+1 ∈ R i+1 : T 1,∞ (X i+1 ) < 0 } .Hence, by Lemma 2.13, there exists k ∗ > 0 such that, for all k ≥ k ∗ , we have for allX i+1 ≠ 0,∂V i+1(X i+1 )[S i+1 X i+1 + B i+1 φ i+1 (X i+1 )] < 0 ,∂X i+1∂V i+1,0(X i+1 )[S i+1 X i+1 + B i+1 φ i+1,0 (X i+1 )] < 0 ,∂X i+1∂V i+1,∞(X i+1 )[S i+1 X i+1 + B i+1 φ i+1,∞ (X i+1 )] < 0 .∂X i+1This implies that the origin is a globally asymptotically stable equilibrium of thesystems (4.4).To construct the function φ n it is sufficient to iterate the construction in Theorem4.1 starting fromφ 1 (X 1) = ψ 1 (X 1) 1α 1 , ψ 1 (X 1) =−k 1∫ X 1with k 1 > 0.0()H |s| α1 r 0,2rr −1 α ∞,20,1, |s|1 r −1 ∞,1ds ,Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1832 V. ANDRIEU, L. PRALY, AND A. ASTOLFIAt the end of the recursive procedure, we have that the origin is a globally asymptoticallystable equilibrium of the systems(4.9)X n = S n X n + B n φ n (X n ) ,X n = S n X n + B n φ n,0 (X n ) ,X n = S n X n + B n φ n,∞ (X n ) .Remark 4.2. Note that if d 0 ≥ 0 and d ∞ ≥ 0, then we can select α i = 1 for all1 ≤ i ≤ n, and if d 0 ≤ 0 and d ∞ ≥ d 0 , then we can select α i = r0,1r 0,i+1. Finally, ifd ∞ ≤ 0 and d 0 ≥ d ∞ , then we can select α i =r∞,1r ∞,i+1.Remark 4.3. As in the observer design, when d 0 ≤ d ∞ , we have r0,i+1+d0r 0,i+1≤for i =1,...,n and we can replace the function ψ i by the simpler functionr ∞,i+1+d ∞r ∞,i+1((4.10) ψ i+1 (X i+1 ) = −k|X αii+1 − φ i(X i ) αi | αi+1 d 0 +r 0,i+1α i r 0,i+1+|X αii+1 − φ i(X i ) αi | αi+1 d∞+r ∞,i+1)α i r ∞,i+1Finally, if 0 ≤ d 0 ≤ d ∞ , then by taking α i = 1 (see Remark 4.2) and φ(X i+1 ) =ψ i+1 (X i+1 ) as defined in (4.10), we recover the design in [1].Example 4.4. Consider a chain of integrators of dimension two with weights anddegrees(r 0 , d 0 ) =()(2 − q, 1), q− 1 , (r ∞ , d ∞ ) =()(2 − p, 1), p− 1 ,with 2 > p > q > 0. Given k 1 > 0, using the proposed backstepping procedure weobtain a positive real number k 2 such that the feedback∫ X 1−φ i(X 1)(4.11) φ 2 (X 1, X 2) =−k 2 H ( |s| q−1 , |s| p−1) ds ,0∫ Xwith φ 1 (X 1) =−k11 H(|s| q−1 p−12−q0 , |s|2−p ) ds, renders the origin a globally asymptoticallystable equilibrium of the closed-loop system. Furthermore, as a consequence ofthe robustness result in Corollary 2.22, there is a positive real number c G such that, ifthe positive real numbers |c 0 | and |c ∞ | associated with δ i in (1.2) are smaller than c G ,then the control law φ 2 globally asymptotically stabilizes the origin of system (1.1).5. Application to nonlinear output feedback design.5.1. Results on output feedback. The tools presented in the previous sectionscan be used to derive two new results on stabilization by output feedback for theorigin of nonlinear systems. The output feedback is designed for a simple chain ofintegrators,(5.1) ẋ = S n x + B n u, y = x 1 ,where x is in R n , y is the output in R, and u is the control input in R. It is thenshown to be adequate to solve the output feedback stabilization problem for the originof systems for which this chain of integrators can be considered as the dominant partof the dynamics..Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

HOMOGENEOUS OBSERVER DESIGN 1833Such a domination approach has a long history. It is the cornerstone of the resultsin [13] (see also [27] and [24]), where a linear controller was introduced to deal withnonlinear systems. This approach has also been followed with nonlinear controllers in[22] and more recently in combination with weighted homogeneity in [35, 26, 28] andthe references therein.In the context of homogeneity in the bi-limit, we use this approach exploiting theproposed backstepping and recursive observer designs. Following the idea introducedby Qian in [26] (see also [27]), the output feedback we proposed is given by(5.2)˙ˆX n = L()S n ˆXn + B n φ n (ˆX n )+K 1 (x 1 − ˆX 1) , u = L n φ n (ˆX n ) ,with ˆX n in R n and where φ n and K 1 are continuous functions and L is a positivereal number. Employing the recursive procedure given in sections 3 and 4, we get thefollowing theorem, whose proof is in section 5.2.1Theorem 5.1. For all real numbers d 0 and d ∞ in (−1,n−1), there exists ahomogeneous in the bi-limit function φ n : R n → R with associated triples (r 0 , 1+d 0 ,φ n,0 ) and (r ∞ , 1+d ∞ ,φ n,∞ ) and a homogeneous in the bi-limit vector field K 1 :R n → R n with associated triples (r 0 , d 0 ,K 1,0 ) and (r ∞ , d ∞ ,K 1,∞ ) such that for allreal numbers L>0 the origin is a globally asymptotically stable equilibrium of thesystems (5.1) and (5.2) and their homogeneous approximations.We can then apply Corollary 2.22 to get an output feedback result for nonlinearsystems described by(5.3) ẋ = S n x + B n u + δ(t), y = x 1 ,where δ : R + → R n is a continuous function related to the solutions as described inthe two corollaries below and proved in section 5.2. Depending on whether d 0 ≤ d ∞or d ∞ ≤ d 0 , we get an output feedback result for systems in feedback or feedforwardform.Corollary 5.2 (feedback form). If, in the design of φ n and K 1 , we selectd 0 ≤ d ∞ , then for all positive real numbers c 0 and c ∞ there exists a real numberL ∗ > 0 such that for every L in [L ∗ , +∞), the following holds:• For every class K function γ z and class KL function β δ , we can find two classKL functions β x and βˆx such that, for each function t ∈ [0,T) ↦→ (x(t), ˆX n (t),δ(t),z(t)),T ≤ +∞, with (x, ˆX n ) C 1 and δ and z continuous, which satisfies (5.3), (5.2), andfor i in {1,...,n} and 0 ≤ s ≤ t

1834 V. ANDRIEU, L. PRALY, AND A. ASTOLFICorollary 5.3 (feedforward form). If, in the design of φ n and K 1 , we selectd ∞ ≤ d 0 , then for all positive real numbers c 0 and c ∞ there exists a real numberL ∗ > 0 such that for every L in (0,L ∗ ], the following holds:• For every class K function γ z and class KL function β δ , we can find two classKL functions β x and βˆx such that, for each function t ∈ [0,T) ↦→ (x(t), ˆX n (t),δ(t),z(t)), T ≤ +∞, with (x, ˆX n ) C 1 and δ and z continuous, which satisfies (5.3), (5.2),and for i in {1,...,n} and 0 ≤ s ≤ t 0 such that for all L in [L ∗ , +∞), the output feedback(5.2) is globally asymptotically stabilizing. Compared to already published results (see[13] and [26], for instance), the novelty of this case is in the simultaneous presence ofthe terms |x j | 1−d 0 (n−i−1)1−d 0 (n−j)and |x j | 1−d∞(n−i−1)1−d∞(n−j).On the other hand, if there exist two real numbers d 0 and d ∞ satisfying −1 0 such that for all L in (0,L ∗ ], the output feedback(5.2) is globally asymptotically stabilizing.⎞⎠ ,⎞⎠ ,Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

HOMOGENEOUS OBSERVER DESIGN 1835Example 5.5. Consider the illustrative system (1.1). The bound (5.6) gives thecondition(5.7) 0 < q < p < 2 .This is almost the least conservative condition we can obtain with the dominationapproach. Specifically, it is shown in [18] that, when p>2, there is no stabilizing outputfeedback. However, when p = 2, (5.6) is not satisfied, although the stabilizationproblem is solvable (see [18]).By Corollary 2.24, when (5.7) holds, the output feedback⎧˙ˆX 1 = L ˆX 2 − Lq ⎪⎨1 (l 1 e 1 ) ,u = L 2 φ 2 (ˆX 1, ˆX 2), ˙ˆX 2 = u L − Lq 2(l 2 q 1 (l 1 e 1 )) ,⎪⎩e 1 = ˆX 1 − y,with l 1 , l 2 , φ 2 , q 1 , and q 2 defined in (3.13) and (4.11) and with picking d 0 in (−1,q−1]and d ∞ in [p − 1, 1), globally asymptotically stabilizes the origin of the system (1.1),with L chosen sufficiently large. Furthermore, if d 0 is chosen strictly negative and d ∞strictly positive, by Corollary 2.24, convergence to the origin occurs in finite time,uniformly in the initial conditions.Example 5.6. To illustrate the feedforward result consider the system 7ẋ 1 = x 2 + x 3 23 + z 3 , ẋ 2 = x 3 , ẋ 3 = u, ż = −z 4 + x 3 , y = x 1 .For any ε> 0, there exists a class KL function β δ such that{}|z(t)| 3 ≤ max β δ (|z(s)|,t− s), (1 + ε) sups≤κ≤t|x 3 (κ)| 3 4 .Therefore by letting δ 1 = x 3 23 +z 3 we get, for all 0 ≤ s ≤ t

1836 V. ANDRIEU, L. PRALY, AND A. ASTOLFIThis yields(5.9)⎧⎪⎨⎪⎩ddτ ̂X n = S n ̂Xn + B n φ n (ˆX n )) + K 1 (e 1 ) ,ddτ E 1 = S n E 1 + K 1 (e 1 )with E 1 =(e 1 ,...,e n ), ̂X n =(ˆX 1,...,ˆX n). The right-hand side of (5.9) is a vectorfield which is homogeneous in the bi-limit with weights (r 0 ,r 0 ), (r ∞ ,r ∞ ).Given d U > max j {r 0,j ,r ∞,j }, by applying Theorem 2.20 twice, we get two C 1 ,proper, and positive definite functions V : R n → R + and W : R n → R + such thatfor each i in {1,...,n}, the functions ∂V∂x iand ∂W∂e iare homogeneous in the bi-limit,with weights r 0 and r ∞ , degrees d U −r 0,i and d U −r ∞,i , and approximating functions∂V 0∂ ˆX j , ∂V∞∂ ˆX j(5.10)and ∂W0∂e jand for all E 1 ≠ 0,(5.11), ∂W∞∂e j. Moreover, for all ̂X n ≠ 0,∂V∂ ̂X (̂X n )∂V 0∂ ̂X(̂X n )n∂V ∞∂ ̂X(̂X n )n[]S n ̂Xn + B n φ n (̂X n ) < 0 ,[S n ̂Xn + B n φ n,0 (̂X n )]< 0 ,[]S n ̂Xn + B n φ n,∞ (̂X n ) < 0 ,∂W(E 1 )(S n E 1 + K 1 (e 1 )) < 0 ,∂E 1∂W 0(E 1 )(S n E 1 + K 1,0 (e 1 )) < 0 ,∂E 1∂W ∞(E 1 )(S n E 1 + K 1,∞ (e 1 )) < 0 .∂E 1Consider now the Lyapunov function candidate(5.12) U(ˆX n ,E 1 ) = V (ˆX n )+c W (E 1 ) ,where c is a positive real number to be specified. Letη(ˆX n ,E 1 )= ∂V ()∂ ̂X(̂X n ) S n ˆXn + B n φ n (ˆX n )+K 1 (e 1 )nγ(E 1 )=− ∂W∂E 1(E 1 )(S n E 1 + K 1 (e 1 )) .These two functions are continuous and homogeneous in the bi-limit with associatedtriples ((r 0 ,r 0 ),d U + d 0 ,η 0 ), ((r ∞ ,r ∞ ),d U + d ∞ ,η ∞ ) and ((r 0 ,r 0 ),d U + d 0 ,γ 0 ),((r ∞ ,r ∞ ),d U + d ∞ ,γ ∞ ), where γ 0 , γ ∞ and η 0 , η ∞ are continuous functions. Furthermore,by (5.11), γ(E 1 ) is negative definite. Hence, by (5.10), we have{} {}(ˆX n ,E 1 ) ∈ R 2n \{0} : γ(E 1 )=0 ⊆ (ˆX n ,E 1 ) ∈ R 2n : η(ˆX n ,E 1 ) < 0 ,{} {}(ˆX n ,E 1 ) ∈ R 2n \{0} : γ 0 (E 1 )=0 ⊆{}(ˆX n ,E 1 ) ∈ R 2n \{0} : γ ∞ (E 1 )=0 ⊆(ˆX n ,E 1 ) ∈ R 2n : η 0 (ˆX n ,E 1 ) < 0 ,{}(ˆX n ,E 1 ) ∈ R 2n : η ∞ (ˆX n ,E 1 ) < 0 .,Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

HOMOGENEOUS OBSERVER DESIGN 1837Consequently, by Lemma 2.13, there exists a positive real number c ∗ such that, for allc > c ∗ and all (ˆX n ,E 1 ) ≠ (0, 0), the Lyapunov function U, defined in (5.12), satisfies∂U()∂ ˆX(ˆX n ,E 1 ) S n ˆXn + B n φ n (ˆX n )+K 1 (e 1 )n+ ∂U∂E 1(ˆX n ,E 1 )(E 1 )(S n E 1 + K 1 (e 1 )) < 0and the same holds for the homogeneous approximations in the 0-limit and in the∞-limit; hence the claim.Proof of Corollary 5.2. We write the dynamics of the system (5.3) in the coordinatesˆX n and E 1 and in the time τ given in (5.8). This yields(5.13)with⎧⎪⎨⎪⎩ddτ ̂X n = S n ̂Xn + B n φ n (ˆX n )) + K 1 (e 1 ),ddτ E 1 = S n E 1 + K 1 (e 1 )+D(L)(δ1D(L) =L ,..., δ )nL n .We denote the solution of this system, starting from (̂X n (0),E 1 (0)) in R 2n at time τ,by (̂X τ,n (τ),E τ,1 (τ)). We have(5.14) x i (t) = L i−1 (ˆX τ,i (Lt) − e τ,i (Lt)) .The right-hand side of (5.13) is a vector field which is homogeneous in the bi-limitwith weights (r 0 ,r 0 ), (r ∞ ,r ∞ ) for (̂X n ,E 1 ) and (r 0 , r ∞ ) for D(L), where r 0,i = r 0,i +d 0and r ∞,i = r ∞,i + d ∞ for each i in {1,...,n}.The time function τ ↦→ δ( τ L) is considered as an input, and when D(L) = 0,Theorem 5.1 implies global asymptotic stability of the origin of the system (5.13)and of its homogeneous approximations. To complete the proof we show that thereexists L ∗ such that the “input” D(L) satisfies the small-gain condition (2.11) of Corollary2.22 for all L>L ∗ . Using (5.8) and (5.14), assumption (5.4) becomes, for all0 ≤ σ ≤ τ < LT and all i in {1,...,n},∣ δi( τL)∣ ∣L i{ (1 ∣∣∣z ( σ)∣ )∣∣≤ maxL i β τ − σδ , ,L L{i∑L −i ∣sup σ≤κ≤τ ∣L (j−1) (ˆX τ,j(κ) − e τ,j (κ)) ∣c 0j=11−d 0 (n−i−1)1−d 0 (n−j)i∑∣(5.15) + c ∞ ∣L (j−1) (ˆX τ,j(κ) − e τ,j (κ)) ∣Note that when 1 ≤ j ≤ i ≤ n, the function s ↦→ 1−(n−i−1) s1−(n−j) sis strictly increas-n−i) in (n+1−j , ij−1 ). As d 0 ≤ d ∞ < 1n−1, we have for all1ing, mapping (−1,1 ≤ j ≤ i ≤ n,n−1j=11−d∞(n−i−1)1−d∞(n−j)}}1 − d 0 (n − i − 1)1 − d 0 (n − j)≤ 1 − d ∞(n − i − 1)1 − d ∞ (n − j)

1838 V. ANDRIEU, L. PRALY, AND A. ASTOLFIHence, selecting L ≥ 1, there exists a real number ɛ> 0 such thatL −ɛThis implies∣ ( {∣δ τ i ∣(L)∣1 ∣∣∣z ( σ)∣ ∣∣L i ≤ maxL i β τ − σδ ,L L{i∑L −ɛ sup σ≤κ≤τOn the other hand, the function(̂X n ,E 1 ) ↦→ c 01−d∞(n−i−1)(j−1)≥ L1−d∞(n−j) −i ≥ L (j−1) 1−d 0 (n−i−1)1−d 0 (n−j) −i .i∑j=1c 0j=1),|(ˆX τ,j(κ) − e τ,j (κ))| 1−d 0 (n−i−1)1−d 0 (n−j)+ c ∞i∑j=1|ˆX j − e j | 1−d 0 (n−i−1)1−d 0 (n−j)+ c ∞i∑|(ˆX τ,j(κ) − e τ,j (κ))| 1−d∞(n−i−1)1−d∞(n−j)j=1|ˆX j − e j | 1−d∞(n−i−1)1−d∞(n−j)}}is homogeneous in the bi-limit with weights (r 0 ,r 0 ) and (r ∞ ,r ∞ ) and degrees 1 −d 0 (n − i − 1) = r 0,i + d 0 and 1 − d ∞ (n − i − 1) = r ∞,i + d ∞ (see (3.2)). Hence, byCorollary 2.15, there exists a positive real number c 1 such thati∑c 0j=1|ˆX j − e j | 1−d 0 (n−i−1)1−d 0 (n−j)+ c ∞i∑j=1(5.16) ≤ c 1 H|ˆX j − e j | 1−d∞(n−i−1)1−d∞(n−j)()|(̂X n ,E 1 )| d0+r0,i(r , |(̂X 0,r 0) n ,E 1 )| d∞+r∞,i(r ∞,r ∞).Hence, by Corollary 2.22 (applied in the τ time-scale), there exists c G such that forany L ∗ large enough such that c 1 L ∗−ε ≤ c G , the conclusion holds.Proof of Corollary 5.3. The proof is similar to the previous one with the onlydifference being that, when i and j satisfy 3 ≤ i +2 ≤ j ≤ n, the function s ↦→1−(n−i−1) s1−(n−j) sis strictly decreasing, mapping (−1,1n−1 ) in ( icondition −1 < d ∞ ≤ d 0 < 1n−11 − d ∞ (n − i − 1)1 − d ∞ (n − j)gives the inequalities≥ 1 − d 0(n − i − 1)1 − d 0 (n − j)j−1 ,>n−in+1−jij − 1 .). Moreover theHence (5.16) holds, and by selecting L

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.

HOMOGENEOUS OBSERVER DESIGN 1841Appendix C. Proof of Lemma 2.13. The proof of this lemma is divided intothree parts.1. We first show, by contradiction, that there exists a real number c 0 satisfyingη 0 (θ) − cγ 0 (θ) < 0 ∀ θ ∈ S r0 , ∀ c ≥ c 0 .Suppose there is no such c 0 . This means there is a sequence (θ i ) i∈N in S r0which satisfiesη 0 (θ i ) − iγ 0 (θ i ) ≥ 0 ∀ i ∈ N .The sequence (θ i ) i∈N lives in a compact set. Thus we can extract a convergentsubsequence (θ il ) l∈N which converges to a point denoted θ ∞ .As the functions η 0 and γ 0 are bounded on S r0 and γ 0 takes nonnegativevalues, 8 γ 0 (θ il ) must go to 0 as i l goes to infinity. Since the functions η 0 andγ 0 are continuous, we get γ 0 (θ ∞ ) = 0 and η 0 (θ ∞ ) ≥ 0, which is impossible.Consequently, there exist c 0 and ε 0 > 0 such that(C.1) η 0 (θ) − cγ 0 (θ) ≤ −ε 0 < 0 ∀ θ ∈ S r0 , ∀ c ≥ c 0 .Moreover, since the functions η 0 and γ 0 are homogeneous in the standardsense (see Remark 2.6), we have the second inequality in (2.4).Following the same argument, we can find positive real numbers c ∞ and ε ∞such that(C.2) η ∞ (θ) − cγ ∞ (θ) < −ε ∞ ∀ θ ∈ S r∞ , ∀ c ≥ c ∞ ,and the third inequality in (2.4) holds.In the rest of the proof, letc 1 = max{c 0 ,c ∞ }, ε 1 = min{ε 0 ,ε ∞ } .2. Since η and γ are homogeneous in the 0-limit, there exists λ 0 such that, forall λ ∈ (0,λ 0 ] and all θ ∈ S r0 , we haveη(λ r0 ⋄ θ) ≤ λ d0 η 0 (θ) +λ d0 ε 14 , λd0 γ 0 (θ) − λ d0 ε 14c 1≤ γ(λ r0 ⋄ θ) ,which readily givesη(λ r0 ⋄ θ) − c 1 γ(λ r0 ⋄ θ) ≤ λ d0 η 0 (θ) +λ d0 ε 12 − c 1λ d0 γ 0 (θ) .Using (C.1), we getη(λ r0 ⋄ θ) − c 1 γ(λ r0 ⋄ θ) ≤ −λ d0 ε 12and therefore, since γ takes nonnegative values,∀ λ ∈ (0,λ 0 ] , ∀ θ ∈ S r0 ,η(λ r0 ⋄ θ) − cγ(λ r0 ⋄ θ) ≤−λ d0 ε 12∀ λ ∈ (0,λ 0 ] , ∀ θ ∈ S r0 , ∀ c ≥ c 1 .8 Indeed, if we had γ 0 (x) < 0 for some x in R n \{0}, by letting ɛ = − γ 0(x), the homogeneity in2the 0-limit of γ would give a real number λ> 0 satisfying γ(λr 0 ⋄x)λ d ≤ γ0 0 (x) +ɛ = γ 0(x)< 0. This2contradicts the fact that γ takes nonnegative values only. Also by continuity we have γ 0 (0) ≥ 0.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1842 V. ANDRIEU, L. PRALY, AND A. ASTOLFISimilarly, there exists λ ∞ satisfyingη(λ r∞ ⋄θ)−cγ(λ r∞ ⋄θ) ≤−λ d∞ ε 12Consequently, for each c ≥ c 1 , the setif not empty, must be a subset of∀ λ ∈ [λ ∞ , +∞) , ∀ θ ∈ S r∞ , ∀ c ≥ c 1 .{x ∈ R n \{0} |η(x) − cγ(x) ≥ 0} ,C = {x ∈ R n : |x| r0 ≥ λ 0 } ⋃ {x ∈ R n : |x| r∞ ≤ λ ∞ } ,which is compact and does not contain the origin.3. Suppose now that for all c the first inequality in (2.4) is not true. This meansthat, for all integers c larger than c 1 , there exists x c in R n satisfyingη(x c ) − cγ(x c ) ≥ 0 ,and therefore x c is in C. Since C is a compact set, there is a convergentsubsequence (x cl ) l∈N which converges to a point denoted x ∗ different fromzero. Also as above, we must have γ(x ∗ ) = 0 and η(x ∗ ) ≥ 0. But thiscontradicts the assumption, namely,{ x ∈ R n \{0} , γ(x) =0} ⇒ η(x) < 0 .Appendix D. Proof of Proposition 2.18. Because the vector field f is homogeneousin the ∞-limit, its approximating vector field f ∞ is homogeneous in thestandard sense (see Remark 2.6). Let d V∞ be a positive real number larger thanr ∞,i for all i in {1,...,n}. Following Rosier [29], there exists a C 1 , positive definite,proper, and homogeneous function V ∞ : R n → R + , with weight r ∞ and degree d V∞ ,satisfying(D.1)∂V ∞∂x (x)f ∞(x) < 0 ∀ x ≠ 0 .From P1 in section 2.2, we know that the function x ↦→ ∂V∞∂x(x)f(x) is homogeneousin the ∞-limit with associated triple ( r ∞ , d ∞ + d V∞ , ∂V∞∂x (x)f ∞(x) ) . Letɛ ∞ = − 1 2 maxθ ∈ S r∞{ ∂V∞∂x (θ)f ∞(θ)and note that, by inequality (D.1), ɛ ∞ is a strictly positive real number. By thedefinition of homogeneity in the ∞-limit, there exists λ ∞ such that∂V ∞∂x(λ r∞ ⋄ θ)f(λ r∞ ⋄ θ)∣ λ d − ∂V ∞V∞ +d∞∂x (θ)f ∞(θ)∣ ≤ ɛ ∞ ∀ θ ∈ S r∞ , ∀ λ ≥ λ ∞ .This yields∂V ∞∂x (λr∞ ⋄ θ)f(λ r∞ ⋄ θ) ≤ λ d V∞ +d∞ ( ∂V∞∂x (θ)f ∞(θ) +ɛ ∞)≤−λ d V∞ +d∞ ɛ ∞ ∀ θ ∈ S r∞ , ∀ λ ≥ λ ∞ ,},Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

HOMOGENEOUS OBSERVER DESIGN 1843or in other words,(D.2)∂V ∞∂x (x) f(x) < 0 ∀ x : |x| r ∞≥ λ ∞ .This establishes global asymptotic stability of the compact setwhere v ∞ is given byC ∞ = {x : V ∞ (x) ≤ v ∞ } ,v ∞ = max|x| r∞ = λ ∞{V ∞ (x)} .Appendix E. Proof of Theorem 2.20. The proof is 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 of the system (2.7) and its homogeneous approximations are negativedefinite.1. As established in the proof of Proposition 2.18, there exist a positive realnumber λ ∞ and a C 1 positive definite, proper, and homogeneous functionV ∞ : R n → R + , with weight r ∞ and degree d V∞ satisfying (D.2). Similarly,there exist 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 , satisfying(E.1)∂V 0∂x (x) f(x) < 0 ∀ x : 0 < |x| r 0≤ λ 0 .Finally, the global asymptotic stability of the origin of the system ẋ = f(x)implies the existence of a C 1 , positive definite, and proper function V m :R n → R + satisfying(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 this, wefollow a technique used by 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 satisfying(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 is C 1 , positive definite, and proper. Moreover, by 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, this proves that∂Vfor each i in {1,...,n},∂x iis 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 of V along f is negative definite.To this 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.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

HOMOGENEOUS OBSERVER DESIGN 1845Appendix F. Proof of Corollary 2.21. Recall (1.6) and consider the functionsη 1 : R n × R m → R and γ 1 : R n × R m → R + defined asη 1 (x, δ) = ∂V∂x (x) [f(x, δ) − 1 2 f(x, 0) ], γ 1 (x, δ) =m∑j=1(d V0 +d 0 d V∞)+d∞rH |δ j | 0,j r, |δ j | ∞,j.These functions are homogeneous in the bi-limit with weights r 0 and r ∞ for x and r 0and r ∞ for δ and degrees d V0 +d 0 and d V∞ +d ∞ . Since the function x ↦→ ∂V∂x(x) f(x, 0)is negative definite, then{(x, δ) ∈ R n+m \{0} : γ 1 (x, δ) = 0} ⊆{ (x, δ) ∈ R n+m : η 1 (x, δ) < 0} .Moreover, since the homogeneous approximations of η are negative definite, we get{(x, δ) ∈ R n+m \{0} : γ 1,0 (x, δ) = 0} ⊆{ (x, δ) ∈ R n+m : η 1,0 (x, δ) < 0} ,{(x, δ) ∈ R n+m \{0} : γ 1,∞ (x, δ) = 0} ⊆{ (x, δ) ∈ R n+m : η 1,∞ (x, δ) < 0} .Hence, by Lemma 2.13, there exists a positive real number c δ such that(F.1)[∂V∂x (x) f(x, δ) − 1 ]2 f(x, 0)≤ c δm∑j=1(d V0 +d 0 d V∞)+d∞rH |δ j | 0,j r, |δ j | ∞,jConsider now the functions η 2 : R n → R + and γ 2 : R n → R + defined as(d V0 +d 0d V∞)+d∞dη 2 (x) =H V (x) V0d,V(x) V∞, γ 2 (x) =− 1 ∂V(x) f(x, 0) .2 ∂xThey are homogeneous in the bi-limit with weights r 0 and r ∞ and degrees d V0 + d 0and d V∞ + d ∞ . Since γ 2 and its homogeneous approximations are positive definite,by Corollary 2.15 there exists a positive real number c V such that(d1 ∂VV0 +d 0d(F.2)2 ∂x (x) f(x, 0) ≤ −c V∞)+d∞dV H V (x) V0d,V(x) V∞.The two inequalities (F.1) and (F.2) yield the claim.Appendix G. Proof of Corollary 2.22. Let d V0 and d V∞ be such that theassumption of Theorem 2.20 holds. For each i in {1,...,m}, let µ i : R + → R + bethe strictly increasing function defined as (see (1.6))(G.1) µ i (s) = H (s qi ,s pi ) ,wherep i = d ∞ + d V∞r ∞,i, q i = d 0 + d V0r 0,i.We first prove that the inequality given by Corollary 2.21 implies that the system(2.8), with δ as input and x as output, is ISS with a linear gain between ∑ mi=1 µ i(|δ i |)and H(|x| d0+d V 0r 0, |x| d∞+d V∞r ∞α(s) = H). To do so we introduce the function α : R + → R + as(d 0 +d V0 d∞+d V∞)ds V0 d,s V∞, s ≥ 0 ..Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1846 V. ANDRIEU, L. PRALY, AND A. ASTOLFIThis function is a bijection, strictly increasing, and homogeneous in the bi-limit withd V0 +d 0d V0d V∞ +d∞dapproximating functions s and s V∞. Moreover, from Proposition 2.10,the function x ↦→ α(V (x)) is positive definite and homogeneous in the bi-limit withassociated weights r 0 and r ∞ and degrees d 0 + d V0 and d ∞ + d V∞ . Moreover, itsd V0 +d 0d V0d V∞ +d∞d V∞approximating homogeneous functions V 0 (x) and V ∞ (x) are positivedefinite as well. Hence, we get from Corollary 2.15 the existence of a positive realnumber c 1 satisfying()(G.2) H |x| d0+d V 0r 0, |x| d∞+d V∞r ∞≤ c 1 α(V (x)) ∀ x ∈ R n .On the other hand, from inequality (2.9) in Corollary 2.21, we have the property{}(x, δ) ∈ R n × R m(G.3): α(V (x)) ≥ 2 c δc Vm∑µ i (|δ i |){i=1⊆ (x, δ) ∈ R n × R m :∂V∂x (x) f(x, δ) ≤ −c V2 α(V (x)) }In the following, let t ∈ [0,T) ↦→ (x(t),δ(t),z(t)) be any function which satisfies (2.8)on [0,T) and (2.10) and (2.11) for all 0 ≤ s ≤ t ≤ T . From [32], we know the inclusion(G.3) implies the existence of a class KL function β V such that, for all 0 ≤ s ≤ t ≤ T ,(G.4)⎧⎨V (x(t)) ≤ max⎩ β V (V (x(s)),t− s) ,sups≤κ≤t⎧ ⎛⎨⎩ α−1 ⎝ 2c δc Vm∑j=1⎞⎫⎫⎬⎬µ j (|δ j (κ)|) ⎠⎭⎭ .With α acting on both sides of inequality (G.4), (G.2) gives, for all 0 ≤ s ≤ t ≤ T ,()H |x(t)| d0+d V 0r 0, |x(t)| d∞+d V∞r ∞⎧⎧⎫⎫⎨(G.5) ≤ max⎩ c 1 α ◦ β V (V (x(s)),t− s) , 2c ⎨1c δm∑ ⎬⎬sup µc V ⎩ j (|δ j (κ)|)⎭⎭ .s≤κ≤tThis is the linear gain property required. To conclude the proof it remains to showthe existence of c G such that a small gain property is satisfied.First, note that the function x ↦→ H(|x| d0+d V 0r 0, |x| d∞+d V∞r ∞) is positive definiteand homogeneous in the bi-limit with weights r 0 and r ∞ and degrees d 0 + d V0 andd ∞ + d V∞( . ( By Proposition 2.10, for i in {1,...,m} the same holds with the functionrx ↦→ µ i H |x|0,i))r 0, |x| r∞,ir ∞ . Hence, by Corollary 2.15, there exists a positive realnumber c 2 satisfying( ((G.6) µ i H |x|r 0,i)) ()r 0, |x| r∞,ir ∞ ≤ c2 H |x| d0+d V 0r 0, |x| d∞+d V∞r ∞∀ x ∈ R n .Let C i for i in {1,...,m} be the class K ∞ functions defined asC i (c) = max{c qi ,c pi } + c p i q iq i +p i + c pi+qi .From (G.1), we get, for each s>0 and c>0,µ i (cs)µ i (s)= c (1 + [ qi sqi )(1 + c pi s pi ) 1+cp(1 + s pi )(1 + c qi s qi ) ≤ i]s pi+qicqi1+c qi s + s qipi+qi 1+c qi s + s cpi piqi+pi 1+s pij=1.,Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

HOMOGENEOUS OBSERVER DESIGN 1847wherec qi 1+cpi s pi+qi1+c qi s pi+qi ≤ max{cqi ,c pi } ,Hence, by continuity at 0, we havec qi s qi1+c qi s qi+pi ≤ c p i q iq i +p i ,c qi c pi s pi1+s pi ≤ c pi+qi .(G.7) µ i (cs) ≤ C i (c) µ i (s) ∀ (c, s) ∈ R 2 + .Consider the positive real numbers c 1 , c 2 , c δ , and c V previously introduced, and selectc G in R + satisfying( )c V(G.8) c G < min1≤i≤m C−1 i.2 mc 1 c 2 c δTo show that such a selection for c G is appropriate, observe that by (G.6) and (G.7)and µ i acting on both sides of the inequality (2.11), we get for each i in {1,...,m}and all 0 ≤ s ≤ t ≤ T ,µ i (|δ i (t)|) ≤ max{µ i ◦ β δ (|z(s)|,t− s) ,C i (c G ) c 2{ (sup Hs≤κ≤tConsequentlym∑{µ i (|δ i (t)|) ≤ max m max {µ i ◦ β δ (|z(s)|,t− s)} ,1≤i≤mi=1{ ((G.9) (m max 1≤i≤m C i (c G ) c 2 ) sup s≤κ≤t HSince (G.8) yields2c 1 c δc V|x(κ)| d0+d V 0r 0|x(κ)| d0+d V 0r 0m max1≤i≤m C i(c G ) c 2 < 1 ,)} }, |x(κ)| d∞+d V∞r ∞.)} }, |x(κ)| d∞+d V∞r ∞.the existence of the function β x follows from (2.10), (G.5), (G.9), and the (proof ofthe) small-gain theorem [11].Appendix H. Proof of Corollary 2.24. First, observe that the continuity off 0 , at least, on R n \{0} implies|d 0 | = −d 0 ≤ min1≤i≤n r 0,i ≤ max1≤i≤n r 0,i < d V0 .Then, let V be the function given in Theorem 2.20 and, since d 0 < 0 < d ∞ , the functionφ(x) =V (x)d V0 +d 0d V0+ V (x)d V∞ +d∞d V∞is homogeneous in the bi-limit with weightsr 0 and r ∞ , degrees d V0 + d 0 and d V∞ + d ∞ , and approximating functions V (x)d V∞ +d∞d V∞d V0 +d 0d V0and V (x) . Moreover, the function ζ(x) = − ∂V∂x(x) f(x) is homogeneous inthe bi-limit with the same weights and degrees as φ. Furthermore, since the functionζ and its homogeneous approximations are positive definite, Corollary 2.15 yields astrictly positive real number c such that(H.1)(d∂VV0 +d 0∂x (x) f(x) ≤ −c dV (x) V0d V∞)+d∞d+ V (x) V∞∀ x ∈ R n .Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1848 V. ANDRIEU, L. PRALY, AND A. ASTOLFILet x ic in R n \{0} be the initial condition of a solution of the system ẋ = f(x), andlet V xic : R + → R + be the function of time given by the evaluation of V along thissolution. Thenfrom which we get{ ˙ {d V∞ +d∞dV xic (t) ≤ −cV xic (t) V∞∀ t ≥ 0 ,V xic (t) ≤(d ∞d V∞1) dct + V (x ic ) − d∞ V∞d∞d V∞≤(d ∞d V∞1ct) d V∞d∞∀ t>0 .Therefore, setting T 1= d V∞cd ∞, we haveV xic (t) ≤ 1 ∀ t ≥ T 1 , ∀ x ic ∈ R nandd{ ˙ {V0 −|d 0 |dV xic (t) ≤ −cV xic (t) V0∀ t ≥ 0 .As a result, we get⎧⎨(V xic (t) ≤ max − |d |d0|0 |c(t − T⎩1 )+V xic (T 1 )d V0⎧⎫⎨(≤ max 1 − |d ) d V 0|d0|0 | ⎬c(t − T 1 ) , 0⎩ d V0 ⎭d V0) d V 0|d 0 |, 0⎫⎬⎭ ,∀ t ≥ T 1 .Therefore, setting T 2 = d V 0c|d 0| yieldsV xic (t) = 0hence the claim.∀ t ≥ T 1 + T 2 = 1 c(dV∞+ d )V 0, ∀ x ic ∈ R n ,d ∞ |d 0 |Acknowledgments. The second author is extremely grateful to Wilfrid Perruquettiand Emmanuel Moulay for the many discussions about the notion of homogeneityin the bi-limit. Also, all the authors would like to thank the anonymousreviewers for their comments, which were extremely helpful in improving the qualityof the paper.REFERENCES[1] V. Andrieu, L. Praly, and A. Astolfi, Nonlinear output feedback design via domination andgeneralized weighted homogeneity, in Proceedings of the 45th IEEE Conference on Decisionand Control, San Diego, 2006, pp. 6391–6396.[2] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, LectureNotes in Control and Inform. Sci. 267, Springer, Berlin, 2001.[3] S. P. Bhat and D. S. Bernstein, Geometric homogeneity with applications to finite-timestability, Math. Control Signals Systems, 17 (2005), pp. 101–127.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

HOMOGENEOUS OBSERVER DESIGN 1849[4] S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems,SIAM J. Control Optim., 38 (2000), pp. 751–766.[5] J.-M. Coron and L. Praly, Adding an integrator for the stabilization problem, Systems ControlLett., 17 (1991), pp. 89–104.[6] J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuousfeedback stabilization, J. Math. Systems Estim. Control, 4 (1994), pp. 67–84.[7] J. P. Gauthier and I. Kupka, Deterministic Observation Theory And Applications, CambridgeUniversity Press, Cambridge, UK, 2001.[8] W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967.[9] H. Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls,in Differential Equations: Stability and Control, Lecture Notes in Pure Appl.Math. 109, S. Elaydi, ed., Marcel Dekker, New York, 1991, pp. 249–260.[10] Y. Hong, Finite-time stabilization and stabilizability of a class of controllable systems, SystemsControl Lett., 46 (2002), pp. 231–236.[11] Z.-P. Jiang, A. Teel, and L. Praly, Small-gain theorem for ISS systems and applications,Math. Control Signals Systems, 7 (1994), pp. 95–120.[12] M. Kawski, Stabilization of nonlinear systems in the plane, Systems Control Lett., 12 (1989),pp. 169–175.[13] H. Khalil and A. Saberi, Adaptive stabilization of a class of nonlinear systems using highgainfeedback, IEEE Trans. Automat. Control, 32 (1987), pp. 1031–1035.[14] S. Lefschetz, Differential Equations: Geometric Theory, 2nd ed., Dover, New York, 1977.[15] W. Liu, Y. Chitour, and E. Sontag, On finite-gain stabilizability of linear systems subjectto input saturation, SIAM J. Control Optim., 34 (1996), pp. 1190–1219.[16] J. L. Massera, Contributions to stability theory, Ann. of Math., 64 (1956), pp. 182–206.[17] F. Mazenc, Stabilisation de trajectoires, ajout d’intégration, commandes saturées, Mémoirede thèse en Mathématiques et Automatique de l’École Nationale Supérieure des Mines deParis, Avril 1996.[18] F. Mazenc, L. Praly, and W. P. Dayawansa, Global stabilization by output feedback: Examplesand counter-examples, Systems Control Lett., 23 (1994), pp. 119–125.[19] P. Morin and C. Samson, Application of backstepping techniques to the time-varying exponentialstabilisation of chained form systems, European J. Control, 3 (1997), pp. 15–36.[20] R. Orsi, L. Praly, and I. Mareels, Sufficient conditions for the existence of an unboundedsolution, Automatica, 37 (2001), pp. 1609–1617.[21] L. Praly, B. d’Andréa-Novel, and J.-M. Coron, Lyapunov design of stabilizing controllersfor cascaded systems, IEEE Trans. Automat. Control, 36 (1991), pp. 1177–1181.[22] L. Praly and Z.-P. Jiang, Stabilization by output feedback for systems with ISS inversedynamics, Systems Control Lett., 21 (1993), pp. 19–33.[23] L. Praly and Z.-P. Jiang, Further results on robust semiglobal stabilization with dynamicinput uncertainties, in Proceedings of the 37th Annual IEEE Conference on Decision andControl, Tampa, 1998, pp. 891–896.[24] L. Praly and Z.-P. Jiang, Linear output feedback with dynamic high gain for nonlinearsystems, Systems Control Lett., 53 (2004), pp. 107–116.[25] L. Praly and F. Mazenc, Design of Homogeneous Feedbacks for a Chain of Integrators andApplications, Internal report E 173, CAS, Ecole des Mines de Paris, Paris, December 11,1995; revised April 15, 1996.[26] C. Qian, A homogeneous domination approach for global output feedback stabilization of a classof nonlinear systems, in Proceedings of the IEEE American Control Conference, Portland,2005, pp. 4708–4715.[27] C. Qian and W. Lin, Output feedback control of a class of nonlinear systems: A nonseparationprinciple paradigm, IEEE Trans. Automat. Control, 47 (2002), pp. 1710–1715.[28] C. Qian and W. Lin, Recursive observer design, homogeneous approximation, and nonsmoothoutput feedback stabilization of nonlinear systems, IEEE Trans. Automat. Control, 51(2006), pp. 1457–1471.[29] L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, SystemsControl Lett., 19 (1992), pp. 467–473.[30] H. Shim and J. Seo, Recursive nonlinear observer design: Beyond the uniform observability,IEEE Trans. Automat. Control, 48 (2003), pp. 294–298.[31] E. D. Sontag, Input to state stability: Basic concepts and results, in Nonlinear and OptimalControl Theory, P. Nistri and G. Stefani, eds., Springer-Verlag, Berlin, 2006, pp. 163–220.[32] E. Sontag and Y. Wang, Lyapunov characterizations of input to output stability, SIAM J.Control Optim., 39 (2000), pp. 226–249.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

1850 V. ANDRIEU, L. PRALY, AND A. ASTOLFI[33] M. Tzamtzi and J. Tsinias, Explicit formulas of feedback stabilizers for a class of triangularsystems with uncontrollable linearization, Systems Control Lett., 38 (1999), pp. 115–126.[34] F. W. Wilson, Jr., Smoothing derivatives of functions and applications, Trans. Amer. Math.Soc., 139 (1969), pp. 413–428.[35] B. Yang and W. Lin, Homogeneous observers, iterative design, and global stabilization ofhigh-order nonlinear systems by smooth output feedback, IEEE Trans. Automat. Control,49 (2004), pp. 1069–1080.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.