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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.
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