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(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.
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