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HOMOGENEOUS OBSERVER DESIGN 1835Example 5.5. Consider the illustrative system (1.1). The bound (5.6) gives thecondition(5.7) 0 < q is 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.
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