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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.
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