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

More documents

Recommendations

Info

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.
Page 1 and 2: SIAM J. CONTROL OPTIM.Vol. 47, No.
Page 3 and 4: 1816 V. ANDRIEU, L. PRALY, AND A. A
Page 31: 1844 V. ANDRIEU, L. PRALY, AND A. A
Page 37: 1850 V. ANDRIEU, L. PRALY, AND A. A

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

Create successful ePaper yourself

Delete template?

Save as template?