Design and Analysis of Control Systems: Case Studies by Humberto Stein Shiromoto (auth.)

By Humberto Stein Shiromoto (auth.)

This booklet offers tips on how to unify diverse techniques to take on balance idea difficulties. particularly, it provides a technique to mixture methods received from degree idea with equipment bought from Lyapunov’s balance conception. the writer summarizes fresh works on how diversified analysis/design equipment might be unified and hired for structures that don't belong to both of domain names of validity.

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Example text

9 are such that M < Mg or N < N g . Let R = cl Ω≤ Mg (Ug ) \ Ω≤ M (U ) . 8). 12 is to show the absence of ω-limit sets in R. This is achieved by exhibiting a contradiction between Fig. 4 Results 45 its existence and the assumption that, for every y ∈ R, div h(y) = 0 and h(y) = 0. 12 is provided in Sect. 3. 9, when Ml < Mg and Nl < N g . 12 are illustrated. 8ρx (x). 3 2 Let the functions R x → V (x) = |x| ∈ R≥0 , R z → W (z) = |z| ∈ R≥0 . The Dini derivative (cf. 39) of V along the vector field f yields the inequality D +f V (x, z) ≤ −ρx (V (x)) + W (z), which holds, for every (x, z) ∈ R × R.

Consequently, the set R = cl Ω≤ Mg (Ug ) \ Ω≤ M (U ) is nonempty. 16. Fourth Part. 8) that converge to the set Z ⊂ R (cf. 16) are shown to have also measure zero. For every t, let Y (t, Z) = {Y (t, z) : t ∈ dom(z), z ∈ Z}, where dom(z) is the maximum time interval where Y (t, z) exists. Since Z is positively invariant, for every t1 , t2 ∈ dom(z), t1 < t2 ≤ 0 ⇒ Y (t2 , Z) ⊂ Y (t1 , Z). This inclusion implies that4 Y := {Y (t, Z)} = t≤0 {Y (t, Z) : t ≤ l}. l∈Z<0 Hence, the set Y is a countable union of images of Z by the flow.

If x ∈ C2,l , then Eq. 11b) implies that ϕ2,l (x) = ϕg (x). 4, the set A is globally practically asymptotically stable for the closed-loop system h (ϕg ). Moreover, Eqs. 36) imply that the inclusion A ⊂ D2,l holds. ii. 11c), g2,l (x) = {(1, l)} and, after the jump, the local hybrid feedback law is selected. Since the value of x does not change during a jump, x ∈ D2,l after a jump. Moreover, Eqs. 36) imply that the inclusion D2,l ⊂ Ω

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