Download Discontinuous Systems: Lyapunov Analysis and Robust by Yury V. Orlov PDF

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By Yury V. Orlov

A significant issue on top of things engineering is strong suggestions layout that stabilizes a nominal plant whereas additionally attenuating the impact of parameter adaptations and exterior disturbances. This monograph addresses this challenge in doubtful discontinuous dynamic structures with precise recognition to electromechanical platforms with hard-to-model nonsmooth phenomena resembling friction and backlash. Ignoring those phenomena may well seriously restrict functionality so the sensible application of current delicate regulate algorithms turns into questionable for lots of electromechanical applications.

With this motivation, Discontinuous Systems develops nonsmooth balance research and discontinuous keep watch over synthesis according to novel modeling of discontinuous dynamic structures, working less than doubtful stipulations. even though it is basically a learn monograph dedicated to the idea of discontinuous dynamic structures, no historical past in discontinuous platforms is needed; such platforms are brought within the publication on the acceptable conceptual point. Being built for discontinuous platforms, the idea is effectively utilized to their subclasses – variable-structure and impulsive platforms – in addition to to finite- and infinite-dimensional structures similar to distributed-parameter and time-delay structures. The presentation concentrates on algorithms instead of on technical implementation even if theoretical effects are illustrated via electromechanical purposes. those particular functions whole the e-book and, including the introductory theoretical ingredients convey a few parts of the educational to the text.

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Additional resources for Discontinuous Systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions

Sample text

Generally speaking, Utkin solutions do not coincide with corresponding Filippov solutions. The following example drawn from [227] illustrates this feature. 3. 57) with the input function u, governed by u= 1 if (x1 + x2 )x1 < 0 . 58), appears to hold on the discontinuity line x1 + x2 = 0, whereas it does not hold on the discontinuity line x1 = 0. Thus, sliding modes of the system in question occur along the line x1 + x2 = 0. 54), which is now specified to (−1 + ueq + 4(ueq)3 + ueq − 1 = 0. 2x1, governing Utkin solutions on the discontinuity line x1 + x2 = 0.

In particular, x D (A) = Ax H if A possesses a growth bound β < 0. , D(A) ⊂ H, D(A) is dense in H and the inequality x H ≤ ω0 x D (A) holds for all x ∈ D(A) and some constant ω0 > 0. If the input function u meets the same smoothness conditions as that imposed on the system nonlinearity f , the above equation locally has a unique strong solution x(t) which is defined as follows. 8. 62) with a continuously differentiable input u(x,t) iff limt↓0 x(t) − x0 H = 0, and x(t) is continuously differentiable and satisfies the equation for t ∈ (0, T ).

L belong to the Filippov set Φ (x,t), but not all these points are the vertices, forming this set. Let us assume now that the function ϕ (x,t) undergoes discontinuities on a smooth surface S only, and let this surface be governed by the equation s(x) = 0. Then the discontinuity set S separates the x space into domains G− = {x ∈ Rn : s(x) < 0} and G+ = {x ∈ Rn : s(x) > 0}. Given t, the Filippov set Φ (x,t) would be a linear segment joining the endpoints of the vectors ϕ − (x,t) = lim (ξ ,t)∈G− , ξ →x ϕ (ξ ,t), ϕ + (x,t) = lim (ξ ,t)∈G+ , ξ →x ϕ (ξ ,t).

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