By Jean Levine
This is the 1st booklet on a scorching subject within the box of keep watch over of nonlinear structures. It levels from mathematical approach thought to sensible commercial keep watch over purposes and addresses primary questions in structures and regulate: the best way to plan the movement of a procedure and song the corresponding trajectory in presence of perturbations. It emphasizes on structural elements and particularly on a category of structures referred to as differentially flat.
Part 1 discusses the mathematical conception and half 2 outlines purposes of this technique within the fields of electrical drives (DC automobiles and linear synchronous motors), magnetic bearings, automobile equipments, cranes, and automated flight keep an eye on systems.
The writer bargains web-based video clips illustrating a few dynamical elements and case reviews in simulation (Scilab and Matlab).
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Extra info for Analysis and Control of Nonlinear Systems: A Flatness-based Approach
2 that an instationary vector field on a manifold X may be seen as a stationary one on the augmented manifold X × R. However, we’ll see that many properties of stationary vector fields do not extend to time-varying ones. In particular, an arbitrary change of coordinates would change (x, t) in (z, s) = ϕ(x, t), s playing the role of a new clock, the sign of s˙ being arbitrary. Therefore, if the sign of s˙ changes along an integral curve, we may lose the possibility of capturing the asymptotic properties of the system (when t → ±∞) in the transformed coordinates.
We immediately deduce that L[ϕ∗ f1 ,ϕ∗ f2 ] h(ϕ(x)) = L[f1 ,f2 ] h(ϕ(x)), which proves the Proposition. The bracket[f, g] has the following geometric interpretation: let us denote def by Xt (x) = exp tf (x), by analogy with the solution of a linear differential equation, the point of the integral curve of f at time t passing through the point x at time 0. This notation allows to precise which vector field is considered when several vector fields may be integrated. Thus, the point of the integral curve of g at time t passing through x at time 0 is noted exp tg(x).
P ) such that ξ ∈ ϕ(V ). The solution y is thus immediately deduced from z by y = z ◦ ϕ. In the case where D is not involutive and if D has constant rank r, we follow the same lines with D in place of D. 18). The supplementary equations generated by the vector fields of D that are not in D are often called compatibility conditions. It results that D cannot be straightened out without simultaneously straightening out D, which implies that z(ξ) = z(ξ) for all ξ = (ξr+1 , . . , ξp ) such that ξ ∈ ϕ(V ).
Analysis and Control of Nonlinear Systems: A Flatness-based Approach by Jean Levine