New PDF release: Advanced Topics in Control Systems Theory: Lecture Notes

By Julio Antonio Loría Perez, Françoise Lamnabhi-Lagarrigue, Elena Viatcheslavovna Panteley

ISBN-10: 1846283132

ISBN-13: 9781846283130

This ebook comprises chosen contributions by means of academics on the 3rd annual Formation d’Automatique de Paris. It offers a well-integrated synthesis of the newest considering in nonlinear optimum keep watch over, observer layout, balance research and structural houses of linear platforms, with no the necessity for an exhaustive literature evaluate. The the world over identified members to this quantity characterize some of the so much respected keep an eye on facilities in Europe.

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Additional info for Advanced Topics in Control Systems Theory: Lecture Notes from FAP 2005

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A sub-Riemannian metric is associated to a metric of the form g = a(q)dx2 + 2b(q)dxdy + c(q)dy 2 . By choosing suitable coordinates, the smooth functions a, b and c can be normalized to a = c = 1 and b = 0. The case g = dx2 + dy 2 is called the Heisenberg case or the sub-Riemannian flat contact case. Heisenberg sub-Riemannian geometry and the Dido problem. We observe that the previous problem can be written x˙ = u1 , y˙ = u2 , z˙ = xy ˙ − yx, ˙ and T 0 (x˙ 2 + y˙ 2 )1/2 dt → min in order that: (i) The length of a curve t → (x(t), y(t), z(t)) is the length of the projection in the xy-plane.

The corresponding solutions are called order zero extremals. From the maximum principle, optimal extremals are contained in the level set {Hr ≥ 0}. Those in {Hr = 0} are exceptional. The following result is standard. 19. The order zero extremals are smooth responses to smooth controls on the boundary of |u| ≤ 1. They are singularities of the endpoint mapping Ex0 ,T : u → x(T, x0 , u) for the L∞ -topology when u is restricted to the unit sphere Sm−1 . In order to construct all extremals, we must analyze the behaviour of those of order zero near the switching surface.

The analysis in the next paragraph shows the connection between the concept of conjugate point and the intrinsic second-order derivative. We derive C 1 -sufficient second order optimality conditions in the smooth case. 28 Bernard Bonnard and Jean-Baptiste Caillau Conjugate Points of Smooth Time-optimal Control Problems Preliminaries. We restrict our presentation to a smooth time optimal control problem x˙ = f (x, u) where u belongs to U , assumed to be an open subset of Rm . The Hamiltonian of the problem is H(x, p, u) = p0 + H(x, p, u) with H(x, p, u) = p, f (x, u) and p0 ≤ 0.

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Advanced Topics in Control Systems Theory: Lecture Notes from FAP 2005 by Julio Antonio Loría Perez, Françoise Lamnabhi-Lagarrigue, Elena Viatcheslavovna Panteley


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