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

ISBN-10: 1846283132

ISBN-13: 9781846283130

This publication contains chosen contributions by means of academics on the 3rd annual Formation d’Automatique de Paris. It presents a well-integrated synthesis of the most recent pondering in nonlinear optimum keep an eye on, observer layout, balance research and structural houses of linear structures, with no the necessity for an exhaustive literature evaluate. The across the world identified participants to this quantity characterize the various so much respected keep an eye on facilities in Europe.

**Read or Download Advanced Topics in Control Systems Theory: Lecture Notes from FAP 2005 PDF**

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

**Example text**

Besides, if the thrust is maximal, maximizing the ﬁnal mass reduces to minimizing the transfer time. If q ∧ q˙ is not zero, the thrust can be decomposed in a moving frame attached to the satellite. A canonical choice consists in the radial-orthoradial frame: F = ur Fr + uor For + uc Fc with Fr = q ∧ q˙ ∂ q ∂ , Fc = r ∂ q˙ |q ∧ q| ˙ ∂ q˙ and For = Fc ∧ Fr . If uc = 0, the state space is the tangent space to the osculating plane generated by q and q˙ and we have a 2D-problem. For the sake of simplicity, we shall restrict our study to this setting which already exhibits all the relevant features of the whole system.

If we set F3 = ∂/∂z, we get [F1 , F2 ] = 2∂/∂z. Moreover, all the Lie brackets of order three or more are zero. The distribution D spanned by F1 , F2 is a contact distribution deﬁned as the kernel of the 1-form α = dz + (xdy − ydx). 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.

The second normalization is performed choosing a reference extremal meeting the generic requirements hereafter: (i) E1 (t) is of codimension one and is generated by the ﬁrst (n − 1) brackets, adk F0 · F1 (γ(t)), k = 0, . . , n − 2, for any t in [0, T ]. (ii) The Lie bracket ad2 F1 · F0 (γ(t)) is not contained in E1 (t) for t in [0, T ]. (iii)The vector ﬁeld F0 restricted to γ is tranverse to E1 (t) on [0, T ]. This has the following implications: ﬁrst, for each 0 < t0 < t1 ≤ T , the singularity of the endpoint mapping at the zero control deﬁned on [t0 , t1 ] is of codimension one and the image of its Fr´echet derivative is E1 (t1 ).

### 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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