Download Advanced Topics in Control Systems Theory: Lecture Notes by Julio Antonio Loría Perez, Françoise Lamnabhi-Lagarrigue, PDF

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

This e-book contains chosen contributions through teachers on the 3rd annual Formation d’Automatique de Paris. It presents a well-integrated synthesis of the most recent pondering in nonlinear optimum keep watch over, observer layout, balance research and structural homes of linear structures, with out the necessity for an exhaustive literature overview. The the world over recognized individuals to this quantity symbolize a number of the so much respected keep watch over facilities in Europe.

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

Example text

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. 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.

Let H (t, z) be a smooth Hamiltonian vector field whose integral curves are the extremals of an optimal control problem with fixed time T and initial manifold M0 . The time tf is a focal time along the BC-extremal z if there is a Jacobi field J such that J(0) is in Tz(0) M0⊥ and J is vertical at tf . Both concepts fit in the same geometric framework: a one parameter family of Lagrangian manifolds obtained by transporting the initial submanifold with the flow. The Jacobi fields span the tangent spaces of the Lagrangian manifolds computed along the reference extremal.

Properties of the model. In this model, we have gathered in one normal form all the information required to evaluate the endpoint mapping (and thus the accessibility set) up to second-order relevant terms. The adjoint covector is oriented by the condition H0 ≥ 0 and normalized to p = (1, 0, . . , 0) The linearized system along the reference trajectory is a constant linear system in Brunovsky normal form. Indeed, x˙ 1 = 1 + q(x1 , x2 , . . , xn ) x˙ 2 = x3 .. x˙ n = u with q(x1 , x2 , . . , xn ) = n i,j=2 aij xi xj .

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