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

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

This ebook comprises chosen contributions through academics on the 3rd annual Formation d’Automatique de Paris. It presents a well-integrated synthesis of the most recent considering in nonlinear optimum keep an eye on, observer layout, balance research and structural houses of linear platforms, with no the necessity for an exhaustive literature assessment. The across the world identified individuals to this quantity signify the various such a lot respected keep watch over facilities in Europe.

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

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Moreover, since the reference trajectory is tranverse, the test is equivalent to det(dΠ(z(t))L(t), f (x(t), u(t))) = 0. 18) In order to estimate the conjugate points we can use the curvature tensor according to the Sturm comparison theorem. 8. (Sturm) Let v be the solution of v¨ + A(t)v = 0 with v(0) = 0, v(0) ˙ = 1, and let w be the solution of w ¨ + B(t)w = 0 with the same initial conditions. Suppose A(t) ≤ B(t). If a ( resp. b) is the first positive zero of v ( resp. w), then a ≤ b . Proof . In accordance with initial conditions, v and w are positive for 0 < t < a and 0 < t < b, respectively.

Xn ) x˙ 2 = x3 .. x˙ n = u with q(x1 , x2 , . . , xn ) = n i,j=2 aij xi xj . Setting x(t) = t + ξ(t) we get ξ˙1 = n aij xi xj i,j=2 ξ˙2 = ξ3 .. ˙ξn = u and the system describing the evolution of ξ is the linearized system. We substitute x1 by t in the quadratic form q. The last diagonal coefficient ann (t) is 32 Bernard Bonnard and Jean-Baptiste Caillau p, ∂ 2 F0 /∂x2n (γ(t)), which is also equal to the opposite of the Poisson bracket {{H1 , H0 }, H1 }(z(t)) involved in the generalized Legendre-Clebsh condition: it is negative in the hyperbolic case, and positive in the elliptic one.

1 Introduction to Nonlinear Optimal Control 43 m x˙ = F0 + ui Fi i=1 with x ∈ Rn , F0 , . . , Fm smooth vector fields, the control u in Rm being m bounded by i=1 |ui |2 ≤ 1. Let the Hi ’s be the usual Hamiltonian lifts p, Fi , i = 0, . . , m, and let Σ be the switching surface {Hi = 0, i = 1, . . , m}. The maximization of the m Hamiltonian H = H0 + i=1 ui Hi outside Σ implies that ui = Hi m i=1 Hi2 , i = 1, . . , m. 22) into H, one defines the Hamiltonian function m Hr = H 0 + 1 2 Hi2 . 13. The corresponding solutions are called order zero extremals.

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