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

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

This booklet contains chosen contributions via academics on the 3rd annual Formation d’Automatique de Paris. It presents a well-integrated synthesis of the newest pondering in nonlinear optimum keep an eye on, observer layout, balance research and structural homes of linear platforms, with out the necessity for an exhaustive literature overview. The across the world identified participants to this quantity signify a few of the so much respected keep watch over facilities in Europe.

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Observe that the generalized Legendre-Clebsch condition is only satisfied in the hyperbolic case. It is now crucial to notice that since the reference curve is a one-dimensional manifold, we can normalize any independent family of Lie brackets to form a frame along it. Our assumptions allow us to pick coordinates preserving the previous normalizations and defining a moving frame defined by: adk F0 · F1 (γ(t)) = ∂ , k = 0, . . , n − 1, t ∈ [0, T ]. ∂xn−k Moreover, since the feedback is chosen so that u is zero along γ, we can impose the linearization condition adk F0 · F1 (γ(t)) = 0 for k > n − 2 and t ∈ [0, T ].

The first conjugate time to 0, t1c , is the smallest t such that λt,1 = 0. If t < t1c , the only minimizer of Qt on Ct is y = 0. If t > t1c , the infimum of Qt is −∞. Proof . Rather than using the standard Morse theory, we make a simple proof of the loss of optimality after the first conjugate time based on the geometric argument of the Riemannian case [7]. Indeed, let t1c be the first conjugate time along the reference trajectory γ. There exists a Jacobi field vertical at 0 and t1c corresponding to a variation of γ with δx(0) = δx(t1c ) = 0.

Therefore, P = (L , L) is a symplectic matrix and we have P −1 = − tLJ . t LJ If we make the symplectic change of coordinates x = P y, we get the Hamiltonian equation y˙ = P −1 (AP − P˙ )y and using the notation x˙ = Ax = JS where S is symmetric. Decomposing y = (u, v), we obtain the equations u˙ = 0 v˙ = − tL (SL + J L˙ )u and the solution can be computed by quadrature. Similar tensor analysis can be developed to study the standard LQ problem. 1 Introduction to Nonlinear Optimal Control 25 Geometric analysis of linear quadratic problems.

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