By Jean Levine

This is the 1st publication on a sizzling subject within the box of keep watch over of nonlinear platforms. It levels from mathematical method concept to useful commercial keep watch over purposes and addresses basic questions in structures and keep an eye on: tips on how to plan the movement of a approach and tune the corresponding trajectory in presence of perturbations. It emphasizes on structural facets and particularly on a category of structures referred to as differentially flat.

Part 1 discusses the mathematical conception and half 2 outlines functions of this technique within the fields of electrical drives (DC automobiles and linear synchronous motors), magnetic bearings, automobile equipments, cranes, and automated flight keep an eye on systems.

The writer bargains web-based movies illustrating a few dynamical facets and case stories in simulation (Scilab and Matlab).

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**Additional info for Analysis and Control of Nonlinear Systems: A Flatness-based Approach**

**Example text**

Let U be a differentiable function from Rn to R and ψ(x) an arbitrary function from Rn to Rn . We denote by , the scalar product of Rn and by m a positive real number. 6) have no periodic orbit. 6) converge to x. Clearly, the right-hand side represents the sum of forces deriving from the potential U and the dissipative ones (− ψ(x), x˙ ψ(x)). Proof. Set V (x, x) ˙ = 12 m x, ˙ x˙ + U (x) (mechanical energy). We have d ∂U V (x(t), x(t)) ˙ = m¨ x, x˙ + (x), x˙ dt ∂x ∂U =− (x), x˙ − ψ(x), x˙ ψ(x), x˙ + ∂x ∂U (x), x˙ ∂x = − ψ(x), x˙ 2 ≤ 0.

Here, the tangent linear mapping of P at x ¯ is unstable. The Poincar´e map doesn’t depend on the choice of the transverse submanifold W or on the point x: if we choose another transverse submanifold W and a point x of γ, the corresponding mapping P is the image of P by the diffeomorphism that transforms W in W and x in x . We leave the verification of this property to the reader. Recall that, given a diffeomorphism f on a manifold X, the point x ∈ X is called a fixed point of f if and only if f (x) = x.

25) for all ω, θ ∈ Λ1 (X), since then dω ∧ θ and ω ∧ dθ are 3-forms. This results from the fact that ω ∧ θ = j ωj dxj ∧ ( k θk dxk ) = j,k ωj θk dxj ∧ dxk . 24), we get d(ω ∧ θ) = i,j,k ∂ωj ∂xi θk Moreover, dω ∧ θ = dxj ∧ dxk . Accordingly, ω ∧ dθ ∂θ + ωj ∂xji dxi ∧ dxj ∧ dxk . ∂ωj i,j ∂xi dxi = ∧ dxj ∧ ( j ωj dxj k ∧ θk dxk ) = ∂ωj i,j,k ∂xi θk dxi ∂θk i,k ∂xi dxi ∧ dxk ∧ = 36 2 Introduction to Differential Geometry ∂θk k ωj ∂θ i,j,k ωj ∂xi dxi ∧ dxj ∧ dxk by skew∂xi dxj ∧ dxi ∧ dxk = − symmetry.