Publication

On two isomorphic Lie algebroids associated with feedback linearization

Philippe Müllhaupt
2019
Conference Papers

Abstract

We present two Lie algebroids linked to the construction of the linearizing output of an input affine nonlinear system. The algorithmic development of the linearizing output proceeds inductively, and each stage has two structures, namely a codimension one foliation defined through an integrable 1-form w, and a transversal vectorfield g to the foliation. Each integral manifold of the vectorfield g defines an equivalence class of points. Due to transversality, a leaf of the foliation is chosen to represent these equivalence classes. A Lie groupoid is defined with its base given as the particular chosen leaf and with the product induced by the pseudogroup of diffeomorphisms that preserve equivalence classes generated by the integral manifolds of g. Two Lie algebroids associated with this groupoid are then defined. The theory is illustrated with an example using polynomial automorphisms as particular cases of diffeomorphisms and shows the relation with the Jacobian conjecture. (C) 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

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https://infoscience.epfl.ch/entities/publication/c0c9ad82-0bfa-467d-97ab-44201877381f

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Philippe Müllhaupt
2019
Conference Papers

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/c0c9ad82-0bfa-467d-97ab-44201877381f

About this result

Related concepts (32)

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.

Equivalence class

In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements and belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set and an equivalence relation on the of an element in denoted by is the set of elements which are equivalent to It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of by and is denoted by .

Poisson manifold

In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics. A Poisson structure (or Poisson bracket) on a smooth manifold is a functionon the vector space of smooth functions on , making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra).

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