14 research outputs found

    Non-linear Symmetry-preserving Observer on Lie Groups

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    In this paper we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which reminds of the linear stationary case.Comment: 12 pages. Submitted. Preliminary version publicated in french in the CIFA proceedings and IFAC0

    Smooth and Analytic Normal and Canonical Forms for Strict Feedforward Systems

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    Recently we proved that any smooth (resp. analytic) strict feedforward system can be brought into its normal form via a smooth (resp. analytic) feedback transformation. This will allow us to identify a subclass of strict feedforward systems, called systems in special strict feedforward form, shortly (SSFF), possessing a canonical form which is an analytic counterpart of the formal canonical form. For (SSFF)-systems, the step-by-step normalization procedure of Kang and Krener leads to smooth (resp. convergent analytic) normalizing feedback transformations. We illustrate the class of (SSFF)-systems by a model of an inverted pendulum on a cart

    Explicit Symmetries of Strict Feedforward Control Systems

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    We show that any symmetry of a smooth strict feedforward system is conjugated to a scaling translation and any 1-parameter family of symmetries to a family of scaling translations along the first variable. We compute explicitly those symmetries by finding the conjugating diffeomorphism. We deduce, in accordance with our previous work, that a smooth system is feedback equivalent to a strict feedforward form if and only if it gives rise to a sequence of systems, such that each element of the sequence, firstly, possesses an infinitesimal symmetry whose flow is conjugated to a 1- parameter families of scaling translations and, secondly, it is the factor system of the preceding one, that is, is reduced from the preceding one by its symmetry. We illustrate our results by computing the symmetries of the Cart-Pole system

    Normal Forms for Nonlinear Discrete Time Control Systems

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    We study the feedback classification of discrete-time control systems whose linear approximation around an equilibrium is controllable. We provide a normal form for systems under investigation

    A Separation Principle on Lie Groups

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    For linear time-invariant systems, a separation principle holds: stable observer and stable state feedback can be designed for the time-invariant system, and the combined observer and feedback will be stable. For non-linear systems, a local separation principle holds around steady-states, as the linearized system is time-invariant. This paper addresses the issue of a non-linear separation principle on Lie groups. For invariant systems on Lie groups, we prove there exists a large set of (time-varying) trajectories around which the linearized observer-controler system is time-invariant, as soon as a symmetry-preserving observer is used. Thus a separation principle holds around those trajectories. The theory is illustrated by a mobile robot example, and the developed ideas are then extended to a class of Lagrangian mechanical systems on Lie groups described by Euler-Poincare equations.Comment: Submitted to IFAC 201

    A non-linear symmetry-preserving observer for velocity-aided inertial navigation

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    International audienceA first theory of invariant observers is developed. An invariant observer is an observer which respects the symmetries of the system equations. As an illustration of the theory, a nonlinear invariant observer for velocity-aided inertial navigation is proposed and analyze

    Invariant tracking

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    The problem of invariant output tracking is considered: given a control system admitting a symmetry group G, design a feedback such that the closed-loop system tracks a desired output reference and is invariant under the action of G. Invariant output errors are defined as a set of scalar invariants of G; they are calculated with the Cartan moving frame method. It is shown that standard tracking methods based on input-output linearization can be applied to these invariant errors to yield the required “symmetry-preserving” feedback

    Weighted canonical forms of nonlinear single-input control systems with noncontrollable linearization

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    We propose a weighted canonical form for single-input systems with noncontrollable first order approximation under the action of formal feedback transformations. This weighted canonical form is based on associating different weights to the linearly controllable and linearly noncontrollable parts of the system. We prove that two systems are formally feedback equivalent if and only if their weighted canonical forms coincide up to a diffeomorphism whose restriction to the linearly controllable part is identity

    Groupe de Lie et observateur non-linéaire

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    On se place sur un groupe de Lie et on considère une dynamique invariante à gauche. On montre, moyennant une hypothèse sur la sortie, qu'il est possible dans ce cas de construire des observateurs non-linéaires pour lesquels l'équation d'erreur est autonome. La théorie est illustrée par un exemple emprunté à la navigation inertielle
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