The Averaging Homotopy

Classical averaging is an asymptotic theory in the sense that it con- siders the problem of finding a limit for a sequence of differential systems. We present here another formulation that is based on an homotopy be- tween any two vectorfields. This formulation accounts both for classical averaging and for regular perturbations. It can also be used to justify the use of numerical windowed averaging schemes without any asymptotic imbedding

Averaging and Deterministic Optimal Control

International audienceAveraging is often used in ordinary differential equations when dealing with fast periodic phenomena. It is shown here that it can be used efficiently in optimal control. As the period tends to zero, a limit or "averaged" problem is defined. The open loop optimal control of the limit problem induces a cost which is optimal up to the second order when evaluated through the original dynamics. The definition of the averaged problem is then generalized to the nonperiodic case. It is shown that the Bellman function of the original "fast" problem tends uniformly on any compact set to that of the averaged problem

Redundant wavelet filter banks on the half-axis with applications to signal denoising with small delays

International audienceA wavelet transform on the negative half real axis is developed using an average-interpolation scheme. This transform is redundant and can be used to perform causal wavelet processing, such as signal denoising, without delay. Nonetheless, in practice some boundary effects occur and thus a small amount of delay is required to reduce them. The theory is implemented on a challenging signal with large noise and sharp transients. Results from the experimental implementation of the proposed algorithm for the denoising of a feedback signal for controlling a three-phase permanent-magnet synchronous brushless DC motor are also presented

Redundant wavelet processing on the half-axis with applications to signal denoising with small delays: Theory and experiments

International audienceA wavelet transform on the negative half real axis is developed using an average-interpolation scheme. This transform is redundant and can be used to perform causal wavelet processing, such as on-line signal denoising, without delay. Nonetheless, in practice some boundary effects occur and thus a small amount of delay is required to reduce them. The effect of this delay is studied using a numerical example of a signal with large noise and sharp transients. It is shown that the delay required to obtain acceptable denoising levels is decreased by using the proposed redundant transform instead of a non-redundant one. We also present results from the experimental implementation of the proposed algorithm for the denoising of a feedback signal during the control of a three-phase permanent-magnet synchronous brushless DC motor

Thermal building model identification using time-scaled identification methods

International audienceThe aim of this paper is to propose a robust and accurate method for the parametric identification of the thermal behaviour of low consumption buildings. These buildings are known to have a two-time scale structure, which, if not handled properly, results in poor conditioning of the parametric identification. We compare three identification methods, one uses the data on the whole frequency domain (ARX) when the other methods use the same data but separated on local frequency domain (time scaled methods). All three methods identify a reduced second order model. Robustness is tested by corrupting the input and output before the identification, and comparing the simulation results for the various models and the original uncorrupted input. The numerical results clearly show that the time scaled methods are superior both in accuracy (noise free identification and simulation) and robustness (when identification is performed on corrupted data)

Generalization of the interaction between the Haar approximation and polynomial operators to higher order methods

International audienceIn applications it is useful to compute the local average of a function f(u) of an input u from empirical statistics on u. A very simple relation exists when the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation

Averaging on simple windows in deterministic optimal control

A windowed averaged scheme is defined for general control systems. The same method is used to average costs in optimal control problems (OCPs). A numerical parameter α can be computed, which expresses the distance between the original system and the averaged system in a weak sense. Then, if we use the optimal control of the averaged OCP in the original OCP, the suboptimality of the control is bounded by an expression of the form Cα 2

Averaging et commande optimale déterministe

On considère un problème de contrôle optimal, déterministe dont la dynamique dépend de phénomènes "rapides", modélisés sous la forme d'un temps rapide t/epsilon , epsilon petit. On étudie le cas particulier ou le temps rapide intervient de manière périodique dans la dynamique. On étudie le problème moyenne dans les cas périodique et non périodique.No summar