In this paper, we proposed a novel approach for nonlinear state estimation, named π-IUKF (Invariant Unscented Kalman Filter), which is based on both invariant filter estimation and UKF theoretical principles. Several research works on nonlinear invariant observers have been led and provide a geometrical-based constructive method for designing filters dedicated to nonlinear state estimation problems while preserving the physical properties and systems symmetries. The general invariant observer guarantees a straightforward form of the nonlinear estimation error dynamics whose properties are remarkable. The developed π-IUKF estimator suggests a systematic approach to determine all the symmetry-preserving correction terms, associated with a nonlinear state-space representation used for prediction, without requiring any linearization of the differential equations. The exploitation of the UKF principles within the invariant framework has required the definition of a compatibility condition on the observation equations. As a first result, the estimated covariance matrices of the π-IUKF converge to constant values due to the symmetry-preserving property provided by the nonlinear invariant estimation theory. The designed π-IUKF method has been successfully applied to some relevant practical problems such as the estimation of Attitude and Heading for aerial vehicles using low-cost AH reference systems (i.e., inertial/magnetic sensors characterized by low performances).
In this paper, it will be shown that open-channel hydraulic systems can be suitably represented for control purposes by using input delay linear parameter-varying (LPV) models. The physical equations on which this work is done are Saint-Venant equations applied to a non-rectangular cross section channel. These later are two coupled non-linear hyperbolic partial differential equations which are linearized and transformed into irrational transfer functions. An accurate model approximation procedure, denoted IPTFA (Irrational Proper Transfer Function Algorithm) is developed in order to obtain a rational transfer function plus input delays which is then parameterized by one single parameter: the initial steadystate discharge. Frequency domain responses of the irrational and reduced-order transfer functions are shown to match for a large range of discharge.
In this paper, the realization-free model approximation problem, as stated in [1,2], is revisited in the case where the interpolating model might be time-delay dependent. To this aim, the Loewner framework, initially settled for delay-free realization, is firstly generalized to the single delay case. Secondly, the (infinite) model approximation H 2 optimality conditions are established through the use of the Lambert functions. Finally, a numerically effective iterative scheme, named dTF-IRKA, similar to the TF-IRKA [2], is proposed to reach a part of the aforementioned optimality conditions. The proposed method validity and interest are assessed on different numerical examples. * I. Pontes Duff is with ISAE, Onera -The French Aerospace Lab,
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