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A lot of procedures for estimating the spool position in linear electromagnetic actuators using voltage and current measurements only, can be found in the literature. Subject to the accuracy of the estimated spool position some achieve better, some worse results. However, in almost every approach hysteresis has a huge impact on the estimation accuracy that can be achieved. Regardless whether these effects are caused by magnetic or mechanical hysteresis, they will limit the accuracy of the position estimate, if not taken into account. In this paper, a model is introduced which covers the hysteresis effects as well as other nonlinear ities occurring in estimated position-dependent parameters. A classical Preisach model is deployed first, which is then adjusted by using novel elementary preceding Relay-Operators. The resulting model for the estimated position-dependent parameters including the adjusted Preisach model can be easily applied to position estimation tasks. It is shown that the considered model distinctly improves the accuracy for the spool position estimate, while it is kept as simple as possible for real-time implementation reasons.

An approach for an adaptive position-dependent friction estimation for linear electromagnetic actuators with altered characteristics is proposed in this paper. The objective is to obtain a friction model that can be used to describe different stages of aging of magnetic actuators. It is compared to a classical Stribeck friction model by means of model fit, sensitivity, and parameter correlation. The identifiability of the parameters in the friction model is of special interest since the model is supposed to be used for diagnostic and prognostic purposes. A method based on the Fisher information matrix is employed to analyze the quality of the model structure and the parameter estimates.

A constructive method for the design of nonlinear observers is discussed. To formulate conditions for the construction of the observer gains, stability results for nonlinear singularly perturbed systems are utilised. The nonlinear observer is designed directly in the given coordinates, where the error dynamics between the plant and the observer becomes singularly perturbed by a high-gain part of the observer injection, and the information of the slow manifold is exploited to construct the observer gains of the reduced-order dynamics. This is in contrast to typical high-gain observer approaches, where the observer gains are chosen such that the nonlinearities are dominated by a linear system. It will be demonstrated that the considered approach is particularly suited for self-sensing electromechanical systems. Two variants of the proposed observer design are illustrated for a nonlinear electromagnetic actuator, where the mechanical quantities, i.e. the position and the velocity, are not measured