Optimal sensor placement and estimator-based temperature control for a deep drawing process

Abstract

Deep drawing is one of the most important forming processes for the forming of flat sheet blanks, where the formation of wrinkles and the appearance of cracks can be a problem, especially in areas of high geometric complexity. The local increase of the temperature in these critical areas can help to improve the formability of the material and thus reduce defects. The present paper aims at a targeted temperature control of the die of a deep-drawing mold. For this sensors are placed systematically to develop an estimator for the spatial–temporal temperature evolution to subsequently realize tracking control using the embedded actuation devices. A continuum representation of the temperature distribution in the die is derived and transferred to a high order finite element (FE) approximation to take the complex-shaped geometry of the tool into account. Parameter identification is performed based on measurement data to improve the accuracy of the FE approximation and model order reduction (MOR) techniques are applied to determine a sufficiently low order system representation. A mixed-integer optimization problem is formulated and solved making use of different formulations of the observability Gramian to determine the optimal sensor locations and a Kalman filter is designed as an estimator based on a reduced order model. Moreover, a linear-quadratic regulator with integral part combined with the Kalman filter is developed to react efficiently towards disturbances. Finally this theoretical framework is tested in a real experiment.