Trajectory planning and tracking control for the temperature distribution in a deep drawing tool

Abstract

The deep drawing process and the resulting product quality essentially rely on the temperature distribution inside the tool. For temperature manipulation and control a flatness-based design technique for thermal trajectory planning and feedforward control for a deep drawing tool is developed based on a distributed parameter system description. Heating cartridges, that are embedded into the tool structure, serve as actuators to insert energy into the system with the desire to transfer the spatial-temporal temperature distribution from an initial to a desired final stationary profile. To address the complex-shaped geometry of the tool a high-order finite element (FE) approximation is deduced and combined with model-order reduction techniques to determine a sufficiently low order system representation that is applicable for optimal actuator placement. For this, a mixed-integer optimization problem is formulated based on a particular reduced-order formulation of the controllability Gramian. The resulting actuator configuration is exploited for flatness-based trajectory planning by constructing a virtual output that differentially parametrizes any system state and input. This implies a particularly intuitive approach to solve the thermal trajectory planning problem. Convergence of the differential parametrization is analyzed in the continuous limit as the finite element approximation approaches the continuum model. Re-summation techniques are integrated into the design to enhance the domain of applicability of the approach. The feedforward control is combined with industry-standardized proportional-integral-derivative (PID) output error feedback control within the so-called two-degree-of-freedom (2DOF) control concept. Simulation and experimental results obtained for a fully equipped forming tool are presented and confirm the applicability of the proposed design technique and the tracking performance. In addition, the results of this paper present a first experimental validation of flatness-based trajectory planning for thermal systems with three-dimensional spatial domain.