Distributed parameter modeling and finite-frequency loop-shaping of electromagnetic molding machine

We derive a mathematical model for an electromagnet inside a molding machine, and propose a novel loop-shaping method of the proportional–integral (PI) controller design for the system based on the generalized KYP (GKYP) lemma. The behavior of the molding machine is difficult to capture by using finite-dimensional models owing to eddy currents spatially distributed throughout the electromagnet. To analyze fundamental properties of the system both theoretically and experimentally, we first derive a mathematical model of the machine in terms of a partial differential equation (PDE). An analysis using the PDE model shows that a low-dimensional approximation performed by standard spatial discretization results in a spillover effect, which makes the behavior of the closed-loop system oscillatory. Then, to develop an easily tunable and implementable control system, we propose a novel loop-shaping method for PI control on the basis of the GKYP lemma. In this control system design, we use multiple low-dimensional models, which work simultaneously in specified finite frequency ranges. The proposed method successfully suppresses the spillover effect despite the use of low-dimensional approximants. Finally, we show the efficiency of the proposed control design method through numerical and experimental verification and discuss a performance limitation of the PI control.