A Conjugate Gradient Method for the Solution of the Inverse Heating Problem in Thermoforming

Finding the setpoints of heater temperatures such that the sheet will achieve the desired temperature at the end of the heating cycle is known as the inverse heating problem (IHP) in thermoforming. Although a major portion of the heat is transferred from the oven to the sheet by radiation, conduction and convection also have a significant contribution in sheet heating. This makes the inverse heating problem more complex. In addition, the IHP is often ill-posed in a thermoforming process, making the solution unstable. In this paper, a conjugate gradient method is used to solve the IHP in the control of sheet temperature. In developing the method, computational cost is considered such that it can be implemented as a real-time algorithm in the controller. The performance of the proposed method for solving IHP, and thereby the corresponding sheet temperature controller, are tested in simulation at different operating conditions and compared with the conventional method of solving IHP based on the pseudo-inverse of the view factor matrix. The efficiency of the proposed method for the estimation of heater setpoints and the accuracy of the controller are evidenced by the results, showing promise for prospective real-time applications.