Modelling, Identification and Control of Thermal Deformation of Machine Tool Structures, Part 3: Real-Time Estimation of Heat Sources

Compensation of thermal deformation of machine tools requires real-time estimation of the heat input to the structure in order to fully describe its thermoelastic response. Available solutions of the inverse heat conduction problem IHCP are not suitable for real-time feedback control applications, since they are too slow and/or rely on future data to stabilize the solution. A new real-time IHCP solver is derived in the form of a convolution integral of the inverse thermal transfer function G−1(s) and the measured temperature difference at two points near the heat source. An expression for G−1(s) is derived for multi-dimensional structural components. To transform G−1(s) to the time domain, a special consideration is given to the treatment of its complex singularity functions. Analytical approach was followed to identify these functions and to determine their time-domain representation. Computer-simulation test cases were conducted using a finite element model of a three-dimensional structure. The random temperature measurement errors, which can lead to non-uniqueness and instability problems, have also been simulated. The test results showed that the computation time can significantly be improved to achieve a control cycle of less than one second, without compromising the accuracy and stability requirements.