ARX model identification for thermal virtual sensors in a vacuum heating furnace

When heating a heterogeneous, aluminum load of several cubic meters in an industrial vacuum furnace, autoregressive with exogenous inputs (ARX) models can be used as virtual sensors to estimate the temperature at certain points on the load. The optimal orders of ARX models are selected via minimization of the corrected Akaike information criterion ($AICc$), supplemented by the moderate contribution condition (MCC), which detects hidden model instability. The best ARX model inputs are a weighted average of heating powers via particle swarm optimization (PSO) to avoid input-power correlation, as well as the temperature at the center of the load to complete the boundary conditions. This paper reaches the identification of parsimonious and robust ARX models on industrial data for most cases, with a measurement-model deviation of less than 2% of the temperature variation to be calculated.