Prompted by the need to determine a unique tuning parameter, the paper proposes a conceptual model for a high temperature industrial furnace in which the notion of an unmeasurable "global temperature" plays a key role. The developed approach is designed to accommodate a priori knowledge of system nonlinearities, as well as the effects of an externally triggered burner firing cycle which gives rise to an oscillatory response present on the locally measured zone temperature. It is assumed that the variation in the global temperature is lower than that observed at the locally measured temperature at a point. The developed model is able to accommodate, and effectively separate, the effects of the combustion burner firing cycle and the underlying bilinear characteristic behaviour relating fuel combustion and temperature of each zone. It is assumed that these physical phenomena can be represented by a linear model, modelling the effects of the combustion burner firing cycle, cascaded with a bilinear model, where an intermediate variable is the global temperature. The result of employing a regularisation technique together with the cascade multiple model produces an optimal value for the tuning parameter of a four-term bilinear PID control system.