A spatial multivariable SVR method for spatiotemporal fuzzy modeling with applications to rapid thermal processing

Many industrial processes have significant spatiotemporal dynamics and they are usually called distributed parameter systems (DPSs). Modeling such system is challenging due to its nonlinearity, time-varying dynamics, and spatiotemporal coupling. Using model reduction techniques, traditional DPS modeling methods usually reduce an infinite-dimensional system to a finite-dimensional system, which leads to unknown nonlinearity and unmodeled dynamics. The modeling method and the established model are hard to understand. Here, we propose a spatial multivariable support vector regression (SVR) based three-domain (3-D) fuzzy modeling method for complex nonlinear DPSs. The proposed 3-D modeling method integrates the time-space separation and time-space synthesis into a 3-D fuzzy model. Therefore, it does not require model reduction and owns the capability of linguistic interpretability. A spatial multivariable SVR with spatial kernel functions is proposed to deal with spatiotemporal data. The spatial fuzzy basis functions from a 3-D fuzzy model are spatial kernel functions for a spatial multivariable SVR, which satisfy Mercy theorem. Hence, the spatial multivariable SVR can be directly employed to build up a complete 3-D fuzzy rule-base of the 3-D fuzzy model. The proposed modeling method integrates the merits of learning ability from a spatial multivariable SVR and fuzzy space processing and fuzzy linguistic expression from a 3-D fuzzy model. The proposed 3-D fuzzy modeling method is successful applied to a simulated rapid thermal processing system. In comparison with several newly developed modeling methods for DPSs, the simulation results validate the superiority of the proposed modeling method.