Fuzzy grey cognitive maps and nonlinear Hebbian learning in process control

Fuzzy Grey Cognitive Maps (FGCM) is an innovative Grey System theory-based FCM extension. Grey systems have become a very effective theory for solving problems within environments with high uncertainty, under discrete small and incomplete data sets. In this study, the method of FGCMs and a proposed Hebbian-based learning algorithm for FGCMs were applied to a known reference chemical process problem, concerning a control process in chemical industry with two tanks, three valves, one heating element and two thermometers for each tank. The proposed mathematical formulation of FGCMs and the implementation of the NHL algorithm were analyzed and then successfully applied keeping the main constraints of the problem. A number of numerical experiments were conducted to validate the approach and verify the effectiveness. Also, the produced results were analyzed and compared with the results previously reported in the literature from the implementation of the FCMs and Nonlinear Hebbian learning algorithm. The advantages of FGCMs over conventional FCMs are their capabilities (i) to produce a length and greyness estimation at the outputs; the output greyness can be considered as an additional indicator of the quality of a decision, and (ii) to succeed desired behavior for the process system for every set of initial states, with and without Hebbian learning.