Dynamic Nonlinear Hebbian Learning Constrained Optimization in Fuzzy Cognitive Maps: Application to a Process Control Benchmark

Abstract

Fuzzy Cognitive Maps (FCMs) are commonly used to model complex systems where variables interact through causal relationships and feedback. In practical control environments, some variables must remain within defined bounds to ensure safe and stable operation. However, many existing learning algorithms for FCMs do not consider the operational limits that often exist for important system outputs. This study addresses this limitation by proposing a Dynamic Nonlinear Hebbian Learning (D-NHL) algorithm that incorporates constraint awareness directly into the weight update process. The proposed method modifies the traditional NHL rule by introducing conditional updates based on whether selected output concepts satisfy predefined operational bounds. The algorithm was evaluated using a benchmark chemical process control problem involving the regulation of liquid height and specific gravity in a mixing tank. Results from the experiments show that both the conventional NHL algorithm and the proposed D-NHL method achieve convergence during learning. However, the D-NHL algorithm consistently guides the system toward equilibrium states that keep the critical output variables within the specified limits. Quantitative evaluation also shows a slight improvement in predictive performance, with the mean squared error (MSE) reduced from 0.072 to 0.069. These findings suggest that embedding constraint handling within the learning process improves the suitability of FCM models for control-oriented applications where operational limits must be maintained. This makes the approach relevant for areas such as industrial process monitoring and other decision-support environments. Future work will focus on applying the method to larger FCM structures and validating its performance using real-world datasets and operational systems.