Time is considered as an important factor in modeling and operation of dynamic systems. However, few studies have considered time factor in modeling and inference of fuzzy cognitive maps (FCMs), besides, no studies have dealt with time delay in learning of FCMs. Therefore, we propose a learning rule for temporal FCMs involving post- and pre-delay time by extending Oja's learning rule. We show the effectiveness of the proposed rule through simulations which solve a time-delayed chemical plant control problem.

Fuzzy cognitive maps (FCMs) are used to support decision-making, and the decision processes are performed by inference of FCMs. The inference greatly depends on activation functions such as sigmoid function, hyperbolic tangent function, step function, and threshold linear function. However, the sigmoid functions widely used for decision-making processes have been designed by experts. Therefore, we propose a method for designing sigmoid functions through Lyapunov stability analysis. We show the usefulness of the proposed method through the experimental results in inference of FCMs using the designed sigmoid functions.

A system based on application of Fuzzy Cognitive Map (FCM) to perform on-line fault diagnosis is presented. The diagnostic part of the system is composed of two diagnostic schemes. The first one (basic diagnostic algorithm) can be considered as a simple transition of Shiozaki's signed directed graph approach to FCM framework. The second one is an extended version of the basic diagnostic algorithm where an important concept, the temporal associative memories (TAM) recall of FCM, is adopted. In on-line application, self-generated fault FCM model generates predicted pattern sequence through the TAM recall process, which is compared with observed pattern sequence to declare the origin of fault. As the resultant diagnosis scheme takes short computation time, it can be used for on-line fault diagnosis of large and complex processes, and even for incipient fault diagnosis. In practical case, since real observed pattern sequence may be different from predicted one through the TAM recall owing to propagation delay between process variables, the time indexed fault FCM model incorporating delay time is proposed. The utility of the proposed system is illustrated in fault diagnosis of a tank-pipe system.