Towards Smooth Monotonicity in Fuzzy Inference System based on Gradual Generalized Modus Ponens

Monotonic relationship between input and output often is an inherent property of fuzzy systems. For example, for a cart-pole system, there is a gradual relationship between the pole angle and the cart velocity of the form: “The more the pole deviates from the upright position, the higher velocity the cart must gain”. Thus, we propose an approach, that we call Gradual GMP, which is able to integrate the graduality, when this underlying hypothesis exists, into the generalized modus ponens. In this paper, we proposed to study in a simple case (crisp observations) the monotonic response of what could be the adaptation to this case of GGMP-based FIS. We showed that for a single rule FIS, the GGMP induces strict monotonocity, to be compared with static response in the standard case (here Mamdani controller). Further experimental results, on single and double input FIS, show that the proposed method implies a smooth monotone behavior even when in presence of several linguistic values for each variable. Finally, this work reveals that the only way to guarantee strict monotonicity is to have an inference that has a gradual behavior.