Fuzzy feedback system analysis using transition matrices

Parameter Identification of Recurrent Fuzzy Systems With Fuzzy Finite-State Automata Representation

“…where i C and j B are multi-dimensional fuzzy sets like before, and k A is a fuzzy set defined on the state variables The output of the linguistic recurrent fuzzy system is computed as an ordinary fuzzy system, choosing the sumproduct inference and singleton defuzzification [19]- [21]. The result of sum-product inference is computed as:…”

“…Fernándes and Lópes [21] presented a similar fuzzy recurrent system, called by the authors Fuzzy Feedback System (FFS). In their system, non-weighted rules are used to compute the fuzzy membership values that are directly fed back into the inputs, without a deffuzification step as in the approach by Adamy and Kempf [19].…”

“…Although related to slightly different models, the results presented by Kempf and Adamy [20] and Fernándes and Lópes [21] show that RFS models described by nonweighted rules as (18) can be used to adjust stable, oscillatory or even chaotic systems. Theoretical results to characterize the RFS model described by weighted rules as (21) are still an open issue.…”

“…Adamy and Kempf [19] show that recurrent fuzzy systems can be directly encoded into a FFA and their dynamical behavior may be deduced from their rule base configuration [20]. Fernándes and Lópes [21] show that, under certain assumptions, recurrent fuzzy systems are asymptotically stable, independently of the initial state. These results, when combined with the universal approximation property of general fuzzy systems [22] [23], yield to the perspective that recurrent fuzzy systems identification and FFA representation may provide an efficient, accurate and sound procedure for identification and analysis of nonlinear dynamic systems.…”