A highly adaptive algorithm for fuzzy modelling of systems (Q2701905)

The study is concerned with the optimization of fuzzy models governed by fuzzy relational equations of the form NEWLINE\[NEWLINE\begin{multlined} Y(k)= Y(k-1)\bullet Y(k-2)\bullet\cdots\bullet Y(k-p)\bullet U_1(k- t_1- 1)\bullet\cdots\bullet U_1(k- t_1- q_1)\bullet\\ U_r(k- t_r- 1)\bullet\cdots\bullet U_r(k- t_r- q_r)\bullet R,\end{multlined}NEWLINE\]NEWLINE where ``\(\bullet\)'' denotes a max-\(t\) composition operator applied to input fuzzy sets \((U_1,U_2,\dots, U_r)\) and state \((Y)\) while \(R\) is an unknown fuzzy relation of the model. The general structure of the model is the one introduced by the reviewer and involves three functional units, namely an input interface, processing module (realized here by the relational equation) and an output interface. An identification of the fuzzy model concentrates on an optimization of the fuzzy relation \((R)\) of the processing module for which the authors developed an iterative optimization algorithm. The study comes with a number of illustrative numerical examples including discrete-time series.