Monotone Fuzzy Rule Interpolation for Practical Modeling of the Zero-Order TSK Fuzzy Inference System

Abstract

Formulating a generalized monotone fuzzy rule interpolation (MFRI) model is difficult. A complete and monotone fuzzy rule-base is essential for devising a monotone zero-order Takagi–Sugeno–Kang (TSK) fuzzy inference system (FIS) model. However, such a complete and monotone fuzzy rule-base is not always available in practice. In this article, we develop an MFRI modeling scheme for generating a monotone zero-order TSK FIS from a monotone and incomplete fuzzy rule-base. In our proposal, a monotone-ordered fuzzy rule-base that consists of the available fuzzy rules from a monotone and incomplete fuzzy rule-base and those derived from the MFRI reasoning is formed. We outline three important properties that the MFRI's deduced fuzzy rules should satisfy to ensure a monotone-ordered fuzzy rule-base. A Lagrangian function for the MFRI scheme, together with its Karush–Kuhn–Tucker optimality conditions, is formulated and analyzed. The key idea is to impose constraints that guide the MFRI inference outcomes. An iterative MFRI algorithm that adopts an augmented Lagrangian function is devised. The proposed MFRI algorithm aims to achieve an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\boldsymbol{\varepsilon }}$</tex-math></inline-formula> -optimality condition and to produce an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\boldsymbol{\varepsilon }}$</tex-math></inline-formula> -optimal solution, which is geared for practical applications. We apply the MFRI algorithm to a failure mode and effect analysis case study and a tanker ship heading regulation problem. The results indicate the effectiveness of MFRI for generating monotone TSK FRI models in tackling practical problems.