Back-propagation fuzzy system as nonlinear dynamic system identifiers

The authors develop a training algorithm, similar to the backpropagation algorithm for neural networks, to train fuzzy systems to match desired input-output pairs. The key ideas in developing this training algorithm are to view a fuzzy system as a three-layer feedforward network, and to use the chain rule to determine gradients of the output errors of the fuzzy system with respect to its design parameters. It is shown that this training algorithm performs an error backpropagation procedure: hence, the fuzzy system equipped with the backpropagation training algorithm is called the backpropagation fuzzy system (BP FS). An online initial parameter choosing method is proposed for the BP FS, and it is shown that it is straightforward to incorporate linguistic if-then rules into the BP FS. Two examples are presented which demonstrate (1) how the fuzzy system learns to match an unknown nonlinear mapping as training progresses and (2) that performance is improved by incorporating linguistic rules