Multi-scale fuzzy modeling of nonlinear dynamical systems

Multi-scale fuzzy modeling of multivariable discrete-time nonlinear dynamical systems is approached analytically in this paper. The underlying idea of this approach is to introduce the idea of multi-scale fuzzy quantization. More globally supported membership functions are used at coarser scales to capture global characteristics of the unknown system, whereas more locally supported membership functions are used at finer scales to fill in the local features. Starting from the coarsest scale down to finer ones, a projection method is used to extract a partial fuzzy-model (in terms of fuzzy membership functions and fuzzy rules) from the residual of the system at each scale until the residual drops below a prescribed modeling error bound. The multi-scale fuzzy model is then obtained by assembling the partial fuzzy models together. To reach a near optimal fuzzy model, a rule reduction technique has also been developed to delete some fuzzy rules. An illustrative example is used to demonstrate the effectiveness of the scheme.