Multiscale fuzzy Kalman filtering

Measured data are usually contaminated with errors which sometimes mask their important features. Therefore, data filtering is needed for effective utilization of such measurements. For nonlinear systems which can be described by a Takagi–Sugeno (TS) fuzzy model, several fuzzy Kalman (FK) filtering algorithms have been developed to extend Kalman filtering to such systems. Also, multiscale representation of data is a powerful data analysis tool, which has been successfully used to solve several data filtering problems. In this paper, a multiscale fuzzy Kalman (MSFK) filtering algorithm, in which multiscale representation is utilized to improve the performance of fuzzy Kalman filtering, is developed. The idea is to apply FK filtering at multiple scales to combine the advantages of the FK filter with those of the low pass filters used in multiscale data representation. Starting with a fuzzy model in the time domain, a similar fuzzy model is derived at each scale using the scaled signal approximation of the data obtained by stationary wavelet transform (SWT). These multiscale fuzzy models are then used in FK filtering, and the FK filter with the least cross validation mean square error among all scales is selected as the optimum filter. Also, theoretically, it has been shown that applying FK filtering at a coarser scale than the time domain is equivalent to using a time-averaged FK filter. Finally, the performance of the developed MSFK filtering algorithm is illustrated through a simulated example.