Multiscale Fuzzy State Estimation using Stationary Wavelet Transforms

Multiscale representation of data is a powerful data analysis tool, which has been successfully used to solve several data filtering problems. For nonlinear systems, which can be represented by a Takagi-Sugeno fuzzy model, several Fuzzy Kalman filtering algorithms have been developed to extend Kalman filtering for such systems. 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 fuzzy Kalman filtering at multiple scales to combine its advantages 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 fuzzy Kalman filtering, and the fuzzy Kalman filter with the least cross validation mean square error among all scales is selected as the optimum filter. Finally, the performance of the developed MSFK filtering algorithm is illustrated through a simulated example.