Fractional-Order Calculus Filter Theory,Application And Fractioanal-Order System Identification

Fractional-order calculus is a generalization of the integer-order calculus. Its order can be arbitrary complex number. Because the fractional-order calculus extends the power of the integer-order calculus with the continuous order differentiation and integration, Fractional-order calculus, as a new tool, have been applied in many research fields.With the need of development of signal processing, we take the fractional-order calculus as the new research tool to generalize or improve some theories and methods in signal processing.Firstly, the characteristics of the ideal fractional-order calculus filters are analyzed and many kinds of implementation methods of the fractional-order calculus filters are summarized. In order to select or design an appropriate fractional-order calculus filters for the particular case, performances of the fractional-order calculus filters designed in different ways are analyzed and compared.Secondly, a kind of fractional-order difference filters are designed, whose theory is worked out and amplitude—frequency characteristic are analyzed. When the frame of the fractional-order calculus filters is given, coefficients of these filters still can change with its orders. The appropriate orders filters have the capability of smooth. At the same time, it overcame the shift of the edge using fractional-order calculus filter directly. In order to evaluate the performances of the fractional-order difference filters, these filters are used to detect the edge of the objective. Results show that there are the same results for the objective without noise, but it can obtain better results for the objective with noise.Thirdly, using fractional differential as a new tool, a new resolution method for the overlapped peak is presented. The estimation model I and the estimation modelⅡof three kinds of peak signals are modeled based on the relationship of the zero-crossing vs. its corresponding differential order and the extremum vs. its corresponding differential order of three kinds of peak signals. Characteristic paramters of the Gaussian peak, Lorentzian peak, and Tsallis peak can be estimated using EstimatorⅠand EstimatorⅡ. To validate the correctness and validity of the proposed methods, parameters of the individual peak signal and the overlapping peaks signal which include two individual peaks are estimated using the data with noise and the data without noise respectively. In additional to, the influence of peak-height ratio, peak-width ratio, separate degree, and SNR on parameter estimation are also analyzed for resolving the overlapping peaks signal. The universality of the proposed method is validated by resolving the overlapping peaks signal including three individual peaks and four individual peaks. The proposed method simplifies the resolution process of the overlapping peaks by combining two steps of traditional methods, by determining positions of the overlapped peaks and by decomposing the overlapped peaks into component peaks. With the help of the fractional differential, the proposed method not only holds the advantages of derivative method, but also reduces the influence of the noise and obtains the parameters of the individual peaks.At last, the least squares algorithm, weighted iteration method algorithm, recursive least squares algorithm and total least squares algorithm for identification of the fractional-order systems are presented in frequency domain. The correctness and validity of these algorithms are verified with the identification of the known systems. Results show that the algorithm is not only fit for identification of the fractional-order systems, but also fit for identification of integer-order systems. Further more, these algorithms are numerically more stable than in the case of integer-order using the same model structure.