The article focuses on the fractional-order backward difference, sum, linear time-invariant equation analysis, and difficulties of the fractional calculus microcontroller implementation with regard to designing a fractional-order proportional integral derivative (FOPID) controller. In opposite to the classic proportional integral derivative (PID), the FOPID controller is defined by five independent parameters. Hence, it is more customizable and, potentially, more precise on condition that the values of fractional integration and differentiation orders are properly selected. However, a number of operations and the time required to calculate the output signal continuously increase. This can be a significant problem considering the limitations of a microcontroller, including memory size and a constant sampling time of the set-up analog-to-digital (ADC) converters. In the article, three solutions are considered, and results obtained in the experiments are presented.
The optimal design of excitation signal is a procedure of generating an informative input signal to extract the model parameters with maximum pertinence during the identification process. The fractional calculus provides many new possibilities for system modeling based on the definition of a derivative of noninteger-order. A novel optimal input design methodology for fractional-order systems identification is presented in the paper. The Oustaloup recursive approximation (ORA) method is used to obtain the fractional-order differentiation in an integer order state-space representation. Then, the presented methodology is utilized to solve optimal input design problem for fractional-order system identification. The fundamental objective of this approach is to design an input signal that yields maximum information on the value of the fractional-order model parameters to be estimated. The method described in this paper was verified using a numerical example, and the computational results were discussed.
The paper presents an interpretation of fractional calculus for positive and negative orders of functions based on sampled measured quantities and their errors connected with digital signal processing. The derivative as a function limit and the Grünwald-Letnikov differintegral are shown in chapter 1 due to the similarity of the presented definition. Notation of fractional calculus based on the gradient vector of measured quantities and its geometrical and physical interpretation of positive and negative orders are shown in chapters 2 and 3.