Abstract
This paper is concerned with the discretization of the fractional-order differentiator and integrator, which is the foundation of the digital realization of fractional order controller. Firstly, the parameterized Al-Alaoui transform is presented as a general generating function with one variable parameter, which can be adjusted to obtain the commonly used generating functions (e.g. Euler operator, Tustin operator and Al-Alaoui operator). However, the following simulation results show that the optimal variable parameters are different for different fractional orders. Then the weighted square integral index about the magtitude and phase is defined as the objective functions to achieve the optimal variable parameter for different fractional orders. Finally, the simulation results demonstrate that there are great differences on the optimal variable parameter for differential and integral operators with different fractional orders, which should be attracting more attentions in the design of digital fractional order controller.
Highlights
Fractional order calculus has a history of more than 300 years, which extends the order of the classical calculus from integer number to arbitrary real number and even complex number
This paper is concerned with the discretization of the fractional-order differentiator and integrator, which is the foundation of the digital realization of fractional order controller
The parameterized Al-Alaoui transform is presented as a general generating function with one variable parameter, which can be adjusted to obtain the commonly used generating functions (e.g. Euler operator, Tustin operator and Al-Alaoui operator)
Summary
Fractional order calculus has a history of more than 300 years, which extends the order of the classical calculus from integer number to arbitrary real number and even complex number. Some kind of generating function ω z−1 is used to discretize the ( ) differentiator s , i.e., sr = ωr z−1 , where r is the order of the fractional( ) order differentiator and ω z−1 is usually expressed as a function of the complex variable z or the shift operator z−1 , and some kind of expansion method is applied to generate the approximate digital filter of the differentiator. Zhu and Zou propose an improved recursive algorithm for fractional-order system solution based on PSE and Tustin operator [6] Miladinovic and his colleagues use genetic algorithm to minimize the deviation in magtitude and phase responses between the original fractional order element and the rationalized discrete time filter in IIR structure [7]. The discretization methods for fractional-order differentiator are compared based on Tustin operator and three different expansion algorithms in [8].
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