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.
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