Abstract
This paper deals with the study and analysis of several rational approximations to approach the behavior of arbitrary-order differentiators and integrators in the frequency domain. From the Riemann–Liouville, Grünwald–Letnikov and Caputo basic definitions of arbitrary-order calculus until the reviewed approximation methods, each of them is coded in a Maple 18 environment and their behaviors are compared. For each approximation method, an application example is explained in detail. The advantages and disadvantages of each approximation method are discussed. Afterwards, two model order reduction methods are applied to each rational approximation and assist a posteriori during the synthesis process using analog electronic design or reconfigurable hardware. Examples for each reduction method are discussed, showing the drawbacks and benefits. To wrap up, this survey is very useful for beginners to get started quickly and learn arbitrary-order calculus and then to select and tune the best approximation method for a specific application in the frequency domain. Once the approximation method is selected and the rational transfer function is generated, the order can be reduced by applying a model order reduction method, with the target of facilitating the electronic synthesis.
Highlights
Arbitrary-order calculus is an important issue that has attracted the attention of research institutes, academic associations, industrial areas and foundry companies
To date, there is no robust passive element with arbitrary characteristics so that it can be used to describe the behavior of a system of arbitrary-order differential equations that models natural phenomena or human-made systems more realistically
Note that in each application, a different arbitrary-order approximation method was used; once the rational approximation is deduced, this can be synthesized using analog or digital platforms. This survey describes and analyzes the main approximation methods reported to date, but two model order reduction methods are reviewed in order to obtain rational transfer functions of reduced order and with the target of simplifying the electronic synthesis process
Summary
Arbitrary-order calculus is an important issue that has attracted the attention of research institutes, academic associations, industrial areas and foundry companies. To date, there is no robust passive element with arbitrary characteristics so that it can be used to describe the behavior of a system of arbitrary-order differential equations that models natural phenomena or human-made systems more realistically In response to this problem, integer-order rational approximations in the frequency domain were developed to approximate the arbitrary order with low level of error and wide bandwidth, such as Oustaloup’s [4], refined Oustaloup’s [5], Charef’s [6], Fractal Fract. Note that in each application, a different arbitrary-order approximation method was used; once the rational approximation is deduced, this can be synthesized using analog or digital platforms This survey describes and analyzes the main approximation methods reported to date, but two model order reduction methods are reviewed in order to obtain rational transfer functions of reduced order and with the target of simplifying the electronic synthesis process.
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