Frequency Response Based Curve Fitting Approximation of Fractional–Order PID Controllers

Abstract

Fractional-order PID (FOPID) controllers have been used extensively in many control applications to achieve robust control performance. To implement these controllers, curve fitting approximation techniques are widely employed to obtain integer-order approximation of FOPID. The most popular and widely used approximation techniques include the Oustaloup, Matsuda and Cheraff approaches. However, these methods are unable to achieve the best approximation due to the limitation in the desired frequency range. Thus, this paper proposes a simple curve fitting based integer-order approximation method for a fractional-order integrator/differentiator using frequency response. The advantage of this technique is that it is simple and can fit the entire desired frequency range. Simulation results in the frequency domain show that the proposed approach produces better parameter approximation for the desired frequency range compared with the Oustaloup, refined Oustaloup and Matsuda techniques. Furthermore, time domain and stability analyses also validate the frequency domain results. stable performance, especially for higher-order systems. Moreover, the controller can easily attain the iso-damping property (Monje et al., 2010; Shah and Agashe, 2016). It should be noted that seven different configurations can be achieved with the PI λ D μ controller (i.e., P, PI, PI λ , PD, PD μ , PID and PI λ D μ ) (Xue et al., 2007;Xue, 2017;Monje et al., 2010;Kishore et al., 2018; Shah and Agashe, 2016). However, a key issue with the practical realization or equivalent circuit implementation of such controllers in a finite-dimensional integer-order system is the approximation of the fractional-order parameters. This has generated a lot of interest among researchers recently.For effective approximation of a fractional-order integrator and differentiator in the PI λ D μ controller,

The Frequency and Real-Time Properties of the Microcontroller Implementation of Fractional-Order PID Controller

“…The FOBD approximation described by (13) with coefficients computed using formula (15) will be used in microcontroller implementation presented in the next section. Its length L should be relatively long to obtain good accuracy.…”

“…The authors focused on examining control performance, disturbance rejection properties, and the trajectory tracking ability of the system. The frequency properties are presented in the article [13]. It compares Oustaloup, refined Oustaloup, and Matsuda approximations.…”

Beyond the particular case of circuits with geometrically distributed components for approximation of fractional order models: Application to a new class of model for power law type long memory behaviour modelling

“…In most of these methods the fractional model is replaced by a classical integer model under various forms: continuous time model, discrete time model, electrical network (it is often simple to go from one form to another). Some apply to the full fractional model such as frequency domain fitting based methods [1] . But, as a fractional integrator chain appears in a fractional model, a large part of the proposed methods concentrates on the approximation of the fractional integrator of transfer function .…”

Design of Cascaded and Shifted Fractional-Order Lead Compensators for Plants with Monotonically Increasing Lags

“…In recent years, researchers proposed several other techniques, among which one may consider [61][62][63][64][65][66][67]. Namely, it was pointed out that fractional-order lead compensators found difficult application in industry, because it is difficult to obtain the desired functions with the commercially available electronic components, such that special methods that are based on operational amplifiers and field programmable analog arrays should be used [61].…”

“…Other peculiar techniques could take advantage from orthonormal rational basis functions, which can be used to improve fitting and approximation of frequency response, as shown in [64]. Some other authors proposed curve fitting techniques (with built-in Matlab functions) that use frequency response data of fractional-order operators to improve the ORA, the modified ORA, or the Matsuda approximation [65]. A Scilab Based Toolbox is also available for fractional-order operators, transfer functions, filters, and controllers [66].…”