Fractional-order systems have been applied in many engineering applications. A key issue with the application of such systems is the approximation of fractional-order parameters. The numerical tools for the approximation of fractional-order parameters gained attention recently. However, available toolboxes in the literature do not have a direct option to approximate higher order systems and need improvements with the graphical, numerical, and stability analysis. Therefore, this chapter proposes a MATLAB-based GUI for the approximation of fractional-order operators. The toolbox is made up of four widely used approximation techniques, namely, Oustaloup, refined Oustaloup, Matsuda, and curve fitting. The toolbox also allows numerical and stability analysis for evaluating the performance of approximated transfer function. To demonstrate the effectiveness of the developed GUI, a simulation study is conducted on fractional-order PID control of pH neutralization process. The results show that the toolbox can be effectively used to approximate and analyze the fractional-order systems.
TopIntroduction
The modeling and controlling possibilities of fractional-order systems and controllers gained a lot of attention in the last two decades. However, a key issue with the implementation of such systems is the approximation of the fractional-order parameters (Kar & Roy, 2018; Machado, Kiryakova, & Mainardi, 2011; Matušů, 2011; Shah & Agashe, 2016). For an effective approximation of such parameters, researchers have proposed several time domain and frequency domain techniques (Deniz, Alagoz, Tan, & Atherton, 2016; Du, Wei, Liang, & Wang, 2017; Krishna, 2011; Li, Liu, Dehghan, Chen, & Xue, 2017; Monje, Chen, Vinagre, Xue, & Feliu-Batlle, 2010; Vinagre, Podlubny, Hernandez, & Feliu, 2000). However, it is very difficult to select the best method among these proposals. This is because, each of these methods has its strength and weakness which could be in terms of the order of approximation, the accuracy of frequency and time responses (Djouambi, Charef, & BesançOn, 2007; Tepljakov, 2017b). Furthermore, researchers have developed few toolboxes such as Commande Robuste d'Ordre Non Entier (CRONE) (Malti, Melchior, Lanusse, & Oustaloup, 2011, 2012; Oustaloup, Melchior, Lanusse, Cois, & Dancla, 2000), Non-Integer (Ninteger) (de Oliveira Valério, 2005; Valério & Da Costa, 2004), Fractional Order Modeling and Control (FOMCON) (Tepljakov, 2017b; Tepljakov, Petlenkov, & Belikov, 2011; Tepljakov, Petlenkov, Belikov, & Finajev, 2013) and fractional-order PID (Lachhab, Svaricek, Wobbe, & Rabba, 2013) for fractional-order modeling and control applications. A common feature of these toolboxes is that they are based on MATLAB/SIMULINK software.
The CRONE toolbox was developed by CRONE research team and is dedicated to the application of fractional-order derivatives in science and engineering. The toolbox allows for mathematical modeling, system identification and CRONE control design (Lanusse, Malti, & Melchior, 2013; Malti et al., 2011, 2012; Oustaloup et al., 2000). On the other hand, the Ninteger toolbox developed by Valerio presents the implementation of non-integer PID controllers and CRONE controllers (Valério & Da Costa, 2004). The toolbox also allows for the approximation of non-integer order derivatives, functions for model development and frequency response plots. Unlike the aforementioned toolboxes, the FOMCON toolbox incorporates advanced features such as system identification, fractional-order PID control, real-time implementation etc. The toolbox is an extension of the fractional-order transfer function (FOTF) toolbox for fractional-order system identification and PID controller design. Similarly, the authors in (Lachhab et al., 2013) developed a fractional-order PID (FOPID) toolbox for designing and tuning the controller using the steepest descent method. A summary of these toolboxes explaining the features and limitations is given in Table 1.