Stability analysis for a class of fractional-order nonlinear systems with time-varying delays
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METHODOLOGIES AND APPLICATION
Stability analysis for a class of fractional-order nonlinear systems with time-varying delays Pourya Rahmanipour1 • Hamid Ghadiri1
Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper presents the stability analysis problem of fractional-order nonlinear systems with time-varying delay. After formulating the problem and selecting the nonlinear model as the system under study, stability analysis and expression of the sufficient conditions for fractional-order nonlinear systems with time-varying delay are obtained using two different methods. In these methods, sufficient conditions for stability of fractional-order nonlinear systems are found in the form of satisfying some inequalities based on norms of nonlinear functions in the system and in terms of linear matrix inequality according fractional-order and nonlinear functions. In each case, despite the presence of time-varying delay, the system stability is ensured by meeting the stability sufficient conditions in terms of an inequality of functions and system parameters. Finally, numerical examples are given to determine the effectiveness of the proposed theorem. Keywords Nonlinear systems Fractional-order systems Time-varying delay Stability analysis
1 Introduction Fractional-order calculations have been used in mathematics many years ago. This mathematical phenomenon makes it possible to describe a real phenomenon in a more accurate manner compared to a rational-order form. Today, it has been proved that the dynamic and real model of most systems is not in the conventional rational-order form and it can be described in fractional-order form (Chen et al. 2005; Petras 2009; Matignon 1996; Petras et al. 2004). Overall, fractional-order systems are useful in modeling diverse stable physical phenomena. In the area of modeling in the continuous-time domain, fractional-order differentiations have shown their efficacy in linear viscoelasticity, rheology, chemistry, biophysics, and so on (Matignon 1996; Petras et al. 2005). Fractional-order systems have
Communicated by V. Loia. & Hamid Ghadiri [email protected] Pourya Rahmanipour [email protected] 1
Faculty of Electrical, Biomedical and Mechatronics Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran
also been studied from various aspects, including stability analysis (Tavazoei and Haeri 2009; Shu and Zhu 2018), system identification (Hartley and Lorenzo 2008), control (Tavazoei and Haeri 2008a; Wang et al. 2019), synchronization (Tavazoei and Haeri 2008b; Mohammadzadeh et al. 2018), dynamic behavior analysis of systems (Tavazoei et al. 2008), and some other related fields. Due to the freedom of action available in integrator and differentiator in fractional calculus, it is viable to model physical systems with a very high precision. On the other hand, the delays virtually affect all practical systems and they may interfere with system performance or even cause system instability. In th
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