Integer-fractional decomposition and stability analysis of fractional-order nonlinear dynamic systems using homotopy sin
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Integer‑fractional decomposition and stability analysis of fractional‑order nonlinear dynamic systems using homotopy singular perturbation method Mahnaz Abolvafaei1 · Soheil Ganjefar1,2 Received: 7 February 2020 / Accepted: 15 October 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract Achieving a simplified model is a major issue in the field of fractional-order nonlinear systems, especially large-scale systems. So that in addition to simplifying the model, the outstanding features of the fractional-order modeling, such as memory feature, are preserved. This paper presented the homotopy singular perturbation method (HSPM) to reduce the complexity of the model and use the advantages of both models of the fractional order and the integer order. This method is a combination of the fractional-order singular perturbation method (FOSPM) and the homotopy perturbation method (HPM). Firstly, the FOSPM is developed for fractional-order nonlinear systems; then, a modification of the HPM is proposed. Finally, the HSPM is presented using a combination of these two methods. fractional-order nonlinear systems can be divided into two lower-order subsystems such as nonlinear or linear integer-order subsystem and linear fractional-order subsystem using this hybrid method. Convergence analysis of tracking error to zero is theoretically presented, and the effectiveness of the proposed method is also evaluated with two examples. Next, the number and location of equilibrium points are compared between the original system and the subsystems obtained from the proposed method. In the end, we show that the stability of fractional-order nonlinear system can be determined by investigating the stability of the subsystems using Theorem 3 and Lemma 2. Keywords Integer–fractional-order system · Singular perturbation method · Homotopy perturbation method · Stability analysis · Model simplification · Convergence analysis
* Soheil Ganjefar [email protected] Extended author information available on the last page of the article
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Mathematics of Control, Signals, and Systems
1 Introduction Fractional calculus, as a mathematical field, has a history of more than 300 years. But its application in the modeling of industrial systems and natural phenomena has attracted particular attention in the last three decades [1–4]. The behavior of many systems can be described better and more precisely by using the fractionalorder modeling [5–7]. So that, adding the fractional-order parameter to the dynamic model can be caused more space and more flexibility in the system [8]. Hence, fractional calculus is applied to the modeling of many systems and natural phenomena, for example control theory [9], chaotic equations in control engineering [10], robotic [11], wind turbine [12, 13], diffusion process modeling [14], the nonlinear oscillation of earthquake [15], signal processing [1], mathematical finance [16], the heat transfer process [17], etc. In the field of engineering, most industrial s
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