Integer-fractional decomposition and stability analysis of fractional-order nonlinear dynamic systems using homotopy singular perturbation method

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.