Abstract
In this paper, we consider a fractional-order model of a brushless DC motor. To develop a mathematical model, we use the concept of the Liouville–Caputo noninteger derivative with the Mittag-Lefler kernel. We find that the fractional-order brushless DC motor system exhibits the character of chaos. For the proposed system, we show the largest exponent to be 0.711625. We calculate the equilibrium points of the model and discuss their local stability. We apply an iterative scheme by using the Laplace transform to find a special solution in this case. By taking into account the rule of trapezoidal product integration we develop two iterative methods to find an approximate solution of the system. We also study the existence and uniqueness of solutions. We take into account the numerical solutions for Caputo Liouville product integration and Atangana–Baleanu Caputo product integration. This scheme has an implicit structure. The numerical simulations indicate that the obtained approximate solutions are in excellent agreement with the expected theoretical results.
Highlights
The newly emerging field has many applications to model the real-world phenomena such as electrode–electrolyte, diffusion wave, electromagnetic waves, dielectric polarization, and superdiffusion equations [1,2,3]
We introduce brushless DC motor (BLDCM) model of noninteger order, which displays the chaotic behavior too
12 Brushless DC motor model via AB–Caputo fractional derivative The AB–Caputo fractional order brushless DC motor model is defined by equation (11)
Summary
The newly emerging field has many applications to model the real-world phenomena such as electrode–electrolyte, diffusion wave, electromagnetic waves, dielectric polarization, and superdiffusion equations [1,2,3]. 12 Brushless DC motor model via AB–Caputo fractional derivative The AB–Caputo fractional order brushless DC motor model is defined by equation (11) We will use it with the initial conditions given in (30). Let us consider ABC initial value problem of fractional (noninteger) order. 14 Numerical implementation for ABC-PI method on equation (11) The following recursive formulas are obtained by applying the computational algorithm (72)–(73) to system (11): hτ τ Z(τ ). Example 8 Taking the iterative arrangement (74), we consider the following values of the parameters: σ = 0.875, β = 0.786, and γ = 4 with initial conditions ud(0) = 10, uq(0) = 10, and ua(0) = 10.
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