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Improving the performance of a chaotic nonlinear system of fractional-order...
R ∞ n−1 −t
It should be noted that considering fractional- where Γ (n) = t e dt, is the Gamma func-
0
order dynamics in many systems can significantly tion. The GL definition is a fundamental ap-
improve stability and increase accuracy in sys- proach for defining fractional derivatives and in-
tem behavior. The numerical solution of the tegrals. The fractional derivative of order n for a
6
fractional-order BLDC system is formulated us- function f(t) using this method is expressed as :
ing the GL fractional derivative, ensuring a more
precise representation of the system’s nonlinear n n
d −j X j n
dynamics. The governing fractional-order equa- ( n f(t) = lim h (−1) f(t − jh)) (7)
tions are expressed as follows: 5,6 dt h→0 j=0 j

This approach provides a discretized represen-
tation of fractional derivatives, making it compu-

x 1 (t k ) = −µx 1 (t k−1 ) + x 2 (t k−1 ) x (t k−1 )
 3 tationally efficient for numerical simulations. It

 P k (q 1 )
q 1
 +u d (t k−1 ) h − c x 1 (t k−j )

 j=v j is particularly suitable for systems with mem-

x 2 (t k ) = −x 2 (t k−1 ) − x 1 (t k ) x (t k−1 )


 3 ory effects and is widely used in electrical and
+ γ x 3 (t k−1 ) + u q (t k−1 ) h q 2
 P (q 2 ) mechanical applications, including BLDC motors.
 k
 − c x 2 (t k−j )
 j=v j

 The GL approach provides a powerful framework
 x 3 (t k ) = σ x 2 (t k ) − x 3 (t k−1 ) − T L (t k−1 )

f


 P k (q 3 ) for modeling fractional-order systems, particu-
+ v x 1 (t k ) x 2 (t k ) h − j=v j x 3 (t k−j )
c
 q 3
(5) larly those with memory effects and long-range
dependencies. Utilizing direct discretization elim-
inates the need for complex transformations, mak-
ing it an efficient and straightforward method for
where T sim is the simulation time, N =
numerical implementation. Moreover, its ability
[T sim /h] , and (x 1 (0) , x 2 (0) , x 3 (0) , ) are the
to capture hereditary system properties ensures
initial conditions. The numerical solution of the
accurate dynamic modeling, particularly in ap-
fractional-order BLDC system follows the GL def-
plications such as BLDC motors, where precise
inition, where the system’s discrete representation
control over chaotic behavior is essential. Ad-
depends on the total number of computational
ditionally, its computational stability makes it
steps, denoted as N. This parameter is essen-
ideal for real-time control applications, reinforc-
tial for ensuring numerical accuracy and stability.
The estimation of N is based on the total simula- ing its superiority over other fractional-order nu-
merical techniques. These advantages make the
tion time T sim and the chosen time step, defined
GL method preferred for solving fractional-order
as N = T sim /h. The selection of h plays a cru-
differential equations, ensuring theoretical rigor
cial role in balancing accuracy and computational
and practical feasibility in control design.
efficiency as a smaller h improves precision but in-
Since we are interested in the chaotic dynam-
creases the computational load, whereas a larger
ics of the system, we need to focus on the equilib-
h reduces complexity but may lead to numerical
rium points and parameter ranges where we ob-
errors or instability in chaotic systems.
serve chaos. To obtain the fixed points of the sys-
The time step should be chosen carefully to
tem (3), we examine it for µ = 1 and v = 0 under
maintain stability in fractional-order systems. A
no-load conditions ( u = u q = T L = 0 ) : 3,8
f
smaller step size ensures better differentiation ac- d
curacy but can significantly slow down computa-

tions, while a larger step size may result in ap-  ˙x 1 = −x 1 + x 2 x 3

proximation errors. Moreover, the memory effect
˙ x 2 = −x 2 − x 1 x 3 + γx 3 (8)
in fractional calculus requires a sufficient number 
 ˙ x 3 = −σ(x 3 − x 2 )
of computational steps to capture the long-term
system behavior properly. To avoid numerical di- Therefore, we have the following in equation :
vergence, the stability constraints and eigenvalues (
of the system must be analyzed, ensuring that the ˙ x 1 = 0, x 1 = 0 (9)
selected h allows proper system convergence. x 1 = x 2 x 3
(q)
The binomial coefficients c are computed
j
(
6
using the GL definition : x 2 = 0
˙ x 2 = 0, (10)
x 2 = x 3 (γ − x 1 )

(q) j q j Γ(q + 1) (
(c j = (−1) = (−1) ) x 3 = 0
j Γ(j + 1)Γ(q − j + 1) ˙ x 3 = 0, (11)
(6) x 2 = x 3
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