Page 10 - IJOCTA-15-3

A. Kaveh, M. Vahedi, M. Gandomkar / IJOCTA, Vol.15, No.3, pp.379-395 (2025)
DC motors that rely on mechanical commuta-
tors, BLDC motors offer several advantages, in-  di d = 1 [v d − Ri d + ωL q i q ]
 dt L d

cluding a higher torque-to-weight ratio, improved di q 1
efficiency, greater torque output per watt, in- dt = L d [v q − Ri q − ωL d i d − ωψ r ]

 dω 1
creased reliability, reduced noise, and extended dt = J [n p ψ r i q + n p (L d − L q )i d i q − T L − βω]
lifespan due to the elimination of brush and com- (1)
mutator wear. Additionally, removing ionizing
where (v , i d ) and (v , i q ) are the d-q voltage
q
d
sparks reduces electromagnetic interference, mak-
and current of the electric motor, L q and L d are
ing BLDC motors ideal for precision applications.
the stator inductances, and R is the stator re-
Their efficiency is notably superior in no-load and
sistance. ψ r , β, and J are the fixed magnetic
low-load regions, primarily due to the absence of flux, friction coefficient, and polar moment of
friction losses associated with brushes. 26,27
inertia. The number of pairs of poles is repre-
sented by n p . T L is the external load torque,
This study makes the following key assump- and ω is the rotor angular velocity. Based on an
tions to ensure a realistic and practical evaluation agreement, Equation (1) is simplified using the
of the FO-SMC for BLDC motors: following transformations. 26
 
bk 0 0
• Bounded motor parameter uncertainty: Assuming T =  0 k 0  , where b =
The motor parameters (resistance, induc- 0 0 R/L q
tance, and back electromotive force coef- βR ψ r β L q
L q /L d , k = , a γ = , σ = ,
ficients) are assumed to be within a rea- L q n q ψ r kL q R J
2 2
n pb L q k ((L d −L q ))
sonable uncertainty range, aligning with u d = v d , u q = v q , v = ,
R k R k J R 2
2
manufacturer-provided tolerances. This L q T L ′
T L = 2 , and t = (R t)/L q . State variables
f
ensures feasibility without assuming unre- J R −1
are obtained as (g. ) = T (.). Using these
alistic, completely unknown parameters.
transformations, Equation (1) is obtained as the
• Fractional-order dynamics for accuracy: 26
following set of dimensionless equations :
Since BLDC motors exhibit memory-
dependent behaviors, fractional-order

modeling provides a more precise rep-  d e i d ′ = u d − µ i d + eωi q
e
e
 dt

resentation than integer-order models, d e i q ′ = u q − i q − eωi d + γeω
e
e
ensuring better energy dissipation and  dt h i
 deω

hereditary effects. dt = σ i q i q − eω + i d + v i d i q − T L
e
f
e e
e e
• Bounded external disturbances and load (2)
variations: Practical disturbances like
such that σ, γ, µ, and v are the structural pa-
friction and load torque changes are nat-
rameters of the dynamic system of the motor
urally bounded. Assuming finite limits
after transformations and are the d-q reference
ensures the stability of the FO-SMC, as
voltage and current of the electric motor. f
T L
real-world controllers cannot compensate
is the load torque after transformation and eω is
for unbounded disturbances.
the rotor angular velocity after transformation.
• Grunwald–Letnikov (GL) definition for
f
f
fractional derivatives: The GL method is Assuming i q = x 1 , i d = x 2 , and eω = x 3 are
similar to the Lorentz system, the dynamic equa-
chosen for fractional derivative computa-
tions of the BLDC system are converted to the
tions due to its computational efficiency,
state space form:
direct discrete approximation, and real-
time feasibility, making it widely used in

electromechanical systems.  ˙x 1 = −µx 1 + x 2 x 3 + u d

• Negligible sensor noise and quantization ˙ x 2 = −x 2 − x 1 x 3 + γx 3 + u q (3)
effects: High-precision sensors and the in-  ˜

˙ x 3 = −σ(x 3 − x 2 ) − T L + vx 1 x 2
herent robustness of FO-SMC minimize
the impact of sensor noise, making de- The fractional order form of such a system is
16,28
tailed noise modeling unnecessary for this as follows :
study.

q 1
 D x 1 = −µx 1 + x 2 x 3 + u d
0 t
q 2 (4)
The dynamic equations of a BLDC electric 0 D x 2 = −x 2 − x 1 x 3 + µx 3 + u q
t
 q 3 ˜
26,27
motor are given in the following Equation (1) :  0 D x 3 = −σ(x 3 − x 2 ) − T L + vx 1 x 2
t
382