electronics

Article
A Robust Controller Based on Extension Sliding Mode Theory
for Brushless DC Motor Drives
Kuei-Hsiang Chao * , Chin-Tsung Hsieh and Xiao-Jian Chen

Department of Electrical Engineering, National Chin-Yi University of Technology, Taichung 41170, Taiwan;
fred@ncut.edu.tw (C.-T.H.); 4b112113@gm.student.ncut.edu.tw (X.-J.C.)
* Correspondence: chaokh@ncut.edu.tw; Tel.: +886-4-2392-4505 (ext. 7272); Fax: +886-4-2392-2156

Abstract: This paper presents the design of a robust speed controller for brushless DC motors
(BLDCMs) under field-oriented control (FOC). The proposed robust controller integrates extension
theory (ET) and sliding mode theory (SMT) to achieve robustness. First, the speed difference between
the speed command and the actual speed of the BLDCM, along with the rate of change of the speed
difference, are divided into 20 interval categories. Then, the feedback speed difference and the
rate of change of the speed difference are calculated for their extension correlation with each of
the 20 interval categories. The interval category with the highest correlation is used to determine
the appropriate control gain for the sliding mode speed controller. This gain adjustment tunes the
parameters of the sliding surface in the SMT, thereby suppressing the overshoot of the motor’s speed.
Because a sliding surface reaching law of the sliding mode controller (SMC) adopts the exponential
approach law (EAL), the system’s speed response can quickly follow the speed command in any
state and exhibit an excellent load regulation response. The simplicity of this robust control method,
which requires minimal training data, facilitates its easy implementation. Finally, the speed control
of the BLDCM is simulated using Matlab/Simulink software (2023b version), and the results are
compared with those of the SMC using the constant-speed approach law (CSAL). The simulation
and experimental results demonstrate that the proposed robust controller exhibits superior speed
command tracking and load regulation responses compared to the traditional SMC.

Keywords: extension theory; robust controller; brushless DC motor; field-oriented control; sliding
mode controller; constant-speed approach law; exponential approach law
Citation: Chao, K.-H.; Hsieh, C.-T.;
Chen, X.-J. A Robust Controller Based
on Extension Sliding Mode Theory for
Brushless DC Motor Drives.
1. Introduction
Electronics 2024, 13, 4028. https://
doi.org/10.3390/electronics13204028
In recent years, the increasing demand for motor drive performance and efficiency
in industrial applications has led to the widespread adoption of permanent-magnet syn-
Academic Editor: Omid Beik chronous motors (PMSMs) [1]. Within PMSMs, brushless DC motors (BLDCMs) [2] are
Received: 8 June 2024
widely utilized in industry due to their high torque, compact size, and high efficiency.
Revised: 5 October 2024 Given the stringent requirements for precision in speed and position control, field-oriented
Accepted: 10 October 2024 control (FOC) [3] is commonly employed for speed and position regulation.
Published: 13 October 2024 In the traditional FOC architecture, three proportional–integral (P-I) controllers [4]
are required, including a speed controller, a d-axis current controller, and a q-axis current
controller. Although traditional P-I controllers can meet control performance requirements
at specific operating points, their performance is adversely affected by load variations,
Copyright: © 2024 by the authors. motor parameter changes, and external disturbances, leading to a degradation in control
Licensee MDPI, Basel, Switzerland.
performance. This makes them unsuitable for applications demanding a high control
This article is an open access article
performance. Additionally, the BLDCM drive system is a nonlinear, strongly coupled
distributed under the terms and
multivariable system. To address the robustness and self-adaptive limitations of traditional
conditions of the Creative Commons
P-I controllers [5], numerous intelligent control strategies have been proposed [6–15].
Attribution (CC BY) license (https://
Among these, the sliding mode controller (SMC) [6–10] is notable for its variable structure
creativecommons.org/licenses/by/
4.0/).
control, which has a low dependency on the motor model and exhibits strong resistance

Electronics 2024, 13, 4028. https://doi.org/10.3390/electronics13204028 https://www.mdpi.com/journal/electronics

Electronics 2024, 13, 4028 2 of 26

to external load variations, system disturbances, and internal parameter changes. This
capability to enforce system motion along a predefined sliding trajectory has led to its
widespread application in speed command tracking and load disturbance rejection in
BLDCM drive systems. However, sliding mode controllers (SMCs) based on exponential
approach laws (EALs) [8] can cause overshoot in speed responses, while those based on
constant-speed approach laws (CSALs) [9,10] tend to have slower speed command tracking
responses and longer recovery times under load variations. On the other hand, traditional
extension controllers [11,12] may face stability concerns if their operating point falls outside
the neighborhood domain. Additionally, fuzzy controllers [13] often suffer from excessively
long computation times. In addition, from a practical implementation perspective, existing
intelligent controllers are relatively difficult to realize and often fail to achieve the expected
control performance. For example, controllers that combine fuzzy theory with sliding mode
theory (SMT) [14] are particularly challenging to implement effectively.
The control strategy proposed in this paper is based on extension theory (ET) [15,16]
and determines the appropriate SMC gains, thereby altering the structure of the sliding
mode function for a faster and more stable sliding mode control response. This gain
modifies the sliding mode controller’s sliding surface function so that the original sliding
surface function no longer has a fixed slope. Thus, by using ET to determine the gain of
the sliding surface function and replacing the CSAL of the SMC with an EAL, rapid speed
command tracking can be achieved. This approach not only optimizes the stability of the
sliding surface but also mitigates the overshoot caused by the SMC. As a result, the motor’s
speed response is not only faster but also more stable. In addition, the controller proposed
in this paper requires relatively less computation, thereby reducing the computational
load on the digital signal processor. Additionally, due to its simple program structure, the
controller’s parameters can be easily modified through programming to enhance system
stability and robustness. This makes it well-suited for the speed control of BLDCMs under
varying speed commands and significant load changes.

2. BLDCM System
The primary distinguishing feature of BLDCMs compared to brushed DC motors is
the absence of brushes for commutation. Instead, BLDCMs use electrical methods to detect
the rotor’s position and control its speed. Therefore, magnetic components (Hall sensors)
or optical encoders [17] must be integrated into the motor’s axis to provide feedback on
the rotor’s position to the controller. This allows the controller to determine the current
position of the motor rotor, which serves as the basis for commutation control.

2.1. Mathematical Model of BLDCM

Establishing a mathematical model is a crucial step in the design of a controller
for BLDCMs. Using the mathematical model of the motor as the core foundation, we
can perform an analysis and judgment using ET and the modeling of the SMC. This
mathematical model allows for the estimation of the controller parameter design values to
determine whether the expected control performance can be achieved, thereby facilitating
the design of the controller’s parameters.
Figure 1 depicts the three-phase equivalent circuit of a BLDCM within a stationary
reference frame.
Electronics 2024, 13, 4028 3 of 28
Electronics 2024, 13, 4028 3 of 26

ian Ra La ean
van →

+ −
Lab ebn
ibn Rb Lb
vbn →

Lac
+ −
Lbc ecn
icn Rc Lc
vcn →

+ −
Figure 1.
Figure Equivalent circuit
1. Equivalent circuit of
of aa three-phase
three-phase BLDCM
BLDCM in
in aa stationary
stationary frame.
frame.

According to Kirchhoff’s voltage law (KVL), the relationship between the phase
According to Kirchhoff’s voltage law (KVL), the relationship between the phase volt-
voltages and phase currents of a BLDCM can be expressed as Equation (1).
ages and phase currents of a BLDCM can be expressed as Equation (1).
        
v an  van  R a Ra 0 0 0 0   LLaa MM ab M
M ac   iianan  ean ean
vbn  =  dd ab ac
 0  R 0 
+ M
vbn  =  0 b Rb 0  +dt  Mabab L M   ibn +  ebn  (1)
Lbb bc   ibn  +  ebn 
Mbc
 
 (1)
vcn 0 0 Rc dt M M Lc icn ecn
 vcn   0 0 Rc   Macac Mbcbc Lc   icn   ecn 
where
where
v an , vbn , vcn : the phase voltages of phases a, b, and c, respectively.
van、vbn、vcn : the phase voltages of phases a, b, and c, respectively.
R a , Rb , Rc : the stator resistances of phases a, b, and c, respectively.
iRana、 ,R b ,R
、
ibn c : : the
icn the stator
phase resistances
currents of of phases
phases a, and
a, b, b, and c, respectively.
c, respectively.
iLan、
a, Li 、 i : the phase currents of phases a, b, and c, respectively.
Lc : the self-inductances of phases a, b, and c, respectively.
bnb , cn

L、
M a abL, 、
b
ML c : the
bc , Mac self-inductances
: the mutual inductances
of phasesbetween
a, b, andphases a, b, and c.
c, respectively.
eMan 、 , eM bnbc、M ac : the mutual inductances between phases a, b,a,and
, e cn : the back electromotive forces (EMF) of phases b, and
c. c, respectively.
ab
Assuming the motor is three-phase balanced, then
ean、ebn、ecn : the back electromotive forces (EMF) of phases a, b, and c, respectively.
Assuming the motor is three-phase L a = balanced,
Lb = Lc =then L (2)
L =L =L =L (2)
Mab = aMbc b= Mcac = M (3)
By substituting Equations (2) and
M (3)
=M into=Equation
M = M (1), we obtain (3)
ab bc ac
        
v an Equations
By substituting 0 (3) into LEquation
R a 0(2) and M M(1), we i an obtainean
vbn  =  0 Rb 0  + d  M L M ibn  + ebn  (4)
vcn  van  0  Ra 0 0 Rc 0  d M
dt L M M  ian   ean 
M L i e
v  =  0 R 0  +  M L M  i cn +  e  cn
 bn   b  dt    bn   bn  (4)
 vcn   0 Rc  ianM+ ian L =0icnand
M + ian   ecn 
Since the motor is three-phase0 balanced, we obtain


Since the motor is three-phase an + Mibn i=

Mibalanced, − Micn
an + ian + ian = 0 and we obtain
(5)

Substituting Equation (5) into Equation Mian + Mi = −obtain

(4),bn we Micn the state equation of the BLDCM, (5)
which is as shown in Equation (6).
Substituting Equation (5) into Equation (4), we obtain the state equation of the
BLDCM,v an which isRas a shown 0in Equation (6).L − M
         
0 i an 0 0 i an ean
d
vbn  =  0 Rb 0 ibn  +  0 L−M 0 i + e  (6)
 van   Ra 0 0  ian  dt  L − M 0  ian  bn ean  bn
   
0
vcn 0
v  =  0 R 0 R i d 0 0 L − M i e
0  ibn  +  0 0  ibn  +  ebn 
c cn cn cn
 bn   b L−M (6)
dt
Since the speed
vcn  of
 0the BLDCM
0 Rc   iis
cn 
 
proportional
 0 to 0
the L
back − M
EMF  iand 
the
  cn   cn  e 
input current
is proportional to the torque, the electromagnetic torque Te can be derived using the
Since the
three-phase speed of
currents, theEMF,
back BLDCM andisrotor proportional
speed, asto the back
shown EMF and(7).
in Equation the input current
is proportional to the torque, the electromagnetic torque Te can be derived using the three-
phase currents, back EMF, and Trotor an i an +as
espeed, ibn + ein
ebnshown cn iEquation
cn (7).
e = (7)
ωm
ean ian + ebn ibn + ecn icn
Te = (7)
where ωm
Te : the electromagnetic torque generated by the motor.
Electronics 2024, 13, 4028 4 of 26

ωm : the mechanical speed of the motor.

The mechanical equation of the BLDCM can be expressed as follows:

dωm
Te = J + Bωm + TL (8)
dt
where
TL : The load torque.
J : The rotational inertia of the motor and load.
B : The coefficient of the viscous friction of the motor and load.

2.2. Dynamic Equations of the FOC System for BLDCMs

FOC finds widespread application in motor drive systems, enabling both motor speed
and position control with the same architecture. To implement FOC, it is necessary to
detect the rotor position of the motor first, and then provide switching control signals
to the inverter [18] through coordinate transformation. However, in the implementation
process, analyzing the three-axis spatial coordinate system of a three-phase motor directly
can be computationally challenging. Therefore, it is necessary to perform a coordinate
transformation [19] to convert the original three-axis spatial coordinates into two-axis
spatial coordinates, making the calculations simpler and easier to implement. Hence,
this section will focus on spatial coordinate transformation to explain the principles of
FOC systems.
Under ideal conditions, the voltage equation of a BLDCM in a three-phase synchronous
stationary coordinate system is represented by Equation (9) [20].
      
v an Rs 0 0 i an ϕa
vbn  =  0 d
Rs 0 ibn  + ϕb  (9)
dt
vcn 0 0 Rs icn ϕc

where
Rs : stator resistance of the three phases.
ϕa , ϕb , ϕc : magnetic fluxes of phases a, b, and c, respectively.
The magnetic flux of a BLDCM is generated by the combination of the current passing
through the stator windings and the permanent magnets on the rotor. Under ideal condi-
tions, the magnetic field generated by the permanent magnets has a constant amplitude,
indicating that the rotor’s relative position is fixed. Therefore, this magnetic field can be
represented by a vector ϕ f . The position of the stator is determined by the angle θ between
the direction of the magnetic field’s orientation and the stator coordinate system. The flux is
the projection of a constant flux vector ϕ f along the a, b, c axis direction. The magnetic flux
equation of a three-phase stationary coordinate system is represented as in Equation (10).
      
ϕa L Mab Mac i an cos θ
 ϕb  =  Mba L Mbc  ibn  + ϕ f  cos(θ − 23 π ) 
(10)
ϕc Mca Mcb L icn cos(θ + 23 π )
R
θ ≜ ωe dt

where
M : the mutual inductance between phase windings.
L : the self-inductance of each phase winding.
ϕ f : the flux produced by the permanent magnets.
θ : the angle between the rotor N pole and the axis of the a phase winding.
ωe : the synchronous speed of the motor.
Electronics 2024, 13, 4028 5 of 26

The three-phase stationary coordinate system can be transformed into a two-phase sta-
tionary coordinate (α, β) system through the Clarke transformation, as shown in Equation (11).
      
vα Rs 0 iα d ϕα
= + (11)
vβ 0 Rs iβ dt ϕβ

The magnetic flux equation can be expressed as Equation (12).

         
ϕα iα ϕαm iα cos θ
=L + =L + ϕf (12)
ϕβ iβ ϕβm iβ sin θ

By differentiating the magnetic flux equation, Equation (12), and substituting it into
Equation (11), the voltage equations for the two-phase stationary coordinate (α, β) system
can be obtained, as shown in Equation (13).
          
vα Rs 0 iα L 0 d iα ϕβm
= + − ωe (13)
vβ 0 Rs iβ 0 L dt i β −ϕαm

After applying the Park transformation to the two-phase stationary coordinate (α, β)
system, two dependent equations can be derived using the two-phase synchronous rotating
coordinate (d, q) system, as shown in Equation (14). The (d, q) magnetic flux equations are
then represented as shown in Equation (15).
        
vd Rs − ωe Lq id d ϕd 0
= + + (14)
vq ωe Ld Rs iq dt ϕq ωe ϕ f
      
ϕd L 0 id ϕ
= d + f (15)
ϕq 0 Lq iq 0
From Equation (15), the torque equation for the two-phase synchronous rotating (d, q)
system can be derived, as shown in Equation (16).

3P 3P
Te = (ϕ iq − ϕq id ) = [ ϕ i q + ( L d − L q )i d i q ] (16)
22 d 22 f
The torque equation for the two-phase synchronous rotating (d, q) system can be
formulated as Equation (17).
P dωm
Te = TL + J (17)
2 dt
If the FOC method is adopted, we can set id = 0. Then, the voltage equations can be
simplified to Equations (18) and (19).

diq
vq = Rs iq + L + ωe ϕ f (18)
dt
vd = −ωe Liq (19)
From Equation (16), the torque equation can be expressed as Equation (20).

3P
Te = ϕ iq = Kt iq (20)
22 f

where Kt ≜ 32 P2 ϕ f is the torque constant and P is the number of poles. Using Equation (8),
the motion equation for the BLDCM can be expressed as Equation (21):

dωm Kt B 1
= iq − ωm − TL (21)
dt J J J
 where 𝐾 ≜ 𝜙 is the torque constant and P is the number of poles. Using Equation (8),
the motion equation for the BLDCM can be expressed as Equation (21):
dωm K t B 1
Electronics 2024, 13, 4028 = iq − ω m − TL (21)
6 of 26
dt J J J

The equation derived from Equation (20) reveals that when employing FOC, control-
iq can
ling The regulate
equation the torque’s
derived magnitude.
from Equation (20)Moreover,
reveals that aswhen
shownemploying
in Equation (19),control-
FOC, the d
axis ivoltage
ling q can regulate
is solelythe torque’s
related to i qmagnitude.
, effectivelyMoreover,
simplifyingasthe shown
controlin requirements
Equation (19),for d
thethe
axis voltage is solely related to iq , effectively simplifying the control requirements for the
BLDCM system’s architecture. When i = 0 , it can be regarded as a separately excited DC
BLDCM system’s architecture. When idd= 0, it can be regarded as a separately excited DC
motor,where
motor, wherethethestator
statorhashasonly
onlyaaquadrature
quadrature(q( qaxis)
axis) component,
component, andandthethe spatial
spatial vector
vector of
of the
the stator
stator magnetic
magnetic flux
flux coincides
coincides orthogonallywith
orthogonally withthethespatial
spatialvector
vectorof ofthe
thepermanent
permanent
magnet field.
magnet field. Figure
Figure 22 illustrates
illustrates the
the block
block diagram
diagram ofof the
the FOC
FOC for
for the
the BLDCM.
BLDCM.

Inverse Park
2e to 2s ia
vd* vα*
=0 + _PI vα , β
-− ib
SVPWM BLDCM
Speed iq*
vq* vd , q vβ* ic
+ _PI
controller −-

-
− θe
+

ωref
∗ iα
iα , β ia ,b ,c
id
iq id ,q iβ iα , β
Park Inverse Clarke
2s to 2e 3s to 2s
Hall and
position
sensors

Figure 2.
Figure 2. FOC
FOC block
block diagram
diagram of
of the
the BLDCM.
BLDCM.

3.
3. Proposed
Proposed Intelligent
Intelligent Algorithm
Algorithm forfor Drive
Drive Control
Control
Due
Due to the fixed parameter values of traditional P-I
to the fixed parameter values of traditional P-I controllers, their control
controllers, their control perfor-
perfor-
mance
mance deteriorates when the speed command of the FOC system changes or when load
deteriorates when the speed command of the FOC system changes or when load
variations
variations occur,
occur,potentially
potentiallyleading
leadingto tosystem
systemdivergence.
divergence.
Therefore,
Therefore,this
thispaper
paperproposes
proposes a speed
a speedcontroller thatthat
controller integrates ET with
integrates SMT,SMT,
ET with enabling
ena-
the system
bling to achieve
the system a faster
to achieve speedspeed
a faster response and possess
response a self-adaptive
and possess capability
a self-adaptive [21].
capability
The design process of the proposed intelligent controller, which combines
[21]. The design process of the proposed intelligent controller, which combines ET with ET with SMC
(ETSMC),
SMC (ETSMC),is described below.below.
is described
3.1. Extension Theory
3.1. Extension Theory
Extension theory (ET) [15,16] was proposed by Chinese scholar Professor Cai Wen in
Extension theory (ET) [15,16] was proposed by Chinese scholar Professor Cai Wen in
1983. It primarily explores the variability of things, examining the principles and methods
1983. It primarily
for resolving explores the
contradictory variability
problems fromofboth
things, examining
qualitative andthe principles and
quantitative methods
perspectives.
for resolving
The two core contradictory
components of problems from both
ET are matter qualitative
element theoryand
andquantitative perspectives.
extension mathematics.
The two core components of ET are matter element theory and extension mathematics.
Matter element theory describes the possibilities of changes in things and the characteristics
Matter
of matterelement
element theory describes thewhile
transformations, possibilities
extensionof mathematics
changes in things
reliesand
on antheextension
characteris-
set
tics of matter element transformations, while extension
and correlation function as the core of its calculations. mathematics relies on an extension
set and correlationinformation
ET represents function as about
the core of its
things calculations.
through the matter element model. It expresses
the transformation relationships between the qualitative and quantitative aspects of things
via matter element transformations. By utilizing correlation functions for discrimination,
this theory helps us understand the influence of qualitative and quantitative factors on
things, thereby clearly presenting their degree of impact on the characteristics of things.
Electronics 2024, 13, 4028 7 of 26

3.1.1. Extension Matter Element Model

In ET, the representation of the information about things is achieved by expressing
the things in terms of a matter element model, using the mathematical function shown in
Equation (22).
R = ( N, c, v) (22)
where
R : the fundamental element describing an entity, referred to as the matter element.
N : the name of the entity.
c : the characteristics or features of the entity.
v : the value of the feature of the entity.
In ET, if the feature of the matter element is not a single feature, it is represented by n
features and their corresponding n feature values. Therefore, a feature can be expressed
as c = [c1 , c2 , . . . , cn ], and its feature values can be represented by v = [v1 , v2 , . . . , vn ].
Therefore, the extensional function of matter elements can transform Equation (22) into a
matrix vector form, as represented in Equation (23).
  
R1 R, c1 , v1
 R2   c2 v2 
R= . = (23)
   
.. .. 
 ..   . .
Rn cn vn

3.1.2. Definition of Classical Domain and Neighborhood Domain in ET

If the value range of a feature is defined as its classical domain C0 =< a, b >, and it is
contained within a neighborhood domain C =< d, e >, then C0 ∈ C. If point ĉ lies within
the interval C, the C0 =< a, b > corresponding to the matter element can be expressed as
Equation (24).
 
C0 , c 1 , < a1 , b1 >
 c 2 , < a2 , b2 > 
R0 = (C0 , ci , vi ) =  (24)
 
.. .. 
 . . 
c n , < a n , bn >
where ci is a feature of C0 and vi is the feature value of ci . As for C, the corresponding matter
element RC can be expressed as Equation (25). In this expression, c j is the characteristic
value of C and v j is the characteristic quantity of c j .
 
C, c1 , < d 1 , e1 >
 c2 , < d 2 , e2 > 
RC = (C, c j , v j ) =  (25)
 
.. .. 
 . . 
cn , < dn , en >

3.1.3. Distance and Rank Value

In classical mathematics, the terms distance and rank value refer to the relationship
between two points. However, in ET, these terms represent the distance relationship
between a point ĉ in the real domain and an interval C0 =< a, b >. Mathematically, this
relationship can be expressed as Equation (26).

a+b b−a
ρ(ĉ, C0 ) = ĉ − − (26)
2 2

In addition to considering the relationship between a point and an interval, it is also

necessary to consider the relationship between a point and two intervals. Thus, assuming
C0 =< a, b > and C =< d, e > are two intervals in the real domain, and the interval C0 is
 0
2 2

In addition to considering the relationship between a point and an interval, it is also

necessary to consider the relationship between a point and two intervals. Thus, assuming
C0 =< a, b > and C = < d , e > are two intervals in the real domain, and the interval C0 is
Electronics 2024, 13, 4028 8 of 26
contained within C , the rank values of point cˆ , interval C0 , and interval C can be ex-
pressed as Equation (27).
contained within C, the rank values of point ĉ, interval C , and interval C can be expressed
 ρ (cˆ, C ) − ρ (cˆ, C0 ) , 0cˆ ∉ C0
as Equation (27). D(cˆ, C0 , C ) =  (27)
( −1 , cˆ ∈ C0
ρ(ĉ, C ) − ρ(ĉ, C0 ) , ĉ ∈ / C0
D (ĉ, C0 , C ) = (27)
−1 , ĉ ∈ C0
3.1.4. Correlation Function
3.1.4. C0 =< a, b > ,Function
If Correlation C = < d , e > , and C0 ∈ C , when the two intervals do not intersect at
a common =< a, b >their
If C0endpoint, , C =< d, e >, and
correlation C0 ∈ C,can
function when the two intervals
be expressed do not
as Equation intersect at a
(28).
common endpoint, their correlation function can be expressed as Equation (28).
ρ (cˆ, C0 )
K (cˆ) = (28)
D (ρcˆ(,ĉ,
C0C,0C))
K (ĉ) = (28)
D (ĉ, C0 , C )
In Equation (28), when ˆ
c = ( a + b) / 2 , the function value reaches its maximum. There-
In correlation
fore, this Equation (28), when
function = (be
canĉ also a+ b)/2, the
referred to asfunction value reaches
the elementary its maximum.
correlation function
Therefore, this correlation function can also be referred
[15,16]. A schematic diagram is shown in Figure 3. Additionally, when to as the K (cˆ
elementary) < − 1
correlation
, it indi-
function [15,16]. A schematic diagram is shown in Figure 3. Additionally, when K (ĉ) < −1,
cates that point cˆ is outside the interval C. Conversely, when K(cˆ) > 0 , it signifies that
it indicates that point ĉ is outside the interval C. Conversely, when K (ĉ) > 0, it signifies
point cˆ is within the interval C0 . C
However, if −1 < K (cˆ) < 0 , it implies that point cˆ is
that point ĉ is within the interval 0 . However, if −1 < K ( ĉ ) < 0, it implies that point ĉ is
located
located within
within the extension
the extension domain.
domain.

K (cˆ)

Extension Extension
domain domain

d a b e ĉ

( a + b) / 2

−1

Figure 3. Schematic diagram of the elementary correlation function.

Figure 3. Schematic diagram of the elementary correlation function.
3.2. Sliding Mode Controller Design
While traditional P-I controllers can meet control performance requirements at specific
operating points, their performance response is affected by load variations. Moreover, they
are susceptible to changes in motor parameters and external disturbances, making them
unsuitable for applications requiring high control performance. Furthermore, BLDCMs
are nonlinear and strongly coupled multivariable systems. Consequently, to address the
robustness [22] issues of traditional P-I controllers and their insufficient self-adaptive
capability, several intelligent control algorithms have emerged.
The variable structure control of SMCs exhibits a lower dependence on motor models
and demonstrates robustness against external load variations, system disturbances, and
internal parameter changes. As a result, it can effectively drive the system along a prede-
fined sliding mode trajectory, making it widely applicable in the speed regulation and load
disturbance rejection control of BLDCMs.
Electronics 2024, 13, 4028 9 of 26

3.2.1. State Variable Design

In order to replace the traditional field-oriented speed control loop with a P-I controller,
the mathematical equations for the d-q axis of the BLDCM can be derived from the matrix
Equations (14) and (15), as shown in Equation (29). Since the BLDCM used in this study is
a surface-mounted (SM) rotor type BLDCM, its d and q-axis inductance values are equal
and denoted as Ld = Lq = Ls .
(
vd = Rs id + Ls didtd − ωe Ls iq
diq (29)
vq = Rs iq + Ls dt + ωe Ls id + ωe ϕ f

Due to the viscous friction coefficient being very small, it can be neglected. Therefore,
Equation (21) allows the motion equation for the SM rotor of a BLDCM to be simplified to
Equation (30).
dωm
J = Te − TL (30)
dt
However, to achieve FOC, the id = 0 control strategy must be adopted. Therefore,
Equation (31) can be derived from Equations (20), (29) and (30).
diq
(
1 P
dt = Ls (− Rs iq − 2 ωm ϕ f + vq )
(31)
dωm 1 3P
dt = J (− TL + 2 2 ϕ f iq )

Since the goal is to apply the SMC to speed loop control, the signal input to the
controller is the difference between the speed command ωm ∗ and the actual feedback signal

ωm (i.e., the speed difference). The controller output is the q-axis command current iq∗ ,
aiming to achieve ωm ∗ − ω = 0 and ensure that the rate of change of the speed difference
m
.
ω m = 0. Based on this, the state variables of the BLDCM system can be defined as shown
in Equation (32).
∗ −ω

x1 = ω m m
. . (32)
x2 = x 1 = − ω m
By differentiating Equations (31) and (32), the rate of change of the state variables can
be derived, as shown in Equation (33).
. . 3P
x1 = −ω m = 1J ( TL −
(
2 2 ϕ f iq )
. .. 3P fϕ . (33)
x2 = −ω m = − 2 2 J iq

3 P ϕf .
Let D = 22 J , then substituting it into Equation (33) for x2 yields Equation (34).

. .. 3 P ϕf . .
x2 = −ω m = − iq = − Diq (34)
22 J

3.2.2. Sliding Surface Design

In SMC, the design objective of the controller is to ensure x1 = 0 and x2 = 0. Therefore,
the sliding surface function [7] can be expressed in the form of Equation (35).

s = Cx1 + x2 (35)

where
s : sliding surface function.
C : control gain.
x1 , x2 : state variables.
Electronics 2024, 13, 4028 10 of 26

.
When s = 0, let x2 = x1 ; Equation (35) can be rewritten as Equation (36) and x1 and x2
can be solved, as shown in Equation (37).
.
Cx1 + x2 = Cx1 + x1 = 0 (36)

x1 = x1 (0)e−ct

. (37)
x2 = x1 = −cx1 (0)e−ct
Equation (37) reveals that over time, the values of the state variables x1 and x2 decay
exponentially to 0. Therefore, when s= 0, which represents the designed sliding surface,
and the sliding surface function s = Cx1 + x2 reaches the sliding surface s = 0, the system’s
state variables will approach 0, thus achieving the goal of state variable control.

3.2.3. Sliding Mode Approach Design

To ensure that the sliding surface function s reaches 0 at a certain time point (i.e.,
reaching the sliding surface) and remains stable, it is necessary to design an approaching
law function [8].
From the design of the sliding surface, it is evident that, to ensure s = 0, the output
function u of the BLDCM speed controller must be designed to meet the control require-
ments. Hence, Equation (35) can be reformulated as Equation (38).
. . . . .
s = C x1 + x2 = Cx2 + x2 = Cx2 − Diq (38)
.
where s represents the sliding mode approaching law function.
Since iq∗ is the output of the speed controller, the control force function is defined as
.
u = iq , thereby transforming Equation (38) into Equation (39).
.
s = Cx2 − Du (39)
.
Among them, s is the sliding mode approaching law function.
According to Lyapunov’s second stability criterion [10], if there exists a continuous
function V, it must satisfy the following three conditions:

(1)V (0) = 0
(2)V ( s ) > 0 (40)
.
(3)V ( s ) < 0

Therefore, when the system is stable at the equilibrium point s = 0, this ensures that
limt→∞ s(t) = 0. Furthermore, if we let V = 21 s2 , not only can we satisfy Conditions (1) and
(2) in Equation (40), but we can also deduce the third condition through analysis, as shown
in Equation (41).
. .
V ( x ) = ss (41)
. . .
Next, we design the approaching law function s such that V ( x ) = ss < 0. The common
approaching law functions include the CSAL function and the EAL function, which can be,
respectively, represented by Equations (42) and (43).
.
s = −εsgn(s), ε > 0 (42)
.
s = −εsgn(s) − qs, ε > 0, q > 0 (43)
where 
1, s>0
sgn(s) =
−1, s<0
.
According to the two approaching law functions mentioned above, when s = −εsgn(s),
ε > 0, the controller’s u = −Cx2 − εsgn(s) can be obtained from Equation (39). This implies
Electronics 2024, 13, 4028 11 of 26

applying the control force function u to the motor model. Thus, the final response will
stabilize at the origin of the designed sliding surface.

3.2.4. Controller Output Design

From the aforementioned reaching law design, it can be inferred that the first two
conditions of the Lyapunov function ensure limt→∞ s(t) = 0 [23]. However, if the CSAL
function given by Equation (42) is adopted, regardless of whether t = 1s or t = 100s,
s = 0 (reaching the sliding surface) can satisfy the requirements of the Lyapunov function.
However, due to its consistent response speed, attempting to expedite the response would
result in oscillations on the sliding surface. Conversely, stabilizing the motor on the sliding
mode surface would make it take a significantly longer time to reach the sliding mode
surface. This intricate delay in response time certainly diminishes its practical applicability
in real-world applications. From Equation (43), it can be observed that the EAL [9] differs
from the former by the addition of an exponential approach term qs. When the s value
.
of the exponential approach term is small (i.e., closer to the sliding surface), s = −qs is
approximately 0. Therefore, it is dominated by −εsgn(s). Conversely, if this value is large
.
(i.e., the approach distance is much greater than the sliding surface), the value of s = −qs
will also be larger. In this case, it is dominated by the qs exponential approach term, thereby
enhancing the speed response of the motor approaching the sliding surface, enabling the
system to approach the sliding surface more quickly. Therefore, the EAL is more suitable
for systems experiencing significant parameter disturbances or large load variations.
In summary, compared to CSALs, the EAL offers superior characteristics. Hence, the
controller proposed in this paper adopts the EAL function for sliding surfaces. However,
this necessitates solving the exponential approach term (−qs) from Equation (43). In
.
Equation (43), where s = −qs represents the exponential approach term, the solution is
given by
s = s(0)e−qt (44)
In Equation (44), q represents the exponential term, thus Equation (43) is referred to as
the EAL function.
The EAL function is the approaching law function selected in this paper. From
Equation (39), it can be inferred that iq∗ is the output of the controller. Therefore, defining
.
the controller function as u = iq , Equation (39) can be rewritten as Equation (45).
.
s = Cx2 − Du = −εsgn(s) − qs (45)

Consequently, the expression for the control force function u can be written as shown
in Equation (46).
1
u = [Cx2 + εsgn(s) + qs] (46)
D
From Equation (46), it can be inferred that the commanded current iq∗ of the q axis can
be represented as Equation (47).

1
Z
iq∗ = [Cx2 + εsgn(s) + qs]dt (47)
D

3.3. Feature Selection for the ET-Integrated SMC of the Drive System
In order to achieve a faster and more stable control response for the AM2200H
BLDCM [24], ET is adopted in this paper for speed control. This control method involves
partitioning the speed difference (e ≜ ωr − ω̂r∗ ) between the motor’s actual speed and com-
.
manded speed, as well as the rate of change of the speed difference (e ≜ e(n + 1) − e(n)),
within the speed range (0 ∼ 2000rpm) into 20 intervals (i.e., 20 states). The relationship
between these intervals is illustrated in Figure 4. From Figure 4, it can be observed that
intervals A1 to A4 exhibit larger oscillations due to the significant differences in speed
commands, while categories A17 to A20 show smaller oscillations as the speed difference
Electronics 2024, 13, 4028 12 of 26

.
is smaller. For interval A1, it is observed that e > 0, e > 0, and e increases continuously.
. .
Although e > 0, its value decreases over time, and, at point m1 , e = 0. However, e reaches
its maximum value at this point. This indicates that when the speed difference e is larger
.
and the rate of change of the speed difference e is smaller, the control effort required for
speed control needs to be significantly reduced. This trend can be observed in other in-
tervals as well. Therefore, according to ET, the dynamic analysis chart in Figure 4, which
is characterized by the speed difference and the rate of change in the speed difference,
is used to establish classical domain matter element models for 20 intervals (as shown
in Table 1). The control gain values C corresponding to these 20 classical domain matter
element models in the SMC are used to modify the dynamic model of the SMC. This
adjustment alters the sliding surface function, enabling a faster transient response and
suppressing the overshoot caused by the EAL. Furthermore, the neighborhood domain is
established using the maximum and minimum values of each feature’s classical domain, as
shown in Equation (48).
 
C e < −2000, 2000 >
Electronics 2024, 13, 4028 Rc = (C, cn , vn ) . 13 of(48)
28
e < −120, 480, 120, 480 >

m1

Category
interval A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20

e + + + + + + + + + +
e + + + + + + + + + +

4. Dynamic
Figure 4.
Figure Dynamicanalysis
analysischart of of
chart speed difference
speed and variance
difference rate ofrate
and variance speed
of difference for BLDCM.
speed difference for
BLDCM.
Table 1. Extension matter element models and variation in control gain for 20 intervals of SMC.

Interval matter element

Table 1. Extension Extensionmodels

Matter and Element
variationModel
in control

Variation
gain of Control
for 20 intervals SMC.∆C
of Gain
A1 C1 e < 0, 2000 > 0
R1 = .
e < 0, 120, 480 > Variation of Control Gain Δ C
Interval Extension Matter 
C2 Element
e < 0,Model
2000 >
A2 R2 = . 0
 e < −120, 480, 0 >
A3 C e
C e3 <. 0,2000 > 
< −2000, 0 > 0
R =
R13=  1
 < 0,120480 >
A1 e < −120, 480, 0 > 0
 eC e < −2000, 0 >
A4 R4 = 4 . 0
 e < 0, 120, 480 >
A5 C2 e C5 < 0,2000e <> 0, 1500 > 15
A2 R2 =R5 = e
.
< 0,>90, 360 >  0
  
e < − 120480,0 
A6 C6 e < 0, 1500 > 15
R6 = .
e < −90, 360, 0 >
C eC7 < e−2000,0 > 1500,


A7 <− 0> 15
A3 R3 R=7 =3 .  360, 0 > 0
 
e < − e <
120480,0− >
90,

C e < −2000,0 > 

R4 =  4
A4
 e < 0,120480 > 0

C5 e < 0,1500 > 

R5 =
Electronics 2024, 13, 4028 13 of 26

Table 1. Cont.

Interval Extension Matter Element Model Variation of Control Gain ∆C

 
A8 C8 e < −1500, 0 > 15
R8 = .
 e < 0, 90, 360 >
A9 C9 e < 0, 1000 > 30
R9 = .
 e < 0, 60, 240 > 
A10 C10 e < 0, 1000 > 30
R10 = .
 e < −60, 240, 0 >
A11 C11 e < −1000, 0 > 30
R11 = .
 e < −60, 240, 0 >
A12 C12 e < −1000, 0 > 30
R12 = .
 e < 0, 60, 240 >
A13 C13 e < 0, 500 > 45
R13 = .
 e < 0, 30, 120 > 
A14 C14 e < 0, 500 > 45
R14 = .
 e < −30, 120, 0 >
A15 C15 e < −500, 0 > 45
R15 = .
 e < −30, 120, 0 >
A16 C16 e < −500, 0 > 45
R16 = .
 e < 0, 30, 120 >
A17 C17 e < 0, 100 > 60
R17 = .
 e < 0, 6024 > 
A18 C18 e < 0, 100 > 60
R18 = .
 e < −6024, 0 >
A19 C19 e < −100, 0 > 60
R19 = .
 e < −6024, 0 >
A20 C20 e < −100, 0 > 60
R20 = .
e < 0, 6024 >

3.4. Integration of ET and SMC for Speed Control

To achieve both stability and a fast response in motor speed control, this study utilizes
ET to calculate the correlation of the speed difference between the speed command of
motor and the actual motor speed, as well as the rate of change of the speed difference.
Simultaneously, it completes the calculation of the sliding surface function of the SMC.
Initially, ET is employed to identify the feature with the highest correlation, categorizing it
into the most appropriate feature category interval. This process is crucial for determining
the optimal control gain function for the SMC. The SMC utilizes an EAL to enhance the
motor speed response. The control process involves computing the correlation using ET to
classify the motor speed features, thereby determining the stable control gain of the sliding
surface. This is crucial for suppressing the overshoot caused by the EAL of the SMC.
The test motor used in this study is the AM2200H permanent-magnet BLDCM [24].
Due to its rated speed of 2000rpm, the speed range was set as 0 ∼ 2000rpm. The state
variable characteristics of the ET and SMC are the speed difference between the commanded
speed and the actual feedback speed, and the rate of change of the speed difference, denoted
.
as e = ωr − ω̂r∗ and e = de/dt.
Therefore, based on the above analysis, the motor operating state is first determined
using ET, which then decides the SMC gain C. The steps of the control process are described
as follows:
Step 1: Establish an extension matter element model using the interval categories of each
speed difference and speed difference variation rate as features.
 
C e ( a1 , b1 )
R g = (C, c. , v) = . , g = 1, 2, 3, . . . , 20 (49)
e ( a2 , b2 )
Electronics 2024, 13, 4028 14 of 26

Step 2: Input the two features of the speed difference e and the speed difference variation
.
rate e that is to be classified and establish a matter element model.
 
Cnew e vnew1
Rnew = . (50)
e vnew2

Step 3: Calculate the correlation function K gj between the input features and each inter-
val category using Equation (28) and based on the speed difference e and speed
.
difference variation rate e.
Step 4: Set the weight values W1 and W2 for each feature to represent their importance.
According to Equation (32), the SMC adjusts the sliding surface functions based on
the speed difference variation rate. Therefore, the weights are set to W1 = 90% and
W2 = 10%, with W1 + W2 = 1 (i.e., 100%).
Step 5: Calculate the correlation degree between the feature values and each interval category.

2
λg = ∑ Wj Kgj , g = 1, 2, 3, . . . , 20 (51)
j =1

Step 6: Normalize the correlation degrees for each interval category using Equation (52),
ensuring that the correlation degrees fall within the range of <−1,1>. This increases
the sensitivity of the correlation degrees, facilitating category classification.

 λ′ = λg
, i f λg > 0
g |λmax |
(52)
 λ′g = λg
, i f λg < 0
|−λmax |

In Equation (52), λmax and −λmax represent the maximum and minimum correlation
degrees for each interval category, respectively.
Step 7: Identify the interval category to which the speed difference e and speed difference
.
variation rate e belong by determining the maximum correlation degree from the
calculations. Based on the identified category, determine the change △C in the
sliding mode controller’s control gain and output it to the SMC to adjust the sliding
surface function. The new control gain Cnew can be expressed by Equation (53).

Cnew = Cold + ∆C (53)

Step 8: After determining the operating condition category using ET, the control gain C of
the sliding surface in Equation (47) can be determined to adjust the sliding surface
function s.
Step 9: Adjusting the sliding surface function ensures that the EAL does not result in exces-
sive overshoot, and the system approaches the sliding surface at a rate determined
by the EAL.
Step 10: Once the system has tracked onto the sliding surface, the final iq∗ output is deter-
mined by Equation (47).

4. Simulation Results
A control block diagram of the proposed BLDCM drive system is shown in Figure 5.
The BLDCM used in this paper is the ADLEE_POWER match-servo-motor-AM2200H [24],
which is a concentrated winding-type BLDCM. The motor specifications and detailed
parameters are shown in Table 2. Table 3 shows the parameter values of the controller
proposed in this paper, which combines extension theory and a sliding mode controller
(ETSMC), as well as the parameters for the exponential approach law sliding mode con-
troller (EALSMC) and the constant-speed approach law sliding mode controller (CSALSMC).
Figure 6 shows the speed response waveforms obtained from Matlab/Simulink simulations
of the different controllers as the speed command rises from 0 rpm to 2000 rpm. Under the
Electronics 2024, 13, 4028 15 of 26

same conditions (control gain C : 60, exponential approach parameter q : 2000, constant-
speed approach parameter ε: 2000), it can be observed that the sliding surface function
determined using ET changes with the speed difference and speed difference variation rate.
This enables the suppression of the overshoot caused by the SMC adopting the EAL. From
the simulation results, it is evident that only the SMC adopting the EAL exhibits instances
of overshoot. However, by utilizing ET to calculate the gain of the SMC, it is possible to
effectively suppress the overshoot caused by the EAL. Therefore, this paper will compare
the control performance of speed control for a BLDCM using two different controllers: one
Electronics 2024, 13, 4028 17 of 28
combining ET with the EAL of SMC (EALSMC) and the other utilizing the CSAL of SMC
(CSALSMC). The sampling period for the simulations in this paper was 0.1 µs.

Inverse Park
2e to 2s iU
+ v*
d
vα*
=0  _PI vα , β
−- iV
SVPWM BLDCM
ET&SMT
ET and SMT iq* + vq* vd ,q vβ* iW
speed
speed  _PI
controller
controller −-

 -− θe
+
ωref
∗ iα
iα ,β iU ,V ,W
id
iq id ,q iβ iα , β
Park Inverse Clarke
2s to 2e 3s to 2s
Hall
sensors

Figure 5.
Figure 5. Control
Control block
block diagram
diagramof
ofthe
theproposed
proposedBLDCM
BLDCMdrive
drivesystem.
system.

Table 2. Brushless DC motor (BLDCM) specifications and parameters [24].

Table 2. Brushless DC motor (BLDCM) specifications and parameters [24].
Electrical Specifications Value
Electrical rated
Three-phase Specifications
voltage ACValue
220 V
Three-phase rated current AC 9.8 A
Three-phase
Rated apparentrated voltage
power AC 220
2156 VA V
Rated speed
Three-phase rated current 6000 rpm
AC 9.8 A
Operating frequency range 0~200 Hz
Rated
Numberapparent
of polespower 2156
4 VA
Stator resistance
Rated speed 6000 Ω
0.15 rpm
Stator inductance 1.235 mH
Operating frequency
Excitation flux range 0~200 Hz
0.0002024 Wb
Rotational inertia 0.0015 kg-m 2
Number of poles 4
Stator resistance 0.15 Ω
Table 3. Parameter values for various sliding mode controllers.
Stator inductance 1.235 mH
Parameterflux
Excitation Name Parameter of Parameter of Wb
0.0002024
CSAL EAL Control Gain
Rotational inertia
Type of Controller (ε) 0.0015 kg-m
(q) 2 (C)

Determined by
ETSMC 2000 2000 extension theory
Table 3. Parameter values for various sliding mode controllers.
0 < C < 60
ParameterEALSMC
Name 2000 2000 60
CSALSMC 2000 0 60
Parameter of CSAL Parameter of EAL Control Gain
(ε) (q) (C)
Type of Controller
Determined by exten-
ETSMC 2000 2000 sion theory
0 < C < 60
EALSMC 2000 2000 60
 Electronics 2024, 13, 4028 18 of 28
Electronics 2024, 13, 4028 16 of 26

2500

2000

Electronics 2024, 13, 4028 18 of 28

Electronics 2024, 13, 4028 18 of 28
1500

speed(rpm) 1000
2500
2500

500
2000
2000
EALSMC
ETSMC
Speed command
15000
speed(rpm)

1500 0
speed(rpm)

0.05 0.1 0.15 0.2 0.25 0.3 0.35

time(s)
Figure
1000
6. Comparison of the speed response between the proposed ETSMC and the EALSMC (speed
1000 6. Comparison of the speed response between the proposed ETSMC and the EALSMC (speed
Figure
command 0 → 2000rpm , and, at 0.25 s, the load changes from 0to16 N-m).
command 0 → 2000rpm , and, at 0.25 s, the load changes from 0to16 N-m).
500
500
2500
Figures 7–10 illustrate the speed control response waveforms obtained from simula-
EALSMC
EALSMC
ETSMC
ETSMC
tions,0 under different speed variations and
Speed load disturbances, of the CSALSMC (simulation
Speed command
command
200000 0.05 0.1 0.15 0.2 0.25 0.3 0.35
parameters:
0 0.05 control
0.1 gain
0.15
time(s)C0.2= 60; 0.25constant-speed
0.3 0.35 approach parameter ε = 2000) and the
time(s)
proposed
Figure
Figure 6. ETSMC (simulation
the speed response parameters:
6. Comparison of the speed response between the proposed ETSMC and
Comparison of between the exponential
proposed ETSMC approach
and the EALSMCparameter
the EALSMC (speed
(speed q = 2000;
1500
0 → 2000rpm
speed(rpm)

command
command 0 → 2000rpm
constant-speed approach
,, and, at parameter
and, at 0.25 s, the load = 2000).
changes
0.25 s, the loadε changes from 0to16 N-m).
from 0to16 N-m).

1000
2500
2500

500
2000
2000
CSALSMC
ETSMC
Speed command
15000
speed(rpm)

1500 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

speed(rpm)

time(s)

Figure
10007. Comparison of speed control response between the proposed ETSMC and the CSALSMC
1000
(speed command 0 → 2000rpm , and, at 0.25 s, the load changes from 0to16 N-m).
500
500
2500 CSALSMC
CSALSMC
ETSMC
ETSMC
Speed command
0 Speed command
00 0.05 0.1 0.15 0.2 0.25 0.3 0.35
2000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
time(s)
time(s)

Figure 7.
Figure
Figure 7. Comparison
Comparison of
of speed
7. Comparison speed control
of speed response
controlcontrol between
between the
responseresponse proposed
between
the ETSMC
the
proposed and
and the
proposed
ETSMC CSALSMC
ETSMC
the and the CSALSMC
CSALSMC
1500
0 → 2000rpm 0to16
speed(rpm)

(speed command 0 → 2000rpm , and, at 0.25 s, the load changes from 0to16 N-m).
(speed command , and, at 0.25 s, the load changes from
(speed command 0 → 2000 rpm , and, at 0.25 s, the load changes from 0 to 16 N-m).N-m).

1000
2500
2500

500
2000
2000
CSALSMC
ETSMC
Speed command
15000
speed(rpm)

1500 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

speed(rpm)

time(s)

10008. Comparison of speed control responses between the proposed ETSMC and the CSALSMC
Figure
1000
(speed command of 1000 rpm increases to 2000 rpm after 0.15 s, and at 0.25 s the load changes
0to16 N-m).
from 500
500
CSALSMC
CSALSMC
ETSMC
ETSMC
Speed command
0 Speed command
00 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
time(s)
time(s)

Figure
Figure 8.
8. Comparison
Comparison of
of speed
speed control responses
controlcontrol
responses between
between the
the proposed
proposed ETSMC
ETSMC and
and the
the CSALSMC
CSALSMC
Figure 8. Comparison
1000 of speed
rpm responses
2000 rpm between the proposed ETSMC and the CSALSMC
(speed command of 1000 rpm increases to 2000 rpm after 0.15 s, and at 0.25 s the load changes
(speed command of increases to after 0.15 s, and at 0.25 s the load changes
(speed command of 1000 rpm increases to 2000 rpm after 0.15 s, and at 0.25 s the load changes from 0
from 0to16
from 0to16 N-m).
N-m).
to 16 N-m).
 Electronics 2024, 13, 4028 19 of 28
Electronics 2024, 13, 4028 17 of 26

speed(rpm)

Figure 9. Comparison
Figure of speed
9. Comparison control
of speed responses
control between
responses the proposed
between ETSMC and
the proposed the CSALSMC
ETSMC and the CSALSMC
(speed command of 2000 rpm decreases to 1000 rpm after 0.15 s, and at 0.25 sload
(speed command of 2000 rpm decreases to 1000 rpm after 0.15 s, and at 0.25 s the the changes
load changes from
0 to0to16
from N-m).
16 N-m).
speed(rpm)

Figure 10.10.
Figure Comparison of speed
Comparison control
of speed response
control between
response the proposed
between ETSMC and
the proposed the CSALSMC
ETSMC and the CSALSMC
(speed command
(speed commandof 2000
of 2000rpmrpmwith
witha load change
a load change 0to16
fromfrom N-mN-m
0to16 at 0.15 s, ands, at
at 0.15 0.25ats 0.25
and the s the load
load changes
changes from 16 16to4
from N-m).
to 4 N-m).

From
From thethe
simulation
simulationresults in Figures
results 7–10, it7–10,
in Figures is observed that the robust
it is observed that the controller
robust controller
proposed in this paper, which combines ET with SMC (ETSMC)
proposed in this paper, which combines ET with SMC (ETSMC) using the exponential using the exponential
approach law, exhibits a better dynamic response and steady-state response in tracking
approach law, exhibits a better dynamic response and steady-state response in tracking
speed commands compared to the SMC with the CSAL (CSALSMC). Additionally, under
speed commands compared to the SMC with the CSAL (CSALSMC). Additionally, under
load variations, the speed recovery response of the proposed robust controller not only
load
has variations,
a smaller the amplitude
recovery speed recovery but alsoresponse
a shorterofrecovery
the proposed
time, androbust controller
it reaches a steadynot only has
a smaller recovery amplitude but also a shorter recovery time, and
state. Therefore, compared to the traditional SMC using the CSAL, the robust controller it reaches a steady state.
proposed in this paper achieves a better speed control response due to its self-adaptive proposed
Therefore, compared to the traditional SMC using the CSAL, the robust controller
in this paper achieves a better speed control response due to its self-adaptive capability.
capability.
InIn order
order to present
to present the control
the control performance
performance of the proposed
of the proposed ETSMC moreETSMC more
clearly, theclearly, the
tracking (load 2 N-m, speed command 1000 →
speed command tracking (load 2 N-m, speed command 1000 → 2000 → 1000 rpm ) and
speed command 2000 → 1000 rpm) and
load
loadregulation
regulation (speed
(speed 3000
3000 rpm , load
rpm, loadchange
change2→23→N-m) 3 N-m)response of three
response different
of three different SMCs
SMCs are shown in Figures 11–13. The comparison results of different
are shown in Figures 11–13. The comparison results of different speed command tracking speed command
tracking
and loadandregulation
load regulation response
response performances,obtained
performances, obtained through
through simulations
simulations usingusing three
three different SMCs, are listed in Tables 4–6.
different SMCs, are listed in Tables 4–6.
Electronics 2024, 13, 4028 20 of 28

Electronics 2024, 13, 4028 20 of 28

Electronics 13,4028
2024,13,
Electronics 2024, 4028 20 of 28 18 of 26

2500

2500
2500
2000

2000
2000
1500

1500
1500
1000
CSALSMC
1000
EALSMC
1000 CSALSMC ETSMC
EALSMC
CSALSMC Speed command
ETSMC
EALSMC
0.7 0.8 0.9 1 1.1 1.2 1.3 command
Speed 1.4 1.5
ETSMC
0.7 0.8 0.9 1 time(s) 1.2
1.1 1.3 1.4 command
Speed 1.5
0.7 0.8 0.9 1 time(s)
1.1 1.2 1.3 1.4 1.5
Figure 11. Comparison of the simulated
time(s) speed command tracking response waveforms of the pro-
Figure 11. Comparison of the simulated speed command tracking response waveforms of the pro-
Figure
posed Comparison
11. the
ETSMC, EALSMC, of thethe
and simulated
CSALSMC speed
(load command
2 N-m, tracking
speed response
command waveforms
1000→2000 of the
FigureETSMC,
posed 11. Comparison of the and
the EALSMC, simulated speed command
the CSALSMC tracking
(load 2 N-m, response
speed waveforms
command of
1000→2000 the pro- rpm).
rpm).
proposed
posed ETSMC,ETSMC, the EALSMC,
the EALSMC, and the CSALSMC
and the CSALSMC (load 2 N-m,(load
speed2 command
N-m, speed command
1000→2000 1000→2000 rpm).
rpm).
2500
2500
2500

2000
2000
2000

1500
1500
1500

1000
1000
1000

500
500
500 CSALSMC
EALSMC
CSALSMC
ETSMC CSALSMC
EALSMC
EALSMC
Speed command
0 ETSMC
2.3 2.4 2.5 2.6 2.7 2.8 ETSMC
2.9 command
Speed
0 Speed command
0 2.3 2.4 2.5 2.6time(s) 2.7 2.8 2.9
2.3 2.4 2.5 2.6
time(s) 2.7 2.8 2.9
Figure 12. Comparison of the simulated time(s)
speed command tracking response waveforms of the pro-
Figure
posed 12. Comparison
ETSMC,
Figure 12. the of the and
EALSMC,
Comparison simulated
ofthe
the speed command
CSALSMC
simulated(load tracking
2 N-m,
speed speed
commandresponse
commandwaveforms of the
2000→1000
tracking response pro-
rpm).
waveforms of the
Figure 12. Comparison
posed ETSMC, of the
the EALSMC, andsimulated
the CSALSMCspeed command
(load tracking
2 N-m, speed response
command waveforms
2000→1000 rpm). of the pro-
proposed
posed ETSMC,ETSMC, the EALSMC,
the EALSMC, and
and the the CSALSMC
CSALSMC (load 2(load
N-m,2 N-m,
speedspeed command
command 2000→rpm).
2000→1000 1000 rpm).
3000

3000

3000
2950

2950

2950
2900

2900

2900
2850

2850

CSALSMC
2800
2850 EALSMC
CSALSMC
ETSMC
2800
EALSMC
Speed command
ETSMC
0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 command
Speed 1.4
time(s)
0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
CSALSMC
2800
time(s) EALSMC
ETSMC
Figure 13. Comparison of the simulated load speed recovery responses between the proposed Speed command
Figure 13.
ETSMC, the Comparison
EALSMC,
0.95 1 andof the
1.05 the simulated
1.1CSALSMC
1.15 load1.25
(speed
1.2 speed recovery
30001.3rpm, load responses
1.35 between the proposed
21.4N-m→3 N-m).
ETSMC, the EALSMC, and the CSALSMC time(s) (speed 3000 rpm, load 2 N-m→3 N-m).

Figure
Figure 13.13. Comparisonofofthe
Comparison the simulated
simulated load
loadspeed
speedrecovery responses
recovery between
responses the proposed
between ETSMC,
the proposed
the EALSMC,
ETSMC, and the
the EALSMC, andCSALSMC (speed(speed
the CSALSMC 3000 rpm,
3000 load N-m2→
rpm,2load 3 N-m).N-m).
N-m→3
 Electronics 2024, 13, 4028 21 of 28

Electronics 2024, 13, 4028 19 of 26

Table 4. Comparison of the simulated speed command tracking response performance of the three
SMCs (load 2 N-m, speed command 1000 → 2000 rpm).
Table 4. Comparison of the simulated speed command tracking response performance of the three
Controller Type SMCsETSMC
(load 2 N-m, speed command 1000 EALSMC
→ 2000 rpm ). CSALSMC

Controller Type Tracking time is 0.204 s, with no Tracking time EALSMC

ETSMC is 0.492 s, with a Tracking time is 0.747 s, with no
CSALSMC
Speed response
overshoot. positive overshoot. overshoot.
Tracking time is 0.204 s, with Tracking time is 0.492 s, with a Tracking time is 0.747 s, with
Speed response
no overshoot. positive overshoot. no overshoot.
Table 5. Comparison of the simulated speed command tracking response performance of the three
Table 5.(load
SMCs 2 N-m, speed
Comparison simulated 2000
of thecommand speed→command
1000 rpm).tracking response performance of the three
SMCs (load 2 N-m, speed command 2000 → 1000 rpm ).
Controller Type ETSMC EALSMC CSALSMC
Controller Type ETSMC EALSMC CSALSMC
Tracking time is 0.113 s, with no Tracking time is 0.424 s, with a Tracking time is 0.456 s, with no
Speed response Tracking time is 0.113 s, with Tracking time is 0.424 s, with a Tracking time is 0.456 s, with
Speed response overshoot. negative overshoot. overshoot.
no overshoot. negative overshoot. no overshoot.

Table 6. Comparison of the simulated load recovery response performance of the three controllers
Table 6. Comparison of the simulated load recovery response performance of the three controllers
(load change 2 → 3 N-m, speed 3000 rpm).
(load change 2 → 3 N-m, speed 3000 rpm).

Controller Type
Controller Type ETSMC
ETSMC EALSMC
EALSMC CSALSMC
CSALSMC
Speed
Speed drop drop 7is rpm,
is about aboutand
7 rpm, Speedisdrop
Speed drop is about
about 8 rpm,
8 rpm, Speed
and Speed dropisisabout
drop about 220
220 rpm,
rpm,
Speed response
Speed response and recovery time is 0.065 s. and recovery time is 0.238 s. and recovery time is 0.511 s.
recovery time is 0.065 s. recovery time is 0.238 s. and recovery time is 0.511 s.

5. Experimental Results
5. Experimental Results
In this study, a TMS320F28335 digital signal processor (DSP) from Texas Instruments
In this study, a TMS320F28335 digital signal processor (DSP) from Texas Instruments
was rebuilt to realize the inverter and different SMCs used for performance comparisons in
was rebuilt to realize the inverter and different SMCs used for performance comparisons
this study. The appearance of the overall hardware circuitry of the motor drive is shown in
in this study. The appearance of the overall hardware circuitry of the motor drive is shown
Figure 14 and the experimental test bench used in this study is shown in Figure 15. IN both,
in Figure
the inverter14isand thetoexperimental
used drive the BLDCM test bench used
and the in this study is
dynamometer is adopted
shown inforFigure 15. IN
a BLDCM
loading test. A digital storage oscilloscope is used to measure the output waveform offor
both, the inverter is used to drive the BLDCM and the dynamometer is adopted thea
BLDCM loading test. A digital storage oscilloscope is used to measure
inverter and the speed response waveform of the BLDCM under different test conditions. the output wave-
form
In of the inverter
addition, and theisspeed
the DC source used response
to supplywaveform
power toof the
the BLDCM underwhile
dynamometer, different
the DCtest
conditions. In addition, the DC source is used to supply power to the dynamometer,
electronic load is used to carry out discharging for the DC link of the inverter in order to while
the DCthe
repeat electronic load as
test as soon is possible.
used to carry out discharging
A notebook computer foristhe DCtolink
used of thethe
modify inverter
controlin
order to repeat the test as soon as
programs online and transmit them to the DSP. possible. A notebook computer is used to modify the
control programs online and transmit them to the DSP.

Figure14.
Figure 14.The
Theappearance
appearanceof
ofthe
theoverall
overallhardware
hardwarecircuitry
circuitryofofthe
themotor
motordrive.
drive.
Electronics 2024,13,
Electronics2024, 13,4028
4028 22 of 26
20 of 28

DC source
DC electronic load

Oscilloscope

Computer
Inverter

Dynamometer
Brushless DC motor

Figure15.
Figure 15.Experimental
Experimentaltest
testbench
benchof
ofthe
theproposed
proposedarchitecture
architectureofofthe
theBLDCM
BLDCMdrive
drivesystem.
system.

Toverify
To verifythe
thefeasibility
feasibilityofofthetheproposed
proposedETSMC, ETSMC,we weemployed
employedthree threetypes
typesof ofSMCs
SMCs
to conduct
to conduct tests of of its
itsspeed
speedcommand
commandtracking tracking andandload regulation
load regulation responses
responses under
underdif-
different operatingconditions
ferent operating conditionsand andcompare
comparethe thecontrol
controlperformance
performance of of the
the three controllers.
controllers.
Figure
Figure1616showsshows thetheexperimental
experimental speed response
speed responsewaveforms
waveforms of theofthree
the controllers when
three controllers
the
whenspeedthe command
speed command changes from 1000
changes fromrpm 1000torpm2000torpm 2000under a loadaofload
rpm under 2 N-m,
of 2 with
N-m,
their
with response performances
their response performancescompared
compared in Table 7. 7.
in Table Subsequently,
Subsequently, thethespeed
speedcommand
command
was
wasreduced
reducedfromfrom2000 2000rpm rpmto to1000
1000rpm rpmwhile
whilemaintaining
maintainingthe the22N-m
N-mload.load.TheThespeed
speed
command
command tracking
tracking response
response waveforms
waveforms of of the
the three
three controllers
controllers areare shown
shown in in Figure
Figure 17,17,
with their response performances
with their response performances compared in Table 8. From Figures 16 and 17,
in Table 8. From Figures 16 and 17, as well as as well
as Tables
Tables 7 and
7 and 8, it8,can
it can be observed
be observed thatthat the proposed
the proposed ETSMC ETSMC demonstrates
demonstrates a superior
a superior speed
speed command tracking performance under diverse operating
command tracking performance under diverse operating conditions compared to tradi- conditions compared to
traditional
tional SMCs. SMCs.
The The sampling
sampling periodperiod for experiments
for the the experiments in this
in this paperpaper
waswas alsoalso 0.1 µs.
0.1 µs.
In
Inthe
themotor
motorloading
loadingtest,
test,the
thetesting
testingcondition
conditionwas wasaaspeed
speedof of3000
3000rpm,
rpm,and andthe
theload
load
was increased from 2 N-m to 3 N-m to perform a speed recovery
was increased from 2 N-m to 3 N-m to perform a speed recovery response test. The speed response test. The speed
recovery
recoveryresponses
responsesof ofthe
theproposed
proposedETSMC ETSMCwere werecompared
comparedwith withthose
thoseof ofthe
theSMCs
SMCsthatthat
used
used only a CSAL and those that used only an EAL. From the speed recoveryresponse
only a CSAL and those that used only an EAL. From the speed recovery response
waveforms
waveformsmeasured
measuredininFigure Figure18, 18,ititcan
canbebeobserved
observedthat thatthe
theproposed
proposed ETSMC
ETSMC exhibits
exhibitsa
smaller speed drop and faster recovery under actual loading conditions.
a smaller speed drop and faster recovery under actual loading conditions. In contrast, the In contrast, the
SMCs
SMCswith withonly
onlyaaCSALCSALor orEAL,
EAL,being
beinglesslessrobust
robustin intheir
theircontrol,
control,showed
showedaaslowerslowerspeed
speed
recovery after loading.
recovery after loading.
 Electronics 2024, 13, 4028

F28335_N1
Electronics 2024, 13, 4028
Electronics 2024, 13, 4028 21 of 26 23 o

3000
F28335_N1

2500

3000

Speed(rpm)
2000

2500

1500
Speed(rpm)
2000
: CSALSMC

1000 : EALSMC
1500
: ETSMC
: CSALSMC
500
2 3 4 5 6 7 8
1000 : EALSMC
Time (s)

: ETSMC
500
2 Figure
3 16. Comparison
4 of the measured
5 speed command
6 tracking
7 response 8waveforms of
posed ETSMC, the EALSMC, and Time the
(s) CSALSMC (load 2 N-m, speed command 1000 → 200

Figure 16. Comparison of the measured speed command tracking response waveforms of the
Figure7.16.
Table Comparisonofofthe
Comparison themeasured
measuredspeed
speed command
command tracking
trackingresponse
responsewaveforms of the
performance of tp
proposed ETSMC, the EALSMC, and the CSALSMC (load 2 N-m, speed command 1000 → 2000 rpm ).
posed ETSMC,
controllers the2 EALSMC,
(load and
N-m, speed the CSALSMC
command (load 2 N-m,
1000 → 2000rpm ).speed command 1000 → 2000rpm

Table 7. Comparison of 7.
Table the measured of
Comparison speed commandspeed
the measured tracking response
command performance
tracking responseof the three of the th
performance
Controller Type ETSMC EALSMC CSALSMC
controllers (load 2controllers
N-m, speed command 1000 → 2000 rpm ).
(load 2 N-m, speed command 1000 → 2000rpm ).

Controller Type
Tracking time is 0.864 s, with no EALSMC
ETSMC ETSMC
Tracking time is 1.23 s, withCSALSMC
a Tracking time is 1.37 s, w
Speed response
Controller Type EALSMC CSALSMC
overshoot. positive overshoot. overshoot.
Tracking time is 0.864 s, with Tracking time is 1.23 s, with a Tracking time is 1.37 s, with
Speed response Tracking time is 0.864 s, withpositive
no Tracking time is 1.23 s, with ano Tracking
no overshoot. overshoot. overshoot.time is 1.37 s, with
Speed response
overshoot. positive overshoot. overshoot.
F28335_N1
2500
6.5K

F28335_N1
2500
6.5K
6K
2000

6K
2000
1500
5.5K
Speed(rpm)
3500

1500
5.5K
Speed(rpm)

1000
5K
3500

: CSALSMC
1000
5K
500
4.5K
: EALSMC
: CSALSMC
500
4.5K : ETSMC
04K : EALSMC
4 5 6 7 8
Time (s) : ETSMC
04K
4 5 6 7 8
Figure 17. Comparison of the measured speed command
Time (s) tracking response waveforms of the pro-
Figure 17. Comparison of the measured speed command tracking response waveforms of
posed ETSMC, the EALSMC, and the CSALSMC (load 2 N-m, speed command 2000 → 1000 rpm ).
posed ETSMC, the EALSMC, and the CSALSMC (load 2 N-m, speed command 2 0 0 0 → 1 0
Figure 17. Comparison of the measured speed command tracking response waveforms of the p
).posed ETSMC, the EALSMC, and the CSALSMC (load 2 N-m, speed command 2 0 0 0 → 1 0 0 0 r
Table 8. Comparison of the measured speed command tracking response performance of the three
controllers (load 2).N-m, speed command 2000 → 1000 rpm ).

Controller Type ETSMC EALSMC CSALSMC

Tracking time is 1.11 s, with Tracking time is 1.33 s, with a Tracking time is 1.73 s, with
Speed response
no overshoot. negative overshoot. no overshoot.

In the motor loading test, the testing condition was a speed of 3000 rpm, and the load
was increased from 2 N-m to 3 N-m to perform a speed recovery response test. The speed
recovery responses of the proposed ETSMC were compared with that of the CSALSMC
and that of the EALSMC. From the speed recovery response waveforms measured in
 Speed response
overshoot. negative overshoot. overshoot.

In the motor loading test, the testing condition was a speed of 3000 rpm, and the load
Electronics 2024, 13, 4028 was increased from 2 N-m to 3 N-m to perform a speed recovery response test. The 22 speed
of 26
recovery responses of the proposed ETSMC were compared with that of the CSALSMC
and that of the EALSMC. From the speed recovery response waveforms measured in Fig-
ure 18, 18,
Figure andand
with the the
with response performance
response compared
performance in Table
compared 9, it 9,
in Table can it be
canobserved that
be observed
the proposed
that ETSMC
the proposed exhibits
ETSMC a smaller
exhibits speedspeed
a smaller drop drop
and faster recovery
and faster underunder
recovery actualactual
load-
ing conditions.
loading In contrast,
conditions. the CSALSMC
In contrast, the CSALSMCor EALSMC, being less
or EALSMC, robust
being less in their control,
robust in their
showed showed
control, a sloweraspeed
slowerrecovery after loading.
speed recovery after loading.

F28335_N1
3.2K

3.1K
Speed(rpm)

3K

2.9K
: CSALSMC

: EALSMC

: ETSMC
2.8K
4 5 6 7 8
Time (s)

Figure 18. Comparison of the measured load speed recovery responses between the proposed ETSMC,
Figure 18. Comparison of the measured load speed recovery responses between the proposed
the EALSMC, and the CSALSMC (speed 3000 rpm, load 2 N-m → 3 N-m).
ETSMC, the EALSMC, and the CSALSMC (speed 3000 rpm, load 2 N-m → 3 N-m).

Table 9.9.Comparison
Table Comparisonofofspeed
speed recovery
recovery response
response performance
performance of three
of three controllers
controllers (load (load
2 N-m,2 speed
N-m,
speed command 3000 rpm, load 2
command 3000 rpm, load 2 N-m →3 N-m). N-m→3 N-m).

Controller Type
Controller Type ETSMCETSMC EALSMC
EALSMC CSALSMC
CSALSMC
Speed drop is about 15 rpm, Speed drop is about 30 rpm, Speed drop is about 60 rpm,
Speed response Speed drop is about 15 rpm,
Speed response and recovery time is 0.79 s.Speedand
drop is about
recovery 30isrpm,
time 1.09 s. Speed
anddrop is about
recovery time is601.28
rpm,
s.
and recovery time is 0.79 s. and recovery time is 1.09 s. and recovery time is 1.28 s.

Because the inertia and viscosity of the motor drive system in the simulation are
Because
different fromtheits inertia and viscosity
actual values, of the
the speed motor drive
response system inshown
performances the simulation
in Tables are
4–6 dif-
are
ferent from its actual values, the speed response performances shown in Tables
slightly different from those in Tables 7–9. However, the simulation and experimental 4–6 are
results show that the control performance of the proposed ETSMC is better than that ofre-
slightly different from those in Tables 7–9. However, the simulation and experimental a
sults show EALSMC
traditional that the control performance of the proposed ETSMC is better than that of a
and CSALSMC.
traditional EALSMC and CSALSMC.
6. Discussion
6. Discussion
An intelligent controller, designed by integrating extension theory with sliding mode
Anbased
theory intelligent
on thecontroller, designed
EAL, enhances by integrating
the performance of aextension
BLDCM theory
beyondwiththat sliding mode
of traditional
theory
P-I based on
controllers theSMCs.
and EAL, enhances thein
This results performance
superior speedof a command
BLDCM beyondtrackingthat of faster
and tradi-
tional recovery
speed P-I controllers and SMCs.
under varying This
loads. results traditional
Although in superiorP-I speed command
controllers tracking
are still and
commonly
used
fasterinspeed
commercially
recoveryavailable motor drivers
under varying due to costtraditional
loads. Although and stability
P-Iconsiderations,
controllers arethese
still
experimental results demonstrate that the proposed intelligent robust controller, when
applied to high-speed BLDCMs, offers exceptional performance. It provides rapid speed
command tracking and load regulation responses with no overshoot, thereby mitigating
the vibration issues in BLDCM drive systems. Moreover, the extension theory used to
automatically adjust the SMC gain does not require prior knowledge of BLDCM parameters,
involves simple rule calculations, and does not necessitate training data, ensuring the
controller’s robustness.
The quantitative performance comparison results of different controllers are presented
in Tables 7–9, as well as in Figures 16–18, highlighting that the intelligent controller, which
combines extension theory with sliding mode theory based on an EAL, significantly en-
hances the response of the BLDCM drive system. This leads to superior speed command
tracking and improved speed recovery under load variations compared to traditional P-I
Electronics 2024, 13, 4028 23 of 26

controllers and SMCs. However, in practical applications, due to cost and stability consid-
erations, most commercially available BLDCM drive systems still rely on traditional P-I
controllers and have yet to adopt intelligent controllers. Therefore, it is anticipated that
the controller designed in this paper will serve as a foundation for future developments,
making intelligent control strategies more accessible, easier to design, and simpler to im-
plement. This would help reduce costs, promote the widespread adoption of intelligent
controllers, and ultimately improve the safety and performance of BLDCM drive systems.

7. Conclusions
This paper combines ET with the EAL of an SMC, replacing the traditional P-I speed
controller used in FOC for BLDCM speed control. The proposed ETSMC determines the
control gain of the SMC by integrating ET, thereby adjusting the parameters of the sliding
surface function. This approach suppresses the overshoot caused by the SMC using only
EAL while also addressing the issue of the slower response speed in traditional SMCs.
Therefore, it improves their performance in speed command tracking and load regulation
response. Additionally, the proposed control method does not require extensive compu-
tation or learning data, making it easy to implement. The simulation and experimental
results demonstrate that the proposed controller not only addresses the issues of excessive
overshoot in speed command tracking observed in SMCs with only an EAL and the slower
speed command tracking response of SMCs with only a CSAL, but that it also exhibits a
superior speed recovery performance under loading compared to these traditional sliding
mode controllers. The SMC with only an EAL, while offering a faster speed command
tracking response, suffers from significant overshoot, raising concerns about its suitability
for motor drive systems requiring precise speed control in industrial applications. On
the other hand, the SMC with only a CSAL achieves speed command tracking without
overshoot but responds more slowly, failing to meet the performance requirements for
applications that demand a rapid speed command tracking response. In contrast, the
proposed ETSMC not only mitigates the overshoot issue associated with the EAL but also
overcomes the slower speed command tracking and load regulation responses of the CSAL,
thereby offering a more stable and robust solution.

Author Contributions: K.-H.C. managed the project and completed the formal analysis of the
extension controller. K.-H.C. also planned the project and wrote, edited and reviewed the manuscript.
C.-T.H. completed the formal analysis of the sliding mode controller. X.-J.C. was responsible for the
software program and validation of the simulation and experimental results. All authors have read
and agreed to the published version of the manuscript.
Funding: The authors gratefully acknowledge the support and funding of this project by the
Industrial Technology Research Institute, Taiwan, under the Grant Number NCUT23TCE09 and
NCUT23TCE021.
Data Availability Statement: The original contributions presented in the study are included in the
article, further inquiries can be directed to the corresponding author.
Conflicts of Interest: The authors of the manuscript declare no conflicts of interest.

Nomenclature

Acronyms
BLDCM : brushless DC motors
FOC : field-oriented control
ET : extension theory
SMT : sliding mode theory
SMC : sliding mode controller
PMSM : permanent-magnet synchronous motor
P-I : proportional–integral
EMF : electromotive forces
Electronics 2024, 13, 4028 24 of 26

EAL : exponential approach law

CSAL : constant-speed approach law
ETSMC : sliding mode controller with extension theory
EALSMC : sliding mode controller with an exponential approach law
CSALSMC : sliding mode controller with a constant speed approach law
Symbols
v an , vbn , vcn : phase voltages of phases a, b, and c
R a , Rb , Rc : stator resistances of phases a, b, and c
i an , ibn , icn : phase currents of phases a, b, and c
L a , Lb , Lc : self-inductances of phases a, b, and c
Mab , Mbc , Mac : mutual inductances between phases a, b, and c
ean , ebn , ecn : back electromotive forces (EMFs) of phases a, b, and c
Te : electromagnetic torque generated by the motor
ωm : mechanical speed of the motor
TL : load torque
J : rotational inertia of the motor and load
B : coefficient of viscous friction of the motor and load
Rs : stator resistance of the three phases
ϕa , ϕb , ϕc : fluxes of phases a, b, and c
M : mutual inductance between phase windings
L : self-inductance of each phase winding
ϕf : flux produced by the permanent magnet
θ : angle between the rotor pole and the axis of the phase winding
ωe : synchronous speed of the motor
α, β : two-phase stationary coordinate
vα , v β : two-phase stationary coordinate voltage
ϕα , ϕβ : two-phase stationary coordinate flux
d, q : two-phase synchronous rotating coordinate
vd , vq : two-phase synchronous rotating coordinate voltage
ϕd , ϕq : two-phase synchronous rotating coordinate flux
id : d-axis current
iq : q-axis current
Kt : torque constant
P : number of poles
R : fundamental element describing an entity is referred to as a matter element
N : name of an entity
c : characteristics or features of an entity
v : value of the feature of an entity
C0 : classical domain
ci : feature of C0
vi : feature value of ci
cj : characteristic value of C
vj : characteristic quantity of c j
D : distance
ρ : rank value
K (ĉ) : correlation function
Ld , Lq : d- and q-axis inductance
ωm ∗ : speed command
ωm : actual feedback speed
x1 : speed difference
iq∗ : q-axis command current
x2 : rate of change of the speed difference
s : sliding surface function
C : control gain
Electronics 2024, 13, 4028 25 of 26

x1 , x2 : state variables
V : Lyapunov’s second stability criterion continuous function
.
s : approaching law functions
u : control force function
e : speed difference
.
e : rate of change of the speed difference
A1~A4 : intervals
W1 , W2 : weight values
λg : correlation degree between the feature values and each interval category
λmax , −λmax : maximum and minimum correlation degrees for each interval category
Cnew : new control gain

References
1. Wang, M.; Chen, Z. Research on Permanent Magnet Structure of Permanent Magnet Synchronous Motor for Electric Vehicle. In
Proceedings of the 2nd International Conference on Electrical Engineering and Control Science (IC2ECS), Nanjing, China, 16–18
December 2022; pp. 990–993.
2. Lee, C.I.; Jang, G.H. Experimental Measurement and Simulated Verification of the Unbalanced Magnetic Force in Brushless DC
Motors. IEEE Trans. Magn. 2008, 44, 4377–4380. [CrossRef]
3. BV, P.; Balamurugan, A.; Selvathai, T.; Reginald, R.; Varadhan, J. Evaluation of Different Vector Control Methds for Electric Vehicle
Application. In Proceedings of the 2nd International Conference on Power and Embedded Drive Control (ICPEDC), Chennai,
India, 21–23 August 2019; pp. 273–278.
4. Bhatti, S.A.; Malik, S.A.; Daraz, A. Comparison of P-I and I-P Controller by Using Ziegler-Nichols Tuning Method for Speed
Control of DC Motor. In Proceedings of the International Conference on Intelligent Systems Engineering (ICISE), Islamabad,
Pakistan, 15–17 January 2016; pp. 330–334.
5. Errouissi, R.; Al-Durra, A.; Muyeen, S.M. Experimental Validation of a Novel PI Speed Controller for AC Motor Drives with
Improved Transient Performances. IEEE Trans. Contr. Syst. Techn. 2018, 4, 1414–1421. [CrossRef]
6. Murali, S.B.; Rao, P.M. Adaptive Sliding Mode Control of BLDC Motor Using Cuckoo Search Algorithm. In Proceedings of the
2nd International Conference on Inventive Systems and Control (ICISC), Coimbatore, India, 19–20 January 2018; pp. 989–993.
7. Guo, X.; Huang, S.; Peng, Y. Sliding Mode Speed Control of PMSM Based on A Novel Hybrid Reaching Law and High-Order
Terminal Sliding-Mode Observer. In Proceedings of the IEEE 4th International Electrical and Energy Conference (CIEEC), Wuhan,
China, 28–30 May 2021; pp. 1–6.
8. Zhang, J.; Liu, B.; Jiang, Z.; Hu, H. The Application of Sliding Mode Control with Improved Approaching Law in Manipulator
Control. In Proceedings of the 13th IEEE Conference on Industrial Electronics and Applications (ICIEA), Wuhan, China,
31 May–2 June 2018; pp. 807–812.
9. Wang, Q.; Chen, J.; Zhou, Z. Simulation of Sliding Mode Control for PMSM on Account of Variable Exponential Approach Law.
In Proceedings of the Chinese Automation Congress (CAC), Hangzhou, China, 22–24 November 2019; pp. 3518–3521.
10. Bodur, F.; Kaplan, O. A Novel Sliding Mode Control Based on Super Twisting Reaching Law for PMSM Speed Controller with
Fixed-Time Disturbance Observer. In Proceedings of the 12th International Conference on Renewable Energy Research and
Applications (ICRERA), Oshawa, ON, Canada, 29 August–1 September 2023; pp. 1–6.
11. Chao, K.-H.; Chang, L.-Y.; Hung, C.-Y. Design and Control of Brushless DC Motor Drives for Refrigerated Cabinets. Energies 2022,
15, 3453. [CrossRef]
12. Chao, K.H.; Chen, P.Y. A Robust Controller Based on Extension Theory for Switched Reluctance Motor Drives. In Proceedings of
the International Conference on Machine Learning and Cybernetics, Xi’an, China, 15–17 July 2012; pp. 845–850.
13. Khanke, P.K.; Jain, S.D. Comparative Analysis of Speed Control of BLDC Motor Using PI, Simple FLC and Fuzzy-PI Controller. In
Proceedings of the International Conference on Energy Systems and Applications, Pune, India, 30 October–1 November 2015;
pp. 296–301.
14. Longfei, J.; Yuping, H.; Jigui, Z.; Jing, C.; Yunfei, T.; Pengfei, L. Fuzzy Sliding Mode Control of Permanent Magnet Synchronous
Motor Based on the Integral Sliding Mode Surface. In Proceedings of the 22nd International Conference on Electrical Machines
and Systems (ICEMS), Harbin, China, 11–14 August 2019; pp. 1–6.
15. Zhang, L.; Li, Z.; Ma, H.; Ju, P. Power Transformer Fault Diagnosis Based on Extension Theory. In Proceedings of the International
Conference on Electrical Machines and Systems, Nanjing, China, 27–29 September 2005; pp. 1763–1766.
16. Qi, C.; Xie, J.; Mao, H.; Xie, Q. Condition Assessment of Valve-side Bushing of Converter Transformer Based on Extension Theory.
In Proceedings of the IEEE International Conference on High Voltage Engineering and Application (ICHVE), Beijing, China, 6–10
September 2020; pp. 1–4.
17. Wang, J.; Ye, J.; Yan, R. The Magnetostatic Simulation and Determination of Magnetic Components for an External Ventricular
Drainage Device. In Proceedings of the 40th Chinese Control Conference (CCC), Shanghai, China, 26–28 July 2021; pp. 6386–6391.
Electronics 2024, 13, 4028 26 of 26

18. Kishore, N.; Shukla, K.; Gupta, N. A Novel Three-Phase Multilevel Inverter Cascaded by Three-Phase Two-Level Inverter and
Two Single-Phase Boosted H-Bridge Inverters. In Proceedings of the IEEE PES Innovative Smart Grid Technologies-Asia (ISGT
Asia), Singapore, 1–5 November 2022; pp. 330–334.
19. Xie, T.; Kuai, Z.; Ye, X.; Zhu, L.; Guo, K.; Shi, S. Coordinate Conversions and Deviation Analysis in Multi-source Data Fusion. In
Proceedings of the 2nd China International SAR Symposium (CISS), Shanghai, China, 3–5 November 2021; pp. 1–3.
20. Yadunandan; Naik, B.; Konduru, R.; Ilkal, B.B.; Kalligudd, S. Terminal Voltage Control of BLDC Motor. In Proceedings of the
International Conference on Smart Systems for Applications in Electrical Sciences (ICSSES), Tumakuru, India, 7–8 July 2023;
pp. 1–5.
21. Wu, H.-Q.; Chen, Y.-W.; Liu, R. College Education Outcome Expectation and Proactive Personality as Predictors of Chinese
College Students’ Learning Motivation, Career Adaptability as a Mediator. In Proceedings of the International Conference on
Modern Education and Information Management (ICMEIM), Dalian, China, 25–27 September 2020; pp. 716–721.
22. Luo, N.; Zhang, L. Chaos Driven Development for Software Robustness Enhancement. In Proceedings of the 9th International
Conference on Dependable Systems and Their Applications (DSA), Wulumuqi, China, 4–5 August 2022; pp. 1029–1034.
23. Zeng, X.; Zeng, X.; Hong, Y. Distributed Computation of Common Lyapunov Functions. In Proceedings of the 37th Chinese
Control Conference (CCC), Wuhan, China, 25–27 July 2018; pp. 2418–2423.
24. AM2200H BLDC Motor. Available online: https://www.adlee.com/zh-tw/product-552515/%E5%B7%A5%E6%A5%AD%E7%9
4%A8%E7%9B%B4%E6%B5%81%E7%84%A1%E5%88%B7%E9%A6%AC%E9%81%94.html (accessed on 17 February 2023).

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.
Reproduced with permission of copyright owner. Further reproduction
prohibited without permission.