Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 7, No. 2, 2023 

 

52 

Research on Active Disturbance Rejection Control of 
Brushless DC Motor Based on Vector Control 
Yuxin Wang 
School of Electrical Engineering, Southwest Minzu University, Chengdu, Sichuan 610041, China 

 

Abstract: A Active Disturbance Rejection Control strategy based on vector control is proposed to address the problems of 
poor speed overshooting immunity in the control system of brushless DC motor. The motor magnetic chain and torque are 
decoupled by vector control method to simplify the control, and the traditional PI controller in the speed loop is replaced by a 
active disturbance rejection controller. The transition process is arranged by tracking differential (TD) to smooth the given speed, 
which overcomes the contradiction between speed overshoot and rapidity, and improves the response capability of the system; a 
second-order extended state observer (ESO) is introduced, which estimates and compensates the total system disturbance and 
improves the system's immunity to disturbance; and the nonlinear law state error feedback (NLSEF) is utilized by the Through 
the nonlinear law state error feedback (NLSEF), the control accuracy of the system is improved by utilizing the nonlinear control 
principle of "large gain with small error and small gain with large error". The simulation results show that the control strategy 
proposed in this paper is characterized by fast response, no overshooting, strong resistance to load disturbance, and strong 
robustness to changes in load and speed, which proves the feasibility and advancement of this strategy. 

Keywords: Brushless dc motor, Active disturbance rejection control, Vector control, Speed-current double closed loop. 
 

1. Introduction 
As a big country of rare earth resources, with the current 

continuous development of power electronics technology, 
computer control technology, materials science and other 
fields, brushless DC motors (BLDCM), which are made of 
permanent magnet materials, are being more and more widely 
used in industrial manufacturing and daily life. However, 
BLDCM is a complex system with many internal variables 
and obvious nonlinear characteristics, which makes it difficult 
to establish a high-precision dynamic mathematical model for 
it. [1] It is difficult to establish a high-precision dynamic 
mathematical model for it. Therefore, in order to meet the 
demand for more accurate control, more advanced motor 
control strategies are needed. The traditional PID control 
algorithm has become one of the most commonly used control 
algorithms in the field of motor control due to its simple 
structure and ease of application in engineering practice. [2] 
However, the traditional PID control algorithm has been used 
as one of the most common control algorithms in motor 
control. However, the traditional PID control algorithm is 
mainly based on linear model, strong dependence on 
parameters, and poor anti-interference ability, and can not 
meet the high precision requirements of the control system. 
[3] The traditional PID control algorithm is mainly based on 
linear model, which has strong dependence on parameters and 
poor anti-interference ability, and cannot meet the high 
precision requirements of control system. Therefore, the 
research of a higher performance brushless DC motor control 
system has become the research direction of many scholars 
around the world. Because the brushless DC motor control 
system is a multivariable coupling system in the three-phase 
natural coordinate system, it is difficult to control the complex 
relationship between the magnetic chain and the torque of the 
motor. [4] The vector control technology can combine the 
magnetic chain and torque with the magnetic chain and the 
torque. Vector control technology can decouple the complex 
relationship between magnetic chain and torque, which not 

only makes the system have faster response speed, but also 
makes the control process simple to a certain extent. [5] 
Therefore, this paper applies it to the control of brushless DC 
motor system. 

With the research carried out in recent years, adaptive 
control [6] and fuzzy control [7] and neural network control 
[8] and other theories have been fully developed. However, 
although these methods have achieved better control effects 
in theory, they are difficult to be applied in practice due to the 
complexity of their realization. 

Active Disturbance Rejection Control (ADRC) is a new 
type of nonlinear control theory that does not depend on the 
exact mathematical model of the object, proposed by Kyung-
Ching Han [9] ADRC is a new nonlinear control theory that 
does not depend on the exact mathematical model of the 
object. Many results have been achieved in the application as 
well as improvement of the active disturbance rejection 
controller. Among them, the literature [10] model-
compensated self-oscillatory controller is used to replace the 
traditional self-oscillatory controller, but it is easy to lead to 
the saturation of the control volume and does not achieve the 
optimal compensation effect. Literature [11] A method of 
speed-loop smooth-mode active disturbance rejection control 
based on internal-mode observation is proposed to improve 
the problem of poor speed regulation performance due to 
motor load disturbance. Literature [12] A feed-forward 
compensation based active disturbance rejection control 
algorithm is used to control a brushless DC motor and applied 
to an electric valve, resulting in a significant improvement in 
position control accuracy and immunity. Literature. [13] A 
fuzzy control strategy combining fuzzy control and self-
resistant control is proposed to control the yarn constant 
tension of a brushless DC motor, which makes the fluctuation 
range of the yarn tension smaller and the response of the 
system faster. Literature [14] A parameter active disturbance 
rejection control method based on the improved particle 
swarm algorithm is designed and applied to the linear active 
disturbance rejection controller, which greatly simplifies the 



 

53 

steps of parameter tuning. 
In this paper, for the problems of poor anti-interference 

ability, speed overshooting and large torque fluctuation of the 
brushless DC motor under the control of traditional PID 
algorithm, a control strategy based on active disturbance 
rejection control is proposed to apply the active disturbance 
rejection controller to the speed loop of the brushless DC 
motor control system. On the basis of establishing a 
mathematical model of the brushless DC motor, the general 
derivation process of the speed loop active disturbance 
rejection controller is given. The performance of the brushless 
DC motor control system under the active disturbance 
rejection controller is investigated by modeling simulation in 
Simulink. The simulation results show that the system 
performance under this control strategy is excellent, with fast 
response, no overshoot, etc., and can effectively resist load 
perturbations and is robust to load variations and speed 
variations. This further confirms the effectiveness and 
feasibility of the strategy. 

2. Mathematical Model of Brushless 
DC Motor 

The mathematical model of the motor can be represented 
as follows in the d-q synchronous rotating coordinate system: 

The motor stator voltage equation is: 
 

d
d s d d e q

q
q s q q e d

di
u R i L

dt
di

u R i L
dt

               (1) 

 
Where: ud , uq are d, q-axis voltage; id , iq are d, q-axis 

current; ψd , ψq are d, q-axis magnetic chain; Ld , Lq are d, q-
axis inductance; ωe is the motor rotor electric angular velocity. 

 
The motor stator magnetic chain equation is: 
 

d d d f

q q q

L i

L i
                  (2) 

Where: ψf is the magnetic chain of the permanent magnet. 
The motor torque equation is: 
 

n1.5 ( )

1.5 ( )
e d q q d

n f q d q d q

T P i i

P i L L i i
            (3) 

 
Where: Te is the electromagnetic torque; Pn is the number 

of motor pole pairs. 
Taking the brushless DC motor with surface-mounted rotor 

structure as the object of study, there is Ld =Lq , and the 
control strategy of id =0 is used, at this time, Eq. (3) can be 
rewritten approximately as: 

 
1.5e n f qT P i                        (4) 

 
The motor equation of motion is: 

 
m

e L m
d

T T J B
dt

                 (5) 

 
Where: TL is the load torque; J is the rotational inertia; B is 

the viscous coefficient; ωm is the mechanical angular velocity. 

3. Active Disturbance Rejection 
Controller Design 

3.1. Introduction to active disturbance 
rejection control 

As a kind of nonlinear control algorithm, the core idea of 
ADRC is to avoid overshooting by arranging the transition 
through the tracking differentiator (TD), to compensate the 
total disturbances inside and outside the system as the state 
quantities for the observation and estimation by the expansion 
observer (ESO), and to obtain the final output quantity 
through the nonlinear feedback law of nonlinear error 
(NLSEF) by nonlinearly feeding back the errors of the first 
two phases. The structure is shown in Fig. 1. 

 

 
Figure 1. Block diagram of ADRC structure 

 
The TD module helps in smoothing the transition of the 

input signal v where v1 is used for fast tracking of the input 
signal and v2 is the differential signal of the input signal. the 
ESO module is capable of observing the state variables of the 
system where z1 is the tracking signal of the output y, z2 is the 
approximate differential of the output y and z3 is used for 

observing the total system perturbation. The error signals e1 
and e2 , which are one of the input signals to the NLSEF 
module, are obtained by differencing v1 , v2 and z1 , z2 
respectively. After the NLSEF has processed the input error 
signals, the output signal u0 is output and the system is 
converted to an integrator series system. 

TD NLSEF plant

ESO

1/b0 b0

v
v1

v2

e1

e2
u0 u

z3

z2

z1

y



 

54 

for a typical first-order system: 

( , ( ), )y f y t t buy f y( ,f ( ,                 (6) 
 

where ω(t) is the value of external disturbance; f is the 
value of total disturbance; u is the control quantity of the 
system. At this point, by making y=x, the state equation of the 
system can be obtained: 

 
( , ( ), )x f x t t bu

y x
x f x( ,f ( ,

            (7) 

 
The core of ADRC is to utilize the ESO module to observe 

and estimate the total disturbance of the system, while 
compensating for it by using the error feedback control law 
module to turn the system into a standard integrator series 
type system for simplified control. 

3.2. Speed loop ADRC design 
3.2.1. Velocity loop TD design 

Arranging the transition process for the input speed signal 
of the BLDC control system, the following mathematical 
model is obtained [15] : 

 
1 11

11 1, 1 1

*
( , )

e z y
z rfal e11z rfal11

             (8) 

 
The fal function in Eq. (11) is defined as follows: 

 
( )     

( , )
              

e sign e e
fal e e e

         (9) 

 
In this equation, the variable r represents the speed of the 

control tracking signal, which is proportional to the tracking 
speed. z11 is the speed signal after smoothing transition, y* is 
the given speed signal, and δ1 is the filtering factor, α is 
usually taken as 0.5 or 0.25. 
3.2.2. Velocity loop ESO design 

Mathematical modeling of ESO for BLDC velocity loops 
[16] for: 

 
2 21

21 22 21 2 2 2

22 22 2 3 2

( , , )
( , , )

e z y
z z fal e bu
z fal e

21 22z z21 22zz22

22 22z f22 f22

       (10) 

 
In this equation, z21 represents the observed value of the 

actual output y, z22 is an estimate of the total system 
perturbation, and the filtering factor δ3 . The parameters to be 
adjusted are β21 and β22 , and e2 is the value of the error 
between the observed value z21 and the actual output signal y. 
The filtering factor δ is a parameter of β and β . 
3.2.3. Velocity loop NLSEF design 

The NLSEF mathematical model of BLDC speed loop is 

designed as follows: 

3 11 21

0 33 3 4 3

22
0

( , , )
e z z
u fal e

zu u
b

         (11) 

 
In this equation, e3 represents the error between the velocity 

signal z11 after arranging the transition process and the 
observed value z21 of the actual output y, while -z22 /b denotes 
the component of the compensating perturbation. 

From Eqs. (3) and (5), we have that under the control 
strategy of id =0: 

 
1.5

1 11.5 ( )

n f q L mm

n q f m L

P i T Bd
dt J

P i B T
J J

         (12) 

 
From the equation (12) it can be seen that iq , TL , ωe and 

other factors have influence on the motor speed. Taking iq as 
the variable to be controlled, the other factors can be regarded 
as the total perturbation, then the total perturbation is: 

 
1( ) ( )m La t B T
J

            (13) 

 
Then equation (6) can be rewritten as: 

 
1 1.5 ( )m

n f q
d

P i a t
dt J

       (14) 

 
According to Section 2.1, the current iq is used as the 

control quantity and the given signal ωm * and the actual 
rotational speed are used as inputs to the ADRC controller, 
then the BLDCM speed loop can be regarded as a first-order 
system. 

4. Simulation Study and Result 
Analysis 

This chapter will compare the performance of the speed 
loop of the brushless DC motor control system under the self-
immobilizing controller and the PI controller, including motor 
performance indexes such as rated speed response, load-
disturbed speed response, and response robustness under 
different speed settings. The effectiveness and feasibility of 
the self-immobilizing controller for improving the 
performance of the brushless DC motor control system will 
be verified through comparative analysis. 

4.1. Simulink model construction 
The model is built by MATLAB/Simulink to verify the 

validity and reliability of the self-immunity controller. The 
basic block diagram is shown in Fig. 2 and Fig. 3



 

55 

 
Figure 2. Block diagram of brushless DC motor control system based on self-immunity control 

 

 
Figure 3. Simulation diagram of self-immunity controller 

The parameters of the brushless DC motor in the simulation 
are as follows: the number of pole pairs Pn = 4; stator 
inductance Ls = 8.5 mH, stator resistance Rs = 0.96 Ω, 
permanent magnet chain ψf = 0.183 Wb, rotational moment of 
inertia J = 0.003 kg- m2 , hysteresis coefficient B = 0.008 N- 

m2 , rated speed of 1000 r/min, and rated voltage of 311V. 

4.2. Simulation results and analysis 
4.2.1. Rated speed response 

 

 
Figure 4. Motor speed response waveform at rated speed 

 
A comparison of the simulated waveforms of ADRC and 

PI control at rated speed is shown in Fig. 4. At a given rated 
speed of 1000 r/min, the ADRC-controlled motor reaches the 

specified speed in about 0.025 s with no overshoot. 
While the motor controlled by the traditional PI controller 

has a significant overshoot of 17% although the start-up 



 

56 

response is slightly higher than that of the motor controlled 
by the ADRC, this is because the ADRC has set up a tracking 
differentiator (TD) to arrange the transition process in order 
to balance the contradiction between the rapidity of the 
system's start-up response and the overshoot. 
4.2.2. Resistance to load torque disturbances 

Without changing any parameter, a 5N-m load is added at 

0.2s, as shown in Fig. 5, the amplitude of speed fluctuation 
and the time taken to recover to the rated speed of the motor 
controlled by the PI controller are much larger than that of the 
motor controlled by ADRC, indicating that the ADRC 
algorithm has better anti-interference ability under loaded 
conditions. 

 
Figure 5. 0.2s sudden load motor speed response curve 

4.2.3. Speed Response at Different Speed Settings 
In order to continue to verify the performance of the ADRC 

strategy, under the condition of keeping the system 
parameters unchanged, the rotational speed was set to 0.8 

times and 1.2 times the rated speed, as well as the forward and 
reverse rotation, acceleration and deceleration experiments 
were carried out at the rated speed, respectively, and the 
resultant waveform graphs are shown in Figs. 6 to 9. 

 

 
Figure 6. Motor speed response for set value 800r/min 

 

 
Figure 7. Motor speed response for set value 1200r/min 



 

57 

 
Figure 8. Motor acceleration and deceleration speed response 

 

 
 

Figure 9. Forward and reverse speed response at rated speed 
 

The above experimental results surface that when the 
rotational speed is 0.8 and 1.2 times of the rated speed, the 
rotational speed waveform response results in a better control 
effect compared to the traditional PI control. In the case of the 
initial speed set to 600r/min, the speed is increased to the rated 
speed of 1000r/min in 0.15s, and then reduced to 600r/min in 
0.35s, the experimental results show that the system is fast 
and free of overshooting, and has a better control performance 
compared with the traditional PI control. At 0.25s, the speed 
is set to -1000r/min, and the motor under ADRC control is 

still free of overshooting and has good rapidity. 
4.2.4. Comparison of other indexes under rated speed 

with load 
The simulated waveforms of d-axis current, torque, and 

three-phase current of a brushless DC motor at rated speed 
with load are investigated to gain a deeper understanding of 
the control performance of the two controllers through 
comparison. 

 



 

58 

 

 
Figure 10. Comparison of motor d-axis current waveforms at rated speed with load 

 

 
 
 

 
Figure 11. Comparison of electromagnetic torque waveforms of motors at rated speed with loads 

 



 

59 

 

 
Figure 12. Comparison of three-phase current waveforms of motor at rated speed with load 

 
Fig. 10 shows the comparison of the d-axis current 

waveforms at the rated speed of the motor with load, through 
which it can be seen that the d-axis current under the ADRC 
strategy is active in the range of about [-2.3,2.1], and the d-
axis current under the PI strategy is active in the range of 
about [-5.8,6.9]. The above results indicate that the 
performance of the d-axis current loop of the brushless DC 
motor under the ADRC strategy performs better as compared 
to the PI strategy. 

Fig. 11 shows the comparison of the electromagnetic torque 
waveforms under the rated speed of the motor with load, and 
it can be seen by comparison that the starting maximum 
electromagnetic torque of the motor under the ADRC strategy 
is about 33 N-m, and the starting maximum torque of the 
motor under the PI strategy is about 39 N-m, and the starting 
maximum torque of the motor under the control of the ADRC 
strategy is smaller, and the torque waveforms are smoother as 
a whole, which shows a better starting performance. 

Fig. 12 shows the comparison of the three-phase current 
waveforms of the motor at rated speed with load. It can be 
seen from the comparison that the three-phase current peaks 
of the motor under the ADRC strategy are smaller as 
compared to that of the PI strategy, and it also shows a better 
control performance. 

Based on the comparison of the above simulation 
waveforms, the vector control-based active disturbance 
rejection control strategy for brushless DC motor proposed in 
this paper is feasible and effective, balancing the 
contradiction between starting rapidity and overshooting, and 
improving the system's immunity to interference, while 
possessing stronger robustness. 

5. Conclusion 
The research objective of this paper is to address the 

problems of large speed response overshoot and poor 
interference immunity in the traditional PI-controlled 
brushless DC motor system, and to propose a vector control-
based active disturbance rejection control strategy, which 
utilizes vector control to decouple the complex relationship 
between the magnetic chain and torque to simplify the control, 
and then introduces the active disturbance rejection controller 
into the motor speed ring, gives the general design method of 
the speed ring active disturbance rejection controller, and 
Comparison is made with the traditional PI control strategy. 
The results show that the control strategy proposed in this 
paper solves the problem of speed overshooting, and has 
better anti-interference ability and improves the robustness of 
the system, which proves the effectiveness and advancement 
of the system. However, this strategy still has room to 
improve the response performance of the system under 
disturbances such as sudden load change, and the anti-
disturbance ability needs to be strengthened. 

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