Comparative study of Disturbance Observer-Based

Control and Active Disturbance Rejection Control

in Brushless DC motor drives
Hung V. Nguyen, Thanh Vo-Duy, Minh C. Ta
CTI Laboratory for Electric Vehicles, Hanoi University of Science and Technology, Hanoi, Vietnam
Email: hung.nguyenvanbkhn@gmail.com, thanh.voduy@hust.edu.vn, minh.tacao@hust.edu.vn

Abstract—Disturbance Observer-Based Control (DOBC) and choice of filter cut o frequency depends on the nature of
Active Disturbance Rejection Control (ADRC) are effective con- actual disturbances and uncertainties [4]. Since the parameter
trol techniques to deal with disturbances. This paper introduces variation is a part of the equivalent disturbance, DOBC is
and applies both DOBC and ADRC to Brushless DC (BLDC)
motor drives system. The system is described by Energetic expected to improve the robustness of system.
Macroscopic Representation (EMR) method which has recently ADRC is the new approach of disturbance observation-
become powerful approach for the systems related to energy cancelation control technique [5]. The plant can be described
transformation. By using EMR, the system can be described in a by extended state space model, in which the disturbances and
comprehensive and visual way, thus the control design becomes uncertainties are additional variable states. A nominal system
simple and easier. The EMR of BLDC motor drives is presented;
the inversion-based control system is deduced from EMR; both parameters are used to design an Extended State Observer
DOBC and ADRC are employed based on the basis of inversion- (ESO). The output of ESO includes the equivalent disturbance
based control system to improve performance of motor drive which is possibly compensated by the control signal. Similar to
system. The simulation in Matlab/Simulink is realized to evaluate DOBC, the estimated equivalent disturbance delay depends on
the control effect of DOBC, ADRC and conventional method in the parameters of ESO. The effect of ADRC is demonstrated in
some scenarios: an unexpected change load and a parameters
variation. many applications such as motion control [6], process control
Index Terms—Disturbance Observer-Based Control, Active [7], and power electronics control [8].
Disturbance Rejection Control, Energetic Macroscopic Repre- In this paper, the authors compare the control effect of
sentation, BLDC Motor Drives. DOBC and ADRC in term of performance improvement of
BLDC motor drives. Such drives are more and more popular
I. I NTRODUCTION in industrial and house appliance application thank to their
Improving the performance of motor drives system is one constructional advantages. One of the highlights of the paper is
of the main objectives of motor control design. The presence description of the drive by using Energetic Macroscopic Rep-
of disturbances (e.g. an unexpected change load,. . . ) and un- resentation (EMR) [9], [10] that is an energy-based method.
certainties (e.g. a parameters variation,. . . ) is a challenge that This paper is structured as follows. The modelling, repre-
causes an unexpected behavior of the system. The issues can sentation and inversion-based control of BLDC motor drives
be solved by an appropriate method. Disturbance Observer- are described in Section II. Section III introduces both DOBC
Based Control (DOBC) and Active Disturbance Rejection and ADRC which are then applied to control a BLDC motor
Control (ADRC) are among the most powerful techniques that drive. Simulation results are given in Section IV to compare
are employed in wide range of applications. The general idea the control effect of DOBC, ADRC and conventional method.
of both DOBC and ADRC is that an equivalent disturbance, Conclusions are summarized in Section V.
which includes disturbances and uncertainties, is estimated by
disturbance observer and compensated by a control signal. But II. MODELLING, REPRESENTATION AND
the procedure to construct the disturbance observer is different INVERSION-BASED CONTROL SYSTEM OF BLDC
for each method. MOTOR DRIVES BY USING EMR
The DOBC has been originally applied for motion control A. Configuration of BLDC motor drives
[1]. The disturbances and uncertainties need to be cancelled Fig. 1 shows the BLDC motor drives system. In which, the
to maintain the required performance of high-precision system motor is powered by DC voltage source through a three phase
such as position servo controller or industrial robots [2], voltage inverter. Shaft of motor is connected to load.
[3]. A Disturbance Observer (DO) estimates the equivalent
disturbance through an inverse plant model. But almost all B. EMR of BLDC motor drives
plants are causal, so the inverse models of plant are non-causal. EMR describes subsystems by using basic elements and
A filter can be added to the DO to guarantee the causality organizes the representation system based on basic principles.
and feasibility. Such filter makes the estimated equivalent The EMR of BLDC motor drives is shown in upper part of
disturbance delayed as compared to the actual one. The Fig. 2.

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 ݅݅݊‫ݒ‬ with kE is the phase back-EMF coefficient and f e (θ) depicts
the relationship between em and θ.
ܵͳͳ ܵʹͳ ܵ͵ͳ Shaft is an accumulation element with motor speed ωm as
݅ܽ 
state variable which is obtained from the motor torque Tm and
Battery ߱݉ ܶ݉
‫݅ ܿܽݑ‬ load torque Tl by using the dynamic equation:
ܾ BLDC
‫݀ܽ݋ܮ‬
‫ݐܾܽݑ‬ ‫ܾܿݑ‬ MOTOR

݅ܿ dωm
= Tm − Tl
J (6)
ܵͳʹ ܵʹʹ ܵ͵ʹ dt
where J is moment of inertia of BLDC motor and equivalent
Fig. 1: Configuration of BLDC motor drives load.
Load is considered as source element (Load oval) which
generates the load torque Tl .
DC voltage source (Battery) supplies the motor drive system
by the battery voltage ubat and receives the inverter current C. Inversion-based control system of BLDC motor drivers
iinv as a reaction input. In EMR, battery is a source element The control system is deduced from EMR by using inversion
(Bat oval). rule. In such method, the relationship without time dependence
Three-phase voltage inverter converts  the battery voltage can be directly inversed . For the accumulation elements, the
T
ubat to the inverter voltages uinv = ua ub uc and controller will take place. The control structure of BLDC
receives the inverter current i from motor phase currents motor drives using EMR is shown in Fig. 2.
T inv
im = i a ib ic as follows, with assumption of no According to the EMR pictogram, the tuning path is deter-
losses:  mined:
uinv = minv .ubat
(1) minv → uinv → im → Tm → ωm
iinv = S T .im
Then, control path is deduced:
where modulation vector minv is defined by:
⎡ ⎤ ⎡ ⎤⎡ ⎤ minv ref ← uinv ref ← im ref ← Tm ref ← ωm ref
m1 2 −1 −1 S11
1
minv = ⎣ m2 ⎦ = ⎣ −1 2 −1 ⎦ ⎣ S21 ⎦ (2) Following the control path, the control structure is formu-
3
m3 −1 −1 2 S31 lated as follows:
 T Speed controller is the indirect inversion of accumulation
and switching functions S = S11 S21 S31 are: element which represents the dynamic relationship (6). The
 command motor torque Tm ref is defined from the motor
Sij = 0 : when switch Sij is opened
(3) speed set-point ωm ref and real speed ωm as:
Sij = 1 : when switch Sij is closed
kiω
with i ∈ {1, 2, 3} is the number of the leg and j ∈ {1, 2} is the Tm ref(ωm ref − ωm ) − kpω ωm
= (7)
number of switches in a leg. In EMR, the three-phase voltage s
inverter is described by a mono-physical conversion element where kpω , kiω are coefficients of speed controller.
(square) in which modulation vector is the tuning vector. A direct inversion
 of (5) generates the command
T phase cur-
BLDC motor is represented by two elements in EMR. The rent im ref = ia ref ib ref ic ref from the Tm ref
first one, accumulation element (rectangle with an oblique bar) as:
corresponds to stator winding which accumulates electrical Tm ref
im ref = f i (θ) (8)
energy. The relationship between the motor phase currents im , kT
which are state variable, the inverter voltages uinv and the in which kT is a torque coefficient and f i (θ) indicates the
 T
motor phase back-EMFs em = ea eb ec is expressed dependence of im ref to the rotor position θ.
by electrical equations as follows:
 d
Ls dt im = uinv − em − Rs im
(4)
0 = ia + ib + ic Battery Inverter BLDC Motor Shaft Load
ubat uinv im Tm ω
where R and Ls are the phase resistance and phase inductance Load EMR
Bat
of stator windings, respectively. The second one is multi- iinv im em ω Tl
physical conversion element (circle) that illustrates an electro-
magnetic relationships of BLDC motor. The electromagnetic minv ref
Inversion
torque Tm is generated by the motor phase currents and the – Based
uinv ref im ref Tm ref ω ref
phase back-EMFs which depends on motor speed ωm and rotor Control
position θ as: 
Tm = ω1m eTm im
(5) Fig. 2: Control structure of BLDC motor drives using EMR
em = kE ωm f (θ)

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 Current controller is required for the indirect inver- ݀
sion of accumulation element that represents the electri-
‫ݑ‬ Plant
‫ݕ‬
cal relationship (4) in stator winding. The output of cur- ‫݂݁ݎݕ‬ +
_ C(s) ‫ ܿݑ‬+ + + P(s)
rent controller are command inverter voltages uinv ref = +
 T
ua ref ub ref uc ref which are obtained from im ref Controller ‫ ܨܩ‬ሺ‫ݏ‬ሻ ‫ ܨܩ‬ሺ‫ݏ‬ሻ
and im according to the control law: Disturbance
Observer
kii +_
uinv = (kpi + )(im ref − im ) (9) ܲ݊െͳ ሺ‫ݏ‬ሻ
݂መ
ref
s
with kpi , kii are coefficients of current controller.
A direct inversion of (1) gives the reference modulation Fig. 3: System configuration with DOBC
vector minv ref as follows:
1
minv ref = u (10) If GF = 1 then
ubat meas inv ref uc
u= −d (16)
with ubat meas is the measured battery voltage. Pn−1 P
III. DISTURBANCE OBSERVER-BASED CONTROL By using the control signal (16), the parameter variation
uc
AND ACTIVE DISTURBANCE REJECTION is compensated through the component P −1 and the distur-
n P

CONTROL bance is rejected by component (−d) .

Unfortunately, GF = 1 is not applicable. In fact, GF is the
A. Disturbance Observer-Based Control (DOBC)
function that indicates the delay of the estimated equivalent
The system configuration using DOBC is shown in Fig. 3. disturbance with regard to the actual disturbance through a
In which, P (s) is the actual plant with control signal input u, time constant.
output y, and d as the disturbance applied to the system. The
control signal u is the combination of uc which is a output B. Active Disturbance Rejection Control (ADRC)
of feedback controller C(s) and the estimated equivalent In the ADRC approach, the disturbance as well as the
disturbance f . parameter variation are considered as the added state variables.
The plant can be described by: We can then proactively estimate them and reject their effect
on the response of the system. With the system represented by
y = P.(u + d) (11)
using EMR, the plants are either integral or first-order plants.
The disturbance d can be evaluated by using the inverse Considering integral or first-order plants, which can be
model of the actual plant: modelled by:
−d = u − P −1 .y (12) dy 1
= ( .y +
1
.d + Δb.u) + bn .u = f + bn .u (17)
dt T1 T2
The actual plant P (s) is difficult to be known exactly but the
nominal plant Pn (s) is really defined. The difference between where y is the output, u is the control input, and f = ( T11 .y +
1
P (s) and Pn (s), which is caused by parameter variations, T 2 .d + Δb.u), which includes disturbance d and parameter
can be expressed through ratio P.Pn−1 . So the inverse model variation Δb.u, is the equivalent disturbance applied to the
of nominal plant Pn−1 is utilized to estimate the equivalent system. bn is the nominal
T parameter of system.
disturbance which includes both the disturbance and parameter Let x = y f as state variables, (17) can be obtained
variation. Since Pn−1 is non-causal so the filter GF is added in the state space as follows:
to assure its causality. And since GF makes the delay of 
ẋ = An x + Bn u + E df
component Pn−1 .y, the same GF need to be added to the dt (18)
y = Cx
component u to synchronize the time delay.

The estimated equivalent disturbance is obtained as follows: 0 1
in which, An = is nominal system matrix, Bn =

 0 0

fˆ = GF (u − Pn−1 y) (13)
bn 0
is nominal distribution vector, E = is distri-
Note that u = uc + fˆ and y = P (uc + fˆ + d) , after some 0   1
manipulations, (13) becomes: bution vector of disturbance, C = 1 0 is observation

 vector.
ˆ 1 GF Pn−1 P
f= − 1 u c − d The Extended State Observer (ESO) can be designed as:
1 − GF (1 − Pn−1 P ) 1 − GF (1 − Pn−1 P ) 
(14) x̂˙ = An x̂ + bn u + K(y − ŷ)
The control law is deduced: (19)
ŷ = C x̂
uc GF Pn−1 P  T
u = uc + fˆ = − d where K = k1 k2 is the observer gain vector and x̂ =
1 − GF (1 − Pn P ) 1 − GF (1 − Pn−1 P )
−1
 T
(15) ŷ fˆ are observed state variables.

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 ݀ Battery Inverter BLDC motor Shaft Load
ubat uinv im Tm ω
‫ ܿݑ‬+ ‫ݑ‬
Plant ‫ݕ‬
‫݂݁ݎݕ‬ +
_ C(s) P(s) Bat Load
EMR
+
iinv im em ω Tl
Controller + fˆi fˆω
ͳ K _
െ
‫ݔ‬ሶ෠
Estimation
ܾ݊ ‫ݔ‬ො
 න C minv
Observer
ref
Observer
‫ݕ‬ො
Extended State Inversion-
An uinv im Tm ω
݂መ Observer ref ref ref ref Base
Control
Current Controller Speed Controller S

Fig. 4: System configuration with ADRC

Fig. 5: Control structure of BLDC motor drives using
estimation-cancellation techniques
According to the principle of separation, the design of the
observer and the controller can be performed independently.
The characteristic polynomial of the observer is defined by: The command inverter voltages uinv are given by the
2 current controller:
det(S.I − (An − K.C)) = s + k1 s + k2 (20)
 T uinv = C[im − im ]+ki .f̂ i (25)
The observer gains K = k1 k2 are calculated by ref ref
choosing poles of ESO which are decided by the nature of the
disturbance and parameter variation. where C[im ref − im ] is current feedback controller; f̂ i is the
The closed-loop function of ESO can be expressed as: estimated equivalent disturbance applied to current; ki = 1 in
DOBC and ki = (−Lsn ) in ADRC with Lsn as the nominal
fˆ k2 value of phase inductance of the BLDC motor.
GESO = = 2 (21)
f s + k1 s + k2
Equation (21) indicates the delay of the estimated equivalent IV. SIMULATION AND RESULTS
disturbance behind the actual disturbance.
The control law is realized: Simulation is realized in Matlab/Simulink to evaluate the
1 control effect of DOBC, ADRC and conventional method in
u = uc − .fˆ (22) some scenarios. Table I shows the parameters of BLDC motor
bn
which is employed in simulation.
In the steady state of observation closed-loop, which implies
f ≈ fˆ, (17) will be reduced to approximate plant: TABLE I: BLDC MOTOR PARAMETERS
ẏ = f + bn u ≈ bn .uc (23)
Parameters Nominal Value
Battery voltage ubat 36 [V ]
where the control signal uc is given by feedback controller Phase resistance R 0.8 [Ω]
C(s). The system configuration with ADRC is shown in Fig. 4. Phase inductance Ls 0.00214 [H]
Torque constant kT 0.1715
C. Application of DOBC and ADRC to BLDC motor drives Moment of inertia J 0.0002 [kg.m2 ]
Rated speed ωrated 1500 [rpm]
In this paper, DOBC and ADRC are applied for speed
Rated torque Trated 0.343 [N.m]
controller and current controller to improve the performance
of BLDC motor drives. Base on the EMR and inversion-
based control system performed in Section II, the disturbance The filter in Disturbance Observer (DO) of DOBC and the
observers (a purple square) are designed to estimate the closed-loop function of Extended State Observer (ESO) in
equivalent disturbance and then their effect is rejected by ADRC are provided in Table II.
controllers. The control structure of BLDC motor drive using
the disturbance estimation-cancelation techniques (DOBC or TABLE II: DO and ESO DESIGN
ADRC) is shown in Fig. 5.
The speed controller defines the command motor torque Speed Control Current Control
1
Tm ref : GF = 1+τ1 s GF = 1+τ s
ω i
DO
Tm ref = C[ωm ref − ωm ]+kω .fˆω (24) τω = 0.002[s] τi = 0.0002[s]
GESO = (1+τ1 s)2 GESO = (1+τ1 s)2
with C[ωm ref − ωm ] is speed feedback controller; fˆω is the ESO
ω i

estimated equivalent disturbance applied to speed; kω = 1 in τω = 0.002[s] τi = 0.0002[s]

DOBC and kω = (−Jn ) in ADRC with Jn as the nominal
value of moment of inertia.

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 Fig. 6: Speed responses in the case of Fig. 7: Speed responses in the case of
unexpected change load parameters variation J = 2Jn

TABLE III: COMPARISON OF THE SPEED RESPONSES TABLE IV: COMPARISON OF THE SPEED RESPONSES

Conv.method DOBC ADRC Conv.method DOBC ADRC

tre 0.08 [s] 0.02 [s] 0.02 [s] ts 0.17 [s] 0.13 [s] 0.13 [s]
Δdrop% 6.5 [%] 1.2 [%] 2.5 [%] Δover% 10 [%] 9 [%] 13 [%]

A. Case 1: The unexpected change load • Δover% (percentage of overshoot): the maximum speed
The simulation condition is: ωm ref = ωrated ; at time t = (current) value minus the reference value divided by the
0.2 [s], the load torque (step load) Tl = Trated is applied to reference value.
the closed system. ωrated and Trated are rated speed and rated The percentage of drop speed Δdrop% in Table III and the
torque of motor, respectively. percentage of overshoot Δover% in Table IV show that the
The simulation results are provided in Fig. 6. DOBC and performance of DOBC is better than the one of ADRC.
ADRC give better performance than conventional method. By
using DOBC or ADRC, the recovery time tre is shorter and the C. Case 3: Parameter variations R = Rn + ΔR and Ls =
percentage of drop speed Δdrop% is smaller as compared to Lsn + ΔLs
that of conventional method. It demonstrates the effectiveness
of DOBC and ADRC. The detailed comparison is given in In this case, the nominal parameters Rn = 0.8 [Ω] and
Table III, in which: Lsn = 0.00214 [H] are utilized to design the current controller
• tre (recovery-time): the time required for the speed
and disturbance observer. The actual plant is considered with
response to recover the reference value from the instant parameters: R = 2Rn and Ls = 0.5Lsn .
when the load torque is applied to the system. Torque responses and phase current responses of BLDC
• Δdrop % (percentage of drop speed): the reference speed
motor are expressed in Fig. 8 and Fig. 9, respectively.
value minus the minimum speed value (after load torque
is applied to the system) divided by the reference value.

B. Case 2: Parameter variations J = Jn + ΔJ

In this case, the nominal moment of inertia Jn =
0.0002 [kg.m2 ] is used to design speed controller and dis-
turbance observer. The value J = 2Jn is considered as the
actual moment of inertia.
When the step reference value is applied to the system,
DOBC and ADRC impose the speed response to the steady
state in the shorter time with regard to the conventional
method. Simulation results are illustrated in Fig. 7 and Ta-
ble IV, in which:
• ts (settling-time): the time required for the speed response
(current response) to reach and stay within a range of
certain percentage (5%) of the reference speed (current
reference). Fig. 8: Torque responses of BLDC motor

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 formance of motor drives system. The simulation for BLDC
motor drive systems is realized to compare the control effect of
DOBC, ADRC and conventional method in some cases. The
simulation results indicates that DOBC and ADRC improve
the system responses (speed response and current response)
as compared to that of conventional method. DOBC gives the
slightly better performance than ADRC in the same simulation
condition, however DOBC’s design procedure is remarkably
simpler than ADRC’s.
R EFERENCES
[1] K. Ohnishi, “Disturbance Observation-Cancellation Technique,” in The
Industrial Electronics Handbook (J. D. Irwin, ed.), ch. 33, pp. 524 –
528, CRC Press - IEEE Press, 1997.
[2] W. H. Chen, D. J. Ballance, P. J. Gawthrop, S. Member, J. O. Reilly,
and S. Member, “A Nonlinear Disturbance Observer for Robotic Ma-
nipulators,” IEEE Transactions on Industrial Electronics, vol. 47, no. 4,
pp. 932–938, 2000.
[3] E. Sariyildiz and K. Ohnishi, “Stability and robustness of disturbance-
observer-based motion control systems,” IEEE Transactions on Indus-
trial Electronics, vol. 62, no. 1, pp. 414–422, 2015.
[4] M. Zheng, S. Zhou, and M. Tomizuka, “A Design Methodology for
Disturbance Observer with Application to Precision Motion Control :
An H-infinity Based Approach,” in 2017 American Control Conference,
pp. 3524–3529, 2017.
[5] J. Han, “From PID to Active Disturbance Rejection Control,” IEEE
Transactions on Industrial Electronics, vol. 56, no. 3, pp. 900–906, 2009.
[6] Q. Zheng and Z. Gao, “On practical applications of active disturbance
rejection control,” in Proceedings of the 29th Chinese Control Confer-
ence, no. January, pp. 6095–6100, 2010.
[7] M. Pizzocaro, D. Calonico, C. Calosso, C. Clivati, G. A. Costanzo,
F. Levi, and A. Mura, “Active disturbance rejection control of tempera-
ture for ultrastable optical cavities,” IEEE Transactions on Ultrasonics,
Ferroelectrics, and Frequency Control, vol. 60, no. 2, pp. 273–280, 2013.
[8] J. Yang, H. Cui, S. Li, and A. Zolotas, “Optimized Active Disturbance
Rejection Control for DC-DC Buck Converters With Uncertainties Using
a Reduced-Order GPI Observer,” IEEE Transactions on Circuits and
Systems I: Regular Papers, vol. 65, no. 2, pp. 832–841, 2018.
[9] K. Chen, A. Bouscayrol, and W. Lhomme, “Energetic Macroscopic
Representation and Inversion-based control: Application to an Electric
Vehicle with an Electrical Differential,” Journal of Asian Electric
Vehicles, vol. 6, no. 1, pp. 1097–1102, 2008.
Fig. 9: Phase current responses in the case of parameter [10] A. Bouscayrol, W. Lhomme, P. Delarue, and S. Aksas, “Hardware-in-
variation: R = Rn + ΔR and Ls = Lsn + ΔLs the-loop simulation of electric vehicle traction systems using Energetic
Macroscopic Representation,” in IECON 2006 - 32nd Annual Conference
on IEEE Industrial Electronics, pp. 5319–5324, 2006.
TABLE V: COMPARISON OF THE PHASE
CURRENT RESPONSES

Conv.method DOBC ADRC APPENDIX: SYNOPTIC OF ENERGETIC

ts 0.1 [s] 0.05 [s] 0.05 [s]
Δover% 15 [%] 20 [%] 25 [%] MACROSCOPIC REPRESENTATION

Accumulation element Mono-physical

By using DOBC and ADRC, the settling time ts is shorter conversion element Coupling element

but the percentage of overshoot Δover% is larger as compared

to the conventional method. The comparison is summarized Estimation Multi-physical Source element
(Any possible shape) conversion element
in Table V. Based on Table V, DOBC gives the better
performance than ADRC.
Indirect inversion Direct inversion
Direct inversion
(Closed–loop control) (Open–loop control)
V. CONCLUSION Strategy
Mandatory link
Measurement
Based on the BLDC motor drive system described by EMR Strategy block

and inversion-based control system, both the Disturbance- Facultative link

Observer Based Control (DOBC) and the Active Disturbance
Rejection Control (ADRC) are applied to improve the per-

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