Hardware-in-the-loop based comparative analysis of speed controllers for a two-mass system using an induction motor drive with ideal stator current performance

Bulletin of Electrical Engineering and Informatics
Vol. 10, No. 2, April 2021, pp. 569~579
ISSN: 2302-9285, DOI: 10.11591/eei.v10i2.2370  569
Journal homepage: http://beei.org
Hardware-in-the-loop based comparative analysis of speed
controllers for a two-mass system using an induction motor
drive with ideal stator current performance
Vo Thanh Ha1
, Tung Lam Nguyen2
, Vo Thu Ha3
1
Faculty of Electrical and Electrical Engineering, University of Transport and Communications, Vietnam
2
School of Electrical Engineering, Hanoi University of Science and Technology, Vietnam
3
Faculty of Electrical Engineering, University of Economics-Technology for Industries, Vietnam
Article Info ABSTRACT
Article history:
Received Sep 5, 2020
Revised Dec 12, 2020
Accepted Jan 5, 2021
A comparative study of speed control performance of an induction motor
drive system connecting to a load via a non-rigid shaft. The nonrigidity of the
coupling is represented by stiffness and damping coefficients deteriorating
speed regulating operations of the system and can be regarded as a two-mass
system. In the paper, the ability of flatness based and backstepping controls
in control the two-mass system is verified through comprehensive hardware-
in-the-loop experiments and with the assumption of ideal stator current loop
performance. Step-by-step control design procedures are given, in addition,
system responses with classical PID control are also provided for parallel
comparisons.
Keywords:
Backstepping
Field oriented control
Flatness-based control
Hardware-in-the-loop
Speed control This is an open access article under the CC BY-SA license.
Corresponding Author:
Tung Lam Nguyen
School of Electrical Engineering
Hanoi University of Science and Technology
1 Đại Cồ Việt, Bách Khoa, Hai Bà Trưng, Hà Nội, Vietnam
Email: lam.nguyentung@hust.edu.vn
1. INTRODUCTION
Asynchronous electrical drive system is found in enormous applications and is considered as well-
established thank to the advancement in current loop control [1]. However, facing mechanical loads, speed
control loop design takes a decisive role, especially when the non-rigidity and backlash of the motor-to-load
connection is taken into account [2-5]. Flexible coupling phenomenon is studied in [6], the authors develop
an elegant control scheme integrated with a state observer for estimating motor and load torques. Brock et al.
[7] propose a filter accompanied with a neural adaptive controller to eliminate high resonance frequencies.
The controlled system is analysed through extensive experiments proving the effectiveness of the proposed
solution. For a class of multi-mass electrical drive systems where measurement is not always available, a
system parameter identification method is presented in [8]. The authors employ pseudo random binary signal
and chirp wave to excite necessary information. The approach might be difficult to apply in a wide range of
practical systems. A 2DOF PI control is systematically designed in [9] for a two-mass drive coupled via a
toothed belt. The tracking results are compared with 1DOF and 1DOF with feedforward controls confirm that
the method exhibits robustness and load torque rejection. Due to flat property of the system, Thomsen et al.
[10] formulate a flatness based control for a finite stiffness coupling drive system fed by an inverter with
speed feedback signal. Backstepping control is developed for a multi-motor system which is vastly found in

 ISSN: 2302-9285
Bulletin of Electr Eng & Inf, Vol. 10, No. 2, April 2021: 569 – 579
570
rewinding systems in [11] suppressing flexible coupling phenomenon in linear speed and tension control
system. In order to deal with nonlinear stiffness, the authors in [12] implement adaptive backstepping to
reduce connecting shaft oscillations. However, system stiffness characteristic must be known to carry out the
control, this requirement might be very challenging in practice. Recently, a active disturbance rejection
control (ADRC) is considered as an alternative to classical PID control, an application of ADRC to two-mass
system can be found in [13]. A fuzzy based control for a three-mass system is presented in [14, 15], where
fuzzy term is selected to modify IPD control. Other control approaches can be found in [16-22].
In general, there is a lack of comprehensive study on how popular control methods react when
applying to two-mass drive system operating in nominal range. In the paper, we carry out a hardware-in-the-
loop based comparative analysis of speed responses using backstepping and flatness-based controls in
nominal and field-weakening modes. Initially, we present the two-mass drive system in a view of ideal
current loop. Subsequently, PID, backstepping, and flatness control design steps are given in details. Finally,
hardware-in-the-loop results of three controls are compared and some important conclusions are drawn.
2. TWO-MASS MODEL SYSTEM WITH IDEAL CURRENT LOOP RESPONSE
Typical configuration of a two-mass system is shown in Figure 1.
Figure 1. Typical configuration of a two-mass system
When operating in field weakening region, the model of the system can be given as [2].
𝑑𝑖𝑠𝑑
𝑑𝑡
= − (
1
𝜎𝑇𝑠
+
1 − 𝜎
𝜎𝑇𝑟
) 𝑖𝑠𝑑 + 𝜔𝑠𝑖𝑠𝑞 +
1 − 𝜎
𝜎𝑇𝑟
+
1
𝜎𝐿𝑠
𝑢𝑠𝑑
𝑑𝑖𝑠𝑞
𝑑𝑡
= −𝜔𝑠𝑖𝑠𝑑 − (
1
𝜎𝑇𝑠
+
1 − 𝜎
𝜎𝑇𝑟
) 𝑖𝑠𝑞 −
1 − 𝜎
𝜎
𝜔𝑖𝑚 +
1
𝜎𝐿𝑠
𝑢𝑠𝑞
𝑑𝜓𝑟𝑑
𝑑𝑡
= −
1
𝑇𝑟
𝜓𝑟𝑑 +
𝐿𝑚
𝑇𝑟
𝑖𝑠𝑑
𝑑𝜓𝑟𝑞
𝑑𝑡
=
𝐿𝑚
𝑇𝑟
𝑖𝑠𝑞 − (𝜔𝑠 − 𝜔)𝜓𝑟𝑑𝑖𝑠𝑑 −
1
𝑇𝑟
𝜓𝑟𝑞
𝜓𝑟𝑞
𝐿𝑚
𝜑̈1 = −
𝑑
𝐽1
𝜑̇1 −
𝑐
𝐽1
𝛥𝜑 +
𝑑
𝐽1
𝜑̇2 +
1
𝐽1
𝑚𝑀
𝛥𝜑̇ = 𝜑̇1 − 𝜑̇2
𝜑̈2 =
𝑑
𝐽2
𝑑/𝐽2𝜑̇1 +
𝑐
𝐽2
𝛥𝜑 −
𝑑
𝐽2
𝜑̇2 −
1
𝐽2
𝑚𝐿
(1)
In which, 𝑖𝑠𝑑; 𝑖𝑠𝑞 are𝑑𝑞 components of the stator current; 𝜔, 𝜔𝑠 rotor and synchronous speeds,
respectively; 𝜓𝑟𝑑
,
, 𝜓𝑟𝑞
,
are rotor flux dq components;𝜎 is total leakage factor; 𝑇𝑟is rotor time constant:𝑢𝑠𝑑,
𝑢𝑠𝑞are stator voltage 𝑑𝑞 components; 𝐿𝑠is stator inductance, 𝜑̇1, 𝜑̇2are motor and load speeds; 𝜑̈1, 𝜑̈2are
motor and load angle accelerations; 𝜑is rotor angle; 𝑑is the coupling shaft damping coefficient; 𝑐is shaft

Bulletin of Electr Eng & Inf ISSN: 2302-9285 
Hardware-in-the-loop based comparative analysis of speed controllers… (Vo Thanh Ha)
571
stiffness. It can be seen that the system of dynamical (1) is of 7th
order. With the assumption of t perfectly
designed current controller, the system model can be reduced as:
𝑑𝑖𝑚
𝑑𝑡
=
1
𝑇𝑟
𝑖𝑠𝑑 −
1
𝑇𝑟
𝑖𝑚
𝜑̈1 = −
𝑑
𝐽1
𝜑̇1 −
𝑐
𝐽1
𝛥𝜑 +
𝑑
𝐽1
𝜑̇2 +
1
𝐽1
𝑚𝑀
𝛥𝜑̇ = 𝜑̇1 − 𝜑̇2
𝜑̈2 =
𝑑
𝐽2
𝜑̇1 +
𝑐
𝐽2
𝛥𝜑 −
𝑑
𝐽2
𝜑̇2 −
1
𝐽2
𝑚𝐿
(2)
where: 𝑖𝑚 =
𝜓𝑟𝑑
𝐿𝑚
. The state (2) is of 4nd
order, stator current isd is used to control the motor flux and isq is
dedicated to speed control. For control design purpose (2) is rewritten in the following state-space form:
[
𝑖̇𝑚
𝜑̈1
𝛥𝜑̇
𝜑̈2
] =
[
−1
𝑇𝑟
0 0 0
0
−𝑑
𝐽1
−𝑐
𝐽1
𝑑
𝐽1
0 1 0 −1
0
𝑑
𝐽2
𝑐
𝐽2
−𝑑
𝐽2 ]
[
𝑖𝑚
𝜑̇1
𝛥𝜑
𝜑̇2
] +
[
1
𝑇𝑟
0
0
𝑘𝜔𝑖𝑚
𝐽1
0 0
0 0 ]
[
𝑖𝑠𝑑
𝑖𝑠𝑞
] +
[
0
0
0
−1
𝐽2 ]
𝑚𝐿 (3)
where we have defined,
𝑦𝑇
= [𝑖𝑚, 𝜑̇2]; [
𝑖𝑚
𝜑̇2
] = [
1 0 0 0
0 0 0 1
] [
𝑖𝑚
𝜑̇1
𝛥𝜑
𝜑̇2
] (4)
3. SPEED CONTROL DESIGN
3.1. PI control
PI controller is designed according to the symmetric optimal standard, so we have the PI control
structure shown as Figure 2. The design of PI for the system is well-established, readers can refer to [23, 24]
for more details.
Figure 2. Block diagram of PI control algorithm of a two-mass system
3.2. Backstepping control
For designing step, it is assumed that all feedback signals from the motor, the shaft and the load
sides are available. As in Figure 2, the backstepping control is divided into two steps. In each step, a
subsystem will be studied and controlled. Selecting a proper Lyapunov candidate function for each design
step can guarantees the asymptotic stability of the corresponding subsystem. After completing all two steps,
the asymptotic stability of the whole system can be achieved [25]. The speed control equation is given:

 ISSN: 2302-9285
572
𝑑𝜑
̇ 2
𝑑𝑡
= −
𝑑
𝐽2
𝜑̇2 +
𝑐
𝐽2
𝛥𝜑 +
𝑑
𝐽2
𝜑̇1 −
1
𝐽2
𝑚𝐿
𝑑𝜑
̇ 1
𝑑𝑡
= −
𝑑
𝐽1
𝜑̇1 −
𝑐
𝐽1
𝛥𝜑 +
𝑑
𝐽1
𝜑̇2 −
𝑘𝜔.𝜓𝑟𝑑
′
𝐽1
𝑖𝑠𝑞
(5)
where 𝑚𝑀 = 𝑘𝜔(𝜓𝑟𝑑/𝐿𝑚)𝑖𝑠𝑞; 𝑘𝜔 = (3/2)𝑧𝑝(𝐿𝑚/𝐿𝑟) is motor torque. For ease of deploying
control design, we define:
𝑥1 = 𝜑̇2
𝑥2 = 𝜑̇1
𝑢 = 𝑖𝑠𝑞
𝑦 = 𝑥1 = 𝜑̇2
(6)
where 𝑥1 and 𝑥2are state variables; u is the control input and y is the output (the load rotating speed). Since
then the system is expressed in the strict feedback form as (7):
𝑥̇1 = (
𝑑𝜑̇2
𝑑𝑡
) = 𝑓1(𝑥1, 𝑥2)
𝑥̇2 = (
𝑑𝜑
̇ 1
𝑑𝑡
) = 𝑓2(𝑥1, 𝑥2, 𝑢) (7)
𝑦 = 𝑥1 = 𝜑̇2
In order to implement backstepping control we define:
𝑧1 = 𝑥1 − 𝑥1𝑑 = 𝜑̇2 − 𝜑̇2𝑑
𝑧2 = 𝑥2 − 𝑥2𝑑𝑘𝑎 = 𝜑̇1 − 𝜑̇1𝑑𝑘𝑎 (8)
where 𝑥1𝑑 = 𝜑̇2𝑑is the set value of the output variable and 𝑥2𝑑𝑘𝑎 = 𝜑̇1𝑑𝑘𝑎is the virtual control. Selecting the
Lyapunov candidate function as (9):
𝑉1 =
1
2
𝑧1
2
(9)
Taking derivative of (9) results in:
𝑉̇1 = 𝑧1𝑧̇1 (10)
Because of 𝜑̈1𝑑 = 0 so that (8) rendered as 𝑧̇1 = 𝜑̈2, substitution 𝑧̇1from (5) into (10) yields:
𝑉̇1 = 𝑧1 [−𝑘1𝑧1 +
𝑑
𝐽2
𝜑̇1 + (𝑘1𝑧1 +
𝑐
𝐽2
𝛥𝜑 −
1
𝐽2
𝑚𝐿 −
𝑑
𝐽2
𝜑̇2)] (11)
Considering𝜑̇1 as the virtual control variable in such a way that 𝜑̇1 = 𝜑̇1𝑑𝑘𝑎so that:
𝑑
𝐽2
𝜑̇1𝑑𝑘𝑎 = −(𝑘1𝑧1 +
𝑐
𝐽2
𝛥𝜑 −
1
𝐽2
𝑚𝐿 −
𝑑
𝐽2
𝜑̇2) (12)
If 𝑉̇1 = −𝑘1𝑧1
2
< 0, with ∀𝑘1 > 0 then the stable condition is satisfied. Defining:
𝐵𝑓 =
𝑐
𝐽2
𝛥𝜑 −
1
𝐽2
𝑚𝐿 −
𝑑
𝐽2
𝜑̇2 (13)
Substitution of (13) into (12) results in:
𝜑̇1𝑑𝑘𝑎 = −
𝐽2
𝑑
[𝑘1𝑧1 + 𝐵𝑓] (14)
The control specified in (14) ensures that the load speed tracks the desired value. Next, the control for the
motor side is designed. It is noted that𝑥1𝑑 = 𝜑̇2𝑑is constant, so that from (5), (8), and (13) can be shortened as:
𝑧̇1 =
𝑑𝜑
̇ 2
𝑑𝑡
= 𝜑̈2 =
𝑑
𝐽2
𝜑̇1 + 𝐵𝑓 (15)

573
Select the Lyapunov candidate function as (16):
𝑉2 = 𝑉1 +
1
2
𝑧2
2
(16)
Taking derivative of (16) gives:
𝑉̇2 = 𝑉̇1 + 𝑧2𝑧̇2 (17)
From (11) and (12), we can write as:
𝑉̇2 = −𝑘1𝑧̇1
2
+
𝑑
𝐽2
𝑧1(𝜑̇1 − 𝜑̇1𝑑𝑘𝑎) + 𝑧2𝑧̇2 (18)
After some basic operations, it can be shown that:
𝑉̇2 = −𝑘1𝑧̇1
2
− 𝑘2𝑧2
2
+ 𝑘2𝑧2
2
+
𝑑
𝐽2
𝑧1(𝜑̇1 − 𝜑̇1𝑑𝑘𝑎) + 𝑧2𝑧̇2 (19)
where we have defined 𝜑̇1 − 𝜑̇1𝑑𝑘𝑎 = 𝑧2, so that:
𝑉̇2 = −𝑘1𝑧1
2
− 𝑘2𝑧2
2
+ 𝑧2(𝑘2𝑧2 +
𝑑
𝐽2
𝑧1 + 𝑧̇2) (20)
At this point in order to render𝑉̇2 < 0, the following conditions must be fullfied:
𝑘2 > 0 and 𝑘2𝑧2 +
𝑑
𝐽2
𝑧1 + 𝑧̇2 = 0 (21)
Conditions specified in (21) imply that:
𝜑̈1 − 𝜑̈1𝑑𝑘𝑎 = −𝑘2(𝜑̇1 − 𝜑̇1𝑑𝑘𝑎) −
𝑑
𝐽2
𝑧1 (22)
From (14), it indicates that:
𝜑̈1𝑑𝑘𝑎 = −
𝐽2
𝑑
(𝑘1𝑧̇1 + 𝐵̇𝑓) and 𝜑̇ − 𝜑̇1𝑑𝑘𝑎 = 𝜑̇1 +
𝐽2
𝑑
(𝑘1𝑧1 + 𝐵𝑓) (23)
Substituting (15) and (22) into (23) gives:
𝜑̈1 = −(𝑘1 + 𝑘2)𝜑̇1 − (
𝐽2
𝑑
𝑘1𝑘2 +
𝐽2
𝑑
)𝑧1 −
𝐽2
𝑑
(𝑘1 + 𝑘2)𝐵𝑓 −
𝐽2
𝑑
𝐵̇𝑓 (24)
To proceed to calculate the final expression of the control current, first of all, we substitution from the second
equation of the system (5) into (24) we have:
𝑘𝜔𝜓𝑟𝑑
′
𝐽1
. 𝑖𝑠𝑞 = [
𝑑
𝐽1
− (𝑘1 + 𝑘2)]𝜑̇1 −
𝑑
𝐽1
𝜑̇2 +
𝑐
𝐽1
𝛥𝜑 − (
𝐽2
𝑑
𝑘1𝑘2 +
𝑑
𝐽2
)𝑧1 −
𝐽2
𝑑
(𝑘1 + 𝑘2)𝐵𝑓 −
𝐽2
𝑑
𝐵̇𝑓
(25)
From the expression (25) and using fundamental calculation, we will get the final expression of isq is:
𝑖𝑠𝑞 =
𝐽1
𝑘𝜔𝜓𝑟𝑑
′ {
[
𝑑
𝐽1
+
𝑑
𝐽2
− (
𝑐
𝑑
+ 𝑘1 + 𝑘2)](𝜑̇1 − 𝜑̇2) + [
𝑐
𝐽1
+
𝑐
𝐽2
−
𝑐
𝑑
(𝑘1 + 𝑘2)]𝛥𝜑
−(
𝐽2
𝑑
𝑘1𝑘2 +
𝑑
𝐽2
)(𝜑̇2 − 𝜑̇2𝑑) +
𝑚𝑤
𝑑
(𝑘1 + 𝑘2 −
𝑑
𝐽2
)
} (26)
Based on (26), the proposed control structure for the two-mass system driven by the induction motor is
shown in Figure 3.

 ISSN: 2302-9285
574
Figure 3. Block diagram of backstepping control algorithm of a two-mass system
3.3. Flatness-based control (FBC)
3.3.1. Feedforward control design
Feedforward control can be considered as the inverse mathematical model of the control object and
plays a decisive role in the FBC structure. From (5), the feedforward control can be derived as:
𝑖𝑠𝑑
𝑓𝑓
= 𝑖𝑚
∗
+ 𝑇𝑟
𝑑𝑖𝑚
∗
𝑑𝑡
𝑖𝑓𝑓
𝑠𝑞 =
𝐽1𝜑
̈ 1+𝐽2𝜑
̈ 2+𝑚𝐿
𝑘𝜔𝑖∗
𝑚
(27)
Based on equation 𝜑̈1 = −
𝐽2
𝐽1
𝜑̈2 +
1
𝐽1
(𝑚𝑀 − 𝑚𝐿) [26], it can be deduced that:
𝐽1𝜔̇1 + 𝐽2𝜔̇ 2 = 𝑚𝑀 − 𝑚𝐿 (28)
Assume that the load torque is not varying, then the load observer can be developed as:
𝑑𝑚
̂𝐿
𝑑𝑡
= −𝑙1(𝜔
̂2 − 𝜔2)
𝑑𝜔
̂2
𝑑𝑡
=
1
𝐽2
(𝑚𝑀 − 𝑚
̂𝐿 − 𝐽1𝜔̇
̂1) + 𝑙2(𝜔
̂2 − 𝜔2) (29)
where 𝑚𝑀 = 𝑘𝜔𝑖𝑚𝑖𝑠𝑞. then from (28) and (29), it is straightforward to show that:
𝑑(𝑚
̂𝐿 − 𝑚𝐿)
𝑑𝑡
= −𝑙1(𝜔
̂𝐿 − 𝜔𝐿)
𝑑(𝜔
̂𝐿−𝜔𝐿)
𝑑𝑡
= −
1
𝐽2
(𝑚
̂𝐿 − 𝑚𝐿 + 𝐽1𝜔̇
̂𝑀 − 𝐽1𝜔̇ 𝑀) + 𝑙2(𝜔
̂𝐿 − 𝜔𝐿) (30)
Defineing 𝜀𝑚 = 𝑚
̂𝐿 − 𝑚𝐿; 𝜀𝜔 = 𝜔
̂𝐿 − 𝜔𝐿 and substituting into.(30) give:
𝑑𝜀𝑚
𝑑𝑡
= −𝑙1𝜀𝜔
𝑑𝜀𝜔
𝑑𝑡
= −
1
𝐽2
𝜀𝑚 + 𝑙2𝜀𝜔 + 𝑓(𝜔̇
̂𝑀, 𝜔̇ 𝑀) (31)
The error model is:

575
[
𝑑𝜀𝑚
𝑑𝑡
𝑑𝜀𝜔
𝑑𝑡
] = [
0 −𝑙1
−
1
𝐽2
𝑙2
] [
𝜀𝑚
𝜀𝜔
] + [
0
𝑓(𝜔̇
̂𝑀, 𝜔̇ 𝑀)
] (32)
The characteristic equation of (32) is:
𝑑𝑒𝑡[𝑠𝑰 − 𝑨] = 𝑠2
− 𝑙2𝑠 −
𝑙1
𝐽2
= 0 (33)
By selecting 𝑙1 and𝑙2 such that:
𝑙1 = −𝐽2𝑠1𝑠2;𝑙2 = 𝑠1 + 𝑠2 (34)
3.3.2. Design of the reference trajectories
The equation 𝒙 = 𝑃 (𝒚,
𝑑𝒚
𝑑𝑡
, … ,
𝑑𝑟𝒚
𝑑𝑡𝑟) ; 𝑟 ∈ 𝑁; 𝒖 = 𝑄 (𝒚,
𝑑𝒚
𝑑𝑡
, … ,
𝑑(𝑟+1)𝒚
𝑑𝑡(𝑟+1)) is also called the “inverse”
process model of the system corresponding to the output given in equation 𝒚 = [
𝑦1
⋮
𝑦𝑚
] = 𝐹 (𝒙, 𝒖,
𝑑𝒖
𝑑𝑡
, … ,
𝑑𝑙𝒖
𝑑𝑡𝑙
) ; 𝑙 ∈
𝑁Trajectories of flux reference:
𝑖𝑚
∗
+ 2𝑇1
𝑑𝑖𝑚
∗
𝑑𝑡
+ 𝑇1
2 𝑑2𝑖𝑚
∗
𝑑𝑡2 = 𝑖𝑚
𝑑
⇔
𝑑2𝑖𝑚
∗
𝑑𝑡2 =
1
𝑇1
2 (𝑖𝑚
𝑑
− 𝑖𝑚
∗
− 2𝑇1
𝑑𝑖𝑚
∗
𝑑𝑡
) (35)
‒ Speed reference trajectories
Trajectories of speed reference are given as:
𝜔∗(𝑡) = {
𝜔0 + 𝑎11𝑡 + 𝑎21𝑡2
+ 𝑎31𝑡3
+ 𝑎41𝑡4
0 ≤ 𝑡 ≤ 2𝑡0
𝜔0 + 𝐴𝑡0 + 𝐴(𝑡 − 2𝑡0) 2𝑡0 < 𝑡 ≤ 𝑡𝐸 − 2𝑡0
𝑎02 + 𝑎12𝜏 + 𝑎22𝜏2
+ 𝑎32𝜏3
+ 𝑎42𝜏4
𝑡𝐸 − 2𝑡0 < 𝑡 ≤ 𝑡𝐸
(36)
where:
𝐴 =
𝜔𝐸−𝜔0
𝑡𝐸−2𝑡0
;𝜏 = 𝑡 − (𝑡𝑐 − 2𝑡0);𝑎11 = 𝑎21 = 0;𝑎31 =
1
4𝑡0
2 𝐴;𝑎41 = −
1
16𝑡0
3 𝐴
3.3.3. Feedback control design
To handle the flux and the speed of the motor, the well-known PI controller is chosen. The detail
equations of the flux and speed controller are described by (37) and (38), respectively:
𝑅𝜓 = 𝑉𝜓
1+𝑑𝜓𝑧−1
1−𝑧−1 𝑤𝑖𝑡ℎ𝑉𝜓 ≈
1
3(1−𝑒
−𝑇𝜓
𝑇𝑟 )
; 𝑑𝜓 ≈ 𝑒
−𝑇𝜓
𝑇𝑟 (37)
𝛥𝑖𝑠𝑞 =
𝐽1𝐽2𝛥𝜔
⃛ 2+(𝑑𝐽1+𝑑𝐽2)𝛥𝜔
̈ 2
𝑘𝜔𝑖𝑚𝑎𝑐
+
(𝐽1𝑐+𝐽2𝑎𝑐)𝛥𝜔
̇ 2
(38)
where: 𝛥𝑖𝑠𝑞 = 𝑖𝑠𝑞
∗
− 𝑖𝑠𝑞
∗𝑓𝑓
; 𝛥𝜔2 = 𝜔 2 − 𝜔2
∗
. Laplace transform of (38) provides the transfer function of
error speed and flux:
𝛥𝜔2(𝑠)
𝛥𝑖𝑠𝑞(𝑠)
=
𝑠[𝐽1𝐽2𝑠2+(𝑑𝐽1+𝑑𝐽2)𝑠 +𝑐𝐽1+𝑎𝑐𝐽2]
(39)
Based on feedforward and feedback control design and trajectories of speed and flux reference the
proposed control scheme for the two-mass system driven by the induction motor with perfect stator current
controller is shown in Figure 4.

 ISSN: 2302-9285
576
Figure 4. Block diagram flatness-based control of two-mass system
4. RESULT AND DISCUSSION
In this section, the proposed control strategy is verified by hardware-in-the-loop (HIL) simulation.
The system hardware is simulated in real time on the HIL platform with a time step of 1μs that means very
close to physical models, while the pulse width modulation carrier frequency is 5 kHz. Voltage and current
controllers are implemented in DSP TMS320F2808. The simulation parameters are: rated power=2.2 kW;
rated phase voltage=380/220 VRMS; rated frequency=50 Hz; d=0.015 Nm/rad; c=0.28 Nm/rad, J1=7.455
x10-5
Kgm2
; J2=5.5913x10-5
Kgm2
; with load torque is 1.5 Nm.
Simulation scenario is: at t=0s, the magnetic current is formed. Then at t=0.5s, the motor starts to
speed up to 1000 rpm. At t=2s, the reverse speed, the motor starts to reverse to -1000 rpm. In order to test the
system ability, low speed range of 1 rpm is also applied in the simulation. This section provides evaluation of
HIL results of speed response. The evaluation of difference speed control strategies for the two-mass system
is shown in Figure 5 and Table 1.
PI control Flatness-based control Backstepping method
(a)
(b)
Figure 5. Speed response of Simulation/HIL platform at, (a) ±1 rpm, (b) ±1000 rpm
w1
w2
+1rpm
-1rpm
+1rpm
-1rpm
+1rpm
-1rpm
At  1 rpm
w1
w2
+1000rpm
-1000rpm
+1000rpm
-1000rpm
+1000rpm
-1000rpm
At  1000 rpm
(a) (b) (c)
w1
w2
+1rpm
-1rpm
+1rpm
-1rpm
+1rpm
-1rpm
At  1 rpm
w1
w2
+1000rpm
-1000rpm
+1000rpm
-1000rpm
+1000rpm
-1000rpm
At  1000 rpm
(a) (b) (c)
w1
w2
+1rpm
-1rpm
+1rpm
-1rpm
+1rpm
-1rpm
At  1 rpm
w1
w2
+1000rpm
-1000rpm
+1000rpm
-1000rpm
+1000rpm
-1000rpm
At  1000 rpm
(a) (b) (c)
w1
w2
+1rpm
-1rpm
+1rpm
-1rpm
+1rpm
-1rpm
At  1 rpm
w1
w2
+1000rpm
-1000rpm
+1000rpm
-1000rpm
+1000rpm
-1000rpm
At  1000 rpm
(a) (b) (c)
w1
w2
+1rpm
-1rpm
+1rpm
-1rpm
+1rpm
-1rpm
At  1 rpm
w1
w2
+1000rpm
-1000rpm
+1000rpm
-1000rpm
+1000rpm
-1000rpm
At  1000 rpm
(a) (b) (c)
w1
w2
+1rpm
-1rpm
+1rpm
-1rpm
+1rpm
-1rpm
At  1 rpm
w1
w2
+1000rpm
-1000rpm
+1000rpm
-1000rpm
+1000rpm
-1000rpm
At  1000 rpm
(a) (b) (c)

577
Table 1. HIL results of the speed response
Speed control
At the speed of ±1 rpm At the speed of ±1000 rpm
Deadbeat PI Deadbeat
Backstepping
Deadbeat
Flatness
Deadbeat PI Deadbeat
Backstepping
Deadbeat
Flatness
Accelerating time (s) 50 50 0 50 50 0
Speed settling time (s) 2.5 2.0 1.0 3.0 2.5 1.0
It is found that the nonlinear control method based on the flatness-based control principle gives
better performance than the PI and backstepping controls, as the referenced load speed match with motor
speed after a short period of time, 0.3 seconds at (±) 1 rpm and 0.5 seconds at (±) 1000 rpm (at up and
reverse speed), without overshoot. That implies the speed controller based on the flatness-based control can
suppress resonance oscillation over the entire operating range of the IM motor.
From the results, it can be seen that the two-mass system with flexible couplings when operating at
the speed of (±) 1000 rpm and even at the low speed range (±) 1 rpm, the nonlinear control method, based on
the flatness-based control, has proved its advantage by solving the problem of resonant vibration suppression
at the spindle compared to the PI controller and backstepping. Besides of at times of transition with fast
setting time, without overshoot and load speed match with motor speed. In the following simulation scenario,
the motor is set in field-weakening region. The magnetizing current responses are shown in Figure 6 and
Figure 7. The three control methods show stable operation not only in nominal but also field weakening
range. However, it can be seen that the flatness-based control gives better transient response, i.e., without
overshoot, as seen in Figure 7, and shorter settling time (0.2s in comparison with 0.5s for PI control and
backstepping method).
(a) (b)
Figure 6. (a) Torque responses, (b) Speed responses
PI control Flatness-based control Backstepping control
(a)
(b)
Figure 7. Controlled system performance, (a) Flux responses, (b) speed responses

 ISSN: 2302-9285
578
(c)
Figure 7. Controlled system performance, (c) torque responses (continued)
5. CONCLUSION
In this paper, the two-mass system with flexible couplings comprising of an induction motor and a load
is considered. At the motor side, the current response is assumed to be ideal leading to a simplified system
model. PI control, Flatness-based control, and backstepping control are employed to solve flux and speed
control problems of the system. The HIL based test shows better control performances of flatness control
compared to the other two schemes. This conclusion implies possibility of practical uses of flatness control in
dealing with systems with flexible couplings. In the future research, in order to give a concrete evaluation,
robustness experiments against varying system parameter and load disturbances will be carried out.
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