Nonlinear Control of the Permanent Magnet Synchronous Motor
PMSM using Backstepping Method
YOUSSEF CHAOU1, SAID ZIANI2, HAFID BEN ACHOUR1, ABDELKARIM DAOUDIA1
1Laboratory of engineering sciences and techniques (STI), Department of Physic,
Faculty of Science and Technology Errachidia (FSTE),
Moulay Ismail University, Meknes,
BP-509 Boutalamine Errachidia 52 000,
MOROCCO
2Laboratory of Networks, Computer Science, Telecommunication, Multimedia (RITM), Department
of Electrical Engineering, High School of Technology (ESTC),
Hassan II University, Casablanca,
KM 7 route d'el jadida BP 8012Oasis Casablanca
MOROCCO
Abstract: - This paper presents a nonlinear control of (PMSM) using backstepping. We will study the different
performances and robustness of each type of control, by introducing a new Lyapunov function candidate with a
large possibility of parameter choice. Simulation results clearly show that the speed and current tracking errors
asymptotically converge to zeros. Compared with neural networks control schemes, we do not require the
unknown parameters to be linear parametrizable. No regression matrices are needed, so no preliminary
dynamical analysis is needed.
Key-Words: - Permanent magnet synchronous motor (PMSM), Backstepping control, Lyapunov fonction
Received: March 13, 2021. Revised: November 16, 2021. Accepted: December 22, 2021. Published: January 17, 2022.
1 Introduction
Permanent magnet synchronous motor (PMSM) is
widely used in industrial applications compared to
other electric motors. Mainly, due to its compact
design, high efficiency, high torque-inertia ratio,
excellent reliability, high robustness, and low
maintenance [1, 2] PMSM is used in wind and
photovoltaic renewable energy, transportation
(electric cars), railway traction and ship propulsion
also make extensive use of these machines and in
other fields. On the other hand, the non-linearity of
the PMSM dynamic model produces a great
difficulty of specific control. The parameters and
load torque variations also the coupling between
motor speed and electrical quantities, such as d-q
axis currents, making this system obviously difficult
to control [1, 3]. This motor can be controlled by the
conventional PI controller but cannot guarantee
satisfactory performance such as stability and
control against disturbances [4]. To solve this
problem, various nonlinear control methods have
been developed and proposed to control and
command the PMSM, such as input-output
linearization control [5], sliding mode control [6],
backstepping control [7] and DTC [8]...etc.
Recently, Backstepping control is developed a
technique for controlling uncertain nonlinear
systems, in particular systems that do not satisfy the
adaptation conditions [9, 10]. The most interesting
point is to use the virtual control variable to simplify
the original high-order system, so that the final
control outputs can be derived systematically by
appropriate Lyapunov functions. A robust and
adaptive nonlinear controller, directly derived from
this control method, is proposed for the speed
control of PMSMs [11, 12]. The controller is robust
against stator resistance, viscous friction, load
torque uncertainties and unknown disturbances.
However, this approach uses feedback linearization,
the use of which can cancel out some useful non-
linearity [13]. Of all the nonlinear adaptive control
methods in the literature, the backstepping design on
the control of highly nonlinear and uncertain
systems has excellent performance in terms of its
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DOI: 10.37394/23203.2022.17. 7
Youssef Chaou, Said Ziani,
Hafid Ben Achour, Abdelkarim Daoudia
E-ISSN: 2224-2856
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Volume 17, 2022
ability to adapt to parameter uncertainties, transient
and steady-state performance, disturbance rejection
capability, and suitability for real time
implementation [10].
2 Mathematical Model of the PMSM
The PMSM model in the reference frame (d-q) is
shown as follows:
(1)
In the above equations, we denote by:
is the stator voltages in (d-q) reference
frame,
is the stator currents in (d-q) reference
frame,
is the d-axes and q-axes stator inductance,
Electrical pulse,
Rotor speed,
is rotor inertia,
Torque resistance,
is the magnet flux,
is the stator resistance,
Viscous friction coefficient,
Number of pole paire of the PMSM,
2.1 The General Equations of State of the
PMSM in the (d-q) Reference Frame
The state writing depends on the chosen reference
frame, we see that the state representation is not
unique. Any linear combination of the components
of a state vector is called state variables. By
developing the system of equations (1) we can
deduce the final form of the PMSM equations in the
reference frame (d-q):
+
(2)
The equation (2) represents the dynamic model of a
nonlinear system whose general form is the
following:
(3)
3 Backstepping Control Design
The backstepping controller is considered a very
useful tool when some states are controlled by other
states. This technique uses one state as a virtual
controller to another state since the system is in
triangular feedback form. It also overcomes the
problem of finding a Lyapunov control function as a
design tool. The design of backstepping control,
nonlinear systems or subsystems of the form (4).
(4)
Where:
We wish to make the output follow the
reference signal supposed to be known. The
system being of order n, the design is done in n
steps.
4 Designed of Backstepping
Controller
The basic idea of the backstepping control is to
make the looped system into cascaded subsystems
of order one stable in the Lyapunov sense, which
gives it robustness qualities and an asymptotic
global stability. The objective is to control the speed
by choosing as subsystems the expressions of
,
and as intermediate variables the stator currents
(,). These last variables are considered as virtual
commands, from these variables (,), we calculate
the voltage commands ( and ) necessary to
ensure the speed control of the PMSM and the
stability of the global system.
4.1 Step 1: Control of
Define the error as:
(5)
From the equations (1) and (5), the dynamic
equations of the error are:
(6)
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(7)
Consider the first Lyapunov function as:
(8)
And the derivative of is:
(9)
We choose
(10)
Where is a positive scalar,
Then
and the backstepping
control law is designed as :
(11)
4.2 Step 2 : Control of Rotor Speed
As the rotor speed is the main control variable, its
trajectory is defined as the reference value and the
control error as :
(12)
(13)
Define the second Lyapunov function as:
(14)
And
In order to obtain , we can choose
Where is a positive scalar
Then
(15)
Considering that this leads to define
the command necessary to determine the
voltage
(16)
4.3 Step 3: Control of
Define the error as:
(17)
Then
(18)
The Lyapunov function can be defined as:
(19)
Then
In order to obtain , we can choose
Where is a positive scalar
(20)
We deduce the final backstepping control law
is designed as:
(21)
Finally, we have defined from the backstepping
control, the reference variables necessary to control
the speed of the PMSM, while requiring a stability
of the cascaded subsystems to ensure an asymptotic
stability of the overall system.
5 Simulation Results and Discussion
5.1 Simulations Results
The adopted control is based on the Backstepping
method applied to a PMSM, whose model is non-
linear and multi-variable, is tested by numerical
simulation for the following parameter values:
and .
The motor parameters used for the simulations are
given in Table 1.
Table 1. The motor parameters.
4
0.6
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0.0014
0.0028
0.2
0.02
0.0014
5.2 Results
The simulation tests are carried out during a
simulation time of 2 seconds. In the first test, a load
torque of 5 Nm was applied at time t = 1 second at
constant speed. In contrast, in the second test, a
variable load torque was applied at variable speed,
in order to test and simulate the tracking of the
reference speed variation under load torque
disturbance variations.
(a)
(b)
Fig. 1: Speed tracking response for reference under load torque disturbance variations for (a) constant speed ,
(b) variable speed
(a) (b)
Fig. 2: Electromagnetique torque tracking a laod torque variation for (a) load torque 5N.m at time t = 1 second
at constant speed, (b) variable load torque and variable speed
(a)
(b)
Fig. 3: Current and for (a) constant speed, (b) variable speed
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
10
20
30
40
50
60
70
80
90
100
Times(s)
Speed(rad/s)
Rotor speed
Wref
W
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-100
-80
-60
-40
-20
0
20
40
60
80
100
Times(s)
Speed(rad/s)
Rotor speed
Wref
W
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-20
0
20
40
60
80
100
120
140
Times(s)
Cem,Cr(Nm)
Electromagnetic and load torque
Cem
Cr
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-250
-200
-150
-100
-50
0
50
100
Times(s)
Cem,Cr(Nm)
Electromagnetic and load torque
Cem
Cr
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-20
0
20
40
60
80
100
120
140
160
180
Times(s)
id,iq(A)
Curent id and iq
id
iq
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-200
-150
-100
-50
0
50
100
150
Times(s)
id,iq(A)
Curent id and iq
id
iq
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Fig. 4: Speed tracking response for reference under
different values of K1, K2 and K3
5.3 Discussion of the Results
The figure 1 shows the results of the simulation of
the speed control by backstepping, in figure 1(a) the
curves show that during the no-load start-up, the
quantities stabilise after a response time of 0.02 sec,
the rotation speed is the reference speed without any
overshoot. Also in The figure 1(b) shows the results
of the simulation with a change of set point and a
speed reversal, we notice that this control presents
very satisfactory results with good tracking
dynamics and a relatively acceptable rejection of the
disturbance. On the other hand, we notice that the
speed is established at its nominal value with good
dynamics and without static error, at the moment
when the load torque is applied, the speed is reduced
but it is re-established again without static error.
The figure 2 shows the behaviour of load torque
and electromagnetic torque. The latter oscillates
during power-up reaching a maximum value and
disappears once the steady state is reached. When
the load is applied, the electromagnetic torque
increases so as to instantly compensate the load
torque with some additional ripples in the
electromagnetic torque.
The figure 3 shows the characteristics of the
stator currents id and iq at start-up the machine
draws a large current afterwards we notice a
decrease as the machine has the normal operating
regime. The stator current components i and i
show the decoupling introduced by the PMSM
Backstepping control (i= 0). The electromagnetic
torque follows well the current Iq as shown in figure
2(b) and figure 3(b) with a peak related to the start-
up, which is reached in the steady state, which
shows the objective of the Backstepping control the
stabilization of PMSM operation with presence of
disturbances.
In order to test the robustness against parametric
variations, the simulation results of the dynamic
behaviour are presented as shown in Figure 4 for
different values of K1, K2 and K3. The table 2
below gives the minimum, maximum and optimum
values of K1, K2 and K3, it can be seen that the
variation of these parameters influence the dynamics
of the velocity ordered by Backstepping. This is
mainly due to the recursive nature of the latter,
which makes it possible to this is mainly due to the
recursive nature of the latter, which allows the
global system to be considered in cascaded
subsystems, to guarantee the stabilisation of the
measurements.
Table 2. The minimum, maximum and optimum
values of K1,K2 and K3 .
minimum value
optimum value
maximum value
K1=300
K2=300
K3=20
K1=1000
K2=1000
K3=100
K1=2000
K2=2000
K3=300
6 Conclusions
The permanent magnet synchronous motor PMSM
is an electric actuator of great industrial interest, due
to its compactness, low inertia, efficiency,
robustness and high power density, but its non-
linear structure makes its control more complex,
which led us to use the non-linear control model that
can provide good performance. Thus, the work
presented in this paper is essentially a contribution
to the backstepping control. The results of the
simulation show that the backstepping controller
was successfully designed a good response of the
PMSM, in pursuit the response time is low and a
high control performance regarding the rapidity, the
stability and robustness in relation to applied loads
and parametric variations vis-à-vis. this way, she
presents very satisfactory results with a good
tracking dynamics as well as a good rejection of the
disturbance. On the other hand, we notice a very
good dynamics when applying the load torque.
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
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120
Times(s)
Speed(rad/s)
Rotor speed
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w(K1=1000,K2=100,K3=100)
w(K1=1000,K2=100,K3=1000)
w(K1=1000,K2=100,K3=10000)
Wref
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