1. SLIDING MODE CONTROL OF A PERMANENT
MAGNETS SYNCHRONOUS MACHINE
A. Attou
A. Massoum
E. Chiali
ICEPS, University of Djilali Liabes
Sidi Bel-Abbes, Algeria.
ICEPS, University of Djilali Liabes
Sidi Bel-Abbes, Algeria.
ICEPS, University of Djilali Liabes
Sidi Bel-Abbes, Algeria.
attouamine@yahoo.fr
ahmassoum@yahoo.fr
ameroufel@yahoo.fr
Abstract— This paper presents a speed sliding mode controller
for a vector controlled (FOC) of the permanent magnet
synchronous machine (PMSM) fed by a pulse width modulation
voltage source inverter. The sliding mode control (SMC) is used
to achieve robust performance against parameter variations and
external disturbances. The problem with this conventional
controller is that it has large chattering on the torque and the
drive is very noisy. In order to reduce a torque ripple, the sign
function is used. The proposed algorithm was simulated by
Matlab/Simulink software and simulation results show that the
performance of the control scheme is robust and the chattering
problem is solved.
Keywords— PMSM, field oriented control, sliding mode control.
I. INTRODUCTION
Permanent magnet synchronous motor (PMSM), which
possesses such advantages as high efficiency, high torque
ratio and the absence of rotor losses .
But, if we compare the PMSM with DC motors, PMSM
are more difficult in speed control and not suitable for high
dynamic performance applications because of their complex
inherent nonlinear dynamics and coupling of the system. So
PMSM commonly run at essentially constant speed, whereas
dc motors are preferred for variable-speed drives [8],[14].
Robustness as a desirable property of the automatic control
systems is defined as the ability of the control system to yield
a specified dynamic response to its reference inputs despite
uncertainties in the plant mathematical model and unknown
external disturbances. In its basic form, the sliding mode
control (SMC) is a kind of nonlinear robust control using a
systematic scheme based on a sliding mode surface and
Lyapunov stability theorem. It features disturbance rejection,
strong robustness and fast response [9],[8],[12].
The design of this control law consists of two phases:
- Depending on the initial conditions of a trajectory is
precomputed. This trajectory is used to modify the sliding
surface so that the trajectories of the change on the surface for
all t 0.
- A discontinuous control is designed to ensure that the
system evolves on the sliding surface, despite the presence of
a certain class of uncertainties and disturbances.[6]
The main advantages of this strategy are:
A prior knowledge of the convergence time and setting of
the control law independent of that time.
- Establishment of the sliding mode at the initial time, which
gives the control law robust behavior throughout the system
response.
- The control strategy is applicable regardless of the order of
sliding mode (equal to or higher relative degree of the
system).
- The generation of the path enabling the convergence in
finite time [5],[6].
In this work is composed of PMSM modeling in the Park
frame and an overview of the sliding mode control of the
PMSM supplied with PWM inverter. In the last step, a
comment on the results obtained in simulation and a
conclusion where we emphasize the interest of this method of
control.
-
II. MACHINE EQUATIONS
With the simplifying assumptions, the model of a 3-phase
permanent magnet synchronous motor can be expressed in the
so-called dq frame by application of the Park transformation,
in the form of state is written [10][15].
x = F(x) +
m
i=1
g i (x)U i
(1)
With
x1 Id
x = x2 = Iq
(2)
x3
U U
Ui = 1 = d
U2 Uq
(3)

2. high frequency which is generally triangular shaped hence the
name triangular-sinusoidal [5].
1
0
Ld
g1 = 0 ; g2 = 1
Lq
0
0
f1 x
F x = f2 x =
f3 x
(4)
− Rs x1 + pLq x2 x3
Ld
Ld
pLd x1 x3 − p f x3
− Rs x2 −
Lq
Lq
Lq
p Ld − Lq
p f
− f x3 +
x1 x2 +
x2 − Td
J
J
J
J
(5)
The variables to be controlled are current Id and
mechanical speed Ω.
Y (x ) =
y1 (x )
y 2 (x )
=
h1 (x )
h2 ( x )
=
x1
x2
=
Id
(6)
The parameters used in these equations are defined as
Follows:
Ud Uq : Stator voltages in the dq axes;
I d I q : Stator current in the dq axes;
Rs : Stator resistance ;
Ld Lq : Stator inductances in the dq axes;
Φ : flux created by the rotor magnets;
f
Ω : Mechanical speed of motor;
P : Number of pole pair;
J : Inertia momentum;
f : The damping coefficient;
F : Frequency.
III. MODELING OF THE VOLTAGE INVERTER
The voltage inverter can convert the DC power to the AC
(DC / AC). This application is widespread in the world of
power conversion today. The connection matrix is given by
(7).
VaN
2 - 1 - 1 Sa
E
VbN = ⋅ − 1 2 - 1 Sb
6
VcN
− 1 - 1 2 Sc
(7)
The inverter is controlled by the technique Pulse Width
Modulation (PWM) generated by a carrier which is triangular.
It is used for generating a signal which controls the switches,
the PWM control signal delivers a square-wave, it is generated
by the intersection of two signals, the reference signal, which
is generally sinusoidal low frequency, and the carrier signal
IV. FIELD ORIENTED CONTROL (FOC)
By analyzing the system of equations (1), we can observe
that the model is nonlinear and it is coupled. The objective of
the field oriented control of PMSM is transformed machine
three-phase axis variables into two-phase axis in order to
obtain the same decoupling between the field and torque that
exists naturally in dc machines; that is to say a linear and
decoupled. This strategy is to maintain the flow of armature
reaction in quadratic with the rotor flux produced by the
excitation system [5].
Since the main flow of the PMSM is generated by the rotor
magnets, the simplest solution to a permanent magnet
synchronous machine is to keep the stator current in quadratic
with the rotor flow ( Id is zero and reduce the stator current to
the only component Iq ), that gives a maximum torque
controlled by a single current component Iq and to regulate
the speed by the current Iq through the voltage Vq . This
verifies the principle of the DC machine [4].
V. SLIDING MODE CONTROL (SMC)
The switching of the variable structure control is done
according to state variables, used to create a "variety" or
"surface" so-called slip. The sliding mode control is to reduce
the state trajectory toward the sliding surface and make it
evolve on it with a certain dynamic to the point of balance.
When the state is maintained on this surface, the system is
said in sliding mode. Thus, as long as the sliding conditions
are provided as indicated below, the dynamics of the system
remains insensitive to variations of process parameters, to
modeling errors [1],[4],[5].
The objective of the sliding mode control is:
• Synthesize a surface, such that all trajectories of the
system follow a desired behavior tracking, regulation
and stability.
• Determine a control law which is capable of attra-cting
all trajectories of state to the sliding surface and keep
them on this surface.
The behavior of systems with discontinuities can be
formally described by the equation:
x ( t ) = f ( x , t, U )
(8)
Where:
x : Vector of dimension n, x ∈ ℜ n .
t : Time.
U : Control input of a dynamical system, u ∈ ℜ .
m
f : The function describing the system evolution over
time.

3. So we seek that the two functions f
+ and f
−
converge
towards the surface of commutation S and which have the
characteristic to slip on it. We say that the surface is attractive
[5],[13] .
+
x( t )= f ( x,t ,U ) = f ( x ,t ) if S( x ,t )>0
f −( x,t ) if S( x,t )<0
A. The choice of desired surface
We take the form of general equation given by J.J.Slotine
to determine the sliding surface given by:
S ( x) = (
∂
+ λx ) r −1e( x )
∂t
(14)
(9)
Where:
e( x ) : Error vector; e( x) = x ref − x .
λ x : Vector of slopes of the S.
r : Relative degree, equal to
the number of times he
derives the output for the command to appear.
B. Convergence condition.
The Lyapunov function is a scalar function positive for the
state variables of the system, the control law is to decrease this
function, provided it makes the surface attractive and
invariant. En defining the Lyapunov function by:
Fig.1 Trajectories of f + and f - in the case of sliding mode.
S=0 mathematically,
this represented as: [2]
lim S > 0
lim S < 0
et
s → 0−
s → 0+
(10)
(11)
The function is used, generally, to ensure stability of
nonlinear systems. It is defined, like its derivative as follows
[1],[3],[4],[13] :
x = f ( x,t )+ g( x,t )U
x ,u
S ( x)S ( x) < 0
(16)
This can be expressed by the following equation :
lim S > 0
s →0
et
−
lim S < 0
s → 0+
(17)
C. DETERMINATION OF THE CONTROL
In sliding mode, the goal is to force the dynamics of the
system to correspond with the sliding surface S(X) by means
of a command defined by the following equation:
u(t ) = ueq (t ) + u N
(18)
f and g are n- dimensional continuous functions
Where
in
(12)
(15)
For the Lyapunov function decreases, it is sufficient to
ensure that its derivative is negative. This is verified by the
following equation:
V ( x) < 0
Hence, the condition of attractiveness to obtain the sliding
mode:
S( x ).S( x ) < 0
1 2
S ( x)
2
The vector f is in a direction towards
V ( x) =
and t.
t ,x
Is an n- dimensional column vector, (
x ∈ ℜn ) and ( u ∈ ℜm ).
When we are in the sliding mode, the trajectory remains on
the switching surface. This can be expressed by [1],[3],[4],
[7],[11],[13] :
S( x,t ) = 0
et
S( x,t ) = 0
(13)
The design of sliding mode control requires mainly the
three following stages:
In which:
U: is called control magnitude, Ueq :is called the equivalent
components which is used when the system states are in the
sliding mode; Un: is called the switching control which drives
the system states toward the sliding mode, the simplest
equation is in the form of relay:
un = ksgnS(x) ;k 0
(19)
k : High can cause the ‘chattering‘ phenomenon.
When the switching surface is reached, (13) we can write:

4. U =Ueq with u N =0.
(20)
VI. THE ‘CHATTERING’ PHENOMENON ELIMINATION
The high frequency oscillation phenomenon can be
reduced by replacing the function ‘sgn’ by a saturation
function [15].
k
S(x)
si
u n = ksgn(S(x))
si
S(x)
S(x)
(21)
Fig. 4 Results of simulation of the adjustment by sliding-mode during
variation parametric
0
IX. INTERPRETATION
Different simulations allow us to see that: disturbance
rejections are very good and a very low response time. This
control strategy provided a stable system with a practically
null static error and a decoupling for the technique suggested
by maintaining Id to zero.
X. CONCLUSION
Fig. 2 Schematic of the overall simulation
VII. SIMULATION RESULTS
For the validation of the structure of the sliding mode
control we made simulations using MATLAB / Simulink.
Figure (3) shows the results obtained with the strategy of
three surfaces:
In this paper, we presented the performances of the sliding
mode control for the PMSM. The decoupling technique is
based on the Field oriented control to PWM tension inverter.
The proposed controller provides high-performance dynamic
characteristics and is robust with regard to plant parameter
variations. So the controller works well with robustness in a
large extent.
MACHINE PARAMETERS
Rs = 0.6
; Ld = 1.4mH ; Lq = 2.8mH ; f = 0.12wb ;P = 4;
J = 1.1⋅10 − 3 kgm 2 ; f = 1.4 ⋅10 − 3 Nm/rds − 1;F = 50 HZ .
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Fig. 3 Results of simulation by sliding-mode control
At t = 0.3 (s), the reference speed varies from
Wr = 100 (rad / s) to Wr = -100 (rad /s), followed by an
external load torque disturbance Td=8 (Nm) for periods [0.1s]
between t = 0.1 (s) to t = 0.2 (s) and t = 0.4 (s) to t = 0.5 (s).
VIII. ROBUSTNESS TESTING
To highlight the importance of the technique of sliding
mode control, we will test the robustness of our machine.
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