This paper presents a comparison between two control strategies, field oriented control (FOC) and sliding mode control (SMC). The first study has been done on controlling the Permanent Magnets Synchronous Machine (PMSM) by using field oriented control technique using a speed sensor to detect the rotor position of the motor. After we present the theory of sliding mode control .We can see from the simulation results that the second proposed control system (SMC), is a robust nonlinear time optimal controller, that's will be more explained in the appendix.

1. International Conference on Electromechanical Engineering (ICEE'2012) Skikda, Algeria, 20-22 November 2012
Comparison Between Two Different Speed Controls
Of The Permanent Magnets Synchronous Machine
A. Attou A. Massoum A. Meroufel
ICEPS, University of Djilali Liabes ICEPS, University of Djilali Liabes ICEPS, University of Djilali Liabes
Sidi Bel-Abbes, Algeria. Sidi Bel-Abbes, Algeria. Sidi Bel-Abbes, Algeria.
attouamine@yahoo.fr ahmassoum@yahoo.fr ameroufel@yahoo.fr
Abstract—This paper presents a comparison between two control vary linearly with the q-axis current component, and the
strategies, field oriented control (FOC) and sliding mode control maximum torque per ampere is achieved which is similar to
(SMC). The first study has been done on controlling the the control of a separately excited DC motor [3][5][6].
Permanent Magnets Synchronous Machine (PMSM) by using
Robustness as a desirable property of the automatic control
field oriented control technique using a speed sensor to detect the
systems is defined as the ability of the control system to yield
rotor position of the motor. After we present the theory of sliding
mode control .We can see from the simulation results that the
a specified dynamic response to its reference inputs despite
second proposed control system (SMC), is a robust nonlinear uncertainties in the plant mathematical model and unknown
time optimal controller, that's will be more explained in the external disturbances. In its basic form, the sliding mode
appendix. control (SMC) is a kind of nonlinear robust control using a
systematic scheme based on a sliding mode surface and
Keywords-PMSM, PI-controller, field oriented control, sliding Lyapunov stability theorem. It features disturbance rejection,
mode control. strong robustness and fast response [6],[10],[13].
 INTRODUCTION I. MACHINE EQUATIONS
Compared with DC motors, PMSM are more difficult in Take into consideration the simplifying assumptions, the
speed control and not suitable for high dynamic performance model with two axes of the PMSM is performed by a real
applications because of their complex inherent nonlinear transformation of the three axis in a fictitious two-axis frame,
dynamics and coupling of the system. So PMSM commonly which is actually based on a change of physical quantities
run at essentially constant speed, whereas DC motors are (voltages, fields, and currents), it leads to relations
preferred for variable-speed drives. The situation has changed independent of the angle θ and reduced-order equations of the
dramatically with the advent of field-oriented control (FOC) machine. The transformation best known by electricians is
or vector control theory and the advances in power electronics that of Park.
and modern microcomputers[6],[11],[12].
The Field Oriented Control (FOC), also called Vector A. Electrical equations
control, the goal of this method is to perform real-time control 
Vd  RsId  dφd - pωr φq

of torque variations demand, to control the rotor mechanical 

speed and to regulate phase currents in order to avoid current 


dt

dφd  pωr φd (1)
spikes during transient phases. It is possible to set up a Vq  RsIq 


coordinate system to decompose the vectors into how much



dt
electromagnetic field is generated and how much torque is
φd  Ld Id  φ f

generated. This coordinate system is generally called d-q 


reference coordinate system (Park transformation). The vector 
(2)
control technique is employed in order to obtain high torque




φq  Lq Iq
capability of the PMSM drive through the decoupling control
of d-q axes stator currents in the rotor reference frame. For a B. Electromagnetic equation
PMSM, the Permanent Magnets provides the flux linkage. By
keeping d-axis current equals to 0, the PMSM torque may Tem  p[ (Ld - Lq )I d  Iq  φ f  Iq ] (3)

2. International Conference on Electromechanical Engineering (ICEE'2012) Skikda, Algeria, 20-22 November 2012
C. Mechanical equation III. FIELD ORIENTED CONTROL (FOC)
The induction machine is a nonlinear model and it is
dr coupled; FOC strategies transform machine three-phase
J  f  r  Tem  Td (4) variables into two-phase axes in order to obtain the same
dt decoupling between flux and torque that exists naturally in dc
machines. That is to say maintain the flow of reaction induced
Consider the voltages ( Vd, Vq ) and the excitation flux in the quadratic rotor flux produced by the excitation system.
φ f as control vector, the stator currents ( Id, Iq ) as state The presented control strategy is based on a rotor field
oriented control (RFOC) and the stator three phase current is
vector. From (1), (2), we can write the system of equations as transformed into dq components of a rotating reference frame.
follows [3],[8] : In this case, the d component of stator current controls the
d [ X ]  [A][X] +[B][U] (5) magnetic state of the machine (cf. equation (2)), while the q
dt component is in charge of generating the electromagnetic




torque (cf. equation (3)) [3][5].
 1
   0 0   


   Rs p r Lq     Ld  
V 
  d  According to (1), Vd and Vq are coupled, to decouple
d  Id  

 Ld Ld   Id    0 1 0  Vq 
dt  Iq   p r Ld
      (6) the system, we will introduce the compensation terms ed et eq.

 Rs   Iq   Lq  φ 


 Lq Lq  
 0 0 p
 r   f 
The voltages Vd, Vq depends only I d (respectively for) I q .




Lq 
Figure 1 shows the decoupling system with compensation
Where: terms:
[A ] : State matrix
[X ] : State vector (where [X]  [ Id Iq ]T )
[B ] : Input matrix
[U] : Control vector (where [U]  [ Vd Vq φ f ]T )
II. MODELLING OF VOLTAGE SOURCE INVERTER
The voltage inverter can convert the DC power to the AC
(DC / AC). An AC electric power conversion based on Fig. 1 decoupling system
technique Pulse Width Modulation PWM inverter is very
popular in these days. This technique is used to control the
ed   r. Lq Iq


magnitude and frequency of the AC output voltages of an Where: 

(8)
eq   r .(Ld Id φ f )


inverter. This technique is widely used in industrial 


applications such as variable-speed electric drives and has
been the focus of research interests in power electronics
Since the Controllers are sized classics from the
applications for many years [14]. The connection matrix is
given by (7). parameters of the machine. So if they vary over a wide range
of operation, performance deteriorated, then it is best to see
VaN   2 - 1 - 1  Sa 
VbN   E   1 2 - 1  Sb  (7)
other tuning techniques. Where controllers are known for their
  6    robustness.
VcN 
   1 - 1 2   Sc 
  
IV. SLIDING MODE CONTROL (SMC)
The inverter controlled by the technique PWM generated Slide-mode control is one of variable structure control
by a carrier which is triangular. It is used for generating a strategies (VSC). The essential difference between VSC and
signal which controls the switches, the PWM control signal the common control strategies is the variable structure of
delivers a square-wave, it is generated by the intersection of control-led system, namely, a kind of switch characteristic of
two signals, the reference signal, which is generally sinusoidal making systematic structure change continually. The
low frequency, and the carrier signal high frequency which is switching of the variable structure control is done according to
generally triangular shaped hence the name triangular- state variables, used to create a "variety" or "surface" so-
sinusoidal [4]. called slip. The sliding mode control is to reduce the state

3. International Conference on Electromechanical Engineering (ICEE'2012) Skikda, Algeria, 20-22 November 2012
trajectory toward the sliding surface and make it evolve on it 
x  f ( x,t ) g( x,t )U (13)
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 Where f and g are n- dimensional continuous functions
are provided as indicated below, the dynamics of the system
in x, u and t. x is an n- dimensional column vector .
remains insensitive to variations of process parameters, to
( x   ) and ( u   ).
n m
modeling errors [2],[3],[7].
The objective of the sliding mode control is:
 Synthesize a surface, such that all trajectories of When we are in the sliding mode, the trajectory remains on
the system follow a desired behavior tracking, the switching surface. This can be expressed by [2],[4],[9] :
regulation and stability. 
 Determine a control law which is capable of
S( x,t )  0 et S( x,t )  0 (14)
attracting 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)
 (9)
Where:
x : Is a vector of dimension n, x  n .
t : Time.
Fig. 2 sliding mode in the intersection
U : control input of a dynamical system, u   .
m
f : The function describing the system evolution over The design of sliding mode control requires mainly the
three following stages [2][3]:
time.
 and f A. The choice of desired surface

So we seek that the two functions f converge We take the form of general equation given by J.J.Slotine
towards the surface of commutation S and which have the to determine the sliding surface given by:
characteristic to slip on it. We say that the surface is attractive 
[15] . S ( x)  (  x )r 1e( x ) (15)
t

x( t ) f ( x,t ,U ) f ( x,t ) if S( x,t )0

 

f ( x,t ) if S( x,t )0 (10)


 Where:


e( x ) : Error vector; e( x)  x ref  x .
The vector f is in a direction towards S0
mathematically, this represented as [1],[15] :  x : Vector of slopes of the S.

lim S  0 et 
lim S  0 r : Relative degree, equal to the number of times he
(11) derives the output for the command to appear.
s  0 s  0 B. Convergence condition.
The Lyapunov function is a scalar function positive for the
Hence, the condition of attractiveness to obtain the sliding
state variables of the system, the control law is to decrease this
mode:
function, provided it makes the surface attractive and

S(x).S(x)  0 invariant. En defining the Lyapunov function by:
(12)
1 2
V ( x)  S ( x) (16)
The function is used, generally, to ensure stability of 2
nonlinear systems. It is defined, like its derivative as follows:
[2],[7],[15]:

4. International Conference on Electromechanical Engineering (ICEE'2012) Skikda, Algeria, 20-22 November 2012
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  S ( x)S ( x)  0 (17)
This can be expressed by the following equation :

lim S  0 et 
lim S  0
(18)
s  0 s  0
C. The insurance of the conditions of convergence
In sliding mode, the goal is to force the dynamics of the
system to correspond with the sliding surface S(X) by means Fig. 4 field oriented control of PMSM with classical control (PI).
of a command defined by the following equation:
The variation of the reference speed is Wr = 100 (rad / s) to
u(t )  ueq (t )  uN (19) Wr = -100 (rad /s) at t = 0.6 (s), followed by an external load
torque disturbance is Td=8 (Nm) at a period [0.2s] between
t = 0.2 (s) and t = 0.4 (s).
In which:
For the sliding mode control, we will replace the PI
U: is called control magnitude; Ueq :is called the equivalent
controllers with sliding mode controllers (SMC), Figure (5)
components which is used when the system states are in the
shows the results obtained with the strategy of three surfaces:
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 (20)
k: high can cause the ‘chattering ‘ phenomenon.
When the switching surface is reached, (14) we can write:
U  Ueq avec u N  0. (21)
VI. SIMULATION RESULTS
To validate the structure of the sliding mode control, we Fig. 5 Results of simulation by sliding-mode
made simulations using MATLAB / Simulink. VII. ROBUSTNESS TESTING
To highlight the importance of the technique of sliding
mode control, we will test the robustness of our machine.
Fig. 3 Schematic of the overall simulation
MSAP is supplied with the voltage variable frequency and
amplitude by an inverter PWM voltage, the driving circuit is
powered by a constant voltage source.
Fig. 6 Results of simulation of the adjustment by sliding-mode during
For the first order, we will implement PI type controller. variation parametric

5. International Conference on Electromechanical Engineering (ICEE'2012) Skikda, Algeria, 20-22 November 2012
VIII. INTERPRETATION Rs : Stator resistance
Different simulations allow us to see that: Ld : Inductance of direct axis
Disturbance rejections are good and a weak response time Lq : Inductance of quadratic axis
for traditional controller PI. This control strategy provided a
φ f : Field flux
stable system with a practically null static error and a
decoupling excel for the technique suggested by maintaining p : Number of pole pairs
Id to zero. The sliding mode control provided a more stable J : Inertia momentum
system with satisfactory performance compared to traditional f : The damping coefficient
control as well as loadless system or during the load variation F : Frequency
and also as the disturbance rejection. The results of robustness MACHINE PARAMETERS
testing shows that the SMC is well known to be insensitive to R 0.6Ω;Ld 1.4mH;Lq  2.8mH;φ f 0.12wb P  4;
;
inner parameters variations and outer disturbance, so the
J 1.1103kgm2; f 1.4103 Nm/rds 1;F  50HZ.

controller works well with robustness in a large extent which
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