International Journal of Applied Engineering Research
ISSN 0973-4562 Vol.2, No.1 (2007), pp. 125–138
© Research India Publications
http://www.ripublication.com/ijaer.htm
Application of Nonlinear Sliding-Mode
Control to Permanent Magnet Synchronous Machine
(Study of the robustness)
A. Kechich1, B. Mazari2 and I. K. Bousserhane3
1
Institut d’électrotechnique, B.P417 C .U. Béchar, Tel/Fax : 049815244
2
Institut d’Electrotechnique, U.S.T.O. B.P1505,
Oran El M’naouer, Algérie
3
Institut d’électrotechnique, B.P417 C .U. Béchar
E-mail: akechich@yahoo.fr
Abstract
This article is devoted to the study of the performances of the nonlinear
order by sliding-mode with the integration of an integrator regulator, applied
to the synchronous permanent magnet machine. In a first stage, we presented
the design of the order by this new approach followed by a structure of the
optimal ordering of the adjustment with the law of commutation against
reaction of state with integrator. The second stage is devoted to the
development of an order to variable structure with a strategy of three surfaces,
which allow the elimination of the overloads that can result in a destruction of
the system to regulate.
Keywords: Synchronous permanent magnet machine, sliding-mode, slip
surface, nonlinear, decoupling, simulation, robustness.
Introduction
In the case of linear systems with constant parameters, the traditional laws of ordering
of type PI and PID give good results. On the other hand, the nonlinear systems or
those with non-constant parameters, these traditional laws of order can be insufficient
since they are non-robust, especially when the requirements on the precision and other
dynamic characteristics of the system are severe. In this case one must call upon laws
of order insensitive to the variations, the nonlinearities and the disturbances [ 1 ].
The order with variable structure (OVS) is of nonlinear order by nature; it
constitutes a good solution with the problems quoted above. This order with variable
structure has a very great advantage, since its law of order changes in a discontinuous
way.
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A. Kechich et al
In order to create a " hyper surface " called as slip, to force the dynamics of the
system to correspond to that definite by the equation of "the hyper surface", then the
commutations of the order are carried out according to the variables of states [ 2 ].
The system is in sliding-mode when the state is maintained on this hyper surface
[7][4]. In this condition the dynamics of the system remains insensitive to the
variations of the parameters of the process, to certain parasitic disturbances and to the
simplifying assumptions at the time of modelling.
The order (OVS) is well adapted, to treat the systems which generate ill-defined
models. It has several advantages such as the precision desired, simplicity, stability, a
very weak response time and a good robustness.
Definition
A system is known as variable structure if it admits a representation by differential
equations of the type:
dx
= f (t , x )
dt
(1)
where : x is a vector of dimension n : 1, 2, 3, … n
x = (x1 , x 2 , x3 .....x n )
t
f i (t , x1 ,.......x n ) t
Are continuous functions per pieces and having discontinuities on a surface S.
The structure is known as variable because of the discontinuous change between
two or several structures [3]. The study of such systems is very advantageous
particularly in mechanics, in physics, in electricity and in chemistry... This advantage
is due to the properties of stability acquired by the total system independent of each
subsystem f i (x ) . The circuits of conversion of power constitute an example typical
practice of a system with variable structure. Indeed, for each position of the switch,
the system is controlled by a system of differential equation.
Theory of the Sliding-Modes
The theory of the sliding-modes finds an adequate application in the theory of order of
the systems to relay and in the circuits of electronics of power. The systems with
variable structure are characterized by the choice of a function and a suitable logic of
commutation. This choice at any moment ensures switching between these structures
and makes it possible to impose the behavior wished for the total system [ 9 ] [ 13].
The hyper surface S(x) = 0 of dimension
n-1 divides space into two parts [ 5 ] [ 6 ].
if
S (x ) > 0
G = G+
−
if
S (x ) < 0
G=G
Around the hyper surface S (x ) = 0 , the function
Permanent Magnet Synchronous Machine
127
f (x, t ) is discontinuous. The latter takes two values on each side of surface in the
vicinity of a point (x) :
in G +
f = f+
f = f−
in G −
Where : S (x, t ) is the switching function.
The variety of commutation is of size equal to (n) deducting the number of the
switching function available. In this case it is the number of exits which one must
stabilize.
Trajectories associated with function f are summarized in three configurations
where they are described by the temporal evolutions:
∗ The first configuration represents the trajectories of f + and f − that highlight
the repulsion for f − (respectively for f + ).
∗ The second represents the trajectories where the phenomena of attraction exist
for
f + (respectively for f − ) and of
f − (respectively for f + ).
∗ The third represents trajectories of
−
which converge towards the surface of commutation S 0 and which
f + and f
have the characteristic to slip on it. This phenomenon is called « sliding-mode » [ 4 ].
f
−
S0
Figure 1: Trajectories of f + and f
−
in the case of sliding-mode.
Advantages and Validity
The advantages of the order by mode of slip are very significant. They are appreciable
and well-known since the eighties [11]. This type of order gives good precisions,
stability in a very short response time, simplicity towards the design and guaranteed a
good robustness. The major advantage resides in the characteristic of adaptation as
concerns the systems which have vague models. This inaccuracy can be due to the
following reasons:
• Inaccuracy of the parameters;
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A. Kechich et al
• Variations of the parameters;
• Simplifications of the dynamic model.
This type of order can be used in an analogical way not only in the problems of
continuation of the models but also in regulation. It is affirmed [5] that the simplicity
of the implementation and the adaptation are applicable in the linear and nonlinear
processes [7].
Design and Trajectory in the Plan of Phase
The recent design of the regulators by sliding-mode deals with the problems of
stability and desired performances in a systematic way. The new technique requires
mainly the three following stages [ 12 ] :
1 - The choice of desired surface;
2 - The determination of the suitable order;
3 - The insurance of the conditions of convergence.
The technique of this order consists in bringing back the trajectory of state of a
system towards slip surface then to make commutation by the intermediary of a logic
of adapted commutation, up to the point of balance [ 5 ].
The trajectory consists of three distinct parts as the figure (2) shows.
¾Mode of convergence (MC): where the variable to be regulated moves
starting from the point of initial balance.
¾Mode of slip (MS): where the variable of state reaches the slip surface.
¾Mode of the permanent mode (MPM): Where the behavior of the system is
around the point of balance.
Direct Sampling Function
It is the first condition of convergence. It is proposed by V.I. Utkin [1]. This
function is expressed in the following form:
S (x )S (x ) < 0
(2)
It is necessary to introduce, under this condition, the exact values on the left and on
the right of commutation for S (x ) and its derivative. The conditions of convergences
ensure a dynamics of the system corresponding to the slip surfaces [ 8 ].
Function of Lyaponov
It is a matter of formulating a positive scalar function V (x ) > 0 for the variables of
states of the system, and to choose the law of commutation which will decrease this
function. That means mathematically: V (x ) < 0 .
The function suggested is used, generally, to ensure stability of nonlinear systems
[10]. It is defined, like its derivative as follows:
V (x ) =
1 2
S (x )
2
V (x ) = S (x ).S (x )
(3)
(4)
Permanent Magnet Synchronous Machine
129
xi
MG
MC
xn
M RP
S (x)
Figure 2: Modes of trajectory in the plan of phase.
S (x )
e (x )
r
+
-
∫
e
r −1
(x )
e( x )
∫
e( x )
λ r −1
λ0
Figure 3: Exact linearization of the difference.
So that the function of Lyaponov decreases, its derivative need be negative, i.e., to
ensure the following condition:
S (x ).S (x ) < 0
(5)
This equation shows that the square of the distance towards the surface measured
by
S 2 (x ) decrease all the time; whereas, the trajectory of the system is obliged to
move towards the surface on the two sides as figure (1) shows. It is an ideal slidingmode.
Calculation of the Order
Once the surface slip as well as the criterion of convergence are chosen, it remains to
determine the corresponding order to bring back the variable to be controlled towards
the surface and then towards its point of balance by maintaining the condition of
existence of the sliding-modes. The order must commutate between the values u max
and u min in a manner instantaneously. It is an essential assumption in the systems
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A. Kechich et al
design with variable structures controlled by the sliding-modes. Alternation depends
on the sign of the slip surface, as the figure (4) shows. In this case, the oscillations of
very high frequencies called "chattering" appear in the mode of slip.
u
u max
S (x )
u min
Figure 4: Order applied to the systems with variable structure.
The structure of a controller consists of two parts, the first relates to the exact
linearization, the second relates to stability. The second part is very significant in the
technique of the order by sliding-mode, because it is used to eliminate the effects of
inaccuracy from the model and to reject the disturbances caused by the phenomena
external. This behavior is the cause of the variation of the profit ( K ). If this last is
very small, the response time will be very long. On the other hand, if it is large, then
response time will be fast, but the phenomenon of "Chattering " is likely to appear in
permanent mode. The size of our order is of the form:
u (t ) = u eq (t ) + u n
(6)
u eq (t )
corresponds to the equivalent order suggested by V.U. Utkin and A.F.
Filipov. This order is the simplest and the most direct. It is calculated according to
the recognition of the behavior of the system during the sliding-mode where:
S (x ) = 0
(7)
The equivalent order can be interpreted as the modulated average value that the
order takes during the very fast commutation between
u max and u min , as shown in figure ( 4 ).
Order with Threshold
During rapid commutations of the order, at the time of the sliding-modes, the
phenomenon of “chattering” appears. High frequency components, are add to the
spectrum of the order. These components are undesirable because, they can damage
the actuators, or deteriorate the system by causing the excitement of the high modes
which one did not consider at the moment of modelling. In order to reduce
Permanent Magnet Synchronous Machine
131
“chatterings” one imposes a variation of the value of the order u n , according to the
distance between the variable of state and the slip surface [ 4 ]. Other methods
introduce thresholds (dead zone) on the commutation of the sign function.
This new order is characterized by a threshold ( ε ) as shown in the figure (5). The
continuous component ( u eq ) acts alone in the band which surrounds surface S (x ) of
slip. Since the discontinuous part u n is null, then the oscillations on the results will
be attenuated in a considerable way. On the other hand, there will be a static
difference on the result, at the moment of the regulation, when ( ε ) increase.
The expression of the discontinuous order would be:
un = 0
if
S (x ) < ε
(8)
u n = KSign(S (x ))
if
S (x ) > ε
(9)
Oscillations can persist in permanent mode, from where a softening of the order
u n is need.
e(x )
u
S (x )
u eq
+K
−ε
+ε
e (x )
S (x )
-K
F1gure 5: The Sign function. The band surrounding surface in the plan of phase.
0
0
S ur face : S (x )
u n : k sign (S (x ))
Figure 6: a- Slip surface ; b- Discontinuous order.
132
A. Kechich et al
Order Softened
In order to decrease the value of the order u n , the softened order is characterized by a
threshold
( ε 1 ) or two ( ε 1 , ε 2 ). As the figure( 7 ) shows, one distinguishes three zones which
depend on the distance from the point considered on the slip surface:
ƒZone 1 : The distance is higher than the threshold ( ε 2 )in this case the function
sign is effective.
ƒZone 2 : The distance is lower than the threshold ( ε 1 ) and consequently u n is
null. It is the dead zone.
ƒZone 3 : the point is in the band [ ε 1 , ε 2 ], u n is, then, a linear function of the
distance from the right-hand side of the slope equals
K
ε 2 − ε1
.
e(x )
un
S (x ) = 0
u eq + u n
a do uc ie
un
− ε2
+K
− ε1
+ ε1 + ε 2
-K
S (x )
P1
P2
e (x )
Figure 7: The Sign function; the band which surrounds the surface in the plan of phase.
0
Surface :
(a)
(b)
Figure 8: a- Slip surface; b- softened order.
Permanent Magnet Synchronous Machine
133
It is noticed that the larger the threshold is, the less the commutations are. In this
case the problem of precision arises. The system evolving in the band is thus likely
never to reach the origin of the plan of phase (desired point). Figure ( 8 ), illustrates
the slip surface as well as the softened order in the case of a control speed with the
(OVS) of the synchronous permanent magnet considered machine. The example
shows us that the introduction of the threshold induces a static error since the surface
S (x ) = 0 is never reached.
Solution Suggested
Whatever the method of softening used to eliminate or limit the oscillations, the
undesirable phenomenon always exists. This last, can be solved by a method which
leads to increase slightly the order of surface S (x ) of slip [ 6 ]. The suggested method
needs to introduce the derivative of the error in the calculation of the area. One
introduces an acceleration term into the case of the control speed. The new equation
which governs surface will be:
S (x ) = λ x e(x ) + e(x )
( 10 )
With :
λ x : Positive constant which interprets the band-width of the desired control.
The use of the new surface known as "increased surfaces" thus involves an
increase in the frequency of commutation of the order and consequently one will have
a strong reduction in the oscillations;
e(x ) = Ω ref − Ω r : Error between the considered speed instruction and the measured
speed;
e(x ) = eref − e(x ) : Error of acceleration.
Application to the Synchronous Permanent Magnet Machine
Model of reference, based on the orientation of flow
The model of the synchronous permanent magnet machine is defined as follows:
did
= −a1 + a 2 iq Ω + a 3 v d
dt
diq
dt
= −b1i q − b2 id Ω − b3 Ω + b4 v q
( 11 )
dΩ
= −c3 − c 4 C r + (c1id + c 2 )iq
dt
Where the coefficients are defined as follows:
a1 =
R
Ld
;
a2 = p
Lq
Ld
;
a3 =
1
Ld
;
134
A. Kechich et al
b1 =
R
Lq
; b2 = p
Ld
Lq
;
b3 = p
ϕf
Lq
c3 =
1
Lq
;
b4 =
Fc
J
; c4 =
; c1 = p
1
J
Ld − Lq
J
;
c2 = p
ϕf
J
;
.
Synthesis of the Regulators
(Strategy of adjustment on three surfaces)
The technique of the sliding-modes requires a choice of surfaces which ensures the
constraints of the order. The error of the adjustment is selected like a surface because,
on one hand, the continuation speed is imposed by the order i ∗ q and in the other
hand the surface S (Ω ) must be of a relative degree of order 1. [ 4][ 5 ]. The error is
defined as follows: S (Ω ) = Ω ref − Ω
The new technique requires that surface S (Ω ) should be invariant and
gravitational. This solution is proposed by the following function of Lyapunov:
v1 =
1 2
S (Ω )
2
The derivative is thus:
v1 = S (Ω ) S (Ω )
( 12 )
To meet the requirement imposed before, one imposes the following dynamics
on S (Ω ) :
S (Ω ) = − K 1 SignS (Ω )
( 13 )
With :
K 1 : Positive constant
Thus the surface S (Ω ) converges asymptotically towards the zero value, in the
same way the speed Ω converges towards its reference Ω ref since the temporal
derivative of surface S (Ω ) is given by the following equation:
S (Ω ) = Ω
ref + c3 Ω + c 4 C r − (c1i d + c 2 )i q
( 14 )
While imposing on S (Ω ) the suggested dynamics, the exit of the speed regulator
will be thus:
iqref =
Ω
ref + c 3 Ω + c 4 C r + K 1 SignS (Ω )
c1id + c 2
( 15 )
Since the current i qref exists the denominator need be different from zero.
i qref ≠ 0 ⇒ iqref ≠ −
c2
c1
The condition (16) is always realised because the current is maintained null.
( 16 )
Permanent Magnet Synchronous Machine
135
The strategy of the adjustment is illustrated by the diagram described by the figure
(9). The nonlinear adjustment by sliding-mode uses, in this case, the method of
adjustment in cascade. The new structure includes a loop of speed regulation. The
latter generates the reference of the current i * q which imposes the order v * q . The
order v * d is imposed by the regulation of current id .
E
Ωr e f
S (Ω)
S (i q
i d r ef
Ω
v ∗q
)
dq
v ∗a
v ∗b
S (i d
)
v ∗d
v ∗c
va
vb
O nd .
à
M .L .I
vc
abc
M SAP
θ
abc
ic
ib
ia
iq
dq
Figure 9: Total diagram of the adjustment by slipping mode using the strategy of
three surfaces
Simulation
For the validation of the structure of the order suggested, one simulated the strategies
of order of three surfaces. The simulation is made in the following way:
One made a loadless starting with an instruction of 200 [Rad/s] with application of
the nominal load between 1 and 2 [sec] then an inversion of the rotation direction of
(200 to -200) [Rad/s].
One notes:
•
The rejection of disturbance is very fast;
•
A very weak response time;
•
A practically null static error;
•
A decoupling excel for the technique suggested.
The curves prove that speed follows its reference in the two directions of rotation.
The stator current remains in an acceptable interval.
Between t = 1 [s] and 4 [s], speed remains insensitive with the variations of the
parameters and the disturbances due to the application of the nominal couple.
For a performance evaluation, of our machine, with the technique of order by the
sliding mode, one did a robustness test of this adjustment vis-à-vis the machine
parameters. One increases resistance of 200% and a reduction of 80% of inductances.
As for constant mechanics, it is increased to 80%. See figure (11).
136
A. Kechich et al
It is noted that the speed of our machine remains practically insensitive with the
variations caused by the disturbances made, what confirms the robustness of the
adjustment suggested.
Conclusion
We presented, in the first part, a method of analysis and synthesis of the nonlinear law
of order by the continuation of a basic discontinuous model. The order uses the
method of Lyapunov. The second part was devoted to the application of the various
algorithms to the order of the synchronous machine to permanent magnet. The
nonlinear order by sliding mode using vectorial decoupling and nonlinear decoupling
shows its effectiveness even if the parameters (mechanical and electric) of the
machine undergo variations. Various simulations made show that the nonlinear
system of regulation gives very good performances. The obtained results, enables us
to conclude that our system of regulation proposed is robust even if the disturbances
are not known. One also adds that the regulation suggested can be applied in fields
requiring high performances such as the field of robotics, the field of the machine
tools...
Results of Simulation
10
200
8
150
6
2
Cem [rad/sec]
4
50
ωr [rad/sec]
100
0
-50
0
-2
-4
-100
-6
-150
-8
-200
0
1
2
3
temps [Sec]
4
5
-10
6
0
1
2
3
temps [Sec]
4
5
6
15
20
15
10
10
5
Ias [A]
Ids [A]
5
0
0
-5
-5
-10
-15
-10
-20
-25
-15
0
1
2
3
temps [Sec]
4
5
6
0
1
2
3
temps [Sec]
4
Figure 10: Results of simulation of the adjustment by sliding-mode.
5
6
Permanent Magnet Synchronous Machine
137
200.3
200.25
Wr [rad/sec]
200.2
200.15
200.1
200.05
200
199.95
199.9
0.998
0.999
1
1.001
1.002
temps [Sec]
1.003
1.004
200
ωr [rad/sec]
150
100
50
a
0
0
0.2
0.4
0.6
0.8
1
temps [Sec]
1.2
1.4
1.6
1.4
1.6
1.8
2
200
ωr [rad/sec]
150
100
50
b
0
0
0.2
0.4
0.6
0.8
1
temps [Sec]
1.2
1.8
2
200
180
160
ωr [rad/sec]
140
120
100
80
60
40
c
20
0
0
0.2
0.4
0.6
0.8
1
temps [Sec]
1.2
1.4
1.6
1.8
2
Figure 11: Results of simulation of the adjustment by sliding-mode during variation
parametric; a- increase of 50%, 100% and 150% of RS; b- reduction 200% of L ; cincrease of 80% of J.
138
A. Kechich et al
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