A New Fuzzy Sliding Mode Controller for Induction Motor Speed
Control.
A Hazzab (1), I.K. Bousserhane (1), M. Kamli (2) & M. Rahli (2)
(1) University Center of Bechar B.P 417 BECHAR (08000) ALGERIA
(2) University of Sciences and Technology of Oran, ORAN (31000) ALGERIA
E-mails: a_hazzab@yahoo.fr, bou_isma@yahoo.fr, kamlimk@hotmail.com and rahlim@yahoo.fr
Abstract— The drawbacks of sliding mode control in terms of
high control gains and chattering are overcome by merging of
the FLC with the variable structure of the SMC to form a fuzzy
sliding mode controller (FSMC). However, the major
drawback of fuzzy control is the lack of design techniques.
Hence this hybrid system increases the complexity in design
and, at present, there exists no effective design tools due to the
lack of analytical and numerical approaches. This paper
develops an automated design approach to this design problem,
using a genetic algorithm. The method is illustrated through
the design of a near-optimal fuzzy sliding mode controller for
induction motor speed control. The control strategy gives a
relatively low overshoots with smooth control action and
retains robustness of the sliding mode approach.
D
I. INTRODUCTION
Indirect field-oriented techniques are now widely used
for the control of induction motor in high-performance
applications. With this control strategy, the decoupled
control of IM is guaranteed, and can be controlled and
provide the same performance as achieved from a separately
excited DC machine. However, the control performances of
the resulted linear system are still influenced by the
uncertainties, which usually are composed of unpredictable
parameter variations, external load disturbances. Therefore,
in order to solve some of the problems of field-oriented
control, the motor drive must be techniques that are
appropriate to discontinuous operation of the switching
devices and allow the robustness of the algorithm, with
regard to changing parameters and external disturbances.
This common drawback can be overcome by using variable
structure control (VSC). The variable structure strategy
using the sliding mode (SMC) has been the focus of many
studies and research for the control of the AC drive. The
goal of the VSC is to constrain the system trajectory to the
sliding surface via the use of the appropriate switching logic.
The sliding mode control can offer good properties, such as
insensitivity to parameter variations, external disturbance
rejection, and fast dynamics response. However, in SMC,
the high frequency chattering phenomenon that results from
the discontinuous control action is a severe problem when
the state of the system is close to the sliding surface.
In various nonlinear system control issues, fuzzy controller
is recently a popular method to combine with sliding mode
control method that can improve some disadvantages in this
issue. Comparing with the classical control theory, the fuzzy
control theory does not pay much attention to the stability of
system, and the stability of the controlled system cannot be
so guaranteed. In fact, the stability is observed based on
following two assumptions: First, the input/output data and
system parameters must be crisply known. Second, the
system has to be known precisely. The fuzzy controller is
weaker in stability because it lacks a strict mathematics
model to demonstrate, although many researches show that
it can be stabilized anyway [3, 4]. Nevertheless, the concept
of a sliding mode controller (SMC) can be employed to be a
basis to ensure the stability of the controller. The feature of a
smooth control action of FLC can be used to overcome the
disadvantages of the SMC systems. This is achieved by
merging of the FLC with the variable structure of the SMC
to form a Fuzzy Sliding Mode Controller (FSMC) [2, 5]. In
this hybrid control system, the strength of the sliding mode
control lies in its ability to account for modeling imprecision
and external disturbances while the FLC provides better
damping and reduced chattering. However, the major
drawback of fuzzy control is the lack of design techniques.
Most of the fuzzy rules are human knowledge oriented and
hence rules will deviate from person to person in spite of the
same performance of the system. The selection of suitable
fuzzy rules, membership functions and their definitions
along the universe of discourse always involve a painstaking
trial-and-error process.
In this paper, the genetic algorithm (GA) is applied for the
automatic design of a fuzzy-sliding mode control system. In
this GA based approach, the genetic algorithm (GA) is
applied to determine the parameter set, consisting of the
width of boundary layer ( φ ) and control gain ( k ) of the
fuzzy sliding mode controller. A near-optimal fuzzy sliding
mode controller has been achieved, fulfilling the robustness
criteria specified in the sliding mode control and yielding a
high performance in implementation to induction motor
speed control.
II. INDIRECT FIELD-ORIENTED CONTROL OF THE IM
The induction motor model expressed in terms of the state
variables is given by equation (1):
⎧ disd
⎛ R
Lm
Lω
1- σ ⎞
1
ψ + m r ψ +
V
= -⎜ s +
⎪
⎟ isd + ωe isq +
σ Ls Lrτ r rd σ Ls Lr rq σ Ls sd
⎪ dt
⎝ σ Ls στ r ⎠
⎪
⎪ disq = -ω I - ⎛ Rs + 1- σ ⎞ I - Lmωr ψ + Lm ψ + 1 V
⎟ sq
e sd ⎜
⎪ dt
σ Ls Lr rd σ Ls Lrτ r rq σ Ls sq
⎝ σ Ls στ r ⎠
⎪
1
⎪ dψ rd Lm
isd - ψ rd + (ωe - ωr )ψ rq
=
⎨
τr
τr
⎪ dt
⎪ dψ
L
1
⎪ rq = m isq - (ωe - ωr )ψ rd - ψ rq
τr
τr
⎪ dt
⎪
2
⎪ d ωr = p Lm i ψ - i ψ - f c ω - p T
sq rd
sd rq
r
l
⎪ dt
JLr
J
J
⎩
(
Lf
Cf
(1)
)
Where σ is the coefficient of dispersion and is given by
(2):
L2
σ =1− m
Ls Lr
(2)
The main objective of the vector control of induction motors
is, as in DC machines, to independently control the torque
and the flux; this is done by using a d-q rotating reference
frame synchronously with the rotor flux space vector [6, 7]
as shown in fig. 1, the d axis is aligned with the rotor flux
*
= 0 and
space vector. Under this condition we have: ψ rq
ψ rd* = ψ r* .
For the ideal state decoupling the torque equation become
analogous to the dc machine as follows:
3 P ⋅ Lm ⋅ψ r
(3)
Te =
Lr
2
And the slip frequency can be given as follow:
*
1 isq
ωsl =
τ r isd*
(4)
Consequently, the dynamic equations (1) yield:
⎧ disd
⎛ R
Lm
1- σ ⎞
1
= -⎜ s +
Vsd
ψ rd +
⎪
⎟ isd + ωe isq +
dt
L
L
L
Ls
σ
στ
σ
τ
σ
r ⎠
s r r
⎪
⎝ s
⎪ di
⎪ sq = -ω i - ⎛ Rs + 1- σ ⎞ i - Lm ωr ψ + 1 V
⎟ sq
e sd ⎜
rd
sq
⎪⎪ dt
σ Ls Lr
σ Ls
⎝ σ Ls στ r ⎠
⎨
1
⎪ dψ r Lm
⎪ dt = τ isd - τ ψ rd
r
r
⎪
⎪ d ωr 3 p 2 Lm
f
p
=
isqψ rd* − c ωr − Tl
⎪
J
J
2 JLr
⎪⎩ dt
(5)
The decoupling control method with compensation is to
choose inverter output voltages such that:
1⎞
⎛
(6)
Vsd* = ⎜ K p + K i ⎟ isd* − isd − ωe σ Ls isq*
s⎠
⎝
(
)
L
1⎞
⎛
Vsq* = ⎜ K p + K i ⎟ isq* − isq + ωe σ Ls isd* + ωe m ψ rd (7)
s⎠
Lr
⎝
Fig. 2 shows the implemented diagram of an induction
motor indirect field-oriented control (IFOC) [4,6].
(
)
PWM
Inverter
PARK-1
IFOC: Indirect Field Oriented
isq*
+
-
KP + Ki
1
s
IM
V sq*
+
PARK
ω e*
-
isq isd
ω eσ L s
ψ rd*
ω eσ L s
1
Lm
i *sd
-
KP + Ki
1
s
Slip ω
calc.
++
ψ rd
V sd*
+
ωe
Lm
Lr
*
sl
ω e*
+
+
ωr
Fig. 1: bloc diagram of IFOC for the IM
III. SLIDING MODE CONTROL
A Sliding Mode Controller is a Variable Structure Controller
(VSC). Basically, a VSC includes several different
continuous functions that can map plant state to a control
surface, and the switching among different functions is
determined by plant state that is represented by a switching
function [2]. Without lost of generality, consider the design
of a sliding mode controller for the following second order
system: Here we assume b > 0 . u (t ) is the input to the
system. The following is a possible choice of the structure of
a sliding mode controller [1, 5, 9]:
u = − k ⋅ sgn( s) + ueq
(8)
Where ueq is called equivalent control which is used when
the system state is in the sliding mode [1]. k is a constant
and it is the maximal value of the controller output. s is
called switching function because the control action
switches its sign on the two sides of the switching surface
s = 0 . s is defined as [1,9]:
(9)
s = e& + λ e
Where e = x* − x and x* is the desired state. λ is a
constant. sgn( s ) is a sign function, which is defined as:
if s < 0
⎧−1
(10)
sgn( s ) = ⎨
if s > 0
⎩ 1
The control strategy adopted here will guarantee the system
trajectories move toward and stay on the sliding surface
s = 0 from any initial condition if the following condition
meets:
(11)
ss& ≤ −η s
Where η is a positive constant that guarantees the system
trajectories hit the sliding surface in finite time [1, 2, 5].
Using a sign function often causes chattering in practice.
One solution is to introduce a boundary layer around the
switch surface [5]:
(12)
u = us + ueq
Where: us = −k ⋅ sat( s / φ ) and constant factor φ defines
the thickness of the boundary layer. sat( s / φ ) is a
saturation function that is defined as:
⎧s
⎪
⎪φ
s
sat
φ =⎨
⎪sgn s
⎪
φ
⎩
( )
if
( )
if
The function between us and
k
s
φ
≤1
BN
(13)
s
MN
µ
JZ MP
BP
>1
φ
s / φ is shown in the fig. 3:
−φ
−φ 2
0
φ 2
s
φ
Fig. 4: The input membership function of the FSMC
µ
SMALLER SMALL MEDIUM BIG BIGGER
k/2
0
-k/2
-k
−1.5φ
−φ
−0.5φ 0 0.5φ
φ
1.5φ
us
− 3k 2 −k − k 2 0 k 2 k 3k 2
Fig. 5: The output membership
function of FSMC
Fig. 6 is the result of output us for a fuzzy input s:
k
Fig. 3: The first part (us) of the SMC
This controller is actually a continuous approximation of the
ideal relay control [1, 5, 9]. The consequence of this control
scheme is that invariance of sliding mode control is lost. The
system robustness is a function of the width of the boundary
layer.
k/2
0
-k/2
IV. FUZZY SLIDING MODE CONTROLLER
In this section, a fuzzy sliding surface is introduced to
develop a sliding mode controller. Which the expression
− k ⋅ sat( s / φ ) is replaced by an inference fuzzy system for
eliminate the chattering phenomenon.
The if-then rules of fuzzy sliding mode controller can be
described as [2, 5]:
R1 : if s is BN then us is BIGGER
R2 : if s is MN then us is BIG
R3 : if s is JZ then us is MEDUIM
R4 : if s is MP then us is SMALL
R5 : if s is BP then us is SMALLER
Where BN, MN, JZ, MP and BP are linguistic terms of
antecedent fuzzy set, they mean Big Negative, Medium
Negative, Just Zero, Medium Positive, and Big Positive,
respectively. We can use a general form to describe these
fuzzy rules:
(14)
Ri : if s is Ai then us is Bi , i = 1,…,5
Where Ai and Bi are a triangle-shaped fuzzy number, see fig.
4 and fig. 5.
Let X and Y be the input and output space, and A be an
arbitrary fuzzy set in X. Then a fuzzy set, A o Ri in Y, can be
determined by each Ri of (14). We use the sup-min
compositional rule of inference [2,5]:
( (
µ Ao R = sup min µ A ( s), min ( µ A ( s ), µ B (us ) )
i
s∈ X
i
i
))
(15)
-k
−φ
−1.5φ
−0.5φ
0 0.5φ
φ
1.5φ
Fig. 6: The control signal of fuzzy
sliding mode controller
.
V. GENETIC ALGORITHMS APPLIED TO FSMC DESIGN
Different approaches have been proposed to automate the
design of fuzzy systems [3, 11, 12]. Many of these
approaches take the genetic algorithm as a base of the
learning process. A GA was used to optimize the fuzzy logic
input membership functions, the fuzzy rules, the output
membership functions and universe of discourse [3, 4].
A. Membership parameters optimization
GA are applied to modify the membership functions. When
modifying the membership functions, these functions are
parameterized with one to four coefficients (fig. 7), and each
of these coefficients will constitute a gene of the
chromosome for the GA.
1
0
a b
c d
a
a
b
c
x
b
Fig. 7: Some parameterized membership functions
B. Fuzzy rule base optimization
Different methods are defined to apply GA to the rule base
optimization, depending on its representation. For example,
GA are used to modify the decision table of an FLC, which
is applied to control a system with two input (trial-and-error)
and one input (command action) variables. A chromosome is
formed from the decision table by going row-wise and
coding each output fuzzy set as an integer in 0, 1,…, n,
where n is the number of membership functions defined for
the output variable of the FLC. Value 0 indicates that there
is no output, and value k indicates that the output fuzzy set
has the k-th membership.
The mechanism of the optimization of the FL controllers can
be represented in fig. 8.
Population of knowledge Bases
Evolution
operators
Know1
Base
Fuzzy Controller
Input
Scaling
Fuzzification
FC inputs
System status
and outputs
Known
Base
KB under
evaluation
Knowledge
Base
Inference
Engine
Know2
Base
Defuzzification
Controlled
System
Output
Scaling
Performance
index
FC Outputs
System Outputs
System status
and outputs
Evaluation
System
Fig. 8: Evolutionary learning of an FLC
We propose a genetic learning method for the Data Base
(DB) of Mamdani fuzzy rule base system that allows us to
define:
• The numbers of labels for each linguistic variable.
• The universe of discourse.
• The form of each fuzzy membership function.
VI. DESIGN OF THE FUZZY SLIDING MODE CONTROLLER BY
GENETIC ALGORITHMS
In general, the performance of sliding mode controller is
influenced by two important factors: chattering phenomenon
and hitting time. The chattering phenomenon of sliding
mode controller usually occurs when the system state gets
close to the sliding surface, and it will affect the stability of
the controlled system. Furthermore, if we can shorten the
time that the state hit the sliding surface, the system with the
desired dynamic character will be faster, and it can also
decrease the uncertainty of the system. In order to improve
the performance of fuzzy sliding mode controller, we try to
adjust the parameters of input and output membership
functions and rule base of the FLC so we adjust indirectly
φ and k in the control law. We use GA to search the
appropriate values of the parameters of the FLC [3, 9].
In GA, we only need to select some suitable parameters,
such as generations, population size, crossover rate,
mutation rate, and coding length of chromosome [9, 11],
then the searching algorithm will search out a parameter set
to satisfy the designer's specification or the system
requirement. In this paper, GA will be included in the design
of sliding mode fuzzy controller.
The parameters for the GA simulation are set as follows:
Initial population size = 30, Maximum number of generation
= 100, Crossover is uniform crossover with probability = 0.8
and mutation probability = 0.01.
The fitness chosen as the minimum integral of squared
which is given as follows:
t
t
0
0
J = ∫ e 2 dt = ∫ (ωr* − ωr ) dt
2
(16)
1. Results and discussion
To prove the efficiency of the proposed method, we apply
the designed controller to the control of the induction motor.
The induction motor is a three phase, Y connected, four
pole, 1.5 kW, 1420min-1 220/380V, 50Hz. The
configuration of the overall control system is shown in fig.
9. It mainly consists of an induction motor, a ramp
comparison current-controlled pulse width modulated
(PWM) inverter, a slip angular speed estimator, an inverse
park, an outer speed feedback control loop and a fuzzy
sliding mode speed controller optimized by genetic
algorithm. The machine parameters are given in appendix.
Fig. 10 shows the disturbance rejection of FSMC controller
when the machine is operated at 200 [rad/sec] under no load
and a load disturbance torque (10 N.m) is suddenly applied
at 1sec, followed by a consign inversion (-200 rad/sec) at
2sec. The FSMC controller rejects the load disturbance very
rapidly with no overshoot and with a negligible steady state
error.
A comparison between the speed control of the IM by a
SMC and a FSMC is presented in fig. 11. This comparison
shows clearly that the FSMC gives good performances.
The same tests applied for FSMC no optimized are applied
with the FSMC optimized by the GA. The results of
membership functions optimisation of the input (s) and the
output (iqsn) is shown in fig. 12 and the rule base optimised
in
fig. 13. Fig. 14 shows the disturbance rejection of
FSMC controller optimised by GA when the machine is
operated at 200[rad/sec] under no load and a load
disturbance torque (10 N.m) is suddenly applied at 1sec,
followed by a consign inversion (-200 rad/sec) at 2sec. The
FSMC controller rejects the load disturbance very rapidly
with no overshoot and with a negligible steady state error.
This controller rejects the load disturbance very rapidly with
no overshoot and with a negligible steady state error more
than the FSMC which is shown clearly in fig. 15.
VII. CONCLUSION
This paper has reported the development of an automated
design approach to soft switched fuzzy sliding mode
controllers using a genetic algorithm. This controller has
been implemented for induction motor speed control.
Moreover, a GA is implemented for tuning of the fuzzy
system parameters. First, the dynamic response of the fuzzy
sliding mode controller was studied. It has been shown that
the proposed controller can provide the properties of
VIII. APPENDIX
ωn[tr/min]
0.258 P
1428
Lm [H]
Ls [H]
ƒn [Hz]
Jn [kg/m2]
ƒc [N.m.s/rd]
Time [sec]
0.274
50
0.031
0.008
2
Torque [N.m]
Ia [A]
6.31
4.85
3.805
0.274
Time [sec]
Rotor speed [rad/sec]
ψrd and ψrq [Wb]
Induction motor parameters:
1.5
Ian [A]
Pn [kW]
Vn [V]
220
Rs [Ω]
0.78
Rr [Ω]
η
0.8
L
Cosϕn
r [H]
Te [N.m]
ωr
insensitivity to uncertainties and external disturbance. Then,
the GA is designed to tuning the fuzzy parts of the fuzzy
sliding mode controller to enhance the control performance
of the induction motor. The theories of the fuzzy slidingmode controller and the implementation of the GA are
described in detail. Finally, the effectiveness of the proposed
controllers has been demonstrated by simulation and
successfully implemented in an induction motor drive.
REFERENCES
[1]
A. Derdiyok, M. K. Guven, Habib-Ur Rahman and N. Inane, Design
and Implementation of New Sliding-Mode Observer for SpeedSensorless Control of Induction Machine, IEEE Trans. on Industrial
Electronics, Vol. 1. N°3, 2002.
[2] Spyros G. Tzafestas and Gerosimos G. Rigatos, Design and stability
analysis of a new Sliding mode Fuzzy logic Controller of reduced
Complexity, Machine Intelligence & Robotic Control, Vol. 1 N°1,
1999.
[3] Kim Chwee NG and Yun LI, Design of sophisticated fuzzy logic
controllers using genetic algorithms, Proc. 3rd IEEE Int. Conf. on
Fuzzy Systems, Orlando, 1994.
[4] LI Zhen and Longya XU, On-Line Fuzzy Tuning of Indirect FieldOriented Induction Machine Drives, IEEE Trans. on Power
Electronics, Vol. 13 N°1, 1998.
[5] JI-CHANG LE and YA-HU KUO, Decoupled fuzzy sliding-mode
control, IEEE Trans. on Fuzzy Systems, Vol. 6 N°3, 1998.
[6] L. Baghli, Contribution to Induction Machine Control: Using Fuzzy
Logic, Neural Networks and Genetic Algorithms, Doctoral Thesis
Henri Poincaré University, January 1999. (Text in French.).
[7] R. D. Lorenz and D. B. Lawson, A Simplified Approach to
Continuous On-Line Tuning of Field-Oriented Induction Machine
Drives,
IEEE
Trans.
On
Industry
application,
Vol.
N°26, Issue/3, May-June 90.
[8] J.P. CARON and J.P. HAUTIER, Modeling and Control of the
Asynchronous Machine, Edition Technip., 1995. (Text in French).
[9] K.C. Ng, Y. Li, D.J. Murray-Smith and K.C. Sharman, Genetic
Algorithms Applied to Fuzzy Sliding Mode Controller Design, Proc.
1st IEE/IEEE Int. Conf. on GA in Eng. Syst., Innovation, Appl.
Scheffield 1995.
[10] D. E. Goldberg, Genetic algorithms, Exploration, Optimization and
Machine Learning, Addison-Wesley Edition, France, SA 1994. (Text
in French).
[11] J. R. Velasco and L. Magedalena, Genetic Algorithm in Fuzzy Control
Systems, Editor J. Periaux and G.Winter, Genetic algorithms in
engineering and computer science, John Wiley and Sons, 1995.
[12] O. Cordon and F. Herrera, A General Study on Genetic Fuzzy
Systems, Editor J. Periaux and G.Winter, M. Galan, P. Custa (Eds.),
Genetic algorithms in engineering and computer science, John Wiley
and Sons, 1995.
Time [sec]
ψ rd and ψ rq [Wb]
Time [sec]
Phase current ia [A]
Fig. 10: Simulated results of the FSM controller of IM
Fig. 11: Simulated results comparison between
the SM and FSM controllers of IM.
Fig. 12a: membership functions of s and isqs
before optimization
Fig. 12b: membership functions of s and isqs
after optimization
Rule5
Rules
Rule4
Rule3
Rule2
Rule1
Fig. 15: Simulated results of the comparison between
the FSMC and FSMC optimized by GA of IM.
Time [sec]
Ia [A]
ψrd and ψrq [Wb]
Te [N.m]
ωr [rad/sec]
Fig. 13: Rule base evaluation during the optimization
Time [sec]
Time [sec]
Time [sec]
Torque [N.m]
Rotor speed [rad/sec]
ψ rd and ψ rq [Wb]
Phase current ia [A]
Fig. 14: Simulated results of fuzzy sliding mode control optimized by GA (FSMC+Ga) of IM.
Lf
PWM
Inverter
Cf
FSMC: FSM Speed
ω
*
r
+
-
S (e)
s
FLC
isqeq
++
isqs
PARK-1
IFOC: Indirect Field Oriented Control
*
sq
i
-
K P + Ki
1
s
*
sq
V
+
IM
PARK
ωe*
+
ωeσ Ls
ωeσ Ls
ψ rd* 1
Lm
isd*
-
K P + Ki
1
s
Vsd*
+
ψ rd
+
ωe
Lm
Lr
Slip ωsl
calc
*
Optimization Mechanism
by Genetic Algorithm
Fig. 9: Optimized Fuzzy sliding mode control of IM.
ωe*
+
+
ω
r
isq isd