Design of a Fuzzy Sliding Mode Controller by Genetic

International Journal of Emerging Electric Power Systems Volume 1, Issue 2

2004

Article 1008

Design of a Fuzzy Sliding Mode Controller by Genetic Algorithms for Induction Machine Speed Control Abdeldjebar Hazzab∗

Ismail Khalil Bousserhane†

Mokhtar Kamli‡

∗

University Center of Bechar, Algeria, a [email protected] University Center of Bechar, Algeria, bou [email protected] ‡ University of Sciences and Technology Of Oran, Algeria, [email protected] †

c Copyright 2004 by the authors. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, bepress, which has been given certain exclusive rights by the author. International Journal of Emerging Electric Power Systems is produced by The Berkeley Electronic Press (bepress). http://www.bepress.com/ijeeps

Design of a Fuzzy Sliding Mode Controller by Genetic Algorithms for Induction Machine Speed Control Abdeldjebar Hazzab, Ismail Khalil Bousserhane, and Mokhtar Kamli

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. KEYWORDS: Fuzzy Control, Sliding Mode Control, Genetic Algorithm, Vector Control, Induction Motor.

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1. INTRODUCTION Sliding mode control (SMC) has for long been known for its capabilities in accounting for modeling imprecision and bounded disturbances. It achieves robust control by adding a discontinuous control signal across the sliding surface, satisfying the sliding condition. 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 [1, 2]. Fuzzy logic control (FLC) has excelled in dealing with systems that are complex, ill-defined, non-linear, or time-varying [3]. FLC is relatively easy to implement, as it usually needs no mathematical model of the controlled system. This is achieved by converting the linguistic control strategy of human experience or experts' knowledge into an automatic control strategy. 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. Ng et al. [11] are developed a FSM controller where the control gain k of the switching signal ( −k ⋅ sgn ( s φ ) ) is adapted by using a fuzzy inference which have an optimized parameters by GA (scaling factors, membership functions, rule base). They are presented in this practical approach a controller SMC whose equivalent component is similar of an adaptive controller with structure PID [13].

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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. 2. INDUCTION MOTOR MODEL Fig. 1 bellow gives three different reference frames: stator reference frame ( α − β ), rotor reference frame (D-Q) and arbitrary reference frame (d-q). q

Q

ωe d

β

 is

 ψr θd

ωr D

ξ

Stator direct-axis

α

Fig. 1: Reference frames and space vector representation The induction motor mathematical model, in space vector notation [6, 7, 8], established in d-q co-ordinate system rotating at speed ωe is given by the following equations.   dψ s  (1) Vs = Rs is + + jωeψ s dt  dψ r  (2) 0 = Rr ir + + j (ωe − ωr )ψ r dt The stator and the rotor fluxes are given by:    ψ s = Ls is + Lm ir (3)    ψ r = Lm is + Lr ir (4) The produced electromagnetic torque is given by: 3 pLm   Te = (ψ r ⊗ is ) (5) 2 Lr Using the d-q co-ordinate system, as illustrated in fig. 1, and separating the machine variables state vectors into their real and imaginary parts, the wellknown induction motor model expressed in terms of the state variables is obtained from (1)-(5).

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This model is given by equation (6):  disd  R Lm Lω 1- σ  1 = - s + ψ rd + m r ψ rq + Vsd   isd + ωeisq + σ Ls Lrτ r σ Ls Lr σ Ls  σ Ls στ r   dt  di  sq = -ωe I sd -  Rs + 1- σ  I sq - Lmωr ψ rd + Lm ψ rq + 1 Vsq  dt σ Ls Lr σ Ls Lrτ r σ Ls  σ Ls στ r   1  dψ rd Lm = isd - ψ rd + (ωe - ωr )ψ rq  τr τr  dt  dψ rq L 1  = m isq - (ωe - ωr )ψ rd - ψ rq τr τr  dt  2  dωr = p Lm ( isqψ rd - isdψ rq ) - f c ωr - p Tl  dt JLr J J  Where σ is the coefficient of dispersion and is given by (7): L2m σ = 1− Ls Lr Ls , Lr , Lm stator, rotor and mutual inductances; Rs , Rr stator and rotor resistances; ωe , ω r electrical and rotor angular frequency; ωsl

(6)

(7)

slip frequency (ωe − ωr ) ;

τr

rotor time constant ( Lr / Rr ) ;

p

pole pairs

3. INDIRECT FIELD-ORIENTED CONTROL OF AN INDUCTION MOTOR

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 space vector. Under this condition we have: ψ rq* = 0 and 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 (8) Te = 2 Lr

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And the slip frequency can be given as follow: * 1 isq ω sl = (9) τ r isd* Consequently, the dynamic equations (6) yield:  disd  R Lm 1- σ  1 = - s + ψ rd + Vsd   isd + ωeisq + σ Ls Lrτ r σ Ls  σ Ls στ r   dt  di  sq = -ωeisd -  Rs + 1- σ  isq - Lmωr ψ rd + 1 Vsq  dt σ Ls Lr σ Ls  σ Ls στ r  (10)  L ψ d 1  r m  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 The decoupling control method with compensation is to choose inverter output voltages such that: 1  (11) Vsd* =  K p + K i  ( isd* − isd ) − ωeσ Ls isq* s  L 1  (12) Vsq* =  K p + K i  ( isq* − isq ) + ωeσ Ls isd* + ωe m ψ rd s Lr  Fig. 2 shows the implemented diagram of an induction motor indirect fieldoriented control (IFOC)[4,6]. Lf PWM Inverter

Cf

PARK-1

IFOC: Indirect Field Oriented Control

* sq

i

+ -

KP

1 + Ki s

IM

V

+

* sq

PARK

ω

* e

-

isq isd

ω eσ L s

ω eσ Ls ψ rd*

1 Lm

i sd*

-

K

P

+ Ki

Slip calc.

1 s

ω sl*

++

ψ rd

V sd*

+ ωe

Lm Lr

ω e* +

+

Fig. 2: bloc diagram of IFOC for an induction motor.

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ωr

Hazzab et al.: Design of a Fuzzy Sliding Mode Controller by Genetic Algorithms f

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4. 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, 11]: u = − k ⋅ sgn( s ) + ueq (13)

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 [1,11]: s = eD + λ e (14) * * 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 (15) 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: ssD ≤ −η s (16) 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]: u = us + ueq (17) 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

s ≤1 φ

s if >1 φ

(18)

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The function between us and s / φ is shown in the fig. 3: k

k/2 0 -k/2

-k

−1.5φ

−φ

−0.5φ

0 0.5φ

φ

1.5φ

Fig. 3: The first part (us) of the SMC This controller is actually a continuous approximation of the ideal relay control [1, 5, 11]. 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. 5. 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: Ri : if s is Ai then us is Bi , i = 1,…,5 (19) 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  Ri in Y, can be determined by each Ri of (19). We use the sup-min compositional rule of inference [2,5]:

( (

(

µ A Ri = sup min µ A ( s ), min µ Ai ( s ), µ Bi (us ) s∈ X

)))

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(20)

Hazzab et al.: Design of a Fuzzy Sliding Mode Controller by Genetic Algorithms f

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µ BN

MN

JZ

BP

MP

0 φ 2 φ −φ 2 Fig. 4: The input membership function of the FSMC −φ

s

µ SMALL MEDIUM BIG

SMALLER

BIGGER

k2 3k 2 us 0 −k k −k 2 Fig. 5: The output membership function of FSMC

− 3k 2

Fig. 6 is the result of defuzzified output us for a fuzzy input s. k

k/2 0 -k/2

-k

−1.5φ

−φ

−0.5φ

0 0.5φ

φ

1.5φ

Fig. 6: The control signal of fuzzy sliding mode controller 6. GENETIC ALGORITHMS APPLIED TO FSMC DESIGN

6.1. The Genetic Algorithms GA [11] are search algorithms that use operations found in natural genetic to guide through a search space. GA use a direct analogy of behavior. They work with a population of chromosomes, each one representing a possible solution to a

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given problem. Each chromosome has assigned a fitness score according to how good solution to the problem it is. GA are theoretically and empirically proven to provide robust search in complex spaces, giving a valid approach to problem requiring efficient and effective searching [13, 14]. Any GA starts with a population of randomly generated solutions, chromosomes, and advances toward better solutions by applying genetic operators, modeled on the genetic processes occurring in nature. In these algorithms we maintain a population of solutions for a given problem; this population undergoes evolution in a form of natural selection. In each generation, relatively good solutions reproduce to give offspring that replace the relatively bad solutions which die. An evaluation or fitness function plays the role of the environment to distinguish between good and bad solutions. The process of going from the current population to the next population constitutes in the execution of GA. Although there are many possible variants of simple GA, the fundamental underlying mechanism operates on a population of chromosomes and consists of three operations: • Evaluation of individual fitness, • Formation of gene pool (intermediate population) • Recombination and mutation. The next procedure shows the structure of a simple GA [13, 14]. Structure of standard genetic algorithm Begin (1) t=1 Initialize Population(t) Evaluate fitness Population(t) While (Generations < Total Number) do Begin (2) Select Population(t+1) out of Population(t) Apply Crossover on Population(t+1) Apply Mutation on Population(t+1) Evaluate fitness Population(t+1) t=t+1 End (2) End (1) A fitness function must be devised for each problem to be solved. Given a particular chromosome, a solution, the fitness function returns a single numerical fitness, which is supposed to be proportional to the utility or adaptation of the individual which that chromosome represents. There are a number of ways of making this selection. We might view the population as mapping onto a roulette wheel, where each chromosome is

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represented by a space that proportionally corresponds to its fitness. By repeatedly spinning the roulette wheel, chromosomes are chosen using stochastic sampling with replacement to fill the intermediate population. The selection procedure proposed in [12], and called stochastic universal sampling is one of the most efficient, where the number of offspring of any structure is bound by the floor and ceiling of the expected number of offspring. After selection has been carried out the construction of the intermediate population is complete, then the genetic operators, crossover and mutation, can occur. A crossover operator combines the features of two parent structures to form two similar offspring. It is applied with a probability of performance, the crossover probability (Pc). A mutation operator arbitrary alters one or more components of a selected structure so as to increase the structural variability of the population. Each position of each solution vector in the population undergoes a random change according to a probability defined by a mutation rate, the mutation probability (Pm). 6.2. Design of fuzzy-genetic system Different approaches have been proposed to automate the design of fuzzy systems [3, 13, 14]. 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

b

a

a

b

c

x

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-anderror) and one input (command action) variables. A chromosome is formed from

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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 application of the GA in the optimization of the FL controllers can be reformulated as follows: 1. Start with an initial population of solutions that constitutes the first generation (P(0)). 2. evaluate P(0): a) Take each chromosome (KB) from the population and introduce it into the FLC, b) Apply the FLC to the controlled system for an adequate evaluation period, c) Evaluate the behavior of the controlled system by producing a performance index to the KB. 3. While the termination condition is not met, do a) create a new generation (P(t+1)) by applying the evolution operators (selection, crossover and mutation) to the individuals in P(t), b) Evaluate P(t+1) c) t = t+1. 4. End. The mechanism of the optimization can be represented in fig. 8. Evolution operators

Population of knowledge Bases Know1 Base

Fuzzy Controller

Input Scaling

FC inputs System status and outputs

Inference Engine

Known Base

KB under evaluation

Knowledge Base

Fuzzification

Know2 Base

Defuzzification

Controlled System

Output Scaling

Performance index

FC Outputs System Outputs

System status and outputs

Fig. 8: Evolutionary learning of an FLC

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Evaluation System

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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. 7. DESIGN OF THE FUZZY SLIDING MODE CONTROLLER ALGORITHMS

BY

GENETIC

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, 11]. In GA, we only need to select some suitable parameters, such as generations, population size, crossover rate, mutation rate, and coding length of chromosome [11, 13], 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: (1) Initial population size: 30; (2) Maximum number of generation: 100; (3) Crossover: Uniform crossover with probability 0.8; (4) Mutation probability: 0.01. In this paper, the performance is measured using the following criteria. (5) Minimum integral of squared which is given as follows: t

t

J = ∫ e2 dt = ∫ (ωr* − ωr ) dt 0

2

(19)

0

8. 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

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(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. L f

PWM Inverter

Cf

FSMC: FSM Speed

ω

* r

+ -

S (e)

s

isqeq

FLC

++

isqs

PARK-1

IFOC: Indirect Field Oriented Control

isq* -

1 K P + Ki s

* sq

V

+

IM

PARK

ω

* e

+ ωeσ Ls ωeσ Ls

ψ rd* 1 Lm

isd* -

1 K P + Ki s

Vsd*

+

ψ rd

+

ωe

Lm Lr

Slip ωsl calc *

ωe* +

+

ωr

Optimization Mechanism by Genetic Algorithm

Fig. 9: Optimized Fuzzy sliding mode control of IM.

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isq isd

13

Te [N.m]

ωr [rad/sec]

Hazzab et al.: Design of a Fuzzy Sliding Mode Controller by Genetic Algorithms f

Time [sec]

Time [sec]

Torque [N.m]

Ia [A]

ψrd and ψrq [Wb]

Rotor speed [rad/sec]

Time [sec]

ψ rd and ψ rq [Wb]

Time [sec]

Phase current ia [A]

Fig. 10: Simulated results of fuzzy sliding mode control (FSMC) of IM

Fig. 11: Simulated results comparison of sliding mode control (SMC) and fuzzy sliding mode control (FSMC) of IM.

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Fig. 12a: membership functions of s and isqs before optimization

Fig. 12b: membership functions of s and isqs after optimization

Rule5

Rules evolution

Rule4

Rule3

Rule2 Rule1

Fig. 13: Rule base evaluation during the optimization

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ωr [rad/sec]

Te [N.m]

Hazzab et al.: Design of a Fuzzy Sliding Mode Controller by Genetic Algorithms f

Time [sec]

Time [sec]

Torque [N.m]

Ia [A]

ψrd and ψrq [Wb]

Rotor speed [rad/sec]

Time [sec]

ψ rd and ψ rq [Wb]

Time [sec]

Phase current ia [A]

Fig. 14: Simulated results of fuzzy sliding mode control optimized by GA (FSMC+Ga) of IM.

Fig. 15: Simulated results comparison of fuzzy sliding mode control (FSMC) and (FSMC+Ga) of IM.

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9. 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 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 sliding-mode 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. 10. APPENDIX

Induction motor parameters: 1.5 Ian [A] 6.31 Pn [kW] Vn [V] 220 Rs [Ω] 4.85 0.78 Rr [Ω] 3.805 η 0.8 Lr [H] 0.274 Cosϕn -1 ωn[min ] 1428 Lm [H] 0.258

Ls [H] fn [Hz] Jn [kg/m2] fc [N.m.s/rd] p

0.274 50 0.031 0.008 2

11. REFERENCES

1. A. Derdiyok, M. K. Guven, Habib-Ur Rahman and N. Inane, “Design and Implementation of New Sliding-Mode Observer for Speed-Sensorless 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 Field-Oriented 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.).

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