A robust integrator algorithm with genetic based fuzzy controller feedback for direct vector control

A robust integrator algorithm with genetic based fuzzycontroller feedback for direct vector control

Erhan Akın, Mehmet Kaya, Mehmet Karakose *

Department of Computer Engineering, Firat University, Computer Engineering Institution, 23119 Elazig, Turkey

Received 5 April 2001; accepted 3 May 2001

Abstract

The voltage model used for direct vector control has in the flux calculation process an open integration

problem, which is generally solved with a feedback loop. In this paper, a new design method is developed

for the feedback loop of the integrator. The method, as apart from standards in the literature, uses a fuzzy

controller. Fuzzy controllers are knowledge-based systems that include fuzzy rules and fuzzy membership

functions to incorporate human knowledge into their knowledge base. The determination of these rules and

membership functions is one the key problems when designing fuzzy controllers, and is generally affected by

subjective decisions. In this study, a fuzzy controller with rules and membership functions determined by

genetic algorithms (GAs) in this study is designed and tested on various motors of different power ratings.The proposed method is simulated by using MATLAB/SIMULINK and implemented on an experimental

system using a TMS320C31 digital signal processor.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Vector control; Flux estimation; Fuzzy controller; Genetic algorithm; Induction motor

1. Introduction

In modern electrical drives, factors such as robustness, size, cost and reliability factors havehigh priority. Nowadays, these new concepts are supported by powerful DSPs, new power elec-tronics switching elements and artificial intelligence techniques including fuzzy logic, artificialneural networks and genetic algorithm. Using vector control is another important side of AC

Computers and Electrical Engineering 29 (2003) 379–394www.elsevier.com/locate/compeleceng

*Corresponding author.

E-mail addresses: [email protected] (E. Akın), [email protected] (M. Kaya), [email protected] (M.

Karakose).

0045-7906/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0045-7906(01)00041-6

mail to: [email protected]

motor control. In recent studies, many real-world applications using artificial intelligence tech-niques and vector control are presented in the literature [1].The performance of the vector control is related to the accuracy of the resultant flux phase and

magnitude information of the induction motor. There are two common methods to estimate thestator or rotor flux vector of the induction motor. These are called the current model and thevoltage model. The current model does not contain an open integration, but requires the rotorparameters and motor speed measurement during the flux calculation process. In this method, theflux vector is also estimated at standstill due to the lack of an open integration process. On theother hand, the voltage model is sensitive to stator parameters and requires voltage and currentmeasurements, but it does not require speed measurement. Therefore, in sensorless vector control,the voltage model is the preferred model. There are various methods of stator flux estimation usingthe voltage model. Rotor flux can also be estimated by using stator flux and motor parameters. Thestator flux equations known as the voltage model, in the stator reference frame, are as follows:

wsa ¼Z

ðusa � isaRsÞdt ð1aÞ

wsb ¼Z

ðusb � isbRsÞdt ð1bÞ

By using Eqs. (1a) and (1b), wsa and wsb can be estimated. The angle of the rotor flux orientationcan be calculated via rotor fluxes. This angle is used for axes transformation between ab and dq,and is called the transformation angle, hs.

hs ¼ arctgwrb

wra

ð2Þ

The block diagram in Fig. 1 shows the flux calculation process described in Eqs. (1a) and (1b).In this process, the integrator has no feedback. This open integration process causes some drift inthe integration. This integration process can be analog or digital. Digital flux integration dependson the integration method and integration step size, errors in the current and voltage measure-ments, variations in the stator resistance due to temperature and frequency of the current, initialvalue of the integrator, error introduced by the finite bit length of the processor, execution time ofthe commands etc.It is clear from the above open integrator that the performance of the voltage model depends on

the stator resistance. This resistance gains further importance in low-speed operation where the

Fig. 1. Block diagram of the voltage model.

380 E. Akın et al. / Computers and Electrical Engineering 29 (2003) 379–394

voltage drop on this resistance becomes significant. At zero speed, it is not possible to estimate thestator flux by using this algorithm.Numerous papers exist in the literature on the integration problem, especially sensorless direct

vector-controlled drives. For example, Ohtani et al. [2] proposed to add a feedback to the inte-gration algorithm to solve the problem. In Ohtani’s study, a steady-state analysis of the issue ispresented. In Ref. [3] another approach is studied. This is based on two-phase voltage and currentmeasurements, and stability analysis is made under dynamic conditions. In this study, flux mag-nitude and flux derivative are used in the feedback loop. However, the gain of the flux derivativefeedback is a function of the load and speed. Therefore, the performance of the system dependson the correct value of these feedback gains. A study by Bose and Patel [4] employs a cas-cade connected and automatically adjusted low-pass filter instead of the integrator. Kubota andMatsuse [5] used a stator voltage compensation method to eliminate the dc error output of theintegrator and for improving the control characteristics in the very low-speed region for vector-controlled drives. An algorithm proposed by Akın [6] eliminates the dc component of the inte-grator output, but it only works in the steady state.Some of the algorithms mentioned here are a d feedback integration algorithm to solve the

stability problem of the integrator and a PI feedback integration algorithm. The block diagramand transfer function of the d feedback integrator are given in Fig. 2 and Eq. (3) respectively.

F ðsÞ ¼ 1

sþ dð3Þ

In this integrator, the d feedback path is used for stability of the integrator. When the frequencyis high ðjw � dÞ, the integrator function behaves well. However, when jw becomes comparable tod, both magnitude and phase errors appear and affect the performance both at steady state and intransient conditions. The d feedback integrator, shown in Fig. 2, eliminates the offset problemsubstantially. However, integration output still involves phase shift and error in magnitude [7].The block diagram of the implemented direct vector-controlled drive is shown in Fig. 3. This

drive does not use any speed sensor. RS (stator resistance) variation is important, especially at lowspeed. This parameter can be easily adopted [8].In this study, a fuzzy controller is developed for the feedback loop of the integrator. This fuzzy

controller has the advantages of robustness, ease of design and good transient response. Fuzzycontrollers are being widely and successfully applied to different areas [9]. One of the mainproblems of a fuzzy controller design is the process of determining membership functions andfuzzy rules. Usually such a determination is made by human experts. However, if this is notpossible, or the obtained knowledge is not good enough the definition or the refinement of theknowledge requires a learning or adaptation process.

Fig. 2. Block diagram of d feedback integrator.

E. Akın et al. / Computers and Electrical Engineering 29 (2003) 379–394 381

GA was employed first by Karr [10] in designing fuzzy controllers. Karr has applied GA to thedesign of a fuzzy logic controller for the cart pole problem. Meredith et al. [11] has applied GA tothe fine tuning of membership functions in a fuzzy logic controller for a helicopter. Initial guessesfor the membership functions were made by a control engineer, and the GA adjusted the definingparameters through use in order to minimize the movement of a hovering helicopter. Triangularmembership functions were used. Lee and Takagi [12] have also tackled the cart problem. Theyhave taken a holistic approach by using GA to design the whole system determining the optimalnumber of rules as well as the membership functions, which are again triangular. Herrera et al.[13] have presented a new learning algorithm for fuzzy control rules using GAs. Recently, Arslanand Kaya [14] have proposed a new method for determination of fuzzy logic membership func-tions using GAs.The rest of the paper is organized as follows. Section 2 introduces some general ideas on GAs

and fuzzy controllers. The design of a fuzzy controller using GAs is presented in Section 3. Ex-perimental setup is contained in Section 4, to be used as a platform for experiments through thisstudy. The experiments and the results are given in Section 5. The final section is devoted toconclusions.

2. A brief overview of fuzzy controller and genetic algorithms

Fuzzy logic is a technology based on engineering experience and observations. In fuzzy logic,an exact mathematical model is not necessary, because linguistic variables are used to define systembehavior rapidly. It is a very recent technology relative to conventional controllers; its areas ofapplication are increasing very quickly. Fuzzy PID, fuzzy PI, fuzzy PD and fuzzy mixed con-trollers are fuzzy controller design approaches, but unlike conventional controllers the focus is notin the modeling [15].Some of the problems, such as stability and performance, are encountered both in fuzzy

controllers and conventional controllers. Unlike conventional control design, which uses math-

Fig. 3. Block diagram of direct vector control.


ematical models to solve these problems, fuzzy controller design involves IF–THEN rules definedby an expert to tackle these problems.There are two methods commonly used to design fuzzy controllers: the trial and error method

and the theoretical method. In trial and error, IF–THEN rules are defined by using expertknowledge and experience. Then, these rules are applied to the actual system. Unlike the theo-retical approach, where the parameters are adjusted to guarantee the desired performance, in thefuzzy method the IF–THEN rules are modified until the desired performance is achieved. Inpractice, both methods can be used to obtain better performance [16].The fuzzy controller has four components as shown in Fig. 4. These are:(a) Fuzzifier: the input values are scaled and grouped into fuzzy sets. In other words, the inputvalues are labeled and transformed into linguistic variables.(b) Inference mechanism: the inference mechanism uses a database and a rule base. The data-base involves membership functions that are used by the inference mechanism to make fuzzydecisions.(c) Rule base: the rule base is a set of IF–THEN rules defined by an expert. The inference mech-anism uses these rules.(d) Defuzzifier: the linguistic variables manipulated by the inference mechanism are convertedback to real values.In a fuzzy controller design, the knowledge and observations of an expert are more important

than the underlying mathematical model. This expert knowledge and observation is used while thesystem is being designed. This kind of approach provides an opportunity to easily embed into acontroller experience, which has been gained over a long time. However, automation is notpossible during controller design.A GA is an iterative procedure that consists of a constant-size population of individuals, each

one represented by a finite string of symbols, known as the genome, encoding a possible solutionin a given problem space. This space, referred to as the search space, comprises all possible so-lutions to the problem at hand. Generally speaking, the GA is applied to spaces that are too largeto be exhaustively searched. The standard GA proceeds as follows: an initial population of in-dividuals is generated at random or heuristically. In every evolutionary step, known as a gener-ation, the individuals in the current population are decoded and evaluated according to somepredefined quality criterion, referred to as the fitness function. To form a new populationindividuals are selected according to their fitness. Many selection procedures are currently in use,

Fig. 4. Block diagram of fuzzy control architecture.


one of the simplest being Holland’s original fitness-proportionate selection [17]. Selection alonecannot introduce any new individuals into the population. These are generated by geneticallyinspired operators, of which the best known are crossover and mutation.

3. Design of a fuzzy controller using genetic algorithms

In this paper, a fuzzy controller is used in the feedback loop of the integrator in the voltagemodel as shown in Fig. 5. The proposed fuzzy controller is based on rules and can be adopted todifferent machines easily. The membership functions of the fuzzy controller used are determinedusing GAs. Unlike conventional controllers, fuzzy controllers are less sensitive to sensor errorsand small variations of the parameters [18].A fuzzy control rule has the following structure.

If xi is Mia and . . . and xj is Mjb

Then yk is Mkq and . . . yl is Mlr;

where is xi is an input variable, Mia is a fuzzy set associated with this variable, yl is an outputvariable, and Mkq is a fuzzy set associated with this variable. All fuzzy inputs are connected by thefuzzy connective ‘‘and’’.In our systems, each rule generates a code of two strings of bits: one string of length Lx for the

antecedent (a bit for each possible linguistic term related to each input variable) and one string oflength Ly for the consequent.

Lx ¼Xn

i¼1Mi; Ly ¼

Xmk¼1

Nk ð4Þ

where n is the number of input variables, m is the number of output variables, Mi is the number oflinguistic terms associated with the ith input variable, and Nk is the number of linguistic termsassociated with the kth output variable.To encode the antecedent, it is started with a string of Lx bits, all with an initial value 0. If the

antecedent of the rule contains a fuzzy input like ‘‘xi is Mia’’, a 1 will replace the 0 at a certainposition (p) in the string:

Fig. 5. Block diagram of the genetic fuzzy controller for stator flux estimation.


p ¼ aþXi�1r¼1

Mr ð5Þ

This process is repeated for all the fuzzy inputs of the rule. It is important to point out that usingthis code, an input variable for which all the corresponding bits have the value 0 is an inputvariable whose value has no effect over the rule. The process to encode the consequent is quitesimilar to that described above, only replacing M with N .With this coding scheme, a fuzzy rule described by our system can have the following code:

00100000100000-100000000

composed of a substring of 7þ 7 bits, and a substring of 9 bits. Each rule of this fuzzy controller isrepresented by a string of 23 bits, where this fixed length has been obtained from the total numberof the membership functions of the input and output variables. While generating each population,only one 1 is allowed in each substring representing input or output variables, because all fuzzyinputs are connected by the fuzzy connective ‘‘and’’. The rule base contains a variable number ofrules, with a maximum of 49 (i.e., 7� 7). Then the rule base is encoded into a string with variablelength, composed of up to 49 substrings of 23 bits each.As shown in Fig. 5, the core of the estimation of the stator flux is the discrete integration of the

difference between ðusa � isaRsÞ and the feedback signal. Rs is assumed to be constant during thesimulation. In this figure, first the flux is compared to the zero reference in the feedback loop.Next, this difference and the derivative of the difference are given as inputs to the fuzzy logiccontroller’s tuned membership functions using GAs. Each variable of the fuzzy controller isrepresented by using seven membership functions at the input, as shown in Fig. 8a and b, and ninemembership functions at the output in Fig. 8c. Initially, the base values and intersection points arechosen randomly. The ranges of the input and output variables are assumed to be [�1, 1], [�1.5,1.5] and [�1.5, 1.5], respectively.GA process is executed in two steps. The first goal expected from the GA is to obtain the rule

base of the fuzzy controller. Its second goal is to find the base lengths and intersection points oftriangles corresponding to the rule base (Fig. 6).

Fig. 6. The base lengths of the membership functions for input variable ‘‘error’’.


Each base length has minimum and maximum values available. For example, the GA searchesbase value bNB between the minimum value of error (i.e., �1) and the maximum value of ac(i.e., 1). The search intervals of some of the base values and intersection points for variable errorare as follows:

bNB : �1; 1 ðminðerrorÞ �maxðerrorÞÞR1 : �1; 1 ðminðerrorÞ �maxðerrorÞÞbNM1 : �1;R1 ðminðerrorÞ � R1ÞbNM2 : R1; 1 ðR1 �maxðerrorÞÞ

These base values and intersection points must be reflected in the definite range of the system,because their values depend on bit length. This is formulated as follows:

b ¼ bmin þd

ð2L � 1Þ ðbmax � bminÞ ð6Þ

where d is the decimal value of a gene, L is the length of this gene, bmin is the minimum value of thearea reflected, and bmax is the maximum value of that area.Two different gene pools are created in the GA process. While one of these pools codes the base

lengths and intersection points of the input variables for the fuzzy system, the other pool does thatof the output. Each chromosome in a gene pool consists of the values of the bases and the in-tersection points. The length of chromosomes in each pool depends on the definite ranges of theinput and output variables. For example, the chromosome encoding the base lengths and theintersection points for the input variables of the fuzzy system consists of genes in the formbNBbNM1R1bNM2 . . . bPM1R5bPM2bPB.The bit length of each gene may be different. In this study, when the bit length is chosen, care

has been taken that the sensitivity should be between 0.2 and 0.3. For example, because the rangeis �1, 1, if the gene of bNB is represented by 3 bits, a sensitivity of 0.28 is achieved.ðð1=ð23 � 1ÞÞð1� ð�1ÞÞ ¼ 0:28Þ.This value is equal to the decimal value of a change in the smallest bit of gene bNB. The GA

process in each pool, therefore, includes the following steps:

1. specify string length ls and population size N ;2. evaluate each chromosome with respect to the fitness function;3. perform selection, crossover and mutation;4. if not (end-test) go to step (2), otherwise stop and return the best chromosome.

The fitness function of this procedure is calculated as follows:

Fitness function ¼ Max: error� Total error ð7ÞThe maximum error is made large enough to prevent the value of the fitness function from beingnegative. The maximum error is found as follows:

X9i¼1

ðactionGAi � 1:5Þ2 ð8Þ


From the equation above, the maximum error is equal to 20.25. Total error is calculated asfollows:

X9i¼1

ðactioni � actionGAiÞ2 ð9Þ

where actionGAi is the output found by the GA in the current cycle and actioni is the outputobtained in the previous cycle. To find the desired results, first of all, corresponding membershipfunctions are found for the ith value of the input variables. Then it is determined whether or notthese two membership functions may be put in a rule. If there is a rule between these twomembership functions, the grades of the membership functions are calculated and analyzed todetermine if the rule contains AND or OR. If these two membership functions are ANDed,outputs are determined from lower grades of membership functions, but if two membershipfunctions are ORed, outputs are determined from higher grades of membership functions.If there is more than one membership function intersecting with the ith input of any input

variable, then the outputs of these membership functions are both evaluated, and the one whichhas less error is used.The ith inputs intersect with NB and NM for error and with NB for cerror. After obtaining the

intersection situation from Fig. 7, grades of membership are determined for each membershipfunction. Then, from the rule bases that were obtained from both NB and NB, the output iscalculated from NVB. While doing calculations, l2 is taken as 0.2, because the rule involves AND(l2 < l1). Since there is also a rule between NM and NB, the output is calculated for l1 ¼ 0:1. Theoutput is calculated from both rules, and the one which has less error compared to the desiredoutput is used. This situation is depicted for three membership functions in Fig. 7. In other words,

if ½ðactioni � actionGA11Þ2 < ðactioni � actionGAi2Þ

2 then

actionGAi ¼ actionGAi1 else actionGAi ¼ wtGAi2

Two important points should be noted here. First, if the intersection between membershipfunction and reference input occurs on the left-hand side of the intersection point (R1) in a non-right triangle, and the output occurs in the range of a non-right triangle for the rule, then the

Fig. 7. The method of finding appropriate outputs of the GA for the ith inputs.


output is also taken from the left-hand side of the intersection point. The same rule is valid for theright-hand side as well. The second important point is that if there is no rule between any twomembership functions for input variables, the output is equal to zero. However, the rule base usedhere has complete rules.The fuzzy rule base and membership functions found by using GAs for this fuzzy controller is

shown in Fig. 8. For better precision, the number of membership functions can be increased at theexpense of computational cost.

Fig. 8. Membership functions and rule table of genetic based fuzzy controller. (a) Membership functions of input

variable ‘‘error’’ found by using GAs. (b) Membership functions of input variable ‘‘cerror’’ found by using GAs. (c)

Membership functions of output variable ‘‘action’’ found by using GAs. (d) Rule table of the fuzzy controller found by

using GAs.


4. Experimental setup

To verify the proposed compensation algorithm, an experimental setup with an inductionmotor drive has been constructed. Fig. 9 shows the block diagram of the drive system where aconventional field oriented control is implemented. An IGBT inverter, which is controlled byspace vector PWM (SVPWM), is used to drive the motor. The controller board is a DS1102 fromdSPACE GmbH [19]. The processor on the controller board is a Texas Instruments TMS320C3132-bit floating-point processor with a 60 ns instruction cycle. The DS1102 is also equipped witha four channel analog-to-digital converter (two 16-bit and two 12-bit channels), a four-channel12-bit DAC and two incremental encoders.In this experimental setup, LEM sensors are used to measure two-phase currents. The voltage

information is obtained from the inverter switching position including dead time effects. Thevector control algorithm has an execution cycle time of 250 ls. The new algorithm with the fuzzycontroller consumes 80 ls of execution time. In this experimental setup the speed control is im-plemented by using speed estimation. However, the speed estimation algorithm has been checkedwith the encoder output for confirmation.

5. Simulation and experimental results

Various simulations were carried out by using MATLAB/SIMULINK to assess the perfor-mance of the integrator with a fuzzy controller on the feedback. The fuzzy controller used forestimating the flux was developed with the MATLAB Fuzzy Toolbox. Simulations were per-formed to investigate transient state and steady state performance of the proposed flux estimator.In this way a general design procedure is presented for fuzzy controller feedback parameters.

The trial and error method is very time consuming.

Fig. 9. Block diagram of the experimental setup.


In the simulations, we tested the fuzzy controller by using two induction motors of differentpower ratings. The stator flux waveform of the fuzzy controlled integrator for a 3-HP motor isgiven in Fig. 10. In Fig. 11 experimental results of the fuzzy controller are given. The differencebetween Figs. 10 and 11 are negligible. The simulation results were obtained for the 3-HP motor

Fig. 10. Simulation results of the stator flux a–b components at transient state with the genetic based fuzzy controller

for a 3 hp motor.

Fig. 11. Experimental results of the stator flux a–b components at transient state with the genetically designed fuzzy

controller for the 3 hp motor.


at 5 Hz, and it has been shown that the fuzzy controller designed using the GA gives good results.The same simulations mentioned above were also performed for a 500-hp motor as shown in Fig.12.In the sensorless drive, the speed estimation depends on flux estimation. In this study, speed

estimation was realized using the following equations:

x̂xsðkÞ ¼bWWdsðk � 1Þ bWWqsðkÞ � bWWqsðk � 1Þ bWWdsðkÞ

T bWW2ds þ bWW2

qs

� � ð10Þ

xsl ¼1

sr

isqisd

ð11Þ

x̂xrðkÞ ¼ x̂xsðkÞ � x̂xsl ð12Þ

Fig. 13 depicts the estimated and measured results. These results are valid for 2 Hz using thevoltage model. At start up, a current model was used.The experimental results mentioned above were obtained with TRACE31TRACE31 software. Experiments

and simulations were also performed in different torque reference and load conditions for thethree motors used in the previous simulations. In these simulations the fuzzy controller performedbetter than other algorithm in most of the torque load conditions.The membership functions and the rule base of the fuzzy controller were determined by using a

GA in an offline run. Although the GA decreases a little the performance of the system, to the userdisappears in the system. The settling time of the induction motor at 500 Hz is as good as can beexpected, because the motor has so much inertia.

Fig. 12. Simulation results of the fuzzy controller at transient state for a 500 hp motor with great inertia.


The results obtained from the experimental setup are satisfactory for settling time of the statorflux at low speed. At startup, another flux model was used. At approximately 1 Hz, a passingalgorithm was used.The performance of the integrator algorithm is also affected by inverter switching control

techniques. In this study, bang–bang control was used first. Then SVPWM was preferred. The

Fig. 13. Experimental results of motor speed. (a) With speed estimation using flux calculation. (b) With encoder.

Table 1

The parameters of the motor used in the simulations and experiments

Motor parameters

Rated power (hp) 3 500

Rated line voltage (V) 220 2300

Number of poles 2 4

Stator resistance (X) 0.435 0.262

Referred rotor resistance (X) 0.816 0.187

Stator leakage reactance (X) 0.754 1.206

Rotor leakage reactance (X) 0.754 1.206

Stator magnetizing reactance (X) 26.13 54.02

Rotor inertia (kg/m2) 0.089 11.06


bang–bang control strategy can be applied simply, but it has some disadvantages. For example, ithas nonconstant switching frequency and cannot be implemented digitally.

6. Conclusions

In this paper, a digital integrator employing genetically designed fuzzy controller feedback ispresented to calculate the stator flux vector. The rule base and membership functions of the fuzzycontroller have been determined by using a GA. Implementation of the proposed algorithm hasbeen performed using a TMS320C31 processor. The most important advantage of this new systemis to provide a robust structure and simple design. Moreover, the proposed genetic fuzzy con-troller method has a shorter settling time than other integration methods. The experimental re-sults show that the proposed design procedure is more efficient and has less development timethan the other methods (Table 1).

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2001;118(2):297–306.

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Erhan Akın was born in Erzincan, Turkey, 1963. He received the BS and MS degrees in electrical engineering

from Firat University, Elazig, Turkey in 1984 and 1987 respectively, and the PhD degree in the area of ac

drives from Firat University, Elazig, Turkey in 1994. Since 1984, he has been with the Department of

Electrical Engineering, Firat University, Elazig, Turkey. His main research interest are in power electronics,

digital control of variable-speed ac drives and fuzzy control.

Mehmet Kaya received the BS degree in electrical and electronics engineering and the MS degree in computer

engineering, both from Firat University, Turkey in 1996 and 1998, respectively. He is currently a PhD student

in the Department of Electrical and Electronics Engineering at Firat University, Turkey. His research interests

include artificial intelligence, genetic algorithms, fuzzy systems, multi-agent systems and machine learning.

Mehmet Karak€oose was born in Elazi�gg, Turkey, 1976. He received the BS degree in electrical–electronic

engineering from Firat University, Elazig, Turkey in 1998. He is currently working toward MS degree in

Department of Computer Engineering, Firat University, Elazig, Turkey. His main research interest are in

fuzzy control and digital control of variable-speed ac drives.


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A robust integrator algorithm with genetic based fuzzy controller feedback for direct vector control

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