Open access peer-reviewed chapter

Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors

By José Luis Azcue, Alfeu J. Sguarezi Filho and Ernesto Ruppert

Submitted: November 24th 2011Reviewed: May 15th 2012Published: September 27th 2012

DOI: 10.5772/48388

Downloaded: 3354

1. Introduction

The electric drives are very common in industrial applications because they provide high dynamic performance. Nowadays exist a wide variety of schemes to control the speed, the electromagnetic torque and stator flux of three-phase induction motors. However, control remains a challenging problem for industrial applications of high dynamic performance, because the induction motors exhibit significant nonlinearities. Moreover, many of the parameters vary with the operating conditions. Although the Field Oriented Control (FOC) [16] schemes are attractive, but suffer from a major disadvantage, because they are sensitive to motor parameter variations such as the rotor time constant, and an incorrect flux estimation at low speeds. Another popular scheme for electric drives is the direct torque control (DTC) scheme [15][8], and an another DTC scheme based on space vector modulation (SVM) technique that reduces the torque ripples. This scheme does not need current regulators because its control variables are the electromagnetic torque and the stator flux. In this chapter we use the DTC-SVM scheme to analyze the performance of our proposed fuzzy controllers.

In the last decade, there was an increasing interest in combining artificial intelligent control tools with conventional control techniques. The principal motivations for such a hybrid implementation were that fuzzy logic issues such as uncertainty (or unknown variations in plant parameters and structure) can be dealt with more effectively. Hence improving the robustness of the control system. Conventional controls are very stable and allow various design objectives such as steady state and transient characteristics of a closed loop system. Several [5][6] works contributed to the design of such hybrid control schemes.

However, fuzzy controllers, unlike conventional PI controllers do not necessarily require the accurate mathematic model of the process to be controlled; instead, it uses the experience and knowledge about the controlled process to construct the fuzzy rules base. The fuzzy logic controllers are a good alternative for motor control systems since they are well known for treating with uncertainties and imprecision�’s. For example, in [1] the PI and fuzzy logic controllers are used to control the load angle, which simplifies the induction motor drive system. In [7], the fuzzy controllers are used to dynamically obtain the reference voltage vector in terms of torque error, stator flux error and stator flux angle. In this case, both torque and stator flux ripples are remarkably reduced. In [10], the fuzzy PI speed controller has a better response for a wide range of motor speed and in [3] a fuzzy self-tuning controller is implemented in order to substitute the unique PI controller, present in the DTC-SVM scheme. In this case, performance measures such as settling time, rise time and ITAE index are lower than the DTC-SVM scheme with PI controller.

The fuzzy inference system can be used to modulate the stator voltage vector applied to the induction motor [18]. In this case, unlike the cases mentioned above, the quantity of available vectors are arbitrarily increased, allowing better performance of the control scheme and lower levels of ripple than the classic DTC. However, it requires the stator current as an additional input, increasing the number of input variables. In this chapter we design and analyze in details three kinds of fuzzy controllers: the PI fuzzy controller (PI-F), the PI-type fuzzy controller (PIF) and the self-tuning PI-type fuzzy controller (STPIF). All of these fuzzy controllers are applied to a direct torque control scheme with space vector modulation technique for three-phase induction motor. In this DTC-SVM scheme, the fuzzy controllers generate corrective control actions based on the real torque trend only while minimizing the torque error.

2. Background

2.1. The three-phase induction motor dynamical equations

By the definitions of the fluxes, currents and voltages space vectors, the dynamical equations of the three-phase induction motor in stationary reference frame can be put into the following mathematical form [17]:

us=Rsis+dψsdtE1
0=Rrir+dψrdt-jωrψrE2
ψs=Lsis+LmirE3
ψr=Lrir+LmisE4

Where usis the stator voltage space vector, isand irare the stator and rotor current space vectors, respectively, ψsand ψrare the stator and rotor flux space vectors, ωris the rotor angular speed, Rsand Rrare the stator and rotor resistances, Ls, Lrand Lmare the stator, rotor and mutual inductance respectively.

The electromagnetic torque teis expressed in terms of the cross-vectorial product of the stator and the rotor flux space vectors.

te=32PLmLrLsσψr×ψsE5
te=32PLmLrLsσψrψssin(γ)E6

Where γis the load angle between stator and rotor flux space vector, Pis a number of pole pairs and σ=1-Lm2/(LsLr)is the dispersion factor.

The three-phase induction motor model was implemented in MATLAB/Simulink as is shown in [2].

2.2. The principle of direct torque control

In the direct torque control if the sample time is short enough, such that the stator voltage space vector is imposed to the motor keeping the stator flux constant at the reference value. The rotor flux will become constant because it changes slower than the stator flux. The electromagnetic torque (6) can be quickly changed by changing the angle γin the desired direction. This angle γcan be easily changed when choosing the appropriate stator voltage space vector.

For simplicity, let us assume that the stator phase ohmic drop could be neglected in (1). Thereforedψs/dt=us. During a short timeΔt, when the voltage space vector is applied it has:

ΔψsusΔtE7

Thus the stator flux space vector moves by Δψsin the direction of the stator voltage space vector at a speed which is proportional to the magnitude of the stator voltage space vector. By selecting step-by-step the appropriate stator voltage vector, it is possible to change the stator flux in the required direction.

3. Direct torque control scheme with space vector modulation technique

In Fig. 1, we show the block diagram for the DTC-SVM scheme [14] with a Fuzzy Controller, the fuzzy controller will be substitute for the three kind of proposed Fuzzy Controller one for time. The DTC-SVM scheme is an alternative to the classical DTC schemes [15], [8] and [9]. In this one, the load angle γ*is not prefixed but it is determinate by the Fuzzy Controller. Equation (6) shows that the angle γ*determines the electromagnetic torque which is necessary to supply the load. The three proposed Fuzzy Controllers determine the load angle using the torque error eand the torque error changeΔe. Details about these controllers will be presented in the next section. Figure 1 shows the general block diagram of the DTC-SVM scheme, the inverter, the control signals for three-phase two-level inverter is generated by the DTC-SVM scheme.

Figure 1.

Direct torque control with space vector modulation scheme and fuzzy controller.

3.1. Flux reference calculation

In stationary reference frame, the stator flux reference ψs*can be decomposed in two perpendicular components ψds*andψqs*. Therefore, the output of the fuzzy controller γ*is added to rotor flux angle ψrin order to estimate the next angle of the stator flux reference.

In this chapter we consider the magnitude of stator flux reference as a constant. Therefore, we can use the relation presented in equation (8) to calculate the stator flux reference vector.

ψs*=|ψs*|cos(γ*+ψr)+j|ψs*|sin(γ*+ψr)E8

Moreover, if we consider the stator voltage usduring a short timeΔt, it is possible to reproduce a flux variationΔψs. Notice that the stator flux variation is nearly proportional to the stator voltage space vector as seen in the equation (7).

3.2. Stator voltage calculation

The stator voltage calculation uses the DC link voltage (Udc) and the inverter switch state (SWa, SWb,SWc) of the three-phase two level inverter. The stator voltage vector usis determined as in [4]:

us=23(SWa-SWb+SWc2)+j32(SWb-SWc)UdcE9
(9)

3.3. Electromagnetic torque and stator flux estimation

As drawn by Fig. 1 the electromagnetic torque and the stator flux estimation depend on the stator voltage and the stator current space vectors,

ψs=(us-Rsis)dtE10

The problem with this kind of estimation is that for low speeds the back electromotive force (emf) depends strongly of the stator resistance, to resolve this problem is used the current model to improve the flux estimation as in [13]. The rotor flux ψrdqrepresented in the rotor flux reference frame is given by:

ψrdq=Lm1+sTrisdq-j(ωψr-ωr)Tr1+sTrψrdqE11

Notice that Tr=Lr/Rris the rotor time constant, andψrq=0. Substituting this expression in the equation (11) yields:

ψrd=Lm1+sTrisdE12

In the current model the stator flux is represented by:

ψsi=LmLrψri+LsLr-Lm2LrisE13

Where ψriis the rotor flux according to the equation (12). Since the voltage model is based on equation (1), the stator flux in the stationary reference frame is given by

ψs=1s(vs-Rsis-Ucomp)E14

With the aim to correct the errors associated with the pure integration and the stator resistance measurement, the voltage model is adapted through the PI controller.

Ucomp=(Kp+Ki1s)(ψs-ψsi)E15
The Kpand Kicoefficients are calculated with the recommendation proposed in [13]. The rotor flux ψrin the stationary reference frame is calculated as:
ψr=LrLmψs-LsLr-Lm2LmisE16

The estimator scheme shown in the Fig. 2 works with a good performance in the wide range of speeds.

Figure 2.

Stator and rotor flux estimator, and electromagnetic torque estimator.

Where LPF means low pass filter. In the other hand, when equations (14) and (16) are replaced in (5) we can estimated the electromagnetic torque teas:

te=32PLmLrLsσψr×ψsE17

4. Design of fuzzy controllers

4.1. The PI fuzzy controller (PI-F)

The PI fuzzy controller combines two simple fuzzy controllers and a conventional PI controller. Note that fuzzy controllers are responsible for generating the PI parameters dynamically while considering only the torque error variations. The PI-F block diagram is shown in Fig. 3, this controller is composed of two scale factorsGe, GΔeat the input. The input for fuzzy controllers are the error (eN) and error change (eΔN), and theirs outputs represent the proportional gain Kpand the integral time Tirespectively. These parameters Kpand Tiare adjusted in real time by the fuzzy controllers. The gain KPis limited to the interval[Kp,min,Kp,max], which we determined by simulations. For convenience, KPis normalized in the range between zero and one through the following linear transformation.

Kp'=Kp-Kp,minKp,max-Kp,minE18

Then, considering that the fuzzy controller output is a normalized valueKp', we obtain Kpby:

Kp=(Kp,min-Kp,max)Kp'+Kp,minE19

However, for different reference values the range for the proportional gain values is chosen as[0,Kp,max],

Kp=Kp,maxKp'E20

Due to nonlinearities of the system and in order to avoid overshoots for large reference torquer, it is necessary to reduce the proportional gain. We use a gain coefficient ρ=1/(1+0002*r)that depends on the reference values. In order to achieves real time adjustment for the Kpvalues. Therefore, Kp,max=ρKp,max0where the value Kp,max0=1.24was obtained through various simulations. Note that both ρand Kp,maxdecreases as the reference value increases. Consequently, the gain Kpdecreases. The PI-F controller receives as input the torque error eand as output the motor load angleγ*.

Figure 3.

PI Fuzzy Controller block diagram.

4.1.1. Membership Functions (MF)

In the Fig. 3, the first fuzzy controller receives as inputs the errorseN,ΔeN, each of them has three fuzzy sets that are defined similarly, being only necessary to describe the fuzzy sets of the first input. The first input eNhas three fuzzy sets whose linguistic terms are N-Negative, ZE-Zero and P-Positive. Each fuzzy set has a membership function associated with it. In our particular case of, these fuzzy sets have trapezoidal and triangular shapes as shown in Fig.4. The universe of discourse of these sets is defined over the closed interval[-1.5,1.5].

Figure 4.

Membership functions for the inputseN,ΔeN.

The output has two fuzzy sets whose linguistic values are associated with them are S-small and B-Big, respectively. Both have trapezoidal membership functions as shown in Fig.5. The universe of discourse of the fuzzy sets is defined over the closed interval[-0.5,1.5]. The fuzzy controller uses: triangular norm, Mamdani implication, max-min aggregation method and the center of mass method for defuzzification [11].

Figure 5.

Membership functions for the first fuzzy controller outputKp.

Similarly, the second fuzzy controller has the same fuzzy sets for its two inputs, however, its output is defined by three constant values defined as1.5, 2and 3which linguistic values associated with them are S-Small, M-Medium and B-Big. This controller uses the zero-order Takagi-Sugeno model which simplifies the hardware design and is easy to introduce programmability [19]. The defuzzification method used for this controller is the weighted sum.

4.1.2. Scaling Factors (SF)

The PI-F controller has two scaling factors, Geand GΔeas inputs, while the fuzzy controllers outputs are the gain Kp'and the integral time TIrespectively. From these values we can calculate the parameterKI=Kp/TI.

The scale factor ensures that both inputs are within the universe of discourse previously defined. The parameters Kpand KIare the tuned parameters of the PI controller. The inputs are normalized, by:

eN=GeeE21
ΔeN=GΔeΔeE22

4.1.3. The rule bases

The rules are based on simulation that we conducted of various control schemes. Fig.6 shows an example for one possible response system. Initially, the error is positive around the pointa, and the error change is negative, then is imposed a large control signal in order to obtain a small rise time.

To produce a large signal control, the PI controller should have a large gain Kpand a large integral gain KI(small integral timeTI), therefore,

Rx:   if   eN   is   S   and   ΔeN   is   N   then   Kp   is   GE23

Figure 6.

Response system.

The rule base for the first fuzzy controller is in Table 1, also, the rule base for the second fuzzy controller is in Table 2.

eN/ΔeNNZEP
NBBB
ZESBS
PBBB

Table 1.

Rule base for fuzzy controller (I), output KË'p

eN/ΔeNNZEP
NSSS
ZEBMB
PSSS

Table 2.

Rule base for fuzzy controller (II), output TI

Fig. 7 shows the control surface for the first and second fuzzy controllers. This figure clearly shows the non-linear relationship between (e, Δe,Kp) and (e, Δe,TI)

Figure 7.

Control surface for: (a) surfaceU fuzzy controller (I) and (b) surfaceA fuzzy controller (II)

4.2. The PI-type fuzzy controller (PIF) and The self-tuning PI-type fuzzy controller (STPIF)

The PI-type fuzzy controller (PIF) is a fuzzy controller inspired by a digital PI controller, which is depicted in Fig. 8. It is composed by two input scale factors "Ge, GΔe" and one output scale factor "Gγ*". Finally it uses saturation block to limit the output.

This controller has a single input variable, which is the torque error "e" and one output variable which is the motor load angle "γ*" given by:

γ*(k)=γ*(k+1)+Δγ*(k)E24

In (23), kis the sampling time and Δγ*(k)represents the incremental change of the controller output. We wish to emphasize here that this accumulation (23) of the controller output takes place out of the fuzzy part of the controller and it does not influence the fuzzy rules.

Figure 8.

PI-type fuzzy controller.

Fig.9 shows the self-tuning PI-type fuzzy controller (STPIF) block diagram, its main difference with the PIF controller is the gain tuning fuzzy controller (GTF) block.

Figure 9.

Self-tuning PI-type fuzzy (STPIF) controller.

4.2.1. Membership Functions (MF)

The MF for PIF controller are shown in Fig. 10(a). This MF for input variables "eN, ΔeN" and output variable "ΔγN*" are normalized in the closed interval [-1,1].

The MF’s for GTF controller are shown in Fig. 10(a) and in Fig. 10(b) for input and output variables respectively. Input variables "eN, ΔeN" are defined in the closed interval [-1,1] and the output variable "α" is defined in the closed interval [0,1].

Most of the MF variables have triangular shape [Fig. 10] with 50% overlapping neighbor functions, except the extremes which are trapezoidal. The linguistic variables are referred to as: NL-Negative Large, NM-Negative Medium, NS-Negative Small, ZE-Zero, VS-Very Small, S-Small, SL-Small Large and so on as shown in Table 3 and in Table 4.

Figure 10.

Membership functions for: (a) membershipFPI error eN and change of error ΔeNis the same for PIF and GTF controllers as well as for the output ΔγN*of the PIF controller. (b)membershipAf the gain updating factor α

4.2.2. Scaling factors

The two inputs SF "GΔe, Ge" and the output SF "Gγ*" can be adjusted dynamically through updating the scaling factor "α". "α" is computed on-line, using a independent fuzzy rule model defined in terms of "e, Δe". The relationship between the SF and the input/output variables of the STPIF controller are shown bellow:

eN=GeeE25
ΔeN=GΔeΔeE26
Δγ*=(αGγ*)ΔγN*E27

4.2.3. The rule bases

The incremental change in the controller output ΔγN*to PIF controller is defined as,

Rx:   if   eN   is   E   and   ΔeN   is   ΔE   then   ΔγN*   is   ΔΓN*E28

Where E=ΔE=ΔΓN*={NL,NM,NS,ZE,PS,PM,PL}. The output αof the GTF controller is determined by the following rules:

Rx:   if   eN   is   E   and   ΔeN   is   ΔE   then   α   is   χE29

Where E=ΔE={NL,NM,NS,ZE,PS,PM,PL}andχ={ZE,VS,S,SL,ML,L,VL}. The rule base for ΔγN*and αare shown in Tab. 3 and Tab. 4 respectively.

ΔeN/eNNLNMNSZEPSPMPL
NLNLNLNLNMNSNSZE
NMNLNMNMNMNSZEPS
NSNLNMNSNSZEPSPM
ZENLNMNSZEPSPMPL
PSNMNSZEPSPSPMPL
PMNSZEPSPMPMPMPL
PLZEPSPSPMPLPLPL

Table 3.

Fuzzy rules for the computation of ΔγN*

ΔeN/eNNLNMNSZEPSPMPL
NLVLVLVLLSLSZE
NMVLVLLLMLSVS
NSVLMLLVLVSSVS
ZESSLMLZEMLSLS
PSVSSVSVLLMLVL
PMVSSMLLLVLVL
PLZESSLLVLVLVL

Table 4.

Fuzzy rules for the computation of α

4.2.4. Gain tuning fuzzy

The purpose of the GTF controller is update continuous the value of αin every sample time. The output αis necessary to control the percentage of the output SF "Gγ*", and therefore for calculating new "Δγ*",

Δγ*=(αGγ*)ΔγN*E30

The GTF controller rule base is based on knowledge about the three-phase IM control, using a DTC type control according to the scheme proposed in [14], in order to avoid large overshoot and undershoot, e.g., when "e" and "Δe" have different signs, it means that the estimate torque "te" is approaching to the torque reference "te*", then the output SF "Gγ*" must be reduced to a small value by "α", for instance, if "e" is "PM" and "Δe" is "NM" then "α" is "S".

On the other hand, when "e" and "Δe" have the same sign, it means that the torque estimate "te" is moving away from the torque reference "te*", the output SF "Gγ*" must be increased to a large value by "α" in order to avoid that the torque depart from the torque reference, e.g., if "e" is "PM" and "Δe" is "PM" then "α" is "VL".

The nonlinear relationship between "e,Δe,ΔγN*" and "e,Δe,α" are shown in Fig. 11.

Figure 11.

Surface of: (a) surfaceU PIF controller and (b) surfaceA GTF controller

The inference method used in PIF and GTF controllers is the Mamdani’s implication based on max-min aggregation. We use the center of area method for defuzzification.

5. Simulation results

We have conducted our simulation with MATLAB simulation package, which include Simulink block sets and fuzzy logic toolbox. The switching frequency of the pulse width modulation (PWM) inverter was set to be 10kHz, the stator reference flux considered was 0.47 Wb. In order to investigate the effectiveness of the three proposed fuzzy controllers applied in the DTC-SVM scheme we performed several tests.

We used different dynamic operating conditions such as: step change in the motor load (from 0 to 1.0 pu) at 90 percent of rated speed, no-load speed reversion (from 0.5 pu to -0.5 pu) and the application of a specific load torque profile at 90 percent of rated speed. The motor parameters used in the tests are given in Table 5.

Fig. 12, shows the response of the speed and electromagnetic torque when speed reversion for DTC-SVM with PI-F controller is applied. Here, the rotor speed changes its direction at about 1.8 seconds. Fig. 13 shows the stator and rotor current sinusoidal behavior when applying reversion.

Figure 12.

Speed reversion for DTC-SVM with PI-F controller.

Figure 13.

Stator and rotor current for speed reversion for DTC-SVM with PI-F controller.

Figure 14.

Sudden torque change for DTC-SVM whit PI-F controller

Figure 15.

Stator and rotor currents for sudden torque change for DTC-SVM with PI-F controller.

Fig. 14 and Fig. 15 show the torque and currents responses respecively, when step change is applied in the motor load for DTC-SVM with the PI-F controller. This speed test was established at 90 percent of rated speed.

In Fig. 16, we demonstrate the speed response when applying a speed reversion for DTC-SVM with PIF controller. In this case the speed of the rotor changes its direction at about 1.4 seconds. Fig. 17 shows the electromagnetic torque behavior when the reversion is applied.

Fig. 18 and Fig. 19 show the response of the electromagnetic torque and phase astator current respectively, when applying a step change in the motor load for DTC-SVM whit PIF controller. In this test the speed of the motor was set to 90 percent of rated speed.

Figure 16.

Speed reversion for DTC-SVM with PIF controller.

Figure 17.

Torque bahavior for speed reversion for DTC-SVM with PIF controller.

Figure 18.

Sudden torque change for DTC-SVM whit PIF controller.

Figure 19.

Phase astator current for sudden torque change for DTC-SVM with PIF controller.

Fig. 20 shows the behaviors of the electromagnetic torque, phase astator current and the motor speed, when applying speed reversion from 0.5 pu to -0.5 pu in the DTC-SVM scheme with STPIF controller. The sinusoidal waveform of the current shows that this control technique also leads to a good current control.

Fig. 21 presents the results when a specific torque profile is imposed to DTC-SVM scheme with STPIF controller. In this case the electromagnetic torque follow the reference.

Fig. 22 illustrates the response of the electromagnetic torque for the DTC-SVM scheme with STPIF controller, when applying step change in the motor load. In this test we used the rise timetr=5.49mS, the settling time ts=12mSand the integral-of-time multiplied by the absolute magnitude of the error indexITAE=199.5.

Figure 20.

Speed, torque and phase astator current for speed reversion for DTC-SVM with STPIF controller.

Figure 21.

Torque profile for DTC-SVM whit STPIF

Figure 22.

Step change in torque for DTC-SVM scheme with STPIF

Rated voltage (V)220/60Hz
Rated Power (W)2238
Rated Torque (Nm)11.9
Rated Speed (rad/s)179
Rs,Rr(Ω)0.435, 0.816
Lls,Llr(H)0.002, 0.002
Lm(H)0.0693
J(Kgm2)0.089
P2

Table 5.

Induction Motor Parameters [12]

6. Conclusion

In this chapter we have presented the DTC-SVM scheme that controls a three-phase IM using three different kinds of fuzzy controllers. These fuzzy controllers were used in order to determinate dynamically and on-line the load angle between stator and rotor flux vectors. Therefore, we determine the electromagnetic torque necessary to supply the motor load. We have conducted simulations with different operating conditions. Our simulation results show that the all proposed fuzzy controllers work appropriately and according to the schemes reported in the literature. However, the STPIF controller achieves a fast torque response and low torque ripple in a wide range of operating conditions such as: sudden change in the command speed and step change of the load.

Acknowledgement

The authors are grateful to FAPESP and CAPES for partially financial support.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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José Luis Azcue, Alfeu J. Sguarezi Filho and Ernesto Ruppert (September 27th 2012). Three Types of Fuzzy Controllers Applied in High-Performance Electric Drives for Three-Phase Induction Motors, Fuzzy Controllers - Recent Advances in Theory and Applications, Sohail Iqbal, Nora Boumella and Juan Carlos Figueroa Garcia, IntechOpen, DOI: 10.5772/48388. Available from:

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