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An Efficient Direct Torque Control Based on Fuzzy Logic

Technique

Jibo Zhao1

and Hong Wang2

1Electrical Engineering and Computer Science Department, University of Toledo , Toledo, OH, USA2Engineering Technique Department, University of Toledo, Toledo, OH, USA

Abstract - Conventional Direct Torque Control (CDTC)

system of Induction Motor (IM) faces the problem of high

torque ripples, and has difficulty in improving the

performance of dynamic torque response and controlling flux

locus at very low speed. In this paper, a DTC control system

for induction motor based on Fuzzy Logic technology

(FLDTC) is proposed. The proposed system aims to make less

torque ripples, faster dynamic response, and higher

performance of flux control at very low speed by introducing

some new fuzzy variables and rescheduling the fuzzy switcherrules. The model of the proposed system is built on

simulink/matlab. Simulation results show that the proposed

technique FLDTC is more efficient than CDTC.

Keywords-Direct torque control, induction motor, fuzzy logic,

and fuzzy switcher rules.

1 IntroductionConventional direct torque control is a simple and

efficient control technique to provide quick torque and flux

control. The major advantages of direct torque control

technique are its simple structure and robust control schemewithout the complex mathematical transforms. However,

CDTC also has some drawbacks like high electromagnetic

torque ripple, high stator current distortion, relatively slow

transient response to torque step changes of load and flux

locus attenuation at very low speed [1-3].

To improve the performance of dynamic response of

CDTC, some studies have been carried out in the past [4] [5]

to increase the response speed of torque step change. One

research has developed a methodology of optimizing the

selection of the voltage vectors to give a maximum rate of

torque increase or decrease to meet the torque step change [6],

and dramatically improved the CDTC responding speed, butthe expense is the low performance of flux locus.

By introducing CDTC technique to induction motor, the

controller uses voltage vectors to control the flux or torque

according to three elements: hysteresis of flux error, torque

error and flux location. However, sometimes the flux locus or

the torque only needs the voltage vectors to last for a very

short time during a switching period in steady-state, or

perhaps it needs the control signal to last for several switching

period in dynamic-state. The hysteresis of CDTC system can

only judge the situation by positive and negative error values,

but it does not have the ability to adjust the flux and torque

according to exact error values. For the purpose of handling

this problem, a method to reduce torque ripple in DTC of

induction motor by using fuzzy mode duty cycle [7] is applied

to control the duty cycle of the switches according to the exact

value of the torque and flux errors and has successfully

decreased the torque ripples. On the other hand, lots of

attempts based on fuzzy logic technique are shown to be

efficient in many researches. For instance, in the fuzzy logic

control method proposed in [8] and [9], the fuzzy logic

controller can recognize how big the error is and makes an

optimal adjustment; Moreover, a stator resistance estimator

using fuzzy logic at low speed applied in [10], can help to

improve the performance of torque ripple by making the

mathematical model of CDTC more accurate. Some other

researches [11-13] also provide several useful applications of

fuzzy logic in DTC. All these methods have proven that fuzzy

logic technique can make great contributions to DTC.

This paper aimed to take advantage of fuzzy logic

technique to solve the problems mentioned above. A group of

new FL switcher rules will be introduced. This FL controller

can detect the steady and dynamic states of induction motor

automatically, and control the flux and torque with optimal

vectors according to fuzzy switcher rules. The FL controller

has also solved the problem of flux attenuation at very low

speed. Both the steady and dynamic performance of torque

error and torque response to step changes can be improved by

the proposed methodology. The simulation results of CDTC

and FLDTC will be studied and compared.

2 Proposed technology with fuzzy logicTo improve the performance of CDTC, we apply a

Mamdani-type fuzzy logic system based on DTC principles.

The torque hysteresis in CDTC is substituted by this FL-

controller. Different from commonly used controller, the

proposed FL-controller has six input variables: Torque error

(Te), flux error (Fe), flux position (SE), angle difference (A),

rotor speed (SP) and working state (WS).

The membership function of flux error (Fe) has four

fuzzy sets: negative (N), zero (Z), positive (P) and positive

large (PL). The fuzzy variable torque error (Te) is

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represented by five fuzzy sets: negative large (NL), negative

(N), zero (Z), positive (P) and positive large (PL). The fuzzy

membership function of sector (SE) which stands for the

location of flux is represented by six fuzzy sets: sector (1-6)

S1, S2, S3, S4, S5 and S6 as shown in Fig. 1. The

membership functions of the three fuzzy variables are shown

in Fig. 2 (a-c). The other three fuzzy variables will be

introduced separately in the following paragraphs.

S2

S1

S3

S6

S4

S5

Figure 1. Spatial distribution of six sectors

Flux error

(a). Fuzzy membership functions for Fe.

Torque error (N.m.)

(b). Fuzzy membership functions for Te.

Sectors

(c). Fuzzy membership functions for SE.

Figure 2. Fuzzy membership functions for Fe, Te and SE.

The electromagnetic torque of IM can be expressed as

follows,

3sin

2

n

e s r

PT

L (1)

where 2( ) /s rL L L M M (2)

s and

r are the stator and rotor flux vectors,

sL is the

stator inductance, rL is the rotor inductance, M is themagnetizing inductance, is the angle between the stator and

rotor fluxes andn

P is the number of the pole pairs.

When 90 ,3

2

n

e s r

PT

L (3)

If the system is ideal no-load, then the average torque is zero.

If assuming =s r , then we can get the maximum value of

dynamic torque:

2

max

3

2

n

e s

PT

L

(4)

Thus, the electromagnetic torque can be written as

max sine eT T (5)Because of the assumption that the IM is ideal no-load, the

average torque is zero. The torque ripple can be written as:

maxsin

e eT T (6)

Assuming the rotating speed of rotator flux is a short-term

constant value, then,

( )s r t (7)

In condition that is very small,

max( )

e e s r T T t (8)

The rotating speed ofs

when the reference torque is ideal

no-load can be expressed as:

2a

aT

(9)

aT is the time period for the stator flux. Merge (8) and (9), we

can get the increasing time of torque:

max (1 )

e a

ir

e

a

T Tt

T

(10)

Similarly, we can get the torque decreasing time:

max

e a

d

re

a

T Tt

T

(11)

Formula (11) can be written as:

max

2e

d

e r

Tt

T

(12)

From (12) we can get the conclusion that the time

required to decrease the torque gets longer when the rotating

speed is very low. It is clear that the torque decreases slower

at low speed than that at high speed if the controller still uses

zero voltage vectors. Thus, replacing zero voltage vectors

with reversed voltage vectors may increase the response speed

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because reversed voltage vectors can produce bigger negative

torque change. Moreover, if we only use zero vectors to

reduce the negative torque error without the usage of reversed

vectors at very low speed, the flux locus would attenuate and

even result in failure start. On the other hand, if using

reversed vectors too often at very low speed may result in

bigger toque ripple in steady state than zero vectors. To

balance the problem, another fuzzy variable SP is used in

the FL controller. The membership function of SP as

indicated in Fig. 3 is divided into low speed mode (L) and

high speed mode (H). When working at low speed (the speed

less than 30% of rated speed is defined as low speed), the FL

controller will test the flux error. If the flux error is N, Z or P,

and the torque error is in the range of P, Z and N, the

controller will work exactly in the same way as it does in high

speed mode. However, if the flux error is PL, which means

the flux locus is attenuating, the controller will enable the

reversed voltage vectors to justify the flux locus. The

switching rules at high speed and very low speed are shown in

table 1 and table 2. The fuzzy variable A in the two tables

will be introduced later. This method can take advantages of

both zero vector and reversed vector.

Rotating speed (as a fraction of rated speed)

Figure 3. Memberships function for SP.

TABLE I. FUZZY CONTROL RULES OF FLDTC IN HIGH SPEED (K IS THENUMBER OF SECTOR)

Te Fe A V

NL/N/Z N/Z/P/PL L/S V0/V7

P N L/S V(k+2)

Z/P/PL L/S V(k+1)

PL N/Z/P/P L V(k+1)

S V(k+2)

In order to detect the working state, another variable

WS is added to the FL controller. The variable WScarries the information of whether or not there is a toque step

change in the load. The system can decide which working

mode of the FL controller should be taken based on this

information. The system automatically tests the change of the

torque in the load and compares it with the output toque. Once

the difference between them reaches the predefined threshold,

the state of the system will change into dynamic working state.

In this working state, the fuzzy rules will allow the controller

temporarily neglects the regulation of flux locus. Because the

dynamic state lasts only for a very short time, the transient

change will not influence flux locus significantly, and the

locus will recover as soon as the system turn back to steady

state. The variable WS is composed of three fuzzy sets:

dynamic work state whose step change is negative (DN),

dynamic work state whose step change is positive (DP) and

steady work state (S). The membership function is shown in

Fig. 4.

TABLE II. FUZZY CONTROL RULES OF FLDTC IN LOW SPEED (K IS THENUMBER OF SECTOR)

Te Fe A V

NL

N/Z L V(k-2)

S V(k-1)

PL L/S V(k-1)

N

N/Z L/S V0/V7

P L/S V(k)

PL L/S V(k-1)

Z N/Z/P L/S V0/V7

PL L/S V(k+1)

P N L/S V(k+2)Z/P/PL L/S V(k+1)

PL

N/Z L/S V(k+2)

P/PL L V(k+1)

S V(k+2)

Working state (as a fraction of rated torque)

Figure 4. Memberships function for WS.

We can directly get the conclusion from [6] that the

optimal voltage vectors giving the fastest response can be

simplified as a problem of maximization:

max{sin( )}k ro

kfn (13)

where k is the order of voltage vector,k

andro

are the stator

voltage vector angle and initial rotor flux angle, respectively.

From (13) we know that the voltage vector which creates

the largest sine value with rotor flux has the ability to produce

the largest torque change. In order to take advantage of the

conclusion, another fuzzy variable A is added to the fuzzy

controller. The rotor flux can be approximately equivalent to

stator flux because the slip angular velocity is actually very

small. When torque needs to be increased, the fuzzy variable

A is the angle between rotor flux and voltage vector V (k+1).

When torque needs to be decreased, variable A becomes the

angle between rotor flux and voltage vectors V (k-2).

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For instance, as shown in figure 5, when the flux is at

point P1, the angle between the vector V (k+1) and flux is /2,

which means that V (k+1) can create the biggest torque

change according to (13), so A is L at this moment and V

(k+1) is chosen to produce the biggest torque increase. As

soon as the flux moves to point P2, the angle variable A

decreases to /3, which means that V (k+1) will not produce

the fastest torque response in the following time and variable

A becomes S at this moment. Hence V (k+2) will be taken

instead of V (k+1). The way to produce the fastest torque

decrease is similar. This conclusion also explains the reason

to use the fuzzy variable A in table 1 and table 2.

V(k+2)

S(k)V(k+1)

S(k-1)

S(k+1)

V(k)S(k-2)

S(k+2)

S(k+3)

V(k+3)

V(k-1)

V(k-2)

P1

P2

A=/3

Directionof

rotation

Figure 5. Rotor flux vectors and six voltage vectors.

Whether A is Large (L) or Small (S) is defined in the

following way. When the torque needs to be increased quickly,

if the difference between the angles of rotor flux and the

voltage vector V (k+1) is in the range of (/3, /2), then A is

large (L), otherwise A is small (S). Conversely, when the

torque needs to be decreased quickly, if the difference

between the angles of rotor flux and voltage vector V (k-2) is

in the range of (/2, 2/3), then A is large (L), otherwise A is

small (S).The results will be transferred to the FL controller

which can analyze the composite conditions and give an

optimal voltage vector selection according to the expert

knowledge. The membership function of A shown in figure 6

is represented by two fuzzy sets: large (L) and small (S).

Flux angle (Degree)

Figure 6. Membership function of A.

From the fuzzy variables and fuzzy rules introduced

above, we can get the flow chart of the FL controller in Fig. 7.

To sum up, the fuzzy switching rules can be summarized

to table 3. Each rule in table 3 can be written as: Ri: if WS is

Ai, SP is Bi, Te is Ci, Fe is Di, SE is Ei and A is Fi, then V is

Vi, where Ri is the ith fuzzy rule. Ai, Bi, Ci, Di, Ei and Fi are

the values of fuzzy sets of the fuzzy variables WS, SP, Te, Fe,

SE and A.

Negative,positive, ornone?Flux angleis large? Flux angleis large?

Positive

Speed isvery low?

Negative

None

V=V(k+1) V=V(k-2)=V(k+2) V=V(k-1)

Get torquestep changeGet fluxangle Ge trotatingspeed

Using Fuzzylogic rulesfor low speed

Using Fuzzylogic rulesfor normalspeed

Yes No

Ye s Ye so No

Figure 7. Flow chart of the FL controller

TABLE III. FUZZY CONTROL RULES OF FLDTC(K IS THE NUMBER OFSECTOR)

WS SP Te Fe A V

S

L

NL

N/Z L V (k-2)

S V (k-1)

PL L/S V (k-1)

N

N/Z L/S V0/V7

P L/S V(k)

PL L/S V (k-1)Z N/Z/P L/S V0/V7

PL L/S V (k+1)

P N L/S V (k+2)

Z/P/PL L/S V (k+1)

PL

N/Z L/S V (k+2)

P/PL L V (k+1)

S V (k+2)

H

NL/N/Z N/Z/P/PL L/S V0/V7

P N L/S V (k+2)

Z/P/PL L/S V (k+1)

PL N/Z/P/P L V (k+1)

S V (k+2)

DP L/H N/P/Z N/Z/P/PL L V (k+1)

S V (k+2)

DN L/H N/P/Z N/Z/P/PL L V (k-2)

S V (k-1)

3 Simulation resultsTo verify the efficiency of the proposed system, the

model is tested on matlab tool. The induction motors

parameters are given as follows:

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E

lectromagneticTorque(N.m.)

ElectromagneticTorque(N.m.)

ElectromagneticTorque(N.m.)

ElectromagneticTorque(N.m.)

Rated Voltage: 380 V

Pole pairs: 2

Stator Resistance 1.111

Rotor Resistance 1.083

Stator Inductance: 0.5974 H

Rotor Inductance: 0.5974 H

Mutual Inductance: 0.2037 H

Moment of inertia J: 0.02 kg.m^2

Friction factor: 0.0057 N.m.s

Sampling period of the system: 50 s

Time (s)

(a). Electromagnetic torque for CDTC

Time (s)

(b). Electromagnetic torque for FLDTC

Figure 8. Electromagnetic torque

Fig. 8 (a) and (b) show the performances of the torque

ripples of the motor at 300 rad/sec and no load for CDTC and

FLDTC, respectively. It is clearly shown that the toque ripple

in Fig. 8(b) is approximately 40% smaller than that in Fig. 8

(a). Hence we can conclude that FLDTC produce less torqueripple than CDTC in steady state.

Keeping the speeds unchanged, and adding a step torque

change as big as 12N.m. at 0.08sec in the load, we can get

the curves of torque response illustrated in Fig. 9. The torque

response of FLDTC is significantly faster than that of CDTC.

Time (s)

(a). Response of the step torque change for CDTC

Time (s)

(b). Response of the step torque change for FLDTC

Figure 9. Electromagnetic torque responses with a step change of 12N.m

at 0.08 sec for CDTC and FLDTC

The flux locus of FLDTC in both steady state and

dynamic state with a torque step change of 12N.m at 0.08 sec

are given by Fig. 10. From Fig. 10(a), we can see that the flux

locus is not significantly different from the flux locus of

CDTC in steady state. Nevertheless, the flux locus shown in

Fig. 10(b) has a transient change when a step change is

applied in the load. That is because the FL controller

temporarily ignores the flux locus when working in dynamic

state. In this state, the controller only imposes the voltage

vectors producing the biggest torque change rate. Hence the

flux locus moves toward the same direction as the voltages

vector. This is why the flux locus rotates along a hexagon

track at that moment, and then recovers as soon as the torque

reaches the reference value.

Fig. 11 shows the flux locus of FLDTC at 20rad/sec and

no load. We know that CDTC has the disadvantages such as

flux locus distortion in very low speed. Simulation result

proves that the flux locus can be improved by using the

proposed controller. The success can be attributed to the

rational selection between reversed voltage vectors and zero

voltage vectors.

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(a). Flux locus of FLDTC in steady state

(b). Flux locus of FLDTC in dynamic state

Figure 10. Flux locus of FLDTC

Figure 11. Flux locus of FLDTC at 20 rad/sec

4 ConclusionA fuzzy logic based direct torque control system is

implemented in this paper to improve the performance of

conventional DTC system. The FL controller enables thesystem to choose optimal stator voltage vectors producing the

most suitable rate of torque change according to the six fuzzy

variables. Simulation results have shown the effectiveness of

the proposed method. Through the comparison between

CDTC and FLDTC, we have shown that the FLDTC design in

this paper keeps all the advantages of CDTC, and makes some

improvement in reducing torque ripples, faster torque

response, and stability at very low speed.

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