icg.isep.pw.edu.plicg.isep.pw.edu.pl/pdf/phd/pawel_grabowski.pdf · contents contents pages 1....

Contents

Contents

Pages

1. INTRODUCTION 1

2. VOLTAGE SOURCE INVERTER FED INDUCTION MOTOR DRIVE 4

2.1. Introduction ` 4

2.2. Mathematical model of induction motor 8

2.3. Voltage Controlled Power Converter 13

2.4. Induction Motor Controllers 19

2.4.1 Rotor Flux Oriented Control 19

2.4.2 Stator Flux Oriented Control 24

2.4.3 Feedback Linearization Control 28

3. INDUCTION MOTOR VARIABLE ESTIMATORS 33

3.1. Introduction 33

3.2. Voltage model based estimator with low pass filter 35

3.3. Voltage model based estimator with new integration algorithm 38

3.4. The improved voltage model based estimator in polar coordinates 44

3.5. Torque estimators 49

3.6. Rotor speed estimators 49

3.7. Stator resistance measurement, calculation and compensation

methods 50

4. PRINCIPLES OF NEURO-FUZZY CONTROL 56


4.2. Fuzzy logic control system 56

4.3. Adaptive Neuro-Fuzzy Inference System 59

5. DIRECT FLUX AND TORQUE CONTROL 64


5.2. Basics of Direct Torque and Flux Control 65

5.3. Direct Torque Control 69

5.3.1 Direct Torque Control - Takahashi's method (circular flux loci) 69

5.3.2 Direct Self Control - Depenbrock's method (hexagonal flux

loci) 89

5.3.3 Direct Torque Control with constant switching frequency 92

5.3.4 Sliding mode approach for DTC as a low speed problem

solution 96

5.3.5 Intelligent methods in DTC 100

5.4. Direct Torque Neuro-Fuzzy Controller 107

5.4.1 Introduction 107

5.4.2 DTNFC scheme 108

5.4.3 Design and investigation of DTNFC 113

5.4.4 Self tuned DTNFC 118

5.4.5 Characteristic futures, advantages and disadvantages

of DTNFC 130

Contents

6. DSP SETUP AND CONTROL ALGORITHM OF DTNFC 131


6.2. Laboratory setup 132

6.3. Control algorithm 133

7. SIMULATION AND EXPERIMENTAL VERIFICATION OF DTNFC 141


7.2. DTNFC simulation comparison with classical DTC 142

7.3. Experimental verification 149

8. CONCLUSIONS 158

REFERENCES 159

SYMBOLS EMPLOYED 166

APPENDIX 171

Introduction

1

1. Introduction

Throughout the twentieth century, the fast development of industry caused

simultaneous increasing of demand for electrical drives. For many years most of drives

for industrial processes, commercial equipment, and domestic appliances have been

design to operate at constant speed. It has been known that the variable-speed drives

will effect in performance, productivity and efficiency improvement. However, until

80th, the variable frequency drives has been used only for special applications like

elevators, cranes, extruders, conveyors, mil drives and many others. It has been caused

generally by complicate and so expensive control structure.

It has been proposed many different control structures to operate in the variable

speed. The most popular industrial used method was U/f constant. However, the

demand for electrical drives still increases very fast. The fast torque responses, precise

operation in every speed region, absence of sensors, and self-tuned controller have

become the main property of variable speed drives. The manufacturer race against each

other to invent and produce best and best cheap variable frequency drives. The most

popular high performance induction motor drive control method is Field Oriented

Control proposed by Hasse [25] and Blaschke [5]. The method allows control not only

amplitude and frequency, like in U/f constant control, but also the phase of the voltage,

current and flux vectors, what further significantly improve dynamic behavior of the

system. Still however, the proposed method has been complicated and difficult to tune.

There have been proposed the next famous industry used method of induction motor

control, called Direct Torque Control, has been presented in [19, 75, 69] for the first

time in the mid 80s. The authors propose to replace motor decoupling via nonlinear

coordinate transformation with hysteresis controllers. The bang-bang controller's suite

well to on-off operation of inverter semiconductor power devices. The method allows

controlling torque and flux directly without current controllers, what considerably

simplify designing and tuning of the system.

Such a fast development of variable frequency drives have not been possible without

simultaneous fast expansion of microelectronics. The processor calculation power and

memory become many times cost-performance-effective. It results that designers do not

have to be so much concerned with the computation effect of complex algorithm or

Introduction

2

their increase to the system cost. The discrete realization of the Direct Torque Control

method has not been implemented until 1995 when the ABB FINLAND Company has

introduced the first industrial DTC induction motor drive [78, 69]. The controller

guarantee very fast torque responses and precise operation in every speed range.

However the system has still few disadvantages like variable switching frequency

(because of hysteresis controllers) and very complicated structure which is realized in

multi-processor controller board. Whereas, the world industrial tendency is to build the

systems based on modules, which introduce that the systems are simple in service and

not failure.

In the light of these facts the author has undertaken a task to introduce the new

control method, which guarantee:

• dynamical property as in the DTC,

• constant switching frequency,

• low flux and torque ripple,

• uni-polar voltage PWM,

• self-tuning,

• single-processor board based controller.

The theses contain eight sections. The first is the present introduction. The second

introduce and put in order the current knowledge about modern voltage controlled

power converters and induction motor controllers. In the third the flux, torque and speed

estimators are presented. Simulation and experimental results support the theory. The

fourth section contains the principles of the neuro-fuzzy controller. The fifth section at

least contains wide consideration on the Direct Torque Control method. There is

presented theoretical basis for the DTC based hysteresis controllers. The modified DTC

control structures, which are presented in current literature, are also described. And

finally the proposed new controller based on neuro-fuzzy structure is presented. The

theoretical considerations are supported by simulation. In the sixth section the DSP

setup and the Direct Torque Neuro-Fuzzy Controller algorithm is presented. The

seventh section contains the simulation and experimental verification of the proposed

controller. Finally the conclusions are in eight section.

Introduction

3

The author would like to stress that the work on this thesis results with the registered

patent nr WP/19/99, P-000020 know-haw.

The proposed structure thanks to the advantages can be implemented in more than

90% of industrial applications, like pumps, funs, mixers, conveyors, elevators, extruders

and many others. Especially thanks to the direct flux and torque control the method can

be successfully used in electrical vehicles.

The author consider that the main self-achievements of the thesis are as follows:

1. elaboration of new Direct Torque Controller based on neuro-fuzzy controller

structure,

2. elaboration of modified stator flux estimator worked in wide speed range,

3. building a simulation control algorithms in C language and verification of the

proposed control algorithm by comparison with the classical DTC,

4. building the experimental setup based on floating-point DSP-TMS320C31 and

practical verification of the proposed controller,

5. practical implementation of a close-loop speed sensorless controller.

Voltage source inverter fed induction motor drive

4

2. Voltage source inverter fed induction motor drive

2.1. Introduction

Electric motors are the most commonly used prime mover in industry. For each

application, the mechanical system to be driven has a set of criteria for torque and speed

as follow [71]:

• two or four-quadrant operation,

• maximum short-term torque maintained from zero up to a base speed,

• short time relation between the applied voltage or current and the resultant torque

(frequency bandwidth),

• torque-to-rotor inertia ratio,

• energy efficiency,

• power to mass ratio,

• torque ripple,

• acoustic noise,

• shape, volume, acceptability in hazardous environments, reliability,

manufacturability, fail-safe features, initial cost.

Three phase electric motors are the largest prime mover in all of the industry. They

are offered in ranges from 0.35 up to 4300 kW. A squirrel cage induction motors fill the

large percentage of the total motor industry. It is thanks to they simple construction what

further leads to low price and no failure. They are more robust and more reliable than

others motors. They require little maintenance. It can be design to work in almost every

outdoor condition (high and low temperature, high humidity, dirt, vibration, explosive

environments and atc.).

The most popular induction motor speed control methods are based on frequency

converters. They contain rectifier and inverter. Thanks to very fast semiconductors

technological progress in the most popular inverter is voltage inverter with modulate

pulse width (PWM - Pulse Width Modulation). (detailed described in Section 2.3.).

Very high switching frequency (3 - 50 kHz) provides close to sinusoidal current


5

waveforms, low loses in the motor and high power density. For motor power up to

150kW there are offered Intelligent Power Modules (IPM), which contain high power

transistors, current and temperature sensors, short circuit and over-voltage protections

etc. There are also proposed power modules, which contain rectifier and high power

transistors in one module, i.e. PowIRtrein (International Rectifier). This power density

tendency leads to simple and not failure structure of the voltage inverter. But the main,

still unsolved, problem is how to control the inverter power transistors to reach desired

criteria for torque and speed of the motor. The high-performance induction motor drive

is characterized by:

• very fast flux and torque response,

• available maximum output torque in full operation region,

• low flux and torque ripple,

• simple tuning method (or even auto-tuned),

• uni-polar voltage PWM,

• constant switching frequency,

• robustness for parameter variation,

• four-quadrant operations,

• simplicity (simple construction, simple tuning and operation and small controller

dimension leads to low final product price).

The induction motor control methods are divided into a scalar and vector control.

The general classification of the frequency controllers is presented in Fig.2.1.1.

The most simple and popular in industry method is Voltage/Frequency control.

However the method, except simplicity, does not perform any high-performance drive

requirements.

Other very popular in present time control methods is known as Field Orientated

Control (FOC) proposed by Hasse [25] and Blaschke [5]. In this method the motor

equation are transformed in a coordinate system that rotates with the rotor flux vector. In

such way created new field coordinates, when the rotor flux amplitude is constant, there

is a linear relationship between control variables and speed. The method allows control


6

not only amplitude and frequency, like in Voltage/Frequency control, but also the phase

of the voltage, current and flux vectors, what further significantly improve dynamic

behavior of the system.

Frequency

controled

methods

Scalar based

controllers

Vector based

controller

U/f=const

Direct

(Blaschke)

Field orientedDirect Torque

Control

Feedback

linearisationis=f(w

r)

Indirect

(Hasse)

Circle flux

trajectory

(Takachashi)

Hexagon flux

trajectory

(Depenbrock)

Stator flux

oriented

Rotor flux

oriented

Natural Field

Orientation

(Jonsson)

Direct Torque

Neuro-Fuzzy

Controller

Fig.2.1.1. Induction motor control methods classification.

The FOC method guarantees flux and torque decoupling. However the induction

motor equations are still nonlinear. The method known as Feedback Linearisation

Control (FLC) [57, 58] introduce a new nonlinear transformation of the motor state

variables, so that in the new coordinates, the speed and the flux are decoupled by

feedback.

Another field oriented induction control method proposed in mid 80s is called

Natural Field Orientated Controller [36, 37]. In the NFO instead of rotor flux, like in

last method, the stator EMF vector is applied as basis for the currents and voltage

transformation. Thanks to this, the additional integration for stator flux calculation is

avoided. As a control signals are used voltages Ud and Uq in EMF oriented coordinate

system. Consequently these components, as in DC motor, separately control flux and

torque signals.


7

The next famous industry used method of induction motor control, called Direct

Torque Control, has been presented in [19, 75, 69] for the first time in the mid 80s. The

authors propose to replace motor decoupling via nonlinear coordinate transformation

with hysteresis controllers. The bang-bang controllers suite well to on-off operation of

inverter semiconductor power devices. The method allows controlling torque and flux

directly. Current and voltage in this method are controlled indirectly.

Induction

Motor

PWM

inverter

Sa

Sb

Sc

ControllerVector

Modulator

Reference

signals

Control

signals

Feedback signals

Fig.2.1.2. General control structure of PWM inverter-fed induction motor drive.

It can be notice, that almost all of the nowadays induction motor systems are based

on the structure as in Fig.2.1.2, which contain:

• high voltage part

� induction motor,

� inverter,

• low voltage part

� controller,

� vector modulator.

The modulator is separate from the controller, because accept classical DTC

methods, all of the structure contain this block.


8

All of the mentioned control methods are presented in the theses. The Field Oriented

Control methods are briefly presented in this section. The Direct Torque Controller is

widely described in Section 5.0.

2.2. Mathematical model of induction motor

There are made several assumptions to simplify thinking over the three-phase

induction motor [43]:

• the three-phase motor is symmetrical,

• only a basic harmonics is taking in to account,

• the spatially distributed stator and rotor windings are replaced by a concentrated

coil,

• an anisotropy effects, magnetic saturation, iron loses and eddy currents are not

taking into considerations,

• the coil resistance's and reactance's are taking to be constant,

• in many cases, especially when considering steady states, the currents and

voltages are taking to be sinusoidal.

There can be written a set of equations for such idealized motor model as follows:

dt

dRIU A

sAA

Ψ+= , (2.2.1a)

dt

dRIU B

sBB

Ψ+= , (2.2.1b)

dt

dRIU C

sCC

Ψ+= . (2.2.1b)

For further simplification of the mathematical considerations the motor model

equations can be written in terms of space vectors, what give the equations of the motor

as follows [43]:


9

dt

dTR Ns

s

ss

ΨIU += , (2.2.2a)

dt

dTR Nr

r

rr

ΨIU += , (2.2.2b)

rss IIΨ mj

s MeLγ+= , (2.2.3a)

srr IIΨ mj

r MeLγ−+= , (2.2.3b)

where the state space vector is defined as:

( ) ( ) ( )[ ]tktktk CBA ⋅+⋅+⋅= 2

s aa1k3

2, (2.2.4)

where: 1, a, a2 -

complex vectors, ( ) ( ) ( )tktktk CBA ,, - temporary effective value of

phase currents, voltages or fluxes, 2/3 - normalization constant.

However the most popular motor mathematical model is received by further

transformations. The set of equations (2.2.2-3) is transformed into a common rotating

coordinate system [43], what leads to the vector equilibrium equations (in per unit

system [43]):

KKK

NKsK jdt

dTr s

s

ss ψψ

iu ω++= , (2.2.5a)

( )rK

rK

r ψψ

i mKNKr jdt

dTr ωω −++=0 , (2.2.5b)

KMKs xx rssK iiψ += , (2.2.6a)

KMKs xx srrK iiψ += , (2.2.6b)

( )[ ]LK

M

m mTdt

d−= s

*

sK iψIm1ω

. (2.2.7)


10

Based on the above equations there are built many different block schemes of the

induction motor, where the differences between them are depending on:

• reference frame rotation speed,

• input signals,

• output signals.

The popular induction-cage motor systems is based on fixed coordinate system

(ωk=0), what lead to the equations:

dt

dTr Ns

s

ss

ψiu += , (2.2.8a)

r

r

r ψψ

i mNr jdt

dTr ω−+=0 , (2.2.8b)

rss iiψ Ms xx += , (2.2.9a)

srr iiψ Ms xx += , (2.2.9b)

( )[ ]L

M

m mTdt

d−= s

*

s iψIm1ω

. (2.2.10)

The complex state-space vectors can be resolved into components α and β:

βα ss uu j+=su , (2.2.11a)

βα ss ii j+=si , (2.2.11b)

βα rr ii j+=ri , (2.2.11c)

βα ψψ ss j+=sψ , (2.2.11d)

βα ψψ rr j+=rψ . (2.2.11e)


11

By taking above (2.2.11) formulae into account the set of machine equations (2.2.8-

10) can be write as follows:

dt

dTiru s

Nsssα

αα

ψ+= , (2.2.12a)

dt

dTiru

s

Nsss

βββ

ψ+= , (2.2.12b)

βα

α ψωψ

rmr

Nrrdt

dTir ++=0 , (2.2.13a)

αβ

β ψωψ

rm

r

Nrrdt

dTir −+=0 , (2.2.13b)

αααψ rMsss ixix += , (2.2.14a)

βββψ rMsss ixix += , (2.2.14b)

αααψ sMrsr ixix += , (2.2.15a)

βββψ sMrsr ixix += , (2.2.15b)

[ ]Lssss

M

m miiTdt

d−−= αββα ψψ

ω 1. (2.2.16)

The above equations create a complete set of induction-cage motor equations in fixed

coordinate system.

Fare considerations can lead to many different block schemes of an induction motor

in fixed coordinate system, where differences will depend on chosen input signals. For

instance, if it is assumed that the input signals to the motor are the voltage, than the

equations (2.2.12-16) after rotor current elimination can be transformed to the equations

set, as follows:


12

αααψ

ssss

N irudt

dT −= , (2.2.17a)

βββψ

sss

s

N irudt

dT −= , (2.2.17b)

ααβα ψψω

ψs

r

Mrr

r

rrm

rN i

x

xr

x

r

dt

dT +−−= , (2.2.18a)

ββαβ ψψω

ψs

r

Mrr

r

rrm

r

N ix

xr

x

r

dt

dT +−= , (2.2.18b)

ααα ψσ

ψσ r

rs

Ms

s

sxx

x

xi −=

1, (2.2.19a)

βββ ψσ

ψσ r

rs

Ms

s

sxx

x

xi −=

1, (2.2.19b)

[ ]Lssss

M

m miiTdt

d−−= αββα ψψ

ω 1, (2.2.20)

what can be presented on the block diagram as in Fig. 2.2.1.

Predominantly the controlled values are the flux and output torque or speed of the

motor. The control system described by equations (2.2.12-16) is not best, because the

output signals are depended on both inputs. From the control view the system is

complicated. That is why there are proposed few methods to decouple the flux and

torque control. It is achieved, for instance, by orientation of the system coordinate to the

vectors:

• the rotor flux vector (it is achieved for coordinate speed rsk Ψ

=ωω ),

• the stator flux vector (it is achieved for coordinate speed ssk Ψ

=ωω ),

The above control systems are widely described in Section 2.3 and 2.4.


13

βψ r

mω

∫

αsi

sr

r

Mr

x

xr

∫

αψ s

αψ r

sxσ1

rs

M

xx

x

σ

-

+

-

-r

r

x

r

∫

βsi

r

Mr

x

xr

∫ βψ s

βψ r

rs

M

xx

x

σ

-

+

r

r

x

r

sxσ1

sr

αψ r

-

+

-

αsu

βsu

∫ mω

Lm

αsi

βsi

αψ s

βψ s

-

+ -

em

em

αψ s

βψ s

Fig.2.2.1. Block diagram of an induction motor in the fixed coordinate system.

2.3. Voltage controlled power converter

As it has been mentioned before, the induction motor supply frequency can be

changed thanks to the frequency converter. The most popular power converter, used for

induction motor supply, is a three-phase bridge voltage-source inverter with transistor

switches. A schematic representation of the voltage-source inverter is presented in

Fig.2.3.1.


14

Three-phase load

T3

T1

T5

T2

T4

T6

D1

D2

D3

D4

D6

D5

CF

Ud

ua

ub

uc

Fig.2.3.1. Voltage-sources inverter basic schema.

The inverter is supply by a voltage source composed of a line-commuted phase-

controlled ac-to-dc converter. The capacitor is chosen to be large enough to obtain

adequately low voltage source impedance for the alternating current component in the

dc circuit. Such a position of the switches in the inverter allows receiving the

symmetrical rectangular three-phase voltages as in Fig.2.3.2.

There are possible eight positions of the switches (as in Fig.2.3.3) in the inverter. The

six of them (Fig.2.3.3a-f) produce an output phase voltage equal 3

1 or

3

2 of the dc

voltage (Ud). The last two (Fig.2.3.3g-h) give zero output voltage. The output voltage

can be represent by space vectors defined as (see Fig.2.3.4):


15

dU3

2

dU3

2−

0

0

0

Ua

Ub

Uc

ωt

ωt

ωt

2π

2π

2π

1 2 3 4 5 6

Fig.2.3.2. Phase voltages waveforms.

0

13

2 j

deuu = , (2.3.1a)

32

3

2πj

deuu = , (2.3.1b)

3

2

33

2π

j

deuu = , (2.3.1c)

πjdeuu

3

24 = , (2.3.1d)

3

2

53

2π

j

deuu−

= , (2.3.1e)

36

3

2πj

deuu−

= . (2.3.1f)

The non-zero vectors are named active vectors.


16

u0(000)

a b c

Ud

+

-

u1(100)

a b c

Ud

+

-

u3(010)

a b c

Ud

+

-

u2(110)

a b c

Ud

+

-

u4(011)

a b c

Ud

+

-

u6(101)

a b c

Ud

+

-

u5(001)

a b c

Ud

+

-

u7(111)

a b c

Ud

+

-

Fig. 2.3.3. Switching states for the PWM VSI inverter.

u1(100)

u3(010)

u4(011)

u5(001)

u2(110)

u6(101)

u0(000)

u7(111)

jIm

Re

Fig. 2.3.4. Switching state vectors in the complex plane.


17

There are proposed many different methods to receive any output voltage [4, 17, 26,

27, 28, 56] (of course limited to the amplitude, because of dc voltage). However the

general idea is based on sequential switching of active and zero (u0 or u7) vectors. The

average value of the vectors should give desired vector. For instance, to receive a voltage

vector usc, as in Fig.2.3.5a, which

u1

u3

u4

u5

u2

u6

β

αu

7

u0

usc

γsc

a)

Ua

Ub

Uc

t

t

t

u0

u1

u2

u7

u7

u2

u1

u0

T0/4 T

1/2 T

2/2 T

0/4

Ts/2 T

s/2

b)

Fig.2.3.5. Principle of vector modulation: a) output voltage space vector, b) pulse

pattern.


18

is placed between u1 and u2 vectors, there can be chosen a vectors sequence as

follows (Fig.2.3.5b):

u0 � u1 � u2 � u7 � u2 � u1 � u0 , (2.2.1a)

or simply:

u0 � u1 � u2 , (2.2.1b)

and others. The amplitude of the vector is depended on the duration times, which for

discrete system with sampling time Ts, can be calculated by the equations [70]:

s

sc

TT

−=

3sin

3sin

u

u3

d

c

1 π

γπ

, (2.3.2a)

( )s

sc TT

=

3sin

sin

u

u3

d

c

2 πγ

, (2.3.2b)

)( 212 TTTT s +−= . (2.3.2c)

Recently there are introduced other modern PWM methods. The general demand of

the vector modulation can be write as in [56]:

• the maximum linear operation range,

• the minimum switching frequency,

• voltage and current harmonic minimization,

• low frequency harmonic elimination,


19

• low computation requirements.

Nowadays, thanks to the properties, there are become more and more popular mixed

method [26], where algorithm witch employ the conventional space-vector PWM

method work in the low modulation range and the Generalized Discontinuous PWM

(GDPWM) [26] method in the higher modulation range. The method guarantees best

satisfying of above requirements.

The vector modulation method used in the setup system is widely described in

Section 6.3.

2.4. Induction motor controllers

2.4.1. Rotor Flux Oriented Control (RFOC)

Through many years a DC motors has been used. One of the properties of these

motors is that the flux and torque can be controlled separately. This independence

allows using very simple control methods, like PI controllers. However, because of

complicated construction the cost of the DC motors is extremely high. An invention of

Field Oriented Control (FOC) has created a possibility to replace DC motors by high

performance induction motors with new control methods.

The independently control of flux and torque in induction motor is possible when

coordinate speed is equal to rotor flux vector angular speed. Than the vector voltage

equation (in p.u.) (2.2.5-7) are transformed to:

ss

ss ψψ

iursNs j

dt

dTr Ψ++= ω , (2.4.1)

( ) rr

r ψψ

i mrsNr jdt

dTr ωω −++= Ψ0 . (2.4.2)

The flux-current equations and equation of motion remain unchanged, and are as

follows:


20

rs ii Ms xx +=σψ , (2.4.3)

srr iiψ Ms xx += , (2.4.4)

( )s*

s iψIm=m . (2.4.5)

Notice, that for rotor flux oriented coordinate system the flux vector ψψψψr=ψrd=ψr. The

equations can be resolved to components:

sqΨrssd

Nsdssd ωdt

dTiru ψ

ψ−+= , (2.4.6a)

sdΨrs

sq

Nsqssq ωdt

dTiru ψ

ψ++= , (2.4.6b)

dt

dTir r

Nrdr

ψ+=0 , (2.4.7a)

rrrqr ωir ψ+=0 , (2.4.7b)

rdMsdssd ixix +=ψ , (2.4.8a)

rqMsqssq ixix +=ψ , (2.4.8b)

sdMrdrr ixix +=ψ , (2.4.9a)

sqMrqr ixix +=0 , (2.4.9b)

rsq

r

m ix

xm ψ= . (2.4.10)

The equations (2.4.7a, 2.4.9a) can be easy transformed to:

sd

r

Mrr

r

rrN i

x

xr

x

r

dt

dT −= ψ

ψ, (2.4.11)

what together with equation (2.4.10) give the equations for current-controlled

induction motor.


21

If it is assumed that the magnetizing current, in agreement with Yamamura definition

[82] is equal:

srMr iii +=M

r

x

x, (2.4.12)

than the equation (2.4.9a) can be transformed to:

MrMr ix=ψ . (2.4.13)

It can be seen that for constant rotor flux amplitude there is a linear relationship

between control variables and speed. The block diagram of the induction motor in d-q

field coordinates is presented in Fig.2.4.1.

ψr

xω

mm

mL

ψr

r

Mr

x

xr

r

M

x

x

r

r

x

r

-

-

isd

isq

m

MxiMr

Fig.2.4.1. The block diagram of the induction motor in d-q field coordinates


22

There are two different FOC methods. They are distinguished by different rotor flux

angle calculation methods. The rotor flux angle is indispensable for the coordinate

transformation.

For the direct FOC the angle is calculated by the estimator or observer, which input

are stator voltage, stator current and rotor speed. An example of direct FOC system for

PWM inverter-fed induction motor is presented in Fig.2.4.2.

For the indirect FOC the angle is calculated based on reference torque and rotor flux

and the measured rotor speed, based on the equation:

rc

sqc

r

Mrr

i

x

xr

ψω −= , (2.4.14)

which is received from equations (2.4.7b and 2.4.9b).

By simple transformation of the equations (2.4.10 and 2.4.11) the reference field

oriented currents can be calculated as:

−=

dt

dT

r

x

xi rc

N

r

rrc

M

sdc

ψψ

1, (2.4.15a)

rc

c

M

rsqc

m

x

xi

ψ= , (2.4.15b)

while the reference voltage is calculated based on equations:

dt

d

x

xTix

dt

dixTiru rc

r

MNsqcs

sdcNsdcssdc

ψω σσ +−+= , (2.4.16a)

rc

r

Mssdcs

sqc

Nsqcssqcx

xix

dt

dixTiru ψωω σσ +++= , (2.4.16b)


23

which can be worked out from set of equations (2.4.6-9).

An example of indirect FOC system for PWM inverter-fed induction motor is

presented in Fig.2.4.3.

The property of the FOC can be summarized as follows:

• the method is based on DC motor control philosophy,

• the method with control variables isd and isq do not guarantee the exactly

decoupling of the motor speed and rotor flux control in both dynamic and steady

states,

• in field coordinates, for the constant rotor flux amplitude, there is a linear

relationship between control variables and speed,

• full information about motor state variable and load torque is required (the

method is very sensitive to rotor time constant parameter),

• the current controllers are required,

• coordinate transformation required,

• the PWM vector modulator is required, what further guarantee constant switching

frequency,

• in direct FOC flux and torque estimator is required,

• the stator currents are sinusoidal.

Induction

Motor

Flux Vector

Estimation

Sa

Sb

Sc

is

us

PWM

Inverter

Vector

Modulator

Control

Signals

Transfor-

mations

Flux

Controller

Torque

Controller

Feedback

Signals

Transfor-

mation

rΨ

rcΨ

mcω

csu α

csu β

isd

controller

isq

controller

αrΨ βrΨ

αsi

βsi

αψ r

βψ r

sdcu

sqcu

sdci

sqci

sxi

syi

rΨ-

-Torque

Controller

em

mω

Fig. 2.4.2. Structure of direct RFOC system for PWM inverter-fed induction motor.


24

Induction

Motor

Sa

Sb

Sc

PWM

Inverter

Vector

Modulator

Control

Signals

Transfor-

mationsSpeed

Controllercm

rcΨ

mcω

csu α

csu β

sdcu

sqcu

sdci

sqci

-

Slip

calculation

∫

Reference

voltage

calculation

Reference

current

calculation

rcΨ sqci

rω

mω

mω

+

sω

sγ

Fig.2.4.3. Structure of indirect RFOC system for PWM inverter-fed induction motor.

2.4.2. Stator Flux Oriented Control (SFOC)

There are many different control methods proposed to achieve desired property.

However, instead of very fast microelectronic advancement and high-density tendency,

there is probably (there can be produced reserved industrial controller which is disable

for others) only one induction motor industrial ready controller realized in one chip. The

ASIC is called NFO controller and is developed by NFO Drives Swedish company [36,

66]. The complete algorithm used in the processor is secret. However, the theoretical

considerations lead to conclusion that NFO contain algorithm, which is based on SFOC.

The SFOC equations are received for coordinate orientation to the stator flux vector,

what give coordinate rotation speed ssk Ψ

=ωω . Than the vector voltage equation (in p.u.)

(2.2.5-7) are transformed to:

ss

ss ΨΨ

iussNs j

dt

dTr Ψ++= ω , (2.4.17a)

( )r

rr Ψ

Ψi mssNr j

dt

dTr ωω −++= Ψ0 . (2.4.17b)

The flux-current equations and equation of motion remain unchanged, and are as

follows:

rss iiΨ Ms xx += , (2.4.18a)


25

srr iiΨ Ms xx += , (2.4.18b)

( )s

*

s iΨIm=m . (2.4.19)

Notice, that for stator flux oriented coordinate system the flux vector ΨΨΨΨs=ψsx=Ψs. The

equations can be resolved to components:

dt

dTiru s

Nsxssx

ψ+= , (2.4.20a)

sΨsssyssy ωiru ψ+= , (2.4.20b)

rymΨssrx

Nrxr ωdt

dTir ψω

ψ)(0 −−+= , (2.4.21a)

rxmΨss

ry

Nryr ωdt

dTir ψω

ψ)(0 −++= , (2.4.21b)

rxMsxss ixix +=ψ , (2.4.22a)

ryMsys ixix +=0 , (2.4.22b)

sxMrxsrx ixix +=ψ , (2.4.23a)

syMrysry ixix +=ψ , (2.4.23b)

sysim ψ= . (2.4.24)

If it is assumed that the magnetizing current, as in Fig. 2.4.2, in agreement with

Yamamura definition [66] is equal:

rsMs iiis

M

x

x+= , (2.4.25)

and the stator flux is constant than the motor equations (2.4.20, 2.4.22 and 2.4.24)

can be write as follows:


26

sxssx iru = , (2.4.26a)

sΨsssyssy ωiru ψ+= , (2.4.26b)

MssrxMsxss ixixix =+=ψ , (2.4.27a)

syMsssys iixim ==ψ . (2.4.28)

Than the current-controlled induction motor is given by 2.4.27 and 2.4.28 equations.

The block diagram of the induction motor in x-y field coordinates is presented in

Fig.2.4.4. It can be seen that for constant flux amplitude there is linear relationship

between control variables and speed.

The above consideration has been used in the available industrial ASIC controller

developed by NFO Drives company [36, 66, 37]. The general structure of SFOC is


ψs

xω

mm

mL

iMx

sx

-

iMx

isy

m

sx

sx

1esx

Fig.2.4.4. The block diagram of the induction motor in x-y field coordinates


27

Induction

Motor

EMF

calculation

Sa

Sb

Sc

is

us

PWM

Inverter

Vector

Modulator

Control

Signals

Transfor-

mationsSpeed

Controller

Feedback

Signals

Transfor-

mation

mcω

csu α

csu β

se

sγ

ssΨ

ω

syci

sxe

sye

-

∫

Msci

Reference

Voltage

Calculation

sxcu

sycu

Fig.2.4.5. Structure of SFOC system for PWM inverter-fed induction motor.

The property of the SFOC can be summarized as follows:

• the method is based on DC motor control philosophy,

• the method with control variables isx and isy do not guarantee the exactly

decoupling of the motor speed and rotor flux control in both dynamic and steady

states,

• in field coordinates, for the constant stator flux amplitude, there is a linear

relationship between control variables and speed,

• there are not required full information about motor parameters (the method is

mainly sensitive to stator resistance and inductance),

• coordinate transformation is required,


frequency,

• if there is assumed that the stator flux is constant than the system does not require

additional integration for flux calculation,



28

2.4.3. Feedback Linearization Control (FLC)

Transformation of the induction motor equations in the field coordinates has a good

physical basis because it corresponds to the decoupled torque production in separately

excited DC motor. However, from the theoretical point of view other type of

coordinates can be selected to achieve decoupling and linearization of the induction

motor equations.

In [44] a controller based on multiscalar motor model has been proposed. The new

state variables (different than in FOC methods) have been chosen. In results, the motor

speed is fully decoupled from the rotor flux. In [7, 16] authors has developed a

nonlinear control system based input-output linearization which allow fully decoupled

flux and rotor speed. However, the system uses the transformation in field coordinates.

In [57, 58] authors has proposed a nonlinear transformation of the motor state variables,

so that in the new coordinates, the speed and rotor flux amplitude are decoupled by

feedback. There are proposed also modified methods based Feedback Linearization

Control like in [72, 41].

The induction motor equations can be write (in p.u. system) in the following form:

( ) ββαα gg ss uuxfx ++=& , (2.4.29)

where

f x( )

( )

=

− − +

− +

+ −

− + −

− −

αψ ω αω αψ ααβψ βω γβω αβψ γ

µτ

α β α

α β β

α β α

α β β

α β β α

r m r M s

m r r M s

r m r s

m r r s

r s r sL

M

ψψ

ψψ

ψ ψ

x i

x i

i

i

i im

,

(2.4.30)

gα = [0, 0, 1

σxs, 0, 0]T, (2.4.31)


29

gβ = [0, 0, 0, 1

σxs, 0]T, (2.4.32)

x = [ψrα, ψrβ, isα, isβ, ωm]T (2.4.33)

and

α = rr

xrβ =

xM

σ xs xr,

γ = xr2 rs + xM2

rr

σ xs xr2

,

µ = xM

τM xr,

σ = 1 - xM2

xs xr.

Because ωm, ψrα, ψrβ are not depended on usα, usβ there is possible to choose

variables depended on x:

φ1(x) = ψrα2 + ψrβ2 = ψr2, (2.4.34)

φ2(x) = ωm. (2.4.35)

If it is assumed that φ1(x), φ2(x) are outputs variables, the full definition of new

coordinates can be given by:

z1 = φ1(x),

z2 = Lf φ1(x),

z3 = φ2(x), (2.4.36)


30

z4 = Lf φ2(x),

z5 =

arctan

ψ

ψr

r

β

α .

It should be remembered that the goal of control is to obtain constant flux amplitude

and to follow reference angular speed.

Because the fifth variable cannot be fully linearizatied and is not controllable (the

fifth variable correspond to slip in the motor) there is not considered last equation. Than

the dynamic of the system can be given by:

+

=

β

α

φ

φ

s

s

f

f

u

u

L

L

z

z

22

12

3

1D

&&

&&, (2.4.37)

where

=

22

11

φφφφ

βα

βα

fgfg

fgfg

LLLL

LLLLD . (2.4.38)

If φ1 ≠ 0 (the amplitude of flux is not zero) than det(D) ≠ 0 and it is possible to

define linearization feedback as follows:

+

−

−=

−

2

1

2

11

v

v

L

L

u

u

2f

2f

s

s

φ

φ

β

αD .

(2.4.39)

Than the result system is described by the equations:

&z1 = z2,


31

&z2= v1, (2.4.40)

&z3 = z4,

&z4 = v2,

and the final block diagram of the induction motor with new defined control signals

can be shown as in Fig.2.4.6.

v1

v2

z2

z4 τ

m

ωm

m

mL

ψr

ψr2

Fig.2.4.6. Block diagram of the induction motor with new control signals.

The control signals v1 , v2 are calculated by using linear feedback as follows:

v1= k11 (z1 - z1ref) - k12z2, (2.4.41)

v2= k21 (z3 - z3ref) - k22z4, (2.4.42)

where coefficients k11, k12, k21, k22 are chosen to receive reference close loop

system dynamic.


32

An example of FLC system for PWM inverter-fed induction motor is presented in

Fig.2.4.7.

Induction

Motor

Flux Vector

Estimation

Sa

Sb

Sc

is

us

PWM

Inverter

Vector

Modulator

Control

Signals

Transfor-

mations

Flux

Controller

Speed

Controller

Feedback

Signals

Transfor-

mation

2

rψ2

rψ&

mω

mω&

rγ

2

rcψ

mcω

1v

2v

csu α

csu β

ψψψψs

is

mω

Fig.2.4.7. Feedback Linearization Control of PWM inverter-fed induction motor.

The property of the FLC can be summarized as follows:

• guarantee the exactly decoupling of the motor speed and rotor flux control in both

dynamic and steady states,

• the method is implemented in a state feedback fashion and needs complex signal

processing,

• full information about motor state variable and load torque is required,

• there are no current controllers,


frequency,


Induction motor variable estimators

33

3. Induction motor variable estimators

3.1. Introduction

The principle of the DTC operation is based on driving the stator flux vector towards

its reference value, defined by input commands that are torque and flux level. There is

indispensable knowledge about actual value of the stator flux and torque. Also, the flux

is needed to calculate the actual rotor speed for sensorless drives.

There are many different methods to calculate the flux, torque and speed of the

induction motor. The classification of the methods is presented in [68]. The

mathematical models are divided in four main groups:

• state variable simulators (voltage model, current model. Lorenz estimator, MRAS

model),

• state variable observers (linear, nonlinear, extended model, and others),

• Kalman filter,

• neural-network based estimator.

The choosing of the model type is strongly depended on the used control method and

the properties like speed range of proper activity or robustness for motor parameter

changes.

There are few well-know methods used in the DTC. Most of them are based on the

voltage model of induction motor [43], where the current and voltage are only needed to

calculate the flux and torque. The method is less sensitive to the parameter variations

and does not require motor speed or position signals. Therefore the method is also more

preferable to use for sensorless drives.

However, the method uses the stator voltage for calculation, which is practically

difficult to measure aseptically in low speed operations.

The most known classical voltage model obtains the flux by integrating the motor

back emf, as follows:


34

∫ −=t

s

N

dtrT

0

)(1

ssse iuψ . (3.1.1)

The block diagram of the voltage estimator is presented in Fig.3.1.1. The method is

sensitive for only one motor parameter, stator resistance. However, the application of

pure integrator is difficult because of its dc drift and initial value problems [43]. The

Fig.3.1.2 presents the estimated stator flux trajectory calculated by the voltage model

with pure integrator. There is constant error of the estimated flux, which is caused by

wrong initial state chosen.

rs

∫ ψsαe

rs

∫

isβ

usβ

usα

_

_

1

TN

1

TN

isα

ψsβe

Fig.3.1.1. The block diagram of the voltage model based estimator with pure

integrators.

a) b)ψsαe ψ

sβe ψsβ(ψsα)

ψsβe(ψsαe)

Fig.3.1.2. Steady state operations for voltage model based estimator with pure

integrators with wrong initial values.


35

There are proposed many improvements of the classical voltage model. Some of

them are presented in the next sections.

3.2.Voltage model based estimator with low pass filter

A well-known solution to the dc-offset problem is to use a low pass (LP) filter to

replace the pure integrator. The method has been proposed in [74], and the equation

3.1.1 is transformed then to the equation as follows:

sessse ψiu

ψ

F

s

N Tr

Tdt

d 1)(

1−−= . (3.2.1)

The block diagram of the proposed estimator is presented in Fig.3.2.1.

rs

Ψsαe

rs

isα

isβ

usβ

usα

_

_

1

TN

1

TN

∫_

∫_

TF

1

TF

1

Ψsβe

Fig.3.2.1. The block diagram of the flux estimator based on the voltage model with LP

filter.

a) b)ψsαe ψ

sβe ψsβ(ψsα)

ψsβe(ψsαe)

Fig.3.2.2. The simulation results of the estimated stator flux behavior for wrong chosen

initial conditions for voltage model based estimator with LP filter.


36

ψsc

eψ eγ

ψse

ωm

Fig.3.2.3. The estimated stator flux behaviors during the speed reverse for voltage

model with LP filter (TF=0.1).

ψsc (0,4 Wb/div)

ψse

ωm

ωse (3600 rpm/div)

Fig.3.2.4. The stator flux amplitude behavior for speed reverses for the voltage model

with LP filter, the experimental results.


37

eψ eγ

ψse

ψsc

Fig.3.2.5. The voltage model with LP filter transient to the flux amplitude steps.

ψse (0,2 Wb/div)

ψsc

ωse (360 rpm/div)

ωm

Fig.3.2.6. The voltage model with LP filter transient to the flux amplitude steps, an

experimental results.


38

The stabilization time is depended on the LP filter constant time TF. An example of

the estimated flux trajectory with chosen wrong initial states is presented in Fig. 3.2.2.

Obviously, the LP filter will introduce some errors (depended on LP filter time

constant), especially when the motor frequency is lower than the filter cutoff frequency.

As a result, the proposed voltage estimator with LP filter can be used successfully only

in a limited speed range. There is presented, in the Fig.3.2.3, the estimator behavior

during speed reverse. Its noticeable a big distortion in the estimated flux when the rotor

speed cross the zero frequency band. The practical result of the speed reverse is


The simulation results for the flux transient to the step change is presented in

Fig.3.2.5. The amplitude trajectory contains oscillations, which influences the estimated

synchronous speed as in Fig.3.2.6. The Fig.3.2.6 present the experimental results for

Voltage/Frequency controlled system with voltage estimator with LP filter.

3.3.Voltage model based estimator with new integration algorithm

Other solution to the dc-offset problem has been proposed in [29]. The expression of

the pure integrator ( xs

y1

= ) can be rewritten as:

ys

xs

yc

c

c ωω

ω ++

+=

1, (3.3.1)

where x and y are the system input and output signals, and ωc is the cutoff frequency.

The first part of the equation represents a LP filter. The second part realizes a feedback,

which is used to compensate the error in the output. The block diagram of the new

integration with saturation block is presented in Fig.3.3.1. The new integration

algorithm contains saturation block, which stop the integration when the output signal

exceeds the reference stator flux amplitude. The simulation results of the system

behavior for wrong chosen initial conditions are presented in Fig.3.3.3. The complete

block diagram of the new flux estimator is presented in Fig.3.3.2.


39

+

ωωc

cs+

1

s c+ω

Ograniczenie

x y

yLPF

ydc

filtr dolnoprzepustowy

Fig.3.3.1. The new integration algorithm block scheme.

rs

Ψsαe

rs

Ψsβe

isα

isβ

usβ

usα

_

_

1

TN

1

TN

+1

s c+ω

ωωc

cs+

+

1

s c+ω

ωωc

cs+

ψsαe

ψsβe

Fig.3.3.2. The block diagram of the flux estimator with new integration algorithm.

a) b)ψsαe

ψsβe

ψsβ(ψsα) ψ

sβe(ψsαe)

Fig.3.3.3. The simulation results of the system behavior for wrong chosen initial

conditions for voltage model with improved integration.


40

There are two parameters of the improved estimator. It is a cutoff frequency and a

saturation level. The first of them is set to the sampling frequency. This set ensures

minimal steady state error of the phase and amplitude. However, than the system is less

sensitive for a distortion.

The second parameter usually is set to the reference flux amplitude. The wrong, too

high or too low saturation level, causes offset or distortion of the estimated flux

respectively, as in Fig.3.3.4. The proposed estimator in [29] could not be used

successfully in the control system with flux controllers. There is existing a big over-

regulation in the flux response as in Fig.3.3.5, which is caused by initially wrong

saturation level (too low). It is proposed, in this work, an improvement of the estimator

to have a possibility to use the estimator in the DTNFC system. There has been made a

few sample times delay of the setting in the saturation level. Such a simple correction

effects full elimination of the over-regulation, as in Fig.3.3.6.

The voltage estimator with the improved integrator is characterized by very small

phase and amplitude error, even in the low speed operation. The estimated flux behavior

during the speed reverse is presented in Fig.3.3.7 and 3.3.8. Note that the chosen scales

are much smaller than in the Fig.3.2.3. There are existing oscillations in the real flux,

which effect the phase and amplitude oscillations in the estimated flux.


41

ψsβe(ψsαe)

ψsαe

ψsc

ψsβe

Fig.3.3.4a. Steady state operations of the improved flux estimator with correctly tuned

saturation

level.

ψsβe(ψsαe) ψsc

ψsαe ψsβe

Fig.3.3.4b. Steady state operations of the improved flux estimator with too high

saturation level.

ψsβe(ψsαe)

ψsβeψsαe

ψsc

Fig.3.3.4c. Steady state operations of the improved flux estimator with too low

saturation level.


42

eψ eγ

ψse

ψsc

Fig.3.3.5. The stator flux transient to the step change for the flux estimator with new

integration algorithm (with initially wrong saturation level)

eψ eγ

ψse

ψsc

Fig.3.3.6. The stator flux transient to the step change for the flux estimator with new

integration algorithm (with few sample times delay of the settings in saturation level).


43

ψs

eψ eγ

ψse

ωm

Fig.3.2.7. The estimated stator flux behaviors during the speed reverse (flux estimator

with saturation level).

ψsc (0.4 Wb/div)

ψse

ωm

ωse (360 rpm/div)

Fig.3.2.8. The stator flux amplitude behavior for speed reverses for the improved flax

estimator with saturation level, the experimental results.


44

3.4.The improved voltage model based estimator in polar coordinates

The DTC and SFOC methods operate in stator flux oriented polar coordinates.

Therefore, it is covet a flux estimator also oriented in polar coordinates [78].

The induction motor voltage equation in stator flux oriented polar coordinates

(2.4.1a) can be resolved to the components (2.4.4):

dt

dTiru Nsxssx

sψ+= , (3.4.1a)

sψΨsssyssy ωiru += , (3.4.1b)

than after simple transformation the equations can be write as follows:

sxssxN irudt

dT −=s

ψ, (3.4.2a)

sψ

syssy

Ψss

iruω

−= , (3.4.2b)

what lead to the block scheme of the stator flux estimator in polar coordinates is

presented as in Fig.3.4.1.

∫αβ

xy rs

isα

isβ

usβ

usα

isx

usx

usy

isy

rs _

_

1

TN

÷

∫sin

cos

esx

esy

sx

sy

s

e

ψ=ω

e

sγ

e

sγ

ψsxe=ψse

Fig.3.4.1. The block diagram of the stator flux estimator in polar coordinates.


45

The estimator calculates the stator flux amplitude in natural way. It reduces number

of coordinate transformation blocks. Furthermore, there is calculating in a simple way

the synchronous speed, which is used to calculate the rotor speed. After synchronous

speed integration the actual stator flux angle can be received.

However, the considered estimator, similarly like voltage model with pure

integration, contains the same problems with the convergence from initial conditions.

The problem is solved in the same way, by adding a LP filter instead of the pure

integration in the flux amplitude calculation line as in Fig.3.4.2.

∫αβ

xy rs

isα

isβ

usβ

usα

isx

usx

usy

isy

rs _

_

1

TN

÷

∫sin

cos

esx

esy

sx

sy

s

e

Ψ=ω

e

sγe

sγ

_TF

1

ψsxe=ψse

Fig.3.4.2. The block diagram of the stator flux estimator in polar coordinates with LP

filter instead of pure integration.

The property, simulation and experimental results are the same as for the voltage

model with LP filter presented in Section 3.2.

Further, the estimator in Fig.3.4.2 can be improved by replacement of LP filter by

new integration algorithm as in Section 3.3. Because the integration algorithm is used to

calculate the flux amplitude there is needed only one pole saturation function. The final

improved stator flux estimator in polar coordinates is presented in Fig.3.4.3.

The proposed new flux estimator, thanks to the new integration algorithm rapidly

reach the steady stade when the wrong initial conditions are set. The simulation results

of the estimated flux trajectory are presented in Fig.3.4.4.


46

αβ

xy rs

isα

isβ

usβ

usα

isx

usx

usy

isy

rs _

_

1

TN

÷

∫sin

cos

esx

esy

ωs

sy

sx

e=Ψe

sγ

ω

ωc

cs +

1

sc

+ω

e

sγ

ψsxe=ψse

Fig.3.4.3. The block scheme of the new flux estimator in polar coordinates.

a) b)ψsαe

ψsβe ψ

sβ(ψsα)ψsβe(ψsαe)

Fig.3.4.4. The simulation results of the estimated stator flux behavior for wrong chosen

initial conditions for voltage model based estimator in polar coordinates.

Thanks to the one pole saturation the flux distortion caused by wrong saturation level

are smaller than in estimator with two-pole saturator. The flux trajectory is the entire

time circular. The experimental results for estimator with wrong saturation level are


The cut-off frequency is set to the sampling frequency what ensure minimal phase

and amplitude error as in Fig.3.4.6. Also the estimator work correctly during crossing

the zero speed band (Fig.3.4.6 and 3.4.7). The proposed estimator also contains a few

sample times delay of the setting in the saturation level. The flux transient to the step

change is the same as for improved voltage estimator with saturation (as in Fig.3.3.6.).


47

ψ sβe (ψ sαe)

ψ sαe ψ sβe

ψ sc

Fig.3.4.5a. Steady state operations of the improved flux estimator in polar coordinates

with correctly tuned saturation level.

ψ sβe (ψ sαe)

ψ sβeψ sαe

ψ sc

Fig.3.4.5b. Steady state operations of the improved flux estimator in polar coordinates

with too high saturation level.

ψ sβe (ψ sαe) ψ sc

ψ sαe ψ sβe

Fig.3.4.5c. Steady state operations of the improved flux estimator in polar coordinates

with too low saturation level.


48

ψs

eψeγ

ψse

ωm

Fig.3.4.6. The estimated stator flux behaviors during the speed reverse (improved flux

estimator with saturation level in stator flux oriented coordinates).

ψ sc (0,4 Wb/div)

ψ se

ωse (360 rpm/div)

ωm

Fig.3.4.7. The stator flux amplitude behavior for speed reverses for the improved stator

flax estimator with saturation level in stator flux oriented coordinates - the experimental

results.


49

3.5.Torque estimators

The induction motor output torque is calculated based on the equation 2.1.10, which

for stator oriented coordinate system can be written as follows:

αββα ψψ sssse iim −== ss iψ*

. (3.5.1)

It can be seen that the calculated torque is depended on the current measurement

accuracy and stator flux estimation method.

3.6. Rotor speed estimators

The speed estimation method used in the simulation and experimental model is based

on the new proposed stator flux estimator in polar coordinates. It is because, the

synchronous speed is calculated there naturally, without any special additional

mathematical functions. Thanks to this, there is only slip calculation block needed to

receive the rotor speed.

The slip is calculated based on the equation as follows:

2ψ r

err

mr=ω , (3.6.1)

where the rotor flux amplitude is calculated based on the equations:

ααα

σψψ s

M

rs

s

m

r

r ix

xx

x

x−= , (3.6.2a)

βββ

σψψ s

M

rs

s

m

r

r ix

xx

x

x−= , (3.6.2b)

222ψ βα ψψ rrr += . (3.6.2c)


50

Father the rotor speed is calculated based on the equation as follows:

rsm ωωω −= , (3.6.3)

where the synchronous speed is calculated directly in the flux estimation model.

The most important advantage of the model is its simplicity, what was the main

reason of its choosing. The disadvantage of the method is their sensitive for almost all

induction motor parameters. Also the mathematical equation of the model has been

work out for steady state operation assumptions [43].

However, most of the proposed methods to calculate actual rotor speed are very

complicated, which are based on observers, Kalman filters, Furrier transformations or

neural-networks. Such complicated structures make the whole system practically

inapplicable.

The speed estimation problem is still open, especially in low and zero speed

operations.

3.7.Stator resistance measurement, calculation and compensation methods

One of the limitations of the DTC based methods is the use of the stator resistance to

stator flux estimation. The variation of the resistance due to temperature changes in the

machine makes the controller operation difficult (especially in low speed operation) or

even can cause an instability of the system. The stator resistance value changes

practically in the range 0.75 to 1.7. The wrong resistance knowledge gives improper

stator flux and torque calculations what further can cause wrong voltage setting in the

inverter. The mathematical consideration of the stator flux resistance influence to the

system behavior is presented in [49, 60].

The stator resistance can be initially measured by applying a DC voltage between

two of stator phases and leaving the third one disconnected. The current is than

measured. The resistance is obtained by dividing the applied voltage by the measured


51

current. The precisely described method, how to measure the resistance with the

disconnected inverter, is presented in [62].

The well known method to compensate the initial value of the stator resistance when

the temperature is changing, is thermal model as follows:

−∆+=

−T

t

SSS eRRR 10 , (3.7.1)

where Rs0 is initial value of Rs at t=0 and ∆R is the change in Rs when the

temperature is changing. The T is the time constant of the variation of Rs. However, the

T and ∆R are strictly depended in many motor conditions like: actual current, speed,

cooling method (if any), cover type and many others, what makes that the method is not

general. There is proposed other method based on the thermal model with the fuzzy

structure to improve classical method [83]. However, the method seems to be to

complicate for practical implementations.

The most popular stator resistance compensation method is based on the calculation

the reference current and comparing them with the measured value [49, 60]. The

method is based on the principle that the error between the measured stator feedback

current phasor magnitude is and its commend isc is proportional to the stator resistance

variation. The proposed in [49] diagram schematic of the adaptive stator resistance

compensator is presented in Fig. 3.7.1.

siτ+11

srτ+11i

sc

is

∆rs

rs0

rse

Low Pass

Filter

PI Controller

and Limiter

Low Pass

FilterLimiter

Fig.3.7.1. Block diagram of the stator resistance compensator.

The increment value of stator resistance for correction is obtained through a PI

controller and limiter. The current error goes through a low pass filter, which has very


52

low cutoff frequency in order to remove high frequency components contained in the

stator feedback current. The incremental stator resistance is continuously added to the

previously measured stator resistance. The final estimated value of the resistance is

obtained as the output of another low pass filter and limiter.

The stator feedback current phasor magnitude is is obtained from the x and y axis

measured current as:

22

sysxs iii += . (3.7.2)

The reference stator current is calculated also in the synchronously rotating with

stator flux reference frame. The x component is calculated as follows:

scψ

csxc

mi = . (3.7.3)

The y component is calculated based on the quadrate equation:

0ψ1ψ2

sc2

2

2sc

2 =−

−+

−−−

rsm

rsycssxc

rsm

rssxcs

xxx

xixi

xxx

xxix . (3.7.4)

Finally, the stator reference current is calculating based on the equation:

22

sycsxcsc iii += . (3.7.5)

It has been shown in the paper that the method work correctly, also in the field

weakening region.


53

The other method, where the stator resistance is compensated based on the

calculation the reference current and comparing them with the measured value was

presented in [60]. The author instead of classical PI controller proposed to use the fuzzy

compensator. It has been also proved the effectiveness of the proposed method.

Though the both methods are very effective it seems to be mathematically

complicated for wide practical implementations.

There are proposed two simple methods in this work to calculate and compensate the

stator resistance.

It can be proved (see in the Appendix 3) that for the constant flux amplitude the

equation:

( ) ( ) 0=−+− essscsessscs iruiru βββααα ψψ , (3.7.6)

is fulfilled.

The equation will not be fulfilled when the stator resistance changes. Than the above

equation can be written as:

( ) ( ) εψψ βββααα =−+− essscsessscs iruiru , (3.7.7)

where the ε error is measurement of the stator resistance change. The proposed block

scheme of the method is presented in Fig. 3.7.2. The simulation results of the resistance

tuning are presented in Fig.3.7.3.


54

∆rs

rse

rse

isα

ψsα

ψsβ

isβ

usβc

usαc

∫cT

1

rse

ε

Fig.3.7.2. Block scheme of the proposed resistance estimation method without

coordinate transformation.

rs

rse

ε

Fig.3.7.3. The stator resistance tuning simulation results.

The most important advantage of the method is that there is no coordinate

transformation.

The other proposed method is based on the comparison of the reference and

calculated actual voltage x component in stator flux oriented frame, as in the Fig.3.7.4.


55

∆rs

rse

isx

usxc

∫cT

1

ε

Fig.3.7.4. Block scheme of the proposed resistance estimation method with coordinate

transformation.

For constant flux amplitude and for correct stator resistance an equation:

0=− sxssxc iru , (3.7.8)

is fulfilled. If the stator resistance change than the error ε will occur.

The simulation results for the method are similar as for the last method. The most

important advantage of the method is its simplicity.

It should be remembered that both of the methods are correct only for constant flux

amplitude. Such an assumption is fulfilled mainly in low speed operation when the

reference voltage is low, and consequently the stator flux ripples are unique. Also it is

assumed that the current measurement is precis and the reference voltage is performed

correctly.

Adaptive Neuro-Fuzzy Inference System

56

4. Principles of neuro-fuzzy control

4.1. Introduction

During the last years, fuzzy logic and neural network have been intensively used in a

lot of domains, and in particular in control applications. The reason of this trend is

possibly caused by increasing requirements of the controllers proprieties what whereas

usually causes control system much more complicated. Even more sometimes the

system is impossible to define in classical mathematical way.

The artificial intelligence methods do not require exact knowledge about the system

and is not restricted by a many assumptions used in control theories. In the fuzzy logic

system the design is based directly on expert knowledge and is formulate in easy human

language definitions, like “if ... then...” rule. In the neural network system the controller

can be trained. The training is generally based on information available during design

(collected data and information about system – off-line tuning) or operation (response

data – off line tuning). Furthermore the artificial intelligence systems exhibits good

noise rejection properties, robust to the parameter variation and fault tolerant. Finally all

advantages of the artificial intelligence can lead to faster controller system design.

4.2.Fuzzy logic control system

There are exists many different types of fuzzy logic controllers, but generally all of

them are based on model as on Fig.4.2.1.

fuzzifier

database

defuzzifier

decision-making unit

rule baseinput output

knowladge base

(fuzzy) (fuzzy)

(crisp) (crisp)

Fig. 4.2.1 Fuzzy inference system


57

The fuzzy system is composed of five functional blocks.

• fuzzifier

• defuzzifier

• database

• rule base

• decision making unit

The fuzzifier performs measurement of the input variable, scale mapping and

fuzzification. As a result of operation in this block, are degrees of matching expressed

in linguistic values.

The fuzzy set A, in not empty universe X, can be characterized by the function µA,

which values are in [0,1] partition. The function µA is called fuzzy membership

function. There are three most used shape of the membership function:

• triangular,

( )

( )

( )

( ) 0an th if

11

than if

11

than if

0an th if

=+>

+−=+<<

+=<<

=−<

xµbcx

xb

xbcxci

xb

xcxc-b

xµbcx

A

A

A

A

µ

µ (4.2.1)

• exponential,

( )

−−=

b

Aa

cxx

2

expµ ; (4.2.2)

• Gaussian shape,

( )bA

a

cx

x

−+

=2

1

1µ ; (4.2.3)

where a, b, c are the membership function parameters.


58

Each of function in the universe has own linguistic value i.e. NEGATIVE SMALL,

POZITIVE ZERO and so on. The user initially determines the number of the

membership function and the shape. The membership functions of the fuzzy sets used in

the fuzzy rules are defined database.

The rule base contains linguistic control if-than rules. The rules can be set by using

the experience and knowledge of en expert for the application and the control goals and

next modeling the process manually or automatically.

µA1(x1)

x

1

0.5

x

1

0.5

µA2(x

1)

x1

µB1(x2)

x

1

0.5

x

1

0.5

µB2(x

2)

x1

µC1(y)

y

1

0.5

y

1

0.5

µC2(y)

y

1

0.5

F(y)

y*

max

(center of area)

Fig. 4.2.2. Commonly used fuzzy if-then rules and fuzzy reasoning mechanism.

The decision making unit is the most important part of the fuzzy logic controller. It

performs the inference operation on the rules. In general for the controllers, the

linguistic rules are in the form IF-THEN. For instance for the PI controller it take the

form:

IF (e is A and ce is B) than (cu is C), (4.2.4)


59

where A, B, C are fuzzy subsets for the universe of discourse of the error, change of the

error and change of the output respectively.

The defuzzifier transforms the fuzzy results of the inference into a crisp output.

There are many defuzzyfication methods. The most used and known is center of gravity

method.

The commonly used fuzzy if-then rules and fuzzy reasoning mechanism is shown on

Fig. 4.2.2.

4.3. Adaptive Neuro-Fuzzy Inference System

As it was mention in the Section 4.1, fuzzy logic is well suited for dealing with ill-

defined and uncertain systems. Fuzzy interface system employing fuzzy if-then rules,

which are very familiar to human thinking method. It is possible to build complete

control system without using any precise quantitative analyses. However, to conceive a

fuzzy controller, it is necessary to choose a lot of parameters, like number of

membership functions in each of input and output, shape of this functions, fuzzy rules,

and other.

On the other hand, neural networks have proved theoretically and experimentally

their capacity to modelling large classes of non-linear structures. Nevertheless, it is

often necessary to run quite a long learning procedure, which can be obstacle to get the

on-line control of the process.

Combining both, fuzzy logic and neural network allows as achieving good

advantages. Human expert knowledge can be used to build initial structure of the

regulator. Underdone parts of the structure can be improved by on- or off-line learning

process.

Adaptive Neuro-Fuzzy Inference System (ANFIS) has been proposed for the first

time in [33, 34, 35]. For bigger lucid there is presented reduced ANFIS on Fig. 4.3.1 to

two inputs with two membership function for each input and one output.


60

min

x1

x2

N f1(x

1,x2)

Σ

x1 x2

y*

min

min

min

N

N

N

Layer 1 Layer 2 Layer 3 Layer 5Layer 4

µA22(x)

µA21(x)

µA12(x)

µA11(x)

o21

o22

o23

o24

o34

o33

o32

o31

o44

o43

o42

o41

f2(x

1,x2)

f3(x

1,x2)

f4(x

1,x2)

Fig. 4.3.1. Two input Adaptive Neuro-Fuzzy Inference System scheme.

For the presented structure the rule base contains four fuzzy if-then rules of Takagi

and Sugeno’s type, which are as follow:

Rule 1: If x1 is A11 and x2 is A21, than f1=p1x1+q1x2+r1,



Rule 4: If x1 is A12 and x2 is A22, than f4=p4x1+q4x2+r4, (4.3.1)

where x1, x2 - input values

A11, A12, A21, A22 - linguistic label,

p, q, r - consequent output function f parameters

The ANFIS structure contains five network layers:

Layer 1:

Every node in this layer contains membership function. Usually there are

chosen triangular or bell shape functions as in equation 4.2.1-3, where the

number of them are depend on control object. The parameters of the


61

functions, called premise parameters, can be tuned by back propagation

algorithm.

The first phase generally can be write as:

( )iAk xOij

µ=1, (4.3.2)

where i - input number,

j - membership function number in ith input,

k - node number in present layer,

xi - input signal,

Ok1 - first layer output,

µAij(xi) - membership function.

Generally the nodes number (K) is:

K=IJ, (4.3.3)

where I – number of inputs,

J – number of membership functions.

Layer 2:

The second part in the ANFIS corresponds to the MIN calculation in

classical fuzzy logic system. It can be write as:

( ) ( )[ ]2

2 ,min xxOjiij AiAk′′

= µµ , (4.3.4)

where O2k – second layer output, with condition ii ′≠ .

There are not every nodes are connected together, as in neural network

classical structure. The connections are between outputs of membership

functions with different inputs.


62

Layer 3:

Every node of this layer calculates the weight, which is normalised firing

strengths. The output results are in range [0,1]. It can be write as:

∑=

=K

k

k

k

k

O

OO

1

2

2

3, (4.3.5)

where O3k – third layer output.

Layer 4: .

The fourth phase can be call decision layer. Every node in this layer is a

connection point with the node function:

( ) ∑=

==I

i

iikkkkk xpOxxfOO1

3

21

34, , (4.3.6)

where O4k – fourth layer output.

pik - consequent parameters

The linear class of function has been chosen to simplify learning process.

The consequent parameters of the functions can be tuned by back

propagation algorithm. Also, thanks to the linear functions the parameters

can be identified by the least square estimate.

The decision layer can be best presented on graphical example for the

ANFIS from Fig. 4.3.1. on Fig. 4.3.2., where the numbers inside X1X2

surface are the decision numbers (consequent function numbers).

Layer 5:

The last phase of ANFIS is summation of all incoming signals. The result of

this node is the control signal. The calculation can be write, as:

∑=

=K

k

kOO1

45 , (4.3.7)


63

where O3k – fifth layer output.

11

1

2 4

3

A11

A12

A21

A22

X1

X2

X2

X1

Fig. 4.3.1. Graphical example of decision layer.

The presented neuro-fuzzy structure was initially tested and employed to model non-

linear functions, identify non-linear components, predict a chaotic time series [33] and

for stabilisation of the inverted pendulum [34].

It has been shown in [35] that the Adaptive Neuro-Fuzzy Inference System can be

used successfully instead most of all neural networks or fuzzy logic based systems. The

ANFIS advantage over them are:

• the human expert knowledge can be used to build initial structure of the regulator

(faster designing than pure neural network)

• the underdone parts of the structure can be improved by on or off line learning

process (impossible in classical fuzzy logic based systems).

Direct Flux and Torque Control

64

5. Direct Flux and Torque Control

5.1. Introduction

Since Blaschke, Hasse and Leonhard have presented for the first time vector control

drives, field oriented control [5, 25, 46] methods have become a standard in the drives

industry. German company Siemens has the most significant contributions in this field.

However during the last years the signal processing technology is improved very fast.

It makes possible to develop new, sometimes more complicated, time consuming or fast

computing required control methods.

One of the most recently investigated methods is Direct Torque Control (DTC). This

type of torque and flux control was initially proposed as Direct Self Control [19] and

Direct Torque Control [75]. Every year the interest of this subject is growing very fast.

However, at present, there is only one industrial company ABB Finland, who have

introduce a commercially available Direct Torque Controlled Inverter (ASC 600) [77,

1].

Such a big interest of DTC is caused mainly by some of advantages like:

• no co-ordinate transformation (which are required in most of the vector

controlled drive methods),

• no separate voltage modulation block (in classical Direct Torque Control),

• no current control loops (reduces difficulties of current controllers tuning),

• fast torque response (excellent dynamic performance)

• robustness for rotor parameter variation.

• However the conventional Direct Torque Control has some disadvantages:

• possible problems during starting and low speed operation,

• requirement for flux and torque estimators,

• variable switching frequency (because of hysteresis controllers),


65

Several solutions have been proposed to improve the conventional DTC. First there

have been modified the controllers and the switching table [12, 51, 40] to decrease

torque ripples. There has been also added modulation block to achieve constant

switching frequency and to remove problems during starting and low speed operations

[50, 21, 23]. To improve steady state performance (low torque ripple) the effective

methods of control such for example neural networks, neuro-fuzzy or fuzzy control

[61, 59] has been used successfully. Sliding mode approach, also has been proposed

[55]. Some of proposed modification of conventional DTC will be presented in the next

sections.

5.2. Basics of Direct Torque and Flux Control

As it was mention before, during the last years the most used drive control strategy

was field oriented method. The produced torque in this method is calculated from the

equation:

δsinψψ sr

r

m

syr

r

m

e ix

xi

x

xm == (5.2.1)

α

β

ΨΨΨΨs

δΨ - DTC ΨΨΨΨr

δ - FOC

is

Fig. 5.2.1 Angles between the stator current and rotor flux vectors in field oriented

control and between stator and rotor fluxes vector for direct torque control.


66

For constant rotor flux amplitude, the control quantity is stator current. It makes the

current source inverters (CSI) and voltage source inverters (VSI) with internal current

control loops very convenient for the practical implementation of this method.

The equation (5.2.1) can be easy transformed to the equation:

Ψ= δσ

sinψ1

ψ srxx

xm

r

m

e (5.2.2)

It can be noticed that in this situation the stator flux vector, instead of stator current,

can be used as a torque control quantity. The stator flux, whereas can be easy expressed

by simple conversion of the voltage equation of the induction motor, under assumption

that the stator resistance (rs) is zero, to:

ss uψ

=dt

dTN (5.2.3)

There are six non-zero possible voltage vectors and two zero vector (as on Fig.5.2.2),

which can be generate by the inverter (uνννν). The possible switching stages for the inverter

are presented on Fig.2.1.2.

The equation (5.2.3) can be write as:

∫=t

N

dtT

0

1νuψs (5.2.4)

The inverter voltage vectors can be described mathematically as complex vector:

( )

=

==−

7 0, kfor 0

6 ..., 2, 1,kfor 3

23

1π

ν

kj

duu (5.2.5)


67

uβ

uα

u1(100)

u3(010)

u4(011)

u5(001)

u7(111)

u2(110)

u0(000)

u6(101)

Fig.5.2.2. Switching voltage space vectors represented on α-β plane.

It can be seen from the equation (5.2.4) that the stator flux is directly impressed by

the inverter voltage (5.2.5). By introduction of any active voltage vector the stator flux

vector movies to the direction and sense of the voltage vector as on Fig.5.2.3. The above

consideration can be observed on simulation results for the inverter six-step operation

(cycling sequence of active vectors) on Fig.5.2.4.

Under sinusoidal PWM operation the stator flux trajectory become a circle as on

Fig.5.2.5. The zoom part of the circle trajectory is presented in Fig. 5.2.6.

The rotor time constant of standard squirrel-cage induction motor is large. It causes

that the rotor flux vector inertia is much larger than the stator flux vectors. Thanks to

this, the increasing and decreasing of the output torque is possible by applying

respectively adequate active voltage vector and zero voltage (widely described in

Section 5.3.1.).

uβ

uα

u1(100)

u3(010)

u4(011)

u5(001)

u7(111)

u2(110)

u0(000)

u6(101)

voltage u4 applied

voltage u3 applied

Fig.5.2.3. Forming of the stator flux trajectory.


68

usα ψ

sαusβ ψ

sβ

α

β

ψψψψs

Fig. 5.2 4. Simulated six-step operation.

usα ψ

sαusβ ψ

sβ

α

β

ψψψψs

Fig. 5.2.5. Simulated sinusoidal PWM operation.


69

β

uβ

uα

u1(100)

u3(010)

u4(011)

u5(001)

u7(111)

u2(110)

u0(000)

u6(101)

voltage u4 applied

voltage u2 applied

voltage u3 applied

voltage u2 applied

voltage u3 applied

voltage u3 applied

voltage u4 applied

α

Fig.5.2.6. Circular stator flux loci.

5.3. Direct Torque Control

5.3.1. Direct Torque Control - Takahashi’s method (circular flux)

Principle of the method

The basic structure of the direct flux and torque control for a voltage source PWM

inverter-fed induction motor is presented in Fig. 5.3.1. The reference stator flux

amplitude ψsc and torque mc are compared with actual estimated values of ψs and me.

The flux eΨ and torque em errors are delivered to two and three level compactors

respectively.

The digitised output signals dΨ and dm and the stator flux vector position sector

decide which appropriate voltage vector is chosen from the selection table. The signal

dΨ is defined as:

dΨ = 1 for eΨ > HΨ, (5.3.1)

dΨ = 0 for eΨ < - HΨ, (5.3.2)


70

and signal dm as:

dm = 1 for em > Hm (5.3.3)

dm = 0 for em = mc (5.3.4)

dm = -1 for em < -Hm (5.3.5)

Induction

motor

Switching

table

Sector

detection

Flux and

torque

estimator

flux

controller

torque

controller

mc

me

ψsc

ψs

eψ

em

dΨ

dm

SaSbSc

is

us

ψs

me

ψsα ψsβ

+-ud

-

-

Fig 5.3.1. Direct Flux and Torque Control – Takahashi’s method

The outputs from the selection table are the switching states (SA, SB, SC) for the VSI.


71

Switching table construction

The DTC is generally based on controlling the stator flux vector position. It can be

seen, from the equation 5.2.2., that the output torque is depend on the stator and rotor

flux amplitude. However, the most important control value is the angle between both

flux vectors. Thanks to the long rotor time constant the angle is easy to change by

applying the adequate voltage vector which causes very fast stator flax vector position

change.

For the DTC proposed by Takahashi [75] the inverter output voltage plain is divided

into six sectors (as in Fig. 5.3.2):

Sector 1:

+−∈6

,6

ππα ,

Sector 2:

+∈2

,6

ππα ,

Sector 3:

++∈6

5,

2

ππα ,

Sector 4:

−+∈6

5,

6

5 ππα ,

Sector 5:

−−∈2

,6

5 ππα ,

Sector 6:

−−∈6

,2

ππα ,

Let make an assumption that the stator flux vector position is as in Fig. 5.3.3. If the

angle between stator and rotor flux δΨ is defined as:

δΨ = δΨs − δΨr, (5.3.6)

than it can be increase by applying u1 and u2 and decrease by applying u4 and u5

voltage vectors. The increasing of the considered angle causes changes the output torque

to the clockwise direction and decreasing to the reverse direction. Note however, that


72

sector 4

uβ

uα

sector 1

sector 2sector 3

sector 5 sector 6

u1(100)

u3(010)

u4(011)

u5(001)

u7(111)

u2(110)

u0(000)

u6(101)

Fig. 5.3.2. Division of the voltage plane into six sectors in the DTC method.

the magnitude of the angle change is depend on the rotor speed. Generally for the

middle and high-speed operation (ωm>0.2ωN) the increasing of the angle δΨ is slower

than the decreasing. It can be very easy notice on Fig.5.3.3. and Fig.5.3.4. (See also the

Appendix 2)

By applying any zero voltage vector (u0 or u7) the integration in Eq.5.2.4 is stopped,

what means that the stator flux is not change. However the rotor flux vector, thanks to

the long rotor time constant, will still rotate and as a result the angle δΨ will be changed.

When the rotor flux rotates with the clockwise direction than the zero voltage vector

applied causes decrease of the motor torque. However, this kind of torque reduction is

only good for the middle and high speeds operations (Fig.5.3.3). In low speed range the

rotor flux motion is too slow to achieve rapid torque reduction (Fig.5.3.4). In this

situation the first proposed active vector selection is more adequate.


73

me

me

Fig. 5.3.3. Influence of speed level on torque production (ωm=ωN), a) load increasing b)

load decreasing.

me

me

Fig. 5.3.4. Influence of speed level on torque production (ωm=1/2ωN), a) load increasing

b) load decreasing..


74

Table 5.3.1. Flux and torque variation due to the applied voltage vectors.

uk-2 uk-1 uk uk+1 uk+2 uk+3 u0

Ψs ↓ ↑ ↑↑ ↑ ↓ ↓↓ ↑↓

me

(ωm>0)

↓↓ ↓↓ ↓ ↑ ↑ ↓ ↓

me

(ωm<0)

↓ ↓ ↑ ↑↑ ↑↑ ↑ ↑

Under assumption that a sector number is k, the Table 5.3.1 can summarise the above

consideration regarding influence of voltage vectors on stator flux and torque [11].

The above considerations lead to the selection table of the control system as

presented in Tab. 5.3.2. The proposed table is destined for a medium and high-speed

operation.

uβ

uα

sector 1

u2

u3

u5 u

6

ΨΨΨΨs

Fig. 5.3.5. Selection of the optimum voltage vectors for stator flux vector in sector 1.


75

Table 5.3.2. Optimum voltage switching vector look-up table.

dΨ dm sector 1 sector 2 sector 3 sector 4 sector 5 sector 6

1 u2 u3 u4 u5 u6 u1

1 0 u7 u0 u7 u0 u7 u0

-1 u6 u1 u2 u3 u4 u5

1 u3 u4 u5 u6 u1 u2

0 0 u0 u7 u0 u7 u0 u7

-1 u5 u6 u1 u2 u3 u4

There also are some papers, which proposed the method to compose different

switching tables for different operation region [53]. The method is based on the same

conclusions as from Table 5.3.1 but with precisely consideration of the flux ant torque

change

The zero vectors are selected, in a way, to minimise a switching frequency. The

switching frequency is depended on the flux and torque hysteresis band (more in further

part of this section). However, it is more depended on the torque then the flux hysteresis

band for an accepted range of them ( { }01.0;05.0∈ΨH and { }1.0;04.0∈mH ). That is

why the zero vectors are selected to minimise the switching frequency in the same flux

digitized output signal dΨ. For instance for the sector 1 and dΨ=1 the best proper zero

vector is u7, because its required only one switch to change the vector from zero to one

of the active vector (u2 or u6) or reverse from the active to zero vector.

Also, another notation of selection table is known. It is called selection strategy [12].

The voltage vectors are chosen in respect of behavior in terms of torque and current

ripple, switching frequency, dynamic performance and two- or four-quadrant operation

capability. For instance, the selection strategies for two-quadrant and not near zero

speed operation can be noted as in Table 5.3.3.


76

Table 5.3.3. Selection strategies for two-quadrant and not near zero speed operation

me↑ me↓

strategy 1 Strategy 2 strategy 3

Ψs↑ uk+1 u0 uk uk

Ψs↓ uk+2 u0 u0 uk+3

The selection strategies for four-quadrant and all speed operation can be noted as in

Table 5.3.3.

me↑ me↓

Ψs↑ uk+1 uk-1

Ψs↓ uk+2 uk-2

Finally, the consideration leads to the speed-dependent selection strategy for four-

quadrant operation [12] as presented in Table 5.3.4.

Table 5.3.4. Speed-dependent selection strategy.

me↑ me↓ me↑ me↓ me↑ me↓

Ψs↑ u0 uk-1 Ψs↑ uk+1 uk-1 Ψs↑ uk+1 u0

Ψs↓ u0 uk-2 Ψs↓ uk+2 uk-2 Ψs↓ uk+2 u0

-wm -wlim 0 +wlim +wm

Unfortunately, in spite of that, the selection Table 5.3.2 or even selection strategy

from Table 5.3.4 is optimal in the configuration with 6 sectors and with hysteresis as in

Fig.5.3.1. some behaviours of DTC are not satisfactory. For instance when the flux

vector is close to a one of sector boundary, two of four possible active vectors are wrong


77

(see Fig.5.3.6.). These wrong vectors can only change the torque error without

correction of the flux error. It causes a distortion witch are easy visible on a current and

a torque (Fig. 5.3.6). Not optimal vectors selection is also the main reason of problems

in low speed operation (if not consider the flux and torque estimation problems).

uβ

uα

sector 1

ΨΨΨΨs

u2

u3

u5 u

6

Fig. 5.3.6. “Not optimal” selected vectors (u3 and u6) from the selection tables.

A few methods to improve the DTC behaviour in the sectors borders have been

proposed. One of them is to compose next six active vectors by adding two nearest exist

active vectors [51] as in Fig.5.3.8. It allows build next selection table with more optimal

configuration. Another method is to add more sectors and hysteresis levels with double

parallel PWM inverter [76].


78

eψ

isα

ψsβ(ψsα)

Fig.5.3.7. A current and a flux distortion (in circle) caused by “not optimal” vectors

from selection table.

V2

V1

1/2V1

1/2V2

V12

Fig. 5.3.8. Voltage vectors synthesis using two neighboring vectors.


79

Hysteresis controllers

The controllers proposed by Takahashi in classical DTC are two and three level

comparators for the stator flux and torque errors respectively, like in Fig. 5.3.9a and

5.3.9b.

a) b) c)

HΨ Hm

Hm1

Hm2

Fig.5.3.9. Hysteresis controllers, a) two level, b) classical three level c) modified three

level.

The two level hysteresis controller for the stator flux can be written structurally as:

if eΨ > HΨ than dΨ = 1, (5.3.7)

if eΨ < -HΨ than dΨ = 0. (5.3.8)

And the three level hysteresis controller for the torque can be writes as a sum of two

two level hysteresis, as:

(first hysteres)

if em > 2Hm than dm = 1, (5.3.9)

if em < 0 than dm = 0, (5.3.10)

(second hysteres)

if em > 0 than dm = 1, (5.3.11)

if em < -2Hm than dm = 0, (5.3.12)


80

However, the classical torque controller has a disadvantage. During the steady state

operation the torque error operate in only one hysteres region. It causes the constant

steady state torque error, as in Fig. 5.3.10.

eψ

isα

Fig.5.3.10. Constant steady state torque error for the torque controller from Fig.5.3.9b.

There is proposed another kind of three level hysteresis controller as for the current

controllers [42, 39], in Fig.5.3.9c. The three level modified hysteresis controllers have

been used also in DTC [40, 38]. This controller for the torque can be written as a sum of

two level hysteresis, as:

(first hysteres)

if em > Hm1 than dm = 1, (5.3.13a)

if em < -Hm1 than dm = 0, (5.3.13b)

(second hysteres)

if em > Hm2 than dm = 1, (5.3.14a)

if em < -Hm2 than dm = 0, (5.3.14b)


81

The hysteresis modification effects on steady state torque error cancellation, as

shown in Fig.5.3.11.

eψ

isα

Fig.3.5.11. Steady state operation with modified hysteresis controller (Fig.3.5.9c).

The hysteresis band amplitudes are chosen by consideration switching losses in the

inverter and low harmonic copper losses in the machine [11]. Small hysteresis band of

flux or torque causes very high switching frequency, which leads to high inverter losses.

On the other side, increasing the flux hysteresis band amplitude causes flux vector locus

degeneration (up to a hexagon) and as result higher low harmonic copper losses. To

wide torque hysteresis band causes increasing of torque pulsation. The flux hysteresis

band has no influence on torque pulsation and the torque hysteresis band has slight

effect on the harmonic copper losses. More information about analytical investigation of

torque and flux ripple in DTC can be found in [10].

The example of the stator flux loci and phase current waveforms for three different

values of the stator flux band are presented in Fig.5.3.12-14.


82

isα ψ

sβ(ψsα)

Fig.5.3.12. Phase current waveform and stator flux vector loci for HΨ=0.005 and

Hm=0.06 (fsw=7.6kHz).

isα ψ

sβ(ψsα)


Hm=0.06 (fsw=6.1kHz).

isα

HΨ

ψsβ(ψsα)


Hm=0.06 (fsw=5.8kHz).


83

It is possible to create a switching frequency and torque pulsation surfaces depended

on flux and torque hysteresis band amplitudes [38]. There are also existing

characteristics of THD2 (harmonic distortion factor) as a function of switching

frequency fsw [11]. Finally, it is possible to create the characteristic of harmonic copper

losses and switching losses as a function of flux and torque hysteresis bands [11] as in

Fig.5.3.15. Note that for different motor parameters and the characteristics of the

inverter high-power semiconductors the characteristics can be changed. However, it

illustrates clearly the tendency.

100

80

60

40

20

0.2 0.4 0.6 0.8 1.0

∆P (W)

Hm (pu)

HΨ = 0

HΨ = 0.058

HΨ = 0.112

** *

*

*

*

*

* *

*

*

*

*

*

**

*

*

Fig.5.3.15. Example of sum of harmonic copper losses and switching losses as a

function of torque and flux hysteresis band (from [20]).

One of the interesting thing in the losses characteristic is that the loses for reduced

hysteresis band close to zero does not caused considerable switching losses. There are

some applications where the work with low acoustic noise level is more important than

the losses. However, increasing the switching frequency is limited by the time delay in

the feedback signals.

Start and low speed operation problems

The main problem directly connected with the classical DTC scheme is imperfect

control of stator flux at low speed and during start up.


84

In the low speed operation the voltage drop on the stator resistance become

significant. When analysing a stator voltage equation 2.2.5a (Equation (5.3.15)) and

concerning the zero voltage applied during close to zero or zero speed operation it can

be seen from the equation (5.3.16) that the stator flux amplitude can not be keep

constant and it decreases. It effects at the demagnetisation of the machine.

ssss

N jrdt

dT ψ

ψω−−= ss iu , (5.3.15)

( )∫ +−=t

ss

N

dtjrT

0

ss ψ1

ψ ωsi . (5.3.16)

One of the solution for this problem is elimination of the zero voltage vectors during

low speed operation (like for instance in speed-dependent selection strategy).

The effect of the demagnetisation is amplified during the close to sector boundary

operation. As it has been shown in Fig. 5.3.6, in classical DTC, the chosen “optimal”

voltage vector is orthogonal to the stator flux vector. It does not allow controlling

(increase or decrease) the stator flux amplitude. The stator flux distortions at low speed

caused by close to sector boundary operation and the demagnetisation effect is presented

in Fig.5.3.16.

The not optimal voltage selection during close to sector boundary operation causes

also the starting problem. If the motor is not magnetised and there is output torque

needed to the clock-wise direction, the hysteresis controllers chooses for stator flux

vector close to sector k boundary (for instance as in Fig. 5.3.17) the active vector uk+1. In

this situation it is not possible to increase the flux amplitude because the selected

voltage vector is perpendicular to the stator flux vector.

However, the most important problem connected with starting implies from other

reason. When the loaded motor reaches (after problems described above) the reference

stator flux, the rotor flux is still small, because of the long rotor time constant.

Furthermore, there is chosen such a voltage vector sequence to produce the maximum

reference output torque. It means that the stator and rotor fluxes are orthogonal to each


85

other. This situation does not allow producing the rotor flux anymore. Simulated

oscillograms of start problem are presented in Fig.5.3.18.

Another reason of starts and low speeds operation problems come also from (the

popular low speed problems):

• the flux and torque estimation errors – the properties of the system behaviour are

strongly depended on precise stator flux and torque estimation (different estimation

method for DTC are described in Section 3). However, generally it is not possible

know precisely all needed parameters for estimation. Than caused stator flux and

torque constant errors.

eψ

isα

ψsβ(ψsα)

Fig. 5.3.16. The stator flux distortions at low speed caused by close to sector boundary

operation and the demagnetization effect (ωref = 0.045, HΨ = 0.01, Hm = 0.05, ML = 0).


86

uβ

uα

sector 1

ΨΨΨΨs

u2

u3

u5 u

6 ΨΨΨΨs

*

Fig. 5.3.17. The voltage vector selection for the not magnetized and loaded motor.

eψ em

ωc

ωm

isα

ψsβ(ψsα)

ψrβ(ψrα)

Fig.5.3.18. The simulation of the start problem.


87

• dead-time effect – it essential to insert the dead-time in PWM control signal of an

inverter in order to avoid the short-circuit of two power switches. It cause the current

wave-form distortion and even system instability [28, 63].

• discrete effect of control and torque ripple (if digital systems) [50]- because of

hysteresis regulation type, which is naturally analog, even very high sampling time

causes error in torque regulation. It can be seen as low harmonics in output torque.

• a noise in the measured speed (if the system with speed loop with sensors) [18]- this

low speed problem is naturally depended on speed measurement quality. If the speed

measurement distortion exist, the speed controller (predominantly PI controller)

amplifies the noise. Than there is needed low pass filter to cut off the high frequency

distortion.

Industrial application - ACS 600

The first commercial DTC method has been realised in. However, this application

did not find many customers, especially in Europe.

The next commercial application of the DTC has been realised by the ABB Finland

company. This application made a huge revolution in the DTC research and industry

opinion. The method has returned to research laboratories.

The ASC 600 product family (ASC 601, 603, 604, 607) has found application in 95%

of industrial demand, like in: funs, pumps, mixers, conveyors, lifts, elevators, cranes,

hoists, winders, centrifuges, extruders and etc. The ASC 600 has a selection in 380-

690V ranges, 2.2kW-630kW power, and many different enclosure ratings (IP 00, IP20,

IP21, IP22, IP54).

Thanks to the very strong 40Mhz Toshiba processor with ASIC hardware the ASC

600 can every 25µs calculate the modulus of the stator flux space vector, its position,

the electromagnetic torque and the rotor speed. It allows precisely control of motor rated

speed, for sensorless speed control - error between ±0.1% and ±0.5% and for control

with pulse encoder – error in the ±0.01% range. The torque response is less than 2ms.

There is possible flux optimisation, flux braking and field weakening operation.

The ASC600 during initialisation can automatically identify motor parameters, like:

stator resistance, stator inductance, magnetising inductance and saturation coefficient for


88

theses two inductance’s, inertia of the motor and some others. Furthermore, there is

realised an auto tuning of the most significant parameters like stator resistance.

The results and more precisely description of the ASC 600 family can be found in

[77, 1, 78, 64].

Characteristic futures, advantages and disadvantages of DTC

The DTC is characterised by nearly sinusoidal stator flux and current waveform

which harmonic contents as a switching frequency is depended on flux and torque

hysteresis band. Only PWM operation is possible, which require adequate supply

voltage reserve.

Advantages:

• no co-ordinate transformation (which are required in most of the vector controlled

drive methods),

• no separate voltage modulation block,

• no current control loops (reduces difficulties of current controllers tuning),

• minimal torque response time (excellent dynamic performance)

• robustness for rotor parameter variation.

Disadvantages:

• possible problems during starting and low speed operation,

• requirement for flux and torque estimators,

• not constant switching frequency (because of hysteresis controllers),

• high torque ripples

• flux and current distortion caused by stator flux sector position change .


89

5.3.2. Direct Self Control - Depenbrock’s method (hexagonal flux loci)

The Direct Self-Control (DSC) proposed by Depenbrock [19], was mainly

constructed to reduce the switching frequency of inverter for high power application

with simultaneous very high torque dynamic keeping. En example of the DSC for

control of the high power traction drives is presented in [32, 14].

A block diagram of the DSC is presented in Fig. 5.3.19.

Induction

motor

Flux and

torque

estimator

is

us

+-ud

me

Ψs

flux

controllers

torque

controller

Ψsc dΨA

εΨA

εΨB

εΨC

εmc

dΨB

dΨC

md

Sa

Sb

Sc

me

ΨsAΨsB

ΨsC

Fig. 5.3.19. A block diagram of the Direct Self-Control.

The construction deference between the DTC is that the stator flux digitized

variables dA, dB and dC calculated as:

dA = 1 for eψΑ > HΨ, (5.3.17)


90

dA = 0 for eΨA < - HΨ, (5.3.18)

dB = 1 for eΨB > HΨ, (5.3.19)

dB = 0 for eΨB < - HΨ, (5.3.20)

dC = 1 for eΨC > HΨ, (5.3.21)

dC = 0 for eΨC < - HΨ, (5.3.22)

directly set up the switching states of the inverter.

sector 4

uβ

uα

sector 1

sector 2

sector 3

sector 5

sector 6

u1

u3

u4

u5

u7

u2

u0

u6

Fig. 5.3.20. The stator flux sectors in DSC.

The torque is controlled based on hysteresis controller which generates the digitized

signal dm. For constant flux region, the control algorithm is as follows:

If dm = 1 than SA=dB, SB=dC, SC=dA.(active vector selected) (5.3.23)

If dm = 0 than SA=0, SB=0, SC=0. (zero vector selected) (5.3.24)


91

In this way, the trapezoidal shape for the stator flux is obtain. The received shape

come from that the torque in the same flux sector, as in Fig.5.3.23, is controlled by

using only one active vector with zero vectors. In such way the stator flux vector can

move only toward the active voltage vector turn or can stop thanks to the zero vector.

Such method of control much reduces a switching frequency of the inverter.

isα u

sβ

ψsα m

e

ψsβ(ψsα) i

sβ(isα)

Fig.5.3.21. Simulated steady-state operation for the DSC method (mc=0.5,

fsw=3.8kHz, Hm=0.06).


92

Furthermore, the method does not require the switching table as in the DTC.

However there are needed the estimation of the output toque and the stator flux.

The typical steady-state operation waveforms are presented in Fig.5.3.21. The

dynamic performances for the DSC are similar as for the DTC.

The DSC is characterized by:

• the stator flux vector moves along the hexagonal trajectory for PWM

operation,

• non-sinusoidal current waveforms,

• no switching selection table required,

• no supply voltage reserve is necessary,

• a low inverter switching frequency (depended on the hysteresis torque band),

• very good torque control dynamic.

There are proposed many papers to improve the conventional DSC behaviors,

especially by taking into consideration the current distortion caused by the hexagonal

flux shape trajectory. The improvement has been achieved by: introduction 12 stator

flux sectors [80] or by processing of not only the stator flux value but also the stator

flux angle [79] or by taking into account the third harmonic flux component [6]. There

has been also fuzzy logic used to improve the DSC [45].

5.3.3. Direct Torque Control with constant switching frequency.

There have been proposed few different methods to keep constant switching

frequency [13,52,21]. However the Direct Torque and Flux control method proposed by

Habetler [23, 30], because of the mathematical approach to the problem, seems to be

more interesting.

The control system proposed in [23] are presented in Fig.5.3.25. The stator flux

vector can be calculated from the equation:

( )dtrT

t

s

N∫ −=0

1sss iuψ (5.3.25)


93

and is determined from the stator voltage vector us, the stator resistance rs and the

stator current vector is.

The electromagnetic torque can be written as a function of stator flux and current

vectors:

αββα ψψ sssse iim −=×= ss iψ . (5.3.26)

It can be proof [Appendix 1] that the inverter-fed induction machine can be analyzed

by employing the circuit [43] as in Fig.5.3.23.

ψs

Flux and torque

estimator

s

s

i

u

Induction

motor

SA

+-ud

SB

SC

me

Eq. 5.3.36

Eq. 5.3.38Eq. 5.3.39

Vector

Modulation

vc

ϕVc

ψsc

mc

esα

esβ

Fig.5.3.22. The DTC control system with the voltage modulator.

which farther can be described mathematically as:


94

rss eiiu +

′+= ssNs x

dt

dTr , (5.3.27)

us

rs

x's

er

Fig.5.3.23. A circuit of the inverter-fed induction motor drive in the stator fixed

system of coordinate.

By taking into account the equation (2.2.8a) and the formal substitution dt

dj s =ω ,

the equation (5.3.27) can be transformed to:

( )ssr iψe ss xj ′−= ω . (5.3.28)

It can be noticed that for constant period Ts the equation (5.3.27) can be

transformed also to:

s

s

Tx

′−

=∆ rs

s

eei . (5.3.29)

Than the change in electromagnetic torque over the period Ts can be written from

(5.3.26) and (5.3.29) as:

( )[ ]αββααββα ψψψψ rsrsssss

s

ss

s

e eeeex

TT

xm −−−′=′

−×=∆×=∆ rs

sss

eeψiψ ,

(5.3.30)


95

and the stator flux change as:

sTss eΨ =∆ , (5.3.31)

Let tn be the time at the beginning of the arbitrary Ts period. The reference stator flux

and torque controlled by dead-beat controller over the Ts period can be written as:

( ) enec mtmm ∆+= , (5.3.32)

( )ss ψψψsc ∆+= nt . (5.3.33)

If the Kc be defined as:

rs eψ ×+∆′

= e

s

s

c mx

TK , (5.3.34)

the equation (5.3.30) can be transformed to:

ss eψ ×=cK , (5.3.35)

and farther the internal voltage esα can be calculated as:

α

αββ ψ

ψ

s

ssc

s

eKe

+= . (5.3.36)

By taking into account the equation (5.3.33) the equation (5.3.31) can be written as:


96

( ) sn Tt ss e+= ψψsc , (5.3.37)

or as:

( ) ( )222

scψ ββαα ψψ ssssss eTeT +++= (5.3.38)

The internal voltage esβ can be calculated from the equation (5.3.38) and (3.3.36).

The reference voltage usc imply to the voltage modulator is calculated from:

sssc ieu sr+= . (5.3.39)

The direct torque and flux controller proposed in [23] is characterized by the

advantages as:

• the torque and the flux are controlled twice per switching period,

• the space vector PWM results the torque and current ripple,

• very good dynamic performance,

• uni-polar voltage PWM.

• However, the method bring the disadvantages as:

• mathematically complicated,

• no robustness for stator and rotor parameter changes.

5.3.4. Sliding mode approach for DTC as a low speed problem solution.

There have been proposed few different methods to improve the control method in

low speed operation. One of the simplest proposed methods, as it has been mention

before, was avoiding switching on the zero vectors in low speed operation [11]. Another

simple method has been tackled in [40] by adding a carrier signal to the torque

reference. However, all this methods lead to very high switching frequency. The method


97

proposed in [23] has been improved by adding better stator flux and torque estimators’

[22]

Another method proposed in [55] has solved the problem by sliding mode approach

implementation. The block scheme of the control system is presented in Fig.5.3.27.

To derive the control a reference frame fixed to the stator flux has been used, and the

following sliding modes have been chosen:

scsyy

scsxx

iiS

S

−=

−= ψψ, (5.3.40)

where the current has been calculated based on the equation:

sysxe im ψ= , (5.3.41)

The sliding mode is guaranteed by selecting the stator voltages, which fulfill the

condition:

0

0

<

<

yy

xx

SS

SS

&

&

, (5.3.42)

The equation (5.3.42) can be fulfilled by selecting:

( ) ( )( ) ( )

ysy

xsx

Su

Su

sgnsgn

sgnsgn

−=

−=, (5.3.43)

since the derivatives in the equation (5.3.42) can be written as:

( )

( )s

sy

yy

sxxx

x

uxfS

uxfS

σ+=

+=

&

&

, (5.3.44)


98

Flux and torque

estimator

s

s

i

u

Induction

motor

SA

+-ud

SB

SC

me

ψs

Sliding mode

controller

ψsc

mc

Sector detection

Ψsα Ψ

sβ

Selection table

for

sliding mode

controller

si

Sx

Sy

Fig.5.3.24. The block scheme of the sliding mode control system in DTC.

where fx(x) and fy(x) are nonlinear function of the motor state x.

The equations (5.3.43) leads to the switching table 5.3.5.

Table 5.3.5. Switching table for the sliding mode approach.

dm sector 1 sector 2 sector 3 sector 4 sector 5 sector 6

ua -sgn(Sx) sgn(Sy) sgn(Sy) sgn(Sx) -sgn(Sy) -sgn(Sy)

ub -sgn(Sy) -sgn(Sy) -sgn(Sx) sgn(Sy) sgn(Sy) sgn(Sx)

uc sgn(Sy) sgn(Sx) -sgn(Sy) -sgn(Sy) -sgn(Sx) sgn(Sy)


99

where the sector are defined as in classical DTC.

The method leads to reduction of the switching frequency. However because the

method does not use the zero vectors, still the reduced frequency is not minimal. The

author of [55] has proposed the equivalent control method for inactive vectors

introduction. The sliding modes has been selected as:

cey

scsxx

mmS

S

−=

−=

′

ψψ, (5.3.45)

In this situation the solution, precisely described in [19], leads to:

( ) ( )

( ) ( ) ( )

−+−−=

−−=

′ syeqsy

s

sxsxeqsxsyy

sxeqsxx

uux

uuiS

uuS

σψ

sgnsgn

sgnsgn

, (5.3.46)

where:

sxssyssyeq

sxsxssxeq

iru

iru

ψω

ψ

+=

+= &

. (5.3.47)

The sgn(Sx) and sgn(Sy) in table 5.3.5 can be replaced with the (5.3.46) equations.

The sliding mode approach proposed in [55] is characterized by:

• reduction of the switching frequency, mainly in low speed operation,

• better low speed operation behavior,

• achievement of the classical DTC property,

• increased complexity if compare to classical DTC controller.


100

5.3.5. Intelligent methods in Direct Torque Control

The intelligent control methods become more and more popular. It is probably

because the mathematical methods for nowadays quality expectation became very

complicated. The neural networks, in spite of often difficult structure “offers” auto

tuning for designers. It eliminates, often very difficult, system tuning. Fuzzy logic based

system however, offers to designers simple creation of controllers for the systems which

very often are mathematically undefined or just strongly nonlinear.

There are not many papers about intelligent control methods used in DTC. Many

scientists or industry engineering’s [2] think that the DTC is unequivocally defined and

there is not needed to use fuzzy logic based method. However it has been proved in few

papers [45, 59, 61] that there are some disadvantages in the DTC, which are possible to

improve thanks to the fuzzy logic based method. Of course, it can be discuss if this

method is only one exists for these problems.

Fuzzy Controller for DTC

The authors of the [61] paper have made an assumption that it can not be good

quality control if there are no distinguishing between very large and relatively small

errors. The switching states for the large errors mainly during the start up or during step

change in command torque or command flux are the same as the switching states

chosen for fine control during steady state operation. Therefore the authors have

proposed the fuzzy controller for the DTC as in Fig.5.3.25.

The fuzzy controller contains three inputs and one crisp output. The numbers of

membership functions for each variable are chosen to have the best response with

minimum number of rules.

The universe of the first input, torque error εm, is divided into five overlapping fuzzy

sets: positive large error (PLEm), positive small error (PSEm), zero error (ZEm) negative

small error (NSEm), negative large error (NLEm). The shapes of the membership

function are presented in Fig. 5.3.26a.

The second input εψ error universe has been divided into three overlapping fuzzy

sets: positive error (PEψ), zero error (ZEψ), negative error (NEψ) as in Fig.5.3.26b.


101

The third fuzzy state variable is the angle between the stator flux vector and the

reference axis α for stator fix system. For this variable the universe is divided into 12

overlapping fuzzy sets as in Fig. 5.3.26c.

Induction

motor

Flux and

torque

estimator

SaSbSc

is

us

+-ud

-

em

-

me

Fuzzy

controller

ψsc

Ψs

eΨ

mc

me

Ψs

ΘΨ

ΘΨ

Fig.5.3.25. Fuzzy controller for direct torque control of induction motor.

There is no needed the output membership function set because the control signal

(switching states) is crisp.

εm

ε Ψ

NEΨPEΨ ZEΨPLEm

PSEm

ZEmNSE

mNLE

m

a) b)

c)Θ3Θ2 Θ4 Θ5 Θ6

Θ7 Θ8 Θ9 Θ10 Θ11 Θ12 Θ1

12

π12

3π12

5π12

7π12

9π12

11π12

13π12

15π12

17π12

19π12

21π

12

23π0

Fig.5.3.26. Membership functions sets for fuzzy controller for DTC.


102

The rules for the controller has been chosen based on the conclusion method

presented in [23]. The sets of fuzzy rules are presented in Fig. 5.3.27. The inference is

based on Mamdani’s inference method [47].

εΨεmεΨ

εmεΨ

εmεΨ

εmεΨ

εmεΨ

εm

εΨεmεΨ

εmεΨ

εmεΨ

εmεΨ

εmεΨ

εm

Θ1

Θ2

Θ3

Θ4

Θ5

Θ6

N Z P N Z P N Z P N Z P N Z P N Z P

NL 1 2 2 NL 2 2 2 NL 2 3 3 NL 3 3 4 NL 3 4 4 NL 4 4 5

NS 2 2 3 NS 2 3 3 NS 3 3 4 NS 3 4 4 NS 4 4 5 NS 4 5 5

ZE 0 0 0 ZE 0 0 0 ZE 0 0 0 ZE 0 0 0 ZE 0 0 0 ZE 0 0 0

PS 4 0 6 PS 5 0 6 PS 5 0 1 PS 6 0 1 PS 6 0 2 PS 1 0 2

PL 5 5 6 PL 5 5 5 PL 6 6 1 PL 6 1 1 PL 1 1 2 PL 1 2 2

Θ7

Θ8

Θ9

Θ10

Θ11

Θ12

N Z P N Z P N Z P N Z P N Z P N Z P

NL 4 4 4 NL 5 5 6 NL 5 6 6 NL 6 6 1 NL 6 1 1 NL 1 1 2

NS 5 5 6 NS 5 6 6 NS 6 6 1 NS 6 1 1 NS 1 1 2 NS 1 2 2

ZE 0 0 0 ZE 0 0 0 ZE 0 0 0 ZE 0 0 0 ZE 0 0 0 ZE 0 0 0

PS 1 0 3 PS 2 0 3 PS 2 0 4 PS 3 0 4 PS 3 0 5 PS 4 0 5

PL 2 2 3 PL 2 2 2 PL 3 3 4 PL 3 4 4 PL 4 4 5 PL 4 5 5

Fig. 5.3.27. The set of fuzzy rules for fuzzy control of induction motor.

The system presented in [61] is characterized by:

• precisely control in steady state operation,

• the fastest stator flux and torque step response (the authors proved this by a

simulation experiments),

• extremely more complicated control structure, mainly caused by big number

of membership functions.

There is proposed [81] paper to increase the fuzzy reasoning speed by the stator flux

angle mapping table technique.

Duty Ratio Controller

As it has been proved in Section 5.3.1, the voltage vectors selected from the

switching table are not optimal in some region of stator flux vector positions. The

method proposed in [59] to improve this disadvantage is based on fuzzy logic and is

called duty ratio controller. The block diagram of the proposed method is presented in

Fig.5.3.28. In this control the selected inverter switching state is applied for a portion of

switching period, defined as a duty ratio δ, and the zero switching state is applied for the

rest of the period. The duty ratio is chosen to give average voltage vector over a


103

switching cycle, which gives the desired change in the torque, but with reduced ripple

as in Fig. 5.3.29.

Induction

motor

Switching

table

Flux vector angle

calculation and sector

detection

Flux and

torque

estimator

flux

controller

torque

controller

dΨ

dm

SaSbSc

is

us

ψs

+-ud

Ψsα Ψsβ

mc

me

ψsc

Ψs

em

-

-

me

Fuzzy Duty

Ratio Controllerem

ΘΨ

ΘΨ

eΨ

eΨ

δ

δ

Fig.5.3.28. Block diagram of the duty ratio control

T 2T 3T 4T 5T 6T 7T

T 2T 3T 4T 5T 6T 7T

T 2T 3T 4T 5T 6T 7T

T 2T 3T 4T 5T 6T 7T

Activ vector

Zero vectors

δ δ δ δ δ δ

time time

Fig.5.3.29. Influence of duty ratio on the DTC operation.


104

ΘΨ ΘΨ εm εm

The fuzzy controller contain two inputs (torque error εm and flux vector position Θ),

and one output (duty ratio δ). There are three membership functions for each input and

output, as in Fig.5.3.30.

The rules ware formulated using data obtain from the simulation of the system using

different switching states and are presented in Tables 5.3.6a and 5.3.6b. One set of rule

is used when the stator flux is less than the reference value of the flux and the other

when the stator flux is greater than the reference value.

µ

Θ

µ

m

µδ

06

π3

π

SΘ MΘ LΘ Sme

Mme

Lme

Sδ Mδ Lδ

0 0.5 1

a) b)

c)

Fig.5.3.30. Membership functions sets for fuzzy duty ratio controller: a) flux position

angle, b) torque error, c) duty ratio.

Table 5.3.6. Fuzzy rules for duty ratio control.

SΘ MΘ LΘ SΘ MΘ LΘ

Sme Mδ Sδ Sδ Sme Sδ Sδ Mδ

Mme Mδ Mδ Mδ Mme Mδ Mδ Lδ

Lme Lδ Lδ Lδ Lme Mδ Lδ Lδ


105

Product operation rule of fuzzy implication is used as inference method [48]:

( ) ( )Θ⋅= Θkmmji µεµα (5.3.48)

and the membership function µδ of the control decision is calculated as:

( ) ( )( )( )δµαδµ δδ ,supmax9

1i

i== (5.3.49)

The center of gravity is used as a defuzzification method.

The system presented in [59] is characterized by:

• reduced current and stator flux ripple,

• more complicated system, mainly for tuning,

• larger average switching frequency per one sample.

Neural Network Approach

Consider the number of papers about the applications of neural network in control of

induction machines, it looks that still neural network can not find practical application

in this field. It is probably caused by still huge neural network ASIC price.

There are existing only one paper [9] about the neural network application in the

DTC. Unfortunately, the paper was rather devoted to demonstrate the potential

application of neural network to control ac drives. The different training algorithms

have been stressed like: back-propagation, adaptive neuron model, extended Kalman

filter and the parallel recursive prediction error. To train, the neural network the set up

as in Fig. 5.3.31 has been used.

After teaching the neural network got the same results as the reference DTC scheme.


106

Induction

motor

Switching

table

Flux vector angle

calculation and sector

detection

Flux and

torque

estimator

flux

controller

torque

controller

dΨ

dm

is

us

ψs

+-ud

ψsα ψsβ

mc

me

ψsc

ψs

em

-

-

me

Neural

Networkem

ΘΨ

ΘΨ

eΨ

eΨ

uaubuc

Error

uaubuc

tatbt c

Fig. 5.3.31. Set up to train the neural network.

It has to be stressed that the neural network in this application does not give

advantages over the conventional DTC. It has just shown that the neural network can

solve even strong nonlinear problems. This motivates to pursue further research in the

application of neural networks to new types of controllers in the motor drive industry.

Specially that it is anticipated that the processors prices will fall down still rapidly [24].


107

5.4. Direct Torque Neuro-Fuzzy Controller

5.4.1. Introduction

"There are two central issues and problems in motion control. One is to make the

resulting system of controller and plant robust against parameter variations and

disturbances. The other is to make the system intelligent" [24].

One of the advantages of the DTC is robustness against variation of almost all motor

parameters. There is not robustness only for stator resistance, which is easy measurable

if comparing with other induction motors parameters. The only one problem connected

with the stator resistance is its variation due to the motor temperature. However, there

are some solution for this problem. For example in [61], there is proposed the fuzzy

resistance estimator inferenced base on the stator current error and change of current

error. The other method proposed in [54] is based on the estimated stator flux changes

observation caused by the stator resistance changes or finally the method used in ABB

DTC, where the stator resistance is updated by using a thermal model of the induction

motor [78]. All the methods are widely described in Section 3.

The fuzzy structure of the control system gives widest robustness for the

uncertainties [47, 48]. Furthermore the neural networks give possibilities to make the

system “more intelligent”. There is learning process possible.

There are only few papers that present this kind of a motion control [9, 45, 59, 61,

81]. However, most of the proposed methods despite important improvements introduce

some disadvantages. For example the approach proposed in [61] gives excellent torque

response and also very low torque distortions in steady state, but the fuzzy structure of

this regulator is very complicated. It is because of many membership functions,

especially in flux angle fuzzyfication line. This leads to a lot of fuzzy rules. Similar

problem is in [45]. In [59] the ripple of the torque were reduced by applying selected

inverter switching state for a portion of switching period and the zero switching state for

the rest of the period. The fuzzy structure in this regulation type is very simple.

However, the switching frequency is still not constant.

One of the fuzzy logic based systems objections is their more complicated structure.

This leads to self-tuning methods expectation. However, non-of the proposed fuzzy

controlled-based method has the auto tuning.


108

The Direct Torque Neuro-Fuzzy Controller except the advantages like: constant

switching frequency, uni-polar voltage PWM, no current and torque distortion caused

by sector changes, no low speed operation problems, can be on- and off-line tuned

automatically.

5.4.2. DTNFC scheme

The basic scheme of the Direct Torque and flux Neuro-Fuzzy Control (DTNFC)

method for a voltage source PWM inverter-fed induction motor is presented in Fig.

5.4.1. The classical hysteresis controllers have been changed to the controller based

neuro-fuzzy structure and a vector modulator. The rest of the whole system is the same

like the structure presented in section 5.3.1.

Induction

Motor

Flux and

Torque

Estimation

eΨ

em

Sa

Sb

Sc

is

us

ψs

me

+-u

d

γs

-

NFC

Input M

ember

ship

Funct

ions

Outp

ut Funct

ions

Vector

Modulator

Vc

ϕVc

mc

me

ψsc

ψs

-

Fig. 5.4.1. Direct Torque Neuro-Fuzzy Controller basic scheme

The reference stator flux Ψsc and torque mc are compared with actual estimated

values of flux Ψs and torque ms. The results, the stator flux (εΨ) end torque (εm) errors,


109

are the controller inputs. The outputs of the regulator are the reference voltage phase

(ϕVc) and amplitude (|Vc|) which are directly delivered to the voltage modulator.

Neuro-Fuzzy layers construction

The precisely example scheme of the DTNFC structure is presented in Fig.5.4.2.

Vector

Adder

min

min

min

min

min

min

Norm

alisation

eΨ

em

γs

o1

o2

o4

o8

min

min

min

o3

o5

o6

o9

o7

Udc

∆γi

Table

ϕVc

w11

w21

w31

w41

w51

w61

w71

w81

w91

Vc

1Layer 2 Layer 3 Layer 4 Layer 5 Layer

wΨ

wm

Fig.5.4.2. An example of the Direct Torque Neuro-Fuzzy Controller scheme.

The system is constructed based on the ANFIS structure presented in section 4.3 and

is build of five layers:

Layer 1:

Sampled flux εΨ and torque εm errors multiply by wΨ and wm weights as:

ΨΨ =′ ewe ψ , (5.4.1)

mmm ewe =′ , (5.4.2)

are delivered to the membership functions. The functions are triangular shape like

shown in Fig. 5.4.3.


110

µ(wmem)

wmem

µ(wΨeΨ)

wΨeΨ

NEGATIV ZERO POSITIVE NEGATIV ZERO POSITIVE

Fig. 5.4.3. The input membership functions for flux and torque error.

The first part outputs are calculated based on equation:

( )mmAmi ewOmi

µ=1, (5.4.3)

( )ΨΨΨ Ψ= ewO

jAj µ1, (5.4.4)

where 1

miO , 1

jOΨ - first layer output signals,

i=1,2,3 - node number for the torque error,

j=1,2,3 - node number for the flux error,

( )mmA ewmi

µ - triangular membership function for the torque error,

( )ΨΨΨew

jAµ - triangular membership function for the flux error,

wΨ - stator flux error input weight,

wm - torque error input weight,

The number of membership function are I and J for torque and flux error

respectively.

Layer 2: .

The second layer calculates the minimum what correspond to the classical fuzzy

logic system. The calculation can be writes as:

( ) ( )[ ]ΨΨΨ= ewewO

jmi AmmAk µµ ,min2

, (5.4.5)

where 2

kO - second layer output signals,

k=IJ - node number for present layer,


111

There are not every node connected together. The connections are between

outputs of membership functions with different input (as in ANFIS controller).

Layer 3: .

In the third layer the output values are normalized, in that way that the following

equation is fulfilled:

∑=

K

k

k

kO

Oo

2

2

3, (5.4.6)

where 3

kO - third layer output signals,

Layer 4:

The weight calculated in this layer as:

dkk uOO ⋅= 34, (5.4.7)

where 4

kO - fourth layer output signals,

Layer 5:

The reference voltage vector vc is a vector sum of the reference voltage vector

components that:

∑ ′=K

k

cc vv , (5.4.8)

where 44kj

k eOϕ=′cv -reference voltage component vector as:

4

kϕ - reference voltage component vector phase.

A graphical example of adding vectors is presented in Fig.5.4.4.

The reference voltage component angle fulfill the equation:


112

α

β

ψψψψs

v'c3

v'c1

v'c2

|vmax

|=ud

vmax

vmax

vc

γ γs i+ ∆

Fig.5.4.4. Reference voltage vector calculation method.

44

ΨΨ ∆+= γγϕ k , (5.4.9)

where 4

kϕ - reference voltage k component angle,

Ψγ - actual stator flux angle,

4

Ψ∆γ - angle increment value from incremental selection table.

En example for the incremental selection table for neuro-fuzzy structure with

three membership functions in each input is presented in Table 5.4.1.

Tab. 5.4.1. An example of the increment selection table.

εΨ P Z N

εm P Z N P Z N P Z N

∆γΨ 4

π 0 -

4

π

2

π

2

π -

2

π

4

3π π -

4

3π


113

5.4.3. Design and investigation of DTNFC

Increment selection table construction

The selection table construction can be based on the analysis presented in Section

5.3.1. However mathematical consideration can give much better results.

Let consider the induction motor voltage equation in stator flux fixed system

(2.4.1a):

ss

ss ψψ

iu sNs jdt

dTr ω++= . (5.4.10)

The equation can be resolve into components x and y (as in Fig.5.4.5):

dt

dTiru Nsxssx

sψ+= , (5.4.11)

sssyssy ωiru ψ+= , (5.4.12)

Furthermore, let take into consideration the torque equation (2.4.3):

( )ss iψ*

Im=em , (5.4.13)

what in stator flux coordinate system can be write as:

sye im sψ= . (5.4.14)

α

β

ψψψψs

δΨψψψψr

ϕ

us

x

y

ux

uy

Fig.5.4.5. Fluxes and voltage angles in stator flux fixed system.


114

By taking into account equations 5.4.11, 5.4.12 and 5.4.13 and assuming that:

sxssx iru >> , (5.4.15)

it can be received:

0s

0

s ψ1

ψ += ∫t

sx

N

dtuT

, (5.4.16)

s

sssy

er

um

ψψs

ω−= , (5.4.17)

what further for:

ϕcosus=sxu , (5.4.18)

ϕsinu ssyu = , (5.4.19)

can be write as:

( ) 0s

0

ss ψcosu1

ψ += ∫t

N

dtT

ϕ , (5.4.20)

s

sse

rm

ψsinuψ s

s

ωϕ −= . (5.4.21)

It should be notice that the increment angle Ψ∆γ = ϕ.

The equation 5.4.21 confirms the conclusions in section 5.3.1, that the output torque

is depended on the speed. Moreover, it can be seen that the x and y components of the

stator voltage can control the stator flux and torque respectively. Unfortunately, the

torque is also depended on the stator flux amplitude, what further mean that the torque

is not decoupled from the stator flux. However, the DTNFC method controls the stator

flux precisely what further ensure almost full decoupling.


115

During the POSITIVEm or NEGATIVEm torque error and ZEROΨ flux error there is

needed only torque control. For the increment angle Ψ∆γ =2

π and Ψ∆γ =-

2

π (see Table

5.4.1) the equations 5.4.20 and 5.4.21 will transform than to:

0ss ψψ = , (5.4.22)

s

se

rm ss

s

ψuψ

ω−±= , (5.4.23)

what give the control of torque without changes of the stator flux.

During the ZEROm torque error and POSITIVEΨ or NEGATIVEΨ flux the angle

Ψ∆γ is set to 0 or 2π. The flux is control than in accordance with the equation:

0s

t

0

ss ψdtu1

ψ += ∫NT

. (5.4.24)

However, as it has been mention before, it has also an influence on the output torque

where the value of this influence is:

s

se

rm

2

sψω= . (5.4.25)

The increment angles for both not zero errors of the flux and torque (pairs:

POSITIVEΨ and POSITIVEm, POSITIVEΨ and NEGATIVEm, NEGATIVEΨ and

POSITIVEm, NEGATIVEΨ and NEGATIVEm) is chosen as a compromise between

increments for separate flux and torque errors (last two points). For example, for the

POSITIVEΨ flux error and ZEROm torque error and for POSITIVEm torque error and

ZEROΨ flux error there has been chosen respectively Ψ∆γ =0 and Ψ∆γ =2

π. That means

that for the POSITIVEΨ flux error and POSITIVEm torque error there is chosen middle

value between 0 and 2

π increments and is equal

4

π.


116

For the ZEROm torque error and ZEROΨ flux error there is needed the output torque

to keep the reference speed without flux changes. The increment angle is than Ψ∆γ =2

π

and the torque should be equal as in equation (5.4.25).

Selectio of the membership function width

The proposed and considered system structure contains three membership functions

for each input. The extension of number of membership function is not considered. Its

because the extension can only little improve the system behavior, but the computing

time increase a lot. The number of rules, what suites to the system complication, depend

on the numbers of membership functions I (torque error functions) and J (flux error

functions) and is equal to product of them. For instance for five membership functions

for each input there would be twenty five rules, whereas for the proposed system (three

membership functions) there is only nine rules.

The shape of membership function has been chosen by consideration the calculation

time. The best functions in this situation are the triangular-shape functions as in Fig.

5.3.3.

The sampled flux eΨ and torque em errors are multiply by wΨ and wm weights (as in

Fig. 5.4.2), which in fact decide about membership functions width. It is possible to

make tuning surfaces of the flux or torque RMS errors in function of the weights wΨ

and wm, as in Fig.5.4.6 and Fig.5.4.7.

It is easy to see that the tuning curve for both flux and torque RMS errors has the

distinctive minimum. This kind of shape allows as implementing easy minimum search

method, which is described in next section. The system can be also easy and effectively

tuned manually. However, the designer should remember about the following points:

� The controller should be tuned for low speed of the motor ( 05.0≈mω in p.u.) and no

loaded system. The speed condition comes from that the flux and torque errors for the

high speed can be more reduced than in low speed operation. If the system would be

tuned for high-speed operation, than in the low speed some instability could appear.

� The flux weight should be tuned first, and than after reaching the minimum of the flux

and torque error, there should be torque error tuned. It comes from that the stator flux is

fully decoupled from the torque, as result of equations 5.4.20. (see also the surfaces

shape in Fig.5.4.6-7)


117

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

1

2

3

4

wm

wΨ

em

a)

Fig.5.4.6 Torque RMS error membership function width parameter tuning surface.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.5

1

1.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

wΨ

wm

eΨ

b)

Fig.5.4.7 Flux RMS error membership function width parameter tuning surface.


118

� The optimal weights should be search from the right side of the curve (the weights too

big). Too narrow membership functions cause that the controllers work like the

comparator what whereas causes very high flux and torque ripples. Too wide

membership function cause choosing the wrong amplitude of the reference voltage.

However the error ripples caused by that are not such intensive. See Fig.5.4.8-11.

� To avoid working in a region when the reference voltage angle is changed rapidly

(like in comparator controller working), what causes high frequency output torque

ripples, the weight parameter are taken from the right side of the minimum

The experimental results of the tuned system are presented in Fig.5.4.12.

5.4.4. Self Tuned DTNFC

There are many methods regarding tuning of fuzzy systems and neural networks. The

Adaptive Neuro-Fuzzy Inference System (ANFIS) [34, 35], used for inverted pendulum

stabilization is very similar to the presented DTNFC controller. The controller has been

tuned automatically by least square estimation method (output membership function)

and back propagation method (output and input membership function). However the

DTNFC does not need such a complicated and calculation time-consuming method to

tune the system. The proposed general DTNFC structure with auto tuning is presented

in Fig.5.4.13.

Input weights off-line tuning

As it has been mention in section 5.4.3 the tuning surfaces as in Fig.5.4.6 and 5.4.7

have not local minimums. It allows us to use search method to find the extremes. There

are two proposed methods to tune the input weights.

In the first method the minimum is found by measuring the flux and torque error

RMS value and iteratively changing the weights in small steps until the errors minimum

is detected. The tuning rules are based on the manual tuning rules as in last section. The

weak side of the method is that the used minimized signals are the flux and torque

errors, which are practically very distorted, what further causes difficulties in finding

the minimum. The first proposed search algorithm is presented in Fig. 5.4.14.


119

eψ em

isα u

sα

usβc(usαcc)

Fig.5.4.8. The steady state operation with flux membership function weight too low

(wΨ=0.13, wm=0.7).


120

eψ em

isα u

sα

usβc(usαcc)

Fig.5.4.9. The steady state operation with flux membership function weight too high

(wΨ=0.6, wm=0.7).


121

eψ em

isα u

sα

usβc(usαcc)

Fig.5.4.10. The steady state operation with torque membership function weight too low

(wΨ=0.2, wm=0.38).


122

eψ em

isα u

sα

usβc(usαcc)

Fig.5.4.11. The steady state operation with torque membership function weight too high

(wΨ=0.2, wm=1.5).


123

eψ em

isα u

sα

usβc(usαcc)

Fig. 5.4.12. Steady state operation with properly tuned DTNFC (wΨ=0.2, wm=0.7).


124

vsc

Vector

Adder

min

min

min

min

min

min

Norm

alization

γs

min

min

min

∆γi

Table

m.f. tuning o5 weight tuning

eΨ

em

ud

ϕVc

w12

w22

w32

w42

w52

w62

w72

w82

w92

1Layer 2 Layer 3 Layer 4 Layer 5 Layer

o1

o2

o4

o8

o3

o5

o6

o9

o7

wΨ

wm

Fig.5.4.13. The DTNFC with auto-tuning – general structure.

As it has been mention before, when the weights are taken from the left side of the

tuning surface, the controller started to work as a comparator. The reference voltage is

changed than rapidly. It is noticeable on the flux and output torque. The inputs weights

influence on the reference voltage, flux and torque errors is presented in Fig.5.4.8-12.

The second algorithm, presented in Fig. 5.4.15 is based on searching the weights when

the controller started to work as a comparator. There are defined another tuning curves,

presented in the Fig. 5.4.16 and 5.4.17, where the value of the input weight function is a

RMS filtered (high pass filter) reference voltage. It can be seen, in the figures, that the

functions have precisely defined point when the RMS filtered voltage become fast

increasing. Let name the left sides of the tuning curves as avoided region (Fig.5.4.16-

17). The proposed algorithm searches the band point (as in Fig.5.4.16 and 5.4.17) when

the controller enters the avoided region.


125

Start

flag_AT

Yes

flag_Ψ

Yes

eΨ(k)<eΨ(k-1)

wΨ-=∆wΨ wΨ+=∆wΨ

flag_Ψ=1 flag_Ψ=0

flag_m

Yes

em(k)<e

m(k-1)

wm-=∆w

mw

m+=∆w

m

flag_m=1 flag_m=0

Stop

Yes

Yes

No

No

No

No

No

The flux membership weight wΨoptimalization algorithm part

The torque membership weight wm

optimalization algorithm part

Fig.5.4.14. An auto-tuning algorithm for DTNFC (variant A).


126

Start

flag_AT

flag_Ψ

Yes

Urms

<Urmsref

wΨ-=∆wΨ wΨ+=∆wΨ

flag_Ψ=1

flag_m=0

flag_Ψ=0

flag_m=1

flag_m

Yes

Urms

<Urmsref

wm-=∆w

mw

m+=∆w

m

flag_Ψ=0

flag_m=1

flag_Ψ=1

flag_m=0

Stop

Yes

Yes

No

No

No

No

No

Ufilt=HPF(U

ref)

High Pass Filter

Urms=RMS(U

filt)

flag_filt

Yes

flag_Ψ=0

flag_m=1

Calculation of the RMS filtered

reference voltage

The flux membership weight wΨoptimalization algorithm part

The torque membership weight wm

optimalization algorithm part

Fig. 5.4.15 An auto-tuning algorithm for DTNFC (variant B).


127

0

0,05

0,1

0,15

0,2

0,25

0,4 0,6 0,8 1 1,2 1,4 1,6

Band point

Avoided region

Fig.5.4.16. RMS filtered reference voltage as a function of the torque membership

function width parameter wm.

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,00000 0,20000 0,40000 0,60000 0,80000 1,00000 1,20000

Band point

Avoided region

Fig.5.4.17. RMS filtered reference voltage as a function of the flux membership

function width parameter wΨ.


128

The search rules are also based on the manual tuning principles presented in last

section. However, the minimized parameter in this method is the RMS filtered (high

pass filter) reference voltage. The method requires the RMS voltage urmsref (see Equation

6.3.1) as a comparison signal, which is reached in first few steps of the tuning process

(see the algorithm in Fig.5.4.15).

The urms for too big input weights (right side of the tuning curve) is comparable with

the urms in the tuning curve band point (see tuning curve in Fig.5.4.16-17):

MINrmsRrms uu ≈ , (5.4.26)

where urmsR - RMS filtered value in the right side of the tuning curve

urmsMIN - RMS filtered value in the minimum of the tuning curve.

However the urms for too low input weights (left side of the tuning curve) is

characterized by huge increasing (see tuning curve in Fig.5.4.16-17):

MINrmsLrms uu >> (5.4.26)

where urmsL - RMS filtered value in the left side of the tuning curve

Thanks to this property it is easy to find the searched band point.

The second proposed method is not sensitive for the imprecisely measured signals as

in the first algorithm. It is because the voltage changes caused by measured distortion

are small in comparison with the voltage changes in the avoided working region.

Output weight off-line tuning

The DTNFC structure presented in Section 5.4.3 guarantee very fast flux and torque

response time. It is thanks to the lack of the integration. Unfortunately this property

causes constant torque error in steady state operation.

The classical solution for this problem is adding the integration block. However, we

would like to avoid this solution to achieve very important response time advantage.

The proposed solution for this problem is to use instead of calculate output weight o5

the value calculated from the equation:


129

( )sc5 ψ1−++= Ψkmkko cmmωω , (5.4.27)

where the kω, kΨ and km parameters are chosen to compensate the constant torque

steady state error. The simulation result of the flux and torque error without and with

compensation is presented in Fig. 5.4.18 and Fig.5.4.19 respectively.

em

eψ

Fig.5.4.18. Flux and torque error without the output weight compensation (with

properly tuned input weights).

em

eψ

Fig.5.4.19. Flux and torque error with the output weight compensation (with properly

tuned input weights).


130

5.4.5 Characteristic futures, advantages and disadvantages of DTNFC

The DTNFC is characterized by sinusoidal stator flux and current waveforms. The

whole control system is based on the only one controller with PWM vector modulator.

There are no hysteresis controllers what allow digressing of the sampling time.

Moreover, the DTNFC system is characterized by the advantages:

� constant switching frequency and uni-polar voltage PWM voltage thanks to separate

PWM block,

� low torque and current distortion,

� no current and torque distortion caused by sector changes,

� very fast torque and flux response,

� no problems low speed operation,

� lower sampling time,

� simple tuning procedure,

� possible on-line tuning,

and disadvantages:

� more complicated control structure,

� additional PWM vector modulation block required (as in FOC methods)

DSP setup and control algorithm of DTNFC

131

6. DSP setup and control algorithm of DTNFC

6.1. Introduction

Most of nowadays laboratory motor control setups are based on DSP. This kind of

laboratory allows developing and fully verifying any proposed digital control

algorithms very fast. The tested method can be easy adopted and used in many other

similar DSP setups but industry oriented. It is mainly easy, when DSP algorithms are

written in high level language C.

To verify the proposed DTNFC method the DSP based laboratory setup as in Fig.

6.1.1, has been build. The used structure is divided into three main parts:

• control part – DSP board and control cockpit (PC computer),

• low voltage part – optic isolation and measurement block,

• high voltage part – inverter and induction motor.

Such kind of configuration is very flexible and allows simple changes of all parts of

the system.

control algorithm

TMS320C31

(Fig. 6.3.1)

Ψsc

ωmc

PC

reference inputs

(keyboard control)

duty time

conversion

TMS320P14inverter

optic

isolation

interface

measuremant

box(galwanic isolation)

Udm

IA

m

IBm

ωm

m

sampled inputs

D_R

D_S

D_T

induction

motor

low voltage part

optic

fiber

s

IA

m' IBm'

ωm

m'

Udc

m'

Tacho-generator

loadDC

motor

converter

high voltage part

control part

Fig.6.1.1. Laboratory control setup used to verify the DTNFC.


132

6.2 Laboratory setup

The core of the setup is based on the dSPACE DSP board DS1102. The board is

equipped with:

• two signal processors Texas Instrument TMS320C31 and TMS320P14,

• four analog to digital converters (two 16-bit and two 12-bit),

• four digital to analog converters and the input for the n-coder.

The block diagram of the board is presented on Fig.6.2.1.

128Kx32

Static RAM

zero wait states

TMS320C31

JTAG

Interface

TMS 320P14

Digital I/O

16-bit ADC 1

16-bit ADC 2

12-bit ADC 3

12-bit ADC 4

12-bit DAC 3

Incr. encoder 1

Incr. encoder 2

12-bit DAC 1

12-bit DAC 2

12-bit DAC 4

Noise filter

Noise filter

Analo

g/d

igital I/O

connecto

r

JTAG connector

Host

Interface

PC/AT Expansion Bus

DS1102 DSP-board

Fig.6.2.1. Block diagram of the DS1102 board.

The TMS320C31 processor is the master processor where all of the control algorithm

blocks are realized. The control loop (interrupt procedure) in master processor is

triggered by the negative slope of the PWM signal, which whereas is provided by the

slave processor TMS320P14. So the sampling time of the system is depended on the

fixed timer period in the slave processor.


133

A PC Pentium 100 is used for software development and results visualization. The

algorithms are done in C language. The software is supported by very powerful

software utilities: Cockpit and Trace. The first of them is used for graphical debugging

of users chosen parameters and variables. The Trace, whereas, is used as a computer

oscilloscope which allow easy graphical visualization of time process.

The galvanic and optic isolation are used as interface between the PWM-VSI and the

DSP board.

The DC motor loads the four-pole 3.00 kW induction machine. The system allows

making four-quadrant operation tests.

6.3. Control algorithm

The DTNFC control algorithm, as it has been said, has been realized in one DSP

master processor. The structure of the algorithm is presented in Fig.6.3.1.

vector

modulatorwith dead time

compensation

Direct Torque

Neuro-Fuzzy

Controller

error

calculation

flux, torque

and speed

estimation

scaling and

transformation

of measured

variable

PI controllerwith anti wind-up

integration

auto-tuning

algorithm

Udm

IΑm

voltage

calculation

IΑm

ωm

m

Ud

ωme

Iα

Iβ

ωm

me

Ψse

mc Ψ

sc

Ψsc

ωmc

eΨ

em

Uα

Uβ

Vsc

ϕVc

D_R

∆wm∆wΨ V

sc

DSP TMS320C31

sam

ple

d inputs

reference inputs

D_S

D_T

duty

times

(connected

with

TM

S320P14)

D_R

D_S

D_T

Fig. 6.3.1. General structure of DTNFC algorithm realized in the DSP master

processor.


134

The DSP algorithm contains six key blocks:

• Direct Torque Neuro-Fuzzy Controller,

• auto-tuning algorithm,

• voltage calculation block,

• vector modulator,

• flux, torque and speed estimation block,

• PI speed controller with anti-windup integration.

Direct Torque Neuro-Fuzzy Controller

The DTNFC algorithm is based on the structure presented in Fig.5.4.2 and is wide

described in the Section 5.4.

The experimental results of the DTNFC operation are presented in the Section 7.0.

Auto-tuning algorithm

The membership functions and reference voltage increment angles of the DTNFC

can be tuned manually very easy. However, the tuning can be done automatically. The

auto-tuning algorithm, which is described in the Section 5.4, can be activated in the

beginning of the operation. The tuning should be done for no-loaded motor and for low

speed operation.

The variant B, as in Fig.5.4.15, contains high pass filter part, which is used to

calculate the reference voltage Urmsref (see Section 5.4.4 for details).

The algorithm of reference voltage Urmsref calculation is based on the equation:

+= sc

F

rmsref V

sT

sRMSU

1, (6.3.1)

where s - Laplace variable,

TF - low-pass filter time constant


135

The low-pass filter transfer function:

( )s

T

ssF

F

LP

+=

1, (6.3.2)

is converted to the discrete equation as:

( ) ( ) ( ) ( )[ ]111 −−+

−−= kxkxT

Tkyky

F

s . (6.3.3)

The experimental results of the system tuning are presented in the Section 7.0.

Voltage calculation block

The stator voltage is required for flux, torque and speed estimation. There are few

methods to receive required voltage. The first, most known, method is based on direct

measurement of the line-to-line voltage. However, this method requires double analog

to digital converters, transformers and anti-aliasing filters.

The method used in the DTNFC algorithm is based on the DC link voltage

measurement and the reference duty-times. The stator voltage can be calculated as

follows:

( )[ ]TSRd DDDUU +−= 5.03

2α , (6.3.4a)

( )TSd DDUU −=3

3β . (6.3.4b)

The disadvantage of the used method is that, the calculated voltage is equal to the

reference voltage. The reference voltage, however, is modified by the dead time and

reduced by the voltage drop in the diodes of inverter. Thus, there should be used an


136

algorithm to compensate the dead time and the voltage drop, mainly when precise low

speed operation is required. There are proposed some methods to compensate both dead

time and voltage drop [3, 28].

Vector modulator

The reference voltage amplitude and phase calculated by neuro-fuzzy controller are

put to the vector modulation block. The block calculates the duty times DR, DS and DT

of the modulated signal, which are transferred to the slave processor. The principle and

theory of the vector modulation is described in the Section 2.2.

The used algorithm realizes the vector modulator with third harmonic. The switch

times are calculated based on the equations:

( )Vc

dc

sc

U

VT ϕsin

32 = , (6.3.5a)

( )2

cos2

3 2

1

T

U

VT

Vc

dc

sc −= ϕ , (6.3.5b)

( )210 1 TTT +−= . (6.3.5c)

The switch times are further transformed to the duty times, which are calculated

based on equations for each sector as follows:

Sector 1 (3

0π

ϕ <<Vc

):

0

20

210

2

1

2

1

2

1

TD

TTD

TTTD

T

S

R

=

+=

++=

(6.3.6a)

Sector 2 (3

2

3

πϕ

π<<

Vc):


137

0

210

10

2

1

2

1

2

1

TD

TTTD

TTD

T

S

R

=

++=

+=

(6.3.6b)

Sector 3 ( πϕπ

<<Vc3

2):

20

210

0

2

1

2

1

2

1

TTD

TTTD

TD

T

S

R

+=

++=

=

(6.3.6c)

Sector 4 (3

4πϕπ <<

Vc):

210

10

0

2

1

2

1

2

1

TTTD

TTD

TD

T

S

R

++=

+=

=

(6.3.6d)

Sector 5 (3

5

3

4 πϕ

π<<

Vc):

210

0

20

2

1

2

1

2

1

TTTD

TD

TTD

T

S

R

++=

=

+=

(6.3.6e)

Sector 6 ( πϕπ

23

5<<

Vc):

10

0

210

2

1

2

1

2

1

TTD

TD

TTTD

T

S

R

+=

=

++=

(6.3.6f)


138

where DR – duty time which correspond to the switch time of inverter upper R

phase leg, DS – duty time which correspond to the switch time of inverter upper S phase

leg, DT – duty time which correspond to the switch time of inverter upper T phase leg.

Flux, torque and speed estimation block

The proposed method directly controls the stator flux and torque. It causes that the

used estimation method should be fast, precise and not sensitive for parameter

variations. The most known estimation methods and modifications of them are

presented in the Section 3.0.

The DSP algorithm use improved stator flux oriented method (see Section 3.4).

One of nowadays industry requirement is to realize sensorless applications, if it is

possible. The proposed method can work as a system with sensors and also as a

sensorless system.

For the sensorless method, the rotor speed of the motor is calculated based on the

equation:

rsm ωωω −= , (6.3.7)

The used flux estimation method calculates synchronous speed of the motor in

natural way. The slip of the motor is calculated based on the equation:

2

r

re

r

Rm

Ψ=ω , (6.3.8)

where the amplitude of the rotor flux is calculate based on the equations:

ααα

σs

m

srs

m

rr I

L

LL

L

L−Ψ=Ψ , (6.3.9a)


139

βββ

σs

m

sr

s

m

r

r IL

LL

L

L−Ψ=Ψ . (6.3.9b)

Unfortunately, the method requires knowledge of almost all of the motor parameters.

The sensorless DTNFC operation results are presented in the Section 7.0.

PI speed controller with anti-windup integration

Because of the simple structure, the most popular and used in the industry speed

controller is a PI type controller. The base structure of this controller is presented in Fig.

6.3.2.

Xref

Reference signal

∫X

act

-Y

ref

Fee

dbac

k sig

nal

Controller output

i

p

T

K

pK

Fig. 6.3.2. Base structure of digital PI controller.

The structure in Fig.3.6.2 correspond to the digital PI controller algorithm used in the

speed control loop in DTNFC, as follows:

( ) ( ) ( ) ( )

−

−++−= 111 ke

T

TkeKkyky

i

sP . (6.3.10)

The practical application of the PI controller, without any additional blocks, should

not cause any problem. However, it is always necessary since a really system has

always limitations. The output of the controller practically must be limited.

Nevertheless, adding of the limiter always causes the oscillations in the output. It is

caused by the integrator part which is integrate the error all the time, even when the


140

output signal reach the limited level. There are proposed many different methods to

stop the integration when limited level is achieved [31, 65].

Xref

Reference signal

∫X

act

-

YL

Fee

dbac

k sig

nal

Controller output

i

p

T

K

pKY

NL

iT

1

-

YE

Actuator

Fig.6.3.3. The PI regulator with anti-windup integrator.

One of the anti-windup integration methods proposed in the literature is as in

Fig.6.3.3. In this system an extra feedback path is provided by measuring the actuator

output and forming an error signal (YE) as a difference between actuator output (YL)

and the controller output (YNL) and feeding this error back to the integrator through the

gain 1/TI. The error signal YE is zero when the actuator is not saturated. The method

correspond to the digital algorithm, as follows:

( ) ( ) ( ) ( ) ( ) ( )[ ]kykyT

Tke

T

TkeKkyky

NLLLL

T

s

i

sP −+

−

−++−= 111

, (6.3.11)

which is used in the presented DSP PI controller algorithm.

The experimental results of speed step response are presented in the Section 7.0.

Simulation and experimental verification of DTNFC

141

7. Simulation and experimental verification of DTNFC

7.1 Introduction

The DTNFC method has been verified by comparison with the DTC (with modified

hysteresis). There have been used the simulation program DSIM [73] written in C

language.

The simulation structure of whole system was simplified. The inverter and the

induction motor were ideal model. Actual values of the flux and torque are taken

directly from induction motor model. They are not estimated. The stator voltage is taken

also as not anti-aliasing filtered value. The simulation structure has been based on the

DSP structure presented in Section 6.0.

The DTNFC method has been verified also experimentally. The simulation and

practical tests have been done for the same induction motor which parameters as given

in the Appendix 3.

There have been considered performance criteria to compare the DTC and DTNFC

methods as follows:

� system behaviour during steady state operation

� uni-polar voltage PWM,

� constant switching frequency,

� flux and torque ripple and distortion,

� influence of the sampling time for the system behavior,

� system behavior during dynamic state operation

� torque transients to the step changes,

� stator flux transients to the step changes,

� system behavior during magnetization,

� control variable decoupling,

� tuning of the system,

� low speed operations possibilities,


142

7.2. DTNFC simulation comparison with classical DTC

The simulation of the DTC and DTNFC has been done for the same operation

conditions, which are discussed below.

The induction motor was not loaded and the DC voltage was set to the 1.5 of nominal

value. The results have been taken for rotor speed equal 0.5 of nominal speed.

The important criteria of the induction motor behaviours are the steady state flux and

torque errors. That is why, in the beginning of tests, the controllers has been set to the

same flux and torque amplitude errors, eY=0.2 and em=4. Such errors for the DTC has

been received for the hysteresis band hY=0.018 and hm=0.035. The DTNFC input

weights has been set to wY=1.2 and wm=1.3. The tuned systems had the switching

frequency fsw=3.2 kHz and fsw=4.0 kHz for the DTC and DTNFC respectively.

The steady state operations for the DTC and the DTNFC are presented in Fig.7.2.1

and 7.2.2 respectively.

The flux and torque errors for steady state operation are presented in Fig. 7.2.1d-e

and 7.2.2d-e for DTC and DTNFC respectively. The error band, as it has been mention

before, is the same for both methods. It can be notice, that the DTC flux and torque

error ripples have much larger frequency. It is an advantage of the conventional method.

Such a frequency is easy filtered by the mechanical time constant of the motor. In the

DTNFC case, the low frequency of the flux and torque errors causes speed disturbances,

mainly noticeable in low speed operation.

It can be notice that both of systems are characterised by uni-polar voltage PWM.

The shape of the current and flux are more distorted for the DTC method. The

distortions, as it has been discussed in the Section 5.0, are mainly caused by sector

changes of the stator flux vector. There are six distortions, as sector bands, for one

period.

There is noticeable the difference in the trajectory of the stator flux. Instead of the

same flux amplitude error the flux in the DTC looks more distorted. However, this

trajectory is more optimal if concern the switching frequency. The switching frequency

for the DTC is 20% lower than in the DTNFC. However the big disadvantage of the

DTC method is that the switching frequency is not constant, what can be seen in the

Fig.7.2.3.


143

isα

isβ

usβ

ψsβ(ψsα)

eψ

em

Fig. 7.2.1. Steady state operation of the DTC with modified hysteresis.

isα

isβ

usβ

ψsβ(ψsα)

eψ

em

Fig. 7.2.2. Steady state operation of the DTNFC.


144

a)

b)

Fig.7.2.3. The transistor switches during steady state operation, a) switches b) stator

current.

Every pin in the Fig.7.2.3a represents one switch. It can be notice that the positions

of the pins are irregular. The DTNFC method uses the vector modulator what guarantee

constant switching frequency.

As it has been presented in Section 5.0 the switching frequency, and consequently

the flux and torque error amplitude in the DTC are proportionally depended on the

hysteresis bands. It can be created a tuning surface, as in Fig.7.2.4, which allow

choosing optional switching frequency for acceptable flux and torque errors.

Hψ

Hm

fsw

Fig.7.2.4. The switching frequency dependence on the hysteresis band in the DTC.


145

In DTNFC case, the switching frequency is depended on the chosen sampling

frequency, which for the symmetrical vector modulator is two times bigger. The input

weights should not be associated with the hysteresis bands. There can be chosen

different input weights for the same switching frequency. However, the input weight

changes influence the flux and torque error. There is only one optimal selection of the

input weights for the chosen switching frequency. It has been created tuning curves,

which demonstrate the flux and torque amplitude errors in function of the switching

frequency (or sampling frequency), as in Fig.7.2.5, for optimal flux and torque input

weights.

0 0.5 1 1.5 2

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

eΨ

fsw

a)

0 0.5 1 1.5 2

x 104

0

2

4

6

8

10

12

14

em

fsw

b)

Fig.7.2.5.The flux (a) and torque (b) error dependence on the switching frequency in

the properly tuned system.

It should be notice a big general difference between the DTC and the DTNFC. The

DTC method is based on hysteresis controllers which are naturally continuos. The

discrete realization of the hysteresis is possible only for huge sampling time. It is main

difficult in practical realization of classical DTC. Too low sampling time causes

additional flux, torque and current distortions, what can be seen in Fig.7.2.6 (compare

with the Fig.7.2.1). The voltage PWM is also not uni-polar. The sampling time was set

to 5ms and 50ms for the results in Fig. 7.2.1 and 7.2.6 respectively.

As it has been mention in Section 5.0, the DTC controller has problem with the low

speed operation. Because the voltage drop at the stator resistance is omitted, in low

speed operation the motor is demagnetized, what causes further classical DTC starting

problems. The simulation results of the system behavior are presented in Fig. 7.2.7.


146

eψ

em

isα i

sβ

usa

Fig.7.2.6.The DTC system behavior for too low sampling time.

isα

isβ

eψ

em

ωm

ψsβ(ψsα)

Fig.7.2.7 The DTC system behavior for low speed operation (wm=0.015).

The demagnetization effect does not exist in the DTNFC method (see Fig. 7.2.8). It

is mainly because the DTNFC calculate the optimal output voltage to compensate both

flux and torque errors (the flux vector position is not sectorizated). Absence of the

demagnetization effect causes that there are no problems with a restarting of the motor.

As it has been mention before, the low frequency torque ripples influence rotor

speed. It can be seen in Fig.7.2.8d that the speed is oscillating with the torque error

frequency.


147

isα

isβ

eψ

em

ωm

ψsβ(ψsα)

Fig.7.2.8 The DTNFC system behavior for low speed operation (wm=0.015).

ψs

me

mc

eψ e

m

isβ

isα

usβ

Fig. 7.2.9. Torque transients to the step changes in the DTC.

The most important advantage of the DTC method, if compare with other different

control methods of induction motor, is very big dynamic of the stator flux and torque.

For the tested induction motor the full torque can be received in about 1.0 ms, as in

Fig.7.2.9. Of course, as it has been discussed in Section 5.0, the torque response


148

depends on the actual operation speed. It can be also notice that the flux of the motor is

decoupled from the torque.

ψs

me

mc

eψ e

m

isβ

isα

usβ

Fig. 7.2.10. Torque transients to the step changes in the DTNFC.

ψse

ψs

me

eψ e

m

isα

isβ

usβ

Fig. 7.2.11. The flux transients to the step changes in the DTC.


149

ψse

ψs

me

eψ e

m

isα

isβ

usβ

Fig. 7.2.12. The flux transients to the step changes in the DTNFC.

The DTNFC method, in spate of vector modulation block, is characterized by the

same big dynamic. The torque transient to the step changes is presented in Fig. 7.2.10.

There is noticeable unique influence of the torque to the flux amplitude.

The flux transient to the step changes in the DTC and DTNFC is presented in

Fig.7.2.11 and 7.2.12 respectively. It can be notice that the flux response for the

DTNFC is faster than in the DTC. It is because, that in the DTNFC for zero torque

error, the reference voltage vector apply to the motor can be parallel to the reference

flux. It whereas assure the shortest possible flux response time. However, such a control

philosophy does not assure full decoupling of the controlled value. There is noticeable

in the DTNFC, a small torque ripple during the flux step, as in Fig. 7.2.12b.

7.3. Experimental verification

The DTNFC system has been verified also experimentally. The description of the

DSP algorithm is presented in Section 6. The used induction motor had the same

parameters as taking to the simulations.


150

eΨΨΨΨ (2.5%/div) 20s/div

A

em (5%/div)

B

wΨΨΨΨ (0.1/div)

C

wm (8.0/div)

D

Fig.7.2.13. Off-line experimental tuning of the input weights wΨ and wm.

All control systems before continuous operation has to be tuned. As it has been

mention in the Section 5.4.4. the DTNFC system can be tuned effectively automatically

as well as manually. The speed controller has been tuned by modulus and symmetry

criterion.


151

The experimental results for the auto-tuning of the system is presented in Fig.7.2.13.

In the beginning, the input flux and torque weights have been set to the nominal flux

and torque values. Such setting results in big flux and torque errors. When the algorithm

is calculating the filtered reference tuning voltage, the behavior of the system is not

changing (first part in Fig. 7.2.13). The second parts present the searching of the

optimal flux weights. It results in decreasing of the flux error. After receiving optimal

flux weight, the torque weight is tuned (third part in Fig.7.2.13). The whole system is

tuned after about 3 min. The tuning time can reduced, however it results in lower

precisely of the tuning.

The sensorless steady state operation of the tuned system is presented in Fig.7.2.14

and 7.2.15. The sampling time has been set to the 500µs, what gave the flux and torque

error equal 1% and 3.5% respectively, as in Fig.7.2.16. The results are taking for half of

nominal speed. It is verified that the uni-polar voltage PWM characterizes system. The

current is not distorted by the sector change as in the DTC. The stator flux trajectory is

circular.

5ms/div)

uΑΒΑΒΑΒΑΒ (400V/div)

isαααα (2A/div)

Fig.7.2.14. Experimental results for the steady state operation for the tuned DTNFC

system.


152

ΨΨΨΨsαααα

0.5Wb/div ΨΨΨΨsββββ

Fig.7.2.15. Experimental results of the stator flux trajectory for the steady state

operation.

eΨΨΨΨ (0.5%/div)

5ms/div

em(2.0%/div)

Fig.7.2.16. The flux and torque errors for the steady state operations (experimental

results).


153

The motor magnetization process is presented in Fig. 7.2.17. It can be seen that

magnetization process last about ten sampling times (about 5ms). The chosen reference

stator voltage is parallel to the stator flux vector. It results in short torque distortion

what is visible in the Fig.7.2.17d.

The DTNFC controller has been precisely tested for dynamic state operations. The

first most important property of the controller is its big torque dynamic. The torque

transients to the step changes are presented in Fig. 7.2.18. It can be seen that the flux

and torque are fully decoupled, and the flux amplitude is not changed during torque

steps. The stator current response is also presented in the figure. The Fig.7.2.19 presents

an enlargement of the torque step. The response time is about 3ms, what give the same

dynamic as in conventional DTC method.

The Fig. 7.2.20 presents the speed trajectory for open-loop speed controlled system.

It can be seen that the speed is almost linear. The soft curves are caused by variable DC

voltage during dynamic states.

ΨΨΨΨs (0.2Wb/div)

ΨΨΨΨc (0.2Wb/div)

isαααα (2A/div)

me (8Nm/div)

2ms/div

Fig.7.2.17. Experimental oscillograms for sensorless motor magnetization process.


154

ΨΨΨΨs (1Wb/div)

isαααα (10A/div)

me (10Nm/div)

mc (10Nm/div)

50ms/div

Fig.7.2.18. The torque transients to the step changes.

2ms/div

me (5Nm/div)

mc (5Nm/div)

Fig. 7.2.19. Experimental result of the torque transients to the step change.


155

0.2s/div me (2Nm/div)

ωωωωm (250rpm/div)

mc (2Nm/div)

Fig. 7.2.20. The speed response experimental results for the torque steps.

Ψs (0.2Wb/div)

5ms/div)

Ψc (0.2Wb/div)

isα (2A/div)

me (8Nm/div)

Fig.7.2.21. Experimental oscillograms for the stator flux transients to the step changes.


156

As it has been mention in last sections the DTNFC system property is characterized

by big stator flux dynamic, which is bigger then in conventional DTC. Such a property

makes the DTNFC controller useful for energy efficient systems, where big dynamic of

the flux is required. The flux response for small step change is presented in Fig. 7.2.21.

The constructed DTNFC system can work with speed sensors as well as without

them. The used speed estimation algorithm is presented in Section 6.3. It can be seen in

Fig. 7.2.22 and 7.2.23 that the controller can operate successfully also in low speed. The

slow speed reversal in Fig. 7.2.22 shows that the induction motor is do not

demagnetized in low speed region and the torque is controlled correctly. There is

noticeable the torque reversal when the motor change the rotation. The speed ramp for

fast speed reversal is presented in Fig. 7.2.23.

To test the PI speed controller, torque transient to the small speed steps changes has

been performed. It can be seen in the Fig.7.2.24, that the speed response contains small

over-regulation, which is visible also at the estimated torque. The speed response time is

about 10ms.



me (5Nm/div)

ωωωωc (100rpm/div)

2s/div

Fig.7.2.22. Sensorlless experimental oscillogram for slow speed reversal.


157



ΨΨΨΨs (1Wb/div)

ωωωωc (100rpm/div)

0.2s/div

Fig.7.2.23. Experimental oscillograms for four quadrant sensorless operations.

ωωωωm(50rpm/div)

me (10Nm/div)

ωωωωmc (50rpm/div)

0.1s/div

Fig.7.2.24. Speed transient to a small step changes (sensorless operation).

Conclusions

158

8. Conclusions

The application of neuro-fuzzy approach for Direct Torque Control of PWM

inverter-fed induction motor has been investigated though simulation and experimental

implementation. The design and tuning procedure has been described. Also, the

improved stator flux estimation algorithm, witch guarantee eccentric estimated flux has

been proposed.

The proposed DTNFC scheme has the following features and advantages:

� only one controller which can be realize in single-processor system,

� very fast torque dynamic, comparable with classical DTC,

� very fast flux dynamic what allow to use the controller for energy efficient systems,

� constant switching frequency and uni-polar voltage thanks to separate PWM block,

� absence of distortions caused by sector change as in classical DTC,

� low torque and current distortions,

� simple auto-tuning procedure based on gradient algorithm,

� possible simple and no time consuming manual tuning,

� no problems during low speed operation thanks to the separate PWM modulation

block,

� low required sampling time if compared with the ASC600 drive where the variable

are sampled with 25µs,

� possible on-line tuning thanks to neuro-fuzzy control structure,

Thanks to all of these advantages a drive with the described control structure is

suitable to almost all-industrial applications. Especially thanks to the direct flux and

torque control the method can be successfully used in electrical vehicles (for example

hybrid cars).

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Symbols employed

166

Symbols employed

dΨ - digitised flux controller output signal,

dm - digitised torque controller output signal,

eΨ - stator flux amplitude error,

em - torque error,

eγ - phase error,

e - induced internal voltage, p.u. system,

es - internal voltage vector induced by stator flux,

esα, esβ - internal voltage components induced by stator flux, in

stationary coordinate system,

esx, esy - internal voltage components induced by stator flux, in stator

flux vector oriented coordinate system,

er - internal voltage induced by rotor flux,

fsw - switching frequency of VS inverter, per leg,

HΨ - flux hysteresis band amplitude,

Hm - torque hysteresis band amplitude,

Hm1 - modified first torque hysteresis band amplitude,

Hm2 - modified second torque hysteresis band amplitude,

I - current, absolute value,

IA, IB, IC - instantaneous values of the stator phase currents,

IAM, IB

M, IC

M - instantaneous values of the measured stator phase currents,

Is - stator current space vector,

Ir - rotor current space vector,

i - current, p.u. value,

is - stator current vector,

ir - stator current vector,

iMs - magnetizing current in stator flux vector oriented coordinate

system,

iMr - magnetizing current in rotor flux vector oriented coordinate

system,

Symbols employed

167

irα, irβ - rotor current components of the induction motor in the

stationary α, β coordinate system,

ird, irq - rotor current components of the induction motor in the rotor

flux oriented coordinate system,

isd, isq - stator current components of the induction motor in the rotor

flux oriented coordinate system,

isdc, isqc - reference stator current components of the induction motor in

the rotor flux oriented coordinate system,

irx, iry - rotor current components of the induction motor in the stator

flux oriented coordinate system,,

isx, isy - stator current components of the induction motor in the stator

flux oriented coordinate system,,

isxc, iscy - reference stator current components of the induction motor in

the stator flux oriented coordinate system,,





isα, isβ - stator current components of the induction motor in the


iA, iB, iC - instantaneous values of the stator phase currents,

k - sector number,

L - inductance, absolute value,

Ls - stator winding self-inductance,

Lr - rotor winding self-inductance,

M - mutual inductance, absolute value,

mc - reference torque,

me - electromagnetic torque,

mL - load torque,

ms - estimated torque,

T - time constant, absolute value

TF - low pas filter time constant,

Symbols employed

168

TN - nominal time constant,

Ts - sampling time,

U - voltage, absolute value

Ud - DC link voltage in converter,

UdM - measured DC link voltage in converter,

UA, UB, UC - - instantaneous values of the stator phase voltages,

Us - stator voltage space vector,

Ur - rotor voltage space vector,

u - voltage, p.u. value,

uc - reference voltage,

ud - DC link voltage in converter,

us - stator voltage vector,

uν - inverter voltage vector,

usd, usq - stator voltage vector components in the rotor flux vector

oriented coordinate,

usdc, usqc - reference stator voltage vector components in the rotor flux

vector oriented coordinate,

usx, usy - stator voltage vector components in stator flux vector oriented

coordinate,

usα, usβ - stator voltage vector components in stationary α, β coordinate

system,

usαc, usβc - reference stator voltage vector components in stationary α, β

coordinate system,

SA, SB, SC - switching states for the voltage source inverter,

R - resistance, absolute value,

Rr - rotor phase windings resistance,

Rs - stator phase windings resistance,

r - resistance, p.u. value,

rr - rotor phase windings resistance,

rs - stator phase windings resistance,

rse - estimated stator phase windings resistance,

wψ - input flux membership function weight,

wm - input torque membership function weight,

Symbols employed

169

x - reactance, p.u. value,

xM - magnetizing (main) reactance,

xr - rotor winding self reactance,

xs - stator winding self reactance,

xσ - total leakage reactance,

γk - angle between real axis α of the fixed system and the real axis

of the rotating system,

γs - stator flux vector angle,

ω - angular speed, p.u. value,

ωc - cut-off frequency,

ωk - angular speed of the coordinate system,

ωm - rotor angular speed,

ωN - nominal rotor speed,

ωsΨr - rotor flux vector angular frequency,

ωsΨs - stator flux vector angular frequency,

ωse - estimated rotor speed,

ωr - rotor angular (slip) frequency,

δ - angle between stator current and rotor flux vectors,

δΨ - angle between stator and rotor fluxes vectors (torque angle),

δΨs - stator flux vector angle in stator oriented coordinates,

δΨr - rotor flux vector angle in stator oriented coordinates,

σ - total linkage factor,

Ψ - flux linkage, absolute value,

ΨΨΨΨs - space vector of the stator flux linkage,

Ψsc - reference stator flux amplitude,

ΨΨΨΨr - space vector of the rotor flux linkage,

ΨA, ΨB, ΨC - flux linkages of the stator phase windings,

ψ - flux linkage, p.u. value,

ψψψψs - space vector of the stator flux linkage,

ψsc - reference stator flux amplitude,

ψψψψr - space vector of the rotor flux linkage,

ψr - rotor flux vector amplitude,

Symbols employed

170

ψrα, ψrβ - space vector rotor flux components in fixed α, β coordinate

system,

ψsα, ψsβ - space vector ststor flux components in fixed α, β coordinate

system,

ψrd, ψrq - space vector rotor flux components in rotor flux vector


ψsd, ψsq - space vector stator flux components in rotor flux vector


ψrx, ψry - space vector rotor flux components in stator flux vector


ψsx, ψsy - space vector stator flux components in stator flux vector


ψsc - reference flux amplitude,

ψse - estimated stator flux amplitude,

ψsc - reference stator flux amplitude,

ψrc - reference rotor flux amplitude,

Rectangular coordinate systems

d, q – rotor flux oriented (rotated) coordinates,

x, y – stator flux oriented (rotated) coordinates,

α, β – stator oriented (stationary) coordinates,

Appendix

171

Appendix 1

(drawing a conclusion of circuit equation replacing the inverter-fed induction

machine standard voltage equation)

By putting the equation:

rs ψiψr

m

ssx

xx +=σ (A1.1)

which can be calculated from the equations (2.2.9a) and (2.2.9b), into the voltage

equation (2.2.8a) in stator fixed system of coordinates:

dt

dTr Ns

s

ss

ψiu += (A1.2)

it can be received:

++= rss ψiiu

r

m

ssNsx

xx

dt

dTr σ (A1.3)

The equation (A1.3) can be written as:

rss ei

iu ++=dt

dxTr s

sNs σ (A1.4)

where: dt

d

x

xT

r

m

N

r

r

ψe =

Appendix

172

and finally the equation (A1.4) can be written as:

rss eiiu +

′+= ssNs x

dt

dTr (A1.5)

where: ss xx σ=′

The equation (A1.5) describe the circuit as in Fig.A1.1.

us

rs

x's

er

Fig.A.1.1. A circuit of the inverter-fed induction motor drive in the stator fixed system

of coordinate.

Appendix 2

(drawing a conclusion of torque change dependence on synchronous speed, angle δΨ

and the angle between stator flux and stator voltage vectors)

The output torque can be calculated as in equation (5.2.2):

Ψ= δσ

sinψ1

ψ r s

r

m

exx

xm (A2.1)

and the change of the torque, as:

Appendix

173

( )Ψ= δσ

sinψψ sNr

r

me

dt

dT

xx

x

dt

dm (A2.2)

For the constant stator and rotor flux amplitude the change of the torque can be write

as:

ΨΨ= δ

δ

σ

cosψψ srdt

dT

xx

x

dt

dmN

r

me (A2.3)

In the rotor flux fixed system for the Us>>rs is the equation (2.4.1) can be write as:

ss

s ψψ

u sn jdt

dT ω+= (A2.4)

The real axis equation for the above equation can be write:

( ) ( ) ( )ΨΨΨ −+= δωϕδδ cosψsinsinψ ss ssN udt

dT (A2.5)

where angles are as in Fig. A2.1.

Appendix

174

α

β

ΨΨΨΨs

δΨ ΨΨΨΨr

ϕ

es

x

y

Fig.A2.1. The voltage, stator and rotor flux vectors angles.

The equation A2.5 can be put into the equation A2.2, what further give the equation:

( )( )ΨΨ −+= δωϕδσ

cosψsinψ sr ss

r

me uxx

x

dt

dm (A2.6)

It can be seen from the equation that the torque change decrease proportionally to the

synchronous speed ωs for a constant load and stator flux amplitude.

The change of the torque in function of angle ϕ between the stator flux and stator

voltage vectors is maximum for the 2

πϕ = , because:

( )0cosψ

/r == ϕ

ϕ σs

r

me uxx

x

d

dtdmd (A2.7)

what give the solution only for the angle 2

πϕ = .

Appendix

175

Appendix 3

(drawing a conclusion of equation for stator resistance compensation)

Let’s take in to the consideration the stator flux equations as follows:

dtes ∫= ααψ (A3.1a)

dtes ∫= ββψ (A3.1b)

where the eα and eβ are calculated as follows:

ααα sss irue ˆ−= (A3.2a)

βββ sss irue ˆ−= (A3.2a)

The quadrate of the stator flux amplitude can be calculated as follows:

[ ] [ ]222dtedtes ∫∫ += βαψ (A3.3)

The two side derivation of the above equation lead to the equation as follows:

[ ] [ ] ββαα

ψψ edteedte

dt

d ss ∫∫ += 222 (A3.4)

If it is assumed that the stator flux is constant the above equation can be write as

follows:

dteedtee ∫∫ += ββαα0 (A3.5)

Appendix

176

By taking into account the equations (A3.1), (A3.2) and (A3.5) the final equation can

be write as follows:

( ) ( ) 0=−+− βββααα ψψ ssssssss iruiru (A3.6)

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