Predictive Torque Control of Interior Permanent

Magnets Synchronous Motors in Stator Co-ordinates

Vanja Ambrožič, Klemen Drobnič, and Mitja Nemec

University of Ljubljana, Faculty of Electrical Engineering, Tržaška 25, 1000 Ljubljana, SLOVENIA

e-mail: vanjaa@fe.uni-lj.si

Abstract - This paper proposes a possible approach to control the synchronous machines with interior (buried) permanent magnets (IPM SM). If properly controlled, this construction allows for an increase in torque through a reluctance component that adds to the main torque caused by the permanent magnets. Since the overall torque depends on the load angle and stator current magnitude, the optimal relationship between these two variables can be pre-calculated prior to the machine start. The inverse procedure allows a determination of the unique load angle and current reference, which, in turn, form a stator flux. The latter is calculated from discretized basic voltage equation in each sampling interval. This procedure is called Predictive Torque Control (PTC). An output variable (reference stator flux) is then passed to the block for Immediate Flux Control (IFC), already tested on induction machines. Its aim is to generate the actual stator flux by selecting proper inverter voltage vectors and determining their application time. The results have been verified on experimental model of a real machine.

I. INTRODUCTION

In recent years, the implementation of synchronous

machines in servo drives has increased due to several

advantages compared to induction machines (IM). This is

mainly owed to the mathematical modeling of the motor in

rotor coordinates, thus obtaining a simple torque equation

(so-called Field Oriented Control – FOC) [1, 2]. The

advantage becomes even more obvious in IPM SM, where, in

order to obtain optimal torque, a proper ratio between both

current components has to be determined.

In the past few years, a revival of the approach developed

in the eighties is competing with FOC. Direct Torque Control

(DTC) is a very simple approach to simultaneous control of

torque and flux through the selection of a proper inverter

voltage space vector [2]. DTC has evolved since including

several modifications. However, the basic idea is still to

perform the control in a stationary (stator) reference frame,

which is very useful in sensorless drives, and to obtain the

fast torque response, without classical cascade structure.

While DTC is very simple to implement in IM and

synchronous machines with surface mounted magnets (PM

SM), its implementation in IPM SM is more complicated [3-

5].

Another important aspect in control of electrical drives is

the predictive control, where discrete states of the control or

estimated variables can be calculated in advance, by utilizing

very fast modern digital processors. In the field of permanent

magnet synchronous machine drives, predictive control is

used either to complement existing approaches based upon

DTC or FOC [6, 7], or for complete control of the drive [8].

A simple approach of predictive control of the (stator) current

has also been implemented in PM SM in field coordinates,

[9]. A very comprehensive overview of the latest

developments in the field is presented in [10].

In this paper, a further development of the approach

developed for IM, called Predictive Torque Control, is

proposed. It has been successfully implemented in IM [10].

According to the peculiarities of IPM SM, especially

concerning the maximum torque to be obtained, PTC had to

be modified. After a brief theory, a basic idea of PTC with

necessary flux generation algorithm IFC is presented,

followed by the experimental results of the startup, steady

state and speed reversal.

II. TORQUE EQUATIONS FOR IPM SM

The main control quantity in electrical drives is the

electrical torque. The contribution of synchronous torque (as

a consequence of a permanent magnet) and reluctance torque

(due to different inductances in direct and quadrature axes) in

IPM SM is most obvious in rotor/field coordinates d and q:

( )( )SqSdSqSdSqPMel iiLLip −+= λτ

2

3, ( 1)

where iSd, iSq denote stator current components and LSd, LSq

(LSd < LSq ) inductances in corresponding coordinates. λPM is

the flux of the permanent magnet and τel the electrical torque.

Of course, the issue of choosing the correct relationship

between both current components is of great importance in

order to find the balance between torque and total current

drawn from the rectifier.

The control in field coordinates has some peculiarities.

Apart from the necessary coordinate transformation, this type

of control is structured as a cascade, usually with PI

controllers, where each inner loop adds to the time delay [1,

2].

In recent approaches, the control is performed in stator (a,

b) coordinates, with some advantages such as the suitability

for sensorless control and faster dynamics. Direct Torque

Control (DTC), as a representative of this approach,

implements the simultaneous control of both the flux and

torque by choosing the proper inverter voltage vector to be

impressed. Thus, the current loop is omitted and the transient

is very fast. However, at least in the classical approach, the

hysteresis control used for selecting the instant when to

978-1-4244-9312-8/11/$26.00 ©2011 IEEE 1823

switch between different voltage vectors (inverters’

transistors combination) leads to either high switching

frequency or high ripple. Therefore, the selection of a proper

hysteresis band plays an important role in drive’s

performance but it also depends on the operating point [11].

In the following chapters, first the IPM SM equations will

be rewritten in a stationary reference frame, after which a

new concept of control will be presented.

III. PREDICTIVE TORQUE CONTROL

The aim of Predictive Torque control (PTC) is to generate

the reference stator flux, which is then actually produced by

Immediate Flux Control (IFC) algorithm. The calculation is

performed in stator reference frame.

Stator flux in IPM SM is easiest to define in d,q rotor

coordinates as

( )FC

S Sd Sd PM Sq SqL i jL iλ= + +λ , ( 2)

being λPM the flux of the permanent magnet, while in a

stationary reference frame it takes the following form:

FC j

S S Sa Sbe jε λ λ= = +λ λ ,

where ε is the rotor (flux) angle.

Considering Park transformations, the solution of the above

equation yields stator flux components in stator reference

frame a,b:

( )( )ελ

εεελ

cos

sincossin 2

PM

SaSbSqSdSaSdSa iiLLiL

+

+−−+=

( 3)

( )( )ελ

εεελ

sin

sincossin 2

PM

SbSaSqSdSbSqSb iiLLiL

+

++−+=

Stator current components in the stationary reference frame

are defined as

αcosSSa Ii =

( 4)

αsinSSb Ii = ,

where IS is the current vector magnitude and α is the stator

current vector angle.

The general equation for electrical torque in a stationary

reference frame is

( )SaSbSbSael iip λλτ −=2

3. ( 5)

Considering (2) – (5), as well as the trigonometric rules, we

can rewrite the torque equation in its final form

( )( )

−+= δδλτ 2sin

2sin

2

32

SSqSd

PMSel

ILLIp , ( 6)

where

δ = α − ε. ( 7)

The relationship between vectors is shown in Fig. 1.

In surface mounted PMSM, the second term in the brackets

equals zero, yielding a known torque equation [9]:

( ) ( ).2

3sin

2

3SqPMPMSel ipIp λδλτ ==

Equation (6) states that the same torque can be obtained for

different combinations of (IS, δ). In order to obtain maximum

torque, the optimal angle δ has to be calculated by

0=∂

∂

δ

τ el .

The solution of the equation gives [5]

( )( )

−

−−+=

SqSdS

PMSqSdSPMopt

LLI

LLI

4

8arccos

222 λλδ . ( 8)

Again, for the surface mounted PMSM, the optimal angle

is π/2. As already established for field coordinates [5], the

optimal angle δopt is not constant but depends also on the

instantaneous stator current magnitude. The above equation

gives a set of pairs where current magnitudes IS (e.g. from 0

to maximum value) are linked to unique angle values δopt.

The mesh in Fig. 2 shows the relationship between all

combinations of pairs IS [0 A – 50 A] and angles δ [0 – π] and

α ε

δ λλλλPM

iS

a

b

Fig. 1. Vector diagram for IPM SM.

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a corresponding torque from (6) for a machine used in testing

(Appendix I). The solid curve shows the maximum possible

torque, which is obtained by first calculating the δopt for every

possible current value from (8) and then inserting the

corresponding pairs (ISopt

, δopt) into (6).

The inverse procedure also holds true: if maximum torque

has to be generated, a unique set of data containing (ISopt

,

δopt), corresponding to a curve in Fig. 2, can be obtained. This

relationship can be established prior to the real time control

and stored as a table for an easy and fast application or, which

is more suitable to DSP applications, as polynomials.

For a machine used in testing, a third order polynomial was

calculated

( )( ) 0123 aaaa elelel

opt +++= τττδ

( 9)

( )( ) 0123 bbbbI elelel

opt

S +++= τττ

For this particular machine, the influence of the reluctance

torque is relatively small and therefore the load angle is

around π/2 for all torques:

a3 = 0; a2 = 0; a1 = 0.0085; a0 = 1.5708;

b3 = 0; b2 = –0.009; b1 = 5.6746; b0 = 0;

IV. CONTROL STRATEGY

A. Predictive Torque Control

In this paper, the main control strategy always generates

the maximum possible torque, depending on the demands

from the speed control loop. In previous chapter, the unique

relationship between each electrical torque value and a

corresponding pair (ISopt

, δopt) has been established. The aim

of the PTC [12] is to calculate the reference flux at the

beginning of the sampling interval n∆t that would produce the

desired torque at the end of the sampling interval (n+1)∆t.

Therefore, from the previous relationship, the values for

current magnitude and angle can be calculated. After

inserting them into the flux equations, considering (3) and (4)

and some simplifications, the reference stator flux

components at the end of the interval are obtained as follows:

( ) ( ) ( )( ) ( )( )( )( ) ( )( ) ( )( )

* 1 1 cos cos 1

sin sin 1 cos

opt opt

Sa S Sd

opt

Sq PM

n I n L n n

L n n n

λ ε δ

ε δ λ ε

+ = + + −

− + +

( 10)

( ) ( ) ( )( ) ( )( )( )( ) ( )( ) ( )( )nnnL

nnLnIn

PM

opt

Sq

opt

Sd

opt

SSb

ελδε

δελ

sin1sincos

1cossin11*

+++

+++=+

Due to the fact that the motor’s mechanical time constant is

several orders of magnitude larger than the sampling interval

(being ∆t in order of few tens of microseconds), we can

assume that the rotor position does not change during the

sampling interval:

)1()( +≈ nn εε . ( 11)

Therefore, the angle can be measured (or estimated) at the

beginning of the interval. Equation (10) shows that the flux at

the end of sampling interval depends on the known

variables/parameters: flux of the permanent magnet and its

angle. It also depends on the current magnitude Is and load

angle δ, which, in order to obtain the optimal torque, are

associated to the reference torque. The output variables of

PTC are stator flux components that should be generated in

such a way to get the actual torque as close as possible to the

reference one. This desired flux values are then passed to the

IFC, whose only task is to generate the desired flux.

After calculating the sum of squares from (10), it is easy to

prove that in steady state, where rotor angle is the only

variable, flux magnitude remains constant.

B. Immediate Flux Control

The concept of Immediate Flux Control has already been

presented and implemented in control of induction machines

[12]. Unlike with PTC, where the control strategy depends on

the machine being used, IFC can be implemented on all AC

machines. Its aim is to generate the stator flux as close as

possible to the reference one (e.g. provided by PTC), with

fastest possible dynamics and a good trade-off between

switching frequency (losses) and ripple.

IFC has already proven to work fine in induction machine

drives. The basis for IFC is the stator voltage equation, from

which it is obvious that the flux vector will change into the

direction of the applied active inverter voltage vector (taking

into account the resistive drop too):

SSSS R

dt

div

λ−= .

Fig. 2. Electrical torque as a function of stator current and load angle

(mesh) and maximum torque (solid line).

1825

If discretized, due to the implementation of the control

algorithm in microcomputer, the previous equation can be

rearranged into the form, here written in stator coordinates

a,b (depending on the index, the equation is valid for each

coordinate)

( ) ( ) ( ) ( ) ,1 ,,,, onbSaonbSaSbSabSa tnvtniRnn +−=+ λλ ( 12)

where n denotes the sampling instant and ton the duration of

the application of an active vector. If a zero vector is applied,

the initial flux “naturally” decreases into the direction of

instantaneous current, due to the, usually small, resistive

drop.

Basically, there are two approaches to IFC, depending on

the calculation of the application time of the active voltage

vector. In the first variant, the voltage vector (either active or

zero) is applied for the whole sampling interval (ton = ∆t). In

this case, a simple algorithm decides whether it is better to

apply the active or a zero vector, in order to obtain the

smallest possible flux error at the end of the sampling

interval.

In the second variant, the application time of the active

inverter voltage vector is actually calculated through a simple

and fast algorithm, again to obtain the smallest flux error.

Usually, ton is smaller than ∆t, unless a substantial reference

flux change has occurred. Therefore, for the remaining time

within the sampling interval, a zero vector is applied and the

flux decreases naturally.

Fig. 3 shows the vector diagram for both variants of IFC.

Subscript “0” denotes already decreased flux vector due to

immanent resistive voltage drop. Also, in this figure, various

flux error vectors are denoted with εεεελ.

In the presented case, if using first variant, the zero vector

is preferred instead of an active voltage vector (here, v3),

since εεεελ0I < εεεελV

I. In the second variant, the flux is brought to

the smallest distance (given the active voltage vector v3) from

the reference flux.

The selection of the variant depends on whether smaller

(and variable, such as in DTC [11]) switching frequency

(variant I) or smaller ripple (with higher and constant

switching frequency – variant II) is desired.

The complete control scheme of PTC and IFC, as a

synthesis of the aforementioned procedure is shown in Fig. 4.

V. RESULTS

The performance of PTC with IFC has been verified by

applying the scheme in Fig. 4 to a laboratory model of a

permanent magnet motor. The motor data are given in

Table I. As already seen from Fig. 2 and machine data, due to

very small inductances, their difference is also small

(although relatively high). Therefore, the machine has a very

small reluctance torque and is operating basically as a PMSM

with δopt ≈ π/2 throughout the entire operating range. The

control has been performed by a TI DSP 320F2808 operating

at a sampling frequency of 15 kHz. In this experiment, only

second variant of IFC (II), namely the one with modulated

voltage, has been tested [12].

Fig. 5 shows one of the stator flux components in a,b

reference frame during start-up to 1700 rpm. The shape of the

flux is as expected; magnitude is constant and a constant

frequency is reached within about 50 ms, as the speed reaches

steady state (Fig. 6).

Current shape is shown in Fig. 7. As expected, since using

IFC II, current ripple is pronounced due to extremely low

inductances. Also, since the switching frequency in IFC II

compared to SVM is at least one third lower, the ripple is

even more visible.

This effect can be observed more in detail if enlarging the

torque transient for a current step change and comparing it

with the one from FOC with SVM (Fig. 8). The first obvious

difference is much higher dynamics of PTC, which always

produces maximum torque without any delay. In contrary, PI-

controllers in FOC with SVM, whose parameters are a

compromise between transition time and dynamic error, are

more conservative regarding dynamics.

)1(* +nSλ

)1( +nI

Sλ

)1( +nII

Sλ

)1(0 +nSλ

)1(0 +nI

λε

)1( +nII

Vλε

ontn ⋅)(3v

tn ∆⋅)(3v )1( +nI

Vλε

1v

6v

a

b

5v

4v

3v 2v

Fig. 3. Vector diagram for the first (superscript I) and second

(superscript II) variant.

Q1 Q3 Q5

Q2 Q4 Q6

vDC

2

iS

PI ctrl.

6

to

Q1 … Q6

IPM

SM

ω PTC

IFC

motor

model

λλλλS∗ τel

∗

ω∗

λλλλS

IE εεεε

iS

Fig. 4. Block scheme of PTC with IFC for IPM SM.

1826

0 0.01 0.02 0.03 0.04 0.05 0.06-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

λ (Wb)

t (s)

Fig. 5. One phase stator flux component (in a, b reference frame) during

start-up.

The results in Fig. 8 were taken as internal values from the

processor prior to modulation (SVM for FOC and IFC II for

PTC) and demonstrate the peculiarity of the control of PTC

with IFC. Namely, the output of the PI-controller in FOC has

a distinctive shape for a step response (here the initial step is

negative), with a slight overshoot and a moderate dynamics.

Nevertheless, the shape is smooth and static error negligible.

On the other side, IFC with PTC exhibits the highest

possible dynamics, but the ripple is much accentuated due to

the nature of IFC. As stressed in [12], the second variant of

IFC calculates a switch-on time based on the predictive

algorithm. Due to the very high current slope owed to low

inductances, this switch-on time is mostly very short, in fact

very frequently shorter than a safety dead-time designed to

prevent short circuits in the inverter leg. As a consequence,

for the entire sampling interval zero voltage vector is applied.

This causes the current to slowly decrease from one sampling

interval to another until the pre-calculated switch-on time

becomes higher than the dead-time and an active voltage is

impressed causing high ripple.

VI. CONCLUSION

In this paper, a method for predictive control applicable to

the IPM SM, has been presented. The control is performed in

stator coordinates. The torque at the end of the sampling

interval is obtained through a two-step process. First, the

appropriate flux is determined using PTC. Then, its actual

generation is generated by the predictive algorithm IFC.

The experimental results, presented in the paper, encourage

further development of the approach and its implementation

to other types of PMSM. Special attention is to be paid to the

analysis of the efficiency of the drive, depending on the IFC

variant and sensibility to the parameters’ variations.

0 0.01 0.02 0.03 0.04 0.05 0.060

200

400

600

800

1000

1200

1400

1600

1800

2000ω (Rpm)

t (s) Fig. 7. Rotor speed at start-up.

0 0.01 0.02 0.03 0.04 0.05 0.06-25

-20

-15

-10

-5

0

5

10

15

20

25i (A)

t (s)

Fig. 6. One-phase (in a, b reference frame) stator current during start-up.

3 3.5 4 4.5 5 5.5 6-25

-20

-15

-10

-5

0

5

i (A)

t (ms)

Fig. 8. One-phase (in a, b reference frame) stator current during start-up;

comparison between SVM (dark) and IFC (grey).

1827

TABLE I IPM SM DATA

P = 3 kW ωmax = 4000 rpm

Vph=16.3 V Rs = 0.04 Ω

λPM = 0.039 Wb LSd = 45 µH

p = 3 LSq = 103 µH

τel = 3.6 Nm J = 16.82 kg cm2

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