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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.
1824
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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