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A new adaptive high speed control algorithm used for a FOC or a DTC PMSM
drive strategies
Conference Paper · October 2012
DOI: 10.1109/IECON.2012.6389465
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Aymen Flah
Sbita Lassaad
National Engineering School of Gabes
University of Gabès
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A new adaptive high speed control algorithm used
for a FOC or a DTC PMSM drive strategies
Flah Aymen
National school of engineering of Gabes
University of Gabes
Gabes, Tunisia
flahaymening@yahoo.fr
Abstract-This paper presents a new adaptive high speed control
algorithm (AHSC) for the permanent magnet synchronous
motor (PMSM) drives. This AHSC algorithm has been built by
using the Model Reference Adaptive System (MRAS) and with
reference to the fed inverter variables limitation. The proposal
control technique is mathematically formulated and
implemented in the two conventional motor control strategies
such as the Field Oriented Control (FOC) in one hand and the
Direct Torque Control (DTC) on the other hand. The proposed
AHSC is tested through computer simulation using MATLAB/
Simulink, where the system efficiency and robustness is verified
for a wide speed range and under the two cited drive strategies.
The obtained simulations results are satisfactory and confirmed
the usefulness of the overall proposed control algorithm.
LIST OF SYMBOLS
AHSC
vd, vq
λd, λq
id, iq
Ld, Lq
Ls
λm
Rs
Tl
P
J
Adaptive high speed control
Direct and quadrature stator voltage
Direct and quadrature stator field
Direct and quadrature stator current
Direct and quadrature stator inductance
Stator inductance
Magnet flux
Stator resistance
Load torque
Poles number
Rotor inertia coefficient.
I.
INTRODUCTION
The permanent magnet synchronous motor (PMSM) is the
most popular in the traction and electrical vehicle applications
and in the generator mode especially in the wind turbine
sector. Due to its superior advantages face to the other motor
such as induction and reluctance motors; the high efficiency,
low inertia, high torque to current ratio, high power factor and
almost no need for maintenance, and especially for its smaller
size, make this motor type the most adapted for high
performance applications [1]. Generally, these applications
can operate the motor in a speed value higher than the
nominal one, where this phenomenon is commonly called the
high speed or field weakening modes.
Effectively, the problem occurring in this speed region is the
required high performances and efficient operating mode. The
basic idea refers to operate the motor as a DC one, where the
Sbita Lassâad
National School of engineering of Gabes
University of Gabes
Gabes, Tunisia
lassaad.sbita@enig.rnu.tn
field weakening phase is naturally operated. But, in the
PMSM, the rotor flux cannot be controlled due to the rotor
permanent magnet. Therefore, the researchers are extremely
dealing to work for resolving this problem details.
The conventional PMSM control strategies as the FOC and
the DTC basic drives which are becoming very popular and
manufactured due to their advantages and effectiveness, as
presented in [2] and [3]. Generally, the speed reference is
given to the control algorithm target and this last generates
the desired stator voltage. However, in the FOC strategy, the
direct stator current must be also given as another reference
input signal [2]. In the standard applications, this second
input signal is always fixed to zero. However, in this case, the
high speed mode is unsupported and the maximum speed can
be touched is the rated one. So, refers to the DC motor
principle, the idea is to decrease the total flux by decreasing
the direct stator current to the negative region. However, this
direct stator current decrease must be also controlled for the
motor parameters and the inverter security [4].
Similarly, in the DTC method, the second reference input
signal must be the reference stator flux. This input signal
must be generated according to the high speed operating
mode and the motor and inverter safety [5].
In the literature, the high speed control algorithm is build
based on the inverter constraints as the maximum speed and
voltage. Morimoto, in his papers [4], [5] and [6], was
considered these constraints in proposed high speed control
algorithm.
The proposed Morimoto high speed control algorithm
weakness is the PMSM parameters variation influence.
Where, in the real applications, as the electrical vehicle, many
parameters as temperature, dust or vibrations can influence on
the PMSM parameters [7] that effect on the field weakening
zone. Therefore, an adaptive high speed control algorithm is
built regarding all these PMSM parameters variation and
guarantees the components safety, is extremely necessary in
this application.
In this paper, this problem is resolved and an adaptive high
speed control (AHSC) algorithm is proposed. Based on the
MRAS estimator and the Morimoto constraints, this
algorithm is build and implemented in the two conventional
control strategies, DTC and FOC, where two control schemes
are given.
This manuscript is organized as fellow. After a general
introduction section, the second one is designed for the
MRAS estimator and the third is for describing the high speed
principle. The fourth part investigates the implementation of
the HSCA in the FOC strategy and the sixth part for the DTC
one. Then, the simulation results are presented and discussed
and finally the conclusion is presented.
II.
MRAS ESTIMATOR
The model reference adaptive system (MRAS) estimator is
developed to identify the PMSM parameters, based on the
POPOV stability theory. The proposed estimator needs only
the online measurement of current ‘i’, voltage ‘v’, and rotor
speed ‘ω’ to estimate the stator resistance ‘Rs’, inductance
‘Ls’ and the rotor flux linkage ‘λm’ simultaneously. Started
from equations (1), (2) and (3), the electrical, the
electromagnetic torque ‘Te’ and the mechanical PMSM
equations are respectively [8].
did
⎧
⎪⎪vd = Rs id + Ls dt − ω Ls iq
(1)
⎨
⎪v = R i + L diq + ω L i + ωλ
s q
s
s d
m
⎪⎩ q
dt
⎧
⎛ 3 ⎞⎛ P ⎞
⎪Te = ⎜ ⎟⎜ ⎟ λd iq − λq id
2 ⎠⎝ 2 ⎠
(2)
⎝
⎨
⎪λ = L i + λ and λ = L i
s d
m
q
s q
⎩ d
P
dω
⎞
+ fω⎟
(3)
(Te − Tl ) = ⎛⎜ ⎞⎛
⎟⎜ J
⎝ 2 ⎠⎝ dt
⎠
The stator current components as state variables in the d, q
reference frame are expressed in (4).
(
)
•
X = AX + BU + C
(4)
⎡ 0
⎡ −τ ω ⎤
⎡c 0⎤
; B=⎢
Where: A = ⎢
; C=⎢
⎥
⎥
⎣ −ω −τ ⎦
⎣0 c ⎦
⎣ −e f
⎤
⎥;
⎦
⎧ X = ⎡i i ⎤T
⎪
⎣ d q⎦
;
⎨
T
⎪U = ⎡⎣ vd vq ⎤⎦
⎩
⎧
1
⎪c =
Ls
⎪
⎪
λm
= ωI f .
⎨e f = ω
Ls
⎪
⎪ 1 Ls
⎪ =
⎩τ Rs
The adjustable parameter state system is given by (5):
•
∧
∧ ∧
∧
∧
∧
X = A X + BU + C + G ( X − X )
⎡ ∧
⎤
⎡∧
⎤
−τ ω ⎥ ∧ ⎢ c 0 ⎥ ∧ ⎡ 0 ⎤
⎡k
⎢
A=
,B=
, C = ⎢ ∧ ⎥ and G = ⎢ 1
∧⎥
⎢
⎢ ∧⎥
⎢ω I f ⎥
⎣0
⎣
⎦
⎢⎣ −ω −τ ⎥⎦
⎢⎣0 c ⎥⎦
∧
(5)
0⎤
.
k2 ⎥⎦
G represents the correction gain matrix to be chosen so
as to achieve pre-specified error characteristics, where k1and
k2 are two limited positive real.
By subtracting the adjustable parameter system (5) from
the state system (4), the result is defined in equation (6).
(6)
e = ΔA e + ΔB U + ΔC + G e
From (6) is decomposed a feed forward linear model
“(A+G)” and a non linear feedback system “ w ” as expressed
in (7).
•
e = ( A + G)e + w
(7)
Based on the POPOV stability theory as presented in [9]
and [10], the study of the system stability can be resolved.
After checking the two stability clauses and especially by
decompressing the second POPOV condition the adaptive
parameters equations are expressed in (8). Where kp, ki are
the PI parameters.
⎧ ^
^
⎪R
^
kir ⎞ ⎛ ^
⎞ Rs
⎛
s
⎪ ^ = − ⎜ k pr +
⎜ i d ed + i q eq ⎟ + ^ (0)
s ⎟⎠⎝
⎪L
⎝
⎠ L
s
⎪ s
⎪⎪ 1 ⎛
kil ⎞
1
(8)
⎨ ^ = ⎜ k pl + ⎟ vd ed + vq eq + ^ (0)
s
⎠
⎪ L ⎝
Ls
⎪ s
^
^
⎪
kif ⎞
⎛
λm
λm
⎪
= − ⎜ k pf +
⎟ ω eq + ^ (0)
^
⎪
s ⎠
⎝
Ls
Ls
⎪⎩
(
)
(
III.
)
ADAPTIVE HIGH SPEED CONTROL ALGORITHM
The constant flux produced by the rotor magnet λm,
characterizes the PMSM motor. Refers to the flux equations
presented in (2), the total flux decrease is possible only if the
direct stator current is controlled. Because the stator
inductance Ls and the flux λm are constant and the quadratic
stator current is proportionally to the load torque. At a rated
speed range, the direct stator current is fixed to zero, so the
idea, in the high speed, is to reduce the total flux by
decreasing the direct stator current to the negative region.
But, it is safety to not undergo a limit decreasing value which
avoids the machine demagnetization problems.
Refers to Morimoto limitation [5], the idea is to follow the
system of equation presented in (9).
2
⎧⎪ I d2 + I q2 < I max
(9)
⎨ 2
2
2
⎪⎩Vd + Vq < Vmax
Therefore, the principle of field weakening is described in
Figure.1a, which presents three zones. The first is the circle
that presents the current limitation equation in the first part of
the expression (9). The second is the ellipse form designed
the voltage limitation in the second part of the equation (9).
The field weakening region is presented by the hachured
circle, designed the interconnection between the two last
zones. Generally, the current limitation circle is centered in
zero point and her radius depends on the inverter current
maximum. It is necessary to indicate, that the limitation
voltage contour, can be formed as an ellipse or a circle form:
if Ld≠Lq, or Ld=Lq, respectively. The voltage limitation
contour characteristics are characterized essentially by the
center point, which depends on the magnet flux and stator
inductance values, and the radius value, which depends also
on the speed value. Well, if the speed increases, the radius
value is reduced. The maximum of current and voltage are
usually set by the inverter [5] and [6].
iq
is max
ω1
ω2
ω3
iq
is max
C
o
−λm /Ld
(a)
iq
is max
ω1
ω3
The FOC strategy is characterized by three PI controllers,
the first one is called the PI speed controller, used to generate
the quadrature reference stator current after compared the
desired and the real speed values. The second one is used to
regulate the quadrature stator current for generating the
quadrature reference stator voltage. The last one is used to
control the direct stator current and for producing the direct
reference stator voltage. After obtaining these reference
voltages in the rotating dq frame, Clarck transformation is
used in order to obtain the corresponding three phases
reference stator voltages. Through the PWM bloc the
corresponding IGBT switches states are obtained. Then the
built stator voltages are feeding the motor. The AHSC-FOC
algorithm is placed in this control strategy for generating the
corresponding direct reference stator current. The scheme in
Figure (2) shows the AHSC-FOC algorithm and the overall
FOC PMSM drive scheme as in Figure (3).
ω2
ω3
is max i q
C
o
∧
is max
C
Rs
id
∧
o
−λm /Ld
id*
2
Ls
−λm /Ld
(b)
AHSC ALGORITHM WITH FOC STRATEGY
IV.
iq
is max
ω1
ω2
2 ⎡ 2
2 2⎤
2 2 2
⎧ Δ = 4Vmax
⎣ Rs + ω Ld ⎦ − 4ω Rs λm
⎪
⎪
2
⎨b = 2 Ld ω λm
⎪
2
2 2
⎪⎩a = Rs + ω Ld
id* =
∧
λm
2
2
Vmax
− Rˆ s2 I max
− ⎡⎣ω Lˆsiq ⎤⎦ − 2Rˆ siqλˆmω − ωλˆm
ωLˆs
iq*
ω*
(c)
ω
Fig. 1. Field weakening principle and limitation
Fig. 2. Adaptive high speed control algorithm in the FOC case
(AHSC-FOC)
Where Imax and Vmax are respectively the maximum inverter
phase-current and phase-voltage amplitudes. So, the two
inverter limit conditions in the high speed range, cited in (9)
and all the PMSM parameters are considered, a new direct
stator currents expressions is here presented (10).
2
2
Vmax
− Rˆ s2 I max
− ⎡⎣ω Lˆs iq ⎤⎦ − 2 Rˆ s iq λˆmω − ωλˆm
ω Lˆ
ω*
iq
vd
(10)
v*q
iq
id
2
id* =
ω
v*d
id
vq
s
Figure.1b and 1c illustrate the influence of the PMSM
parameters variation on the field weakening zone especially
the stator inductance and the permanent magnet flux.
To ovoid the demagnetization problems, it is necessary to
define the direct stator current limit condition, which, can be
obtained if the quadrature component is equal to zero. Then,
the direct stator current limit is expressed in equation (11).
id lim
Where
iq = 0
=
−b − Δ
2a
(11)
∫
ω
Fig. 3.FOC strategy in a wide speed range
V.
AHSC ALGORITHM WITH DTC STRATEGY
The DTC principles can be shown in figure (5), where the
errors between the reference and the estimated torque (ΔTe)
and flux (Δλ) are fixed as the input of two-level hysteresis
comparators. Through the switching table, the selection of the
corresponding voltage vector is generated. This switching
table is explained in the reference [2] and [3]. Generally, in
the DTC applications, the reference torque is given. However
in this application, the desired reference torque is obtained
after the appliance of the desired speed goal.
As indicated previously, the stator reference flux value is
extremely important if the high speed running mode is
applied. Many works about this field in this control strategy
are studied as presented in [11] and [12]. However, in the
same that Morimoto, the sensitivity to motor parameter
variation is the weakness of these methods. Additionally, the
chattering phenomena presented in the case of [12], present
another disadvantageous. In other hand, the inverter security
is not taken account. Therefore, a different high speed control
algorithm is applied in this control strategy.
Based on the good performance obtained by the AHSCFOC algorithm, the AHSC-DTC algorithm is implemented on
the DTC technique with a less change. Effectively, a PI
controller is added to regulate the direct stator current and to
produce the reference direct stator flux. Based on system of
equations presented in (12) and characterized the flux vector
and the system (13), where θ is the angular position, the
reference stator flux is obtained in equation (14).
⎧ λ = λ2 + λ2
α
β
⎪⎪ s
⎨
λ
⎛ β⎞
⎪θ s = tan −1 ⎜
⎟
⎪⎩
⎝ λα ⎠
⎡ λ α ⎤ ⎡ cos( θ )
⎢λ ⎥ = ⎢
⎣ β ⎦ ⎣ sin( θ )
λ
*
s
λ *2d
=
(12)
− sin( θ ) ⎤ ⎡ λ d ⎤
.⎢ ⎥
cos( θ ) ⎥⎦ ⎣ λ q ⎦
+ λ *2q
(13)
(14)
ω
Te*
iq*
⎛ 3 ⎞⎛ P ⎞
Te = ⎜ ⎟⎜ ⎟(λmiq )
⎝ 2 ⎠⎝ 2 ⎠
λq = Ls iq
λq*
λs*
∧
Rs
∧
Ls
∧
λm
ΔTe
τT
Δλ
τλ
Te
λs
*
Te* λs
→
Vs =
∧
Rs
∧
Ls
∧
λm
2
UDC Sa +Sb ei2π/3+Sc ei4π/3
3
(
)
vd
vq
id
iq
*
ω ω
θ
∫
Fig. 5: High speed PMSM control scheme based on the DTC strategy
VI.
SIMULATION RESULTS
In the simulation results section, the AHSC algorithms are
tested and verified in the two conventional control strategies
FOC and DTC. The motor parameters used in these
applications are given in the table (1).
TABLE: 1 PERMANENT MAGNET SYNCHRONOUS MOTOR SPECIFICATIONS
The AHSC-DTC is then illustrated in figure (4).
ω*
(λs
λd*
2
2
2
− Rˆ s2 I max
− ⎡⎣ω Lˆsiq ⎤⎦ − 2 Rˆ siq λˆmω − ωλˆm
Vmax
id* =
ω Lˆs
id
Fig. 4: Adaptive high speed control algorithm in the DTC case
(AHSC-DTC)
The overall control scheme illustrates the DTC strategy in
the high speed mode can be then presented in figure (5).
Parameters
Rs
5.2Ω
Ld=Lq=Ls
21.5m.H
λm
0.24Wb
J
85e−6kg.m2
f
0.00036Nm.s.rad-1
Rated characteristics
Power
400W
ω
2500rpm/min
Torque
1.5N.m
Current
I=2.5A
Maximum Current
5A
In order to show the performance of the proposed AHSC
algorithm, a hardly test in a wide speed region is applied.
Effectively, the given variable reference speed is started from
0 rpm to the rated values. Then, from the rated one to exceed
it up to 4000 rpm. In this application, the load torque applied
on the motor is similar to the electric vehicle approach. Well,
the load torque decreases if the desired speed exceeds the
rated one.
In figure 6 and 7, the present results are illustrated in the
FOC case. These figures show respectively the evolution of
the speed and the electromagnet torque in the case of the
100% of the rated speed and in the case of 160% of the rated
one. The corresponding direct stator current, generated from
the AHSC-FOC is shown in the same figure. Figure.7 makes
clear the field weakening principle. It is important to indicate
that the maximum speed can be more increased but only if we
decrease more the lad torque and unsure the maximum
current condition.
R1 and R2 are the radius of the corresponding flux
trajectory, when the reference speed is 100% of the rated
speed and when the desired one is equal to 160% of the rated
speed.
R1
R2
Fig. 9: Flux trajectory in the case of DTC with field weakening
VII.
Fig. 6: The corresponding FOC Speed, torque and direct stator
current results in 100% and 160% of the rated speed
R1
R2
Fig. 7: Flux trajectory in the case of FOC with field weakening
The DTC case is also applied and the AHSC-DTC is
implemented as presented in the figure. 5. The corresponding
results in figure. 8 and 9 show respectively, the speed, torque,
direct stator current and the flux components. The satisfaction
of the proposed AHSC-DTC algorithm is verified under the
same condition as the FOC case.
Fig. 8: The corresponding DTC Speed, torque and direct stator
current results in 100% and 160% of the rated speed
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CONCLUSION
An adaptive high speed control algorithm has been
presented. MRAS estimator and the Morimoto constraints are
used to build the proposed high speed control algorithm. This
proposed algorithm is formulated and applied in the two
conventional control strategies as the FOC and the DTC. The
effectiveness of the proposed high speed control algorithm is
proven over simulation results where a wide speed region
equal to 160% of the rated speed is highlighted. The drive
system has operated satisfactory in a wide speed range
suitable for electrical vehicle applications.
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