The main purpose of this paper is to describe a DTC (direct torque control) method for fourswitch brushless dc motor (BLDCM) drive. In the method, a novel voltage space vector modulation scheme, an optimal switching table, and a flux observation method are proposed. Eight voltage vectors are summarized, which are selected to control BLDCM in SVPWM pattern, and an optimal switching table is proposed to improve the torque distortion caused by midpoint current of the split capacitors. Unlike conventional flux observers, this observer does not require speed adaptation and is not susceptible to speed estimation errors, especially, at low speed. Global asymptotic stability of the flux observer is guaranteed by the Lyapunov stability analysis. DCoffset effects are mitigated by introducing a PI component in the observer gains. This method alleviates the undesired current and torque distortion which is caused by uncontrollable phase. The correctness and feasibility of the method are proved by simulation and experimental results.
1. Introduction
As BLDCMs have such good features as simple construction, high reliability, light electromagnetic pollution, and high power density, they are used extensively in servo systems and lowpower drive systems
[1
,
2]
.
FSTPI (fourswitch three phase inverter) as a reconfiguration topology for fault condition of traditional six switch inverter has been widely used
[3

5]
. But the uncontrollable phase current causes unsymmetrical voltage vector in FSTPI, and its waveform is much of distortion from rectangular
[3]
. A new speed control method using the acceleration feed forward compensation is proposed to improve the speed response characteristic for a fourswitch threephase BLDCM
[4]
. The disturbance torque estimation method is adopted to improve the robustness of the method. But it is just verified to be economical and efficient in some occasions with light load, such as robot arm. A study on the generated torque ripples due to phase commutation is presented in the fourswitch threephase BLDCM
[6]
, in which a current control technique is developed to minimize commutation torque for the entire speed range. But Hall sensor is used in this method and the cost of the system is raised. A novel direct torque control scheme
[7]
including the actual prestored backEMF constants vs. electrical rotor position lookup table is proposed for BLDCM drive with two phase conduction scheme using fourswitch inverter. And lowfrequency torque ripples and torque response time are minimized. But the control method is more complicated, and does not discuss flux change characteristics.
In this paper, a novel voltage space vector modulation strategy and an optimal switching table are proposed to provide a basic condition for improve torque control. A single neuron adaptive PID method is proposed to observe flux for providing the accurate control information. The operating principle of FSTPI for the threephase BLDCM drive and the proposed control scheme are explained. The validity of the proposed system is verified by simulation and experimental results.
2. Phase Current Analysis of Traditional Four Switching Patterns
The FSTPI topology consists of 4 power switches that provide two of the inverter output phases. The third phase is fed by the dc link from the center of a splitcapacitor, as shown in
Fig. 1
.
Power Circuit of FSTPI
In FSTPI system, there are four possible switching patterns to generate threephase currents, as shown in
Fig. 2
with ideal switches; these four switching patterns are V(0101), V(1010), V(1001), and V(0110), where “0” means the switch is turned off and “1” the switch is turned on in inverter. In the same figure, the free wheeling diodes are ignored. In sixswitch inverter, two zero voltage space vectors cannot supply the dclink to the load, therefore no current flows through the load. The main difference of the fourswitch inverter compared to its counterpart sixswitch one is that one phase is always connected to the center tap of the split capacitors, so that there will always be current flowing through that phase even with voltage vectors V(0101) and V(1010), as shown in
Fig. 2
. Under balanced load condition with fourswitch topology, there will be no current flow through the phase which is connected to the midpoint of the split capacitors using two possible nonzero voltage vectors, V(1001) and V(0110), as shown in
Fig. 2
. When voltage vectors V(1001) and V(0110) are used and the load is not completely balanced, only the resultant current of the other two phases flow through the phase connected to the midpoint of the split capacitors. This phenomenon will cause the ripple of current and torque. The simulation curves of current under 15rpm and 5Nm load using fourswitching pattern is shown in
Fig. 3
.
Conventional fourswitch voltage vector topology
Current curves using fourswitching pattern
In addition, three different cases may be found at the commutation moment:
Case A
: decaying current (
i_{d}
) vanishes at the same time rising current (
i_{r}
) reaches to its final value +I in
Fig. 4(a)
.
Case B
: decaying current (
i_{d}
) vanishes before rising current (
i_{r}
) reaches to it final value +I in
Fig. 4(b)
.
Case C
: rising current (
i_{r}
) reaches the value +I before decaying current (
i_{d}
) vanishes in
Fig. 4(c)
.
Current behaviors during commutation
In case B and case C, current disturbance flowing through the phase connected to the midpoint of the split capacitors will cause distortion of magnetic linkage and pulsation of torque.
To solve above problems, this paper proposed a novel space vector modulation method and an optimal switching table.
3. DTC of BLDCM Using FSTPI
The typical mathematical model of a threephase BLDCM is described as:
Where,
ν_{ao}
,
ν_{bo}
, and
ν_{co}
are phase voltages;
R_{s}
is stator resistance;
L_{s}
is stator inductance ;
i_{a}, i_{b}
, and
i_{c}
are phase currents;
e_{ao}, e_{bo}
, and
e_{co}
are phase backEMFs.
 3.1. Novel space vector modulation method
Generating a 120 electrical degree current conduction is inherently difficult with the conventional fourswitch topology, because a BLDCM with nonsinusoidal back EMF (i.e. trapezoidal) requires a quasisquare wave current profile to generate constant output torque compared to that of a permanent magnet synchronous motor with sinusoidal backEMF requiring sinewave current. These currents which have 120 electrical degrees conduction period are synchronized with the flat portion of the corresponding phase backEMFs, therefore a smooth electromagnetic torque can be obtained. As a result, at every instant of time only two phases conduct and the other phase is supposed to be inactive.
Although four voltage vectors in conventional fourswitch inverter system are sufficient enough to control the threephase ac motors using PWM techniques, it will cause the ripple of current in BLDCM drive, which is shown in
Fig. 3
. So, additional voltage vectors are required for BLDCM with two phase conduction mode in order to control the midpoint current of the split capacitors at a desired value
[8
,
9]
. Since the conventional method cannot provide a two phase conduction method completely, a new modulation scheme with new switching patterns should be developed such that only two of the three motor phases conduct.
To obtain the switch modes of operation in fourswitch BLDCM drive, a novel voltage vector selection lookup table is designed as shown in
Table 1
and the
Table 1
is concluded by the fourswitch voltage vector topology which is shown in
Fig. 6.
Based on the
Table 1
, implementation of the voltage space vectors is depicted in
Fig. 5
The controlling voltage space vector and switch combination
The controlling voltage space vector and switch combination
Voltage space vector chart (V_{x}=V_{ao}+V_{bo}+V_{co}, x= 0,1,2,3,4,5,6,7)
 3.2. Optimal switching table
Traditional DCC (direct current control) technique employs current hysteresis control
[3
,
10]
. Normally, sixpossible voltage space vectors of fourswitch topology are supposed to be used in
Table 2
as shown in
Figs. 6(a)

(f)
similar to the sixswitch version, however two of the voltage vectors V2 and V5 as shown in
Fig. 6
create problems in the current control. When they are directly used in the voltage vector selection table (
Table 2
), back EMF of the uncontrolled phase (phasea) generates undesired current and distortions occur in each phase current. As a result, undesired electromagnetic torque is inevitable. Therefore, when the rotor position is in the sec.1, 2, 4, 5, 7, 8, 10 and 11, special switching pattern should be adapted
Fourswitch voltage vector topology
FourSwitch voltage space vector selection for BLDCM drive
FourSwitch voltage space vector selection for BLDCM drive
Additional two voltage vectors V
_{0}
and V
_{7}
which are unused in conventional fourswitch PWM scheme are included in the optimal switching table to obtain torque in twophase conduction fourswitch BLDCM drive. The reason is there will be always current trying to flow in phase –a due to its backEMF and the absence of switches controlling its current. As a result, there will be a distorted current in phase–
a
as well as in phase–
b
and –
c
. Therefore, voltage space vectors of phase–
b
and –
c
conduction can be difficult to implement for BLDCM drive unless some modifications are applied to overcome the backEMF effect of the phase –
a
in these conditions. Selecting the right switching pattern to control the current on phase–
b
and –
c
independently will reduce the distorted currents on those phases and result in a smoother overall current and electromagnetic torque.
In order to improve the above problems, a DTC method has been proposed as shown in
Fig. 7
and explained in detail below :
Schematic of BLDCM control system
The DTC method controls instantaneous torque and flux to achieve highperformance operation
[11

13]
. For this purpose, an optimized switching table must be defined based on the output states of the instantaneous stator flux magnitude hysteresis controller and the electromagnetic torque hysteresis controller, together with the equivalent sector, in which instantaneous stator flux space vector is located
[14
,
15]
. The outputs of hysteresis controller for torque and flux are shown in (3) and (4).
where Δ
λ_{s}
is the stator flux hysteresis bandwidth and Δ
T_{e}
is the torque hysteresis bandwidth. Since there are eight available switching states in FSTPI, stator flux (
αβ
) plane is divided into twelve different sectors spaced by 30 electrical degrees and an optimized lookup table (see
Table 3
) is used to translate these two control states, together with the stator flux position information defined in equivalent sector, to the inverter gate drive signals
[16
,
17]
.
Voltage vector selection in sectors II and V for fourswitch BLDCM drive (CCW)
Voltage vector selection in sectors II and V for fourswitch BLDCM drive (CCW)
In the following, how to eliminate flux disturbance and torque pulsation is explained, when rotor position is in sec. 2 and in CCW direction.
Case A
(
τ_{T}
=
τ_{λ}
= 1 ): in this case,
and
, it should increase flux and torque, and V
_{5}
is selected.
Case B
(
τ_{T}
= 1 and
τ_{λ}
= 0): in this case,
and
, it should increase torque and decrease flux, and V
_{3}
is selected.
Case C
(
τ_{T}
= 0 and
τ_{λ}
= 1): this case is in contrast to the case B, and it should select V
_{6}
.
Case D
(
τ_{T}
= 0 and
τ_{λ}
= 0): this case is in contrast to the case A, and it should select V
_{2}
.
In the same way, it can be analyzed, when rotor position in other sectors.
In addition, the direction of the rotor is important to define the specific switching pattern. If the rotor direction is CW, then the above claims are reversed.
This technique has the advantage of not requiring an integration step in the estimation calculation, thus removing problems associated with drift and integral windup
[18
,
19]
. However, it does not rely on the motor parameters and an expensive position sensor.
In the next section, stator flux observation using single neuron adaptive PID is proposed and the observation principle will be analyzed in detail.
4. Flux Calculation
In the
αβ
reference frame, the model of flux can be described as:
where
λ_{s}
= [
λ_{sα} λ_{sβ}
]
^{T}
is the stator flux matrix.
The flux errors are given by:
Where
e_{λ} =[ e_{λα} e_{λβ}]^{T}
is the stator flux error matrix, denotes the estimated quantities,
ν_{s}
=[
ν_{sα} ν_{sβ}
]
^{T}
is the stator voltage matrix,
i_{s}
= [
i_{sα} i_{sβ}
]
^{T}
the stator current matrix.
Based on (6), flux observer is build using single neuron adaptive PID in
Fig. 8
.
Schematic of observer of λ_{s}
In
Fig. 8
, the meaning of each part is explained as follows:
After discretization of (6), the input of single neuron adaptive PID is
Assuming the Quadratic performance index function is
In order to make modification of weighted coefficient
w
_{i=1,2,3}
decrease along the reduce direction of
E
(
t
) , and (9) can be concluded.
Learning algorithm is as follows
Where, K is neuron proportional coefficient,
w
_{i=1,2,3}
is the weighting efficient of
x
_{i=1,2,3}
, 0 <
w
_{i=1,2,3}
<
U
_{max}
, and
U
_{max}
is maximum amplitude.
Single neuron weighting efficient can be expressed
where,
η_{p}, η_{I}
, and
η_{D}
are proportional, integral and differential weights of learning rate, respectively.
PID parameters are
5. Stability Analysis and Parameter Selection
 5.1 Lyapunov stability analysis
Define a Lyapunov candidate function
In the learning process, the change of
ν
(
t
) can be expressed as follows
Assuming
e
(0) = 0 , (15) can be concluded.
The change of
ν
(t) can also be expressed as follows
Assuming
H
= [
∂e
(
k
) /
∂w
(
k
)]
^{T}
= (
∂e
(
k
) /
∂w
(
k
))[
∂u
(
k
) /
∂w
(
k
)]
^{T}
, (17) can be concluded.
Substituting (17) in (15) yields
For global asymptotic stability, Δ
ν
(
k
) < 0 . Hence, the following equations can be deduced:
So, the scope of the learning rate is
For Δ
ν
(
k
) < 0 , (21) can be concluded from (14)
So learning algorithm is convergence, and control system is stability.
 5.2 Parameter selection
When the error is large, proportional coefficient
K
should large for having fast startup speed. And when the error is small,
K
should small for preventing overshoot. Generally, nonlinear adjustment formula for K is shown in (20)
[20]
.
Where k
_{0}
is steadystate proportional coefficient,
A
is regulation coefficient.
But a critical problem of the lowspeed operation of the flux observer is the dcmeasurement offset. In reality,
ν_{α}
and
ν_{β}
are reconstructed from the dcbus voltage Vdc and the switching status. Any dcmeasurement error, switch voltage, and deadtime voltage are reflected in both
ν_{α}
and
ν_{β}
, especially in lowspeed performance. In
Fig. 9
, torque and flux estimation error curves are provided under given speed 1500rpm and 3Nm load conditions. In
Fig. 9(a)
, it is obvious that the operational performance of control system is poor.
The effects of the dc offset can be alleviated by introducing a variable PI component in regulation coefficient as follows:
Where
k_{p}
=
R_{s}
/
L_{s}
, and
k_{i}
= 1/
L_{s}
.
A
≈
k
_{0}
/10 is practical for a real drive.
The advantages of (23) is the function of last item can be obviously increased, when the error is large at the beginning operation or flux error is large, especially at lowspeed. With the increasing of speed or flux error becoming small, the function of last item is obviously weakened, and the overshoot is limited. After compensation, the result is shown in
Fig. 9(b)
.
Torque and flux estimation error
6. Simulation and Experiment Results
According to above analysis, it can get the expressions of rotor position, speed and electromagnetic torque as follows.
Direct torque control system structure of BLDCM is shown in
Fig. 7
. Based on
Fig. 7
, simulation model has been built using Matlab/Simulink, and parameters of BLDCM are shown in
Table 4
. Simulation results are shown in
Fig. 10
.
Parameter of BLDCM
Fig. 10
describes the simulation results which is control system starts up with 5Nm load, and load sudden drops from 5Nm to 1Nm. From
Fig. 10(a)
, it can be seen flux observation curve is smooth, and good observation effect of flux can provide accurate information for system control. In
Fig. 10(b)
and
(e)
, current curve of DTC is smoother than that of DCC, and DTC can improve the current distortion causing by uncontrolled phase. In
Fig. 10(c)
,
(d)
,
(f)
, and
(g)
, it is shown that the DTC can effectively alleviate commutation torque ripple, and improve the torque and speed response.
Simulation curve
Based on the above research content, BLDCM control system has been established with IPM (Intelligent Power Module) devices, and dsPIC6010A is used as main controller. Experimental results are shown in
Fig. 11
and
12
under torque changed from 5Nm to 1Nm at high speed (1500rpm) and low speed (15rpm).
Fig. 13
shows the experimental results under traditional DCC, when torque is changed from 5Nm to 1Nm at high speed (1500rpm) and low speed (15rpm).
In
Fig. 11(a)
and
Fig. 12(a)
, since the dc voltage compensation has been introduced, flux observation precision has been improved and it can provide accurate flux size and position information for the DTC.
From
Fig. 11(b)
,
Fig. 12(b)
and
Fig. 13(a, b)
, it can be seen that the DTC can improve current distortion causing by uncontrolled phase, and the current response speed is faster than that of traditional DCC.
From
Fig. 11(c, d)
,
Fig. 12(c, d)
and
Fig. 13(c, d, e, f)
, it can be seen that the DTC can improve torque pulsation, and the DTC has better dynamic and static performance characteristics of torque and speed than that of traditional DCC.
Experimental curves for DTC under high speed
Experimental curves for DTC under low speed
Experimental curves for traditional DCC
7. Conclusion
In this paper, a novel voltage space vector modulation scheme, an optimal switching table and a single neuron stator flux observer for direct torque control were proposed. The flux observer does not require speed adaptation and is not susceptible to speed estimation errors, especially, at low speed. The stability of the proposed observer is proven by the Lyapunov stability analysis. The effects of dcmeasurement offsets are mitigated by incorporating an integral compensating term in the observer gain. The proposed single neuron flux observer is capable of delivering high performance over a wide speed range, including very low speeds. As a consequence, current distortion and torque pulsation are effectively improved, and the response speed of speed and torque are significantly improved. The proposed DTC can also be used in servo control for BLDCM.
Acknowledgements
This work was supported by Universities Science and Technology Fund Planning Project of Tianjin (20130419); National science and technology support plan (2013 BAJ09B03).
BIO
Lei Pan He received the Ph.D. degrees from HeBei University of Technology, Tianjin, China, in 2014. He is currently a lecturer of Tianjin Chengjian University. His research interests are power converters and motor drives.
Beibei Wang He received Bachelor and Master degree of Engineering in electrical engineering and power electronics from Liaoning Technical University, Liaoning, China in 2006 and 2009 respectively. He is currently a lecturer of Tianjin Chengjian University. His research interests include power converter system and electrical motor drives.
Gang Su He received the Ph.D. degrees from Nankai University, Tianjin, China, in 2005. He is currently a professor of Tianjin Chengjian University. His research interests are control of power converters and motor drives.
Baohua Cheng He received the Ph.D. from Tianjin University, Tianjin, China, in 2014. He is currently a lecturer of Tianjin Chengjian University. His research interests are active power filter and refrigeration system energy saving technology.
Guili Peng He received the Master degrees from Southwest University of Science and Technology, Mianyang, Sichuan, China, in 2007. He is currently a lecturer of Tianjin Chenjian University. His research interests are automatic control theory and Electrical and electronic technology.
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