Robot and Servo Drive Lab.
Implementation Of Finite-State Model Predictive Control
For Commutation Torque Ripple Minimization
Of Permanent-Magnet Brushless DC Motor
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS,
VOL. 60, NO. 3, Page.896~Page.905, MARCH 2013,
By Changliang Xia, Senior Member, IEEE, Yingfa Wang, and Tingna Shi
Professor： Ming-Shyan Wang
Student ： Shih-Yu Wu
Department of Electrical Engineering
Southern Taiwan University
2016/7/15
Outline






Analysis Of Commutation Process
Reduction Of Commutation Torque Ripple
Numerical Simulations And Analysis
Experimental Result And Analysis
Conclusion
References
2016/7/15
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Abstract

This method proposes a unified approach for suppressing commutation torque
ripple over the entire speed range without distinguishing high speed and low
speed and overcomes the difficulties of commutated-phase-current control,
avoiding complex current controllers or modulation models.

A discrete-time noncommutated-phase-current predictive model of BLDCM
during commutation is established. According to the predefined cost function,
the optimal switching state is directly selected and applied during the next
sampling period so as to make the slope rates of incoming and outgoing phase
currents match in the course of commutation, thus ensuring the minimization
of commutation torque ripple.
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Department of Electrical Engineering
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Robot and Servo Drive Lab.
Mathematical Model Of Commutation Torque


Permanent-magnet BLDCM conventionally operates in two phase 120
(electrical) conducting mode, and this conducting mode includes a
commutation region and a noncommutation region.
This paper focuses on the commutation region, aiming at reducing
commutation torque ripple
2016/7/15
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Department of Electrical Engineering
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Robot and Servo Drive Lab.
Mathematical Model Of Commutation Torque

The mathematical model of BLDCM can be expressed as :
d
ia  ea  un
dt
d
ud 0  Rib  Ls ib  eb  un
dt
d
uc 0  Ric  Ls ic  ec  un
dt
ua 0  Ria  Ls

3 phase balanced :
ia  ib  ic  0
(1)
(2)
ua 0 , ub 0 , and uc 0 are the terminal voltages of the threephase winding
ia , ib , and
ic are the phase currents of the three-phase winding
ea , eb , and e c are the back EMFs of the three-phase winding
un is the neutral point voltage
R and Ls are the phase resistance and equivalent phase inductance
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Mathematical Model Of Commutation Torque



The commutation of the motor from phase A → C conduction to phase B
→ C conduction is taken as an example of commutation process
A phase is floating phase
C phase is noncommutated phase
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Mathematical Model Of Commutation Torque

Generally, the electromagnetic torque developed by BLDCM is given by :
Te 

1
r
(ea ia  ebib  ec ic )
(3)
r is the rotor mechanical angular velocity
Te is the electromagnetic torque.
Supposing that back EMF maintains constant value E during commutation,
the commutation torque of BLDCM can be expressed as
Te 
1
( Eia  Eib  ( E )ic )
r
2 Eic E
2 Eic
 Te 

(ia  ib  ic ) 

r

 Te 
2016/7/15
2 Eic

E
(ia  ib  ic )

r
2 Eic
 Te  
  kT ic

Department of Electrical Engineering
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(4)
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7
Mathematical Model Of Commutation Torque
where E is the flap top value of the trapezoidal back EMF,
kT is the torque constant,
ic is the noncommutated phase current.


It can be seen from the aforementioned analysis that the commutation
torque developed in the commutation process is proportional to
noncommutated phase current.
Therefore, noncommutated phase current can be taken as the evaluation
criterion of commutation torque.
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Robot and Servo Drive Lab.
Mathematical Model Of Commutation Torque

As known in [11], the relationship between the slope rate of falling current and that of
rising current indicates that there exist three different cases of commutating current in the
winding
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Noncommutated-Phasecurrent Constant u d = 4E

Known from the analysis in [11], the necessary and sufficient condition for
eliminating commutation torque ripple is
dic dia dib


0
dt
dt dt

(5)
The following result is derived:
d
0  Ria  Ls ia  ea  un (5.1)
dt
ud  Rib  Ls
d
ib  eb  un (5.2)
dt
0  Ric  Ls
d
ic  ec  un (5.3)
dt
ud  R(ia  ib  ic )  Ls
d
(ia  ib  ic )  (ea  eb  ec )  3un (5.4)
dt
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Noncommutated-Phasecurrent Constant u d = 4E


3 phase balance、phase resistance neglected 、 ea  E 、 eb  E 、ec   E
代入公式(5.4)  ud  (ea  eb  ec )  3un
ud  E
(5.5)
3
u E
d
0  Ls ia  E  d
dt
3
(5.6)
ud  2 E
d
 ia 
dt
3Ls
u E
d
ud  Ls ib  E  d
dt
3
2u  2 E
d
(5.7)
 ib  d
dt
3Ls
 un 

(5.5)代入(5.1)

(5.5)代入(5.2)

(5.5)代入(5.3)
2016/7/15
u E
d
ic  E  d
dt
3
u  4 E
d
 ic  d
(5.8)
dt
3Ls
0  Ls
Department of Electrical Engineering
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11
Noncommutated-Phasecurrent Constant u d = 4E


Taking the beginning of the commutation as the time origin, the initial phase
currents are given by
ia (0)  I ， ib (0)  0 ， ic (0)   I
ia (t ) 
ud  2 E
tI
3Ls
(5.9)
ib (t ) 
2ud  2 E
t
3Ls
(5.10)
ic (t ) 
ud  4 E
tI
3Ls
(5.11)
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Noncommutated-Phasecurrent Constant u d = 4E

Current
ia vanishes at the same time the current ib reaches its final value I
ia (t1 )  0
ib (t1 )  I
0
ud  2 E
t1  I
3Ls
 t1 
3ILs
ud  3E
2ud  2 E
t1
3Ls
3ILs
 t1 
2ud  2 E
I 
( A  C)  (B  C)
(a) Noncommutated-phasecurrent constant
3ILs
3ILs

ud  3E 2ud  2 E
2016/7/15
ud = 4E
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(
d
d
ia =
ib ; u d = 4E)
dt
dt
13
Robot and Servo Drive Lab.
Noncommutated-Phase-Current Dips

Noncommutated-phase-current dips: With the presence of
d
d
ia >
ib during commutation, the conduction status of the inverter
dt
dt
ud  Ria  Ls
d
ia  ea  un
dt
d
ib  eb  un
dt
d
0  Ric  Ls ic  ec  un
dt
ud  Rib  Ls
2ud  R(ia  ib  ic )  Ls
d
(ia  ib  ic )  (ea  eb  ec )  3un
dt
2016/7/15
floating Phase A
14
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Noncommutated-Phase-Current Dips
3 phase balance、phase resistance neglected 、 ea  E 、 eb  E 、 ec   E
 un 

2ud  E
3
The following result is derived:
u  2E
d
ia  d
dt
3Ls
u  2E
d
ib  d
dt
3Ls
2ud  4 E
d
ic 
dt
3Ls
2016/7/15
u  2E
d
d
ia  ib  2 d
0
dt
dt
3Ls
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(6)
15
Noncommutated-Phase-Current Dips



The sum of the slope rates of phase A current and phase B current can
satisfy condition (5) by correctly switching power switch T1 between
conduction and shutdown.
Hence, commutation torque ripple can be reduced.
As shown in Figure, improved commutating phase current is given in
broken lines.
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Noncommutated-Phase-Current Spikes

Noncommutated-Phase-Current Spikes : With the presence of
d
d
ia <
ib during commutation, the conduction status of the inverter
dt
dt
0  Ria  Ls
d
ia  ea  un
dt
d
ib  eb  un
dt
d
ud  Ric  Ls ic  ec  un
dt
0  Rib  Ls
ud  R(ia  ib  ic )  Ls
d
(ia  ib  ic )  (ea  eb  ec )  3un
dt
2016/7/15
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Robot and Servo Drive Lab.
Noncommutated-Phase-Current Spikes
3 phase balance、phase resistance neglected 、 ea  E、 eb  E 、 ec   E
 un 

ud  E
3
The following result is derived:
u  2 E
d
ia  d
dt
3Ls
u  2 E
d
ib  d
dt
3Ls
2u  4 E
d
ic  d
dt
3Ls
2016/7/15
u  2E
d
d
ia  ib  2 d
0
dt
dt
3Ls
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(7)
18
Noncommutated-Phase-Current Spikes



Therefore, the sum of the slope rates of phase A current and phase B
current can satisfy condition (5) by correctly switching power switches T2
and T3 between conduction and shutdown.
Hence, commutation torque ripple can be reduced.
As shown in Figure, improved commutating phase current is given in
broken lines.
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Analysis Of Conduction Status

It is known from the aforementioned analysis that, under the
condition of constant dc-link voltage, by adopting a proper
control strategy that switches the inverter from three different
conduction statuses.
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FS-MPC



Making full use of the inherent discrete nature of power inverters, by
computation of a discrete model, FS-MPC predicts the future behavior of the
system in the next sampling period under each possible state of power
switches.
In addition, it makes use of predictive behavior to evaluate the predefined
cost function and finally selects and generates the optimal state of power
switches.
A control objective, cost function which is a scalar criterion measuring
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FS-MPC

The cost function of noncommutated phase current during commutation is
defined as:
*
G  i (k  1)  inon cmt (k  1)



(8)
i*
usually the output of the speed loop, is the reference current
inon cmt is the predictive value of noncommutated phase current during
commutation
k + 1 is the (k + 1)th sampling time
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Noncommutated-Phase-Current Predictive Model




As known from the preceding section, noncommutated phase current is
taken as the evaluation criterion of commutation torque.
Hence, only noncommutated phase current needs to be measured, while
there is no need to consider the slope rates of falling current and rising
current.
Noncommutated-phase-current predictive model in different commutation
processes is established as follows.
1. Phase C as noncommutation phase: From (1) and (2), the following
equations are derived:
ua 0  ub 0  (2ib  ic ) R  Ls (2
ub 0  uc 0  (ib  ic ) R  Ls (
dib dic
 )  (ea  eb ) (9)
dt dt
dib dic
 )  (eb  ec )
dt dt
2016/7/15
(10)
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Noncommutated-Phase-Current Predictive Model


From (9) and (10), the following expression is obtained:
di
ua 0  ub 0  2uc 0  3Ric  3Ls c  (ea  eb  2ec ) (11)
dt
dic
dic (ic ( k )  ic ( k  1))
Approximating the derivative dt by dt 
and then
Ts
replacing it in (11), the discrete model of (11) is obtained as
(ua 0  ub 0  2uc 0 )Ts (ea  eb  2ec )Ts  3Ls ic (k  1)

3RTs  3Ls
3RTs  3Ls
is the sampling time
ic (k )  

Ts
2016/7/15
(12)
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Noncommutated-Phase-Current Predictive Model

The discrete model (12) needs to be iterated for the (k + 1)th prediction. Then,
the predictive current of phase C is given as
ic (k  1)  
(ua 0 (k  1)  ub 0 (k  1)  2uc 0 (k  1))Ts (ea (k  1)  eb (k  1)  2ec (k  1))Ts  3Lsic (k )

3RTs  3Ls
3RTs  3Ls
(13)

2. Phase B as noncommutation phase: Similarly, the predictive current of phase
B is given as
ib (k  1)  
(ua 0 (k  1)  uc 0 (k  1)  2ub 0 (k  1))Ts (ea (k  1)  ec (k  1)  2eb (k  1))Ts  3Lsib (k )

3RTs  3Ls
3RTs  3Ls
(14)
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Noncommutated-Phase-Current Predictive Model

3. Phase A as noncommutation phase: Similarly, the predictive current of
phase A is given as
ia (k  1)  
(ub 0 (k  1)  uc 0 (k  1)  2ua 0 (k  1))Ts (ea (k  1)  ec (k  1)  2ea (k  1))Ts  3Ls ia (k )

3RTs  3Ls
3RTs  3Ls
(15)
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Voltage Selection in Commutation Interval



The optimal switching state is selected by the cost function so as to ensure that
falling current and rising current are matched and noncommutated phase current
is kept constant.
The switching states of the inverter are defined as TT
1 2T3T4T5T6 ,
with 0 denoting shutdown and 1 denoting conduction.
From phase A → C conduction to phase B → C conduction .The three
conduction statuses mentioned previously are defined as Status(1) (011000),
Status(2) (111000), and Status(3) (000000), respectively.
Table
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Voltage Selection in Commutation Interval


In each commutation mode, terminal voltages are selected from different
conduction statuses.
Then, according to the noncommutated-phase-current predictive models
(13)–(15), the future prediction of noncommutated phase current in the
next sampling period is obtained.
 ua 0   0 
   
ub 0   ud   u (1)
uc 0  0 
ua 0   ud 
u   u   u (2)
 b0   d 
uc 0  0 
ua 0   0 
   
ub 0   0   u (3)
uc 0  ud 
2016/7/15
(16)
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Commutation Interval Measurement

In commutation process, the correctness of commutation interval
measurement is critical to commutation torque ripple reduction

The rising edge or falling edge of Hall sensor signals H1 , H 2 , and H 3 , is
considered as the starting moment of commutation.

The time when the falling current during commutation reduces to zero is
considered as the end moment of commutation.
Table
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Commutation Interval Measurement









Gop is the target value of the cost
function
Gop max is the initial set value of the
cost function
j is the variable in the loop
jop is the selected number of the
switching state
status ( jop ) is the selected switching
state given in Table
u( j ) represents the three-phase terminal
voltages given in (16)
e is the back EMF of the three-phase
winding
ke is the coefficient of back EMF
k is the kth sampling time
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Commutation Interval Measurement




The proposed algorithm is set in interrupt program, and the interrupt timer
period is 40 kHz. While the commutation interval condition given in Table
is detected.
*
First, reference current i and phase C current ic are sampled, and the
current back EMF is computed.
Second, according to noncommutated-phase-current predictive model (13),
predictive currents of phase C for the next time are computed under three
possible switching states given in Table
Third, according to the cost function, the optimal switching state, which
minimizes the cost function, is selected and applied to the inverter. Then,
the interrupt program returns.
2016/7/15
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Numerical Simulation Results-Low Speed

when the motor runs at low speed, the
rising current ib increases quickly.
Consequently, noncommutated phasecurrent spikes are generated. Hence,
commutation torque increases, and
commutation torque ripple is generated.

In Fig. 8(b), phase B current is regulated
by power switch T3 at the moment of
commutation, and the rise of phase B
current slows down. Then, the rising
current and falling current are matched,
and constant noncommutated phase
current is ensured.
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Numerical Simulation Results-High Speed

which leads to the appearance of
noncommutated-phase-current dips and
the decrease of commutation torque,
hence generating commutation torque
ripple.

In Fig. 9(b), the slope rate of phase A
current decreases via regulating power
switch T1 at the moment of commutation.
Consequently, the rising current and
falling current are matched, and constant
noncommutated phase current is ensured
during commutation.
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Experimental Result And Analysis




Texas Instruments floatpoint DSP TMS320F28335 with 32-bit CPU
System clock rate is 150 MHz
Switching frequency and ADC sampling frequency 40 kHz
The most relevant prototype parameters of BLDCM are presented in Table
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Experimental Result And Analysis
CH1:phase A current CH3:Hall signal H2
CH2:phase B current CH4:commutation interval signal measured

With the commutation interval measurement, it is ensured that the approach
of commutation torque ripple reduction proposed in this paper can be
applied correctly during commutation.
2016/7/15
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Experimental Result And Analysis At Low Speed

Fig. 13(a), at low speed n = 200 r/min,
the rising current in phase B goes
faster, and noncommutated-phasecurrent spikes occur in phase C.

In Fig. 13(b), when the proposed FSMPC approach is adopted at the speed
n = 200 r/min, phase B current is
adjusted by power switch T3 , and
consequently, the rising current and
falling current are matched. Hence,
noncommutated phase current is kept
constant.
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CH1:phase A current CH4:control signal of power switching T3
CH2:phase B current CH5:control signal of power switching T1
CH3:phase C current A(T1)C→B(T3)C
36
Robot and Servo Drive Lab.
Experimental Result And Analysis At Low Speed



Fig. 14(a), at high speed n = 2000 r/min,
the slope rate of rising current in phase B
gets lower, and the rising current and
falling current are mismatched.
noncommutated phase current ic
generates current dips. Furthermore,
commutation torque ripple is generated.
In Fig. 14(b),when the proposed FS-MPC
approach is adopted at the speed n = 2000
r/min, the actions of T1 during
commutation properly adjust the current
slope of phase A.
CH1:phase C current CH4:control signal of power switching T3
CH2:phase A current CH5:control signal of power switching T1
CH3:phase B current A(T1)C→B(T3)C
37
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Experimental Result And Analysis In Dynamic Process

Figure shows the three-phase current
and speed response at low speed in
dynamic process

Figure(a), when the motor runs at low
speed in dynamic process, the slope rate
of rising current goes higher, and
noncommutated-phase winding
generates current spikes. Then,
commutation torque increases.
(a) Without the proposed approach.
CH1:speed response CH4: phase C current
CH2:phase A current CH3: phase B current
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Experimental Result And Analysis In Dynamic Process

Figure shows the three-phase current
and speed response at low speed in
dynamic process

Fig. 15(b), the rising current and
falling current are matched when the
proposed FS-MPC approach is adopted
at low speed in dynamic process.
(b) With the proposed approach
CH1:speed response CH4: phase C current
CH2:phase A current CH3: phase B current
2016/7/15
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Experimental Result And Analysis In Dynamic Process

Figure shows the three-phase current
and speed response at high speed in
dynamic process

In Fig. 16(a), when the speed is greatly
increased, noncommutated phase
current generates current dips, and
commutation torque decreases.
(a) Without the proposed approach.
CH1:speed response CH4: phase C current
CH2:phase A current CH3: phase B current
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Department of Electrical Engineering
Southern Taiwan University
Robot and Servo Drive Lab.
Experimental Result And Analysis In Dynamic Process

Figure shows the three-phase current
and speed response at high speed in
dynamic process

The proposed method gives a unified
approach for suppressing commutation
torque ripple over the entire speed
range without distinguishing high
speed and low speed.
(a) With the proposed approach.
CH1:speed response CH4: phase C current
CH2:phase A current CH3: phase B current
2016/7/15
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Department of Electrical Engineering
Southern Taiwan University
Robot and Servo Drive Lab.
Conclusion

This paper has proposed a unified approach, which can effectively reduce
commutation torque ripple over the whole speed range, without considering
different current cases at high speed and low speed, respectively, and
overcome the difficulties of commutated-phase-current control, avoiding
complex current controllers or modulation models.

The study in this paper has demonstrated that commutation torque ripple can
be effectively reduced by properly switching three conduction statuses of the
power inverter during commutation.

The results indicate that the proposed approach exhibits improved
performance in both dynamic and steady states for reducing commutation
torque ripple over the entire speed range.
2016/7/15
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Department of Electrical Engineering
Southern Taiwan University
Robot and Servo Drive Lab.
References
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[17] T. N. Shi, Y. T. Guo, P. Song, and C. L. Xia, “A new approach of minimizing commutation
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Department of Electrical Engineering
Southern Taiwan University
Robot and Servo Drive Lab.
Thanks for listening!
2016/7/15
44
Department of Electrical Engineering
Southern Taiwan University
Robot and Servo Drive Lab.