Energy and Power En gi neering, 2011, 3, 69-78
doi:10.4236/epe.2011.32010 Published Online May 2011 (http://www.SciRP.org/journal/epe)
A Repetitive-PI Current Controller for Boost Single Phase
PFC Converters
Turki Kahawish Hassan
Electrical Engineering Department, University of AL-Mustansiriya, Baghdad, Iraq
Received November 27, 2010; revised January 4, 2011; accepted January 26, 2011
Abstract
In this paper, the Author presents the theory and application of repetitive proportional integral current con-
troller for boost single phase ac-dc converter with power factor correction (PFC). A repetitive controller
which is inserted in series with the proportional integral (PI) controller shows very low crossover distortion
of input current, low total harmonic distortion and very low tracking error when is compared with the con-
ventional proportional integral controller. Full analysis of proposed controller is given and Matlab/Simulink
is used for simulation. The simulation results show the validity of the proposed control method.
Keywords: Boost Converter, Unity Power Factor Correction, Repetitive Control
1. Introduction
The developments in power factor correction technology
in the past two decades have enabled the design and im-
plementation of single phase ac-dc boost converters with
close to unity power factor and much less input current
distortion that generated by simple diode rectification
circuits.
Typical PFC schemes use two cascaded loops: the first
loop aims to regulate the boost dc output voltage and
provide the amplitude for the reference to the second
controller, which controls sinusoidal current provided
from grid. This current always contains some residual
distortion, especially in the vicinity of zero crossing of
the input voltage. The reasons of this zero crossing dis-
tortion (also referred to as cross distortion) are discussed
by authors in [1,2] but no any effective method is sug-
gested to reduce or eliminate this phenomenon. Feed
forward current control methods for boost single phase
PFC converters are suggested for robust control against
parameters variation and to reduce the total harmonic
distortion [3,4] .These methods need additional circuits
and the cost is increased .
Receptive control theory [5-7] originating from inter-
nal model principle provides a solution for eliminating
periodic errors which occur in a dynamic system. A re-
petitive controller can be viewed as a periodic waveform
generator augmented within the control loop of a control
system, which is closed loop regulated by a feed back
controller so that the periodic errors can be eliminated. A
number of repetitive control schemes have been devel-
oped and applied to eliminate periodic distortions in
PWM inverters [8,9].
In this paper, a repetitive controller is proposed and
designed to eliminate the zero crossing distortion of in-
put current of the converter and to obtain good tracking
performance. The design of the repetitive controller is
performed by analyzing the frequency domain, and Ny-
quist plot play a central role throughout the design phase.
The feasibility of the proposed design technique is
shown by simulation. The simulation results of the pro-
posed system show a highly improvement compared to
the conventional PI controlled system.
The reminder of this paper is organized as follows.
Section 2 gives operation principle of boost single phase
PFC converter and formulates the problem. In Section 3,
the principle of repetitive control is reviewed and a re-
petitive controller for PFC boost converter is proposed.
In section 4, the repetitive controller is designed and
several simulation results for proposed and conventional
systems are presented. Section 5 concludes the paper.
2. Operation Principe and Problem
Formulation
Power circuit configuration of a boost single-phase
power factor correction (PFC) converter and its control
scheme are shown in Figure 1(a) and (b) respectively.

70 T. K. HASSAN
V
o
in
v
_
L
i
in
v
in
+
_
_
+
i
L
D
C
+
d(t)
SW L
O
A
D
(a)
d(t)

V
o
V
ref
+
_ G
cv
e
v

÷
×
Peak
ABS
v
in
s(t)
I
ref
i
ref
i
L
G
ci
+
_
e
i
v
tri
+
_
V
cont
in
v
ˆin v
(b)
Figure 1. Boost single phase PFC converter (a) power circuit and (b) control block diagram.
The circuit consists of a full-bridge diode rectifier and
a boost dc/dc converter. The dc output voltage of the
boost converter Vo is compared with the reference Vref
and the error ev is applied to the voltage controller Gcv(s)
which is a PI controller. A current command iref is
yielded by multiplying the output of voltage controller
Iref with a rectified unity sine wave (s(t)) which is:

sin
ˆ
in
in
v
s
t
vt

 (1)
where ˆsin
in in
vv t

, is the amplitude of input
voltage.
ˆin
v
The current command is compared with the inductor
current iL and the error ei is applied to the PI current con-
troller Gci(s) to obtain Vcont. A pulse width modulation
control signal d which is obtained by comparing Vcont
with the triangular wave Vtri is applied to gate of power
MOSFET transistor to achieve the sinusoidal input cur-
rent iin.
In order to make the following control system analysis
and design, some reasonable assumptions are made:
1) the losses of converter components are neglected;
2) the dc output voltage is well regulated such that Vo
equals to reference value Vref.
Accordingly one can derive the following equation
using the state-space averaging method.
 
dˆsin 1
d
L
in ref
i
Lvt dtV
t
  (2)
where L is the boosting inductance and d(t) is the duty

T. K. HASSAN
71
ratio function of the PWM modulator which can be ex-
pressed as

ˆ
cont
tri
V
dt V  (3)
where is the amplitude of the triangular wave.
From (2) and (3), one can find
ˆtri
V
ˆsin
d
ˆ
d
ref ref in
L
cont
tri
VV vt
iV
tLL
LV

 (4)
As shown in Figure 1(b), Vcont can be written in laplace
form as
   
contrefL ci
Vs isisGs (5)
The inductor current iL can be derived from (4) and (5)
to be
 
   

 
21
11
ref in
L
V vs Gs
isi s
Gs
sLG ssLG s
 

 r
ef
(6)
where
 
ˆ
ref
ci
tri
V
GsG s
sLV
 (7)
and

i
ci p
K
Gs K
s
 (8)
where Kp and Ki are proportional and integral gains re-
spectively.
According to (6), the absolute value of input voltage
can be viewed as a disturbance to the control loop. Also
Vref can be considered as a constant disturbance. The
objective of control is to make the inductor current tracks
a sinusoidal reference current, in face of these distur-
bances. In the conventional control method, a propor-
tional integral (PI) controller Gci(s) is used in design and
implementation, and also the most commonly used one
in practical applications. However, it fails to yield excel-
lent control performances in both tracking and distur-
bance rejection issues. To solve these problems, a repeti-
tive controller is proposed and applied to the current loop
of boost converter. This controller is discussed in next
section.
3. Repetitive Control
3.1. Principle of Repetitive Control
Any periodic signal with period Ts can be generated by
the free time delay system. The block diagram of time
delay system including unity positive feedback is shown

in Figure 2(a). The resulting transfer function is [5]:



1
S
s
sT
R
s
T
us e
9)
where y(s) is the output and u(s) is the input. Due to this
ys e
Gs

 (
delay, the transfer function has infinitely many poles on
the imaginary axis: jkωs (see Figure 2(b)).The poles can
be found from 1.0
s
sT
sjks
e


 for every k = 0, ±1,
±2,, where ωs = 2л/Ts.
e model in (9) is said to be a
he of the proposed repetitive control
A controller including th
repetitive controller and a system with such controller is
called a repetitive control system.
The basic concept of the repetitive controller origi-
nates from the internal model principle [7]. This princi-
ple state that the controlled output tracks a set of refer-
ence inputs without steady state error if the model which
generates these references is included in the stable closed
loop system. For example, no steady state error occurs
for step reference commands in type-1 stable feedback
system that has an integrator (1/s) in the loop, i.e., the
generator of step function .However ,stand-alone repeti-
tive controller cannot yield good transient performance,
therefore the repetitive controller is often used together
with another controller such as P-I controller to give
quick transient response.
.2. PFC Boost Converter with Repetitive 3
Controller
block diagramT
system is shown in Figure 3. The repetitive controller
CRP(s) is located in series with conventional PI controller
Gci(s) of current loop of boost converter. The transfer
function of the repetitive controller is:
u(s) y(s) +
+
S
sT
e
(a)
Re
I
m
0
jω
s
2
s
j


2π
s
S
T  
2jω
s
(b)
Figure 2. Generaf periodic signal. tor o

T. K. HASSAN
72
C
RP
(s)





ref
in
V
vs
s



S
s
T
qse

i
p
K
K
s

ref
V
1
s
L
1
ˆtri
V

L
is
i
ref
(s)
G
ci
(s)
+
+
+
+ +
+
+
_
Figure 3. Block diagram of a repetitive control system.
 
1
1S
RP
s
T
Cs qse

 (10)
Note that CRP(s) is equivalent to the modified repeti-
tiv
ntrol system, (6) can be
re
e controller with a(s) = 1 as proposed by Hara et al.
[5]. q(s) is a low pass filter that should be appropriately
chosen so that good tracking performance is obtained
without resultant system instability. The choice of q(s) is
not straightforward and simulations are needed to choose
the most appropriate filter [10].
For the proposed repetitive co
written as:
 
   

 
2
11
1
ref in
L
RP RP
RP
ref
RP
V vs
is
s
LGssLGs
Gs
is
Gs



(11)
where
  
ˆ
ref
RPci RP
tri
V
Gs GsCs
sLV
 (12)
Let Us discuss the effectiveness of the repetitive con-
troller according to (11) and (12). The transfer function
CRP(s) is large enough for s = jkωs, k = 0, 1, 2,  If the
proportional integral controller Gci(s) of currentoop of
boost converter is chosen such that

l
R
Ps
Gjk

is suf-
ficiently large for the selective range
are multiples of frequency of reference current, then
 
1
of harmonics which
R
PsRPs
Gjk Gjk

 . Since the denominator is
jected and the inductor cur-
rent tracks the sinusoidal reference current with very
small steady state error.
The stability condition t
large, the disturbances are re
hat is obtained in [5] is applied
to
ing proposition:
ystem without repetitive controller, i.e.,
proposed repetitive control system to give the follow-
In the repetitive control system shown in Figure 3 if
the closed loop s






1s Gs, is stable and

G


1qj Gj

 
0


 where

2ˆ
ref pi
tri
VKs

K
Gs sV L 
then the system is exponentially stable.
The system without repetitive controller is stable since
e control system
is
G(s) has no unstable poles. The repetitiv
therefore exponentially stable if the Nyquist plot of
G(s) [11] does not encircle the (–1, j0) point and lies
outside of the circle of radius

qj

centered at the
(–1, j0) point in the complex plane. The low pass filter
q(s) should be chosen so that tuist plot of G(s)
remains outside the circle for all frequencies.
4. Controller Design and Results
he Nyq
verter with power factor
orrection has outer voltage loop and inner current loop.
is shown in Figure 3. The
pa
for stable system
th
4.1. Controller Design
The boost single phase ac-dc con
c
Some nominal values and circuit parameters of the con-
verter are shown in Table 1.
Our investigation is focused on the inner current loop
with repetitive controller that
rameters of P-I controller of current loop are chosen as
Kp = 0.8 and Ki = 300. The repetitive controller is in-
serted in series with the PI controller.
Figure 4 shows the Nyquist plot of G(s). The plot
does not encircle the (–1, j0) point and
e plot must be outside the circle of radius of


qj

.
To ensure the stability for all frequencies the value of

T. K. HASSAN
73

qj

Figure 4. Stability circle and Nyquist plot of G(s).
Table 1. Simulated converter parameters.
Input line voltage (peak) ˆin
v= 170 V
Input line frequency f =
e F
ncy
V
50 Hz
Smoothing capacitancC = 1000 µ
Smoothing inductance L = 1 mH
Rated power 500 W
Carrier freque25 KHz
Output voltage Vo = 300

qj

between
must be less than unity to prevent any contact
obtain the cutoff frequency
of
the plot of G(s) and the circle (see the stability
condition in the previous section). The gain of low pass
filter q(s) is chosen as 0.98 to give stable system and
good tracking performance.
No systematic method to
q(s), therefore simulation is required to find the best
value from the point of view of good tracking and dis-
turbances rejection. Accordingly, the cutoff frequency is
chosen as 1000 HZ. It seems that 1000 HZ is reasonable
choice since most periodic disturbances are expected to
lie within this band. Let the filter q(s) be

0.98
1 2000π
qs s  (13)
In our applications, we require a delay of Ts = 10 ms
fo
function
r compensation of harmonics of fs = 100 Hz. Finally,
we get a repetitive controller CRP(s ), which has transfer

/100
0.98
1
RP
s
Cs
e


(14)
1
1 2000πs
The effects that the designed repetiti
CRP(s) has on the PFC boost converter system can be
investigated by the bode plots of loop gains G(s) and
G
in, are clearly seen from the
Bo
ve controller
RP(s) (without and with repetitive controller respec-
tively) shown in Figure 5.
The major characteristics of the repetitive controller,
such as effective rejection of periodic disturbances and
reduction of stability marg
de plots. Note that the added repetitive controller in-
creases the loop gain at particular frequencies, integral
multiples of 100 Hz, maintaining a relatively unchanged
gain at other frequencies. Let us considerer the sensitive-
ity function

1
1RP Gs which is the transfer function
from disturbance input to tracking error, which is of pri-
mary importaning the performance of feedback ce in judg
control systems. Gain increase at particular frequencies
imulation is performed by Matlab/Simulink to verify
leads to decrease in sensitivity at these frequencies. This
implies that zero crossing distortion in inductor current
which is considered as the periodic disturbances appear-
ing at multiple of 100 Hz are more strongly attenuated by
the repetitive controller than non periodic disturbances.
4.2. Results
S

74 T. K. HASSAN
Figure 5. Bode plots of G(s) with (dash) and GRP(s) with solid.
the proposed PFC boost con
oller. The converter parameters in Table 1 and the de-
co
r current iL waveforms with 100 W (Rl = 900 Ω)
w
put voltage and input current
for Rl = 450 Ω (output power equals 200 W) with repeti-
t current is canceled compared with Figure
8(
current when load
re
racking is achieved with repetitive controller (very
sm
verter with repetitive con-Figure 8(a) shows the in
tr
signed repetitive controller in the previous section are
used in simulation. For all simulation results, the output
voltage of the converter is constant and equals to 300 V.
Figure 6(a) and (b) show the input current iin and the
input voltage vin waveforms with and without repetitive
ntroller respectively for 100 W output power and load
resistance (Rl) of 900 Ω. The input current is in phase
with input voltage and has an amplitude of 1.18 A with
no zero switching distortion when repetitive controller is
used.
Figure 7(a) shows the reference current iref and the
inducto
hen the repetitive controller is applied. Comparison
with waveforms in Figure 7(b) when only PI-controller
is used indicates that current tracking performance is
significantly improved by the proposed repetitive con-
troller (reference and inductor current waveforms are
coincident.
tive controller. It is noticed that the zero crossing distor-
tion in inpu
b) when only P-I controller is used.
Zero crossing distortion in input current shown in Fig-
ure 6(b) is more than the distortion shown in Figure 8(b)
because of increment in load current with load resistance
of 450 Ω.More leading phase of input
sistance is increased (less load current) which causes
more zero crossing distortion [1]. This leading phase is
not appeared when the proposed repetitive controller is
used.
Figure 9(a) and (b) show the reference and inductor
current waveforms for load resistance of 450 Ω with and
without repetitive controller respectively. It is clear that a
good t
all steady state error between reference and inductor
currents).
Figure 10(a) and (b) show the input voltage and input

T. K. HASSAN
75
(a)
(b)
Figure 6. vin and iin waveforms for Rl = 900 Ω (a) With re-
petitive controller; (b) Without repetitive controller. Hori-
zontal axis 10 ms/div), vertical axes: voltage (50 V/div),
current (2.5 A/div).
(a)
(a)
(b)
Figure 8. vin and iin waveforms for Rl = 450 Ω. (a) With re-
petitive controller; (b) Without repetitive controller. Hori
zontal axis (10 ms/div).Vertical axes: voltage (50 V/div),
current (2.5 A/div).
-
(a)
(b)
Figure 7. iref and iL waveforms for Rl = 900 Ω. (a) With re-
petitive controller; (b) Without repetitive controller. Hori-
zontal axis (5 ms/div).Vertical axis (0.2 A/div).
(b)
Figure 9. iref and iL waveforms for Rl = 450 Ω. (a) With re-
petitive controller; (b) Without repetitive controller. Hori
zontal axis (5 ms/div). Vertical axis (0.5 A/div).
-

76 T. K. HASSAN
(a)
(b)
Figure 10. vin and iin waveforms for Rl = 225 Ω. (a) With
repetitive controller; (b) Without repetitive controller.
Horizontal axis (10 ms/div). Vertical axes: voltage (5
V/div), current (2.5 A/div).
Figure 11(a) shows the reference current and inductor
ident.
factor (PF) and the total harmonic distor-
tio
900 Ω to 180 Ω. The input cur-
re
0
current waveforms with and without repetitive controller
respectively with Rl = 225 Ω (output power equals 400
W).
current waveforms for Rl = 225 Ω when the repetitive
controller is applied. It is noticed that the waveforms are
coinc
Figure 11(b) is similar to Figure 11(a) but no repeti-
tive controller is used. No good tracking between the
reference and inductor current is observed.
The power
n (THD) of input current with and without repetitive
controller are calculated under various load and are listed
in Table 2 and Table 3.
As shown in Table 2, a highly reduction in total har-
monic distortion of input current is shown when a repeti-
tive controller is used compared with Table 3 (without
repetitive controller).
The waveforms of input voltage and input current for
the proposed repetitive controlled PFC boost converter
are plotted in Figure 12 when the load resistance Rl is
suddenly changed from
nt magnitude is increased from 1.18 A to 5.8 A.
We can find that the input current iin always in phase
with the input voltage vin even though under transient
response. Consequently, the proposed repetitive controller
(a)
(b)
Figure 11. (a) iref and iL waveforms for Rl = 225 Ω with
repetitive controller. Horizontal axis (5ms/div).Vertical axis
(0.5A/div); (b) iref and iL waveforms for Rl = 225 Ω withou
repetitive controller. Horizl axis (5ms/div). Vertical
All the previous results are obtained for input voltage
ontroller.
Rl (Ω) Output power (w) THD (%) PF
t
onta
axis (0.5A/div).
can keep good performance under the condition of load
change.
Table 2. Calculated THD and PF for various loads with
repetitive c
1800 50 2.1 0.9992
900 100 0.9 0.9998
0.9999 450 200 0.41
225 400 0.22 1
Table 3. Calculated THD and PF arious loads with
repetit controller.
Rl (Ω) Output power (w) THD (%) PF
for v
ive
1800 50 34.16 0.9965
900 100 14.99 0.9977
0.9992 450 200 6.8
225 400 3.5 0.9998

T. K. HASSAN
77
of 1(peak). der to valie repeon-
troller performance foinput voltagariation, u-
lated steady state eforms of inductor currLnd
fee current or the case nput voith
70 V In ordate thtitive c
r
wav
e vthe sim
ent i a
re renciref fof iltage w
20% larger than the nominal value and Rl = 450 Ω (out-
put power equals 200 W) are plotted in Figure 13. The
reference current and inductor current are coincident and
good tracking is also maintained and is not affected with
the variation of input voltage.
Figure 14 shows the input voltage and input current
Figure 12. vin and iin waveforms with repetitive controller.
Rl is changed from 450 Ω to 180 Ω. Horizontal axis (50
ms/div).Vertical axes: voltage (50 V/div), current (5 A/div).
Figure 13. iref and iL waveforms for Rl = 450 Ω without re-
petitive controller. Horizontal axis (5 ms/div). Vertical axis
(0.2 A/div). Amplitude of input voltage equals 203 V.
Figure 14. vin and iin waveforms for Rl = 450 Ω. with repeti-
tive controller. Horizontal axis (5 ms/div). Vertical ax
voltage (50 V/div), current (5 A/div). Amplitude of input
voltage equals 203 V.
waveforms for the same conditions in Figure 13. No
zero crossing distortion in input current is observed and
the input current is in phase with the input voltage.
5. Conclusions
In this paper, the control approach to solve the problem
of zero crossing distortion of input current of PFC boost
converter has been presented. To achieve this goal, a
repetitive controller is inserted in series with the PI con
y. Because our approach is based on
bility
ndwidth is included. Several simula-
to verify the validity of the proposed
es:
-
troller of current loop. We presented a graphical design
technique based on the frequency domain analysis of
linear system to achieve a repetitive controller that pre-
serves system stabilit
graphical inspection of the Nyquist envelop, the design
procedure was simple and intuitive. A low pass filter
with gain lower than one to ensure the system sta
and to limit the ba
tions are performed
repetitive controller. The results obtained with and with-
out repetitive controller are compared. The results with
repetitive controller shows very low total harmonic dis-
tortion of input current, good tracking of reference cur-
rent and inductor current (very low steady state error)
and no zero crossing distortion of input current. Transit
responses to step change in load are presented to exhibit
the robustness of the proposed repetitive controller
against load variations. The performance of the system
with the proposed controller has not affected with the
variation of input voltage.
6. References
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PFC Converters,” IEEE Transactions on Power Elec-
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[2] H. C. Chen, “Duty Phase Control for single-Phase Boost-
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Current Waveform Controls,” IEEE Transactions on
[4] H. C. Chen, S
Rectifier with
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[5] S. Hara, Y. Yamamoto, T. Omata and M. Nak
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ano, “Re-
Type Servo System for
Periodic Exogenous Signals,” IEEE Transaction on
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T. K. HASSAN
78
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Nomenclature
o Dc output voltage of boost converter.
ref Reference voltage for voltage loop.
v Voltage error (the difference between refer-
ence and output voltage).
cv(s) Voltage controller transfer function.
voltage controller.
e wave.
t)).
n current
ction.
Vcont Output of current controller.
V Triangular wave.
Amplitude of triangular wave.
in Ac input current to the rectifier.
L Boosting inductance.
d(t) Duty ratio.
Kp Proportional gain of current controller.
ller.
on.
troller.
out repetitive
unction with repetitive
.
tri
ˆin
v
i
V
V
e
G
Iref Output of
(t) Unity sin
Ki Integral gain of current contro
snsfer functis q() Low pass filter tra
CRP siref Current command (Iref multiplied by s( () Transfer function of repetitive con
vin Ac input voltage to the rectifier G(s) Open loop transfer function with
ˆin vAmplitude of ac input voltage.
iL Inductor current.
controller).
transfer f GRP(s) Open loop
ei Current error (the difference betwee
t).
controller.
/sec) ωs Double of input line frequency (radcommand and inductor curren
s) Cu transfer funGci(rrent controller