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Input Voltage and Current Sensorless Control of a Single Phase ACDC Boost Converter
Mehdi Tavan and Kamel Sabahi

Abstract— It is well known that measuring the input voltage
and current, as the feedforward and feedback terms, are vital
for the controller design in the problem of power factor
compensation (PFC) of an AC-DC boost converter. Traditional
adaptive scenarios corresponds to the simultaneous estimation
of these variables are failed because the system dynamics is not
in the classical adaptive form. In this paper, the system
dynamics is immersed to a proper form by a new filtered
transformation to overcome the obstacle. The phase and
amplitude of the input voltage along with the input current is
exponentially estimated from the output voltage with global
convergent. An application of the proposed estimator is
presented in conjunction with a well-known dynamic
controller.
and power factor quality [6]. It is worth noting that, the ACDC boost converter belongs to second-order bilinear systems
group and both of its states are not observable when the
control signal is zero. These features pose an interesting state
and parameter estimating problem.
I. INTRODUCTION
Consider the single-phase full-bridge boost converter
shown in Fig. 1. The circuit of the converter combines two
pair of transistor-diode switches in two legs to form a
bidirectional operation. The switches in each leg operate in
complementary way and are controlled by a PWM circuit.
The dynamic equations describing the average behavior of
the converter can be obtained using the Kirchhoff’s laws as
(1)
( )
(2)
where
describes the current flows in the inductance ,
and
is the voltage across both the capacitance and
the load conductance . The continuous signal
operates as a control input and is fed to the PWM circuit to
generate the sequence of switching positions . The switch
position function takes the values in finite set
[1].
(
) represents the voltage of the
Finally, ( )
AC-input.
The problem of power factor compensation (PFC) for an
AC-DC boost converter has widely been studied by many
researchers due to its applications for interfacing renewable
energy sources to hybrid microgrids [2], flexible AC
transmission systems [3], motor drive systems [4], LED drive
systems [5] etc. Knowledge about the phase and amplitude of
AC-source is vital to provide the feedforward control signal
in the problem of PFC of an AC-DC boost converter; see
control signals in [6] and [7]. Besides an accurate measuring
of these parameters, feedback from input current improves
output voltage regulation, total harmonic distortion (THD)
M. Tavan is with the Department of Electrical Engineering,
Mahmudabad Branch, Islamic Azad University, Mahmudabad, Iran (e-mail:
[email protected]).
K. Sabahi is with the Department of Electrical Engineering, Mamaghan
Branch, Islamic Azad University, Mamaghan, Iran.
Figure 1. AC-DC full-bridge boost converter circuit [8].
In this paper the effect of the phase difference between
the input voltage and current on the power quality is
investigated. Specially, the DC error and the amplitude of
harmonic in the output voltage increase for big values of the
phase shift. Hence, the problem of phase difference
estimation is interested in this paper. A new nonlinear,
globally convergent, and robust estimator is designed via
immersion and invariance (I&I) based filtered
transformation (see [9-11] for some applications of the
method). The input current and voltage is exponentially
estimated from the output voltage via a fifth- dimensional
estimator. An application of the estimator is presented in
conjunction with a well-known dynamic controller.
II. PROBLEM STATEMENT
For the system, the control objective is to regulate in
average the output (capacitor) voltage in some constant
desired value
with a nearly unity power factor in
input side. The objective can be achieved indirectly by means
of stabilizing input (inductor) current in phase with the source
voltage. This is done due to unstable zero dynamic with
respect to the output to be regulated which imposed a secondorder harmonic on the output in steady-state [8].
Let us to assume that the input current has the following
steady-state form
( )
(
)
(3)
for some constant
as the phase difference
between input voltage and current, and some
yet to be
specified. Replacing (3) into (2) yields
(
)
(
)
(4)
where
and
are the control input and output voltage in
steady-state. Now, replacing (3) and (4) into (1) yields the
following dynamics in steady-state
(
̇
)
(
(
)
(
(
)
(6)
)
(
)
(
(
)
(
))
(7)
(
)
(
(
)
(
))
(8)
Finally, doing a basic trigonometric simplification (6) takes
the form
)
(
)
(9)
with
(
)
(
)
√
(10)
(
)
(11)
(16)
which indicates a steady state error in output voltage.
Remark 2: The minimum amplitude of the second-order
harmonic ( ) does not occur for
. From (10) it can
be computed that for
which satisfies the following
inequality
)
(
)(
(
(
(15)
√
(
with
( )
( ))
(5)
)
The steady-state solution of (4) can be calculated using
Fourier series and is given by
( )
(
√
with
for some constant
. Replacing the
amplitude and phase of the current in (12) yields to
)
(
( )
)
(
)
(
(
))
(
))
(17)
the magnitude of ( ) is less than ( ). From (17) it can
be concluded that when the converter operates in leading
form, i.e.
, the magnitude of ( ) increases, but, it
can be decreased when the converter operates in lagging form
with a small
near to zero.
For instance consider the experimental system used in [8]
with the set of parameters;
,
,
,
, and
. The ratio of
(
) to (
) is shown in Fig. 2 for
(
)
(
) and
. Fig. 2 justifies
the above statements and shows that a small lagging phase
shift can be beneficial to reduce the harmonic amplitude. This
may be induced by time delays or unmodeled dynamics in
the actual system. On the other hand, a big phase difference,
in leading or lagging form, causes high harmonic amplitude
as shown in Fig. 2.
The new representation, given by (9), demonstrates the
relationship between the phase difference and the steady-state
amplitude of the input current and the second-order harmonic
on the output.
Remark 1: The DC-component of
concluded from (9) and is given by
(
)
√
(
in steady state can be
)
(12)
As a result, in order to provide the desired output voltage
the input current amplitude must be forced to achieve
(
)
(
)
,
(13)
which justifies the fact that for any phase shift (positive or
negative) the input current and then the converter losses
increases. For
, the minimum value of
can be
achieved and is defined as
(14)
As an example, consider the static control law in [8]
which causes the following input current in steady-state
Figure 2. The influence of
and on the amplitude of the second-order
harmonic.
With respect to the remarks above which cite the
importance of phase difference in the power quality, in this
paper we are interested to estimate the parameter vector
( )
( )
(18)
from the output voltage . The phase and the amplitude
can be obtained from the above vector as
with
( )
( )
(20)
)
(
)
(
̇
(21)
(
( )
(
(22)
)
In this section an estimator is designed for the system (1),
(2) which requires the knowledge about , , , and . It
should be noted that the only measurable state is the output
voltage . This cancels the use for input current sensor and
makes the practical implementation of the proposed
procedure attractive. In the design procedure the following
assumptions are considered.
where
Assumption 1: The control signal
indicated as
.
in
(23)
where
is an auxiliary dynamic vector, which its
dynamics are to be defined. The equation above admits the
global inverse
(24)
( ) is the new unavailable vector. The
where
dynamics of the new variable can be obtain as
( )
[
]
(25)
)
̅
̅
which ̅
( ̅ ̅)
̅,
(29)
(
̅
(30)
̅
(
and
, and
)
with
)
.
To obtain the dynamics of ̅ differentiate (29), yielding
̇̅
A. Design procedure
In order to form a proper adaptive structure the following
input-output filtered transformation is considered
̇
(
̇̅
is bounded and
Assumption 1 is needed so that the input current become
observable from the output voltage . The assumption is
satisfied in operation mode when the system is forced to track
a positive constant as the desired output. Although
Assumption 2 seems to be somewhat restrictive, a dynamic
control law, like the one introduced in Proposition 8.9 of [8],
satisfies the assumption. The final assumption is standard in
the estimator design and is extremely milder compared to the
boundedness of trajectories.
( )
(28)
is the estimator state, and the mapping
yet to be specified. From (24)
and (29), the current estimation error can be obtained as
(
Assumption 3: The system (1), (2) is forward complete,
i.e. trajectories exist for all
.
( )
)
̅
is PE, which will be
(27)
Now, under the inspiration of the I&I technique, let us to
define the estimation error as
III. ESTIMATOR DESIGN
Assumption 2: The time derivative of
available.
)
where
to get
( )
(26)
where
and
are the zero matrix and zero
vector, respectively. The dynamics of the output voltage (1)
can be rewritten in terms of the new variable as
and the correspond regressor can be defined as
(
]
(19)
√
( )
̇
[
)
[
̇
]
̇
̇
(31)
Let
̇
)(
(
[
)
]
̇
̇
(32)
which cancels the known terms in the error dynamics and
yields to
̇̅
(
) ̅
(33)
The following proposition put forward a selection of
and ̇ , which renders the origin of the system (33) attractive.
Proposition 1: Consider the system (33) verifying
Assumption 1-3, with
[
̇
( )
]
(
(34)
)
(35)
where
is the
identity matrix,
and
are constant and positive definite matrixes, and
is a
constant value. Then the system (33) has a uniformly
globally stable equilibrium at the origin, ̅
and ̇ ̅
asymptotically converges to zero. Moreover, the regressor
is PE and then ̅ converges to zero exponentially.
Proof 1: The proof is constructed in three steps. In Step 1
the convergence of ̇ ̅ is proved. Step 2 shows that
.
Finally, the convergence of ̅ is proved in Step 3.
Step 1: Substituting (34), (35) into (33) yields the error
dynamics
̇̅
( ) [
] ̅
(36)
Consider now the following Lyapunov function candidate
( ̅ ̅)
̅
(̅
̅)
̅
( ) (̅
( ) (̅
̅)
̅) )
(
(38)
Step 2: Notice that, for the asymptotic stable filter (35),
if
[13]. The PE condition implies the
existence of a constant
such that
∫
(39)
‖ ‖
for all
and all
. With respect to
(21), a straightforward check for the inequality above
( ( )
( )) with
obtains by parameterizing
) [13], which yields to
(
∫
for all . This implies that
)
(40)
.
Step 3: As shown in Step 1, ̅ converges to zero. Let us to
assume that after the convergence time of ,̅ there exists a
time instance
such that
‖ ̅‖
‖ ̅‖
(41)
for any constant
. By virtue of the assumption
above, using (41) and (37) in (38) and applying some basic
bounding, results in
̅)
(
‖̅‖
̅
‖ ̅‖
(42)
̅)
(
‖̅‖
for some constant
,̅ (42) can be written as
. After the convergence time of
̇
̅
(
Integrating (43) when
( )
‖̅ ‖
)
(43)
yields the solution
∫ ( ( )
( )
∫ ( ( )
( )
( )
)
(44)
)
̅ ‖ ̅ ‖ and it is well known that ‖ ‖
with
.
Considering (39), (40) and invoking Proposition 2 in [13] for
some constant
and all
implies that
( ( ) ( ) )
∫
for some constant
. This implies that the system
(36) has a uniformly globally stable equilibrium at the
origin, ̅
and ̅
then ̅
, and finally ̅
̅
and
then ̇ ̅
. It is clear that, the dynamic (35) is
input-to-state stable with respect to
which is given by
(21). Hence, due to
then
and ̇
. Now
with respect to Assumption 2 and (36) it can be concluded
that ̇ ̅
and then ̈
. Therefore, the uniformly
̅ and then ̇ ̅ to zero follows directly by
convergence of ,̅
Lemma 1 in [12].
̅
( )
(37)
whose time-derivative along the trajectories of (36) satisfies
̇
̇
(45)
due to
. Therefore, there exists some time instants
, such that (44) satisfies
( )
(46)
for some constant
, which implies that
‖ ̅‖
‖ ̅‖
(47)
This contradicts with (41) and induces attractiveness of the
equilibrium ̅
by contradiction. This property is uniform
with respect to . Therefore the equilibrium ̅
is
uniformly asymptotically stable. Indeed, due to linear form
of the time-varying system (36), the convergence property is
exponential by Theorem B.1.4 in [14].

Remark 3: With respect to (29) and (30), the equilibrium
point ̅
admits to estimating and by
̂
(48)
̂̇
(
)
Indeed, from (19) and (20) an estimate of
obtained by
and
̂
( )
̂
̂
̂
(49)
√̂
̂
can be
(50)
(51)
IV. OUTPUT FEEDBACK CONTROL
The PFC main function is to achieve unity power factor.
This happens if the input current to be in the phase with the
AC-source. This can be achieved by the following dynamic
control law [6]
̇
where
(
)
(52)
( ) is derived from the filter
( )
( )
with some positive constants ,
(53)
phase shift. A new nonlinear, globally convergent, robust
estimator is designed via I&I based filtered transformation.
The input current and voltage is exponentially estimated
from the output voltage via a fifth-dimensional estimator. An
application of the estimator is presented in conjunction with
a well-known dynamic controller.
, , and
(
REFERENCES
)
(54)
(
)
with positive constant value , where
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is given by (14). For sufficiently large , the closed
loop system is asymptotically stable, and then converges to
zero and converges to
[6]. Note that, from analysis in
Remark 1, for
in (12) with zero phase difference, i.e.
, the output voltage converges in average to . On
the other hand,
and in (54) are sinusoid signals, hence
we can conclude that is sinusoid in the steady state. As a
result
and Assumption 1 is satisfied.
From (54), it is clear that an accurate measuring of the
input voltage and current is vital to achieve control
objective. Hence, the state and parameter estimation given
by (49)-(51) can be used in conjunction with the
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
(55)
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̂( )
̂
(
̂)̇
̂(
̂)
̂
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[8]
(56)
[9]
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√( ̂
̂
(
̂
)
(
̂
)
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(57)
[10]
(58)
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̂̇
̂)
(59)
[13]
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[12]
̂( )
is related to in (53) with
(60)
.
V. CONCLUSION
In this paper the effect of the phase difference between
the input voltage and current on the power quality is
investigated. Specially, the DC error and the amplitude of
harmonic in output voltage increase for big values of the
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