Systems theory has seen significant progress over the years.
Most analytical and synthetic techniques are based on linear
models of the controlled processes. Nevertheless, the nonlin-
ear nature of physical systems and due to the increasingly
increasing performance demanded in industrial applications,
so the use of advanced control techniques (adaptive control,
optimal control, sliding mode control,... ) becomes essential.
Nowadays, advanced control techniques are becoming one
of the most active areas of research. At the same time, we
have powerful calculators and a variety of software tools. This
facilitates the synthesis of advanced control laws and their
execution, without difficulties in real time.
It takes electronic components to achieve the desired volt-
age. The recent components of the power converter are of high
quality and high efficiency. One of them is the DC-DC step-
up converter which allows to increase the output voltage [1].
In order to create rapid changes in response of the DC-DC
boost converter, it must operate at high frequency [2]. In this
condition, the DC-DC boost converter requires a controller to
handle the desired value.
Adaptive controls are widely used by researchers to
solve dynamic problems [3] and some of them use
Proportional-Integral-Derive as its control structure. Conven-
tional Proportional-Integral-Derive is based on a mathematical
model, such that it has stability, reliability and control capa-
bilities. Conventional Proportional-Integral-Derive controllers
are effective in linear systems, but it is not suitable for
non-linear systems and high-order systems. Determination
of Proportional-Integral-Derive parameters has been used in
many ways [6]. Several methods have their own advantages
and disadvantages for determining the baseline adaptive con-
trol parameters of the model to achieve a stable system. The
fixed parameter in Proportional-Integral-Derive controller is
not quite robust or not able to adapt and therefore the adaptive
controller techniques is required improve system response
[7]. Several adaptive control techniques are used to solve
this problem and one of them uses Direct Model Reference
Adaptive Control (DMRAC) [8]. Model Reference Adaptive
Control performances are provided by model as reference,
this means that the plant’s response must follow the model’s
response. The following parameter adjustment mechanism is
calculated by using Massachusetts Institute of Technology
(MIT) rule [9].
DC-DC boost converter is commonly used in DC systems
and also known as DC boost converter. The output voltage
demand must be greater than the input voltage and continuous.
It uses two semiconductors such as a controlled power device
and an uncontrolled device. They basically consist of a series
inductor and a parallel capacitor. The electric circuit of the
Boost converter is presented by the figure 1.
s
i
C
i
L
i
L
V
L
M
V
E
D
D
V
C
V
C
w
S
R
S
V
Fig. 1. Diagram of the Boost converter.
The equivalent circuit of the Boost converter presented by
the figure 1 is :
Adaptive Linearizing Control with MRAC Regulator for DC-DC Boost
Converter
AHMED CHOUYA
Department of Genie Electrical, University of Djilali Bounaama, Khemis-Miliana
ALGERIA
Abstract: In this paper; we treat the converter boost DC-DC by an adaptive linearizing controller. Where regulators located
at the feed forward and feedback. Small signal model is used as a linearizing technique. Massachusetts Institute of
Technology (MIT) rule is applied as an adaptive mechanism to determine the optimal control parameters in some
conditions. The used adaptive control technique is Model Reference Adaptive Control (MRAC), this method as able to
control system in various output voltage. The proposed method has a stable response and able to reach the model reference
smoothly. However, the response of the system has instantaneously overshoot and follows the response back of model
reference.
Keywords: Automatic Control, Power Electronics, Feedback, Converters
Received: March 21, 2022. Revised: October 16, 2022. Accepted: November 19, 2022. Published: December 31, 2022.
1. Introduction
2. DC-DC Boost Converter Model
2.1 Basic Modeling
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2022.1.4
Ahmed Chouya
E-ISSN: 2945-0489
25
Volume 1, 2022
s
i
C
i
L
iL
V
L
EC
V
C
RS
V
s
i
C
i
L
iL
V
L
E
D
D
V
C
V
C
RS
V
Fig. 2. Diagram of the Boost converter with
Ë
Û
closed (left) and
Ë
Û
opened
(right)
On the interval,
, the switch
is closed
and the diode
is blocked. The linear model which represents
the left configuration of the circuit describes in figure 2 is
given by :
(1)
On the interval,
,
is opened and the
diode
is busy. The linear model which represents the right
configuration of the circuit describes in figure 2 is given by :
(2)
The general equation of the model of instantaneous state which
governs the operation of the Boost converter is (see [14]):
(3)
In order to define the small signal model (SSM) of the
boost converter, it is necessary to substitute each variable.
Each parameter is presented in steady state and in small signal
variation as follows (see [12])
One first of all will study the equilibrium state of the system.
One thus has (see [16]):
¾
(4)
To obtain the small-signals model of the Boost, we will lin-
earize the model of average state around the equilibrium state
. One then uses a development limited of TAYLOR
to order 1. After one immediate calculation, the system of
linearized state is written :
(5)
¾
¾
(6)
Since
; we finds the first transfer function binding the
output voltage
with the duty cyclic
:
¾
¾
(7)
We deduces the transfer function binding the inductor current
with the duty cyclic
:
¾
(8)
the transfer function binding the output voltage
with the
inductor current
:
¾
(9)
From equation (5), we can deduce the law control according
to:
(10)
In Massachusetts Institute of Technology (MIT) rule is :
(11)
Where
represents the error between the plant and model
output. The
is adjustable parameter and it is set in such a
way such that
is minimized to zero.
2.2 Small-Signal Modeling of the Boost Converter
3. Synthesis of Adaptive Control with
MRAC Regulator
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2022.1.4
Ahmed Chouya
E-ISSN: 2945-0489
26
Volume 1, 2022
We will present the synthesis of each closely related regu-
lator separately to clarify the synthesis methodology of each
of them.
We want to obtain in closed loop a response of the first
order type. To achieve this objective, we take a MRAC of the
type:
(12)
The closed loop system can be represented by the figure 3.
Adaptation law
V
e
m
V
ref
V
)(sG m
RC
s
L
R
s
RLC
e
e
2
1
.
1
1
2
D
D
6
s
V
~
s
I
~
sVVref
~
21
UU
Fig. 3. Diagram block in closed loop output voltage of MRAC regulator.
The reference model of the closed loop system is selected
with a first order transfer function :
That is to say the optimality criterion
of the adjustment
loop is expressed by the absolute value in [15] and [10]:
(13)
Its derivative is :
(14)
The out-put is written:
(15)
With
¾
,
and
.
The error
, its derivative compared to the
parameters gives :
(16)
(17)
For
then
,
and
.
(18)
(19)
Taking into account (14), (18) and (19), one can write the
equation of gradient
and
:
with
(20)
(21)
And
with
(22)
(23)
The search is based on the output voltage generated by
the step-up converter which has not been properly regulated.
This problem occurs when there are changes in the reference
voltage. This research which will be carried out in a boost
converter using an adaptive controller when the regulator
located in feed-forward and feedback and a boost converter
using MRAC.
To examine practical utility, the proposed regulator has been
simulated for a boost (see [13]), whose parameters are shown
in the table I.
TABLE I
DC-DC BOOST CONVERTER PARAMETERS.([13])
Parameters Notation Value Unit
Input Supply Voltage
Inductor
Resistor Load
Capacitor
!
Normal switching frequency
"
#
Switch off
$
Switch off
$
Duty cycle
Desire Output Voltage
Inductor steady-state current
%
By using these parameters, the model of DC-DC boost
converter (3) is utilized as a plant of the system. The deriva-
tion MRAC based on MIT rule for inductor output voltage
Regulator obtain (21) and (23). The value of
is specified to
achieve the appropriate response.
4. Results and Simulations
We show a detailed scheme general of the adaptive control
with MRAC regulator in figure 4. The performance of boost
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2022.1.4
Ahmed Chouya
E-ISSN: 2945-0489
27
Volume 1, 2022
Fig. 4. Diagram general of adaptive control with MRAC regulator for DC-DC
boost converter.
converter in proposed controller is proven in simulation such
that any changed responses are able to be observed. The input
voltage and resistor load of the boost converter are
and
, respectively. Reference voltage is set to be
and of
and
. As regulation parameters
and
&'(
are initialized at
and
respectively.
Figure (Fig.5) shows the evolution of the voltages (desired
and output), where after the transient state; the output voltage
follows the desired voltage
which is double the supply
voltage
. Because of the duty cycle equal to
,
The voltage error is shown in figure (Fig.6(a)). It cancels out
at
)
, with a voltage mean error
. The histogram
(figure Fig.6(b)) shows more than 1000 samples centralizing
at
.
The forms of induction current and desired current appear
in Figure (Fig.7) where after the transient state the induction
current follows the desired current of
%
. The current error is
shown in figure (Fig.8(a)) with a current mean error
%
.
It cancels out at
)
.Over
samples has
%
error and the
rest of the samples are around
%
(see Fig. 8(b)).
.
00.5 11.5
x 10
−3
0
5
10
15
20
25
30
Time[s]
Voltage[V]
Vd
V
Fig. 5. Output voltage for change in reference output voltage.
00.5 11.5
x 10−3
−20
−15
−10
−5
0
5
10
(a)
Time[s]
Voltage error[V]
−20 −10 010
0
200
400
600
800
1000
(b)
Gaussian
Histogram
Fig. 6. Output voltage error with histogram and Gaussian distribution for
change in reference output voltage.
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2022.1.4
Ahmed Chouya
E-ISSN: 2945-0489
28
Volume 1, 2022
0 0.5 1 1.5
x 10
−3
0
1
2
3
4
5
6
7
8
Time[s]
Inductor current [A]
id
i
Fig. 7. Inductor current for change in reference output voltage.
0 0.5 1 1.5
x 10
−3
−4
−3
−2
−1
0
1
2
3
4
(a)
Time [s]
Current error [A]
−2 0 2
0
100
200
300
400
500
600
700
(b)
Gaussian
Histogram
Fig. 8. Inductor current error with histogram and Gaussian distribution for
change in reference output voltage.
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2022.1.4
Ahmed Chouya
E-ISSN: 2945-0489
29
Volume 1, 2022
In this paper, MRAC with MIT rule is chosen to control
the DC-DC boost converter, this method is satisfied for
its controller structure and good performance in various
output voltages. The proposed system is stable and able
to perfectly reach the model reference with a shorter
recovery time. Adaptive gains determine the success of
adaptive control. The adaptation gains of the proposed
controller are obtained by empirical gains.
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(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
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5. Conclusion
References
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2022.1.4
Ahmed Chouya
E-ISSN: 2945-0489
30
Volume 1, 2022
