Design and Implementation of An Observer Controller For A Buck Converter
Design and Implementation of An Observer Controller For A Buck Converter
Design and Implementation of An Observer Controller For A Buck Converter
http://journals.tubitak.gov.tr/elektrik/
c TUB
doi:10.3906/elk-1208-41
Research Article
Received: 12.08.2012
Accepted: 02.12.2012
Printed: 18.04.2014
Abstract: An observer controller for a buck converter is presented. A state feedback gain matrix is derived in order
to achieve the stability of the converter and to ensure the robustness of the controller. A load estimator is designed to
estimate the unmeasurable variables and to obtain the zero output voltage error. A pulse-width modulation scheme is
adopted to obtain the output voltage regulation. In order to improve the transitory response and dynamic constancy of
the converter, the controller parameters are designed based on the current mode control. The design is evaluated and
verified using MATLAB/Simulink. An experimental set-up is done to evaluate the controller platform.
Key words: Buck converter, control law, LabVIEW, load estimator, observer controller, state feedback matrix
1. Introduction
DC-DC converters are widely used in personal computers, computer peripherals, communication systems,
medical electronics, and adapters of consumer electronic devices to provide the required level of DC voltages
[1]. The switched-mode power supplies that are used for telecommunication and computer systems require a
high switching frequency, high efficiency, high power density, small size, low weight, low voltage stress, and low
component count [2]. The feedback loops are designed in order to obtain the stability of the converter system.
Due to issues such as component deprivation or input voltage changes, conventional designs may lead to the
degradation of the closed-loop performance, resulting in poor dynamic stability due to a change in the operating
point [3]. This leads to the design of a robust controller that achieves good dynamic performance.
Among DC-DC converters, the buck converter plays a vital role. In portable consumer electronics, buck
converters are used the most. A buck converter is the simplest one, requiring only 1 switch, and is more than
90% efficient. In low-power DC-DC converters, overload protection, increased efficiency, and improved dynamic
response are obtained by current sensing or measurement. The measurement methods are generally the voltage
drop method and observer-based method. In the voltage drop method, the major drawback is that it decreases
the efficiency and requires an amplifier with a wide bandwidth, which is very difficult to implement [4]. Hence,
there is a need for the design of an observer controller.
The control techniques implemented through pulse-width modulation can be categorized into voltagemode control or current-mode control. Current-mode control is advantageous over voltage-mode control because
the system responds quickly to the disturbances [5]. However, this technique suffers from an inherent instability
Correspondence:
562
lakshmi amrith@yahoomail.com
and subharmonic oscillations at constant frequency operation; hence, a dynamic compensation has to be
designed. The major constraint in the design of control based on the frequency domain is the presence of
a zero in the right-hand side of the plane in many of the averaged models. The average value of the inductor
current is inversely proportional to the location of this zero; therefore, any increase in the value of the inductor
current may reallocate this zero to the lower frequency side of the right-hand side of the plane. This results in
considerable phase lags, thereby restraining the existing bandwidth for a constant operation of the converter
[6]. This enables the design to be carried out in the time domain.
The main control objective in the design of a controller for a buck converter is to impel the semiconductor
switch with switching pulses that make the system able to track the desired reference value of the given voltage
at the output. The output voltage regulation should be maintained consistently, regardless of the deviation in
the load or in the input voltage. Furthermore, the limitations in the design of the controller result from the
duty cycle, which is circumscribed between 0 and 1. This problem can be solved by modeling the buck converter
using the state-space averaging technique [7]. Using this technique, the converter can be described by a single
equation, approximately, over a number of switching cycles. The averaged model makes the simulation and
control design much faster.
The main objective of this work is to design a robust compensator based on the observer approach
that overcomes the above-mentioned problems. The design is based on the time domain, in which converter
specifications such as the rise time, settling time, maximum peak overshoot, and steady state error are met.
The modeling of the buck converter is done using the state-space averaging method and the observer controller
is designed using the pole placement technique and separation principle. MATLAB/Simulink is used to perform
the simulation. The experimental set-up is carried out using LabVIEW program as a controller platform and a
convenient USB data acquisition (DAQ) gadget, the results of which are illustrated. The subsequent sections are
organized as follows: Section 2 gives the design of a buck converter, Section 3 discusses the modeling, Sections
4 and 5 explain the design of the state feedback matrix and the observer controller, Sections 6 and 7 give the
simulation and experimental results, and, finally, the conclusion is given in Section 8.
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The output voltage of the buck converter is always less than the input voltage and is given by:
VO = dVS ,
where d =
Ton
T
(1)
is the duty cycle ratio, Ton is the on time of the semiconductor switch, and T is the switching
period. To ensure the continuous current mode of conduction, the selected value of inductance should be
greater than the critical value of the inductance LC , which acts as a boundary condition for the continuous
and discontinuous current modes of the operations.
The critical value of inductance is given by:
LC = (1 d)
R
,
2fS
(2)
VS T d(1 d)
,
L
(3)
1
.
8fS C
(4)
The following are the parameters considered for design: V S = 48 V, V O = 12 V, f S = 100 kHz, L = 720 H,
C = 8.667 10 7 F, and R = 14.4 .
3. Modeling of the buck converter
The state space analysis for the converter is carried out. The unique feature of this method is that the design
can be carried out for a class of inputs, such as the impulse, step, or sinusoidal function, in which the initial
conditions are also incorporated. This technique is expedient to use but it presents a low frequency estimate
of the accurate dynamics, where the discontinuous results initiated by the switching are disregarded [8]. The
state space analysis is discussed below.
The switch is turned on and off by a sequence of pulses with a constant switching frequency, f S [10,11].
[
]
il (t)
The state vector for this converter is defined as x (t) =
, where il (t) is the current through the
Vc (t)
inductor and VC (t) is the voltage across the capacitor. For the given duty cycle d(k) for the k th period, the
systems are illustrated by the following set of state-space equations in the continuous time domain:
= A1 x (t) + B1 VS (t) , s = 1
x (t)
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= A2 x (t) + B2 VS (t) , s = 0,
x(t)
(5)
where s = 1 represents the condition at which the switch is conducting and s = 0 represents the off condition
of the switch. Matrices A1 , A2 , B1 , and B2 for the buck converter are given by:
[
A1 = A2 =
1
C
[
B1 =
and
1
L
1
RC
1
L
0
0
(6)
B2 =
(7)
(8)
0 1
x(t)
(9)
r(t)
DC-DC
CONVERTER
y(t)
K
State feedback
control
The necessary condition for the arbitrary pole placement is that the system should be absolutely state
controllable. The state feedback matrix reduces the chattering of the input when the system attains its steady
state and further variations in the duty cycle. The system with the control law is defined as:
= (A Bk) x(t).
x (t)
(10)
The system under consideration is completely state controllable; hence, all of the eigenvalues of (A Bk) are
placed in the left half of the s-plane, causing the system to become asymptotically stable.
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The buck converter under consideration is of the second order and the desired poles can be easily placed
by assuming the following converter specifications:
(11)
2
(12)
1.4
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
0.002
0.004
0.006 0.008
Time (sec)
0.01
0.012
0.014
(13)
where k1 is the element of the state feedback matrix and r is the step input. The dynamic equation describing
the state observer is given by:
x
(t) = (A ke C) x
+ Bu (t) + ke y(t),
(14)
where ke is the observer gain matrix.
Using the separation principle, a dynamic compensator can be obtained by combining the control law and
the observer poles. The main advantage of this principle is that the design of the control law and observer can
be carried out independently, and when both are used concurrently, the roots remain unchanged. The observer
controller thus designed for the buck converter is discussed below.
The transfer function of the observer controller for the buck converter for a continuous time system is
obtained as follows:
U (s)
3.624 103 s + 4.22 1010
.
(15)
= 2
Y (s)
s + 2.113 105 s + 4.467 1010
Input
Voltage(V)
It is obvious from Figure 4 that the output thus obtained for this converter shows a much lower settling time
and no overshoots or undershoots, with zero steady-state error.
48
46
44
0.5
0.5
0.5
Output
Voltage(V)
20
1.5
1.5
1.5
1.5
10
0
4
Inductor
Current(A)
Time(s)
Time(s)
2
0
Load
Current(A)
Time(s)
4
2
0
0.5
Time(s)
6. Simulation results
The design and performance of the buck converter are accomplished in continuous conduction mode and
simulated using MATLAB/Simulink. The ultimate aim is to achieve a robust controller in spite of uncertainty
and large load disturbances. The converter specifications under consideration are the rise time, settling time,
maximum peak overshoot, and steady-state error, which are shown in Table 1. The results thus obtained are
in concurrence with the mathematical calculations. The simulation is also carried out by varying the load, not
limited to the R load, and it is illustrated in Table 2. It is evident from Table 2 that the controller tracks the
reference voltage in spite of the load variations. The output of the buck converter shows some steady-state
error, which is of appreciable order. The output voltage, load current, and inductor current are obtained by
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varying the input voltage as 44 V, 46 V, and 48 V, respectively, and are shown in the Figure 4. For all of the
variations, the controller is robust enough in tracking the reference voltage. It is evident that the controller
improves the dynamic performance of the system irrespective of the variations in the input voltages and load
values.
Table 1. Performance parameters of the buck converter.
Parameters
Settling time (s)
Peak overshoot (%)
Steady state error (V)
Rise time (s)
Output ripple voltage (V)
Values
0.015
0
0.05
0.0125
0
Buck converter
R () L (H)
3.6
15
9
10
9
35
20
10
9
20
E (V)
10
6
7. Hardware implementation
7.1. LabVIEW package
The buck converter with the observer controller is implemented using LabVIEW as a controller platform. LabVIEW is primarily used as a platform for implementing any closed-loop system and it can be used for the
improvement of a control system. It is extensively used for analyzing projects experimentally with a shorter
duration due to its programming flexibility along with integrated tools designed especially for testing, measurements, and control. The key feature of LabVIEW is that it extensively supports accessing the instrumentation
hardware. Drivers and abstraction layers are provided for almost all types of instruments. The buses are
also accessible for addition. The abstraction layers and drivers act as graphical nodes and enable effective
communication with the hardware devices, thereby offering standard software interfaces [14].
This software is used to build up the virtual instrumentation, which comprises the front panel and a
functional block diagram. The front panel shown in Figure 5 is mainly used for user interactions. It is through
the front panel that the desired transfer function of the observer controller is entered, as well as the corresponding
parameters of the closed-loop control, and hence the restructured condition of the system is obtained. The block
diagram, data acquisition, transfer function, and signal generation are built using the functional block diagram,
as shown in Figure 6. It provides wide varieties of small icons to perform the desired task. The LabVIEW package
provides many libraries with a large number of tasks for data acquirement, signal production, arithmetical and
statistical analysis, signal conditioning, and investigation, along with many graphical interface elements. These
features make it superior when compared with other development environments.
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generated within a much shorter duration of time without any delay or time lag. The experimental results thus
obtained are in concurrence with the simulation results and mathematical calculations. The prototype model
is developed using the values shown in Table 3.
Figure 12. Output voltage measured using an oscilloscope (scale: y-axis: 5divX1V/div).
570
Description
L
C
R
VS
fS
D
M
Experimental values
15 mH
1 F
20
10 V
20 kHz
1N4007
IRF840
8. Conclusion
A state feedback control approach has been designed for the buck converter in the continuous time domain using
the pole placement technique and separation principle. The load estimator was designed by deriving a full-order
state observer to ensure robust control for the converters. The separation principle allows the designing of a
dynamic compensator, which looks very much like a classical compensator since the design is carried out using the
simple root locus technique. The observer controller thus designed for the buck converter was implemented using
LabVIEW as a control platform and the results were illustrated. The mathematical analysis, simulation study,
and experimental study showed that the controller thus designed achieves tight output voltage regulation, good
dynamic performances, and higher efficiency. This method is topology-independent and can also be extended
for any of the applications such as power factor preregulation, photovoltaic cell, or speed control applications.
References
[1] S. Chander, P. Agarwa, I. Gupta, Auto-tuned, discrete PID controller or DC-DC converter for fast transient
response, International Conference on Power Electronics, pp. 17, 2011.
[2] B.R. Lin, C.L. Huang, H.K. Chiang, Analysis, design and implementation of an active snubber zero-voltage
switching cuk converter, IET Power Electronics, Vol. 1, pp. 5061, 2007.
[3] J. Morroni, R. Zane, D. Maksimovic, Design and implementation of an adaptive tuning system based on desired
phase margin for digitally controlled DC-DC converters, IEEE Transactions on Power Electronics, Vol. 24, pp.
559564, 2009.
[4] Z. Lukic, Z. Zhao, S.M. Ahsanuzzaman, A. Prodic, Self-tuning digital current estimator for low-power switching
converters, Applied Power Electronics Conference and Exposition, pp. 529534, 2008.
571
[5] K. Tse, M.D. Bernardo, Complex behavior in switching power converters, Proceedings of the IEEE, Vol. 90, pp.
768781, 2002.
[6] C. Sreekumar, V. Agarwal, A hybrid control algorithm for voltage regulation in DC-DC boost converter, IEEE
Transactions on Industrial Electronics, Vol. 55, pp. 25302538, 2008.
[7] T. Geyer, G. Papafotiou, R. Frasca, M. Morari, Constrained optimal control of the step-down DC-DC converter,
IEEE Transactions on Power Electronics, Vol. 23, pp. 24542464, 2008.
[8] N. Gonzalez Fonseca, J. De Leon Morales, J. Leyva Ramos, Observer-based controller for switch mode DC-DC
converters, Proceedings of the 44th IEEE Conference on Decision and Control, and European Control Conference,
pp. 47734777, 2005.
[9] S. Mariethoz, S. Almer, M. Baja, A.G. Beccuti, D. Patino, A. Wernrud, J. Buisson,
[10] H. Cormerais, H. Fujioka, T. Geyer, U.T. Jonsson, C.Y. Kao, M. Morari, G. Papafotiou, Rantzer, P. Riedinger,
Comparison of hybrid control techniques for buck and boost DC-DC converters, IEEE Transactions on Control
Systems Technology, Vol. 18, pp. 11261145, 2010.
[11] R.D. Middlebrook, Small-signal modeling of pulse-width modulated switched-mode power converters, Proceedings
of the IEEE, Vol. 96, pp. 768781, 1988.
[12] J. Sun, M.D. Mitchell, F.M. Greuel, T.P. Krein, M.R. Bass, Averaged modeling of PWM converters operating in
discontinuous conduction mode, IEEE Transactions on Power Electronics, Vol. 16, pp. 482492, 2001.
[13] M. Gopal, Digital Control and State Variables Methods Conventional and Intelligent Control Systems, New Delhi,
McGraw-Hill, 2009.
[14] H.S. Bae, J.H. Yang, J.H. Lee, H.B. Cho, Digital state feedback current control using pole placement technique
for the 42V/14V bi-directional DC-DC converter application, Proceedings of the Applied Power Electronics
Conference, pp. 37, 2007.
[15] C.M. Lee, Y.L. Liu, H.W. Shieh, C.C. Tong, LabVIEW implementation of an auto-tuning PID regulator via
grey-predictor, IEEE Conference on Cybernetics and Intelligent Systems, pp. 15, 2006.
572