Download International Electrical Engineering Journal (IEEJ) Vol. 6 (2015) No.3, pp. 1815-1821

International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.3, pp. 1815-1821
ISSN 2078-2365
http://www.ieejournal.com/
Reference Signal Limiter in the
Control of Buck Power Converters
with Improved Stability
Prasanth Sai1, M. Narendra Kumar2, S. Sunisith3
[email protected], [email protected], [email protected]

Abstract—This paper presents the solution for controlling the
dc-dc switched power of Buck type converter. This technique
states that the control signal can take the values in the range (0,
1) only. Such a control limitation is generally not considered into
account when designing the converter regulators. This is only
dealt with in the control implementation stage, placing a limiter
(controls the range of input signal) between the controller and
the controlled system. In this, the control signal limitation is
provided by using a non-linear regulator that consists of an
internal limiter. The final closed-loop control system is nothing
but the non-linear feedback involving a linear dynamics block in
closed-loop with a non-linear static element. If these conditions
are satisfied when choosing the control design parameters then
the regulator improves closed-loop stability and output
reference tracking.
tracking. However, in these studies it was never accounted for
the limitation of the control signal. As a matter of fact, the
control is not allowed to be outside the interval (0, 1), due to
the technological nature of the controlled circuits. Therefore,
the usual practice consists, when it comes to implementing the
controllers, in simply placing an isolated limiter between the
designed controller and the controlled system (Fig. 1.1).
Keywords: Buck Converters, Reference Signal Limiter,
Switched Power Converter.
Fig 1.1 Basic control setup of Buck converters
I. INTRODUCTION
There are three main types of switched power
converters, namely boost, buck, and buck–boost. These have
recently received an increasing deal of interest both in power
electronics works and in automatic control applications. This
is due to their wide applicability domain, e.g., domestic
equipments, communication systems, computers, industrial
electronics, battery-operating, embedded equipments,
uninterruptible power sources. From an automatic control
viewpoint, a switched power converter constitutes a
challenging case study as it is variable-structure and
nonlinear. Its rapid structure variation is generally coped with
using averaged models. Based on these average models,
different nonlinear controllers have been developed using
Passivity techniques, feedback linearization, flatness
methods, sliding mode control and back stepping control
technique. In all works the proposed controllers are designed
to achieve closed-loop global stability and voltage reference
Unfortunately, the nonlinear effect of such a limiter
is never taken into account when analyzing the closed-loop
system. As a consequence, the aforementioned stability
results may lose their global nature, the controller transient
performances may deteriorate and the output-reference
tracking objective may not be achieved. On the other hand,
the problem of controlling linear systems subject to input
saturation constraint has received a great deal of interest over
the last two decades. During the time intervals of control
saturation, the closed-loop system is no longer linear and the
controlled system is steered in open loop. Then, global
asymptotic stabilization is only possible for stable systems.
The problem of output-reference (in presence of control input
limitation) has not been so deeply investigated. When the
controlled system is of type-1 and the reference signal is
step-like, then the tracking problem can be transformed into a
disturbance-free regulation problem and existing solutions
can be applied. For non-type-1 systems, an integrator should
be incorporated in the regulator to make the tracking
objective achievable. But, the presence of an integrator
1815
Prasanth
et. al.,
Reference Signal Limiter in the control of Buck Power Converters with Improved Stability
International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.3, pp. 1815-1821
ISSN 2078-2365
http://www.ieejournal.com/
generally results in large control actions which, due to the
control limitation, may lead to undesirable oscillatory
behavior. To avoid such behavior, many authors have
proposed linear regulators together with ad-hoc
“anti-windup” devices.
Our approach consists in first designing a linear
control law that achieves the tracking objective in the absence
of control limitation. Then, an adequate anti-windup device is
incorporated, in the above control law, leading to a nonlinear
regulator. It is shown that the resulting closed-loop control
system is equivalent to a nonlinear feedback consisting of a
linear dynamics block in closed-loop with a nonlinear static
element. Sufficient conditions for -stability of this feedback
are then established using tools from the absolute stability
theory (circle criterion, Barbalat’s lemma). The main
condition is the real positivity (RP) of a specific transfer
function that involves, on one hand, the poles of the controlled
system and, on the other hand, those of the underlying linear
closed-loop system. In fact, the RP condition defines a
neighborhood of the controlled system poles in which should
be assigned those of the closed loop. Such a requirement can
always be satisfied through an adequate choice of the
controller parameters. Finally, it is shown that if the reference
signal is slowly varying then the proposed nonlinear
controller stops saturating after finite transient-periods and
the system output tracks asymptotically its reference.
consists of two parts that is D1TS and D2TS (D1+D2=1).
During D1TS, inductor current increases linearly and then in
D2TS it ramps down that is decreases linearly.
2.1.2 Discontinuous Conduction Mode:
When the inductor current has an interval of time
staying at zero with no charge and discharge then it is said
to be working in Discontinuous Conduction Mode (DCM)
operation and the waveform of inductor current is illustrated.
At lighter load currents, converter operates in DCM. The
regulated output voltage in DCM does not have a linear
relationship with the input voltage as in CCM. In DCM,
each switching cycle is divided into of three parts that is D1TS
, D2TS and D3TS (D1+D2+D3=1). During the third mode i.e. in
D3TS, inductor current stays at zero.
II. CONVERTER MODELING
A Buck converter is constituted of power electronic
components connected together, as shown in Fig. 2.1.
Figure 2.2: Inductor current waveform of PWM converter (a) CCM (b)
boundary of CCM &DCM (c) DCM
Fig 2.1 Buck converter circuit
2.1. Modes of Operation:
The operation of dc-dc converters can be
classified by the continuity of inductor current flow. So
dc-dc converter has two different modes of operation that
are (a) Continuous conduction mode (CCM) and (b)
Discontinuous conduction mode (DCM). A converter can be
design in any mode of operation according to the requirement.
2.1.1 Continuous Conduction Mode:
When the inductor current flow is continuous of
charge and discharge during a switching period, it is called
Continuous Conduction Mode (CCM) of operation shown in
figure 2.1 The converter operating in CCM delivers larger
current than in DCM. In CCM, each switching cycle T S
The switching converters convert one level of
electrical voltage into another level by switching action. They
are popular because of their smaller size and efficiency
compared to the linear regulators. Dc-dc converters have a
very large application area. These are used extensively in
personal computers, computer peripherals, and adapters of
consumer electronic devices to provide dc voltages. The
wide variety of circuit topology ranges from single
transistor buck, boost and buck-boost converters to
complex configurations comprising two or four devices and
employing soft-switching or resonant techniques to control
the switching losses.
The non-isolated dc/dc converters can be classified as
follows:
• Buck converter (step down dc-dc converter),
• Boost converter (step up dc-dc converter),
•Buck-Boost converter (step up-down dc-dc converter,
opposite polarity), and
•Cuk converter (step up-down dc-dc converter).
1816
Prasanth
et. al.,
Reference Signal Limiter in the control of Buck Power Converters with Improved Stability
International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.3, pp. 1815-1821
ISSN 2078-2365
http://www.ieejournal.com/
2.2 A Pulse-Width-Modulation Based Sliding Mode
Controller for Buck Converters:
Sliding mode (SM) controllers are well known for
their robustness and stability. Most of the previously
proposed SM controllers for switching power converters are
hysteresis modulation (HM) based. Naturally, they inherit the
typical disadvantages of having variable switching- frequency
operation and being highly control sensitive to noise. Possible
solutions are to incorporate constant timer circuits into the
hysteretic SM controller to ensure constant switching
frequency operation or to use adaptive hysteresis hand that
varies with parameter changes to control and fixate the
switching frequency. However, these solutions require
additional components and are unattractive for low cost
voltage conversion applications.
An alternative solution to this is to change the
modulation method of the SM controllers from HM to
pulse-width modulation (PWM). The idea is based on the
assumption that at a high switching frequency, the control
action of a sliding mode controller is equivalent to the duty
cycle control action of a PWM controller. Hence, the
migration of a sliding mode controller from being HM based
to PWM based is made possible.
III. DC-DC BUCK CONVERTER
The operation of the tri-state dc-dc buck
converter with hysteretic current-mode control scheme is
discussed in this subsection. Figure 3.6 shows the tri-state
buck converter topology. It consists of two controlled
switches S1 and S2, an uncontrolled switch D , an inductor L
and a capacitor C, a load resistance R . Switch S2 is the
additional switch which is connected across the inductor.
The operation of the tri-state converter includes three
different configuration or structures that are show in figure.
At the start of the clock period, the switch S1 is
turned on and the switch S2 is turned off. During this interval
(mode 1), inductor current increases with a slope of
(vin-v0/L) When iL reaches a peak value (upper bound), S1
turns off. Then iL starts falling with a slope of (-v0/L) until it
reaches some lower threshold. This interval is denoted as
mode 2. During this interval, diode is forward biased and
both switch S1 and S2 are turned off When the inductor
current reaches lower threshold, it stays constant at lower
boundary, because the switch S2 shorts the inductor L and
voltage across the voltage across the inductor is thus equal
to zero. During this interval S2 is turned on while S1 and diode
are off. This is the additional interval, denoted as mode 3.
The inductor current waveform showing the switch conditions
for a tri-state buck converter is shown in figure 3.3.
Fig. 3.1. A tri-state buck converter configuration
Fig. 3.2. Equivalent circuits under different modes of operation
These three modes of operation can be described as follows:
Mode 1: when S1 is on and, S2 is off, the state space equation
of buck converter is derived as
---(3.1)
Where x=[v0 iL] , v0 is the output voltage , iL is the inductor
current.
Mode 2: when S1 and S2 is off , the equation derived as,
T
..(3.2)
Mode 3: when S1 is off and S2 is on the state-space equation
is
…(3.3)
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Prasanth
et. al.,
Reference Signal Limiter in the control of Buck Power Converters with Improved Stability
International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.3, pp. 1815-1821
ISSN 2078-2365
http://www.ieejournal.com/
Where iL represent the inductor current, v0 and vref represent
the output voltage and reference voltage respectively. Here
the switching state of the switch is either 1 or 0.
Then by taking the derivative of (3.4) with respect to time,
Fig. 3.3. Inductor current waveform of tri-state buck converter showing
the switch conditions
3.1 Mathematical Analysis of Proposed Controller:
The operation of a hysteretic current-mode
controller for tri-state dc-dc buck converter is proposed and
the schematic diagram of proposed controller is shown figure
3.4.
…(3.5)
Considering the buck converter when the switch S1 is on, S2 is
off.
…(3.6)
…(3.7)
Substituting these equations we get the results as shown
below:
Fig 3.4. Schematic diagram of the hysteretic controller for tri-state buck
converter
The digital logic blocks generates required switching
pulses for controlling the Switches S1 and S2 .This block
consists of two SR flip-fops and some logic gates that can be
shown in figure 3.5.
…(3.8)
The dynamics of the converter circuit, when S1 and S2 are off,
can be expressed as,
…(3.9)
Substituting these equations we get the results as:
Figure 3.5. Schematic diagram of pulse generator circuit
The state-space description of the system in terms of
the desired control variables (i.e, voltage, current etc) is
developed. The proposed current controller employs both
the output voltage error x1 and the inductor current x2 as the
controlled state variables, which are expressed as
…(3.4)
…(3.10)
The dynamics of the converter circuit in mode 3, when S2 is
on, S1 is off can be expressed as,
…(3.11)
Since in this mode of operation, inductor current stays at a
constant value, so we get the derivative of a constant value is
zero.
1818
Prasanth
et. al.,
Reference Signal Limiter in the control of Buck Power Converters with Improved Stability
International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.3, pp. 1815-1821
ISSN 2078-2365
http://www.ieejournal.com/
…(3.12)
Then, the state-space modeling of the converter circuit with
state variables x1 and x2 is given by,
(3.13)
Where x=[x1 x2] is the state vector and A‟ s and B‟ s are
the system matrices.
The state matrices and the input vectors for the three periods
are,
T
As studied from the previous discussion that the basic
principles of a hysteresis control is based on the two
hysteresis bands (upper and lower bands), whereby the
controller turns the switch on when the output current
falls below the lower band and turns the switch off when
output is beyond the upper bound. The switching action
can be determined in the following way:
1. If iL< lower bound, u=1(ON)
2. If iL> upper bound, u=0(OFF) , where u is the control
input.
In this case, the hysteretic current controller shown in
figure 3.2 consists of two control loops. One is the current
control loop and the other is the voltage control loop. The
difference between the actual output voltage and reference
voltage generates the error voltage.
A controller may a P or PI type use the voltage error
signal to provide a control voltage which is consider as the
upper boundary or the upper threshold vhys+ Here a simple
proportional controller is considered for easy analysis. The
control voltage vcon is a function of the output voltage v0 and a
reference signal vref in the form,
Hence, there are two switching actions u1 and u2 are
occurring on the two switching surfaces in the region of
phase plane. The control law determines the switching
action. This can be represented mathematically as,
…(3.15)
Where, h1 and h2 are the two switching surfaces (manifolds) in
sliding mode phase plane that can be expressed as the
combination of state variables,
..(3.16)
Where, x1 and x2 are the state variables, k is a
constant factor that transforms the current signal to voltage
signal, kp is the proportional gain, which is known as sliding
coefficient. In these equations u1 and u2 can be known as the
discrete control inputs to the system and this equations forms
the basis of control law for the hysteretic control system.
Generally a SM control design approach consists of
two components. The first involves the design of a sliding
surface and the second is concerned with the selection of a
control law that directs the system trajectory towards system
stable operating point. If we assume the width of the
hysteresis band Δ→0, then there exists only a single switching
surface h1 in phase plane. And for sliding mode to be exist on
the switching surface, then h1=0. This is said to be as ideal SM
operation. Hence for ideal SM operation dh1/dt= h1=0. Also
the equivalent control produces trajectory whose motion is
exactly equivalent to the trajectory motion in an ideal SM
operation. In equivalent control, the discrete control input is
substituted by continuous equivalent control signal ueq and is
obtained by solving the equations:
…(3.17)
…(3.14)
where, kp is the gain of proportional controller. The lower
threshold vhys- , is determined by subtracting a small constant
term Δ , which is given as, vhys- = vhys+ - Δ.
1819
Prasanth
et. al.,
Reference Signal Limiter in the control of Buck Power Converters with Improved Stability
International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.3, pp. 1815-1821
ISSN 2078-2365
http://www.ieejournal.com/
Then substituting the values, we get
converters can be found out by considering both the switching
surfaces for analysis. From these equations we can find out
the values of control parameters such that it will satisfy the
existence condition.
IV. RESULTS
…(3.18)
4.1 The Tracking behavior of the controller in presence of
a time varying output reference signal y*(t) switching
between 9 and 15 V:
…(3.19)
From above equation ueq can be found as
…(3.20)
Where ueq is a smooth function of the discrete input function u
and bounded by 0 and 1.
Then by substituting the values into 0< ueq <1, the function
can be expressed as,
(3.21)
Multiplying the above equation by vin,
(3.22)
The SM existence region provides a range of sliding
area where the sliding motion takes place. The condition for
the existence of sliding mode is achieved, if the following
inequalities are satisfied.
Fig. 4.1. Tracking behavior of the controller in presence of a time
varying output reference signal
4.2 Comparison between the computed duty ratio v(t) and
the applied value μ(t) in presence of a time-varying input
reference:
….(3.23)
(3.23)
By solving the above equations, we get,
…(3.24)
The above two equations represents two lines in
phase plane with same slope. As there are two switching
surfaces, so the SM existence region for hysteresis controlled
Fig. 4.2. Comparison between the computed duty ratio v(t) and the applied
value μ(t)
1820
Prasanth
et. al.,
Reference Signal Limiter in the control of Buck Power Converters with Improved Stability
International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.3, pp. 1815-1821
ISSN 2078-2365
http://www.ieejournal.com/
4.3 Robustness of the controller with respect to load
resistance changes:
Fig. 4.3. Robustness of the controller with respect to load
resistance
4.4 Comparison between the computed duty ratio v(t) and
the applied value μ(t) in presence of load resistance
changes:
Fig. 4.4. Comparison between the computed duty ratio v(t) and the applied
value μ(t) in presence of load resistance
4.5. Real part of c(jῳ)/A(jῳ) versus:
Fig. 4.5. Real part of c(jῳ)/A(jῳ) versus
V. CONCLUSION
The control limitation is important to all power
converters operating according to the PWM principle,
whatever the control design approach. In the present paper,
such issue has been dealt with, for converters of the Buck
type, using the nonlinear controller. It contrasts with previous
solutions that consist in first designing a controller as if there
was no control limitation and then incorporating an isolated
limiter, placed between the controller and the controlled
system. The point is that the effect of such a limiter is also
ignored, in previous works, when analyzing the closed-loop
control system.
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et. al.,
Reference Signal Limiter in the control of Buck Power Converters with Improved Stability