Mismatch between switching frequency and circuit parameters often occurs in industrial applications, which would lead to instability phenomena. The bifurcation behavior of V
^{2}
controlled buck converter is investigated as the pulse width modulation period is varied. Nonlinear behavior is analyzed based on the monodromy matrix of the system. We observed that the stable period1 orbit was first transformed to the period2 bifurcation, which subsequently changed to chaos. The mechanism of the series of period2 bifurcations shows that the characteristic eigenvalue of the monodromy matrix passes through the unit circle along the negative real axis. Resonant parametric perturbation technique has been applied to prevent the onset of instability. Meanwhile, the extended stability region of the converter is obtained. Simulation and experimental prototypes are built, and the corresponding results verify the theoretical analysis.
I. INTRODUCTION
The V
^{2}
controlled buck converter has been extensively used because of its advantages of rapid load transient response and easy implementation
[1]

[4]
. The exact small signal model of the system and the design guidelines have been presented in
[5]
. The mathematical model of the compensated V
^{2}
controlled buck converter has been derived in
[6]
, which verified the stability characteristics of the regulator. A quasiV
^{2}
adaptive ontime controller for the buck converter has been proposed in
[7]
, which can provide stable operation with small equivalent series resistance of the output capacitor. With the addition of an inductor current ramp estimator, an enhanced V
^{2}
type constant ontime controller, which is used to manipulate the buck converter, has been proposed in
[8]
to suppress unstable behavior. However, the abovementioned studies mainly focus on technology improvements instead of nonlinear dynamics analysis.
Power regulators are inherently nonlinear circuits with extensive instability phenomena. Therefore, a range of bifurcation behaviors have been observed, such as flip bifurcation
[9]
, Hopf bifurcation
[10]
, border collision
[11]
, quasiperiodicity
[12]
, and chaos
[13]
.
In
[14]
, the authors reported that subharmonic oscillations occur if the duty cycle is greater than 0.5. Slope compensation technology is applied to solve the problem. The describing function model of the V
^{2}
constant ontime controlled buck converter was built in
[1]
. The subharmonic oscillation point is also predicted successfully. The averaged model based on the Krylov–Bogoliubov–Mitropolsky ripple estimation technique was built in
[15]
, and ripple oscillation instability is solved based on the presented fixed compensation ramp method. In
[16]
, the subharmonic oscillations of V
^{2}
and V
^{2}
I_{C}
controlled buck converters were analyzed as the varied output voltages resulted in the alteration of the duty cycle. In
[17]
, the authors studied the instability characteristics of the converter based on the Floquet method as the feedback factor varied and stabilized the entire system with the application of sinusoidal compensation.
However, none of the abovementioned works investigated the mechanism of bifurcation and chaos of the system as the switching frequency varies. Typically, along with variation in the switching frequency, the dynamics of the converter show a series of bifurcations. Thus, whether the object will exhibit a similar phenomenon is still unknown.
Generally speaking, instability phenomena, such as bifurcation, subharmonic oscillation, and intermittent oscillation, are detrimental to industrial electronics. Efforts have been made to suppress unstable behavior in electronic circuits. Control of chaos was first conceived in
[18]
by injecting temporal parameter perturbations into the system to force the trajectory to approach the preset stable periodic orbit. A newly developed sliding mode controller with a timevarying manifold dynamic to compensate for the external excitation in chaotic systems was proposed in
[19]
. The resonant parametric perturbation methods in
[20]
and
[21]
have been applied to control chaos in power converters. In
[22]
, a conventional feedback controller was designed to drive the chaotic Duffing equation to one of its inherent multiperiodic orbits. The features of these methods involve the use of an extra state feedback or parameter disturbance to force the unstable periodic orbit back to the stable period1 limit cycle.
Undoubtedly, instability phenomena exist in the buck converter under V
^{2}
manipulation. The extensive application of these circuits enables people to have a deep understanding of the essence of the phenomenon. Such knowledge could be used to improve the performance of these circuits as well as determine new fields of applications. In this study, the existence of bifurcation and chaos is described by calculating the monodromy matrix of the entire switching period. The mechanism of bifurcation and chaos control is described as well as the extended stable region of the stabilized system.
The paper is organized as follows: The mathematical model of the V
^{2}
controlled buck regulator is described in Section II. The derived monodromy matrix of the closedloop converter is analyzed in Section III. The mechanism of period2 bifurcation and chaos exhibited by the system is investigated in Section IV. In Section V, the resonant parametric perturbation technique is applied to suppress the instability. The simulation and experimental results for the verification are given in Section VI, and conclusions are drawn in Section VII.
II. MATHEMATICAL MODEL OF THE CONVERTER
The circuit diagram of the V
^{2}
controlled buck converter is shown in
Fig. 1
(a), and steadystate waveforms for the continuous conduction mode are shown in
Fig. 1
(b), where
U_{r}
,
R_{E}
, and
G
_{1}
are the reference voltage, Equivalent Series Resistance (ESR) of the output capacitor, and feedback amplification factor, respectively. Given that the capacitance is large, voltage
u_{C}
is essentially constant, and the output voltage is expressed as follows:
V^{2} controlled buck converter. (a) Schematic circuit diagram of the system. (b) Illustrated steadystate waveforms.
The mathematical model of the V
^{2}
switching technique can be expressed as follows:
Substituting Eq. (2) and
into Eq. (1) obtains
and
The value of
R_{E}
is small. Therefore,
G
_{1}
≫ 1 leads to
u_{C}
≈
u_{O}
≈
U_{A}
. The relationships between
u_{O}
,
U_{r}
, and
G
_{1}
in Eq. (5) are shown in
Fig. 2
.
Relationship of u_{0}(U_{r}, G_{1})
Supposing the system begins a new operating period at the time instant of
t
=
nT
, the switch turns on and the trajectories of the state variable
x
= [
i_{L}
,
u_{C}
]
^{T}
run in subsystem [
A
_{1}
,
B
_{1}
]. When the switch is turned off at
t
=
nT
+
dT
, the converter runs in subsystem [
A
_{2}
,
B
_{2}
]. Afterward, the state variable
x
crosses the switchoff state and then turns back to
x
(
nT
). At the moment when the switch is turned off, the switching surface
h
_{2}
is expressed by Eq. (2). Given that the capacity of
C
is large,
u_{C}
is kept almost constant at the switching frequency. As
i_{O}
=
u_{O}
/
R
≈
u_{C}
/
R
, the switching surface, where
u_{O}
is considered the controlled object, can be rewritten from Eq. (2) as follows:
The state matrices of the subsystems that belong to the converter are shown in
Table I
.
SYSTEM MATRICES OF V2CONTROLLED BUCK CONVERTER
SYSTEM MATRICES OF V^{2} CONTROLLED BUCK CONVERTER
The system state vectors can be expressed as follows:
With respect to a current conduction mode buck converter, the iterated equation of the switching point can be expressed as follows
[23]
:
where
d
is the duty cycle and
d′
= 1 −
d
.
Numerically solving the previously presented equations using MATLAB with the Newton–Raphson method, the values of
x
and duty cycle
d
can be obtained for the periodic orbit.
III. MONODROMY MATRIX
The state variable trajectory of the closedloop regulator forms a period1 limit cycle in the phase space over a complete switching period. Based on theory of monodromy matrix
[24]
,
[25]
, if the Floquet multipliers of the monodromy matrix are all within the unit circle, then the system is stable. If the maximum Floquet multiplier equals 1, then bifurcation occurs; otherwise, it is unstable.
Based on Eq. (6), the normal vector and derivative of
h
_{n2}
with respect to time
t
are expressed as follows:
Substituting Eqs. (7) and (8) into Eq. (10) obtains
Based on Eqs. (10), (11), (12), and (13), the saltation matrix can be expressed as follows:
Evidently, the following inequality is true:
Therefore, only one Filippov solution for the switching surface of the system is obtained
[26]
, and the monodromy matrix over one switching period can be expressed as follows:
IV. MECHANISM OF THE NONLINEAR BEHAVIOR
The converter is stable and the maximum Floquet multiplier is less than 1 if the switching frequency
f
is high. With decreasing
f
, one of the Floquet multipliers goes beyond the unit circle through the negative real line, which marks the onset of instability through a period2 bifurcation. If
f
decreases continuously, then the state trajectory of the system will continue to double until it becomes chaotic.
The selected parameters for the system are shown in
Table II
. The output voltage
u_{O}
= 4.76 V was obtained by using Eq. (5). The equilibrium points of subsystems
S
_{1}
and
S
_{2}
are
E
_{1}
(4, 12) and
E
_{2}
(0, 0), respectively. Calculating Eq. (9) obtains the Floquet multipliers of monodromy matrix
M
, whose evolution diagram is shown in
Fig. 3
. To predict the critical value of the bifurcation point and the type, we assume that the absolute value of the maximum Floquet multiplier is 1, that is,
CIRCUIT PARAMETERS OF V2CONTROLLED BUCK CONVERTER
CIRCUIT PARAMETERS OF V^{2} CONTROLLED BUCK CONVERTER
Evolution diagram of the maximum Floquet multiplier.
Substituting Eq. (16) into Eq. (17) results in
f
= 18.4 kHz. Thus, the eigenvalues are −1.000 and 0.422, respectively. One of the Floquet multipliers is located at the cross point between the unit circle and the negative real line, which predicts the onset of period2 bifurcation.
 A. Period1
The closedloop converter exhibits a periodic steady state at
f
= 25 kHz, as shown in
Fig. 4
.
Stable period1 waveforms and the switching surface. (a) Phase diagram of i_{L}–u_{O}. (b) Output voltage and switching surface.
Considering the effect of
R_{E}
on the filter capacitor,
u_{C}
approximates but is not equal to
u_{O}
. The state variable that we selected for use in this study is
x
= [
i_{L}
,
u_{C}
]
^{T}
.
The stable period1 waveforms are shown in
Figs. 4
(a) and
4
(b). At the beginning of a period, the switch S turns on and the system runs in subsystem
S
_{1}
starting from point A. Given that point A does not locate exactly on the periodic orbit of subsystem
S
_{1}
, the system will gradually approach the limit cycle that encompasses equilibrium point
E
_{1}
(4, 12) along the trajectory of ACB, which is the stable periodic orbit of subsystem
S
_{1}
. After a time interval
t
=
dT
, the trajectory reaches the switching surface
h
_{2}
, namely, point B. As a result, the system switches to subsystem
S
_{2}
and approaches the equilibrium point
E
_{2}
(0, 0) starting from point B. The system trajectory is shown as BDA.
 B. Period2 Bifurcation
The period2 bifurcation of the system with switching frequency at 17 kHz is shown in
Fig. 5
. Supposing switch tube S turns on at the beginning of the operating period, the converter runs in subsystem
S
_{1}
and the starting point is
C
_{1}
. At this time subinterval, the running trajectory is shown as the curve of
C
_{1}
–
D
_{1}
. As soon as the state trajectory reaches the switching surface
h
_{2}
at the time instant
t
=
d
_{1}
T
, the switch tube S turns off and the converter switches to subsystem
S
_{2}
. The running trajectory is shown as the curve of
D
_{1}
–
C
_{2}
. At time instant
t
=
T
, a period is over and the switch S turns on again, with the trajectory approaching
C
_{2}
instead of
C
_{1}
. At this time, the running trajectory is shown as the curve of
C
_{2}
–
D
_{2}
. At the time instant
t
=
T
+
d
_{2}
T
, the state trajectory arrives at the switching surface
h
_{2}
, the switch turns off, and the converter switches to and runs in subsystem
S
_{2}
. The running trajectory is shown as the curve of
D
_{2}
–
C
_{1}
. When
t
= 2
T
, the system trajectory goes back to point
C
_{1}
and then approaches the next period. Repeating these processes constantly forms the period2 oscillation.
Period2 waveforms and the switching surface. (a) Phase diagram of i_{L}–u_{O}. (b) Output voltage and switching surface.
From the previously discussed analysis, we can conclude that, the steadystate solutions of subsystems
S
_{1}
and
S
_{2}
remain unchanged when switching frequency
f
changes. However, with the value of
f
decreasing, the distance between the switching equilibrium point
x
(
dT
) and the switching surface
h
_{2}
increases. As a result, the trajectory of subsystem
S
_{1}
starting from point
C
_{1}
takes a longer time interval,
t
=
d
_{1}
T
, to reach the switching surface
h
_{2}
in the first period, which causes the system trajectory to pass point
D
_{1}
and reach point
C
_{2}
during time (1 −
d
_{1}
)T when switch S turns off. Then, the second period, starting from point
C
_{2}
, begins. Consequently, the system trajectory returns to starting point
C
_{1}
after two periods 2
T
.
 C. Period4 Bifurcation and Chaos
When
f
= 11.5 kHz, the period4 bifurcation occurs in the system, as shown in
Fig. 6
. The mechanism of period4 bifurcation is similar to period2 bifurcation and is not discussed here.
Period4 waveforms and the switching surface. (a) Phase diagram of i_{L}–u_{O}. (b) Output voltage and switching surface.
As the switching frequency
f
continues to decrease, period2 bifurcation correspondingly occurs until chaos is achieved. The phase diagram is shown in
Fig. 7
, with switching frequency equal to 9.5 kHz and
u_{C}
is aperiodic. Many collision points exist between the state trajectory of the system and switching surface
h
_{2}
. This mechanism can be interpreted as follows: Given that the system trajectory sets off from subsystem
S
_{1}
, it collides with the switching interface after some time, thereby enabling the system to enter subsystem
S
_{2}
. Considering the effect of the periodic pulse signal, the system goes back to subsystem
S
_{1}
. However, the collision points between the state trajectory of the system and switching surface differ with time. As such, if we consider the collision point as the initial value, an infinite number of trajectories run in the subsystem and approach the equilibrium point of
S
_{2}
gradually, which generates chaos and oscillation in the system.
Chaos waveforms and the switching surface. (a) Phase diagram of i_{L}–u_{O}. (b) Output voltage and switching surface.
V. STABILIZATION CONTROL
The resonant parametric perturbation method is applied to control the aforementioned chaotic system and expand the stability boundary
[20]
. A sinusoidal signal
U_{e}
with the same frequency as driving signal
f
is added to
U_{r}
. The sinusoidal signal
U_{e}
takes the form asin(2
πft
). After compensation, the reference voltage can be expressed as follows:
 A. Analysis of the SteadyState Error
Adding the reference voltage signal changes the expression of the switching surface
h
_{2}
in the system, thereby changing the duty cycle
d
. The duty cycle of the system before and after sinusoidal voltage compensation is denoted as
d
and
, respectively. The change ratio Δ
d
of the duty cycle is calculated as follows:
As the value of
a
is small, apparently, it could have the possibility Δ
d
≈ 0. As such, after adding the sinusoidal compensation voltage
U_{e}
, the duty cycle of the system, the output voltage, and inductor current hardly vary. Thus, no significant changes were observed for the steadystate error of the system.
 B. Monodromy Matrix
With the resonant parametric perturbation added to the system, the switching surface transforms to
The derivative of the switching surface and the saltation matrix are shown as follows:
where
n_{e}
= [
R_{E}
1 
R_{E}
/
R
]
^{T}
.
The transmission matrices are denoted as
and
The monodromy matrix is expressed as follows:
The minimum boundary of
a
_{min}
is shown in
Table III
and
Fig. 8
based on Eq. (23).
STABILITY BOUNDARY WHEN max λMe = 1
STABILITY BOUNDARY WHEN max λ_{Me} = 1
Region of stability of the period1 orbit in the a–f parameter space.
To verify the stable region shown in
Fig. 8
, the simulation results around the boundary conditions are given as follows: Based on
Fig. 8
, the stable boundary at
f
= 9.5 kHz is
a
= 0.07839. At the time instant
t
= 0.08 s, two sinusoidal compensation voltages,
U_{e}
= 0.078sin(2
πft
) and
U_{e}
= 0.08sin(2
πft
), are added to
U_{r}
. The values of the switching point and characteristic multiplier are shown in
Table IV
. The phase diagrams of the two simulations are shown in
Fig. 9
. The system is stabilized to period2 bifurcation at
a
= 0.078 and enters the stable period1 trajectory at
a
= 0.08. The transient responses of the output voltage and switching surface are shown in
Fig. 10
.
FLOQUET MULTIPLIER ATf= 9.5 kHz
FLOQUET MULTIPLIER AT f = 9.5 kHz
Phase diagram after stabilization control at f = 9.5 kHz. (a) Stable period1 at a = 0.08. (b) Period2 bifurcation at a = 0.078.
Waveforms of output voltage. (a) Transient responses of the output voltage. (b) Output voltage u_{O} and switching surfaces h_{e} and h_{n2}.
 C. Mechanism of Chaos Control
Before and after perturbation is applied, the switching surfaces shown in
Fig. 9
(b) are expressed as follows:
Compensation enables the output voltage of the converter
u_{O}
to intersect point P (which belongs to
h_{e}
) instead of point Q, which shortens the distance between the equilibrium point
x
(
d_{e}T
) and the switching surface
h
_{2}
, which decreases the duty cycle of the system. The converter was forced back to stable period1 orbit.
VI. EXPERIMENTAL DETAILS
The experimental circuit diagram shown in
Fig. 11
is set up based on
Fig. 1
. In the experiment, the IRF640 type MOSFET and MBR3045PT are selected as the power switch and diode, respectively. The pulse signal and the sine compensation voltage signal are generated by a dual output function signal generator. We use LM358 and LM393 as the amplifier and comparator, respectively. The RS flipflop consists of CD4001, whereas the drive circuit is equipped with the TLP250 and is supplied by an independent +12 power source.
Circuit diagram of V^{2} controlled buck converter.
In the converter, the ESR of the output capacitor significantly influences the instability behavior of the regulator. To facilitate integration, the ESR is shown in
Fig. 1
. However, in industrial application, the ESR will not actually be implemented. As a result,
u_{C}
and
u_{O}
are equal to each other.
The experimental waveforms of the closedloop system are shown in
Figs. 12
(a) to
12
(f). After sinusoidal signal compensation,
U_{e}
= 0.078sin(2
πft
) and
U_{e}
= 0.08sin(2
πft
) are added to the reference voltage
U_{r}
. Then, the system will be stabilized to the period2 bifurcation or the stable state of period1. Considering the parasitic parameters and measurement errors, the experiment results slightly deviate from the simulation. In particular, when the compensation voltage
U_{e}
= 0.8sin(2
πft
) is injected to the system, period2 bifurcation was achieved instead of the stable state of period1. By contrast, the system achieves the stable state of period1 when
U_{e}
= 0.2sin(2
πft
) is applied to the experiment. However, the state deviation is small and the analytical method is proven to be effective.
Experimental waveforms of V^{2} controlled buck converter. (a) The waveforms of period1 when f = 25 kHz. (b) Period2 bifurcation when f = 17 kHz. (c) Period4 bifurcation when f = 11.5 kHz. (d) Chaos when f = 9.5 kHz. (e) Controlling chaos to period2 bifurcation when a = 0.078. (f) Controlling chaos to stable period1 when a = 0.08.
VII. CONCLUSIONS
The switching frequency of the converter is an important design objective in the power supply field. An improper switching frequency would render the system unstable, which is often encountered by engineers. A detailed analysis of a buck converter manipulated using the V
^{2}
technique with variations in the operating period is conducted based on the resulting monodromy matrix. The switching frequency taken as the bifurcation parameter reveals some interesting nonlinear phenomena in the converter. With the decrease in
f
, the series of period2 bifurcation up to the chaotic state is observed. The mechanism of bifurcation and chaos is analyzed in detail in this study. To suppress the instability behavior, the socalled resonant parametric perturbation technique is applied, which helps control the chaos phenomenon. The extended stable region of the stabilized system is obtained, which is proven to be true by the simulation and experimental results.
Acknowledgements
This project is supported by the Twelfth FiveYear Educational Science Plan Project of Guangzhou City (grant no. 2013A060 and 2013A038).
BIO
Wei Hu was born in Ningxiang, China, in 1980. He received his B.S. and M.S. degrees from the College of Electrical and Information Engineering, Hunan University, Changsha, China, in 2003 and 2006, respectively. He is currently working toward a Ph.D. degree. Since 2006, he has been with Guangzhou University, Guangzhou, where he is currently an experimentalist in the Lab Center. His current research interests include modeling and nonlinear control of power converters and stability analysis of DC–DC converters.
Fangying Zhang was born in Hubei, China, in 1981. She received her B.S. degree in Electronics and Information Engineering from Hubei University of Technology, China, in 2003 and her M.S. degree in Geodetic Survey from Wuhan University, Wuhan, China, in 2006. She is currently an experimentalist in the Lab Center at Guangzhou University. Her current research interests include modeling and nonlinear control of power converters and stability analysis of DC–DC converters.
Xiaoli Long was born in Xiangtan, China, in 1966. She received her M.S. degree from South China University of Technology in 2001. Her current research interests include robotic control and stability analysis of DC–DC converters.
Xinbing Chen was born in Henan, China, in 1978. His current research interests include modeling and nonlinear control of power converters and stability analysis of DC–DC converters.
Wenting Deng was born in Hunan, China, in 1983. Her current research interests include modeling and nonlinear control of power converters and stability analysis of DC–DC converters.
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“Differential equations with discontinuous righthand side,”
American Mathematical Society Translations
42
(2)
199 
231