International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.2, pp. 1787-1791
ISSN 2078-2365
http://www.ieejournal.com/
Control of Chaos in Positive Output Luo
Converter by means of Time Delay
Feedback
Nagulapati Kiran
nkiran.ped@gmail.com

Abstract— Faster development in Dc to Dc converter
techniques are undergoing very drastic changes due to
that major advancements like low voltage, high power
density in electronic industry. Chaos is a kind of quasistochastic behaviors of determinate nonlinear system. It
is a non-linear phenomenon specifically found in all nonlinear systems. The dc-dc converters exhibit a wide
range of bifurcation and chaotic behavior under certain
operating conditions. This paper analyses the behavior
of Luo converter under current control mode. As the
reference current is increased, the system becomes
unstable to chaos. This is shown by means of bifurcation
diagram. This is stabilized by means of time delayed
feedback control method. The simulation is done using
PSIM.
Index Terms— Chaos, Bifurcation, Positive Output Luo
Converter,Time Delayed Feedback Control
I. INTRODUCTION
Switched dynamical systems such as dc-dc converters are
known to exhibit non-linear behavior such as bifurcation and
chaos due to cyclic switching of circuit topology [1]. Chaotic
motion occurs frequently in the DC-DC converters, for the
performance of the harsh electromagnetic noise, the control
system of the intermittent unstable and critical operation of
the collapse, and so on. Chaos movement studies have
identified in the DC-DC converters, most of them because of
bifurcation and chaos caused in the system [2]. In most
practical situations, required stable operation is a period-1
operation. The interaction of non-linear components with
certain range of operating parameters can cause qualitative
changes or bifurcations in a power converter [3]. Thus, any
effective design automatically has to avoid the occurrence of
chaos for the range of variations of the parameters. In this
paper current mode control of positive output Luo converter
is implemented in
MATLAB/SIMULINK. Also Time Delayed Feedback
Control is implemented in order to eliminate chaos
phenomenon occurring in dc-dc converters. Profound study
and analysis about these non-linear phenomena is mandatory
so that this non-linear phenomenon can be controlled. In
most practical situations, required stable operation is a
period-1 operation. Thus, any effective design automatically
has to avoid the occurrence of chaos for the range of
variations of the parameters.
II. BASIC OPERATION OF POSITIVE OUTPUT
LUO CONVERTER
The Basic Circuit Diagram of Positive Output Luo
Converter is shown in Figure below.
The MOSFET is driven by PWM signal with frequency fs
and duty ratio D. The converter is assumed to be operated in
continuous conduction mode. Analysis of positive output
Luo Converter is explained below.
Figure 1: Basic Circuit of Luo Converter
A. Mode 1 Operation: ( 0  t  DT )
When switch is ON, the inductor L1 absorbs energy from
source and inductor L2 absorbs energy from both source and
capacitor. Both iL1 and iL2 increases during this mode and
source current is sum of inductor currents.
1787
Kiran
Control of Chaos in Positive Output Luo Converter by means of Time Delay Feedback
International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.2, pp. 1787-1791
ISSN 2078-2365
http://www.ieejournal.com/
Fig 2. Waveforms of Luo Converter
Equations governing Mode 1 operation are:
III. CHAOTIC BEHAVIOR
di L1 E

dt
L1
dVc  iL2

dt
C1
di L 2
E Vc1 Vc 2



dt
L2 L2 L2
dVC 2 i L 2 Vc 2


dt
C 2 Rc 2
B. Mode 2 Operation: ( 0  t
(1)
(2)
(3)
(4)
 DT )
When the switch S is turned off, inductor L1 transfers
stored energy to capacitor C1 through freewheeling diode D.
At the same time current iL2 flows through C2-R-D. Both
currents iL1 and iL2 decreases during this mode of operation
and the switch current is zero.
The equations governing this mode of operation are:
d iL1
dt
d iL 2

 VC1
L1
(5)
 VC 2
dt
L2
dVC1 i L1

dt
C1
dVC 2 i L 2 VC 2


dt
C 2 RC 2

Current mode control is considered here to analyze the
chaotic behavior in positive output LUO converter. Here the
switch is turned ON periodically by the clock and OFF
according to the output of a comparator that compares the
inductor current IL with a current reference Iref.
When the switch is on, the inductor current climbs up and
as it reaches Iref, the switch is turned off, thereby causing the
inductor current to ramp down until the next clock comes. To
analyze the chaotic behavior in positive output LUO
converter, current mode control is considered.
For this analysis the converter parameters are chosen as
follows.
Supply Voltage E = 12v
Output Voltage Vo = 12v
Inductor Current L = 100μH
Capacitor C = 10μF
Load Resistance R = 10W
Switching frequency Fs = 50 KHz
Load Current IL = 1.2A
DC Voltage conversion M Ratio = V0/Vin = d/(1-d)
PSIM circuit diagram of Current Mode Control in
Positive Output Luo Converter is shown below.
(6)
(7)
(8)
Here the converter is assumed to operate in continuous
conduction mode. The waveforms for voltage across two
inductors and current passing through both inductors are
shown below:
Fig 3. PSIM Circuit Diagram of Current Mode Control in Positive Output
Luo Converter
The route to chaos can be observed by varying reference
current in the range of (3-5) A. For values of reference current
lower than 3 A, the system is periodic. As reference increases
above 3 A, system enters chaotic region.
A. Phase-1 Operation
By the principle operation of current mode controlled
positive Luo converter as iL1+ iL2 approaches the value of
Iref, the switch is turned off, and remains off until the next
cycle begins. For the reference current of 3A, the fundamental
waveform is shown in Figure 4.
1788
Kiran
Control of Chaos in Positive Output Luo Converter by means of Time Delay Feedback
International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.2, pp. 1787-1791
ISSN 2078-2365
http://www.ieejournal.com/
Fig. 4: Simulated Fundamental Waveform of Inductor Current (IL1+IL2)
and Capacitor Voltage for Iref=3A
B. Phase Operation of Period-1
The phase portrait drawn between inductor current and
capacitor voltage when the reference current is about 3 A. The
portrait diagram of period-1operation is shown in Fig. 5.
Fig. 7: Phase Portrait of Period-2 Operation
E. Chaos Operation
Chaos is nothing but transition from period to aperiodic
state. Chaos occurs when reference current increases to 5A.
Simulation waveform of inductor current and capacitor
voltage is shown in Fig. 8.
Fig. 5: Phase Portrait of Period-1 Operation
C. Phase-2 Operation
If Iref is further increase beyond 3A, the period doubling
stage is reached. For the reference current of 4A the inductor
current waveform and capacitor voltage waveform is as
shown:
Fig.8: Simulated Chaotic Waveform of Inductor Current and Capacitor
Voltage for Iref =5 A
F. Phase Operation of Chaos Operation
The phase plot drawn between inductor current and
capacitor voltage when the reference current is about 5 A. The
portrait diagram of chaotic operation is shown in Fig 9.
Fig. 6: Simulated Period Doubling Waveform of Inductor Current (IL1+IL2)
and Capacitor Voltage for Iref =4 A
D. Phase Operation of Phase-2
The phase portrait drawn between inductor current and
capacitor voltage when the reference current is about 4 A. The
portrait diagram of period-2 operation is shown in fig. 7.
Fig 9. Phase Plot of Chaotic Operation
1789
Kiran
Control of Chaos in Positive Output Luo Converter by means of Time Delay Feedback
International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.2, pp. 1787-1791
ISSN 2078-2365
http://www.ieejournal.com/
IV. CHAOS CONTROL BY TIME DELAYED FEEDBACK
CONTROL
Time Delayed feedback control (DFC),proposed by
Pyrags[10] is one of useful method for chaotic systems. Here
the reference current is calculated using the formula,
Iref = Ic+ k(i(t)-i(t-T))
(9)
The control input Iref is fed by the difference between the
current state and the delayed state. The delay time is
determined as the period of the unstable periodic orbit to be
stabilized. Hence the control input vanishes when the
unstable periodic orbit is stabilized. In addition this method
requires no preliminary calculations of the unstable periodic
orbit. Hence it is simple and convenient for controlling
chaos. The use of TDF control technique increases operating
region of current controlled Luo converter.
Without TDF control Luo converter goes into unstable
periodic orbits, when Iref is above 3 A. After applying TDF
control, results will be periodic one operation for the same
Iref of period doubling and chaos. The circuit of TDFC is
shown in Fig. 10.
Fig. 11: Simulated Waveform of Controlled Capacitor Voltage with Iref=4 A
With Iref= 4A and 5A , it was observed that the system
enters in to chaotic operation without TDFC. With TDFC for
the same Iref result will be period 1 operation and is shown in
Fig.11. and Fig 12.
Fig.12: Simulated Waveform of Controlled Capacitor Voltage with Iref=5 A
Fig. 8 Experimental results during repetitive operation.
V. CONCLUSION
Fig. 10: PSIM Circuit Diagram of Current Mode Control in Positive Output
Luo Converter with Time Delay Feedback Control
The most common, only acceptable operating regime
employed in practical power supplies is the fundamental
operating regime, which demonstrates the stable and periodic
nature of the system. With Iref=4A and Iref=5A, it was
observed that the system enters in to period doubling
operation without TDFC. With TDFC for the same Iref , result
will be period 1 operation and is shown in Fig.11 and Fig 12.
The Chaotic behavior of the current programmed Luo
converter has been successfully controlled by using feedback
control namely Time Delayed Feedback Control. The
simulation results for period-2 to period-1 and chaotic to
period-1 operation has been presented for the same reference
current where period-2 and chaotic operation are resulting.
The stability range of the Luo converter was increased.
In this work, the analysis of chaos of a current mode
controlled Luo converter has been performed. It was shown
that as the reference current is varied, the nominal periodic
orbit undergoes a flip bifurcation and finally enters into the
chaotic regime. The simulated results using PSIM is
presented. The results obtained reveals that the current mode
controlled Luo converter becomes unstable, when Iref is
increased beyond 3 A. By using Time Delayed Feedback
Control, results will be period one operation for the same I ref
of period doubling and chaos.
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[1] Ned Mohan and M. Undeland, "Power Electronics Converters,
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[2] C.K. Tse, Mario DC Bernardo, May 2002 "Complex behavior in
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[4] Chaos in power electronics: An overview, Mario di Bernardo and Chi.k.
[5] S.K. Mazumender, A.H. Nayfeh and D. Borojevich, March 2001,
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1790
Kiran
Control of Chaos in Positive Output Luo Converter by means of Time Delay Feedback
International Electrical Engineering Journal (IEEJ)
Vol. 6 (2015) No.2, pp. 1787-1791
ISSN 2078-2365
http://www.ieejournal.com/
[7] F.L. LUO, July 1999, "Positive output LUO converters: Voltage Lift
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Kiran
Control of Chaos in Positive Output Luo Converter by means of Time Delay Feedback