Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 7439137 |

Cheng-Biao Fu, An-Hong Tian, Kuo-Nan Yu, Yi‐Hung Lin, Her-Terng Yau, "Analyses and Control of Chaotic Behavior in DC-DC Converters", Mathematical Problems in Engineering, vol. 2018, Article ID 7439137, 13 pages, 2018.

Analyses and Control of Chaotic Behavior in DC-DC Converters

Academic Editor: Francesco Riganti-Fulginei
Received11 Apr 2018
Revised26 Sep 2018
Accepted21 Oct 2018
Published08 Nov 2018


In this study the nonlinear behavior of a buck converter was simulated and the responses of Phases 1 and 2 and the chaotic phase were investigated using changes of input voltage. After a dynamic system model had been acquired using basic electronic circuit theory, Matlab and Pspice simulations were used to study system inductance, resistance, and capacitance. The characteristic changes of input voltage, and phase plane traces from simulation and experiments showed nonlinear behavior in Phases 1 and 2, as well as a chaotic phase. PID control and Integral Absolute Error (IAE) were used as adaption coefficients to control chaotic behavior, and particle swarm optimization (PSO) and the genetic algorithm were used to find the optimal gain parameters for the PID controller. Simulation results showed that the control of chaotic phenomena could be achieved and errors were close to zero. Fuzzy control was also used effectively to prevent chaos. The experimental results also showed nonlinear behavior from Phases 1 and 2 as well as the chaotic phase. Laboratory experiments conducted using both PID and fuzzy control echoed the simulation results. The fuzzy control results were somewhat better than those obtained with PID.

1. Introduction

DC-DC converters are very common and can be found in almost all the electronic devices in everyday use: PCs, Pads, mobile phones, TVs, and other types of equipment. The need for the delivery of finely tuned stable voltages by these devices makes it necessary for them to be of high quality. However, a review of the literature showed that abnormal or irregular behavior is often a characteristic of DC-DC converters [1] and this includes electromagnetic noise and critical operation breakdown. A previous study [2] using nonlinear dynamic theory, numerical computation, and experimental circuitry revealed the presence of nonlinear bifurcation behavior in many of these converters.

In a power conversion system, DC-DC converters are basic modules that include inductors (L), capacitors (C), switches (S), and diodes (D). Different connection arrangement gives different topological structure to achieve the various purposes of power treatment or the format of the transformation required. The DC-DC converters discussed here include buck converters, boost converters, and buck-booster converters.

Control of a DC-DC converter is achieved by a voltage or current signal. The operation of such circuits is often associated with irregular and unstable chaotic behavior similar to noise [3]. These phenomena are often ignored, but these nonlinear oscillations consume system energy and lower the efficiency of the converter.

DC-DC converters operate within a certain range of input voltage, and a phase plane diagram will show the changes that occur over the range [46]. This limits the conditions of use to a significant degree. To improve this situation, the converter was viewed as a controlled body and various control theories were added so that it could be used to return the system to stability when chaos arose.

Matlab and Pspice simulations of the behavior of the buck converter chosen for this study showed that the stable dynamic trace of output in the phase plane diagram against changes of input voltage later becomes chaotic.

We cut in from a control perspective. In the past, there were many commonly used control rules. Traditionally, these rules have used PID system controllers [7, 8], but recently many rules using optimization to implement automatic parameter settings have been presented. However, most optimization algorithms fall into the local optimum trap. To avoid this type of situation some investigators have used a chaos logistic map to improve optimization [9, 10].

To better understand and achieve control of this phenomenon, optimization algorithms were used: the particle swarm optimization algorithm (PSO) [11], the Genetic algorithm (GA) [12], and fuzzy control (Fuzzy control) [13]. Phase plane diagrams, power spectra, and the Lyapunov exponent were used to display the results of system status analysis.

Section 2 of this paper describes system structure and analyzes and derives system equations. In Section 3 the equations derived in Section 2 were used to construct a block diagram and conduct a Pspice simulation. Changes in input voltages are clearly reflected in changes in the phase plane diagrams. Simulation results were then examined and the power spectrum and Lyapunov exponent were introduced to demonstrate the presence of chaos in the system. After the confirmation of the existence of chaos, optimization algorithms were used to look for the best PID parameters to control the chaotic behavior. Fuzzy control was included for comparison. Section 5 is the conclusion.

2. Buck Converters

Buck converters were invented at the beginning of the 20th Century. DC-DC converters were the earliest type and the principle used can be easily understood.

The structure of a buck DC-DC converter is shown in Figure 1 and the equations can be derived as follows.

Model 1. In both ON and OFF condition, Kirchhoff’s Voltage () and Current () laws can be used to derive the following equations:In this study of DC-DC converters, both simulation and laboratory experiments were done using basic buck converter specifications. A feedback system was added to the voltage output that was used to send a 0 or 1 signal via the error amplifier after amplification had been done times. A comparison was made using and the sawtooth wave to feed back to (on/off) to drop the output voltage to a certain level. Figure 2 shows a schematic of the buck converter. To start with it is assumed that all circuit components are in an ideal state. When S is ON and is OFF, the input voltage flows to and and current passes the inductance and charges the capacitor. When is OFF and is ON, energy stored by the inductor and capacitor via D provides feedback and releases energy to . The feedback, connected to . can be regarded as gain . After is less than and amplified , (4) is valid:When connects to compare the , sawtooth wave, > gives a high signal to switch on; conversely, if < , the output signal of the comparator is Low, is switched off. This nonlinear switching can result in several different kinds of system behavior.

3. Simulation and Analyses

3.1. Matlab Simulation

A mathematical model of a buck converter was made with Matlab/Simulink using the equations derived in Section 2, (1)-(3), and (4). The parameters used are shown in Table 1. The symbol table of this study is shown in Table 2. To start with the input voltage was set to 11, and the buck converter behavior was stable. Figure 4(a) shows the phase plane diagram of Phase 1. In the next step was gradually increased to 30, and the phase plane diagram (Phase 2) changes as shown in Figure 4(b). Finally, after the voltage was increased to 40, the converter entered an irregular phase as shown in Figure 4(c), the irregular phase plane diagram.


L2 m
R35 Ω
f1 K

NameSymbol NameSymbol NameSymbol

SSwitchRResistanceOutput voltage
DDiodeInductance currentInductance voltage
LInductanceCapacitance currentCapacitance voltage
CCapacitanceInput voltage

When the parameters of a nonlinear circuit are changed, an initial stable phase gradually changes and eventually become irregular. In this study the parameters selected for study and observation were those most likely to change, such as the input voltage . All the parameters with the exception of had a fixed value [1416].

Figure 3 clearly shows how the system changes from Phase 1, to Phase 2, and then to irregular dynamic behavior. We used input voltages of 11, 30, and 40 for comparison.

The phase plane diagrams in Figures 4(a), 4(b), and 4(c) clearly show how the dynamic behavior of the buck converter becomes nonlinear with an increase of voltage after Phase 1.

3.2. Pspice Simulation

A MOSFET switch at 15 V was used for the buck converter simulation using Pspice. A Zener was added at the back of A2 to increase the signal by 15 V to ensure proper operation of the switch. Figure 5(a) shows the phase 1 phase plane diagram with set as 11 V in a Pspice simulation. When was increased to 30, the phase plane diagram changed to Phase 2 behavior as shown in Figure 5(b). Finally, with a voltage of 40, the converter phase plane diagram displayed the irregular behavior illustrated in Figure 5(c).

Figures 5(a), 5(b), and 5(c) clearly show distinct changes from Phase 1 to Phase 2, and then to nonlinear dynamic behavior, in response to an increase of voltage.

3.3. Experimental Results

Circuit experiments are less convenient than computer simulations. For example, current cannot be directly detected. It was necessary to use a Fluke or Hall device to measure current. For resistance measurement, a high value ceramic resister was used. Figure 6 shows the buck converter circuit used in the experiments.

Figures 7(a), 7(b), and 7(c) are V0–IL plane diagrams of an oscilloscope display showing voltage as 11 V in Phase 1, 30 V in Phase 2, and 40 V in the irregular phase. The experimental setup is shown in Figure 8 and the results coincided very well with those of the simulation.

3.4. Chaos Analyses

While the phase plane diagrams clearly showed whether the behavior of the buck converter was regular or not, they did not provide information about chaos. Therefore, power spectra and the Lyapunov exponent were used in addition to phase plane diagrams to investigate chaotic behavior.

Figure 9 shows the power spectra of the converter in Phase 1, 11 V (see Figure 9(a)), Phase 2, 30 V (see Figure 9(b)), and irregular phase, 40 V (see Figure 9(c)).

Figure 10(a) shows the power spectrum diagram of Phase 1 with an input voltage of 11V. In Figure 10(b) the input voltage was 40 V and circuit behavior was nonlinear.

The Lyapunov exponent diagrams in Figures 11 and 12 show the state of the converter in Phase 1 (11V) and in a chaotic state (40V). From Figure 11 it can be seen that after stabilization the curve is close to 0 and shows stability.

Figure 12 shows a Lyapunov diagram of the chaotic phase with the curve lying above zero after stabilization indicating the system is chaotic.

4. Optimized Control

In this study, Integral Absolute Error (IAE) was used as the adaption coefficient, as shown in Equation (5). The genetic algorithm was introduced to identify the parameters of the PID controller, , , and . Figure 19 is a diagram of conditions after the establishment of PID control. The PID control parameters were acquired using the genetic algorithm: = -19.67, = -16.98, and = -1.02. The genetic algorithm convergence curve in Figure 20 identifies , , and after several iterations. Figure 21 shows and equivalent IAE curve of , , and .

Figure 13 is control block diagram of the system where u is input control source of the system, is the system error, is system control option, and is system output.

The genetic algorithm [17, 18] finds the optimized parameter values after several iterations and then converges (see Figure 14). Figure 15 shows the IAE convergence curve.

The particle swarm optimization algorithm [19, 20] finds the optimized parameter values after several iterations: = -20.6112, = -25.3143, and = -0.514; see Figure 16.

Figures 15 and 17 show that the particle swarm optimization algorithm works better than the gene algorithm, and the PID parameter values identified by PSO are closer to optimal.

Figure 18 is the voltage wave diagram acquired after introducing and acquired from the PID controller optimization algorithm, where PID control starts after 2 seconds. It is clear from the diagram that the voltage becomes stable soon after the addition of control. Figure 19 shows the phase plane diagram after the start of control and the red line shows the curve after control had been established.

An optimization algorithm was used to identify the best parameters for PID and fussy control. Figure 20(a) shows a block diagram after fuzzy control had been applied and Figure 20(b) shows the membership functions screen of the fuzzy controller [2123].

Figure 21 shows the output voltage curve 2 seconds after fuzzy control had been added. It is clear from the diagram that the voltage becomes stable soon after the addition of control. Figure 22 is the phase plane diagram after the start of control, and the red line shows the curve after control had been added.

Figure 23(a) shows a trace of the circuit output voltage at 40V, and the situation is clearly chaotic. Figure 23(b) shows the voltage trace after the addition of PID control, and a comparison with Figure 24 shows that fuzzy control is more effective than traditional PID control.

5. Conclusion

In this study, phase plane diagrams were used to verify the system status cycle and chaos phenomena. At the same time, stricter mathematical definitions and the largest Lyapunov exponent were used for system verification. This proved that, under special conditions, a DC-DC converter and the system will exhibit chaotic phenomena. Although the dynamic status falls within the stable strange attractor status, this is not desirable behavior for such a physical system. Logistic mapping was used to improve PSO, inhibit chaos, and control PID parameters. The experimental results showed that the method used here is very effective and can achieve faster convergence than other methods. In this study, optimization algorithms were used to identify gain values of PID controller parameters and PSO and GA were included. IAE was used as an adaption coefficient. A comparison of two algorithm methods showed that PSO works more effectively than GA. After fuzzy control was added the buck converter simulations showed that all three methods would allow the buck converter to reach stability. However, DSpace experiments with PID and fussy control showed fuzzy control to be better than PID.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.


Financial support of this research from the Ministry of Science and Technology, Taiwan, under Projects nos. 106-2218-E-167-001 and 106-2622-E-167-020-CC3 is gratefully acknowledged. The authors also would like to acknowledge, with thanks, comments from ICCPE&ICI for All MNHTE Workshop SPINTECH. This study has also gained from comments made at the poster workshop. This research project also received financial support from the Yunnan Province Science and Technology Department and Education Department Project of China (2017FH001-067, 2017FH001-117, and 2016ZDX127)


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