Research Article | Open Access
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. https://doi.org/10.1155/2018/7439137
Analyses and Control of Chaotic Behavior in DC-DC Converters
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.
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  and this includes electromagnetic noise and critical operation breakdown. A previous study  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 . 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 [4–6]. 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) , the Genetic algorithm (GA) , and fuzzy control (Fuzzy control) . 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.
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 [14–16].
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.
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).
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.
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.
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 [21–23].
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.
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.
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)
- E. Fossas, “Study of chaos in the buck converter,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 43, no. 1, pp. 13–25, 1996.
- C. K. Tse, Y. M. Lai, and H. H. C. Lu, “Hopf bifurcation and chaos in a free-running current-controlled Cuk switching regulator,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, no. 4, pp. 448–457, 2000.
- S. Parui and S. Banerjee, “Bifurcations due to transition from continuous conduction mode to discontinous conduction mode in the boost converter,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 50, no. 11, pp. 1464–1469, 2003.
- K. Chakrabarty, G. Poddar, and S. Banerjee, “Bifurcation behavior of the buck converter,” IEEE Transactions on Power Electronics, vol. 11, no. 3, pp. 439–447, 1996.
- B. Allaoua and A. Laoufi, “Application of a robust fuzzy sliding mode controller synthesis on a buck-boost DC-DC converter power supply for an electric vehicle propulsion system,” Journal of Electrical Engineering & Technology, vol. 6, no. 1, pp. 67–75, 2011.
- W. C. Chan and C. K. Tse, “Study of bifurcations in current-programmed dc/dc boost converters: from quasi-periodicity to period-doubling,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 44, no. 12, pp. 1129–1142, 1997.
- A. S. Oshaba, E. S. Ali, and S. M. Abd Elazim, “ACO based speed control of SRM fed by photovoltaic system,” International Journal of Electrical Power & Energy Systems, vol. 67, pp. 529–536, 2015.
- A. S. Oshaba, E. S. Ali, and S. M. Abd Elazim, “MPPT control design of PV system supplied SRM using BAT search algorithm,” Sustainable Energy, Grids and Networks, vol. 2, pp. 51–60, 2015.
- J. Chen, H. Yau, and J. Lu, “Implementation of FPGA-Based Charge Control for a Self-Sufficient Solar Tracking Power Supply System,” Applied Sciences, vol. 6, no. 2, p. 41, 2016.
- C. Hsieh, H. Yau, C. Wang, and Y. Hsieh, “Particle swarm optimization used with proportional–derivative control to analyze nonlinear behavior in the atomic force microscope,” Advances in Mechanical Engineering, vol. 8, no. 9, p. 168781401666727, 2016.
- J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks (ICNN ’95), vol. 4, pp. 1942–1948, Perth, Western Australia, November-December 1995.
- J. H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence, MIT Press, 1992.
- L. A. Zadeh, “Outline of new approach to the analysis of complex systems and decision processes,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 3, pp. 28–44, 1973.
- Y. Zhou, C. K. Tse, S.-S. Qiu, and F. C. M. Lau, “Applying resonant parametric perturbation to control chaos in the buck dc/dc converter with phase shift and frequency mismatch considerations,” International Journal of Bifurcation and Chaos, vol. 13, no. 11, pp. 3459–3471, 2003.
- Y.-L. Zou, X.-S. Luo, and G.-R. Chen, “Pole placement method of controlling chaos in DC-DC buck converters,” Chinese Physics, vol. 15, no. 8, pp. 1719–1724, 2006.
- F. Angulo, J. E. Burgos, and G. Olivar, “Chaos stabilization with TDAS and FPIC in a buck converter controlled by lateral PWM and ZAD,” in Proceedings of the 2007 Mediterranean Conference on Control and Automation, MED, Greece, July 2007.
- N. Hussain, J. V, and M. Abraham, “Improving the Performance of Buck Converter PID Controller using Genetic Algorithm,” in Proceedings of the 8th SAEINDIA International Mobility Conference & Exposition and Commercial Vehicle Engineering Congress 2013 (SIMCOMVEC), 2013.
- X. Wang, M. Wu, L. Ouyang, and Q. Tang, “The application of GA-PID control algorithm to DC-DC converter,” in Proceedings of the 29th Chinese Control Conference, CCC'10, pp. 3492–3496, China, July 2010.
- X. Li, M. Chen, and Y. Tsutomu, “A method of searching PID controller's optimized coefficients for Buck converter using particle swarm optimization,” in Proceedings of the 2013 IEEE 10th International Conference on Power Electronics and Drive Systems, PEDS 2013, pp. 238–243, Japan, April 2013.
- S. A. Emami, M. B. Poudeh, and S. Eshtehardiha, “Particle swarm optimization for improved performance of PID controller on buck converter,” in Proceedings of the 2008 IEEE International Conference on Mechatronics and Automation, ICMA 2008, pp. 520–524, Japan, August 2008.
- M. Ayati and Z. Sharifi, “Analysis and fuzzy control of chaotic behaviors in buck converter,” in Proceedings of the 4th International Conference on Control, Instrumentation, and Automation, ICCIA 2016, pp. 319–323, Iran, January 2016.
- A. Abbasi, M. Rostami, J. Abdollahi, and H. R. Abbasi, “An analytical discrete model for evaluation the chaotic behavior of buck converter under current control mode,” in Proceedings of the 2009 IEEE Symposium on Industrial Electronics and Applications, ISIEA 2009, pp. 112–116, Malaysia, October 2009.
- A. Perry, G. Feng, Y. Liu, and P. Sen, “A new design method for PI-like fuzzy logic controllers for DC-to-DC converters,” in Proceedings of the 2004 IEEE 35th Annual Power Electronics Specialists Conference, pp. 3751–3757, Aachen, Germany, 2004.
Copyright © 2018 Cheng-Biao Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.