# Analyses and Control of Chaotic Behavior in DC-DC Converters

Authors: Cheng-Biao Fu; An-Hong Tian; Kuo-Nan Yu; Yi‐Hung Lin; Her-Terng Yau
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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.


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## Body

## 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 [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) [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.Section2 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 Figure1 and the equations can be derived as follows.Figure 1
The basic circuit of a buck converter.Model 1.
In both ON and OFF condition, Kirchhoff’s Voltage (KVL) and Current (KCL) laws can be used to derive the following equations:(1)CdVCdt=iL-VoR(2)LdiLdt=uVin-Vo(3)u=0,whenSoff1,whenSon
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 donea times. A comparison was made using A2 and the sawtooth wave Vramp to feed back to S (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 D is OFF, the input voltage Vin flows to L and R and current passes the inductance and charges the capacitor. When S is OFF and D is ON, energy stored by the inductor and capacitor via D provides feedback and releases energy to R. The feedback, Vo connected to A1. A1 can be regarded as gain a. After Vo is less than Vref and amplified a, (4) is valid:(4)Vo1t=aVot-Vref
WhenVo1 connects A2 to compare the Vramp, sawtooth wave, Vramp>Vo1 gives a high signal to switch MP on; conversely, if Vramp < Vo1, the output signal of the comparator is Low, MP is switched off. This nonlinear switching can result in several different kinds of system behavior.Figure 2
Buck converter feedback circuit.


## 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 Section2, (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 Vin was set to 11, and the buck converter behavior was stable. Figure 4(a) shows the phase plane diagram Vo-Io of Phase 1. In the next step Vin 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 Vo-Io phase plane diagram.Table 1
Parameter specification.
Parameter Value Vin 11~40V L 2 m C 47u A1 5.6 R 35 Ω Vref 9.4V Vramp 4-10V f 1 KTable 2
Symbol table.
Name Symbol Name Symbol Name Symbol S Switch R Resistance V O Output voltage D Diode I L Inductance current V L Inductance voltage L Inductance I C Capacitance current V C Capacitance voltage C Capacitance V S Input voltageWhen 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 voltageVin. All the parameters with the exception of Vin had a fixed value [14–16].Figure3 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.Figure 3
A buck converter bifurcation trace.Figure 4
(a)Vo-IL phase plane diagram of Phase 1. (b) Vo-IL phase plane diagram of Phase 2. (c) Vo-IL phase plane diagram of the irregular phase.
(a) (b) (c)The phase plane diagrams in Figures4(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. Figure5(a) shows the phase 1 Vo-IL phase plane diagram with Vin set as 11 V in a Pspice simulation. When Vin was increased to 30, the Vo-IL 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).Figure 5
(a).Vo-IL phase plane diagram of Phase 1. (b). Vo-IL phase plane diagram of Phase 2. (c). Vo-IL phase plane diagram of the irregular phase.
(a) (b) (c)Figures5(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. Figure6 shows the buck converter circuit used in the experiments.Figure 6
Physical buck converter circuit board.Figures7(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.Figure 7
(a)Vo-IL phase plane diagram on oscilloscope display in Phase 1. (b) Vo-IL phase plane diagram on oscilloscope display in Phase 2. (c) Vo-IL phase plane diagram on oscilloscope display in the chaotic phase.
(a) (b) (c)Figure 8
Laboratory experimental setup.


### 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.Figure9 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 9
Power spectrum diagram of buck converter: (a) Phase 1; (b) Phase 2; (c) irregular phase.
(a) (b) (c)Figure10(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.Figure 10
Power spectrum diagrams of buck converter: (a) Phase 1; (b) irregular phase.
(a) (b)The Lyapunov exponent diagrams in Figures11 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 11
Diagram of Phase 1 Lyapunov exponent.Figure 12
Diagram of the chaotic phase Lyapunov exponent.Figure12 shows a Lyapunov diagram of the chaotic phase with the curve lying above zero after stabilization indicating the system is chaotic.


## 3.1. Matlab Simulation

A mathematical model of a buck converter was made with Matlab/Simulink using the equations derived in Section2, (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 Vin was set to 11, and the buck converter behavior was stable. Figure 4(a) shows the phase plane diagram Vo-Io of Phase 1. In the next step Vin 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 Vo-Io phase plane diagram.Table 1
Parameter specification.
Parameter Value Vin 11~40V L 2 m C 47u A1 5.6 R 35 Ω Vref 9.4V Vramp 4-10V f 1 KTable 2
Symbol table.
Name Symbol Name Symbol Name Symbol S Switch R Resistance V O Output voltage D Diode I L Inductance current V L Inductance voltage L Inductance I C Capacitance current V C Capacitance voltage C Capacitance V S Input voltageWhen 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 voltageVin. All the parameters with the exception of Vin had a fixed value [14–16].Figure3 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.Figure 3
A buck converter bifurcation trace.Figure 4
(a)Vo-IL phase plane diagram of Phase 1. (b) Vo-IL phase plane diagram of Phase 2. (c) Vo-IL phase plane diagram of the irregular phase.
(a) (b) (c)The phase plane diagrams in Figures4(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. Figure5(a) shows the phase 1 Vo-IL phase plane diagram with Vin set as 11 V in a Pspice simulation. When Vin was increased to 30, the Vo-IL 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).Figure 5
(a).Vo-IL phase plane diagram of Phase 1. (b). Vo-IL phase plane diagram of Phase 2. (c). Vo-IL phase plane diagram of the irregular phase.
(a) (b) (c)Figures5(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. Figure6 shows the buck converter circuit used in the experiments.Figure 6
Physical buck converter circuit board.Figures7(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.Figure 7
(a)Vo-IL phase plane diagram on oscilloscope display in Phase 1. (b) Vo-IL phase plane diagram on oscilloscope display in Phase 2. (c) Vo-IL phase plane diagram on oscilloscope display in the chaotic phase.
(a) (b) (c)Figure 8
Laboratory experimental setup.


## 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.Figure9 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 9
Power spectrum diagram of buck converter: (a) Phase 1; (b) Phase 2; (c) irregular phase.
(a) (b) (c)Figure10(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.Figure 10
Power spectrum diagrams of buck converter: (a) Phase 1; (b) irregular phase.
(a) (b)The Lyapunov exponent diagrams in Figures11 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 11
Diagram of Phase 1 Lyapunov exponent.Figure 12
Diagram of the chaotic phase Lyapunov exponent.Figure12 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, KP, KI, and KD. Figure 19 is a diagram of conditions after the establishment of PID control. The PID control parameters were acquired using the genetic algorithm: KP= -19.67, KI= -16.98, and KD= -1.02. The genetic algorithm convergence curve in Figure 20 identifies KP, KI, and KD after several iterations. Figure 21 shows and equivalent IAE curve of KP, KI, and KD.(5)IAE=∫0∞etdtFigure13 is control block diagram of the system where u is input control source of the system, e is the system error, u is system control option, and y is system output.Figure 13
Block diagram of buck converter circuitry.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.Figure 14
The genetic algorithm convergence curve ofKP, KI, and KD.Figure 15
The genetic algorithm IAE curve.The particle swarm optimization algorithm [19, 20] finds the optimized parameter values after several iterations: KP= -20.6112, KI= -25.3143, and KD= -0.514; see Figure 16.Figure 16
The particle swarm optimization algorithm convergence curve ofKP, KI, and KD.Figures15 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 17
The particle swarm optimization algorithm IAE curve.Figure18 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.Figure 18
V o after adding PID control.Figure 19
The phase plane diagram after adding PID control.Figure 20
Block diagram after the addition of fuzzy control. (b). Membership functions screen.
(a) (b)Figure 21
Voltage wave diagram after fuzzy control had been added.An optimization algorithm was used to identify the best parameters for PID and fussy control. Figure20(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].Figure21 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 22
Phase plane diagram after fuzzy control had been added.Figure23(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.Figure 23
(a) The chaotic output voltage of the circuit. (b). Output voltage after control has been added.
(a) (b)Figure 24
Comparison of GA, PSO, and fuzzy 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.



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