Dynamic Analysis of the Switched-Inductor Buck-Boost Converter Based on the Memristor

: The direct current (DC)–DC converter presents abundant nonlinear phenomena, such as periodic bifurcation and chaotic motion, under certain conditions. For a switched-inductor buck-boost (SIBB) converter with the memristive load, this paper constructs its state equation model under two operating statuses, investigates its chaotic dynamic characteristics, and draws and analyzes the bifurcation diagrams of the inductive current and phase portraits, under some parameter changing by the MATLAB simulation based on the state equation. Then, by applying certain minor per-turbations to parameters, the chaotic phenomenon suppression method is explored by controlling peak current in continuous current mode (CCM) to keep the converter run normally. Finally, the power simulation (PSIM) verifies that the waveforms and the phase portraits controlling the corre-sponding parameters are consistent with those of the MATLAB simulation.


Introduction
Power electronic technology has developed rapidly in recent years. Like other power electronic devices, the DC-DC converter has also penetrated into many fields, including energy science [1], physics [2], industry [3], automation [4][5][6], and so on.
The DC-DC buck-boost converter presents nonlinear physical phenomena, such as chaos and periodic motion, which are affected by system parameters, topological structure, load, and pulse period. The converter may produce results that cannot be predicted, and which affect the normal operation of the system. Thus, researchers from all over the world in this field pay lots of attention on these nonlinear phenomena. The bifurcation and chaos of a current-mode buck-boost converter was studied by using the input voltage, the reference current, and the load resistance as variable parameters [7]. The chaotic behavior in a Buck-Boost converter depending on the circuit parameters and the inductive load has been explored [8]. The chaos suppression for a Buck converter with the memristive load has also been investigated [9].
In recent years, some scientists have proposed a new switched-inductor topology [10]. It can be embedded into the DC-DC converter to replace the traditional inductor, forming a converter with stronger capacity of lifting and reducing voltage. One paper [11] studied the nonlinear phenomena in the switched-inductor buck-boost (SIBB) converter in continuous current mode (CCM). Another [12] investigated the various nonlinear behaviors of the SIBB converter under the resistive load in discontinuous current mode (DCM). A further study [13] has discussed chaos and its control of the current-mode switched-inductor converter. Lastly, another part of the literature [14] indicated that the SIBB converter can shift from DCM to CCM, and can operate in steady period-one state by utilizing ramp compensation.
Recently, replacing the traditional load in boost and buck-boost converters with the memristors, and analyzing the nonlinear dynamic characteristics of converters, has become a hot topic [33]. However, few researchers consider the dynamic characteristics of the SIBB converter with the memristive load. The purpose of this paper is to investigate the bifurcation and chaotic behavior of the SIBB converter with the memristive load when the peak current changes. Furthermore, the harmful bifurcation and chaotic behavior are suppressed by controlling the peak current.

Working Principle of the Converter and the Memristor
Terminology description is given as follows.

Working Principle of the Converter
The current-mode-controlled SIBB converter is a kind of DC-DC converter controlled by the inductive current. It is formed by applying a switched-inductor structure to the traditional converter. Its basic circuit is shown in Figure 1. The feedback loop consists of an RS flip-flop and a comparator. An RS flip-flop has a reset input R and a set input S, as well as an output Q. The output Q directly controls the state of the switch S. The set input S comes from the clock, while the reset input R comes from the comparison of the reference current and the inductive current in the comparator. This converter has two operating modes: continuous current mode (CCM) and discontinuous current mode (DCM). This paper studies the dynamic characteristics of the converter in CCM. In this mode, the circuit has two statuses, as shown in Figure 2. To make the analysis easier, let two inductors of the switched-inductor = and is approximately equal to , and the inductive current can be written as . Thus, the system with the resistive load can be simplified to a second-order model, and its state equations are as follows is the input voltage, and the coefficient matrices are

The Simulation Model of the Memristor
About 50 years ago, professor Cai theoretically predicted that the existence of a nonlinear, passive, two-terminal electronic element that could describe the relationship between charge and flux, and called it a memristor [34]. Cai also generated the concept of a generalized memristor. In 2008, the feasibility of memristors was reported in Nature. This report shocked the international electrical and electronic world, and made the theoretical basis of circuits more perfect. The memristor was finally unveiled in 2009 after a flurry of research by many scientists. Its symbol is shown in Figure 3. Memristors are similar to resistors, capacitors, and inductors as a nonlinear, passive, two-terminal electronic element. It has two forms: the charge-controlled memristor and the magnetron-controlled memristor. The relationship between its magnetic flux and accumulated charge is where ( ) is called memristance and ( ) is called memductance [34]. A simulator of the magnetron-controlled memristor is designed by using circuit elements, as shown in Figure 4. Figure 4. The simulator of the magnetron-controlled memristive load.
Referring to [33], the simulation circuit consists of two operational amplifiers, and ; three resistors, , , and ; a capacitor ; and a multiplier M, which is g = −0.1. In addition, u and are expressed as the input voltage and input current of the simulation circuit, respectively. According to the volt-ampere relationship of the memristor, ( ) = ( ) ( ) and ( ) = ( ) ( ). The mathematical model of the memristor simulator can be obtained as follows:

The Modeling of the SIBB Converter with the Memristive Load
We have described the operating principle of the SIBB converter and obtained its state equations under the resistive load. Now we have replaced the resistive load with the memristive load.
In CCM, the SIBB converter with its memristive load still has two statuses: switch S is on, is off; and switch S is off, is on, as shown in Figure 2. At this time, the voltage on the capacitor C is the input voltage of the simulation circuit, and is the input current of the simulation circuit. The state equations of the system with the memristive load can be obtained as follows: where = [ ] is the state vector; is the input voltage; and y is the product of two state variables, such that y = u . The coefficient matrices are

The Simulation of the SIBB Converter with the Memristive Load
The circuit parameters are selected as shown in Table 1. Taking as the bifurcation parameter, the bifurcation diagrams of the inductive current with changing can be obtained, as shown in Figure 5. Figure 5a is the bifurcation diagram of the SIBB converter with the resistive load, and Figure 5b is the bifurcation diagram of the SIBB with the memristive load.  According to the comparison of the bifurcation diagrams, we can see that the memristive load does not affect the bifurcation structure. Both of them have chaos and bifurcation of two-period, four-period, eight-period, etc., but the bifurcation point of the period doubling can be moved. The results are listed as follows: (1) The two-period bifurcation of the converter with the resistive load occurs when = 3 A, while two-period bifurcation of the converter with the memristive load occurs when = 3.85 A. Therefore the normal working area of the system is widened under the memristive load.
(2) At the same time, it can be found that the SIBB converter with the resistive load occurs chaos when = 5.8 A, while the SIBB converter with the memristive load occurs chaos when = 8.9 A. The point where the chaos occurs moves back significantly, which reduces the probability of chaos of the converter. The eight-period waveform and phase portrait of the converter with the memristive load are shown in Figure 6. (3) The behavior of the SIBB converter with the memristive load is much richer. When ϵ [9.87, 9.95], the system no longer exhibits a chaotic state, but a bifurcation. At this time, the system would occur the tangent bifurcation after chaos [35]. When = 9.9 A, the waveform and the phase portrait of are as shown in Figure 7. The bifurcation diagrams of the SIBB converter with the resistive load and memristive load were obtained by MATLAB simulation. Next, the PSIM software was used for simulation to verify the accuracy of the bifurcation and chaotic behavior of the converter with the memristive load.

Verification by the PSIM Simulation
The PSIM software is used for simulation to verify the accuracy of the bifurcation and chaotic behavior of the converter with the memristive load. According to the schematic diagram shown in Figure 1, this section builds the simulation model of the system in the PSIM circuit software, as shown in Figure 8. The reference current is selected as the variable, and the parameters of each circuit element are the same as those shown in It can be seen from Figures 9-12 that when is 3.0, 5.0, 8.0, and 9.5 A, the converter is in the state of one-period, two-period, four-period, and chaos, respectively. It is not difficult to see that the simulation results of this PSIM circuit are consistent with those of the MATLAB simulation.

Suppression of the Chaos
The resonant parametric perturbation method takes advantage of the characteristics that the chaotic state is very sensitive to-the minor perturbation of the parameters. By applying certain minor perturbation to the parameters, the system can be controlled from the chaotic state to the stable one-period state.
According to the above results of the PSIM simulation, the SIBB converter with the memristive load is in a chaotic state when = 9.5 A. The resonant parametric perturbation method is used to control the chaos of the SIBB converter on this condition.
It is relatively simple that is used as the control parameter. Using = + to replace , according to this method, where = (2 + ) and is the switching frequency (f = 20 kHz). Referring to [13], let A equal 0.3 and = 1.2 rad. A minor perturbation is added when t = 0.05 s. The circuit diagram of the controller is shown in Figure 13b, compared with the original simple controller in Figure 13a. Then, the waveform of u and , as well as the phase portrait of − under the minor perturbation, are obtained, as shown in Figure 14.

Conclusions
In this paper, the nonlinear behavior of the SIBB converter in CCM is observed by using the memristive load instead of the resistive load. In order to study the dynamic effect of the SIBB converter with the memristive load, this paper investigates the working principle of the converter by using the simulation circuit, and obtains the bifurcation diagrams, waveforms, and phase portraits of the inductive current by controlling the peak current. The results are listed as follows: (1) It was found that the system has abundant dynamic behavior, including periodic motion, period-doubling motion, and chaotic motion; (2) The memristive load does not affect the bifurcation structure, but expands the normal working area of the system and suppresses the occurrence of chaos; (3) At the same time, the SIBB converter with the memristive load has richer behaviors, and appears to demonstrate the behavior of the tangent bifurcation; (4) The simulation results of PSIM are consistent with the numerical results of MATLAB. It is also easy to see that the converter has been working in CCM from the waveforms of ; (5) The resonant parametric perturbation method has a good effect on suppressing chaotic phenomenon. The system can be controlled from the chaotic state to the stable one-period state by this method.
In further research, we would extend the memristor model as a nonlinear part into block-oriented nonlinear systems [36][37][38], to explore the system's performance.

Conflicts of Interest:
The authors claim that there are no conflicts of interest involved in publishing this article.