Complex Dynamics and Hard Limiter Control of a Fractional-Order Buck-Boost System

Chaos and control analysis for the fractional-order nonlinear circuits is a recent hot topic. In this study, a fractional-order model is deduced from a Buck-Boost converter, and its discrete solution is obtained based on the Adomian decomposition method (ADM). Chaotic dynamic characteristics of the fractional-order system are investigated by the bifurcation diagram, 0-1 test, spectral entropy (SE) algorithm, and NISTtest. Meanwhile, the control of the fractional-order Buck-Boost model is discussed through two diﬀerent ways, namely, the intensity feedback and the hard limiter control. Speciﬁcally, the hard limiter control can be realized using a current limiter in the circuit, where the current limiter device is applied to control the branch current. The results show that the proposed fractional-order system has complex dynamic behaviors and potential application values in the engineering ﬁeld.


Introduction
e Buck-Boost converters are strongly nonlinear circuits [1][2][3][4] which can produce subharmonic, bifurcation, and chaos under certain conditions. At present, many researchers have investigated the chaos of the Buck-Boost converter systems [5][6][7][8][9][10][11]. For example, Wang et al. [10] investigated the mechanisms that lead to chaotic behaviors in the Buck-Boost power converter, and the existence of chaos is verified successfully by the Smale horseshoe, while Demirbaş et al. [11] investigated the bifurcation of the system with the material parameters and found a number of periodic windows. Obviously, chaos will lead to the instability of the converter via increasing the oscillation and producing the excessive irregular electromagnetic noise, which directly affects the operation quality and reliability of the converter. In fact, the Buck-Boost is a switched model, and control of the nonlinear switched systems is a challenging topic [12]. As a result, there are also many other reports about the chaos control of the Buck-Boost converters [13][14][15]. For instance, Sriramalakshmi et al. [15] analyzed the nonlinear phenomena and control of the current mode-controlled Buck-Boost converter. Meanwhile, complexity of nonlinear dynamical systems has aroused much interest of researchers [16,17]. erefore, an in-depth study of the complexity with the circuit parameters is still significant for the design and application of the Buck-Boost converter systems. e fractional calculus was proposed at about the same time as the regular calculus. At present, since it can better describe those processes with time and the space memory effect, it has been accepted as a novel tool for building the mathematical models. Nowadays, the fractional calculus has extensive applications in the field of physics, engineering, and biology economics [18][19][20]. Generally, the nonlinear systems have more complex dynamical behaviors after the fractional calculus is introduced [21][22][23][24]. In fact, dynamics analysis of different kinds of fractional-order Buck-Boost systems has aroused concern from scholars [25 -28]. However, these studies mainly focused on the modeling of fractional-order Buck-Boost systems. Currently, dynamics in different kinds of nonlinear systems have been investigated, such as memristor-based systems [29,30] and chaotic systems [31,32]. It is indicated that those systems have abundant dynamic characteristics including the multistability and can be used for the image encryption applications. However, dynamics of the fractional-order Buck-Boost systems still need further study. Meanwhile, how to build an effective model is necessary for the applications of the fractional-order Buck-Boost system. In fact, He et al. [33] shows that, compared with other solution algorithms, the Adomian decomposition method (ADM) is more accurate, and the deduced solution can be used in the engineering applications. us, it can be employed to solve fractionalorder Buck-Boost systems for further analysis and applications.
In 2008, Wu and Zhang [34] proposed an improved exponential delayed feedback controller to a Buck-Boost converter in the current mode.
is exponential delayed feedback controller provides a new strategy for chaos control and for the switching law. As a matter of fact, control of chaos is important for the application of nonlinear systems [35]. On the other hand, although the chaotic systems can be applied in secret communications and information encryption [36,37], there are few reports on applications of the Buck-Boost system in information security field. Motivated by the above discussions, we will explore the dynamics and control the potential application values of the fractionalorder Buck-Boost system based on Wu and Zhang's model [34] by employing the ADM. e rest of the study is organized as follows. In Section 2, the fractional-order Buck-Boost model which is a switching system is solved by employing the ADM. In Section 3, complexity of the fractional-order Buck-Boost system is investigated, and the existence of chaos is verified by the 0-1 test. A chaotic pseudorandom sequence generator is designed. In Section 4, control of chaos in the fractionalorder Buck-Boost system is studied. Two different approaches are introduced. Finally, conclusion is made in Section 5.  [34] designed a Buck-Boost converter in the current mode and found chaos in the proposed model. e circuit of the modified Buck-Boost converter is shown in Figure 1, and the description of the parameters is given in Table 1. In this enhanced Buck-Boost converter, there is an adjustment coefficient k which can be used to control the stability of the converter.

The Fractional-Order Buck-Boost Model
Let the Buck-Boost converter work in the continuous conductive mode, and the inductance current x 1 (i L ) and the capacitance voltage x 2 (u c ) be the system variables. According to on and off states of the power switch in the converter and based on Kirchhoff's law, the Buck-Boost converter system can be expressed as [34] _ Now, the Caputo fractional calculus, which is given by [38] is introduced to the system, where D q t 0 is the Caputo derivative operator of order q(0 < q ≤ 1), t 0 is the initial time, and Γ(·) is the Gamma function. us, the fractional-order Buck-Boost system is defined as Although there are some research studies regarding the fractional-order Buck-Boost system, it is necessary to give more interpretations of the fractional-order Buck-Boost system [25][26][27][28]. Usually, we hold the opinion that the fractional-order calculus provides a more effective way for modeling of real systems, and the analysis results can reflect the dynamics of real systems better. Here, we hope that the fractional-order system could provide more information about the Buck-Boost circuit.

e Adomian Decomposition
Method. As shown above, the systems have only linear items and a constant. To solve this system, the ADM [39][40][41] is employed, and the description of this algorithm is illustrated as follows.
For a given fractional-order linear system, where D q t 0 is the Caputo definition, Υx is the linear terms in the system, and q ∈ (0, 1]. Its solution is given by [39] x Meanwhile, those decomposition items x i (i � 0, 1, . . . , ∞) are calculated as [40,41] x 0 � J q t 0 g + x 0 , When an infinite number of items is used, the solution is the exact solution. In this study, 11 decomposition items, namely, x i (i � 0, 1, . . . , 10), are used.

Solution of the Fractional-Order
Model. Let us focus on the n th sampling period; thus, t ∈ (t n , t n+1 ]. e initial value of this interval is x 1 (t n ) and x 2 (t n ). Case 1. By employing the above ADM, the solution of equation (3) is given by When t � t n+1 , we can obtain the following discrete solution of the system, which is defined as Case 2. By employing the above ADM, the solution of equation (4) is denoted as So the following discrete solution is obtained: Based on x 1 (t n ) and x 2 (t n ), the reference current for the inductance I ref (t n ), the boundary value of the inductance current I b (t n ), and the duty cycle d(t n ) are calculated by [34] When x 1 (t n ) < I b (t n ), the power switch S is on during the whole sampling period, and the system is defined as Case 1. However, if x 1 (t n ) ≥ I b (t n ), the power switch S is on when t ∈ (t n , t n + d n T], while the power switch S is off when t ∈ (t n + d n T, t n+1 ]. As a result, in this sampling period, the system is defined as Case 1 first and then become Case 2 in the rest part. Finally, the discrete model is built as follows.
x t n+1 � where T � t n+1 − t n is the length of the sampling period. Obviously, this is a switched discrete system. When the initial conditions x 1 (t 0 ) and x 2 (t 0 ) are given, states of the system can be observed.

Dynamics of the Fractional-Order Model. Let the simulation parameters be
Phase diagrams with different derivative order q are shown in Figure 2. It is shown in Figure 2 that the states of the system are different with a different derivative order q. Especially, when q � 0.95, the system becomes a periodic system because there are only two points in the phase diagram. Bifurcation diagrams of the system with different system parameters include the inductance reference current I ref , the Mathematical Problems in Engineering inductance reference current L, the intensity of feedback k, and the derivative order q. Figure 3 shows the bifurcation diagrams with the inductance reference current I ref under a different derivative order q. In Figure 3, I ref varies from 0 to 4 with step size of 0.008 and k � 0. As shown in Figure 3, dynamics of the system with the variation of I ref are different with a different derivative order q. Meanwhile, bifurcations versus different parameters are analyzed. Fix k � 0, q � 0.9, and I ref � 3 A, and vary L from 2 mH to 5 mH with step size of 0.006. e analysis result is shown in Figure 4(a). Fix q � 0.9, I ref � 3 A, and L � 0.3 mH, and vary parameter k from 0 to 5 with step size of 0.01. e bifurcation diagram with k is shown in Figure 4 and T � 50 μs, and vary the derivative order q from 0.85 to 1, where the step size is 3 × 10 − 4 . As shown in Figure 4, the system has rich dynamics with those parameters. e systems can be chaotic and periodic with different parameters. For instance, when the intensity of feedback k is larger than 2, the system is periodic. It means that the system is nonchaotic, and k can be treated as a control parameter of the circuit.

0-1 Test.
e proposed system is a typical switched system, and Lyapunov exponents of the system cannot be calculated directly. In this study, the 0-1 test is employed to verify the existence of chaos.
If a set of one-dimensional observable data obtained from the iterative is represented by a set of multiscale discrete time series x(n): n � 1, 2, 3, . . . , { }, then the following two real valued sequences can be defined as [42,43] Usually, the state of the system can be identified by plotting the trajectories in the (p, s) plane. Figure 5 shows the (p, s) plots of the system with a different derivative order q, where the parameters are set as same as Figure 1, respectively. It shows that the system is chaotic when q �1, 0.95, and 0.9, while the system is periodic when q � 0.85. us, the existence of chaos is verified.
However, (p, s) plots can only be used to illustrate the state of the system under given parameters but cannot be used to show dynamics with the variation of parameters.
us, it is necessary to deduce an index from the (p, s) plots. Fortunately, Gottwald and Melbourne [43] proposed a test which can be used to distill a binary quantity (K) from the power spectrum. Namely, if K � 0, the time series has regular dynamics, but if K � 1, the time series is chaotic.  First, the mean square displacement is defined as [42,43] where n ≪ N. In real applications, n ≤ (N/10) yields good results. Second, the asymptotic growth rate K is defined as [42,43] Plots of K versus different parameters for the fractionalorder Buck-Boost system are shown in Figure 6.  Figure 4 with the variation of parameters L, k, and q, respectively. It is shown in Figure 6 that chaos in the system is verified since K reaches to 1, and the K curves agree well with the bifurcation diagrams.
Moreover, K-value based contour plots of the fractionalorder Buck-Boost system in different parameter planes is shown in Figure 7, where those parameter planes are divided as the 100 × 100 grid. e yellow color means that the system is chaotic when it takes parameters in those regions, while the rest parts mean that the system is nonchaotic. Meanwhile, it also shows that the system has wide regions for chaos in those parameter planes.

Complexity Analysis.
Bifurcation diagrams, phase diagrams, (p, s) plot, and K curves are mainly used to analyze dynamics of the system but cannot show how the complexity of the system changes with the parameters. e spectral entropy (SE) [33] algorithm is employed to analyze complexity of the fractional-order Buck-Boost system, and details of this algorithm are presented as follows.
Given a time sequence of length N {x (n), n � 0, 1, 2, . . ., N − 1}, let x(n) � x(n) − x, where x is the mean value of time series. Its corresponding DFT is defined by  where k � 0, 1, . . . , N − 1, and j is the imaginary unit. If the power of a discrete power spectrum with the k th frequency is |X(k)| 2 , then the "probability" of this frequency is defined as When the DFT is employed, the summation runs from k � 0 to k � (N/(2 − 1)). e normalization entropy is denoted by [33]   where ln(N/2) is the entropy of a completely random signal. Obviously, the more balanced the probability distribution is, the higher complexity (the larger entropy) the time series is. e larger measuring value means higher complexity and vice versa.
Using the same parameters, SE complexity of the fractional-order Buck-Boost system with different parameters is analyzed, and the results are shown in Figures 8 and 9. First, the SE analysis results have a high positive correlation with the corresponding bifurcation diagrams and 0-1 test results. As shown Figures 8 and 9, higher complexity is observed when the system is chaotic, but the nonchaotic states have lower complexity analysis results. Second, compared with the other methods in this study, SE complexity shows the variation tendency when the parameters vary.

Chaotic Pseudorandom Sequence Generator.
Chaotic behaviors such as randomness, sensitive dependence on initial conditions, and ergodicity are important issues for the real applications. At present, there are many researchers focusing on the applications of chaos, such as chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors [44], chaotic artificial neural networks for model memory in the brain [45], and a fractional-order chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola [46]. Currently, designing chaos-based pseudorandom sequence generators has aroused much interest of researchers [33,[47][48][49]. Since the fractional-order Buck-Boost system has high complexity, it is a good model for designing the chaotic pseudorandom sequence generator.
In this section, by modifying our previously designed pseudorandom sequence generator [33], a pseudorandom bit generator is designed based on the fractional-order Buck-Boost system. e specific steps are given as follows.
Step 2. Iterate the system one time and then obtain the new value of data � x 2 (n + 1) for the further calculation, where Data � round data × 10 8 .  us data are converted to as a 64-bit binary number DB 63 − DB 0 .
Step 4. Set n � n + 1. If n% 1000 � 0, we define 100000 , otherwise, Δ 1 � 0 and Δ 2 � 0, where [·] represents the round function. Now, set the initial condition for the next iterative as Step 5. Do Steps 2-4 in a loop until n > M. As a result, a txt file which contains 10 8 more bits of "0" and "1" is obtained. In this section, the statistical test suite of NIST [50] is applied to test the randomness of the sequence. e two indicators of the uniformity of p values and the proportion of passing sequences are used to check whether the targeting sequence passes the standard test or not. Here, the minimum uniformity of the p value is set as 0.0001. If all p values are larger than 0.0001 and the confidence interval satisfies where m is the sample size and α is the given significance level, then the pseudorandom bit generator is determined to pass the standard test successfully.
One hundred pseudorandom sequences are tested, all of which are 10 6 bits long. For most tests, the confidence  interval is [96.015%, 1] when α � 0.01. For those items tested more than once (for example, N. O. Temp. is tested 148 times), we only chosen the worst results. e results are shown in Table 2. It can be seen that all p values are larger than 0.0001, and the computed proportion for each test lies inside the confidence interval. Obviously, the obtained chaotic pseudorandom sequence is random and can be used in the information security fields. And the fractional-order Buck-Boost system provides a good model for engineering applications.

Control of the Buck-Boost System
In the real applications, sometimes, it is deleterious when there appears chaos; then, we need to control the chaos [51]. However, in many cases, we need to use the chaos [36,37]. In this section, the control of chaos in the fractional-order Buck-Boost system is discussed.

Control by the Intensity of Feedback k.
In fact, there is no need to design an extra controller for the Buck-Boost circuit as shown in Figure 1. As shown in the above analysis results, in real applications, the values of different parameters should be chosen carefully. According to Figures 7 and 9, if parameters are chosen in those blue regions, the system is nonchaotic, and the behaviors of the circuit is under control. Moreover, if the parameters of circuit are given but it is chaotic, the system can be controlled by increasing the value of the intensity of feedback k. Here, a numerical simulation is carried out on the control of chaos of the circuit by changing the value of k. When U in � 8 V, R � 10 Ω, L � 0.3 mH, C � 40 μF, T � 50 μs, I ref � 3 A, and q � 0.9, the system is chaotic for k � 0. In this case, we change the value of k as 0.1 (for the case q � 0.9) and 0.4 (for the case q � 1) when t � 0.005 s. As shown in Figure 10, the system is chaotic at the beginning, but it becomes periodic when the value of k is changed.
As shown above, the control parameter can change the state of the system from chaos to periodic state. But it depends on how does the parameter affects dynamics of the system. As for the other parameters, they can be also used to control the state of the fractional-order Buck-Boost system. It should be noted out that the parameter control cannot make the system stable or make the system to a target state. us, we still need another reliable control method.

Hard Limiter Control.
Hard limiter control is originally designed to control the dynamics of one-dimensional discrete maps. Specifically, Gueron [52] designed this controller to reformatting the dynamical behaviors of the discrete dynamical systems effectively. Based on the limiter control method, a one-dimensional chaotic map where ρ is the value of the limiter. Recently, He et al. [53] introduced this method to control the dynamics of a fractional-order SIR epidemic model which is a two-dimensional model. For the Buck-Boost circuit, we can introduce a current limiter as shown in Figure 11 to control the dynamics of the whole circuit.
In Figure 11, the current limiter is actually a hard limiter controller. As a result, this is also a new application of the hard limiter controller. e new circuit with a hard limiter controller is rebuilt as follows.
First, let the model of Case 1 be given by In this case, the result is obtained based on the sample length of T. Second, the model of Case 2 is defined as where a hard limiter is used for t ∈ (0t n , t n + d n T]. ird, the hard limiter controlled model with limited variable x 1 is denoted as Obviously, the model is proposed based on the numerical simulation model as given by equation (15). ere are two major reasons for this. One is that this numerical simulation model is a kind of equivalence of the original fractional-order Buck-Boost system. e second reason is that the hard limiter controller is proposed based on the discrete map.
and q � 0.9, and the system is chaotic. Hard limiter control results of the fractional-order Buck-Boost system with ρ varying are shown in Figure 12. Bifurcation diagrams and the mean densities of the variables x 1 and x 2 show that the system has different states with the increase of the value of the limiter ρ. Here, the mean densities are the mean values of the variables. When the bifurcation diagram and its mean densities are overlapped, the system is convergent. Otherwise, the system is chaotic or periodic. It is shown in Figure 12 that the controlled system is convergent to the limiter ρ when ρ < 2.4. Meanwhile, it is verified in Figures 13(a) and 13(b) that the system is stable when ρ � 1 and ρ � 2. As for the variable x 1 , it stabilizes at x 1 � 1 and 2, while the variable x 2 is convergent to zero. As shown in Figure 12, the system is periodic or chaotic when ρ > 2.4. It is also shown in Figures 13(c) and 13(d) that the controlled system is periodic for ρ � 2.5 and is chaotic for ρ � 4.5.   According to above analysis, it is found that the hard limiter controller is effective for controlling the fractionalorder Buck-Boost system. First, it is easy to realize in the real applications since we only need to add a current limiter in the circuit. Second, compared with the control by the intensity of feedback k, the hard limiter control is more flexible. e system can be controlled to convergent state, periodic state, and chaotic state by properly choosing a limiter ρ. ird, the hard limit controller is designed based on the obtained numerical model. In our opinion, the numerical solution is obtained from the system equation; thus, it also provides a model for analysis of the fractional-order Buck-Boost system. As for the hard limiter controller, it is designed for the discrete system, originally.
us, it is reasonable to introduce the hard limiter controller to the numerical model of the system. Meanwhile, the proposed model can well explain the proposed circuit as shown in Figure 11. In conclusion, the hard limiter control can be related to the real situations and has a strong application prospect for controlling the Buck-Boost circuit.

Conclusions
In this study, a fractional-order Buck-Boost system is deduced and analyzed based on the Adomian decomposition method. Chaos, complexity, and application of this fractional-order Buck-Boost system are investigated, and methods including bifurcation diagram, the 0-1 test, and SE are introduced to carry out the study. According to the simulations and analyses, the following conclusions are obtained.
(1) e deduced fractional order has rich dynamics with different parameters, which means that the original Buck-Boost converter could be chaotic if a set of proper parameters is given.
(2) As shown in those contour plots, the system has wide regions for chaos. e 0-1 test is an effective tool for detecting chaos, but SE shows more information on the variation tendency of the system with the variations of different parameters. Meanwhile, it shows that the generated pseudorandom sequence is random since it passes all the NIST tests. (3) Chaos of the circuit can be controlled by adjusting the intensity of feedback k or by introducing a hard limiter controller.
us, this study extended the model of the Buck-Boost converter by introducing the fractional calculus. And it provides a good model for engineering applications in the information security field. Our next work is to investigate the hard limiter control of the circuit practically and to use this system for the information encryption.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.