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

Authors: Bo Yan; Shaojie Wang; Shaobo He
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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 NIST test. 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.


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## Body

## 1. Introduction

The Buck-Boost converters are strongly nonlinear circuits [1–4] which can produce subharmonic, bifurcation, and chaos under certain conditions. At present, many researchers have investigated the chaos of the Buck-Boost converter systems [5–11]. 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 [13–15]. 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]. Therefore, 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.The 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 [18–20]. Generally, the nonlinear systems have more complex dynamical behaviors after the fractional calculus is introduced [21–24]. 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. Thus, it can be employed to solve fractional-order 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. This 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 fractional-order Buck-Boost system based on Wu and Zhang’s model [34] by employing the ADM.The rest of the study is organized as follows. In Section2, 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 fractional-order Buck-Boost system is studied. Two different approaches are introduced. Finally, conclusion is made in Section 5.


## 2. The Fractional-Order Buck-Boost Model

### 2.1. Building the Fractional-Order Buck-Boost System

Wu and Zhang [34] designed a Buck-Boost converter in the current mode and found chaos in the proposed model. The 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.Figure 1
Circuit of the Buck-Boost converter.Table 1
Description of the parameters in the Buck-Boost converter.
SymbolExplanationUinThe input voltage of the converterSThe power switchDThe diodeLThe inductanceCThe capacitanceRThe load resistanceiLThe inductive currentucThe capacitor voltageIrefThe inductance reference currentx1 and x2The inductive currentiL and the capacitor voltage uckThe intensity of feedbackLet the Buck-Boost converter work in the continuous conductive mode, and the inductance currentx1iL and the capacitance voltage x2uc 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](1)x˙1=1LUin,x˙2=−1RCx2,x˙1=−1Lx2,x˙2=1Cx1−1RCx2.Now, the Caputo fractional calculus, which is given by [38](2)Dt0qft=1Γ1−q∫t0tt−τ−qf′τdτ,is introduced to the system, where Dt0q is the Caputo derivative operator of order q0<q≤1, t0 is the initial time, and Γ⋅ is the Gamma function. Thus, the fractional-order Buck-Boost system is defined as(3)Dt0qx1=1LUin,Dt0qx2=−1RCx2,(4)Dt0qx1=−1Lx2,Dt0qx2=1Cx1−1RCx2.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 [25–28]. 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.


### 2.2. The Adomian Decomposition Method

As shown above, the systems have only linear items and a constant. To solve this system, the ADM [39–41] is employed, and the description of this algorithm is illustrated as follows.For a given fractional-order linear system,(5)Dt0qxt=ϒx+gx.t0+=x0,,where Dt0q is the Caputo definition, ϒx is the linear terms in the system, and q∈0,1. Its solution is given by [39](6)x=∑i=1∞xi=Jt0qϒ∑i=0∞xi+Jt0qg+x0.Meanwhile, those decomposition itemsxii=0,1,…,∞ are calculated as [40, 41](7)x0=Jt0qg+x0,xi=Jt0qϒxi−1,i=1,2,…,∞.When an infinite number of items is used, the solution is the exact solution. In this study, 11 decomposition items, namely,xii=0,1,…,10, are used.


### 2.3. Solution of the Fractional-Order Model

Let us focus on thenth sampling period; thus, t∈tn,tn+1. The initial value of this interval is x1tn and x2tn.Case 1.
By employing the above ADM, the solution of equation (3) is given by(8)x1t=x1tn+UinLt−tnqΓq+1,x2t=x2tn∑i=010κ2it−tniqΓiq+1,where(9)κ2i=−1RCκ2i−1,κ20=1.When t=tn+1, we can obtain the following discrete solution of the system, which is defined as(10)xtn+1=f1xtn,h,q.Case 2.
By employing the above ADM, the solution of equation (4) is denoted as(11)x1t=x1tn+∑i=110c1it−tniqΓiq+1,x2t=x2tn+∑i=110c2it−tniqΓiq+1,where(12)c1i=−1Lc2i−1,c2i=1Cc1i−1−1RCc2i−1,c10=x1tn,c20=x2tn.
So the following discrete solution is obtained:(13)xtn+1=f2xtn,h,q.
Based onx1tn and x2tn, the reference current for the inductance Ireftn, the boundary value of the inductance current Ibtn, and the duty cycle dtn are calculated by [34](14)Ireftn=Iref−ekun−un−1+1,Ibtn=Ireftn−UinTL,dtn=LIreftn−x1tnUinT.
Whenx1tn<Ibtn, the power switch S is on during the whole sampling period, and the system is defined as Case 1. However, if x1tn≥Ibtn, the power switch S is on when t∈tn,tn+dnT, while the power switch S is off when t∈tn+dnT,tn+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.(15)xtn+1=f1xtn,T,q,ifx1tn<Ibtn,f2f1xtn,dtnT,q,1−dtnT,q,ifx1tn≥Ibtn,where T=tn+1−tn is the length of the sampling period. Obviously, this is a switched discrete system. When the initial conditions x1t0 and x2t0 are given, states of the system can be observed.


## 2.1. Building the Fractional-Order Buck-Boost System

Wu and Zhang [34] designed a Buck-Boost converter in the current mode and found chaos in the proposed model. The 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.Figure 1
Circuit of the Buck-Boost converter.Table 1
Description of the parameters in the Buck-Boost converter.
SymbolExplanationUinThe input voltage of the converterSThe power switchDThe diodeLThe inductanceCThe capacitanceRThe load resistanceiLThe inductive currentucThe capacitor voltageIrefThe inductance reference currentx1 and x2The inductive currentiL and the capacitor voltage uckThe intensity of feedbackLet the Buck-Boost converter work in the continuous conductive mode, and the inductance currentx1iL and the capacitance voltage x2uc 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](1)x˙1=1LUin,x˙2=−1RCx2,x˙1=−1Lx2,x˙2=1Cx1−1RCx2.Now, the Caputo fractional calculus, which is given by [38](2)Dt0qft=1Γ1−q∫t0tt−τ−qf′τdτ,is introduced to the system, where Dt0q is the Caputo derivative operator of order q0<q≤1, t0 is the initial time, and Γ⋅ is the Gamma function. Thus, the fractional-order Buck-Boost system is defined as(3)Dt0qx1=1LUin,Dt0qx2=−1RCx2,(4)Dt0qx1=−1Lx2,Dt0qx2=1Cx1−1RCx2.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 [25–28]. 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.


## 2.2. The Adomian Decomposition Method

As shown above, the systems have only linear items and a constant. To solve this system, the ADM [39–41] is employed, and the description of this algorithm is illustrated as follows.For a given fractional-order linear system,(5)Dt0qxt=ϒx+gx.t0+=x0,,where Dt0q is the Caputo definition, ϒx is the linear terms in the system, and q∈0,1. Its solution is given by [39](6)x=∑i=1∞xi=Jt0qϒ∑i=0∞xi+Jt0qg+x0.Meanwhile, those decomposition itemsxii=0,1,…,∞ are calculated as [40, 41](7)x0=Jt0qg+x0,xi=Jt0qϒxi−1,i=1,2,…,∞.When an infinite number of items is used, the solution is the exact solution. In this study, 11 decomposition items, namely,xii=0,1,…,10, are used.


## 2.3. Solution of the Fractional-Order Model

Let us focus on thenth sampling period; thus, t∈tn,tn+1. The initial value of this interval is x1tn and x2tn.Case 1.
By employing the above ADM, the solution of equation (3) is given by(8)x1t=x1tn+UinLt−tnqΓq+1,x2t=x2tn∑i=010κ2it−tniqΓiq+1,where(9)κ2i=−1RCκ2i−1,κ20=1.When t=tn+1, we can obtain the following discrete solution of the system, which is defined as(10)xtn+1=f1xtn,h,q.Case 2.
By employing the above ADM, the solution of equation (4) is denoted as(11)x1t=x1tn+∑i=110c1it−tniqΓiq+1,x2t=x2tn+∑i=110c2it−tniqΓiq+1,where(12)c1i=−1Lc2i−1,c2i=1Cc1i−1−1RCc2i−1,c10=x1tn,c20=x2tn.
So the following discrete solution is obtained:(13)xtn+1=f2xtn,h,q.
Based onx1tn and x2tn, the reference current for the inductance Ireftn, the boundary value of the inductance current Ibtn, and the duty cycle dtn are calculated by [34](14)Ireftn=Iref−ekun−un−1+1,Ibtn=Ireftn−UinTL,dtn=LIreftn−x1tnUinT.
Whenx1tn<Ibtn, the power switch S is on during the whole sampling period, and the system is defined as Case 1. However, if x1tn≥Ibtn, the power switch S is on when t∈tn,tn+dnT, while the power switch S is off when t∈tn+dnT,tn+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.(15)xtn+1=f1xtn,T,q,ifx1tn<Ibtn,f2f1xtn,dtnT,q,1−dtnT,q,ifx1tn≥Ibtn,where T=tn+1−tn is the length of the sampling period. Obviously, this is a switched discrete system. When the initial conditions x1t0 and x2t0 are given, states of the system can be observed.


## 3. Complexity of the Fractional-Order Buck-Boost System

### 3.1. Dynamics of the Fractional-Order Model

Let the simulation parameters beUin = 8 V, R=10Ω, L = 0.3 mH, C = 40 μF, and T = 50 μs. Set the inductance reference current as Iref = 3 A. 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.Figure 2
Phase diagrams of the fractional-order model. (a)q=1, (b) q=0.95, (c) q=0.9, and (d) q=0.85.
(a)(b)(c)(d)Bifurcation diagrams of the system with different system parameters include the inductance reference currentIref, the 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 Iref under a different derivative order q. In Figure 3, Iref 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 Iref are different with a different derivative order q. Meanwhile, bifurcations versus different parameters are analyzed. Fix k=0, q=0.9, and Iref=3A, and vary L from 2 mH to 5 mH with step size of 0.006. The analysis result is shown in Figure 4(a). Fix q=0.9, Iref=3 A, and L = 0.3 mH, and vary parameter k from 0 to 5 with step size of 0.01. The bifurcation diagram with k is shown in Figure 4(b). Let Uin = 8 V, L = 0.3 mH, R=10Ω, C = 40 μF, k=0, 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. The 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.Figure 3
Bifurcation diagrams with the inductance reference currentIref under different derivative orders q. (a) q=1, (b) q=0.95, (c) q=0.9, and (d) q=0.85.
(a)(b)(c)(d)Figure 4
Bifurcation diagrams with the variation of different parameters. (a)L varying, (b) k varying, and (c) q varying.
(a)(b)(c)


### 3.2. 0-1 Test

The 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 seriesxn:n=1,2,3,…,, then the following two real valued sequences can be defined as [42, 43](16)pn=∑j=1nxjcosθj,sn=∑j=1nxjsinθj,where θj=jη+∑i=1jφσi, and η∈π/5,4π/5. 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. Thus, the existence of chaos is verified.Figure 5
p,s plots of the fractional-order model. (a) q=1, (b) q=0.95, (c) q=0.9, and (d) q=0.85.
(a)(b)(c)(d)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. Thus, 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. There are two steps for this quantity which are expanded as follows.First, the mean square displacement is defined as [42, 43](17)Mn=limN⟶∞1N∑j=1Npj+n−pj2+sj+n−sj2,where n≪N. In real applications, n≤N/10 yields good results.Second, the asymptotic growth rateK is defined as [42, 43](18)K=limn⟶∞logMnlogn.Plots ofK versus different parameters for the fractional-order Buck-Boost system are shown in Figure 6. Figures 6(a)–6(d) show the dynamics corresponding to the bifurcation diagrams as shown in Figure 3. And Figures 6(e)–6(g) are the corresponding analysis results of 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.Figure 6
Plots ofK versus different parameters for the fractional-order Buck-Boost system. (a) q=1 and Iref varying, (b) q=0.95 and Iref varying, (c) q=0.9 and Iref varying, (d) q=0.85 and Iref varying, (e) L varying, (f) k varying, and (g) derivative order q varying.
(a)(b)(c)(d)(e)(f)(g)Moreover,K-value based contour plots of the fractional-order Buck-Boost system in different parameter planes is shown in Figure 7, where those parameter planes are divided as the 100×100 grid. The 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.Figure 7
K value-based contour plots of the of fractional-order Buck-Boost system. (a) q−Iref, (b) q−L, (c) q−k, and (d) Iref−L.
(a)(b)(c)(d)


### 3.3. 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. The 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 lengthN {xn, n = 0, 1, 2, …, N−1}, let xn=xn−x¯, where x¯ is the mean value of time series. Its corresponding DFT is defined by(19)Xk=∑n=0N−1xne−j2πnk/N,where k=0,1,…,N−1, and j is the imaginary unit. If the power of a discrete power spectrum with the kth frequency is Xk2, then the “probability” of this frequency is defined as(20)Pk=Xk2∑k=0N/2−1Xk2.When the DFT is employed, the summation runs fromk=0 to k=N/2−1. The normalization entropy is denoted by [33](21)SExN=1lnN/2∑k=0N/2−1PklnPk,where lnN/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. The 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 Figures8 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.Figure 8
Plots of SE versus different parameters for the fractional-order Buck-Boost system. (a)q=1 and Iref varying, (b) q=0.95 and Iref varying, (c) q=0.9 and Iref varying, (d) q=0.85 and Iref varying, (e) L varying, (f) k varying, and (g) derivative order q varying.
(a)(b)(c)(d)(e)(f)(g)Figure 9
SE-based contour plots of the fractional-order Buck-Boost system. (a) q−Iref, (b) q−L, (c) q−k, and (d) Iref−L.
(a)(b)(c)(d)


### 3.4. 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, 47–49]. 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. The specific steps are given as follows.Step 1.
SetUin = 8 V, R=10Ω, L = 0.3 mH, C = 40 μF, T = 50 μs, Iref = 4 A, k=0,and q=0.95. The initial value x0=1,2, M=0.125×108+100, and iterate the system 1000 times; then n=1000, and thus, x0=x1n,x2n. Let n=1.Step 2.
Iterate the system one time and then obtain the new value ofdata=x2n+1 for the further calculation, where(22)Data=rounddata×108.
Thusdata are converted to as a 64-bit binary number DB63−DB0.Step 3.
Definedata1 = DB7−DB0. Then, the 8-bit 0-1 sequence data1 are saved in the target txt file.Step 4.
Setn=n+1. If n%1000=0, we define(23)Δ1=x1n×108%255100000,Δ2=x2n×108%255100000,otherwise, Δ1=0 and Δ2=0, where ⋅ represents the round function. Now, set the initial condition for the next iterative as(24)x0=x1n+Δ1,x2n+Δ2.Step 5.
Do Steps2–4 in a loop until n>M.
As a result, a txt file which contains108 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.
The two indicators of the uniformity ofp 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(25)1−α−31−ααm,1−α+31−ααm,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 are106 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. The 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.Table 2
NIST test result of the generated pseudorandom sequence.
p valueProportion (%)SuccessFrequency0.77918898√B. Frequency0.514124100√C. Sums (2)0.31908498√Runs0.41902198√Longest run0.05194297√Rank0.36691899√FFT0.102526100√N. O. Temp. (148)0.00344796√O. Temp.0.53414699√Universal0.657933100√App. Entropy0.16260699√R. Excur. (8)0.32785498.6√R. Excur. V. (18)0.01791297.3√Serial (2)0.43727499√L. Complexity0.574903100√


## 3.1. Dynamics of the Fractional-Order Model

Let the simulation parameters beUin = 8 V, R=10Ω, L = 0.3 mH, C = 40 μF, and T = 50 μs. Set the inductance reference current as Iref = 3 A. 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.Figure 2
Phase diagrams of the fractional-order model. (a)q=1, (b) q=0.95, (c) q=0.9, and (d) q=0.85.
(a)(b)(c)(d)Bifurcation diagrams of the system with different system parameters include the inductance reference currentIref, the 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 Iref under a different derivative order q. In Figure 3, Iref 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 Iref are different with a different derivative order q. Meanwhile, bifurcations versus different parameters are analyzed. Fix k=0, q=0.9, and Iref=3A, and vary L from 2 mH to 5 mH with step size of 0.006. The analysis result is shown in Figure 4(a). Fix q=0.9, Iref=3 A, and L = 0.3 mH, and vary parameter k from 0 to 5 with step size of 0.01. The bifurcation diagram with k is shown in Figure 4(b). Let Uin = 8 V, L = 0.3 mH, R=10Ω, C = 40 μF, k=0, 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. The 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.Figure 3
Bifurcation diagrams with the inductance reference currentIref under different derivative orders q. (a) q=1, (b) q=0.95, (c) q=0.9, and (d) q=0.85.
(a)(b)(c)(d)Figure 4
Bifurcation diagrams with the variation of different parameters. (a)L varying, (b) k varying, and (c) q varying.
(a)(b)(c)


## 3.2. 0-1 Test

The 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 seriesxn:n=1,2,3,…,, then the following two real valued sequences can be defined as [42, 43](16)pn=∑j=1nxjcosθj,sn=∑j=1nxjsinθj,where θj=jη+∑i=1jφσi, and η∈π/5,4π/5. 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. Thus, the existence of chaos is verified.Figure 5
p,s plots of the fractional-order model. (a) q=1, (b) q=0.95, (c) q=0.9, and (d) q=0.85.
(a)(b)(c)(d)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. Thus, 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. There are two steps for this quantity which are expanded as follows.First, the mean square displacement is defined as [42, 43](17)Mn=limN⟶∞1N∑j=1Npj+n−pj2+sj+n−sj2,where n≪N. In real applications, n≤N/10 yields good results.Second, the asymptotic growth rateK is defined as [42, 43](18)K=limn⟶∞logMnlogn.Plots ofK versus different parameters for the fractional-order Buck-Boost system are shown in Figure 6. Figures 6(a)–6(d) show the dynamics corresponding to the bifurcation diagrams as shown in Figure 3. And Figures 6(e)–6(g) are the corresponding analysis results of 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.Figure 6
Plots ofK versus different parameters for the fractional-order Buck-Boost system. (a) q=1 and Iref varying, (b) q=0.95 and Iref varying, (c) q=0.9 and Iref varying, (d) q=0.85 and Iref varying, (e) L varying, (f) k varying, and (g) derivative order q varying.
(a)(b)(c)(d)(e)(f)(g)Moreover,K-value based contour plots of the fractional-order Buck-Boost system in different parameter planes is shown in Figure 7, where those parameter planes are divided as the 100×100 grid. The 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.Figure 7
K value-based contour plots of the of fractional-order Buck-Boost system. (a) q−Iref, (b) q−L, (c) q−k, and (d) Iref−L.
(a)(b)(c)(d)


## 3.3. 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. The 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 lengthN {xn, n = 0, 1, 2, …, N−1}, let xn=xn−x¯, where x¯ is the mean value of time series. Its corresponding DFT is defined by(19)Xk=∑n=0N−1xne−j2πnk/N,where k=0,1,…,N−1, and j is the imaginary unit. If the power of a discrete power spectrum with the kth frequency is Xk2, then the “probability” of this frequency is defined as(20)Pk=Xk2∑k=0N/2−1Xk2.When the DFT is employed, the summation runs fromk=0 to k=N/2−1. The normalization entropy is denoted by [33](21)SExN=1lnN/2∑k=0N/2−1PklnPk,where lnN/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. The 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 Figures8 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.Figure 8
Plots of SE versus different parameters for the fractional-order Buck-Boost system. (a)q=1 and Iref varying, (b) q=0.95 and Iref varying, (c) q=0.9 and Iref varying, (d) q=0.85 and Iref varying, (e) L varying, (f) k varying, and (g) derivative order q varying.
(a)(b)(c)(d)(e)(f)(g)Figure 9
SE-based contour plots of the fractional-order Buck-Boost system. (a) q−Iref, (b) q−L, (c) q−k, and (d) Iref−L.
(a)(b)(c)(d)


## 3.4. 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, 47–49]. 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. The specific steps are given as follows.Step 1.
SetUin = 8 V, R=10Ω, L = 0.3 mH, C = 40 μF, T = 50 μs, Iref = 4 A, k=0,and q=0.95. The initial value x0=1,2, M=0.125×108+100, and iterate the system 1000 times; then n=1000, and thus, x0=x1n,x2n. Let n=1.Step 2.
Iterate the system one time and then obtain the new value ofdata=x2n+1 for the further calculation, where(22)Data=rounddata×108.
Thusdata are converted to as a 64-bit binary number DB63−DB0.Step 3.
Definedata1 = DB7−DB0. Then, the 8-bit 0-1 sequence data1 are saved in the target txt file.Step 4.
Setn=n+1. If n%1000=0, we define(23)Δ1=x1n×108%255100000,Δ2=x2n×108%255100000,otherwise, Δ1=0 and Δ2=0, where ⋅ represents the round function. Now, set the initial condition for the next iterative as(24)x0=x1n+Δ1,x2n+Δ2.Step 5.
Do Steps2–4 in a loop until n>M.
As a result, a txt file which contains108 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.
The two indicators of the uniformity ofp 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(25)1−α−31−ααm,1−α+31−ααm,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 are106 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. The 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.Table 2
NIST test result of the generated pseudorandom sequence.
p valueProportion (%)SuccessFrequency0.77918898√B. Frequency0.514124100√C. Sums (2)0.31908498√Runs0.41902198√Longest run0.05194297√Rank0.36691899√FFT0.102526100√N. O. Temp. (148)0.00344796√O. Temp.0.53414699√Universal0.657933100√App. Entropy0.16260699√R. Excur. (8)0.32785498.6√R. Excur. V. (18)0.01791297.3√Serial (2)0.43727499√L. Complexity0.574903100√


## 4. 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.


### 4.1. Control by the Intensity of Feedbackk

In fact, there is no need to design an extra controller for the Buck-Boost circuit as shown in Figure1. 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 Uin = 8 V, R=10Ω, L = 0.3 mH, C = 40 μF, T = 50 μs, Iref = 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=0.005s. As shown in Figure 10, the system is chaotic at the beginning, but it becomes periodic when the value of k is changed.Figure 10
Plots of the fractional-order Buck-Boost system with regulablek. (a) q=0.9 and k becomes to 0.1; (b) q=1 and k becomes to 0.4.
(a)(b)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. Thus, we still need another reliable control method.


### 4.2. 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 xn+1=fxn becomes(26)fxn,ρ=minfxn,ρ,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.Figure 11
The Buck-Boost circuit with a limiter controller.In Figure11, the current limiter is actually a hard limiter controller. As a result, this is also a new application of the hard limiter controller. The new circuit with a hard limiter controller is rebuilt as follows.First, let the model of Case1 be given by(27)x˜1tn+1,Tx˜2tn+1,T=f1xtn,T,q.In this case, the result is obtained based on the sample length ofT.Second, the model of Case2 is defined as(28)x^1tn+1x^2tn+1=f2minx˜1tn+1,dtnT,ρx˜2tn+1,dtnT,1−dtnT,q,where a hard limiter is used for t∈0tn,tn+dnT.Third, the hard limiter controlled model with limited variablex1 is denoted as(29)xtn+1=minx˜1tn+1,T,ρx˜2tn+1,ifx1tn<Ibtn,minx^2tn+1,ρx^2tn+1,ifx1tn≥Ibtn.Obviously, the model is proposed based on the numerical simulation model as given by equation (15). There 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. The second reason is that the hard limiter controller is proposed based on the discrete map.LetUin = 8 V, R=10Ω, L = 0.3 mH, C = 40 μF, T = 50 μs, Iref = 3 A, k=0, 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 x1 and x2 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 x1, it stabilizes at x1=1 and 2, while the variable x2 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.Figure 12
Hard limiter control results of the fractional-order Buck-Boost system withρ varying. (a) Bifurcation diagram and mean densities of x1. (b) Bifurcation diagram and mean densities with x2.
(a)(b)Figure 13
Hard limiter control results of the fractional-order Buck-Boost system with differentρ. (a) ρ=1, (b) ρ=2, (c) ρ=2.5, and (d) ρ=4.5.
(a)(b)(c)(d)According to above analysis, it is found that the hard limiter controller is effective for controlling the fractional-order 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 feedbackk, the hard limiter control is more flexible. The system can be controlled to convergent state, periodic state, and chaotic state by properly choosing a limiter ρ. Third, 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. Thus, 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.


## 4.1. Control by the Intensity of Feedbackk

In fact, there is no need to design an extra controller for the Buck-Boost circuit as shown in Figure1. 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 Uin = 8 V, R=10Ω, L = 0.3 mH, C = 40 μF, T = 50 μs, Iref = 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=0.005s. As shown in Figure 10, the system is chaotic at the beginning, but it becomes periodic when the value of k is changed.Figure 10
Plots of the fractional-order Buck-Boost system with regulablek. (a) q=0.9 and k becomes to 0.1; (b) q=1 and k becomes to 0.4.
(a)(b)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. Thus, we still need another reliable control method.


## 4.2. 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 xn+1=fxn becomes(26)fxn,ρ=minfxn,ρ,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.Figure 11
The Buck-Boost circuit with a limiter controller.In Figure11, the current limiter is actually a hard limiter controller. As a result, this is also a new application of the hard limiter controller. The new circuit with a hard limiter controller is rebuilt as follows.First, let the model of Case1 be given by(27)x˜1tn+1,Tx˜2tn+1,T=f1xtn,T,q.In this case, the result is obtained based on the sample length ofT.Second, the model of Case2 is defined as(28)x^1tn+1x^2tn+1=f2minx˜1tn+1,dtnT,ρx˜2tn+1,dtnT,1−dtnT,q,where a hard limiter is used for t∈0tn,tn+dnT.Third, the hard limiter controlled model with limited variablex1 is denoted as(29)xtn+1=minx˜1tn+1,T,ρx˜2tn+1,ifx1tn<Ibtn,minx^2tn+1,ρx^2tn+1,ifx1tn≥Ibtn.Obviously, the model is proposed based on the numerical simulation model as given by equation (15). There 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. The second reason is that the hard limiter controller is proposed based on the discrete map.LetUin = 8 V, R=10Ω, L = 0.3 mH, C = 40 μF, T = 50 μs, Iref = 3 A, k=0, 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 x1 and x2 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 x1, it stabilizes at x1=1 and 2, while the variable x2 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.Figure 12
Hard limiter control results of the fractional-order Buck-Boost system withρ varying. (a) Bifurcation diagram and mean densities of x1. (b) Bifurcation diagram and mean densities with x2.
(a)(b)Figure 13
Hard limiter control results of the fractional-order Buck-Boost system with differentρ. (a) ρ=1, (b) ρ=2, (c) ρ=2.5, and (d) ρ=4.5.
(a)(b)(c)(d)According to above analysis, it is found that the hard limiter controller is effective for controlling the fractional-order 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 feedbackk, the hard limiter control is more flexible. The system can be controlled to convergent state, periodic state, and chaotic state by properly choosing a limiter ρ. Third, 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. Thus, 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.


## 5. 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)
The 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. The 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 feedbackk or by introducing a hard limiter controller.Thus, 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.



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*Source: 5572840-2021-03-19.xml*