Quantifying the Nonlinear Dynamic Behavior of the DC-DC ... - MDPI

energies Article

Quantifying the Nonlinear Dynamic Behavior of the DC-DC Converter via Permutation Entropy Zhenxiong Luo, Fan Xie *, Bo Zhang and Dongyuan Qiu School of Electric Power, South China University of Technology, Guangzhou 510641, China; [email protected] (Z.L.); [email protected] (B.Z.); [email protected] (D.Q.) * Correspondence: [email protected]; Tel.: +86-020-8711-1764 Received: 19 September 2018; Accepted: 10 October 2018; Published: 13 October 2018







Abstract: Quantifying nonlinear dynamic behaviors, such as bifurcation and chaos, in nonlinear systems are currently being investigated. In this paper, permutation entropy is used to characterize these complex phenomena in nonlinear direct current-direct current (DC-DC) converter systems. A mode switching time sequence (MSTS), containing the information from different periodic states, is obtained in a DC-DC converter by reading the inductor current when altering the switching mode. To obtain the nonlinear characteristics of this system, the concept of permutation entropy of symbolic probability distribution properties is introduced and the structure of the chaotic system is reproduced based on the theory of phase space reconstruction. A variety of nonlinear dynamic features of the DC-DC converter are analyzed using the MSTS and permutation entropy. Finally, a current-mode-controlled buck converter is reviewed as a case to study the quantification of nonlinear phenomena using permutation entropy as one of the system parameters changes. Keywords: nonlinear behaviors; symbol sequence; operate mode; border collision bifurcation; period-doubling bifurcation; permutation entropy

1. Introduction 1.1. Background Direct current-direct current (DC-DC) switching converters are a typical nonlinear system that enable the observation of nonlinear behaviors, such as period-doubling bifurcation, border collision bifurcation, frequency-locking phenomenon, quasi-period, and chaos [1–5]. The study of nonlinear behaviors is helpful for improving the performance of DC-DC converters. 1.2. Formulation of the Problem of Interest for This Investigation Generally speaking, these nonlinear phenomena can be observed by phase plane, bifurcation diagrams, and Poincaré sections in DC-DC converters [6–8], but these only provide qualitative analyses. To examine nonlinear phenomena in a timely manner and identify them accurately, quantitative analysis is necessary. At present, some quantitative analysis methods have been proposed to process these nonlinear features, and the largest Lyapunov exponent (LLE) is the most common [9,10]. 1.3. Literature Survey In practical applications, analyzing the diverse nonlinear phenomena in LLE is challenging due to the complex of the phase space trajectory and its solution process is complicated [7,11]. Researchers [12] found the symbolic time sequence contains a large amount of DC-DC converter system state information, including nonlinear stability, bifurcation information, and degree of confusion. Both symbolic sequence analysis and entropy are utilized to quantitatively analyze DC-DC converters. Energies 2018, 11, 2747; doi:10.3390/en11102747

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Multiscale entropy and sample entropy are used to characterize nonlinear phenomena and fault features [13–16]. The symbolic time sequence method has been combined with block entropy to quantify bifurcation and chaos state in boost converters [12,17]. Switch information sequences and the ´ concept of joint entropy have been used to quantify the complex state of the Cuk converter [18,19]. 1.4. Scope and Contribution of This Study When the circuit parameters or the control parameters of the DC-DC converter are changed, the system tends to exhibit processes from bifurcation to chaos or other nonlinear behaviors. The conventional time sequence of a DC-DC converter is based on discrete mapping modeling with the control period as the sampling period, so the information of the switching state time is ignored. With nonlinear behavior, the switching state of a DC-DC converter may differ greatly in adjacent control periods. Therefore, this paper introduces the mode switching time sequence (MSTS) by sampling the value of the state variables at the mode switching moment. The MSTS not only obtains the modality of each period, but also distinguishes the modalities presented in different periods. Hence, two characteristics of the DC-DC converter can be simultaneously obtained through a time sequence period. In traditional symbolic sequence analysis, the process of hierarchical symbolization is prone to excessive symbol in a certain layer, and the overlapping symbols indirectly lead to the loss of information. Permutation entropy [20,21] is a special symbol sequence method based on information entropy [22] and phase space reconstruction theory. It can effectively quantify the complexity of a time sequence. Permutation entropy does not need to stratify the amplitude of the time sequence, which is different from the general symbol sequence analysis method. Permutation entropy is based on sorting the coordinates of the reconstructed phase space, and then the generated symbol space is compared with all possible symbol sequences to achieve classification. Permutation entropy can accurately obtain the autocorrelation degree of the time sequences [23,24]. When the sequence representation is completely random, the self-similarity of the sequence is zero. This implies that the probability of occurrence of each sequence is the same, and the entropy value is the largest. The entropy value is the smallest for the sequence based on a period-1 orbit because the sampling data in each period remain unchanged. 1.5. Organization of the Paper This paper is organized as follows: Section 2 introduces the MSTS to describe the characteristics of the DC-DC converter. In Section 3, the identification of permutation entropy in periodic signals and the selection of parameters of permutation entropy are discussed. In Section 4, MSTS combined with permutation entropy is proposed. In Section 5, an example of a DC-DC converter is described in detail, and the proposed method is applied to analyze nonlinear behaviors. Finally, through discussion and summary, the feasibility of the method is demonstrated by comparison with other methods and the nonlinear dynamic behaviors of the DC-DC converter are better quantified. 2. MSTS of the Modeled Converter Current symbolic time sequence uses the control period as the sampling period in DC-DC converters. However, the switching states or the change in the system topology may be different in one control period, and the sampling control period ignores these details. For example, continuous conduction mode (CCM) and discontinuous conduction mode (DCM), both working in period-1, orbit at the stable state of the converter. So, the MSTS of the DC-DC converter is introduced to solve this issue and is shown as follows.

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Energies 2018, 3 of 15 Set X11,=x [FOR X XPEER X ·REVIEW · · X ]T to the state vector (with p being the numbers of variables and superscript 2

1

p

3

T denoting transposition) of a DC-DC converter. So, in the ith control period, the discretized expression of X can be expressed as:

 X(ti  ti1  ti 2   tik )  Sk ( X(ti  ti1  ti 2   ti ( k 1) ), tik ), · · · + t ) = Sk (X(ti + ti1 + ti2 + · · · + ti(k−1) ), tik ), X(ikti + Sti1k (+Xti i2( k + X 1) , tik ), ik

(

(1)

(1)

Xik = Sk (Xi(k−1) , tik ),

where Sk represents the relationship between the nitial and final terative alues in the circuit topology Sk represents between the nitial and final terative in topology the circuitand topology as where the operating mode the k is relationship presented (with its expression being only relatedalues to the the as the operating mode k is presented (with its expression being only related to the topology and selected state variable). ti and tik represent the start time of the ith period and the mode k occurs in thethe selected tikrepresent the start time of the ith period and the mode k occurs in the period, andstate Xik variable).  X(ti  ti1 ti tand tik ) . i2  period, and Xik = X(ti + ti1 + ti2 + · · · + t ik ). Suppose the system has ji (1 ≤ ji ≤ 2m –1, where m indicates the number of switching devices) Suppose the system has ji (1 ≤ ji ≤ 2m –1, where m indicates the number of switching devices) modes in the ith control period. As the system stabilizes, we take xL in the state variable X as the object modes in the ith control period. As the system stabilizes, we take xL in the state variable X as the object of observation and sample it when the mode changes from each control period. Hence, we obtain the of observation and sample it when the mode changes from each control period. Hence, we obtain the MSTS, i.e., MSTS, i.e.,

 {x1 , x, 11 , xx , , . ., .x,1xj1 , x,21x , , . ,. x. ,( Nx1) j( N 1) , xN1 ,xxN 2,,x ,,x. Nj }, xxLL−-MSTS MSTS = { x 1 x11 , 12 12 1j1 21 ( N −1) j( N −1) , N1 N2 . .N, x NjN },

(2) (2)

x(ti x1 () ti+x1(t)i = ti1 x (ti + tij )t,i1the Considering thethe equation be left Equation Considering equation + .x. i+1 . +has tiji )to , the xi+1out hasfrom to be left out (2). from Equation (2). an Figure 1 shows an switching example ofdevices two switching existing system, in the converter system, Figure 1 shows example of two existing devices in the converter which means which means it can express up to three modes, where T is the control period and t is the time of the ith i it can express up to three modes, where T is the control period and ti is the time of the ith switching switching moment (t − t = T). i i-1 moment (ti−ti-1 = T). i

t1

T

t2

t3

t4

t5

t6

t7

t8

t9

t10

x21

xL

x11

x1

x22 t11

t21

t31

t22

t41

t32 t23

t51

t42

t61

t52

t71

t81

t72

t91

t82

t53

Figure 1. Sampling at the mode switching time to build a mode switching time sequence (MSTS). Figure 1. Sampling at the mode switching time to build a mode switching time sequence (MSTS.)

3. Permutation Entropy 3. Permutation Entropy 3.1. Phase Space Reconstruction Theory 3.1. Phase Space Reconstruction Theory The low-dimensional coordinate system cannot reflect the complex nonlinear states, as demonstrated The low-dimensional coordinate system cannot reflect the complex nonlinear states, as by Packard et al. [25]. Packard et al. [25] proposed two methods of reconstructing phase space in demonstrated by Packard et al. [25]. Packard et al. [25] proposed two methods of reconstructing phase a time sequence. The coordinate delay reconstruction method is widely used. In this method, the space in a time sequence. The coordinate delay reconstruction method is widely used. In this method, one-dimensional time sequence is constructed by embedding the dimension m and the delay time τ the one-dimensional time sequence is constructed by embedding the dimension m and the delay time to form a new m-dimensional vector sequence. The phase space of the discrete time sequence x(n) of τ to form a new m-dimensional vector sequence. The phase space of the discrete time sequence x(n) length N can be obtained: of length N can be obtained: Y((ii)) = ), x x ((ii + ), . . .,,xx((ii  +τ×(m (m− 1))}i , i= 1, 2, .N ..N m1) −1) × τ Y  {{xx((ii),  τ), 1))}, 1, 2,  (−m(

(3) (3)

Afterward, Takens embedding theorems proves rationality phase space Afterward, thethe Takens embedding theorems [26[26] ] proves thethe rationality of of phase space reconstruction and proposes thatthe theembedding embeddingdimension dimensionofofphase phasespace space should should satisfy m ≥ 2D ++ 1, reconstruction and proposes that ≥ 2D 1, where D is the singular attractor dimension of the system and it is a geometric invariant. A dynamic system with correlation dimension D can be described through 2D + 1 independent variables at most [7]. This means that the geometry of the dynamic system is completely open when m is greater than

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where D is the singular attractor dimension of the system and it is a geometric invariant. A dynamic system with correlation dimension D can be described through 2D + 1 independent variables at most [7]. This means that the geometry of the dynamic system is completely open when m is greater than the minimum embedding dimension 2D + 1. At this time, the geometric invariant properties of the system, such as the Lyapunov exponent, are independent of the embedding dimension m. In addition, the delay time can be obtained by the autocorrelation method. The time sequence of a state variable can reconstruct the phase space of the system, for it contains information about all variables participating in the dynamic system. According to the Takens embedding theorem, the m-dimensional state space, expressed in Equation (4), has the same meaning as the original state space topologically; that is, the original state variable information can be reproduced.       Y=    

Y(1, :) Y(2, :) .. . Y(i, :) .. . Y( N − (m − 1) × τ, :)





          =        

x1 x2 .. . xi .. .

x 1+ τ x 2+ τ .. . xi +τ .. .

... ...

x N −(m−1)×τ

x N −(m−2)×τ

...

...

x1+τ ×(m−1) x2+τ ×(m−1) .. . xi+τ ×(m−1) .. . xN

          

(4)

3.2. Symbol Permutation Method Entropy can quantify the periodic states and chaos by time sequence. Its concept is sourced in the second law of thermodynamics. After Shannon introduced it into information theory, Shannon entropy became an important quantitative means to measure the probability distribution of a sequence signal [22]. For any discrete sequence { x (i ), i = 1, 2, . . . , n}, the corresponding probability distribution { p(i )|i = 1, 2, . . . , n } can be calculated, where p(i) is equal to the proportion of x(i) in the entire sequence. The Shannon entropy HP can be defined as HP = −∑ p(i ) ln p(i ). The larger the HP , the more uneven the signal distribution. This indicates that the system is highly complex, and the time sequence is difficult to predict. On the contrary, the information contained in the signal is easy to find for small HP , and the prediction can be realized with minimal error. Particularly, in the case where HP = 0, the sequence signal could be theoretically predictable, such as the step signal at t > 0. In order to accurately quantify the nonlinear features, we first need to acquire certain statistical characteristics of the signal. In 2001, Bant et al. proposed the concept of permutation entropy based on phase space reconstruction theory, and successfully applied it to the complex quantization process of time sequences [23]. Equation (5) was generated by sorting Equation (4) in ascending order: Y(i ) = { x (i + k i1 τ ) ≤ x (i + k i2 τ ) ≤ . . . ≤ x (i + k im τ )}.

(5)

We determined the corresponding symbol permutation: Ki = {k i1 , k i2 , . . . , k im }, k i· = 0, 1, . . . , m − 1. For m-dimensional phase space reconstruction, we studied all n! (1, 2, . . . , m − 1) s( g) = { g1 , g2 , . . . , gm }, 1 < g < m!,

(6) permutations s( g) of (7)

where gi ∈ (0, 1, . . . , m − 1)( gi 6= g j , 1 ≤ i ≤ m, 1 ≤ j ≤ m). Then, comparing Ki logically with the sequence of symbols s( g), we define p(g) as the probability that s(g) appears in n o K1 ; K2 ; . . . ; K N −( M−1)τ :

p( g) =

Card{i |1 ≤ i ≤ N − (m − 1)τ , Ki == s( g)} , m!

(8)

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p( g) 

Card{i 1  i  N  ( m  1) , Ki ==s( g)}

(8) , where Card is a function that represents the number m of! elements in a collection According to the definition of the Shannon entropy, the permutation entropy can be defined as: where Card is a function that represents the number of elements in a collection According to the definition of the Shannon entropy, the permutation entropy can be defined as: m! PE(m) = −m∑ p ( g ) ln p ( g ). (9) ! ( m) g = 1 PE =- p( g ) ln p( g ). (9) g 1

The whole -process -can be illustrated by the special example in Figure 2. In this graph, a The whole ‐process be as illustrated special example in the Figure 2. In thisdimension graph, a sequence of period-6 was-can chosen the objectbyofthe analysis below. We set embedding sequence period-6 was chosen the assumed object of analysis Weisset the embedding m m = 3 andofthe delay time τ = 2. Itaswas that the below. sequence sampled from thedimension first 13 data =points 3 andonto the delay time τ = 2. It was assumed that the sequence is sampled from the first 13 data points the curve shown in Figure 2. In this case, the state space is divided into 3! (=6) parts, onto the curve in Figure 2. In this case, thesymbols. state space is dividedto into (=6) parts, in so Equation there are so there are 6 shown mutually different arrangement According the3!definition 6(3), mutually different arrangement symbols. According to the definition in Equation (3), first the first sample point has a three-dimensional (3D) coordinate of (1.3660,–1.0000,−the 0.3660). sample point has a three-dimensional (3D) coordinate of (1.3660,–1.0000, −0.3660). Following Following Equation (5), we found that x(i + τ) ≤ x(i + 2τ) ≤ x(i), i = 1. Then, the corresponding Equation (5), we that−0.3660,1.3660) permutation x(i   )  x(i  2and )  x(the i)，i corresponding  1 . Then, the corresponding permutation is found (−1.0000, symbol sequence {120} is is (−1.0000,−0.3660,1.3660) and the corresponding symbol sequencewith {120}Equation is obtained referring to obtained by referring to Equation (6). Therefore, combined (3),bythe sequence Equation (6). Therefore, combined with Equation (3), −0.3660,3.0000,1.3660) the sequence (1.3660,−0.6340, −1.0000, −2.3660,−0.3660,3.0000,1.3660, −0.6340, −1.0000,−2.3660, (1.3660,−0.6340,−1.0000,−2.3660,−0.3660,3.0000,1.3660,−0.6340,−1.0000,−2.3660,−0.3660,3.0000,1.3660) can be converted into 9 symbol sequences for representation ({120}, {102}, {012}, {021}, {201}, {210}, can be{102}, converted {120}, {012}).into 9 symbol sequences for representation ({120}, {102}, {012}, {021}, {201}, {210}, {120}, {102}, {012}).

Figure 2. thethe permutation symbols from the the sample period-6 signal as delay time 2. Procedure Proceduretotoobtain obtain permutation symbols from sample period-6 signal as delay τtime = 2.τ = 2.

symbol sequences areare different fromfrom eacheach other.other. Starting from As analyzed analyzed above, above,the theprevious previoussixsix symbol sequences different Starting the seventh symbol sequence, the periodic extension is performed in the in which the the firstfirst six from the seventh symbol sequence, the periodic extension is performed in order the order in which sequences appeared. This means that all possible symbol sequences are included. The remaining other six sequences appeared. This means that all possible symbol sequences are included. The remaining parameters are unchanged, the analysis is performed again withagain m = 4,with and the calculation is other parameters are unchanged, the analysis is performed m =above 4, and the above repeated to is obtain 7 symbol sequences ({1230},{1032},{3012},{3021},{2031},{2130},{1230}). considered calculation repeated to obtain 7 symbol sequences ({1230},{1032},{3012},{3021},{2031}, We {2130},{1230}). k ij > k i( j+1) as xkij(i+kik( jij1)) = entropyentropy calculated by Equation (8) is ki ( j 1)permutation ) . The permutation We considered as x (xi(i+ kiji()j+1)x)(.i The calculated by Equation approximately 1.7918, provided the sampling sequence is long enough. The comparison shows that (8) is approximately 1.7918, provided the sampling sequence is long enough. The comparison shows the distribution of symbol sequences in m = 3 is consistent with m = 4. That means if the cycle count that the distribution of symbol sequences in m  3 is consistent with m  4 . That means if the cycle is less than m!, the m would not affect the amount of information acquired. So, the information that count is less than m!, the m would not affect the amount of information acquired. So, the information can be obtained is also uniform. that can be obtained is also uniform. The permutation entropy is a method proposed according to the spatial characteristics of The permutation entropy is a method proposed according to the spatial characteristics of time. time. We can obtain the new state space Y by reconstructing the phase space. After sorting each We can obtain the new state space Y by reconstructing the phase space. After sorting each mm-dimensional vector in Y, N − (m − 1)τ sorts symbol sequence K can be obtained. For the embedding dimensional vector in Y, N  (m  1) sorts symbol sequence K can be obtained. For the embedding

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dimension m, the total number of corresponding order permutations is m ! . To ensure the degree of dimension m, the total number of corresponding order permutations is m!. To ensure the degree of uniformity of quantization, N  (m  1)  m! is generally required, and N  m! is adopted in uniformity of quantization, N − (m − 1)τ >> m! is generally required, and N >> m! is adopted in practical application. Furthermore, if mifism small, the signal withwith a higher multiple periodic cannot be practical application. Furthermore, is small, the signal a higher multiple periodic cannot identified. Hence, the information volume read is correspondingly reduced, eventually resulting in be identified. Hence, the information volume read is correspondingly reduced, eventually resulting a loss of information. However, if the selection of m is oversized, it often leads to N  m! . As a in a loss of information. However, if the selection of m is oversized, it often leads to N < m!. As a consequence, thethe identification accuracy decreases and thethe error trend increases. consequence, identification accuracy decreases and error trend increases. TheThe noise-free chaotic sequence studied by Taken’s theorem of embedding indicates thatthat there noise-free chaotic sequence studied by Taken’s theorem of embedding indicates there is no specific limitation on the delay time τ. τ does not take the optimal delay time only affects the is no specific limitation on the delay time τ. τ does not take the optimal delay time only affects the Euclidean geometry of the reconstructed attractor when reconfiguring thethe phase space. So,So, it affects Euclidean geometry of the reconstructed attractor when reconfiguring phase space. it affects thethe calculation of the correlation dimension. However, it will not affect the reconstructive attractor's calculation of the correlation dimension. However, it will not affect the reconstructive attractor’s unambiguous representation the system systemdynamics. dynamics. In addition, a completely independent unambiguous representationof of the In addition, a completely independent situation situation of the two adjacent coordinate components could occur if the value of τ is too large, in which of the two adjacent coordinate components could occur if the value of τ is too large, in which case, case, projection thechaotic chaoticattractor’s attractor's trajectory would become irrelevant. thethe projection ofofthe trajectoryin inthe thesame samedirection direction would become irrelevant. Hence, smaller values of τ should be chosen according to the specific situation; Bandt and Pompe Hence, smaller values of τ should be chosen according to the specific situation; Bandt and Pompe suggested choosing a τavalue of 1of[20]. Therefore, thethe embedding dimension m is to be a a suggested choosing τ value 1 [20]. Therefore, embedding dimension mconsidered is considered to be key parameter of spatial reconstruction, and it is commonly recommended to select m within the key parameter of spatial reconstruction, and it is commonly recommended to select m within the range range for systems three dimensions. of 3of to37to for7 systems withwith lessless thanthan three dimensions. The traditional symbol sequence divides the time sequence into n regions by amplitude coarse The traditional symbol sequence divides the time sequence into n regions by amplitude coarse granulation to obtain the symbol sequence. After the window with length L is selected, the block granulation to obtain the symbol sequence. After the window with length L is selected, the block L entropy (BE) of of thethe symbol sequence isisobtained Forthe entropy (BE) symbol sequence obtainedand andits itsmaximum maximumvalue valueisisnot not more more than than lnln(2 (2 L )). .For thepermutation permutationentropy, entropy,after afterthe them-dimensional m-dimensionalphase phasespace spacereconstruction, reconstruction,the themaximum maximumvalue valueofofthe L L  3Compared thepermutation permutationentropy entropydoes doesnot notexceed exceedln(ln( . That is to say, as =mL>L 3. . Compared m!  m!m ).!)That is to say, m! >> 2 2as m with the traditional symbol sequence method, the permutation entropy can read more information and does with the traditional symbol sequence method, the permutation entropy can read more information need toneed perform amplitude layering.layering. The method combination of the permutation entropy and andnot does not to perform amplitude The of method of combination of the permutation MSTS and is elaborated Section 4. Combined MSTS, wewith compare permutation entropy and sample entropy MSTS isinelaborated in Section with 4. Combined MSTS, we compare permutation entropy in Section 5. Given the logical calculation used in the method, the permutation entropy entropy and sample entropy in Section 5. Given the logical calculation used in the method, thehas the advantages of high robustness andof fast calculation speed. permutation entropy has the advantages high robustness and fast calculation speed.

4. Combination of MSTS Permutation Entropy 4. Combination of MSTS andand Permutation Entropy In common DC-DC switching converters that include both inductor a capacitor, trend In common DC-DC switching converters that include both an an inductor andand a capacitor, thethe trend in the current variable or voltage variable reflect operating mode of the system. a switched in the current variable or voltage variable cancan reflect thethe operating mode of the system. ForFor a switched capacitor DC-DC converter [27], since it does not contain an inductor, the capacitor voltage capacitor DC-DC converter [27], since it does not contain an inductor, the capacitor voltage cancan be be used object extracting MSTS. Peak current single loop control is characterized used as as an an object forfor extracting thethe MSTS. Peak current single loop control is characterized by by thethe inability to stabilize the output voltage and produces a rich nonlinear behavior. We used the current inability to stabilize the output voltage and produces a rich nonlinear behavior. We used the current variable as the observation signal common converter (Figure procedure ofas Km as =m3,= 3, variable as the observation signal forfor thethe common converter (Figure 1). 1). TheThe procedure of K 1 can be observed in Figure τ =τ1=can be observed in Figure 3. 3. Embedded dimension m=3 (as τ=1)

MSTS

x1 x11 x21 x22 x23 x31 x32 x41 x42 x51 x52 x53 x61 x71 x72 x81 x82 x91 x.... {120} {201}

K

{012}

s

{102} {021}

{012}

+2

+1

{012}

{120}

+1

{021}{102}

......

......

{021}

+2

......

{201} {210}

+1

+2

{210} {201}

Figure 3. Generating thethe K from MSTS as m Figure 3. Generating K from MSTS as =m3,=τ3,= τ1.= 1.

Combined with the above analysis, the permutation entropy has the following characteristics when applied to a DC-DC converter:

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(1) Assuming is analysis, random, such as white noise, permutation is expected to be close Combined withthe thesignal above the permutation entropy has theentropy following characteristics to the value ln(m!) . Considering that there are at least two operation modes in when applied to maximum a DC-DC converter:

(1)

(2)

(3)

period-1, the recurring number of the MSTS is greater than or equal to 2. Hence, the minimum Assuming the signal is random, such as white noise, permutation entropy is expected to be value of PE is ln(2)  0.6931 . close to the maximum value ln(m!). Considering that there are at least two operation modes in (2) Whenthe therecurring system number does notofenter the chaotic state byorchanging m,the acquired permutation period-1, the MSTS is greater than equal to 2. Hence, the minimum entropy value of PE isremains ln(2) =uniform. 0.6931. This means that increasing or decreasing m as appropriate does not When thethe system not enter the chaotic affect valuedoes of the permutation entropy.state by changing m, the acquired permutation entropy remains uniform. This means that increasing orworking decreasing m as appropriate not (3) DCM and CCM can be further divided into different patterns according todoes the different affect the value of the permutation entropy. switching numbers of modes. Border collision occurs when the working pattern combination DCM and CCM can be further divided into different working patterns according to the different changes. The complexity the systems in DCM working patterns is usually greater than that of switching numbers of modes.ofBorder collision occurs when the working pattern combination the CCM working pattern. Therefore, the more DCMpatterns patterns system contains, the higher changes. The complexity of the systems in DCM working is the usually greater than that of the CCM working pattern. Therefore, the more DCM patterns the system contains, the higher the value of the permutation entropy. After entering chaos, the permutation sequence K shows the value of the permutation entropy. After entering the permutation K shows a certain disorder. Because the chaotic motion chaos, is pseudo-random, thesequence permutation entropy a certain disorder. Because the chaotic motion is pseudo-random, the permutation entropy obtained in the chaotic state is not able to reach the maximum value. obtained in the chaotic state is not able to reach the maximum value.

5. Application Example 5. Application Example In section, this section, the peak current control DC-DC converter is analyzed the above In this the peak current control buckbuck DC-DC converter is analyzed usingusing the above method. By quantifying its complexity, the behaviors of period-doubling the period-doubling bifurcation, border method. By quantifying its complexity, the behaviors of the bifurcation, border collision bifurcation, and chaos present in the converter are quantitatively described. The circuit collision bifurcation, and chaos present in the converter are quantitatively described. The circuit model model shown4.inRelevant Figure 4.parameters Relevant parameters were 20 Ω, V, L R == 3.3 19 Ω, L = 3.3 is shown inisFigure were selected asselected follows:asE follows: = 20 V, RE== 19 mH, mH, C = 1000 uF, T = 400 us, and the variation range of the control variable I ref was controlled within C = 1000 uF, T = 400 us, and the variation range of the control variable Iref was controlled within 0.12–1.32. As shown in Figure 4, there are switching two switching devices incircuit: the circuit: the fully controlled 0.12–1.32. As shown in Figure 4, there are two devices in the the fully controlled device V and the uncontrollable diode Combined the principle ofbuck the buck converter, device V and the uncontrollable diode VD. VD. Combined withwith the principle of the converter, we we obtained three working modes of the circuit, as shown in Figures 5 and 6. These figures depict obtained three working modes of the circuit, as shown in Figures 5 and 6. These figures depict the switchingmode mode(V,VD) (V,VD)==(on,off (on,off)→(V,VD) (off,on)→(V,VD) the switching )→(V,VD) = (off,on) →(V,VD)==(off,off) (off,off )of ofeach each switching switching period. period. The The inductor 0 during mode-1 andand mode-2 andand xL =x0L during mode-3. In Figure 6, Ib16,and inductorcurrent currentisisxLxL>＞ 0 during mode-1 mode-2 = 0 during mode-3. In Figure Ib1 and Ib2 are the borderlines in the buck converter with peak current control. Analyzing the characteristics of Ib2 are the borderlines in the buck converter with peak current control. Analyzing the characteristics the buck converter and using the precise discretization method to model the system, we determined the of the buck converter and using the precise discretization method to model the system, we workdetermined pattern of each control cycle. of After inductor current xthe numerically L was the work pattern eachmultiple control iterations, cycle. Afterthe multiple iterations, inductor current xL sampled under the overall steady state, and MSTS was obtained. Here, m = 4 and τ = 1 were was numerically sampled under the overall steady state, and MSTS was obtained. Here,selected. m = 4 and τ Figure shows the permutation entropy forpermutation the MSTS asentropy a function theMSTS reference current for = 17were selected. Figure 7 shows the forof the as a function of different the reference nonlinear feature. For the data of the chaotic state part, since the irregularity was relatively strong, was current for different nonlinear feature. For the data of the chaotic state part, since the irregularity cyclerelatively extensionstrong, was performed to homogenize the order. cycle extension was performed to homogenize the order. xC

L Voltage Probe

E

V

xL

VD

C

R

MSTS

Comparator + Q R Q S

Iref

-

Figure 4. Buck converter current control. Figure 4. Buck converter with with peak peak current control.

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R R

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EE

Model-2 Model-2

V V

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R R

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Figure5. 5.Three Threemodes modesof ofaaabuck buckconverter. converter. Figure 5. Three modes of buck converter. Figure

TT

ref IIref

IITT

TT

/Iref ref II11/I

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/Iref ref II11/I II00

Ib22 Ib IITT II00

Mode-2 Mode-2

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Ib11 Ib

II00 00

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tt11 (a) FF11 (a)

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Mode-2 Mode-2

/0 IITT/I/I22/0

00 tt11

tt22

tt11

(b) FF22 (b)

tt33

tt22 (c) FF33 (c)

Figure (b) F2FF2means mode-1 Figure6. 6.(a) (a)FFF111 working workingpattern patternmeans meansthat thatonly onlymode-1 mode-1exists existsinin inone oneperiod; period; (b) 2 means meansthat that modeFigure 6. (a) working pattern means that only mode-1 exists one period; (b) that modeis performed first in the period, then the switch state changes once and enters into mode-2; and (c) F(c) is performed performed first first in in the the period, period, then then the the switch switch state state changes changes once once and and enters enters into into mode-2; mode-2; and and (c) 3 11 is represents thethe mode in the period change from mode-1 to mode-2 to mode-3. represents the mode in the the period change from mode-1 to mode-2 mode-2 to mode-3. mode-3. FF33 represents mode in period change from mode-1 to to

As Asshown shownin inFigure Figure8, 8,the theinductor inductorcurrent currentxxxLLLis issampled sampledat atthe theoperating operatingmode modeinterval intervaland andthe the As shown in Figure 8, the inductor current is sampled at the operating mode interval and the bifurcation diagram of the system with I as a changing parameter can be obtained. ref bifurcation diagram diagram of of the the system system with with IIref ref as as aa changing changing parameter parameter can can be be obtained. obtained. bifurcation By Bycomparing comparingFigures Figures777and and8, 8,the thedynamic dynamicbehavior behaviorof ofthe thebuck buckconverter converterin inthe thepeak peakcurrent current By comparing Figures and 8, the dynamic behavior of the buck converter in the peak current control can be clearly analyzed. Similarly, each state of the border collision bifurcation and the standard control can be clearly analyzed. Similarly, each state of the border collision bifurcation and the control can be clearly analyzed. Similarly, each state of the border collision bifurcation and the bifurcation are represented in Table 1. In this table, S and MSTS-PE indicate the permutation symbol of standard bifurcation bifurcation are are represented represented in in Table Table 1. 1. In In this this table, table, SS and and MSTS-PE MSTS-PE indicate indicate the the permutation permutation standard the MSTS and the corresponding permutation entropy, respectively. symbol of of the the MSTS MSTS and and the the corresponding corresponding permutation permutation entropy, entropy, respectively. respectively. symbol Table Table1.1. 1.Permutation Permutationsequence sequenceand andpermutation permutationentropy entropyof ofthe thebuck buckconverter converteras asreference referencecurrent current Table Permutation sequence and permutation entropy of the buck converter as reference current IIref changes. ref changes. changes. Iref Iref /A ref/A /A IIref [0.1200,0.2778] [0.1200,0.2778] [0.1200,0.2778] [0.2779,0.8295] [0.8296,0.9006] [0.2779,0.8295] [0.9007,1.1577] [0.2779,0.8295] [1.1578,1.1946] [0.8296,0.9006] [1.1947,1.2624] [0.8296,0.9006] [1.2625,1.3200] [0.9007,1.1577] [0.9007,1.1577]

S MSTS-PE MSTS-PE MSTS-PE SS ({1203},{0231},{0231})∞ 1.0986 ({1203},{0231},{0231})∞ 1.0986 ({1203},{0231},{0231})∞ 1.0986 ({1203},{0213})∞ 0.6931 ({3102},{2013},{1302},{0213})∞ 1.3863 ({1203},{0213})∞ 0.6931 ({3102},{2301},{1203},{0132},{0213})∞ 1.6094 ({1203},{0213})∞ 0.6931 ({2301},{1230},{0123},{0312})∞ 1.3863 ({3102},{2013},{1302},{0213})∞ 1.3863 ({2031},{1203},{3012},{2301},{1230},{0123},{0312})∞ 1.9459 ({3102},{2013},{1302},{0213})∞ 1.3863 No period: ({ . . . }, . . . ) >2.2 ({3102},{2301},{1203},{0132},{0213})∞ 1.6094 ({3102},{2301},{1203},{0132},{0213})∞ 1.6094

State State State Period-1 Period-1 Period-1 Period-1 Period-2 Period-1 Period-2 Period-1 Period-2 Period-2 Period-4 Period-2 Chaos Period-2 Period-2

[1.1578,1.1946] [1.1578,1.1946]

({2301},{1230},{0123},{0312})∞ ({2301},{1230},{0123},{0312})∞

1.3863 1.3863

Period-2 Period-2

[1.1947,1.2624] [1.1947,1.2624]

({2031},{1203},{3012},{2301},{1230},{0123},{0312})∞ ({2031},{1203},{3012},{2301},{1230},{0123},{0312})∞

1.9459 1.9459

Period-4 Period-4

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Figure Permutation entropy of buck converter with ref as the changing parameter. Figure 7.7.7. Permutation entropy ofof buck converter with Iref asas the changing parameter. Figure Permutation entropy buck converter with IIref the changing parameter.

Figure Bifurcation diagram converter with IIref Figure Bifurcation diagram of abuck buck converter with ref as the changing parameter. parameter. Figure 8.8.8. Bifurcation diagram ofof a abuck converter with Iref as the changing parameter.

∈ [0.1200,0.2778], 1. As IIref the system works in the F 3 pattern of period-1. In this pattern, there are [0.1200,0.2778], the system works the pattern period-1. this pattern, there are 1.1. AsAs Iref ref ∈∈[0.1200,0.2778], the system works inin the F3F3pattern ofof period-1. InIn this pattern, there are three modes in each period of the system, which means the system is in the DCM state. Then, the threemodes modesinineach eachperiod periodofofthe thesystem, system,which whichmeans meansthe thesystem systemis isininthe theDCM DCMstate. state.Then, Then, three first border collision bifurcation occurs in the system at at IrefIref= =0.2779 A. Because the inductor the first border collision bifurcation occurs in the system 0.2779 A. Because the inductor the first border collision bifurcation occurs in the system at Iref = 0.2779 A. Because the inductor current the state collides withwith Ib , the2,operating mode changes and the and working patterns currentinin inthe thesteady steady state collides theoperating operating modechanges changes theworking working current steady state collides with2 IbIb 2, the mode and the of the converter system change from F to F . 3from 2F3Fto patterns the convertersystem system change from 3 to patterns ofof the converter change F2F. 2. 2. During I ∈ [0.2779,0.8295], the system still works the period-1 state, but the operating mode 2. ref ∈[0.2779,0.8295], the system still works in the period-1 state, but the operating mode ref 2. During Iref ∈[0.2779,0.8295], the system still works inin the period-1 state, but the operating mode is F . At this time, the system has two modes in each period. When I = 0.8296, the system systemhas hastwo twomodes modesinineach eachperiod. period.When WhenIrefref Iref= =0.8296, 0.8296,the thesystem system is F2. 22AtAtthis time, the system generates period doubling bifurcation and enters period-2. there is no border collision period-2. Since no border collision generates period doubling bifurcation and enters period-2. Since there is no border collision bifurcation,the theworking workingpatterns patternsare areextended extendedtoto toF2FFF222F. 22.. bifurcation, When thesystem systemworks worksininperiod-2. period-2. We found that two bounder collisions ∈[0.8296,1.1946],the 3. When ∈[0.8296,1.1946], period-2. We found that two bounder collisions ref 3.3. IrefIref ∈[0.8296,1.1946], We found that two bounder collisions of the system occur in this area because the inductance current of the system collides with the system occur in this area because the inductance current of the system collides with the of the system occur in this area because the inductance current of the system collides with the the borderline of Ib and Ib at I = 0.9007 and I = 1.1578 A, respectively. Hence, the borderline of Ib 2 and Ib 1 at I ref = 0.9007 and I ref = 1.1578 A, respectively. Hence, the operating ref and Iref = 1.1578 ref A, respectively. Hence, the operating borderline of Ib2 and 2Ib1 at Iref 1= 0.9007 operating pattern of the converter system changes from F[0.8296,0.9006] at Iref ∈[0.8296,0.9006] pattern of the converter system changes from F 2 F 2 2∈ 2 [0.8296,0.9006] 2 F∈ 3 pattern of the converter system changes from F2F2 atatIrefIrefF∈ totoF2FF23F3atatto IrefIFref∈ at I [0.9007,1.1577] and to F F at I [1.1578,1.1946]. From the entropy value of three ∈ ∈ [0.9007,1.1577] and to F 1 F 3 at I ref ∈[1.1578,1.1946]. From the entropy value of three period-2 orbits 1 3 ref ref [0.9007,1.1577] and to F1F3 at Iref ∈[1.1578,1.1946]. From the entropy value of three period-2 orbits period-2 orbits shown in Figure 7, entropy the permutation entropy value of period-2 is greater than the shown Figure 7, the permutation entropy value period-2 greater than the corresponding shown inin Figure 7, the permutation value ofof period-2 is is greater than the corresponding corresponding entropy valuepattern of anyin working pattern inperiod-1. the entirety of period-1. The sample entropyvalue valueofof anyworking working pattern inthe theentirety entirety Thesample sample entropy, used for entropy any ofofperiod-1. The entropy, used for entropy, used for assessing the complexity of the time sequence shown in Figure 10, is the assessing the complexity of the time sequence shown in Figure 9, is the negative logarithm assessing the complexity of the time sequence shown in Figure 9, is the negative logarithm ofof negative logarithm oftwo the probability of if twodata sets ofpoints simultaneous data points of lengthsmaller m have theprobability probability sets simultaneous datapoints length have distance smaller the ofofif iftwo sets ofofsimultaneous ofoflength mmhave a adistance a distance smaller than r, then two sets of simultaneous data points of length m + 1 also have a

border collisions occur in the chaotic state, as depicted in Figure 7. Then, the value of permutation entropy is no longer stable, exhibiting similar characteristics to the LLE, and its corresponding value is greater than any period. The characteristics described are consistent with Bandt C et al. [20]. Energies 2018,Moreover, 11, 2747 10 of 15 the fourth border collision bifurcation occurs at Iref = 1.1947 A. Since the system has

unstable characteristic roots at the same time, the system generates period-doubling bifurcation and enters the period-4 state. The working pattern is extended while F2 switches to F3.

4.

distance smaller than r [28]. However, with the change in sample entropy, we could not find the bifurcation information of the system under period-1 and period-2 states. As Iref ∈[1.1947,1.2624], the system works in the period-4 state with the working pattern F1 F2 F1 F3 , and there are seven modes in this state. Then, the fifth time border collision bifurcation of a fixed point and Ib1 curve collision occur once more in the system at Iref = 1.2625 A. At this time, LLE shown in Figure 9 is greater than 0 and the mode switching shows disorder, which means that the system is beginning enter the chaotic state. When Iref ≥ 1.2625 A, a number of border collisions occur in the chaotic state, as depicted in Figure 7. Then, the value of permutation entropy is no longer stable, exhibiting similar characteristics to the LLE, and its corresponding value is greater than any period. The characteristics described are consistent with Bandt C et al. [20].

Moreover, the fourth border collision bifurcation occurs at Iref = 1.1947 A. Since the system has unstable characteristic roots at the same time, the system generates period-doubling bifurcation and Energies 2018, 11, x FORthe PEER REVIEW 10 of 15 enters period-4 state. The working pattern is extended while F2 switches to F3 . Figure 9. Sample Entropy of buck converter with Iref as the change parameter

4.

than r, then two sets of simultaneous data points of length m + 1 also have a distance smaller than r [28]. However, with the change in sample entropy, we could not find the bifurcation information of the system under period-1 and period-2 states. As Iref ∈ [1.1947,1.2624], the system works in the period-4 state with the working pattern F1F2F1F3, and there are seven modes in this state. Then, the fifth time border collision bifurcation of a fixed point and Ib1 curve collision occur once more in the system at Iref = 1.2625 A. At this time, LLE shown in Figure 10 is greater than 0 and the mode switching shows disorder, which means that the system is beginning enter the chaotic state. When Iref ≥ 1.2625 A, a number of border collisions occur in the chaotic state, as depicted in Figure 7. Then, the value of permutation entropy is no longer stable, exhibiting similar characteristics to the LLE, and its corresponding value is greater than any period. The characteristics described are consistent with Bandt C et al. [20].

Moreover, the fourth border collision bifurcation occurs at Iref = 1.1947 A. Since the system has unstable characteristic roots at the same time, the system generates period-doubling bifurcation and enters the period-4 state. pattern is (LLE) extended F2 switches to Fthe 3. changing parameter. Figure 9. The Largest Lyapunov exponent of of a buck converter with asthe refas Figure 10. working Largest Lyapunov exponent (LLE) a while buck converter with IIref changing parameter.

Figure 9. Sample Entropy of buck converter with Iref as the change parameter Figure 10. Sample Entropy of buck converter with Iref as the change parameter

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xL/A

xL/A

To further explain permutation entropy, the time-domain simulating waveforms with phase To further explain permutation entropy, the time-domain simulating waveforms with phase plane, and the experimental results at the typical values of Iref, are illustrated in Figures 11 and 12, plane, and the experimental results at the typical values of Iref, are illustrated in Figures 11 and 12, respectively. We found that the values of the permutation entropy correspond to the complexity of the respectively. We found that the values of the permutation entropy correspond to the complexity of working waveforms. In summary, the presented MSTS-PE can quantify the complexity of the operation the working waveforms. In summary, the presented MSTS-PE can quantify the complexity of the states of the buck converter, and the established MSTS can characterize the bifurcation phenomena of operation states of the buck converter, and the established MSTS can characterize the bifurcation the system. phenomena of the system.

t/s

xC/A

xC/A

t/s

xL/A (b)

xL/A

xL/A

xL/A (a)

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xL/A (c) Figure 11. Cont.

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xL/A (f)

xC/A

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xL/A (g) Figure 11. phase plane at (a) Iref = 0.20 A; (b) Iref = 0.50 A; (c) Iref Figure 11. Waveforms Waveformsofofinductor inductorcurrent currentxLxand L and phase plane at (a) Iref = 0.20 A; (b) Iref = 0.50 A; = 0.85 A; (d) I ref = 1.00 A; (e) I ref = 1.17 A; (f) I ref = 1.20Iref A;=and = 1.27 In (a) to (g), xC and xL (c) Iref = 0.85 A; (d) Iref = 1.00 A; (e) Iref = 1.17 A; (f) 1.20(g) A;Iref and (g) IA. ref = 1.27 A. In (a) to (g), represent the inductor current and capacitor voltage, respectively, and the red marker dot x C and xL represent the inductor current and capacitor voltage, respectively, and the red marker dot corresponds the the mode mode switching switching moment. moment. corresponds

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xL

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(a)

(b)

(c)

xL

xL

xL

clock

clock

clock

(d)

(e)

(f)

xL

clock (g)

Figure 12. 12.Experimental Experimentalwaveforms waveformsofofinductor inductor current switching signal at (a) = 0.20 A; Figure current xLxand switching signal at (a) Iref =Iref 0.20 A; (b) L and = A; 0.50 0.85 Iref = (e)1.17 Iref A; = 1.17 A;=(f) IrefA;= and 1.20 (g) A; Iand I(b) ref =I ref 0.50 (c)A;Iref(c)= I0.85 (d)A; Iref(d) = 1.00 A;1.00 (e) IA; ref = (f) Iref 1.20 ref = (g) 1.27Iref A.= 1.27 A. ref =A;

6. Discussion Discussionand andConclusions Conclusions 6. Phasespace spacereconstruction reconstruction is is aamethod methodof ofreconstructing reconstructingattractors attractorsbased basedon onfinite finitedata datato tostudy study Phase the dynamic behavior of a system. Its basic idea is that the evolution of any component in the system the dynamic behavior of a system. Its basic idea is that the evolution of any component in the systemis determined by its with other In DC-DC the dynamic information is determined by interactions its interactions with components. other components. In converters, DC-DC converters, the dynamic of one component is implicit in any related component; that is, the original dynamic system can information of one component is implicit in any related component; that is, the originalmodel dynamic be reconstructed one variable observation in the observation system. system model canwith be reconstructed with one variable in the system. The quantization process of the sequence information basedon onthe thepermutation permutationentropy, entropy, and and The quantization process of the sequence information based comparison between permutation entropy and the traditional symbol sequence method, are described comparison between permutation entropy and the traditional symbol sequence method, are in Sectionin 3. Section In addition, permutation entropy and sample entropy ofentropy buck converter shown described 3. In the addition, the permutation entropy and sample of buck are converter in Figures 7 and 10. The permutation entropy seems to be a better indicator in some circumstances are shown in Figures 7 and 9. The permutation entropy seems to be a better indicator in some because it has because high robust performance anti-interference ability for analysis theanalysis mixed signal. circumstances it has high robust and performance and anti-interference abilityoffor of the Permutation overcomes shortcomings, like the poor relative of sample entropy mixed signal.entropy Permutation entropy overcomes shortcomings, like theconsistency poor relative consistency of and time consuming calculation of multi-scale entropy, which is easily affected byisthe non-stationarity sample entropy and time consuming calculation of multi-scale entropy, which easily affected by andnon-stationarity outliers of time and sequences. better description dynamicof behaviors of dynamic a system the outliersFor of atime sequences. Forofa nonlinear better description nonlinear from a time sequence, MSTS was proposed to describe the dynamic features of a DC-DC converter. behaviors of a system from a time sequence, MSTS was proposed to describe the dynamic features of addition, distinguished from joint entropy, permutation entropy combined with MSTS only needs aInDC-DC converter. In addition, distinguished from joint entropy, permutation entropy combined one sequence to identify thesequence system features. Thisthe concept is features. also consistent with theisconcept of phase with MSTS only needs one to identify system This concept also consistent space reconstruction. with the concept of phase space reconstruction. Based on on the the theory theory of of phase phase space space reconstruction, reconstruction, the the permutation permutation entropy entropy of of the the DC-DC DC-DC Based converter under the determined parameters is obtained by sorting each group of m-dimensional converter under the determined parameters is obtained by sorting each group of m-dimensional vectors. In Inthis thispaper, paper, we we found found that that the the higher higher the the frequency frequency of of mode mode switching, switching,the themore more complex complex vectors. theobtained obtainedMSTS. MSTS.This Thismeans meansthat thatthe thehigher higherthe thecomplexity complexityof ofthe thecorresponding correspondingsystem, system,the thelarger larger the the permutation entropy value. With the fluctuation in the permutation entropy value of different the permutation entropy value. With the fluctuation in the permutation entropy value of different control parameters, parameters, the the nonlinear nonlinear dynamic dynamic behaviors behaviors of of aa DC-DC DC-DC converter converter can can be be better better quantified. quantified. control

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Author Contributions: Z.L. developed the methodology, carried out the research, analyzed the numerical, and wrote the manuscript; F.X. provided the idea, performed the experiment and revised the article. B.Z. and D.Q. guided and revised the paper. Funding: This project was supported by the National Natural Science Foundation of China (Grant No. 51507068) and the Team Program of Natural Science Foundation of Guangdong Province, China (Grant No. 2017B030312001). Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

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