Piecewise Smooth Dynamical Systems Modeling Based on Putzer and Fibonacci-Horner Theorems: DC-DC Converters Case

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1 International Journal of Automation and Computing 13(3), June 2016, DOI: /s Piecewise Smooth Dynamical Systems Modeling Based on Putzer and Fibonacci-Horner Theorems: DC-DC Converters Case Abdelkader Khoudiri 1 Kamel Guesmi 1,2 Djillali Mahi 2 1 Department of Sciences and Techniques, Djelfa University, Djelfa 17000, Algeria 2 CReSTIC, IUT de Troyes, 09 rue de Quebec, Troyes, France 3 LEDMaSD, Laghouat University, Laghouat 03000, Algeria Abstract: The paper deals with the problem of switched dynamical systems modeling especially in DC-DC converters case study consideration. It presents two approaches to describe accurately the behavior of this class of systems. To clarify the paper s contribution, the proposed approaches are validated through simulations and experimental results. A comparative study, between the obtained results and those of other techniques from the literature, is given to evaluate the performances of the studied approaches. Keywords: Switched systems, DC-DC converters, modeling, Cayley-Hamilton theorem, Putzer s theorem, Fibonacci-Horner decomposition theorem. 1 Introduction Switched dynamical systems exhibit a wealth of nonlinear phenomena due to their components that are considered as the principal sources of nonlinearities in this class of systems and also due to the control methods. In recent years, many studies focused on the modeling of these systems to describe accurately their nonlinear behaviors with a minimum of simplifying assumptions and validity conditions. In this area, one research direction is to develop models that take into account as much as possible real system considerations [1 3]. Studies showed that small signals model and averaged one are unable to describe fast scale nonlinearities of this class of systems and fail to explain their abnormal behaviors [1 4]. Besides, it is proven that the detailed model, based on solving analytically the differential equations of the system, is very consuming in terms of processing time and memory space [5]. Moreover, to keep the solution compact, condition like the invertibility of the state matrices is required, which is not verified in all cases. To deal with this problem, the discrete mapping method can be used. It allows the system normal and abnormal behaviors in both slow and fast scales. Other research direction is based on refining existing models in order to overcome limitations related to the system structure, computing time and accuracy. Power electronics devices, as an important class of dynamical systems, have gained more attention due to the fact they are simple circuits and can exhibit nearly all the non- Research Article Manuscript received November 16, 2013; accepted May 27, 2014 Recommended by Associate Editor Zi-Qiang Zhu c Institute of Automation, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016 linear phenomena observed in dynamical systems. This fact justifies research activities on modeling these devices with enhanced algorithms, more accuracy and reduced computing time and memory space especially with developments in calculators and power electronics components. In the case of DC-DC converters, the current controlled boost converter and the voltage controlled buck converter are the most studied; indeed, the first one is wealthy of nonlinear phenomena whereas the second one gives an explicit relation between the successive samples of the system state [6 9]. Nevertheless, the model is obtained under simplifying assumptions and is usually valid for a specific functioning mode (continuous conduction mode CCM or discontinuous conduction mode DCM). Furthermore, in techniques such as detailed model and discrete one, the Taylor series expansion method is, generally, used which leads to approximation errors that cannot be neglected in all cases. Hence, these approaches can ensure only a qualitative description of the converter behaviors [2, 3, 10, 11]. Authors in [12 15], proposed a refined models with minimum of approximations to eliminate the aforementioned problems. They are based on mathematical approaches like Cayley-Hamilton theorem [14] and spectral decomposition theorem [15]. In fact, the proposed modeling technique based on Cayley-Hamilton theorem is general, but it has relatively complicated flowcharts which make it consuming in terms of memory space and computing time. To deal with this problem, the method given in [15] is based on spectral decomposition theorem, it reduces both memory space and computing time; however, it suffers from the limitation related to the system matrix diagonalization. In this context, two discrete modeling techniques are presented in the paper, these techniques allow obtaining an accurate de-

2 A. Khoudiri et al. / Piecewise Smooth Dynamical Systems Modeling Based on 247 scription of the converters normal and abnormal behaviors with better calculation performances and validity regardless of the functioning or control modes. The evaluation of the proposed approaches is carried out by a simulation on boost and buck converters, followed by a comparative study between the proposed approaches and those given in [14, 15] in terms of accuracy, computing time and allocated memory space for data processing and storage. 2 Proposed modeling methods Let us consider a switched dynamical system that toggles between m possible configurations. In each configuration, the system can be described by a system of m linear first order differential equations of the form: Ẋ(t) =A ix(t)+b iu, i=1, 2,,m (1) where X(t) and U are the state and input vectors respectively. A i and B i are the system matrices in the i-th configuration. The analytical exact solution of (1) is X i(t) =e (t t i0)a i (X i(t 0i)+A 1 i B iu) A 1 i B iu (2) with t i0 as the initial time in the configuration (i). This analytical model gives a detailed exact description of the system behavior within the configuration, and is obtained by considering the final state value of each configuration as an initial value for the next one and so on until building the system response. Nevertheless, this exact analytical solution needs an important computing time and a large memory space due to the need of computing the state matrices inverses and exponentials many times in each configuration of the system. Furthermore, it needs also that all the state matrices to be nonsingular which cannot be ensured in all cases. A possible solution to this problem is to add a small perturbation term η j to each singular state matrix A j, however, this gives approximated values of the transition matrices but it introduce additional small time-constants which can cause inherent numerical problems that cannot easily be sidestepped. A second solution is to approximate the transition matrix by the first N terms of its series expansion, also, numerical approximation can be used to calculate the inverse of each singular matrix A j. Indeed, this can simplify the model of the system, however, a trade-off between accuracy and simplicity must be made in this case. To bypass this trade-off, the discrete modeling technique is used; it is based on the transition matrix that has the following property: t 0 Φ i(τ)dτ = t 0 e τa i dτ = [ ] e ta i I A 1 i. (3) Let us consider that, during a switching period T, the system dwells in m configurations, with: t n0 as the beginning functioning time of the system, and t i is the i-th configuration duration. Then, T = m t i. i=1 And the discrete model expression is x(nt )= m Φ i(t i) i=2 m Φ i(t i) i=3 m Φ i(t i) x((n 1)T )+ i=1 tn0 +t 1 t n0 tn0 +t 1 +t 2 t n0 +t 1 τ) B 2 U dτ + +Φ m(t m) Φ 1((t n0 + t 1) τ) B 1 U dτ+ Φ 2((t n0 + t 1 + t 2) i=m 1 ((t n0 + t i) τ) B 2 U dτ+ i=1 tn0 +T i=m 1 t n0 + i=1 i=m 2 tn0 + t i i=1 Φ i=m 1 m 1 t n0 + t i i=1 t i Φ m((t n0 + T ) τ) B m U dτ. (4) Accuracy of this model depends on the calculation method of the two terms: the transition matrix and its integral term. Authors in [14] proposed a method based on Cayley-Hamilton theorem to calculate exact values of the transition matrix and its integral; they obtained an accurate discrete model without restrictions on state matrices singularity [7,9,10,14] ; however, the given flowcharts are relatively complicated. Based on spectral decomposition theorem, authors in [15] obtained the same accuracy degree with less complicated flowcharts which reduced considerably the computing time and the needed memory space for data processing and storage; nevertheless, this approach requires that the state matrices have to be diagonalizable. In order to overcome the abovementioned problems in [14 16], two discrete modeling methods based on Putzer and Fibonacci-Horner decomposition theorems will be presented in the next sub-sections, these approaches present straightforward methods that can be used to calculate the exact values of transition matrix and its integral term with minimum of assumptions and less computing time and memory space. 2.1 Putzer s theorem based modeling Let us consider a switched system with state matrices A i(n xn)withs distinct eigenvalues λ ie,e {1,,S}. The calculation of each transition matrix by applying Putzer sformula [17] is given, in general, by the following steps: First, a sequence of polynomials in A i is considered: P 0(A i)=i n P k (A k i)= (A i λ il I n) for k =1, 2,,n. (5) l=1

3 248 International Journal of Automation and Computing 13(3), June 2016 Then, the transition matrix is given by n 1 Φ i (t i)= r k+1 (t i)p k (A i) (6) k=0 where the functions r 1(t i),,r n(t i) are defined by the differential system: ṙ 1(t i)=λ i1r 1(t i), r 1(0) = 1 ṙ k+1 (t i)=λ m(k+1) r k+1 (t i)+ r k (t i) (7) r k+1 (0) = 0, for k =1, 2,,n 1. Flowcharts for computing the transition matrices and its integral terms in the case of (2 2) state matrices A i are givenbyfigs.1and2respectively. 2.2 Fibonacci-Horner decomposition based modeling In order to calculate the state transition matrix and its integral using the Fibonacci-Horner decomposition theorem [18],let s consider, the general case of switched Fig. 1 Transition matrix computation using Putzer theorem Fig. 2 Integral term computation using Putzer theorem

4 A. Khoudiri et al. / Piecewise Smooth Dynamical Systems Modeling Based on 249 system with (n n)a i state matrices that has S distinct eigenvalues λ il,l {1,,S}, with multiplicities L 1,L 2,,L S and P (z) its characteristic polynomial: P (z) =z n a 0z n 1 a n 1 = S (z λ il ) L l. l=1 The transition matrix can be expressed as Φ i (t i)=φ (n 1) (t i)a i0 + φ (n 2) (t i)a i1 + + φ(t i)a i(n 1) (8) Using (10) and (11) respectively, the following expression can be obtained R 1(t i)=γ [1] 0 + γ[1] 1 ti, R2(ti) =γ[2] 0 γ [1] 1 0 = (λ i1 λ = 1 i2) 2 4, 1 γ[1] 1 = (λ i1 λ = 1 i2) 2 and γ [2] 1 0 = (λ i1 λ = 1 i2) 2 4. Hence, with A i0 = I n A ij = A j i a0aj 1 i a j 1I n for j =1, 2,,n 1 φ(t i)=( ti)e3t i et i φ(t i)=( ti)e3t i et i φ(t i)=( ti)e3t i et i. and φ(t i)= S R i(t i)e t iλ il (9) l=1 is the solution of the associated ordinary differential equations to A i. φ (k) (t i)isthek-th time derivative of φ(t i) that satisfies φ (k) (0) = 0 for k =0, 1,,n 2andφ (n 1) (0) = 1. The associated polynomials R i(t i) are expressed as R i(t i)= L p 1 k=0 γ [p] k k! With the coefficients γ [p] k given by γ [p] k =( 1)n Lp t k i, 1 p S. (10) nj =L p k 1 1 j p S (n j + L j 1)! 1. (11) n j!(l j 1)! (λ ij λ ip) n j +L j Example 1. The following state matrix is considered: A i = And Φ i(t i) becomes ( Φ i(t i)= ( ) 2 ti)e3t i et i A 2 i + ( ( 3 2 2ti)e3t i 3 ) 2 et i A i+ ( ( ti)e3t i + 9 ) 4 et i I. For example, if t i =1,then Φ i(t i)= In the case of (2 2)A i state matrices, the flowcharts of computing the transition matrix and its integral term are givenbyfigs.3and4respectively: This matrix has two distinct eignvalues λ i1 =3, λ i2 =1 and multiplicities L 1 =2,L 2 =1respectively. The characteristic polynomial is P (z) =z 3 7z 2 +15z 9. According to (8) and (9), Φ i (t i)= φ(t i)a i0 + φ(t i)a i1 + φ(t i)a i2 with { A i0 = I 3,A i1 = A i 7I 3,A i2 = A 2 i 7A i +15I 3 φ(t i)=r 1(t i)e t iλ i1 + R 2(t i)e t iλ i2 where R 1(t i) is a polynomial of degree 1 and R 2(t i)isa constant polynomial. Fig. 3 Transition matrix computation using Fibonacci-Horner decomposition theorem

5 250 International Journal of Automation and Computing 13(3), June 2016 differential equation of the form Ẋ(t) =A ix(t)+b iu, i=1, 2, 3. (12) Where X(t) =[v C i L] T is the state vector, A i and B i are the system matrices in the i-th configuration. For buck converter: 1 A 1 = C(R + r C) R L(R + r C) R C(R + r C) 1 (rl + rsw + RrC L R + r C ). For boost converter: 1 0 A 1 = C(R + r C) (rc + rsw) 0 L Fig. 4 Integral term computation using Fibonacci-Horner decomposition theorem 3 Case study presentation As cases study, DC-DC buck and boost power converters givenbyfigs.5and6areconsidered. r L, r SW, r VD and r C are the internal series resistances (ESRs) of inductor, switch sw, diode and capacitor, respectively. U and i L are, respectively, the supply voltage and the inductance current. R and u 0 designate the load and the output voltage respectively. The clock period T is imposed by an external pulse width modulation (PWM) block. According to the control mode, the reference to be attained may be voltage (V ref ) or current (i ref ). And for both converters: 1 A 2 = C(R + r C) R L(R + r C) A 3 = 1 0 C(R + r C) 0 0 B 1 = B 2 = 0 [ 1, B 3 = L R C(R + r C) 1 (rl + rv D + RrC L 0 0 ]. R + r C ) In the two cases, the third configuration of the each converter has a singular state matrix (A 3), this gives approximated values of the transition matrix which effect the accuracy of model (1). Using (4), the exact discrete model of the converter is Fig. 5 Buck converter with ESRs x(nt )=Φ 3(t 3)Φ 2(t 2)Φ 1(t 1)((n 1)T )+ Φ 3(t 3)Φ 2(t 2) Φ 3(t 3) (n+d1 )T nt (n+d1 +d 2 )T (n+d 1 )T (n+1)t (n+d 1 +d 2 )T Φ 1((n + d 1)T τ) B 1 U dτ+ Φ 2((n + d 1 + d 2)T τ) B 2 U dτ+ Φ 3((n +1)T τ) B 3 U dτ with t 1 = d 1T, t 2 = d 2T, t 3 = d 3T =(1 d 1 d 2)T, T = t 1 +t 2 +t 3, and d i is the duty cycle of the i-th configuration of the system. Fig. 6 Boost converter with ESRs The converters toggle between two configurations in continuous conduction mode (CCM) and three configurations in discontinuous conduction mode (DCM). In each configuration, the system can be described by a linear first order 4 Simulations and practical results To show the ability of the proposed modeling approaches to describe normal and abnormal behaviors exhibited by DC-DC converters, the following examples are considered:

6 A. Khoudiri et al. / Piecewise Smooth Dynamical Systems Modeling Based on Current controlled boost converter In the case of a current controlled boost converter, the same parameters given in [14, 15] are chosen: E =5V, r L = r C =0Ω, L =1.5mH r VD = r SW =0.001 Ω, R =40 Ω, C = T (γr), T =0.1ms. Under this control mode (Fig. 7), a control law can be expressed by ( ) L E d(n) = T (r L + r ln (rl + r SW ) i L(n). (13) SW) E (r L + r SW ) I ref (b) Fig. 7 Current controlled boost converter T RC The reference current I ref and γ = are chosen to be the first and the second bifurcation parameters respectively to explore the nonlinear phenomena exhibited by the Boost converter [19]. Using the bifurcation diagram, the obtained results are given by Figs. 8 and 9, respectively. As can be seen in above, the current controlled DC-DC boost converter changes its behavior with the changing in the 2nd bifurcation parameter (γ); indeed, period one behavior part in Figs. 8 and 9 decrease with (γ) increasing, also, it can be remarked that the bifurcation maps contain both of period doubling and border collisions bifurcations as described in [3, 20]; with an increasing of period doubling bifurcation when (γ) increase; a slight difference between the results of the two approaches has been remarked only where the Fibonacci-Horner based model ensures smoother diagrams than the Putzer based model (see Fig. 8 (b) and Fig. 9 (b)). (c) (d) Fig. 8. Boost converter routes to chaos using Putzer based model: (a) γ = 0.15, (b) γ = 0.35, (c) γ = 0.5, (d) perioddoubling (γ = 0.65) 4.2 Voltage controlled buck converter In this example, a voltage mode buck converter with PWM bloc is considered, the parameters of the converter are the same as those given in [21]: L =20mH, C =47μF, r L = r C = 0Ω, r VD = r SW = Ω, R = 22Ω and T =0.4 ms, the reference is V ref =11.3V. The control of the buck converter (see Fig. 10) is performed by means of a control voltage V con(t) givenby V con(t) =α(v ref v c(t)) (14) (a) with α as the gain of the error amplifier.

7 252 International Journal of Automation and Computing 13(3), June 2016 The external saw tooth PWM is given by ( ) t V ramp = V L +(V U V L) T mod 1 (15) where V L and V U are the upper and lower limits of the PWM signal. The binary control voltage is u = { 1, if V con(t) <V ramp 0, otherwise. (16) (a) (b) Fig. 10 Voltage controlled buck converter (c) In order to depict the abnormal behavior of the buck converter under this control mode, the input voltage U is considered to be varying from 12 V to 34 V and V L =3.8V, V U =8.2V and α =8.4 values are chosen. Using the proposed approaches, the simulation results are given in Fig. 11. Fig. 11 shows that the route to chaos of the buck converter contains a period-doubling bifurcation followed by border collisions bifurcations [22], the first bifurcation occurs when U =24.5 V, the second and the third bifurcation occur when U =32.2VandU =31.4 V, respectively, and the buck converter enters to chaotic region with supply voltage U V, the obtained results prove also that the two modeling approaches have the same accuracy to describe the converter behavior. (d) Fig. 9. Boost converter routes to chaos using Fibonacci-Horner based model: (a) γ =0.15, (b) γ =0.35, (c) γ =0.5, (d) perioddoubling (γ = 0.65) (a)

8 A. Khoudiri et al. / Piecewise Smooth Dynamical Systems Modeling Based on 253 (c) (b) Fig. 11 Routes to chaos of the buck converter using (a) Putzer based model and (b) Fibonacci-Horner based model In order to obtain more information about the converter behavior, the proposed approaches are used to calculate the buck converter response within a clock cycle. Fig. 12 gives the system response with the use of PWM signal and input voltage U variation. 4.3 Comparative study In this part, the proposed approaches are evaluated in terms of accuracy, computing time and data size. Without loss of generality, we consider as example, the current controlled boost converter functioning in period one. In order (d) (e) (a) Fig. 12 Buck converter behavior using proposed approaches: (a) U =22V, (f PWM/f) =1,(b)U =26V, (f PWM/f) =2,(c) U =31.5V, (f PWM/f) =4,(d)U =32.2V, (f PWM/f) =8, (e) Chaos (U =32.4V) to evaluate the performances of the proposed approaches and to give an idea about the error between them and approximation based method (second order Taylor series approximation); the second configuration is used as test case. Indeed, this configuration is a critical step between CCM and DCM with a full rank state matrix and highly depending on switching frequency, duty cycle and converter parameters. The relative error between the approximated and the ex- (b)

9 254 International Journal of Automation and Computing 13(3), June 2016 act value of the transition matrix is given by E R(%) = Φ Taylor Φexact 100. (17) Φ exact The word exact refers to one of the proposed methods since that both of them have the same accuracy. By the same way, the relative error in the integral calculation is given by IE R(%) = I b a Taylor I b a exact 100. (18) I b a exact (b) In order to explore the effect of duty cycle and switching frequency on the relative errors (18) and (19), let us consider a boost converter with switching frequency range (f <30 khz). In this range, the simulation results (E R(%) and IE R(%)) with duty cycle and switching frequency variation are given by Figs. 13 and 14, respectively. From these results, it is noticed that the approximation errors increase with the decrease of the switching frequency and the duty cycle. In fact, in all cases, the calculation error cannot be neglected due to the variation of the duty cycle and frequency in real functioning conditions, which presents an additional motivation to use the proposed methods. As an example, Table 1 presents the relative errors in the calculation of the transition matrix and its integral in the second configuration for two operating points of the converter. To situate the proposed models compared to other models, let us calculate the averaged computing time and the allocated data size used for the calculation of the second configuration for each model in both cases of buck and boost converters. The obtained results are summarized in Tables 2 and 3 for computing, respectively, the transition matrix and its integral term. Fig. 13 (c) (d) Approximation errors in the transition matrix elements (a) (a)

10 A. Khoudiri et al. / Piecewise Smooth Dynamical Systems Modeling Based on 255 assumptions is considered as the better choice. Table 1 Relative errors in transition matrix and integral calculation Errors Case E R(%) IE R(%) D(%) = 30 % f =3kHz (b) D(%) = 60 % f =6kHz Note that for switched systems with higher-order states matrices, the computing time will be increased; however, this is not discussed in the paper. 4.4 Practical results (c) In order to validate the proposed modeling, a PWM based hysteric controller for DC-DC boost converter is considered, practical results are carried out to show the efficiency of the proposed methods in terms of output voltage regulation and error. Fig. 15 presents the boost converter with the proposed controller. The difference between the reference voltage and the output voltage is injected to a hysteresis band that limits its value, the output of the hysteresis block is then injected to a PWM bloc to control the switching frequency; the choice of the hysteresis bandwidth is related to the accepted output voltage and current ripples and also to the tolerated frequency and switching losses. Approximation errors in the integral term matrix ele- Fig. 14 ments (d) Tables 1 and 2 show that the proposed models are less consuming in terms of computing time and memory space compared to the Cayley Hamilton based model and the exact detailed model, they are in the same range of computing time with the spectral decomposition based modeling given in [20], also, they can be classified in terms of the occupied memory space and computation limitations. Indeed, the spectral decomposition based method is less consuming in memory space, however, as mentioned before, this approach diagonalizable state matrices which cannot be ensured in all cases. Hence, the Fibonacci-Horner based model with less computing time, less data size and minimum validity Fig. 15 Boost converter with the proposed control Fig. 16 gives overall practical circuit. The details about practical circuit blocs are given in the Appendix. Simulation results of boost converter closed loop control using Fibonacci-Horner based model and PWM based hysteric controller are compared to the obtained practical results of [23]. The same circuit parameters are: L =10mH,C = 400 μf,r L = r C = 0 Ω,r VD = r SW = Ω, R = 100 Ω, and input voltage U =5V.

11 256 International Journal of Automation and Computing 13(3), June 2016 Table 2 Computing time and data size in the case of transition matrix calculation Models Data size (byte) Computing time (s) Boost converter Buck converter 2nd order Taylor series based model Spectral decomposition based model Fibonacci-Horner based model Putzer based model Cayley-Hamilton based model Table 3 Computing time and data size in the case of integral term calculation Computing time (s) Models Data size (byte) Boost converter Buck converter 2nd order Taylor series based model Spectral decomposition based model Fibonacci-Horner based model Putzer based model Cayley-Hamilton based model Fig. 17 gives the system response in case of voltage reference variation from 7 V to 12 V at t =5susingtheproposed modeling, the system presents a fast response with zero steady state error and peak-to-peak output voltage ripple less then (10%) even with reference changing. With the same circuit and controller parameters, the model is validated via experimental results given in Fig. 18. From the results, it is remarked that the practical response of the boost converter has the same form of simulation results given in Fig. 17 with neglected steady state error, which validate the proposed modeling and shows its ability to describe the normal behavior of the converter; however, peak-to-peak output voltage ripple is about (15 %) of the voltage reference in this situation, also the converter ESRs values are different from those given previously, indeed, this can affect the bifurcation diagram of the boost converter. Note that the obtaining of the bifurcation diagrams was impractical due to experimental limitations with the hardware used in this case. Fig. 17 Simulation results Fig. 16 Practical circuit Fig. 18 Practical results

12 A. Khoudiri et al. / Piecewise Smooth Dynamical Systems Modeling Based on Conclusions In this paper, two exacts discrete techniques for switched systems modeling have been proposed and applied in case of DC-DC power converters. The two techniques can describe normal and abnormal behaviors of the DC-DC converters with the same accuracy of the detailed model regardless of the control mode with no validity conditions or simplifying assumptions and also can be extended to switched systems with higher-order states matrices. To show their utility, the validation is carried out throughout simulations of buck and boost DC-DC converters using time response and bifurcation diagram tools. Experimental results confirm the simulation results and the conclusions about the proposed approaches. Appendix instabilities of a DC-DC converter. IEEE Transactions on Power Electronics, vol. 16, no. 2, pp , [6] M. DiBernardo, F. Vasca. Discrete-time maps for the analysis of bifurcations and chaos in DC/DC converters. IEEE Transactions on Circuits and Systems, vol. 47, no. 2, pp , [7] L.Benadero,A.ElAroudi,G.Olivar,E.Toribio,E.Gómez. Two-dimensional bifurcation diagrams. Backround pattern of fundamental DC-DC converters with PWM control. International Journal of Bifurcation and Chaos, vol. 13, no. 2, pp , [8] K. Guesmi, N. Essounbouli, A. Hamzaoui, J. Zaytoon, N. Manamanni. Shifting nonlinear phenomena in a DC- DC converter using a fuzzy logic controller. Mathematics and Computers in Simulation, vol. 76, no. 5 6, pp , [9] K. Guesmi, A. Hamzaoui, J. Zaytoon. Control of nonlinear phenomena in DC-DC converters: Fuzzy logic approach. International Journal of Circuit Theory and Applications, vol. 36, no. 7, pp , Fig. 19 Clock circuit [10] S. Maity, D. Tripathy, T. K. Bhttacharya, S. Banerjee. Bifurcation analysis of PWM-1 Voltage-mode-controlled buck converter using the exact discrete model. IEEE Transactions on Circuits and Systems, vol. 54, no. 5, pp , [11] G. H. Zhou, J. P. Xu, F. Zhang. Unified pulse-width modulation scheme for improved digital-peak-voltage control of switching DC-DC converter. International Journal of Circuit Theory and Applications, vol. 42, no. 7, pp , [12] S. Banerjee, K. Chakrabarty. Nonlinear modeling and bifurcations in the boost converter. IEEE Transactions on Power Electronics, vol. 13, no. 2, pp , [13] W. Aloisi, G. Palumbo. Efficiency model of boost DC-DC PWM converters. International Journal of Circuit Theory and Applications, vol. 33, no. 5, pp , [14] K. Guesmi, A. Hamzaoui. On the modeling of DC-DC converters: An enhanced approach. International Journal of Numerical Modelling, vol. 24, no. 1, pp , References Fig. 20 PWM bloc [1] R. Erickson, D. Maksimovic. Fundamentals of Power Electronics, 2nd ed., Berlin, Germany: Springer, [2] S. Banerjee, G. C. Verghese. Nonlinear Phenomena in Power Electronics, USA: IEEE Press, [3] C. K. Tse. Complex Behavior of Switching Power Converters, Boca Raton, USA: CRC Press, [4] A. El Aroudi, M. Debbat, G. Olivar, L. Benadero, E. Toribio, R. Gira. Bifurcations in DC-DC switching converters, review of methods and applications. International Journal of Bifurcation and Chaos, vol. 15, no. 5, pp , [5] S.K.Mazumder,A.H.Nayfeh,H.D.Boroyevich.Theoretical and experimental investigation of the fast-and slow-scale [15] A. Khoudiri, K. Guesmi, D. Mahi. DC-DC converters modelling approach using spectral decomposition theorem. In Proceedings of Power Plants and Power Systems Control, IFAC, Toulouse, France, pp , [16] C. Moler, C. Van Loan. Nineteen dubious ways to compute the exponential of a matrix twenty-five years later. SIAM Review, vol. 45, no. 1, pp. 3 49, [17] E. J. Putzer. Avoiding the Jordan canonical form in the discussion of linear systems with constant coefficients. The American Mathematical Monthly, vol. 73, no. 1, pp. 2 7, [18] R. Ben Taher, M. Mouline, M. Rachidi. Fibonacci-Horner decomposition of the matrix exponential and the fundamental system of solutions. Electronic Journal of Linear Algebra, vol. 15, pp , [19] C. K. Tse, O. Drangea. Bifurcation analysis with application to power electronics. Bifurcation Control, Berlin Heideberg, Germany: Springer-Verlag, vol. 293, pp , 2003.

13 258 International Journal of Automation and Computing 13(3), June 2016 [20] M. di Bernardo, F. Garofalo, L. Glielmo, F. Vasca. Switchings, bifurcations and chaos in DC/DC converters. IEEE Transactions on Circuits Systems I, vol. 45, no. 2, pp , [21] Y. Zhou, J. N. Chen, H. C. I. U. Herbert, C. K. Tse. Complex intermittency in switching converters. International Journal of Bifurcation and Chaos, vol. 18, no. 1, pp , [22] G. H. Yuan, S. Banerjee, E. Ott, J. A. Yorke. Border collision bifurcation in the buck converter. IEEE Transactions on Circuits System I, vol. 45, no. 7, pp , [23] M. Boudiaf, K. Guesmi. Implementation of PWM based hysteric controller for boost converter. In Proceedings of the 3rd International Conference on Systems and Control, Algiers, Algeria, Abdelkader Khoudiri graduated from Djelfa University, Algeria in He received the M. Sc. degree from Polytechnic Military school (EMP), Algeria in He is currently an associate professor at Djelfa University, Algeria. His research interests include intelligent control, power electronics and renewable energy. (Corresponding author) ORCID id: Kamel Guesmi graduated from Djelfa University, Algeria, in He received the M. Sc. degree from Reims University, France in 2003, and the Ph. D. degree from Reims University, France in He is currently an associate professor at Djelfa University, Algeria. His research interests include automatic control, power electronics and dynamical systems. Djillali Mahi received the B. Sc. degree from the School of Frenda, Algeria in 1978, the M. Sc. degree from University of Sciences and Technology of Oran, Algeria in 1983, and the Ph. D. degree from Paul Sabatier University, France in Professor Mahi s teaching activities involve courses at the bachelors, masters, and Ph.D. levels at University of Laghouat, Algeria. To date, he has given a number of courses in the fields of electromagnetic compatibility, high voltage engineering, high voltage discharge physics and insulating coordination. He has also been director of the masters program in engineering. He is currently the director of Materials Dielectrics Group for study and development of dielectrics and semiconductors laboratory. His research interests include dielectrics materials, flashover of polluted insulators and electromagnetic compatibility. d.mahi@mail.lagh-univ.dz