Bifurcations and chaos produced by the modulation signal in a PWM buck converter

Chaos, Solitons and Fractals 42 (2009) 2260–2271

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Bifurcations and chaos produced by the modulation signal in a PWM buck converter Manuel Miranda *, Joaquin Alvarez Center for Scientific Research and higher Education of Ensenada (CICESE), Electronics and Telecommunications Department, Ensenada, B.C., Mexico

a r t i c l e

i n f o

Article history: Accepted 30 March 2009 Communicated by Prof. G. Iovane

a b s t r a c t We present an analysis of the complex dynamics displayed by the classical buck converter, controlled with a Pulse-Width-Modulation technique. We show the conditions to make the circuit display a sliding mode and a null steady-state error. Also, some conditions for the existence of a periodic orbit with the same period as the modulation signal are established. Finally, taking the period of the modulation signal as a bifurcation parameter, we describe a situation where the controlled circuit exhibits chaotic behavior. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The study of nonlinear dynamics of power converters is an actual research subject due to the growing demand for more efficient use of energy and new systems with stricter energy requirements. The switched converters constitute a special class of power electronic devices. They are frequently used to convert the electrical energy to a suitable voltage type and level, due to their high efficiency and simple structure. Among these converters, the buck circuit is the simplest one; however, in spite of its simple topology, this circuit can exhibit nonlinear and complex dynamics. The only nonlinear element of the buck circuit is the switch, which is described by a discontinuous function that makes the buck dynamics discontinuous also. A common approach to describe the dynamics of switched converters is the average model. The average model of the buck circuit has a linear structure; in consequence, this model can not reproduce the nonlinear phenomena that may appear when feedback is used to control the device. These nonlinear phenomena should be analyzed using other tools rendering a more accurate representation of the circuit, such as the discontinuous theory developed in [29], which is used in the present paper. Some popular control algorithms for the buck circuit may produce the dynamical phenomenon known as ‘‘sliding mode”. This phenomenon constraints the behavior of the circuit, forcing the system to move on a surface given by S ¼ fx 2 Rn : rðx; tÞ ¼ 0g, where r : Rn  R ! R, and x is the system state, called the discontinuity surface. Once it reaches the surface S, the system can be guided to the desired equilibrium point lying on this surface. The major advantage of this control is its good robustness performance, even under parameter variations or model uncertainties. The order of the sliding mode regime corresponds to the number of time-derivatives of the function rðx; tÞ before the appearance of the signal control u [33]. For the buck circuit, if r depends on the time-derivative of the inductor current or the capacitor voltage, the controlled system exhibits a so-called sliding mode regime of first order [21,23–25]. Under this scenario, the theory developed in [30] can be applied. In practice, the system moves around the discontinuity surface. This behavior generates a large-frequency signal which, being directly related to the commutation frequency of the switch, may harm the device. In this matter, several approaches for control design using the robustness characteristics of the sliding mode control, but free from high frequency components, have been proposed [8,22]. * Corresponding author. Address: UABC, Facultad de Ingeniería, Ensenada, B.C., Mexico. E-mail address: [email protected] (M. Miranda). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.133

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Another approach to fix the high commutation frequency is the use of an external periodic signal gðtÞ, called ‘‘modulation signal”, used in the so-called Pulse Width Modulation (PWM) controller, widely used in practice. In this structure, a sawtooth-shaped modulation signal is introduced, which is compared to the control to generate the PWM signal. The simplest PWM controller [27] is shown in Fig. 1. Several authors have reported the existence of complex dynamics arising as a consequence of applying control laws similar to those referred before. Some examples of these nonlinear behaviors are subharmonic, quasi-periodic, and chaotic oscillations [1], which in practice the designer tries to avoid. A common element in all these controllers is the modulation signal gðtÞ, whose influence on the controlled circuit dynamics is rarely analyzed. In fact, these studies normally focus on the effects produced by other circuit parameters, like the input voltage V in (see Fig. 1). A technique to study the nonlinear dynamics of the controlled buck circuit has been introduced in the seminal paper of Hamill and Deane [20]. This technique utilizes a discrete map to describe the complex behavior of a PWM-controlled buck. Later on, some authors complemented this study [9,10,14–19]. Also, Kraein and Bass [28] analyzed the system stability through a sequence of the so-called successor points. They analyzed the stability by establishing a relation with a decreasing distance between these points. Moreover, Benadero et al. [2] proposed the study of stability using a graphical tool called characteristic curve. Angulo et al. [4] proposed a new technique for the analysis of complex behavior in a Zero Average Dynamics (ZAD) controlled buck. The suggested controller uses a PWM block to adequate the control signal. In this work, the appearance of a cascade of period doubling bifurcations can be achieved when the gain of the controller is considered as a bifurcation parameter. Other analysis technique of the complex behavior in the buck circuit can be found in [11]. This technique, however, are oriented to other applications of the circuit different to CD–CD conversion. Suppression of complex dynamics, particularly chaotic, has been a research subject since several years ago. Given the easy way a switching converter can display chaos, special emphasis has been put to eliminate this behavior [13,5–7,12]. The proposed control algorithms are rather complex, which makes difficult to implement such algorithms. An alternative approach is to analyze some controllers used in practice, and give some directions to design or tune a practical algorithm to suppress, modify, even produce, a complex behavior. This is the approach taken in the present paper. An important remark is that none of the mentioned works discusses the way the modulation signal gðtÞ affects the dynamics of a PWM-controlled buck. In this paper, a systematic and integral analysis of the effect of the modulation signal gðtÞ on the dynamics of the buck circuit, controlled with the simplest classical feedback controller to regulate the output voltage, is presented (see Fig. 1). A particular analysis of the dynamics exhibited by the circuit, for some characteristics of the modulation signal such as frequency, is performed. We show that the circuit can display several behaviors, from equilibrium to chaotic dynamics, and exhibits also a so-called sliding mode regime of second order, derived from varying the frequency of the modulation signal. The results reported here establishes a sufficient condition to be satisfied by the modulation signal to achieve a 1-period dynamic behavior of the controlled buck circuit. The paper is organized as follows. In Section 2 we present the discontinuous and the normalized models of the buck converter, which will be used in the rest of the paper. In Section 3, we discuss the presence of sliding modes exhibited by the controlled system, as a result of including a modulated signal gðtÞ, and the corresponding stability of equilibrium points. Section 4, depicts a condition for the existence of a periodic orbit with the same frequency as the modulation signal gðtÞ; this case corresponds to the PWM-controller. In Section 5, we analyze the dynamical behavior for different parameter values of the modulation signal gðtÞ, where the system may exhibit complex behavior. Finally, in Section 6 we give some conclusions and final comments. 2. Model Fig. 1 shows the analyzed buck converter, whose model is given by

Fig. 1. Simplified scheme of the controlled buck converter.

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V in u  z2 ; L z1  z2 =R z_ 2 ¼ ; C z_ 1 ¼

ð1Þ ð2Þ

where z1 is the inductor current, z2 is the capacitor voltage, and u 2 f0; 1g defines the switch position. In this paper we consider the control input proposed in [26]

u¼ where

1 þ sgnðrÞ ; 2

ð3Þ

r defines the switching strategy, to be specified later, and ‘‘sgn” is the sign function defined in Filippov sense [29] 8 > < 1 sgnðv Þ ¼ s > : þ1

v < 0; v ¼ 0; v > 0;

ð4Þ

where s 2 ½1; 1. The model given by equations (1) and (2) is known as the switching model. This representation has four parameters; however, it can be reduced to a one-parameter model, called the normalized model, using the transformations [31]

pffiffiffi L pffiffiffi z1 ; V in C 1 z2 ; x2 ¼ V in x1 ¼

ð5Þ ð6Þ

and

1

s ¼ pffiffiffiffiffiffi t:

ð7Þ

LC

Using these transformations in (1) and (2), the normalized model is obtained as

x_ 1 ¼ x2 þ u; x2 x_ 2 ¼  þ x1 ; Rn

ð8Þ

pffiffiffiffiffiffiffiffi where Rn ¼ R C=L, and the upper dot now means the derivative with respect to s, that is, x_ i ¼ dxi =ds. The parameter Rn is considered the quality factor of the RLC circuit. 3. A sliding-mode controller The control law (3) can be classified as a sliding mode controller, a popular technique proposed to control systems with a good robustness performance with respect to parameter uncertainty [33]. It has been successfully applied to control power circuits [22–26,32]. A common approach in these works has been to use the inductor current, z1 , or the time-derivative of the capacitor voltage, z_ 2 , to generate first-order sliding modes to yield a robust control design. In this section we analyze this same type of controllers; however, we use a sliding surface characterized directly in terms of the capacitor voltage, z2 , which is the main variable to regulate. For that we use the control law (3) with r given by

r ¼ z2  z2 ;

ð9Þ

z2

where ¼ z2  gðtÞ, z2 2 ð0; V in Þ is the desired average value for the capacitor voltage z2 , and gðtÞ is the modulation signal. Furthermore, we consider two cases; a) gðtÞ ¼ 0, which is equivalent to regulate the capacitor voltage around the constant value z2 , and b) gðtÞ is a smooth, low frequency signal. 3.1. A modulation signal gðtÞ ¼ 0 For this case, the discontinuous surface to control the output voltage is characterized by

r ¼ z2  z2 ;

ð10Þ

where z2 ¼ V ref is the reference for this output voltage (see Fig. 1). The buck converter (1)–(2), with the control law Eqs. (3)– (10), has an equilibrium point when the next conditions are fulfilled

zp1 ¼

zp2 ; R

zp2 ¼ V in

1 þ sgnðz2  zp2 Þ : 2

From condition (11) we conclude that the buck converter has only one equilibrium point at

ð11Þ

M. Miranda, J. Alvarez / Chaos, Solitons and Fractals 42 (2009) 2260–2271



 p

zp1 ; z2

8 z2 6 0; > < ð0; 0Þ;  1  ¼ z2 R ; 1 ; 0 < z2 < V in ; > : 1   V in R ; 1 ; z2 P V in :

2263

ð12Þ

Because of this limitation, in what follows we consider the voltage reference to be restricted as 0 < z2 < V in . A first approximate analysis of the stability of this equilibrium can be performed if the sign function (4) is defined as the limit

sgnðrÞ ¼ lim ½tanh ðfrÞ: f!1

In this case, the poles of the linear approximation around the equilibrium point are the roots of the polynomial

  1 1 1 V in kþ þ f ; RC C C 2L

pA ðkÞ ¼ k2 þ

which have a negative real part for positive parameters R, L, C, V in , and f, as is the case in practice. In fact, the equilibrium point (12) is globally asymptotically stable for any positive parameter values and disturbances satisfying some conditions. To show this fact it is more convenient to use the normalized model (8), for which the discontinuous surface (10) is given by

r ¼ V in ðx2  x2 Þ:

ð13Þ

By defining the new variables

y1 ¼ r;

y2 ¼ r_ ;

ð14Þ

the closed-loop system can be described by

y_ 1 ¼ y2 ; y_ 2 ¼ ay1  by2 þ e  csgnðy1 Þ;

ð15Þ

where

a ¼ 1; Note that, if

 A¼

1 ; Rn

b¼

c¼

V in ; 2





1 e ¼ V in x2  : 2

e ¼ c ¼ 0, then this system is linear and asymptotically stable, with a system matrix given by 0

1

a b

 :

Let us denote by

P¼



p11

p12

p12

p22



the positive definite matrix that is the solution of the Lyapunov equation AT P þ PA ¼ I, and suppose the ‘‘disturbance” bounded by q0 , j e j6 q0 . Then system (15) has the same form given in [3], where the next theorem is proved.

e is

Theorem 1 [3]. For system (15), if

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kmax ðPÞaq0  kmin ðPÞ h

c > 2kmax ðP Þ

ð16Þ

for some 0 < h < 1, then the origin of the state space is a globally asymptotically stable equilibrium point in the Lyapunov sense. Furthermore, it is also possible to show that the convergence is exponential in a region close enough to the origin. This is stated in the next theorem, which can be applied even when the term e is a time-varying function, eðtÞ. Theorem 2 [3]. If c > jeðtÞj, c > jð2=bÞdeðtÞ=dt  eðtÞj for all t > 0, then the origin of system (15) is an exponentially stable equilibrium point; i.e., there exists a neighborhood K of the origin, and positive constants aþ , a , Lþ , L such that, for any ðy1 ð0Þ; y2 ð0ÞÞ 2 K,



L exp ða t Þ jy1 ð0Þj þ y22 ð0Þ < jy1 ðtÞj þ y22 ðtÞ

< Lþ exp ðaþ t Þ jy1 ð0Þj þ y22 ð0Þ

for all t P 0. The proofs of these two theorems can be found in [3]. For the buck converter we have

ð17Þ

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kðPÞ ¼ rðRn r  1=2Þ;

r¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1=ð4R2n Þ;

and from (16), the next inequality must be satisfied to have asymptotic stability of y ¼ 0,

    h x2  1 qffiffiffiffiffiffiffiffiffiffiffiffiffi <  2max 4r ðR r þ 1=2Þ Rn rþ1=2 n Rn r1=2 for some h 2 ð0; 1Þ. It is evident that this inequality can be always satisfied with a voltage reference close enough to 1=2. Unx2 < 1, then c ¼ V in =2 P V in j  x2  1=2 j. Hence, Theorem 1 guarantees der this condition, because 0 < z2 < V in , that is, 0 <  global asymptotical stability of the origin, and Theorem 2 ensures exponential convergence once the state is near the origin. Moreover, from (14), and (15) it can be shown that the system displays a sliding mode of second order [33]. 3.2. A smooth low frequency modulation signal It is worth mentioning that, under the above circumstances, if the control objective is to make the voltage z2 follow a reference signal z2 ¼ z2  gðtÞ or, in normalized variables,

x2 ¼ x2  hðtÞ; where

x2

¼

z2 =V in ,

ð18Þ

 x2 ¼ z2 =V in is a constant and hðtÞ ¼ gðtÞ=V in is a smooth function, then

r ¼ V in ½x2  x2  hðtÞ;

ris defined as ð19Þ

and the control law given by Eqs. (3)–(19) guarantees that the capacitor voltage converges to the reference signal. This means that the control law yields a zero tracking error for the capacitor voltage, and the closed-loop system can be described by (15), with the same values for a, b, c, and





1 1 e ¼ V in x2   h  h_  h€ : 2

Rn

For this case, let  x2 be given by  x2 ¼ d þ 0:5, suppose h satisfies

    € < q; h þ 1 h_ þ h   Rn

ð20Þ

and define a ¼ maxfj d j; qg. Then the tracking error will be zero at steady-state if

a<



h ffi; qRffiffiffiffiffiffiffiffiffiffiffiffi 1 n rþ1=2

4r Rn r þ 2

ð21Þ

Rn r1=2

for some h 2 ð0; 1Þ. Note that, for any h 2 ð0; 1Þ, there is always a value of a satisfying (21). Note also that, conditions (20) and (21) define a bound on hðtÞ, and on its frequency contents. For example, let us consider the special case hðtÞ ¼ b sinðxtÞ. Then









e ¼ V in d þ b x2  1 sinðxtÞ 

x Rn

 cos ðxt Þ :

Then the tracking error will be zero at steady-state if

  x h <  jdj þ b j 1  x2 j þ qffiffiffiffiffiffiffiffiffiffiffiffiffi ; Rn 4r Rn r þ 12 RRnn rþ1=2 r1=2

ð22Þ

for some h 2 ð0; 1Þ. As in the previous case, there are always values of d, x, and b satisfying (22). In particular, d ¼ 0 (i.e.,  x2 ¼ 0:5) and x ¼ 1 give the best conditions to have a perfect tracking. Fig. 2 shows the response of system (8) with the control (13) and the reference (18), with  x2 ¼ 0:5, b ¼ 0:05, x ¼ 1, Rn ¼ 0:7071, for which the inequality (21) is satisfied. This figure shows that x2 converges to the reference signal and the error converges to zero. In summary, this controlled buck circuit will deliver the desired voltage, with zero steady-state error, if conditions (20) and (21) are satisfied. This means that the tracking signal must be smooth enough and with a bandwidth not very large. Furthermore, the system will exhibit a second-order sliding mode illustrated in Fig. 2.c), which means that at steady-state the switch will have a non-bounded commutation frequency, which may have harmful effects. 4. Periodic behavior of a PWM-controller In the previous section we showed that the control given by equations (3), and (10), when applied to the buck converter, guarantees a global convergence to the desired output, provided some conditions were satisfied (see Theorem 1). However,

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Fig. 2. Response of the buck converter; (a) output voltage and the reference signal, (b) error x2  x2 , (c) y2 ðy1 Þ.

this controller produces also an infinite commutation frequency, which has the obvious disadvantage of decreasing the life of the electronic components. A usual way used in practice to limit the commutation frequency of this electronic device is to use a high frequency modulation signal hðtÞ. A standard procedure to calculate this frequency as a function of the circuit parameter and the ripple factor is described in [27]. In particular, the smaller this parameter the higher should be the frequency of this signal. In this case condition (21) will not be satisfied anymore. Therefore, Theorems 1 and 2 will not be satisfied anymore, and there will be a frequency value for which the circuit will not follow the signal hðtÞ, neither will converge to an equilibrium point. A typical modulation signal used to limit the commutation frequency is a sawtooth-shaped, which may be characterized as

hðtÞ ¼ F ðV u  V l Þ½t  intðtÞ þ V l ;

ð23Þ

where V l and V u are the lower and upper values, respectively, F ¼ 1=T is the frequency, T is the period, and intðhÞ is a function that returns the integer part of h. The control law given by (3)–(19), with a signal hðtÞ given by (23), used to limit the commutation frequency, is known as a PWM controller. A convenient steady-state of the buck converter with this controller is that the capacitor voltage exhibits a periodic behavior with the same period as hðtÞ, a mean value  x2 2 ð0; 1Þ, and a voltage ripple inside some design limits. This means that the voltage must be a 1-period signal with frequency F ¼ 1=T and, besides a commutation at instant t k ¼ kT, k 2 Zþ , the switch must have only one commutation in the interval ðtk ; tk þ TÞ. This is illustrated in Fig. 3, where t sk denotes the (unique) switch commutation in this interval. Let us analyze the scenario where the system displays this behavior. The evolution of the buck converter with the PWMcontroller can be modeled by the equation

x_ ¼ where



rðx; tÞ > 0; Ax þ B2 u; rðx; tÞ 6 0;

Ax þ B1 u;

ð24Þ

r is given by (19),

Fig. 3. Behavior corresponding to a 1-period voltage output.

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" A¼

0

1

1

 R1n

# ; B1 ¼

1 0 ; B2 ¼ : 0 0

In the k-th interval, system (24) takes the form

x_ ¼



Ax þ B1 ; t k 6 t < t sk ;

ð25Þ

t sk 6 t < tk þ T;

Ax;

where t sk ¼ t k þ ssk , ssk 2 ð0; TÞ corresponds to the commutation time (see Fig. 3). The action of the feedback control in system (25) determines the instant of commutation tsk . At this instant, the switch strategy rðx; tÞ will be





rðx; tsk Þ ¼ V in x2  x2 ðtsk Þ  hðtsk Þ ¼ 0:

ð26Þ

If the output voltage shows a 1-period signal with period T at steady-state, the system must fulfill the condition

xðt k þ TÞ ¼ xðtk Þ:

ð27Þ

Note from (25) that, at instant tsk , the state is given by

xðt sk Þ ¼ eAðtsk tk Þ xðt k Þ þ

Z

t sk

eAðtsk nÞ B1 dn:

ð28Þ

tk

Also, the state at instant tk þ T is given by

xðt k þ TÞ ¼ eAðtk þTtsk Þ xðt sk Þ:

ð29Þ

Taking into account (28) in (29) we arrive at

xðt k þ TÞ ¼ eAT xðt k Þ þ eAðtk þTÞ ¼ eAT xðtk Þ þ eAT

Z

Z ss k

t sk

eAn dnB1

ð30Þ

tk

eA/ d/B1 ;

ð31Þ

0

where / ¼ n  t k . By denoting xðtk Þ ¼ xk , xðtk þ TÞ ¼ xkþ1 , and noting that matrix A is nonsingular, the state at instant t k þ T is given by

h i   xkþ1 ¼ eAT xk  eAssk  I A1 B1 ;

ð32Þ

where I is identity matrix. At steady-state, in a free disturbance scenario and constant parameters, if the system shows a 1-period behavior with period T, then ssk is constant (denoted as ss ) and condition (27) must be satisfied. Then we will have

xk ¼ MðA; T; ss ÞeAðTss Þ A1 B1 ;

ð33Þ

where

 1   MðA; T; ss Þ ¼ I  eAT I  eAss : Given any constant matrix A, some parameters T and ss , expression (33) gives the steady-state value, at instant t ¼ kT, of the state of the open loop circuit displaying a 1-period orbit with period T. It is easy to see that the state will have a value for x2 ðkTÞ in the interval ð0; 1Þ. The PWM controller calculates the commutation instant ssk of the closed-loop circuit, such that expression (26) is fulfilled. Therefore, from (26), (28), and (33) we arrive at



x1 ðkT þ ss Þ ¼ MðA; T; ss ÞA1 B1 : x2  hðkT þ ss Þ

ð34Þ

MðA; T; ss Þ is a matrix depending (nonlinearly) on the period T and the commutation instant ss . Since we consider that condition (26) is fulfilled, given the matrix A and the period T, the expression (34) becomes a system of two equations with two unknown variables, ss and x1 ðss Þ. By solving (34), we can find values of ss , x1 ðss Þ and x2 ðss Þ, corresponding to a 1-period orbit with period T in steady-state, under the feedback given by expressions (3), (19) and (23). Note that, since (34) is a transcendental equation an analytical solution does not exist. Nevertheless, given a circuit charx2 2 ð0; 1Þ, and a sawtooth signal hðtÞ with some acterized by some parameter values R, L, C, V in , a desired output voltage  values V l , V u calculated by design, it is always possible to find a numerical solution for (34), for any arbitrary period T of the modulation signal hðtÞ, such that x1 > 0 and ss 2 ð0; TÞ.

M. Miranda, J. Alvarez / Chaos, Solitons and Fractals 42 (2009) 2260–2271

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In practice, the design of controllers for power electronic circuits is traditionally performed using the average model [27]. With this technique, the steady-state output voltage is set to

x2 ¼ D;

ð35Þ

where D is known as the duty cycle, defined as

D¼

ss T

ð36Þ

:

Therefore, in steady-state, the expected commutation instant is

ss ¼ T x2 :

ð37Þ

Given the load/unload characteristics of the capacitor, a difference between the commutation instants ss and ss always exists, except for some value of  x2 near 1=2, which we will see later. If T corresponds to the design frequency it can be expected that the difference Ds ¼ ss  ss should be small. However, if the frequency of the signal hðtÞ is decreased or the average output voltage  x2 is varied from 1/2, the difference Ds increases and there will be a value for which the periodic orbit becomes unstable and the 1-period scenario is lost. We illustrate this situation in what follows. For that, we will use three parameters; Di is the ideal duty cycle designed with the average technique, given by (35) Ds is the duty cycle determined by the PWM controller, given by

Ds ¼

ss T

ð38Þ

;

where ss is given by the solution of (34), and Dx is the average of the capacitor voltage x2 once in steady-state. We consider a numerical example calculated from a practical design. Suppose a circuit is designed with pffiffiffi the parameters given in Table 1. With these values of the physical circuit, the normalized parameter has the value r ¼ 2, the time scaling is 7071.07, and the period of the sawtooth signal is T ¼ 0:35355, which corresponds to a normalized frequency of 17.77153 rad/s. x2 2 ½0:05; 0:95 and We illustrate the steady-state behavior with the aid of Figs. 4 and 5. Fig. 4 shows Di , Ds and Dx for  x2 and T. We observe from Fig. 4, that for values of T ¼ 0:35355. Fig. 5 displays the difference Ds  Dx as a function of   x2 2 ð0:18; 0:76Þ, Dx follows Ds . This correspond to an output voltage displaying a 1-period signal, given place to the flat rex2 < 0:18 and  x2 > 0:76, Dx does gion of the surface shown in Fig. 5. Beyond this interval, the difference Ds  Di grows and, for  not follow Ds anymore. This corresponds to the irregular zone of the surface shown in Fig. 5, where the output voltage does not exhibit a stable 1-period signal any longer. We conclude that the output voltage exhibits a 1-period behavior when the difference j Ds  Dx j is small enough. Furthermore, Fig. 5 shows that the limits of the flat zone corresponding to this scenario is determined by the values of ½T;  x2 . From Fig. 5 we can conjecture that the lost of the stability of the 1-period behavior of the output voltage can be produced by variations in the period T and the desired average output voltage  x2 . 5. Chaotic behavior of the PWM-controlled buck Beyond a value of the period T larger than 0.544, the stable 1-period scenario is lost. However, some other periodic orbits may born, as it can be seen in Fig. 6, which shows a stable 2-period orbit for T ¼ 0:57. This period duplication leads to wonder if other k-period orbits can be generated for other values of this parameter. The dynamics beyond the breaking point of the 1-period orbit can be analyzed in a more convenient way using the Feigenbaum bifurcation diagram with respect to period T, which is shown in Fig. 7. The starting value of the period T corresponds in this diagram to the design value for the frequency of the sawtooth signal (20 KHz, see Table 1). Note a ‘‘safe” range of the period where the circuit exhibits a good response, that is, a 1-period behavior with a voltage ripple inside the design limits; this corresponds to the flat zone of the surface shown in Fig. 5. However, there are also values of this period where a complex behavior is displayed by the circuit while the ripple staying approximately inside the margin limits T 2 ð0:548; 0:64Þ; this corresponds to the practical frequency interval ð12:9; 11:05Þ KHz. For larger values of period T the dynamics becomes more complex. In particular, in the interval T 2 ð0:64; 1:4Þ ! ð11:05; 5:05Þ KHz, where cascades of period-doubling bifurcations can be observed, as well as regions where 3-period orbits Table 1 Parameter values of the buck converter. Parameter

Value

Unit

R L C V in DV c f Vl Vu

50 10 2 20 10 20 0.4 0.4

ohms mH lF V % KHz V V

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M. Miranda, J. Alvarez / Chaos, Solitons and Fractals 42 (2009) 2260–2271

Fig. 4. Di ,Ds and Dx for  x2 2 ½0:05; 0:95.

Fig. 5. Error jDs  Dx j for T 2 ½1189=3363; 1053=1787 and  x2 2 ½0:05; 0:95.

are displayed by the circuit. It can be observed also that the voltage range where this dynamics is exhibited is outside the ripple margins x2 2 ð0:48; 0:535Þ ! z2 2 ð9:6; 10:7Þ V. This dynamic behavior is different from that reported elsewhere [1,20] in which V in , used as the bifurcation parameter, is increased. It is possible to show that this is equivalent to increase the gain of a proportional controller used to control the circuit. When the period of the signal hðtÞ is further increased, the dynamics of the circuit is even more complex, as it can be seen in Fig. 8. A common test to verify the chaotic nature of a signal is the Lyapunov exponent. This test is by no way sufficient to establish a complex behavior, but a positive first exponent is a useful indication of a possible chaotic dynamics. Fig. 9 shows the average first Lyapunov exponent in the same parameter range than the previous Feigenbaum diagrams. This exponent was calculated according to Sprott [34]. Note the concordance between a complex dynamics indicated by these later diagrams and the positive values of this parameter. Finally, a common property displayed by a chaotic system is the sensitivity to initial conditions. Together with the bounded nature of orbits, this property causes the repeated process of folding and stretching of the flow. To show this property in the buck converter, some trajectories starting from a set of initial conditions are shown in Fig. 10. These trajectories,

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Fig. 6. Time response of the normalized PWM-controlled buck for three values of the normalized period T; (a) T ¼ 0:3536, (b) T ¼ 0:545 and (c) T ¼ 0:57.

Fig. 7. Feigenbaum bifurcation diagram of the PWM-controlled buck, T 2 ð0:35; 1:4Þ.

Fig. 8. Feigenbaum bifurcation diagram of the PWM-controlled buck, T 2 ð1:4; 3:5Þ.

calculated with MatlabÒ, were obtained for a period T ¼ 1:2023 of the modulation signal hðtÞ. The set of initial conditions forms a circle, shown in the Fig. 10b, where two separate initial conditions are labeled. Note that the circle geometry is lost rapidly and that the separate initial conditions move and are very close at the s ¼ 1:5 (see Fig. 10c); at s ¼ 7:5 these two

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Fig. 9. Averaged Lyapunov Exponent.

Fig. 10. (a) Evolution of trajectories with different initial conditions; (b) a set of different condition initial conditions (s ¼ 0), (c) states at s ¼ 1:5, (d) states at s ¼ 7:5.

points are separate again (see Fig. 10d). This behavior shows the existence of a stretching and folding mechanism, which is a positive indication of the existence of chaotic dynamics. 6. Conclusions In this paper we have presented an analysis of the dynamics of the buck converter controlled with a very simple control algorithm. We have shown that, under certain conditions of the modulation signal hðtÞ, the buck will deliver the desired average output voltage. Furthermore, if this signal is smooth enough, with a bandwidth not very large, the system will exhibit a second-order sliding mode. When a sawtooth-shape signal with period T is used as the modulating signal hðtÞ, we have shown that, if the period T is small enough, the output voltage of the controlled circuit will be a 1-period signal with period T. However, for some values of this parameter, the output voltage will present a complex behavior. The complex behavior described here, where we consider the period T of the modulating signal hðtÞ as the bifurcation parameter, is different to those reported in previous works, where the input voltage V in is taken as the bifurcation parameter. This can be seen from the corresponding Feigenbaum diagrams. In this sense, the results presented in this paper complement the analysis presented elsewhere.

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