Research Article...

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2008, Article ID 798395, 6 pagesdoi:10.1155/2008/798395

Research Article

Chaotic Behavior in a Switched Dynamical System

Fatima El Guezar1, 2 and Hassane Bouzahir3

1 Laboratoire Toulousain des Technologies et Ingenierie des Systemes (LATTIS), INSA, 135 Avenue de Rangueil,31077 Toulouse Cedex 4, France

2 LabSIV-FS, University Ibn Zohr, P. O. Box 28/S, Agadir 80000, Morocco3 ESSI, ENSA, University Ibn Zohr, P. O. Box 1136, Agadir 80000, Morocco

Correspondence should be addressed to Fatima El Guezar, fatima.el [email protected]

Received 4 September 2007; Revised 1 January 2008; Accepted 6 February 2008

Recommended by Igor Kotenko

We present a numerical study of an example of piecewise linear systems that constitute a class of hybrid systems. Precisely, we studythe chaotic dynamics of the voltage-mode controlled buck converter circuit in an open loop. By considering the voltage input asa bifurcation parameter, we observe that the obtained simulations show that the buck converter is prone to have subharmonicbehavior and chaos. We also present the corresponding bifurcation diagram. Our modeling techniques are based on the new Frenchnative modeler and simulator for hybrid systems called Scicos (Scilab connected object simulator) which is a Scilab (scientificlaboratory) package. The followed approach takes into account the hybrid nature of the circuit.

Copyright © 2008 F. El Guezar and H. Bouzahir. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

1. Introduction

Hybrid dynamical systems (HDSs) have attracted consider-able attention in recent years. HDS arise from the interactionbetween continuous variable systems (i.e., systems thatcan be described by a difference or differential equation)and discrete event systems (i.e., systems where the statetransitions are initiated by events that occur at discrete timeinstants). Switched piecewise linear systems are an importantclass of hybrid systems that are simple and can have veryrich and typical nonlinear dynamics such as bifurcationsand chaos. As example, DC-DC switching converters areswitched piecewise linear systems [1]. The three basic powerelectronic converters buck, boost, and buck-boost are vari-able structure systems that are highly nonlinear. This kind ofpiecewise model may present nonlinear phenomena such asbifurcations and chaos. The study of nonlinear dynamics ofDC-DC converters started in 1984 by Brockett’s and Wood’sresearch [2]. Since then, chaos and nonlinear phenomenain power electronic circuits have stolen the spotlight andhave attracted the attention of different research groups.Different nonlinear phenomena were investigated such asflip bifurcation or period doubling and its related routeto chaos [3–5] or quasiperiodicity route to chaos [6, 7] aswell as border collision bifurcation [1–3, 6–16]. There are

many modeling techniques, programming languages, anddesign toolsets for HDS. To model and simulate our HDS,we use Scicos (Scilab connected object simulator) which isa Scilab package for modeling and simulation of dynamicalsystems including both continuous and discrete time sub-systems [17, 18]. Scilab (scientific laboratory) is a scientificsoftware package for numerical computations that provides apowerful open computing environment for engineering andscientific applications [10]. It has been developed at INRIAand ENPC and is freely available for download. This paperaims to study and analyze some dynamic phenomena thatcan occur in the voltage-mode controlled buck converter.We also show from Scicos simulations that variation of thevoltage input can lead to a particular route to chaos. InSection 2, the general equation of a hybrid dynamical systemis briefly recalled. In Section 3, we explain the operationof the voltage-mode controlled buck converter. Then, weintroduce the state equations of the circuit in question. InSection 4, we comment on the obtained Scicos simulations.We end by some concluding remarks.

2. Hybrid Dynamical System

The evolution of an autonomous hybrid dynamical systemcan be described by [19]

mailto:[email protected]

2 Modelling and Simulation in Engineering

QVramp

vconComp

+

− aVref

S L iL

E+−

DC R vC

Figure 1: Voltage-mode controlled buck converter.

x(t) = f(x(t), q(t)

), x

(t0) = x0,

q(t) = e(x(t), q(t−)

), q

(t0) = i0,

(1)

where x(t) is the continuous state vector, q(t) ∈ Q ={1, . . . ,nq} denotes the discrete state, and q(t−) is theprevious discrete state. The state space is H = Rn × Q, andthe initial state is supposed belonging to the set of initialconditions (x0, i0) ∈ H0 ⊆ H . The function e : Rn × Q→Qdescribes the change of the discrete state. The change fromone distinct discrete state to another is called a transitionor a switch. A transition between two states i and j occursif x(·) reaches the switch set Si, j : Si, j = {x : e(x, i) =j}. Among important classes of hybrid systems, there arepiecewise linear systems that are described by

x(t) = fq(x) = A(q)x(t) + B(q), (2)

where A(q) ∈ Rn×m and B(q) ∈ Rn are matrices dependingon q.

3. Voltage-mode Controlled Buck Converter

3.1. Operation of Voltage-mode ControlledBuck Converter

A voltage feedback buck converter is represented in Figure 1.It consists of a basic RLC circuit, a diode, and a switchingelement S. The aim of the circuit is to maintain a desiredvoltage, across the load resistance R, lower than the inputvoltage E. This can be realized by the relieve of feedbackPWM control. The PWM control of a switched converter isachieved by obtaining a control voltage vcon(t), as a linearcombination of the output capacitor voltage vC(t) and areference signal Vref in the form of

vcon(t) = a(vC(t)−Vref

), (3)

where a is the gain of the error amplifier. The control voltageis compared with an externally generated sawtooth waveVramp(t) given by

vramp(t) = VL +(VU −VL

) t

Tt ∈ [0,T]. (4)

The output of this comparator is used to determine the stateof the switch S, in such way that S is off when vcon(t) ≥vramp(t) and S is on when vcon(t) < vramp(t).

3.2. State Equations

When operating in continuous conduction mode (CCM),two switch states can be identified as follows:

(i) switch off and diode on;(ii) switch on and diode off.

Whether the switch is on or off, the buck converter can alwaysbe described as a second-order linear system, whose states arethe voltage vC across the capacitor, and the current iL alongthe inductor. The general equation that models operation ofthe buck converter takes the form

x(t) = fq(x) = A(q)x(t) + B(q), with q ∈ Q = {1, 2}.(5)

For q = 1 and q = 2, we obtain the following two systems ofdifferential equations:

Soff : x(t) = f1(x) = Ax(t) + B1,

Son : x(t) = f2(x) = Ax(t) + B2,(6)

where

A =

⎛

⎜⎜⎝

−1RC

1C

−1L

0

⎞

⎟⎟⎠ , B1 =

(00

)

, B2 =

⎛

⎜⎝

0

E

L

⎞

⎟⎠ , (7)

and x = ( vciL ) is the vector of the state variables.

The border function is given by

β(x, t) = vcon(t)− vramp(t)

= avc(t)− aVref−VL−(VU−VL

) t

T, for t ∈ [0,T].

(8)

Therefore, the switching sections of each subsystem Son andSoff are given by

βon,off ={

(x, t) ∈ R2 ×R : β(x, t) ≥ 0}

,

βoff,on ={

(x, t) ∈ R2 ×R : β(x, t) < 0}.

(9)

The buck converter in CCM switches between two systemsSon and Soff if the state reaches the switching sections βon,off

and βoff,on. Figures 2 and 3 show the corresponding Scicosschematic diagram and the transition diagram, respectively.

3.3. Simulation Results and Comments

We choose the parameter values: L = 30 mH, T = 400microseconds, R = 22Ω, C = 47 μF, a = 8.4, Vref = 11.3 V,VL = 3.8 V, and VU = 8.2 V. We consider the input voltageE = 30∼47 V as a parameter of bifurcation. By varying E, thecircuit changes its qualitative behavior from a stable periodicsystem to another situation that exhibits chaos. At first, using

Modelling and Simulation in Engineering 3

1

Vramp

Relay

−+

aVref

A

B1

B2

iL

1/s

Demux

vC

If in > 0then else

+

−+

Figure 2: Scicos schematic diagram.

β(x, t) < 0 β(x, t) ≥ 0

Soff

Son

Figure 3: Transition diagram of the buck converter.

11.7

11.8

11.9

12

12.1

12.2

12.3

12.4

12.5

12.6

v Cn

30 32 34 36 38 40 42 44 46

E

BCB

Upper border line

Lower border line

Figure 4: Bifurcation diagram.

Scicos we draw the one-parameter bifurcation diagram givenin Figure 4 where the input voltage E is the bifurcationparameter and the sampled vC is the variable. By increasingE, we observe at first glance that the displayed diagram (seeFigure 4) shows a period doubling route to chaos. However,after a clear 4-T periodic operation, instead of appearanceof an immediate 8-T periodic operation, the system followsa 7-T periodic operation. This means that border collisionbifurcation comes into play and interrupts the normalperiod doubling cascade. Here, this type of bifurcation is

12.017

12.0467

12.0765

12.1062

12.136

v C(V

)

0 0.0004 0.0008 0.0012 0.0016 0.002

Time (s)

Figure 5: Fundamental periodic operation. (E = 30 V): timewaveform of the capacitor voltage.

12.0135

12.026

12.0384

12.0509

12.0633

12.0757

12.0882

12.1007

12.1131

12.1256

12.138

v C(V

)

0.494 0.5162 0.5384 0.5606 0.5828 0.605

iL (A)

Figure 6: Fundamental periodic operation. (E = 30 V): phaseplane.

characterized by the intersection of the bifurcation diagramwith the upper border line defined by

vc = Vref +VU

a. (10)


11.9

12.001

12.102

12.204

12.305

v C(V

)

0 0.0006 0.0012 0.0018 0.0024 0.003

Time (s)

Figure 7: 2-T subharmonic operation. (E = 37.5 V): timewaveform of the capacitor voltage.

11.87

11.916

11.962

12.008

12.054

12.1

12.146

12.192

12.238

12.284

12.33

v C(V

)

0.464 0.501 0.538 0.575 0.612 0.649

iL (A)

Figure 8: 2-T subharmonic operation. (E = 37.5 V): phase plane.

11.792

11.953

12.114

12.274

12.435

v C(V

)

0 0.0012 0.0024 0.0036 0.0048 0.006

Time (s)

Figure 9: 4-T subharmonic operation. (E = 41 V): time waveformof the capacitor voltage.

Figure 4 shows clearly the occurrence of this phenomenon ataround the critical value Ec = 41.45 V.

For different increasing values of E, we give the capacitorvoltage wave form vC and its corresponding phase planevC-iL.

By choosing E = 30 V, we get a fundamental periodicoperation. This periodic regime is possible just for small

11.787

11.853

11.919

11.985

12.052

12.118

12.184

12.25

12.317

12.383

12.449

v C(V

)

0.4253 0.4744 0.5234 0.5725 0.6215 0.6706

iL (A)

Figure 10: 4-T subharmonic operation. (E = 41 V): phase plane.

11.63

11.965

12.3

12.635

12.97

v C(V

)

0 0.016 0.032 0.048 0.064 0.08

Time (s)

Figure 11: Chaotic regime. (E = 46.5 V): time waveform of thecapacitor voltage.

11.779

11.893

12.007

12.121

12.235

12.349

12.462

12.576

12.69

12.804

12.918

v C(V

)

0.403 0.4684 0.5338 0.5992 0.6646 0.73

iL (A)

Figure 12: Chaotic regime. (E = 46.5 V): phase plane.

values of E. Figures 5 and 6 show the fundamental periodicoperation. Figure 5 displays the capacitor voltage wave form,and Figure 6 gives the corresponding phase plane.

For E = 37.5 V and E = 41 V, subharmonic operationhas been found. Figures 7 and 8 present 2-T periodicsubharmonic operation, whereas Figures 9 and 10 illustrate4-T periodic subharmonic operation.

Modelling and Simulation in Engineering 5

Table 1

E(V) λ

41 −0.0491

42 1.2341

43 0.9264

44 1.1942

45 1.1973

46 1.8422

47 3.8948

However, the chaotic operation is given for E = 46.5 V.Figure 11 indicates a chaotic signal with infinite order, andFigure 12 shows the phase plane vC-iL that corresponds toa chaotic attractor. In order to qualify this chaotic behaviorfor beyond E = 42 V, we have also computed the Lyapunovexponents λ from time series for a voltage range 41–47 V (seeTable 1).

Actually, it was shown in [9] that most controlled DC-DCconverters like the voltage-mode controlled buck convertercan be represented by piecewise smooth maps and suchtype of maps generates robust chaos, defined by the absenceof periodic windows and coexisting attractors in someneighborhood of parameter space.

4. Conclusion

This article has illustrated a Scicos numerical study of thevoltage-mode controlled buck converter that is modeled bya hybrid system. Variations of the voltage input can leadto a particular route to chaos; the system pursues a perioddoubling bifurcation that is interrupted by border collisionafter a 4-T periodic operation.

The purpose of studying the hybrid aspect of this circuitis to interest people working on hybrid dynamical systemsdomain, which may have some applications, especially,in information transmission. Also, the paper will attractthe attention of readers that work with Scilab/Scicos formodeling and simulation of hybrid dynamical systems. Itis true that [18] given in this paper is a good reference onthis matter. However, our paper is concerned with anothercomputational view which is the numerical study of route tochaos in a hybrid system that is different from the one studiedin [18]; displaying a bifurcation diagram, for instance, is amore complex numerical study that is not included in [18].

Acknowledgments

This work is revised within a Moroccan-Spanish cooperationframework under Grant no. A/6827/06. The authors wouldlike to thank Dr. Abdelali El Aroudi for his helpful discus-sions and the anonymous referees for valuable commentsand suggestions. The first author Ph.D. research program ispartially supported by a grant from CNRST-Morocco.

References

[1] A. El Aroudi, M. Debbat, R. Giral, G. Olivar, L. Benadero,and E. Toribio, “Bifurcations in DC-DC switching converters:

review of methods and applications,” International Journal ofBifurcation and Chaos in Applied Sciences and Engineering,vol. 15, no. 5, pp. 1549–1578, 2005.

[2] R. W. Brockett and J. R. Wood, “Understanding powerconverter chaotic behavior mechanisms in protective andabnormal modes,” in Proceedings of 11th Annual InternationalPower Electronics Conference (Powercon ’84), pp. E-14–E-15,Dallas, Tex, USA, April 1984.

[3] D. C. Hamill and D. J. Jeffries, “Subharmonics and chaos ina controlled switched-mode power converter,” IEEE Transac-tions on Circuits and Systems, vol. 35, no. 8, pp. 1059–1061,1988.

[4] C. K. Tse, “Chaos from a buck switching regulator operating indiscontinuous mode,” International Journal of Circuit Theoryand Applications, vol. 22, no. 4, pp. 263–278, 1994.

[5] C. K. Tse, “Flip bifurcation and chaos in three-state boostswitching regulators,” IEEE Transactions on Circuits andSystems I, vol. 41, no. 1, pp. 16–23, 1994.

[6] A. El Aroudi, L. Benadero, E. Toribio, and S. Machiche,“Quasiperiodicty and chaos in the DC-DC buck-boost con-verter,” International Journal of Bifurcation and Chaos inApplied Sciences and Engineering, vol. 10, no. 2, pp. 359–371,2000.

[7] A. El Aroudi and R. Leyva, “Quasi-periodic route to chaosin a PWM voltage-controlled DC-DC boost converter,” IEEETransactions Circuits and Systems I, vol. 48, no. 8, pp. 967–978,2001.

[8] S. Banerjee, P. Ranjan, and C. Grebogi, “Bifurcations in two-dimensional piecewise smooth maps-theory and applicationsin switching circuits,” IEEE Transactions on Circuits andSystems I, vol. 47, no. 5, pp. 633–643, 2000.

[9] S. Banerjee, D. Kastha, S. Das, G. Vivek, and C. Grebogi,“Robust chaos-the theoretical formulation and experimentalevidence,” in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS ’99), vol. 5, pp. 293–296,Orlando, Fla, USA, June 1999.

[10] S. L. Campbell, J.-P. Chancelier, and R. Nikoukhah, Modelingand Simulation in Scilab/Scicos, Springer, New York, NY, USA,2006.

[11] J. H. B. Deane and D. C. Hamill, “Instability, subharmonics,and chaos in power electronic systems,” IEEE Transactions onPower Electronics, vol. 5, no. 3, pp. 260–268, 1990.

[12] M. di Bernardo, F. Garefalo, L. Glielmo, and F. Vasca,“Switching, bifurcations and chaos in DC/DC converters,”IEEE Transactions Circuits and Systems I, vol. 45, no. 2, pp.133–141, 1998.

[13] C.-C. Fang and E. H. Abed, “Harmonics balance analysis andcontrol of period-doubling bifurcation in buck converters,”in Proceedings of IEEE International Symposium on Circuitsand Systems (ISCAS ’01), vol. 2, pp. 209–212, Sydney, NSW,Australia, May 2001.

[14] E. Fossas and G. Olivar, “Study of chaos in the buck converter,”IEEE Transactions on Circuits and Systems I, vol. 43, no. 1, pp.13–25, 1996.

[15] Y. Ma, C. K. Tse, T. Kousaka, and H. Kawakami, “Connectingborder collision with saddle-node bifurcation in switcheddynamical systems,” IEEE Transactions on Circuits and SystemsII, vol. 52, no. 9, pp. 581–585, 2005.

[16] G. Yuan, S. Banerjee, E. Ott, and J. A. Yorke, “Border-collisionbifurcations in the buck converter,” IEEE Transactions onCircuits and Systems I, vol. 45, no. 7, pp. 707–716, 1998.

[17] R. Nikoukhah, “Scicos: a dynamic systems modeler andsimulator,” in Proceedings of the 23rd IASTED International


Conference on Modelling, Identification, and Control (MIC ’04),pp. 263–268, Grindelwald, Switzerland, February 2004, paper412–133.

[18] R. Nikoukhah, “Modeling hybrid systems in scicos: a casestudy,” in Proceedings of the 25th IASTED International Con-ference on Modeling, Indentification, and Control (MIC ’06), pp.315–319, Lanzarote, Spain, February 2006, paper 500–082.

[19] S. Pettersson and B. Lennartson, “Stability and robustness forhybrid systems,” in Proceedings of the 35th IEEE Conference onDecision and Control (CDC ’96), vol. 2, pp. 1202–1207, Kobe,Japan, December 1996.

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