Chaos Control

8/8/2019 Chaos Control

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Control of Chaos by Resonant Parametric

Perturbation in a Current Mode Controlled Buck-

Boost Dc-Dc Converter

A.KAVITHA1, G.UMA2 1. Lecturer, EEE Dept., CEG, Anna University, E-mail: [email protected]

2. Assistant Professor, EEE Dept., CEG, Anna University, E-mail: [email protected]

Abstract- Resonant parametric perturbation (RPP) methodis an effective non-feedback method for controlling chaos. Inthis paper, the above method is applied for the voltage controlof a current programmed buck-boost DC-DC converter whichhas been known to easily become chaotic for wide parametervariations. The different possible operating regimes leading tochaotic operation of the current mode controlled buck-boostconverter has been discussed and the control of output voltageby RPP method is demonstrated through computersimulations and experimental studies. The converter has beenstabilized to period-1 operation practically.

Keywords: - Resonant parametric perturbation, Chaos,buck-boost DC-DC converter

I. I NTRODUCTION

Chaotic behavior has been identified in many

physical and engineering systems. In particular it has been

observed that a large number of power electronic systems

exhibit chaos. [1-3]. When chaos is found undesirable for a

particular system, as would be the case for most engineering

applications, controlling it becomes necessary to avoid

harmful consequences.

Various methods have been proposed for the

control of chaos. They can be classified into two general

categories [4], namely feedback control methods and non-

feedback control methods. Examples of feedback control

methods include the Ott-Grebogi-Yorke method, Occasional

proportional feedback method and time delayed feedback

method. Examples of non-feedback methods include

adaptive control, resonant parametric perturbation, and

weak periodic perturbation method. Compared to the

feedback type of control, the non-feedback type of control is

simple and suitable for practical implementations.

Attention in this paper has been focused on thenon-feedback type of control for controlling chaos. The

resonant parametric perturbation method has been

considered for controlling chaos in a current mode

controlled buck-boost converter and show that an unstable period-1 orbit in the chaotic attractor can be stabilized.

II. REVIEW OF RESONANT PARAMETRIC

PERTURBATION:

Resonant parametric perturbation can suppress

chaos [5]. In general parametric perturbations can make a

system chaotic, but applying it at appropriate frequencies

and amplitudes can induce the system to stay in periodic

regimes. Thus the resonant parametric perturbation is to

perturb some parameters at appropriate frequencies and

amplitudes, thereby converting a chaotic operation into a

regular one.

Usually a parameter that strongly affects the

system and can be easily varied is chosen. Suppose this

parameter is c. This parameter is then perturbed with the

function (1+ αsin2πft) where α <<1 and f is the perturbation

frequency to be chosen. This approach has been used by

Lima and Pettini for stabilizing a chaotic Duffing-Holmes

system [6]. In particular it has been shown that when the

perturbation frequency f resonates with the periodic driving

frequency f s, chaos subsides and periodic state emerges.

III.APPLICATION OF RESONANT PARAMETRIC

PERTURBATION TO CONTROL CHAOS IN DC-DC CURRENT MODE CONTROLLED BUCK-BOOST

CONVERTER:

3.1 Overview of circuit operation:

The current mode controlled buck-boost converter [7] is shown in Fig.1.The switch is turned on periodically by

the clock and off according to the output of a comparator that compares the inductor current iL with the reference

current iref .

Figure 1. Circuit diagram of current mode controlled buck-boost DC-DCconverter

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While the switch is on, the inductor current climbs

up, and as it reaches iref , the switch is turned off, thereby

causing the inductor current to ramp down till the next clock

comes. This behavior is illustrated in Fig.2. It has beenassumed that the converter operates in continuous current

mode. There are two circuit configurations according towhether the switch is closed or open.

When switch S is closed, the state equations are (1) and (2):

L

E

dt

diL = (1)

cc v

RC

1

dt

dv−= (2)

Figure.2 Inductor current and Capacitor voltage waveforms of the currentmode controlled buck-boost DC-DC converter

When switch S is opened, the state equations are (3) and (4)

cL v

L

1

dt

di−= (3)

cL

c vRC

1-i

C

1

dt

dv= (4)

There are two state variables, inductor current iL

and capacitor voltage vc. Sampled data modeling in the formof stroboscopic map has been adopted, where the state

variables are observed in synchronism with clock. Let the

state variables at a clock instant be in and vn and those at thenext instant is in+1 and vn+1.

There are two ways in which the state can movefrom one clock instant to the next [8]. A clock pulse may

arrive before the current reaches iref . In this case the map

can be obtained by solving the equations for the on state

with in and vn as the initial conditions. These yields,

L

Tvii in

n1n +=+ (5)

RC

T

n1n evv

−

+ = (6)

If the inductor current reaches iref before the arrival of nextclock pulse, the map would include an on and off interval.

The map is obtained by stacking the solutions of on and off time equations:

))tT(sinc)tT(cosc(ei n2n1

)tT(

1nn −β+−β= −α

+

(7)

)]tT(sin)cc(

)tT(cos)cc[(ev

n12

n21

)tT(

1nn

−ββ−α

+−ββ+α= −α+

(8)

Where,2RC

1α −= , 1

L

C4R

2RC

1β

2

−= ,

ref 1 ic = ,

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ += αi

L

ev

β

1-c ref

RC

t

n2

n

3.2 Chaotic behavior:

Based on the map the chaos in buck-boost

converter has been studied. As the parameter iref is variedfrom 1A to 4A, the converter goes through period-1, period-

2, period-4, period-8 and eventually exhibit chaos. To verify

the theoretical analysis and simulation results, an

experimental circuit of a current mode controlled buck- boost converter has been built. For analyzing the chaotic

behavior, the reference current iref is varied. The circuit parameters are chosen as E=12V, L=1mH, C=4µF, R=20Ω,

T= 50µs (f s=20 kHz).The input voltage is kept at 12V. The

various possible operating regimes of the current

programmed buck-boost converter have been analyzed.

A. Period - 1 operation

With iref 0.3A, the buck-boost converter exhibits

fundamental period-1 waveform. The simulated andexperimental capacitor voltage waveforms and thecorresponding phase portrait are shown in Figures 3.11,

3.12 and 3.13 respectively.

Figure 3.11 Simulated capacitor voltage waveform of Period-1 operationwith iref =0.3A

Figure 3.12 Experimental capacitor voltage waveform of Period-1operation with iref =0.3A

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Figure 3.31 Simulated capacitor voltage waveform of Period-4 operationwith iref =2A

Figure 3.13 Experimental phase portrait showing period-1 operation

B. Period - 2 operationWhen the reference current is increased further, the

converter enters into period-2 operation. By increasing the

value of iref to 1.86 A, the capacitor voltage waveform of the

Period-2 operation has been obtained. The simulation andhardware results showing Period-2 operation are shown in

Figures 3.21 and 3.22 respectively and the corresponding phase

portrait is shown in Figure 3.23.

Figure 3.21 Simulated capacitor voltage waveform of Period-2 operation withiref =1. 86A

Figure 3.22 Experimental capacitor voltage waveform of period-2 operationwith

iref =1. 86A

Figure 3.23 Experimental phase portrait showing Period-2 Operation

C. Period - 4 operation

By increasing the value of iref further to 2A, the

converter enters into Period-4 operation. The simulatedcapacitor voltage waveform and the experimental capacitor

voltage waveform of the converter are shown in Figures 3.31

and 3.32 respectively.

Figure 3.32 Experimental capacitor voltage waveform of Period-4 operationwith iref =2A

C. Chaotic operation

When the value of iref is further increased the

converter enters into chaotic regime. The simulated capacitor

voltage waveform and the experimental capacitor voltagewaveform are shown in Figure 3.41 and 3.42 respectively. The

experimental phase portrait showing chaotic operation has been shown in Figure 3.43.

Figure 3.41 Simulated capacitor voltage waveform of chaotic operation withiref =2.21A

Figure 3.42 Experimental capacitor voltage waveform of chaotic operationwith iref =2.21A

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Figure 3.43 Experimental phase portrait showing chaotic operation

The simulation and hardware results show that the buck-boost

converter exhibits a wide range of non-linear behavior.

3.3 APPLICATION OF RESONANT PARAMETRIC

PERTURBATION TO CONTROL CHAOS:

The method of resonant parametric perturbation

is now applied to stabilize the chaotic buck-boost converter

described above. iref has been chosen as the perturbation

parameter[9]. To control the converter to operate in the period-1 orbit, iref has been selected as the perturbing parameter as it

can be changed easily. iref has been replaced by iref (1+ ε sin2πf pt) where ε is the perturbation amplitude set to 0.027 and f p

is the perturbation frequency and it is set to the switching

frequency of 20kHz. The function iref (1+ ε sin 2πf pt) has been

implemented using µA741 OpAmp. The power circuit of theconverter with the resonant parametric perturbation control is

shown in Figure 4.

Figure .4 Power circuit of the current mode controlled buck-boost converter.implementing resonant parametric perturbation

The output of the µA741 Op-Amp which is the perturbedcurrent reference is shown in Figure 4.1. The perturbing signal

has been synchronized with the switching frequency. With this perturbation, the chaotic converter has been stabilized to

operate in Period-1 orbit. The experimental capacitor voltage

waveform showing the controlled Period-1 operation and the

corresponding phase portrait are shown in Figures 4.2 and 4.3respectively.

Figure 4.1 Experimental waveform of perturbed Current reference

Figure 4.2 Experimental capacitor voltage waveform showing controlledPeriod-1 operation with iref =2.21A

Figure 4.3 Phase portrait showing controlled Period-1 operation

II. IV.CONCLUSION

The various possible operating

regimes of the current mode controlled buck-boost converter have been discussed and the simulation and experimental

results are shown. The simulation results show a good

agreement with the experimental waveforms. The resonant parametric perturbation method has been applied to the current

programmed buck-boost dc-dc converter. The converter has been stabilized to period-1 operation practically.

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Chaos Control

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