Chaos Control for the Buck-Boost Converter under Current-mode Control

Lihong He, Meimei Jia, Guangyan Sun

Abstract—Based on chaotic phenomena of the Buck-Boost converter under current-mode control, according to the reference current refI which works as a bifurcation parameter,

the poincare section diagram, the switch logic diagram and the bifurcation map are studied by applying the discrete map; moreover，by employing a nonlinear feedback controller which

is a piecewise-quadratic function in the form of x x in the

chaotic Buck-Boost converter, chaos control of the Buck-Boost converter is achieved. The numerical simulation shows that this method is suitable for the stability control of diversified periodic orbits by adjusting the feedback gain in the nonlinear feedback controller.

I. INTRODUCTION

OME irregular phenomena often emerge when the DC-DC power converter is running, such as a sudden collapse of the running state, the unknown

electromagnetic noise, the instability of the running system, not working properly by design requirements and so on. Existing studies have shown that these irregular phenomena are very popular in chaotic phenomena of the DC-DC power converter. The Buck-Boost converter is more widely used in the DC-DC converter because the Buck-Boost converter which can boost and lower the DC voltage combines advantages of the Buck converter and the Boost converter. In order to improve the performance of the Buck-Boost converter, such as the scope of work, stability, efficiency and so on, the nonlinear dynamic behavior of the Buck-Boost converter is studied. This study will have important applicable value. However, at present, chaotic control and chaotic phenomena of the converter are focused on the Buck converter [1-2,10] and the Boost converter [3-6,10]; chaotic control and chaotic phenomena for the Buck-Boost converter are seen in fewer documents [7-8].

In view of the above fact, firstly, this paper establishes the discrete map for the Buck-Boost converter; secondly, the nonlinear dynamic behavior for the converter which operates in the continuous conduction mode is studied; thirdly, by designing a nonlinear feedback controller which is a

This work was supported in part by National Nature Science Foundation under Grant 50477014.

F. A. Lihong He is with the School of Information Science and Engineering, Northeastern University, Shenyang, China (phone: 13840013335; e-mail: helihong@mail.neu.edu.cn).

S. B. Meimei Jia is with the School of Information Science and Engineering, Northeastern University, Shenyang, China (phone: 15840224833; e-mail: meimeijia14@yahoo.cn).

T. C. Guangyan Sun is with the School of Shenyang Institute of Engineering, Shenyang, China (e-mail: sungy0921@sina.com).

piecewise-quadratic function in the form of X|X| for the Buck-Boost converter, chaos control of the converter is achieved. Simulation experiments show that the control method is quite simple and easily executed on the circuit and its accurate simulation model is easily available.

II. DISCRETE MAP FOR THE BUCK-BOOST CONVERTER

Fig.1. Buck-Boost converter under current-mode control

Fig.2. Sample map of the inductor current and the output voltage

The schema of the Buck-Boost converter under

current-mode control is shown in fig.1.Assuming that the converter works in the continuous conduction mode (CCM), there are two switching modes: when t is 0, the switch H is on and the diode D is off with the inductor current Li rising;

when the inductor current Li equals to the reference

current refI , the switch H is off and the diode D is on. According to the Kirchhoff law and the constraint equations of components, state equations of the converter are as follows:

1 1 inx A x BV= + (1)

2 2 inx A x B V= + (2)

S

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Where, 1

0 010

ARC

⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦

, 1

1

0B L

⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦

, 2

00

B⎡ ⎤

= ⎢ ⎥⎣ ⎦

,

2

10

1 1LA

C RC

−⎡ ⎤⎢ ⎥

= ⎢ ⎥−⎢ ⎥

⎢ ⎥⎣ ⎦

, , 0

T

Lx i V⎡ ⎤= ⎣ ⎦ is the state variable.

By using the stroboscopic mapping method, the state equations (1) and (2) are discreted. Namely, the inductor current Li and the output voltage oV are periodically sampled

at time instants t nT= . We assume that the sampled inductor current and the sampled output voltage are

( )n Li i nT= and ( )n ov V nT= .The T is the clock pulse cycle. By using the solving method of the state equation for the linear system, the solution of equations (1) and (2) are:

( )( )

1 1

1 11( )n

n n n

nT d T

n innT

x d T x

d T nT BV d

φ

φ φ τ τ

+

+

=

+ −∫ (3)

( )

( )( )

1 2 1

2 1 11

1

2 2 2

( )

( ) ( )

( ) ( )

n

n

n n n n

nT d T

n n innT

n T

n n innT d T

x T d T d T x

T d T d T nT BV d

T d T nT d T B V d

φ φ

φ φ φ τ τ

φ φ τ τ

+

+

+

+

= − +

− −

+ − + −

∫∫

(4)

Where ( ) , 1,2i iA ti t e iφ = = , ( )1 1 1, T

n n nx i v+ + += , ( )i tφ

are the state transition matrix of the matrix iA . According to Laplace inverse transformation, the state transition matrix ( )i tφ are:

( ) ( )( )

111

1 1

1 0

0

A tt

RC

tt e L SI A

eφ −−

−

⎡ ⎤⎡ ⎤ ⎢ ⎥= = − =⎣ ⎦ ⎢ ⎥⎣ ⎦

(5)

( ) ( )211

2 2

10

10 1

1

A tt e L SI A

aaL

a a aC RC

φ −− ⎡ ⎤= = −⎣ ⎦⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥− +⎢ ⎥⎣ ⎦

(6)

Where 2 2

1 14LC R C

ω = − , 1sinkte ta ωω

−= ,

0sin coskt ktke t e ta ω ω ω

ω−= − ,

12

kRC

= − .

From the fig.2 which is the sample map of the inductor current and the output voltage, we know the switch H is off at nt . Based on the relationship between nt and T , the discrete map of the Buck-Boost converter is divided into two cases.

Case1: nt T≥ .The switch H is off and the diode D is on.

Namely, the switch H in a clock cycleT is in a state of conduction. The equation (5) is substituted into the equation (3) and we know:

1

1

n n in

TRC

n n

Ti i VL

v v e

+

−

+

= +

=

(7)

Case2: nt T< .During this period 0~ nt , the switch H is on and the diode D is off. The equation (5) is substituted into the equation (3) and we know:

1

1

n n in

tRC

n n

ti i VL

v v e

+

−

+

= +

=

(8)

During this period nt ~T, The switch H is off and the diode

D is on. The nt t= is substituted into the equation (8) and we know the terminal value of the equation (8) is:

( )

( )0

n

L n ref

tRC

n n

i t I

V t v e−

=

= (9)

The equation (6) is substituted into the equation (4) and we know:

( )( )

( )

1 1 2

1 1 2

2 1

cos sin

cos

sin

m

m

m

ktn m m

ktn m

ktm

i e c t c t

v Le c k c t

Le c k c t

ω ω

ω ω

ω ω

+

+

= +

= − +⎡ ⎤⎣ ⎦− −⎡ ⎤⎣ ⎦

(10)

Where m nt T t= − , ( )n ref nin

Lt I iV

= − , 1 refc I= ,

21 ( )

ntRC

nref

v ec kILω

−

= − +

According to the Matlab program and the discrete model (7), (8), (9), (10) which describe the Buck-Boost converter, simulation study can be performed for the Buck-Boost converter.

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III. CHAOTIC PHENOMENA STUDY FOR THE BUCK-BOOST

CONVERTER

When the reference current refI is the bifurcation parameter, circuit parameters [7-8] are as follows:

12inV V= ， 0.5L mH= ， 4C Fμ= ， 20R = Ω ，

50T sμ= ， 0.8 ~ 4.5refI A= .We execute simulation on each specific value of the bifurcation parameter and record state variables of each clock pulse, and then a poincare section can be constituted. Fig.3 is the poincaré section diagrams when the reference current refI equals to 1A, 1.5A,

1.8A, 2A, 4.5A respectively. When refI equals to 1A, the Buck-Boost converter is in the 1-period state, corresponding to one point on the poincare section; when refI equals to 1.5A, the Buck-Boost converter is in the 2-period state, corresponding to two points on the poincare section; when refI equals to 1.8A, the Buck-Boost converter is in the 4-period state, corresponding to four points on the poincare section; when refI equals to 2A, the Buck-Boost converter is in the 8-period state, corresponding to eight points on the poincare section. When refI equals to 4.5A, the Buck-Boost converter is in the chaotic state, corresponding to the strange attractors which have a certain structure and occupy a limited range on the poincaré section. The strange attractors indicate that the converter does oscillation in an unrepeatable way within a certain range.

(a) 1-period poincare section

(b) 2-period poincare section

(c) 4-period poincare section

(d) 8-period poincare section

(e) Chaotic poincare section

Fig.3. Poincare section

Fig.4 is the switch logic diagrams for the switch H when the reference current refI equals to 1A, 1.5A, 1.8A, 2A, 4.5A respectively. When the value of the switch logic equals to 1, the switch H is on; when the value of the switch logic equals to 0, the switch H is off. From the fig.4(a), fig.4(b), fig.4(c) and fig.4(d),we know that the switch cycle of the switch H is 0.05ms, 0.1ms, 0.2ms and 0.4ms under the different reference current refI . Namely, the switch cycle of the switch H is one times, two times, four times, eight times of the clock cycleT respectively. The switch logic is periodic, regular and fixed. When the reference current refI equals to 4.5A, the confusion with the switch logic will occur because the capacitor voltage rises too rapidly with the reference current refI increasing, as shown in fig.4(e).

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(a) 1-period switch logic

(b) 2-period switch logic

(c) 4-period switch logic

(d) 8-period switch logic

(e) Chaotic switch logic

Fig.4. Switch logic

Assuming the bifurcation parameter refI as the abscissa

and the output voltage oV as the ordinate, so the bifurcation diagram for the Buck-Boost converter is obtained with the reference current refI changing, as shown in fig.5. When the

reference current refI ranges from 0.8A to 1.11A, the converter is in the 1-period state; when the reference current refI ranges from 1.11A to 1.65A, the converter is in

the 2-period state; when the reference current refI ranges from 1.65A to 1.96A, the converter is in the 4-period state; when the reference current refI ranges from 1.96A to 2A, the converter is in the 8-period state; when the reference current refI is greater than 2A, the converter is chaotic.

However, when the reference current refI is near to the 4A, the Buck-Boost converter goes into the periodic window [5].

Fig.5. The bifurcation diagram for the Buck-Boost converter

By using the same method, the bifurcation diagrams for

the Buck-Boost converter are obtained when the input voltage, the resistance, the inductance, the capacitance are the bifurcation parameter respectively.

IV. PIECEWISE-QUADRATIC FUNCTION FEEDBACK CONTROL OF THE BUCK-BOOST CONVERTER

By applying a nonlinear feedback controller which is a piecewise-quadratic function in the form of x x [9] to

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achieve chaos control for the Buck-Boost converter under certain conditions. Considering the following definition for the nonlinear chaotic system which is n-dimensional:

( )( ),X F X t t

y cX

=

= (11)

F is the nonlinear smooth vector function; X is the

system state and equals to [ ]1 2, ,..., Tnx x x ; y is the system

output; c is the constant matrix. We assume that the nonlinear feedback controller of the system is as follows:

U Ky y= (12)

K is the feedback gain matrix. The nonlinear negative feedback controller is added to the chaotic dynamical system, thus the controlled system is as follows:

( )( ),X F X t t U= − (13)

According to the above method, the state equation [10] of the Buck-Boost converter can be built as follows:

0 1/ L1/ C 1/ RC

0L| |

C

LL

oo

o in

L LL

iix

VVV V

s Gi ii

−⎡ ⎤ ⎡ ⎤⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦

+⎡ ⎤⎢ ⎥ ⎡ ⎤

+ −⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

(14)

Whether the switch H is on or not is determined by the signal value s in the equation 14. When s equals to 1, the switch H is on; when s equals to 0, the switch H is off. The precise simulation model of the equation 14 is established by applying Matlab's Simulink software. Circuit parameters are as follows: 12inV V= ， 0.5L mH= ， 4C Fμ= ，

20R = Ω， 50T sμ= ， 4.5refI A= .At this moment, the Buck-Boost converter is chaotic. By gradually changing the feedback gain value G, the Buck-Boost converter is controlled into stable periodic orbits. When G is 5700, the chaotic Buck-Boost converter can be controlled to the 3-period orbit steadily, as shown in fig.6; when G is 3600, the chaotic Buck-Boost converter can be controlled to the 6-period orbit steadily, as shown in fig.7; when G is 3260, the chaotic Buck-Boost converter can be controlled to the 9-period orbit steadily, as shown in fig.8.Comparing the inductor current waveforms in fig.6(a), fig.7(a) and fig.8(a), we know that their periods are 0.15ms, 0.3ms and 0.6ms respectively which are three times, six times and nine times of the clock cycle T . Also, inductor current waveform diagrams, phase diagrams and poincare section diagrams after being controlled are stable. Simulation results show that the method can achieve the stability of periodic orbits commendably.

(a) The inductor current waveform

(b) The phase diagram

(c) The poincare section

Fig.6. The 3-period circumstance after being controlled

(a) The inductor current waveform

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(b) The phase diagram

(c) The poincare section

Fig.7. The 6-period situation after being controlled

(a) The inductor current waveform

(b) The phase diagram

(c) The poincare section

Fig.8. The 9-period situation after being controlled

V. CONCLUSIONS

By adding a nonlinear feedback controller which is a piecewise-quadratic function in the form of x x to the chaotic Buck-Boost converter, the stability control of the various periodic orbits and satisfactory results are achieved. This control method is characterized by the tiny changes to the system, so it is easy to implement in the project and has good applicable value. The disadvantage of the method is that a single-period orbit can not be controlled steadily for the converter, so it will be meaningful to construct a new piecewise-quadratic function. If this method is integrated with other control strategies, the operating performance of the converter will be improved much better.

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