Bifurcation Study in Discretized Sliding Mode Controlled DC-DC Buck Converter

Bifurcation Study in Discretized Sliding Mode Controlled DC-DC Buck Converter

Mousumi Biswal, Assistant Professor Sidharth Sabyasachi, Assistant Professor Department of Electrical & Electronics Engineering,

Centurion University of Technology & Management(CUTM), Bhubaneswar, Odisha

Abstract

This paper presents the study of bifurcation and chaos in a discretized sliding mode controlled for buck converters operating in continuous conduction mode. The discretization behaviors of equivalent SMC of dc-dc buck converter are investigated. In particular, one of the most frequently used discretization schemes for digital controller implementation, the zero-order-holder discretization, is studied. The different parameters such as input voltage, load resistance, and inductor are used as bifurcation parameter to observe such unusual dynamic behaviors. Some inherent dynamical properties of the discretized SMC systems are studied. All the simulation work is done in the MATLAB environment.

1. Introduction

Nonlinear dynamics can be described in terms of chaos, bifurcation, fractal, Lyapunov exponent, period doubling, Poincare map, strange attractors. The chaos theory has been applied to all the branches of science. This nonlinear dynamics is applied to the power electronics to explore it completely. Since the power electronics systems are mostly nonlinear systems, it can exhibit four basic states such as fixed point, limit cycle, quasi-periodic state and chaotic state. Chaotic state is the last state appearing after all other three states. The nonlinear system can undergo to chaos state in two ways which are local bifurcation to chaos and global bifurcation to chaos. In local bifurcation, one limit cycle loses its stability whereas in global bifurcation, more than one limit cycles lose its stability. Local bifurcation has three subclasses such as period doubling, quasi-periodicity and intermittency. Global bifurcation has two subclasses such as chaotic transient and the crisis. In our study, we are considering the local bifurcations and the subclass is period doubling. The nonlinear systems can be described by nonlinear differential equations and represents in state space.

This chaos and bifurcation has been studied in different converters by applying different control methods. There are mainly six parameters in a power electronics dc-dc converter system. Those are load resistance, input voltage, inductance, load capacitance, frequency and amplitude of the triangular

wave. The behavior of a power electronic converter is sensitively dependent on the parameter. So detail knowledge about the chaos and bifurcation in the parameter space is required while choosing the parameter values depending upon the desired output behavior.

2. Literature Review

The bifurcation behavior of the buck converter has been studied by applying current feedback control [2]. It has taken load resistance, inductance and frequency amplitude of triangular wave as bifurcation parameters. It has been observed that the system directly undergoes from period-1 behavior to chaos. Chaos has been observed in the buck converter while applying constant-frequency Pulsewidth modulation in continuous conduction mode [3]. In chaotic region, a strange attractor was found. The goal was to design a converter with reliability and predictability in their performance, when operating under instability or even chaotically. Sub-harmonics, bifurcations and chaos in a sliding-mode controlled boost switching regulator have been studied [4]. Hysteresis band theory has been applied to observe the unusual behavior. A discrete time mapping for the analysis of nonlinear phenomena in current and voltage controlled pulse width modulation dc/dc buck converters has been studied [5]. Period-doubling bifurcations have been explained with detailed mathematical analysis. This work is extended by systematic classification of discrete-time models for different dc-dc converter [6]. A Jacobian matrix of the map is used for the analysis of several bifurcations and chaotic behaviors. Complex discretization behaviors have been observed in a simple sliding-mode control system [7]. The system is analyzed with the Eulers discretization. This work is extended into two-dimensional sliding mode control systems to study the dynamic behavior with the help of zero-order discretization [8]. A chattering phenomenon is observed and for certain parameter values arbitrarily, long periodic orbits exist for small discretization steps. Chattering phenomenon is studied in a buck-boost converter by applying fuzzy sliding mode control [9]. Fixed frequency PWM based fuzzy sliding mode control is compared with a classical sliding mode control in terms of start-up behavior and robustness to disturbances. The complex behavior of a voltage feedback buck converter is

experimentally is observed with wide variation of input voltage and different resistances [10].

3. Basic Principle of Buck Converter

Euler method, while a zero-order holder (ZOH) is used in practical implementation of the SMC strategy. Discretization of SMC may cause irregular behaviors such as periodic trajectories and strange attractors. In Euler method, the sampled SMC suffers more from the sampling process, as it would lose the high gain property nearby the vicinity of the switching surface. To compensate for this, disturbance prediction is indispensable, which is feasible under the hypothesis that the disturbance is slow time-varying.

5. DSMC Applied to Buck Converter

We are applying DSMC to the DC-DC buck converter. First the state equations of buck converter have to be formed.

For ON state dynamic,

Fig. 1 Buck converter

dx1

0 1

0

dt

x1 v

• E

  =f

  A buck converter is a basic converter for steeping down the

  dx 1

  1 x

  ref

  input voltage to the desired output voltage. The components

  2

  LC RC

  2 LC

  used in this converter are MOSFET, diode, inductor, capacitor and load resistance. Depending upon this switching state of the MOSFET, the converter changes its mode of operation. There are two mode of operation: Switch ON where input voltage is transferred to the output voltage, and Switch OFF where input

  dt

  For OFF state dynamic,

  1

  dx

  0 1

  x 0

  voltage is disconnected and inductor is discharged to the load.

  dt

  1

  1

  1 v

  =f

  dx

  x

  ref

  During switch ON: u=1 During switch OFF: u=0

  2

  LC RC

  2 LC

  dt

  E L di v, dt

  The system is defined as x Ax Bu D

  di v E ,

  di v 0,

  0 1 0

  0

  dt L L

  dt L

  Where

  A 1 1 , D v , B E

  i C dv v ,

  dv i v

  ref

  dt R

  dt C CR

  LC RC LC

  LC

  dv i v

  Sliding surface S(x) is defined as

  dt C CR

  S (x) c1x1 c2 x2 Cx

  In compact form, it can be written as,

  =c (v

• v) c

i

v .

di 0

1 E

1 ref

2 C CR

dt

L i u

For existence condition,

S(x) CT x 0

dv 1 1 v L

0

S(x) CT Ax CT Bu CT D

dt C RC

Where u(0, 1). E s input voltage, v is output voltage, L is inductor, C is capacitor, i is inductor current, R is resistance.

4. Discretized Sliding Mode Controller

The basic principle of sliding mode control is to design a certain sliding surface in its control law that will direct the trajectory of the state variables toward a desired origin when coincided. SMC systems are usually discretized using the

After a detailed mathematical derivation,

The resulting dynamics can be written as

x Ax B[ueq us ] D.

The system is simulated with the help of MATLAB. The combination of SIMULINK model and program runs the system to study its dynamic behaviors.

6. Simulation Results

0.62

Current(Amps)

0.615

0.61

0.605

x 106

6

4

2

X2

0

-2

0.6

12.1992 12.1994 12.1996 12.1998 12.2 12.2002 12.2004 12.2006 12.2008 12.201

Voltage(Volts)

Fig. 2 Limit cycle with R=50

0.196

0.194

0.192

Current(Amps)

0.19

0.188

0.186

0.184

0.182

0.18

12.19 12.192 12.194 12.196 12.198 12.2 12.202 12.204

Voltage (Volts)

Fig. 3 Limit cycle with R=65

-4

-6

-25 -20 -15 -10 -5 0 5 10 15

X1

Fig. 6 Phase Plane with the two boundary line

Current(Amps)

0.4

0.3

0.2

0.1

0

0 10 20 30 40 50

Load Resistance(Ohm)

Fig. 7 Bifurcation diagram with load resistance as the bifurcation parameter

0.61

Current(Amps)

0.605

0.6

0.595

0.4

Current(Amps)

0.3

0.2

0.1

0.59

12.036 12.037 12.038 12.039 12.04 12.041 12.042 12.043 12.044

Voltage(Volts)

Fig. 4 Limit cycle with Vin=25V

0.612

Current(Amps)

0.61

0.608

0.606

0.604

0.602

0.6

12.108 12.11 12.112 12.114 12.116 12.118 12.12 12.122

Voltage(Volts)

Fig. 5 Limit cycle with L=38mH

0

10 20 30 40 50 60 70 80

Input Voltage(Volts)

Fig. 8 Bifurcation diagram with input voltage as the bifurcation parameter

0.8

Current(Amps)

0.6

0.4

0.2

0

0.02 0.03 0.04 0.05 0.06 0.07 0.08

Inductance(henry)

Fig. 9 Bifurcation diagram with inductor as the bifurcation parameter

The converter is modeled in the discrete domain. The combination of both SIMULINK and program helps to simulate this converter. The sample and hold circuit and zero- order hold block are used for discretization. There are three parameters such as input voltage, load resistance and inductor which are using for the bifurcation parameters. The limit cycles for different parameters are shown in fig. 2, fig. 3, fig. 4, and fig. 5. The boundaries lines for phase plane are shown in fig. 6. The bifurcation diagrams are shown in fig. 7, fig. 8 and fig. 9. The current is compared with the bifurcation parameters. The converter goes under instability which is show by the bifurcation diagram. It has been observed that the system does not go through the period doubling bifurcations, but it is leading from period one to chaos. There are wide regions where the converter behaves chaotically in the parameter space. The results have been studied in the discrete domain with sliding mode control.

7. Conclusion

We have discussed the use of DSMC in this paper. Discretization of SMC using zero-order-hold is studied in details. The simulation results are mentioned in discrete domain by varying parameters to study the behavioral changes in discrete domain. We can observe that by varying parameter, system goes from period one to chaos. The simulations results for different bifurcations parameters are shown. The previous work has been extended and unusual behavior has been discussed. This work is done under continuous conduction mode and this can be extended to discontinuous mode and for different converters.

8. References

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3. Enric Fossas and Gerard Olivar, Study of Chaos in the Buck Converter, IEEE Transactions on Circuits and Systems-I, vol. 43, no. 1, pp. 13-25, January 1996.

4. Javier Calvente, Francisco Guinjoan, Luis Martinez and Alberto Poveda, Subharmonics, Bifurcations and Chaos in a Sliding- Mode Controlled Boost Switching Regulator, IEEE International Symposium on Circuits and Systems (ICAS), vol. 1, pp. 573-576, 1996.

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