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 Total Downloads : 1112
 Authors : Mousumi Biswal, Sidharth Sabyasachi
 Paper ID : IJERTV1IS8351
 Volume & Issue : Volume 01, Issue 08 (October 2012)
 Published (First Online): 29102012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Discretization Effect of Equivalent Sliding Mode Controlled DCDC 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 a simple and systematic approach to use of a practical voltage mode based sliding mode controlled for buck converters operating in continuous conduction mode. In the use of sliding mode control (SMC) in dcdc converter, major problems arise like variable switching frequency. The switching frequency fixation can be achieved by implementing SM controller in discrete domain. The discretization behaviors of equivalent SMC of dcdc buck converter are investigated. In particular, one of the most frequently used discretization schemes for digital controller implementation, the zeroorderholder discretization, is studied. Some inherent dynamical properties of the discretized SMC systems are studied. Based on MATLAB and SIMULINK model, the simulation results of such kind of dynamics behaviors are also addressed.
1. Introduction
Switched mode DCDC converters are being used extensively for the purpose of efficient power conversion. These converters are nonlinear and time variant systems. Due to its nonlinear characteristic in nature, a nonlinear Sliding Mode Control (SMC) is developed. Although the analog control systems have proven successful in SMPS applications, there are several reasons that make digital control attractive. The digital controllers are less sensitive to external influences, re programmable, more flexible and allow implementation of more sophisticated control laws.
The operation of DCDC buck converter can be classified based on the flow of current in the inductor. They are called as continuous conduction mode (CCM) and discontinuous conduction mode (DCM). In CCM, the inductor current never falls to zero where in DCM, the inductor current falls to zero and remains at zero for some portion of the switching cycle. CCM of the buck converter is our study of interest here. Control system comes where some desired value is required. To maintain the output voltage at desired value with some constant value in DCDC buck converter, we need control system. The main aim of the control system is to improve the
stability, reducing the sensitivity to disturbances, improving the efficiency and system performances.
Sliding mode control is one of the best control techniques which were introduced initially for variable structure systems. A variable structure system is a dynamical system whose structure changes in accordance with the current value of its state. Since DCDC converters are inherently variable structured systems, it is suitable to apply sliding mode controller to DCDC buck converter. This control method has several advantages over other control methods such as stability even for large load and line variations, robustness, good dynamic response, and simple implementation. There are two ways of applying sliding mode control to the system. These are analog sliding mode control and digital sliding mode control. Sliding mode control in digital domain has not been studied as much as in analog domain. Here, we have applied both analog and digital sliding mode control to our system. We have studied the behavioral changes in both domains by parameter variations.
Because of its simplicity and low cost, DCDC converters are controlled in analog controllers. In practical application, the analog controller requires a lot of external passive components and this architecture increases the overall size of the system. The analog components are sensitive to the environmental influence, such as temperature, ageing, noise, tolerance of fabrication, which results in lack of flexibility, low reliability. Again because of higher switching frequency operation in the system, the analog controller suffers from the limitation of bandwidth and large gain variation. Therefore, Digital controller came to overcome these problems.
Since the system is nonlinear in nature, it exhibits some nonlinear phenomena. We found this in digital sliding mode controller while applying to our system. So some restrictions are also there to use of digital controller. We have studied the effect of discretization of sliding mode controller of DCDC buck converter. We have derived the mathematical analysis for this system and observed the undesirable phenomena by doing the simulation with help of MATLAB. We have observed that by varying the system parameter up to some extent, the system fulfills the properties of sliding mode control that means the system is robust to parameter variations.
But after crossing certain limits, the stability of the system is affected. Some subharmonic oscillation is coming which is the aim of this paper to study that behavior. This type of behavior brings the system into quasiperiodic state and chaotic state. We can say in one line that chaotic state is unstable but bounded. A lot of research work is going on chaos. The main objective is how to avoid the chaos and all other nonlinear behaviors from the system and make it smooth operation. This chaotic behavior also has some advantages in space applications. But this behavior is not always desirable. We need to optimize the controller to operate the system in an ideal manner.
2. Buck Converter
Buck converter is the most widely used DCDC converter topology in power management and microprocessor voltage regulator applications. Those applications require fast load and line transient responses and high efficiency over a wide load current range. The output voltage of this converter is less than the input voltage. The converter is shown in Fig. 1.
Fig. 1 Buck converter
The theory of operation of buck converter is divided into two states. They are ON state and OFF state. In ON state, the switch is turned ON. The output voltage is equal to the input voltage. In OFF state, the switch is turned OFF. The output voltage is zero. During switch ON condition, the current through the inductor increases and the energy stored in inductor increases. During switch OFF condition, the inductor acts as a source and maintains the current through the load resistor. During this state, the energy stored in the inductor decreases and its current also decreases. Since we are operating it in continuous conduction mode, the current never falls to zero. Current flows continuously in the inductor during the entire switching cycle in steady state operation. Since buck converter is a nonlinear system, we are using nonlinear controller to control this converter.
3. Sliding Mode Control
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. State variables mean minimum number of variables that uniquely specify the state of the dynamic system. Many control laws are used in sliding mode controller. The control
parameters denote the desired state feedback variables to be controlled. A sliding surface can be obtained by making S(x) equals to zero which is shown in Fig. 2.
Fig. 2. State trajectory converging the sliding plane irrespective of its initial condition.
Where x1, x2, and x3 are three state variables of a system. SMC process can be divided into two phases. In the first phase, which is called reaching phase, regardless of the starting position, the controller will perform a control decision that wil drive the trajectory of the state variables to converge to the sliding surface. This is possible through the compliance of the hitting condition. Hitting condition assures that, regardless of the initial condition, the state trajectory of the system will always be directed towards the sliding surface. When the trajectory is within a small vicinity of the sliding surface, it is said to be in sliding mode operation, which is called the second phase of the control process. The sliding mode controller is performing its control decision by utilizing the sliding plane as a reference path, on which the trajectory will track and eventually converge to the origin to achieve steady state operation which is shown in Fig. 3
.
Fig. 3. Phase trajectory being maintained within a small vicinity from the sliding plane and concurrently being directed to converge to the origin.
4. Discretized Sliding Mode Controller
Study of SMC in discretetime can be divided into two parts. One is the discretetime sliding mode control. Here both the system and controller are in discrete domain. Another approach is the discretization of SMC designed in the continuoustime domain. In simulations, SMC systems are usually discretized using the Euler method, while a zeroorder 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 timevarying.
In [17], the properties of equivalent control based two dimensional SMC system discretized using ZOH have been studied in detail. It has been shown that if the discretization step is sufficiently small then in the steady state trajectories spend at most two iterations on each side of the sliding surface. In [21], the zeroorder holder (ZOH) discretization of the double integrator system has been studied. It is shown that if the switching frequency is high enough, then the system converges to a small neighborhood of the origin, while for low switching frequencies, the system sustains periodic oscillations with arbitrarily large magnitudes.
5. Design of Digital Controller
A digital control system uses digital hardware, usually in the form of a programmed digital computer, as the heart of the controller. A typical digital controller has analog components at its periphery to interface with the plant. It is the processing of the controller equations that distinguishes analog from digital control.
One way of designing digital control system for continuous time systems is first to design an analog controller for the system, then to derive a digital counterpart that closely approximates the behavior of the original analog controller. It is shown in the Fig. 4
This type of control design is required when the designer replaces all or part of an existing analog controller with a digital controller. However, even for small sampling periods, the digital approximation usually less well than the analog controller from which it is derived. In Fig. 4, the analog to digital converter (A/D) samples the analog sensor signals and produce equivalent binary representations of these signals. The sampled sensor signals are then modified by the digital controller algorithms, which are designed to produces the necessary digital control inputs.
Consequently, the control inputs are converted to analog signals using digital to analog converters (D/A). The D/A transforms the digital codes to signal samplers and then produces step reconstruction from the signal samples by transforming the binarycoded digital input to voltages. These
voltages are held constant during the sampling period T until the next sample arrives. The process of holding each of the samples is termed zeroorderholder. Then the analog control signals are applied to control the behavior of the plant. The other way of designing digital control system for continuous time systems is first to derive a discretetime equivalent of the system and then to design a digital controller directly to control the discretized system. It is shown in Fig. 5.
Fig. 4. Closed loop system formed by a discretetime controller and a continuoustime system.
Fig. 5. Closed loop system formed by a discretetime controller and a continuoustime system. The D/A and A/D converters have been combined with the continuoustime system illustrating the discretetime description of the system.
Both way of designing is acceptable when the sampling is fast enough. However, first approach has been used more in industrial applications.
6. Stability of DSMC
The properties of discretetime SMC system are different from the continuoustime SMC system. In the continuoustime SMC system, the sliding mode existence condition is SS 0 ,
whereas, it is different for discretetime SMC system. The equivalent form of the continuous sliding mode existence condition is to give a discrete sliding mode existence.
(S(k 1) S(k))S(k) 0 (1)
DSMC can undergo only quasisliding mode control (QSMC). The quasisliding mode is the motion that satisfies the following conditions:

Starting from any initial state, the trajectory will move monotonically towards the switching plane and cross it in finite time.

Once the trajectory of the system first crosses the
where 1
eAh
and
h
eA d B.
0
switching plane, it will cross it again at every successive sampling time, resulting in a zigzag motion about the switching plane.

The size of each zigzagging step is not increasing and
The equivalent control is
uk ueq (k) uS (k) CAx(k) Sk , (8)
hence the trajectory stays within a specified band.
where Sk
sgn(S (x(k))).
Applying (8) to the sampled
The condition in equation (1) is found out not sufficient for the existence of a quasisliding mode. A new condition is that:
 S(k 1)   S(k)  (2)
data system, (7) leads to the following discrete system
x(k 1) x(k) Sk (9)
which is decomposed into the following inequalities:
where
eAh
h
eA d BCA
(10)
(S (k 1)
(S (k 1)
S (k)) sgn(S (k)) 0 (3)
S (k)) sgn(S (k)) 0
and
0
h
eA d B
0
(11)
Using the equivalent form of a Lyapunovtype of continuous time condition, the quasi sliding mode existence condition
8. Analysis of DSMC using Simulation
with the Lyapunov function candidate V (x(k)) S 2 (k) ,
V (k)
S 2 (k 1)
S 2 (k) 2S(k)
S(k)
S 2 (k) 0 (4)
A more suitable approach called reaching law approach for quasi sliding mode is:
S (k 1)
S (k )
qhS (k )
h sgn(S (k )),
0, q
0,1 qh
0 (5)
Where h is the sampling period.
7. DSMC System Model
Let us consider the continuoustime system
Fig. 6 Block diagram of buck converter with DSMC
x Ax Bu D
x Ax Bu
(6)
0 1 0 0
The switching function defined as S (x)
CT x .
Where A
1 1 , D v , B E
ref
where x Rn , u, S R1, A is an n n matrix,
B and C are both ndimensional vectors.
S is the switching surface to represent a desired asymptotically stable dynamics.
The equivalent control based SMC is u ueq us .
LC RC LC LC
Sliding surface S(x) is defined as
S(x) c1x1 c2 x2 Cx
where u
(CB) 1CT Ax,u
(CB) 1 sgn(S(x))
=c (v v) c
i v .
eq S
Under ZOH, this system (6) is converted into the following discrete form, that is u uk over the time interval [kh, (k+1
1 ref
For existence condition,
2 C CR
S(x) T x 0
h)], where h is a sampling period.
The continuoustime system (6) under the ZOH into the discrete form is
S(x)
CT Ax CT Bu CT D
x(k 1)
1x(k)
uk (7)
The condition for the sliding surface to exist results
discrete domain. We vary the parameter i.e. load in both cases.
We have observed that for same parameter variation, the
S(x)
CT Ax CT Bu CT D CT Ax CT Bu CT D
0 for S(x) 0
0 for S (x) 0
system behavior does not change whereas in discrete domain, we found the system behavior changes. Due to discretization effect of SMC, subharmonic2 behavior comes to the picture.
The switching plane is defined as
The limit cycle breaks and enters into period2 cycle. We use the MATLAB environment for simulation. Voltage waveform,
u(t)
1; if S (x) 0,
0; if S (x) 0,
current waveform, phase plane diagram and limit cycle are obtained from the simulation.
Therefore, switching surface is S(x)=0. According to SMC,
dS (x) 0
dt
dx
8. Simulation Results
10.0004
Output Voltage (V)
10.0002
C dt 0 10
C[ Ax Bu D] 0
9.9998
CAx CBu CD 0
CBu CAx CD
9.9996
9.9994
1 1 0.0154 0.0155 0.0156 0.0157
u (CB) (CA)x
(CB) (CD).
Time (sec)
Fig.8(a) Output voltage with R=5 in analog domain
Equivalent control law based SMC is u ueq us
10.0004
ueq
(CB) 1CAx
(CB) 1(CD)
10.0002
Output Voltage (V)
s
10
u (CB) 1 sgn(S (x))
The resulting dynamics can be written as
9.9998
x Ax B[ueq
us ] D.
9.9996
9.9994
We consider the effect of ZOH for digital implementation
0.0274 0.0275 0.0276 0.0277 0.0278 0.0279
during two successive sampling instant tk
and t
k+1
. Control
Time (sec)
Fig.8(b) Output voltage with R=50 in analog
signal is constant. The state trajectory moving on the switching line is defined by the switching logic which is shown in Fig. 7.
domain
10.001
Output Voltage(Volt)
10.0008
10.0006
10.0004
10.0002
10
9.6 9.65 9.7 9.75 9.8 9.85 9.9 9.95 10
Time(sec)
x 103
Fig. 7. State trajectory moving on the switching line
This switching logic helps the state trajectory to follow the desired path. Here, we have done the simulation in analog and
Fig.8(c) Output voltage with R=5 in discrete domain
10.0065
Output Voltage (V)
10.006
10.0055
10.005
10.0045
0.0262 0.0263 0.0264 0.0265 0.0266 0.0267
Time (sec)
Fig.8(d) Output voltage with R=50 in discrete domain
0.2001
Output Current (A)
0.2001
Steady State
0.2001
0.2
0.1999
10 10 10 10 10 10 10
Output Voltage (V)
Fig. 9(a) Limit cycle with R=50 in analog domain
Fig. 9(b) Limit cycle with R=50 in discrete domain
(a)
(b)
Fig. 10(a) Phase plane with R=5 in discrete domain,
(b) State trajectory converging towards origin.
(a)
(b)
Fig. 11(a) Phase plane with R=50 in discrete domain, (b) State trajectory converging towards origin.
The simulations are done for buck converter using SMC. First the system is simulated in continuous SMC. After that, SMC is converted into discrete form using ZOH and applied to buck converter. The output results are shown in the above Figures. In Fig. 8(a), the output voltage is shown in analog domain. By varying load resistance, we observed that the behavior of output voltage shown in Fig. 8(b) does not change. We got the desired output voltage. Now we simulate in discrete environment. The output voltages with two different load resistances are shown in Fig. 8(c) and Fig. 8(d) respectively. We observed from these Figures that, no behavioral change occurs upto certain limits. Beyond certain limits, the system shows behavioral changes. The system enters from period one to period two. The state trajectory along the switching line is plotted which is shown in Fig. 10. The existence region is defined by line 1 and line 2. The state trajectory is starting from the point (vref, 0) with initial condition zero and
converging towards the origin. We can observe that the periodicity breaks while converging towards the origin by parameter variation in discrete domain which is shown in Fig.
11. In the switching converter, the steady state operating condition is a periodic orbit, not an equilibrium point. The periodic state, the socalled limit cycle is obtained for both continuous and discrete domain with varying parameter. In the state of limit cycle, the trajectory keeps circulating along the same closed loop. But due to the discretization effect, limit cycle losses its stability, a period1 orbit becomes period2 orbit as shown in Fig. 9(b).
9. Conclusion
We have discussed the use of DSMC in this paper. SMC is mainly discretized in two ways. Here, discretization of SMC using zeroorderhold is studied in details. There are two methods for designing of digital controller. Both are described using the block diagram. The conditions for stability of DSMC are mentioned. The system model is derived for DSMC. The block diagram of buck converter with controller is shown. We define the control law for equivalent SMC. SMC is discretized using zeroorderhold for the proposed system. The simulation results are mentioned both for analog domain and discrete domain by varying parameters to study the behavioral changes in discrete domain. We can observe that by varying parameter, the limit cycle in discrete domain changes from period one to period two. Voltage waveform and phase plane are shown in the simulation results.
10. References

K. David Young, Vadim I. Utkin, and Umit Ozguner, A Control Engineers Guide to Sliding Mode Control, IEEE Transaction on Control Systems Technology, vol. 7, no. 3, pp. 328342, May 1999.

B. J. Cardoso Fo, A. F. Moreira, B. R. Menezes, and P. C. Cortizo, Analysis of Switching Frequency Reduction Methods Applied to Sliding Mode Controlled DCDC Converters, in Proceeding of IEEE Applied Power Electronics Conf. Expo (APEC), pp. 403410, 1992.

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 (ISCAS), vol. 1, pp. 573576, 1996.

SiewChong Tan, Y.M. Lai, Chi K. Tse, and Martin K. H. Cheung, An Adaptive Sliding Mode Controller for Buck Converter in Continuous Conduction Mode, in Proc. IEEE Applied Power Electronics Conf. Expo (APEC), pp. 13951400, Feb. 2004.

SiewChong Tan, Y.M. Lai, Chi K. Tse, and Martin K. H. Cheung, On the Practical Design of a Sliding Mode Voltage Controlled Buck Converter, IEEE Transactions on Power Electronics, vol. 20, no. 2, pp. 425437, March 2005.

SiewChong Tan, Y.M. Lai, and Chi K. Tse, Implementation of PulsewidthModulation Based Sliding Mode Controller for Boost Converters, IEEE Power Electronics Letters, vol. 3, no. 4, pp. 130135, December 2006.

SiewChong Tan, Y.M. Lai, Chi K. Tse, and Martin K. H. Cheung, Adaptive Feedforward and Feedback Control Schemes
for Sliding Mode Controlled Power Converters, IEEE Transactions on Power Electronics, vol. 21, no. 1, pp. 182191, January 2005.

Eva M. NavarroLopez, Domingo Cortes, and Christian Castro, Design of practical slidingmode controllers with constant switching frequency for power converters, Electric Power Systems Research, vol. 79, no. 5, pp. 795802, 2009.

Yiwen He, Weisheng Xu, and Yan Cheng, A Novel Scheme for SlidingMode Control of DCDC Converters with a Constant Frequency Based on the Averaging Model, Journal of Power Electronics, vol. 10, no. 1, pp. 18, January 2010.

SiewChong tan, Y. M. Lai and Chi K. Tse, General Design Issues of SlidingMode Controllers in DCDC Converters, IEEE Trans. on Ind. Electron., vol. 55, no. 3, pp11601174, March 2008.

J. F. TSAI and Y. P. CHEN, Sliding mode control and stability analysis of buck DCDC converter, International Journal of Electronics, vol. 94, no. 3, pp. 209222, march 2007.

R. Orosco and N. Vazquez, Discrete Sliding Mode Control for DC/DC Converters, CIEP 2000, Acapulco, MEXICO, pp. 231 236.

Mario di Bernardo and Francesco Vasca, DiscreteTime Maps for the Analysis of Bifurcations and Chaos in DC/DC Converters, IEEE Transactions on Circuits and Systems I, Fundam. Theory Appl., vol. 47, no. 2, pp. 130143, Feb. 2000.

A Jafari Koshkouei and A. S. I. Zinober, Sliding Mode Control of DiscreteTime Systems, Journal of Dynamic Systems, Measurement, and Control, vol. 122, pp. 793802, December 2000.

Xinghuo Yu and Guanrong Chen, Discretization Behaviors of Equivalent Control Based SlidingMode Control Systems, IEEE Trans. on Automatic Control, vol. 48, no. 9, pp. 1641 1646, Sept. 2003.

Zbigniew Galias and Xinghuo Yu, Complex discretization Behaviors of a Simple SlidingMode Control System, IEEE Transactions on Circuits and SystemsII, vol. 53, no. 8, pp. 652656, August 2006.

Zbigniew Galias and Xinghuo Yu, Eulers Discretization of Single Input SlidingMode Control Systems, IEEE Trans. on Automatic Control, vol. 52, no. 9, pp. 17261730, Sept.2007.

Zbigniew Galias and Xinghuo Yu, Study of zeroorder holder discretization in single input sliding mode control systems, in Proc. IEEE Int. Symposium on Circuits and Systems, ISCAS08, Seattle, pp. 13201323, May 2008.

Zbigniew Galias and Xinghuo Yu, Analysis of zeroorder holder discretization of twodimensional slidingmode control systems, IEEE Transactions on Circuits and SystemsII, vol. 55, no. 12, pp. 12691273, December 2008.

Zbigniew Galias and Xinghuo Yu, Study of discretization of twodimensional sliding mode control systems, in Proc. European Conference on Circuit Theory and Design, ECCTD07, Sevilla, pp. 727730, 2007.

Zbigniew Galias, Finite Switching Frequency Effects in the Sliding Mode Control of the Double Integrator System, in Proc. IEEE Int. Symposium on Circuits and Systems, ISCAS2006, Kos, Greece, pp. 21452148, May 2006.