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A comparative study of Dc- Dc coverter with dynamic evolution control.

INTRODUCTION

Rapid growth of communication and computer world there is a need for an advanced and sophisticated device. Switching converters are growing rapidly to cope up with this power electronics environment. Synchronous buck converter and Buck converter is a step down switching-mode regulator that has been widely used in most of the power electronics application since its high efficiency and compact size [1] and replacing high output power linear voltage regulators. The sudden load changes introduced by modern processors make the dynamic response of a conventional power-supply system too slow to track the changes, so dynamic response of power supplies is so significant in this respect [2]. Under changing supply and load conditions the power converter requires very tight output voltage regulation. This enforces challenge of very good controller to meet the parameter variations. Conventional PID controllers use single loop voltage control as feedback loop. The demand of parameter variations, large supply and load variations, non-linearity in the converter operation makes PID controllers are not suitable. The ideal characteristic of a controller is to operate at an infinite switching frequency while tracking reference signal to achieve the better performance both dynamic as well as steady state operation [3]. This extreme switching frequency power converters leads to high switching losses, core losses in inductor and transformer, and electromagnetic interference (EMI) problems. The commercially viable analogy controllers use P or PI control with voltage feedback in control loop. The compensator is designed based on phase margin criteria and location of roots depends on the root locus technique. To place the poles in any arbitrary location feedback should be applied on all the state variables [4]. Current mode controllers are independent of current or voltage loops and apply feedback on both states. Hence the pole location is limited to root locus technique. Dc-Dc converters are time-varying structure and contain elements with nonlinearity, parasitic components. This leads to modelling issues such as complexity, nonlinearity, and imprecision which makes fuzzy control is an attractive option for controller design. With recent advances in digital signal handling for control techniques, it is easy to implement fuzzy controllers for power converters. Applying linear control techniques to the small signal model of the converter stability measures and small signal dynamic performance can be assessed and heuristic knowledge can be incorporated into the controller by implementation fuzzy logic [5]. This will improve the nonlinear controller that surpasses its linear controller. The improvement is made in such a way that it will not trade off the stability or performance of the controller for small signals. Moreover, better large signal dynamic performance can be achieved. Classical control techniques involve lot of complex equations and computational burden. If the control strategy is based on soft computing the much of computational burden levitated due to exclusion of arithmetic and computational time is also reduced. Conventional converter operation deviates from theoretical prediction due to the parasitic resistances, stray capacitances and leakage inductances of the components so the dynamics of DC-DC converters is non-linear [7]. The linguistic rules of the fuzzy controller accurate the model is not necessary. So the fuzzy controller is simpler to design. Dc-dc converter's small signal model changes with variations in the operating point but PID control is typically designed for one nominal operating point [8]. In buck converter, the magnitude of the frequency response depends on the duty cycle. The operating points change in dc converter is not tightly tracked by the PID controllers [9]. Expert knowledge based fuzzy controllers' does not depend on precise mathematical model [10]. These Controllers are suitable for particularly those having nonlinear dynamics. Fuzzy logic concepts are blended with other conventional control methods, like sliding-mode control, neuro-control. Reference [11] has given the comparative study of PI, fuzzy and sliding mode controllers.

In this proposed work dynamic evolution control is designed for a synchronous buck converter and is compared with classical controller such as PI and fuzzy logic controller for the synchronous buck as well as buck converter. The performance measures such as peak overshoot settling times are compared.

This paper is organised as section.2 discusses the problem formulation for asynchronous buck with dynamic evolution control. Section. 2.1 is presented with PI controller for above converter. Fuzzy logic controller for a buck converter is given in section 2.3. Results and discussion in section 3 followed by section.4 conclusion.

2. Problem formulation:

The dynamic response of the converter is controlled in such a way that [psi]=0. The dynamic characteristic of system is made to track the evolution path. Evolution path is exponential function whose decay depends on the controller parameter k. The exponential function is given in equation 1.

[psi] = [beta][e.sup.-kt] (1)

Where, [beta] is the initial value of function [psi] and k is a proportionality constant which decides the initial decrease rate of [psi].

The converter state error function is represented as [psi]. The dynamic response of the converter is controlled in such a way that [psi]=0. This exponential function, [psi] decreases exponentially to zero as a function of time. The rate of decay depends on the constant k.

d[psi]/[dt] = -k[beta] [e.sup.-kt] (2)

d[psi]/[dt] + k[psi] = 0k> 0 (3)

The aim of the controller is to find the duty cycle [delta] as function of states of the output voltage [V.sub.o], input voltage [V.sub.in], and the inductor current [I.sub.Lin]. The control algorithm forces the state error [psi] of a converter decreases exponentially to reach zero following the evolution path.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Where L is the inductance, C the capacitance, R the load resistance, [V.sub.o], [V.sub.in] the input and output voltage, [I.sub.Lin] the inductor current, and [delta] duty cycle.

The output voltage [V.sub.o] is obtained by rearranging equation (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The error voltage or error current in a converter is taken as [psi]. It is assumed here [psi] is a linear function of error voltage as

[psi] = m[V.sub.[epsilon]] (7)

[V.sub.[epsilon]] = [V.sub.Ref] - [V.sub.o] (8)

d[psi]/dt = m d[V.sub.[epsilon]]/dt (9)

Combining equation (3),(7) and (9)

m d[V.sub.[epsilon]]/dt - km[V.sub.[epsilon]] (10)

The duty cycle [delta] is calculated after several manipulation of above equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

2.1. PI Controller:

The buck converter can be approximated into two first order systems with voltage and current dynamics. The PI controller can be easily designed for a two first order systems and the design involves two control loops with the outside voltage loop and the inner current loop. The inner current loop is much faster than those of the outer voltage loop, so control can be implemented cascade manner. PI controller design [2] for voltage loop is given in equation (12-15) and equation (16-22) provides the analysis of current loop.

2.1. a.Voltage control loop:

If the inductor current [I.sub.Lin] is taken as the control input to the converter, the transfer function of output regulated voltage [V.sub.o] to the inductor current [I.sub.L] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Let proportional-plus-integral (PI) controller of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[FIGURE 1 OMITTED]

2.1.b. Current control loop:

A current control loop can be designed based on the current dynamics. The dynamics of the inner current loop are very much faster than these of the outer voltage loop, the output regulated voltage [V.sub.o] is taken as a constant disturbance. To track the constant load disturbance, a PI controller is incorporated within the control loop. The form PI controller [2] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

As stated [2] the secondary loop dynamics should be at least four times faster than the primary loop dynamics. Let the un-damped natural frequency of the current loop is N times faster than the voltage loop

[[omega].sub.i]=N[[omega].sub.n] = [square root of ([K.sub.2][V.sub.i]/L)] N >4 (17)

Thus K2 is given by

[K.sub.2] = [N.sup.2][[omega].sup.2.sub.n] L/[V.sub.in] (18)

L=1 mH, C=120[micro]F, nominal supply voltage [V.sub.in]=50 V. The nominal load current was 1A. The output regulated voltage was set to 10V and the nominal loading resistor R1 was 10[ohm].

[K.sub.1] = 1/[R.sub.1]C = 833.3 (19)

The PI controller loop becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Let current control loop, the current loop dynamics were set to 20 times [2] faster than the voltage dynamics with N=20.

[K.sub.2] = [N.sup.2][[omega].sup.2.sub.n] L/[V.sub.in] = 5555 T = 2/N[[omega].sub.n] = 1.2 x [10.sup.4] (21)

The PI controller of current loop is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

3. Fuzzy logic controller for dc/dc converters:

The Fuzzylogic controller for dc-dc converters can be implemented through five major processes: fuzzification, data base, rule base formation, decision making and defuzzification [7-10]. The input to the controller are error and change in error and defined by the following equations

[epsilon] = [V.sub.0]- [V.sub.Ref] (23)

c[epsilon] = [epsilon](n) - [epsilon](n - 1) (24)

Where, [V.sub.o], [V.sub.ref] are regulated and reference output voltage respectively, 'n' represents values sampled at the nth switching cycle.

The duty cycle is the output of the fuzzy controller and is defined as [delta](n) = [delta](n - 1) + [lambda][alpha][delta](n) (25)

where, [lambda] is the gain factor of the fuzzy controller.

[alpha] [delta](n) is the inferred change of duty cycle at the nth sampling time by the fuzzy controller; the effective gain of the controller is adjusted by [lambda].

The fuzzy variables [epsilon] and c[epsilon] are used in the inference system without being fuzzified and described by fuzzy singletons values. The fuzzy rules [7] are framed in the form

Rule i: If [epsilon] is [O.sub.i] and c[epsilon] is [P.sub.i] then [delta] (n) is [M.sub.i] (26)

Where, [O.sub.i], [P.sub.i] are fuzzy subsets in their universes of discourse, and [M.sub.i] is a fuzzy singleton. The Controller rule base table. 1, [7] is obtained from five fuzzy subsets: PB (Positive Big), PS (Positive Small), ZE (Zero), NS (Negative Small), and NB (Negative Big).

[FIGURE 2 OMITTED]

The values in the table are the normalized singleton of the changes of duty cycle. The inference system includes weight factor [w.sub.i] and the change in duty cycle [M.sub.i] of the individual rule [7]. Mamdani's [8]min fuzzy implication operation of [mu][epsilon] ([epsilon][k]) and [mu]c[epsilon](c[epsilon][k]),provides weighting factor [w.sub.i] where [epsilon][k] and c[epsilon][k] are the singleton inputs of [epsilon] and c[epsilon]. The membership functions for error and change in error is given figure.2.

The change in duty cycle inferred by the [i.sub.th] rule is given by [Z.sub.i] = min{[mu][epsilon]([epsilon][k]), [mu]c[epsilon] (c[epsilon][k])} * [M.sub.i] (27)

RESULTS AND DISCUSSION

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The table.2 shows performance measures of three controllers. It is clear that DEC has better settling time with 3% overshoot, followed by PI controller. It has very low over shoot however the settling time is little bit higher. Fuzzy has 13% peak overshoot which is so undesirable and also has sluggish settling time. Figure.3 shows output voltage and inductor current for step load change for DE controller. PI controller dynamic response is given in figure.4. It is obvious from figure.4 inductor current is not smooth for PI controller. From figure.5 the fuzzy controller has the highest level of peak overshoot output of these three controllers.

Conclusion:

Three controllers DEC, PI, FUZZY are designed for a synchronous buck converter. The performance measures of these controllers are compared. The dynamic evolution control has the advantages of low peak over shoot and faster settling time over the other classical PI and fuzzy controllers. The simulations are carried out in Matlab.2010R .a.

REFERENCES

[1.] Samosir, Ahmad Saudi and Abdul Halim Mohd Yatim, 2010. "Dynamic evolution control for synchronous buck DC-DC converter: Theory, model and simulation." Simulation Modelling Practice and Theory, 18(5): 663-676.

[2.] Tsang, K.M. and W.L. Chan, 2005. "Cascade controller for DC/DC buck convertor." Electric Power Applications, IEE Proceedings., 152: 4.

[3.] Tan, S.C., Y.M. Lai, C.K. Tse, M.K. Cheung, 2005. A fixed-frequency pulsewidth modulation based quasi-sliding-mode controller for buck converters. Power Electronics, IEEE Transactions on. 20(6): 1379-92.

[4.] Oliva, Alejandro, R., Simon S. Ang and Gustavo Eduardo Bortolotto, 2006. "Digital control of a voltage-mode synchronous buck converter." Power Electronics, IEEE Transactions on, 21(1): 157-163.

[5.] Perry, A.G., G. Feng, Y.F. Liu, P.C. Sen, 2007. "A design method for PI-like fuzzy logic controllers for DC-DC converter." Industrial Electronics, IEEE Transactions on., 54(5): 2688-96.

[6.] Feng, Guang, Eric Meyer and Yan-Fei Liu, 2007. "A new digital control algorithm to achieve optimal dynamic performance in DC-to-DC converters." Power Electronics, IEEE Transactions on, 22(4): 1489-1498.

[7.] So, Wing-Chi, C.K. Tse and Yim-Shu Lee, 1994. "A fuzzy controller for DC-DC converters." Power Electronics Specialists Conference, PESC'94 Record.25th Annual IEEE. IEEE.

[8.] Guo, Liping, John Y. Hung and R. Mark Nelms, 2009. "Evaluation of DSP-based PID and fuzzy controllers for DC-DC converters." Industrial Electronics, IEEE Transactions on, 56(6): 2237-2248.

[9.] So, Wing-Chi, Chi K. Tse and Yim-Shu Lee, 1996. "Development of a fuzzy logic controller for DC/DC converters: design, computer simulation, and experimental evaluation." Power Electronics, IEEE Transactions on, 11(1): 24-32.

[10.] Gupta, T., R.R. Boudreaux, R.M. Nelms, J.Y. Hung, 1997. "Implementation of a fuzzy controller for DCDC converters using an inexpensive 8-b microcontroller. "Industrial Electronics, IEEE Transactions on. 44(5): 661-9.

[11.] Raviraj, V.S.C. and Paresh C. Sen, 1997. "Comparative study of proportional-integral, sliding mode, and fuzzy logic controllers for power converters." IEEE Transactions on Industry Applications, 33(2): 518-524.

1 A. Thangaraj and 2 S. Arockia Edwin Xavier

1 Assistant Professor, Dept of EEE, GCE, Bargur.

2 Assistant Professor, Dept. of EEE, TCE, Madurai.

Received 25 January 2016; Accepted 28 April 2016; Available 5 May 2016

Address For Correspondence:

A. Thangaraj, Assistant Professor, Dept. of EEE, GCE, Bargur.
Table 1: Rule base for the fuzzy controller

                               [epsilon]

                  NB      NS      ZE      PS      PB

             PB   -0.30   -0.35   -0.45   -0.65   -1.00
             PS   0.00    -0.10   -0.20   -0.35   -0.50
[epsilon]    ZE   0.20    0.10    0.00    -0.10   -0.20
             NS   0.50    0.35    0.20    0.10    0.00
             NB   1.00    0.65    0.45    0.35    0.30

Table 2: Performance measures of DEC, PI, FUZZY controller.

Controller   Peak overshoot   Settling Time

DEC          3.00%            4 msec.
PI           2.92%            7 msec.
Fuzzy        13.0%            15 msec
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Author:Thangaraj, A.; Xavier, S. Arockia Edwin
Publication:Advances in Natural and Applied Sciences
Date:May 15, 2016
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