J Intell Manuf (2012) 23:1023–1034DOI 10.1007/s10845-010-0388-1

A type-2 fuzzy logic controller design for buckand boost DC–DC converters

Ismail Atacak · Omer Faruk Bay

Received: 14 August 2009 / Accepted: 10 February 2010 / Published online: 23 February 2010© Springer Science+Business Media, LLC 2010

Abstract Conventional (type-1) fuzzy logic controllershave been commonly used in various power converter appli-cations. Generally, in these controllers, the experience andknowledge of human experts are needed to decide parame-ters associated with the rule base and membership functions.The rule base and the membership function parameters mayoften mean different things to different experts. This maycause rule uncertainty problems. Consequently, the perfor-mance of the controlled system, which is controlled withtype-1 fuzzy logic controller, is undesirably affected. In thisstudy, a type-2 fuzzy logic controller is proposed for the con-trol of buck and boost DC–DC converters. To examine andanalysis the effects of the proposed controller on the systemperformance, both converters are also controlled using thePI controller and conventional fuzzy logic controller. Thesettling time, the overshoot, the steady state error and thetransient response of the converters under the load and inputvoltage changes are used as the performance criteria for theevaluation of the controller performance. Simulation resultsshow that buck and boost converters controlled by type-2fuzzy logic controller have better performance than the buckand boost converters controlled by type-1 fuzzy logic con-troller and PI controller.

Keywords DC–DC converter · PI controller · Type-1fuzzy logic controller · Type-2 fuzzy logic controller ·Type reduction

I. Atacak · O. F. Bay (B)Department of Electronics and Computer Education,Gazi University, 06500 Ankara, Turkeye-mail: omerbay@gazi.edu.tr

Introduction

Type-1 fuzzy logic (T1FL) controllers have been successfullyused in numerous applications many of which are toocomplex to be analyzed using conventional mathematicaltechniques for years. In design of T1FL controllers, theexperience and knowledge of human experts are neededto determine parameters associated with the rule base andmembership function (Lee 1990; Zimmermann 1987).Because available information usually includes an uncer-tainty, T1FL controllers will also have an uncertainty relatedto rule base and membership functions. Zadeh introducedtype-2 and higher-types fuzzy systems in (1975) to eliminatethe paradox of T1FL systems which can be formulated as theproblem that the membership grades are themselves precisereal numbers. Type-2 and higher-types systems are an exten-sion of T1FL systems. Similar to T1FL systems, type-2 fuzzylogic (T2FL) systems comprise fuzzifier, rule base, inferenceengine and output processor. One of the most important dif-ferences between T1FL and T2FL systems takes place in theoutput processing. The output processing of T2FL systemsincludes type reducer which converts the type-2 fuzzy out-put sets into type-1 sets and defuzzifier which maps type-1fuzzy sets obtained from type reducer into crisp data. There-fore, the type reduction captures more information about ruleuncertainties than that of a crisp number. Another importantdifference between these systems is the membership sets usedin the fuzzifier. A type-2 fuzzy set is characterized by a fuzzymembership function, i.e., the membership value (or mem-bership grade) for each element of this set is a fuzzy numberin [0,1]. Thus, these sets can be used in situations where thereis uncertainty about the membership grades themselves, e.g.,an uncertainty in the shape of the membership function orin some of its parameters (Karnik 1999; Liang and Mendel2000).

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1024 J Intell Manuf (2012) 23:1023–1034

DC–DC converters are employed in a variety of appli-cations, including power supplies for personal computers,office equipment, spacecraft power systems, laptop com-puters, and telecommunications equipment, as well as DCmotor drives (Wu 2006; Cuk and Middlebrook 1981). Thedynamics of DC–DC converters is non-linear, and thepractical converter operation deviates from theoreticalprediction because of problems associated with parasiticresistances, stray capacitances and leakage inductances ofthe components. All these make the design of an optimalcontrol scheme for closed loop operation of the converter dif-ficult (Muhammed 2001). A control scheme should ensurestability in arbitrary operating conditions along with goodresponse in terms of the rejection of load variations, inputvoltage changes and even parameter uncertainties. Severalcontrol techniques for DC–DC converters have been reportedin the literature, such as linear based control techniques, slid-ing mode control technique, and fuzzy logic control tech-nique. Although the structure and design of linear basedcontrol techniques are simple, their performance usuallydepends on the working conditions of the controlled system.Sliding mode control technique needs a system modelto be designed. One of the most important problems indesign of this controller is control chattering (Wan et al.2007; Alvarez-Ramirez 2001; Spiazzi et al. 1995). Tradi-tional fuzzy techniques provide for the output voltage reg-ulation against input voltage and load variations. However,the performance of this controller depends on the experienceand knowledge of human experts. In general, trial-and-errortuning procedure is used to adjust parameters of the rule baseand membership sets (Gupta et al. 1997; So et al. 1996). Thismeans that these parameters will be change from one expertto another expert. The controlled system performance maybe undesirably affected from these uncertainty conditions.Thus, a type-2 fuzzy controller will be highly suitable totackle the uncertainty which occurs in traditional fuzzy logiccontrollers.

In this study, a T2FL controller is proposed for the con-trol of the buck and boost DC–DC converters to achieve agood output voltage regulation and dynamic response againstinput voltage and load variations. To analyze the effects onthe system performance of T2FL controller, the convertersare also controlled using the PI and traditional fuzzy con-troller. The structure of the converters and its mathemati-cal model are given in section “Mathematical modelling of

DC–DC converters”. Section “Control techniques used forthe DC–DC converters” presents the control algorithms usedfor the control of the DC–DC converters. The simulationresults are presented in section “Simulation results”. Finally,the performance of the proposed controller is evaluated byusing the simulation results.

Mathematical modelling of DC–DC converters

Two basic converter topologies known as a buck converterand a boost converter are used to evaluate the effect of type-2fuzzy logic controller on the performance of DC–DC con-verters. A buck converter and a boost converter is basicallycomposed of a power switch Q which transforms the energyof a voltage source Vi, a diode D, a LC filter which reducethe high frequency components, and a resistance RL whichis used as a load. The circuit diagrams of both convertersare shown in Fig. 1. Where rC and rL define the parasiticresistances of the inductor and capacitor in the LC filters,respectively and vO is the output voltage of both converters.

In steady state operation, both converters have two oper-ating mode. In the first operating mode of the buck converter,a positive voltage across the inductor occurs while the powermosfet is turned on and the diode becomes reverse biased.This voltage causes a linear increase in the inductor current.In the second mode, when the power mosfet is turned off andthe diode is forward biased, the inductor current decrementsdue to a negative voltage across the inductor. The differen-tial equations describing the dynamics of the buck converteris obtained through the direct application of Kirchoff’s cur-rent and Kirchoff’s voltage laws for each one of the possiblecircuit topologies arising from both operating modes of thebuck converter.

diL

dt= − 1

L·[(

RL · rc

RL+rc+rL

)· iL+ RL

RL+rC· vC−vi · sw

]

(1)dvC

dt= 1

C·[

RL

RL+rC· iL− 1

RL+rC· vC

](2)

vO =[

RL · rc

RL+rc· iL+ RL

RL+rC· vC

](3)

In the first operating mode of the boost converter, whenthe power mosfet is turned on, it conducts the inductor cur-rent and the diode becomes reverse biased. This results in

Fig. 1 Equivalent circuits oftwo basic DC–DC converters.a Buck converter. b Boostconverter

Q

vi D

L

iL

rL

C

rC

RL

vC+- vO

vi

DL

iL

rL

C

rC

RL

vC+-

QvO

(a) (b)

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J Intell Manuf (2012) 23:1023–1034 1025

the input voltage of the converter across the inductor. Thisvoltage causes a linear increase in the inductor current. In thesecond mode, when the power mosfet is turned off and thediode is forward biased, the inductor current will continueto fall as long as the output voltage is greater than the inputvoltage. In a similar way, the differential equations describ-ing boost converter can be obtained by applying Kirchoff’slaws to the circuit topologies arising from both operatingmodes of the converter.

diL

dt= − 1

L·[(

(1 − sw) · RL · rc

RL + rc+ rL

)· iL

+(1 − sw) · RL

RL + rC· vC − vi

](4)

dvC

dt= 1

C·[(1 − sw) · RL

RL + rc· iL − 1

RL + rC· vC

](5)

vO =[(1 − sw) · RL · rc

RL + rc· iL + RL

RL + rC· vC

](6)

where vC is the capacitor voltage, iL is the inductor current,and sw is the switching function. The switching function swis the parameter which determines the operating mode of theconverters. The value of this parameter is either zero or one.While the first operating mode of the converters representsthat the parameter sw takes the value of 1, the operation ofthe converters on the second operating mode, the parametersw takes the value of zero. The control algorithm used in theconverters assigns the value of this parameter.

Control techniques used for the DC–DC converters

DC–DC converters use closed-loop control algorithms toachieve design objectives for line regulation, load regula-tion and dynamic response. Output voltage of converters iscontrolled with duty cycle rate. There are three basic con-trol techniques for the control of DC/DC converters: Voltagemode control technique, Feed-forward voltage mode controltechnique and Current mode control technique. An optimalmodel of the control circuit does not appear possible to say for

all applications. However, the best way to say this is not true.When compared with each other, the above three topologyhave some unique advantages and disadvantages. While thevoltage mode control is attractive for some applications,the current mode control is crucial for others (Mammano1994). In this study, the voltage-mode control technique isselected to control the buck and boost converters becauseof its simplicity. This technique consists of a voltage loopand a pulse width modulation (PWM) generator as shown inFig. 2.

The voltage loop generates the reference waveform forPWM generator by using the output voltage error and changeof error of converter. In this study, PI, T1FL and T2FL con-trollers are used instead of voltage loop. The PWM generatorobtains the required switching signals by comparing the car-rier waveform with the reference waveform obtained fromcontroller output.

Design of the PI controller

Today, PI controllers are implemented in numerous applica-tions by computer algorithms. This means that the controllerinputs are measured at certain sampling rates. Thus, it isimportant that controller equations are defined as discrete-time before making application of the controlled system. Inthe first step of the design procedure of the discrete PI con-troller, the equation of a conventional PI controller definedin the time domain is needed. Generally, the output of a con-ventional PI controller can be represented by

u(t) = K p ∗ e(t) + Ki ∗t∫

0

e(τ ).dτ (7)

where u(t) is the control signal fed to the process to be con-trolled, e(t) is the error signal: the difference between thedesired and measured process output, and K p, and Ki arethe controller constants A computer implementation of a con-ventional PI controller can be expressed as in the frequencys domain, given by

Fig. 2 Block diagram ofvoltage-mode control techniquefor a DC–DC converter system

-

-

+

DC-DCconverter-

+vi

)()1()( kfc

ukuku Δ+−=

VoltageLoop

-

+vO

dt

d

evref

+-

fcuΔ

vramp

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1026 J Intell Manuf (2012) 23:1023–1034

U (s) =(

K p + Ki

s

)· E(s) (8)

Applying the bilinear transformation,

s = 2

T·(

1 − z−1

1 + z−1

). (9)

where T > 0 is the sampling period (Tang et al. 2001).Thus, results the following discrete form to the conventionalPI controller equation,

U (z) =(

K p − Ki · T

2+ Ki · T

1 − z−1

)· E(z) (10)

Considering K0 = K p − (Ki · T/2) and K1 = Ki · T , andtaking the inverse z transform, we will have

u(k) − u(k − 1) = K0 (e(k) − e(k − 1)) + K1 · T · e(k)(11)

Dividing this equation by T , the control law for the discretePI controller can be obtained as follow.

u(k) = u(k − 1) + K0 ·�e(k) + K1 · e(k) (12)

where K0 ·�e(k) + K1 · e(k) is represented by

�u f c(k) = K0 ·�e(k) + K1 · e(k) (13)

The simplest form of the controller output can be written asfollow.

u(k) = u(k − 1) + �u f c(k) (14)

The equations associated with the output error e andchange in the output error �e of the converters are givenin the design procedure of the T1FL controller. The coeffi-cients K p and Ki of the PI controller can be adjusted usingZiegler–Nichols method and the model based methods suchas frequency response method, root locus and pole assign-ment design methods. In this study, Ziegler–Nichols methodis used to adjust the coefficients of the PI controller becauseit does not require a system model and control parameters

are designed from the plant step response. From Ziegler–Nichols method, the coefficients K p and Ki of the control-lers are obtained as 1,288 for the buck converter and as 1.2,195 for the boost converter, respectively. Then, the coeffi-cients K0 and K1 of the discrete PI controller are computedusing equations K0 = K p −(Ki · T/2) andK1 = Ki · T . Thecoefficients K0 and K1 are found as 0.9928, 0.0144 for thebuck converter and as 1.195, 0.00975 for the boost converter,respectively.

Design of the T1FL controller

Block diagram of the T1FL controller used for the control ofthe DC–DC converters is shown in Fig. 3. The T1FL control-ler is divided into four sections: fuzzifier, rule base, inferenceengine, and defuzzifier. During the fuzzification process, allinput data are classified into suitable linguistic values or sets.According to knowledge of the control rules and the linguis-tic variable definition, a fuzzy control action is derived in theinference section. Then, a crisp control action is obtained byconverting the inferred fuzzy control action in the defuzzifiersection.

The output of the fuzzy controller is the change in the ref-erence waveform which determines the change in the dutycycle rate. The reference waveform u(k) is determined byadding the previous reference waveform u(k − 1) to the cal-culated change in reference waveform �u(k), which can beexpressed as follow.

u(k) = u(k − 1) + �u f c(k) (15)

This representation is an integrating process and decreasesthe steady state error of the controlled system. The calculatedreference waveform u(k) is then sent to the PWM generator.The T1FL controller has two inputs: error e and the changein the error �e. Both inputs are created by using the outputvoltage of converters. The error for the output voltage can begiven as follow.

Fig. 3 Block diagram of theT1FL controller used for thecontrol of the DC–DCconverters

Type-1 fuzzy system

-

+

DC-DCconverter-

+vi

)()1()( kfc

ukuku Δ+−=

-

+vO

dt

d

e+

-

fcuΔ

vramp

defuzzifierinferenceengine

rule base

fuzzifier

vref

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J Intell Manuf (2012) 23:1023–1034 1027

NB NS Z PS PB1

0-1 -0.5 0 0.5 1

ee Δ,

e

e

Δμμ

NB NS Z PS PB

-1 -0.5 0 0.5 1

fcuΔ(b)(a)

Fig. 4 Input and output membership functions of the T1FL controller used for the control of DC–DC converters: a Membership functions forinput variables. b Membership functions for output variable

e(k) = vre f − vO(k) (16)

where vre f is the reference output voltage and vO(k) thesensed output voltage at the kth sampling instant. The changein the error is expressed as follow.

�e(k) = e(k) − e(k − 1) (17)

The universe of the output error and the change in theoutput error are divided into five fuzzy sets as given in Fig. 4a.Triangular membership functions are used for fuzzy setsbecause this is the simplest and the most efficient form formany applications. The membership functions are definedwith linguistic labels: Negative Big (NB), Negative Small(NS), Zero (Z), Positive Small (PS) and Positive Big (PB),respectively. Singleton membership function is used as theoutput membership functions as shown in Fig. 4b due toeasy calculation. The value of the input and output variablesis normalized in [−1,1] by using suitable scale factors.

The control rules associated with the fuzzy input and fuzzyoutput are derived from general knowledge of the converterbehavior. However, the control rules for most applicationsare usually developed using “trial and error”. In this study,the fuzzy control rules for the T1FL controller are obtainedfrom the analysis of the system behavior. The derivation ofthe fuzzy control rules is based on the following criteria: 1)when the output voltage error and the change in the outputvoltage are very big (PB or NB), the corrective response givenby the controller must be robust (duty cycle close to zero orone) in order to have the dynamic response as fast as possi-ble. 2) when the output voltage error and the change in the

output voltage approach zero(NS or PS), the controller mustbe forced to give a lower response (a small change in dutycycle). 3) when the output voltage error and the change in theoutput voltage is reached zero or is very close to this point,the controller response must kept constant (zero change induty cycle) so as to prevent overshoot. Each fuzzy rule isin form: Ri : IF e is Fi

e and �e is Fi�e THEN �u f c is wi .

Where Fie and Fi

�e are fuzzy sets in their universe of dis-course and wi is a fuzzy singleton. Twenty five fuzzy controlrules derived from these criteria are given in Table 1.

After selecting the active rules associated with the inputvalues, the fuzzy inference method whose task is the infer-ence result of each rule is determined. Although there havebeen several fuzzy inference method such as Mamdani,Larsen, and Takagi_Sugeno fuzzy implications, Mamdani’sMIN implication is commonly used in most applicationstudies. According to Mamdani’s MIN implication method,the inference result of each rule consists of two parts: thedegree of change in reference waveform wi and its weightingfactor f i . The weighting factor wi is obtained by means ofMamdani’s MIN fuzzy implication of membership degreesμFi

e(e) and μFi

�e(�e) · wi is retrieved from the control rule

table. The inferred output of each rule is written as follow.

zi = min{μFi

e(e), μFi

�e(�e)

}·wi = f i ·wi (18)

where zi is the fuzzy representation output of change in con-trol effort inferred by the i th control rule. Since zi is a lin-guistic result, a defuzzification operation is required next toobtain a crisp result.

Table 1 Fuzzy control rules ofthe T1FL controller used for theDC–DC converters

�e e

NB NS Z PS PB

NB −1 −1 −1 −0.5 0

NS −1 −1 −0.5 0 0.5

Z −1 −0.5 0 0.5 1

PS −0.5 0 0.5 1 1

PB 0 0.5 1 1 1

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1028 J Intell Manuf (2012) 23:1023–1034

Fig. 5 Block diagram of theT2FL controller used for thecontrol of the DC–DCconverters

vrefType

reducerinference

enginefuzzifier

d/dt

defuzzifierRulebase

vO

vi DC-DCconverter

vramp

)()1()( kukuku fcΔ+−=

Humanknowledge

e

fcuΔ

-0,9 -0,4 0 0,4 0,9

BPSPZBN NS

ee Δ,0-0,25 5,052,05,0-

NB NS Z PS PB

fcuΔ(b)(a)

Fig. 6 Type-2 membership functions for the T2FL controlled system. a Membership functions for the input variables. b Membership functionsfor the output variable

In the last step, the defuzzification method obtained crispresult from linguistic inference result is selected. The mostdesirable way to perform the defuzzification operation isthrough the center of gravity method, where the output ofcontroller can be computed by a logical sum of the inferenceresults of the active control rules.

�u f c =∑N

i=1 f i · wi∑Ni=1 wi

(19)

where N is the maximum number of the firing rules. In thisstudy, this number is equal to 4 since symmetric membershipfunctions is used for the input variables to T1FL controller.

Design of the T2FL controller

The block diagram of the proposed T2FL controller for DC–DC converters is given in Fig. 5. The controller mainly con-sists of five sections which are fuzzifier, rule base, inferenceengine, type-reducer and defuzzifier. In the T2FL control-ler, as in the T1FL controller, the crisp inputs are first fuzz-ified into input fuzzy sets which then activate the inferenceengine and the rule base to produce output type-2 fuzzy sets.The type-2 fuzzy outputs of the inference engine are thenprocessed by the type-reducer which combines the outputsets and then performs a centroid calculation, which leads to

type-1 fuzzy sets called type-reduced sets (Mendel 2001).Then, the defuzzifier defuzzifies the type-reduced type-1fuzzy outputs to produce crisp outputs that are used to gen-erate PWM waveform.

Interval type-2 fuzzy sets are used to fuzzify the inputvariables e and �e as they are simple to use and they dis-tribute the uncertainty evenly among all admissible primarymemberships. As in the T1FL controller, the labels of type-2fuzzy sets are assigned with Negative Big (NB), NegativeSmall (NS), Zero (Z), Positive Small (PS) and Positive Big(PB), respectively. Output membership sets representing sin-gleton upper and lower control actions are shown in Fig. 6.These sets are labelled with the same linguistic terms of theinput membership sets.

In the T2FL controller, the rules remain the same as intype-1 FL controller but the antecedents and the consequentsare represented by type-2 fuzzy sets, each of which has thefollowing form:

Ri : IF e is F̃ ie and �e is F̃ i

�e THEN �u f c is [wil , w

ir ] (20)

where F̃ ie and F̃ i

�e are the type-2 fuzzy sets for which aredefined both input variables, and wi

l and wir are the control

output obtained from system actions. The upper and lowerbound of the type-2 fuzzy sets and the control output in fuzzyrules are determined by using general knowledge of the both

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J Intell Manuf (2012) 23:1023–1034 1029

Table 2 Upper bounds of the fuzzy rules for the T2FL controlled DC–DC converters

�e e

NB NS Z PS PB

NB −0.51 −0.51 −0.51 −0.41 −0.01

NS −0.51 −0.51 −0.41 −0.01 0.41

Z −0.51 −0.41 −0.01 0.41 0.51

PS −0.41 −0.01 0.41 0.51 0.51

PB −0.01 0.41 0.51 0.51 0.51

Table 3 Lower bounds of the fuzzy rules for the T2FL controlled DC–DC converters

�e e

NB NS Z PS PB

NB −0.49 −0.49 −0.49 −0.39 0.01

NS −0.49 −0.49 −0.39 0.01 0.39

Z −0.49 −0.39 0.01 0.39 0.49

PS −0.39 0.01 0.39 0.49 0.49

PB 0.01 0.39 0.49 0.49 0.49

converter behaviors. Since there are five type-2 fuzzy sets fore and five type-2 sets for�e, twenty-five fuzzy rules for boththe upper and lower bound of T2FL controller are needed.Tables 2 and 3 show the upper and lower bounds of the fuzzyrules for the T2FL controlled DC–DC converters.

The inference engine combines the fired rules and gives amapping from input type-2 fuzzy sets to output type-2 fuzzysets. In the inference engine, antecedents in the rules are con-nected using the Meet operation, the membership grades inthe input sets are combined with those in the output sets usingthe extended sup-star composition, and multiple rules arecombined using the Join operation. Similar to type-1 fuzzysystem, the firing strength can be obtained by the followinginference process:

Fi = [ fi , f i ] (21)

where fi

and f i can be written as follow:

fi = μF̃ i

e(e)∗μF̃ i

�e(�e) (22)

f i = μF̃ i

e(e)∗μ

F̃ i�e

(�e) (23)

where μ and μ denote the grade of upper and lower member-ship functions, respectively. Symbol ∗ is the t-norm operator.In this study, the meet operation is carried out using a mini-mum t-norm operator as shown in Fig. 7.

Unlike T1FL controller, in the T2FL controller, the out-put of the inference engine must be type reduced before thedefuzzifier can be used to generate a crisp output. Thereare many various type reduction methods, such as centroid,

NS

Z

eΔ

e

min

mini

f

if

Fig. 7 The meet operation using a minimum t-norm operator in theT2FL controller

center of sets, height, and modified height for realizing typereduction process. In this study, the type-reduction procedureis fulfilled using the center of set type-reduction method, asit has reasonable computational complexity. For the T2FLcontroller, the type-reduced set can be obtained as follows:

�ucos =∫

w1∈[w1l ,w1

u ]. . .

∫

wM ∈[wMl ,wM

u ]

∫

f 1∈[ f1, f 1]

. . .

∫

f M ∈[ fM

, f M ]

1/

∑Mi=1 f i · wi

∑Mi=1 f i

= [�ul ,�ur ] (24)

where �ucos is an interval type-1 fuzzy set determined by itsleft most point �ul and its right most point �ur . To be found

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1030 J Intell Manuf (2012) 23:1023–1034

�ucos, firstly, the two-end-points of the interval type-1 fuzzyset �ul and �ur must be computed. The two-end-points �ul

and �ur can be expressed as follows, in which [wil , w

ir ] is

the consequent centroid set for the i th rule and Fi = [ fi, f i ]

corresponds to the firing strength set for the ith rule (Liangand Mendel 2000).

�ul =∑M

i=1 f il ·wi

l∑Mi=1 f i

l

(25)

and

�ur =∑M

i=1 f ir ·wi

r∑Mi=1 f i

r

(26)

In order to compute �ul and �ur , it is needed to computef il (I = 1 . . . M) and f i

r (I = 1 . . . M). In this study, this isrealized applying the minimum t-norm operator to the activerule as shown in Fig. 7. Here, the computational procedurefor �ul or �ur is briefly provided as follows. Without lossof generality, assume that the pre-computed wi

r are arrangedin ascending order; w1

r ≤ w2r ≤ · · · ≤ wM

r , then,

1) Compute �ur in (26) by initially setting f ir = (( f

i +f i )/2) for i = 1, . . .., M , where f

iand f i have been

previously computed using (22) and (23) and let �u′r =

�ur .2) Find R(1 ≤ R ≤ M −1) such that wR

r ≤ �u′r ≤ wR+1

r .3) Compute �ur in (26) with f i

r = f i for i ≤ R and

f ir = f

ifor i > R and let �u′′

r = �ur .4) If �u′′

r �= �u′r , then go to step 5. If �u′′

r = �u′r , then

set �ur = �u′′r and stop.

5) Let �u′r = �u′′

r and return to step 2.

The procedure for computing�ul is similar to the onejust given for �ur . Replace wi

r by wil and, in Step 2, find

L(1 ≤ L ≤ M − 1) such that wLr ≤ �u′

r ≤ wL+1r . Addi-

tionally, in Step 3, compute �ul in (25) with f il = f i for

i ≤ L and f il = f

ifor i > L .

After �ul and �ur are obtained, the type-reduced set canbe defuzzified to calculate the crisp output. For an intervaltype-reduced set, the defuzzified output can be calculated bygetting the average of �ul and �ur .

�u f c = �ul + �ur

2(27)

Simulation results

In order to analysis the effect on the performance of the DC–DC converters of the proposed T2FL controller, the con-trol of the DC–DC converters have been controlled using

Table 4 Circuit parameters for the buck and boost converters

Parameter Buck converter Boost converter

vi (V) 380 120

vO (V) 120 380

L(μH) 500 400

rL (m�) 10 20

C(μF) 440 1000

rC (m�) 5 12.5

fS(kHz) 20 20

not only T2FL controller but also PI controller and T1FLcontroller, whose design procedures was explained in theabove section. Simulation studies were realized by a pro-gram written C programming language, which contains thedynamic equations of the converters and the control algo-rithm related to the mentioned controllers. The simulationsare performed for the following three cases of the convert-ers: 1) the nominal case (vi = 120V and RL = 100� forthe buck converter, and vi = 380V and RL = 300� forthe boost converter); 2) the load variation case (the outputload changes: either from 100 to 10� or from 10 to 100�

for the buck converter, and either from 300 to 30� or from30 to 300� for the boost converter); and 3) the input var-iation case (falling 30% of the input voltage for both con-verters). The circuit parameters listed in Table 4 for the buckconverter and boost converter were used in the simulationstudies.

The simulation results of the PI controlled buck converterfor the three operating cases are shown in Fig. 8. Figure 8aillustrates that the settling-time, the overshoot and the steady-state error for the converter output are 3.3 ms, 1,67% and1.95 V, respectively. The converter behavior in the case ofstep load changes from full-load to light-load and vice-versais given in Fig. 8b. The controller corrects the voltage collapsewhich happens when the output load value is changed from100 to 10� at 1.71 milliseconds. When the output load valueis changed from 10 to 100�, the time needed by the control-ler to correct the voltage collapse is 1.67 milliseconds. Asunderstood from Fig. 8c, the voltage collapse due to 30% ofchange in the input voltage is corrected by controller at 1.35milliseconds. The simulation results of the T1FL controlledbuck converter are shown in Fig. 9. As shown in Fig. 9a,b and c, the settling time for the output voltage response ofthe converter is 2.6 milliseconds, the steady-state error of theconverter is 0.7 V, the transient responses of the converter forload changes are 1.4 and 1.35 milliseconds, and the transientresponse of converter for input voltage change is 0.9 milli-seconds. There isn’t any overshoot in the output voltage ofthe T1FL controlled buck converter.

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0 5 10 15 200

20

40

60

80

100

120

140

Time ( ms )

Vo(

V )

3.3ms

ess=1.95V

5 10 15 200

20

40

60

80

100

120

140

Time ( ms )

Vo(

V )

7V

1.71ms 1.67ms

6.95V

5 10 15 200

20

40

60

80

100

120

140

Time ( ms )

Vo(

V )

1.35ms

3.8V

(a)

(b)

(c)

Fig. 8 Simulation results of PI controlled buck converter. a Outputvoltage for the nominal case. b Output voltage under the load changes,c Output voltage under the input voltage change

Figure 10 shows the simulation results of the T2FL con-trolled buck converter. As shown in Fig. 10a, the settling timefor the output voltage response of converter is 0.95 millisec-onds, the steady-state error of converter is 0.38 V, and doesn’tappear any overshoot in the output voltage. The transientresponses of the T2FL controlled converter when load value

0 5 10 15 200

20

40

60

80

100

120

140

Time ( ms )

Time ( ms )

Vo(

V )

2.6ms

ess=0.7V

5 10 15 200

20

40

60

80

100

120

140

Vo(

V )

1.4ms 1.35ms

5.5V5.6V

5 10 15 200

20

40

60

80

100

120

140

Time ( ms )

Vo(

V ) 0.9ms

2.1V

(a)

(b)

(c)

Fig. 9 Simulation results of T1FL controlled buck converter. a Outputvoltage for the nominal case. b Output voltage under the load changes.c Output voltage under the input voltage change

changes from 100 to 10� and from 10 to 100� are obtainedas 0.57 and 0.55 milliseconds, respectively, as understoodfrom Fig. 10b. Figure 10c shows the transient response ofthe converter for 30% of the change in the input voltage andthe transient response for this change is obtained as 0.3 mil-liseconds. From these results, it can be seen that the T2 FL

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1032 J Intell Manuf (2012) 23:1023–1034

0 5 10 15 200

20

40

60

80

100

120

140

Time ( ms )

Time ( ms )

Vo(

V )

0.95ms

ess=0.38V

5 10 15 200

20

40

60

80

100

120

140

Vo(

V )

0.55ms0.57ms

3.8V 3.72V

5 10 15 200

20

40

60

80

100

120

140

Time ( ms )

Vo(

V ) 0.3ms

0.8V

(a)

(b)

(c)

Fig. 10 Simulation results of T2FL controlled buck converter. a Out-put voltage for the nominal case. b Output voltage under the loadchanges. c Output voltage under the input voltage change

controlled buck converter has better performance than boththe PI controlled buck converter and T1FL controlled buckconverter.

The simulation results of the PI controlled boost con-verter for the three considered cases are shown in Fig. 11.From Fig. 11a, it can be seen that the settling-time, the

0 5 10 15 200

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100

150

200

250

300

350

400

Time ( ms )

Vo(

V )

4.27ms

ess=2.5V

5 10 15 200

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100

150

200

250

300

350

400

Time ( ms )

Vo(

V )

2.4ms

8.9V

2.3ms

8.4V

5 10 15 200

50

100

150

200

250

300

350

400

Time ( ms )

Vo(

V )

4.9V

1.67ms

(a)

(b)

(c)

Fig. 11 Simulation results of PI controlled boost converter. a Outputvoltage for the nominal case. b Output voltage under the load changes.c Output voltage under the input voltage change

overshoot and the steady-state error for the converter out-put are 4.27 milliseconds, 3.68% and 2.5 V, respectively. Fig-ure 11b shows the converter behavior in the case of stepload changes from 300 to 30� and from 30 to 300�. In thatcase, the transient responses of the PI controlled boost con-verter are obtained as 2.4 milliseconds and 2.3 milliseconds,

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0 5 10 15 200

50

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150

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250

300

350

400

Time ( ms )

Vo(

V )

3.2ms

ess=1.8V

5 10 15 200

50

100

150

200

250

300

350

400

Time ( ms )

Vo(

V )

1.8ms 1.6ms

6.9V7.3V

5 10 15 200

50

100

150

200

250

300

350

400

Time ( ms )

Vo(

V )

4.1V

1.3ms

(a)

(b)

(c)

Fig. 12 Simulation results of T1FL controlled boost converter. a Out-put voltage for the nominal case. b Output voltage under the loadchanges. c Output voltage under the input voltage change

respectively. The transient response under 30% of change inthe input voltage is 1.67 milliseconds, as shown in Fig. 11c.The simulation results of T1FL controlled boost converterare given in Fig. 12. In Fig. 12a, it is shown that the settling-time, the overshoot and the steady-state error of the converter

0 5 10 15 200

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100

150

200

250

300

350

400

Time ( ms )

Time ( ms )

Vo(

V )

1.3ms

ess=0.8V

5 10 15 200

50

100

150

200

250

300

350

400

Vo(

V )

5.3V

0.68ms

5.1V

0.7ms

5 10 15 200

50

100

150

200

250

300

350

400

Time ( ms )

Vo(

V )

0.6ms

2.4V

(a)

(b)

(c)

Fig. 13 Simulation results of T2FL controlled boost converter.a Output voltage for the nominal case. b Output voltage under the loadchanges. c Output voltage under the input voltage change

are 3.2 milliseconds, no overshoot and 1.8 V, respectively.The transient responses of the converter under the same loadchanges are obtained as 1.8 and 1.6 milliseconds, as shownin Fig. 12b. The transient response of the converter underthe input voltage change is shown in Fig. 12c. The controller

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1034 J Intell Manuf (2012) 23:1023–1034

corrects the voltage collapse caused by this change at 1.3milliseconds.

The simulation results of the T2FL controlled boost con-verter are given in Fig. 13. As shown in Fig. 13a, b and c,the settling time for the output voltage response of the con-verter is 1.3 milliseconds, the steady-state error in the con-verter output is 0.8 V, the transient responses of the converterunder the both load changes are 0.7 and 0.68 milliseconds,and the transient response of the converter under the inputvoltage change is 0.6 milliseconds. There isn’t any overshooton the converter output.

From these results, it can be understood that the perfor-mance of the T2FL controller is better than that of both thePI controller and T1FL controller for the control of the boostconverter, as in the buck converter. Thus, the T2FL con-troller design method is highly suitable to apply to DC–DCconverters, compared with the PI controller design and T1FLcontroller design.

Conclusions

In this study, a T2FL controller was proposed for the controlof the two main DC–DC converters: a buck converter anda boost converter to obtain a good performance and dem-onstrated that the T2FL controller could effectively controlthese converters. In order to determine the effectiveness of theT2FL controller on the performance of the DC–DC convert-ers, the converters were also controlled using PI and T1FLcontrollers. To compare with the converter performances, thesettling time, the overshoot, steady-state error, and the tran-sient responses of the converter under the load change andthe input voltage change were taken as the performance cri-teria. Simulation results showed that T1FL controller usedfor the control of the both DC–DC converters had a betterperformance in terms of the mentioned performance crite-ria than the PI controller under the three operating cases ofthe converters: the nominal case, the input voltage changecase and the output load change. When compared with T1FLcontroller and T2FL controller, it was shown that the T2FLcontroller provided a better performance in the control ofthe converters because the T2FL controller was able to han-dle uncertainties in rules and parameters of input member-ship functions occurring T1FL controller. In conclusion, the

simulation result confirmed that the T2FL controller wasmore suitable for application to DC–DC converters than boththe PI controller and T1FL controller.

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