ISA Transactions 51 (2012) 673–681

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ISA Transactions

0019-05

http://d

n Tel.:

E-m

journal homepage: www.elsevier.com/locate/isatrans

Adaptive terminal sliding-mode control strategy for DC–DC buck converters

Hasan Komurcugil n

Computer Engineering Department, Eastern Mediterranean University, Gazi Magusa, North Cyprus, via Mersin 10, Turkey

a r t i c l e i n f o

Article history:

Received 2 January 2012

Received in revised form

22 June 2012

Accepted 17 July 2012Available online 9 August 2012

This paper was recommended for publica-

tion by Jeff Pieper

Keywords:

Sliding-mode control

Terminal sliding-mode control

Finite time convergence

DC–DC buck converter

78/$ - see front matter & 2012 ISA. Published

x.doi.org/10.1016/j.isatra.2012.07.005

þ90 392 6301363; fax: þ90 392 3650711.

ail address: hasan.komurcugil@emu.edu.tr

a b s t r a c t

This paper presents an adaptive terminal sliding mode control (ATSMC) strategy for DC–DC buck

converters. The idea behind this strategy is to use the terminal sliding mode control (TSMC) approach to

assure finite time convergence of the output voltage error to the equilibrium point and integrate an

adaptive law to the TSMC strategy so as to achieve a dynamic sliding line during the load variations. In

addition, the influence of the controller parameters on the performance of closed-loop system is

investigated. It is observed that the start up response of the output voltage becomes faster with

increasing value of the fractional power used in the sliding function. On the other hand, the transient

response of the output voltage, caused by the step change in the load, becomes faster with decreasing

the value of the fractional power. Therefore, the value of fractional power is to be chosen to make a

compromise between start up and transient responses of the converter. Performance of the proposed

ATSMC strategy has been tested through computer simulations and experiments. The simulation

results of the proposed ATSMC strategy are compared with the conventional SMC and TSMC strategies.

It is shown that the ATSMC exhibits a considerable improvement in terms of a faster output voltage

response during load changes.

& 2012 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

DC–DC converters are power electronics devices which arewidely used in many applications including DC motor drives,communication equipments, and power supplies for personalcomputers [1]. The buck type DC–DC converters are used inapplications where the required output voltage is smaller thanthe input voltage. Since buck converters are inherently nonlinearand time-varying systems due to their switching operation, thedesign of high performance control strategy is usually a challen-ging issue. The main objective of the control strategy is to ensuresystem stability in arbitrary operating condition with gooddynamic response in terms of rejection of input voltage changes,load variations and parameter uncertainties. Nonlinear controlstrategies are deemed to be better candidates in DC–DC converterapplications than other linear feedback controllers. Various non-linear control strategies for the buck converters have beenproposed to achieve these objectives [2–15]. Among these controlstrategies, the sliding mode control (SMC) has received muchattention due to its major advantages such as guaranteed stabi-lity, robustness against parameter variations, fast dynamicresponse and simplicity in implementation [2–4,6–8,10,11,14].The design of an SMC consists of two steps: design of a sliding

by Elsevier Ltd. All rights reserve

surface and design of a control law [8]. Once a suitable slidingsurface and a suitable control law are designed, the system statescan be forced to move toward the sliding surface and slide on thesurface until the equilibrium (origin) point is reached.

The SMC introduced in [3] has the advantages of separateswitching action and the sliding action, but the computationrequirement of the inductor’s current reference function increasesthe complexity of the controller. A simple and systematicapproach to the design of practical SMC has been presentedin [6]. The adaptive feedforward and feedback based SMC strategyintroduced in [7] has the advantages of adjusting the hysteresiswidth according to the input voltage change and the slidingcoefficient according to the load change. The indirect SMC viadouble integral sliding surface strategy introduced in [10] reducesthe steady-state error in the output voltage at the expense ofhaving additional two states in the sliding surface function. In[13], a time-optimal based SMC has been introduced aiming toimprove the output voltage regulation of the converter subjectedto any disturbance. The SMC strategy in [14] is based on thealternative model of the buck converter with bilinear terms.

In most SMC strategies proposed for the buck converters so far,the most commonly used sliding surface is the linear slidingsurface which is based on linear combination of the system statesby using an appropriate time-invariant coefficient (commonlytermed as l). The use of such coefficient makes the sliding linestatic during load variations resulting in a poor transient responsein the output voltage. Despite the transient response can be made

d.

H. Komurcugil / ISA Transactions 51 (2012) 673–681674

faster by utilizing a larger valued coefficient in the linear slidingsurface function, the system states cannot converge to theequilibrium point in finite time. Different from the conventionalSMC, the terminal sliding mode control (TSMC) has a nonlinearsliding surface function [18]. The nonlinear sliding surface func-tion has the ability to provide a terminal convergence (finite-timeconvergence) of the output voltage error from an initial point tothe equilibrium point.

In this paper, an adaptive terminal sliding mode control(ATSMC) strategy is proposed for the DC–DC buck converters.The idea behind this strategy is to use the TSMC for assuring finitetime convergence of the output voltage error to the equilibriumpoint and integrate an adaptive law to the TSMC strategy so as tomake the sliding line dynamic during the load variations. Differ-ent from the linear sliding surface function in conventional SMC,the output voltage error (x1) in the nonlinear sliding surfacefunction has a fractional power (commonly termed as g) in theTSMC. Therefore, the convergence time of the output voltagedepends on the parameters l and g. The influence of the fractionalpower on the start up and the transient responses of the outputvoltage, the inductor current and the state trajectory is investi-gated. Since the performance of the TSMC with fixed l (i.e. staticsliding line) during load variations does not exhibit the desiredresponse, an adaptive terminal sliding mode control (ATSMC)strategy has been employed in which sliding line is madedynamic by making l load dependent. Finally, simulation andexperimental results are presented for verification.

The rest of this paper is organized as follows. In Section 2, thedynamic model of buck converter is given. Section 3 reviews theconventional sliding mode control method for the buck converter.In Section 4, the proposed adaptive terminal sliding mode controlwas described for the buck converter. In Section 5, the simulationand experimental results are presented and compared with theconventional sliding mode control method. Finally, the conclu-sions are addressed in Section 6.

2. Dynamic model of the DC-DC buck converter

Fig. 1 shows a DC–DC buck converter. It consists of a DC inputvoltage source (Vin), a controlled switch (Sw), a diode (D), a filterinductor (L), filter capacitor (C), and a load resistor (R). Equationsdescribing the operation of the converter can be written for theswitching conditions ON and OFF, respectively, as

diL

dt¼

1

LðVin�voÞ ð1Þ

dvo

dt¼

1

CiL�

vo

R

� �ð2Þ

and

diL

dt¼�

vo

Lð3Þ

+

C

L

D

+

SW

– –

RvoVin

iL

iRiC

Fig. 1. DC–DC buck converter.

dvo

dt¼

1

CiL�

vo

R

� �ð4Þ

Combining (1), (2), (3) and (4) gives

diLdt¼

1

LðuVin�voÞ ð5Þ

dvo

dt¼

1

CiL�

vo

R

� �ð6Þ

where u is the control input which takes 1 for the ON state of theswitch and 0 for the OFF state. Let us define the output voltageerror x1 and its derivative (rate of change of the output voltageerror) as

x1 ¼ vo�Vref ð7Þ

x2 ¼ _x1 ¼ _vo�_V ref ¼ _vo ð8Þ

where _x1 denotes the derivative of x1, and Vref is the DC referencefor the output voltage.

By taking the time derivative of (6), the voltage error x1 andthe rate of change of voltage error x2 dynamics can be expressedas

_x1 ¼ x2 ð9Þ

_x2 ¼�x2

RC�o2

ox1þo2o ðuVin�Vref Þ ð10Þ

where o2o ¼ 1=LC.

3. Conventional sliding mode control

Let a linear sliding surface function S be expressed as

S¼ lx1þx2, l40 ð11Þ

where l is a time-invariant sliding coefficient. The dynamicbehavior of (11) without external disturbance on the slidingsurface is [16]

S¼ lx1þ _x1 ¼ 0 ð12Þ

In the phase-plane (x1�x2 plane), S¼0 represents a line, calledsliding line, passing through the origin with a slope equal tom¼�l. The sliding mode (S¼0) is described by the followingfirst-order equation:

_x1 ¼�lx1 ð13Þ

During the sliding mode, the output voltage error is expressedas

x1ðtÞ ¼ x1ð0Þe�lt ð14Þ

It should be noted that l must be positive for achieving thesystem stability. This fact can be easily verified by substituting anegative l quantity into (14) which results in x1(t) moving awayfrom zero.

In general, the SMC exhibits two modes: the reaching modeand the sliding mode. While in the reaching mode, a reachingcontrol law is applied to drive the system states to the sliding linerapidly. When the system states are on the sliding line, the systemis said to be in the sliding mode in which an equivalent controllaw is applied to drive the system states, along the sliding line, tothe origin. When the state trajectory is above the sliding line, u¼0(Sw is OFF) must be applied so as to direct the trajectory towardsthe sliding line. Conversely, when the state trajectory is below thesliding line, u¼1 (Sw is ON) must be applied so that the trajectoryis directed towards the sliding line. The control law that adopts

Fig. 2. Regions of existence of the sliding mode for: (a) l41/RC and (b) lo1/RC.

H. Komurcugil / ISA Transactions 51 (2012) 673–681 675

such switching can be defined as

u¼1

2ð1�signðSÞÞ ¼

�1 if So0

0 if S40ð15Þ

However, direct implementation of this control law causes theconverter to operate at an uncontrollable infinite switchingfrequency which is not desired in practice. Hence, it is requiredto suppress the switching frequency into an acceptable range. Inorder to accomplish this, a hysteresis modulation (HM) methodemploying a hysteresis band with suitable switching boundariesaround the switching line is used as [6]

u¼1 when So�h

0 when S4h

(ð16Þ

where h is the hysteresis bandwidth. When S4h, switch Sw willturn OFF. Conversely, it will turn ON when So�h. Such operationlimits the operating frequency of the switch.

When the system is in the sliding mode, the robustness of theconverter will be guaranteed and the dynamics of the converterwill depend on l. In order to ensure that the movement of theerror variables is maintained on the sliding line, the followingexistence condition must be satisfied

S _So0 ð17Þ

The time derivative of (11) can be written as

_S ¼ l _x1þ _x2 ð18Þ

Substituting (9) and (10) into (18) yields the followinginequalities [6]:

l1 ¼ l�1

RC

� �x2þo2

oðVin�Vref�x1Þ40 for So0 when u¼ 1

ð19Þ

l2 ¼ l�1

RC

� �x2�o2

o ðVref þx1Þo0 for S40 when u¼ 0 ð20Þ

Equations l1¼0 and l2¼0 define two lines in the phase-planewith the same slope passing through points P1¼(Vin�Vref,0) andP2¼(�Vref,0) on the x1 axis, respectively. The slope of these linesis given by

m1 ¼m2 ¼o2

o

ðl�1=RCÞð21Þ

The regions of existence of the sliding mode for different l values(l41/RC and lo1/RC) are depicted in Fig. 2. It is clear that thesliding line splits the phase-plane into two regions. In each region, thestate trajectory is directed towards the sliding line by an appropriateswitching action. The sliding mode occurs only on the portion of thesliding line, S¼0, that covers both regions. This portion is within S1

and S2. It can be seen from Fig. 2(a) that the large l value causes areduction of sliding mode existence region. When the state trajectoryhits the sliding line in a point outside the sliding mode existenceregion S1S2, it overshoots the sliding line which leads to an overshootin the output voltage. The x2 intercepts of the lines l1 and l2 areY1 ¼o2

oðVref�VinÞ=ðl�1=RCÞ and Y2 ¼o2oVref =ðl�1=RCÞ, respec-

tively. On the other hand, when l is small, the state trajectory hitsthe sliding line in a point inside the sliding mode existence regionS1S2, and it moves toward the origin as seen in Fig. 2(b). Note thatwhen lo1/RC, the slope of these lines is negative which changes thex2 intercepts of the lines l1 and l2 as Y2 ¼o2

oVref =ðl�1=RCÞ andY1 ¼o2

oðVref�VinÞ=ðl�1=RCÞ, respectively. It is evident from (14)and Fig. 2 that the dynamic response of the buck converter dependson the value of l. In order to ensure that l is large enough for fastdynamic response and low enough to retain a large existence region,

it has been proposed in [6] to set l as

l¼1

RCð22Þ

However, despite the dynamic response can be made faster byutilizing a large l in the sliding surface function, the system statesstill cannot converge to the equilibrium point in finite time.

4. Adaptive terminal sliding mode control for buck converter

Let a nonlinear sliding surface function for the buck convertersystem defined in (9) and (10) be defined as

Sn ¼ lxg1þ _x1 ð23Þ

where l40, and 0o(g¼q/p)o1 where p and q are positive oddintegers satisfying p4q. When the system is in the terminalsliding mode (Sn¼0), its dynamics can be determined by thefollowing nonlinear differential equation:

_x1 ¼�lxg1 ð24Þ

Note that Eq. (24) reduces to _x1 ¼�lx1 for g¼1, which is theform of conventional SMC. It has been shown in [17] that x1¼0 isthe terminal attractor of the system defined in (1). The term‘‘terminal’’ is referred to the equilibrium which can be reached infinite time. Note that Eq. (24) can also be written as

dt¼�dx1

lxg1ð25Þ

Taking integral of both sides of (25) and evaluating theresulting equation on the closed interval (x1(0)a0, x1(ts)¼0)

0

Fig. 3. Simulated start up responses of the output voltage, the inductor current,

and the state trajectories obtained by the TSMC method with different g values:

(a) vo, (b) iL, and (c) state trajectories.

H. Komurcugil / ISA Transactions 51 (2012) 673–681676

gives the following equation [18]:

ts ¼�1

l

Z 0

x1ð0Þ

dx1

xg1¼

x1ð0Þ�� ��1�glð1�gÞ

ð26Þ

Eq. (26) means that when the system enters to the terminalsliding mode at t¼tr with initial condition x1(0)a0, the systemstate x1 converges to x1(ts)¼0 in finite time and stay there fortZts. In other words, when the state trajectory hits the slidingsurface at time tr, the system state cannot leave the sliding linemeaning that the state trajectory will belong to the sliding line fortZtr. However, it is obvious from (26) that the convergence time,ts, still depends on the parameters g and l. Therefore, theseparameters must be carefully selected to ensure the desiredresponse.

4.1. Selection of g

When x1 is near the equilibrium point (9x19o1), the fractionalpower g¼q/p leads to 9xg1949x19. In such a case, the system statewith the nonlinear term xg1 converges toward equilibrium pointmore faster than the linear term x1. Especially, when g is too smallnear the equilibrium point, the state trajectory moves toward theequilibrium point as if it follows the bottom of a bowl rather thanfollowing a straight line which results in a distorted inductorcurrent during load variations. In order to show the influence ofparameter g on the dynamic performance of the TSMC, a samplesimulation study has been performed by using the parametersVin¼10 V, Vref¼5 V, L¼1 mH, C¼1000 mF, R¼10 O, andl¼1/RC¼100.

Fig. 3 shows the start-up responses of the output voltage, theinductor current and the state trajectories of the converter withdifferent g values. It is clear from Fig. 3(a) and (b) that the outputvoltage and the inductor current responses become faster withincreasing the value of g. The main reason of this comes from thefact that the slope of the sliding line with g4¼0.8182 is greaterthan all other sliding line slopes as shown in Fig. 3(c) resulting ina faster response as pointed out in (14). All the state trajectoriesstart from the point �Vref¼�5 V on the x1 axis (as vo(0)¼0 V),which agrees well with the intersection point of l2 shown in Fig. 2.When the trajectory reaches the sliding line, it starts to slidealong it by making a zigzag movement. However, when itapproaches the equilibrium point (9x19o1) it changes its direc-tion and makes a movement similar to the bottom of a bowl. Thisdirection change is inversely proportional with the g value. Thismeans that any change in the direction becomes smaller when gvalues get larger. This fact is clearly visible in Fig. 3(c).

Fig. 4 shows the responses of the output voltage and theinductor current for a step change in R from 10 O to 2 O whichare obtained by the SMC method with g¼1 and the TSMC methodwith different g values. Unlike the start up case, it is interesting tonote that the output voltage and the inductor current responsesbecome faster with decreasing the value of g. Therefore, g value isto be chosen to make a compromise between start up and transientresponses of the converter. It is well known that the currentdynamics is faster than the voltage dynamics. Since the transientresponses of the inductor current for different g values are super-imposed, then only two responses are presented in Fig. 4(b).

4.2. Selection of l

It has been discussed before that l is usually set to 1/RC for agood performance of the system. However, the TSMC withconstant l cannot exhibit the same transient performance whenthe converter is subjected to a load change. The main reason ofthis performance degrade is due to the static sliding line

irrespective of the operating point change caused by the loadchange. It is clear from (21) that l is inversely proportional to R.Therefore, instead of fixing it at l¼1/RC with nominal loadresistance, its value should be adaptively changed when a loadchange occurs. However, it is not possible to measure the loadresistance directly in a practical implementation. Therefore, theinstantaneous value of the load resistance can be estimated bymeasuring the output voltage across and the current passingthrough the load resistance as

R̂¼vo

iRð27Þ

Fig. 4. Simulated transient responses of the output voltage and the inductor

current due to a step change in R from 10 O to 2 O obtained by the SMC and the

TSMC methods with different g values: (a) vo and (b) iL.

Fig. 5. Block diagram of the DC–DC buck converter with the proposed ATSMC method.

Fig. 7. Switching frequency for different input voltages.

Fig. 6. Sliding function with hysteresis band.

H. Komurcugil / ISA Transactions 51 (2012) 673–681 677

Since the value of R is used to vary l in the TSMC, such controlresults in an adaptive terminal sliding mode control (ATSMC). Theadaptation comes from the time variation of the l term in thenonlinear terminal sliding mode. The block diagram of the DC-DCbuck converter with the proposed ATSMC method is depicted inFig. 5.

4.3. Stability analysis and sliding mode dynamics

The sufficient condition for the existence of the terminalsliding mode is given by

Sn_Sno0 ð28Þ

If a control input is designed which ensures (28), then thesystem will be forced towards the sliding surface and remains onit until origin is reached asymptotically. Now, let

VðtÞ ¼ 12S2

n ð29Þ

be a Lyapunov function candidate for the system described in (9)and (10). The time derivative of (29) can be written as

_V ðtÞ ¼ Sn_Sno0 ð30Þ

Differentiating (23) with respect to time and using in (30), onecan obtain

_V ðtÞ ¼ Sn_Sn ¼ Snðlgxg�1

1 x2þ _x2Þo0 ð31Þ

Substituting (10) into (31) and solving for u gives

ueq ¼1

o2oVin

x2

RCþo2

o ðVref þx1Þ�lgxg�11 x2

� �ð32Þ

Eq. (32) is the equivalent control needed to keep the systemmotion on the sliding surface under ideal terminal sliding mode.However, the equivalent control may not be able to move thesystem states from reaching mode to the sliding mode. Therefore,an additional control action known as the switching control isneeded that should be applied to the system together with theequivalent control. Hence, the total control input can be writtenas

u¼1

o2oVin

x2

RCþo2

o ðVref þx1Þ�lgxg�11 x2�KsignðSnÞ

� �ð33Þ

H. Komurcugil / ISA Transactions 51 (2012) 673–681678

where K denotes the switching control gain. Rewriting (10) as

_x2 ¼�x2

RC�o2

ox1þo2oðuVin�Vref ÞþdðtÞ ð34Þ

Fig. 8. Switching frequency for different output voltage references.

SMC

SMC

TSMC

ATSMC

TSMCandATSMC

1.66ms

Fig. 9. Simulated start up and transient responses (due to a step change in R from 10

TSMC and the ATSMC methods with g¼0.2: (a) vo, (b) magnified response of vo and iL o

obtained by the TSMC and the ATSMC.

and substituting (33) into (34) and the resulting equation into(31) yields

_V ðtÞ ¼ Sn_Sn ¼ ð�KSnsignðSnÞþSndðtÞÞo0 ð35Þ

where satisfies K49d(t)9, and d(t) denotes the disturbancescaused by parametric variations in the system. Clearly, for bothSn40 and Sno0, _V ðtÞ is always negative for all values of thesystem states which means that the system has a finite-timeconvergent stability irrespective of the disturbances in the sys-tem. This means that the ATSMC offers a strong robustnessagainst the variations in input voltage and reference outputvoltage during the sliding mode. It should be noted that theATSMC method does not offer a strategy to estimate the domainof attraction around the equilibrium point. Although, it is verydifficult to estimate the domain of attraction analytically, thereare some recent works that try to estimate it [19,20].

Substituting (9) and (10) into _Sn ¼ lgxg�11

_x1þ _x2 and usingEq. (15) give the following inequalities that satisfy the existencecondition given in (28):

lgxg�11 x2�

x2

RCþo2

oðVin�Vref�x1Þ40 for So0 when u¼ 1 ð36Þ

lgxg�11 x2�

x2

RC�o2

oðVref þx1Þo0 for S40 when u¼ 0 ð37Þ

Eqs. (36) and (37) describe the sliding mode dynamics of theclosed-loop system.

TSMCandATSMC

ATSMC

State trajectory dueto the step change

State trajectory dueto the start up

State trajectory dueto the step change

State trajectory dueto the start up

O to 2 O) of the output voltage, and the state trajectory obtained by the SMC, the

btained by ATSMC, (c) state trajectory obtained by the SMC and (d) state trajectory

H. Komurcugil / ISA Transactions 51 (2012) 673–681 679

4.4. Switching frequency

The frequency at which a practical buck converter can beswitched is limited by such factors as hysteresis bandwidth, inputvoltage, output voltage, and LC filter. Therefore, an estimate of theswitching frequency would be helpful in designing the system.Consider a typical trajectory of the sliding function with hyster-esis band depicted in Fig. 6. From the geometry of Fig. 6, we canwrite the ON and OFF periods of switch Sw as

TON ¼2h

_Sþ

n

ð38Þ

TOFF ¼�2h_S�

n

ð39Þ

where _Sþ

n and _S�

n are the time derivatives of Sn for ON and OFFstates of the switch Sw, respectively. The time derivative of (23)can be written as

_Sn ¼ lgxg�11 x2�

x2

RCþo2

o ðuVin�Vref�x1Þ ð40Þ

Assuming that the errors x1 and x2 are negligible in the steady-state, Eqs. (38) and (39) can be written as

TON ¼2h

o2oðVin�Vref Þ

ð41Þ

TOFF ¼2h

o2oVref

ð42Þ

Hence, the expression for the switching frequency can beobtained as

f s ¼1

TONþTOFF¼o2

oVref

2h1�

Vref

Vin

� �ð43Þ

Clearly, the switching frequency is inversely proportional tothe hysteresis bandwidth (h). The hysteresis bandwidth is chosenso as to obtain a suitable switching frequency given in (43).

Fig. 10. Experimental responses of the output voltage and the inductor current for

start up and a step change in R from 10 O to 2 O: (a) start up and (b) step change.

In order to investigate the influence of other parameters, theconverter’s switching frequency is computed by using (43) withL¼1 mH, C¼1000 mF, and h¼100 for different input voltages anddifferent output voltage references. Figs. 7 and 8 show theswitching frequency (fs) for different input voltages (Vin) atVref¼5 V and different output voltage references (Vref) atVin¼10 V, respectively. It can be seen that fs increases withincreasing Vin. As for output voltage reference variation, it isobserved that fs decreases while Vref increases. Note that bychanging the hysteresis bandwidth, it is possible to keep theswitching frequency constant against input voltage variation atthe expense of additional control complexity requirements.

5. Simulation and experimental results

In order to demonstrate the performance of the proposedATSMC strategy, the DC–DC buck converter system has beentested by simulations and experiments. Simulations are carriedout using Simulink of Matlab with a step size of 2 ms. Experi-mental results were obtained from a hardware setup constructedin the laboratory. The parameters of the system are Vin¼10 V,Vref¼5 V, h¼80, L¼1 mH, and C¼1000 mF.

Fig. 9 shows the simulated start up and transient responses ofthe output voltage and the state trajectories obtained by SMC,TSMC and ATSMC strategies. The value of g used in the TSMC and

Fig. 11. Experimental responses of the inductor current and the control input for

the step change in the load resistance from 10 O to 2 O: (a) responses of iL and u,

(b) magnified responses of iL and u for R¼10 O and (c) magnified responses of iLand u for R¼2 O.

Fig. 13. Experimental state trajectory in the steady-state.

H. Komurcugil / ISA Transactions 51 (2012) 673–681680

ATSMC strategies was 0.2 (q¼1 and p¼5). The load resistor wasset to R¼10 O during the start up. The transient responses aredue to a step change in R from 10 O to 2 O at t¼0.08 s. It isobvious from Fig. 9(a) that the output voltage tracks its referencesuccessfully in all cases. The start up response of the outputvoltage obtained by the SMC is faster than that of obtained by theTSMC and ATSMC. Conversely, the transient responses of theoutput voltage obtained by the TSMC and ATSMC are faster thanthat of obtained by the SMC. The ATSMC offers the fastesttransient response which shows that the controller acts very fastin correcting the output voltage. Note that the TSMC and ATSMCexhibit exactly the same response from the start up until the stepchange occurs in the load resistance. The main reason of thiscomes from the fact that both methods employ the same l valueuntil the load change takes place. After the load change occurs,while the TSMC continues to operate with the same l value, theATSMC makes use of the adaptively computed l value. Themagnified response of the output voltage, due to the step changein the load, together with the inductor current obtained by theATSMC method is shown in Fig. 9(b). The output voltage takesapproximately 1.66 ms to track its reference. Fig. 9(c) and (d)shows the state trajectories that correspond to the simulationcase shown in Fig. 9(a). Note that both trajectories start from thepoint �Vref¼�5 V on the x1 axis, since vo(0)¼0 V. In the start up,the slope of the sliding line in Fig. 9(d) is smaller than that of inFig. 9(c). However, after the load is changed, the slope of thesliding line in Fig. 9(d) becomes greater than that of inFig. 9(c) leading to a faster response in the output voltage.

Experimental results were also obtained for testing the dynamicperformance of the proposed ATSMC method under step changes inthe load, in the input voltage and in the reference output voltage.Fig. 10 shows the experimental responses of the output voltage andthe inductor current for the start up and the step change in the loadresistance from 10 O to 2 O. It can be seen from Fig. 10(a) that the

Fig. 12. Experimental responses of the output voltage for step changes in the

reference output voltage and the input voltage: (a) response of vo for a step

change in Vref from 5 V to 7 V and (b) response of vo for a step change in Vin from

10 V to 8 V.

start up responses of the output voltage and inductor current are ingood agreement with the simulation results shown in Fig. 9(a) andFig. 3(b), respectively. Fig. 10(b) shows the experimental result thatcorresponds to the simulation result presented in Fig. 9(b). It tookabout 2 ms for the controller to correct the output voltage at 5 V. Thesmall discrepancy between the simulation and experimental resultscomes from the component tolerances, and non-ideal effects (e.g.,finite time delay) in the practical system which cause the outputvoltage to make slower transitions.

Fig. 11 shows the response of the inductor current and thecontrol input for the step change in the load resistance from 10 Oto 2 O. Fig. 11(a) shows the magnified response of the inductorcurrent that is shown in Fig. 10(b) and the control input. Themagnified responses of the control input for R¼10 O and R¼2 Oare shown in Fig. 11(b) and (c), respectively. The switchingfrequency of the converter is measured approximately as fs¼

1/64 ms¼15.625 kHz. Note that one can obtain the same result byusing (43) with Vin¼10 V, Vref¼5 V, h¼80, L¼1 mH, andC¼1000 mF. This shows that the switching frequency computa-tion is fairly accurate.

Fig. 12 shows the response of the output voltage for stepchanges in the reference output voltage (Vref) from 5 V to 7 V andthe input voltage (Vin) from 10 V to 8 V. It is clear fromFig. 12(a) that the output voltage tracks its reference faster andsuccessfully. Also, as shown in Fig. 12(b), the output voltage isalmost not affected for the step change in the input voltage. Theresults presented in Fig. 12 show that the ATSMC method is veryrobust against input and reference voltage variations.

Fig. 13 shows the state trajectory in the steady-state.

6. Conclusions

An adaptive terminal sliding mode control (ATSMC) strategy,which assures finite time convergence of the output voltage errorto the equilibrium point, has been proposed for DC–DC buckconverters. The influence of the controller parameters on the perfor-mance of closed-loop system is investigated. It is observed that thestart up response of the output voltage becomes faster when thevalue of the fractional power is increased. On the other hand, thetransient response of the output voltage caused by the step change inthe load resistance becomes faster when the value of the fractionalpower is decreased. Therefore, one should consider a compromisebetween start up and transient responses when choosing the value ofthe fractional power. Simulation and experimental results show thatthe ATSMC strategy, as compared to the conventional SMC and TSMCstrategies, is quite successful in obtaining very fast output voltageresponses to load disturbances.

H. Komurcugil / ISA Transactions 51 (2012) 673–681 681

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