Design and analysis of adaptive sliding-mode-like controller for DC-DC converters

Design and analysis of adaptive sliding-mode-likecontroller for DC–DC converters

Y. He and F.L. Luo

Abstract: An adaptive sliding-mode-like controller for DC–DC buck converters is proposed.Constant switching frequency can be achieved with the proposed approach. An adaptive item isincluded to adjust the controller parameter online and eliminate the steady-state error. Theadaptive scheme is based on a feedback policy that suitably modifies the corresponding powerlosses term in the controller equation. In addition, this adaptive term will speed up the systemresponse and will stabilise the error dynamics of the closed-loop control system. The controllerperformance can be examined via traditional small-signal analysis. The implementation of theproposed closed-loop control scheme is illustrated with simulation and experiment.

1 Introduction

DC–DC converters have been widely used in industrialapplications such as DC motor drives, computer systemsand communication equipment [1–4]. Since DC–DCconverters are nonlinear time-varying systems, they repre-sent a big challenge for control design. Traditional studyhas been focused on the application of linear control theorysomehow to derive linearised models [5–7]. These modelsdescribe the system behaviour very well around any fixedoperating point and give simple solutions of feedbackcontrollers. However, the control performance deterioratesunder large disturbances of circuit conditions.

Recently, much attention has been paid to the nonlinearcontrol solutions. DC–DC converter systems under properpulsed nonlinear control are more robust and have fasterdynamic response. They can also provide better rejection ofpower-source perturbation than the same systems underlinear feedback control. Many nonlinear control schemes,such as one-cycle control, current-mode control and HN,and sliding-mode control (SMC), have been proposed forDC–DC converters [8–14]. Among them, SMC hasadvantages such as simple implementation, insensitive toparameter changes and first-order response. For example,in [15] and [16], sliding-mode controllers with offset havebeen proposed to achieve better start-up performance. Theauthors of [17] suggested a time-varying sliding surface sothat the resultant system has better performance underdecreasing load. However, SMC also has some disadvan-tages. For example, the switching frequency of DC–DCconverters under SMC varies depending on a number ofvariables, including the input supply voltage and theequivalent series resistance of the output filter capacitor.Non-zero steady-state error is another disadvantage ofSMC.

Adaptive hysteresis control [18] may be helpful for fixingthe switching frequency of the SMC. However, steady-stateerror cannot be fully removed. In [19], sliding-mode-likecontrol has been proposed to solve problems of the SMCwhile retaining its advantages. However, this controller mayslow down the response speed to achieve the limited dutycycle output so that the constant switching frequency isensured. Another direction is equivalent control. Reference[20] proposed the sliding surface only with the outputvoltage sensed. Although constant switching frequency isachieved, there are low-frequency oscillations on the outputwaveform. A different sliding-mode-like controller imple-mented in analogue control is proposed in [21]. Thecontroller does not need exoteric limit. The controlperformance is exactly like the same version in traditionalSMC with the same parameters. The switching frequencyis fixed by the input sawtooth waveform. However, thecontroller is based on an exact model without consideringpower losses. Power losses can also be considered indesigning SMC for converters [22].

This paper therefore tries to improve the previouscontroller [21] by eliminating the steady-state error andalso retaining the excellent response. A detailed model ofthe converter is derived first. The power losses items areconsidered in the derivation. Then this model is simplifiedto incorporate the proposed simple adaptive algorithm.Finally, the proposed controller is applied. Small-signalanalysis is conducted on the proposed control scheme.Simulation and experimental results are also given.

In this paper, the PWM repeating period is T and theswitching frequency f¼ 1/T. The duty cycle can berepresented as a scalar input u. The instantaneous valuesof current and voltage are iX and vX for the component X.Their corresponding average values are IX and VX. Theinput voltage and input current are Vin and Iin. The outputvoltage and output current are Vout and Iout. The voltagetransfer gain is thus M¼Vout/Vin¼ Iin/Iout.

2 DC–DC converter model with power losses

Generally, power losses in DC–DC converters mainly comefrom three parts: the power dissipation in the MOSFETmodelled as a switch on resistor rS; the power dissipationdue to parasitic effect of the inductor modelled as resistor rLE-mail: [email protected]

The authors are with the Centre for Advanced Power Electronics, School ofEEE, Block S1, Nanyang Technological University, Nanyang Avenue,Singapore 639798

r The Institution of Engineering and Technology 2006

IEE Proceedings online no. 20050222

doi:10.1049/ip-epa:20050222

Paper first received 6th June and in final revised form 28th November 2005

IEE Proc.-Electr. Power Appl., Vol. 153, No. 3, May 2006 401

in series; the diode forward voltage drop modelled as a turn-on voltage source VD. The equivalent series resistance ofa capacitor is very small and thus ignored in this paper.A buck converter with the above-mentioned power losses isshown in Fig. 1.

The state vector is chosen as X ¼ ½x1; x2T ¼ ½iL; voutT .Figure 1b shows the equivalent circuit when switch S is on.The state-space equation is written as:

_X ¼ A1X þ B1

A1 ¼ðrS þ rLÞ

L1L

1

C 1

RC

2664

3775; B1 ¼

Vin

L0

24

35 ð1Þ

When switch S is off, the state-space equation of buckconverter is derived as

_X ¼ A2X þ B2

A2 ¼ rL

L1L

1

C 1

RC

264

375; B2 ¼

VD

L0

24

35 ð2Þ

Combining (1) and (2), the state-space averaged model ofthe buck converter can be derived as

_X ¼ A3X þ B3

A3 ¼ A1uþ A2ð1 uÞ ¼ rSuþ rL

L1L

1

C 1

RC

2664

3775

B3 ¼ B1uþ B2ð1 uÞ ¼Vinu VDð1 uÞ

L

0

264

375

ð3Þ

The steady-state voltage transfer gain M can be obtained

M1 ¼Vout

Vin¼

u VD

Vinð1 uÞ

1þ rSuþ rL

R

ð4Þ

Equation (4) describes how the voltage transfer gain isinfluenced by different practical components. In normaloperating conditions, the power loss is mainly due to theinductor series resistor rL. A simple circuit model, shown inFig. 2, is used to represent the practical buck converter.There is only one power loss component RLOSS in thecircuit. The switch and diode are considered ideal in Fig. 2.By setting other power loss item in (3) to zero, the revisedstate-space averaged equation can be derived:

_X ¼ AX þ Bu

A ¼RLOSS

L1L

1

C 1

RC

264

375; B ¼

Vin

L0

24

35 ð5Þ

and the modified voltage transfer gain is

M2 ¼Vout

Vin¼ R

Rþ RLOSSu ð6Þ

Assuming the circuit parameters in (3) are known, in orderto achieve M1¼M2, one can derive

RLOSS ¼VinuðrL þ rSuÞ þ RVDð1 uÞ

Vinu VDð1 uÞ ð7Þ

Considering practical conditions, it is shown that the mainpart of the power losses in the circuit is due to the inductorseries resistor. The circuit, shown in Fig. 2, will give asimilar response to that in Fig. 1. A controller designedbased on the model shown in Fig. 2 will be suitable forFig. 1 as well. In the later part of the paper, we will focus ondescribing the proposed adaptive sliding-mode-like con-troller based on Fig. 2 and (5). It will be shown that thecontrol performance predicted based on Fig. 2 is almost thesame as that based on Fig. 1.

+

Vin

−

+

−D

S rL L

iL

C

iC

R

+

vout

−

−

+

+

Vin

−

C

ic

iL

rs rL L

R

+

vout

−

+

Vin

−

− +

−+ −

+VD

rL

iL

L

C

iC

R

+

vout

−

a

b

c

Fig. 1 Topology of buck converter with power losses itemsa Buck converterb Equivalent circuit when switch is turned onc Equivalent circuit when switch is turned off

+

+Vin

−

−D

S RLOSS L

iL

C

iC

R

+

vout

−

Fig. 2 Simplified buck converter with one power loss item RLOSS

402 IEE Proc.-Electr. Power Appl., Vol. 153, No. 3, May 2006

3 Proposed controller

3.1 Review of sliding-mode controlFor the DC–DC converter system described in (5),traditional SMC will enforce a sliding manifold as

sðx; tÞ ¼ GX þ j ¼ 0 ð8Þwhere G ¼ g1; g2; . . . . . . gN½ with positive constant com-ponents. j is a reference value calculated from the systemstate-space equation.

The system is in sliding mode if its representing point(RP) moves on the sliding surface sðx; tÞ ¼ 0. Existencecondition and reaching condition are two requirements forthe stability of sliding-mode controller. The existencecondition is represented as

lims!0

s _so0 ð9Þ

This ensures the system RP can slide across the slidingsurface.

The reaching condition means that the system RP willreach the sliding surface despite the RP0 initial position inthe state-space. If the system RP is in one substructureinitially and the switch input is kept constant according tothe sign of sðx; tÞ, the system RP must be able to cross thesliding surface in finite time and the sliding function sðx; tÞcan change its sign correspondingly.

Analysis of a sliding-mode control system on the slidingsurface is often via the equivalent input. When on thesliding surface, the sliding function satisfies

sðx; tÞ ¼ 0; _sðx; tÞ ¼ 0 ð10ÞSubstituting (5) and (8) into (10) gives

G _X þ _j ¼ GAðx; tÞ þ GBðx; tÞueq þ _j ¼ 0 ð11Þwhere the discrete control input is substituted by anequivalent control ueq that has the same mathematicalmeaning as the duty cycle input. In conventional applica-tions, j is a constant value calculated from the circuitparameters. The derivative of j is zero under suchcondition. Thus (11) becomes

G _X ¼ GAðx; tÞ þ GBðx; tÞueq ¼ 0 ð12Þ

Assuming that ½GB1 exists, one can derive the expressionfor the equivalent control:

ueq ¼ ½GB1GAðx; tÞ ð13ÞBy substituting (13) into (1), the closed-loop system state-space equation can be obtained as

_X ¼ ½I BðGBÞ1GAðx; tÞ ð14ÞEquation (14) describes the system dynamics under theSMC. The system should be stable around any operatingpoint. The characteristics around any operating point canbe analysed using the small-signal perturbation method.

3.2 Proposed sliding-mode-like controllerThe traditional SMC described above is often implementedin a discrete version with hysteresis control. Therefore, theswitching frequency is determined by the hysteresis widthand other circuit parameters and the switching frequencywill change undesirably.

From the equivalent control in (13), it is found that ueq isjust a desired duty cycle value under steady state. It can beused as the control input. Then, the switching frequency willbe determined only by the input sawtooth waveform. Theresulting control scheme is as a nonlinear feedback controlsystem. However, there is no input reference informationcontained in (13). In this paper, a new kind of sliding

surface is proposed, as below:

sðx; tÞ ¼ GX þ j

j ¼Z t

0

ðvout VrÞdtð15Þ

where G ¼ ½a; b. The derivative of j contains informationabout the reference voltage and its derivative equals zero inthe steady state.

The existence condition of this sliding surface on thesimplified model is obtained as

sðx; tÞju¼1¼ a RLOSS þ RR

Vr þ Vin

40

sðx; tÞju¼0¼ a RLOSS þ RR

Vr VD

o0

ð16Þ

The reaching condition of this sliding surface on thesimplified model is also obtained as

RVin

RLOSS þ R Vr40

Vro0

ð17Þ

Equations (16) and (17) are based on the simplified model.To ensure that it is also valid for the original converter, it isnecessary to examine the equations of the original circuitwith the same sliding-mode controller. The reachingcondition and existence conditions are derived from (3)and (15). The existence condition for the original circuit is

sðx; tÞju¼1¼ að rS þ rL þ RR

Vr þ VinÞ40

sðx; tÞju¼0¼ að rS þ rL þ RR

Vr VDÞo0

ð18Þ

The reaching condition for the original circuit is

RVin

rS þ rL þ R Vr40

aVD

L Vro0

ð19Þ

Comparisons between (16) and (19) show that the stabilityrequirements for G are the same for the original andsimplified model. Therefore, it is applicable to use thesimplified model for derivation of the sliding-modecontrollers.

The equivalent control becomes

ueq ¼ ½GB1 GAðx; tÞ þ ðvout VrÞ½ ð20ÞIn such a case, the switching frequency can be chosen usingthe open-loop control model according to the requirementsof ripple voltage and current.

Substituting (5) into (20), the proposed controller for thebuck converter is

ueq ¼aLðRLOSSx1 þ x2Þ

bC

x1 x2R

þ ðVr x2Þ

ðaVinÞ=Lð21Þ

The closed-loop control system is also obtained bysubstituting (21) into (5):

_X ¼ baC

b RCaRC

1

C 1

RC

2664

3775X þ

1

a0

" #Vr ð22Þ

The large-signal closed-loop state-space equation is a linearone. Its control performance is not influenced by differentoperating points, which is desirable. The characteristic


equation of (22) is

aRCs2 þ ðbRþ aÞsþ R ð23Þwhere s is the Laplace operator. With positive coefficients aand b, it can be proved that the two poles of (23) are alwayslocated in the left-hand half of the s plane regardless of thecircuit parameter. Therefore, the system is stable if theexistence and reaching condition for slidingmode are satisfied.

The control-to-output transfer function [23] is obtainedfrom (22) as

Gref ðsÞ ¼R

aRCs2 þ ðbRþ aÞsþ Rð24Þ

The output impedance is also obtained from (22) as

ZoutðsÞ ¼aRs

aRCs2 þ ðbRþ aÞsþ Rð25Þ

In practical applications, the second item in the numeratorof (21) is actually the capacitor current iC. Although thecapacitor current can be obtained by differentiating theoutput voltage, it is measured directly in this paper becauseof noise. Thus, the controller can be simplified as

ueq ¼aLðRLOSSx1 þ x2Þ

bC

iC þ ðVr x2ÞðaVinÞ=L

ð26Þ

The controller described by (26) is called the non-adaptivecontroller in this paper. In this controller, there is noinformation about the load resistance. This is desirablebecause the load resistance always changes in manyapplications. In addition, by sensing the capacitor current,the output impedance of the closed-loop system will bedecreased.

3.3 Adaptive version of proposed controllerThe non-adaptive controller combines the advantages oftraditional SMC and PWM control. However, it can onlyoperate well under the assumption that RLOSS is known.

Adaptive control is suitable here because its mainprinciple is to estimate some system parameters via somealgorithms. Reference [24] gave an adaptive controller toestimate the output load resistance. It is also applicableto the controller in this paper. A similar method is used toestimate the loss item RLOSS. To make the analysis easierand clearer, another state variable x3 is introduced as theestimation of RLOSS. The adaptive law is

_x3 ¼ gVrðVr x2Þ ð27Þwhere g is an adaptive coefficient which determines theconverging speed of the x3. The rule for selecting g will betreated in the following. In general, it should not be selectedtoo large, which causes overshoot or rings; and it cannot betoo small, which makes the response slow. Therefore, thecontroller in (26) becomes

ueq ¼aLðx3x1 þ x2Þ

bC

iC þ ðVr x2ÞðaVinÞ=L

ð28Þ

Consequently, the closed-loop state-space averaged equa-tion is obtained by substituting (28) into (5):

_x1 ¼1

Lðx3 RLOSSÞx1

baC

x1 x2R

þ 1

aðVr x2Þ

_x2 ¼x1C x2

RC

_x3 ¼ gVrðVr x2Þ

8>>>>><>>>>>:

ð29Þ

Solving (29), it is found that there is only one equilibriumpoint for this nonlinear system, which is

x1d ¼Vr

R; x2d ¼ Vr; x3d ¼ RLOSS ð30Þ

The small perturbations of the state variables around theoperating point are defined as

Z ¼z1z2z3

24

35 ¼ x1 x1d

x2 x2d

x3 x3d

24

35 ð31Þ

Therefore, the linearised model around the operating pointis

_Z ¼ baC

b RCaRC

Vr

RL1

C 1

RC0

0 gVr 0

26664

37775Z þ

1

a

0

gVr

2666664

3777775DVr ð32Þ

From (32), the system eigenvalues can be calculated asfunctions of parameters a, b, and g. As all the systemeigenvalues in (32) are in the left part of the s plane,the nonlinear system shown in (29) is asymptotical stablearound the operating point shown in (30) [25]. Thecontroller described in (28) is then called the adaptivecontroller in this paper.

A generalised representation of the closed-loop controlscheme for the buck converter is shown in Fig. 3. Althoughthe second-order non-adaptive system is changed into thethird-order adaptive system, the overall closed-loop systemperformance is almost unchanged. This can be explained bythe pole locations [26]. For the non-adaptive system, thepoles are always set as two conjugate poles. For theadaptive system, the three poles will be located as follows:one pole is on the real axis and the other two poles areconjugated. The two conjugate poles will be located inapproximately the same positions as the two in the non-adaptive system. The pole on the real axis will be locatedmuch closer to the imaginative axis by proper selection of g.Therefore, the response corresponding to this real pole willquickly reach its steady state and will not deteriorate thesystem overall performance. Actually, the two conjugatepoles are moved closer to the imaginary axis. The systemresponse is therefore more rapid.

It is possible to choose an appropriate g withoutinfluencing the system response much. The design proce-dure can be described as follows. First, one should selectparameters a and b to satisfy the stability and performancerequirements for a traditional SMC based on the simplifiedcircuit model shown in Fig. 2. Then, g should be chosen to

PWM

S RLOSS L− + Vr

voutR

C

iC

iL

D+

−Vin

deq

∗÷

/C− +

+

++

L

γ

∫

Fig. 3 Closed-loop control scheme for proposed adaptive controller


achieve proper convergence response. The third step is totest the closed-loop system properties.

To avoid the ripple components’ influence on the overallsystem performance, lowpass filters are used for variablessuch as iC, x1 and x2. The crossover frequency for the filterscan be selected to mitigate the high-frequency ripplecomponents, which at the same time achieves an acceptableresponse speed.

4 Numerical example

4.1 Controller designA prototype of the buck converter with parameters asshown in Table 1 was constructed. It was established thatthe desired duty cycle is u ¼ 0.56. From (7), thecorresponding loss resistance RLOSS can be obtained, whichis RLOSS¼ 1.36O. It was verified that rL contributes themost part in RLOSS.

For the circuit in Fig. 2, the existence condition andreaching condition are satisfied with all positive values of aand b. Thus, the next step is to examine the closed-loopperformance in terms of control-to-output transfer functionand output-impedance. The input-to-output transfer func-tion for the proposed control system is zero because of theinput voltage feedforward.

Based on the above considerations, the parameters arechosen as a¼ 0.001 and b¼ 0.0005. The Bode plot of thetransfer function Gref (s) is shown in Fig. 4, and the Bodeplot of the transfer function Zout(s) is shown in Fig. 5. Thetwo poles of the non-adaptive system are 295571127j.

The adaptive part is added to mitigate the steady-stateerror. g can be chosen through the time response of thevirtual system shown in (18). The principle is to let x3rise quickly in the rising period of the output voltage. When

the output voltage reaches its steady-state value, x3 shouldnot be much larger than its approximated value. It can beseen from Fig. 4 that the bandwidth of Gref (s) is about350Hz. The rise time for the closed-loop system to track astep change of the reference voltage is thus approximatelyTr¼ 3ms. Then, the integral of the error term can beapproximated asZ T

0

VrðVr x2Þdt ’ 0:5V 2r Tr ð33Þ

The steady-state value of x3 is RLOSS, which can beapproximated with high accuracy. Thus, the adaptivecoefficient g during step up of the system is chosen as 3–6times the critical value calculated from (33):

gcritical ’RLOSS

0:5V 2r Tr

ð34Þ

The adaptive coefficient is chosen as g¼ 20. At theoperating point, the reference voltage will be kept constant,the adaptive coefficient can be chosen as a larger value,g¼ 60, to make the anti-perturbation performance better.

The poles of this adaptive system can be calculated as28517833j and 206 from (32). Compared with the twopoles for the non-adaptive system, their locations satisfy therequirements of decoupled design described before. Thecontrol-to-output transfer function and the output impe-dance can be calculated for the adaptive system from (32).They are shown as the solid lines in Figs. 4 and 5,respectively. This verifies the effectiveness of the chosenmethod for the adaptive coefficient.

4.2 Simulation resultsThe step response of the system is first simulated when thereference voltage equals 10V. The simulation results areshown in Fig. 6. It is shown that the rise time for the outputvoltage is less than 4ms, which is very fast compared withthe natural oscillation period (about 2ms) for the outputfilter of the buck converter. In Fig. 6c, x3 converges to itssteady-state value (1.36) given above. It is noted that, whenthe output voltage is near the reference voltage, theconverging speed of x3 is very slow. This is the reasonwhy larger g is used when the reference voltage is constantin the steady state. The duty cycle never reaches zero orincreases beyond 1 in the transient process. This ensures aconstant switching frequency operation. Furthermore, theinductor current shown in Fig. 6b has an acceptableovershoot of about 60%, which is relatively small comparedto the traditional SMC.

Table 1: Converter parameters

Vin¼20V a¼ 0.001

R¼11O b¼ 0.0005

C¼100mF g¼ 20

L¼1mH Vr¼10V

rL¼ 1O fS¼20kHz

VD¼ 0.73V fc¼ 2kHz

rS¼ 0.008O

5

0

−5

−10

−15

−20

−25

−30

−35

−40

mag

nitu

de, d

B

101 102

frequency, Hz103 104

Fig. 4 Control-to-output transfer function Gref (s)Dashed line: non-adaptive systemSolid line: adaptive system

5

0

−5

−10

−15

−20

−25

−30

mag

nitu

de, d

B

101 102 103 104

frequency, Hz

Fig. 5 Output impedance Zout(s)Dashed line: non-adaptive systemSolid line: adaptive system


The system behaviour under input voltage variation isshown in Fig. 7. The input voltage is given as a step changefrom 20 to 15V and then recovers in the simulation.Theoretically, the output voltage should not be changed

under such conditions. The duty cycle quickly reaches itsdesired value when the input voltage changes. However, asa short duration of time is needed for the output voltage tofollow the changes in the duty cycle, the output voltage

1.5

1.0

0.5

0

x 1, A

12

10

8

6

4

2

0a

b

c

0 2 4 6 8 10

d

x 2, V

1.5

1.0

x 3, Ω

0.5

0

0.6

0.5

0.4

0.3

0.2

u

time, ms

Fig. 6 Step responsea Output voltage x2b Inductor current x1c Adaptive part x3d Duty cycle u

12

10

8

6

4

x 2, V

2

0a

1.5

1.0

0.5

0

x 3, Ω

c

2.0

1.5

1.0

x 1, A

0.5

0b

0.80

0.75

0.70

u

0.65

0.60

0.550 10 20 30 40 50

time, ms

d

Fig. 7 System response under input voltage variationa Output voltage x2b Inductor current x1c Adaptive part x3d Duty cycle u


changes little, as shown in Fig. 7a. As the input voltage andduty cycle are changed, the adaptive part automaticallyregulates x3 to eliminate the steady-state error. In addition,the inductor current ripple decreases as the difference ofinput voltage and output voltage also decreases.

The system response under load variation was alsoexamined. In the simulation, the load resistance is changedfrom 11 to 22O and then recovers. The system behaviour isgiven in Fig. 8. The overshoot of output voltage is about5% of nominal value. The recovery time is about 5ms. Theinductor current also changes quickly under load variation.When the load and the equivalent duty cycle changes,RLOSS changes as well. It is shown in Fig. 8c that x3 reachesa new value, which mitigates the steady-state error in x2.

The system capability to track the reference voltagechanges was also simulated and the results are shown inFig. 9. The system behaviour is similar to the step responseshown in Fig. 7.

4.3 Comparison with a current-modecontrollerTo show the advantages of the proposed controller, atraditional current-mode controller, shown in (35), was alsobuilt into the simulation:

sðx; tÞ ¼ iL þ pðvout VrÞ þ IZ t

0

ðvout VrÞdt ð35Þ

To allow comparison, the closed-loop bandwidth of thecurrent-mode PI controller is set to about the same asthe two proposed controllers. Comparing (35) with theproposed controller in (15), the coefficients of (35) arechosen as

p ¼ 0:5

I ¼ 1000ð36Þ

A hysteresis for the current-mode control is chosen so thatthe switching frequency is 20kHz, which is the same as theproposed controller. Other circuit parameters are the sameas those shown in Table 1. The output voltage and inputinductor current of the current-mode controller are shownin Fig. 10. It can be seen that both the output voltage andthe input inductor current have larger overshoot than theproposed controller in the paper. Especially for the inputinductor current, the overshoot is significantly reduced inthe proposed controller. In addition, the switchingfrequency of the current-mode PI controller will vary whenconverter operation conditions change.

4.4 Experimental resultsThe proposed adaptive controller was applied to a buckconverter with parameters shown in Table 1 in theexperiments. The MOSFET type used is 2SK2267 [27],the diode D type is MBR6045WT [28]. In the experiments,a 5V sawtooth waveform was used to avoid noise effect.

The system step response is shown in Figs. 11 and 12.The experimental results are almost the same as thesimulation results. The current shown in Fig. 11 is theoutput of the lowpass filter with smaller ripple. It can beseen that their responses also verify the simulation results.

Figures 13 and 14 show the system response under inputvoltage variations. The input voltage is changed from 20to 15V and then recovers. Owing to the practical limitationsof the system response, the theoretical zero change in theoutput voltage cannot be achieved. The practical overshootof the output voltage is about 1V, 10% of its nominalvalue. The current overshoot is about 10% of its nominalvalue.

Figures 15 and 16 show the system response under loadresistance variations. The load resistance changes from 11 to22O and recovers. The Figures also verify the simulationresults.

12

10

8

x 2, V 6

4

2

0a

1.5

1.0

x 1, A

0.5

0b

2.5

2.0

1.5

1.0

x 3, Ω

0.5

0c

0.70

0.65

0.60

u

0.55

0.50

0.450 10 20 30 40 50

times, ms

d

Fig. 8 System response under load resistance variationa Output voltage x2b Inductor current x1c Adaptive part x3d Duty cycle u


The system response under step changes of the referencevoltage is shown in Figs. 17 and 18. The reference voltage isstep changed from 10 to 8V and recovers. The results verifythe theoretical analysis.

12

10

8

6

x 2, V

4

2

0a

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

x 1, A

b

2.5

2.0

1.5

x3, Ω

1.0

0.5

0c

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

u

0 10 20 30 40 50

d

time, ms

Fig. 9 System response under step change of reference voltagea Output voltage x2b Inductor current x1c Adaptive part x3d Duty cycle u

12

10

8

6

4

2

0

v out,

V

a

5

4

3

2

1

00 2

i L, A4 6 8 10

b

time, ms

Fig. 10 Simulation waveforms of a current-mode PI controllerduring start-upa Output voltageb Inductor current

2.00 V 0.00 s Sng1 STOP

vout

iL

1 2 500 2.50 /

1

2

1m v

m s

Fig. 11 Experimental step responseChannel 1: vout

Channel 2: iL

1.00 V1 2 1.00 V 0.00 s Sng1 1 STOP

1

2

x3

u

2.50 /m s

Fig. 12 Experimental step responseChannel 1: x3Channel 2: 5u


5 Conclusion

This paper has studied the constant switching frequencyimplementation of SMC for DC–DC converters. Asimplified model for the buck converter is derived, whichapproximates the power losses term in the detailed model.An adaptive algorithm was then used to modify the controloutput to achieve zero steady-state error.

It has been shown that the proposed adaptive sliding-mode-like controller has following advantages:

The switching frequency is strictly determined by thesawtooth waveform as a traditional duty cycle controller.

The response of the closed-loop system can be shaped bycombining sliding-mode control design in the first step.With the proposed sliding surface, start-up performanceis very good with small overshoot in both the outputvoltage and the input inductor current

The steady-state error is eliminated using a built-inadaptive controller. In addition, the adaptive controllerenhances the system transient performance

The simulation and experimental results verify thetheoretical analysis given in the paper. Future work canbe conducted to study the digital implementation of thiscontroller.

6 References

1 Mohan, N., Undeland, T.M., and Robbins, W.P.: ‘Power electronics:converters, applications and design’ ( John Willey and Sons, Inc.,2004, 3rd edn.)

vout

iL

2

1

1 2.00 V 2 Sng1 1 STOP5.00 /m s−5.00m

s500m v

Fig. 13 Input variationChannel 1: vout

Channel 2: iL

u

x3

1 2 −5.00 5.00 / Sng1 1 STOP

2

1

500m v

m s

m s500m

v

Fig. 14 Input variationChannel 1: 5uChannel 2: x3

1 2.00 V 2 Sng1 1 STOP

1

2

vout

iL

500 m v −5.00m

s 5.00 /m s

Fig. 15 Load variationChannel 1: vout

Channel 2: iL

1

2

u

x3

1 1.00 V 2 Sng1 1 STOP500m v −5.00m

s 5.00 /m s

Fig. 16 Load variationChannel 1: x3Channel 2: 5u

2

1

vout

iL

1 2.00 V 2 Sng1 1 STOP500m v −5.00m

s 5.00 /m s

Fig. 17 Reference stepChannel 1: vout

Channel 2: iL

1 1.00 V2 Sng1 STOP15.00 /

2

1

x3

u

−5.00500m v

m s

m s

Fig. 18 Reference stepChannel 1: x3Channel 2: 5u


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Design and analysis of adaptive sliding-mode-like controller for DC-DC converters