ISA Transactions 52 (2013) 638–643

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ISA Transactions

0019-05http://d

n CorrE-m

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Analysis and design of sliding mode controller gains for boost powerfactor corrector

Abdelhalim Kessal a,n, Lazhar Rahmani b

a Mohammed el bachir el ibrahimi University, Bordj Bou Arréridj, Algeriab Automatic Laboratory of Sétif (LAS), Ferhat Abbas University, Algeria

a r t i c l e i n f o

Article history:Received 26 March 2013Accepted 3 May 2013Available online 2 June 2013This paper was recommended for publica-tion by Jeff Pieper

Keywords:PFCPower factorSliding mode controlSwitching frequency

78/$ - see front matter & 2013 ISA. Publishedx.doi.org/10.1016/j.isatra.2013.05.002

esponding author. Tel.: +213 66404 8090; faxail address: Abdelhalim.kessal@yahoo.fr (A. Ke

a b s t r a c t

This paper presents a systematic procedure to compute the gains of sliding mode controller based on anoptimization scheme. This controller is oriented to drive an AC–DC converter operating in continuousmode with power factor near unity, and in order to improve static and dynamic performances with largevariations of reference voltage and load. This study shows the great influence of the controller gains onthe global performances of the system. Hence, a methodology for choosing the gains is detailed. Thesliding surface used in this study contains two state variables, input current and output voltage;the advantage of this surface is getting reactions against various disturbances—at the power source, thereference of the output, or the value of the load. The controller is experimentally confirmed for steady-state performance and transient response.

& 2013 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Usually, traditional PID controllers are used for the control ofpower converters [1–3]. Simple models of converters are generallyobtained from signals averaging and linearization techniques;these models may then be used for control design [4,5]. On theother hand, PID controllers failed to satisfactorily perform con-strained specifications under large parameter variations and loaddisturbances [2]. Another choice for controlling power converter isto use the sliding control techniques. Sliding mode control (SMC)of variable structure systems such as power converters is particu-larly interesting because of its natural robustness, its capability ofsystem order reduction, and suitability for the nonlinearity aspectof power converters [5–7]. However, despite being a popularresearch subject, SMC is still rarely applied in practical AC–DCconverters. It is mainly due to the fact that no systematicprocedure is available for the design of SMC in practical applica-tions [8]. For example, the influence of the controller gains on theclosed loop system performances for a given application is notproperly clarified, and most of the previous works are limited tothe study of the influence of these parameters only on theexistence and stability of sliding mode [9,10]. In other cases anempirical approach is adopted for selecting these gains of SMC;computer simulation and experiments were performed to studythe effect of the various control gains on the response of theoutput voltage [10]. Therefore in this paper, analysis and design of

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: +213 35 674543.ssal).

SMC for power factor corrector (PFC) are studied. After studyingand analyzing different existing solutions for sliding mode controlof PFC, a control mode that allows a direct control of the voltage ofboost converter is proposed. The performances of the controller interms of robustness and dynamic response will be improved. Mostliterature works are concerned with the study of hitting, existenceand stability conditions of the SMC. The contribution of this papergoes beyond this direction by involving the study of the influenceof control parameters on system performances. In this context, anoptimization algorithm is developed in order to choose thecontroller parameters based on a predefined specification for agiven real application.

Accordingly, this paper is oriented in the application of thesliding modes for control of the bench of the power factorcorrector (PFC). Principle of control by sliding modes is describedbriefly. Thereafter, the application of this principle for the controlof the bench of PFC will be evoked. Based on the choice of thesliding surface, various modes of control will be studied. Then amode of control based on a sliding surface utilizing all thevariables of state are studied; this is in order to improve theperformances of the closed loop. The important concepts asso-ciated to this type of control such as the convergence conditions,existence, or stability of the sliding mode, are considered carefully.

This paper proposes a systematic analysis, design and digitalimplementation of the proposed controller, composed by linearcontroller in the DC voltage loop and sliding mode controller inthe current loop. This controller is verified by detailed MATLAB/Simulink based on simulations through the use of a continuoustime plant model and a discrete time controller. Design iscomprehensive in the sense that it accounts for sampling effects,

.

Fig. 1. Boost converter circuit.

Fig. 2. Boost converter circuit governed by sliding mode controller.

A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643 639

computation delays, hardware filtering for antialiasing, and soft-ware filtering for measurement noise reduction, where necessary.Real-time implementation is done on an experimental prototypeusing the dSPACE DS1104 controller board. This controller isexperimentally compared for steady-state performance and tran-sient response over the entire range of input and load conditionsfor which the system is designed. The paper is organized asfollows. In Sections 2 and 3, a description of converter, and adesign and analysis of controllers are given. The experimentalsetup is detailed in Section 4. Section 5 presents the obtainedresults with discussions.

2. Mathematical model of boost converter

The basic circuit diagram of the DC–DC converter with frontend solid state input power factor conditioner used in theproposed scheme is shown in Fig. 1.

The power circuit is that of an elementary step-up converter.When the boost switch Sw is turned on (u¼1), the inductor currentbuilds up, and energy is stored in the magnetic field of theinductor, whereas the boost diode D is reverse biased, and thecapacitor supplies power to the load. This is the first modeoperation. As soon as the boost switch is turned off (u¼0), thepower circuit changes mode, and the stored energy in theinductor, together with the energy coming from the input ACsource, is pumped to the output circuitry (capacitor–load combi-nation). This is mode 2 of the circuit. Then the state space modelfor the boost PFC in continuous current mode can be found by thecircuit analysis. The output voltage and inductor current dynamicsare governed by the variable structure real switched system.

C dvodt ¼ ð1−uÞiL−io

L diLdt ¼ vin−ð1−uÞvo

8<: ð1Þ

In order to obtain a sinusoidal input current in phase with theinput voltage, the control unit should act in such a way that vinsees a resistive load equal to the ratio of vin and iL. This has beendone by comparing the actual current passing through theinductor with a current reference, which is derived from vin andhas an amplitude determined by the output voltage controller.

3. Design of sliding mode controller

The control objectives of the PFC are twofold: regulate theoutput voltage vo to a reference voltage Vref and give the inputcurrent iL a rectified sine waveform in phase with the rectifiedvoltage vin. The design of sliding mode controller for PFC startswith the choice of sliding surface. As it is shown in [11], it is clearthat direct surface vo−Vref can tend to zero only if the currentincreases continuously. Usually, a cascade control structure is used,which leads to solve the control problem using two control loops[12]: an outer voltage loop which generates the reference currentfrom voltage error and an inner current loop which controls the

inductor current via sliding mode that replace classical hysteresiscurrent control (Fig. 2).

This control of the output voltage of AC–DC converter meets thecriteria of stability and existence of sliding mode. However, it isdifficult to determine the gains of the voltage loop since slidingmode is a highly nonlinear method [2]. Furthermore, since SMC isonly applied to current regulation, the voltage loop will be moresensitive to high frequencies phenomena and to uncertainties inthe reference current. In order to improve the performances of thecontroller, a control mode based on a sliding surface whichinvolves output voltage will be treated.

Let (Vequ, Iequ) be the desired equilibrium point, where Vequ is theoutput voltage, and Iequ is the inductor current peak at equilibriumpoint. The input current peak IL can be expressed as [13]

IL ¼π

2Vo

ð1−αÞR

ð2Þ

So, the equilibrium point becomes ðVequ; Iequ ¼ π=2� �

Vequ

ð1−αÞ=R� �, and the sliding surface shall be given according to the

expression

S¼ λ1ðvo−Vref Þ þ λ2ðiL−iref Þ ð3Þ

where λ1 and λ2∈R+.The control by current imposes the average power passed to

the load with the ideal PFC pre-regulators [13].

P ¼ VSMIref2

¼ voio ð4Þ

The reference current peak depends on the operating point; itcan be taken as

VSMIref2Vref

¼ ioð1−cos 2ωtÞ ¼

voRð1−cos 2ωtÞ ð5Þ

Iref ¼io

ð1−cos 2ωtÞ ¼2voVref

VSMRð1−cos 2ωtÞð6Þ

Sliding surface coefficients (λ1, λ2) should be chosen such thatthe sliding mode exists at least around the desired equilibriumpoint, and the dynamics of the system will reach the surface andlead toward the equilibrium point.

3.1. Existence condition

The existence condition of sliding mode implies that both _S and_S will tend to zero (when t-∞), which means that the systemdynamics remains on the sliding surface. The existence conditionof the sliding mode is S_So0 (when S-0); achieving this

A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643640

inequality guarantees the existence of the sliding mode around theswitching surface.

The model of PFC can be written in a state space where theequilibrium point is the origin, whether

x1 ¼ vo−Vref and x2 ¼ iL−iref where iref ¼ Iref jsin ωtj, so

C dx1dt ¼ ð1−uÞðx2 þ iref Þ−ðx1þVref

R ÞL dx2

dt ¼ vin−ð1−uÞðx1 þ Vref Þ

8<: ð7Þ

Replacing iref by its value from (6) in the expression of thecommutation surface (3), the surface becomes

S¼ λ1 vo−Vref� �þ λ2 iL−

2voVref jsin ωtjVSMR 1−cos 2ωtð Þ

� �

¼ λ1ðvo−Vref Þ þ λ2iL−λ22voVref jsin ωtj

VSMRð1−cos 2ωtÞ

¼ λ1−2λ2Vref jsin ωtj

VSMRð1−cos 2ωtÞ

� �vo−λ2iL−λ1Vref ð8Þ

This equation can be written in the coordinate system (x1, x2):

S¼ λ1−2λ2Vref jsin ωtj

VSMRð1−cos 2ωtÞ

� �x1 þ λ1−

2λ2Vref jsin ωtjVSMRð1−cos 2ωtÞ

� �Vref

þλ2x2 þ λ2iref−λ1Vref ¼ λ01x1 þ λ2x2 ð9Þ

where λ01 ¼ λ1−ð2λ2Vref jsin ωtj=VSMRð1−cos 2ωtÞÞThe state of the switch (u∈{0,1}) imposes two signs of the

derivative of the sliding surface; replacing u in the state system(7), the boundaries of the sliding area are deduced by

−x1RC −λ10

Vref

RC þ λ2 vinL 40 ð_S40Þ

x1ð− λ10

RC −λ2L Þ þ x2

λ10

C þ λ2ðvinL − Vref

L Þ−λ10 ðvrefRC − irefC Þo0 ð_So0Þ

8<: ð10Þ

To ensure that the sliding mode exists at least around theequilibrium point (x1¼x2¼0), the following condition must besatisfied:

λ′1λ2

o RCvinVref L

ð11Þ

3.2. Stability condition

To ensure stability, the system dynamics during sliding mode isdirected to the desired equilibrium point. The goal is to determinethe dynamic of x1 and x2 when the sliding regime is achieved.Taking the state space model in (7) and the commutation surfacein (8), from _S¼ 0, the equivalent average control that must beapplied to the system in order that the system state slides along

Fig. 3. Sliding mode with a

the surface is given by

ueq ¼ 1−vin=L� �

− λ10 ðx1 þ Vref Þ=λ2RC

� �x1 þ Vref =L� �

− λ10 ðx2 þ iref Þ=λ2C� � ð12Þ

Replacing the equivalent control (12) in the state spacemodel (7) and from S¼0, the dynamic of x1 at the sliding regimeis given as

dx1dt

¼vin iref− x1λ10 =λ2

� �� �− x1 þ Vref� �2

=R� �

C x1 þ Vref� �

−L iref− λ10 x1=λ2� �� �

λ10 =λ2� � ð13Þ

Introducing the Lyapunov function V ¼ 1=2x21, its derivative is_V ¼ x1 _x1, so

_V ¼−x21vinðλ1=λ2Þ þ ðv0 þ Vref Þ=R

� �Cðx1 þ Vref Þ−Lðiref− λ10 x1=λ2

� �Þðλ10 =λ2Þ ð14Þ

The condition for _V to be negative is

CðVref þ x1Þ−L iref−λ10 x1λ2

� �λ10

λ2

� �40⇒x14

LV2ref =Rvin

� �ðλ10 =λ2Þ−CVref

C þ Lðλ10 =λ2Þ2ð15Þ

Based on the sliding region defined by (10) and the existencecondition (11), the condition given in (15) is always satisfied alongthe sliding region of the commutation surface. According to thetheorem of Lyapunov stability, the system is globally asymptoti-cally stable.

3.3. Controller parameters and system performance

Inequality (11) provides only general information concerning theexistence of sliding mode. On the other hand, performances of theclosed loop system are influenced by the choice of parameters of thecontroller, especially when the system presents large variationsaround nominal. To choose these parameters, the size of the slidingpart in the switching surface must be taken into consideration.

Really, sliding condition is only satisfied on a subpart of thesurface and not on the entire surface as shown in (10). Therefore,the controller parameters must be carefully chosen to ensure thatthe system dynamic will intercept the commutation surface in thesliding part. For this, precautions against unwanted behaviors thatcause an overshoot response must be taken (Fig. 3).

The two points A(x1A,x2A) and B(x1B,x2B) are assumed which arerespectively the crossing of system dynamic together with thecommutation surface, and the boundary of the sliding parcel(Fig. 3). Supposing at t¼0, the output voltage is Vini (initial voltagevalue) and the input current is null, since the surface S will benegative, then u¼1 such that the space vector becomes

C dx1dt ¼ −ðx1þVref

R ÞL dx2

dt ¼ vin

8<: ð16Þ

nd without overshoot.

A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643 641

Taking into consideration the initial values, the resolution ofthis system gives

x1 ¼ Vinie−Lðx2þiref ÞRC:vin −Vref ð17Þ

This solution represents the dynamic of the system beforeintercepting the surface at point A. From S¼0 and from (16) wededuce the coordinate of point A:

x2A ¼λ10

λ2Vinie

−Lðx2Aþiref Þ

RC:vin þ Vref

� �ð18Þ

From the equation of the commutation surface (9) and the limitof sliding zone defined by (10), the coordinate of point B is

x2B ¼λ10

λ2

λ10 Vref =RC� �

− iref =C� �� �

=λ2� �

− vin−Vref� �

=L� �

λ10 =λ2� �2 þ λ10 =λ2RC

� �þ 1=C� �

!ð19Þ

By deduction, the system intercepts the commutation surface inthe right part if the controller parameters are selected in such a waythat

‖OA‖o‖OB‖ ð20ÞThe theory of sliding mode assumes that the hysteresis band

shall be null, so frequency approaches infinity. It is obvious thatthis assumption could not be made owing to the frequencylimitation caused by the feature of circuit components and losses.Generally, a hysteresis window is added around the surface tomaintain the operating frequency; from Fig. 4, the rise time ton andfall time toff can be expressed as

ton ¼ 2ΔSþ

tof f ¼2ΔS−

f ¼ 1ton þ tof f

ð21Þ

From (9) and (21), the hysteresis band expression is deduced, so

Δ¼ 12f

11= λ2=L� �

vin− λ10 Vref =RC� �� �� �þ 1= λ10 Vref =RC

� �1=vin� �

−1� �þ λ2=L

� �vin−Vref� �� �

!ð22Þ

Consequently, from the expression of Δ(λ1,λ2), a limitation onthe choice of the controller parameters is maintained. In fact,values of these parameters must ensure that the hysteresiswindow is greater than the perturbation generated by the con-verter, and in the same time, the band value should be limited toguarantee the robustness. In addition, the main goal is to regulatethe output voltage, but the commutation surface (3) depends onthe current errors, so an important optimization study of con-troller gains should be the analysis of the sensitivity of thecontroller in front of a measurement or a current referenceestimation error.

Actually, the current reference is unknown; it can be extractedfrom the load current (6). However, this latter can be measured orobserved through an extended Luenberger observer. In both cases,an error can occur which affects the response of the system in

Fig. 4. Hysteresis band of sliding regime.

terms of steady state error. So the choice of controller parametersmust be taken into account to make the closed loop system lesssensitive to an error in the current part of the surface. It issupposed that the measured reference current (IrefMEAS) can beexpressed as the sum of the expression of reference current givenby (4) and an error term (eIref), where e is defined as the errorpercentage, so that

IrefMeas ¼ Iref−eIref ð23Þ

Replacing Irefmeas in the expression of sliding surface (3)

S¼ λ10 þ λ2eVref

vinR

� �ðvo−Vref Þ þ λ2ðiL−Iref Þ þ

λ2eV2ref

vinRð24Þ

The average value of the term λ2eV2ref =vinR is constant, so

S¼ λ10 þ λ2eVref

vinR

� �ðvo−ðVref þ ΔvoÞÞ þ λ2ðiL−ðIref þ ΔiLÞÞ ð25Þ

where Δvo, and ΔiL, are the voltage and the current steady stateerrors respectively.

Error part in the sliding surface is given by

ðλ10 þλ2eVref

vinRÞðΔvoÞ þ λ2ðΔiLÞ ¼ −

λ2eV2ref

vinRð26Þ

The control u will tend S to zero, so vo-Vref þ Δvo andiL-Iref þ ΔiL; from the balance of input/output power

ðVref þ ΔvoÞio ¼ vinðIref þ ΔiLÞÞ ð27Þ

Then, expression of Δvo will be

Δvo ¼Rvin2

−λ1λ2

−Vref

Rvin−eVref

Rvinþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ1λ2

þ Vref

Rvin−eVref

Rvin

� �2

−4eV2

ref

v2inR2

vuut0@

1Að28Þ

4. Experimental setup

In this study, the gains λ1 and λ2 of sliding controller areadjusted employing the off-line iterative genetic algorithm (GA).Thus, GA determines the controller gains which are the mostcompatible and provide optimum performance. The criteria arebased on practical specifications. Thus the objective of the geneticalgorithm is to determine the values of parameters that ensure,regardless of the operating point, that the system will interceptthe sliding part of commutation surface while respecting thefollowing conditions:

λ1λ2ominðRCvinVref L

ÞΔminoMaxðΔðλ1; λ2ÞÞoΔMax

MaxðΔvoÞoΔvoMax⋯f or⋯e¼ eMax

8>><>>: ð29Þ

The power circuit is designed to meet the followingspecifications:

output voltage V0¼160 Voutput voltage ripple o2%

Figreg

A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643642

input voltage VSeff ¼115 V, RMSinput current ripple ≤5%load resistance R¼210 Ω

The experimental prototype was built around the dSPACE1104controller board, which hosts the PowerPC 603e processor, toexamine operating characteristic of the proposed method controlfor PFC. Although the PowerPC and its associated data acquisitioncircuitry can run up to 1 MHz, computation delay and commu-nication overheads only allowed for the control algorithm to beexecuted at 20 kHz. The fourth-order Runge–Kutta solver waschosen to discretize the controller for real-time implementation.One Hall-effect CT's LEM (PR30) and isolation amplifier HAMEG(HZ64) were employed to detect the inductor current, input linevoltage, and the output DC-bus voltage. Control circuits were builtfor offset correction and appropriate scaling. To prevent aliasing inthe sampling process, second-order low-pass Butterworth filterswere used to remove noise and switching frequency ripple in thesensed signals. Step load changes were affected by electronicallyconnecting/disconnecting a parallel load.

Fig. 6. Experimental measurem

1.4 1.6 1.8 2140

160

180

200

220

Vref, vo

Increasing load

Fig. 7. Simulation resul

. 5. Experimental results for steady state, grid voltage, input current andulated output DC voltage.

5. Results

A real-time experimental study was performed to capture theperformance of the proposed method control for PFC. First, thesteady-state performance is evaluated in terms of output voltageregulation, THD, and power factor. Next, the transient performanceis evaluated for output voltage response on application of loadstep changes that are expected in practical applications of thiscircuit. All the data presented here were captured at 20 kHz usingthe control desk user interface for the dSPACE1104.

1.

en

ts fo

Figinp

Steady-state performance: Fig. 5 shows the correspondingexperimental results: the obtained power factor is 0.998% andTHD is 2.95%; it is important to note that at nominal line andload condition, the method control has a THD value below 3%even with the limited bandwidth that is allowed by the digitalimplementation. Line current is very close to sine wave and inphase with the line voltage as shown in Fig. 6; the outputvoltage error is about less than 2 V. These results show that theproposed PFC control method achieves near unity power factorunder steady state, and THD value is much better than theadoption of IEC1000-3-2 as the EN61000-3-2 standard.

ts, THD, PF and phase.

2.2 2.4 2.6 2.8

Decreasing load

r load changes.

[50V/div]

[2A/div]

[100ms/div]

. 8. Experimental results for load disturbances, regulated output DC voltage andut current.

1 1.4 1.8 2.2 2.6 3 3.2160

180

200

220

240Vref, vo

Fig. 9. Simulation results for output voltage reference changes.

[100V/div]

[2A/div]

[100ms/div]

Fig. 10. Experimental results for output voltage reference variations, regulatedoutput DC voltage and input current.

A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643 643

2.

Transient performance: To evaluate performances in transientmode, step load changes are effected by disconnecting (orconnecting) parallel load. The reference current amplitude islimited to 3.5 A in the control method designs. Figs. 7 and 8show, respectively, simulation and experimental results oftransient response for the proposed method control for PFCfor a load resistor step, by 733% of the nominal value of theload (212 Ω). After a short transient (about 150 ms), the DC-busvoltage is maintained close to its reference value with a goodapproximation and stability. The line currents have nearlysinusoidal waveforms.

The dynamic behavior of the proposed method under a stepchange of Vref is presented in Fig. 9 for simulation and Fig. 10 forexperimental results. After a short transient (about 100 ms), theDC-bus voltage is maintained close to its new reference (from180 V to 220 V and vice versa) with good approximation andstability. The line currents have nearly sinusoidal waveforms.

6. Conclusion

In this paper, a practical design of sliding mode control forboost power factor controller is established, using a sliding surfacewhich includes all state variables, output regulated voltage andinput sinusoidal current. An optimization algorithm was

developed in order to calculate the optimal values of the slidingsurface parameters based on a predefined specification. Resultsshow excellent dynamic response of controller and robustness toload and voltage reference with large variations around nominalvalues. Experimental results show excellent dynamic response,good output regulation, low harmonic distortion, and high powerfactor can be achieved with the proposed single-stage converterand control scheme based on the proposed sliding mode con-troller. Finally, to verify the PFC function, the harmonic distortionsare measured and compared to the international standards as EN61000-3-2 and IEEE 519, the power factor is near unity and theTHD is less than 3%.

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