960 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 15, NO. 6, NOVEMBER 2000

On the Design of Sliding Mode Control Schemes forQuantum Resonant Converters

Miguel Castilla, Member, IEEE, Luis García de Vicuña, Member, IEEE, Mariano López, Member, IEEE,Oscar López, Student Member, IEEE, and José Matas, Student Member, IEEE

Abstract—The design of sliding mode control schemes forquantum resonant converters is introduced by means of twodifferent approaches. First, an easy-to-use procedure for devisingnonlinear control structures is established, using Lyapunov’swell-known stability criteria. Second, an alternative method thatprovides linear sliding surfaces is also developed, consideringreaching, existence, and stability conditions. The application ofboth control design techniques is illustrated in detail by means ofthree selected examples. The advantages and drawbacks of theresulting control circuits are examined. Simulation and experi-mental results corroborate the expected features of the close-loopquantum converters.

Index Terms—Linear sliding surfaces, Lyapunov-based control,quantum resonant converters.

I. INTRODUCTION

A T PRESENT, most control schemes for power resonantconverters are based on frequency or phase domain tech-

niques. In both cases, the behavior of the output voltage is highlynonlinear and largely dependent on line and load conditions.In addition, the switching losses become severe when both theswitching frequency deviates from the resonant frequency andthe phase difference becomes large [1].

To overcome these disadvantages, a time domain controltechnique has been studied over the last few years, resulting ina new family of converters named quantum resonant converters(QRC) [1], [2]. The main advantage of these converters isthat zero-current or zero-voltage switching conditions can beguaranteed in all the power devices because the switchingfrequency is always the resonant frequency [3]. However, thediscrete time duration of the control cycles generates quantizedoutput levels and large ripple in the resonant waveforms.

Two main classes of close-loop control schemes can befound in the literature to solve these drawbacks: one employingpredictive techniques, which are based on output voltageerror estimation methods, and the other using sliding modecontrol strategies. In predictive approaches, very accuratediscrete-time models are used both to describe the converterdynamic behavior and to design the estimation control routine[4]–[6]. However, the implementation of these controllers is

Manuscript received April 12, 1999; revised July 14, 2000. This work wassupported by the Spanish Ministry of Education and Culture (CICYT TIC99-0743) and Universidad Politécnica de Cataluña (FIU PR99-02). Recommendedby Associate Editor A. Kawamura.

The authors are with the Departamento de Ingeniería Electrónica,Universidad Politécnica de Cataluña, Barcelona 08800, Spain(e-mail: mcastilla@eel.upc.es; vicuna@eel.upc.es; lopezg@eel.upc.es;oscar@eel.upc.es; matas@eel.upc.es).

Publisher Item Identifier S 0885-8993(00)09807-0.

difficult, since many calculations are required in the real-timecontrol algorithm [7].

Sliding mode control schemes for QRC converters arefrequently designed using continuous reduced-order averagedmodels. Such models provide enough state information to bothcorrectly regulate the output voltage in dc-to-dc applications[8]–[11] and force the output voltage to follow an externalsinusoidal reference in ac waveform generation [12], [13]. Inaddition, control schemes offer robustness against external dis-turbances and parameter variations. However, the applicationof sliding mode control to QRC converters is still only exploredtoday in a short number of studies, which basically involve thecontrol design of the series-type topology.

In a clear-cut contrast, linear and nonlinear control designtechniques have been extensively tested in hard-switchingdc-to-dc conversion cells. On the one hand, the structure of thesliding surface is proposed as a linear combination of all thestate variables, using linear control techniques [14]–[18]. Insuch a case, the design procedure consists in determining theappropriate gain parameters that satisfy a set of conditions forthe existence and reachability of a sliding regime. Althoughthis class of sliding surfaces leads to a good dynamic response,the price to pay is the need for sensing and processing all thestate variables, thus increasing the complexity of the controlcircuit implementation.

On the other hand, several systematic approaches for thesynthesis of nonlinear sliding surfaces with suitable stabilizingproperties have been investigated, using nonlinear controltechniques. Sliding mode control via feedback linearization[19], extended linearization [20], and other nonlinear controlmethods [21]–[23] is just a short list of examples. In thosecases, the sliding surfaces are composed of nonlinear functions,which, in turn, depend on the input voltage, the load, and circuitparameters. Consequently, control circuits generally require theuse of multiplier and divider circuits as well as the sensing ofthe input voltage and the load, thereby increasing the difficultyof their practical implementation and operation.

The aim of this paper is to deduce sliding mode controllers fora wide range of series and parallel QRC converters, includingsingle- and multi-input topologies. Two different control designapproaches are proposed in order to compare the advantagesand drawbacks of the resulting close-loop controllers. In Sec-tion III, an easy-to-use procedure for devising nonlinear controlstructures is established, using the Lyapunov’s well-known sta-bility criteria. An alternative method that provides linear slidingsurfaces is developed in Section IV, considering reaching, ex-istence, and stability conditions. The application of both con-

0885–8993/00$10.00 © 2000 IEEE

CASTILLA et al.: SLIDING MODE CONTROL SCHEMES FOR QUANTUM RESONANT CONVERTERS 961

Fig. 1. Basic topologies of quantum resonant converters: (a) series-type and (b) parallel-type.

trol design techniques is illustrated in detail by means of threeselected examples. Moreover, sliding mode controllers for allQRC converters are proposed. Finally, in Section V, simulationand experimental results corroborate the expected features ofthe close-loop quantum converters.

II. PRINCIPLE OFOPERATION AND LARGE-SIGNAL MODELS

OF QRC

The basic topologies of series and parallel QRCs are shownin Fig. 1. For both converters, the switching frequency is alwaysthe resonant frequency, so that each control-input state presentsa discrete time duration that coincides with an integer numberof half resonant periods. Moreover, control-input changes are al-ways synchronized in the QSRC with the zero-current crossingpoints in the resonant inductor and in the QPRC with the zero-voltage crossing points in the resonant capacitor. Consequently,nearly zero switching conditions are guaranteed in all the powerdevices [1], [2].

The direction of the energy flow between the dc source andthe load can be determined by two independent control inputs(TCI): and . On the one hand, the control inputregulatesthe state of the full-bridge switching network. When ,the pair of switches S1–S4 and S2–S3 are switching on andoff alternatively, thus continuously transferring energy to theresonant tank; whereas if , the full-bridge network isblocked, thus not delivering energy to the resonant circuit. Onthe other hand, the control input commands the state of S5switch; so when , the resonant tank energy is partially

TABLE ISINGLE-INPUT QSRC

TABLE IISINGLE-INPUT QPRC

discharged into the load, and if , the load does not receiveenergy from the resonant circuit [1], [2].

Quantum converters can also operate just like some single-input conventional converters, driving and as shown inTables I and II. In such a case, the state of the full-bridge networkand S5 switch are governed by the control input u only, whichcan take the values 1 or 0.

For control design purposes, the appropriate dynamicdescription of quantum converters is by means of averagedlarge-signal models, which are essential to study their globaldynamic properties. The equivalent circuits of such models

962 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 15, NO. 6, NOVEMBER 2000

Fig. 2. Averaged circuit models for quantum resonant converters: (a) series-type, (b) parallel-type.

Fig. 3. Averaged circuit models for single-input QSRC: (a) Buck, (b) Boost, and (c) Buck-Boost.

for the TCI QSRC and QPRC are shown in Fig. 2(a) and (b),respectively. Details of the model derivation and averagedvariable definition are given in [9] and [25]. The averagingprocedure for resonant converter modeling is also describedin [26]. For the case of single-input converters, the averagedmodels are obtained by replacing the control variablesand as shown in Tables I and II. The equivalent circuits ofsingle-input series- and parallel-type models are represented inFigs. 3 and 4, respectively.

III. L YAPUNOV-BASED CONTROL DESIGN

In this Section, a systematic approach for the control design ofall single- and multi-input QRCs is developed, using the second

method of Lyapunov. In addition, a set of sliding mode con-trollers for such converters is proposed.

A. Synthesis Procedure of Lyapunov’s Sliding Surfaces

Quantum resonant converters are multi-input nonlinear sys-tems whose averaged state models can be represented as fol-lows:

(1)

where the state vector , the vector fields ,, and are the control inputs ( , ).

CASTILLA et al.: SLIDING MODE CONTROL SCHEMES FOR QUANTUM RESONANT CONVERTERS 963

Fig. 4 Averaged circuit models for single-input QPRC. (a) Boost with output filter, (b) buck with input filter, and (c) Cuk.

The sliding mode control structure for this kind of systemsconsists of a set of sliding surfaces and their associatedcontrol laws:

for

for(2)

where and take the values 1 or 0, and verify .For the design of the control structure (2), the Lyapunov func-

tion approach proceeds by first defining a positive-definite func-tion , and then using as a global reachingcondition [21], being the deviation of the state vector withregards to its steady-state value(i.e. ). As theLyapunov function candidate, we chose the following quadraticform:

(3)

where is the transposed of and is a diagonal matrixof positive constant elements. In fact, such a function coincideswith the incremental energy of the converter if the elements ofmatrix are conveniently expressed in terms of the values ofreactive components [22].

The state model (1) can be rewritten in terms of, defininga new set of control inputs as the deviation of with

respect to their average values in steady-state(i.e.):

(4)

where . In that case, the controllaw (2) can be expressed as:

for

for(5)

Note that the two possible values of can be positive () or negative ( ), as must be bounded between their

minimum and maximum limits .By replacing (3) and (4) in the reaching condition, we obtain:

(6)

The term corresponds to the time-derivative of theincremental energy of the converter in open-loop operation. In[24], it is demonstrated that this term is always nonpositive, due

964 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 15, NO. 6, NOVEMBER 2000

to it coincides with the incremental power dissipated in the con-verter with the negative sign. In such a case, the fulfilment ofexpression (7) ensures that the reaching condition is satisfied:

(7)

The sliding mode control structure can be directly obtainedfrom (7). In fact, we could select each control inputaccording to the sign of each value , so that eachproduct can always be negative:

for

for(8)

Finally, when comparing (5) and (8), the sliding surfaces andtheir associated control laws can be identified as

for

for(9)

B. Lyapunov-Based Controllers for QRCs

We begin by illustrating in detail the synthesis of a slidingmode controller for the Buck QSRC shown in Fig. 3(a). Fromthe analysis of the circuit, the state-space model can be ex-pressed as

(10a)

where can take the values 1 or 0, depending on the switchposition. The state-space model can be rewritten as

(10b)

being

and

From (9), the Lyapunov-based control structure for thissingle-input converter has the expression

(11)

then, in order to obtain a practical description of the switchingsurface, the incremental state vector and matrix must bededuced.

First, considering the state variable that is desired to reg-ulate to a constant value , the steady-state solution of (10)in close-loop operation is

(12)

and, consequently, the incremental state vector can bewritten as

(13)

Second, the incremental energy of the Buck QSRC takes theform

(14)

From (3) and (14), the matrix is given

(15)

In using (11), (13), and (15), Lyapunov’s sliding surface forthe Buck QSRC results in

(16)

For the rest of QRCs, Lyapunov’s sliding curves are shownin Table III. These control structures have been deduced byapplying the synthesis procedure to the averaged modelsrepresented in Figs. 2–4. However, the utilization of suchsurfaces is strongly limited by some practical drawbacks. Thefirst problem is the complexity of the hardware required toimplement these nonlinear functions, which usually depend onthe input voltage, the load, and a considerable number of statevariables. The second problem is caused by the appearanceof output voltage steady-state errors, due to the imperfectionsinvolved in different parts of the system, such as relays, losses,etc. (see Section V).

In the next Section, an alternative approach for the synthesisof sliding surfaces is proposed. The focus will be on the useof linear stabilizing terms as a way of generating simple andlow-cost control circuits.

IV. DESIGN OFLINEAR SLIDING SURFACES

In this Section, an alternative approach for the design ofsliding mode controllers for single- and multi-input QRCs ispresented. First, an interesting class of linear sliding surfaces,showing low-cost implementation and absence of steady-stateerrors, is proposed. Second, a set of design constraints for thegain parameters of such surfaces is deduced by using reaching,existence, and stability conditions.

We begin by describing the control design procedure forsingle-input converters, including a detailed example of ap-plication for the Buck QSRC. Next, the case of multi-inputconverters is introduced together with a detailed example forthe TCI QSRC.

A. Linear Sliding Surfaces for Single-Input QRCs

In single-input nonlinear systems, the relative degree of astate variable is defined as the smallest number of differentia-tions of the state variable with regards to time, so that the con-trol input appears explicitly [21].

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TABLE IIISLIDING SURFACES FORQRC (e = V � v , e = V � v )

When considering ( ) the state variable thatis desired to regulate to a reference value , the followinglinear sliding surface is proposed:

(17)

where are constant gains andis the relative degree of( ). Note that surface (17) satisfies the transversalitycondition (i.e. depends explicitly on the control input) andremoves the undesirable steady-state errors of, due to thepresence of the integral term. Moreover, a linear dynamics of

is achieved in sliding motion ( , ), which can beexpressed as follows:

(18)

However, in using surface (17), the equilibrium point of theconverter can be unstable, even when choosing appropriate.For the analysis of such situation, we can examine the stabilityof the equivalent control . In fact, in open-loop operation, aconverter has always stable steady-state behavior, since the con-verter is composed of passive components [22]. Therefore, thestability of ensures the stability of the close-loop converter.

In defining the control-to- transfer function asand using (18), the small-signal

deviation of the equivalent control with regards to itssteady-state value takes the expression:

(19)

From (19), a necessary condition for local stability of the close-loop converter is minimum-phase control-to-transfer func-tion.

If is a nonminimum-phase transfer function, anew minimum-phase state variable( , ) mustbe included in surface (17) to guarantee system stability:

(20)Now depends on the relative degree of variablesand ,defined as and , respectively ( and ).In order to fulfill the transversality condition,must be exactly

if ; whereas if , can take any value from 1to , allowing for some freedom in the choice of the terms of

966 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 15, NO. 6, NOVEMBER 2000

surface (20). Note that the incorporation of in surface (20)causes a moderate increase of the control circuit complexity.

Again, the system stability is investigated by testing the ex-pression of the equivalent control. In such a case, when usingthe control-to- and control-to- transfer functions and theinvariance condition , the equivalent control can be ex-pressed as

(21)

being

Taking into account that has been chosen among the statevariables observing minimum-phase control-to-transferfunction, the stability of expression (21) is conditioned onlyby the position of the roots of . That condition couldbe examined using classical control design techniques such asroot locus diagrams. Because of the poles of are in theleft half-plane, it can be easily shown that a set of values forguaranteeing the stability of will always exist.

For all single-input QRCs, the linear sliding surfaces shownin Table III have been deduced using the previous synthesis pro-cedure. A set of design conditions for the gain parameters ofthese surfaces is represented in Table IV. These constraints havebeen derived testing the existence and reachability of a slidingregime and the local stability of the steady-state solution. In thefollowing Subsection, the application of the control design al-gorithm to a Buck QSRC is illustrated in detail.

B. Example 1: Control Design of a Buck QSRC

By using the state-space model (10), the open-loop dynamicsof can be expressed as

(22)

Because the relative degree of is two and is aminimum-phase transfer function, as can be easily found from(22), a linear sliding surface is obtained using (17)

(23)

where , , and .In sliding motion ( , ), the close-loop dynamics

of results in

(24)

The steady-state solution of (24) shows asymptotic stability ifthe following conditions are fulfilled:

(25)

The attraction domain is depicted in Fig. 5 as function ofand . Such existence region of sliding regime is derived byconsidering the boundaries of the equivalent control (i.e.

). The expression of is obtained by using (10) andthe invariance condition

(26)

In order to make the system show optimum attraction-domainsize, the sliding curve parameters must have the following con-straints:

(27)

From (26), the existence region in steady-state takes the form, which corroborates the step-down behavior of

the Buck QSRC.Finally, the control law is determined by means of the often-

used reaching condition , resulting in

S < 0(28)

C. Linear Sliding Surfaces for Multi-Input QRCs

In multi-input converters, the design of linear surfacesshowing low-cost implementation and absence of steady-stateerrors proceeds as follows. First, a specific control objectivemust be assigned to each sliding surface, which usually involvesthe stabilization of a state variable ( ) to a constantreference value . Second, according to expressions (17)and (20), the composition of the sliding surfaces must beproposed. The concept of the relative degree appearing in suchsurfaces is extended for that situation of multiple control inputs.In that case, the relative degree of a state variable is definedas the smallest number of differentiations of the state variablewith regards to time, so that at least one of the control inputsappears explicitly.

The local stability criteria for selecting between surfaces(17) and (20) in single-input converters cannot be used here.The reason is that the relationship among the control inputs isunknown beforehand, and, therefore, the small-signal transferfunctions ( and ) cannot befound. In such a case, the validity of the proposed solution isshown guaranteeing the existence and reachability of a slidingregime and the local stability of the equilibrium points.

For the TCI QSRC, the control objective of and is toregulate the input current and the output voltage to getand , respectively. Taking into account that the rela-tive degree of and is one for both variables, the followingsurfaces are proposed among other possible candidates:

(29)

being . The choice of these control structuresis due to the fact that the close-loop converter behaves in sliding

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TABLE IVDESIGN CONDITIONS FORLINEAR SURFACES

regime as a linear first-order system, as we will show in the nextSubsection.

For the TCI QPRC, surfaces and are conceived to stabi-lize the intermediate capacitor voltageto its reference value

and to provide output voltage regulation, respectively.Considering that the relative degree of, , and is one forthe first variables and two for the last one, the following surfacesare proposed:

(30a)

(30b)

being .

Finally, a set of design conditions for the gain parameters ofsurfaces (29) and (30) are shown in Table IV.

D. Example 2: Control Design of a TCI QSRC

In this Subsection, the design of the proposed controller forthe TCI QSRC is illustrated in detail. From the analysis of thecircuit, the state-space model can be written as

(31)

968 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 15, NO. 6, NOVEMBER 2000

Fig. 5. Existence region of sliding regime(k = (k E=L C V ) andk = (k =L C ) + (k � k =RC =RC )).

Fig. 6. Transient responses using Lyapunov sliding curves. (a) Boost QPRC.(b) TCI QSRC (I = 1:4 A).

where and can take the values 1 or 0, depending on theswitch position.

Fig. 7. (a) Unstable behavior of Boost QPRC usingS = k � e + k �

e dt�v (k = 0:01,k = 100) and (b) high-ripple steady-state behaviorof TCI QSRC usingS = I � i andS = k � e + k � e dt� i(I = 1:4 A, k = 0, k = 2500).

In considering that a sliding mode exists on the intersectionof the surfaces and using the invariance conditions ,

, the equivalent control and canbe expressed as

(32)

In such situation, the converter behaves as a linear first-ordersystem. In fact, replacing (32) in (31) and considering( ), the converter dynamics in sliding regime is given by

(33)

being . The equilibrium point of (33) shows asymp-totic stability if the following condition is satisfied

(34)

CASTILLA et al.: SLIDING MODE CONTROL SCHEMES FOR QUANTUM RESONANT CONVERTERS 969

Fig. 8. Comparison of (a) simulation and (b) experimental waveforms of BoostQPRC usingS = i �i , beingi = k �e +k � e dt,k = 0:04,andk = 1600. Top: Output voltage (5 V/div). Middle: Currenti (0.2A/div). Bottom: Load control signal (low:47 , high:66 ).

In this multi-input converter, the attraction domain can be de-rived by considering the natural limits of the equivalent controls

and . However, the interactionexpressed in (32) between the equivalent controls restricts thepossible values of such variables as follows:

(35)

Therefore, the actual attraction domain is obtained by replacing(32) in (35). From the analysis of such existence region in thesystem start-up and in steady state, the control parameters musthave the following constraints:

(36a)

(36b)

Fig. 9. Comparison of (a) simulation and (b) experimental waveforms of TCIQSRC usingS = I � i andS = k � e + k � e dt � v

(I = 1:4 A, k = 0, k = 212). Top: Output voltage (5 V/div). Bottom:Load control signal (low:10 , high:20 ).

Finally, the design of the control laws is done by theoften-used reaching condition , resultingin

(37)

V. SIMULATION AND EXPERIMENTAL RESULTS

The proposed control schemes for quantum resonant con-verters are evaluated in this Section. The common power-circuitparameters used in all simulations and experiments are

QSRCs: V H nF

F

QPRCs: V H H

nF H F

970 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 15, NO. 6, NOVEMBER 2000

Fig. 10. Block diagram of the proposed controllers for (a) Boost QPRC and (b) TCI QSRC (k = 0).

First, the control performance of a single-input Boost QPRCand a TCI QSRC are investigated. Fig. 6 shows transient re-sponses using Lyapunov’s sliding curves expressed in Table III.Note that presents a load-dependent steady-state error,thereby causing a poor regulation of the output voltage.

Some examples of linear sliding surfaces not observing theproposed design criteria for the state variableare depictedin Fig. 7. Using , the Boost QPRC has an unstableequilibrium point. In fact, the only state variable having min-imum-phase control-to- transfer function is ; therefore, theBoost QPRC exhibits a stable behavior whenis selected as .The transient response of the TCI QSRC usingand is represented in Fig. 7(b).Using in is not a good choice because the dy-namics of is independent of when sliding mode existson , resulting in a steady-state behavior with an undesirablehigh-ripple.

Figs. 8 and 9 compare simulation and experimental resultsof the Boost QPRC and the TCI QSRC using the linear slidingcurves shown in Table III. Excellent agreement was obtainedfor steady state and large-signal transient responses. Moreover,good output performances (such as nonzero steady-state errors,large-signal stability, low steady-state ripple, fast transient re-sponses, and high robustness) are obtained in relation to resultsdepicted in Figs. 6 and 7.

A block diagram of the control schemes shown in Table III forthe Boost QPRC and the TCI QSRC is represented in Fig. 10,showing the simplicity of the proposed solution. The slidingsurfaces can be built by linear analog circuits, and the controllaws by a comparator and a flip-flop. The goals of the flip-flipare to synchronize the control input changes with the resonantfrequency and to limit the maximum operating frequency, thusguaranteeing nearly zero switching losses and frequency sta-bility.

All the linear sliding surfaces expressed in Table III haveoutput performances as good as those described above for thecase of the Boost QPRC and the TCI QSRC. For the purposeof comparison, simulation results for all QSRCs and QPRCsare shown in Figs. 11 and 12. According to a step-down and astep-up behavior, two results are depicted in such figures foreach multi-input converter. Note that multi-input converterspresent, in all cases, lower voltage overshoot and faster re-covery time for start-up and load step changes than single-inputconverters. Especially interesting is the dynamic behavior ofthe TCI QSRC, which acts as a linear first-order system withvery low sensitivity to external perturbations and parametricvariations (see Figs. 9 and 11). The reason for this is that theconverter dynamics in sliding regime is independent of theinput source, the load, and the power circuit parameters, as itcan be seen in expression (33).

CASTILLA et al.: SLIDING MODE CONTROL SCHEMES FOR QUANTUM RESONANT CONVERTERS 971

Fig. 11. Transient responses of QSRC. (a) Buck (k = 0:4, k = 1000). (b) Step-down TCI (k = 0, k = 2500, I = 1:4 A). (c) Boost (k = 0:4,k = 1500). (d) Buck-Boost (k = 0:4, k = 1500). (e) Step-up TCI (k = 0, k = 2500, I = 7 A).

VI. CONCLUSIONS

The design of sliding mode control schemes for a widerange of quantum resonant topologies is examined by meansof two different approaches. First, an easy-to-use procedure fordevising nonlinear control structures is established, using Lya-punov’s well-known stability criteria. Second, an alternative

method that provides linear sliding surfaces is also developed,considering reaching, existence, and stability conditions.

The operation of Lyapunov’s controllers is strongly limitedby some drawbacks, which makes it unsuitable for theirpractical use. All regulators using the proposed linear surfaces,however, posses good output performances, such as nonzerosteady-state errors, large-signal stability, low steady-state

972 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 15, NO. 6, NOVEMBER 2000

Fig. 12. Transient response of QPRC. (a) Buck (k = 1:6, k = 1500). (b) Step-down TCI (k = 0:03, k = 200,k = 0:15,k = 100,V = 20 V).(c) Boost (k = 0:04,k = 400). (d) Cuk (k = 0:04, k = 400). (e) Step-up TCI (k = 0:03, k = 200, k = 0:5, k = 500,V = 30 V).

ripple, fast transient responses, and high robustness. In usingsuch controllers, multi-input converters improve the dynamicbehavior of single-input converters, showing lower voltageovershoot and faster recovery time for start-up and load stepchanges. Simulation and experimental results corroborate theabove features.

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Miguel Castilla (M’99) received the B.S., M.S., andPh.D. degrees in telecommunications engineeringfrom the Universidad Politécnica de Cataluña,Barcelona, Spain, in 1988, 1995, and 1998, respec-tively.

Since 1992, he has been an Assistant Professorin the Departamento de Ingeniería Electrónica,Universidad Politécnica de Cataluña, where heteaches analog circuits and power electronics. Hisresearch interests are in the areas of modeling,simulation, and control of dc-to-dc power converters

and high-power-factor rectifiers.

Luis García de Vicuña (M’90) received the Inge-niero de Telecomunicación and Dr.Ing. de Telecomu-nicación degrees from the Universidad Politécnica deCataluña, Barcelona, Spain, in 1980 and 1990, re-spectively, and the Dr.Sci. degree from the UniversitéPaul Sabatier, Toulouse, France, in 1992.

From 1980 to 1982, he was an Engineer withControl Aplicaciones Company. He is currentlyan Associate Professor in the Departamento deIngeniería Electrónica, Universidad Politécnica deCataluña, where he teaches power electronics. His

research interests include power electronics modeling, simulation and control,active power filtering, and high-power-factor ac/dc conversion.

Mariano López (M’98) received the M.S. and Ph.D.degrees in telecommunications engineering fromthe Universidad Politécnica de Cataluña, Barcelona,Spain, in 1996 and 1999, respectively.

Since 1996, he has been an Assistant Professor inthe Departamento de Ingeniería Electrónica, Univer-sidad Politécnica de Cataluña, where he teaches mi-croelectronics and power electronics. His main re-search interests are distributed power system, controltheory, and modeling of power converters.

Oscar López(S’99) received the M.S. degrees in physics and electronic engi-neering from the Universidad de Barcelona, Barcelona, Spain, in 1994 and 1996,respectively, and the Ph.D. degree in electronics engineering from the Univer-sidad Politécnica de Cataluña, Barcelona, Spain, in 2000.

Since 1996, he has been an Assistant Professor at the Universidad Politécnicade Cataluña. His research interests are in the area of nonlinear control systems,in particular, in applications to power electronics.

José Matas (S’97) received the B.S. and M.S.degrees in telecommunications engineering fromthe Universidad Politécnica de Cataluña, Barcelona,Spain, in 1988 and 1996, respectively, wherehe is currently pursuing the Ph.D. degree in theDepartamento de Ingeniería Electrónica.

Since 1997, he has been an Assistant Professorat the Universidad Politécnica de Cataluña. Hisresearch interests include power electronics,power-factor-correction circuits, distributed powersystems, and nonlinear control.