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960 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 15, NO. 6, NOVEMBER 2000 On the Design of Sliding Mode Control Schemes for Quantum 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 for quantum resonant converters is introduced by means of two different approaches. First, an easy-to-use procedure for devising nonlinear control structures is established, using Lyapunov’s well-known stability criteria. Second, an alternative method that provides linear sliding surfaces is also developed, considering reaching, existence, and stability conditions. The application of both control design techniques is illustrated in detail by means of three selected examples. The advantages and drawbacks of the resulting control circuits are examined. Simulation and experimental results corroborate the expected features of the close-loop quantum converters. Index Terms—Linear sliding surfaces, Lyapunov-based control, quantum resonant converters. I. INTRODUCTION A T PRESENT, most control schemes for power resonant converters are based on frequency or phase domain techniques. In both cases, the behavior of the output voltage is highly nonlinear and largely dependent on line and load conditions. In addition, the switching losses become severe when both the switching frequency deviates from the resonant frequency and the phase difference becomes large [1]. To overcome these disadvantages, a time domain control technique has been studied over the last few years, resulting in a new family of converters named quantum resonant converters (QRC) [1], [2]. The main advantage of these converters is that zero-current or zero-voltage switching conditions can be guaranteed in all the power devices because the switching frequency is always the resonant frequency [3]. However, the discrete time duration of the control cycles generates quantized output levels and large ripple in the resonant waveforms. Two main classes of close-loop control schemes can be found in the literature to solve these drawbacks: one employing predictive techniques, which are based on output voltage error estimation methods, and the other using sliding mode control strategies. In predictive approaches, very accurate discrete-time models are used both to describe the converter dynamic 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 was supported by the Spanish Ministry of Education and Culture (CICYT TIC990743) and Universidad Politécnica de Cataluña (FIU PR99-02). Recommended by Associate Editor A. Kawamura. The authors are with the Departamento de Ingeniería Electrónica, Universidad Politécnica de Cataluña, Barcelona 08800, Spain (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Publisher Item Identifier S 0885-8993(00)09807-0. difficult, since many calculations are required in the real-time control algorithm [7]. Sliding mode control schemes for QRC converters are frequently designed using continuous reduced-order averaged models. Such models provide enough state information to both correctly regulate the output voltage in dc-to-dc applications [8]–[11] and force the output voltage to follow an external sinusoidal reference in ac waveform generation [12], [13]. In addition, control schemes offer robustness against external disturbances and parameter variations. However, the application of sliding mode control to QRC converters is still only explored today in a short number of studies, which basically involve the control design of the series-type topology. In a clear-cut contrast, linear and nonlinear control design techniques have been extensively tested in hard-switching dc-to-dc conversion cells. On the one hand, the structure of the sliding surface is proposed as a linear combination of all the state variables, using linear control techniques [14]–[18]. In such a case, the design procedure consists in determining the appropriate gain parameters that satisfy a set of conditions for the existence and reachability of a sliding regime. Although this class of sliding surfaces leads to a good dynamic response, the price to pay is the need for sensing and processing all the state variables, thus increasing the complexity of the control circuit implementation. On the other hand, several systematic approaches for the synthesis of nonlinear sliding surfaces with suitable stabilizing properties have been investigated, using nonlinear control techniques. Sliding mode control via feedback linearization [19], extended linearization [20], and other nonlinear control methods [21]–[23] is just a short list of examples. In those cases, the sliding surfaces are composed of nonlinear functions, which, in turn, depend on the input voltage, the load, and circuit parameters. Consequently, control circuits generally require the use of multiplier and divider circuits as well as the sensing of the input voltage and the load, thereby increasing the difficulty of their practical implementation and operation. The aim of this paper is to deduce sliding mode controllers for a wide range of series and parallel QRC converters, including single- and multi-input topologies. Two different control design approaches are proposed in order to compare the advantages and drawbacks of the resulting close-loop controllers. In Section III, an easy-to-use procedure for devising nonlinear control structures is established, using the Lyapunov’s well-known stability criteria. An alternative method that provides linear sliding surfaces is developed in Section IV, considering reaching, existence, 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 Fig. 1. 961 Basic topologies of quantum resonant converters: (a) series-type and (b) parallel-type. trol design techniques is illustrated in detail by means of three selected examples. Moreover, sliding mode controllers for all QRC converters are proposed. Finally, in Section V, simulation and experimental results corroborate the expected features of the close-loop quantum converters. TABLE I SINGLE-INPUT QSRC II. PRINCIPLE OF OPERATION AND LARGE-SIGNAL MODELS OF QRC The basic topologies of series and parallel QRCs are shown in Fig. 1. For both converters, the switching frequency is always the resonant frequency, so that each control-input state presents a discrete time duration that coincides with an integer number of half resonant periods. Moreover, control-input changes are always synchronized in the QSRC with the zero-current crossing points in the resonant inductor and in the QPRC with the zerovoltage crossing points in the resonant capacitor. Consequently, nearly zero switching conditions are guaranteed in all the power devices [1], [2]. The direction of the energy flow between the dc source and the load can be determined by two independent control inputs (TCI): and . On the one hand, the control input regulates , the state of the full-bridge switching network. When the pair of switches S1–S4 and S2–S3 are switching on and off alternatively, thus continuously transferring energy to the , the full-bridge network is resonant tank; whereas if blocked, thus not delivering energy to the resonant circuit. On commands the state of S5 the other hand, the control input , the resonant tank energy is partially switch; so when TABLE II SINGLE-INPUT QPRC discharged into the load, and if , the load does not receive energy from the resonant circuit [1], [2]. Quantum converters can also operate just like some singleand as shown in input conventional converters, driving Tables I and II. In such a case, the state of the full-bridge network and S5 switch are governed by the control input u only, which can take the values 1 or 0. For control design purposes, the appropriate dynamic description of quantum converters is by means of averaged large-signal models, which are essential to study their global dynamic 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 averaged variable definition are given in [9] and [25]. The averaging procedure for resonant converter modeling is also described in [26]. For the case of single-input converters, the averaged models are obtained by replacing the control variables and as shown in Tables I and II. The equivalent circuits of single-input series- and parallel-type models are represented in Figs. 3 and 4, respectively. method of Lyapunov. In addition, a set of sliding mode controllers for such converters is proposed. A. Synthesis Procedure of Lyapunov’s Sliding Surfaces Quantum resonant converters are multi-input nonlinear systems whose averaged state models can be represented as follows: (1) III. LYAPUNOV-BASED CONTROL DESIGN In this Section, a systematic approach for the control design of all single- and multi-input QRCs is developed, using the second , the vector fields where the state vector , 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 systems and their associated consists of a set of sliding surfaces control laws: for for respect to their average values in steady-state ): (i.e. (4) (2) and take the values 1 or 0, and verify . where For the design of the control structure (2), the Lyapunov function approach proceeds by first defining a positive-definite func, and then using as a global reaching tion the deviation of the state vector with condition [21], being ). As the regards to its steady-state value (i.e. Lyapunov function candidate, we chose the following quadratic form: (3) is the transposed of and is a diagonal matrix where of positive constant elements. In fact, such a function coincides with the incremental energy of the converter if the elements of matrix are conveniently expressed in terms of the values of reactive components [22]. , defining The state model (1) can be rewritten in terms of as the deviation of with a new set of control inputs . In that case, the control where law (2) can be expressed as: for for (5) can be positive ( Note that the two possible values of ) or negative ( ), as must be bounded between their . minimum and maximum limits By replacing (3) and (4) in the reaching condition, we obtain: (6) corresponds to the time-derivative of the The term incremental 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 converter with the negative sign. In such a case, the fulfilment of expression (7) ensures that the reaching condition is satisfied: and, consequently, the incremental state vector written as can be (13) (7) The sliding mode control structure can be directly obtained from (7). In fact, we could select each control input according to the sign of each value , so that each can always be negative: product Second, the incremental energy of the Buck QSRC takes the form (14) From (3) and (14), the matrix for is given (8) for (15) Finally, when comparing (5) and (8), the sliding surfaces and their associated control laws can be identified as for for In using (11), (13), and (15), Lyapunov’s sliding surface for the Buck QSRC results in (9) (16) B. Lyapunov-Based Controllers for QRCs We begin by illustrating in detail the synthesis of a sliding mode controller for the Buck QSRC shown in Fig. 3(a). From the analysis of the circuit, the state-space model can be expressed as (10a) where can take the values 1 or 0, depending on the switch position. The state-space model can be rewritten as (10b) being For the rest of QRCs, Lyapunov’s sliding curves are shown in Table III. These control structures have been deduced by applying the synthesis procedure to the averaged models represented in Figs. 2–4. However, the utilization of such surfaces is strongly limited by some practical drawbacks. The first problem is the complexity of the hardware required to implement these nonlinear functions, which usually depend on the input voltage, the load, and a considerable number of state variables. The second problem is caused by the appearance of output voltage steady-state errors, due to the imperfections involved in different parts of the system, such as relays, losses, etc. (see Section V). In the next Section, an alternative approach for the synthesis of sliding surfaces is proposed. The focus will be on the use of linear stabilizing terms as a way of generating simple and low-cost control circuits. IV. DESIGN OF LINEAR SLIDING SURFACES and From (9), the Lyapunov-based control structure for this single-input converter has the expression (11) then, in order to obtain a practical description of the switching and matrix must be surface, the incremental state vector deduced. the state variable that is desired to regFirst, considering , the steady-state solution of (10) ulate to a constant value in close-loop operation is (12) In this Section, an alternative approach for the design of sliding mode controllers for single- and multi-input QRCs is presented. First, an interesting class of linear sliding surfaces, showing low-cost implementation and absence of steady-state errors, is proposed. Second, a set of design constraints for the gain parameters of such surfaces is deduced by using reaching, existence, and stability conditions. We begin by describing the control design procedure for single-input converters, including a detailed example of application for the Buck QSRC. Next, the case of multi-input converters is introduced together with a detailed example for the TCI QSRC. A. Linear Sliding Surfaces for Single-Input QRCs In single-input nonlinear systems, the relative degree of a state variable is defined as the smallest number of differentiations of the state variable with regards to time, so that the control input appears explicitly [21]. CASTILLA et al.: SLIDING MODE CONTROL SCHEMES FOR QUANTUM RESONANT CONVERTERS TABLE III SLIDING SURFACES FOR QRC (e V = When considering ( ) the state variable that , the following is desired to regulate to a reference value linear sliding surface is proposed: 0v ,e 965 =V 0v ) deviation of the equivalent control with regards to its steady-state value takes the expression: (17) are constant gains and is the relative degree of where ( ). Note that surface (17) satisfies the transversality condition (i.e. depends explicitly on the control input ) and removes the undesirable steady-state errors of , due to the presence of the integral term. Moreover, a linear dynamics of is achieved in sliding motion ( , ), which can be expressed as follows: (18) However, in using surface (17), the equilibrium point of the converter can be unstable, even when choosing appropriate . For the analysis of such situation, we can examine the stability . In fact, in open-loop operation, a of the equivalent control converter has always stable steady-state behavior, since the converter is composed of passive components [22]. Therefore, the ensures the stability of the close-loop converter. stability of transfer function as In defining the control-toand using (18), the small-signal (19) From (19), a necessary condition for local stability of the closeloop converter is minimum-phase control-to- transfer function. is a nonminimum-phase transfer function, a If , ) must new minimum-phase state variable ( be included in surface (17) to guarantee system stability: (20) and , Now depends on the relative degree of variables and ). defined as and , respectively ( In order to fulfill the transversality condition, must be exactly if ; whereas if , can take any value from 1 to , 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 expression of the equivalent control. In such a case, when using the control-to- and control-to- transfer functions and the , the equivalent control can be exinvariance condition pressed as The attraction domain is depicted in Fig. 5 as function of and . Such existence region of sliding regime is derived by considering the boundaries of the equivalent control (i.e. ). The expression of is obtained by using (10) and the invariance condition (26) (21) In order to make the system show optimum attraction-domain size, the sliding curve parameters must have the following constraints: being (27) Taking into account that has been chosen among the state transfer variables observing minimum-phase control-tofunction, the stability of expression (21) is conditioned only . That condition could by the position of the roots of be examined using classical control design techniques such as are in the root locus diagrams. Because of the poles of left half-plane, it can be easily shown that a set of values for guaranteeing the stability of will always exist. For all single-input QRCs, the linear sliding surfaces shown in Table III have been deduced using the previous synthesis procedure. A set of design conditions for the gain parameters of these surfaces is represented in Table IV. These constraints have been derived testing the existence and reachability of a sliding regime and the local stability of the steady-state solution. In the following Subsection, the application of the control design algorithm to a Buck QSRC is illustrated in detail. B. Example 1: Control Design of a Buck QSRC of By using the state-space model (10), the open-loop dynamics can be expressed as (22) is two and is a Because the relative degree of minimum-phase transfer function, as can be easily found from (22), a linear sliding surface is obtained using (17) (23) , where In sliding motion ( results in of , and , . ), the close-loop dynamics (24) The steady-state solution of (24) shows asymptotic stability if the following conditions are fulfilled: (25) 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, resulting in used reaching condition S<0 (28) C. Linear Sliding Surfaces for Multi-Input QRCs In multi-input converters, the design of linear surfaces showing low-cost implementation and absence of steady-state errors proceeds as follows. First, a specific control objective must be assigned to each sliding surface, which usually involves ( ) to a constant the stabilization of a state variable . Second, according to expressions (17) reference value and (20), the composition of the sliding surfaces must be proposed. The concept of the relative degree appearing in such surfaces is extended for that situation of multiple control inputs. In that case, the relative degree of a state variable is defined as the smallest number of differentiations of the state variable with regards to time, so that at least one of the control inputs appears 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 is unknown beforehand, and, therefore, the small-signal transfer ( and ) cannot be functions found. In such a case, the validity of the proposed solution is shown guaranteeing the existence and reachability of a sliding regime and the local stability of the equilibrium points. and is to For the TCI QSRC, the control objective of regulate the input current and the output voltage to get and , respectively. Taking into account that the relative degree of and is one for both variables, the following surfaces are proposed among other possible candidates: (29) . The choice of these control structures being is due to the fact that the close-loop converter behaves in sliding CASTILLA et al.: SLIDING MODE CONTROL SCHEMES FOR QUANTUM RESONANT CONVERTERS 967 TABLE IV DESIGN CONDITIONS FOR LINEAR SURFACES regime as a linear first-order system, as we will show in the next Subsection. For the TCI QPRC, surfaces and are conceived to stabilize the intermediate capacitor voltage to its reference value and to provide output voltage regulation, respectively. is one for Considering that the relative degree of , , and the first variables and two for the last one, the following surfaces are proposed: Finally, a set of design conditions for the gain parameters of surfaces (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 for the TCI QSRC is illustrated in detail. From the analysis of the circuit, the state-space model can be written as (30a) (30b) being . (31) 968 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 15, NO. 6, NOVEMBER 2000 Fig. 5. Existence region of sliding regime (k k = (k =L C ) + (k 0 k =RC =RC = (k E =L C V ) and )). 1 1 Fig. 7. (a) Unstable behavior of Boost QPRC using S = k e +k e dt v (k = 0:01, k = 100) and (b) high-ripple steady-state behavior i and S = k e + k e dt i of TCI QSRC using S = I (I = 1:4 A, k = 0, k = 2500). 0 0 1 1 0 In considering that a sliding mode exists on the intersection , of the surfaces and using the invariance conditions , the equivalent control and can be expressed as (32) In such situation, the converter behaves as a linear first-order system. In fact, replacing (32) in (31) and considering ( ), the converter dynamics in sliding regime is given by (33) 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 the switch position. . The equilibrium point of (33) shows asympbeing totic stability if the following condition is satisfied (34) CASTILLA et al.: SLIDING MODE CONTROL SCHEMES FOR QUANTUM RESONANT CONVERTERS Fig. 8. Comparison of (a) simulation and (b) experimental waveforms of Boost QPRC using S i i , being i k e + k e dt, k = 0:04, and k = 1600. Top: Output voltage (5 V/div). Middle: Current i (0.2 A/div). Bottom: Load control signal (low: 47 , high: 66 ). = 0 = 1 1 In this multi-input converter, the attraction domain can be derived by considering the natural limits of the equivalent controls and . However, the interaction expressed in (32) between the equivalent controls restricts the possible values of such variables as follows: 969 Fig. 9. Comparison of (a) simulation and (b) experimental waveforms of TCI QSRC using S = I i and S = 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 ). 0 1 1 0 Finally, the design of the control laws is done by the , resulting often-used reaching condition in (37) (35) V. SIMULATION AND EXPERIMENTAL RESULTS Therefore, the actual attraction domain is obtained by replacing (32) in (35). From the analysis of such existence region in the system start-up and in steady state, the control parameters must have the following constraints: (36a) The proposed control schemes for quantum resonant converters are evaluated in this Section. The common power-circuit parameters used in all simulations and experiments are QSRCs: H nF F QPRCs: (36b) V V nF H H H F 970 Fig. 10. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 15, NO. 6, NOVEMBER 2000 Block diagram of the proposed controllers for (a) Boost QPRC and (b) TCI QSRC (k First, the control performance of a single-input Boost QPRC and a TCI QSRC are investigated. Fig. 6 shows transient responses using Lyapunov’s sliding curves expressed in Table III. presents a load-dependent steady-state error, Note that thereby causing a poor regulation of the output voltage. Some examples of linear sliding surfaces not observing the are depicted proposed design criteria for the state variable , the Boost QPRC has an unstable in Fig. 7. Using equilibrium point. In fact, the only state variable having minimum-phase control-to- transfer function is ; therefore, the Boost QPRC exhibits a stable behavior when is selected as . The transient response of the TCI QSRC using and is represented in Fig. 7(b). in is not a good choice because the dyUsing is independent of when sliding mode exists namics of on , resulting in a steady-state behavior with an undesirable high-ripple. Figs. 8 and 9 compare simulation and experimental results of the Boost QPRC and the TCI QSRC using the linear sliding curves shown in Table III. Excellent agreement was obtained for 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 responses, and high robustness) are obtained in relation to results depicted in Figs. 6 and 7. = 0). A block diagram of the control schemes shown in Table III for the Boost QPRC and the TCI QSRC is represented in Fig. 10, showing the simplicity of the proposed solution. The sliding surfaces can be built by linear analog circuits, and the control laws by a comparator and a flip-flop. The goals of the flip-flip are to synchronize the control input changes with the resonant frequency and to limit the maximum operating frequency, thus guaranteeing nearly zero switching losses and frequency stability. All the linear sliding surfaces expressed in Table III have output performances as good as those described above for the case of the Boost QPRC and the TCI QSRC. For the purpose of comparison, simulation results for all QSRCs and QPRCs are shown in Figs. 11 and 12. According to a step-down and a step-up behavior, two results are depicted in such figures for each multi-input converter. Note that multi-input converters present, in all cases, lower voltage overshoot and faster recovery time for start-up and load step changes than single-input converters. Especially interesting is the dynamic behavior of the TCI QSRC, which acts as a linear first-order system with very low sensitivity to external perturbations and parametric variations (see Figs. 9 and 11). The reason for this is that the converter dynamics in sliding regime is independent of the input source, the load, and the power circuit parameters, as it can be seen in expression (33). CASTILLA et al.: SLIDING MODE CONTROL SCHEMES FOR QUANTUM RESONANT CONVERTERS Transient responses of QSRC. (a) Buck (k = 0:4, k = 1000). (b) Step-down TCI (k = 0, k = 2500, I = 1500). (d) Buck-Boost (k = 0:4, k = 1500). (e) Step-up TCI (k = 0, k = 2500, I = 7 A). Fig. 11. k VI. CONCLUSIONS The design of sliding mode control schemes for a wide range of quantum resonant topologies is examined by means of two different approaches. First, an easy-to-use procedure for devising nonlinear control structures is established, using Lyapunov’s well-known stability criteria. Second, an alternative 971 = 1:4 A). (c) Boost (k = 0:4, method that provides linear sliding surfaces is also developed, considering reaching, existence, and stability conditions. The operation of Lyapunov’s controllers is strongly limited by some drawbacks, which makes it unsuitable for their practical use. All regulators using the proposed linear surfaces, however, posses good output performances, such as nonzero steady-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 using such controllers, multi-input converters improve the dynamic behavior of single-input converters, showing lower voltage overshoot and faster recovery time for start-up and load step changes. Simulation and experimental results corroborate the above features. REFERENCES [1] G. B. Joung, C. T. Rim, and G. H. Cho, “An integral cycle mode control of series resonant converter,” in Proc. IEEE PESC Conf. Rec., 1988, pp. 575–582. [2] G. B. Joung and G. H. Cho, “Modeling of quantum parallel resonant converters controlled by integral cycle mode,” in Proc. IEEE PESC Conf. Rec., 1989, pp. 744–751. CASTILLA et al.: SLIDING MODE CONTROL SCHEMES FOR QUANTUM RESONANT CONVERTERS [3] W. H. Kwon and G. H. Cho, “Optimum quantum sequence control of quantum series resonant converter for minimum output voltage ripple,” IEEE Trans. Power Electron., vol. 9, pp. 74–84, Jan. 1994. 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[25] M. Castilla, “Modelos no Lineales y control en modo deslizamiento de convertidores de estructura resonante,” Ph.D. dissertation, Univ. Politécnica de Cataluña, Barcelona, Spain, 1998. [26] M. Castilla, L. García de Vicuña, M. López, and V. Barcons, “An averaged large-signal modeling method for resonant converters,” in Proc. IEEE IECON Conf. Rec., 1997, pp. 447–452. 973 [27] O. López, L. García de Vicuña, M. Castilla, M. López, and J. Majó, “A systematic method to design sliding mode surfaces by imposing a desired dynamic response,” in Proc. IEEE IECON Conf. Rec., 1998, pp. 381–384. Miguel Castilla (M’99) received the B.S., M.S., and Ph.D. degrees in telecommunications engineering from the Universidad Politécnica de Cataluña, Barcelona, Spain, in 1988, 1995, and 1998, respectively. Since 1992, he has been an Assistant Professor in the Departamento de Ingeniería Electrónica, Universidad Politécnica de Cataluña, where he teaches analog circuits and power electronics. His research 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 Ingeniero de Telecomunicación and Dr.Ing. de Telecomunicación degrees from the Universidad Politécnica de Cataluña, Barcelona, Spain, in 1980 and 1990, respectively, and the Dr.Sci. degree from the Université Paul Sabatier, Toulouse, France, in 1992. From 1980 to 1982, he was an Engineer with Control Aplicaciones Company. He is currently an Associate Professor in the Departamento de Ingeniería Electrónica, Universidad Politécnica de Cataluña, 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 from the Universidad Politécnica de Cataluña, Barcelona, Spain, in 1996 and 1999, respectively. Since 1996, he has been an Assistant Professor in the Departamento de Ingeniería Electrónica, Universidad Politécnica de Cataluña, where he teaches microelectronics and power electronics. His main research interests are distributed power system, control theory, and modeling of power converters. Oscar López (S’99) received the M.S. degrees in physics and electronic engineering from the Universidad de Barcelona, Barcelona, Spain, in 1994 and 1996, respectively, and the Ph.D. degree in electronics engineering from the Universidad Politécnica de Cataluña, Barcelona, Spain, in 2000. Since 1996, he has been an Assistant Professor at the Universidad Politécnica de 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 from the Universidad Politécnica de Cataluña, Barcelona, Spain, in 1988 and 1996, respectively, where he is currently pursuing the Ph.D. degree in the Departamento de Ingeniería Electrónica. Since 1997, he has been an Assistant Professor at the Universidad Politécnica de Cataluña. His research interests include power electronics, power-factor-correction circuits, distributed power systems, and nonlinear control.