Passivity-Based PFC for Interleaved Boost Converter of PMSM drives

11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing July 3-5, 2013. Caen, France

WeS2T1.4

Passivity-Based PFC for Interleaved Boost Converter of PMSM drives Gionata Cimini ∗ Maria Letizia Corradini ∗∗ Gianluca Ippoliti ∗ Giuseppe Orlando ∗ Matteo Pirro ∗ ∗

Dipartimento di Ingegneria dell’Informazione, Universit` a Politecnica delle Marche, Ancona, Italy Email: {gianluca.ippoliti,giuseppe.orlando,m.pirro}@univpm.it ∗∗ Scuola di Scienze e Tecnologie, Universit` a di Camerino, Camerino (MC), Italy Email: [email protected] Abstract In this paper a sensorless passivity-based control with input voltage feedforward has been designed for a boost AC-DC interleaved converter to increase the Power Factor (PF) in Permanent Magnet Synchronous Motor (PMSM) drives. A robust current observer has been adopted. The proposed solution has been experimentally tested on a commercial PMSM drive equipped with a control system based on a microcontroller (MCU) of the Texas Instrument (TI) family C2000T M . Keywords: Power amplifiers; Power control; Power management; Digital control; AC converter machines; Brushless motors; Lyapunov stability; Observers. 1. INTRODUCTION The ever increasing penetration of single-phase power electronic-based appliances into the different segments of end-use demands the establishment of adequate harmonic current limits. The harmonic reduction requirements imposed by regulatory agencies (IEC 1000 − 3 − 2 and EN 61000−3−2) have accelerated interest in active power factor corrected preregulators for switching power supplies. Therefore the implementation of a power factor correction technique is a required feature for a good quality electrical and electromechanical system. There are two general types of Power Factor Control (PFC) methods to obtain a unity power factor: analog and digital PFC techniques. In the past, due to the absence of fast microprocessors and DSPs, analog PFC methods were the only choice for achieving the unity power factor [Erickson and Maksimovic, 2001]. Many control strategies using analog circuits have been explored, including average current control [Sun and Bass, 1999], peak current control [Redl and Erisman, 1994], hysteresis control [Spangler and Behera, 1993] and nonlinear carrier control [Zane and Maksimovic, 1998]. With the recent developments in the microprocessor and DSP technologies, there is the possibility to implement complex PFC algorithms using these fast processors [Zhang et al., 2004]. An important application field of PFC techniques is electric motor control. In fact AC motor drives rejects a high number of harmonics in the line current and the Power Factor Correction (PFC) method is a good candidate for AC-to-DC switched mode power supply in order to reduce the harmonics in the line current, increase the efficiency and capacity of motor drives in particular for autonomous 978-3-902823-37-3/2013 © IFAC

128

vehicles [Armesto et al., 2008, Ippoliti et al., 2005, Fulgenzi et al., 2009], and reduce customers’ utility bills [Li et al., 2010, Ciabattoni et al., 2013, de Angelis et al., 2013]. For the electrical and electromechanical systems considered in this paper, an interesting feedback control methodology has been developed with the aim to modify the closed loop energy dissipation and potential energy properties of nonlinear passive systems [Ortega et al., 1998] and references therein. Passive systems are a class of dynamical systems in which the energy exchanged with the environment plays a central role. In passive systems the rate at which the energy flows into the system is not less than the increase in storage. The purpose of this approach, known as Passivity Based Controller (PBC) design [Ortega et al., 1998] is to render the closed-loop system passive with respect to a desired storage function. Passivity-based method has clear advantages in terms of considering the system physical structure in the control design procedure [Rosa et al., 2011, Wang and Ma, 2010, Ortega et al., 1998]. Furthermore the reduction of the sensor number is of utmost concern to industrial users as it represents a significant cost advantage as well as a reliability issue. An interesting approach is to use observer based techniques in order to estimate the unmeasurable variables [Xu and Rahman, 2007, Corradini et al., 2013]. Besides obvious economic benefits offered by sensor reduction, sensorless approach has several other advantages such as elimination of sensor offsets, insensitivity to noise and size reduction of converter system. Indeed sensorless approach is well known in literature [Guo et al., 2006]; avoiding to use expensive current sensors [Mattavelli, 2004] or voltage sensors [Oettmeier et al., 2009] could be strategic for consumer and mass-produced industrial applications. 10.3182/20130703-3-FR-4038.00128

11th IFAC ALCOSP July 3-5, 2013. Caen, France

Figure 1. System overview block diagram. In this paper a sensorless passivity-based solution for ACDC interleaved boost converter power factor control has been considered for a Permanent Magnet Synchronous Motor (PMSM). The motor drive is equipped with a sliding mode based sensorless cascade speed control scheme [Corradini et al., 2012c], which is computationally simpler with respect to other robust control approaches, thus well suited for low-cost DSP implementation [Utkin et al., 1999, Corradini et al., 2012d,b,a, Cimini et al., 2013]. Tests are based on experimental results. 2. SYSTEM OVERVIEW The system overview is shown in Fig. 1: the first stage is the AC source that is rectified and is the input of the passivity-based controlled interleaved boost converter; the third stage is the inverter properly driven for PMSM control. 2.1 Interleaved configuration In order to extend the power limits, it has been proposed the parallel connection of PFC boost converters operating in interleaved mode. If the input current is shared among the converters, the interleaved mode generates a smaller input ripple current for EMI filter and a smaller RMS ripple current for output capacitors, reduces current stress for each parts (including switches, diodes, inductance), has lower switching frequency for each phase and has simple structure [Olmos-Lopez et al., 2011]. To get equal input current sharing many proposals had been studied: masterslave configuration, control current mode, programming average current technique. Master-slave approaches with a fix on-time control are the most common techniques to synchronize the operation of the converters, but it is necessary that both inductors have equal characteristics; in that way, if this is assurance, current sharing is guarantee. With master-slave technique here adopted, master branch operates freely, driven by output duty-cycle control; whereas slave branch follows master one with a 180◦ phase shift. The interleaving technique has significant advantages compared to the standard configuration, both for the smaller size of the power circuit, but also from the point of view of the controlled variables. Since the currents flowing in the two branches have a phase shift of 180◦ , also the related ripples are out of phases and attenuate each others. This phenomenon is clearly shown in Fig. 2. The first graph shows the control signals of the mosfets; presented is the special case in which the duty cycle is set to 0.5. The ripple on the input current is shown in the second graph where there is a clear trend of attenuation of the ripple. Instead, a light ripple in the output current is visible in the third graph: in fact, during the first lapse of the switching period, inductor currents flow through the mosfets. Furthermore it can be proven that the condition which ensures the largest cancelation of the ripple is to set the value of the duty cycle to 0.5. This result is particularly important for the design of the boost in order to perform a correct dimensioning in the case of interleaved configuration. 129

Figure 2. Interleaved boost converter ripple. 2.2 Power Factor Control Power Factor (PF) is defined as the ratio of the real power to apparent power; it is a measure of how efficiently the current is drawn from the source. Power factor can vary between 0 and 1 and when the current and voltage waveforms are in phase, the power factor is 1. The purpose of making the power factor equal to one is to make the circuit look purely resistive (apparent power equals real power). The 3-phase inverter stage and the motor act as a load to the PFC stage and represent a non linear load that draw harmonic currents from the power stage. Harmonics result in reactive power which causes the real power to be less than apparent power, reducing power factor and resulting in losses. Harmonic current with high THD (Total Harmonic Distortion) can also distort the line voltage. The purpose of power factor control is to drawn a sinusoidal current from the line (with less distortion as possible) in phase with line voltage. PF and THD indexes are related by Eqs. (1) and (2) (distortion factor)

z

}|

  PF =  s 

I √1 2

{

   cos(θ1 − φ1 )  ∞ I2  P n I02 + n=1 2

(1)

1 (2) 1 + T HD2 with I0 the current dc component, I1 the current fundamental component and (θ1 − φ1 ) the difference between fundamental components phase of voltage and current. (distortionf actor) = √

The PFC stage used for experimental tests is an interleaved boost converter as shown in Fig. 3. 3. PFC CONTROL Control scheme of adopted PFC solution is shown in Fig. 3. A cascade structure, with outer voltage control loop, intermediate input voltage feedforward stage and inner

11th IFAC ALCOSP July 3-5, 2013. Caen, France

passivity current loop has been considered and argued in this section. Boost converter mathematical model, used for control design has been discussed too. 3.1 Interleaved boost converter model Referring to Fig. 3 the dynamic model of the interleaved boost converter is derived using the state-space averaging method [Erickson and Maksimovic, 2001]. The variables are averaged over one switching period Ts , leading to the following equivalent low-frequency model of the converter: dhig (t)i 2 RL RDSon = (hvg (t)i − hig (t)i − d(t) hig (t)i− dt L 2 2 Rd − d0 (t)hv(t)i − d0 (t) hig (t)i − d0 (t)Vd ) 2   dhv(t)i 1 1 0 = d (t)hig (t)i − hv(t)i dt C R (3) where hv(t)i and hig (t)i are respectively the averaged voltage and current, L, C and R are respectively the inductance, the capacitance and the load value, Vd is the diode threshold, Rd is the diode resistance, RDSon is the mosfet on-resistance and finally d(t) = 1 − d0 (t) is the duty cycle of the PWM. For interleaved boost modeling and control it is taken into account the relationship between shunt sensed current and inductor branches (see Fig. 3), i.e. ig (t) = il1 + il2 , which is valid with equivalent inductor branches. 3.2 Voltage loop A linear compensator Gcv (s) (see Fig. 3) has been designed, considering the transfer function (4), to regulate the DC output voltage v(t) at the desired value vref , driving the signal vcontrol (t) used in feedforward stage. The control-to-output transfer function (4) has been obtained from the equivalent small-signal ac circuit (Loss Free Resistor (LFR) model) resulting from the perturbation and linearization technique [Singer and Erickson, 1994]: Pav vˆ(s) = (4) ¯ vˆcontrol (s) sC V Vcontrol where s is the Laplace variable, V¯ and Vcontrol represent the DC output voltage and the DC control signal, respectively, Pav is the average rectifier power and C is the boost capacitance. The designed linear compensator Gcv (s) consists of a PI with anti windup technique [Erickson and Maksimovic, 2001]. 3.3 Input voltage feedforward Derivation of inner control reference from input voltage vg (t) makes the current ig (t) following a sinusoidal wave shape, proportional to the input voltage. Multiplier feedforward stage implements the following equation: kv vcontrol (t)vg (t) igref (t) = (5) vg,rms (t)2 where kv is a design constant, vcontrol (t) is the control signal of the compensator Gcv (s), vg (t) is the rectified input voltage and vg,rms (t) is the computed vg (t) RMS value [Erickson and Maksimovic, 2001]. 130

Figure 3. PFC control scheme. 3.4 Current Control Inner control loop has been designed to regulate the input rectified current at the desired value igref , driving the mosfet gating signal. Following PBC methodology of [Ortega et al., 1998], the averaged model (3) has been used in an Euler-Lagrange (EL) matrix equivalent formulation as DB z(t) ˙ − (1 − d(t))JB z(t) + RB z(t) = εB (d(t)) (6) where       ig (t) L/2 0 0 −1 z(t) = ; DB = ; JB = ; v(t) 0 C 1 0     r(d(t)) 0 v (t) − (1 − d(t))Vd RB = ; εB (d(t)) = g ; 0 1/R 0 RL Ron Rd and r(d(t)) = + d(t) + (1 − d(t)) . The storage 2 2 2 function 1 Hd = z˜DB z˜ (7) 2 has been considered, where z˜ = z −zd and zd is the desired trajectory for state variables. The imposed damping can be achieved choosing a Rayleigh error dissipation term such as 1 1 Fd = z˜T RBd z˜ = z˜T (RB + R1B )˜ z (8) 2 2 with   R1 0 R1B = ; R1 > 0; R2 > 0. (9) 0 R2 Error dynamics associated to storage function (7) is DB z˜˙ (t) − (1 − d(t))JB z˜(t) + RB z˜(t) = Ψ(d(t)) (10) where Ψ(d(t)) = εB (d(t)) − (z˙d (t) − (1 − d(t))JB zd (t)+ (11) +RB zd (t) − R1B z˜) is a perturbation term. Unperturbed error dynamics, obtained setting Ψ(d(t)) = 0 in (10), with the injection designed matrix (9) is exponentially convergent; in fact the time derivative of Hd along solutions of unperturbed dynamics is H˙ d = −˜ z T RBd z˜ < 0. (12) Controller dynamics are derived from (6) and unperturbed version of (10), obtaining DB z˙d (t)−(1 − d(t))JB zd (t)+ (13) + RB zd (t) − R1B z˜d (t) = εB (d(t)) thus, the implicit definition of PBC is given as

11th IFAC ALCOSP July 3-5, 2013. Caen, France

L z˙1d + r(d)z1d + d0 z2d − R1 (z1 − z1d ) = vg − d0 Vd (14) 2 z2d C z˙2d − d0 z1d + − R2 (z2 − z2d ) = 0. (15) R Above equations have been discretized and implemented using backward Euler method for voltage equations and a discrete differentiator for current reference. Regulating z1 (t) = ig (t) towards a desired value z1d (t) provided by input voltage feedforward stage, in [Ortega et al., 1998] has been shown that zero-dynamics associated with controller (14) and (15) is locally stable around the only physically meaningful equilibrium point. Figure 4. Experimental setup.

4. CURRENT OBSERVER A model-based observer for the estimation of current ig (t) has been considered and used to generate driving error for the inner control loop (see Fig. 3). In Eq. (3) the parameters Rd , RDSon , RL and Vd which represents losses of real components are well known from their datasheets; changes in capacitor C or inductor L influence the ripple of the state variables but not their mean value. On the contrary load value is frequently unknown or highly variable, thus a robust observer against R parameter variation is proposed. To account for possible model uncertainty, it is assumed that the R parameter may differ from its nominal value ˆ + ∆R for some unknown but bounded quantity, i.e. R = R with |∆R| ≤ ρR . Referring to the structure of system dynamics (Eq. (3)), the observer dynamics equations are:   ˆi˙ g (t) = 2 −d0 (t)ˆ v (t) + vg (t) − d0 (t)Vd − r(d(t))ˆig (t) L (16) ˆ ig (t) vˆ(t) vˆ˙ (t) = d0 (t) − − K(ˆ v (t) − v(t)) + ζ(t) (17) ˆ C RC RL RDSon Rd where r(d(t)) = + d(t) + d0 (t) , K is a design 2 2 2 gain and sign(˜ v (t)) ζ(t) = − (ρ1 |v(t)| + α) (18) C ρR with ρ1 = . Defining ˜ig (t) = (ˆig (t) − ig (t)) ˆ ˆ (R + ρR )R and v˜(t) = (ˆ v (t) − v(t)) as the observer errors, the error dynamics is 0 v (t) 2r(d(t))˜ig (t) ˜i˙ g (t) = − 2d (t)˜ − (19) L L ˜ig (t) v˜(t) ∆1 v(t) v˜˙ (t) = d0 (t) − − K v˜(t) + + ζ(t) (20) ˆ C C RC ∆R with ∆1 = − . Thus choosing the Lyapunov ˆ ˆ (R + ∆R )R function   1 L ˜2 V = ig (t) + C v˜2 (t) (21) 2 2 its time derivative along solutions of observer error dynamics (19) and (20) can be found as: V˙ < −α|˜ v (t)|. (22) Therefore observer errors tend to zero asymptotically, with robustness against load variation. 131

5. EXPERIMENTAL IMPLEMENTATION 5.1 Experimental Overview In this section PFC performance comparison in terms of PF and THD has been reported comparing different scenarios. Simulated and real systems have been taken into exam, using a pure resistor and a motor as a load. Numerical and experimental tests have been performed in order to verify system performance respectively using PSIM (by Powersim Inc), a powerful simulation software designed for power electronics, motor control and dynamic system simulation and real tests on a commercial PMSM drive equipped with a control system based on a microcontroller (MCU) of the Texas Instrument family C2000T M . Control purpose of test scenario is to supply a 20W load or driving a brushless motor at 750RPM (in motor control scenario) with a rectified DC-bus at 24V without degrading AC power quality. In particular power factor and total harmonic distortion improvements with PFC have been reported. In the motor control scenario the PMSM drive is equipped also with a sliding mode based sensorless cascade speed control scheme. The details are reported in [Corradini et al., 2012c].

5.2 Experimental Setup Experiments have been carried out on the TI Dual Motor Control and PFC Developer’s Kit shown in Figure 4. It includes an industry-standard pin-out dual 4-A high speed low-side MOSFET drivers with enable (UCC27424) for boost converter; an Anaheim Automation BLY172S24V-4000 PMSM; a high performance, integrated dual full bridge motor driver (DRV8402) that includes a protection system integrated on-chip for short-circuit protection, overcurrent protection, undervoltage protection, and two-stage thermal protection. The control unit is a highperformance Texas Instruments TMS320F28335 MCU. The command signals for boost and inverter are generated using the PWM unit of the MCU and are applied to the transistors. The MCU has a 32-bit core and operates up to 150-MHz frequency. AC source has been provided by a 50W AC adapter with a peak value of 13V .

11th IFAC ALCOSP July 3-5, 2013. Caen, France

Motor Load Not Interleaved Boost Interleaved Boost

VP F C (V),IP F C (A)

PFC with Not Interleaved Boost VPFC Not Interleaved

10

IPFC Not Interleaved

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

THD 43.70% 20.30%

Table 2. Performance comparison between classical and interleaved boost configuration: PMSM load.

−10 0

PF 94.90% 96.77%

0.05

Time (ms) V

Interleaved

PFC

10

I

PFC

PFC with Motor as Load

Interleaved

1.5

VPFC Motor

0

IPFC Motor

−10 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time (ms)

Figure 5. PFC variables comparison with and without interleaving technique Resistor Load Not Interleaved Boost Interleaved Boost

PF 99.34% 99.58%

THD 10.92% 7%

IP F C ripple 0.448A 0.106A

Measured ig(t) Current Observed ig(t) Current

ig (t) (A)

1 0.55

0.6

0.65

0.7

0.75

0.8

Time (ms) VP F C (V),IP F C (A)

AC Source Variables VPFC

10

I

PFC

0 −10 0.5

0.55

0.6

0.65

0.7

0.75

0.005

0.01

0.015

0.02

Figure 7. PFC measured variables with PMSM as load.

2

0 0.5

0.5

Time (ms)

Observed Current Comparison

3

1

0 0

Table 1. Performance comparison between classical and interleaved boost configuration: resistor load. 4

VP F C (V), IP F C (A)

VP F C (V),IP F C (A)

PFC with Interleaved Boost

0.8

Time (ms)

Figure 6. Observer performance for interleaved boost converter in a perturbed scenario

been measured with a current sensor gain of 2.336 and a voltage sensor gain of 16. The PF value is 96.77% and THD is 20.3%. A variable and non linear load (motor) degrades PFC performance as expected, compared to results obtained with resistor load in Tab. 1. Interleaving technique improves PFC indexes also with a motor as load; numerical results have been summarized in Tab. 2. The DC-bus voltage and speed motor control results have been shown in Fig.8. In particular, DC-bus voltage measure is presented in Fig. 8(a); the correct tracking of output voltage at 24V reference is achieved. Whereas rotor speed has been plotted in Fig. 8(b): reference value of 750RP M is reached within 1s. 6. CONCLUSION

5.3 Test Results Fig. 5 shows PFC related quantities in the resistance load scenario; source voltage and current (VP F C and IP F C ) have been plotted. The figure presents a comparison between PFC system performance with and without the use of interleaving technique, in order to verify performance improvements. PF, THD and current ripple have been calculated over Fig. 5 trends. The results show that interleaving technique improves all critical converter characteristics, in particular source current ripple decreases of about 0.3A: ripple of 0.448A is reduced to 0.106A. PFC indexes have been summarized in Tab. 1. Fig. 6 shows current observer performance during a step load disturbance in resistance load scenario. At 0.6s output resistance R is doubled for 0.1s, then, at 0.7s, it returns to the nominal value. First plot shows comparison between the real ig (t) rectified source current and the observed one during normal operations and during the perturbation period; the observed current properly tracks the real variable. Second plot compares PFC related variables in order to evaluate power factor degradation during perturbation period; indexes calculated over these trends show that PF value during 0.6s − 0.7s interval is 98.07% and THD is 9.37%. Real test results of PMSM motor control with passivity-based interleaved PFC have been reported in Fig. 7 where the reported quantities have 132

In this paper a sensorless passivity-based control with input voltage feedforward has been designed for a boost AC-DC interleaved converter to increase the power factor in electrical and electromechanical systems. The proposed solution has been numerically simulated on PSIM and experimentally tested on a commercial PMSM drive. REFERENCES L. Armesto, G. Ippoliti, S. Longhi, and J. Tornero. Probabilistic self-localization and mapping - an asynchronous multirate approach. IEEE Robotics Automation Magazine, 15(2):77–88, 2008. L. Ciabattoni, A. Freddi, G. Ippoliti, M. Marcantonio, D. Marchei, A. Monteriu’, and M. Pirro. A smart lighting system for industrial and domestic use. In IEEE International Conference on Mechatronics (ICM), pages 126–131, 2013. G. Cimini, M.L. Corradini, G. Ippoliti, N. Malerba, and G. Orlando. Control of variable speed wind energy conversion systems by a discrete-time sliding mode approach. In IEEE International Conference on Mechatronics (ICM), pages 736–741, 2013. M.L. Corradini, V. Fossi, A. Giantomassi, G. Ippoliti, S. Longhi, and G. Orlando. Discrete time sliding mode control of robotic manipulators: Development and

11th IFAC ALCOSP July 3-5, 2013. Caen, France

PMSM DC−bus Motor DC−bus

26

v(t) (Volt)

24

22

20

18

16

14 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (ms)

(a) Regulated DC-bus voltage trend. PMSM Rotor Speed

800

Rotor Speed 700

600

Speed (RPM)

500

400

300

200

100

0

−100 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (ms)

(b) PMSM speed.

Figure 8. PMSM control variables. experimental validation. Control Engineering Practice, 20(8):816 – 822, 2012a. M.L. Corradini, V. Fossi, A. Giantomassi, G. Ippoliti, S. Longhi, and G. Orlando. Minimal resource allocating networks for discrete time sliding mode control of robotic manipulators. IEEE Transactions on Industrial Informatics, 8(4):733 –745, nov. 2012b. M.L. Corradini, G. Ippoliti, S. Longhi, and G. Orlando. A quasi-sliding mode approach for robust control and speed estimation of PM synchronous motors. IEEE Transactions on Industrial Electronics, 59(2): 1096–1104, 2012c. M.L. Corradini, G. Ippoliti, and G. Orlando. An aerodynamic torque observer for the robust control of variablespeed wind turbines. In IEEE 51st Annual Conference on Decision and Control, pages 2483–2488, 2012d. M.L. Corradini, G. Ippoliti, S. Longhi, D. Marchei, and G. Orlando. A quasi-sliding mode observer-based controller for PMSM drives. Asian Journal of Control, 15 (2):380–390, 2013. F. de Angelis, M. Boaro, D. Fuselli, S. Squartini, F. Piazza, and Q. Wei. Optimal home energy management under dynamic electrical and thermal constraints. IEEE Trans. on Ind. Informatics, 2013. R.W. Erickson and D. Maksimovic. Fundamentals of Power Electronics. Springer, 2001. C. Fulgenzi, G. Ippoliti, and S. Longhi. Experimental validation of FastSLAM algorithm integrated with a linear features based map. Mechatronics, 19(5):609 – 616, 2009. Lina Guo, Weisheng Xu, Ying Lu, Zhiyu Xu, and Xingbi Chen. A novel voltage sensor-less control technique synthesis for a boost AC/DC pre-regulator. In Int. Conf. 133

on Cont., Aut., Rob. and Vis., pages 1 –4, 2006. G. Ippoliti, L. Jetto, and S. Longhi. Localization of mobile robots: Development and comparative evaluation of algorithms based on odometric and inertial sensors. Journal of Robotic Systems, 22(12):725–735, 2005. Yun Wei Li, M. Pande, N.R. Zargari, and Bin Wu. An input power factor control strategy for high-power current-source induction motor drive with active frontend. IEEE Trans. on Power Electr., 25(2):352 –359, 2010. P. Mattavelli. Digital control of DC-DC boost converters with inductor current estimation. In Appl. Power Electr. Conf. and Expos., pages 74 – 80, 2004. F.M. Oettmeier, J. Neely, S. Pekarek, R. DeCarlo, and K. Uthaichana. MPC of switching in a boost converter using a hybrid state model with a sliding mode observer. IEEE Trans. on Ind. Electr., 56(9):3453 –3466, 2009. A. Olmos-Lopez, G. Guerrero, J. Arau, C. Aguilar, and J.C. Yris. Passivity-based control for current sharing in PFC interleaved boost converters. In Appl. Power Electr. Conf. and Exp., pages 475 –480, march 2011. R. Ortega, J.A.L. Perez, P.J. Nicklasson, and H.J. SiraRamirez. Passivity-based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications. Communications and Control Engineering. Springer, 1998. R. Redl and B.P. Erisman. Reducing distortion in peakcurrent-controlled boost power-factor correctors. In Appl. Power Electr. Conf. and Exp., pages 576 –583, 1994. A.H.R. Rosa, S.I.S. Junior, L.M.F. Morais, P.C. Cortizo, and M.A.S. Mendes. Passivity-based control of PFC boost converter with high-level programming. In Power Electr. Conf., pages 801 –806, 2011. S. Singer and R.W. Erickson. Power-source element and its properties. IEE Proceedings - Circuits, Devices and Systems, 141(3):220 –226, 1994. J.J. Spangler and A.K. Behera. A comparison between hysteretic and fixed frequency boost converters used for power factor correction. In Applied Power Electr. Conf. and Exposition, pages 281 –286, 1993. J. Sun and R.M. Bass. Modeling and practical design issues for average current control. In Appl. Power Electr. Conf. and Exp., volume 2, pages 980 –986, 1999. V.I. Utkin, J. Guldner, and J. Shi. Sliding Mode Control in Electromechanical Systems. Taylor and Francis Systems and Control. Taylor & Francis Group, 1999. Bingyuan Wang and Yuling Ma. Research on the passivitybased control strategy of buck-boost converters with a wide input power supply range. In Int. Symp. on Power Electr. for Distrib. Generat. Syst., pages 304 –308, 2010. Zhuang Xu and M.F. Rahman. An adaptive sliding stator flux observer for a direct-torque-controlled IPM synchronous motor drive. IEEE Trans. on Ind. Electr., 54(5):2398 –2406, 2007. R. Zane and D. Maksimovic. Nonlinear-carrier control for high-power-factor rectifiers based on up-down switching converters. IEEE Trans. on Power Electr., 13(2):213 –221, 1998. Wanfeng Zhang, Guang Feng, Yan-Fei Liu, and Bin Wu. A digital power factor correction (PFC) control strategy optimized for DSP. IEEE Trans. on Power Electr., 19 (6):1474 – 1485, 2004.