Formal Framework for Nonlinear Control of PWM ACDC Boost Rectifiersamp; Controller Design and -TB7

7/29/2019 Formal Framework for Nonlinear Control of PWM ACDC Boost Rectifiersamp; Controller Design and -TB7

1/13

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 2, MARCH 2010 323

Formal Framework for Nonlinear Control of PWMAC/DC Boost RectifiersController Design and

Average Performance AnalysisFouad Giri, Abdelmajid Abouloifa, Ibtissam Lachkar, and Fatima Zahra Chaoui

AbstractWe are considering the problem of controlling fil-tered AC/DC switched power converters of the Boost type. Thecontrol objectives are twofold: 1) guaranteeing a regulated voltagefor the supplied load and 2) enforcing power factor correction(PFC) with respect to the main supply network. The consideredproblem is dealt with using a double-loop controller developedon the basis of the system nonlinear model. The inner-loop isdesigned by the backstepping technique to cope with the PFCissue. The outer-loop is designed to regulate the converter outputvoltage. Experimental tests show that the proposed controlleractually meets the objectives it is designed for. While similar per-formances have been experimentally demonstrated, for differentconverters and controllers, it is the first time that a complete andrigorous formal analysis, based on averaging theory, is developedto describe the observed performances (PFC and output voltageripples). As a matter of fact, the averaging theory constitutesthe natural framework to analyzing the performances due to theperiodic nature of the system input signals.

Index TermsAC/DC switched power converters, average per-formances, nonlinear control, power factor correction, stability,voltage regulation.

I. INTRODUCTION

THE role of AC-DC power converters is to produce a regu-

lated DC voltage drawing power from an AC supply net.

These are needed to supplying a too wide class of equipments:

personal as well as industrial, fixed as well as imbedded. Their

necessity has considerably increased due to the boom in com-

puterized technology applications (microelectronics, telecom,

etc.). From a control viewpoint, an AC/DC converter is a non-

linear and hybrid system. Then, undesirable current harmonics

may be generated when the converter is connected to an AC

power source. These harmonics may be harmful for both the

converter and the main supply network necessitating additional

protection and over-dimensioning of both the converter compo-nents and the network elements (transformers, condensers,).

These precautions are costly (higher component prices, higher

power consumption).

To avoid the above drawback, the converter should be con-

trolled bearing in mind not only output voltage regulation but

Manuscript received March 28, 2007; revised January 27, 2009. Manuscriptreceived in final form April 23, 2009. First published August 18, 2009; currentversion published February 24, 2010. Recommended by Associate Editor A.Bazanella.

The authors are with the Groupe de Recherche en Informatique, Image, Au-tomatique, Instrumentation de Caen (GREYC), University of Caen Basse-Nor-mandie, 14032 Caen, France (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCST.2009.2022014

also rejection of undesirable current harmonics. The last objec-

tive is referred to power factor correction (PFC). A comprehen-

sive overview on circuits that are able (if well controlled) to

achieve the PFC requirement can be found e.g., in [11], [12],

and [14]. The problem of designing controllers that are able to

achieve simultaneously the PFC requirement and the output reg-

ulation objective has been considered in many places, e.g., [1],

[3], [4], [6], [7], [13], [16], [17]. The solution proposed in [4]

and [7] involves a single-loop controller designed using the pas-sivity technique. The control objective is the enforcement of the

current (entering the considered second-order boost rectifier) to

asymptotically track a sinusoidal reference signal that oscillates

at the same frequency as the supply-net. The constant ampli-

tude of the reference current is a priori computed so that, in

steady-state, the output voltage equals its desired value. That is,

voltage regulation is indirectly achieved through the achieve-

ment of the PFC property. The shortcoming of this solution lies

in the following facts: 1) it applies solely to the case of constant

voltage reference signals; 2) the lack of voltage loop makes the

output voltage regulation extremely sensitive to model uncer-

tainties (mainly those resulting from load changes); and 3) theexperimentally observed output voltage ripples were not shown

to be actually weak through a formal analysis. These drawbacks

are presently overcome using a control strategy that involves the

following two loops:

1) a current loop to enforce the PFC by acting on the switch

duty ratio;

2) a voltage loop to achieve output voltage regulation

through the tuning of the reference input of the current

loop.

In fact, the double-loop idea is not new; it was proposed in the

early 1990s [3], [13], [16], [17]. However, the proposed reg-

ulators were not formally demonstrated to achieve the perfor-

mances they were designed for. As a matter of fact, this is notsurprising as those regulators were linear (typically PID regula-

tors) while the controlled power converters are highly nonlinear

and time-varying. It is only recently that a serious attempt to

build up a formal framework for the above double-loop control

strategy has been made [1], [6], considering filtered boost and

buck-boost diode-based converters. The filtering was introduced

through LC and LCL filters placed at the converter input stage

in order to reduce the pollution of the power supply net. The

complexity of the resulting control problem is twofold:

1) the PFC requirement and the output regulation must be

simultaneously achieved;

2) the converter model is nonlinear and hybrid.

1063-6536/$26.00 2009 IEEE

AlultIXDoM1a1UfIX Ra


2/13

324 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 2, MARCH 2010

Following the usual practice, the hybrid feature was coped with,

in [1] and [6], basing the control design on average models. In-

voking the double-loop control strategy, a nonlinear controller

was developed in two stages. First, a current-loop was designed

using the backstepping technique in order to achieve a unitary

power factor, i.e., enforcing the converter input current to be

sinusoidal and in phase with the supply net voltage. This con-trol issue has been mathematically formulated as a problem of

regulating the ratio input-current/supply-voltage to a desired

value (by acting on the duty ratio, subsequently denoted ).

The value of is allowed to be time-varying (and it will be

so in transient periods) but it must rapidly converge to a pos-

itive constant value. As long as the inner-loop is concerned the

limit value of is not important. The purpose of the voltage

outer-loop is precisely to tune so that the output voltage

tends to its desired value (despite the load changes). The re-

lationship between and was shown to be a linear differ-

ential equation involving (time-varying) periodic parameters. A

linear regulator has then been synthesized, for achieving the de-

sired output reference tracking, based upon the time-invariantlinear average model. The resulting closed-loop system turned

out to be a time-varying and highly nonlinear. The time-varying

feature leads to output voltage ripples and the question is: how

small the ripples amplitude is? It is worth noting that the aver-

aging theory is the natural framework to analyzing such an issue,

due to the periodic nature of the closed-loop signals. However,

the analysis developed in [6] was not complete and, by some as-

pects, not fully rigorous [this will be made clear later (Remark

3.2)]. Furthermore, the function was not shown there to be

convergent which means that the PFC requirement was not re-

ally been guaranteed.

In the present is paper, we aim at developing a complete andrigorous analysis of the closed-loop performances generated by

the double-loop regulator first presented in [6]. Making better

use of the averaging theory [8], [15], it is established that the

PFC requirement is actually achieved (i.e., converges) and

the output ripples are actually insignificant. More precisely, the

output tracking error is shown to be, in steady-state, a harmonic

signal whose amplitude depends on the frequency of the supply

net voltage. The larger the net frequency is, the smaller the

tracking error. It is the first time that the insignificance of the

output ripples is so formally analyzed. Finally, the above PFC

and voltage regulation results are experimentally confirmed

using industrial scale components; the experiments show

further that the proposed controller presents quite interesting

robustness properties especially when facing unknown load

changes.

This paper is organized as follows. The class of converters

under study is presented and modeled in Section II, the con-

troller design and analysis are dealt with in Section III, the con-

trol performances are experimentally illustrated in Section IV, a

conclusion and a reference list end the paper. To help this paper

reading, the main notations are recapitulated in Table V.

II. CONVERTER DESCRIPTION AND MODELING

The full-bridge PWM boost rectifier under study is repre-

sented by Fig. 1. It includes a -filter, on one hand, anda commutation-cell , on the other hand. The circuit

Fig. 1. PWM boost rectifier under study.

operates according to the well known pulse width modulation

(PWM) principle, [9], [2], [14], [5]. Accordingly, the time is

shared in intervals of length ( is referred to cutting period).

Within a given period, the switches are both ON while

are OFF during , for some . During the rest

of time, i.e., , are OFF and are ON. The

value of changes when passing from one period to another

and its variation law determines the trajectory of the outputvoltage . The variable thus defined is called duty ratio

and serves as the control signal for the considered converter.

Mathematical modeling of the converter is completed ap-

plying Kirchhoffs laws. Doing so, one gets

(1a)

(1b)

(1c)

(1d)

Equations (4a)-(4d) involve the internal voltage and the cur-

rent , which need to be expressed in function of the state vari-

ables ( to ). To this end, let us introduce the following bi-

nary variable:

if are ON and are OFF

if are OFF and are ON.

Then, can be expressed as follows:

(2a)

Similarly, the current is given the following expression:

(2b)

Finally, notice that the duty ratio may simply be interpreted

as the mean value of over a cutting period . Then, substi-

tuting (2a)(2b) in (1a)-(1d) yields the (instantaneous) converter

model

(3a)

(3b)

(3c)



3/13


4/13


5/13


6/13


signal should be slower than the network voltage. Nev-

ertheless, the involved class of reference signals is larger

than in earlier relevant works, e.g., [4] where only con-

stant references were considered.

c) The fact that the tracking error is harmonic (Part

2a) proves the existence of output ripples. Then, Part 2b-i

ensures that the effect of ripples is insignificant if issufficiently large.

d) It is the first time that the control performances (PFC re-

quirement and output ripples) are so rigorously described.

In this respect, Theorem 1 constitutes a quite significant

progress with respect to previous works, e.g.,[1], [4], [7],

and [6].

e) Concerning the output tracking objective, the average

analysis presented in this paper generalizes the work

started in[6] where the problem has been improperly

simplified supposing that the variables are

null. Furthermore, the fact that converges to a positive

limit has not been proved in that work.

f) The control strategy presented in this paper involves anexplicit voltage regulation loop unlike in[4] and [7] where

the output voltage regulation objective was indirectly en-

sured as a consequence of the PFC achievement. The lack

of an explicit voltage loop makes the regulator sensitivity

(to the parameter uncertainties and load changes) higher.

Proof of Theorem 1:

Part 1: Equation (31a) guarantees that and its derivatives

(up to the third order) are available. Then, Part 1 of the Theorem

follows directly from Proposition 1 (Part 1).

Note that Proposition 1 also guarantees that (28), in Proposi-

tion 2, holds.

Part 2: In order to prove the second part of Theorem l, letus introduce a state vector, denoted ,

defined as follows:

(34a)

(34b)

Then, it follows from (26), (28), and (31a)-(31b) that under-

goes the following state equation:

(35)with

(36a)

where

(36b)

Stability of the above system will now be dealt with using av-

eraging theory. As is periodic with period , it will

prove to be useful introducing the following auxiliary reference

function:

(37)

This readily implies that is periodic, with period , and

that . Let us now introduce the time-scale

change . Then, the term containing in (36) becomes

(38)

It also follows from (35) and (36) that undergoes the differ-

ential equation

(39)

with

(40a)

where

(40b)

Now, let us introduce the average function

, where . It

follows from (40a)(40b) that

(41)

where the s denote the components of and represents

the mean value of (which is the same as that of ). Note that

the mean value over of the derivative in the first line of

(36) is zero because is periodic with period . In order to get

stability results regarding the system of interest (35)(36), it is

sufficient (thanks to averaging theory) to analyze the following

averaged system:

(42)



7/13

GIRI et al.: FORMAL FRAMEWORK FOR NONLINEAR CONTROL OF PWM AC/DC BOOST RECTIFIERS 329

To this end, notice that (42) has a unique equilibrium at

(43)

On the other hand, as (42) is linear the stability properties of its

equilibrium are fully determined by the state-matrix

where denotes null matrices of appropriate dimensions and

(44)

More specifically, the equilibrium will be globally asymp-

totically stable if the matrix is Hurwitz. It has already noted

(see Proposition 1) that is Hurwitz. So, it is sufficient to

check that is in turn Hurwitz. To this end, note that its eigen-

values are the zeros of the following polynomial:

(45)

where the s are defined by (32). Applying for instance the

well known Rouths algebraic criteria, it follows that all zerosof the polynomial (45) have negative real parts if the coefficients

( to ) satisfy (33a)-(33c). Now, invoking averaging theory,

e.g., Theorem 4.1 in [15], one concludes that there exists a

such that for , the differential (35) -(36) has a harmonic

solution , that continuously depends on , and

that

(46)

Then, one readily gets that and

are in turn harmonic and depend continuously on .

Then, it follows from (31a) that is in turn harmonic and de-

pends continuously on . This proves part 2a of the theorem.

To establish part 2b, note that (46) and (39) imply

(47)

(48)

Using (31a), one gets from (48) that

(49)

where we have used the fact that Part 2b follows from (47) and(49). Theorem 1 is thus established.

TABLE ICONVERTER CHARACTERISTICS

IV. EXPERIMENTAL EVALUATION

A. Experimental Setup

1) Converter Characteristics: The performances of the pro-

posed controller are now experimentally evaluated using a real

PWM rectifier with the characteristics of Table I.The previous choice of is motivated as follows: the R-C

filter at the converter output should reject well the unavoidable

ripples. To this end, the filter time-constant must be suffi-

ciently larger than the inverse ripples frequency, which is equal

to [see e.g., (28)]. This condition should be fulfilled for

all possible values of the load (as this is allowed to be changing).

Let us suppose that the smallest value of the changing load is

. Then, the previous condition amounts to the following:

2) Regulator Parameter Tuning:

Parameters : These are the parameters of the outer

PI regulator (31a). For the sake of simplicity, the third-order

filter in (31a) is ignored (which amounts to suppose be large

enough) and the output voltage (28) is assimilated to a time-in-

variant first-order system. Then, the mentioned equations sim-

plify respectively to

(50)

That is, the last term (including ), that acts as an external dis-

turbance in (28), is simply ignored. It is easily checked that theclosed-loop system (50) has the transfer function

with

(51)

which presents a unit static gain, due to the integral nature of

the regulator. It is not necessary, in the present case, that the

closed-loop system is more rapid than the open-loop. Also, it

is preferable to have a weakly oscillatory closed-loop response.

The above remarks can be formulated as follows:

and . The values ,meet these requirements as they yield ,



8/13


9/13


TABLE IVMAIN CHARACTERISTICS OF THE DIGITAL-ANALOG CARD

TABLE VMAIN NOTATIONS

Owing to , it follows from (55) that

(60)

Fig. 4. Output voltage x in response to a step in the reference voltage.

Fig. 5. Input current x in response to a step reference voltage.

Fig. 6. Zoom on the supply voltage v and input current x in steady state.



10/13


Fig. 7. Variation of the ratio in response to a varying reference voltage.

It was already noted that the computation of the derivatives of

is not an issue as this signal is sinusoidal and known. Fur-

thermore, (56) implies

(61)

which shows that can be computed provided that

and its three first derivatives are available. Finally, we get from

(57)

(62)

The first term on the right side of (62) is computed as follows:

using (63)

The last term in (62) is computed using (56).

In the light of the previous discussion it is clearly seen that

the control law (25) can be implemented, using (52) to (63),

provided that the voltages and currents to are accessibleto measurements and the signal and its three first derivatives

are available. In this study, the first requirement is met using Hall

effect sensors. The second requirement is coped with in the next

point.

Computation of and its Three First Derivatives: The

control law (31a) shows that is obtained filtering the signal

through the third-order filter . The error

is easily computed and is obtained

integrating . The derivatives of are also obtained filtering

as shown in Fig. 3. Note that all filters are realizable

because they are proper or strictly proper.

Digital-Analog Card: The controller is implemented using

an Analog Device AMC401 DSP Motor Control DevelopmentTool Kit. This has the characteristics of Table IV.

Fig. 8. Output voltage x in response to load changes.

Fig. 9. Input current x in response to load changes.

B. Experimental Results

The experiments aim at illustrating the behavior of the con-

troller in response to step changes on both the voltage reference

and the load resistance . More specifically, the voltage ref-

erence goes from 100 V to 120 V and then back to 100 V. The

load resistance steps from its nominal value (40 ) up to no load

condition (load less) and then back to its nominal value.

1) Controller Performances in Presence of Varying Output

Voltage Reference: The output voltage reference is a step likesignal that switches from 100 to 200 V at time 0.4 s and goes



11/13


Fig. 10. Variation of the ratio in response to load changes.

back to 100 V at 1.4 s. As stipulated by theorem 1 (and pointed

out in Remarks 3.2b), the output voltage converges in the

mean to its reference value with a good accuracy (see Fig. 4).

Furthermore, it is observed that the voltage ripples oscillates at

the frequency but their amplitude is insignificant compared

to the average value of the signals, confirming thus Theorem

1 (Part 2b-i). The corresponding input current is shown in

Fig. 4. Comparing Figs. 4 and 5, one particularly notes that the

variation of the input current magnitude is correlatedwith

the (mean) value of the squared output voltage . This con-

firms power conservation through the circuit. Finally, the zoom

in Fig. 6 showsthat the input current and the network voltageare actually in phase in steady state. Hence, the requirement

of unitary power factor is achieved after transient periods fol-

lowing output reference steps. This is further demonstrated by

Fig. 7 which shows that the ratio always takes a constant value,

after those transient periods.

2) Control Performances in Presence of Varying Converter

Load: Figs. 8 and 9 illustrate the behavior of the control system

in presence of load changes that are not accounted for in the con-

troller design. The rest of the converter characteristics are kept

unchanged. It is seen from Fig. 7 that the disturbing effect due

to load changes is well compensated by the controller. Further-

more, Fig. 9 shows the correlation of the current amplitude withthe output voltage. Finally, Fig. 10 shows that the ratio takes

constant values after the (finite) transient periods following the

load changes, confirming thus the achievement of unitary power

factor. Robustness of the proposed controller with respect to

load changes is thus established.

C. Additional Simulation Results

1) Effect of the Filter Capacitance: Such effect is illustrated

by Fig. 11 which shows the input current and output voltage for

two values of the capacitance . All other control parameters

are kept unchanged. It is observed that the sought two control

objectives (output voltage regulation and power factor correc-tion) are achieved in the mean for both capacitances. However,

Fig. 11. Effect of the filter capacitance.

the larger capacitance ensures smaller ripples and a more rapid

transient confirming thus the discussion in Section IV-A1.

2) Comparison Between the Double-Loop and Single-Loop

Control Strategies:Comparison in Presence of Load Uncertainty: The

supremacy of the double-loop strategy over the single-loop

approach, suggested in, e.g., [4] and [7], is now illustrated

using the same rectifier. The single-loop (current) controller

is designed using the backstepping technique. The involved

current reference signal is given a constant value chosen such

that the resulting steady-state output voltage is equal to its de-

sired value. The relation between the current reference and the

corresponding steady-state voltage involves the converter load.

While the load is supposed to be constant (equal to its nominal

value) in both single-loop and double-loop controllers, it is in

fact time-varying during the experiments. More specifically,

the load changes at time 1.5 s, falling from its nominal value(40 ) to half this value (20 ). Figs. 12 and 13 illustrate the



12/13


Fig. 12. Rectifier in closed-loop with single-loop controller: (a) input current x (solid line) and supply net voltage v (dotted line); (b) Output voltage (x ).

Fig. 13. Rectifier in closed-loop with double-loop controller: (a) input current x (solid line) and supply net voltage v (dotted line); (b) Output voltage ( x ).

resulting performances for both controllers in presence of a step

reference signals. More precisely, the output voltage reference

signal, for the double-loop controller, steps from 100 to 120 V

at time 0.5 s. The corresponding current reference signal, for

the single-loop controller, steps at the same time from 7 to 10A. As pointed out previously (e.g., Remarks 3.2-f), the output

voltage regulation in the single loop-controller is achieved

indirectly through input current regulation.

It is observed from (the zoomed curves in) Fig. 12(a) and (b)

that the two controllers perform equally well as long as the PFC

issue is concerned. Note that the PFC requirement is well ful-

filled, in both cases, even after the converter load changes at time

1.5 s.

Figs. 12(b) and 13(b) show that both controllers guarantee

asymptotically a good tracking of the output voltage reference

before the load changes at time 1.5 s. But, this is no longer

the case after the change of the load. Indeed, it is observed inFigs. 12(b) and 13(b) that, after time 1.5 s, only the double-loop

controller proves still the output voltage at its desired value

(120 V). The single-loop controller only regulates well the cur-

rent to its true value (10 A) after the load change. But, such cur-

rent regulation does not correspond to the desired output voltage

(i.e., 120 V) because the load is no longer equal to its nominal

value.

Comparison in Presence of Power Supply Net Voltage Dis-

tortion: Fig. 14 illustrates the behavior of the double-loop and

single-loop control strategies in the presence of distorted power

supply net voltage. To this end, the input voltage is disturbed

by a stochastic noise of significant amplitude. The max-

imum value of the latter is approximately 20% of the nominalvalue V. That is the really applied net voltage is

Fig. 14. Controller behavior in presence of input voltage distortion.

. Such noisy voltage is used in the

simulation model only; in the regulator, the net voltage is still

supposed to be perfectly sinusoidal with con-

stant, equal to its nominal value50 V. It is seen from Fig. 14 that

the double-loop regulator is more robust against such impor-

tant net voltage distortion (than the single-loop controller). As

expected, the supremacy of the double-loop strategy concerns

mainly the output voltage regulation: the transient is much more

rapid and the average steady-state tracking error is null. As long

as the PFC requirement is concerned, the two control strategiesare comparable.



13/13


V. CONCLUSION

In this paper, we have considered the problem of control-

ling a full-bridge rectifier of the boost type. The converter dy-

namics have been described by the averaged 4th order non-

linear state-space model (4a)-(4d). Based on such a model, a

cascade nonlinear controller has been designed. It has been for-

mally established that the obtained controller meets its objec-tives. Specifically, we have the following.

The error (where denotes the converter

input average current) vanishes exponentially fast and the

signal is, in steady-state, a harmonic signal that oscil-

lates around a positive mean value with an amplitude that

depends on the supply net frequency . The larger is

the small the oscillation amplitude and, consequently, the

better the quality of power factor correction.

The (average) output voltage tracks its reference with

an accuracy that depends on the supply net frequency:

the larger is the frequency, the more accurate the output

tracking.

It is the first time that a so complete formal description of the

closed-loop system performances is achieved making use of sta-

bility and averaging theory.

REFERENCES

[1] A. Abouloifa, F. Giri, and I. Lachkar, Nonlinear control of a PWMrectifier output voltageregulation andpower factor correction,in Proc.5th Int. Symp. Adv. Electromech. Motion Syst. (ELECTROMOTION),Marrakech, Morocco, 2003, pp. 27.

[2] C. Andrieu, J. P. Ferrieux, and M. Rocher, The ac/dc stage: A surveyof structures and chopper control modes for power factor correction,

EPE J., vol. 5, pp. 1722, 1996.[3] M. O. Eissa, S. B. Leeb, G. C. Verghese, and A. M. Stankovic, A fast

analog controller for a unity-power factor AC/DC converter, in Proc.IEEE APEC, 1994, pp. 551555.

[4] G. Escobar, D. Chevreau, R. Ortega, and E. Mendes, An adaptive pas-sivity-based controller for a unity power factor rectifier, IEEE Trans.Control Syst. Technol., vol. 9, no. 4, pp. 637644, Jul. 2001.

[5] R. Erickson, M. Madigan, and S. Singer, Design of simple high powerfactor rectifier based on the flyback converter, in Proc. IEEE Appl.Power Electron. Conf. Expo., 1990, pp. 792801.

[6] F. Giri, A. Abouloifa, and I. Lachkar, Nonlinear control of boostAC/DC converters, presented at the IFAC World Congr., Prague,Czech Republic, Jul. 2005.

[7] Karagiannis, D. E. Mendes, A. Astolfi, and R. Ortega, An exper-imental comparison of several PWM controllers for a single-phaseAC-DC converter, IEEE Trans. Control Syst. Technol., vol. 11, no. 6,pp. 940947, Nov. 2003.

[8] H. Khalil, Nonlinear Systems. Englewood Cliffs, NJ: Prentice-Hall,2003.

[9] P. T. Krein, J. Bentsman, R. M. Bass, and B. Lesieutre, On the use of

averaging for analysis of power electronic system, IEEE Trans. PowerElectron., vol. 5, no. 2, pp. 182190, Feb. 1990.

[10] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear andAdaptive Control Design. New York: Wiley, 1995.

[11] A. Mechi and S. Funabiki, Step-up/down voltage PWM AC to DCconvector with one switching device, IEE Proc.-B, vol. 140, no. 1, pp.3543, 1993.

[12] R. Redl, Power-factor correction in single-phase switching-modepower supplies, Int. J. Electron., vol. 77, pp. 555582, 1994.

[13] G. Spiazzi, P. Mattavelli, and L. Rossetto, Power factor preregula-tors with improved dynamic response, in Proc. IEEE Power Electron.Spec. Conf. (PESC), 1995, vol. 1, pp. 150156.

[14] C. K. Tse and M. H. L. Chow, Theoretical study of switchingconverters with power factor correction and output regulation, IEEETrans. Circuits Syst. I, Reg. Papers, vol. 47, no. 7, pp. 104755, Jul.2000.

[15] Z. Zhi-Fen, D. Tong-Ren, H. Wen-Zao, and D. Zhen-Xi, QualitativeTheory of Differential Equations. Rhode Island: American Mathe-matical Society, 1992.

[16] M. J. Zhou, Design trade-offs in continuous current-mode controlledboost power-factor correctioncircuits,in Proc. High Frequency PowerConverter Conf. (HFPC), 1992, pp. 209220.

[17] S. Wall and R. Jackson, Fast controller design for practicalpower-factor correction systems, in Proc. IEEE IECON, 1993,pp. 10271032.

Fouad Giri was born in 1957. He received thePh.D. degree in automatic control from the InstitutNational Polytechnique de Grenoble, Grenoble,France, in 1988.

He is currently with the Groupe de Recherche enInformatique, Image, Automatique, Instrumentationde Caen (GREYCLab).He hasspent long-termvisitsat the Laboratoire dAutomatique de Grenoble, andthe University of Southern California, Los Angeles,and the Ruhr University, Bochum, Germany. Since1982, he has been successively Assistant Professor

and Professor with the Mohammadia School of Engineers, Rabat-Morocco andthe Universit de Caen, France. His research interests include nonlinear systemidentification, nonlinear control, adaptive control, constrained control, powerconverters and electric machine control. He has published over 150 journal/con-ference papers. He has coauthored the books (in French) Feedback Systems in

Control and Regulation: Representations Analysis and Performances (Eyrolles,1993) and Feedback Systems in Control and Regulation: Synthesis,Applicationsand Instrumentation (Eyrolles, 1994).

Abdelmajid Abouloifa received the Aggregationof Electrical Engineering from the Ecole NormaleSuprieure de lEnseignement Technique, Rabat,Morocco, in 1999, the Ph.D. degree in controlengineering from the Universit de Caen Basse-Nor-mandie, Caen, France, in 2008, under the supervisionof Prof. F. Giri and Prof. F. Z. Chaoui.

Currently, he is a Professeur-Agrg with TheLyce Technique, Casablanca-Morocco. His researchinterests include high-frequency power converter

topologies, power-factor-correction techniques,power supplies, and nonlinear control. He has coauthored several papers onthese topics.

Ibtissam Lachkar received the graduate degreefrom the Ecole Normale Suprieure de lEnseigne-ment Technique, Rabat, Morocco, in 1995 and theDiplme dEtudes Suprieures Approfondies fromthe Ecole Mohammadia dIngnieurs (EMI), Rabat,in 2005. She is currently working towards the Ph.D.degree on nonlinear control of power converters fromthe Laboratoire dAutomatique et dInformatiqueIndustrielle (EMI), under the supervision of Prof. F.Giri and Prof. F. Z. Chaoui.

Currently, he is a teacher of electrical engineeringwith the the Lyce Technique, Sal, Morocco.

Fatima-Zahra Chaoui was born in 1969. Shereceived the Ph.D. degree in automatic control fromthe Institut National Polytechnique de Grenoble,Grenoble, France, in 2000.

Since 1995, she has been successively AssistantProfessor and Professor at the Ecole NormaleSuprieure dEnseignement Technique (ENSET),Rabat, Morocco. She has spent long-term visits atthe Laboratoire dAutomatique de Grenoble and theGREYC Lab, University of Caen, both in France.She is also with the Laboratoire dAutomatique

et Informatique Industrielle (LAII). Her research interests include nonlinear

system identification and control. She published several journal and conferencepapers on these topics.

Documents

Formal Framework for Nonlinear Control of PWM ACDC Boost Rectifiersamp; Controller Design and -TB7