Formal Framework for Nonlinear Control of PWM AC/DC Boost Rectifiers—Controller Design and Average Performance Analysis

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

Formal Framework for Nonlinear Control of PWMAC/DC Boost Rectifiers—Controller Design and

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

Abstract—We 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 Terms—AC/DC switched power converters, average per-formances, nonlinear control, power factor correction, stability,voltage regulation.

I. INTRODUCTION

T HE 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. Theirnecessity 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 harmonicsmay be generated when the converter is connected to an ACpower source. These harmonics may be harmful for both theconverter and the main supply network necessitating additionalprotection and over-dimensioning of both the converter compo-nents and the network elements (transformers, condensers,…).These precautions are costly (higher component prices, higherpower 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) toachieve the PFC requirement can be found e.g., in [11], [12],and [14]. The problem of designing controllers that are able toachieve 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 thecurrent (entering the considered second-order boost rectifier) toasymptotically track a sinusoidal reference signal that oscillatesat the same frequency as the supply-net. The constant ampli-tude of the reference current is a priori computed so that, insteady-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 liesin the following facts: 1) it applies solely to the case of constantvoltage reference signals; 2) the lack of voltage loop makes theoutput voltage regulation extremely sensitive to model uncer-tainties (mainly those resulting from load changes); and 3) theexperimentally observed output voltage ripples were not shownto be actually weak through a formal analysis. These drawbacksare presently overcome using a control strategy that involves thefollowing two loops:

1) a current loop to enforce the PFC by acting on the switchduty ratio;

2) a voltage loop to achieve output voltage regulationthrough the tuning of the reference input of the currentloop.

In fact, the double-loop idea is not new; it was proposed in theearly 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 nonlinearand time-varying. It is only recently that a serious attempt tobuild up a formal framework for the above double-loop controlstrategy has been made [1], [6], considering filtered boost andbuck-boost diode-based converters. The filtering was introducedthrough LC and LCL filters placed at the converter input stagein order to reduce the pollution of the power supply net. Thecomplexity of the resulting control problem is twofold:

1) the PFC requirement and the output regulation must besimultaneously achieved;

2) the converter model is nonlinear and hybrid.

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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 controllerwas developed in two stages. First, a current-loop was designedusing the backstepping technique in order to achieve a unitarypower factor, i.e., enforcing the converter input current to besinusoidal and in phase with the supply net voltage. This con-trol issue has been mathematically formulated as a problem ofregulating the ratio “input-current/supply-voltage” to a desiredvalue (by acting on the duty ratio, subsequently denoted ).The value of is allowed to be time-varying (and it will beso in transient periods) but it must rapidly converge to a pos-itive constant value. As long as the inner-loop is concerned thelimit value of is not important. The purpose of the voltageouter-loop is precisely to tune so that the output voltagetends 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. Alinear 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 turnedout to be a time-varying and highly nonlinear. The time-varyingfeature leads to output voltage ripples and the question is: howsmall 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 (Remark3.2)]. Furthermore, the function was not shown there to beconvergent 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 bythe double-loop regulator first presented in [6]. Making betteruse of the averaging theory [8], [15], it is established that thePFC requirement is actually achieved (i.e., converges) andthe output ripples are actually insignificant. More precisely, theoutput tracking error is shown to be, in steady-state, a harmonicsignal whose amplitude depends on the frequency of the supplynet voltage. The larger the net frequency is, the smaller thetracking error. It is the first time that the insignificance of theoutput ripples is so formally analyzed. Finally, the above PFCand voltage regulation results are experimentally confirmedusing industrial scale components; the experiments showfurther that the proposed controller presents quite interestingrobustness properties especially when facing unknown loadchanges.

This paper is organized as follows. The class of convertersunder 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, aconclusion and a reference list end the paper. To help this paperreading, 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 isshared 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 restof time, i.e., , are OFF and are ON. Thevalue of changes when passing from one period to anotherand 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 Kirchhoff’s 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 OFFif 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 interpretedas the mean value of over a cutting period . Then, substi-tuting (2a)–(2b) in (1a)-(1d) yields the (instantaneous) convertermodel

(3a)

(3b)

(3c)

GIRI et al.: FORMAL FRAMEWORK FOR NONLINEAR CONTROL OF PWM AC/DC BOOST RECTIFIERS 325

(3d)

This model is useful to buildup an accurate simulator for theconverter. But, it is not suitable for designing a continuous con-trol law, because it involves a binary control input, namely .To overcome this difficulty, it is usually resorted to model aver-aging over cutting intervals (e.g., [9]). This process gives rise toaverage versions (of the instantaneous model) involving as con-trol input the mean value of which is nothing other than theduty ratio . The average model thus obtained [1], is describedby the following equations, where denotes the average valueof ( ), over cutting periods:

(4a)

(4b)

(4c)

(4d)

III. CONTROLLER THEORETICAL DESIGN AND ANALYSIS

The controller synthesis is carried out in two major steps.First, a current inner loop is designed to cope with the PFC issue.In the second step, an outer loop is built-up to achieve voltageregulation.

A. Current Loop Design

The PFC objective means that the input current must besinusoidal and in phase with the AC supply voltage. We there-fore seek a regulator that enforces to track a reference signal

of the form

At this point the parameter is any function of time. The (inner)control loop will now be designed using the backstepping tech-nique [10], [8], based on the partial model (4a)-(4c).

1) Step 1: Output Regulation of Subsystem (4a): Let us in-troduce the current tracking error

(5)

Using (4a), time-derivation of (5) yields the following error dy-namics:

(6)

In (6), stands for a (virtual) control variable. Then,can be regulated to zero if , where , called

stabilizing function, is defined by

(7)

where is a design parameter. If was actuallythe control input in (6), letting would implythat: which clearly establishes asymptotic stabilityof (6) with respect to the Lyapunov function

(8)

Then, time-derivation of would be

(9)

As is not the actual control input in (6), one cannotlet . However, we retain the expression of thestabilizing function and introduce a new error variable, de-noted , between the virtual control and its desired value ( ):

(10)

Then, (6) becomes, using (7) and (10)

(11)

Also, the derivative of Lyapunov function (9) becomes

(12)

2) Step 2: Stabilization of the Subsystem: Achieving the PFCobjective amounts to enforcing the error vector to vanishasymptotically. To this end, one needs to know the dynamics of

. Deriving (10), it follows from (4b) that:

(13)

In the previous equation, the quantity stands as a newvirtual control input. We now need to select a Lyapunov function

for the -system. As the objective is to drive the statesto zero, it is natural to choose the following function:

(14)

Using (12)–(14), one gets the following derivative:

(15)

For the -system to be globally asymptotically stable, itis sufficient to choose the virtual control input so that

(for any ). If this holds, (15) implies

(16)

Now, comparing (16) and (13) one gets thatwith

(17)

However, is not the actual control input. Then, aspreviously a new error variable between the above virtualcontrol and the stabilizing function is introduced

(18)


Then, (13) becomes, using (17) and (18)

(19)

Also, the Lyapunov function derivative (15) becomes

(20)

3) Step 3: Stabilization of the Subsystem: Time-derivation ofgives, using (18) and (4c)

(21)

The actual control variable, namely , appears for the first timein (21). An appropriate control law for generating has now tobe found for the system (11), (19), and (21) whose state vectoris . Let us consider the Lyapunov function

(22)

Using (20), the time-derivative of can be rewritten as

(23)

This shows that, for the -system to be globallyasymptotically stable, it is sufficient to choose the control sothat which, due to (23), amountsto ensuring that

(24)

Comparing (24) and (21) yields the following backstepping con-trol law

(25)

It can be easily checked that the previous control law involvesthe reference signal and its three first derivatives.Then, it follows from (17) and (7) that the ratio should bederivable up to the third order and its three first derivativesshould be available. The results thus established are summarizedin the following proposition.

Proposition 1: Consider the control system, next called innerclosed-loop, consisting of the subsystem (4a)-(4d) and the con-trol law (25). If and its three first derivatives are availablethen one has the following.

1) The inner closed-loop system undergoes the followingequation in the -coordinates

(26)

Furthermore, (26) is globally asymptotically stablewith respect to the Lyapunov function

becauseis negative definite. As (26) is linear, then the error vector

converges exponentially fast to zero, whateverthe initial conditions. It follows in particular that the

(average) input current tends asymptotically (andexponentially fast) to its reference value .

2) If in addition converges (to a positive limit value), thenthe PFC requirement is fulfilled.

Remarks 3.1:a) As mentioned previously, the signal is arbitrary at this

point. This provides a degree of freedom that will be used,in Section III-B, to enforce the achievement of the outputvoltage regulation. Specifically, the signal will be gener-ated by the external loop in order to achieve output voltageregulation. Then, it becomes obvious that Part 1 of Propo-sition 1 is not sufficient to conclude that the PFC objectiveis realized; the additional statement in Part 2 is in effectnecessary. It will be established later (see Theorem 1) that

actually converges to a positive limit value, proving thusthe PFC achievement.

b) The control strategy based upon in [4] involved a singleloop namely the current loop. The latter was designedfor the achievement of the current tracking objective, i.e.,the current must tracks a reference signal of the form

. The amplitude , taken constant, that is apriori computed in such a way that the steady-state outputvoltage equals its desired value. Since the computed ex-pression of contains the model parameters, it is ex-pected that perfect output voltage regulation cannot beachieved unless all model parameters are perfectly known.The point is that some model parameters may be (and gen-erally are) ill-known or time-varying. This is especiallythe case of the converter load and converter supply netvoltage (which may not be rigorously sinusoidal).

B. Voltage Loop Design

The aim of the outer loop is to generate a tuning law for thesignal so that the output voltage is steered to a given ref-erence value .

1) Relation Between and : The first step in designingsuch a loop is to establish a relationship between (the controlsignal) and the output voltage . This is the subject of the fol-lowing proposition.

Proposition 2: Consider the power converter described by(4a)-(4d) augmented with the inner control law defined by (25).Under the same assumptions as in Proposition 1, one has thefollowing.

1) The output voltage varies in response to the signalaccording to the following equation:

(27)

where denotes the amplitude of the (sinusoidal) supplynet voltage and is defined by (5).

2) Then, the squared-voltage varies, in response tothe tuning ratio , according to the following first-ordertime-varying linear equation

(28)

with .


Fig. 2. Equivalent current generator as seen by the circuit �� .

Proof:Part 1: The first step consists in replacing the circuit part

located above the set , by an equivalent current gener-ator, as shown by Fig. 2. In view of (4d), the underlying currentvalue coincides with . So, (4d) becomes

(29)

The equivalent current will now be expressed in functionof the signal , using power conservation arguments. From (5)one has . Then, the instantaneous power injectedto the converter turns out to be the following:

(30)On the other hand, the power that is actually consumed by theload is . But, the injected power is integrallytransmitted to the load (which is the only dissipative element).Then, the quantity does coincide with . This yields:

, which to-gether with (29) establishes (27).

Part 2: Deriving with respect to time and using (27),yields the first-order differential (28) and completes the proofof Proposition 2.

2) Squared Output Voltage Regulation: The ratio standsas a control input in the first-order system (28). The problem athand is to design for a tuning law so that the squared voltage

tracks a given reference signal . Ignoringthe linear time-varying feature of the first-order system, a PIcontrol is invoked. Bearing in mind the fact that and its threefirst derivatives should be available (Proposition 1), a filtered PIcontrol law is resorted to

(31a)

(31b)

where may denote as well the Laplace variable or the deriva-tive operator , depending on the context. At thispoint, the regulator parameters are any positive real constants.The next analysis will make it clear how these should be chosenfor the control objectives to be achieved.

C. Control System Analysis

In the following Theorem, it is shown that, for a specific classof reference signals, including periodic signals, the control ob-jectives are achieved (in the mean) with an accuracy that de-pends, among others, on the network frequency . The fol-lowing notations are needed to formulate the results:

(32)

Theorem 1 (Main Result): Consider the AC/DC Boostpower converter, shown by Fig. 1, represented by its averagemodel (4a)-(4d), in closed-loop with the controller consistingof the inner-loop regulator (25) and the outer-loop regulator(31a)-(31b). Then, the closed-loop system has the followingproperties.

1) The error vanishes exponentially fast (where).

2) Let the reference signal be nonnegative and periodicwith period , where is any positive integer.Let the converter components , the supply netvoltage frequency and the regulator positive parame-ters satisfy the following inequalities:

with

(33a)

with

(33b)

with

(33c)

Then, there exists a positive real such that if (i.e.,if ), one has the following.

a) The tracking error and are harmonic signals thatcontinuously depend on , i.e.,

b) Furthermore, the above signals satisfy:

where denotes the mean value of the referencesignal .

Before embarking in the proof of the previous theorem, thereare some comments pointing out the originality of the result.

Remarks 3.2:a) Part 1 and Part 2b-ii of the Theorem demonstrate that,

under conditions (33a)-(33c), the PFC requirement is ac-tually fulfilled with an accuracy that depends on the valueof . The smaller is, the more accurate the PFCquality. It will be seen, in the next experimental study thatconditions (33a)-(33c) are not an issue.

b) Part 2b-i demonstrates that tracking objective is achievedin the mean for the output squared voltage with an ac-curacy that depends on the voltage network frequency .The class of admissible reference signals includes peri-odic signals with period . That is, the reference


signal should be slower than the network voltage. Nev-ertheless, the involved class of reference signals is largerthan in earlier relevant works, e.g., [4] where only con-stant references were considered.

c) The fact that the tracking error is harmonic (Part2a) proves the existence of output ripples. Then, Part 2b-iensures 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 significantprogress with respect to previous works, e.g.,[1], [4], [7],and [6].

e) Concerning the output tracking objective, the averageanalysis presented in this paper generalizes the workstarted in[6] where the problem has been improperlysimplified supposing that the variables arenull. Furthermore, the fact that converges to a positivelimit 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] wherethe output voltage regulation objective was indirectly en-sured as a consequence of the PFC achievement. The lackof 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 Theoremfollows 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 willprove to be useful introducing the following auxiliary referencefunction:

(37)

This readily implies that is periodic, with period , andthat . Let us now introduce the time-scalechange . 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 representsthe mean value of (which is the same as that of ). Note thatthe mean value over of the derivative in the first line of(36) is zero because is periodic with period . In order to getstability results regarding the system of interest (35)–(36), it issufficient (thanks to averaging theory) to analyze the followingaveraged system:

(42)


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

(43)

On the other hand, as (42) is linear the stability properties of itsequilibrium 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 tocheck 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 thewell known Routh’s 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 harmonicsolution , that continuously depends on , andthat

(46)

Then, one readily gets that andare 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 realPWM rectifier with the characteristics of Table I.

The previous choice of is motivated as follows: the R-Cfilter at the converter output should reject well the unavoidableripples. To this end, the filter time-constant must be suffi-ciently larger than the inverse ripples frequency, which is equalto [see e.g., (28)]. This condition should be fulfilled forall 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-orderfilter in (31a) is ignored (which amounts to suppose be largeenough) 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 ofthe regulator. It is not necessary, in the present case, that theclosed-loop system is more rapid than the open-loop. Also, itis 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 ,


TABLE IICOEFFICIENTS � TO �

TABLE IIICOEFFICIENTS � TO �

13.3 rd s, (the last value is close to.

Parameter b: This is tuned so that the filter in (31a) canbe made much more rapid than the closed-loop system definedpreviously and represented by the transfer function (51). To thisend, it is sufficient that , i.e., . This moti-vates the value used in the present experiment.

The choices made until now for the different parametersmakes it possible to compute the coefficients to using thedefinitions (32). Doing so, one obtains the values of Table II.Then, one readily gets the values of the coefficients to

involved in (33a)-(33c) (see Table III). Hence, conditions(33a)-(33c) are all fulfilled.

Inner-Loop Design Parameters: The parametersand should be chosen so that the inner loop (characterizedby the state vector ) is much more rapid than theouter voltage loop (the latter being approximately representedby the transfer function (51)). It was already shown (see Propo-sition 1) that the time-derivative of the Lyapunov function

is . This impliesthat with and, consequently

. Then, the inner-loop will be much morerapid than the outer-loop if , i.e., if . Thisjustifies the values used in the experiment.

3) Control Implementation: The controller we have to im-plement involves two control laws, (25) and (31a).

Implementation of (25): In order to implement (25), weneed the quantities and .

The voltages are accessible to measurement. From(10), (7) and (5) one has

and (52)

which show that can be computed, using the measures of theinput current and the net voltage , provided that and itsderivative are available. These are needed to get

(53)

Note that is a known sinusoidal signal and so the computationof the derivative is not an issue.

From (18) and (17), it is readily seen that

with (54)

Fig. 3. Generation of � and its derivatives.

To compute we need, in addition to the previously men-tioned signals, the measurement of and the computation ofthe derivative . The latter can be computed as follows:

using (55)

where

using (56)

using and

(57)

Note that because is sinusoidal with frequency. Then, it is seen from (55) that can be computed, using

(56) and (57), provided that and its derivatives and areavailable. To be able to implement (25), it only remains to com-pute the derivative . It readily follows from (54) that

(58)

From (4a), it is easily seen that the first term, on the right sideof (58), can be computed by . Thethird term is given by (57). So, we will focus on the second andfourth term. First, it readily follows from (10) that

(59)

From (4b), it is seen that . That is,the measurements of and are needed to compute .The computation of is performed using (55).


TABLE IVMAIN CHARACTERISTICS OF THE DIGITAL-ANALOG CARD

TABLE VMAIN NOTATIONS

Owing to , it follows from (55) that

(60)

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

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

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


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

It was already noted that the computation of the derivatives ofis not an issue as this signal is sinusoidal and known. Fur-

thermore, (56) implies

(61)

which shows that can be computed provided thatand 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 derivativesare available. In this study, the first requirement is met using Halleffect sensors. The second requirement is coped with in the nextpoint.

Computation of and its Three First Derivatives: Thecontrol law (31a) shows that is obtained filtering the signal

through the third-order filter . The erroris easily computed and is obtained

integrating . The derivatives of are also obtained filteringas 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 � in response to load changes.

Fig. 9. Input current � 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. Theload resistance steps from its nominal value (40 ) up to no loadcondition (load less) and then back to its nominal value.

1) Controller Performances in Presence of Varying OutputVoltage Reference: The output voltage reference is a step likesignal that switches from 100 to 200 V at time 0.4 s and goes


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 pointedout in Remarks 3.2b), the output voltage converges in themean to its reference value with a good accuracy (see Fig. 4).Furthermore, it is observed that the voltage ripples oscillates atthe frequency but their amplitude is insignificant comparedto the average value of the signals, confirming thus Theorem1 (Part 2b-i). The corresponding input current is shown inFig. 4. Comparing Figs. 4 and 5, one particularly notes that thevariation of the input current magnitude is correlated withthe (mean) value of the squared output voltage . This con-firms power conservation through the circuit. Finally, the zoomin Fig. 6 shows that the input current and the network voltage

are actually in phase in steady state. Hence, the requirementof unitary power factor is achieved after transient periods fol-lowing output reference steps. This is further demonstrated byFig. 7 which shows that the ratio always takes a constant value,after those transient periods.

2) Control Performances in Presence of Varying ConverterLoad: Figs. 8 and 9 illustrate the behavior of the control systemin presence of load changes that are not accounted for in the con-troller design. The rest of the converter characteristics are keptunchanged. It is seen from Fig. 7 that the disturbing effect dueto 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 takesconstant values after the (finite) transient periods following theload changes, confirming thus the achievement of unitary powerfactor. Robustness of the proposed controller with respect toload changes is thus established.

C. Additional Simulation Results

1) Effect of the Filter Capacitance: Such effect is illustratedby Fig. 11 which shows the input current and output voltage fortwo values of the capacitance . All other control parametersare kept unchanged. It is observed that the sought two controlobjectives (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 rapidtransient confirming thus the discussion in Section IV-A1.

2) Comparison Between the Double-Loop and Single-LoopControl Strategies:

Comparison in Presence of Load Uncertainty: Thesupremacy of the double-loop strategy over the single-loopapproach, suggested in, e.g., [4] and [7], is now illustratedusing the same rectifier. The single-loop (current) controlleris designed using the backstepping technique. The involvedcurrent reference signal is given a constant value chosen suchthat the resulting steady-state output voltage is equal to its de-sired value. The relation between the current reference and thecorresponding steady-state voltage involves the converter load.While the load is supposed to be constant (equal to its nominalvalue) in both single-loop and double-loop controllers, it is infact 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


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

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

resulting performances for both controllers in presence of a stepreference signals. More precisely, the output voltage referencesignal, for the double-loop controller, steps from 100 to 120 Vat time 0.5 s. The corresponding current reference signal, forthe single-loop controller, steps at the same time from 7 to 10A. As pointed out previously (e.g., Remarks 3.2-f), the outputvoltage regulation in the single loop-controller is achievedindirectly 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 PFCissue is concerned. Note that the PFC requirement is well ful-filled, in both cases, even after the converter load changes at time1.5 s.

Figs. 12(b) and 13(b) show that both controllers guaranteeasymptotically a good tracking of the output voltage referencebefore the load changes at time 1.5 s. But, this is no longerthe 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-loopcontroller 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 nominalvalue.

Comparison in Presence of Power Supply Net Voltage Dis-tortion: Fig. 14 illustrates the behavior of the double-loop andsingle-loop control strategies in the presence of distorted powersupply net voltage. To this end, the input voltage is disturbedby 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 thesimulation model only; in the regulator, the net voltage is stillsupposed to be perfectly sinusoidal with con-stant, equal to its nominal value 50 V. It is seen from Fig. 14 thatthe double-loop regulator is more robust against such impor-tant net voltage distortion (than the single-loop controller). Asexpected, the supremacy of the double-loop strategy concernsmainly the output voltage regulation: the transient is much morerapid and the average steady-state tracking error is null. As longas the PFC requirement is concerned, the two control strategiesare comparable.


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, acascade 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 converterinput average current) vanishes exponentially fast and thesignal is, in steady-state, a harmonic signal that oscil-lates around a positive mean value with an amplitude thatdepends on the supply net frequency . The larger isthe small the oscillation amplitude and, consequently, thebetter the quality of power factor correction.

• The (average) output voltage tracks its reference withan accuracy that depends on the supply net frequency:the larger is the frequency, the more accurate the outputtracking.

It is the first time that a so complete formal description of theclosed-loop system performances is achieved making use of sta-bility and averaging theory.

REFERENCES

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[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. 17–22, 1996.

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[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. 637–644, Jul. 2001.

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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 (GREYC Lab). He has spent long-term visitsat the Laboratoire d’Automatique 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 Université 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 inControl 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 NormaleSupérieure de l’Enseignement Technique, Rabat,Morocco, in 1999, the Ph.D. degree in controlengineering from the Université 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-Agrégé with TheLycée Technique, Casablanca-Morocco. His researchinterests include high-frequency power convertertopologies, 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 Supérieure de l’Enseigne-ment Technique, Rabat, Morocco, in 1995 and theDiplôme d’Etudes Supérieures Approfondies fromthe Ecole Mohammadia d’Ingénieurs (EMI), Rabat,in 2005. She is currently working towards the Ph.D.degree on nonlinear control of power converters fromthe Laboratoire d’Automatique et d’InformatiqueIndustrielle (EMI), under the supervision of Prof. F.Giri and Prof. F. Z. Chaoui.

Currently, he is a teacher of electrical engineeringwith the the Lycée Technique, Salé, 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 NormaleSupérieure d’Enseignement Technique (ENSET),Rabat, Morocco. She has spent long-term visits atthe Laboratoire d’Automatique de Grenoble and theGREYC Lab, University of Caen, both in France.She is also with the Laboratoire d’Automatique

et Informatique Industrielle (LAII). Her research interests include nonlinearsystem identification and control. She published several journal and conferencepapers on these topics.

Documents

Formal Framework for Nonlinear Control of PWM AC/DC Boost Rectifiers—Controller Design and Average Performance Analysis