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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSII: EXPRESS BRIEFS 1

A New Control Law Based on the DifferentialFlatness Principle for Multiphase Interleaved

DCDC ConverterPhatiphat Thounthong, Member, IEEE, and Serge Pierfederici

AbstractThis brief presents an innovative control law for adistributed dc generation supplied by a dc power source, here, afuel cell (FC) generator. Basically, an FC is always connected witha power-electronic converter. This kind of system is a nonlinearbehavior. Classically, to control the voltage, the current, or thepower in the converter, a linearized technique is often used tostudy the stability and to select the controller parameters of thenonlinear converter. In this brief, a nonlinear-control algorithmbased on the flatness property of the system is proposed. Flat-ness provides a convenient framework for meeting a number ofperformance specifications on the power converter. Utilizing theflatness property, we propose simple solutions to the system perfor-mance and stabilization problems. Design controller parametersare autonomous of the operating point. To validate the proposedmethod, a prototype FC power converter (1.2-kW four-phase boostconverters in parallel) is realized in the laboratory. The proposedcontrol law based on the flatness property is implemented bydigital estimation in a dSPACE 1104 controller card. Experimentalresults with a polymer electrolyte membrane FC of 1200 W and46 A in the laboratory corroborate the excellent control scheme.

Index TermsConverters, flatness-based control, fuel cells(FCs), interleaved, nonlinear, power control.

I. INTRODUCTION

POLYMER electrolyte membrane fuel cells (PEMFCs) as

dc generators have become an overwhelming competitor in

the distributed generation due to their insuperable advantages,

such as high energy efficiency, near-zero emissions, ease of

installation, quiet operation, and fewer moving parts and higher

power quality [1], [2].

Manuscript received March 8, 2010; revised May 10, 2010 and July 2, 2010;

accepted August 23, 2010. This work was supported in part by a researchprogram in cooperation with the ThaiFrench Innovation Institute, KingMongkuts University of Technology North Bangkok (KMUTNB), and the In-stitut National Polytechnique de Lorraine, Nancy Universit, under the Franco-Thai Higher Education and Research Joint Project (20092010), by the Facultyof Technical Education under Grant FTE255301, by the Thailand ResearchFund under Grant MRG5380261, by theNationalResearch Council of Thailandunder Grant NRCT40/2553, and by KMUTNB under Grant KMUTNB2553.This paper was recommended by Associate Editor H. S.-H. Chung.

P. Thounthong is with the Department of Teacher Training in ElectricalEngineering and the Renewable Energy Research Centre, King MongkutsUniversity of Technology North Bangkok, Bangkok 10800, Thailand (e-mail:[email protected]; [email protected]).

S. Pierfederici is with the Groupe de Recherche en Electrotechnique et Elec-tronique de Nancy, Institut National Polytechnique de Lorraine, Nancy Univer-sit, 54510 Nancy, France (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCSII.2010.2082830

Parallel dcdc converters are widely used in high-power

applications [2]. They operate under a closed-loop feedback

control to regulate the output voltage and enable load shar-

ing. These closed-loop converters are intrinsically nonlinear

systems. The common method of controlling dcdc chopper

converters still relies on averaged small-signal models and

then facilitating the application of the linear-control theory,

such as the proportionalintegral (PI) controller [3]. Nonethe-

less, there are situations where this technique offers a limitedperformance.

This brief presents a new control law based on the differential

flatness theory for nonlinear power-electronic switching appli-

cations. Here, only the inner fuel cell (FC) power regulation

loop is studied. In this kind of system, the main important

specification is that a power dynamic response must follow

a power reference as fast as possible. By using the nonlinear

flatness property, the power regulation loop will operate at very

high dynamics. This brief is organized as follows. Section II

shows the differential equations describing the ideal multiphase

boost converters with an interleaving switching technique for

FC high-power applications. In Section III, the brief of the dif-ferential flatness principle is introduced. Section IV discusses

the proof of differential flatness of the proposed FC converter

models and the control law and stability. Experimental results

will authenticate the proposed control system in Section V. A

comparative study between the flatness control and a linear PI

control will be presented in Section VI. Finally, this brief ends

with concluding remarks in Section VII.

II. MULTIPHASE INTERLEAVED BOOST CONVERTER FOR

FC HIG H-P OWER APPLICATIONS

A. Power Converter

Fundamentally, low-voltage high-current (power) converters

are needed because of the FC electrical characteristics [1].

A classical boost converter is frequently selected as an FC

converter [2], [4]. However, the classical converters will be

limited when the power increases or for higher step-up ratios.

In this manner, the utilization of paralleling power converters

(multiphase converter in parallel) with an interleaved technique

may offer some better performances [2]. As a general rule, the

interleaved switching technique is composed of phase shifting

the control signals of several converter cells N in parallel [5].Fig. 1 shows the functional diagram of the proposed multiphase

interleaved step-up converter for FC applications.

1549-7747/$26.00 2010 IEEE


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2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSII: EXPRESS BRIEFS

Fig. 1. Multiphase parallel boost converters for FC applications.

B. Boost Converter Average Model

The state equations of the multiphase converters are given as

LKdiLKdt

= vFCRLK iLK (1 dK) vBus (1)

CBusdvBusdt

=

NK=1

(1 dK) iLK vBusRLoad

(2)

where the subscript number K= 1, 2, . . . , N represents the

parameters of each converter module, d is the duty cycle ofthe pulsewidth modulation (PWM) converter, vBus is the dc busvoltage, vFC is the FC voltage, iFC is the FC current, iL is theinductor current, RLoad is the equivalent resistance as a load atthe dc bus, L is the input inductance, CBus is the total outputcapacitance at the dc bus, and RL is the series resistance ofinductor LK . Note that RL represents the static losses in eachboost converter module.

The power of each cell can be written as

pL1 = vFC iL1 , pL2 = vFC iL2 , . . . , pLN = vFC iLN (3)

pFC =N

K=1

vFC iLK (4)

pFC = vFC iFC. (5)

Then, the input powerpL of each cell is given versus vFC andiL by the following differential equation:

dpLdt

=d(vFC iL)

dt= iL

dvFCdt

+ vFC diLdt

(6)

dpLdt

= vFC diLdt

vFCConstant

. (7)

III. BRIEF OF THE DIFFERENTIAL FLATNESS PRINCIPLE

Currently, many modeling and linear- or nonlinear-control

methods, including classical state-space or transfer approaches[6], self-tuning methods or sliding mode control [7], or fuzzy-

Fig. 2. Concept of the flatness-based control, where y is the output variable,yREF is the output setpoint, and u is the control input variable.

logic-based control [8], [9], have been extensively studied fornonlinear power-electronic applications.

The flatness theory was introduced by Fliess et al. [10]in 1995. Lately, these ideas have been used in a variety ofnonlinear systems across the various engineering disciplines,including the following: the control of an inverted pendulumand a vertical takeoff and landing aircraft [10]; the control ofcathode pressure and oxygen excess ratio of a PEMFC system[11]; the reactive power and dc voltage tracking control ofa three-phase voltage source converter [12]; and the current

control for three-phase three-wire boost converters [13]. Sincethe flatness-based control is model based, one expects it tohave some sensitivity to error in model parameters. However,Song et al. [12] have proved that the flatness-based control is ro-bust and provides an improved transient tracking performancerelative to a traditional linear-control (PI controller) method.

A nonlinear system is flat [9], [14], [15] if there exists a setof differentially independent variables (equal in number to thenumber of inputs) such that all state variables x and (control)input variables u can be expressed in terms of those outputvariablesy and a finite number of their time derivatives withoutintegrating differential equations. More specifically, considerthe nonlinear dynamic system of the general form, i.e.

x(t) = f(x(t),u(t))y(t) = h (x(t),u(t))

(8)

where

x = [x1, x2, . . . xn]T, x n (9)

u = [u1, u2, . . . um]T, u m (10)

y = [y1, y2, . . . , ym]T, y m (11)

f() and h() are smooth nonlinear functions, and (n,m) N.Moreover, it is assumed that m n.

As depicted in Fig. 2, the nonlinear flat systems are equiv-alent to the linear controllable systems. A system is denotedflat if an output vector y exists, which fulfills the followingconditions:

1) The output variables yi can be stated as functions ofthe state variables xi, the input variables ui, and a finitenumber of their time derivatives, i.e.

y = x, u, u , . . . , u()

. (12)

2) All state variables xi and all control inputs ui can bestated as functions of the output variables yi and a finitenumber of their time derivatives, i.e.

x =y, y , . . . , y()

(13)

u = y, y , . . . , y(+1) (14)where (), (), and () are smooth mapping functions.


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THOUNTHONG AND PIERFEDERI CI: NEW CONTROL LAW BASED ON THE DIFF ERENTIAL FLATNESS PRINCIPLE 3

If the output variables of interest can be proven to be flatoutputs y, the reference signal yREF becomes straightforward.

IV. POWER CONTROL LOOP OF THE MULTIPHASE

INTERLEAVED BOOST CONVERTER

A. Flatness of the Boost Converter

Controlling an interleaved multiphase converter is the sameas controlling a single boost converter, but the uniqueness of theproposed method is the new control law. Based on the flatnessprinciple introduced above, the input power of each convertercell pL is assumed to be the flat output component. Thus, wedefine a flat output y = pL, a control variable u = d, and a statevariable x = iL. From (3), the state variable x can be written as

x =pLvFC

= (y). (15)

From (1) and (7), the control variable u can be calculated fromthe flat output y and its time derivative y, i.e.

u = 1 +y L

vFC+ RL iL vFC

1vBus

= d = (y).

(16)

It is apparent that x = (y) and u = (y) correspond to (13)and (14), respectively. Consequently, the mathematical modelof the converter can be considered as a flat system.

B. Control Law and Stability

The input-power reference of each module is represented byyREF (= pLREF). A linearizing feedback control law achievingan exponential asymptotic tracking of the trajectory is given by

the following expression [12], [16]:

(y yREF) + K11(y yREF) + K12

t0

(y yREF)d = 0

(17)

where K11 and K12 are the controller parameters. One may setthe following as a desired characteristic polynomial:

p(s) = s2 + 2ns + 2n K11 = 2n K12 =

2n

(18)

where and n are the desired dominant damping ratio andnatural frequency, respectively.

Replacing the term for y into (16) gives the equation forthe closed-loop static-state feedback duty cycle d (called theinverse dynamic equation), where e = y yREF, i.e.

u = 1 +

yREF K11eK12

t0

ed

L

vFC+ RL iL vFC

1

vBus

=(y) = d. (19)

Clearly, the control system is stable for K11, K12 >

0 (, n > 0). Thus, the nonlinear-control law of the inputpower detailed above is portrayed in Fig. 3. The measured

Fig. 3. Power-control loop based on the flatness principle of the multiphaseboost converters.

Fig. 4. Photograph of test-bench system.

TABLE ICONVERTER PARAMETERS AND SEMICONDUCTOR DEVICES

input powers of each module are carried out by means of (3)associated to a first-order filter used to reduce harmonics dueto power-electronic switching. Based on the power-electronicconstant switching frequency S (PWM) and the cascade con-trol structure, the outer control loop (here, the input-powercontrol) must operate at a cutoff frequency P F (a cutofffrequency of the first-order filter) S . Once the flat outputsare stabilized, the whole system is exponentially stable becauseall the variables of the system are expressed in terms of the flatoutputs.

V. EXPERIMENTAL VALIDATION

A. Test-Bench Description

The small-scale test bench was implemented in the labora-tory, as presented in Fig. 4. The four-phase boost converterparameters and semiconductor components are detailed inTable I. The FC system used in this effort was a PEMFC system(1.2 kW, 46 A, and based on Ballard Power Systems Inc.), asillustrated in Fig. 4. The power-control loops were implementedin the real-time card dSPACE DS1104 platform (see Fig. 4)using the fourth-order RungeKutta integration algorithm and a

sampling time of 20 s, through the mathematical environmentof MATLABSimulink.


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4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSII: EXPRESS BRIEFS

Fig. 5. Dynamic performance at an input-power reference pINREF step from150 to 800 W at t = 2 ms and from 800 to 400 W at t = 14 ms.

B. Experimental Results

1) FC Converter Testing With an Ideal Power Supply: Be-

cause the FC has slow dynamics by nature [17], an initial testingwas performed using an ideal 26-V power supply (which hasthe same rated voltage as the FC) in place of the FC in order toconfirm that the converter can operate correctly and to observethe dynamic performance of the proposed control law.

Fig. 5 presents waveforms that are obtained during the largestep of the input-power setpoint. The nonlinear controller gainsused were K11 = 1414 rd s

1 and K12 = 1000000 rd2

s2 (= 0.707; n = 1000 rd s1). As a result, the cutoff

frequency P of the closed-loop input power was equal to1000 rd s1, which is lower than the cutoff frequency F ofthe measured power filter of 10 000 rd s1 (TF = 0.1 ms), sothat the system was exponentially stable.

The data show the input-power reference pINREF (insteadof pFCREF), the input-power response pIN (instead of pFC),the input current iIN (instead of iFC) and the first inductorcurrent iL1. In the initial state, the input-power setpoint isequal to 150 W, the constant input voltage (instead of vFC)is equal to 26 V, and the dc bus voltage is equal to 60 V; asa result, the average input current is equal to around 5.77 A(150 W/26 V), and the first inductor current is equal around1.44 A (150 W/4/26 V). At t = 2 ms, the input-power setpointsteps to the constant power of 800 W (positive transition),and at t = 14 ms, the input-power setpoint steps from 800 to400 W (negative transition). The results reveal that, correspond-ing to the second-order dynamics [see (18)] of the observation

error, the dynamic response is affected by this kind of largeinput command. The overshoot (no oscillations) in the input-power response is due to a large proportional gain K11 of1414 rd s1 and the vast setpoint step. The value of K11 canbe reduced to attenuate the overshoot; however, this leads toa slower transition. Therefore, the present mathematical modelof the power converter precisely predicts the dynamics of thecontrol system.

2) FC Converter Testing With a PEMFC: Because the FChas slow dynamics by nature, only the constant input-powerreference was performed. The oscilloscope waveforms in Fig. 6portray the steady-state characteristics of the interleaved con-verters at the FC power reference of 500 W, the electronic load

at the dc bus being adjusted in order to obtain a constant dc busvoltage of 60 V (here, the rated value). The data illustrate the

Fig. 6. Inductor-current waveforms of the converter at an FC powerreference pFCREF of 500 W (vBus = 60 V, vFC = 33.00 V, iFC =15.15 A, and iL = 3.79 A).

Fig. 7. Steady-state waveforms of the converter at an FC power referencepFCREF of 700 W (vBus = 60 V, vFC = 30.00 V, iFC = 23.80 A, andiL = 5.95 A).

first, the second, the third, and the fourth inductor currents. Onecan observe that the paralleled interleaving-boost approach usesforced current sharing between the power stages to balance thepower that the stages deliver.

As a final test, oscilloscope waveforms obtained during theFC power demand of 700 W are presented in Fig. 7. The datashow the FC current, the dc bus voltage, the FC voltage, and thefirst inductor current iL1. One can observe that the FC current is

the sum of the inductor currents and that the FC ripple current is1/Nof the individual inductor ripple currents. It means that theFC mean current is close to the FC root-mean-square current.

VI. COMPARATIVE STUDY

To compare the performance of the flatness-based control, atraditional linear PI control method is also detailed. However,a current controller is classically implemented [3], [17] inplace of the power controller. An inductor-current reference isrepresented by iLREF. A linear feedback PI control law is givenby the following expression:

d = KP(iLREF iL) + KI

t0

(iLREF iL)d (20)


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THOUNTHONG AND PIERFEDERI CI: NEW CONTROL LAW BASED ON THE DIFF ERENTIAL FLATNESS PRINCIPLE 5

Fig. 8. Power-control loop based on a linear PI principle of the single-phaseboost converters.

Fig. 9. Simulation results. Comparison of the flatness-based control law with

a linear PI control law. (a) Linear PI and (b) flatness converter responses to alarge step of the power reference from 50 to 250 W at t = 0.5 ms.

where KP and KI are the set of controller parameters. Thus,the linear-control law of the power-control loop is portrayed inFig. 8. It is similar to the nonlinear-control law (see Fig. 3),where the PI controller also generates a duty-cycle referenced. The main difference between the nonlinear control based onthe flatness property and the classical linear control is that theinverse dynamic equation, known as the flatness property [see(16) and Fig. 3], appears in the nonlinear control.

To compare the performance of the flatness-based controland the linear PI control laws, the simulation was implemented.

Simulations with MATLAB/Simulink were performed using aswitching model of the boost converter. To give a reasonablecomparison between the methods, the parameters of the linearPI controller KP and KI were tuned to obtain the best possibleperformance, and this result is compared with the flatness-basedcontrol. Then, KP = 0.15 A

1, and KI = 200 (A s)1.

Fig. 9 shows the simulation results obtained for both con-trollers during the large step of the power reference. It is similarto the test-bench results illustrated in Fig. 5. The flatness-basedcontrol shows a better dynamic response. Although the dynamicresponse of the linear-control law could be improved relative tothat shown in Fig. 9, this enhancement came at the expense of areduced stability margin. From these results, we conclude that

the flatness-based control provides a better performance thanthe classical PI controller.

VII. CONCLUSION

The main contribution of this brief is to model and controla nonlinear switching-power converter. Distributed power sys-tems often invoke the need to parallel power converters for avariety of reasons, i.e., enhanced reliability, enabling the useof standardized designs with varying loads, distributing heat

sources, and improved maintainability. The proposed converteris four-phase parallel step-up converters with the interleavedswitching technique. Controlling an interleaved multiphaseconverter is the same as controlling a single boost converter,but the uniqueness of the proposed method is the new controllaw. Using the nonlinear-control approach based on the flatnessproperty, we have proposed a simple solution to the optimizationand stabilization problems in the nonlinear power-electronicsystem. This is the novel concept for this kind of application.

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