Control Engineering Practice ] (]]]]) ]]]–]]]

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Control Engineering Practice

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journal homepage: www.elsevier.com/locate/conengprac

Experimental results applying second order sliding mode control to a PEMfuel cell based system$

Cristian Kunusch a,n, Paul F. Puleston b, Miguel A. Mayosky d, Leonid Fridman c

a Institut de Rob�otica i Inform �atica Industrial (CSIC-UPC), Parc Tecnol �ogic de Barcelona, c/ Llorens i Artigas 4-6, 08028 Barcelona, Spainb LEICI, Facultad Ingenierıa, Universidad Nacional de La Plata and CONICET, Argentina. Marie Curie Fellow, ACES at IRI, CSIC-UPC, Barcelona, Spainc Departamento de Ingenierıa de Control y Robotica, Facultad de Ingenierıa UNAM, Edificio T, Ciudad Universitaria D.F., Mexicod LEICI, Facultad Ingenierıa, Universidad Nacional de La Plata and CICPBA, Argentina.

a r t i c l e i n f o

Article history:

Received 18 March 2011

Accepted 1 August 2012

Keywords:

PEM fuel cell systems

Second order sliding mode control

Experimental results

61/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.conengprac.2012.08.002

horter version of this paper was presente

op on Variable Structure Systems, Nationa

June 26–28, 2010.

esponding author: Tel.: þ34 93 4015786; fax

ail addresses: ckunusch@iri.upc.edu (C. Kunus

n@ing.unlp.edu.ar (P.F. Puleston), mayosky@i

@servidor.unam.mx (L. Fridman).

e cite this article as: Kunusch, C., ed system. Control Engineering Practic

a b s t r a c t

A robust control solution is proposed to solve the air supply control problem in autonomous polymer

electrolyte membrane fuel cells (PEMFC) based systems. A Super Twisting controller is designed using a

nonlinear model of a laboratory fuel cell test station, even a Lyapunov based stability discussion is

included. Subsequently, the proposed control strategy is successfully implemented in the laboratory

test bench. Highly satisfactory results are obtained, regarding dynamic behaviour, oxygen stoichio-

metry regulation and robustness against uncertainty.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Increasing society concerns regarding the fundamental pro-blems of the use of hydrocarbons are soaring research on renew-able power sources (Barbir, 2005). Hydrogen, which is an efficientand clean energy carrier, is a viable solution to mitigate theproblems associated to greenhouse gas emissions and sourcedependence. In future energy schemes, renewable energy sourceswill be fundamental and hydrogen can play a key role forefficiency enhancement. Considering that renewable energysources are often intermittent and difficult to predict, it is usuallydifficult to match the energy production and the energy demand.Thus, the introduction of hydrogen as an energy vector helps thismatching and allows increasing the efficiency and stability of thegeneration systems. Going from a hydrocarbon based energysystem to a new scheme where hydrogen plays a basic role,naturally introduces fuel cells (FC) in the energy conversion chain.These devices produce electrical power through the catalyticreaction of hydrogen oxidation, they are highly efficient and theironly by-products are water and heat. However, high costs, lowreliability and short lifetime of fuel cells are still limiting itsmassive utilization in real applications. In this context, not only

ll rights reserved.

d at the 11th International

l Autonomous University of

: þ34 93 4015750.

ch),

eee.org (M.A. Mayosky),

t al. Experimental resultse (2012), http://dx.doi.org/1

the improvement of the system design, better materials andcomponents, but also new advanced control systems are neces-sary to achieve costs reduction, faster dynamic responses, longerlifetimes and the optimization of the energy conversion.

Improving the dynamic response and efficiency of a fuel cellbased system is a challenging control goal, as the FC itself involvesthe interaction of many nonlinear components. For instance,a system comprising an air supply subsystem of a polymerelectrolyte membrane (PEM) fuel cell stack connected to an aircompressor is described by a seventh order nonlinear model(Kunusch, Puleston, Mayosky, & Husar, 2011), where many inter-nal variables are inaccessible for its use in control algorithms.Besides, there are measurable and non-measurable distur-bances that strongly affect system operation, as well as modeluncertainties.

Then, reliable control systems ensuring stability and perfor-mance, as well as robustness to model uncertainties and externalperturbations are of capital importance for PEMFC success. Inparticular, the oxygen stoichiometry control system (Gruber,Bordons, & Oliva, 2012; Larminie & Dicks, 2003; Pukrushpan,Stefanopoulou, & Peng, 2004) has to be able to optimize thesystem conversion efficiency, avoiding performance deteriorationtogether with eventual irreversible damages in the polymericmembranes, due to oxygen starvation. As a novel solution to thistechnological problem, an air flow nonlinear control design andits implementation in a laboratory fuel cell system are presentedin this work. In the current work, the control problem is solvedthrough a Super Twisting algorithm (STA). Among some of theadvantages of the control solution, it can be highlighted thecapability of system robust stabilization, finite time convergence

applying second order sliding mode control to a PEM fuel cell0.1016/j.conengprac.2012.08.002

Fig. 1. Experimental PEM fuel cells laboratory at IRI (UPC-CSIC).

C. Kunusch et al. / Control Engineering Practice ] (]]]]) ]]]–]]]2

to the sliding surface and chattering reduction even in thepresence of model uncertainties and disturbances (Levant, 1993).

The potentiality of this design technique applied to fuel cellshas been preliminarily explored in Kunusch, Puleston, Mayosky,and Riera (2009), using a well known model of the PEMFC systemfor automotive applications presented in Pukrushpan et al.(2004). While in recent years important and sound works havealso been published. In Talj, Hissel, Ortega, Becherif, and Hilairet(2010) a SOSM control is applied in a cascade configuration toregulate the oxygen flow of a 33 kW fuel cell stack. This paperpresents the regulation of the compressor speed through a SuperTwisting controller and then the air mass flow regulation isachieved based on the compressor static characteristic. In thiswork, an order reduction and transformation of the system modelis performed to design the controllers, even experimental tests innominal operation are presented. In Garcıa-Gabin, Dorado, andBordons (2010) a comprehensive study of sliding mode imple-mentation in PEM fuel cells is addressed. This work considers acommercial 1.2 kW stack and the control development is per-formed using a first order sliding method, while the design isbased on a linear model of the plant and considering an integral–differential sliding surface. The control law also incorporates afeedforward term in order to compensate the stack current effectsand enhance the controller dynamic performance. In Matraji,Laghrouche, and Wack (2010), initial simulation results arepresented to control a MIMO PEM fuel cell plant using the so-called Twisting algorithm and the model is derived in Pukrushpanet al. (2004).

The controller design presented in the current paper is basedon the complete nonlinear model of the plant (seventh order),while the control solution directly tackles the oxygen supplythrough a STA. In this way, avoiding model reduction and cascadecontrol, the controller presents enhanced robustness propertiesand a more direct path between design and implementation.Moreover, a Lyapunov based stability and reaching time analysisare provided. Finally, simulation results show the suitability ofSTA to address the control problem, while its successful imple-mentation in an actual fuel cell based generation system experi-mentally demonstrates the viability of this control solution, evenunder external disturbances.

2. Experimental PEM fuel cell system

Details of the laboratory test station used for controllerdevelopment and testing are shown in Fig. 1. A schematic diagramof the system is depicted in Fig. 2, where sensors and actuatorsare also displayed. The main subsystems are:

�

Pb

Air compressor: 12 V DC oil-free diaphragm vacuum pump.Input voltage to this device is used as the main control action.

� Hydrogen and oxygen humidifiers and line heaters: These are

used to maintain proper humidity and temperature conditionsinside the cell stack, an important issue for PEM membranes.Cellkrafts membrane exchange humidifiers are used in thecurrent set-up. Decentralized PID controllers ensure adequateoperation values.

� Fuel cell stack: an ElectroChems 7-cell stack with Nafion 115s

membrane electrode assemblies (MEAs) is used, with a cata-lyst loading of 1 mg/cm2 of platinum, 50 cm2 of active area and100 W power.

Different sensors are incorporated to measure specific vari-ables, suited for modelling and control. Regarding Fig. 2, theseare: motor shaft angular velocity (ocp), compressor air mass flow(Wcp), hydrogen mass flow (WH2), cathode and anode humidifiers

lease cite this article as: Kunusch, C., et al. Experimental resultsased system. Control Engineering Practice (2012), http://dx.doi.org/1

pressures (Phum,ca and Phum,an), stack pressure drops (Pca and Pan),motor stator current (Icp) and voltage (Vcp), stack voltage (Vst) andload current (Iload). Besides, a number of sensors were included toregister significant temperatures (Tst, Thum,ca, Tlh,ca, Thum,an andTlh,an). It must be noted that in a typical fuel cell application manyof these measurements are not necessary. For instance, thecontroller proposed in Section 5 only requires measurement ofthe stack current and the compressor air flow.

System modelling was performed by the authors combiningtheoretical techniques and empirical analysis. Dynamic models ofthe compressor, cathode and anode humidifier, line heaters, fuelcell stack channels and membrane water transport were devel-oped and experimentally validated. The resulting model, suitablefor SOSM control design, has the following general form:

_x ¼ FðxðtÞÞþG � uðtÞ

where X1 ¼ ½x1 x2 x3 x4 x5 x6x7�AR7, uAR and G is a constant.

�

app0.1

x1 ¼ocp: compressor shaft speed (rad/s).

� x2 ¼mhum,ca: mass of air in the cathode humidifier (kg). � x3 ¼mo2 ,ca: mass of oxygen in the stack cathode (kg). � x4 ¼mN2 ,ca: mass of nitrogen in the stack cathode (kg). � x5 ¼mv2 ,ca: mass of vapour in the stack cathode (kg). � x6 ¼mH2 ,an: mass of hydrogen in the stack anode (kg). � x7 ¼mv2 ,an: mass of vapour in the stack anode (kg).

The complete set of equations and physical parameters is notincluded here for space reasons and can be found in Kunusch et al.(2011).

3. Control objective and sliding surface

As already stated, the main objective of the proposed controlstrategy is the optimization of the energy conversion of the fuel

lying second order sliding mode control to a PEM fuel cell016/j.conengprac.2012.08.002

Fig. 2. PEM fuel cell system schematics.

C. Kunusch et al. / Control Engineering Practice ] (]]]]) ]]]–]]] 3

cell, maximizing the net power generated by the system underdifferent load conditions. Considering that the compressor is alsodriven by the fuel cell (in fact, it can be regarded as a parasiticload), the output net power (Pnet) can be defined as the electricalpower delivered by the stack (Pst ¼ VstIst) minus the electricalpower consumed by the compression subsystem (Pcp ¼ VcpIcp).Optimization of the system efficiency can be achieved by regulat-ing the air mass flow entering to the stack cathode at differentload conditions.

Accomplishing such optimal comburent flow is equivalent tomaintaining the cathode line oxygen stoichiometry (or oxygenexcess ratio) in an optimal value. This becomes evident fromFig. 3. The optimum value of lo2

can be experimentally deter-mined from a thorough off-line analysis of the open loop system,considering changes in the current demanded to the stack and awide set of stoichiometry values.

The oxygen stoichiometry is defined as

lo2¼

Wo2 ,in

Wo2 ,react, ð1Þ

where Wo2 ,in is the oxygen partial flow in the cathode, whichdepends on the air flow released by the compressor Wcp and thevapour injected by the humidifier. Wo2 ,react is the oxygen flowconsumed in the reaction. It can be directly related to the totalstack current (Ist):

Wo2 ,react ¼ Go2

nIst

4F, ð2Þ

Go2 is the molar mass of oxygen, n the total number of cells of thestack and F the Faraday constant.

Once lo2 ,opt is determined, the objective of keeping the oxygenexcess ratio within optimal values can be achieved by controllingthe oxygen mass flow (Wo2 ,in). It is important to highlight thatin the current work the optimal oxygen stoichiometry wasdetermined from an off-line analysis as it is shown in Fig. 3. Forthe current system this optimal operating point ranges from 2 to3. In nominal operation, the experimental system is usually set towork at lo2 ,opt ¼ 2:3. Nevertheless, an enhanced approach can bedone using on-line extremum seeking algorithms, this is still anopen topic where the authors are currently working.

Then, once fixed lo2 ,opt the following mass flow reference canbe obtained from (1) and (2):

Wo2 ,in,ref ¼ lo2 ,optMo2

nIst

4F, ð3Þ

where tracking Wo2 ,in,ref effectively implies lo2¼ lo2 ,opt . Never-

theless, due to the fact that Wo2 ,in is an internal unavailable

Please cite this article as: Kunusch, C., et al. Experimental resultsbased system. Control Engineering Practice (2012), http://dx.doi.org/1

variable of the system, it is not practical to include it in thecontrol algorithm. This problem can be circumvented by inferringinformation of Wo2 ,in from an accessible variable of the system,such as the air mass flow delivered by the compressor Wcp. Notethat once the humidifier transient is extinguished, the relation-ship between Wcp and Wo2 ,in remains constant.

In the framework of the sliding mode theory, this controlobjective can be expressed as follows:

SðxÞ ¼Wcp�Wcp,ref

¼ B00þB10 x2ThumRa

VhumþKhum

� �þB20 x2

ThumRa

VhumþKhum

� �2

þB01x1þB11 x2ThumRa

VhumþKhum

� �x1þB02x2

1�Wcp,ref , ð4Þ

where S is the sliding variable that must be steered to zero, Wcp,ref

the compressor air mass flow and Wcp,ref is the flow reference. Thecompressor parameters B00, B01, B10, B11, B02 and B20 can beobtained from Kunusch et al. (2011), Thum is the humidifiertemperature, Vhum is the humidifier volume, Ra the air gasconstant and

Khum ¼ PsatðThumÞRHhum�PsatðTambÞRHamb,

where PsatðThumÞ is the vapour saturation pressure at Thum, RHhum

the relative humidity of the gas at the humidifier output,PsatðTambÞ the vapour saturation pressure at ambient temperatureand RHhum the relative humidity of ambient air. All these variablesare considered constant.

The expression of Wcp,ref can be readily obtained from the airmass flow reference. Given that the molar fraction of oxygen inthe air (wo2) is known, the desired mass flow of dry air can bedirectly computed from

Wdryair,ref ¼1

wo2

Wo2 ,inref¼

1

wo2

lo2 ,optMo2

nIst

4F: ð5Þ

Then, taking into account the relative humidity of the air(Oatm), the final expression of the air mass reference results:

Wcp,ref ¼ 1þOatmð Þ1

wo2

lo2,optMo2nIst

4F: ð6Þ

Note that for stable ambient conditions, the reference onlydepends on a single measurable variable, i.e. the stack current Ist.

4. Design and analysis of PEMFC-STA controller

It has been established in Section 1 that SOSM techniques presentattractive characteristics that well suit the PEMFC breathing control

applying second order sliding mode control to a PEM fuel cell0.1016/j.conengprac.2012.08.002

1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

35

λo2

Pne

t [W

]

Ist=1 [A]Ist=2 [A]Ist=3 [A]Ist=4 [A]Ist=5 [A]Ist=6 [A]Ist=7 [A]Ist=8 [A]Ist=9 [A]Ist=10 [A]

Fig. 3. System performance in different load conditions (Pnet vs. lo2).

C. Kunusch et al. / Control Engineering Practice ] (]]]]) ]]]–]]]4

requirements. Together with the general features of robustness,enhanced accuracy, finite time convergence and chattering reduction,in particular, the generation of continuous control action signals avoidoutput power quality deterioration and the inconvenience of dis-continuous voltage directly applied to the compressor input.

A battery of SOSM algorithms can be found in the literature,each one of them with its distinctive features. In this work, theSuper Twisting has been chosen to evaluate the viability of theSOSM approach to this fuel cell system. This is because thisalgorithm is specially suited for relative degree one plants and donot require information about the sign of the surface derivative. Tothis aim, the controller has been designed using the detailed modeldescription given in Kunusch et al. (2011) and has been assessed bythorough simulation tests and experimentally. To obtain thecontroller gains, an initial design procedure must be followed. Tobegin with, the first and second time derivatives of the slidingvariable (22) have to be computed. They can be expressed as

_S ¼@

@tSðt,xÞþ

@

@xSðt,xÞ:ðFðxÞþGuÞ, ð7Þ

€S ¼@

@t_Sðt,x,uÞþ

@

@x_Sðt,x,uÞ:ðFðxÞþGuÞ

þ@

@u_Sðt,x,uÞ: _uðtÞ ¼jðt,x,uÞþgðt,x,uÞ _uðtÞ, ð8Þ

where jðt,x,uÞ and gðt,x,uÞ for the PEMFC system are smoothfunctions that have to be bounded as follows:

0oGmrgðt,x,uÞrGM , ð9Þ

9jðt,x,uÞ9rF: ð10Þ

For the PEMFC under study, the bounding values were com-puted by means of a numerical study of the nonlinear system andrefined through a physical analysis. Additionally, uncertaintieswere included in representative parameters such as the motorinertia, torque friction, humidifier volume and cathode air con-stant. The following values were obtained:

F¼ 2:3� 10�5; Gm ¼ 0:002; GM ¼ 0:0083:

Once the bounds have been determined, the stabilizationproblem of system (7) with input–output dynamics (8) can besolved through the solutions of the following equivalent

Please cite this article as: Kunusch, C., et al. Experimental resultsbased system. Control Engineering Practice (2012), http://dx.doi.org/1

differential inclusion by applying SOSM:

€SA ½�F,F�þ½Gm,GM� _u: ð11Þ

Then, the gains of the STA are calculated from F, Gm and GM

guaranteeing that, once the system is steered to the region wherethey hold, the trajectories do not escape and converge to S¼ _S ¼ 0in to1.

The implemented Super Twisting algorithm is insensitive withrespect to Lipschitz perturbations, while the control law iscontinuous.

4.1. Super twisting algorithm

This algorithm is intended to systems with relative degree one,this is the case of the presented PEM fuel cell system. Oneinteresting feature of the STA is that during on-line operation, itdoes not require information of _S. The trajectories converge to theorigin of the sliding plane turning around in a typical way. Thecontrol law comprises two terms. One is the integral of a dis-continuous control action and the other is a continuous function ofS, contributing only during the reaching phase (Levant, 1993):

uðtÞ ¼ u1þu2, ð12Þ

_u1 ¼�g signðSÞ,

u2 ¼�l9S91=2

signðSÞ, ð13Þ

where g and l are design parameters that can be first derived fromthe corresponding sufficient conditions for finite time convergenceof the algorithm (Levant, 1993):

g4 FGm

,

l4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

G2m

ðGmgþFÞ2

ðGmg�FÞ

s: ð14Þ

To begin with, the following set of controller parameters wasselected in order to fulfil and obtain a good trade-off betweendynamic performance and chattering attenuation in the

applying second order sliding mode control to a PEM fuel cell0.1016/j.conengprac.2012.08.002

C. Kunusch et al. / Control Engineering Practice ] (]]]]) ]]]–]]] 5

experimental set-up:

l¼ 20, g¼ 0:08: ð15Þ

4.2. Lyapunov stability analysis and convergence time estimation

Lyapunov functions as basic tools for analysis and design inmodern control theory can provide a powerful methodology toanalyse stability issues in Second Order Sliding Modes. In thissection, a Lyapunov based analysis is presented in order to checkthe system stability, analyse its perturbations and give an upperbound of the algorithm convergence time.

In this context, the first Lyapunov function for the SuperTwisting Algorithm was first reported in Moreno and Osorio(2008). In such work, a strong and robust Lyapunov function isintroduced, but its handling is cumbersome. Recent advances inthe formalization of the generalized Super Twisting algorithm (awider class of algorithms that includes the Super Twistingalgorithm of fixed gains) have provided another approach toobtain quadratic Lyapunov functions that is also useful to esti-mate the algorithm convergence time and robustness properties.

For the PEM fuel cell system under study and consideringð@=@tÞSðt,xÞ ¼ 0, Eq. (7) under the effect of control (13) can bepresented as follows:

_S ¼@

@xSðxÞ FðxÞþG �lF1ðSÞþu1ð Þ

� �,

_u1 ¼�2gF2ðSÞ, ð16Þ

with

F1ðSÞ ¼ 9S91=2signðSÞ,

F2ðSÞ ¼12 signðSÞ: ð17Þ

Equivalently, a useful coordinate transformation can be consideredin order to analyse the problem and size the perturbation that affectthe controlled system (16). Taking into account that the slidingvariable is given by Eq. (5), the control objective can be attained as

x2 ¼ hðx1,SÞ, ð18Þ

where hðx1,SÞ is the solution of the implicit equation (5).After this transformation procedure, the underlying system under

Super Twisting control (implicit system) can be expressed as follows:

_x1 ¼ F1ðx1,x2Þ9x2 ¼ hðx1 ,SÞ þGu

¼ rðx1,SÞþGð�lF1ðSÞþu1Þ, ð19Þ

where rðx1,SÞ can be considered as a bounded disturbance(9rðx1,SÞ9oL) that affects the control performance. Note that ifð@=@tÞ Sðt,xÞa0 but it is bounded, its value can be included in thebound of L.

To provide a better description of the perturbations that can becounteracted by the Super Twisting algorithm, the uncertainty/disturbance can always be written as follows (Gonzalez, Moreno,& Fridman, 2012):

rðx1,SÞ ¼ ½rðx1,SÞ�rðx1,0Þ�|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}r1ðx1 ,SÞ

þrðx1,0Þ|fflfflfflffl{zfflfflfflffl}r2ðx1Þ

¼ r1ðx1,SÞþr2ðx1Þ, ð20Þ

where r1ðx1,SÞ¼0 when S¼0. In accordance with Moreno andOsorio (2012), the Super Twisting controller is robust to generalperturbations rðx1,SÞ satisfying the following conditions:

9r1ðx1,SÞ9rk19S91=2

,

@

@tr2ðx1Þ

��������rk2, ð21Þ

Please cite this article as: Kunusch, C., et al. Experimental resultsbased system. Control Engineering Practice (2012), http://dx.doi.org/1

with

k140, k240:

A detailed stability proof of the STA under perturbations fulfilling(20) and (21) can be found in Moreno and Osorio (2012).

For the considered problem, these perturbation terms can bebounded in the entire range of operating conditions of the PEMfuel cell system, as it is shown in Fig. 4. Moreover, constants ofinequalities (21) can be founded (k1 ¼ 0:8� 103, k2 ¼ 4� 102).

As stated in Moreno (2011, Chap. 4) and Moreno and Osorio(2012), this a strong requirement in order to ensure the completerejection of the disturbance rðx1,SÞ by the Super Twisting con-troller. This allows to develop a general stability proof of theSuper Twisting algorithm subject to the general class of distur-bances in Eq. (16) bounded as (21). The proof is based on thefollowing family of quadratic Lyapunov functions:

Vðz,u1Þ ¼ zT Pz,

z¼ ½9S91=2signðSÞ,u1�

T , ð22Þ

where matrix P¼ PT 40 is a symmetric and positive definitesolution of the following linear matrix inequality (LMI) with someconstant E40:

AT PþPAþEPþR PB

BT�Y

" #r0: ð23Þ

A being the Hurwitz matrix of the controlled system (16)

A¼� l

G 1

�gG 0

" #¼�1:3097� 104 1

�52:38 0

" #, ð24Þ

Q is an arbitrary symmetric and positive definite matrixQ ¼QT 40, B¼ ½1 0�T and G¼ 15:27� 10�4. The matrices R andY take into account the perturbation bounds of the statedproblem and can be considered as parameters for control design.

Then, the function (22) is a global strong Lyapunov function forthe system (16) and (17). From Davila, Moreno, and Fridman(2009) it can be stated that the selection of Q¼ I (identity matrix)is the best choice to estimate the minimal time in the family ofLyapunov functions without perturbations. Using this selection ofQ for the system under study, the matrix P¼ PT 40 from (23)results:

P¼0:0003 �0:0162

�0:0162 4:0560

, ð25Þ

with E¼ 1:8� 10�15.In Moreno (2011) and Moreno and Osorio (2012) it is given the

proof that the system trajectories under Super Twisting controlstarting at S0 ¼ ½Sð0Þ, _Sð0Þ�, to the origin in finite time when theperturbation rðx1,SÞ of Eq. (16) is bounded by (21).

From the Lyapunov function (22) it is also possible to obtain anestimate for the convergence time. From Davila et al. (2009),it can be stated that being l=G40 and g=G40, a trajectory ofthe system starting at S0 converge to the origin at most after atime TðS0Þ:

TðS0Þ ¼2

g1 Q ,lG� �V1=2

ðS0Þ, ð26Þ

with

g1 Q ,lG

� �¼

lG

� �2 1

2

lminfQgl1=2minfPg

lmaxfPg: ð27Þ

In conclusion, the presented approach shows the stability ofthe Super Twisting algorithm when controlling the fuel celloxygen of the presented PEM fuel cell system. Therefore, givencertain controller gains l, g and given the perturbation bounds of

applying second order sliding mode control to a PEM fuel cell0.1016/j.conengprac.2012.08.002

0 0.2 0.4 0.6 0.8 1x 10−3

−4−2024

x 10−5

0

200

400

x1

|ρ1(x1,S)|

S

0 0.2 0.4 0.6 0.8 1

x 10−3−4−2024

x 10−5

0

200

400

x1

k1 |S|1/2

S

Fig. 4. Perturbation analysis of the implicit system.

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

Wcp [slpm]

0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4Control actions [V]

time [sec]

FeedforwardSuper Twisting

Fig. 5. Air flow and control input (simulation results).

C. Kunusch et al. / Control Engineering Practice ] (]]]]) ]]]–]]]6

the system, then the robust finite time stability of the algorithm isassured through a strong Lyapunov function and an upperestimate of the convergence time can be computed. Remark: itis important to highlight that this is a conservative estimation ofthe convergence time, but it is guaranteed for the entire operationrange of the system (Davila et al., 2009).

4.3. Feedforward term

It was previously stated that it is necessary to define an extracontrol action that steers the sliding variable within a region suchthat the bounds on the sliding dynamics given in (9) and (10) aresatisfied (Fridman & Levant, 2002, Chap. 3). With this purpose, ithas been included as a feedforward (FF) action uff based on thestack current, implicitly through the reference Wcp,ref (6). Itprovides the control effort required to reach the surface neigh-bourhood where conditions (9) and (10) hold. Therefore, theimplemented control action (ui) comprises two terms:

uiðtÞ ¼ uff þu, ð28Þ

Please cite this article as: Kunusch, C., et al. Experimental resultsbased system. Control Engineering Practice (2012), http://dx.doi.org/1

where u corresponds to the SOSM control action particularizedabove. The expression of uff is computed via a polynomialobtained from an off-line test covering the entire operation rangeof the experimental PEMFC system and is given by

uff ¼ 0:1014 W6cp,ref�1:1412W5

cp,ref þ4:8303 W4cp,ref�9:3370W3

cp,ref

þ8:1430 W2cp,ref�0:6129 Wcp,ref�0:1934 ð29Þ

The following figures present the simulations results of thecontrolled system using the algorithm designed above. Fig. 5shows that the Super Twisting controller presents a satisfac-tory dynamic response when controlling the air mass flow. InFig. 6, the typical behaviour that imposes the algorithm to thenonlinear system is shown through as S2 _S diagram. It is impor-tant to stress that, after an adequate parameters tuning, thecontroller presents a satisfactory dynamic behaviour, confirmingthe suitability of the STA for the breathing control of this PEMFCsystem.

applying second order sliding mode control to a PEM fuel cell0.1016/j.conengprac.2012.08.002

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

x 10−8

−5

−4

−3

−2

−1

0

1

2

3 x 10−7 Sliding variable

S

dS/d

t

Fig. 6. S vs. _S (simulation results).

Fig. 7. Wcp regulation with SOSMþFF controller (experimental results).

C. Kunusch et al. / Control Engineering Practice ] (]]]]) ]]]–]]] 7

5. Experimental results

The main objective of this section is to present the perfor-mance of the proposed SOSM controller, implemented in the fuelcell test station, considering external disturbances and differentworking conditions.

Among several simulations and experimental tests performed inthe fuel cell system, a first approach to solve the real problem wasimplemented through the designed Super Twisting controller. Con-sidering that the plant has a relative degree one, the Super Twistingalgorithm appears as particularly suitable option for the currentlaboratory implementation. Another implementation advantage ofthis algorithm is that it does not require information of the sign of _S.

To assess the controller (12), (13), (30) performance in realoperation, in Fig. 7 it can be appreciated as the performance of theSOSMþFF controller at different load conditions. ComparingFigs. 7 and 5, it can be stated that the simulation and experimentalresults preserve the same dynamic behaviour, showing the relia-bility, robustness and accuracy of the design methodology.

Please cite this article as: Kunusch, C., et al. Experimental resultsbased system. Control Engineering Practice (2012), http://dx.doi.org/1

Furthermore, another representative set of tests was per-formed in the PEM fuel cell test station, forcing an externalperturbation in the cathode line pressure. In these experiments,the control performance was assessed at a fixed control reference,see Fig. 8, while a pressure disturbance in the air pressure wasincorporated by means of an electronic valve.

It can be noticed that the control objective is satisfactoryaccomplished during the imposed perturbation. Moreover, whenthe system is strongly disturbed (tC195 s), the controller drivesagain the system trajectories to the sliding surface. This effect isachieved because the stability of the closed loop system isguaranteed given that the differential inclusion (11) is satisfied.

Apart from the presented examples, extensive simulation andexperimental analysis have been conducted and, in every case,highly satisfactory results have been obtained using the proposedSuper Twisting controller. It is important to empathise that theproposed STAþFF controller showed very good performance for awide range of operating conditions, proving robustness withrespect to external disturbances and model uncertainties.

applying second order sliding mode control to a PEM fuel cell0.1016/j.conengprac.2012.08.002

Fig. 8. Perturbation test (experimental results).

C. Kunusch et al. / Control Engineering Practice ] (]]]]) ]]]–]]]8

6. Conclusions

After evaluating a second order sliding mode controller solu-tion to globally solve the oxygen stoichiometry problem of aPEMFC generation system, a Super Twisting controller has beendeveloped and implemented in a laboratory test station. Itssuitability was successfully verified through extensive computersimulations, based on a previously designed model of the plantespecially built for nonlinear control purposes and taking intoaccount external disturbances and uncertainties in the systemparameters (Kunusch et al., 2011). Subsequently, highly satisfac-tory experimental results using the aforementioned approach,confirm the feasibility, simplicity and robustness of the solution.Main advantages of the proposed Super Twisting control forPEMFC systems can be summarized as follows:

�

Pb

robust stabilization of the oxygen stoichiometry problemavoiding chattering effects;

� enhanced dynamic characteristics; � robustness to parameter uncertainties and external disturbances; � guaranteed extended range of operation, in spite of the highly

nonlinear nature of plant;

� the control law only depends on two measurable variables,

namely the stack current and the compressor air flow, there-fore no state observer is required;

� simple controller structure, resulting in low real time compu-

tational costs.The resulting controller is relatively simple to design and tune,

only requiring certain measurements of easy and low cost imple-mentation. This represents important issue for fuel cells industrialapplications, where the plant instrumentation is critical.

Acknowledgements

All the experimental tests were performed at the PEM Fuel CellsLaboratory of the Institut de Rob �otica i Inform�atica Industrial(CSIC-UPC, Barcelona, Spain) and only possible due to its advancedequipment and proficient technical staff. This research has beensupported by the Seventh Framework Programme of the EuropeanCommunity through the Marie Curie actions (GA: PCIG09-GA-

lease cite this article as: Kunusch, C., et al. Experimental resultsased system. Control Engineering Practice (2012), http://dx.doi.org/1

2011-293876 and GA: PIIF-911767-ACRES), the CICYT projectDPI2011-25649 (MICINN-Espana), the CSIC JAE-DOC ResearchProgramme, Universidad Nacional de La Plata (UNLP), CONICETand CICpBA from Argentina, Universidad Nacional Autonoma deMexico (UNAM) and projects FONCICyT-93302, PAPIIT-117211 andCONACyT-132125.

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