Sliding-Mode Control of PEM Fuel Cells

Advances in Industrial Control

For further volumes: www.springer.com/series/1412

Cristian Kunusch r Paul Puleston Miguel Mayosky

Sliding-Mode Control of PEM Fuel Cells

r

Cristian Kunusch Institut de Robòtica i Informàtica Industrial CSIC-Universitat Politècnica de Catalunya Barcelona Spain Paul Puleston Depto. Electrotecnia Laboratorio de Electrónica Industrial, Control e Instrumentación (LEICI) Universidad Nacional de La Plata—CONICET La Plata, Buenos Aires Argentina

Miguel Mayosky Depto. Electrotecnia Laboratorio de Electrónica Industrial, Control e Instrumentación (LEICI) Universidad Nacional de La Plata Comisión de Investigaciones Científicas de la Provincia de Buenos Aires (CICpBA) La Plata, Buenos Aires Argentina

ISSN 1430-9491 e-ISSN 2193-1577 Advances in Industrial Control ISBN 978-1-4471-2430-6 e-ISBN 978-1-4471-2431-3 DOI 10.1007/978-1-4471-2431-3 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2012930019 © Springer-Verlag London Limited 2012 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my parents Zulma and Orlando, of course. To Maria and all the people who encourage me everyday, but specially to my tireless co-authors. To Mark, Ana, Tommy, Kate and my parents. To Daniela, Matilde and the memory of Norma and Alejandro.

Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage technology transfer in control engineering. The rapid development of control technology has an impact on all areas of the control discipline. New theory, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies, . . . , new challenges. Much of this development work resides in industrial reports, feasibility study papers and the reports of advanced collaborative projects. The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination. Fuel cells are one of the devices promoted as a potential technological contributor to the ‘green’ energy economy. Fuel cells are an energy conversion device that uses the catalytic oxidation of hydrogen at an anode with the catalytic reduction of oxygen at a cathode to release energy as electricity and heat with the non-polluting by-product of water. So long as hydrogen and oxygen fuel is supplied (at the appropriate operating conditions, and with sufficient purity to avoid possible catalyst poisoning), a fuel cell will provide a continuous energy output. In this way a fuel cell does not suffer the fuel depletion that characterises the energy storage of battery technology. Several different fuel cell types arise from the use of a variety of different electrolyte and electrode media, with differing operating temperatures. To create a power unit, fuel cells are then combined in series or parallel in planar or tubular stacks and supplied with auxiliary equipment. Following on from this are requirements for pressure regulation, temperature control and humidity-level maintenance. The complete assembly involves electrochemical, physical, chemical, and thermal processes and is consequently, a highly nonlinear system. Finally, add to this that some internal variables are difficult to measure, that instrumentation overheads should be kept low for portability’s sake, and that system models have associated model and parametric uncertainty, and the fuel-cell-control problem clearly emerges as a highly challenging one. This technological précis is the starting point for the authors Cristian Kunusch, Paul F. Puleston, and Miguel A. Mayosky who make a very convincing case for vii

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Series Editors’ Foreword

the benefits that modern nonlinear closed-loop control has to offer fuel-cell technology. Their research is with the proton exchange membrane (PEM) fuel cell. Apparently, PEM fuel cells have a high power density, low weight, and volume that make them particularly appropriate for portable or mobile applications. The authors chosen control paradigm is that of sliding-mode control, and in this monograph, they give a comprehensive report of their research; however, what makes this monograph particularly appropriate to the Advances in Industrial Control series is the way in which the authors combine a wide of variety of skills, empirical nous, and technical knowledge to develop a practical demonstration prototype for the fuel-cell-control application. The monograph is self-contained and opens with introductory material on fuel cells, PEM fuel cells, and sliding-mode control (Chaps. 1, 2 and 3). For a first demonstration of their sliding-mode control proposals, the authors use their own reformulation of a benchmark model that was previously reported in an earlier seminal Advances in Industrial Control monograph, Control of Fuel Cell Power Systems by Jay T. Pukrushpan, Anna G. Stefanopoulou, and Huei Peng (ISBN 978-1-85233816-9, 2004). Thus the first test of the sliding-mode control is through a detailed simulation benchmark case study (Chap. 4). The authors’ research with an experimental PEM-fuel-cell test bed is then reported in Chaps. 5 and 6 in the monograph. As so often occurs in process control case studies, the system modelling (Chap. 5) follows a process-based modular structure and the model is derived with a judicious blend of physically based equations and experimental-data-supported empiricism. The trade-off between the actual process behaviour and the accuracy that can be achieved with a tractable model naturally introduces a level of system and parametric uncertainty and leads to requirements for a robust and reliable control scheme. Chapter 6 presents the authors’ second-order sliding-mode control design and the results from the experimental test bed facility. All is fully documented and offers an opportunity for further independent study of the control system, its results, and the control outcomes. The monograph closes with a chapter of conclusions and a discussion of future possible research issues for this system and fuel-cell control per se. A wide range of readers from the control community, and also those involved in new energy technologies, will find much of interest in this research from authors Cristian Kunusch, Paul F. Puleston, and Miguel A. Mayosky. This volume is also a very welcome complement to the earlier fuel-cell monograph mentioned above and now provides the Advances in Industrial Control series with two excellent texts on control for fuel-cell technologies. Industrial Control Centre Glasgow Scotland, UK

M.J. Grimble M.A. Johnson

Acknowledgements

There are many people and institutions to whom we would like to express our gratitude. Too many, in fact. And each one of us has our particular debts of gratitude. Therefore, we are forced to just explicitly mention those that directly contributed to the realisation of this book. To all the others, many thanks. We wish to acknowledge the support of the National University of La Plata (UNLP), CONICET and CICPBA from Argentina, the Technical University of Catalonia (UPC), CSIC and AECID from Spain, and the Seventh Framework Programme of the European Community through the Marie Curie actions. We specially like to thank the Laboratorio de Electrónica Industrial, Control e Instrumentación (LEICI), Faculty of Engineering (UNLP) and its mentor, Professor Carlos F. Christiansen, who created an institution where we could initiate and develop our careers in the field of scientific-technological research. We also wish to express profound thanks: – To Professor Jordi Riera Colomer for his enduring collaboration and generosity, and to each and every member of the IRI’s Fuel Cell Laboratory, whose proficiency and dedication made the experimental phase of this investigation a reality. – To Professor Enric Fossas Colet, without whom this collaboration would not have been at all possible. Thanks him for his continuous support and unbreakable friendship. – To Professor Leonid Fridman for his wise advice on Higher-Order Sliding-Mode control and, moreover, for his unselfish help all the way. – To our friends and colleagues in the sliding mode community, who have supported and encouraged us during the different stages of our research. A final comment. We have made use of material from several books and papers in this work. We are very grateful to their authors, and we do hope that we have cited them abundantly and appropriately, acknowledging the full recognition they deserve.

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Contents

1

Introducing Fuel Cells . . . . . . . . . . . . . . . . . 1.1 An Outlook of the Energy Perspective . . . . . . . 1.2 What is a Fuel Cell? . . . . . . . . . . . . . . . . 1.3 State-of-the-Art on Fuel Cell Systems Technology 1.3.1 Fuel Cell Units . . . . . . . . . . . . . . . 1.3.2 Fuel Cell Stacks . . . . . . . . . . . . . . 1.3.3 Auxiliary Components . . . . . . . . . . . 1.4 A Brief History of Fuel Cells . . . . . . . . . . . 1.5 Closed-Loop Operation of Fuel Cells. Why? . . . 1.6 Scope and Outline of the Book . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 3 3 4 5 6 6 9 10 10

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PEM Fuel Cell Systems . . . . . . . . . . . . . . . . . . . 2.1 An Introduction to PEM Technology . . . . . . . . . . 2.2 Basics of PEM Fuel Cells Operation . . . . . . . . . . 2.3 Efficiency and Power Conversion . . . . . . . . . . . . 2.4 State-of-the-Art in PEM Fuel Cells Technology . . . . 2.5 Components and Associated Devices . . . . . . . . . . 2.5.1 Polymeric Membranes . . . . . . . . . . . . . . 2.5.2 Electrodes . . . . . . . . . . . . . . . . . . . . 2.5.3 Gas Diffusion Layers . . . . . . . . . . . . . . 2.5.4 Sealing Gaskets . . . . . . . . . . . . . . . . . 2.5.5 Bipolar Plates . . . . . . . . . . . . . . . . . . 2.5.6 Auxiliary Devices . . . . . . . . . . . . . . . . 2.6 Available PEM Fuel Cell Models in the Open Literature 2.6.1 Control Oriented Models . . . . . . . . . . . . 2.6.2 Control Objectives and Challenges . . . . . . . 2.6.3 Recent Advances on PEM Fuel Cell Control . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13 13 14 17 18 19 19 20 21 22 22 24 27 28 29 30 31

3

Fundamentals of Sliding-Mode Control Design . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2 Variable Structure Control Preliminaries . . . . . . . . . . . . . 3.3 Fundamentals of Sliding-Mode Control . . . . . . . . . . . . . . 3.3.1 Diffeomorphisms, Lie Derivative and Relative Degree . . 3.3.2 First-Order Sliding Mode . . . . . . . . . . . . . . . . . 3.3.3 Equivalent Control Regularisation Method. Ideal Sliding Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Existence Conditions for the First-Order Sliding Regime . 3.3.5 Extension to Nonlinear Systems Non-affine in Control . . 3.3.6 Filippov Regularisation Method . . . . . . . . . . . . . . 3.3.7 Discontinuous Control Action in Classic Sliding-Mode Control. Chattering Problem . . . . . . . . . . . . . . . . 3.4 Some General Concepts on Higher-Order Sliding Modes . . . . . 3.4.1 Definition of Differential Inclusion . . . . . . . . . . . . 3.4.2 Sliding Modes on Manifolds . . . . . . . . . . . . . . . 3.4.3 Sliding Modes and Constraint Functions. Regularity Condition . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Closing Comments on Higher-Order Sliding Modes in Control Systems . . . . . . . . . . . . . . . . . . . . . . 3.5 Design of Second-Order Sliding-Mode Controllers . . . . . . . . 3.5.1 Second-Order Sliding Generalised Problem . . . . . . . . 3.5.2 Solution of the Control Problem. SOSM Algorithms . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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53 54 54 56 69 69

Assessment of SOSM Techniques Applied to Fuel Cells. Case Study: Electric Vehicle Stoichiometry Control . . . . . . . . . . . . . . . . . 73 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Succinct Description of the Electric Vehicle Fuel Cell System . . . 74 4.2.1 Air Compressor . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.2 Air Supply Manifold . . . . . . . . . . . . . . . . . . . . . 76 4.2.3 Air Humidifier and Temperature Conditioner Subsystems . 77 4.2.4 Cathode Channels . . . . . . . . . . . . . . . . . . . . . . 77 4.2.5 Anode Channels . . . . . . . . . . . . . . . . . . . . . . . 78 4.2.6 Water Model of the Polymeric Membrane . . . . . . . . . 79 4.2.7 Return Manifold . . . . . . . . . . . . . . . . . . . . . . . 80 4.3 Electric Vehicle FCGS State Space Model for SOSM Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.4 Control Objective and Sliding Surface . . . . . . . . . . . . . . . 82 4.5 Design of a SOSM Super-Twisting Controller for the Electric Vehicle FCGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.6 SOSM Super-Twisting Controller Simulation Results . . . . . . . 88 4.7 Comparison with Other Control Strategies . . . . . . . . . . . . . 93 4.7.1 Different SOSM Control Algorithms . . . . . . . . . . . . 93 4.7.2 LQR Controller . . . . . . . . . . . . . . . . . . . . . . . 97 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Contents

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Control-Oriented Modelling and Experimental Validation of a PEMFC Generation System . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Air Compressor Subsystem . . . . . . . . . . . . . . . . 5.2.1 Air Compressor Motor Dynamic Equations . . . . 5.2.2 Diaphragm Vacuum Pump Characteristics . . . . 5.3 Air Supply Manifold Subsystem . . . . . . . . . . . . . . 5.4 Air Humidification Subsystem . . . . . . . . . . . . . . . 5.4.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . 5.5 Fuel Cell Stack Flow Subsystem . . . . . . . . . . . . . 5.5.1 Cathode Channels . . . . . . . . . . . . . . . . . 5.5.2 Anode Channels . . . . . . . . . . . . . . . . . . 5.5.3 Membrane Water Transport . . . . . . . . . . . . 5.6 Electrical Characterisation of the Fuel Cell Stack . . . . . 5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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105 105 106 107 111 112 113 114 115 117 117 120 121 123 126 128

SOSM Controller for the PEMFC-Based Generation System. Design and Implementation . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 State Space Model of the Experimental Fuel Cell System for Control Design . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Control Output . . . . . . . . . . . . . . . . . . . . . 6.2.2 Auxiliary Functions . . . . . . . . . . . . . . . . . . 6.3 Control Objective . . . . . . . . . . . . . . . . . . . . . . . 6.4 Controller Synthesis . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Super-Twisting Algorithm . . . . . . . . . . . . . . . 6.4.2 Twisting Algorithm . . . . . . . . . . . . . . . . . . 6.4.3 Sub-Optimal Algorithm . . . . . . . . . . . . . . . . 6.4.4 Feedforward Term . . . . . . . . . . . . . . . . . . . 6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 6.6 Experimental Set-up of the PEM Fuel Cell System . . . . . . 6.7 Experimental Tests . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Regulation Tests . . . . . . . . . . . . . . . . . . . . 6.7.2 Perturbation Tests . . . . . . . . . . . . . . . . . . . 6.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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129 133 133 134 137 139 140 140 141 141 143 144 144 148 150 152

Conclusions, Open Lines and Further Reading 7.1 Main Results . . . . . . . . . . . . . . . . . 7.2 Further Issues Related to PEMFC Control . 7.2.1 Adaptive Super-Twisting Algorithms 7.2.2 HOSM MIMO Control . . . . . . . 7.2.3 Model Predictive Control . . . . . . 7.2.4 Observers for Internal Variables . . .

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7.3 Further Issues Related to Fuel-Cell-Based Systems 7.3.1 Hybrid Standalone Systems . . . . . . . . 7.3.2 Distributed Generation Systems . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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157 157 157 158

Appendix A Electric Vehicle PEM Fuel Cell Stack Parameters A.1 Return Manifold Polynomial Fitting . . . . . . . . . . . . A.2 Differential Equations Parameters . . . . . . . . . . . . . A.3 Controller Parameters . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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161 161 161 163 164

Appendix B

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Laboratory FC Generation System Parameters . . . . . . . 165

Appendix C Laboratory FCGS State Space Functions and Coefficients C.1 Expression of ϕ(x, u, t) . . . . . . . . . . . . . . . . . . . . . . . C.2 Model Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Feedforward Control Action . . . . . . . . . . . . . . . . . . . . .

169 169 172 173

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Notation and Acronyms

FC FCGS LQR PEM SM SOSM ST a a0 a1 a2 a3 A0 A1 A00 A10 A20 A01 A11 A02 Aapp Afc b B00 B10 B20 B01 B11 B02 Cp C0

Fuel cell Fuel cell generation system Linear Quadratic Regulator controller Proton Exchange Membrane Sliding mode Second-order sliding mode Super-Twisting Polarisation curve coefficient Empirical coefficient of the membrane water content model Empirical coefficient of the membrane water content model Empirical coefficient of the membrane water content model Empirical coefficient of the membrane water content model Compressor model fitting parameter Compressor model fitting parameter Compressor model fitting parameter Compressor model fitting parameter Compressor model fitting parameter Compressor model fitting parameter Compressor model fitting parameter Compressor model fitting parameter Apparent fuel cell area Fuel cell active area Polarisation curve coefficient Compressor model fitting parameter Compressor model fitting parameter Compressor model fitting parameter Compressor model fitting parameter Compressor model fitting parameter Compressor model fitting parameter Specific heat capacity of air Cathode humidifier model fitting parameter xv

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C1 C3 cv,an cv,ca Dw E F Ga G H2 G N2 G O2 Gm,dry Gv i Icp i0 ia Iload Ist J Jg Ksm k, kI Kca,out kdct , kdcv , Rdc Ksm kφ L m mH2 ,an mH2 O,an,liq mH2 O,ca,liq mhum mN2 ,ca mO2 ,ca msm mv,an mv,ca mv2 ,an mv2 ,ca mw,an n

Notation and Acronyms

Cathode humidifier model fitting parameter Cathode humidifier model fitting parameter Water concentration at the membrane surface on the anode side Water concentration at the membrane surface on the cathode side Diffusion coefficient of water in the membrane Nernst voltage of a PEM fuel cell Faraday constant Dry air molar mass Hydrogen molar mass Nitrogen molar mass Oxygen molar mass Membrane dry molecular weight Vapour molar mass Stack current density Armature current of the compressor’s DC motor Exchange current density Armature current of the compressor’s DC motor Load current Fuel cell stack current Combined inertia of the compressor motor and the compression device Compressor and air manifold gathered or equivalent inertia Supply manifold restriction constant Proportional and integral gains of the LQR controller Cathode output restriction Compressor DC motor constants Supply manifold restriction constant DC motor constant Electrical inductance of the DC motor stator winding Polarisation curve coefficient Hydrogen mass in the anode channels Mass of liquid water in the anode channels Mass of liquid water in the cathode channels Cathode humidifier mass of air Nitrogen mass in the cathode channels Oxygen mass in the cathode channels Supply manifold mass of air Vapour mass in the anode channels Vapour mass in the cathode channels Vapour mass in the anode channels Vapour mass in the cathode channels Mass of water inside the anode Number of fuel cells in the stack

Notation and Acronyms

nd nc n0 n1 n2 Pamb Pan Pca Pcp PH2 ,an PH2 ,lh,an Phum Phum,an Phum,ca Phum,diff Pnet PN2 ,ca PN2 ,hum PN2 ,lh PO2 ,ca PO2 ,hum PO2 ,lh Prm Psat Psm Pst Pv,an Pv,ca Pv,hum Pv,lh Pv,lh,an q r1 , r2 RH amb R R0 R1 Ra RH an RH ca RH hum RH lh s s0 Trm

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Electro-osmotic drag coefficient Polarisation curve coefficient Empirical parameter of the electro-osmotic coefficient model Empirical parameter of the electro-osmotic coefficient model Empirical parameter of the electro-osmotic coefficient model Atmospheric pressure Stack pressure drop (anode) Cathode pressure Compressor power demand Hydrogen partial pressure in the anode channels Anode line heater hydrogen partial pressure Cathode humidifier pressure Anode humidifier pressure Cathode humidifier pressure Cathode humidifier pressure drop FCGS net power Nitrogen partial pressure in the cathode channels Cathode humidifier nitrogen partial pressure Cathode line heater nitrogen partial pressure Oxygen partial pressure in the cathode channels Cathode humidifier oxygen partial pressure Cathode line heater oxygen partial pressure Return manifold pressure Vapour saturation pressure of a gas Supply manifold pressure Stack generated power Vapour partial pressure in the anode channels Vapour partial pressure in the cathode channels Cathode humidifier vapour partial pressure Cathode line heater vapour partial pressure Anode line heater vapour partial pressure Integral state variable of the LQR Twisting controller parameters Relative humidity of the air Electrical resistance of the DC motor stator winding Resistance coefficient Resistance coefficient Specific gas constant of atmospheric air Relative humidity of the gas in the anode channels Relative humidity of the gas in the cathode channels Relative humidity of the gas exiting the cathode humidifier Relative humidity of the gas exiting the cathode line heater Sliding variable or output Size of the validity region |s(t, x)| < s0 Return manifold temperature

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Tamb Tcp Te Thum Thum,an Thum,ca Tl Tl,amb Tlh Tsm Tst tm Tp Tl U, α ∗ , β, sM uff ui = u + uff u1 , u2 Vact Vrm Vca Vconc Vcpff Vcpi Vcpp Vcp Vfc Vhum Vnor V0 Vohm Vsm Vst Wa,hum Wan Wan,out Wca Wca,out Wrm,out Wair,ref Wcp Wdry,ref W H2 WH2 ,an,in

Notation and Acronyms

Atmospheric temperature Compressor air temperature Compressor motor torque Cathode humidifier temperature Anode’s humidifier temperature Cathode’s humidifier temperature Load torque Compressor load torque at ambient pressure Cathode line heater temperature Manifold air temperature Stack temperature Thickness of the membrane Unknown torque disturbance Compressor extra load torque at pressures higher ambient Sub-Optimal controller parameters Feedforward control action Implemented control action Super-Twisting control action components Activation losses Return manifold volume Volume of the cathode channels Concentration losses Compressor voltage feedforward term Implemented compressor voltage command Compressor voltage LQR pre-compensation term Input voltage of the compressor DC motor Voltage of a single fuel cell Cathode humidifier volume DC motor voltage normalisation constant Theoretical electrochemical potential of single hydrogen/oxygen fuel cell Ohmic losses Supply manifold volume Stack voltage Mass flow of dry air exiting the cathode humidifier Mass flow entering the anode channels Mass flow exiting the anode channels Cathode input mass flow of air Flow entering the return manifold Outlet mass flow of the return manifold Air mass flow reference Compressor air mass flow Reference dry air flow Hydrogen mass flow Mass flow rate of H2 entering the anode

Notation and Acronyms

WH2 ,an,out WH2 ,react Whum Wl,ca,out WN2 ,ca,in WN2 ,ca,out WO2 ,ca,in WO2 ,ca,out WO2 ,ca,ref WO2 ,react Wsm Wv,an,in Wv,an,out Wv,ca,gen Wv,ca,in Wv,ca,out Wv,hum Wv,inj Wv,mem x XO2 ,ca XO2 ,ca,out α αca αan χ O2 δ xˆ  δx  ηcp ηdc γ , λ, ρ γ Γm , ΓM , Φ λca λan λm λ O2 ν ωamb ωca ωhum

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Mass flow rate of H2 leaving the anode Rate of H2 reacted Cathode humidifier output flow of air Mass flow of liquid water exiting the cathode channels Mass flow rate of N2 entering the cathode Mass flow rate of N2 leaving the cathode Mass flow rate of O2 entering the cathode Mass flow rate of O2 leaving the cathode Cathode oxygen mass flow reference Oxygen flow consumed in the reaction Manifold outlet mass flow Mass flow rate of vapour entering the anode Mass flow rate of vapour leaving the anode Rate of vapour generated in the fuel cell reaction Mass flow rate of vapour entering the cathode Mass flow rate of vapour leaving the cathode Mass flow of vapour entering the cathode humidifier Mass flow of vapour injected by the cathode humidifier Mass flow rate of water transfer across the membrane LQR system state Oxygen mass mole fraction of the air entering the cathode channels Oxygen mass mole fraction of the air exiting the cathode channels Charge transfer coefficient Polarisation curve coefficient related to liquid water accumulation in the cathode Polarisation curve coefficient related to liquid water accumulation in the anode Molar fraction of oxygen in the air Estimate of the linearised LQR system state Linearised LQR system state Compressor efficiency Compressor DC motor mechanical efficiency Super-Twisting controller parameters Ratio of the specific heats of air Bounds of s¨ Membrane water content on the cathode Membrane water content on the anode Membrane water content Oxygen excess ratio or oxygen stoichiometry Ancillary input of the expanded Relative Degree 1 system Rate of vapour in the ambient air flow Humidity ratio of the gas entering the cathode channels Humidity ratio of the ambient air

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ωca,out ωan ωan,out ωcp ϕ(t, x, u), γ (t, x, u) ρm,dry

Notation and Acronyms

Humidity ratio gas exiting the cathode channels Humidity ratio of the gas entering the anode channels Humidity ratio of the gas exiting the anode channels Compressor rotational speed Functions of s¨ Membrane dry density

Chapter 1

Introducing Fuel Cells

1.1 An Outlook of the Energy Perspective Increasing demands on pollution reduction is driving innovation on clean energy sources. Among these, fuel cells (FCs) are regarded as one of the most promising technologies, due to their efficiency, compactness and reliability [3]. FCs are electrochemical devices that generate electrical current from hydrogen and oxygen, with pure water and heat as by-products. Considering that hydrogen is widely available and can be obtained from many renewable sources using solar and wind energy, fuel cells represent an attractive, feasible alternative to reduce fossil fuel dependence. However, the widespread use of hydrogen as combustible—and the resulting “hydrogen economy”—despite its interesting possibilities, has some technological issues to be resolved. Approximately 70 percent of the electrical power we use today is generated from fossil combustibles. Besides, almost all the transportation vehicles have internal combustion engines. Although in a complete hydrocarbon combustion the only by-products are carbon dioxide and water, in practice other substances like carbon monoxide (a poisonous gas), nitrogen oxides (responsible of most of the urban smog) and unburned hydrocarbons are also produced. Greenhouse gases like carbon monoxide are responsible for increasing temperatures of the planet, which is directly related to climate changes. The amount of greenhouse gases in the atmosphere is continuously increasing since the Industrial Revolution, and the final ecological consequences are still hard to foresee. For this reason, most industrialised countries are raising their pollution-reduction standards, in an attempt to slow down the consequences of climate change. However, this is not the only problem with fossil combustibles. There are also pollution hazards related with their transportation (oil spills, pipeline explosion and well fires), as well as economic and political concerns regarding dependence on oil-rich countries. A switch to hydrogen as the main energy carrier could be a practical solution to some of these problems if it can be produced in a clean and sustainable way. Molecular H2 has the highest energy content per unit weight among the known gaseous fuels (143 MJ kg−1 ) and is the only carbon-free fuel which ultimately oxidises to water as a combustion product. C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3_1, © Springer-Verlag London Limited 2012

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Introducing Fuel Cells

Most of the current production of hydrogen is used in the fertiliser (50%) and petroleum (37%) industries. Industrial production of hydrogen is currently accomplished in four main ways: • Thermo-chemical technologies: Steam reforming is the most widely used process to produce hydrogen from natural gas, coal, methanol, ethanol and gasoline, among others [4]. The process uses catalysts in one or more steps of the conversion process. Steam reforming of natural gas accounts for almost 50% of the world feedstock for H2 production. Obviously, this technology does not avoid either pollution (as one of the by-products is carbon monoxide) or fossil oil dependence (hydrocarbon is still required). In spite of these facts, the process is seen by many as an intermediate step in the right direction. For instance, several car manufacturers are currently testing prototypes using a reformer and a fuel cell to drive an electric engine. This allows using the existing fuel distribution network while the alternative fuels infrastructure is developed. • Photo-biological technology: This approach uses the natural photosynthetic activity of bacteria and green algae to produce H2 . A wide range of alternatives are available, including direct and indirect biophotolysis, photofermentation and dark-fermentation [2]. One major limitation of this promising technology is the relatively slow production rates, which precludes its use in massive production. • Photo-electrochemical technology: This process produces H2 in one step, splitting water by illuminating a water-immersed semiconductor with sunlight [5]. This technology is still in early stages of development, and many obstacles must be overcome before being economically viable. • Electrochemical technologies: Electrolysis is a simple process that only requires water and electricity. It can be accomplished securely almost anywhere, even at home. The combination of electrolysis and a fuel cell looks like a virtuous cycle: hydrogen is produced using water and electricity, while electricity and water are obtained in a fuel cell combining hydrogen and oxygen (or air), without any combustion or Carnot cycle involved. However, electrolysis is energy-intensive: it takes approximately 50 kilowatt-hours of electricity per kilogramme of hydrogen produced. As it was already said, most of the electricity is generated using fossil combustibles, and widespread use of electrolyses will increase electricity demands. For instance, if all transportation becomes electrical, electricity demands to the power grid will double. In the light of the above considerations it becomes clear that, in order to put hydrogen economy to work, H2 should be produced using non-pollutant energy sources (e.g. solar or wind). However, renewable energy sources are often intermittent and difficult to predict; therefore it is usually difficult to match the energy production and the energy demand. This naturally introduces fuel cells (FC) in the energy conversion chain.

1.2 What is a Fuel Cell?

3

Fig. 1.1 Basic H2 /O2 fuel cell

1.2 What is a Fuel Cell? A fuel cell is an electrochemical device that produces electricity from hydrogen and oxygen. To be specific, the catalytic oxidation of hydrogen in an anode and the catalytic reduction of oxygen in the cathode provide a voltage drop between electrodes that can be used by an external circuit if the electrolyte isolates both electrodes but allows ions mass and charge exchange (Fig. 1.1). Besides the energy released as electricity, water and heat are also produced. Given that the electrochemical process does not involve a Carnot cycle, efficiency can be very high. In applications designed to capture and utilise the system’s waste heat (co-generation), fuel use efficiencies could reach 80–85%. Fuel cells have no moving parts, make no noise and are scalable. The basic chemical equations in a H2 /O2 fuel cell are the following: O2

2H2 → 4H+ + 4e− + 4e− → H2 O 2H2 + O2 → H2 O

+ 4H+

Anode half reaction Cathode half reaction Overall reaction

(1.1)

Although a fuel cell is similar to a typical battery in many ways, it differs in several aspects. The battery is an energy storage device where all the energy available is stored within the battery itself. It will cease to produce electrical energy (discharge) when the reactants are consumed. A fuel cell, on the other hand, is an energy conversion device to which fuel and oxidant are supplied continuously. It can produce electric current for as long as fuel is supplied.

1.3 State-of-the-Art on Fuel Cell Systems Technology Most fuel cell systems can be characterised according to three main aspects:

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1

Introducing Fuel Cells

• Cell units, the individual devices where electrochemical reactions take place. • Cell stacks, the combination of individual cell units to provide the desired voltage and current. • Auxiliary components necessary to ensure fuel feeding, feed-stream conditioning, stack temperature and humidity control, electrical output conditioning, etc. These ancillary components usually have important dynamics that affect the overall system behaviour.

1.3.1 Fuel Cell Units During its evolution, many fuel cell unit configurations were proposed, considering different electrolyte and electrode materials, catalysts and operation temperatures. Presently, there are six major fuel cell types: Alkaline fuel cell (AFC), Molten Carbonate fuel cell (MCFC), Phosphoric Acid fuel cell (PAFC), Proton Exchange Membrane fuel cell (PEMFC), Solid Oxide fuel cell (SOFC) and Direct Methanol fuel cell (DMFC) [3]. • Alkaline fuel cells were one of the first technologies developed. They were used by NASA for electricity and potable water generation in the manned missions Gemini, Apollo and the Space Shuttle. AFCs use compressed hydrogen and oxygen as fuel and produce potable water as a by-product. The electrolyte used is potassium hydroxide, which is relatively expensive and is susceptible to poisoning by carbon dioxide. The efficiency can reach 70 percent. Operating temperatures of the early AFC were in the range of 100 °C and 250 °C, but newer designs work at much lower temperatures (23 °C to 70 °C). • Molten Carbonate fuel cells: This is a promising technology that uses a molten carbonate-salt-impregnated ceramic matrix as the electrolyte. They are used mostly with natural gas and coal in industry applications. MCFCs operate at relatively high temperatures (650 °C) and can reach efficiencies of 85% when waste heat is used. Due to the high operation temperature, no external hydrocarbon reforming is necessary. Fuel is converted to hydrogen within the fuel cell itself by a process called internal reforming. • Phosphoric Acid fuel cell: PAFCs use liquid phosphoric acid as electrolyte. Acid is contained in a Teflon-bonded silicon carbide matrix. Electrodes are made of porous carbon with a platinum catalyst. This is the first commercial technology developed for stationary generation. Operating temperatures are in the 150– 250 ◦ C range. Efficiency can be as high as 85% in co-generation. • Proton Exchange Membrane fuel cells: Also called Polymer Electrolyte fuel cells (same acronym), are characterised by high-power density and low weight and volume, which make them a natural choice for mobile and portable applications. PEM fuel cells use a solid polymer electrolyte and porous carbon electrodes containing a platinum catalyst. They need only hydrogen, oxygen from the air, and water to operate and do not require corrosive fluids like other fuel cell technologies. PEM cells are typically fuelled with pure hydrogen supplied from storage

1.3 State-of-the-Art on Fuel Cell Systems Technology

5

Fig. 1.2 Planar-bipolar stacking

tanks or on-board reformers, and operate at relatively low temperatures, around 80 °C. • Solid Oxide fuel cell: SOFCs use a hard, non-porous ceramic compound as the electrolyte. Because the electrolyte is a solid, the cells do not have to be constructed in the plate-like configuration typical of other fuel cell types. SOFCs are expected to have efficiencies around 50–60% when converting fuel to electricity and 80–85% in co-generation. Its operation at high temperatures, around 1000 °C, allows SOFCs to reform fuels internally, which enables the use of a wide variety of fuels and reduces the cost associated with adding a reformer to the system. • Direct Methanol fuel cell: DMFCs are powered by pure methanol, which is mixed with steam and fed directly to the anode. They do not have many of the fuel storage problems typical of other technologies, because methanol has a higher energy density than hydrogen but less than gasoline or diesel fuel. It can be produced easily from biomass (corn, sugar-cane, agricultural waste, etc.). Methanol is also easier to transport and supply to consumers using the existing infrastructure because it is a liquid, like gasoline. It is a relatively new technology compared with that of fuel cells powered by pure hydrogen. Operational temperatures are in the 90–120 °C range.

1.3.2 Fuel Cell Stacks Individual fuel cell units are combined as modules in series or parallel configurations to provide desired voltage and output power. The mechanical arrangement must ensure not only electrical contact among units, but also adequate circulation of gases, allowing catalyst reactions to take place at the correct temperatures and humidity levels. For flat fuel cell units like PEM fuel cells, planar-bipolar stacking is the most usual connection (Fig. 1.2). The interconnect is a separator plate, which usually includes channels to distribute gases evenly, separating fuel and oxidant flows of adjacent cells. Specially for high-temperature cells (for instance, Solid Oxide fuel cells), tubular cell stacks are usual (Fig. 1.3). They have some structural mechanical advantages, although ensuring short paths in the electrical interconnect can be difficult. Flattened tubes are sometimes used to reduce packing space.

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Introducing Fuel Cells

Fig. 1.3 Tubular stacking, tangential current flow. Other configurations are possible

1.3.3 Auxiliary Components Fuel cells require a number of ancillary subsystems to ensure proper operation. Among these, the following are present in almost all FC technologies [3]: • Fuel preparation. As it was already mentioned, not all FC types need pure hydrogen as the combustible. Hydrogen can be obtained from almost any hydrocarbon through reforming. This can be accomplished using an external reformer or, in high-temperature FC units, via internal reforming. Reforming allows some FC types to switch between different kind of fuels, an important issue regarding reliability. On the other hand, certain technologies are very sensitive to hydrogen impurities, as they can degrade catalyst performance and lifetime (for instance, PEM fuel cells require less than a few ppm impurities to avoid electrode poisoning). In this case, additional filtering stages are mandatory. • Air supply. This involves compressors, blowers or compressed air tanks, as well as air filters. Usually compressor dynamics are not negligible in the overall fuel cell model. • Thermal management. Precise temperature control is required to ensure proper operation. This includes gas and stack temperature control. • Water management. Besides water being a by-product of fuel cell operation, in certain FC types interacting gases must be pre-humidified. For instance, in PEM fuel cells this is necessary to avoid drying of the membrane. On the other hand, excessive water can affect fuel cell performance, due to the reduction of the effective reaction area. The amount of water that is generated depends on the electrical power demand and must be properly controlled for every operating condition. • Electric power conditioning equipment. This relates to the electronics required to fulfil electric load requirements.

1.4 A Brief History of Fuel Cells Since the discovering of electrolysis in 1800 by Nicholson and Carslisle, the possibility of an “inverse” mechanism for the production of electricity from hydrogen and oxygen was actively investigated [1]. In 1838, William Robert Grove built an

1.4 A Brief History of Fuel Cells

7

Fig. 1.4 Grove’s gas battery (Philosophical Magazine and Journal of Science, 1843)

Fig. 1.5 Mond and Langer fuel cell (Transactions of the American Electrochemical Society, 1905)

experimental set-up using two platinum electrodes partially immersed in a sulphuric acid container (Fig. 1.4). Each electrode had one end immersed in acid and the other end in a sealed container of hydrogen and oxygen, respectively. Grove observed a constant current flow between the electrodes. The containers held water as well as the gases, and he noted an increasing water level in both of them as the current flowed. Grove’s “gas battery” was the first experimental fuel cell. However, the electrochemical reactions that took place in the cell were still poorly known. Until the early twentieth century, many people tried to produce an FC that could convert coal or carbon to electricity directly, without combustion. In 1889, the chemists Ludwig Mond and Charles Langer attempted to build a practical device using air, coal gas and platinum electrodes, but failed because not enough was known about materials or electricity (Fig. 1.5). It was not until 1893 that Nobel laureate Friedrich Wilhelm Ostwald provided much of the theoretical understanding of how fuel cells operate. He experimentally determined the interconnected roles of the various components of the fuel cell: electrodes, electrolyte, oxidising and reducing agents, anions and cations. Ostwald explained Grove’s gas battery operation, relating its physical properties and chemical reactions. His pioneering work was the basis for modern fuel cell research. In his own words, as he wrote in 1834: “If we gave a galvanic element which directly delivers electrical power from coal and oxygen (. . . ), we are facing a technical revolution that must push back the one of the invention of the steam engine. Imagine how (. . . ) the appearance of our industrial places will change! No more smoke, no more soot, no more steam engine, even no more fire” [6].

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Introducing Fuel Cells

Fig. 1.6 Francis Bacon with a prototype fuel cell (Source: IEEE Global History Network)

In 1932, Francis Bacon, an engineer working at Cambridge University in England, made a number of significant modifications to Mond and Langer’s design. He replaced platinum electrodes with less expensive nickel gauze and substituted the sulphuric acid electrolyte for alkali potassium hydroxide, a less corrosive substance. In 1959, Bacon built a 5-kW alkaline fuel cell, capable to power a welding machine (Fig. 1.6). Later in that year, Allis-Chalmers demonstrated the first FC powered vehicle, combining 1008 cells to power a 20 hp tractor [1]. Aerospace applications boosted the research on FC technology. In the late 1950s, US National Aero Space Agency (NASA) needed a compact way to generate electricity for manned space missions. Nuclear generation was too dangerous, conventional batteries were too heavy, and solar power was too expensive and complex at the time. Fuel cells seemed to be an interesting alternative. Therefore, NASA went on to fund 200 research contracts for FC technology. One of the principal achievements of this initiative was the development of Proton Exchange Membrane fuel cells (PEMFCs). Indeed, in 1958 General Electric (GE) scientists Willard Thomas Grubb and Leonard Niedrach devised a way to deposit platinum electrolyte on a sulphonated polystyrene ion-exchange membrane. GE and NASA further developed this technology for the Gemini space project (Fig. 1.7). This was the first commercial use of a PEM fuel cell [7]. In an attempt to reduce FC weight and increase its operational lifetime, Pratt and Whitney (an aircraft engine manufacturer) licensed Bacon patents for the Alkaline fuel cell in the early 1960s. They made a number of modifications to Bacon’s design, which eventually led them to win a contract to provide FC technologies for the Apollo missions, providing electric power and also potable water for the crew in a compact assembly. An Apollo spacecraft carried three fuel cells in the service module. Each unit housed 31 individual fuel cells connected in series, operating at 27 to 31 volts for a typical power output of 563 to 1420 watts, with a maximum of 2300 watts. Each unit measured 105 cm in height, 55 cm in diameter, and weighted 112 kg. Since then, similar cells have been used in most of the NASA manned missions, including the Space Shuttle.

1.5 Closed-Loop Operation of Fuel Cells. Why?

9

Fig. 1.7 Gemini spaceship PEM fuel cell assembly (Source: NASA)

During the last decades, there has been important research on new materials for electrolytes, electrodes and catalysts, in order to reduce costs and improve efficiency and reliability. Commercial applications include stationary systems for energy backup in hospitals and schools, portable devices and vehicles. The first bus powered by a fuel cell was completed in 1993. Daimler Benz and Toyota launched prototype fuel-cell powered cars in 1997. Nowadays, most of the major automotive companies have prototypes of fuel-cell powered cars and utility vehicles.

1.5 Closed-Loop Operation of Fuel Cells. Why? In spite of current advances in fuel-cell-based technologies, their relatively high costs, moderate reliability, and reduced lifetime remain as major limitations. For this reason, together with the continuous improvement of materials and components, the incorporation of advanced control strategies embodies a major technological issue, in order to achieve cost reduction, performance improvement and efficiency optimisation. The design of control systems must be understood as a whole, taking into account sensing devices, actuators and local control schemes for each subsystem, as well as supervisory and fault tolerant strategies, for optimal energy management in each operating condition. By their very nature, fuel cell systems are complex devices. As it has been analysed in former sections, they are made of many interconnected subunits, comprising interdependent electrochemical, chemical and thermal phenomena. For proper operation, precise pressure, temperature and humidity levels are required. Closedloop operation is therefore necessary to fully exploit fuel cells potential. Comburent

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Introducing Fuel Cells

flows, moisture levels and temperatures must be controlled to fulfil load demands, while ensuring safe operation and avoiding internal components damage. Many factors make this a challenge. Firstly, fuel cells are highly nonlinear devices. This precludes the use of conventional linear control techniques, especially if performance under wide operation ranges is required. Besides, important internal variables are difficult to access, making their measurement expensive or cumbersome. Finally, there exist uncertainty in system parameters, modelling errors and disturbances of many types. Reliable control systems ensuring stability and performance, as well as robustness to model uncertainties and external perturbations are of crucial importance for the advance of fuel cell technology.

1.6 Scope and Outline of the Book This book addresses the analysis and design of closed-loop controllers for fuel cell systems. In particular, robust controllers based on the so-called “sliding mode” paradigm are considered for the case of PEM FC systems. It resumes the current state-of-the-art and can be used as a starting point for further research and development. The text is conceptually divided in two parts. The first section (Chaps. 2 and 3) is devoted to theory of fuel cells and Sliding-Mode (SM) controllers. Fuel cell operation principles are discussed, in particular for the case of PEMFCs. A survey of the dynamic models in the literature is presented, with special emphasis on their suitability for control design. The mathematical background relative to Sliding-Mode controllers, and in particular Second-Order Sliding-Mode (SOSM) Controllers is summarised, explaining design issues and discussing performance and dynamic response features. SOSM controllers allow reduction of chattering problems typically associated with standard SM techniques. The second part (Chaps. 4 to 6) presents simulation and experimental results PEM fuel-cell-based systems. A modelling methodology is discussed for the related subsystems: compressor-based air supply, gases humidifiers and line heaters, fuel cell stack thermodynamics and electro chemistry. The proposed framework can be used as a guide for similar FC configurations. A detailed design procedure for different SOSM control strategies is outlined, and their results compared in simulations and experimentally. Performance and robustness issues are discussed in depth. Finally, conclusions and open research lines are presented. The intended audience for this book comprises control engineers, FC systems developers and postgraduate students of Electronics and Control Engineering.

References 1. Collecting the history of fuel cells. Smithsonian National Museum of American History

References

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2. Das D, Veziroglu TN (2001) Hydrogen production by biological processes: a survey of literature. Int J Hydrog Energy 26:13–28 3. EG&G Technical Services, Inc (2004) Fuel cell handbook, 7th edn. U.S. Department of EnergyOffice of Fossil Energy, Morgantown 4. Haryanto A, Sandum F, Murali N, Adhikari S (2005) Current status of hydrogen production techniques by steam reforming of ethanol: a review. Energy Fuels 19:2098–2106. http://americanhistory.si.edu/fuelcells/index.htm. Cited 4 Feb 2011 5. Kotay SM, Das D (2008) Bio-hydrogen as a renewable energy resource—prospects and potentials. Int J Hydrog Energy 33:258–263 6. Kunze J, Stimming U (2009) Electrochemical versus heat-engine energy: a tribute to Wilhelm Ostwald’s visionary statements. Angew Chem Int Ed 48:9230–9237 7. Scott JH (2006) The development of fuel cell technology for electric power generation: from NASA’s manned space program to the “hydrogen economy”. Proc IEEE 94:1815–1825

Chapter 2

PEM Fuel Cell Systems

2.1 An Introduction to PEM Technology Among the many different technologies summarised in Chap. 1, Proton Exchange Membrane (PEM) fuel cells are extensively used for mobile and portable applications. This is due to their compactness, low weight, high power density and clean, pollutant free operation. From the operational point of view, a relevant aspect is their low temperature of operation (typically 60–80 °C), which allows fast starting times. In a PEM Fuel Cell, a hydrogen-rich fuel is injected by the anode, and an oxidant (usually pure oxygen or air) is fed through the cathode. Both electrodes are separated by a solid electrolyte that allows ionic conduction and avoids electrons circulation. Catalytic oxidation of H2 and catalytic reduction of O2 take place in the negative and positive electrodes, respectively. The standard electrolyte used in PEM Fuel Cells is a perfluored solid polymer composed by Teflon-like chains. This material combines mechanical, chemical and thermal stability with a high protonic conductivity, when properly humidified. Electrodes are typically made of a porous carbon compound coated with a catalyst such as platinum or palladium, to improve the efficiency of electrochemical reactions. Catalysts are essential in this technology and constitute one of the most expensive components of the cell. Besides, they are very sensitive to CO contamination, which makes the use of high-purity hydrogen (CO  20 ppm) mandatory. This is a serious limitation when H2 is obtained from hydrocarbon reforming. Alternative alloys, like platinum/ruthenium, which are more resistant to CO poisoning, are currently under development. The output of a PEM Fuel Cell is electric energy, with water and heat as the only by-products. Efficiency can be high, as previously said, due to the absence of a Carnot cycle. From the electrical point of view, the cell can be seen as a voltage source where the output impedance presents a highly nonlinear dependence to operating conditions such as temperature, electric current, partial pressures and humidity levels of the incoming gases. Due to this nonlinear, multi-variable dependent behaviour, precisely controlled conditions must be ensured for proper operation. C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3_2, © Springer-Verlag London Limited 2012

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PEM Fuel Cell Systems

2.2 Basics of PEM Fuel Cells Operation Catalytic reactions of hydrogen oxidation at the anode and oxygen reduction at the cathode produce an electric potential difference between electrodes, that can be used in an external circuit if the electrolyte allows ionic mass transport but isolates electrically both electrodes. On the anode side, the catalyst produces the dissociation of hydrogen molecules into protons (H+ ) and electrons (e− ). Protons cross the polymeric membrane, while electrons are forced to the external electric network. In the cathode’s surface the oxygen molecules react with electrons from the external circuit and protons from the membrane to produce water. In the process, the only byproduct is water, in vapour and liquid phases. The membrane must be properly humidified, because its protonic conductivity depends directly on its water content. To accomplish this, input gases are previously humidified. The amount of energy produced in the electrochemical process can be calculated from changes on the Gibbs free energy (gf ), that is, the difference between the Gibbs free energy of products and reactants. In the particular case of PEM Cells fuelled with pure hydrogen, the product is distilled water (H2 O), the reactants are hydrogen and oxygen, and then Δgf = (gf )prod − (gf )react = (gf )H2 O − (gf )H2 − (gf )O2

(2.1)

The Gibbs free energy represents the energy available for external work. The values of Δgf depend on the reactants temperatures and pressures according to the following expression:  1/2  PH2 PO2 Δgf = Δgfo − RT fc ln PH 2 O

(2.2)

where R is the universal constant for ideal gases, PH2 is the hydrogen partial pressure, PO2 is the oxygen partial pressure, PH2 O the water vapour partial pressure, and Δgfo the change in the process gf at a standard working pressure (1 bar), which in turn changes with the temperature of the fuel cell (Tfc ). Values of the Gibbs free energy for standard pressure at different temperatures are shown in Table 2.1 [17]. A negative Δgfo implies that the reaction releases energy ((gf )reac > (gf )prod ). If the electrochemical processes taking place in the cell were reversible, all the Gibbs free energy could be converted into electrical energy for the external circuit. In this ideal case, for each mol of hydrogen, two moles of electrons circulate by the electric circuit making an electric work (charge × voltage) −2F E, where F is the Faraday’s constant or, equivalently the charge of an electron mol (96485.309 C/mol), and E is the open circuit fuel cell voltage. This amount of electrical work is the net change in the Gibbs free energy: Δgf = −2F E

(2.3)

Therefore, the “reversible” voltage of a PEM cell is expressed as E=−

 1/2  Δgfo PH2 PO2 RT fc Δgf =− + ln 2F 2F 2F PH 2 O

(2.4)

2.2 Basics of PEM Fuel Cells Operation Table 2.1 Changes in Δgf for a standard 1 bar pressure

Water phase

15 Temperature °C

Δgfo (kJ/mol)

liquid

25

−273.2

liquid

80

−228.2

gaseous

80

−226.1

gaseous

100

−225.2

gaseous

200

−220.4

gaseous

400

−210.3

gaseous

600

−199.6

gaseous

800

−188.6

gaseous

1000

−177.4

Expression 2.4 is the so called Nernst voltage of a PEM fuel cell. In practice, however, the open circuit voltage is smaller than what Eq. (2.4) predicts. In fact, the term Δgfo /2F varies with temperature,and differs from its value E0 = 1.229 V at standard conditions (25 °C, 1 atm) according with the following expression:   Δgfo ΔS o = 1.229 + (Tfc − To ) (2.5) − 2F 2F where To is the standard temperature of reference (298 K), and ΔS o is the entropy change for the new operating conditions. Therefore, the last equation can be rewritten as   Δgfo ΔS o 298ΔS o (2.6) = 1.229 − + Tfc − 2F 2F 2F Using the standard thermodynamical relations regarding entropy changes [17], Eq. (2.4) can be written as   E = 1.229 − 0.85 × 10−3 (Tfc − 298) + 4.3 × 10−5 Tfc ln(PH2 ) + 1/2 ln(PO2 ) (2.7) Additionally, the cell voltage varies with electric load conditions. This is due to electric losses, which can be classified as activation, ohmic and concentration or diffusion losses. • Activation losses are important at low currents and reflect the fact that the cell requires a certain amount of energy to start electron circulation and create/break chemical bondings, both in the anode and the cathode [19]. This produces an important voltage drop at low current densities in both electrodes. It is worth noting that hydrogen oxidation at the anode is considerably faster than oxygen reduction at the cathode. Therefore, the dynamics of activation losses are always dominated by the cathode. The relationship between activation losses and current density can be described using the Tafel equation [18]   i (2.8) Vact = A ln i0

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PEM Fuel Cell Systems

where the constant A is higher for slow electrochemical reactions, and i0 is higher for fast reactions. The value of i0 can be considered as the current density from which the voltage drop becomes evident. It is called exchange current density and its typical values are in the 10−2 –10−8 A range. The Tafel equation is only valid for i > i0 . For a pure hydrogen fuel cell, A is given by A=

RT 2αF

(2.9)

The constant α is known as the charge transfer coefficient and represents the amount of electric energy applied that is harnessed in changing the rate of an electrochemical reaction. Its value depends on the reaction involved and the material of the electrode, and falls in the 0–1 range. • Ohmic losses are due to the resistance of the polymeric membrane to proton circulation, and also to the electrical resistance of electrodes and current collectors. Therefore, these losses are proportional to electric current in a wide operational range: Vohm = i · Rohm

(2.10)

The value of Rohm represents the internal resistance of the cell and has a strong dependency with the cell humidity and temperature levels. It depends on the membrane conductivity (σm ) and dry thickness (tm) according to the following expression [26, 30]: Rohm =

tm σm

(2.11)

For standard operating conditions, the following empirical expression is frequently used [30]: Vohm = (R0 − R1 λm )i

(2.12)

where λm is the membrane water content, defined as the number of water molecules per sulfonate group in the ionomer, and R0 and R1 are values to be experimentally determined. • Diffusion losses are the result of changes in the concentration of reactants as they are consumed by the electrochemical reaction. This effect is responsible of an important voltage drop at high current densities. The non-uniform conditions arising at the porous electrodes discourage this as a desirable operation zone. A semi-empirical expression for this effect is [15]   PO2 Vconc = me(n·i) + b ln (2.13) a where m and n are empirical coefficients with typical values close to 3 × 10−5 V and 8 cm2 /A, respectively [17]. The last term is included to take into account concentration losses due to low oxygen stoichiometry values.

2.3 Efficiency and Power Conversion

17

Fig. 2.1 Polarisation curve of a PEM fuel cell

Therefore, taking into account all the losses, the cell voltage can be written as Vfc = E − Vact − Vohm − Vconc

(2.14)

Replacing the values of the individual terms, the expression for (2.14) is     i RT PO2 ln (2.15) Vfc = E − − (R0 − R1 λm )i − me(n·i) + b · ln 2αF i0 a where α, i0 , R0 , R1 , m, n, b and a are empirical parameters that take into account the different polarisation effects and are adjusted for a specific fuel cell stack, without loss of generality. A systematic procedure for the determination of these constants is outlined in Chap. 5. The resulting polarisation curve of a typical PEM fuel cell is shown in Fig. 2.1. It can be seen that the open circuit voltage is close to 1 V. In applications requiring higher voltage and power, several cells can be combined in series/parallel configurations to fulfil load demands.

2.3 Efficiency and Power Conversion The efficiency of any energy conversion device is defined as the ratio between output and input useful energy. In the PEM fuel cell case, the available energy at the input of the device is the hydrogen’s enthalpy (measured as the amount of heat that can be converted to work). If all the Gibbs free energy at the cell output is converted to electric energy, the efficiency results in ηmax =

Δgf 237.2 = = 0.83 ΔH 286

(2.16)

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PEM Fuel Cell Systems

This is the maximum theoretical efficiency that a PEM fuel cell can reach at 25 °C. Using Faraday’s constant and considering that there are two electrons involved in the reaction, the efficiency of a PEM fuel cell can be expressed as a quotient of voltages: ηmax =

−Δgf −Δgf /2F = = 1.229/1.482 = 0.83 −ΔH −ΔH /2F

(2.17)

where −Δgf /2F = 1.229 V is the theoretical voltage of an open circuit cell, and −ΔH /2F = 1.482 V is the value of the thermoneutral voltage (that is, the resulting voltage if all the enthalpy of hydrogen is converted in electric energy). In this way, the efficiency of the cell at any condition can be obtained from the voltage at its output terminals (Vfc ): η=

Vfc 1.482

(2.18)

2.4 State-of-the-Art in PEM Fuel Cells Technology Current research efforts in PEM technology are mainly oriented to three basic directions: new components and materials, modelling of cell dynamics, and control. • Materials: evolution of new components and devices, capable of efficient operation under wider ranges of temperature, humidity and gas purity is required to broaden the spectrum of applications currently devised for PEM fuel cells. This includes aspects related to fabrication and operation of membranes and electrodes, manufacturing processes, design and characterisation of components. Research in membranes is oriented to thermostable polymers (polyetheretherketone, polysulphone, etc.) and composite membranes capable to operate at temperatures above 100 °C and lower humidity levels than the actual commercial membranes [31]. Regarding electrodes, most efforts are centred on reducing the amount of platinum required in catalysts and improving gas diffusion layers. Advances in the production, transport and storage technologies are also necessary, to make hydrogen an economically viable alternative [15]. • Cell Dynamics Modelling: better understanding of the processes involved in fuel cell operation, both at the membrane level and the auxiliary subsystems (compressors, line heaters, humidifiers, etc.), is helping engineers to develop dynamic models suitable for the design of more reliable, compact and efficient devices. This is also widening the spectrum of applications of PEM fuel cells, making them attractive alternatives in fields traditionally reserved to other types of cells, for instance, in stationary, high-power installations. On the other hand, accurate models of gas distribution and fluid dynamics inside cells are improving the predictability of electrochemical processes, with direct impact on design strategies. • From the automatic control perspective, efforts are conducted to the development of robust, nonlinear strategies capable to improve efficiency and reliability of PEM cells, avoiding permanent damage to membranes. Robustness is required

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19

to take into account the intrinsic uncertainty in the system model and to avoid expensive or cumbersome (even impossible) measurement of certain variables. Additionally, nonlinear control strategies usually allow wider operation ranges than “local” approaches that linearise the system around equilibrium points. For commercial success, it is also important to develop models and strategies to diagnose and predict failure situations.

2.5 Components and Associated Devices 2.5.1 Polymeric Membranes Membranes in a PEM fuel cell must have a relatively high protonic conductivity and also act as suitable mechanical barriers to avoid mixing of comburents and reacting gases. Additionally, they must be chemically stable for the entire operation range. Standard membranes are made of perfluorosulfonic acid (PFSA). This material is, essentially, a copolymer of tetrafluoroethylene (TFE) and several sulfonated perfluor monomers. The most popular commercial membrane is Nafion® , made by Dupont, which uses perfluoro-2-(2-fluorosulfonylethoxy) Propyl Vinyl Ether (PSEPVE). Similar materials are produced by several companies, such as Asahi Glass (Flemion® ), Asahi Chemical (Aciplex® ), Chlorine Engineers (C Membrane® ), etc. Dow Chemical has developed a composite membrane called GoreSelect® , using a Teflon-like material. Important features of membranes suited for PEM fuel cells are protonic conductivity, water transport properties, gas permeability, mechanical resistance and dimensional stability. These parameters strongly depend on the membrane water content. The water content is usually expressed as the weight rate of water and dry polymer or, alternatively, the rate between the number of water molecules per sulfonic groups present in the polymer. The maximum amount of water in a given membrane depends heavily on its previous preparation [11]. Regarding its critical role on protonic conduction, it is important to keep a proper membrane water content at all possible operating conditions. Several mechanisms affecting water transport are present in a PEM fuel cell, and their combined effects determine the amount of water present. Among them, the most relevant are the following: • Water generation on the cathode side, at a rate proportional to the electric current produced. • Electro-osmotic drag, produced by water molecules dragged by the proton flow from anode to cathode. • Diffusion, due to the water concentration gradient across the membrane. • Water permeability, due to the pressure difference between anode and cathode channels. In thin membranes, diffusion can be compensated by the drying effect produced by electro-osmotic drag at the anode. In thick membranes this drying effect can be

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Fig. 2.2 PEM fuel cell Membrane and cathode. Effective area is 50 cm2 . IRI (CSIC-UPC)

more important, especially at high current densities. An ideal electrolyte should be impermeable to reactant gases, to avoid gas mixing inside the cell. However, due to its porous structure, water content, and the solubility of hydrogen and oxygen in water, small amounts of reactant gases can pass through the membrane. It is worth noting that hydrogen has a permeability one order of magnitude higher than oxygen [4].

2.5.2 Electrodes Electrodes in PEM fuel cells are basically catalytic layers placed between the polymeric membrane and an electrically conducting substrate. This is where the electrochemical reactions take place. Given that gases, electrons and protons react in the surface of electrodes, their efficiency can be improved increasing its rugosity (effective area), reducing catalyst particle size and/or incorporating ionomeric material (a one-ion polymer) in the catalyst. The later can be accomplished by painting the electrode with a PFSA solution in a mix of alcohol and water, or simply mixing the ionomer and catalyst in the preparation of the catalytic layer. A 30 percent ionomer content in the catalytic layer is typical. The catalyst most usually employed in both electrodes of PEM fuel cells is platinum. In the early stages of PEM fuel cells development, important quantities of platinum were used (more than 28 mg/cm2 ). By the end of the 1990s this amount was reduced to 0.3–0.4 mg/cm2 . Considering that the effective area of catalyst is of paramount importance, it is crucial to achieve a fine dispersion of catalyst particles in the support material, usually carbon powder [29]. The combination of electrodes and polymeric membrane is known as Membraneelectrode assembly (MEA). There are basically two different approaches for its con-

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21

Fig. 2.3 Toray paper diffusion layers. IRI (CSIC-UPC)

struction. In the first one, the catalyst is deposited on a porous substrate called gas diffusion layer, which is typically a carbon fibre paper. These are then placed at both sides of the membrane using heat and pressure to ensure proper contact. In the second approach, the catalyst is deposited directly on the membrane, an arrangement known as catalysed membrane. A porous substrate is then added, resulting in a five-layer MEA (Fig. 2.2). Several techniques can be used for deposition of catalyst on diffusion layers and membranes (spreading, spraying, sputtering, painting, screen printing, decaling, electro-deposition, evaporative deposition, impregnation reduction, etc.), and many other proprietary approaches exist.

2.5.3 Gas Diffusion Layers The main purpose of gas diffusion layers is conducting and spreading reacting gases from bipolar plate channels to the MEA. They are typically made of porous materials. Regarding desirable properties of diffusion layers, the following aspects must be considered: • Porosity must be such that the flux of reactants and water is efficient. Note that both flows are in opposite directions. • Electrical and thermal conductivity must be high. The contact resistance or interface is typically dominant versus the volume conductivity. • Given that the catalyst is a discrete material (small particles), diffusion layer pore size cannot be excessively big. • They must provide proper mechanical support to the membrane. However, some degree of flexibility is required to provide good electrical contact. Although conflicting somewhat, these requirements are typically fulfilled with carbon fibre-based papers and cloths (Fig. 2.3). These materials are usually made hydrophobic to avoid flooding in the structure. To achieve this, diffusion layers are usually treated with polytetrafluorethylene (PTFE), a polymer similar to polyethylene. To improve electrical properties, a microporous layer made of carbon or

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Fig. 2.4 Sealing gasket placed over a current collector. IRI (CSIC-UPC)

graphite particles mixed with PTFE is added, resulting on pore sizes of 0.1–0.5 µm, which are much smaller than the carbon fibre pores (20–50 µm) [21].

2.5.4 Sealing Gaskets These components provide mechanical sealing among bipolar plates and diffusion layers. Their purpose is twofold: on one hand, minimising leaking of gases to the exterior of the cells and, secondly, avoiding the mixing of reactant gases near the catalyst areas (Fig. 2.4). However, it is worth noting that a certain amount of hydrogen is expected to pass through the membrane by diffusion. This can be computed from the number of cells, their width and type, effective area, partial pressures and working temperature. Sealing gaskets avoid direct combination of reactants, which in turn can produce irreversible damages to cell components [12]. Most sealing gaskets are made of silicone, neoprene or plastic polymers, with additional fibre materials to improve their mechanical properties.

2.5.5 Bipolar Plates Bipolar plates, also called collectors or separators, have many functions in a fuel cell system [4]. Among others, the most relevant are the following: • Electrical connection between individual cells of the stack. • Separation of gases among adjacent cells. Thus in PEM fuel cells they must be impermeable to H2 , O2 and N2 .

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Fig. 2.5 Graphite bipolar plate. IRI (CSIC-UPC)

• Structural support to the cell. Robustness and light weight are mandatory. • Efficient heat conduction. In addition, bipolar plates must be corrosion resistant, which make them expensive. In most PEM cells pH is between 2 and 3, and temperatures are in the 60–80 °C range. These environmental conditions discourage the use of traditional choices like aluminum, steel, titanium or nickel. Corrosion produces ions of metal which can diffuse through the membrane, affecting its ionic conductivity, reducing its lifetime and increasing its electrical resistance. For this reason, metallic plates are usually coated with non-metallic conductive materials, such as graphite, diamond carbon, conductive polymers, organic polymers, noble metals, metallic nitrides, tindoped indium, etc. Bipolar plates can also be made of thermoplastic materials, such as polypropylene, polyethylene and polyvinyldene fluoride. Thermosetting resins (phenolic, epoxy, etc.) are also used, with the addition of graphite and fibre reinforcing (Fig. 2.5). An important property of bipolar plates is their electrical conductivity. In graphite composites, typical values range between 50 and 200 S/cm. Although pure graphite has a conductivity of 680 S/cm, metallic plates have values an order of magnitude higher. Note that the overall conductivity of the cell is always lower than the material conductivity, due to the contact resistance among components. In Fig. 2.6, a schematic representation of a typical PEM fuel cells stack is presented, where all the components described above can be visualised.

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Fig. 2.6 PEM fuel cell stack scheme [13]

2.5.6 Auxiliary Devices 2.5.6.1 Air/Oxygen and Hydrogen Supplies In PEM fuel cell stacks, air can be supplied by means of a compressor, a fan or a compressed air source, although the latter is mostly used only in laboratory settings. Fans are customarily used in open-cathode designs working at ambient pressure. Air compressors give autonomy and flexibility to the system, allowing precise control of working pressures. Usually, both compressors and fans are electrically connected to the stack, becoming part of its losses or parasitic loads and thus reducing the amount of energy available to external loads. This results in a significant reduction of the overall performance of the system as an energy conversion device. At the time of this writing, PEM cells require a high-purity hydrogen supply (CO < 20 ppm). Hydrogen storage for PEM cells is a matter of active research, the main options being compressed gas, cryogenic liquids and solid metallic hydrides.

2.5.6.2 Water and Heat Management Water management in individual PEM cells and cell assemblies is a technological challenge. For proper operation, membranes must be totally saturated of vapour but, in order to ensure optimum performance, excess water must be efficiently removed (especially from the cathode line). Ionic conductivity in the polymeric membrane is directly related to its water content, which in turn affects the conversion efficiency

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25

Fig. 2.7 A PEM fuel cell stack, showing the series connection of individual MEAs. IRI (CSIC-UPC)

at each operation point. The amount of liquid water and membrane humidification levels can be modified by controlling the relative humidity of the reacting gases, as well as their individual pressures and temperatures. Temperature is as important as water content, and both magnitudes are closely related. PEM fuel cells are intended to operate at high power densities (>0.5 W/cm2 ), but, apart of new high-temperature membranes, the most widespread technology now prevents operation at temperatures above 100 °C. This small gap between operational and ambient temperatures makes it difficult to remove the 1.3 W of heat produced for each watt of electric power generated.

2.5.6.3 Electrical Conditioning Output power of a fuel cell is not regulated, and its stability is a relevant issue. The small voltage of each individual cell is heavily influenced by changes in electric current, partial gas pressures, reactants humidity level, gas speed and stoichiometry, temperature and membrane water content. According to Eqs. (2.6) and (2.7), the maximum voltage of a single PEM cell is close to 1.229 V. Higher power and voltage is obtained connecting individual cells in series/parallel configurations (Fig. 2.7). Then, an electronic conditioning system is necessary to meet load requirements. For instance, DC/DC converters can be used to extend the range of operating voltages or provide specific voltage values. The combination of a DC/AC stage and a transformer can be used to ensure electric isolation between load and stack, converting DC to an intermediate AC frequency and using a decoupling transformer and a rectifier. Usually an auxiliary power source is necessary to start the stack operation. In the special case of autonomous systems, power converters associated with a supervisory electronic control are required to ensure proper management of batteries or super-capacitors charge/discharge cycles.

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2.5.6.4 Humidification Proper humidification of polymeric membranes is an important matter. At moisture levels below recommended values, ionomer water content falls, reducing conductivity and adversely affecting the kinetics of reduction and oxidation reactions. Water content of an MEA depends on many factors and is closely related with operating conditions. For instance, in open circuit and also at low current densities, the small water production by oxygen reduction causes the MEA humidification levels to decrease, although the reacting gases are saturated with vapour. This is because water absorption in perfluoro-sulfonated membranes is lower when the membrane has been equilibrated with vapour instead of liquid water. Besides, at higher current densities, the electro-osmotic dragging of water through the membrane tends to dry the anode. An additional problem in this case is the eventual cathode flooding due to the excess of water produced by the electrochemical reactions. Under certain conditions, the amount of water produced by oxygen reduction at the cathode suffices to keep proper membrane hydration. However, this fact does not ensure an equilibrium point in water content. In most cases, a gas humidification system is required, at least in the cathode line, to control this important performance variable. Humidity control is a challenging task, because moisture levels in both channels are coupled and many perturbation (known and unknown) exist. Besides, humidification requirements can be in conflict with other control objectives, such as oxidant stoichiometry. For instance, in certain situations the optimal humidity level could be incompatible with the most efficient reactant flux level required to satisfy electric load demands. In such a case, a problem arises because it is difficult to suspend instantaneously the cathode humidification if the cells start to flood. On the other hand, it is also impossible to increase hydration levels at low currents when the cell starts to dry. Among the many techniques for gas humidification, the main alternatives are the following: • Gas bubbling. This is a method mostly used in laboratory applications for relatively low flows, being seldom used in commercial devices. It involves circulating air or hydrogen through a porous tube immersed in liquid water at a regulated temperature. The resulting bubbles provide a relatively big contact area between gas and water, allowing proper humidity transfer. Moisture level is controlled by varying water temperature. In a well-designed system, emerging gases are saturated of vapour at the water temperature. The main drawback of this approach is the presence of water droplets in the outcoming gas. This affects gas diffusion in the MEAs, reducing the overall efficiency. • Direct vapour injection. This is the most compact, efficient and easy to control method. A fine mist is injected in the gas stream using a small pump. An additional heat source is usually required to produce complete water evaporation if hot water enthalpy solely does not suffice. Moisture level is directly controlled by varying the amount of vapour injected [4].

2.6 Available PEM Fuel Cell Models in the Open Literature

27

• Water exchange through a permeable material. In this approach, gases are circulated on one side of a permeable membrane such as Nafion® . De-ionised, temperature-controlled water circulates on the other side of the membrane. During the process, a moisture gradient is established, which allows water transfer via diffusion through the membrane. The amount of water transfer is controlled by varying its temperature. • Enthalpy wheel. In this passive method the water and heat content of a gas can be transferred to other gas using a thermal process. It comprises a cylinder that interfaces two parallel conducts, where gases at different humidity and temperature circulate in opposite directions. The cylinder is filled with a permeable material, to provide a large interface area for energy transfer. As the wheel rotates between the ventilation and exhaust gas streams, it takes heat energy and releases it into the colder gas stream. The driving force behind the exchange is the thermal gradient between the opposing gas streams. The enthalpy exchange is accomplished by the use of desiccants, which transfer moisture by adsorption. This effect is predominately driven by the difference in the partial pressure of vapour between the gas streams. 2.5.6.5 Gas Heating Lines These devices are inserted in the gas path to provide temperature control, independently from the humidification process. Basically they are made of heating resistances, with protective stainless steel shielding. In PEM fuel cells, the main objective of the heating lines is keeping gas temperature high enough to avoid condensation inside cell channels. The comprehensive block diagram presented in Fig. 2.8 shows a typical laboratory set up of a PEM-based generation system.

2.6 Available PEM Fuel Cell Models in the Open Literature Accurate mathematical models of PEM cells behaviour are subject of current interest and active research. Dynamics of the electrochemical reactions that take place in the MEAs and the ancillary devices required for operation of the cell constitute a nonlinear, highly coupled multi-variable dynamic system. In control applications, the interest is focused on the development of reduced-order nonlinear dynamical models relating smooth vector fields. This is usually required in the design of many nonlinear control strategies. A suitable model should be capable of predicting the dynamic and stationary behaviour of the fuel cell in a wide range of operating conditions. Important variables to be taken into account are temperature and relative humidity of interacting gases, partial pressures (hydrogen, oxygen, nitrogen and water vapour), velocities of flows in the MEAs channels, electric currents and voltages, etc. Although many models of PEM fuel cells have been reported, only a few are suitable for their use in nonlinear control design.

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Fig. 2.8 PEM fuel-cell-based system scheme

2.6.1 Control Oriented Models PEM fuel cell modelling has been studied by several recognised authors and with different approaches [2, 20, 27, 28]. However, many of these models have not been experimentally validated, and there is still a lack of rigorous studies on parameters identification and their association with performance variables. On the other hand, there are few models and methodologies specifically oriented to control design. For example, the first models from the open literature, as the ones presented in [2] and [30], are essentially electrochemical characterisations based on empirical relationships that do not consider gas dynamics. More recently, works such as [8, 9, 27, 28] have presented extended equations, including gas dynamics and temperature effects inside the cells. However, only [28] and [9] have proposed fully analytical control oriented models. In [9], the model considers only three of the six states of a typical air supply subsystem, the humidification phase is not included, and characterisation of the other subsystems is only briefly described. In [28] it is presented probably the first proposal of a PEM fuel cell stack model, fully validated and especially developed for control engineering, and it is the basis of numerous works such as [5, 14, 27]. Changes in the liquid water and oxygen concentrations, as well as temperature, have significant effects on the PEM fuel cell performance and may even affect its

2.6 Available PEM Fuel Cell Models in the Open Literature

29

durability. All these variables exhibit a spatial dependence along anode and cathode channels, and therefore it is necessary to incorporate control mechanisms to keep them within their nominal values [20].

2.6.2 Control Objectives and Challenges PEM fuel cell systems have many advantages over traditional alternatives such as internal combustion engines. However, a number of technical issues must be faced to make them competitive. Among them, operational costs, lifetime and reliability are of vital importance. From the automatic control perspective, a PEM fuel cell is a nonlinear multipleinput–multiple-output (MIMO) dynamical system with strongly coupled internal variables, external perturbations and parameter uncertainties. Its normal operation is always associated with the generation and transport of liquid water, vapour and gas mixtures, spontaneous electrochemical reactions, exothermic processes and thermal conduction. By their very nature, they are sensitive to changes on operation conditions (power demand, partial pressures and relative humidity of reacting gases, temperatures, etc.) and also susceptible to potential damage. Three basic degradation mechanisms can be clearly distinguished: mechanical, thermal and electrochemical. Among the mechanical processes that produce a significant degradation, the cycles of humidification/drying play a decisive role, as they cause membrane expansion/shrinking. This leads to mechanical stress of the membranes and gaskets. Additionally, the thin polymeric membranes currently used make the system potentially vulnerable to abrupt pressure changes between channels, excessive temperature and low relative humidity conditions. Thermal degradation arises when considerable temperature variations occur in the stack, even within the range usually recommended by manufacturers of PEM fuel cells (60–80 °C). These thermal cycles, which in some cases can be extreme (e.g. those driven by cold starts and sudden high power demands), produce accumulative mechanical damages that affect the resistance contact between the membrane and electrodes, as well as the mechanical resistance, conductivity and permeability of the polymeric membranes. In fact, heat management is recognised as one of the most important issues in high-power PEM cells. This is due to the fact that under normal conditions, the cell produces as much heat energy as electricity. This means that in an automotive 100-kW fuel cell it is necessary to provide a structure capable of dissipating 100 kW of heat, which is particularly difficult if the operating temperature is only 80 °C. This operating temperature is a restriction imposed by the materials employed. Besides, temperature cannot be lower than 60 °C, to avoid water condensation inside cell channels, which would result in a voltage drop, caused by a reduction of gas mass transport to the membrane. Regarding the electrochemical mechanisms of degradation, it is important to state that the chemical reactions on the catalysts produce small amounts of peroxide radicals (HO) or hydroperoxide (HOO) that are responsible for the chemical degrada-

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tion of the membrane and its catalyst. The production of such radicals is accelerated when the fuel cells operate in open circuit or low humidity conditions. Another issue which deserves special attention is oxygen control. If the oxygen flow is too low, undesirable hot spots appear in the membrane, and output power decreases because of the lack of reactants, a situation called cathode starvation. On the other hand, if oxygen flow is too high, an excessive amount of water is pushed to the cathode outlet, which in turn results in membrane drying, which affects its ionic resistance. Besides, an increase of the air flow results in higher power demand to the compressor that supplies it, reducing the overall system performance. Thus, an efficient control system must be capable of regulating air flow properly, avoiding irreversible damages to the membrane and delivering enough oxygen to meet the electric power demand in a reliable and efficient way.

2.6.3 Recent Advances on PEM Fuel Cell Control During the last years, several control proposals have been made for PEM fuel-cellbased systems. Many examples can be mentioned. For instance, in [38] fuel cell power output is directly regulated by limiting its hydrogen feed. This is achieved using a PID control that varies the internal resistance of the membrane-electrode assembly in a self-draining fuel cell with the effluents connected to water reservoirs. In [35] cathode oxygen is regulated through a feedforward loop, and temperature is controlled using a proportional control, to ensure stack performance around an optimal operation point, where net power is maximised. In [23] a MIMO system is considered, with hydrogen and coolant as inputs and power density and temperature as outputs. Those variables were selected from a steady-state analysis using a relative gain array (RGA) technique. Two PID controllers were used, and simulation results suggest that the design can be accomplished from two decoupled SISO systems. In [34] and [3] predictive control approaches are considered, allowing improvements on the response of the air supply and efficiency optimisation in fuel cell stacks. In [28] a dynamic model of the air supply subsystem of a PEM cell is presented. Based on this model, an LQR controller was designed to decouple the air mass flow from cathode pressure. Reported results favourably compare against a standard PI controller. A substantial improvement was made in [37] where, from a linear identification of a fuel cell system, a H∞ controller was designed to regulate the cell output resistance and control output voltage, manipulating input gases flows. Alternatively, several proposals have been made regarding hybrid power generation configurations. In these systems, a fuel cell stack is usually combined with different energy storage devices, to provide a more reliable power source. Ultracapacitors [33] can be included to improve power transients using, for instance, PID controllers to regulate DC bus voltage. In [18] and [7] a hybrid system made from a fuel cell and batteries is proposed. In this approach, fuel cells are used for low power demands, while battery banks supply additional energy for higher power requirements, when the cell tends to reduce its output voltage. In such cases power

References

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converters can also be used to improve efficiency and extend operating ranges of the stack. For instance, in [36] resonant soft-switching techniques are used to adjust the output voltage of a 250-W PEM fuel cell. Most of the above approaches are based on linear models, providing interesting solutions that efficiently solve different control objectives [10, 27, 39]. However, an important issue such of robustness against parametric uncertainty and external perturbations have not been addressed in depth. Besides, the validity range of these controllers is local, and results do not extend to the entire operation range. Solutions to this problem can be found in the nonlinear control field, such as [1] and [24, 25], where strategies based on parametric cerebellar model articulation and exact linearisation were addressed, respectively. More recently, in [6] a methodology based on an energetic macroscopic representation of the fuel cell stack is proposed. Good global performances were attained with these nonlinear techniques, but, regretfully, their applicability in real systems is still limited because the algorithms demand considerable computational burden. Therefore, despite these advances in fuel cell controllers, it is evident that to meet the expected enhanced capabilities, a substantial R&D work is still necessary, and a control approach particularly suitable to cope with their challenging features is of high interest. At this point, Sliding-Mode control emerges as an especially apt technique to tackle the complex characteristics inherent to fuel cell systems (e.g. high nonlinear dynamics, inaccessible variables, model uncertainties and disturbances). In addition, the on-line computational burden of the resultant algorithms can be conveniently low. Promising results have been obtained with sliding mode controllers for fuel cell systems [16, 22, 32], strongly encouraging the prosecution of research in this direction.

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Chapter 3

Fundamentals of Sliding-Mode Control Design

3.1 Introduction This chapter provides an introduction to Variable Structure Control (VSC) theory and its extension to the so-called Sliding-Mode (SM) control. Note that the presentation is not intended as a comprehensive survey of the state-of-the-art in the field, but to merely supply the basic concepts on SM control required to understand the developments to come in this book. Readers well acquainted with this subject matter may omit this chapter. On the other hand, first-timers can use this material as a straightforward, but incomplete, introduction to the field of SM control, and are strongly encouraged to search for further and more substantial reading in the seminal works cited in the bibliography (useful introductory material could be, for instance, [5, 12, 14, 25, 26, 38, 44, 45]). The chapter is divided in two parts. In the first one, Sect. 3.3, a general analysis of the classic or first-order SM control is formulated, which is the natural background to the subsequent generalisation known as Higher-Order Sliding-Mode control (HOSM). This section is mainly based on the influential works [38, 42, 44] and, to a lesser extent, on contributions from a series of classic survey papers such as [12, 45]. In the second part, Sects. 3.4 and 3.5, a general study of systems operating in sets of arbitrary sliding-mode order is presented. This section outlines the fundamentals of Higher-Order Sliding-Mode control theory, particularly focusing on Second-Order Sliding-Mode (SOSM) controllers. To a great extent, this part has been inspired in the works and results from [5, 25, 31, 33].

3.2 Variable Structure Control Preliminaries The variable structure control and associated sliding modes were firstly proposed and developed by Stanislav Emelyanov and Vadim Utkin in the early 1950s in the Soviet Union [16, 43]. The most relevant feature of the SM control is its ability to C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3_3, © Springer-Verlag London Limited 2012

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generate robust control algorithms that are invariant under certain conditions. Briefly speaking, the concept of invariance indicates that the system remains completely insensitive to certain types of disturbances and uncertainties [13]. Since the 1990s, the control of systems subject to external disturbances and model uncertainty has been the focus of increasing interest. Among the different existing alternatives, the SM control has proven to be an attractive option to implement in systems electronically controlled, proving to be highly robust and even insensitive against certain system uncertainty and perturbations. The feasibility and benefits of SM control applied to electronically controlled actuators have been extensively demonstrated in the literature, such as [44]. In addition, the SM control allows a relatively simple design approach, even dealing with nonlinear systems, admitting a successful combination with other nonlinear control techniques such as energy shaping and model predictive control. As a result, the research and development of SM control design methods have been greatly accelerated, both in theoretical and practical fields [1, 6, 15, 36, 46]. One of the most distinctive aspects of the SM is the discontinuous nature of its control action. Its primary function consists in performing a switching between two different structures in order to get a desired new dynamics in the system, known as sliding-mode dynamics. This feature allows the system to have an enhanced performance, including insensitivity to parametric uncertainties and rejection to disturbances that verify the so-called matching condition [13, 38]. When the concept of parametric uncertainties is considered, it is referred to both external and internal uncertainties in the parameters as the product of the process of model reduction used in control design [14, 42]. However, a great deal of the success to fulfil the control objectives depends on the capability of the sliding-mode controller design to reduce chattering. The term chattering describes the phenomenon of finite-frequency, finite-amplitude oscillations appearing in many sliding-mode implementations. These oscillations are caused by the high-frequency switching of a sliding-mode controller under practical (non-ideal) operating conditions, such as unmodelled dynamics in the closed-loop or finite switching frequency [9, 22, 23, 44]. A successful alternative to reduce this undesired phenomenon, currently addressed by many control researchers and engineers, is to use the so-called HigherOrder Sliding-Mode control. In this case, from the definition of a continuous control action, the HOSM generalises the notion of sliding surface or manifold while keeping the main advantages of the original approach of SM for Lipschitz continuous uncertainty/perturbations. In particular, there are several promising results related to Second-Order Sliding-Mode control, existing several algorithms that solve the robust stabilisation of nonlinear uncertain systems, while guaranteeing a finite-time convergence of the sliding variable [5, 24, 29, 34].

3.3 Fundamentals of Sliding-Mode Control The SM control is a strategy based on output feedback and a high-frequency switching control action which, in ideal conditions, is infinite. Essentially, this high-speed

3.3 Fundamentals of Sliding-Mode Control

37

control law can lead the system trajectories to a subspace of the state space (commonly associated to a sliding surface or manifold). If a system is forced to constrain its evolution on a given manifold, the static relationships result in a dynamical behaviour determined by the design parameters and equations that define the surface [38]. On average, the controlled dynamics may be considered as ideally constrained to the surface while adopting all its desirable geometrical features. Thus, making an appropriate design of the sliding surface (i.e. embedding the control objectives into the control function that gives rise to such manifold), it is possible to achieve conventional control goals such as global stability, optimisation, tracking, regulation, etc. In the sequel, the basics of the theory of classical sliding-mode control are introduced, focusing on Single-Input Single-Output (SISO) systems. Note that in most sections of this chapter, the possible explicit dependence on time of the dynamical system has been omitted for the sake of clarity and economy of notation. In the present approach, this compacted notation can be used without loss of generality, provided that in the case of a non-autonomous system, it could be rewritten as autonomous by treating t as an additional dependent variable, with its trivial evolution given by the fictitious equation t˙ = 1 (obviously, at the expense of increasing the dimension by one).

3.3.1 Diffeomorphisms, Lie Derivative and Relative Degree Firstly, it is useful to review some mathematical tools and procedures that will be necessary later. Let a control affine nonlinear system be given by x˙ = f (x) + g(x)u y = h(x)

(3.1)

with x ∈ X ⊂ Rn , f : Rn → Rn and g : Rn → Rn smooth vector fields (infinitely differentiable) with g(x) = 0, h(x) smooth scalar field and u : Rn → R possibly discontinuous. These systems are linear in the control, so they are called control affine systems or analytical linear systems. A diffeomorphism is defined as a coordinate transformation of the form z = φ(x) with φ : Rn → Rn vector field with inverse φ −1 . In particular, we only consider transformations such that φ and φ −1 are C n (i.e. with n continuous derivatives). This last condition ensures that the transformed system preserves the original system structure. After making the proposed change of coordinates, the dynamical system (3.1) looks as follows: ∂φ ∂φ ∂φ ˙ z˙ = φ(x) = x˙ = f (x) + g(x)u (3.2) ∂x ∂x ∂x ∂φ(x) ∂φ(x) ∂φ(x) Note that ∂φ ∂x = [ ∂x1 ∂x2 · · · ∂xn ] gives the direction of the gradient vector of φ(x), ∇φ(x). So the system (3.1) can be written in terms of the new variable z:

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z˙ = f˜(z) + g(z)u ˜ ˜ y = h(z)

(3.3)

 ∂φ  f˜(z) = f (x) −1 x=φ (z) ∂x  ∂φ  g(x) −1 g(z) ˜ = x=φ (z) ∂x ˜h(z) = h(x)x=φ −1 (z)

(3.4)

where

To simplify the notation, it is necessary to define the concept of directional derivative or Lie derivative [40], which is expressed as (Lf h)(x) = Lf h(x) : Rn → R

(3.5)

and represents the derivative of a scalar field h(x) : Rn → R in the direction of a vector field f (x) : Rn → Rn , Lf h(x) =

∂h f (x) ∂x

(3.6)

Lf is a first-order differential operator, while the composition Lf ◦Lg , which is usually written as Lf Lg , is a second-order operator. Moreover, the directional derivative can be applied recursively: Lkf h(x) =

 ∂  k−1 Lf h(x) f (x) ∂x

(3.7)

In this way, a compact notation for the derivatives of scalar functions in the direction of vector fields is obtained. Either in the direction of a single vector field (f ) or more (f and g): Lg Lf h(x) =

 ∂  Lf h(x) g(x) ∂x

(3.8)

Finally, assuming a smooth output h(x) of system (3.1), the relative degree of h(x) at the vicinity of a given point x is defined as the smallest positive integer r, if one exists, with the property that Lg Lif h = 0

∀0 ≤ i ≤ r − 2

(3.9)

and Lg Lr−1 f h = 0

(3.10)

Therefore, a system output h(x) with relative degree r implies, in a simplified way, that u explicitly appears for the first time at the rth time derivative of h(x). In short, r gives an idea about how directly the control influences the output.

3.3 Fundamentals of Sliding-Mode Control

39

3.3.2 First-Order Sliding Mode Consider the nonlinear dynamical system (3.1), with control action u : Rm → R (possibly discontinuous), f and g smooth vector fields with g(x) = 0 ∀x ∈ X. Let s be defined as a smooth constraint function s : X → R, designed according to the desired control objectives (i.e. the specifications are fulfilled when s is constrained ∂s to zero), with gradient ∇s = ∂x non-null on X [38]. Then the set   (3.11) S = x ∈ X ⊂ Rn : s(x) = 0 defines a locally regular manifold in X (of dimension n − 1 in the case of a SISO system), called sliding manifold or, simply, switching surface. This order reduction feature is a characteristic of SM control systems (first and HOSM) and indicates that the subspace on which the sliding movements occur have “non-zero co-dimension”, meaning that after reaching the sliding regime, the trajectories of the system will remain within a subspace of lower dimension than the space generated by n states. The results obtained below are of a local nature, restricted to an open neighbourhood of X ⊂ Rn , having a non-empty intersection with the sliding manifold S [16, 28, 42]. In order to attain the sliding motion in such manifold, a variable structure control law can be proposed by imposing a discontinuous control action u, which takes one of two possible feedback values, depending on the sign of s(x). For example,  + u (x) if s(x) > 0 with u+ = u− (3.12) u= u− (x) if s(x) < 0 The upper and lower levels of u (u+ (x) and u− (x), respectively) are smooth functions of x. Moreover, without loss of generality, it can be assumed that u+ (x) > u− (x) holds locally in X. Note that if u+ (x) > u− (x) for any point x, then the inequality holds for every x, given that the functions are smooth and do not intersect. Suppose that, as a result of the control law (3.12), the constraint function locally satisfies the following inequalities in the neighbourhood of S :  s˙ (x) < 0 if s(x) > 0 (3.13) s˙ (x) > 0 if s(x) < 0 Under these conditions, the system will reach the sliding manifold S and thereafter will remain confined in a vicinity of S (see Fig. 3.1). Then, it is considered that a sliding regime is established on S whenever (3.13) holds. Using the notation of the directional derivative, s˙ (x) can be expressed as follows: s˙ (x) = Lf +gu s = Lf s + Lg s · u

(3.14)

Note that the output s(x) must have relative degree 1 with respect to u, i.e. Lg s = 0, to ensure that the discontinuous control action is able to influence the sign of s˙ (x). Expression (3.13) can also be written as follows: ⎧ ⎨ lim Lf +gu+ s < 0 s→+0 (3.15) ⎩ lim Lf +gu− s > 0 s→−0

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Fig. 3.1 Sliding manifold and system trajectories

meaning that the rate of change of the constraint or SM function s(x), evaluated in the direction of the control field, is such that a crossing of the surface is guaranteed from each side of the surface, by using the switching law (3.12). This can be graphically interpreted with the help of Fig. 3.1, analysing the projection of the controlled field f + gu onto the gradient vector ∇s at both sides of S . To conclude this subsection, a succinct final remark regarding the SM control finite reaching time is pertinent. Note then that the explicit condition (3.13) can be condensed as s˙ (x)s(x) < 0. From this it is simple to understand that, to achieve finite reaching time, the control law (3.12) must be designed to fulfil the previous inequality, but in a more strict way, that is according to the scalar sufficient condition s˙ (x)s(x) < κ|s(x)| with κ > 0 (or, similarly, s˙ (x) sign s(x) < κ). This means that the system should always be moving toward the switching surface with non-zero speed. This can be straightforwardly proven by taking V = 12 s 2 (x) as a Lyapunov function.

3.3.3 Equivalent Control Regularisation Method. Ideal Sliding Dynamics From a methodological and systematic point of view, it is convenient to develop a regularisation method for deriving the sliding-mode equations for system (3.1). Assuming that the state vector is in the manifold S (s(x) = 0) and the sliding mode occurs with the state trajectories confined to this manifold for t > 0, one way to define the ideal sliding mode is using the so-called equivalent control method [44]. Since the motion in the sliding mode implies s(x) = 0 for t > 0, it may be assumed that ds/dt = s˙ = 0 as well. Hence, in addition to s(x) = 0, the time derivative s˙ (x) = 0 may be used to characterise the state trajectories during the sliding mode.

3.3 Fundamentals of Sliding-Mode Control

41

In summary, the equivalent control action is defined by the following invariance conditions on the switching manifold S [38]:  s(x) = 0 (3.16) s˙ (x) = Lf s + Lg s · ueq = 0 where ueq (x) is a smooth control law called equivalent control that makes S a local invariant manifold of system (3.1). Therefore, the equivalent control ueq (x) can be obtained from Eq. (3.16):  Lf s  ueq (x) = − (3.17) Lg s s(x)=0 Thus, once s = 0 is attained, ueq (x) would provide the continuous control action required to maintain the system confined in the sliding surface. The ideal sliding-mode dynamics, i.e. the closed-loop dynamics on the manifold S , is obtained by substituting ueq for u into (3.1):  Lf s  (3.18) x˙ = f (x) + g(x)ueq |s(x)=0 = f (x) − g(x) Lg s s(x)=0 Note that the state variables are related by the algebraic equation s(x) = 0, reducing the order of the closed-loop system dynamics to n − 1. Substituting the Lie derivative and operating in (3.18), we have

  ∂s −1 ∂s x˙ = I − g g f (x) = Ψ (x)f (x) (3.19) ∂x ∂x Evaluated in s(x) = 0, (3.19) gives an idealised version of the motions occurring on the sliding manifold S , constituting an “average” description of the trajectories of system (3.1) controlled with the VSC law (3.12). The geometrical representation presented in Fig. 3.2 can be of help for a better understanding. In accordance with ueq being the control action that makes the system remain on S , the vector Ψf = f + gueq must lie in Tx , the tangent plane to S (i.e. normal to the gradient ∇s, as can be seen in Fig. 3.2). Mathematically this is expressed as Ψ (x)f (x) ∈ ker(∇s) ≡ Tx

(3.20)

Consequently, the matrix Ψ (x) can be considered as a projection operator that, applied to the vector f (x), projects it onto the plane tangent to surface S at the point x. To conclude, it is of interest to briefly consider the effect of projector Ψ over any vector collinear with g(x). Let Λ be a general vector of arbitrary amplitude, possibly a function of x, such that Λ ∈ span(g): Λ(x) = g(x)μ(x)

with μ(x) : Rn → R

(3.21)

Note then that the application of the operator Ψ (x) to this vector projects Λ to the origin. In fact,  

∂s −1 ∂s g g(x)μ(x) = 0 (3.22) Ψ (x)Λ(x) = I − g ∂x ∂x

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Fundamentals of Sliding-Mode Control Design

Fig. 3.2 Sliding surface and detail of the vector fields and projections

The cancellation of expression (3.22) can be interpreted in Fig. 3.2 as follows: the operator Ψ (x) projects any vector in the direction of g(x) onto the tangent subspace of S . Hence, the projection of any vector that belongs to span(g) would result in just a point on Tx .

3.3.4 Existence Conditions for the First-Order Sliding Regime 3.3.4.1 Existence of Equivalent Control It can be stated that the equivalent control is well defined if ueq exists and is uniquely determined from the invariance conditions (3.16) [38]. Lemma 3.1 The equivalent control is well defined if and only if the following condition is satisfied locally in S : ∂s(x) g(x) = 0 (3.23) ∂x This condition is known as “transversality condition” and may be inferred from (3.14). The proof of the lemma can be found in [38]. Lg s(x) =

Geometrically, this lemma states that the vector field g cannot be tangential to the sliding manifold (S : g ∈ / ker(∇s)); otherwise it could not force the system to cross the surface. The transversality condition represents just a necessary condition for the existence of a first-order sliding mode.

3.3 Fundamentals of Sliding-Mode Control

43

3.3.4.2 Necessary Conditions for the Existence of a First-Order Sliding Regime Based on the transversality condition, the following necessary condition for the existence of a sliding regime can be stated. Lemma 3.2 A necessary condition for the existence of a local sliding mode in S is that the equivalent control action ueq (x) must be well defined. Indeed, if ueq is not well defined, i.e. Lg s = 0 at some point, the existence conditions of the sliding mode (3.15) cannot be satisfied simultaneously. Lemma 3.3 Assume, without loss of generality, that u+ (x) > u− (x). Then the following condition is necessary for the existence of a sliding regime on S : Lg s(x) =

∂s g(x) = ∇sg(x) < 0 ∂x

(3.24)

The proof, direct from (3.15) and (3.16), is given in [38] and can be easily inferred from Fig. 3.2 by analysing the sign of the projection of g(x) onto ∇s(x).

3.3.4.3 Necessary and Sufficient Condition for the Existence of a First-Order Sliding Regime A necessary and sufficient condition for the local existence of a sliding mode in S is that, for x ∈ S , u− (x) < ueq (x) < u+ (x)

(3.25)

This condition can also be proved from (3.15) and (3.16) [38]. Then, ueq can be interpreted as the averaged control signal resulting from the implementation of the maximum and minimum control actions, with an infinitesimal duty cycle resolution (in ideal sliding mode). However, in practice, several model imperfections and finite switching frequency make the state oscillate in a vicinity of the manifold [44].

3.3.4.4 Robustness of the First-Order SM The behaviour of SM controlled systems under the effect of disturbances is discussed briefly. To this end, consider system (3.1) perturbed as follows: x˙ = f (x) + g(x)u + ζ (x)

(3.26)

with ζ (x) a vector of lumped perturbations that may take into account parametric perturbations of the nominal drift field or unstructured external disturbances

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Fig. 3.3 Sliding surface and detail of the vector fields with perturbations

[38]. The vector ζ (x) can be uniquely decomposed into two components, one in the span(g), g(x)υ(x), and the other, η(x), onto the tangent plane Tx (see Fig. 3.3): ζ (x) = g(x)υ(x) + η(x)

(3.27)

It is said that the disturbances that belong to span(g) satisfy the matching condition, and the SM control is not merely robust to them but exhibits a strong invariance property. Effectively, as it can be observed in Fig. 3.3, if u+ and u− are strong enough, the component g(x)υ(x) can be completely annihilated by the control, simply generating a new infinitesimal duty cycle and, consequently, a new equivalent control for the disturbed system (ueqp (x) = ueq (x) − υ(x)). In addition, the undisturbed or nominal sliding dynamics suffers no modifications. On the other hand, it can be appreciated (in Fig. 3.3) that the tangential component of the disturbances, η(x), cannot be rejected. However, it does not compromise the local existence of the sliding motion, but definitely influences the ideal sliding dynamics. In accordance with this analysis, it can be stated that a necessary and sufficient condition for the local existence of a sliding mode in the perturbed system is u− (x) < ueqp (x) = ueq (x) − υ(x) < u+ (x) A detailed demonstration is provided in [38].

(3.28)

3.3 Fundamentals of Sliding-Mode Control

45

3.3.5 Extension to Nonlinear Systems Non-affine in Control Consider a generic system described by the following differential equation: x˙ = F (x, u)

(3.29)

Using the same SM function s(x), the controlled system can be again decomposed into two subsystems or structures, depending on whether s(x) > 0 or s(x) < 0:  F (x, u+ ) = F + if s(x) > 0 x˙ = F (x, u) = (3.30) F (x, u− ) = F − if s(x) < 0 Along the trajectories of the system, the dynamics of s(x) has the following expression: ⎡ ⎤  x˙1 ∂s ∂s ∂s ∂s ⎢ ⎥ s˙ (x) = x˙1 + x˙2 + · · · = · · · ⎣ x˙2 ⎦ = LF s(x) (3.31) ∂x1 ∂x2 ∂x1 ∂x2 .. . In the same way as in system (3.30), in the time derivative of the SM function two cases can be distinguished: if s > 0 → s˙ = LF + s(x) if s < 0 → s˙ = LF − s(x)

(3.32)

As the trajectory has to converge to the manifold, when s(x) > 0, the states should move towards s(x) = 0 (i.e. s˙ (x) < 0, so s(x) decreases), and conversely in the reciprocal case. This means that the establishment of the sliding mode on s(x) = 0 is fulfilled with a condition similar to (3.13): if s > 0 → s˙ = LF + s(x) < 0 if s < 0 → s˙ = LF − s(x) > 0

(3.33)

3.3.6 Filippov Regularisation Method Besides the equivalent control method presented in Sect. 3.3, at this point it is of interest to introduce another regularisation method, also capable of dealing with discontinuous systems. In particular, the underlying concept behind this method will be of use in the higher-order SM control strategies to come. Recall that conventional theory of differential equations is limited to continuous state functions, hence when dealing with discontinuous systems, it does not answer even fundamental questions, such as the existence and uniqueness of the solution. Strictly speaking, most conventional methods require the right-hand side of the differential equation (3.29) to satisfy the Lipschitz condition   F (x1 ) − F (x2 ) < Lx1 − x2  (3.34)

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Fig. 3.4 Hysteresis of the switching device

with L being some positive value, known as Lipschitz constant, for any x1 and x2 . This condition implies that the function does not grow faster than some linear function [44]. Nevertheless, this is not the case for discontinuous functions if x1 and x2 are close to a discontinuity point. So as previously stated, in situations where conventional methods are not applicable, the common approach is to employ different methods of regularisation like the equivalent control method proposed in [42]. Another useful regularisation method usually applied to general nonlinear systems as (3.29) is the so-called Filippov method [20]. This procedure consists of considering that the discontinuous control is implemented with a switching device with small imperfections. In particular, if a hysteresis loop of width 2Δ is considered, then the state trajectories oscillate in a Δ-vicinity of the switching surface when the control takes one of the two extreme values, u+ (x) or u− (x) (see Fig. 3.4). Δ is considered small enough, so the state velocities F + = F (x, u+ ) and F − = F (x, u− ) are assumed to be constant for some point x on the surface s(x) = 0 within a short time interval [t, t + Δt]. Let the time interval Δt consist of two sets of intervals Δt1 and Δt2 such that Δt = Δt1 + Δt2 , u = u+ during Δt1 and u = u− during Δt2 . Then, the increment of the state vector once Δt is elapsed is found as Δx = F + Δt1 + F − Δt2

(3.35)

and the average velocity of the state vector is given by the convex average of the velocity vectors: Δx x¯˙ = = μF + + (1 − μ)F − (3.36) Δt where the convex average factor μ = Δt1 /Δt can be understood as the percentage of time that the control takes the value u+ , while (1 − μ) is the percentage corresponding to u− , with μ belonging to the closed set [0, 1]. Now, the procedure to get the state vector movement x˙ is to make Δt tend to zero. Nevertheless, this limit is intrinsic to the assumption that the state velocity vector, or equivalently the vector field F (x), is constant within the time interval Δt. Then, for the Filippov regularisation method, the convex expression x˙ = μF + + (1 − μ)F −

(3.37)

3.3 Fundamentals of Sliding-Mode Control

47

Fig. 3.5 Filippov’s regularisation method

represents the motion during the first-order sliding mode (see the convex closure in Fig. 3.5, as a graphical interpretation of the Filippov method). Accordingly, since the trajectories during the sliding mode are on the manifold s(x) = 0, the following equation holds:   s˙ = ∇s(x)x˙ = ∇s(x) μF + + (1 − μ)F − = 0 (3.38) so the parameter μ should take a value that allows the state velocity of the system (3.37) to lie on the tangent plane (see Fig. 3.5). From (3.38) it can be easily inferred that such value of μ must be μ=

∇s(x)F − . ∇s(x)[F − − F + ]

(3.39)

Note For control affine systems, the resultant sliding equations derived from the Filippov regularisation method are the same as those obtained from the Utkin equivalent control.

3.3.7 Discontinuous Control Action in Classic Sliding-Mode Control. Chattering Problem One of the main drawbacks of the first-order sliding-mode control in certain applications is the direct use of discontinuous control actions. In actual implementations, the discontinuous control law, together with unmodelled dynamics and finite switching frequency, may produce fast oscillations in the outputs of the system. This effect is known as “chattering” phenomenon. During the mid-1980s, the following three main approaches to reduce chattering in sliding-mode controlled systems were proposed [8]: • The use of a saturation control instead of the discontinuous action [10, 39]. This well-established approach allows the control to be continuous, restraining the system dynamics not strictly onto the sliding manifold, but within a thin boundary layer of the manifold. This method ensures the convergence to the boundary layer, whose size is defined by the slope of the saturation linear region.

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• The observer-based approach [9, 44]. This method allows bypassing the plant dynamics by the chattering loop. This approach successfully reduces the problem of robust control to the problem of exact robust estimation. However, in some applications it can be sensitive to the plant uncertainties, due to the mismatch between the observer and plant dynamics [45]. • The Higher-Order Sliding-Mode approach (HOSM) [18, 29]. This method allows the finite-time convergence of the sliding variable and its derivatives. This approach was actively developed since the 1990s [2, 4, 5, 29, 32, 37], not only providing chattering attenuation, but also robust control of plants of relative degree one and higher. Theoretically, an r-order sliding mode would totally suppress the chattering phenomenon in the model of the system (but not in the actual system) when the relative degree of the model of the plant (including actuators and sensors) is r. Yet, no model can fully account for parasitic dynamics, and, consequently, the chattering effect cannot be totally avoided. Nevertheless, theoretical results in HOSM, especially Second-Order Sliding-Mode algorithms, have been successfully proven in practice, encouraging the progress of the research activities. Then, it is of natural interest the study of sliding-mode control alternatives that, smoothing the control action, reduce the chattering effects and avoid unnecessary requirements on the actuators. This is particularly relevant in fuel cell control, as there are mechanical actuators involved that may suffer when exposed to control actions of high frequency and amplitude. It should be noted that when using Higher-Order Sliding Modes, it is not possible to maintain the invariance properties against matched disturbances as in the original approach. However, different control schemes that guarantee robust stability of the system can be achieved, satisfying the condition s(x) = 0 (and even zeroing higherorder derivatives of s(x)) in finite time [18, 25]. In the sequel, a brief introduction to Higher-Order Sliding-Mode control applied to uncertain nonlinear systems is presented. Then, Second-Order Sliding-Mode control and in particular three different algorithms are analysed in detail.

3.4 Some General Concepts on Higher-Order Sliding Modes As discussed in Sect. 3.2, first-order sliding-mode control has certain properties that make it particularly attractive to apply to uncertain nonlinear systems. Among them, it can be highlighted finite convergence to the surface, system order reduction and robustness against certain disturbances. In this context, Higher-Order Sliding-Mode control will inherit some of these properties. This control approach generalises the idea of first-order sliding mode, by acting on the higher-order derivatives of the constraint function s(x), instead of influencing the first derivative (as in (3.14)). Keeping the main advantages of the original approach, the HOSM control works with continuous action over s˙ (x), relegating the discontinuous control to operate on the higher derivatives of s(x). This weakens the effect of chattering in the output, providing greater accuracy in realisation. Additionally, in some applications (namely,

3.4 Some General Concepts on Higher-Order Sliding Modes

49

plants with relative degree 1 with respect to s), the resultant physical control input to the plant is continuous, contributing to the longer service life of certain actuators. A significant number of these controller proposals can be found in [2, 7, 18, 21, 25, 29, 30, 35]. An important concept in HOSM is the notion of sliding order. If the goal is to maintain a constraint given by s(x) = 0, the sliding order is defined as the number of continuous time derivatives of s(x) (including the zero-order one) in the vicinity of a sliding point. With these considerations, a sliding mode of order r is determined by the following equalities: s = s˙ = s¨ = · · · = s (r−1) = 0

(3.40)

Expression (3.40) represents an r-dimensional condition in the dynamic system, which implies an order reduction of r (that is, (3.40) specifies r algebraic equations that bond the state variables).

3.4.1 Definition of Differential Inclusion As in first-order sliding-mode control, the HOSM scheme forces a movement on a set of discontinuity, demanding an approach to the problem capable to deal with differential equations with right-hand side single-valued, but discontinuous, functions. Such an approach can be found in the Filippov concepts introduced in Sect. 3.3.6. The basic idea behind Filippov’s method was not to focus on the value of the vector function precisely at the discontinuity point, but on its values in the point’s immediate neighbourhood. Then, the function at the point is replaced with an average function, taken from a set generated by the convex combination of the values at each side of the discontinuity point. This replacement can be interpreted as including or covering the discontinuous right-hand side single-valued function with a more comprehensive set-valued function (as would be the convex closure presented in the case treated in Sect. 3.3.6). This idea of inclusion, or better expressed differential inclusion, will be of help when designing SOSM controllers for dynamical system with uncertainties. To better formalise this mathematical concept, consider a general differential equation of the form z˙ = v(z, t)

(3.41)

where the generic variable z ∈ Rn , and v(z, t) is a piecewise-continuous singlevalued function in a domain G with some points of discontinuity in a set M of measure zero. Note that in the framework of the SM control problem the generic variable z could be particularised to be a suitable variable of the dynamical system under control (for instance, x or, through a diffeomorphism, s, s˙ and any appropriate internal variables), while the discontinuity set M could be the sliding manifold. Next, for each point (z, t) of the domain G, a set-valued function V (z, t) in an n-dimensional space must be considered. Note that just as the single-valued function takes a point in its domain into a single point (direction) in another space, the

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set-valued function take a point in its domain into a set of points (directions) in another space [11]. In this particular case, for points (z, t) where function v(z, t) is continuous, the set V (z, t) trivially consists of one point (direction) which coincides with the single value of v(z, t) at this point. On the other hand, if (z, t) is a point of discontinuity of function v(z, t), then V (z, t) comprises a set of directions rather than a single specific one. Now, in accordance with Filippov’s definition, the discontinuous differential equation (3.41), can be formally replaced by an equivalent differential inclusion of the form z˙ ∈ V (z, t)

(3.42)

The expression above, is called a Filippov differential inclusion if the set V (z, t) is non-empty, closed, convex, locally bounded and upper-semicontinuous. In this way, V (z, t) can cover the situation in which the state derivative belongs to a set of directions, not to a single one. In the simplest case, i.e. when v(z, t) is continuous almost everywhere, V (z, t) is the convex closure of the set of all possible limits of v(t, zcont ) as zcont → z, while zcont are continuity points of v(z, t). Note that this definition verifies the description of V (z, t) previously given. When zcont approaches a continuity point, the limits converge to a single value, so, as expected, V (z, t) effectively coincides with the continuous value of v(z, t). Conversely, when zcont approaches a discontinuity point, limits are different, and V (z, t) comprises a set of directions. It can be stated, then, that a solution z(t) of the differential equation (3.41) is understood as a solution in the Filippov sense, if it is an absolute continuous function in an interval and satisfies the differential inclusion (3.42) almost everywhere on such interval [19, 20]. Summarising, the Filippov definition replaces the discontinuous differential equation (3.41) by the differential inclusion (3.42). Removing sets of zero measure (discontinuity points) from the values taken by v(z, t) corresponds to purposely ignoring possible misbehaviour of the right-hand side in (3.41) on small sets.

3.4.2 Sliding Modes on Manifolds The notion of sliding mode manifold acquired with the first-order SM can be extended to HOSM. The progression that generates the successive sliding manifolds can be described as follows. Let S be the smooth manifold defined from a smooth function s(x) (see Eq. (3.11)). The set of points x for which the set of possible velocities entirely lies in the subspace Tx tangent to S is defined as a second-order sliding set with respect to S (recall that in a first-order SM the set of possible velocities of the system does not lie in Tx , but intersects it. See Fig. 3.2). The former concept means that once S is reached, the Filippov solutions of Eq. (3.41) fall within the tangent space of the manifold S . This set of points is denoted as S2 . Assuming that S2 can be considered as a manifold smooth enough, the same construction can be performed for S2 , calling S3 to the corresponding set of second-order sliding solutions with respect to S2 or third-order sliding set with respect to S . Thus,

3.4 Some General Concepts on Higher-Order Sliding Modes

51

continuing this way, one can find sliding sets of any order [25]. Summarising, it is said that there is an rth-order sliding mode on the manifold S in a neighbourhood of an rth-order sliding point x ∈ Sr if in a neighbourhood of this point x, the set Sr is an integral set, and this means that the set of trajectories is understood in the Filippov sense.

3.4.3 Sliding Modes and Constraint Functions. Regularity Condition 3.4.3.1 Definition of Regularity Condition At this point, it is useful to briefly introduce the definition of the regularity condition and its relation with other concepts, such as the normal form of nonlinear systems. Hence, reconsider the constraint given by s(x) = 0, where s : Rn → R is a function smooth enough. Assume also that the time derivatives of s(x), i.e. s˙ , s¨, . . . , s (r−1) exist and are single-valued functions of x (which is not trivial in discontinuous dynamical systems). Recall that the discontinuity does not appear in the first r − 1 derivatives of the constraint function s, or analogously, s is an output of relative degree r with respect to the discontinuous input, according with (3.9) and (3.10). When these assumptions hold, the sliding set of order r will be unequivocally determined by Eqs. (3.40), implying that the reduced system dynamics has order n − r. Definition 3.1 Consider the non-empty rth-order sliding set (3.40) and assume that it is a set locally integrable in the sense of Filippov (i.e. consisting of Filippov trajectories of the discontinuous dynamical system). Then, the corresponding motion that satisfies (3.40) is called rth-order sliding mode with respect to the constraint function s. To show the relationship of this definition with other control definitions, consider a manifold S given by the equation s(x) = 0. Suppose that s, s˙ , s¨, . . . , s (r−1) are smooth functions of x and   rank ∇s, ∇ s˙ , ∇ s¨, . . . , ∇s (r−1) = r (3.43) holds locally. Then, since all Si , i = 1, . . . , r − 1, are smooth manifolds, Sr is a differentiable manifold determined by (3.40). Recall that the rank of a set of vectors indicates the dimension of the subspace they define. Equation (3.43), together with the requirement that the corresponding time derivatives of s are smooth functions of x, is referred to as the “sliding regularity condition” [25, 31]. This is a useful definition because if condition (3.43) is reached, new local coordinates y1 = s can be taken, and the system can be described through the following set of equations:

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Fundamentals of Sliding-Mode Control Design

⎧ y˙1 = y2 ⎪ ⎪ ⎪ ⎪ y˙2 = y3 ⎪ ⎪ ⎨ .. . ⎪ ⎪ ⎪ ⎪ y˙r = Φ(y, ξ ) ⎪ ⎪ ⎩ ξ˙ = Ψ (y, ξ ) with ξ ∈ Rn−r

(3.44)

Note that this is similar to the normal form of nonlinear systems; the only difference is that in the normal form y˙r = a(y) + b(y)u [31]. Additional Remark It is sometimes mentioned that the higher-order sliding modes differ depending on the number of total derivatives of s which are extinguished when reaching the manifold S . However, this number cannot be considered as a feature of the HOSM, since formally all orders of derivatives are cancelled at S [4]. The most important feature of a sliding mode is the number of successive continuous derivatives of s in the neighbourhood of the manifold. In other words, the value of r is taken from computing the first discontinuous or non-existent time derivative of s. The sliding order r is understood in this sense.

3.4.3.2 Connection with Other Well-Known Results in Control Theory Let the control affine nonlinear system (3.1) be recalled as ⎧ ⎨ x˙ = f (x) + g(x)u s = s(x) ∈ R ⎩ u∈R

(3.45)

with f , g and s sufficiently smooth vector functions. Assuming that the output s(x) has relative degree r, according to (3.9) and (3.10), this means that in the neighbourhood of a given point, Lg s = Lg Lf s = · · · = Lg Lr−2 f s = 0;

Lg Lr−1 f s = 0

(3.46)

hence s (i) = Lif s for i = 1, . . . , r − 1, and the regularity condition (3.43) is automatically satisfied. For this reason, a direct analogy between the relative degree notion and the regularity condition of sliding mode can be established. In general terms, it can be stated that the regularity condition (3.43) means that the relative degree of system output with respect to the discontinuity is at least r. Similarly, the notion of rth-order sliding-mode dynamics is analogous to the zero dynamics concept defined in [27]. The nominal stability of the controlled system can be guaranteed if the stability of (3.44) holds when y = 0, i.e. when the reduced system ξ˙ = Ψ (y, ξ ) with ξ ∈ Rn−r is stable.

3.4 Some General Concepts on Higher-Order Sliding Modes

53

3.4.4 Closing Comments on Higher-Order Sliding Modes in Control Systems 3.4.4.1 An Observation Regarding the Accuracy of Real Sliding Modes It is necessary to clarify that when referring to a system operating in sliding mode, it can be both ideal (nominal) sliding, which takes place when the switching imperfections are neglected and the restriction is maintained accurately, or real sliding, which occurs when the switching imperfections are taken into account. In the latter case the restriction can be satisfied only approximately. The “quality” of the control design is related to the sliding accuracy. It is worth mentioning that in practice, there are no design methods that can ideally maintain the desired constraint s(x) = 0. Therefore, there is a need to introduce some sort of comparison between different control systems. Further details and proofs can be found in [25]. Strictly speaking, any ideal sliding mode should be understood as the limit of movements when the imperfections disappear and the switching frequency tends to infinity. Therefore, if ε is taken as a measure of these imperfections, the accuracy of any sliding-mode control design can be characterised by its asymptotic behaviour as ε → 0 [29]. For example, to obtain a real sliding mode of order r (with discrete switching), it is required to satisfy an order r of ideal sliding (at infinite switching frequency). So, most of the real second-order algorithms come from discretising ideal second-order algorithms [17, 29]. A special discrete switching algorithm that provides second-order real sliding was presented in [41]. Another example of a second-order real sliding controller is the “Drift Algorithm” [29]. Moreover, a real third-order sliding controller that only uses measures of s has been presented in [3].

3.4.4.2 HOSM Convergence Time Prior to entering the section devoted to the design of specific Second-Order SlidingMode (SOSM) algorithms, a final general comment concerning the convergence time is of interest. Convergence in HOSM can be either asymptotic or in finite time. Examples of asymptotically stable sliding-mode algorithms of arbitrary order are well known in the literature [24]. On the other hand, fewer examples can be cited for r-sliding controllers that converge in finite time. For instance, these can be found for r = 1 (which is trivial), for r = 2 [2, 5, 29] and for r = 3 [24, 31]. Despite the fact that some arbitrary-order sliding-mode controllers of finite-time convergence have already been presented [33], its implementation is not yet widespread.

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3.5 Design of Second-Order Sliding-Mode Controllers 3.5.1 Second-Order Sliding Generalised Problem This final section is focused on the design of SISO second-order sliding-mode controllers, aiming to explain the specific algorithms that will be used in this book. To this end, consider the uncertain nonlinear system (initially, not necessarily affine in the control), explicitly defined as ⎧ ⎨ x˙ = F (x, u, t) (3.47) s = s(x, t) ∈ R ⎩ u = U (x, t) ∈ R with x ∈ Rn , u the single control input, and F and s smooth functions. Note that in this section the possible direct dependence on t has been explicitly manifested in system (3.47), in order to better explain the subsequent SOSM design procedure. As always, the ultimate control objective would be steering the sliding output s to zero. However, the SOSM approach enables not only that s = 0 and its time derivative s˙ = 0, but also finite time stabilisation of both, as long as s is of relative degree 1 or 2 with respect to the control input u. Moreover, in the former case the physical control action synthesised by the SOSM algorithm is continuous. The SOSM design procedure depends on the bounds of certain functions that constitute the second time derivative of the sliding output s. Hence, as a first step, s is differentiated twice, and the following general expressions are derived: ∂ ∂ s(x, t) + s(x, t)F (x, u, t) ∂t ∂x

(3.48)

∂ ∂ ∂ s˙ (x, t) + s˙ (x, t)F (x, u, t) + s˙ (x, t)u(t) ˙ ∂t ∂x ∂u

(3.49)

s˙ = s¨ =

Then, two different cases will be addressed, depending on the relative degree of s with respect to input u. Systems with relative degree 1 and relative degree 2 will be considered, respectively. Case 1 Systems with relative degree 1. In relative degree 1 systems, u appears in s˙ , thus in the expression of s¨ the derivative u˙ is explicitly presented in affine form, as in (3.49). Therefore expression (3.49) can be given as follows: s¨ = ϕ(x, u, t) + γ (x, u, t)u(t) ˙

(3.50)

with ϕ(x, u, t) and γ (x, u, t) uncertain but uniformly bounded functions in a bounded domain. In order to specify the control problem, the following conditions must be assumed [29]:

3.5 Design of Second-Order Sliding-Mode Controllers

55

1. There are bounds Γm and ΓM such that within the region |s(x, t)| < s0 the following inequality holds for all t, x ∈ X , u ∈ U : 0 < Γm ≤ γ (x, u, t) =

∂ s˙ (x, t) ≤ ΓM ∂u

(3.51)

The constant s0 defines a region of operation around the sliding manifold, where the bounds are valid. Note that, eventually, an appropriate control action has to be included in the controller, in order to attract the system into this validity region. 2. There is also a bound Φ such that, within the region |s(x, t)| < s0 ,     ϕ(x, u, t) = ∂ s˙ (x, t) + ∂ s˙ (x, t).F (x, u, t) ≤ Φ (3.52)   ∂t ∂x for all t, x ∈ X , u ∈ U . With these bounds at hand, the following differential inclusion can be proposed to replace (3.50) [33]: s¨ ∈ [−Φ, Φ] + [Γm , ΓM ]u˙

(3.53)

This is a very important relation when considering robustness. As it will be demonstrated in the following subsection, many SOSM controllers ensure finitetime stabilisation of both s(x, t) = 0 and s˙ (x, t) = 0, not merely for the nominal original system, but for (3.53). Since inclusion (3.53) does not remember whether or not the original system (3.47) is perturbed (it will include both cases, as far as perturbations had been computed into the bounds), then such a controller will be obviously robust with respect to any perturbation or uncertainty existing in (3.47) and, consequently, translated to (3.50). Case 2 Systems with relative degree 2. In relative degree 2 systems, u is not present in s˙ , hence the derivative u˙ does not appear in s¨ (i.e. the third term of (3.49) is null), resulting in s¨ =

∂ ∂ s˙ (x, t) + s˙ (x, t)F (x, u, t) ∂t ∂x

(3.54)

In this case, we will limit the analysis to affine in the control nonlinear systems of the form: x˙ = F (x, u, t) = f (x, t) + g(x, t)u(t)

(3.55)

Therefore, expressions (3.54) and (3.55) can be combined as follows: s¨ = ϕ (x, t) + γ (x, t)u(t)

(3.56)

Once again ϕ (x, t) and γ (x, t) are uncertain but uniformly bounded functions in a bounded domain.

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Fig. 3.6 System trajectory on the plane (s, s˙ )

As in the relative degree 1 case, analogous conditions must be assumed: 1. There are bounds Γm and ΓM such that, within the region |s(x, t)| < s0 , the following inequality holds for all t, x ∈ X , u ∈ U : 0 < Γm ≤ γ (x, t) ≤ ΓM and s0 defines the region of validity around the sliding manifold. 2. There is also a bound Φ such that, within the region |s(x, t)| < s0 ,   ϕ (x, t) ≤ Φ

(3.57)

(3.58)

for all t, x ∈ X , u ∈ U . In this case, the following differential inclusion can be proposed to replace (3.56):     (3.59) s¨ ∈ −Φ , Φ + Γm , ΓM u Robustness considerations similar to the prior case can be established.

3.5.2 Solution of the Control Problem. SOSM Algorithms There is a wide variety of proposals for second-order sliding-mode controllers that provide solutions to the aforementioned problem of finite-time convergence and robust stability. In this section three of the most widely-known algorithms will be reviewed. Two of them, the Twisting and Sub-Optimal algorithms, are devised for relative degree 2 systems, and the other one, the Super Twisting algorithm, is for relative degree 1 systems.

3.5.2.1 Twisting Algorithm This is one of the first SOSM mode algorithms, and it is primarily intended for relative degree 2 systems. Once the initialisation phase is elapsed (i.e. the region

3.5 Design of Second-Order Sliding-Mode Controllers

57

Fig. 3.7 Expanded system

|s| < s0 is reached by using an appropriate extra control action), it generates system trajectories that encircle the origin of the plane (s, s˙ ) an infinite number of times (Fig. 3.6), converging to it in finite time [29]. Consider system (3.47) under the conditions of Case 2 of Sect. 3.5.1; these are of relative degree 2 with respect to s and affine in the control form, as in (3.55). Additionally, the bounds defined in Case 2, conditions 1 and 2, exist and are available for the designer. Then, the Twisting algorithm can be written as u = −r1 sign(s) − r2 sign(˙s )

(3.60)

where r1 and r2 are the controller parameters, to be tuned based on the system bounds. It will be demonstrated in the sequel that if they simultaneously satisfy the conditions Γm (r1

r1 > r2 > 0 + r2 ) − Φ > ΓM (r1 − r2 ) + Φ

(3.61)

Γm (r1 − r2 ) > Φ

then the Twisting controller (3.60) generates a second-order sliding mode that attracts the trajectories of the system to s = s˙ = 0 in finite time [25]. Twisting Adaptation to Relative Degree 1 Systems Prior to proving the convergence of the Twisting algorithm, it would be useful to explain how to apply it to relative degree 1 systems. It is rather obvious that algorithms intended for relative degree 2 can straightforwardly be adapted for its implementation on relative degree 1 systems. The procedure would be as simple as to artificially increase the relative degree to 2 by expanding the system. To this end, an integrator is incorporated prior to input u, and a new artificial input ν is created (see Fig. 3.7) in accordance with the differential equation below: u˙ = ν(t)

(3.62)

Now, u has become a new internal state variable (xi = u) of the following expanded system with artificial or auxiliary input ν: x˙e = Fe (xe , ν, t) = fe (xe , t) + ge (xe , t)ν(t)

(3.63)

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with xe = [x T xi ]T ∈ Rn+1 , ν the new input of the expanded system, and Fe , fe and ge smooth functions. The expanded system (3.63) is then of relative degree 2 with respect to s and, consequently, fulfils the conditions required for the design of the Twisting algorithm. Following the steps previously described, a Twisting SOSM control signal would be synthesised for the relative degree 2 input: ν = u˙ = −r1 sign(s) − r2 sign(˙s )

(3.64)

To conclude, it is pertinent to remark on the bounds required for the design. It is straightforward to infer, according with (3.56), that the expression of s¨ in terms of the expanded system becomes s¨ = ϕ (xe , t) + γ (xe , t)ν(t)

(3.65)

In turn, if the expression of s¨ of the original system, i.e. the expression for relative degree 1 systems (3.50), is also written in terms of the expanded system variables, it results in s¨ = ϕ(x, u = xi , t) + γ (x, u = xi , t)u(t) ˙ = ϕ(x, xi , t) + γ (x, xi , t)ν(t)

(3.66)

Comparing (3.65) and (3.66), it can be appreciated that in this case, in which the relative degree 2 system comes from an artificially expanded relative degree 1 system, the functions to be bounded are exactly the same to those of the latter (i.e. ϕ (xe , t) = ϕ(x, u, t) and γ (xe , t) = γ (x, u, t)). So, in this particular case, this would allow the use of the same bounds for designing either relative degree 1 or 2 SOSM controllers. In this way, the computations involved in the design procedure of this type of systems are significantly alleviated. Proof of Convergence of the Twisting Algorithm Auxiliary System To analyse the convergence of the algorithm, consider a suitable auxiliary system, namely a double integrator s¨ = u. This system has been chosen given that it will behave as a “majorant” of the system under consideration (3.56), i.e. it will give the worst case for the region of convergence in the phase plane (s, s˙ ). Then, controlling the auxiliary system with a control law of the Twisting form u = −b1 sign(s) − b2 sign(˙s ), yields s¨ =

d s˙ d s˙ = s˙ = −b1 sign(s) − b2 sign(˙s ) dt ds

(3.67)

with constant parameters b1 and b2 satisfying b1 > b2 > 0. Now, the task in this first stage will be to show that the solutions of (3.67) converge to the origin (˙s = s = 0) in finite time. This result will be used at the end of the subsection to prove the convergence of the uncertain system under study.

3.5 Design of Second-Order Sliding-Mode Controllers

59

Fig. 3.8 Vector field

Analysing the differential equation it can be noticed that ⎧ −b1 −b2 if s > 0, s˙ > 0 ⎪ s˙ ⎪ ⎪ ⎪ −b +b d s˙ ⎨ 1s˙ 2 if s > 0, s˙ < 0 = b −b 1 2 ds ⎪ if s < 0, s˙ > 0 ⎪ ⎪ s˙ ⎪ ⎩ b1 +b2 if s < 0, s˙ < 0 s˙

(3.68)

Then, the system vector field will take the shape shown in Fig. 3.8. Taking the initial conditions P1 = (0, s˙0 ), the solution of (3.67) for the first quadrant can be found as follows:  s  s˙ s˙ d s˙ = −(b1 + b2 )ds (3.69) 0

s1

1 2 s˙ = (b1 + b2 )(s1 − s) 2 s˙2 2(b1 + b2 ) For the second quadrant, a similar expression can be found: ⇒

s = s1 −

(3.70)

(3.71)

s˙2 (3.72) 2(b1 − b2 ) Taking two fixed points of this trajectory, P1 = (0, s˙0 ) and P2 = (0, s˙1 ) (see Fig. 3.9), it can be concluded that s = s1 −

s˙02 = s1 2(b1 + b2 ) s˙12 = s1 2(b1 − b2 )

(3.73)

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Fundamentals of Sliding-Mode Control Design

Fig. 3.9 Phase trajectory of the Twisting algorithm

|˙s1 | = |˙s0 |

⇒



b1 − b2 =q <1 b1 + b2

(3.74)

Extending this reasoning to the rest of the trajectory, it is verified that the following inequality always holds: |˙si+1 | =q <1 |˙si |

(3.75)

Therefore, the algorithm converges to the origin. Time of Convergence Considering t1+ as the time that takes the auxiliary system trajectories to go from the point s˙0 to s1 . Integrating Eq. (3.67) in the first quadrant, it results in  t  s˙ d s˙ = −(b1 + b2 )dt (3.76) s˙0

⇒ Knowing that

s˙ (t1+ ) = 0,

0

s˙ (t) = −(b1 + b2 )t + s˙0

(3.77)

we get that t1+ =

s˙0 b1 + b2

(3.78)

The computation of t1− (time that takes the system to go from s1 to s˙1 ) is similar to the previous step: s˙ (t) = s1 −

s˙ 2 [−(b1 − b2 )t]2 (b1 − b2 )t 2 = s1 − = s1 − 2(b1 − b2 ) 2(b1 − b2 ) 2

(3.79)

3.5 Design of Second-Order Sliding-Mode Controllers

61

Taking into account that s(t1− ) = 0 and using (3.73), we can state that   2s 1 1 = t1− = s˙0 b1 − b2 (b1 − b2 )(b1 + b2 )

(3.80)

This means that the time interval t1 of the trajectory s˙0 s1 s˙1 is t = t1+ + t1− = η˙s0 where

(3.81)

 1 (b1 − b2 )(b1 + b2 )

1 + η= b1 + b2

(3.82)

More generally, we can state that ti = η|˙si−1 | = ηq i−1 s˙0

(3.83)

So the total convergence time of the auxiliary system is given by T=

∞ 

ti =

i=1

∞ 

η|˙si−1 | =

i=1

∞ 

ηq i−1 s˙0 =

i=1

η˙s0 1−q

(3.84)

In the sequel, it will be demonstrated that, with an adequate selection of b1 and b2 , the auxiliary system effectively acts as a majorant of (3.56) and (3.84) is a bound for its convergence time. Convergence of the Uncertain System Once the convergence of the auxiliary system has been demonstrated, the last stage consists of using this result to prove the convergence of the relative degree 2 uncertain system under study. To this end, reconsider the differential equation s¨ = ϕ (x, t) + γ (x, t)u with the aforementioned bounds   ϕ (x, t) ≤ Φ ,

0 ≤ Γm ≤ γ (x, t) ≤ ΓM

(3.85)

(3.86)

and the differential inclusion     s¨ ∈ −Φ , Φ + Γm , ΓM u

(3.87)

provided that the solutions of (3.85) are understood in the Filippov sense. The Twisting law (3.60) with r1 and r2 simultaneously satisfying Γm (r1

r1 > r2 > 0 + r2 ) − Φ > ΓM (r1 − r2 ) + Φ

is used to control the system.

Γm (r1

− r2

) > Φ

(3.88)

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Fig. 3.10 Twisting algorithm and majorant trajectories

Then, to prove that the auxiliary system (3.67) is a majorant for (3.85) controlled with the Twisting algorithm, the following appropriate selection for constants b1 > b2 > 0 is proposed: b1 + b2 = Γm (r1 + r2 ) − Φ ,

b1 − b2 = ΓM (r1 − r2 ) + Φ

(3.89)

With this parameter selection, trajectories s˙0 s1 s˙1 and s˙0 sM s˙M in Fig. 3.10 correspond to (3.85) and (3.67), respectively, with common initial conditions s = 0 and s˙ = s˙0 > 0 for t = 0. In particular, s˙0 sM s˙M is the so-called “majorant curve” of the system. One of its points, P (sp , s˙p ), in the first quadrant is considered. If the trajectories of the system (3.85) pass through this point, controlled by (3.60), the system velocity would have the coordinates (˙sp , s¨p ). Note that the horizontal component of velocity (˙sp ) is the y-coordinate of the point P (positive value). On the other hand, using the control (3.60) in system (3.87) on the first quadrant, the following will be fulfilled:     s¨ ∈ −Φ , Φ + Γm (−r1 − r2 ), ΓM (−r1 − r2 ) (3.90) Moreover, due to the fact that inequalities (3.88) hold, the vertical component is kept within the following limits: −Φ − ΓM (r1 + r2 ) ≤ s¨ ≤ Φ − Γm (r1 + r2 ) < 0

(3.91)

This implies that the velocity of system (3.85), (3.60) at P (sp s˙p ), will always “point” to the interior of the region bounded by the axes s = 0, s˙ = 0 and the surrounding curve (3.67), (3.89). Then, the trajectory of system (3.85), (3.60) intersects + . the axis s˙ = 0 in a point s1 ≤ sM in finite time t1+ ≤ tM

3.5 Design of Second-Order Sliding-Mode Controllers

63

Considering the trajectories s1 s˙1 and sM s˙M of systems (3.85), (3.60) and (3.67) in the second quadrant (s ≥ 0, s˙ ≤ 0), a point C(sC , s˙C ) where similar properties are verified can be taken. As it can be inferred from inequality (3.88), in this quadrant the module of the vertical component of velocity vector (3.85), (3.60) is smaller than the surrounding system (3.67), while the horizontal component is s˙C : −Φ − ΓM (r1 − r2 ) ≤ s¨ ≤ Φ − Γm (r1 − r2 ) < 0

(3.92)

This means that the trajectories of system (3.85), (3.60) are inside the surrounding system. On the other hand, the time required to cover the vertical segment (0, s˙1 ) is the same, but the surrounding trajectory must also cover the vertical segment − s˙1 , s˙M . This is the reason why t1− ≤ tM , where t1− is the time it takes the system to − cover the trajectory s1 s˙1 , and tM is that for sM s˙M . Then, the trajectory of system (3.85), (3.60) intersects s = 0 at s˙1 . Moreover, |˙s1 | ≤ |˙sM |, and therefore the evolution time of the uncertain system is bounded by the majorant as t1 ≤ tM .

3.5.2.2 Super Twisting Algorithm The Super Twisting algorithm, one of the most widely used algorithms of the family, is particularly intended for systems with relative degree 1 [25, 29]. Highly suitable for real implementation, with a proper choice of parameters, this algorithm converges in finite time after describing a trajectory similar to the one of the Twisting algorithm (Fig. 3.11). The most distinctive features of the Super Twisting algorithm are the aforementioned direct applicability to relative degree 1 systems, the synthesis of continuous control actions and the absence of a measurement of s˙ in the control law. This makes it more immune to output measurement noise and possible errors in the estimation of s˙ . The control action u(t) of the Super Twisting algorithm is composed of two continuous terms, even though the first one is given by the integral of a discontinuous action. Once the validity region |s| < s0 is attained, with the help of an appropriate reaching control action, the Super Twisting control is given by u(t) = u1 (t) + u2 (t)

(3.93)

with u˙ 1 (t) = −α sign(s) u2 (t) = −λ|s|ρ sign(s)

(3.94)

where α > 0, λ > 0 and ρ ∈ (0, 1/2] are the parameters of the controller. The restrictions for their design are based on the bounds defined in Case 1 and conditions 1 and 2 in Sect. 3.5.1. The following are sufficient conditions for convergence in finite

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Fig. 3.11 Phase trajectory of the Super Twisting algorithm

time to the sliding manifold [25, 29]: Φ Γm 2 (Γm α + Φ)2 λ2 > 2 Γm (Γm α − Φ) α>

(3.95) if ρ = 0.5

Note that if ρ = 1 and α and λ/α are large enough, it can be even proven that there will be a stable second-order sliding mode. In this case, |s| + |˙s | would tend to zero with exponential upper and lower bounds. Super Twisting Adaptation to Relative Degree 2 Systems In similar way that an algorithm intended for relative degree 2 can be adapted for its implementation on relative degree 1 systems, the Super Twisting algorithm can be adapted for application to relative degree 2 systems. In this case a differentiator should be incorporated into the system. This is not at all a specific subject matter of this book, and therefore it will not be addressed here. Nevertheless, to give an idea of the procedure, consider that the artificial input ν is created in this case such that u = ν˙ , i.e. differentiating the new input instead of integrating it (as it was done in the Twisting analogous case). Then, substituting ν˙ for u in the expression of s¨ for the relative degree 2 system (3.55), it would become affine with respect to the derivative of the input, specifically, the artificial input ν. Displaying that form, the differential inclusion proposed for relative degree 1 is applicable to (3.56), hence the standard Super Twisting algorithm could be used to synthesise the control signal for the artificial input ν. The actual implementation method and the proper use of differentiators are much more elaborate and, as previously stated, far exceed the scope of this book. The interested reader is strongly encouraged to read the specialised literature on this topic (e.g., [4, 32]).

3.5 Design of Second-Order Sliding-Mode Controllers

65

Proof of Convergence of the Super Twisting Algorithm Consider now the controlled process (3.47) with output s of relative degree 1 with respect to u. As was said, if the control problem satisfies conditions (3.51), (3.52), the system solutions can be understood in the sense of Filippov, and Eq. (3.50) can be replaced by the differential inclusion (3.53). Then, substituting (3.94) into the differential inclusion (3.53), the overall system performance and the majorant curves that limit the evolution of the system trajectories can be evaluated. Consider the case where |s| < |s0 | and the trajectory of the system is within the first quadrant (s > 0 and s˙ > 0):   (3.96) s¨ ∈ [−Φ, Φ] + [Γm , ΓM ] −λρs ρ−1 s˙ − α Due to the fact that in this quadrant s˙ > 0, in order to decrease the value of s˙ and ensure that the system trajectories cross s˙ = 0, the condition s¨ < 0 must be achieved in the entire quadrant. The worst possible scenario is when ϕ(x, t) = Φ (maximum positive value that the vector field ϕ can take) and γ (x, t) = Γm (lower dominance of control in the system dynamics):   (3.97) s¨ = Φ + Γm −λρs ρ−1 s˙ − α < 0 To keep the sign of s¨ negative, even when λρs ρ−1 s˙ → 0, the relation Φ − Γm α < 0 must be satisfied. This imposes the first convergence condition of the algorithm: α>

Φ Γm

(3.98)

To improve clarity in the analysis, the second convergence condition of the algorithm will be obtained with the help of the following change of notation: s = z1 and s˙ = z2 . Then, the dynamics of the planar system in the first quadrant is given by  z˙ 1 = z2   (3.99) ρ−1 z˙ 2 = ϕ + γ −λρz1 z2 − α Considering the worst-case scenario for this quadrant (ϕ = Φ and γ = Γm ), system (3.99) will have a limit trajectory in the solutions of the following planar system:  z˙ 1 = z2   (3.100) z˙ 2 = Φ + Γm λρ(−z1 )ρ−1 z2 + α The analytical solutions of this nonlinear system cannot be straightforwardly found, but numeric tools can be used to predict the solution from an initial condition. Figure 3.12 presents the results of a numerical evaluation of the limit trajectories of system (3.99) with a set of parameters arbitrarily chosen as an example (Φ = 10, Γm = 1, ΓM = 1.7, α = 50, λ = 15 and ρ = 0.5). The case of slower decrease of s¨ takes place as λρs ρ−1 s˙ → 0. Thus, in the first quadrant, the majorant curve is governed by the following expression: s¨ = Φ − Γm α

(3.101)

Integrating successively this equation, the following general expression with initial conditions (0, s˙0 ) can be found, which represent the majorant curve of the sys-

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Fig. 3.12 Example of convergence of the Super Twisting algorithm

tem trajectories in the first quadrant: s = (˙s − s˙0 )2

1 s˙0 + (˙s − s˙0 ) 2K K

(3.102)

with K = Φ − Γm α < 0

(3.103)

In the example of Fig. 3.12, the external line shows the majorant curve of the system trajectories (3.99), with the corresponding parameters listed above. It is important to note that, although the trajectories can be closer to the majorant curve at certain points, the majorant curve does not represent a possible path of the system, which would imply λρs ρ−1 s˙ = 0 for all (s, s˙ ). Using an analogous analysis on the fourth quadrant, similar equations for the system dynamics can be found:  z˙ 1 = z2   (3.104) z˙ 2 = ϕ + γ λρ(−z1 )ρ−1 z2 + α In this case, there are no restrictions on the sign of s¨ , but it can be known from the continuity of trajectories that, in the first section, s¨ < 0. Then, from an isocline analysis, the area where s¨ is zero can be determined. Taking s¨ = 0 in the worst-case scenario of the quadrant (ϕ = −Φ and γ = Γm ), we obtain that s˙ = −βs 1−ρ

(3.105)

with β=

Φ + Γm α Γm λρ

(3.106)

3.5 Design of Second-Order Sliding-Mode Controllers

67

Therefore, when crossing the curve (3.105), (3.106), the system trajectories will do it with s¨ = 0, yielding a local minimum at s˙ and a change of sign in s¨ (see Fig. 3.12). Since in this quadrant s˙ < 0, the function s(x) decreases monotonically. This ensures an intersection point with the curve (3.105), (3.106). To define a majorant curve of the system, the minimum possible value of s˙ can be analysed, which is determined by the value taken by (3.105), (3.106) when s = sμ (see Fig. 3.12):

1−ρ s˙02 Φ + Γm α 1−ρ − s˙μ = −βsμ = − (3.107) Γm λρ 2(Φ − Γm α) Finally, to ensure the algorithm convergence to the origin, it is necessary to satisfy that |˙sμ | < |˙s0 |:

1−ρ s˙02 |˙sμ | Φ + Γm α 1 <1 (3.108) = |˙s0 | Γm λρ 2(Γm α − Φ) s˙0 If ρ = 0.5, this condition is reduced to λ2 ≥

2 (Γm α + Φ)2 Γm2 (Γm α − Φ)

(3.109)

For the case where |s| > s0 , a simplified expression similar to (3.102) can be found, which ensures that the controlled system trajectories arrive at the area |s| < s0 in finite time.

3.5.2.3 Sub-Optimal Algorithm Similarly to the Twisting algorithm, this SOSM controller is primarily intended for relative degree 2 systems. It was developed as a Sub-Optimal feedback implementation of the classical time-optimal control for the uncertain double integrator problem [2]. In this case, the system trajectories on the plane (s, s˙ ) are confined within limit parabolic arcs that include the origin. So both twisting and leaping behaviours are possible. A most important feature is that the coordinates of the successive trajectory intersection with axis s decrease in geometric progression (see Fig. 3.13). Given that the Sub-Optimal algorithm is a controller primarily intended for relative degree 2 systems, we consider similar conditions to the ones established for Twisting algorithm (i.e., conditions of Case 2 of Sect. 3.5.1). Then, once the region of validity is reached, the Sub-Optimal algorithm is defined by the following control law: u = −α(t)U sign(s − βsM )  1 if (s − βsM )sM ≥ 0 α(t) = ∗ α if (s − βsM )sM < 0

(3.110)

where α ∗ > 1 is the so-called modulation factor, 0 ≤ β < 1 is the anticipation factor, and U > 0 is the minimum control magnitude. sM corresponds to the last singular

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Fig. 3.13 Phase trajectory of the Sub-Optimal algorithm

value of the function s, i.e. the last value of s at the time its derivative s˙ reaches zero (see Fig. 3.13). This value must be refreshed every time s˙ zeroes, thus the SubOptimal algorithm is particularly suitable for plants that incorporate a special device or software code to detect the singular values (sM ) (i.e. peak detector). The parameters of the controller, α ∗ , β and U , should be tuned based on the bounds defined in conditions 1 and 2 of Case 2, Sect. 3.5.1. The sufficient conditions for the finite-time convergence and robust stabilisation of the system are U>

Φ Γm

 Φ + (1 − β)ΓM U ; +∞ α ∈ [1; +∞) ∩ βΓm U ∗



(3.111)

which are known as the dominance condition and the convergence condition, respectively. The former ensures that the control has sufficient authority to affect the sign of s¨ . The latter guarantees the sliding-mode stability and determines the rate of convergence. A detailed analysis and proof of convergence can be found in [2] and [5]. Sub-Optimal Algorithm Adaptation to Relative Degree 1 Systems Similarly to the Twisting algorithm case, the application of the Sub-Optimal algorithm to relative degree 1 systems can be achieved by artificially increasing the relative degree, with the incorporation of an integrator prior to the system input u. The procedure to be followed is exactly the same as that described in the Twisting algorithm Sect. 3.5.2.1.

3.6 Conclusions

69

3.6 Conclusions Fundamentals of sliding-mode control have been introduced in this chapter. This control theory has proven to be capable of successfully dealing with nonlinear systems, presenting several attractive characteristics. Among them, finite convergence, system order reduction and robustness against certain disturbances are the most relevant. In this context, the extension known as Higher-Order Sliding Modes adds chattering reduction to the list of positive features, improving accuracy in realisation and, in several plants, contributing to extending the service life of the actuators. These, together with relatively low on-line computational cost, make the HOSM technique specially suitable for implementation. To this end, several algorithms have been developed, particularly the SecondOrder Sliding-Mode ones. In this chapter, three of the most widely used SOSM controllers have been reviewed, namely Twisting, Super Twisting and Sub-Optimal. There are many others that robustly solve the problem of convergence, each one with its own advantages and features (e.g., Drift algorithm [18, 29], Global algorithm [5] and Prescribed convergence law algorithm [29]). Then, the challenge now is to tackle the specific problem of PEM fuel cell control and assess the applicability of this control technique, with the objective of enhancing fuel cell efficiency and increasing their service life. This will be the topic of the next chapter.

References 1. Barbot JP, Perruquetti W (eds) (2002) Sliding mode control in engineering. Dekker, New York 2. Bartolini G, Ferrara A, Usai E (1998) Chattering avoidance by second order sliding mode control. IEEE Trans Autom Control 43(2):241–246 3. Bartolini G, Levant A, Pisano A, Usai E (1999) 2-Sliding mode with adaptation. In: Proceedings of the 7th IEEE Mediterranean conference on control and systems, Haifa, Israel 4. Bartolini G, Levant A, Pisano A, Usai E (2002) Higher-order sliding modes for outputfeedback control of nonlinear uncertain systems. In: Variable structure systems: towards the 21st century. Lecture notes in control and information sciences, vol 274, Springer, Berlin, pp 83–108 (Chap 6) 5. Bartolini G, Pisano A, Punta E, Usai E (2003) A survey of applications of second-order sliding mode control to mechanical systems. Int J Control 76(9/10):875–892 6. Bartolini G, Fridman L, Pisano A, Usai E (eds) (2008) Modern sliding mode control theory. New perspectives and applications. Lecture notes in control and information sciences, vol 375. Springer, Berlin 7. Boiko I, Fridman L (2005) Analysis of chattering in continuous sliding mode controllers. IEEE Trans Autom Control 50(9):1442–1446 8. Boiko I, Fridman L, Pisano A, Usai E (2007) Analysis of chattering in systems with secondorder sliding modes. IEEE Trans Autom Control 52(11):2085–2102 9. Bondarev A, Bondare S, Kostyleva N, Utkin V (1985) Sliding modes in systems with asymptotic state observers. Autom Remote Control 46(5):679–684 10. Burton J, Zinober A (1986) Continuous approximation of variable structure control. Int J Syst Sci 17:875–885

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11. Cortes J (2008) Discontinuous dynamical systems—a tutorial on solutions, nonsmooth analysis, and stability. IEEE Control Syst Mag 28(3):36–73 12. DeCarlo R, Zak S, Matthews G (1988) Variable structure control of nonlinear multivariable systems: a tutorial. Proc IEEE 76(3):212–232 13. Drazenovic B (1969) The invariance conditions in variable structure systems. Automatica 5:286–295 14. Edwards C, Spurgeon S (1998) Sliding mode control: theory and applications, systems and control book series. Taylor & Francis, London 15. Edwards C, Fossas Colet E, Fridman L (eds) (2006) Advances in variable structure and sliding mode control. Lecture notes in control and information sciences, vol 334. Springer, Berlin 16. Emelyanov S (1957) Variable structure control systems. Nauka, Moscow 17. Emelyanov S, Korovin S (1981) Applying the principle of control by deviation to extend the set of possible feedback types. Sov Phys Dokl 26(6):562–564 18. Emelyanov S, Korovin S, Levantovsky L (1986) Higher order sliding regimes in the binary control systems. Sov Phys Dokl 31(4):291–293 19. Filippov A (1960) Differential equations with discontinuous right-hand side. Mat Sb (in Russian). English translation: Trans. Am Math. Soc. 62 (1964) 20. Filippov A (1988) Differential equations with discontinuous righthand sides. Mathematics and its application. Kluwer Academic, Dordrecht 21. Floquet T, Barbot J-P, Perruquetti W (2003) Higher-order sliding mode stabilization for a class of nonholonomic perturbed systems. Automatica 39(6):1077–1083 22. Fridman L (2001) An averaging approach to chattering. IEEE Trans Autom Control 46(8):1260–1265 23. Fridman L (2003) Chattering analysis in sliding mode systems with inertial sensors. Int J Control 76(9/10):906–912 24. Fridman L, Levant A (1996) Higher order sliding modes as a natural phenomenon in control theory. In: Robust control via variable structure and Lyapunov techniques. Lecture notes in control and information sciences, vol 217. Springer, London, pp 106–133 (Chap 7) 25. Fridman L, Levant A (2002) Higher order sliding modes. In: Sliding mode control in engineering. Dekker, New York, pp 53–101 (Chap 3) 26. Hung JY, Gao W, Hung JC (1993) Variable structure control: a survey. IEEE Trans Ind Electron 40(1):2–22 27. Isidori A (1995) Nonlinear control systems. Springer, Berlin 28. Khalil H (2002) Nonlinear systems, 3rd edn. Prentice Hall, New York 29. Levant A (1993) Sliding order and sliding accuracy in sliding mode control. Int J Control 58(6):1247–1263 30. Levant A (1998) Arbitrary-order sliding modes with finite time convergence. In: Proceedings of the 6th IEEE Mediterranean conference on control and systems, Alghero, Sardinia, Italy. 31. Levant A (2001) Universal SISO sliding-mode controllers with finite-time convergence. IEEE Trans Autom Control 46(9):1447–1451 32. Levant A (2003) Higher order sliding modes, differentiation and output feedback control. Int J Control 76(9):924–941 33. Levant A (2005) Homogeneity approach to high-order sliding mode design. Automatica 41:823–830 34. Levant A (2007) Construction principles of 2-sliding mode design. Automatica 43(4):576– 586 35. Plestan F, Glumineau A, Laghrouche S (2008) A new algorithm for high order sliding mode control. Int J Robust Nonlinear Control 18(4–5):441–453 36. Sabanovic A, Fridman L, Spurgeon S (eds) (2004) Variable structure systems: from principles to implementation. IET control engineering series. IET, London 37. Shtessel Y, Shkolnikov I, Brown M (2003) An asymptotic second-order smooth sliding mode control. Asian J Control 5:498–503 38. Sira-Ramirez H (1988) Differential geometric methods in variable structure control. Int J Control 48(5):1359–1390

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39. Slotine J (1991) Applied nonlinear control. Prentice Hall, New York 40. Sontag E (1998) Mathematical control theory: deterministic finite dimensional systems. Springer, Berlin 41. Su W, Drakunov S, Ozguner U (1994) Implementation of variable structure control for sampled-data systems. In: Proceedings of IEEE workshop on robust control via variable structure and Lyapunov techniques, Benevento, Italy, pp 166–173 42. Utkin V (1978) Sliding modes and their application in variable structure systems. Mir, Moscow 43. Utkin V (2002) First stage of VSS: people and events. Springer, Berlin, pp 1–33 (Chap 1) 44. Utkin V, Gulder J, Shi J (1999) Sliding mode control in electro mechanical systems. Taylor & Francis, London 45. Young K, Utkin V, Ozguner U (1999) A control engineer’s guide to sliding mode control. IEEE Trans Control Syst Technol 7(3):328–342 46. Yu X, Xu JX (eds) (2002) Variable structure systems: towards the 21st century. Springer, Berlin

Chapter 4

Assessment of SOSM Techniques Applied to Fuel Cells. Case Study: Electric Vehicle Stoichiometry Control

4.1 Introduction As it was introduced in Chap. 2, the design of proficient control algorithms for the fuel-oxidant coordination problem in PEM fuel cells under variable load operation can reduce the fuel consumption while preventing performance deterioration, oxygen starvation and, eventually, irreversible damage in the polymeric membranes. However, the stoichiometry problem is complex and cannot be successfully tackled by using traditional techniques. A number of reasons turn fuel cell stacks into a major control challenge, e.g. high-order nonlinear equations are required to describe hydrogen-air fed stack dynamics, experimental data-based models usually incorporate look-up tables and piecewise functions, as well as a wide variety of model uncertainties [8]. Besides, many internal variables are inaccessible for measurement, and there exist disturbances that affect the system operation. It is clear that, in order to attain an efficient controller for the FC system, a special control technique capable of coping with such challenges is required. Therefore, the main objective of this chapter is to evaluate the feasibility of SOSM techniques for oxygen stoichiometry control in PEM fuel-cell-based systems. According to the features reviewed in Chap. 3, SOSM algorithms are especially suited candidates to deal with the aforementioned challenges. In general terms, they are capable of solving the nonlinear robust stability of the fuel cell system, converging to the reference in finite time and avoiding chattering effects, due to the absence of direct discontinuous components in the first derivative of the output. Moreover, if the system is of relative degree 1, then the control action applied to the actual input of the plant will be continuous. Another feature of relevance for fuel cell applications is the control design based on nonlinear models. Given the highly nonlinear nature of the stack, this feature guarantees robust operation and performance in a wider range than those achieved by control methods based on model linearization [8, 9]. In addition, the SOSM control laws are relatively simple, relying on a reduced set of parameters and few measurable variables, rather than on the knowledge of the full state vector. Thus, the implementation requirements and on-line computational burden are considerably alleviated. C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3_4, © Springer-Verlag London Limited 2012

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The assessment of the SOSM controllers suitability is performed here through computer simulations, using a benchmark model of a fuel cell system for automotive applications. To this end, the model developed by J. Pukrushpan et al. at the Mechanical Department of the Michigan University [8] has been selected, since it is one of the most complete and accurate models available in the open literature (Sect. 4.2). Therefore, the case study under consideration in this chapter is a 75-kW PEM FC stack where the oxygen is fed by an air turbo compressor used in the Ford P2000 fuel cell electric vehicle [8].

4.2 Succinct Description of the Electric Vehicle Fuel Cell System The first stage in the controllers design procedure is to reformulate the model description according to the design requirements of the SOSM control techniques. A nonlinear state model of the form x˙ = F (x, u, t) with F a smooth vector function, at least C 1 , is required (see Sect. 4.3). The original benchmark model comprises look-up tables and switched piecewise functions, continuous but not differentiable. Hence, in the proposed design model, they are replaced with smooth functions using adequate polynomial approximations. Additionally, an order reduction based on control considerations of the system physics can be done. Any minor degree of error or uncertainty generated by the aforementioned model adaptation procedure can be added to the inherent uncertainty of the original model. So, such lumped uncertainties, together with external disturbances, are taken into account in the design of the proposed controllers. Then, the aim of this section is twofold: on the one hand, to present a sixth-order nonlinear state space model of the FC flow system in the SOSM control design form; on the other hand, to review its constituent subsystems, giving a mathematical characterization in accordance with the approach followed for the model construction in [7, 8]. This will be particularly useful to readers unacquainted with such models. However, it is not the intention of this section to give an exhaustive description, but to briefly provide the necessary background to understand the subsequent developments in this book. Therefore, the readers interested in a detailed description and discussions on the benchmark model are referred to the original works of Pukrushpan et al. The fuel cell generation system (FCGS) roughly comprises five main subsystems: the air flow (breathing), the hydrogen flow, the humidity, the stack electrochemistry and the stack temperature subsystem, respectively. In [8], it is assumed that the input reactant flows are efficiently humidified and the stack temperature is well regulated by dedicated controllers. In addition, it is considered that sufficient compressed hydrogen is available, and therefore the main attention is focused on the air management. In Fig. 4.1 a schematic view of the FCGS under consideration is represented. The most relevant components related to the FC flow system are succinctly characterized in the sequel.

4.2 Succinct Description of the Electric Vehicle Fuel Cell System

75

Fig. 4.1 Schematic diagram of a PEM fuel-cell-based generation system (FCGS)

4.2.1 Air Compressor A 14-kW Allied Signal air turbo compressor is considered in the case under study. Its control input is the command voltage of the DC motor (Vcp ), while the shaft rotational speed is the only dynamic state variable. The model is divided into two parts. One part is characterized by a static compressor map, which determines the air flow rate through the compressor. The compressor air flow Wcp is computed using the rotational speed and the pressure ratio across the compressor. The inlet air is atmospheric, so constant pressure and temperature (Pamb = 1 atm, Tamb = 25 °C) are assumed. The other part represents the dynamics of the air compressor subsystem. Neglecting the armature inductance, a dominant first-order mechanical dynamic equation can be obtained by applying Newton’s second law to the rotating parts, assuming lumped rotational parameters: dωcp 1 (4.1) = (Te − Tl ) dt J where ωcp is the compressor rotational speed, J is the combined inertia of the compressor motor and the compression device, Te is the compressor motor torque, and Tl the load torque. Te can be computed using the simplified DC motor equation ηdc kdct (Vcp − kdcv ωcp ) (4.2) Rdc the compressor DC motor constants, and ηdc its mechanical Te =

with kdct , kdcv and Rdc efficiency.

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The load torque Tl depends on the compressor speed, the supply manifold pressure Psm , the compressor efficiency (ηcp ) and the compressor air flow (Wcp ). Its expression can be obtained from turbine thermodynamics equations given by [2, 3] Tl =

Cp Tamb ωcp ηcp



Psm Patm

 γ −1 γ

 − 1 Wcp

(4.3)

with Cp the specific heat capacity of air, and γ the ratio of the specific heats of air. Wcp is first given as a flow map determined from the pressure ratio across the compressor and the motor shaft speed. Then, the look-up table is replaced by an interpolation given by the Jensen and Kristensen nonlinear curve fitting method [6]. Thus, in the current model, Wcp is given in the form of a continuously differentiable bivariate function of Psm and ωcp . In turn, the compressor efficiency was originally represented as a double input look-up table depending on Wcp and Psm . However, it has been proved that the use of the mean value of ηcp is a sufficiently good approximation for modeling and control design purposes. Under these conditions, the right-hand side of (4.1) is in the whole operation range, at least, a C 1 function.

4.2.2 Air Supply Manifold The model of the air supply manifold takes into consideration the piping between the compressor and the stack (cooler and humidifier included). Two dynamic equations are required. The first one is given by the mass conservation principle: dmsm (4.4) = Wcp − Wsm dt where msm is the mass of air accumulated in the supply manifold, and Wcp and Wsm are the manifold inlet and outlet mass flows, respectively. The former comes directly from the compressor subsystem, while the latter, assuming a small pressure gradient between the cathode and the manifold, can be taken as Wsm = Ksm (Psm − Pca )

(4.5)

with Ksm the supply manifold constant or restriction. The second dynamic equation results from applying the energy conservation principle and the ideal gas law. Assuming that the air temperature changes inside the manifold, the pressure dynamic equation is given by Ra dPsm (Wcp Tcp − Wsm Tsm ) = dt Vsm

(4.6)

with Ra the gas constant of air, Vsm the supply manifold volume, and Tsm the manifold air temperature that can be computed using (4.4), (4.6) and the ideal gas law: Tsm =

Psm Vsm msm Ra

(4.7)

4.2 Succinct Description of the Electric Vehicle Fuel Cell System

77

4.2.3 Air Humidifier and Temperature Conditioner Subsystems Typically PEM fuel cell systems require air temperature and humidity conditioning before entering the cathode. According to [8], in this electric vehicle FC stack, the temperature of the air entering the supply manifold is high (>90 °C), and thus it has to be cooled to prevent damage to the fuel cell membranes. Then, it is assumed that an ideal cooler conditions the air temperature to the stack operating temperature (80 °C), without producing an appreciable pressure drop. Additionally, the cathode air must have a high humidity level to maintain the hydration of the polymeric membranes, but excessive water amounts should be avoided. Dry membrane and flooded fuel cells are both undesirable situations that produce an efficiency reduction and may cause irreversible damage. Hence, in this system a humidifier injects the proper amount of water into the stream, to adjust the cathode inlet flow relative humidity to stay close to 100%. It is assumed that the temperature of the air flow is constant and the humidifier volume is negligible compared to the supply manifold volume. In order to compute the change in air humidity, due to the additional injection of water, a static model was used. Based on the air flow conditions released by the cooler and by using thermodynamic equations, the mass flows of dry air and vapor can be determined (please refer to [7]). In particular, the vapor saturation pressure of gases is calculated from the flow temperature using the following equation: log10 (Psat ) = −1.69 × 10−10 T 4 + 3.85 × 10−7 T 3 − 3.39 × 10

−4

T + 0.143T − 20.92 2

(4.8) (4.9)

where the units of Psat are [kPa] and [K] for T .

4.2.4 Cathode Channels The cathode model represents the lumped volume of the cathode channels of the 381 stacked cells. Using the mass conservation principle and thermodynamic properties of the air, the model has been developed under the following assumptions: (1) gases are ideal, (2) the stack and the cathode flow temperatures are constant and uniformly distributed (80 °C), (3) the flow rate of liquid water leaving the cathode is zero, and (4) temperature, pressure, humidity and oxygen mole fraction are equal inside and exiting the cathode. Then, the mass balance of oxygen, nitrogen and water inside the cathode gives dmO2 ,ca = WO2 ,ca,in − WO2 ,ca,out − WO2 ,react dt mO2 ,ca : mass of O2 inside the cathode. WO2 ,ca,in : mass flow rate of O2 entering the cathode. WO2 ,ca,out : mass flow rate of O2 leaving the cathode.

(4.10)

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WO2 ,react : rate of O2 reacted. dmN2 ,ca = WN2 ,ca,in − WN2 ,ca,out dt

(4.11)

mN2 ,ca : mass of N2 inside the cathode. WN2 ,ca,in : mass flow rate of N2 entering the cathode. WN2 ,ca,out : mass flow rate of N2 leaving the cathode. dmw,ca = Wv,ca,in − Wv,ca,out − Wv,ca,gen + Wv,mem dt

(4.12)

mw,ca : mass of water inside the cathode. Wv,ca,in : mass flow rate of vapor entering the cathode. Wv,ca,out : mass flow rate of vapor leaving the cathode. Wv,ca,gen : rate of vapor generated in the fuel cell reaction. Wv,mem : mass flow rate of water transferred across the membrane. The partial pressures of the oxygen, nitrogen and vapor inside the cathode channels can be computed using the ideal gas law for each species. At this point, an additional analysis can be performed to simplify the cathode model. After a thorough simulation study, it can be established that in this modeled system an important amount of water is transferred across the membrane and, consequently the cathode gas is saturated at almost every operating condition. Therefore, the dynamics of the mass of water inside the cathode (mw,ca ) can be neglected, turning Eq. (4.12) into an algebraic relationship. Thus, the model dynamics is reduced by one order [4].

4.2.5 Anode Channels Several assumptions are made in [8] to develop the model of the anode subsystem. The most relevant are: (1) a compressed hydrogen tank and a valve instantaneously ensure sufficient hydrogen flow for the fuel cell reaction, (2) the stack and the anode flow temperatures are the same, (3) one lumped volume is considered, and (4) temperature, pressure and humidity of the hydrogen outlet flow are similar to those of the gas inside the anode channels. In the same way as in the cathode, the gas humidity and partial pressures are computed by balancing hydrogen and water mass flows inside the anode. dmH2 ,an = WH2 ,an,in − WH2 ,an,out − WH2 ,react dt mH2 ,an : mass of H2 inside the anode. WH2 ,an,in : mass flow rate of H2 entering the anode. WH2 ,an,out : mass flow rate of H2 leaving the anode.

(4.13)

4.2 Succinct Description of the Electric Vehicle Fuel Cell System

79

WH2 ,react : rate of H2 reacted. dmw,an = Wv,an,in − Wv,an,out − Wv,mem dt

(4.14)

mw,an : water mass inside the anode. Wv,an,in : mass flow rate of vapor entering the anode. Wv,an,out : mass flow rate of vapor leaving the anode. Wv,mem : mass flow rate of water transfer across the membrane. In this model the dynamics of the mass of water (mw,an ) can also be neglected, given that in the model it only affects the hydration of the membrane, which is assumed to be 100% humidified. Then, Eq. (4.14) becomes a static relationship, reducing another order of the model dynamics.

4.2.6 Water Model of the Polymeric Membrane The hydration model of the membrane allows the computation of the membrane water content and the mass flow rate of water transfer across the membrane. Both are functions of the stack current and the relative humidity of the cathode and anode flows. The dynamics involved in the PEM subsystem are considerably faster than those of the gas channels and the gas diffusion layers, so they have been modeled with static equations. One possible approximation to obtain the membrane water content is to calculate the average between the water content in the anode flow and the cathode flow. On the other hand, the total stack mass flow rate across the membrane (Wv,mem ) depends on two different phenomena: the electro-osmotic drag, i.e. the water molecules dragged across the membrane by the hydrogen proton, and the back diffusion of water from cathode to anode, caused by the concentration gradient [10]:   i (cv,ca − cv,an ) (4.15) Wv,mem = n · Af c · Gv · nd − Dw F tm n: number of fuel cells in the stack. Af c : fuel cell active area. Gv : vapor molar mass. nd : electro-osmotic drag coefficient. F : Faraday constant. Dw : diffusion coefficient of water in the membrane. cv,ca : water concentration in the cathode. cv,an : water concentration in the anode. tm : thickness of the membrane. More details and general ideas about PEM fuel cell modeling can be found in [8] and in Chap. 5 of this book.

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4.2.7 Return Manifold The return manifold model takes into consideration the pipeline dynamics at the fuel cell stack exhaust. The temperature of the air leaving the stack is relatively low (close to 80 °C), so the changes of air temperature in this manifold can be neglected. Therefore, the manifold dynamics follow the isothermic relation dPrm Ra Trm (Wca,out − Wrm,out ) = dt Vrm

(4.16)

Prm : return manifold pressure. Trm : return manifold temperature. Vrm : return manifold volume. Wca,out : flow entering the return manifold. Wrm,out : outlet mass flow of the return manifold. The relation between the multiple output mass flow, Wrm,out , and the pressure within its volume, Prm , is modeled through the following piecewise continuous function [8]: ⎧ γ −1 ⎨ A · P ( Pamb )1/γ · [1 − Pamb γ ]1/2 if Pamb > A 1 rm Prm 2 P Prm rm (4.17) Wrm,out = ⎩ Pamb A3 · Prm if Prm ≤ A2 where A1 , A2 , A3 , Pamb and γ are model constants. Expression (4.17) is not C 1 , so it cannot be included in the model for secondorder sliding-mode design. Therefore, in order to include the return manifold in the control design model of Sect. 4.3, Eq. (4.17) was replaced by the following fifthorder polynomial [4]: 2 3 4 5 Wrm,out = p0 + p1 Prm + p2 Prm + p3 Prm + p4 Prm + p5 Prm

(4.18)

The values of the polynomial constants in Eq. (4.18) can be found in Appendix A, Table A.1.

4.3 Electric Vehicle FCGS State Space Model for SOSM Control Design To conclude the first stage of the control design procedure, the dynamic equations required for oxygen stoichiometry control must be rearranged in the SOSM design form. Considering that in this model the anode subsystem is decoupled from the cathode subsystem and does not enter in the oxygen stoichiometry control loop, its dynamics can be neglected, and the system order is reduced by one. As a result of the reduction and rebuilding work performed on Pukrushpan et al.’s model, the following sixth-order control design model is proposed: x˙ = F (x, u, t) = f (t, x) + g(t, x, u) x ∈ R6 ;

u ∈ R;

f : R6 → R6 ;

g : R6 → R6

(4.19)

4.3 Electric Vehicle FCGS State Space Model for SOSM Control Design

81

with f and g sufficiently smooth vector functions, where the coordinates of the state vector x = [ωcp

Psm

msm

mO2 ,ca

mN2 ,ca

Prm ]T

(4.20)

can be summarized as follows: • x1 = ωcp : angular speed of the compressor motor that feeds the stack cathode through the supply manifold. • x2 = Psm : total pressure inside the supply manifold, consisting of the sum of the partial pressures of the gases that constitute the air (oxygen, nitrogen and water vapor). • x3 = msm : total mass of air in the supply manifold, consisting of the sum of the instantaneous masses of oxygen, nitrogen and water vapor. • x4 = mO2 ,ca : instantaneous oxygen mass in the stack’s cathode channels. This state is affected by the oxygen consumed in the reaction, the amount of oxygen coming from the supply manifold and the oxygen mass outgoing through the return manifold. • x5 = mN2 ,ca : instantaneous mass of nitrogen inside the stack’s cathode channels. It only relies on the incoming nitrogen from the supply manifold and the outgoing nitrogen that leaves the stack through the return manifold. • x6 = Prm : total pressure inside the return manifold, consisting of the sum of the partial pressures of the gases that constitute the air. The input u(t) is the normalized input voltage of the compressor DC motor Vcp . The normalization constant (Vnor = 180 V) is the maximum value of Vcp in the operating region. The detailed expressions of the system model are given by



x˙1 = B3 n(x) 1 − d(x) + B1 u2 x1−1 − B2 u



2

−1 x˙2 = B6 1 − d(x) x1 1 + n(x)n−1 cp − x2 B7 − x2 B8 − x2 x5 B9 − x2 x4 B10 x3

x˙3 = B11 1 − d(x) x1 − x2 B12 + B13 + x5 B14 + x4 B15 x˙4 = (x2 B59 − B60 − x5 B61 − x4 B62 )(x2 − x2 B21 )−1 e(x)

+ (x2 B63 − B64 − x5 B65 − x4 B66 )e(x) k(x) − x4 (B25 + x5 B26

−1 (4.21) + x4 B27 − x6 B24 )j (x)x4−1 j (x)B67 + GN2 m(x) − Ist B32 −1 x˙5 = (x2 B50 − B51 − x5 B52 − x4 B53 )(x2 − x2 B21 ) e(x) + (x2 B54



−1

− B55 − x5 B56 − x4 B57 )e(x) k(x) − 1 − j (x)B30 j (x)B68 + GN2 × (B35 + x5 B36 + x4 B37 − x6 B34 )m(x) x˙ 6 = B47 + x5 B48 + x4 B49 − x6 B46 − B39 c(x)5 − B40 c(x)4 − B41 c(x)3 − B42 c(x)2 − B43 c(x) − B44

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with the auxiliary functions a(x) = a1 (x) + a21 (x)a22 (x)x1 a1 (x) = 1 − d(x) a21 (x) = d(x) a22 (x) = −2B69 x1−3 n(x) b(x) = b1 (x)b2 (x)x1 b1 (x) = a21 (x) b2 (x) =

B69 B4 x2B4 −1

B4 x12 Pamb c(x) = x6 − B45 B ((

x2

)B4 −1)x −2 −β

1 d(x) = e 69 Pamb −1  x2 B20 e(x) = 1 + x2 − x2 B21 x4 j (x) = x5 B28 + x4 B29 −1  B22 k(x) = 1 + x2 − x2 B21 + B23

−1

−1 m(x) = 1 + B58 j (x)B68 + GN2 j (x)x4−1   x2 B4 n(x) = −1 Pamb

(4.22)

The compressor air flow can be written in terms of the states as follows:

(4.23) Wcp = B11 1 − d(x) x1 The constants are given in Appendix A, Table A.2.

4.4 Control Objective and Sliding Surface The second stage in the SOSM control design procedure is to establish the control objective and, accordingly, define the sliding surface. In this case, the proposed objective is the optimization of the energy conversion of the FCGS, maximizing the net power (Pnet ) generated by the system under different load conditions. We assume that the system net power results as follows: Pnet = Pst − Pcp

(4.24)

Pst : stack generated power. Pcp : compressor power demand. It can be shown that the efficiency optimization can be achieved by regulating the oxygen mass flow entering to the stack cathode. If an adequate comburent flow is

4.4 Control Objective and Sliding Surface

83

Fig. 4.2 Steady state analysis of the system performance under different load conditions

ensured through the stack, the load demand is satisfied with minimum fuel consumption. Additionally, oxygen starvation and irreversible membrane damage are averted. Accomplishing such an optimal comburent flow is equivalent to maintaining the oxygen excess ratio of the cathode at an optimal value. The oxygen excess ratio or oxygen stoichiometry is defined as λO2 =

WO2 ,ca,in WO2 ,react

(4.25)

where WO2 ,ca,in is the aforementioned oxygen partial flow entering the cathode, which depends on the air flow released by the compressor Wcp . Recall that WO2 ,react is the oxygen flow consumed in the reaction, so it is directly related to the stack current: WO2 ,react = GO2

nIst 4F

(4.26)

with GO2 the molar mass of oxygen. The optimum value of λO2 is determined from a thorough off-line analysis of the open-loop system, considering changes in the current demand and a wide range of oxygen stoichiometry values. In Fig. 4.2 it can be observed that the optimum

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value of λO2 (λO2 ,opt ) depends on Ist . However, in the electric vehicle FCGS under consideration, λO2 ,opt presents minor variations over the operation range. Therefore, a constant value of λO2 ,opt was used when computing the system reference. This condition may not hold for other PEM fuel cell systems, and hence the variable λO2 ,opt could be easily expressed as a function of Ist . Alternatively, a possible solution would be to apply an extremum seeking algorithm. In that case, the proposed problem is solved by a local minimum algorithm, which does not rely on a detailed model description, and it is considered as a nonlinear on-line optimization for unconstrained problems [1]. Another important advantage of stoichiometry control is the avoidance of oxygen starvation on the cathode channels, which would occur if λO2 is allowed to go below 1. Oxygen starvation represents a significant problem in PEM fuel cell systems because if it persists for a long time, it may produce hot spots and consequent irreversible damage in the catalyst layers and polymeric membranes. Once λO2 ,opt is determined, the objective of keeping the oxygen excess ratio within optimal values can be written in terms of controlling the oxygen mass flow (WO2 ,ca,in ). Then, the following cathode oxygen mass flow reference can be obtained from (4.25) and (4.26): WO2 ,ca,ref = λO2 ,opt GO2

nIst 4F

(4.27)

where tracking WO2 ,ca,ref effectively implies λO2 = λO2 ,opt . Nevertheless, due to the fact that WO2 ,ca,in is an inaccessible internal variable of the FCGS, it is not practical to include it in the control algorithm. This problem can be successfully circumvented by inferring information of WO2 ,ca,in from an accessible variable of the system, such as the air mass flow delivered by the turbo compressor (Wcp ). Under the aforementioned fixed humidification assumption, this variable is directly related to WO2 ,ca,in through the supply manifold dynamics. Furthermore, once the manifold transient is finished, the relation between Wcp and WO2 ,ca,in remains fixed in all operating conditions. Therefore, the operation of the stack close to its maximum efficiency points can be successfully achieved by posing the control objective in terms of a tracking control problem of Wcp (in this way, λO2 = λO2 ,opt is ensured for every load condition, once the supply manifold transient expires) [5]. As it has been reviewed in Chap. 3, in the framework of the sliding-mode theory, such a control objective is formalized defining the sliding variable s(x, t) = Wcp − Wair,ref

(4.28)

and steering s to zero. The expression of the air mass flow reference Wair,ref can be readily obtained from the cathode oxygen mass flow reference (4.27). Given that the molar fraction of oxygen in the air is known (χO2 = 0.21), the desired mass flow of dry air can be directly computed from (4.26): Wdry,ref =

nIst 1 1 WO2 ,ca,ref = λO ,opt GO2 χO2 χO2 2 4F

(4.29)

4.5 Super-Twisting Controller Design

85

Then, taking into account the relative humidity of the air (RH amb ), the final expression of the air mass reference results in Wair,ref = (1 + ωamb )

nIst 1 λO2 ,opt GO2 χO2 4F

(4.30)

ωamb being the rate of vapor in the ambient air flow, ωamb =

Psat (Tamb )RH amb Gv Ga Pamb − Psat (Tamb )RH amb

(4.31)

Ga : molar mass of dry air. Gv : molar mass of vapor. Note that for stable ambient conditions, the reference Wair,ref only depends on one single measurable variable, the stack current (Ist ).

4.5 Design of a SOSM Super-Twisting Controller for the Electric Vehicle FCGS Once the FCGS dynamic model has been re-modeled in the SOSM design form and the sliding variable s has been defined, the final stage of the design procedure, i.e. the synthesis of the SOSM control law, can be completed in terms of λO2 ,opt and the measurable system variables Wcp and Ist . According with the SOSM design procedure presented in Chap. 3, the controller design requires the computation of the time derivative of s and global bounds for the second derivative. To this end, using (4.23) and (4.28), we can write the sliding variable as s(x, t) = Wcp − Wair,ref

nIst 1 = B11 1 − d(x) x1 − (1 + ωamb ) λO ,opt GO2 χO2 2 4F

(4.32)

and directly derive, through standard mathematical computations, the expressions of the first and second time derivatives of s:

∂ ∂ s(x, t). f (x, t) + g(x, u, t) (4.33) s˙ (x, t) = s(x, t) + ∂t ∂x

∂ ∂ s˙ (x, t). f (x, t) + g(x, u, t) s¨(x, t) = s˙ (x, t) + ∂t ∂x ∂ + s˙ (x, t).u(t) ˙ = ϕ(x, u, t) + γ (x, u, t)u(t) ˙ (4.34) ∂u The vector fields f and g are smooth enough in the stack operation range, and information about s is assumed to be available. Besides, note that s does not explicitly depend on the control input, but u does appear in s˙ via the expression of x˙1 ; consequently, the sliding variable s has relative degree 1 with respect to u.

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Assessment of SOSM Techniques Applied to Fuel Cells

Then, analyzing the system equations, it is verified that the conditions required for SOSM control design are satisfied by the fuel cell system model (4.19) with output s given by (4.32). In accordance with the general framework established in Chap. 3, the system is under the conditions established in Case 1 of Sect. 3.5.1, which for the particular FCGS under study are: 1. Given the total second time derivative s¨ , there are bounds Γm and ΓM such that, within the region |s(x, t)| < s0 , the following inequality holds for all t, x ∈ X , u∈U : ∂ (4.35) s˙ (x, t) ≤ ΓM ∂u For the fuel cell stack under consideration, such bounds were obtained from a detailed analysis of the system structure together with comprehensive simulation studies, considering the addition of an appropriate feedforward control that leads the system to the validity region. As a result, the following bounds were determined for s0 = 5e−3 : 0 < Γm ≤ γ (x, u, t) =

Γm = 0.5,

ΓM = 0.9

2. Analogously, a positive constant (Φ = 0.01 for this FCGS) can also be calculated, such that, within the region |s| < s0 , the following inequality holds for all t, x ∈ X , u ∈ U :   ∂

  s˙ (x, t) + ∂ s˙ (x, t). f (x, t) + g(x, u, t)  ≤ Φ (4.36)   ∂t ∂x with Ist covering the whole operation range of the stack (1 A to 300 A). Therefore, the stabilization problem of the electric vehicle FCGS (4.19) with input–output dynamics (4.34) is solved through the solutions of the following equivalent differential inclusion by applying SOSM: s¨ ∈ [−Φ = −0.01, Φ = 0.01] + [Γm = 0.5, ΓM = 0.9]u˙

(4.37)

and the final parameters of the robust controller can be designed based on Φ, Γm and ΓM . It is interesting to stress that the bounds for functions ϕ(x, u, t) and γ (x, u, t) were calculated considering the bounded perturbations and uncertainties existing in the FCGS. In this way, (4.37) covers their effects, and hence the design based on these values results in controllers which are naturally robust to such disturbances. In the sequel, a control law based on the Super-Twisting (ST) algorithm is developed. As it was explained in Chap. 3, this algorithm only requires the knowledge of the sign of s during on-line operation, and it is specially suited for plants of relative degree 1, like the FC system under consideration. According to (3.93) and (3.94), the control u is given as the sum of two components: u(t) = u1 (t) + u2 (t)

(4.38)

4.5 Super-Twisting Controller Design

87

Fig. 4.3 Schematic block diagram of the FCGS and the proposed SOSM control set up

u˙ 1 (t) = −α sign(s) u2 (t) = −λ|s|ρ sign(s)

(4.39)

where the controller parameters γ , λ and ρ are designed to fulfill the sufficient conditions for convergence to s(x) = s˙ (x) = 0 in finite time, i.e. Φ Γm 2 (Γm α + Φ)2 λ2 > 2 Γm (Γm α − Φ)

α>

(4.40)

ρ = 0.5 In this way it is ensured that the control has sufficient authority to affect the sign of s¨ , guaranteeing the establishment of a SOSM. To complete the control set up to be implemented, as previously mentioned, an appropriate extra control action is added, to steer the sliding variable within the validity region. In the FCGS, a feedforward action (uff ) proved to be effective to speed up the reaching phase. Therefore, uff was included into the control scheme, and the implemented control action (ui (t)) comprises two terms: ui (t) = u(t) + uff (t)

(4.41)

where u corresponds to the SOSM control action particularized by Eqs. (4.38) and (4.39). The expression of uff is computed via a simple polynomial, a function of the measurable current Ist , obtained from an off-line test along the entire operation range of the FCGS [8]. A schematic block diagram of the control set-up proposed for implementation can be appreciated in Fig. 4.3. To refine the final tuning of the Super-Twisting controller, it is recommended to consider not only the behavior of the controlled variable (in this case the oxygen stoichiometry), but also any other involved variable that can affect the overall performance of the plant. For instance, in this particular system, the electric quality of the FCGS net power should be also taken into account. This is due the direct effect of the control on the electric power (a relative degree 0 output). Therefore, the Super-Twisting controller parameters were selected aiming to smooth the control

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4

Table 4.1 Variations of system parameters

Assessment of SOSM Techniques Applied to Fuel Cells

Parameter

Variation

Stack temperature (Tst )

+10%

Cathode volume (Vca )

+5%

Supply manifold volume (Vsm )

−10%

Atmospheric pressure (Patm )

+10%

Ambient temperature (Tamb )

+10%

Return manifold volume (Vrm )

−10%

Motor constant (Kv )

−10%

Electric resistance of the motor (Rcm )

+5%

Compressor diameter (dc )

+1% +10%

Motor inertia (J )

action and to have a low content of high-frequency components. After an iterative refining procedure, the most adequate set of parameters resulted as follows: α = 2,

λ = 3,

ρ = 0.5

(4.42)

In the next section it will be shown that, under the influence of the control (4.41), the phase portrait ((s, s˙ ) plane) of the controlled system presents the characteristic non-monotonous behavior of the homogeneous Super-Twisting algorithm. The trajectories converge to the origin in finite time, twisting around the center during the reaching mode.

4.6 SOSM Super-Twisting Controller Simulation Results To evaluate the efficiency of the proposed controller (4.41) dealing with model uncertainties, external disturbances and a wide range of current demand, a number of simulation studies were performed. To assess the SOSM controller performance under realistic operating conditions, simulation tests were conducted using the comprehensive ninth-order nonlinear model developed in [8]. The simulation model incorporates the original look-up tables representing parameter characteristics obtained from experimental data. In addition, extra uncertainty has been incorporated in several parameters of the system (see Table 4.1). Moreover, an unknown torque disturbance, modeled as a noisy quadratic function of the angular speed (ωcp ), was included in the same test (see Fig. 4.4). This friction term is set to start at t = 15 sec, and its expression is Tp = ωcp (t)2 B1 + ωcp (t)B2 + e(t) 10−9

[N m/s2 ],

10−6

(4.43)

B2 = 20 × [N m/s] and e(t) corresponding to a with B1 = band-limited noise component. Then, the features of the designed controller are examined through simulation tests, which aim to demonstrate its nominal performance and its robust tracking

4.6 SOSM Super-Twisting Controller Simulation Results

89

Fig. 4.4 Torque disturbance

Fig. 4.5 Load current

characteristics in the presence of the aforementioned unknown disturbances and uncertainties. To this end, series of load current (ranging from 60 A to 300 A) were designed in order to illustrate the air regulation performance in a wide range of operation. The sequence of current variations, generated from a filtered steps series, is shown in Fig. 4.5. Note that abrupt and significant changes in the amplitude of the load demand were considered to test the proficiency of the controller under exacting operating conditions.

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Assessment of SOSM Techniques Applied to Fuel Cells

Fig. 4.6 Oxygen excess ratio (oxygen stoichiometry)

In Fig. 4.6 and the sequel, the black line depicts the system variables evolution in the presence of uncertainties and friction disturbances, while the grey line shows the corresponding variables of the undisturbed system. Note that in Figs. 4.6, 4.8 and 4.9 both lines are superimposed during the entire test. The behavior of the oxygen stoichiometry (λO2 ) under different load conditions is depicted in Fig. 4.6. In accordance with the explanation of Sect. 4.4, the values of λO2 that enable the system to work in its maximum net power points were determined performing off-line simulations (Fig. 4.2). In this electric vehicle FCGS, due to the relatively small variation of λO2 ,opt for different values of stack current, a unique value λO2 ,opt = 2.05 was adopted. When λO2 ,opt is reached, it can be assured that the system is working in the neighborhood of its maximum power generation points. It is observed in Fig. 4.6 that the excess oxygen ratio satisfactorily follows that reference in spite of uncertainties and disturbances. Figures 4.7 and 4.8 present the dynamic behavior of the system net power (Pnet ) and the stack voltage (Vst ). It can be observed that, effectively, the former displays a desirable low content of high-frequency components. Besides, remark that the noticeable difference between the net power of the nominal and the disturbed system (after t = 15 sec) is not a controller flaw, but the expected consequence of the addition of the torque disturbance of Fig. 4.4. To compensate this spurious load, the compressor must increase its consumption, hence for given Ist , the resultant net power decreases. In spite of all disturbances, the ST robustly controls the output s, as attested by Fig. 4.9. Figure 4.9 shows the evolution of the sliding output s. Once the brief reaching mode elapses, the controller maintains the system operating on the sliding surface (s = 0, i.e. regulating λO2 at its optimum value), despite the coexistence of parameters uncertainties, external perturbations and important load variations. Recall that

4.6 SOSM Super-Twisting Controller Simulation Results

91

Fig. 4.7 Net power delivered by the system

Fig. 4.8 Stack output voltage

the stability of the closed-loop system is guaranteed because the differential inclusion (4.37) has been computed to hold in the presence of such disturbances. Note that s exhibits the typical overshoots of a system controlled by an ST algorithm (Fig. 4.9). Effectively, the characteristic twisting behavior imposed by the Super-Twisting controller can be better appreciated in Fig. 4.10, where the phase portrait of the system is plotted for a representative lapse, ranging in time from 24.2 sec to 24.3 sec.

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Assessment of SOSM Techniques Applied to Fuel Cells

Fig. 4.9 Sliding variable

Fig. 4.10 s vs. s˙

Finally, in Fig. 4.11 the time evolution of the control signals are presented, in terms of the actual input compressor voltage units (i.e. denormalizing the control actions by multiplying by Vnor ). The implemented control action ui that drives the fuel-cell-based system, together with the two constituent components u and uff , are depicted. Note the increase in the control effort after the appearance of the friction disturbance. To counteract its effect, the SOSM controller is in charge of providing the necessary action to achieve the regulation objective.

4.7 Comparison with Other Control Strategies

93

Fig. 4.11 Control signal

4.7 Comparison with Other Control Strategies 4.7.1 Different SOSM Control Algorithms In this section, the electric vehicle FCGS performance is evaluated under the action of other control algorithms. Given the encouraging results obtained in the previous section that proved the suitability of a SOSM ST controller for oxygen stoichiometry control, it is naturally of interest to explore the capabilities of other SOSM algorithms. Therefore, two SOSM controllers based on Twisting and Sub-Optimal algorithms, respectively, are synthesized and assessed in the present subsection. According to the description in Chap. 3, these SOSM algorithms are originally intended for relative degree 2 systems, and thus the FCGS must be expanded with an integrator, taking the ancillary input ν = u(t) ˙ as the control action for the design (see Fig. 4.12). 4.7.1.1 Twisting Algorithm As it was previously established, the Twisting controller is given by ν = u˙ = −r1 sign(s) − r2 sign(˙s )

(4.44)

with sufficient conditions for finite-time convergence r1 > r2 > 0 (r1 + r2 )Γm − Φ  > (r1 − r2 )ΓM + Φ  Φ (r1 − r2 ) >  Γm

(4.45)

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Assessment of SOSM Techniques Applied to Fuel Cells

Fig. 4.12 Schematic block diagram proposed for Twisting and Sub-Optimal algorithms in relative degree one systems

Note that the control law (4.44) requires the knowledge of s and s˙ and, in the relative degree 1 fuel cell system, synthesizes a continuous control voltage Vcp , due to the integrator. Recall that in this particular case, in which the relative degree 2 system is artificially obtained from expanding a relative degree 1 plant, the bounds of the former are equal to those of the latter (see Chap. 3, Sect. 3.5.2.1). Therefore, the Twisting controller gains can be directly computed using the same bounds calculated in (4.35) and (4.36) (i.e., Γm = Γm = 0.5, ΓM = ΓM = 0.9 and Φ  = Φ = 0.9). Then, the selected parameters for the controller were r1 = 2.25 and r2 = 0.75.

4.7.1.2 Sub-Optimal Algorithm The Sub-Optimal control law, detailed in Chap. 3, is given by ν = u˙ = −α(t)U sign(s − βsM ) 1 if (s − βsM )sM ≥ 0 α(t) = ∗ α if (s − βsM )sM < 0

(4.46)

It requires the knowledge of s and of the last extremal value of the sliding variable, sM , that is the value of s at the last local maximum, minimum or horizontal inflexion point, which has to be updated on-line (see Chap. 3, Sect. 3.5.2.3 for details). To ensure the finite-time convergence, the minimum control magnitude U > 0, the modulation factor α ∗ > 1 and the anticipation factor 0 ≤ β < 1 must be selected in accordance with U>

Φ Γm



 Φ  + (1 − β)ΓM U α ∈ [1; +∞) ∩ ; +∞ βΓm U ∗

(4.47)

Then, the Sub-Optimal controller parameters for the electric vehicle fuel cell were chosen as U = 3, β = 0.3 and α ∗ = 5.

4.7 Comparison with Other Control Strategies

95

Fig. 4.13 Time evolution of the sliding variable under the action of different SOSM controllers

4.7.1.3 SOSM Comparison Simulation Tests To assess and compare the performance of the aforementioned SOSM controllers, simulation tests start considering the undisturbed FCGS, and, later, several sources of disturbances are incorporated. Similarly to the Super-Twisting case, in Sect. 4.6, the complete ninth-order nonlinear model developed in [8] is used to model the plant in the simulations. Additionally, the same load profile utilized in Sect. 4.6 is used for these tests (see stack drained current in Fig. 4.5). Results of the three SOSM algorithms (namely Super-Twisting, Twisting and Sub-Optimal) controlling the FCGS are presented in Figs. 4.13, 4.14, 4.15 and 4.16. Numerical values of the controllers parameters are summarized in Appendix A, Table A.3. The former corresponds to the undisturbed system and depicts a magnified view of the time response of the sliding variable after the first step of current demand occurred (a brief lapse between 2 and 2.5 sec is displayed). Figure 4.14 shows the associate trajectories in the phase plane. The typical reaching behaviors of the three algorithms are manifested in this figure. It is observed that the three SOSM controllers display the expected finite-time convergence during this first part of the tests, with neither disturbances nor uncertainties present in the system. Next, prior to the incorporation of the whole model uncertainties and external disturbances, it is of interest to appreciate the behavior of the control algorithms under the effect of measurement noise. To this end, between t = 4 sec and t = 11 sec important measurement noise is added to Wcp . After that, from t = 15 to t = 30 sec, the FCGS is strongly perturbed by incorporating the compressor disturbance torque (given in (4.43), Fig. 4.4), together with all the parameter uncertainties of Table 4.1. Figure 4.15 presents the actual and measured (noise included) Wcp time evolution of the FCGS controlled with the SOSM algorithms (for the sake of clarity, in this

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Fig. 4.14 s(x) vs. s˙ (x)

Fig. 4.15 Compressor air mass flow

figure only the Super-Twisting and Sub-Optimal algorithms are depicted. In fact, the response of the Twisting controller would almost overlap the Super-Twisting one). Their performance can be assessed from the actual Wcp curves, proving all of them to have an excellent behavior. This is specially manifested after t = 15 sec, with the perturbations disturbing the system simultaneously. Note that there is an output that can be considerably influenced by the measurement noise in certain cases. This is the net power of the generation system. Even though the latter has not been defined as an output from the strict control viewpoint,

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97

Fig. 4.16 FCGS net power

it is still an important physical output. Its sensitivity to high-frequency variations in the control input is because Pnet has relative degree 0 with respect to u. In Fig. 4.16, it can be seen that the Super-Twisting algorithm may produce a more marked noise transmission to Pnet than the other algorithms. This is clearly due to the fact that this algorithm contains a term related to s, not integrated, which directly injects the fluctuations of s into u. Nevertheless, this is not necessarily a serious drawback for the algorithm application, since that negative effect can be considerably ameliorated with a proper tuning of λ. To illustrate this feature, the value of λ has been changed from 3 to 1.5 in the interval 8 sec < t < 10 sec. In the zoom window in Fig. 4.16, a substantial reduction of the noise effect can be appreciated. In the other two algorithms, measurement noise in s has been attenuated more in the Pnet value since, in the relative degree 1 FCGS under consideration, the control terms that directly depend on s are integrated before entering to the plant.

4.7.2 LQR Controller It is also interesting to establish comparisons between the proposed SOSM solutions and other more widely accepted techniques, such as state feedback linear control. To this end, a Linear Quadratic Regulator (LQR) controller proposed in [7, 8] is considered and analyzed in this subsection. As it can be appreciated in the cited bibliography, satisfactory results controlling the FCGS under study have been attained with this approach. In particular, this linear controller combines feedforward with

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Fig. 4.17 Compressor air mass flow

feedback (state estimated plus integral) control. The form of the proposed controller is [8] Vcpi = Vcpff + VcpLQR = Vcpff + Vcpp − k · δ xˆ − kI · q

(4.48)

The implemented control input Vcpi is the compressor voltage command, while Vcpff and Vcpp are feedforward and pre-compensation terms, respectively, which only depend on the measurable stack current Ist . The third term is a state feedback control action, where δ xˆ is the estimate of the linearized system state δx  = x  −x  o . The constant vector x  o represents the state value at the nominal operating point of linearization, in this case computed at optimal stoichiometry and a net power of 40 kW, with Ist = 191 A and Vcp = 164 V. Note that in this controller, the system state x  has order eight, due to the additional presence of mH2 ,an and mw,an , the mass of hydrogen and water inside the anode, respectively: x  = [mO2 ,ca

mH2 ,an

mN2 ,ca

ωcp

Psm

msm

mw,an

Prm ]T

(4.49)

The state estimation δ xˆ is obtained using the Kalman-based observer presented in [7]. The fourth term in (4.48) provides the integral control action, where the state equation of the integrator is given by the compressor flow error: q˙ = Wair,ref − Wcp

(4.50)

Finally, the gains of the feedback control terms have been designed using LQR optimal control techniques, resulting in [7]

k = −28.59 −1.6 × 10−13 −60.57 7.57 579.74 2.55  (4.51) −3.6 × 10−14 −189.97 kI = −0.18

4.7 Comparison with Other Control Strategies

99

Fig. 4.18 Oxygen excess ratio (oxygen stoichiometry)

Fig. 4.19 Control action (in terms of the input voltage)

4.7.2.1 LQR Comparison Simulation Tests For comparisons, the simulation tests presented in Figs. 4.17, 4.18 and 4.19 have been conducted under operating conditions similar to the ones in the previous subsections. Numerical values of the LQG controller parameters are summarized in Appendix A, Table A.4. They show the time evolution of the LQR and the SuperTwisting controlled system variables Wcp (actual and measured, with added noise), λO2 and the control actions (in terms of the input voltage), respectively.

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Fig. 4.20 Oxygen excess ratio (small excursion)

In the former figure, it can be appreciated that, during undisturbed operation, both controllers display very good results, although the Super-Twisting better regulates the actual Wcp at its desired value, given that it maintains s = 0 (recall Fig. 4.9), except for the abrupt steps of current Ist . Respectively, with regard to the undisturbed behavior of the oxygen stoichiometry λO2 , it is observed in Fig. 4.18 that the LQR presents an initially faster dynamic response after the steeps, but in the end it requires longer times to reach the immediate vicinity of the desired value. On the other hand, the Super-Twisting controller attains less than 1% error in 3 sec. It is worthwhile to mention that the linear LQR improves considerably its performance in the neighborhood of the linearization point. Moreover, in that region and considering small excursions, it could even surpass the performance of the SuperTwisting regarding the control of λO2 (see Fig. 4.20). However, as it can be observed in Figs. 4.17 and 4.18, the SOSM Super-Twisting excels the linear controller when working in a wide operation range. Nevertheless, in this fuel cell application, it is in the presence of perturbations and uncertainties that the SOSM controller demonstrates its superiority over the LQR. This can be attested by comparing the response of both controllers under disturbed conditions (see t > 15 sec in Figs. 4.17 and 4.18). To emphasize this feature, showing that the robustness of the Super-Twisting algorithm surpasses that of the LQR approach, a test conducted at constant Ist = 191 A (i.e., the value used for the linearized design of the LQR) is presented in Fig. 4.21. In this magnified view, the excellent robustness of the SOSM proposed controller can be clearly appraised. To farther prove its advantageous robustness, a more exacting condition is introduced in Figs. 4.22 and 4.23, where the controllers are facing an extreme torque

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Fig. 4.21 Oxygen excess ratio (oxygen stoichiometry)

Fig. 4.22 Torque disturbance

disturbance (four times greater than the original). Even when subjected to this bulk disturbance, the Super-Twisting controller exhibits a remarkable robustness. Comparable satisfactory results have been obtained with the other SOSM controllers. To conclude, it is interesting to stress that the Super-Twisting controller does not require of a state observer depending only on measurements, and its on-line computational burden is considerably lower.

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Fig. 4.23 Oxygen excess ratio (oxygen stoichiometry)

4.8 Conclusions In this chapter the feasibility of SOSM techniques to control PEM fuel cells has been assessed. The evaluation has been conducted using a benchmark model of a fuel cell system for an electric vehicle. The analysis has established the viability of SOSM techniques for oxygen stoichiometry control, aiming to improve the overall energy efficiency. Taking into account several features, such as the controlled system performance, robustness and implementation simplicity, the SOSM controllers are shown to be a highly efficient solution for this challenging problem, proving to be capable to robustly tracking the optimal air mass flow, even in the presence of severe external perturbations and model uncertainties. Among them, the Super-Twisting has been considered as a very suitable algorithm for the FCGS, given that it is specially intended for relative degree 1 systems and only requires the real-time knowledge of the sliding variable and not of its derivative. Compared to the standard LQR control, the SOSM controllers demonstrate better robustness features, in a wide range of operation. Additionally, no state observers are required, resulting in a simple and low-computational-cost algorithm. Now that the suitability of SOSM control for FCGS has been confirmed, the following stage will be the development and implementation of these controllers in an experimental fuel cell plant. This will be the subject matter of the following two chapters.

References 1. Arce A (2010) Advanced control for fuel cell systems. PhD thesis, Universidad de Sevilla

References

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2. Boyce M (1982) Gas turbine engineering handbook. Gulf, Houston 3. Gravdahl J, Egeland O (1999) Compressor surge and rotating stall. Springer, London 4. Kunusch C (2006) Second order sliding mode control of a fuel cell stack using a twisting algorithm. Master thesis, Electrical Department, National University of La Plata, Argentina (in Spanish) 5. Kunusch C, Puleston P, Mayosky M, Riera J (2009) Sliding mode strategy for PEM fuel cells stacks breathing control using a super-twisting algorithm. IEEE Trans Control Syst Technol 17:167–174 6. Moraal P, Kolmanovsky I (1999) Turbocharger modeling for automotive control applications. SAE Technical Paper 1999-01-0908 7. Pukrushpan J, Stefanopoulou A, Peng H (2004) Control of fuel cell breathing. IEEE Control Syst Mag 24(2):30–46 8. Pukrushpan J, Stefanopoulou A, Peng H (2004) Control of fuel cell power systems. Springer, Berlin 9. Rodatz P (2003) Dynamics of the polymer electrolyte fuel cell: experiments and model-based analysis. PhD thesis, Swiss Federal Institute of Technology Zurich 10. Springer T, Zawodzinski T, Gottesfeld S (1991) Polymer electrolyte fuel cell model. J Electrochem Soc 138(8):2334–2342

Chapter 5

Control-Oriented Modelling and Experimental Validation of a PEMFC Generation System

5.1 Introduction In the previous chapter, the viability of second-order sliding-mode (SOSM) techniques to control fuel cell (FC) stacks has been theoretically investigated and analysed through computer simulations. After an extensive assessment of their features, their suitability has been well established. Therefore, the natural next step is to face the challenge of designing and implementing an efficient SOSM controller for an actual FC stack. Particularly, the practical case under study in this chapter is an FC-based generation workbench used for laboratory experimentation. Like in most control design procedures, the first and decisive stage is to obtain a reliable and adequate mathematical description of the system. Specifically, the control-oriented nonlinear model is a key requirement for the development of the sliding-mode control algorithms, capable to robustly improve the efficiency of the generation system, avoid irreversible damage in cell membranes and extend the device lifetime. Then, the main goal of the current chapter is to present an experimentally validated, fully analytical model of the laboratory FC flow dynamics, specially developed for SOSM control design, testing and implementation. The modelling process is conducted following a modular methodology combining a theoretical approach, analogous to the one outlined in Chap. 4, together with an empirical analysis based on experimental data. The proposed semi-empirical model is capable to adequately describe the interaction between the different subsystems, while retaining parameters that have physical significance. The systematic procedure developed in this chapter is organised in a way such that it can be used as a general modelling guideline, being straightforward to adapt to different fuel cell systems with few modifications. Concisely, the laboratory test plant under consideration mainly comprises a central fuel cell stack and ancillary units: air compressor, hydrogen storage tank, humidifiers and line heaters (see schematic representation in Fig. 5.1). In addition, to measure the required experimental data, different sensors are incorporated into the system: an air mass flow meter (range 0–15 slpm) at the end of the supply manifold, a piezoresistive pressure transducer (range 0–6 bar) to measure the manifold pressure, a piezoresistive differential pressure transducer (range 0–250 mbar) to measure C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3_5, © Springer-Verlag London Limited 2012

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Fig. 5.1 Fuel cell system diagram

the stack pressure drop, a tachometer (range 0–3000 rpm) on the motor shaft, a current clamp (range 0–3 A) and a voltage meter (range 0–15 V) to measure the motor stator current and voltage, respectively. Besides, temperature sensors are arranged in order to register the different operation conditions. In the sequel the following modelling assumptions have been considered: • A mass flow control device ensures a constant hydrogen supply. • An auxiliary control system efficiently regulates gas temperatures at five points of the plant (cathode and anode humidifiers, cathode and anode line heaters and stack). • A humidity control loop regulates moisture to a relative level close 100%. • The fuel cell model is one-dimensional, so the gases and reactions are considered uniformly distributed in the cell. • The electrochemical properties are evaluated at the average stack temperature (60 °C), so temperature variations across the stack are neglected. • The water entering to the cathode and anode is only in vapour phase. • The effects of liquid water creation are negligible at the gas flow model level. • The water activity is uniform across the membrane and is in equilibrium with the water activity at the cathode and anode catalyst layers. General physics constants used in calculations and typical operating conditions are summarised in Appendix B, Tables B.1 and B.5.

5.2 Air Compressor Subsystem The air compressor is a 12-V DC oil-free KNF® diaphragm vacuum pump, which is based on a simple principle: an elastic diaphragm, fixed on its perimeter, moves up and down its central point by means of an eccentric. On the down-stroke it draws

5.2 Air Compressor Subsystem

107

Fig. 5.2 Schematic diagram of the compressor subsystem

the air or gas being handled through the inlet valve. On the up-stroke the diaphragm forces the gas through the exhaust valve and out of the head. The compression chamber is hermetically separated from the drive mechanism by the diaphragm. The pump transfers, evacuates and compresses completely oil-free gas and is driven by a 15-W DC motor. The equations that describe the behaviour of the system are obtained by analysing the air compressor as two coupled subsystems. The first one embodies the permanent magnet DC motor dynamics, and the second one represents the compressing diaphragm nonlinear characteristics (Fig. 5.2).

5.2.1 Air Compressor Motor Dynamic Equations The following equations summarise the dynamic model of the compressor DC motor [7]: dia (t) Vcp (t) = L + Ria (t) + kφ ωcp (t) dt   dωcp = Te (t) − Tl ωcp (t), Pcp (t) J dt

(5.1) (5.2)

with Te (t) = kφ ia (t)

(5.3)

where Vcp is the armature voltage, ia the armature current, L and R the electrical inductance and resistance of the stator winding, kφ the motor constant, ωcp the shaft angular speed, Pcp the absolute pressure at the compressor output, J the inertia, Te the electrical torque, and Tl is a nonlinear function that groups together the effects of the motor and diaphragm friction and the pneumatic load. The computations of the electrical and mechanical parameters of the compressor and the load torque function Tl are systematically developed in the sequel. To start with, the electrical resistance and inductance of the stator winding can be directly measured through an electronic impedance meter. Then, the kφ value is determined using the electrical equation of the motor (5.1) in steady-state operation: L

dia (t) = 0 = Vcp − Ria − kφ ωcp dt

V −R.i

(5.4)

From (5.4) kφ = cpωcp a can be computed by measuring ia and ωcp at different equilibria. Figure 5.3 shows that for various compressor pressures (Pcp ), the value of kφ remains constant.

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Fig. 5.3 kφ vs. ωcp (experimental data)

The next step deals with the characterisation of the load torque function Tl that lumps the friction and the pneumatic loads. In a first approach, it can be modelled as a nonlinear static function of ωcp and Pcp . For modelling purposes, this load torque expression was divided into two terms: Tl (ωcp , Pcp ) = Tl,amb (ωcp ) + Tl (ωcp , Pcp )

(5.5)

The former corresponds to the load torque of the system operating at ambient pressure. The latter takes into account the extra torque that appears when the diaphragm vacuum pump operates at pressures higher than the ambient [7]. The experimental values of the load torque can be computed using data obtained from steady-state operation tests. Under these conditions, ω(t) ˙ is zero, and thus Eq. (5.2) gives Te (t) = kφ ia (t) = Tl (ωcp , Pcp )

(5.6)

and Tl can be readily inferred from direct measurement of the current ia . Then, with the assistance of (5.2), the values of the first term of (5.5), Tl,amb , are obtained conducting experiments at ambient pressure and different shaft speeds. Analysing the data (see Fig. 5.4), it can be concluded that Tl,amb can be well approximated by a linear expression, such as Tl,amb (ωcp ) = A0 + A1 ωcp

(5.7)

The fitting parameters A0 and A1 can be found in Appendix B, Table B.2. To find the expression of the second term of (5.5), Tl , a new set of experiments is required, with the compressor working at different speeds and compressor pressures, covering its entire range of operation (60 rad/s < ωcp < 360 rad/s, 1 bar < Pcp < 2.5 bar). Then, from (5.5), (5.6) and (5.7), Tl can be written in terms of the current ia and the speed ωcp , both measurable variables:

5.2 Air Compressor Subsystem

109

Fig. 5.4 Tl,amb vs. ωcp : measured data and linear approximation

Tl (ωcp , Pcp ) = Tl (ωcp , Pcp ) − Tl,amb (ωcp ) = kφ ia − A0 − A1 ωcp

(5.8)

Then using (8) in combination with data gathered in the experiments conducted with different values of < ωcp < and < Pcp <, the following bivariate quadratic function can be obtained by means of polynomial regression modelling tools (e.g. polyfitn [8]): 2 2 + A01 Pcp + A11 ωcp Pcp + A02 Pcp Tl = A00 + A10 ωcp + A20 ωcp

(5.9)

with constant coefficients Aij (see Appendix B, Table B.3). In Fig. 5.5, two surfaces are displayed. The first one (coloured surface) results from the interpolation of experimental data (dots and crosses). The second one (grey surface) depicts the quadratic approximation (9) that best fits the obtained data. In fact, the approximation error of ωcp remains below 3% in all the tested working conditions. Note that with this approach, losses due to deviations from the isentropic compression and unmodelled friction terms are now incorporated into the model [13]. The next parameter to be estimated is the motor inertia. Considering the practical impossibility to directly measure J , its value can be determined by adjusting the dynamic response of the model variable ωcp (t) to the experimental data. In the validation test presented in Fig. 5.6, step variations are applied to the compressor input voltage. It can be seen that the angular speed matches the experimental values. This result is verified in a broad span of working conditions, indicating that the developed model allows for predicting the motor behaviour in an extended range of

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Fig. 5.5 Tl surface interpolation from real data (coloured surface) and polynomial approximation (grey surface)

Fig. 5.6 Validation of the DC motor model dynamics

5.2 Air Compressor Subsystem

111

Fig. 5.7 Wcp surface interpolation from real data (coloured surface) and polynomial approximation (grey surface)

operation. In fact, the approximation error of ωcp attained with proposed approach is less than 3% in all tested working conditions.

5.2.2 Diaphragm Vacuum Pump Characteristics The next modelling stage required to complete the compressor model is the characterisation of the map that relates the output flow Wcp with the internal variable ωcp and the compressor pressure Pcp . To this end, several steady-state tests exploring different operating conditions have been conducted in order to gather data from the mass flow meter, the tachometer and the pressure transducer, respectively. Then, following a fitting procedure similar to the one performed with Tl , an approximating bivariate function is obtained: 2 2 Wcp = B00 + B10 ωcp + B20 ωcp + B01 Pcp + B11 ωcp Pcp + B02 Pcp

(5.10)

with Bij constant values. In Fig. 5.7 the Wcp approximated surface corresponding to Eq. (5.10) is shown (grey surface), as well as the spline interpolation of the real data coloured surface. The parameters and coefficients of the compressor under study can be found in Appendix B, Table B.3.

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Fig. 5.8 Compressor dynamics

5.3 Air Supply Manifold Subsystem The air supply manifold of the laboratory fuel cell system under study is relatively small. From the control viewpoint, a reasonable good model approximation can be attained by neglecting its dynamics, assuming that the compressor dynamics are dominant. Thus, the effect of the supply manifold can be taken into account by merely readjusting the value of J determined in Sect. 5.2.1. The new value of this equivalent inertia Jg is computed from experimental tests (Jg = 1.2 × 10−6 N m). Figure 5.8 presents the time evolution of the air flow of the compressor (including manifold). It can be observed that, even during transients, the model provides a very good approximation of the experimental data. This result is also verified at different air compressor flows and pressures, covering the whole operating range. The previous approximated model proved to be sufficiently accurate to be used in the subsequent SOSM control design procedure. Nevertheless, in the case that certain application would require a more precise description, the supply manifold air dynamics could be fully modelled using the principles of mass and momentum conservation [5]. Similarly to the case study modelled in Chap. 4, the general equations that govern the supply manifold are: dmsm = Wcp − Wsm dt

(5.11)

dPsm Ra (Wcp Tcp − Wsm Tsm ) = dt Vsm

(5.12)

together with the nozzle equation Wsm = Ksm (Psm − Phum )

(5.13)

5.4 Air Humidification Subsystem

113

where msm is the supply manifold mass of air, Wsm the supply manifold output flow, Ra the atmospheric air gas constant, Vsm the supply manifold volume, Tcp the compressor air temperature, Tsm the supply manifold air temperature, Psm the supply manifold air pressure, Phum the humidifier pressure, and Ksm is the restriction constant of the output nozzle. In the general case, the temperatures Tsm and Tcp are different; however in the laboratory FC stack this process is practically isothermal, so Eqs. (5.11) and (5.12) become equivalent, reducing the order by one. Additionally, in this system, due to the presence of a humidifier before the cathode, Phum appears instead of Pca in the output nozzle equation. Note that Ra is a known constant and Vsm is easily measurable, so the single parameter left to be estimated is Ksm . This goal can be achieved by setting the compressor flow Wcp at different steady-state operating points and collecting data of the differential pressure Psm − Phum . Then, Ksm can be obtained by adjusting the restriction constant value such that the experimental data satisfies Eq. (5.13). Note that this value does not exactly correspond to the actual output nozzle. It can be interpreted as the restriction constant of an equivalent air supply manifold, which is taking into account the effect of any possible remaining dynamics, disregarded in the model of the compressor subsystem developed in the previous subsection.

5.4 Air Humidification Subsystem Cellkraft® P-series humidifiers based on exchange membranes are used to maintain proper humidity conditions inside the cells, which is crucial to ensure the optimal operation of PEM membranes. Gas humidification at flows rates up to 10 slpm and pressures close to 10 bar can be achieved with this sort of devices. The line heaters and stack temperatures are controlled by a power station via decentralised PID controllers, allowing for independent gas conditions to the stack. The operating principle of the membrane humidification technique is feeding deionised temperature-controlled water and the gas to be humidified to each side of a polymeric membrane, respectively. During the process, a humidity gradient is established across the membrane, allowing the transference of water towards the circulating gas by diffusion. The degree of humidification is regulated by adjusting the water temperature within the humidifier. The closed-loop control of moisture can be performed by two different methods, i.e. controlling either the dew point of the gas or the water temperature [3]. Prior to entering the stack, the humidified gas circulates through a line heater. The purpose of this device is twofold. On the one hand, by rising the gas temperature, condensation inside the cathode channels is prevented. On the other hand, regulation of the differential temperature (between the humidifier and line heater) allows one to control the relative humidity of the cathode inlet gas flow. To obtain the equations that govern the humidification subsystem, the modelling process is divided into two steps. First, it is assumed that the subsystem dynamics is dominated by the air humidifier pressure change, without taking into account the

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Fig. 5.9 Schematic diagram of the humidification subsystem

effect of the vapour injected to the gas (see Fig. 5.9). Then, as a second stage, the effect of the vapour flow injected to the gas, Wv,inj , is taken into consideration by adding a nonlinear static term, function of the gas temperature, pressure and speed conditions [7].

5.4.1 Step 1 Under the aforementioned assumptions and considering that the humidifier pressure is similar to the compressor load pressure (Pcp = Phum ), the equations of the cathode humidifier dynamics are given by the following expressions: dmhum = Wcp − Whum dt

(5.14)

Ra dPhum = (Wcp Tcp − Whum Thum ) dt Vhum

(5.15)

Whum = f (Thum , Pca , Phum,diff )

(5.16)

Phum,diff = Phum − Pca

(5.17)

with

where mhum and Phum are the mass and pressure of air inside the humidifier, Whum is the flow of air that leaves the humidifier, Vhum is the volume of the humidifier, Thum is the gas temperature inside the humidifier, Phum,diff is the humidification subsystem pressure drop, and Pca is the cathode inlet pressure. Note that in the approximated model proposed in Sect. 5.3 the effect of the manifold has been incorporated into the compressor model hence, in (5.14), Wcp has been considered as the inlet flow into the humidifier. The right-hand side of (5.16) corresponds to a nonlinear nozzle function, strongly dependent on the humidifier gas temperature, the cathode pressure and the humidifier pressure drop (see Fig. 5.10). It could be approximated by a trivariate function or a family of bivariate functions (parameterised by Thum ), obtained through experimental data gathered from tests performed at different operating temperatures and pressures. In particular, the current laboratory test plant is set to operate at constant stack temperature of 60 °C and humidifier temperature of 55 °C, respectively, regulated through external controllers. The former is a recommended operation temperature for the equipment, whereas the latter is adjusted to obtain high relative

5.4 Air Humidification Subsystem

115

Fig. 5.10 Humidifier characteristics (fits and real data)

humidity of the cathode inlet flow (> 79%) while preventing vapour condensation. Then, considering a constant humidifier working temperature, the nozzle function can be well approximated by the following bivariate function: Whum = C0 + C1 Phum,diff Pca + C2 Pca

(5.18)

where the coefficients C0 , C1 and C2 are experimentally determined from the tests (see thick solid lines in Fig. 5.10). Coefficient values are given in Appendix B, Table B.4.

5.4.2 Step 2 In this step, the vapour injected to the air stream is incorporated to the model. Then, the total humidified air flow entering the cathode (Wca ) is given by Wca = Whum + Wv,inj

(5.19)

Assuming that the humidifying closed-loop system of the device efficiently regulates the gas relative humidity, the computation of the injected water to the air flow can be described by Wv,inj =

Gv RH hum Psat (Thum )Wa,hum − Wv,hum Ga Pa

(5.20)

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Gv being the vapour molar mass, Ga the dry air molar mass, and Psat (Thum ) the vapour saturation pressure at the humidifier temperature. RH hum is the relative humidity of the gas exiting the humidifier, which in normal operating conditions can be considered a known value, in accordance with the humidifier technical specifications (usually, close to 100%). The dry air partial pressure (Pa ), the dry air output flow of the humidifier (Wa,hum ) and the flow of vapour due to ambient moisture entering the humidifier (Wv,hum ) are variables that depend on the ambient conditions and can be directly computed using the following relationships: Wa,hum =

1 Whum,out 1 + ωhum

Wv,hum = Whum − Wa,hum

(5.21)

(5.22)

with ωhum =

Psat (Tamb )RH amb Gv Ga Pamb − Psat (Tamb )RH amb

(5.23)

where ωhum is the ambient humidity ratio, Tamb the ambient temperature, Pamb the ambient pressure, and RH amb is the ambient relative humidity. At this point, there is only one parameter left to be estimated to complete the humidification subsystem model. This is the humidifier volume (Vhum ), present in the dynamic equation (5.15). Instead of using a measured value of the volume, a convenient estimation of this parameter can be attained by adjusting the transient response of the modelled Phum to match the experimental data (tests varying the compressor air flow at fixed humidifier temperature are considered). The estimated value does not exactly correspond to the actual humidifier volume. It rather represents the volume of an equivalent humidification subsystem, which takes into account modelling errors and unmodelled dynamics. Figure 5.11 shows that highly satisfactory dynamic validation results are achieved (refer to Appendix B, Tale B.4 for estimated parameters). A final remark is pertinent to close this section. It was previously mentioned that regulation of the line heater temperature allows controlling the relative humidity of the gas. Then, in accordance with the Dalton law, the effect of the increase of temperature (from Thum to Tlh ) on the partial pressures and the relative humidity of the cathode inlet gas flow can be easily computed through Tlh Pi,hum Thum Pv,lh RH lh = Psat (Tlh ) Pi,lh =

(5.24) (5.25)

where i stands for O2 , N2 and vapour, respectively, and Psat (Tlh ) is the vapour saturation pressure at the line heater temperature Tlh .

5.5 Fuel Cell Stack Flow Subsystem

117

Fig. 5.11 Humidifier dynamic validation

5.5 Fuel Cell Stack Flow Subsystem The stack is an EFC50-ST ElectroChem® , which is a laboratory PEM fuel cell system designed for the study of membrane electrode assemblies and fuel cell operation. The system consists of a seven-cell stack with Nafion® 115 membranes, platinum catalyst loading of 1 mg cm−2 , Toray carbon fibre papers as gas diffusion layers and 50 cm2 of active area. This unit generates up to 100 W.

5.5.1 Cathode Channels The dynamic mass balance within the stack channels depends on the gases partial pressures, on the water transported by the membrane and on the electric current drained from the stack (Ist ). The cathode state variables are the masses of the circulating elements, i.e. oxygen (mO2 ,ca ), nitrogen (mN2 ,ca ) and vapour (mv,ca ). Then, the dynamic equations that govern the behaviour of the gases inside the cathode are given by dmO2 ,ca = WO2 ,ca,in − WO2 ,ca,out − WO2 ,react dt

(5.26)

dmN2 ,ca = WN2 ,ca,in − WN2 ,ca,out dt

(5.27)

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dmv,ca (5.28) = Wv,ca,in − Wv,ca,out + Wv,ca,gen + Wv,mem dt while the following relationships hold for the cathode input and output flows (Wca and Wca,out ): Wca = WO2 ,ca,in + WN2 ,ca,in + Wv,ca,in

(5.29)

Wca,out = WO2 ,ca,out + WN2 ,ca,out + Wv,ca,out

(5.30)

where WO2 ,ca,in , WN2 ,ca,in , Wv,ca , WO2 ,ca,out , WN2 ,ca,out and Wv,ca,out are the input and output flows of oxygen, nitrogen and vapour, WO2 ,react the flow of oxygen that reacts in the cathode, Wv,ca,gen the flow of vapour generated in the reaction, and Wv,mem is the flow of water transferred across the membrane (comprising the electro-osmotic drag term and the back-diffusion term). Subsequently, the calculation of the flow terms that constitute the right-hand sides of Eqs. (5.26)–(5.28) must be addressed. To begin with, the amount of reduced oxygen and generated vapour in the cathode reaction is computed from the stack current, according to the following two electrochemical principles: nIst (5.31) 4F nIst Wv,ca,gen = Gv (5.32) 2F where n is the number of cells of the stack, and GO2 and Gv are the molar masses of oxygen and vapour, respectively, and F is the Faraday constant. Next, the components of the cathode input and output flows are considered. First, assuming knowledge of Wca from (5.19), the partial input flows WO2 ,ca,in , WN2 ,ca and Wv,ca are readily calculated using (5.29): WO2 ,react = GO2

1 Wca 1 + ωca 1 Wca WN2 ,ca,in = (1 − XO2 ,ca ) 1 + ωca Wv,ca,in = Wca − WN2 ,ca,in − WO2 ,ca,in

WO2 ,ca,in = XO2 ,ca

(5.33) (5.34) (5.35)

where ωca is the humidity ratio, and XO2 ,ca is the mass mole fraction of the input air flow, given by Gv Pv,lh Ga (PO2 ,lh + PN2 ,lh ) χO2 GO2 XO2 ,ca = χO2 GO2 + (1 − χO2 )GN2 ωca =

(5.36) (5.37)

χO2 being the oxygen mole fraction of the ambient air. Second, using (5.30), the partial output flows WO2 ,ca,out , WN2 ,ca,out and Wv,ca,out can be obtained following a similar procedure:

5.5 Fuel Cell Stack Flow Subsystem

119

1 Wca,out 1 + ωca,out 1 Wca,out WN2 ,ca,out = (1 − XO2 ,ca,out ) 1 + ωca,out Wv,ca,out = Wca,out − WO2 ,ca,out − WN2 ,ca,out WO2 ,ca,out = XO2 ,ca,out

(5.38) (5.39) (5.40)

with the output humidity ratio and mass mole fraction ωca,out = XO2 ,ca,out =

Gv Pv,ca Ga (PO2 ,ca + PN2 ,ca )

χO2 ,out GO2 χO2 ,out GO2 + (1 − χO2 ,out )GN2 χO2 ,out =

PO2 ,ca Pca

(5.41) (5.42) (5.43)

where χO2 ,out is the cathode oxygen mole fraction. However, for these computations, the cathode output flow Wca,out is not yet available, given that it is not measurable due to its high vapour content. It must be indirectly obtained, making use of the pressure drop measurement. The relationship between the output flow and the pressure drop can be modelled as a linear nozzle equation: Wca,out = Kca,out (Pca − Pamb )

(5.44)

where Pamb is the ambient pressure, and Kca,out is the cathode output restriction that can be also governed through a mechanical back pressure regulator. Then, to compute Wca,out , it is necessary to determine Kca,out (with a bypass in the cathode back pressure regulator). To estimate this parameter, experimental data of the pressure drop and the cathode output flow are required. The former is available from the differential pressure transducer, but, as it was previously said, no direct measurement of Wca,out is feasible due to its high relative humidity. However, under appropriate experimental conditions, its steady-state values can be inferred from measurements of the compressor flow Wcp . The estimation test conditions are: (a) steady-state operation, (b) equally humidified reactant gases and (c) nil stack current. On the one hand, Ist = 0 guarantees that the liquid water (Wl,ca,out ) and the reaction flows (WO2 ,react and Wv,ca,gen ) remain zero. On the other hand, considering anode and cathode gases at similar relative humidities ensures that at steady-state operation there is no water concentration gradient across the membrane, so the effect of Wv,mem can be neglected. Therefore, under these testing conditions, Wca,out is equal to Wca (see (5.26)–(5.28)). Then, using (5.14), (5.19) and (5.20), the data of Wcp allow the computation of Wca,out and, consequently, the estimation of the nozzle restriction [7]. Note that the partial pressures of the gases inside the cathode, required in (5.36), can be obtained from the stack temperature and the masses of oxygen, nitrogen and vapour. Using the Dalton law, the cathode partial pressures and relative humidity are:

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mi,ca Ri Tst Vca Pv,ca RH ca = Psat (Tst ) Pi,ca =

(5.45) (5.46)

where subscript i stands for O2 , N2 and v, respectively, and Vca is the cathode volume. The last flow term of (5.26)–(5.28) to be computed is the water transferred across the membrane. To this end, the anode relative humidity is required, so the anode flow model will be first addressed, and, subsequently, the calculation of Wv,mem will be resumed.

5.5.2 Anode Channels In this type of PEMFC systems the input hydrogen flow is independently regulated; thus the input Wan is assumed to be a known constant. Moreover, due to the fact that Wan remains constant and the anode humidifier and line heaters are controlled, it is not necessary to model such gas conditioning devices in the anode line. Under this conditions, the dynamics of the anode channel can be modelled by dmH2 ,an = WH2 ,an,in − WH2 ,an,out − WH2 ,react dt dmv,an = Wv,an,in − Wv,an,out − Wv,mem dt while the following equations hold for the anode input and output flows:

(5.47) (5.48)

Wan = WH2 ,an,in + Wv,an,in

(5.49)

Wan,out = WH2 ,an,out + Wv,an,out

(5.50)

where WH2 ,an,in , Wv,an,in , WH2 ,ca,out and Wv,an,out are the input and output flows of hydrogen and vapour, respectively, WH2 ,react is the flow of hydrogen consumed in the reaction, and Wv,mem is the aforementioned flow of water transferred to the cathode. In this particular case, no liquid water is supposed to be condensed in the anode channels, given that in normal working conditions the relative humidity of the anode is always below 100%. On the other hand, the hydrogen consumed in the reaction is nIst (5.51) 2F where GH2 stands for the molar mass of hydrogen. Analogously to the cathode channel, the components of the anode input and output flows must be calculated. The partial input flows WH2 ,an,in and Wv,an,in are obtained through WH2 ,react = GH2

5.5 Fuel Cell Stack Flow Subsystem

121

WH2 ,an,in =

1 Wan 1 + ωan

Wv,an,in = Wan − WH2 ,an,in ωan =

Gv Pv,lh,an GH2 PH2 ,lh,an

(5.52) (5.53) (5.54)

where ωan is the humidity ratio of the anode input gas, PH2 ,lh,an the anode input H2 partial pressure, and Pv,lh,an is the anode input vapour partial pressure that can be obtained using the Dalton law. Besides, the partial output flows WH2 ,an,out and Wv,an,out are computed as follows: 1 Wan,out (5.55) WH2 ,an,out = 1 + ωan,out Wv,an,out = Wan,out − WH2 ,an,out ωan,out =

Gv Pv,an GH2 PH2 ,an

(5.56) (5.57)

where ωan,out is the humidity ratio of the gas inside the anode, PH2 ,an the anode H2 partial pressure, and Pv,an is the anode vapour partial pressure.

5.5.3 Membrane Water Transport Now the calculation of Wv,mem can be taken up again. The flow of water across the membrane is modelled assuming linear concentration gradients from channels inlet to outlet and across the membrane thickness. Then, it can be expressed as [15]   i cv,ca − cv,an (5.58) Gv Afc n Wv,mem = nd + Dw F tm where i is the stack current density, Afc the fuel cell active area, tm the membrane dry thickness, and cv,ca and cv,an are the water concentrations at the membrane surfaces on the cathode and anode sides, respectively. The term nd is the electro-osmotic drag coefficient (number of water molecules carried by each proton), and Dw is the back-diffusion coefficient of the membrane. The water concentration terms are determined from the membrane water contents on the cathode (λca ) and anode (λan ) sides: ρm,dry λca (5.59) cv,ca = Gm,dry ρm,dry cv,an = λan (5.60) Gm,dry

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Fig. 5.12 Experimental and estimated membrane water diffusion characterisation

where ρm,dry is the membrane dry density, and Gm,dry is the membrane dry molecular weight. The water content in the membrane is defined as the ratio of water molecules to the number of charge sites. When no liquid water is present in the channels, the ratio can be estimated at both sides using the following equation [15]: λj = a0 + a1 RH j + a2 RH 2j + a3 RH 3j

(5.61)

RH j being the gas relative humidity, and subscript j referring to cathode or anode (j = ca, an), respectively. The next step is to estimate the apparent diffusion coefficient Dw of expression (5.58). Two different experiments can be set-up to compute this parameter, either a cathode or an anode drying test. In both cases, the stack current must be set to zero (Ist = 0) in order to cancel the stack current density i in (5.58). For the former, a long-term cathode drying procedure is conducted, decreasing the cathode humidifier temperature from 55 °C to 40 °C, while setting the temperatures of the anode humidifier, both line heaters and the stack at 60 °C. With this test, a water concentration gradient is established between the channels, and an increasing extra flow in the cathode output can be detected due to the membrane contribution [7]. The second test is conducted analogously to the first one, but in this case the anode channel is dried out, keeping the other variables at similar stationary conditions. Following this simple procedure, the Dw coefficient can be directly determined without using humidity sensors or a more specific equipment. Figure 5.12 displays the data gathered from both tests (the average value obtained for the backdiffusion coefficient is given in Appendix B, Table B.4).

5.6 Electrical Characterisation of the Fuel Cell Stack

123

Finally, the electro-osmotic coefficient nd is characterised through the widely accepted expression developed in [4] and reported in [11] and [9]: nd = n0 + n1 λm + n2 λ2m

(5.62)

where λm is the average membrane water content, which can be derived from equation (5.61) considering RH m = (RH ca + RH an )/2

(5.63)

5.6 Electrical Characterisation of the Fuel Cell Stack Up to this point, the equations and parameters required to build the model of the laboratory FC flow dynamics have been obtained. To complete the system model, it is of interest to provide an electrical characterisation of the fuel cell stack behaviour. To this end, a semi-empirical approach, derived from the physical processes that occur in the MEA and the phenomena that take place in the channels, is used to model the output electrical properties of the stack under different operating conditions, covering the whole operation range of air flow rates and pressures [6]. As explained in Sect. 2.2, combining the open circuit voltage E with the effects of activation, ohmic resistance and concentration, the operation voltage of a PEM fuel cell (Vfc ) can be described by the following expression: Vfc = E − Vact − Vohm − Vconc

(5.64)

The parameters that define expression (5.64) can be computed with the help of appropriate experimental data. Under these conditions, the term E can be substituted by V0 = E0 = 1.229 V, the theoretical electrochemical potential of a single hydrogen/oxygen fuel cell. Then, the following particular expression is valid:     RTst PO2 ,ca i (nc ·i) Vfc = V0 − − (R0 − R1 λm )i − me + b · ln ln (5.65) 2αF i0 a where i is the current density, and α, i0 , R0 , R1 , m, nc , b and a are empirical parameters obtained from the experiments, which are required to characterise the general expression (5.64) (see the explicit equation in Chap. 2). Hereinafter the section is devoted to describe the procedure followed to obtain such parameters. This procedure can be used as a general guideline applicable to different PEM fuel-cell-based generation systems. First, considering that the activation losses do not depend on the pressure and oxygen stoichiometry, the coefficients α and i0 are calculated. To this end, a logarithmic plot of the V–I curve is a helpful tool to better determine the electrical behaviour of the MEAs at low current densities (see Fig. 5.13). In this particular system, α = 0.7 and i0 = 4.5 × 10−6 A/cm2 are the values that better adjust the experimental data when the cathode air flow rates (Wca ) is varied from 1.108 × 10−5 kg/s to 7.707 × 10−5 kg/s and Pan = Pca = 1 bar. It is important to stress that in all the experiments presented in this section the air and hydrogen

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Fig. 5.13 V–I logarithmic curve

dew points are set at 55 °C, while the line heaters and stack temperatures are fixed at 60 °C in order to avoid water condensation inside the stack. Considering these externally regulated temperatures, the relative humidity of the inlet gases is 0.75. The stack current profile is set to vary from 0 A to 12 A, and the anode hydrogen flow is fixed at 6.483 × 10−6 kg/s (2 slpm of dry H2 ). Under these testing conditions, the hydrogen stoichiometry remains above 5, even at the highest current (12 A), and thus the losses due to hydrogen concentration can be neglected [6]. At this stage, is important to remark that liquid water occupies the pore volume and reduces the gas diffusion layer surface, resulting in a decrease in voltage and fuel cell efficiency. This phenomenon is unimportant in the cathode and anode dynamics because it does not interfere with the channels gas transport as the water excess is dragged by the flows. Nevertheless, it affects the system efficiency because the amount of liquid water inside the diffusion gas layers increases, decreasing the membrane effective area (Afc ), resulting in an increased current density and hence a drop in voltage [1, 10, 14]. Consequently, for model purposes, an apparent fuel cell area (Aapp ) can be defined, to take into account the surface reduction due to the liquid film at the interface of the gas diffusion layers and membrane. In the present test bench there is not a purging schedule, and cathode and anode flows are kept constant in every operating conditions, so the aforementioned effect can be modelled with the incorporation of a scaling factor for Afc , which takes into account the membrane effective area decrease due to liquid water generation: i=

Ist Ist = Aapp Afc (1 − αca mH2 O,ca,liq − αan mH2 O,an,liq )

(5.66)

5.6 Electrical Characterisation of the Fuel Cell Stack

125

Fig. 5.14 Membrane water content

where mH2 O,ca,liq and mH2 O,an,liq are the cathode and anode liquid water masses in the channels, respectively. This terms were not approximated in this modelling approach, due to the fact that the model presented in this book is mainly oriented to be used in gases flow control issues. Besides, note that the parameters αca and αan can be adjusted in order to take into account the modulation effects of cathode and anode water accumulation in the stack voltage (low-frequency phenomena). Nevertheless, for steady-state computations done in this section, the approximated value of the apparent fuel cell area for the current stack was Aapp = 20.74 cm2 for all working conditions. Subsequently, the ohmic polarisation is determined in base of the membrane average water content λm (Eqs. (5.61)–(5.63)). The following figure (Fig. 5.14) depicts the profile described by λm in the validated model of the stack presented in Sect. 5.5, at different current densities and air flow rates. Considering that above a certain value of i the relationship between λm and i remain fixed, it is natural to assume that the ionic conductivity of the membrane remains practically unaltered at different air flow rates. This assumption conducts to select a small value of R1 and R0 close to the average resistance of the MEA, according to the data. In this particular case, the most representative values for the stack cells are R1 = 0.005 /cm2 and R0 = 0.22 /cm2 . The next pair of parameters to be estimated are a and b, which consider the effect of low oxygen stoichiometry in the cathode. As it can be seen in Fig. 5.15, in all polarisation curves there is a significant voltage drop at certain current densities. Moreover, the values of current densities where voltage drops start vary with the air mass flow. Then, in the model, the parameter a can be associated to the value

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Control-Oriented Modelling and Experimental Validation

Fig. 5.15 Polarisation curve of the fuel cell stack at constant pressure and different levels of air flow

of oxygen partial pressure that produces the steep effect of the concentration loss. On the other hand, parameter b represents the sensitivity of the polarisation curve at different air flow rates, so it can be adjusted depending on the data obtained from the polarisation curves at the minimum and maximum air flow rates (red line and yellow line in Fig. 5.15). For the laboratory fuel cell stack, the estimated values of these parameters were a = 0.2 bar and b = 0.06 V. A standard curve fitting procedure was followed to estimate the parameters [6]. Lastly, values m and nc are adjusted in order to verify the general gas diffusion loss at high stack currents. The obtained values from the fitting were m = 8.1 × 10−5 V and nc = 15.2 cm2 /A. Figure 5.15 presents the polarisation curves obtained at constant pressure (1 bar) and different levels of air flow (1.108 × 10−5 kg/s to 7.707 × 10−5 kg/s), while Fig. 5.16 depicts the polarisation curves obtained at constant air flow (7.707 × 10−5 kg/s) and increasing levels of working pressures (1–2.35 bar). In these figures it can be appreciated that the model based on Eq. (5.65) with the proposed parameters satisfactorily predicts the fuel cell V–I behaviour under different tested conditions of air flow rates and cathode pressures.

5.7 Conclusions The control-oriented modelling of an actual PEM fuel cell stack has been approached in this chapter. The proposed procedure tackles the modular modelling of

5.7 Conclusions

127

Fig. 5.16 Polarisation curve of the fuel cell stack at constant air flow and increasing levels of working pressure

an experimental complex system that combines mechanical, electrical, pneumatic and electrochemical subsystems. It provides a nonlinear characterisation that satisfactorily describes the steady-state and dynamical behaviour, successfully covering the entire operation range of the fuel-cell-based system under study. Due to the fact that the model has been primarily developed for model-based control studies, a system level approach has been considered and only dynamics in the range of 10−2 to 100 seconds has been taken into account. The semi-empirical methodology followed in this chapter is not an example on identification nor a theoretical exercise. Guided by the knowledge of the processes and reactions that take place in the real fuel cell, the different components were modelled using available general information and particular experimental data, gathered from simple tests. Therefore, the proposed procedure can be used as a guide for control-oriented modelling of PEM fuel cell systems with similar features. Important control problems found in PEM fuel cells such as the ones presented in [2, 11, 13] and [12] (H2 /O2 stoichiometry regulation, total and partial pressures control, H2 consumption minimisation, etc.) can be approached using the developed control model. In particular, taking advantage of the continuous, smooth dynamic equations presented here, in the beginning of next chapter a seventh-order nonlinear state space model, primarily focused on the fuel cell fluid dynamics, will be formulated as the first stage for SOSM control design and practical implementation.

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References 1. Arce A (2010) Advanced control for fuel cell systems. PhD thesis, Universidad de Sevilla, Spain 2. Arce A, del Real AJ, Bordons C, Ramírez DR (2010) Real-time implementation of a constrained MPC for efficient airflow control in a PEM fuel cell. IEEE Trans Ind Electron 57(6):1892–1905 3. Cellkraft (2007) P-10 humidifier manual: v 2.0 4. Dutta S, Shimpalee S, Van Zee JW (2001) Numerical prediction of mass-exchange between cathode and anode channels in a PEM fuel cell. Int J Heat Mass Transf 44:2029–2042 5. Helvoirt J, Jager B, Steinbuch M, Smeulers J (2005) Modeling and identification of centrifugal compressor dynamics with approximate realizations. In: IEEE conference on control applications, Toronto, Canada 6. Kunusch C, Puleston PF, Mayosky MA, More J (2010) Characterization and experimental results in PEM fuel cell electrical behaviour. Int J Hydrog Energy 35:5876–5881 7. Kunusch C, Puleston PF, Mayosky MA, Husar A (2011) Control-oriented modelling and experimental validation of a PEMFC generation system. IEEE Trans Energy Convers 26(3):851– 861 8. D’Errico J (2006) Polyfitn (N-d polynomial regression model). http://www.mathworks.com/ matlabcentral/fileexchange/10065 9. McKay DA, Ott WT, Stefanopoulou AG (2005) Modeling, parameter identification, and validation of reactant and water dynamics for a fuel cell stack. In: ASME international mechanical engineering congress & exposition 10. McKay DA, Siegel JB, Ott WT, Stefanopoulou AG (2008) Parameterization and prediction of temporal fuel cell voltage behavior during flooding and drying conditions. J Power Sources 178:207–222 11. Pukrushpan JT, Stefanopoulou AG, Peng H (2004) Control of fuel cell power systems. Springer, Berlin 12. Ramos-Paja C, Bordons C, Romero A, Giral A, Martinez-Salamero L (2009) Minimum fuel consumption strategy for PEM fuel cells. IEEE Trans Ind Electron 56(3):685–696 13. Rodatz PH (2003) Dynamics of the polymer electrolyte fuel cell: experiments and modelbased analysis. PhD thesis, Swiss Federal Institute of Technology Zurich 14. Siegel JB, McKay DA, Stefanopoulou AG (2008) Modeling and validation of fuel cell water dynamics using neutron imaging. In: American control conference 15. Springer TE, Zawodzinski TA, Gottesfeld S (1991) Polymer electrolyte fuel cell model. J Electrochem Soc 138(8):2334–2342

Chapter 6

SOSM Controller for the PEMFC-Based Generation System. Design and Implementation

6.1 Introduction The main objective of this chapter is to present the actual development and experimental validation of SOSM controllers, previously reviewed and analysed in this book. As anticipated, the controllers were designed and implemented in the real FC-based generation workbench introduced and modelled in Chap. 5. Three control set-ups based on Super-Twisting, Twisting and Sub-Optimal algorithms were developed. As detailed in Chap. 4, power conversion optimisation of the laboratory PEM fuel cell system is sought via oxygen stoichiometry regulation. In the implementation, the effect of practical problems such as saturation and possible wind-up have been also taken into account and counteracted. There is also a second intention for this chapter. Taking advantage of the consecutive steps required to present the development of the aforementioned actual FC controllers, it is aimed to provide in parallel a condensed recapitulation of the concepts and design procedures reviewed and elaborated along the previous chapters. The objective is twofold. On the one hand, this endows the chapter with a certain degree of self-containment, improving its readability by sparing the reader from awkward goings and comings. On the other hand, the goal is to summarise the successive stages of the FC-SOSM control design process presented in this book into one concise format. This would offer a unified abridged design guideline to develop proficient practical SOSM controllers for efficiency optimisation of FCGS.

6.2 State Space Model of the Experimental Fuel Cell System for Control Design A necessary phase, previous to the controller design stage, deals with the rearrangement of the equations presented in Sects. 5.2–5.5, in order to obtain the state space model suitable for nonlinear control design purposes. This procedure involves coupling all the presented differential equations with their auxiliary equations in order C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3_6, © Springer-Verlag London Limited 2012

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6 SOSM Control Design and Implementation

to represent the system only in terms of the space states, external inputs (Ist , Wan,in and Vcp ) and constants, as well as taking into account any possible issue leading to order reduction. In this particular case, the latter involves taking Tcp = Thum and dia /dt = 0 [4]. In fact, experimental tests revealed that Eqs. (5.14)–(5.15) are linearly coupled within the PEMFC operation range. Besides, the time constant of the variable ia can be neglected with respect to the rest of the system dynamics. It must be taken into account that the system is operating under the assumptions stated in Sect. 5.1. Taking the state x ∈ R7 , the control input u = Vcp ∈ R1 and the operating conditions given in Table B.5 of Appendix B, the experimental PEMFC generation system under study can be described as a seventh-order nonlinear system with C 1 vector fields, accordingly represented by the following state space relations (6.2)–(6.8): x˙ = f (x, t) + gu(t) x∈R ; 7

u ∈ R;

f : R7 → R 7

(6.1)

where g = [K 0 . . . 0]T with K = m1 , and the state variables are as follows: • • • • • • •

x1 = ωcp : motor shaft angular velocity. x2 = mhum,ca : air mass inside the cathode humidifier. x3 = mO2 ,ca : oxygen mass in the cathode channels. x4 = mN2 ,ca : nitrogen mass in the cathode channels. x5 = mv,ca : vapour mass in the cathode channels. x6 = mH2 ,an : hydrogen mass in the anode channels. x7 = mv,an : vapour mass in the anode channels.

The physical input to the system is the armature voltage of the compressor’s DC motor (Vcp ). The measured outputs selected to evaluate the control performance are the compressor current, the stack voltage, the stack cathode pressure and the compressor air mass flow. The stack current is an external perturbation that can also be measured in real time. In what follows, each differential equation is analysed, reviewing the corresponding physical relations that support them [3]. The first expression comes from Eq. (5.1) and relates the motor shaft angular velocity, the inertia, the electric torque delivered by the motor and the compressor’s load torque. This last variable value comes from the empirical relation that links the compressor working pressure and the shaft angular velocity (x1 ) (see Eq. (5.10)), considering both friction and pneumatic load terms:  x˙1 = m1 (u − m2 x1 ) − x1 m3 + A0 + A00 + A10 (x2 m5 + m6 ) + A20 (x2 m5 + m6 )2 + A01 x1 + A11 (x2 m5 + m6 )x1  + A02 x12 m4

(6.2)

The dynamics of the second state variable is obtained considering the mass conservation principle inside the humidifier. It takes into account the mass flow of air injected by the compressor, given by the relation obtained from (5.10), which includes the working pressure of the compressor and the shaft angular velocity:

6.2 State Space Model of the Experimental Fuel Cell System

131

x˙2 = B00 + B10 (x2 m5 + m6 ) + B20 (x2 m5 + m6 )2 + B01 x1 + B11 (x2 m5 + m6 )x1 + B02 x12 − b1 (x)3 C3 − b1 (x)2 C2 − b1 (x)C1 − C0

(6.3)

The remaining states of the model are related to the PEM fuel cell stack variables. As already stated in Chap. 5, these are given by three states of the cathode channels and two of the anode. Thus, the third state space equation describes the behaviour of the oxygen mass within the cathode channels that comes from Eq. (5.26):    −1 x˙3 = XO2 ,ca m9 b1 (x)3 C3 + b1 (x)2 C2 + b1 (x)C1 + C0 G−1 a (x2 m5 − m10 )  × 1+

m11 x2 m5 − m10

−1

  + b1 (x)3 C3 + b1 (x)2 C2 + b1 (x)C1 + C0

−1  −1  m14 m11 1+ × 1+ x2 m5 − m10 (x3 RO2 + x4 RN2 + x5 Rv )m8 − m12   − Kca (x3 RO2 + x4 RN2 + x5 Rv )m8 − Pamb x3 RO2 GO2   −1   x3 RO2 GO2 x3 RO2 GN2 × 1 + Gv x5 Rv + 1− x3 RO2 + x4 RN2 x3 RO2 + x4 RN2 −1 −1 × (x3 RO2 + x4 RN2 ) (x3 RO2 + x4 RN2 )−1  ×

 −1  x3 RO2 GO2 x 3 RO 2 GO nIst GN2 + 1− − 1/4 2 x3 RO2 + x4 RN2 x3 RO2 + x4 RN2 F (6.4)

The first term of Eq. (6.4) corresponds to the incoming oxygen mass fraction, while the second models the cathode output flow using a linear nozzle equation (5.44). The third term takes into account the oxygen mass consumed per second by the cathodic reaction (5.31). In accordance with Sect. 5.5.1, the dynamics of the fourth state variable, the mass of nitrogen in the cathode, is similar to that obtained for x3 , apart from the fact that there is no nitrogen consumption in the reaction. Then, from the analysis of Eq. (5.27), there are only two terms involved, relating the nitrogen input and output flow fractions:    −1 x˙4 = m9 b1 (x)3 C3 + b1 (x)2 C2 + b1 (x)C1 + C0 G−1 a (x2 m5 − m10 )  × 1+  × 1+

m11 x2 m5 − m10 m11 x2 m5 − m10

−1

  + b1 (x)3 C3 + b1 (x)2 C2 + b1 (x)C1 + C0

−1   (1 − XO2 ,ca,in ) 1 +

  x3 m8 GO2 − 1 − x3 m8 GO2 b3 (x)−1 b3 (x)

Gv m12 Ga (b2 (x) − m12 )   −1  x3 m8 + 1− GN2 b3 (x)

−1

132

6 SOSM Control Design and Implementation

 −1    x3 m8 GO2 x3 m8 × 1 + Gv x5 Rv m8 + 1− GN2 b3 (x) b3 (x) −1   × (x3 RO2 m8 + x4 RN2 m8 )−1 Kca,n b2 (x) − Pamb

(6.5)

The vapour mass equation in the cathode takes into account four different flows (5.28), namely the input and output partial vapour flows, the water generated by the reaction (which depends on the stack current Ist ) and the water flow transported by the polymeric membrane (Sect. 5.5.3):   −1 x˙ 5 = Gv m12 b1 (x)3 C3 + b1 (x)2 C2 + b1 (x)C1 + C0 G−1 a (x2 m5 − m10 )  −1 Gv m10 b4 (x) × 1+ + Ga (x2 m5 − m10 ) b5 (x)   −1 Gv m12 b4 (x) b4 (x) Gv m12 − + 1+ Ga (x2 m5 − m10 )b5 (x) b5 (x) Ga (b2 (x) − m12 )     − Kca,out b2 (x) − Pamb + Kca,out b2 (x) − Pamb     −1 −1 x3 m8 −1 −1 x3 m8 GO2 + 1− GN2 × 1 + Gv x5 Rv m8 b3 (x) RO2 b3 (x) b3 (x)     Gv nIst + n0 + n1 a0 + a1 b6 (x) + a2 b6 (x)2 + a3 b6 (x)3 + 1/2 F  2  + n2 a0 + a1 b6 (x) + a2 b6 (x)2 + a3 b6 (x)3 Ist /Afc /F  (a0 + a1 x5 m16 + a2 x52 m216 + a3 x53 m316 )ρm,dry − Dw Gm,dry   (a0 + a1 x7 m15 + a2 x72 m215 + a3 x73 m315 )ρm,dry −1 tm Gv Afc n − (6.6) Gm,dry The first anode state (Sect. 5.5.2) is derived from the mass conservation principle applied to the hydrogen in the anode channels. Following Eq. (5.47), its dynamics is computed in a similar way as x˙3 . It relates the input and output partial flows, as well as the hydrogen consumed by the electrochemical reaction on the anode’s catalyst surface:  −1   Gv m17 − Kan,out b7 (x) − Pamb x˙6 = Wan,in 1 + GH2 (b7 (x) − m17 )   Gv x7 m19 −1 GH nIst × 1+ − 1/2 2 (6.7) GH2 x6 m20 F The last state space equation models the vapour mass dynamics in the anode channels. Regarding Eq. (5.48), it only consists of three flow terms, given the assumption that no water is generated at the anode side: −1    Gv m17 − Kan,out b7 (x) − Pamb x˙7 = Wan,in − Wan,in 1 + GH2 (b7 − m17 )

6.2 State Space Model of the Experimental Fuel Cell System







Gv x7 m19 1+ GH2 x6 m20

133

−1

+ Kan,out b7 (x) − Pamb     − n0 + n1 a0 + a1 b6 (x) + a2 b6 (x)2 + a3 b6 (x)3  2  + n2 a0 + a1 b6 (x) + a2 b6 (x)2 + a3 b6 (x)3 Ist /Afc /F  (a0 + a1 x5 m16 + a2 x52 m216 + a3 x53 m316 )ρm,dry − Dw Gm,dry   (a0 + a1 x7 m15 + a2 x72 m215 + a3 x73 m315 )ρm,dry −1 tm Gv Afc n − Gm,dry

(6.8)

6.2.1 Control Output From the model developed in Chap. 5, a wide variety of control objectives can be tackled. Hence, output variables such as partial pressures of oxygen and hydrogen, water content in the membranes, relative humidity in the channels, etc., can be chosen. However, to address the classical problem of oxygen regulation, the compressor air mass flow is considered as the control output in what follows. Note that Wref comes from the Eq. (5.10) and depends on x1 and x2 : y(t) = Wcp = B00 + B10 (x2 m5 + m6 ) + B20 (x2 m5 + m6 )2 + B01 x1 + B11 (x2 m5 + m6 )x1 + B02 x12

(6.9)

6.2.2 Auxiliary Functions b1 (x) = x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8

(6.10)

b2 (x) = (x3 RO2 + x4 RN2 + x5 Rv )m8 x3 RO2 m8 + x4 RN2 m8 b3 (x) = RO2

(6.11)

b4 (x) = (x2 m5 − b2 )3 C3 + (x2 m5 − b2 )2 C2 + (x2 m5 − b2 )C1 + C0 Gv m10 b5 (x) = 1 + Ga (x2 m5 − m10 ) b6 (x) = 1/2x7 m15 + 1/2x5 m16

(6.13) (6.14)

b7 (x) = (x6 RH2 + x7 Rv )m18

(6.16)

(6.12)

(6.15)

Appendix C resumes the values of the parameters used in the model equations presented above. Most of them come directly from physical constants of the experimental system. The proposed model that can be straightforwardly adjusted or scaled to other PEM fuel-cell-based systems.

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Fig. 6.1 Pnet vs. Wcp . Cathode pressure Pca = 1 bar

6.3 Control Objective As previously mentioned, the main objective of the proposed control strategy is the optimisation of the energy conversion of the fuel cell, maximising the net power generated by the system under different load conditions. This is specially important, for instance, in automotive applications, where the extra energy generated by the conversion system (not used by the vehicle electric power train), can be stored in a battery or super-capacitor bank. This improves autonomy and fuel efficiency. Considering that the compressor is also driven by the fuel cell (in fact, it can be regarded as a parasitic load), the output net power (Pnet ) can be defined as the electrical power delivered by the stack (Pst = Vst Ist ) minus the electrical power consumed by the compression subsystem (Pcomp = Vcp ia ). Then, the optimisation of the system efficiency can be achieved by regulating the air mass flow entering to the stack cathode at different load conditions. Let us recall that accomplishing such optimal comburent flow is equivalent to maintaining the cathode line oxygen stoichiometry (or oxygen excess ratio) in an optimal value [5]. In the particular case of the laboratory FC system, the most efficient operating zones were determined from the experimental analysis of the conversion system under different working conditions (see as examples Fig. 6.1 at Pca = 1 bar and Fig. 6.2 at Pca = 1.6 bar). It can be seen that, for different cathode’s working pressures, the system presents variations on the net power generated for different air flows and cell’s currents. This indicates that, for a given load current, the maximisation of the conversion performance can be achieved through the regulation of the input air mass flow as a function of Ist . In this way, the proper comburent flow can be ensured in the membranes in order to satisfy load requirements.

6.3 Control Objective

135

Fig. 6.2 Pnet vs. Wcp . Cathode pressure Pca = 1.6 bar

As stated in Chaps. 4 and 5, a direct relation exists between Pnet and the system conversion efficiency, given that for a constant current demand, the amount of hydrogen consumed by the anodic reaction remains constant. In Figs. 6.3 and 6.4 the experimental static characteristics of the plant are displayed as functions of the cathode’s oxygen stoichiometry (also for Pca = 1 bar and Pca = 1.6 bar, respectively). It becomes evident that setting the air flow at its optimum value for each case is equivalent to maintaining an optimal value of oxygen stoichiometry. This value can be obtained off-line or using on-line trajectory planning. From Chap. 4, oxygen stoichiometry or oxygen excess ratio is defined as follows: λO2 =

WO2 ,in WO2 ,react

(6.17)

where WO2 ,in is the partial oxygen flow that inputs the cathode, and WO2 ,react is the amount of oxygen consumed by reduction at the cathode: nIst (6.18) 4F From these figures it becomes evident that starting at low oxygen stoichiometry values, as λO2 increases (and so does the partial oxygen pressure), the stack power (and therefore Pnet ) is also increased. However, after reaching a region of optimal values, further increments on λO2 cause a decrement on Pnet due to the excess power drained by the compressor. This effect is verified over the entire stack working range (1 bar < Pca < 2.5 bar). Analysing the optimal values for λO2 , Figs. 6.3 and 6.4 suggest that the use of oxygen excess ratio values should be within a range from 2 to 3. It is important to stress that values lower than one (λO2 < 1) must be avoided because, if WO2 ,in < WO2 ,react , there is the risk of cathode starvation. WO2 ,react = GO2

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Fig. 6.3 Pnet vs. λO2 characteristic. Cathode pressure Pca = 1 bar

Once the desired values for the oxygen excess ratio for each load demand are determined, the control strategy can be formulated from concepts already presented in Chap. 4. The immediate way to attain the control goal of λO2 regulation would be to directly control WO2 ,in . However, this is impossible to do in practice, given that it is an inaccessible internal variable. To circumvent this problem, an alternative formulation in terms of control of the compressor air mass flow previously proposed. The compressor output mass flow (Wcp ), which is easily measurable, is directly related with WO2 ,in through the humidifier dynamics. Besides, once this dynamics is extinguished, a constant relationship between Wcp and WO2 ,in exists. Therefore, setting proper Wcp reference values, an operation in the neighbourhood of the maximum efficiency can be achieved, and, after the humidifier transient is elapsed, λO2 = λO2 ,opt can be ensured for each load condition [5]. In the sliding mode framework, this objective can be attained using the following restriction or sliding surface: s(x, t) = Wcp − Wair,ref

(6.19)

where s is the sliding variable that must be steered to zero. The reference value (Wair,ref ) can be directly obtained from the desired dry air flow. Given that the oxygen mass fraction (χO2 ) is known for the current ambient conditions, the required dry air flow (Wdry,ref ) can be found from the following expression: Wdry,ref =

nIst 1 λO2 ,opt GO2 χO2 4F

(6.20)

Then, considering that the relative humidity of the air (ωamb ) is also known, the final expression of the reference to be followed by the control system is Wair,ref = (1 + ωamb )

nIst 1 λO ,opt GO2 χO2 2 4F

(6.21)

6.4 Controller Synthesis

137

Fig. 6.4 Pnet vs. λO2 characteristic. Cathode pressure Pca = 1.6 bar

It is worth noting that for stable ambient conditions (ωamb and χO2 constant), this value depends on a single measurable variable, Ist . Therefore, according to Chap. 3, the control problem can be mathematically stated as follows: ⎧ ⎨ x˙ = f (x, t) + gu(t) (6.22) s = s(x, t) ∈ R ⎩ u = U (x, t) ∈ R where x ∈ R7 , t is time, u is a bounded control action, and f , s are smooth functions. The control task is to keep s ≡ 0.

6.4 Controller Synthesis Analysing the sliding variable and its Lie derivative along the vector field g (Lg s), it becomes clear that s(x, t) has relative degree one (i.e. s(x, t) does not explicitly depend on the control input): s(x, t) = B00 + B10 (x2 m5 + m6 ) + B20 (x2 m5 + m6 )2 + B01 x1 + B11 (x2 m5 + m6 )x1 + B02 x12 − Wair,ref while u explicitly appears in its derivative s˙ (x, t) = ∂s ∂t + Lf s + Lg su:   Lg s(x, t) = B01 + B11 (x2 m5 + m6 ) + 2B02 x1 m1 m4 = 0

(6.23)

(6.24)

Given that s(x, t) has relative degree 1 with respect to the control action u, it could have been possible to implement a classic (first-order) sliding-mode controller. However, as it was discussed in Chaps. 3 and 4, this option does not represent

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6 SOSM Control Design and Implementation

an attractive alternative for this particular type of plant. A switching control action may provoke undesirable oscillations in the compressor DC motor and also affect the net output power, which has a relative degree 0 with respect to the compressor voltage (Vcp ). In this context a second-order sliding-mode controller proves to be an interesting choice to robustly control the system, achieving smoothness on Vcp , thus avoiding deterioration of the quality of Pnet . Differentiating twice the sliding variable with respect to time, we obtain the following expressions:   ∂ ∂ s(x, t) + s(x, t). f (x, t) + gu(t) ∂t ∂x     ∂ = s(x, t) + B01 + B11 (x2 m5 + m6 ) + 2B02 x1 m1 u(t) − m2 x1 ∂t − x1 m3 + B0 + A00 + A10 (x2 m5 + m6 ) + A20 (x2 m5 + m6 )2 + A01 x1   + A11 (x2 m5 + m6 )x1 + A02 x12 m4 + B10 m5 + 2B20 (x2 m5 + m6 )m5  + B11 m5 x1 B00 + B10 (x2 m5 + m6 ) + B20 (x2 m5 + m6 )2 + B01 x1 + B02 x12  3 + B11 (x2 m5 + m6 )x1 − x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3   2 − x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2 − x2 m5   − (x3 RO2 + x4 RN2 + x5 Rv )m8 C1 − C0 (6.25)

s˙ =

  ∂ ∂ ∂ s˙ (x, t) + s˙ (x, t). f (x, t) + g + s˙ (x, t).u(t) ˙ ∂t ∂x ∂u = ϕ(x, u, t) + γ (x, u, t)u(t) ˙

s¨ =

(6.26)

with γ (x, u, t) = Lg s(x, t)

(6.27)

ϕ(x, u, t) = Lf s˙ (x, t) + Lg s˙ (x, t)

(6.28)

See Appendix C for the complete expression of ϕ(x, u, t). The functions ϕ(x, u, t) and γ (x, u, t) can be globally bounded as follows: 0 < Γm ≤ γ (x, u, t) ≤ ΓM

(6.29)

ϕ(x, u, t) ≤ Φ

(6.30)

Particularly, for the PEMFC system under study, the bounding values were computed by means of a numerical study of the nonlinear system and refined through an experimental analysis. Additionally, uncertainties were included in representative parameters such as the motor inertia, torque friction, humidifier volume and cathode air constant. Under the effect of an appropriate control action (a feedforward, as in Chap. 4, proved to be suitable), to lead the system to the vicinity of the desired air mass flow, the following values were obtained: Φ = 2.3 × 10−5 ;

Γm = 0.002;

ΓM = 0.0083

(6.31)

6.4 Controller Synthesis

139

Once the bounds have been determined, the stabilisation problem of system (6.22) with input–output dynamics (6.26) can be solved through the solutions of the following equivalent differential inclusion by applying SOSM:

s¨ ∈ −2.3 × 10−5 ,

 2.3 × 10−5 + [0.002, 0.0083]u˙

(6.32)

It is worth noting that this study is performed off-line and does not burden the real-time controller operation. As mentioned in Chap. 3, a variety of SOSM algorithms can be found in the literature to solve the control problem, each of them with its distinctive features. In this chapter, the SOSM controllers analysed and detailed in Chaps. 3 and 4, namely Super-Twisting, Twisting and Sub-Optimal, have been designed and implemented in the laboratory fuel-cell-based system. Previous to the experimental tests, the controllers have been assessed by thorough simulations. The gains of the SOSM algorithms were calculated from Φ, Γm and ΓM guaranteeing that, once the system is steered to the region where Eq. (6.32) holds, the trajectories do not escape and converge to s = s˙ = 0 in finite time. It is worthwhile to emphasise that the considered algorithms depend only on few parameters, which were computed during the off-line tuning procedure. Thus, the on-line operation of the control algorithms is very simple and consumes scarce computational resources. In particular, the Twisting and Sub-Optimal algorithms are intended for a sliding variable of relative degree 2. Given that the considered output of the PEMFC-based system is of relative degree 1, for these last two cases, the system has been expanded with an integrator, considering ν = u(t) ˙ as the control action for the design. The algorithms structures and the chosen gains for the PEMFC controllers are succinctly recalled below.

6.4.1 Super-Twisting Algorithm As discussed in Chap. 3, this algorithm is specifically intended for systems with relative degree 1. One interesting feature is that, during on-line operation, it does not require information of s˙ . The output converge to the origin of the sliding plane following a characteristic looping trajectory. The control law comprises two terms. One is the integral of a discontinuous control action, and the other is a continuous function of s, contributing only during the reaching phase [6]: u(t) = u1 (t) + u2 (t) u˙ 1 (t) = −γ sign(s) u2 (t) = −λ|s|

1/2

(6.33)

sign(s)

where γ and λ are design parameters that were derived from the corresponding sufficient conditions for finite-time convergence of the algorithm [6]:

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6 SOSM Control Design and Implementation

Φ Γ m 2 (Γm γ + Φ)2 λ> Γm2 (Γm γ − Φ)

γ>

(6.34)

Among the set of gains that fulfil (6.34), the best control performance has been achieved with γ = 1;

λ = 35

(6.35)

6.4.2 Twisting Algorithm Let us recall that this algorithm is characterised by the way in which its trajectories converge to the origin in the sliding plane (s, s˙ ). Knowledge of the signs of s and s˙ is needed, and the control law is given by [2] ν(t) = u(t) ˙ = −r1 sign(s) − r2 sign(˙s )

(6.36)

Sufficient conditions for finite-time convergence can be summarised as r1 > r2 > 0 (r1 + r2 )Γm − Φ > (r1 − r2 )ΓM + Φ Φ (r1 − r2 ) > Γm

(6.37)

Therefore, the chosen gains resulted in r1 = 0.7;

r2 = 0.1

(6.38)

6.4.3 Sub-Optimal Algorithm Trajectories in the (s, s˙ ) plane are confined within parabolic arcs which include the origin, so the convergence behaviour may exhibit “twisting” around the origin, “bouncing” on the s axis or a combination of both. Its control law is given by [1] ν(t) = u(t) ˙ = α(t)U sign(s − βsM )

1 if (s − βsM )sM ≥ 0 α(t) = ∗ α if (s − βsM )sM < 0

(6.39)

where U > 0 is the minimum value of u(t), ˙ α ∗ > 1 is the modulation parameter, 0 ≤ β < 1 is an anticipation factor, and sM is a piece-wise constant function representing the last extremal value of the sliding variable s(t).

6.5 Simulation Results

141

The convergence in finite time is guaranteed as long as U>

Φ Γm

α ∗ ∈ [1; +∞) ∩



 Φ + (1 − β)ΓM U ; +∞ βΓm U

(6.40)

This algorithm requires the detection of s˙ becoming zero and the corresponding values of s at those instants, i.e. sM . The final choice of the controller parameters for the PEMFC system under consideration was U = 1;

α ∗ = 1.2;

β = 0.5

(6.41)

6.4.4 Feedforward Term It was previously stated that it is necessary to define an extra control action that steers the sliding variable within a region such that the bounds on the sliding dynamics given by (6.32) are satisfied [2]. With this purpose, the aforementioned feedforward (FF) action uff has been included. It provides the control effort required to approach surface neighbourhood, reaching the validity region. Therefore, the twoterm control action (ui ) proposed in (4.41) is implemented as ui (t) = u(t) + uff (t)

(6.42)

where u corresponds to the SOSM control action particularised above. The expression of uff is computed via a polynomial obtained from an off-line test covering the entire operation range of the PEMFC system. The complete polynomial expression is given in Appendix C.

6.5 Simulation Results Previous to the implementation and experimental assessment, the controllers were tested through numerical simulations. The following two figures show representative simulation results of the controlled system using the algorithms designed above. Figure 6.5 confirms that the Super-Twisting, Twisting and Sub-Optimal controllers present a satisfactory dynamic response when controlling the air mass flow. In Fig. 6.6, the response of the nonlinear system using the proposed algorithms is shown through an s–˙s diagram. It is important to stress that, after an adequate parameters tuning, the three controllers present comparable satisfactory dynamic behaviours, confirming that the suitability of the SOSM approach for the breathing control of this PEMFC system is verified.

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6 SOSM Control Design and Implementation

Fig. 6.5 Air flow and control input (simulation results)

Fig. 6.6 s vs. s˙ (simulation results)

6.6 Experimental Set-up of the PEM Fuel Cell System

143

Fig. 6.7 Experimental PEM fuel cells laboratory at IRI (UPC-CSIC)

6.6 Experimental Set-up of the PEM Fuel Cell System A detailed description of the laboratory test station used for the controllers development and testing was presented in Chap. 5 (a view is shown in Fig. 6.7). A schematic diagram of the system interconnection is depicted in Fig. 6.8, where sensors and actuators are also displayed. The main subsystems can be summarised as follows: • Air Compressor: 12-V DC oil-free diaphragm vacuum pump. The input voltage to this device is a continuous variable used as the main control action. • Hydrogen and Oxygen humidifiers and line heaters. These are used to maintain proper humidity and temperature conditions inside the cell stack, an important issue for PEM membranes. Cellkraft® membrane exchange humidifiers are used in the current set-up, and decentralised PID controllers ensure adequate operation values. • Fuel cell stack: an ElectroChem® seven-cell stack with Nafion 115® membrane electrodes assemblies (MEAs) is used, with a catalyst loading of 1 mg/cm2 of platinum, 50 cm2 of active area. Different sensors were incorporated to measure specific variables, suited for modelling and control. Regarding Fig. 6.8, these are: motor shaft angular velocity (ωcp ), compressor air mass flow (Wcp ), hydrogen mass flow (WH2 ), cathode and an-

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6 SOSM Control Design and Implementation

Fig. 6.8 PEM Fuel cell system schematics

ode humidifiers pressures (Phum,ca and Phum,an ), stack pressure drops (Pca and Pan ), motor stator current (ia ) and voltage (Vcp ), stack voltage (Vst ) and current (Ist ). Besides, a number of sensors were included to monitor and register relevant temperatures (Tst , Thum,ca , Tlh,ca , Thum,an and Tlh,an ). It must be noted that in a typical fuel cell application many of these measurements are not necessary [5]. For instance, the proposed controllers only require the measurement of the stack current and the compressor air mass flow. The operating conditions for all fuel cell test bench experiments are summarised in Appendix C.

6.7 Experimental Tests 6.7.1 Regulation Tests The main objective of this section is to present the experimental performance of the proposed SOSM controllers, considering actual external disturbances and different working conditions. Comparative results are provided for Super-Twisting, Twisting and Sub-Optimal controllers. To assess the first controller (6.33) performance in real operation, Fig. 6.9 presents the behaviour of the Super-Twisting Algorithm controller at different regulation conditions (without FF action). It depicts the evolution of the compressor air flow versus a series of steps in the reference. Simulation and experimental results are superimposed. Close matching of simulation and experimental results shown in Fig. 6.9 confirm the reliability and accuracy of the design methodology. It can be seen that, although the closed loop succeeds in driving the air flow to its desired value in finite time, the reaching dynamics can be improved. As it was stated in Chap. 4, the control action on the reaching phase of the second-order

6.7 Experimental Tests

145

Fig. 6.9 Control of Wcp using a Super-Twisting algorithm: simulation and experimental results

Fig. 6.10 Control of Wcp with Super-Twisting + FF: experimental results

sliding-mode controllers can be freely designed if it is ensured that u(t) is continuous and drives the sliding variable to a neighbourhood of the surface s = 0 in finite time. Then, Fig. 6.10 shows the improvement in the dynamic response obtained when the SOSM controller is combined with the static FF action given in Appendix C. In Fig. 6.11, the behaviour of each component of the control action (u and uff ) is presented, as well as the armature voltage of the compressor motor (Vcp = ui ). Analogous tests were conducted for the Twisting and Sub-Optimal algorithms. The results are presented in Figs. 6.11 and 6.12, showing that the three controllers present similar dynamic performance when the parameters tuning is performed following similar criteria.

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Fig. 6.11 Control of Wcp with Twisting + FF: experimental results

Fig. 6.12 Control of Wcp with Sub-Optimal + FF: experimental results

In the following analysis, a test of oxygen stoichiometry regulation was conducted to verify its performance. This is shown in Fig. 6.13, where λO2 is displayed versus time, for the same reference signal. This test was performed at constant load conditions. In Fig. 6.14 the dynamical behaviour of Vst , Ist , ia and Pnet for the same test is presented. Afterwards, an experiment under variable load conditions was performed to verify the controlled system response in more demanding (and realistic) situations. Figures 6.15 and 6.16 show that the system exhibits good regulation characteristics for

6.7 Experimental Tests

147

Fig. 6.13 λO2 regulation, Ist = 3 A: experimental results

Fig. 6.14 λO2 regulation, Ist = 3 A: experimental results

changing loads. Figures 6.17 and 6.18 show a similar test, but in this case performed at a standard regulation point of the oxygen stoichiometry (λO2 = 2.5). An important issue from the practical point of view is related with actuator saturation. Given that the compressor’s input cannot exceed the power source voltage (0 V < ui < 12 V), a wind-up effect may appear, due to the integral part of the controller. To correct this, an anti-wind-up algorithm was included which disconnects the integrator’s input when such saturation is detected. The effect of the anti-windup action is shown in Fig. 6.19. Note that the flow value recovers quickly, converging to the desired values once saturation ends and nominal operation is restored.

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Fig. 6.15 λO2 regulation, variable Ist : experimental results

Fig. 6.16 λO2 regulation, variable Ist : experimental results

6.7.2 Perturbation Tests Having verified the control operation in nominal regime, the system was then tested under the influence of external perturbations. In particular, an increment in the cathode pressure was forced using a mechanical back pressure regulator. This effect can be appreciated in Figs. 6.20–6.22, where similar tests were induced to the three algorithms implementations (ST, Twisting and Sub-Optimal). For instance, in Fig. 6.21 it is shown that during the 0–190 sec interval, as the valve is increasingly throttled, reference tracking is successfully preserved. Then, when the valve

6.7 Experimental Tests

149

Fig. 6.17 λO2 regulation: experimental results

Fig. 6.18 λO2 regulation: experimental results

is suddenly bypassed (t = 190 sec), the system abandons the sliding surface, but the SOSM controller provides a quick recovery. It is worth noting that during this experiment, the feedforward action remains constant, because it only depends on the reference. Finally, in Fig. 6.23 a test analogous to the one presented in Fig. 6.20 is depicted. Nevertheless, this last perturbation test is done at λO2 = 3 that is an operating point in the area of maximum power generation.

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Fig. 6.19 Actuator saturation and anti wind-up effect

Fig. 6.20 Regulation of λO2 under external perturbations: experimental results using Super-Twisting

6.8 Conclusions After theoretically and experimentally evaluating different second-order slidingmode controllers that globally solve the oxygen stoichiometry problem of a PEMFC generation system, a set of SOSM controllers have been developed and implemented in a laboratory test station. Its suitability was successfully verified through extensive computer simulations, based on the plant model previously developed in Chap. 5, which was especially built for nonlinear control purposes, taking into account external disturbances and uncertainties in the system parameters [4]. Subsequently, highly satisfactory experimental results using Super-Twisting, Twisting and Sub-

6.8 Conclusions

151

Fig. 6.21 Regulation of λO2 under external perturbations: experimental results using Twisting

Fig. 6.22 Regulation of λO2 under external perturbations: experimental results using Sub-Optimal

Optimal topologies with feedforward action confirm the feasibility, simplicity and robustness of the solutions. The main advantages of the proposed SOSM control for PEMFC systems can be summarised as follows: • robust stabilisation of the oxygen stoichiometry problem avoiding 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;

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Fig. 6.23 Regulation of λO2 under external perturbations: experimental results

• the control law only depends on two measurable variables, namely the stack current and the compressor air flow, and therefore no state observer is required; • simple controller structure, resulting in low real-time computational burden. The resulting controllers are relatively simple to design and require only a few measurements of easy and low-cost implementation. This represents a major advantage for fuel cells in industrial applications, where the cost of the plant instrumentation is critical.

References 1. Bartolini G, Ferrara A, Usai E (1997) Applications of a sub-optimal discontinuous control algorithm for uncertain second order systems. Int J Robust Nonlinear Control 7(4):299–310 2. Fridman L, Levant A (2002) Higher order sliding modes. In: Sliding mode control in engineering, Dekker, New York, pp 53–101 (Chap 3) 3. Kunusch C (2009) Modelling and nonlinear control of PEM fuel cell systems. Phd thesis, Electrical Department, National University of La Plata, Argentina (in Spanish) 4. Kunusch C, Puleston PF, Mayosky MA, Husar A (2011) Control-oriented modelling and experimental validation of a PEMFC generation system. IEEE Trans Energy Convers 26(3):851–861 5. Kunusch C, Puleston PF, Mayosky MA, Dávila A (2010) Efficiency optimisation of an experimental PEM fuel cell system via super twisting control. In: Proceedings of IEEE 11th international workshop on variable structure systems, Mexico City, June 26–28 6. Levant A (1993) Sliding order and sliding accuracy in sliding mode control. Int J Control 58(6):1247–1263

Chapter 7

Conclusions, Open Lines and Further Reading

7.1 Main Results This book addressed the application of Second-Order Sliding-Mode (SOSM) algorithms for robust control of autonomous PEM fuel cells. These techniques allow the optimisation of the energy conversion performance of the system and are robust to external disturbances and model uncertainty. SOSM controllers retain many of the desirable properties of First-Order Sliding-Mode controllers (robustness, finite reaching time to the sliding surface, implementation simplicity, etc.) and exhibit a better behaviour with respect to the chattering effect. The design procedure involved the following steps: • Development of a control-oriented, smooth dynamic model of the plant. This was accomplished using a semi-empirical approach, combining the knowledge of the system and numerical approximations, in order to avoid discontinuities and lookup tables. • Design of the proper sliding surface to meet the performance requirements. The sliding manifold proposed ensures the optimisation of the energy conversion performance. • Calculation of a set of norms for the model. • Choice of the specific algorithm best suited for the application. Twisting, SuperTwisting and Sub-Optimal controllers were tested, and a performance comparison was made. For the specific devised objective, namely the oxygen stoichiometry control, the stack current Ist and the air flux provided by the compressor Wcp are the only measurements required. Simulation and experimental results were presented, showing that excellent performance can be obtained on a wide range of operating conditions. These results encourage the application of SOSM controllers to the remaining subsystems of the PEM fuel cell assembly, in particular the thermal and water subsystems. Several improvements to the existing control scheme and many open directions of research can be proposed. In the following sections, the most promising issues are outlined. C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3_7, © Springer-Verlag London Limited 2012

153

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7 Conclusions, Open Lines and Further Reading

7.2 Further Issues Related to PEMFC Control 7.2.1 Adaptive Super-Twisting Algorithms The SOSM approaches analysed in Chaps. 4 and 6 of this book allow for finitetime convergence to zero of not only the sliding variable but its first derivative too. Among them, the Super-Twisting algorithm has the additional advantage of a simple implementation, requiring only the knowledge of the sliding variable. The resulting continuous controller ensures all the main properties of first-order sliding-mode control for a system with smooth matched bounded uncertainties/disturbances, allowing also reduction of the chattering effect. The classic Super-Twisting algorithm equations (already presented in Chap. 3 and repeated here for convenience) are: u(t) = u1 (t) + u2 (t)

(7.1)

where u˙ 1 (t) = −α sign(s)  −λ|s0 |ρ sign(s) u2 (t) = −λ|s|ρ sign(s)

if |s| ≥ |s0 |

(7.2)

if |s| ≤ |s0 |

where α > 0, λ > 0 and ρ ∈ (0, 12 ) are parameters defined from suitable bounds on the system model. The homogeneous nature of the classic Super-Twisting algorithm does not allow the compensation of uncertainties/disturbances that change with the state variables. This means that it cannot ensure the sliding motion for systems with, for instance, an unknown linear part. Using Lyapunov-based techniques, a non-homogeneous extension of the Super-Twisting algorithm can be proposed, allowing the exact compensation of smooth uncertainties/disturbances with derivatives bounded by known functions [8]. These are called Variable-Gain Super-Twisting (VGST) algorithms. The general form of VGST controllers is  t v = k1 (t, x)Φ1 (s) − k2 (t, x)Φ2 (s) dt (7.3) 0

where 1

(7.4) Φ1 = |s| 2 sign(s) + k3 s, k3 > 0 1 3 1 Φ2 = sign(s) + k3 |s| 2 sign(s) + k32 s (7.5) 2 2 Note that when k3 = 0 and the gains k1 and k2 are constant, the classic SuperTwisting controller results. If k3 > 0, the algorithm can reject perturbations growing linearly in s, i.e. outside of the sliding surface, and the variable gains k1 and k2 can be used to make the sliding surface insensitive to perturbations growing with bounds given by known functions. In these cases, the VGST algorithm exhibits a better behaviour with respect to chattering reduction.

7.2 Further Issues Related to PEMFC Control

155

7.2.2 HOSM MIMO Control Although the techniques analysed in this book are general and can be applied to MIMO systems, the reported results are shown for a nonlinear SISO model of the oxygen subsystem. To this end, in Chap. 6 it was supposed that auxiliary controllers regulate temperature, humidity levels and hydrogen supply at critical points of the system. However, these control goals could be part of a more global strategy to be addressed, for instance, with sliding mode controllers. In fact, a MIMO control would allow fulfilment of complex, combined objectives. For instance, if an electromechanical valve is installed at the cathode’s output, a different control strategy can be proposed, combining regulation of oxygen stoichiometry and control of the oxygen partial pressure. Additionally, the regulation of the hydrogen circulation to the cathode can be accomplished if a mass flux controller at the input of the anode’s humidifier is available.

7.2.3 Model Predictive Control Model Predictive Control (MPC), also known as moving horizon control or receding horizon control, refers to a class of algorithms which make explicit use of a process model to optimise the future predicted behaviour of a plant. More particularly, at each sampling time t , a finite-horizon optimal control problem is solved over a prediction horizon N , using the current state x of the process as the initial state. The online optimisation problem takes account of system dynamics, constraints and control objectives. The consideration of model predictive control strategies in dynamical systems such as fuel-cell-based systems is nowadays a growing topic in the literature. However, the use of those control strategies is becoming more widespread due to the particular capabilities of the technique, which allow solving crucial problems related to the control and management of the fuel cells and ancillary systems dynamics. MPC can deal with system constraints in a systematic and straightforward way [18]. Additionally, it can be applied to MIMO systems with multiple objectives, providing also robustness against system disturbances. Nevertheless, MPC is very sensitive to the model accuracy since the control computation is precisely based on a mathematical model of the plant. This dependence opens several ways on how to design the MPC depending on the system model: from the purely nonlinear MPC [22] to linear approaches [2, 5], piecewise affine models [10] and hybrid systems forms [9]. Regarding the multi-objective control nature of MPC, in [6] it is used for optimising the fuel cell power utilisation and the oxygen flow to the stack, so the strategy is then implemented for global optimisation, improving both objectives. Additionally, the use of real data for updating online the prediction allows to minimise the modelling mismatch, while the control law avoids the oxygen saturation, enhances the transient performance and extends the operating life of the whole system. Regarding the incorporation of actuator limitations and state constraints in the controller

156

7 Conclusions, Open Lines and Further Reading

design, [7] discusses those aspects for the prevention of fuel cell starvation within the MPC framework. Given the advantages mentioned above, model predictive control of fuel cell systems has been analysed not only for the control of the fuel cell stacks in itself, but also in its interaction to other electrical generation system. Research works such as [9, 22] and [14] discuss aspects such as fuel cell management within micro cogeneration systems, decentralised MPC schemes for mixed topologies of fuel cells with ultra capacitors and with Photovoltaic Panels. In particular, [21] propose an adaptive MPC for the network control of fuel cells. All these approaches try to improve the global system performance through MPC controllers (in global and local mode) in order to fulfil the control objectives and reduce the computation burden.

7.2.4 Observers for Internal Variables Observers are of great interest for fuel cell operation, both for monitoring purposes and for feedback control design. They can give information about variables that are difficult, expensive or even impossible to measure. Several publications of the last decade deal with the design of PEMFC state observers. Hydrogen and oxygen partial pressures can be estimated by different kind of observers such as adaptive observers [1] and sliding-mode observers validated using Lyapunov stability analysis methods [16]. The membrane water content can be estimated with a nonlinear observer from its close relation with the resistive voltage drop [12], or by using open-loop and Luenberger observers [20]. However, in the presence of disturbances, Luenberger observers are not accurate, as they can only ensure the convergence to a bounded region near the real value of the state. In these cases, Sliding-Mode-based observers are regarded as an alternative to the problem of observation of perturbed systems [19], being suitable for robust state estimation also in the presence of unknown inputs. Results reported in [3] estimate the internal fuel cell temperature from surface measurements, while [15] propose the design of first-order slidingmode observers for the estimation of the oxygen outflow of the system compressor. There, the inherent robustness of the sliding-mode techniques is exploited despite the resultant chattering phenomenon. Finally, the fast response convergence of robust nonlinear unknown-input observers for the estimation of several variables in a PEMFC-based system is reported in [4]. Higher-Order Sliding-Mode (HOSM) Based Observers represent a useful technique for the state observation of perturbed systems, due to their high precision and robust behaviour with respect to parametric uncertainties. In particular, robust exact differentiators [17] based on the Super-Twisting algorithm can be applied for state reconstruction. The methodology involves the transformation of the system to a regular form (chain of integrators) using a suitable diffeomorphism. Then the state is estimated applying HOSM differentiators to the system output. Even when differentiators appear as a natural solution to the observation problem, the use of the system knowledge for the design of an observation strategy results in reduction of

7.3 Further Issues Related to Fuel-Cell-Based Systems

157

gains for the sliding-mode compensation terms, improving accuracy. Moreover, the complete or partial knowledge of the system model can give place to the application of techniques for parametric reconstruction or disturbance reconstruction [11].

7.3 Further Issues Related to Fuel-Cell-Based Systems 7.3.1 Hybrid Standalone Systems Hybrid alternative energy power generation systems are expected to be an important part of the power generation paradigm of the future [13]. In fact, none of the currently available technologies is cost efficient or reliable enough to operate as independent power sources. Hybrid systems have better potential to provide dependable power than a system comprising a single resource. For this reason, hybrid energy systems have caught worldwide research attention. Of course, each hybrid system needs a proper control strategy to manage and prioritise power generation among the different sources in the system. Hybrid energy systems can be made of many combinations of different sources and storage devices. For instance, a typical standalone hybrid power system can be made of a wind energy conversion system (WECS), a photovoltaic array and a PEMFC-based systems as energy sources. Due to the intermittent nature of wind and solar energy, standalone wind and/or PV energy systems normally require energy storage devices or some other generation sources. The storage devices can be a battery bank, a super-capacitor bank (to handle fast transient power demands), compressed air system, electrolyser (or a regenerative FC) system, etc., or a combination thereof. For standalone applications, the system needs to have sufficient storage capacity to handle the power variations of the alternative energy sources involved. Due to their different operation principles, each of the components of the system should be controlled with different objectives, and a supervisory strategy must be devised to optimise the energy flow, in order to fulfil load demands and ensure reliable operation.

7.3.2 Distributed Generation Systems The work of this book analysed the use of PEMFC systems in standalone applications, that is, systems where the fuel cell is the only energy source available. However, the operational advantages of PEM technology, as well as its cost reduction, suggest that it can be also applied to stationary systems. Control architectures for these systems are of vital importance and regard PEMFC systems as part of a larger, distributed generation systems (DGS). Steady progress in power deregulation and tight constraints over the construction of new transmission lines for long-distance power transmission have created increased interest in DGS. Of particular importance are renewable DGS with free energy resources, such as wind and photovoltaic,

158

7 Conclusions, Open Lines and Further Reading

and alternative energy conversion devices such as fuel cells and microturbines. DGS are modular in structure, and they are normally placed at the distribution level or near load centres. They can be strategically placed in distribution systems for grid reinforcement, reducing power losses and on-peak operating costs. Additionally, they can be used as control devices to improve voltage profiles and load factors. As a result, their installation can defer or eliminate the need for system upgrades and can improve system integrity, reliability and efficiency. However, important control problems arise because many DG devices are intermittent and usually not synchronised with power consumption. Among others, grid balancing, smart power injection, distributed power supervision/monitoring and fault detection/recovery arise as fundamental open issues to be resolved.

References 1. Arcak M, Gorgun H, Pedersen LM, Varigonda S (2004) A nonlinear observer design for fuel cell hydrogen estimation. IEEE Trans Control Syst Technol 12(1):101–110 2. Arce A, Ramirez DR, del Real AJ, Bordons C (2007) Constrained explicit predictive control strategies for PEM fuel cell systems. In: 46th IEEE conference on decision and control, New Orleans, LA 3. Begot S, Harel F, Kauffmann JM (2008) Design and validation of a 2 kW-fuel cell test bench for subfreezing studies. Fuel Cells 8(1):23–32 4. Benallouch M, Outbib R, Boutayeb M, Laroche E (2009) A new scheme on robust unknown input nonlinear observer for PEM fuel cell stack system. In: 18th IEEE international conference on control applications. Part of 2009 IEEE multi-conference on systems and control, Saint Petersburg, Russia 5. Bordons C et al (2006) Constrained predictive control. Strategies for PEM fuel cells. In: Proceedings of the 2006 ACC, Minnesota, USA, June 14–16 6. Chen Q, Gaob L, Dougal RA, Quan S, (2009) Multiple model predictive control for a hybrid proton exchange membrane fuel cell system. J Power Sources 191(2):473–482 7. Danzer MA, Wittmanna SJ, Hofera EP (2009) Prevention of fuel cell starvation by model predictive control of pressure, excess ratio, and current. J Power Sources 190(1):86–91 8. Davila A, Moreno FA, Fridman L (2010) Variable gains super-twisting algorithm: a Lyapunov based design. In: 2010 American control conference Marriott Waterfront, Baltimore, MD, USA, 30 June–02 July 9. Del Real AJ, Bordons Arcea AC (2007) Development and experimental validation of a PEM fuel cell dynamic model. J Power Sources 173(1):310–324 10. Fiacchini M, Alamo T, Alvarado I, Camacho EF (2008) Safety verification and adaptive model predictive control of the hybrid dynamics of a fuel cell system. Int J Adapt Control Signal Process 22:142–160 11. Fridman L, Shtessel Y, Edwards C, Yan X (2008) Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems. Int J Robust Nonlinear Control 18(4–5):399–412 12. Gorgun H, Arcak M, Barbir F (2006) An algorithm for estimation of membrane water content in PEM fuel cells. J Power Sources 157(1):389–394 13. Hashem Nehrir M, Wang C (2009) Modeling and control of fuel cells. Distributed generation applications. IEEE Press, New York 14. Houwing M, Negenborn RR, Ilic MD, De Schutter B (2009) Model predictive control of micro cogeneration fuel cell systems. In: Proceedings of the 2009 IEEE international conference on networking, sensing, and control (ICNSC 2009), Okayama City, Japan

References

159

15. Kazmi IH, Bhatti AI, Iqbal S (2009) A nonlinear observer for PEM fuel cell system. In: IEEE 13th international multitopic conference 16. Park I, Kim S (2006) A sliding mode observer design for fuel cell electric vehicles. J Power Electron 6(2):172–177 17. Levant A (1998) Robust exact differentiation via sliding mode technique. Automatica 34(3):379–384 18. Maciejowski JM (2002) Predictive control with constraints. Prentice Hall, Harlow 19. Spurgeon SK (2008) Sliding mode observers: a survey. Int J Syst Sci 39(8):751–764 20. Thawornkuno C, Panjapornpon C (2008) Estimation of water content in PEM fuel cell. Chiang Mai J Sci 35(1):212–220 21. Tong S, Liu G (2007) Networked control of PEM fuel cells using an adaptive predictive control approach. In: Proceedings of the 2007 IEEE international conference on networking, sensing and control, London, UK, pp 15–17 22. Vahidi A, Stefanopoulou A, Peng H (2006) Current management in a hybrid fuel cell power system: a model predictive control approach. IEEE Trans Control Syst Technol 14(6):1047– 1057

Appendix A

Electric Vehicle PEM Fuel Cell Stack Parameters

A.1 Return Manifold Polynomial Fitting Table A.1 Return manifold polynomial fitting

Parameter

Value

Return manifold parameter (p0 )

0.001248 kg/s

Return manifold parameter (p1 )

−0.001967 kg/s/KPa

Return manifold parameter (p2 )

−0.001524 kg/s/KPa2

Return manifold parameter (p3 )

0.002122 kg/s/KPa3

Return manifold parameter (p4 )

0.02772 kg/s/KPa4

Return manifold parameter (p5 )

0.07804 kg/s/KPa5

A.2 Differential Equations Parameters For further information about the model parameters, structure and reduction, refer to [3], [2] and [1]. Table A.2 State equations parameters Parameter

Expression

Value

B1

ncm /Rcm /Jcp

1.6667 × 104

B2

Kv ncm /Rcm /Jcp

255

B3

−Cp Tatm φmax ρa π/4dc2 KU c δ/(ncp θ 1/2 efmec Jcp )

−7.4154 × 104

B4

(γ − 1)/γ

0.2857

B5

2Cp Tcp,in KU−2c

4.7393 × 107

B6

1/4φmax ρa πdc2 KUc δ/θ 1/2 Tatm γ Ra /Vsm

51.7305

C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3, © Springer-Verlag London Limited 2012

161

162

A Electric Vehicle PEM Fuel Cell Stack Parameters

Table A.2 (continued) Parameter

Expression

Value

B7

Vsm Ksm,out /Ra γ Ra /Vsm

5.081 × 106

B8

B7 pv,ca

0.2391

B9

B7 RN2 Tst /Vca

53.2352

B10

B7 RO2 Tst /Vca

46.5988

B11

1/4φmax ρa πdc2 KU c δ/θ 1/2

8.6437 × 106

B12

Ksm,out

3.6294 × 106

B13

Ksm,out pv,ca

170.815 × 10−3

B14

Ksm,out RN2 Tst /Vca

38.025

B15

Ksm,out RO2 Tst /Vca

33.284

B16

Gv φdes psat,Tcl Ksm,out /Ga

106.734 × 10−3

B17

B16 pv,ca

5.0234 × 103

B18

B16 RN2 Tst /Vca

1.1183 × 106

B19

B16 RO2 Tst /Vca

978.85 × 103

B20

Gv φatm psat,Tatm /Ga /patm

9.6827 × 10−3

B21

φatm psat,Tatm /patm

15.496 × 10−3 )−1

B22

Gv φca,in psat,Tcl (χO2 ,amb GO2 + (1 − χO2 ,amb )GN2

29.4084 × 103

B23

φdes psat,Tcl − φca,in psat,Tcl

0

B24

RO2 Tst GO2 kca,out /Vca

639.069 × 10−3

B25

B24 pv,ca

30.077 × 103

B26

B24 RN2 Tst /Vca

6.6956 × 106

B27

B24 RO2 Tst /Vca

5.8609 × 106

B28

RN2 Tst /Vca

10.4770 × 106

B29

RO2 Tst /Vca

9.1709 × 106

B30

B24 /kca,out

293.470 × 103

B31

RO2 Tst GN2 /Vca

256.786 × 103

B32

1/4GO2 n/F

31.590 × 10−6

B33

1 − χO2 ,amb GO2 /(χO2 ,amb GO2 + (1 − χO2 ,amb )GN2 )

766.990 × 103

B34

kca,out

2.177 × 10−6

B35

kca,out pv,ca

102.489 × 10−3

B36

kca,out RN2 Tst /Vca

22.815

B37

kca,out RO2 Tst /Vca

19.970

B38

Ra Tst /Vrm

20.255 × 106

B39

5.18 × 10−21

B42

B38 pa1 /std5a B38 pa2 /std4a B38 pa3 /std3a B38 pa4 /std2a

B40 B41

−708.05 × 10−18 −47.513 × 10−12 5.729 × 10−6

B43

B38 pa5 /stda

6.482

B44

B38 pa6

1.5807 × 106

B45

meana

250 × 103

A.3 Controller Parameters

163

Table A.2 (continued) Parameter

Expression

Value

B46

B34 B38

44.108

B47

B35 B38

2.075 × 106

B48

B36 B38

462.123 × 106

B49

B37 B38

404.514 × 106

B50

B33 B16

81.864 × 10−3

B51

B50 pv,ca

3.852 × 103

B52

B50 B28

857.694 × 103

B53

B50 B29

750.771 × 103

B54

B33 B12

2.783 × 10−6

B55

B33 B13

131.013 × 10−3

B56

B33 B14

29.164

B57

B33 B15

25.529

B58

Gv pv,ca

848.103

B59

XO2 ,ca,in B16

24.870 × 10−3

B60

XO2 ,ca,in B17

1.1705 × 103

B61

XO2 ,ca,in B18

260.565 × 103

B62

XO2 ,ca,in B19

228.082 × 103

B63

XO2 ,ca,in B12

845.68 × 10−9

B64

XO2 ,ca,in B13

39.801 × 10−3

B65

XO2 ,ca,in B14

8.8602

B66

XO2 ,ca,in B15

7.7557

B67

B30 − B31

36.683 × 103

B68

B29 GO2 − B31

36.683 × 103

A.3 Controller Parameters Table A.3 Super-Twisting, Twisting and Sub-Optimal

Parameter

Value

Super-Twisting algorithm parameter (α)

2

Super-Twisting algorithm parameter (λ)

3

Super-Twisting algorithm parameter (ρ)

0.5

Twisting algorithm parameter (r1 )

2.25

Twisting algorithm parameter (r2 )

0.75

Sub-Optimal algorithm parameter (U )

3

Sub-Optimal algorithm parameter (β)

0.3

Sub-Optimal algorithm parameter

(α∗ )

5

164 Table A.4 LQR controller

A Electric Vehicle PEM Fuel Cell Stack Parameters Parameter

Value

LQR controller parameter (k1 )

28.59

LQR controller parameter (k2 )

−1.6 × 10−13

LQR controller parameter (k3 )

−60.57

LQR controller parameter (k4 )

7.57

LQR controller parameter (k5 )

579.74

LQR controller parameter (k6 )

2.55

LQR controller parameter (k7 )

−3.6 × 10−14

LQR controller parameter (k7 )

−189.97

LQR controller parameter (kI )

−0.18

References 1. Kunusch C (2006) Second order sliding mode control of a fuel cell stack using a twisting algorithm. MSc thesis, University of La Plata, Argentina (in Spanish) 2. Pukrushpan JT, Stefanopoulou AG, Peng H (2004) Control of fuel cell power systems. Springer, Berlin 3. Pukrushpan JT, Stefanopoulou AG, Peng H (2004) Control of fuel cell breathing. IEEE Control Syst Mag 24:30–46

Appendix B

Laboratory FC Generation System Parameters

Table B.1 General physics constants

Table B.2 Compressor’s parameters

Parameter

Value

Dry air molar mass (Ga )

0.029 kg/mol

Oxygen molar mass (GO2 )

32 × 10−3 kg/mol

Nitrogen molar mass (GN2 )

28 × 10−3 kg/mol

Vapour molar mass (Gv )

0.01802 kg/mol

Hydrogen molar mass (GH2 )

2.01 × 10−3 kg/mol

Air specific constant (Ra )

286.9 N m/kg/K

Oxygen specific constant (RO2 )

259.8 N m/kg/K

Nitrogen specific constant (RN2 )

296.8 N m/kg/K

Vapour specific constant (Rv )

461.5 N m/kg/K

Hydrogen specific constant (RH2 )

4.124 × 103 N m/kg/K

Faraday’s constant (F )

96485 C/mol

Parameter

Value

Electric inductance (L)

2.12 mH

Electric resistance (R)

2.03

Torque constant (kφ )

0.0031 N m/A

Motor inertia (J )

0.2 × 10−6 N m

Compressor’s equivalent inertia (Jeq )

1.2 × 10−6 N m

Load torque coefficient (A0 )

4.10 × 10−4 N m

Load torque coefficient (A1 )

3.92 × 10−6 N m s

C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3, © Springer-Verlag London Limited 2012

165

166

B Laboratory FC Generation System Parameters

Table B.3 Polynomial fitting coefficients Parameter

Value

Parameter

Value

A00

0

B00

4.83 × 10−5 kg/s

A10

0.0058 N m s

B10

−5.42 × 10−5 kg/s2

A20

−0.0013

B20

8.79 × 10−6 kg/s3

A01

3.25 × 10−6

B01

3.49 × 10−7 kg/s2 /bar

A11

−2.80 × 10−6

B11

3.55 × 10−13 kg/s

A02

−1.37 × 10−9 N m s/bar2

B02

−4.11 × 10−10 kg/s/bar

N m s2 N m/bar N m s/bar

Table B.4 Humidifier and FC stack parameters Parameter

Value

Air manifold volume (Vsm ) Air manifold constraint (Ksm,out ) Humidifier volume (Vhum ) Humidifier constraint coefficient (C0 ) Humidifier constraint coefficient (C1 ) Humidifier constraint coefficient (C2 ) Number of cells (n) Cathode constraint (Kca,out ) Cathode volume (Vca ) Membrane effective area (Afc ) Dry membrane thickness (tm ) Dry membrane density (ρm,dry ) Dry membrane molecular weight (Gm,dry ) Membrane diffusion coefficient (Dw ) Water content coefficient (a0 ) Water content coefficient (a1 ) Water content coefficient (a2 ) Water content coefficient (a3 ) Electro-osmotic coefficient (n0 ) Electro-osmotic coefficient (n1 ) Electro-osmotic coefficient (n2 ) Charge transfer coefficient (α) Exchange current density (i0 ) Apparent fuel cell area (Aapp ) Resistance coefficient (R0 ) Resistance coefficient (R1 ) Polarisation curve coefficient (a) Polarisation curve coefficient (b) Polarisation curve coefficient (m) Polarisation curve coefficient (nc )

8 × 10−6 m3 0.0486 kg/s/bar 2 × 10−4 m3 1.048 × 10−7 kg/s 2.109 × 10−4 kg/s/bar2 1.562 × 10−5 kg/s/bar 7 0.0094 kg/s/bar 4 × 10−4 m3 50 cm2 0.0127 cm 0.002 kg/cm3 1.1 kg/mol 5.43 × 10−6 cm2 /s 0.043 [H2 O/SO3 ] 17.81 [H2 O/SO3 ] −39.85 [H2 O/SO3 ] 36.0 [H2 O/SO3 ] −3.4 × 10−19 [H2 O/H+ ] 0.05 [H2 O/H+ ] 0.0029 [H2 O/H+ ] 0.7 4.5 × 10−6 A/cm2 20.74 cm2 0.22 /cm2 0.005 /cm2 0.2 bar 0.06 V 8.1 × 10−5 V 15.2 cm2 /A

B Laboratory FC Generation System Parameters Table B.5 Operating conditions

167

Parameter

Value

Humidifier temperature (Thum )

55 °C

Heating line temperature (Tlh )

60 °C

Fuel cell stack temperature (Tst )

60 °C

Humidifier relative humidity (RH hum )

0.95

Ambient relative humidity (RH amb )

0.5

Ambient pressure (Pamb )

1 bar

Ambient temperature (Tamb )

25 °C

Ambient oxygen molar fraction (χO2 ,amb )

0.21

Hydrogen input flow (WH2 ,an )

2 slpm

Appendix C

Laboratory FCGS State Space Functions and Coefficients

C.1 Expression of ϕ(x, u, t)     ϕ(x, u, t) = 2B02 m1 u(t) − m2 x1 − x1 m3 + A0 + A00 + A10 (x2 m5 + m6 )  + A20 (x2 m5 + m6 )2 + A01 x1 + A11 (x2 m5 + m6 )x1 + A02 x12 m4   + B01 + B11 (x2 m5 + m6 ) + 2B02 x1 −m1 m2 − m3 + A01   + A11 (x2 m5 + m6 ) + 2A02 x1 m4 + B11 m5 B00 + B10 (x2 m5 + m6 ) + B20 (x2 m5 + m6 )2 + B01 x1 + B11 (x2 m5 + m6 )x1 + B02 x12   3 − x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3 − x2 m5 − (x3 RO2  2  + x4 RN2 + x5 Rv )m8 C2 − x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C1    − C0 + B10 m5 + 2B20 (x2 m5 + m6 )m5 + B11 m5 x1 B01    + B11 (x2 m5 + m6 ) + 2B02 x1 m1 u(t) − m2 x1 − x1 m3 + A0 + A00 + A10 (x2 m5 + m6 ) + A20 (x2 m5 + m6 )2 + A01 x1      + A11 (x2 m5 + m6 )x1 + A02 x12 m4 + B11 m5 m1 u(t) − m2 x1 − x1 m3 + A0 + A00 + A10 (x2 m5 + m6 ) + A20 (x2 m5 + m6 )2  + A01 x1 + A11 (x2 m5 + m6 )x1 + A02 x12 m4   + B01 + B11 (x2 m5 + m6 ) + 2B02 x1 A10 m5 + 2A20 (x2 m5 + m6 )m5   + A11 m5 x1 m4 + 2B20 m25 B00 + B10 (x2 m5 + m6 ) + B20 (x2 m5 + m6 )2 + B01 x1 + B11 (x2 m5 + m6 )x1 + B02 x12  3 − x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3  2 − x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2    − x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C1 − C0   + B10 m5 + 2B20 (x2 m5 + m6 )m5 + B11 m5 x1 B10 m5 C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3, © Springer-Verlag London Limited 2012

169

170

C Laboratory FCGS State Space Functions and Coefficients

+ 2B20 (x2 m5 + m6 )m5 + B11 m5 x1  2 − 3 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3 m5    − 2 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2 m5 − m5 C1  × B00 + B10 (x2 m5 + m6 ) + B20 (x2 m5 + m6 )2 + B01 x1 + B11 (x2 m5 + m6 )x1 + B02 x12  3 − x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3 2  − x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2    − x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C1 − C0   + B10 m5 + 2B20 (x2 m5 + m6 )m5 + B11 m5 x1 2   × 3 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3 RO2 m8    + 2 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2 RO2 m8 + RO2 m8 C1   3 × m9 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3  2 + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2    + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C1 + C0 G−1 a  −1 m11 × (x2 m5 − m10 )−1 1 + x2 m5 − m10  3 + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3 2  + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2    + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C1 + C0  −1  m11 × 1+ XO2 ,ca,in x2 m5 − m10  −1 m14 × 1+ (x3 RO2 + x4 RN2 + x5 Rv )m8 − m12   − Kca (x3 RO2 + x4 RN2 + x5 Rv )m8 − Pamb x3 RO2 GO2   −1   x3 RO2 GO2 x3 RO2 GN2 × 1 + Gv x5 Rv + 1− x3 RO2 + x4 RN2 x3 RO2 + x4 RN2 −1  x3 RO2 GO2 −1 −1 × (x3 RO2 + x4 RN2 ) (x3 RO2 + x4 RN2 ) x3 RO2 + x4 RN2   −1  x3 RO2 GO2 nIst GN2 + 1− − 1/4 x3 RO2 + x4 RN2 F   + B10 m5 + 2B20 (x2 m5 + m6 )m5 + B11 m5 x1 2   × 3 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3 RN2 m8

C.1 Expression of ϕ(x, u, t)

171

  + 2 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2 RN2 m8    3 + RN2 m8 C1 m9 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3  2 + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2    + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C1 + C0  −1 m11 −1 × G−1 1 + (x m − m ) 2 5 10 a x2 m5 − m10  3 + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3 2  + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2    + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C1 + C0  −1  m11 × 1+ x2 m5 − m10 −1  Gv m12 × 1+ Gac ai n((x3 RO2 + x4 RN2 + x5 Rv )m8 − m12 )  × (1 − XO2 ,ca,in ) − 1 − x3 RO2 m8 GO2 (x3 RO2 m8 + x4 RN2 m8 )−1  −1   x3 RO2 m8 GO2 x3 RO2 m8 GN2 + 1− x3 RO2 m8 + x4 RN2 m8 x3 RO 2 m8 + x4 RN 2 m8   × Kca (x3 RO2 + x4 RN2 + x5 Rv )m8 − Pamb   x3 RO2 m8 GO2 × 1 + Gv x5 Rv m8 (x3 RO2 m8 + x4 RN2 m8 )−1 x3 RO2 m8 + x4 RN2 m8  −1 −1   x3 RO2 m8 GN2 + 1− x3 RO2 m8 + x4 RN2 m8   + B10 m5 + 2B20 (x2 m5 + m6 )m5 + B11 m5 x1 2   × 3 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3 Rv m8    + 2 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2 Rv m8 + Rv m8 C1   3 × Gv m12 x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C3 

×

2  + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C2    + x2 m5 − (x3 RO2 + x4 RN2 + x5 Rv )m8 C1 + C0 G−1 a  −1 Gv m10 × (x2 m5 − m10 )−1 1 + Ga (x2 m5 − m10 )   + (x2 m5 − b2 )3 C3 + (x2 m5 − b2 )2 C2 + (x2 m5 − b2 )C1 + C0  −1   Gv m10 × 1+ − Gv m12 (x2 m5 − b2 )3 C3 Ga (x2 m5 − m10 )

172

C Laboratory FCGS State Space Functions and Coefficients

 + (x2 m5 − b2 )2 C2 + (x2 m5 − b2 )C1 + C0 G−1 a −1  m G v 10 × (x2 m5 − m10 )−1 1 + Ga (x2 m5 − m10 )   + (x2 m5 − b2 )3 C3 + (x2 m5 − b2 )2 C2 + (x2 m5 − b2 )C1 + C0  −1  Gv m10 × 1+ Ga (x2 m5 − m10 ) −1  Gv m12 × 1+ Ga((x3 RO2 + x4 RN2 + x5 Rv )m8 − m12 )   − Kca (x3 RO2 + x4 RN2 + x5 Rv )m8 − Pamb   + Kca (x3 RO2 + x4 RN2 + x5 Rv )m8 − Pamb   x3 RO2 m8 GO2 × 1 + Gv x5 Rv m8 (x3 RO2 m8 + x4 RN2 m8 )−1 x3 RO2 m8 + x4 RN2 m8   −1 −1 x3 RO2 m8 Gv nIst + 1− + 1/2 GN2 x3 RO2 m8 + x4 RN2 m8 F    1 a1 n0 + n1 a0 + (x7 m15 + x5 m16 ) + Af c F 2  a a2 3 + (x7 m15 + x5 m16 )2 + (x7 m15 + x5 m16 )3 4 8  a1 a2 + n2 a0 + (x7 m15 + x5 m16 ) + (x7 m15 + x5 m16 )2 2 4 2  a3 Ist + (x7 m15 + x5 m16 )3 8  (a0 + a1 x5 m16 + a2 x52 m216 + a3 x53 m316 )ρm,dry − Gm,dry    (a0 + a1 x7 m15 + a2 x72 m215 + a3 x73 m315 )ρm,dry Dw G v Af c n − Gm,dry tm

C.2 Model Coefficients m1 = Kφ /R,

m2 = Kφ 30/π,

m5 = Tsm Ra /Vhum , m8 = Tst /Vca ,

m4 = π/30/Jeq

m6 = −Psat (Tsm )RH amb + RH hum,ca Psat (Thum,ca ) m9 = Gv RH hum,ca Psat (Thum,ca ),

m11 = Gv Psat (Tsm )RH amb /Ga , m13 = RO2 Tst GO2 ,

m3 = A1 30/π,

m10 = Psat (Tsm )RH amb

m12 = RH hum,ca Psat (Thum,ca )

m14 = Gv RH hum,ca Psat (Thum,ca )/Ga

m15 = Tst Rv /Van /Psat (Tlh,an ),

m16 = Rv Tst /Vca /Psat (Tlh,ca )

C.3 Feedforward Control Action

m17 = RH an,in Psat (Tlh,an ),

173

m18 = Tst /Van ,

m19 = Tst Rv /Van

m20 = Tst RH2 /Van

C.3 Feedforward Control Action 6 5 4 3 uff = 0.1014Wair,ref − 1.1412 Wair,ref + 4.8303Wair,ref − 9.3370 Wair,ref 2 + 8.1430Wair,ref − 0.6129Wair,ref − 0.1974

Index

A Air compressor, 75, 106 humidifier, 77, 113 supply, 6, 24 supply manifold, 76, 112 Algorithm sub-optimal, 56, 67, 94, 139, 140 super twisting, 56, 63, 86, 88, 139, 144 twisting, 56, 93, 139, 140 Alkaline fuel cell, 4 Anode channels, 78, 120 reaction, 3 Armature current, 107 voltage, 107 B Back-diffusion, 121 Bipolar plate, 22 C Catalyst, 13, 20 Catalytic oxidation, 3, 13 reduction, 3, 13 Cathode channels, 77, 117 reaction, 3 starvation, 30 Charge transfer coefficient, 16 Chattering, 36, 47 Compressor, 24 Control affine system, 37 Current collector, 16

D DC motor, 75 DC/AC converter, 25 DC/DC converter, 25 Degradation, 30 Diaphragm vacuum pump, 106, 111 Diffeomorphism, 37 Differential inclusion, 49, 56, 86, 139 Direct methanol fuel cell, 5 Discontinuous control action, 39, 47 E Efficiency, 17 Electro-osmotic drag, 19, 79, 121 Electrochemical potential, 123 Electrode, 20 Electron, 14 Energy conversion, 17 Enthalpy, 17 Entropy change, 15 Equivalent control, 40, 42 Exchange current density, 16 F Faraday’s constant, 14, 79 Feedforward control, 86, 87, 141 Filippov differential inclusion, 50 method, 45 sense, 51 solution, 51 Finite time convergence, 64 Fuel cell, 3 active area, 79 apparent area, 124 stack, 5, 117

C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI 10.1007/978-1-4471-2431-3, © Springer-Verlag London Limited 2012

175

176 G Gas diffusion layer, 21 Gibbs free energy, 14 H Heat management, 24 Higher order sliding mode, 48 Humidification, 26 Humidifier, 26 Humidity ratio, 118 Hydrogen, 1 Hydrogen supply, 24 I Ideal sliding, 40 Invariance conditions, 41 K Kalman observer, 98 L Lie derivative, 37, 41, 137 Line heater, 27 Load torque, 75, 107 Losses activation, 15, 123 concentration, see diffusion diffusion, 16, 123 ohmic, 16, 123 LQR, 30, 97, 100 Lyapunov function, 40

Index excess ratio, see stoichiometry starvation, 73, 83, 84, 135 stoichiometry, 73, 83, 90, 135, 145 supply, 24 P Parasitic load, 24 Peak detector, 68 PEM fuel cell, 5, 10, 13, 17, 30 Phosphoric fuel cell, 4 PID, 30 Platinum, 13, 20 Polarisation curve, 17, 123, 126 Polymer electrolyte membrane fuel cell, see PEM fuel cell Polymeric membrane, 19, 79 Potential difference, 14 Power conversion, 17 Proton, 14 Proton exchange membrane fuel cell, see PEM fuel cell Protonic conductivity, 14 R Regularity condition, 51 Relative degree, 37, 52, 54, 55 Return manifold, 80 Reversible voltage, 14

M Majorant curve, 62, 65 Mass mole fraction, 118 Matching condition, 44 MEA, 21, 26, 125 Membrane conductivity, 16 dry thickness, 16 electrode assembly, see MEA water content, 16, 125 Modulation factor, 68 Molten carbonate fuel cell, 4 Motor torque, 75, 107

S Scalar field, 37 Sealing gasket, 22 Second order sliding mode, 54 Shaft angular speed, 107 Sliding manifold, see surface mode control, 35 surface, 37, 39, 82, 136 variable, 137 Solid oxide fuel cell, 5 Stack current, 89 voltage, 90 State space model, 130

N Nernst voltage, 15 Net power, 82, 90, 134 Normal form, 52 Nozzle, 112

T Tafel equation, 16 Thermal management, 6 Torque disturbance, 88 Transversality condition, 42

O Oxygen, 3 control, 30

V Vector field, 37 Voltage drop, 3

Index W Water diffusion, 19

177 management, 6, 24 transport, 121 Windup, 147