Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems (Springer Theses) [1st ed. 2023] 3031389239, 9783031389238

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Andreu Cecilia

Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

Andreu Cecilia

Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems Doctoral Thesis accepted by Universitat Politècnica de Catalunya, Barcelona, Spain

Author Dr. Andreu Cecilia Institut de Robòtica i Informàtica Industrial (CSIC-UPC) Universitat Politècnica de Catalunya Barcelona, Spain

Supervisors Dr. Ramon Costa-Castelló Institut de Robòtica i Informàtica Industrial (CSIC-UPC) Universitat Politècnica de Catalunya Barcelona, Spain Dr. Maria Serra Prat Institut de Robòtica i Informàtica Industrial (CSIC-UPC) Universitat Politècnica de Catalunya Barcelona, Spain

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-031-38923-8 ISBN 978-3-031-38924-5 (eBook) https://doi.org/10.1007/978-3-031-38924-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Supervisors’ Foreword

Feedback is one of the automatic control fundamental concepts. It involves the use of sensors to obtain the value of the physical variables to be controlled. Unfortunately, the addition of new sensors increases the cost and complexity of control systems. Moreover, in certain applications, it is not possible to measure the variables of interest. A current trend to address these problems is the use of virtual sensors that combine the use of measured variables with mathematical models in order to estimate the desired variables. In many cases, these virtual sensors take the form of state observers. These methods, typical of automatic control and with a long conceptual trajectory, are being increasingly integrated into control and supervision systems as in the case of the popular digital twins. The thesis presents the application of state observers to two types of energy systems such as fuel cells and energy distribution networks. In fuel cells, the application of state observers to the estimation of PEM fuel cell membrane humidity is presented. This is a very relevant problem for improving the performance of PEM fuel cells, and it is very difficult to solve using conventional techniques. In energy distribution networks, the detection and compensation of attacks caused by external entities is addressed. This is a subject of great interest and current affairs. In addition to the practical applications, the thesis has made novel conceptual proposals that represent an advance in the state of art of estimation algorithms, highlighting the proposal made in the joint estimation of states and parameters. We believe that it is a very actual work and that can contribute significantly to the improvement of the control and supervision of energy systems. Barcelona, Spain February 2023

Maria Serra Prat Ramon Costa-Castelló

v

Parts of this thesis have been published in the following articles: Peer-Reviewed Journals • Cecilia, A., & Costa-Castelló, R. (2022). Addressing the relative degree restriction in nonlinear adaptive observers: A high-gain observer approach. Journal of the Franklin Institute, 359(8), 3857–3882. • Cecilia, A., Sahoo, S., Dragiˇcevi´c, T., Costa-Castelló, R., & Blaabjerg, F. (2021). On Addressing the Security and Stability Issues Due to False Data Injection Attacks in DC Microgrids-An Adaptive Observer Approach. IEEE Transactions on Power Electronics, 37(3), 2801–2814. • Cecilia, A., Serra, M., & Costa-Castelló, R. (2021). Nonlinear adaptive observation of the liquid water saturation in polymer electrolyte membrane fuel cells. Journal of Power Sources, 492, 229641. • Cecilia, A., Sahoo, S., Dragiˇcevi´c, T., Costa-Castelló, R., & Blaabjerg, F. (2021). Detection and mitigation of false data in cooperative dc microgrids with unknown constant power loads. IEEE Transactions on Power Electronics, 36(8), 9565– 9577. • Cecilia, A., & Costa-Castelló, R. (2020). High gain observer with dynamic deadzone to estimate liquid water saturation in pem fuel cells. Revista Iberoamericana de Automática e Informática industrial, 17(2), 169–180. International Conferences • Cecilia, A., & Costa-Castelló, R. On state-estimation in weakly-observable scenarios and implicitly regularized observers. In 60th IEEE Conference on Decision and Control, Austin, Texas, USA, December 2021. • Cecilia, A., & Costa-Castelló, R. (2021, June). Library-based adaptive observation through a sparsity-promoting adaptive observer. In 2021 European Control Conference (ECC) (pp. 2187–2192). IEEE. • Cecilia, A., Serra, M., & Costa-Castelló, R. (2020). PEMFC state and parameter estimation through a high-gain based adaptive observer. IFACPapersOnLine, 53(2), 5895–5900.

vii

Acknowledgements

I would like to dedicate this section to express my gratitude to all people that made this thesis possible. First, I want to thank my supervisor Ramon who motivated me to start this long journey, and has guided me through the technical and personal aspects of this Ph.D. Furthermore, I would like to thank Maria for complementing Ramon’s work and helping me with further insights on fuel cell systems. I also want to thank all my companions of IRI that I had the pleasure to work with during these years. I would like to thank all the past and present members of Office 6 for their valuable assistance, specially during my Ph.D. beginning. Furthermore, I want to express my gratitude to the tupper crew that, each day, let me disconnect from the work. Finally, I want to thank Vicente Roda for his valuable help during the fuel cell experiment design process. I would like to thank Marc Secanell from Energy Systems Design Laboratory, for helping me to develop the fuel cell model and receiving me at University of Alberta. Our collaboration was tampered by the COVID pandemic, nonetheless, ended up as a key part of my Ph.D. work. Moreover, I want to thank Subham Sahoo and Frede Blaabjerg from Aalborg University and Tomislav Dragiˇcevi´c from Technical University of Denmark, for helping me to correctly formulate, solve and validate a valuable cybersecurity problem in microgrids. Finally, I would like to thank Daniele Astolfi for receiving me at Laboratory of Automatic Control, Chemical and Pharmaceutical Engineering in Lyon, for providing key information related to filter design in nonlinear observers and helping me to rigorously formulate the observer design problem. I would like to thank my family and friends who always supported me throughout this project. My father Joan, my mother Fina, and my brother Aleix have always backed my decisions, motivated me and without their support I would have never reached where I am now. My girlfriend Ariadna, and my friends Francesc Ganau, Francesc Antonijuan, and Ricard Hidalgo, who have been there for me, both, during the good times and the bad times.

ix

x

Acknowledgements

Finally, I would like also to acknowledge the funding project that allowed this work to be started and completed. That is, this work has been partially supported by the Spanish State Research Agency through the María de Maeztu Seal of Excellence to IRI (MDM-2016-0656).

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3

2 The Observation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Observability Analysis and Necessary Conditions for the Solvability of the Observation Problem . . . . . . . . . . . . . . . . . . 2.3 The Benefits of Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6 13 14

3 Nonlinear Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Observer Definition and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 High-Gain Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Classic High-Gain Observer . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Transformation to a Triangular Form . . . . . . . . . . . . . . . . . . . 3.2.3 Extension of the High-Gain Observer: Low-Power Peaking-Free Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Parameter Estimation-Based Observer . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Idea and Structure of the Observer . . . . . . . . . . . . . . . . . . . . . 3.3.2 Standard Parameter Estimator . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Implicitly-Regularized Observer . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Transformation to a State-affine Form . . . . . . . . . . . . . . . . . . 3.4 Nonlinear Observers in the Presence of Unmodelled Elements . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 20 22 28 32 38 40 41 42 47 48 51

4 Adding Filters in Nonlinear Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Conflict of Measurement Noise in Nonlinear Observers . . . . . . 4.2 On Adding Filters in Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Low-Pass Filters in Nonlinear Observer . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Cascaded Filters and Iterative Filter Design . . . . . . . . . . . . . 4.3.2 Filter Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 57 60 65 66

xi

xii

Contents

4.4 On Internal-Model Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Main Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Internal-Model Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Combining Both Filters: Low-Pass Internal-Model Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Dynamic Dead-Zone Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 69 70

5 Adaptive Observers: Direct and Indirect Redesign . . . . . . . . . . . . . . . . 5.1 Adaptive Observers and the Conflict of Model Uncertainty . . . . . . . 5.2 Direct Adaptive Observer Redesign . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Constructing a Strict Lyapunov Function . . . . . . . . . . . . . . . 5.2.2 Addressing the Relative Degree Condition: A High-Gain Observer Approach . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Library-Based Adaptive Observation: A Sparsity-Promoting Adaptive Observer . . . . . . . . . . . . . . . 5.3 Indirect Adaptive Observer Redesign . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 94 99

6 PEM Fuel Cell Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction and PEM Fuel Cell Principles . . . . . . . . . . . . . . . . . . . . . 6.2 PEM Fuel Cell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Control Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Membrane Sub-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Porous Media Sub-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Liquid Water Dynamics in the Porous Media . . . . . . . . . . . . 6.2.6 Channel Sub-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.7 Thermal Sub-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.8 Electrochemical Sub-model . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.9 Partial Experimental Validation and Model Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.10 Estimation Objectives and Model Reduction . . . . . . . . . . . . 6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 System Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Observer 1: Low-Power Peaking-Free Dead-Zone Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Adding the Voltage Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Observer 2: High-Gain Observer with Voltage Sensor and Low-Pass Internal-Model Filter . . . . . . . . . . . . .

78 82 88

102 112 119 125 129 129 131 132 133 133 138 141 143 146 149 151 157 160 161 163 164 167 172

Contents

xiii

6.3.6 Observer 3: Direct Adaptive Observer . . . . . . . . . . . . . . . . . . 6.3.7 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.8 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

176 180 182 188

7 Cybersecurity in DC-Microgrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 System Model, Problem Formulation and Proposal . . . . . . . . . . . . . . 7.2.1 DC-Microgrid Model and Control . . . . . . . . . . . . . . . . . . . . . 7.2.2 Cyber-Attack Model and Problem Formulation . . . . . . . . . . 7.3 Proposed Nonlinear Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Estimation of ξ1,i , ξ2,i and Design of Extended Low-Power Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Estimation of η i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Estimation of Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Observer Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Numerical Simulations and Experimental Validations . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 194 194 197 200 202 205 208 211 214 214

8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Future Research Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217 217 219 220

About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Abbreviations, Nomenclature and Notations

CL CPL DC DGU FDIA GDL I&I ISS MPL MSE PEM PM

Catalyst Layer Constant Power Load Direct Current Distributed Generation Unit False Data Injection Attack Gas Diffusion Layer Immersion and Invariance Input-to-State Stable Micro-porous Layer Mean Squared Error Polymer Electrolyte Membrane Porous Media

Operators x∈A A⊆B |·| f −1 Lkf h dh ⊗ | · |[0,t] | · |∞ satk (·) dzk (·)

Element x belongs to set A A is a subset of B Standard Euclidean norm Inverse function of f k-th order Lie derivative of h along the vector field f Differential of function h Kronecker product Supremum norm of the signal at time range [0, t] Maximum value of the norm Saturation function of amplitude k Dead-zone function of amplitude k

xv

xvi

Abbreviations, Nomenclature and Notations

General Convention a a A a(·) aˆ a˜ a˙ a(k)

Scalar Vector Matrix Function Estimation of the factor a Estimation error of the factor a Time derivative of a k-th time derivative of a

Specific Notation λmax (·) λmin (·) σ(·) K∞ R Ik max{·} min{·} blckdiag(·) rank (·)

Maximum eigenvalue Minimum eigenvalue Spectrum of a matrix Radially unbounded function Set of real numbers Identity matrix of dimension k × k Maximum value of a set Minimum value of a set Block diagonal matrix Rank of a matrix

Specific Observer Notation x u y θ ξ X(x0 , u, t) Y(x0 , u, t) ϕ(·) κ(·) (·)  −R (·)

State vector in the original coordinates Controllable inputs Measured outputs Vector of unknown parameters State vector in the target coordinates Value of the state vector at time t with input u and initialized at the state x0 Value of the output at time t with input u and initialized at the state x0 Observer internal dynamics Observer feedback term Map between target coordinates and original coordinates Map between original coordinates and target coordinates

Abbreviations, Nomenclature and Notations

V Ok Φ

Lyapunov function Observability map of order k Fundamental matrix

Specific Fuel Cell Notation εi ρi eff Di,e Mi μl eff Kl Pl Pg Pc EW λ nd Cp KT Vi Dλ j ai kλ α δi DB DrBe f DK k0,l s σH2 O θi n˙ Ii n˙ O i yj,i Ai di I ˙ Q m ˙ Ii

Porosity of phase i Density of element i Effective diffusion of reactant i Molar mass of element i Dynamic viscosity Effective permeability Liquid pressure Gas pressure Capillarity pressure Ionomer equivalent weight Membrane water content Electro-osmotic drag coefficient Specific heat capacity Thermal conductivity Volume of phase i Back-diffusion coefficient Current density Activity of the species i Sorption/desorption rate Sorption fitting parameter Thickness of the phase i Binary diffusion coefficient Reference binary diffusion coefficient Knudsen diffusion coefficient Intrinsic permeability Liquid water saturation Liquid water surface tension Contact angle of phase i Input molar flow of reactant i Output molar flow of reactant i Molar fraction of the species i at phase j Cross-sectional area of the phase i Effective diameter of phase i Exchange current Heat flux Input mass ow of the species i

xvii

xviii

m ˙O i hi Vf c vair Ecell ηi Eo cv ΔS sopt Rionic Ea kb F R ΔHf ΔHv a p ΔG ∗ ncell E0 j0,a n j0,c a,r e f jL Rcom rp kcond ,evap αc

Abbreviations, Nomenclature and Notations

Output mass ow of the species i Convective heat transfer of the volume i Stack voltage Air velocity Cell voltage Overpotential i Open-circuit voltage Entropy change of the reaction Optimal liquid water saturation Ionic resistance of the membrane Activation energy of the evaporation process Boltzmann constant Faraday Constant Universal gas constant Heat formation of water Water heat of vaporization Activation energy for the ORR on Pt Number of cells in the stack Theoretical voltage at standard conditions Anode reference exchange current ref Standard cathode reference exchange current Limiting current Electrical components resistance Primary-particle radius Phase change rate Cathode transfer coefficient

Specific DC Microgrid Notation Iti Vi Ik Lti Ci Ri Rk lK ui Pi B G V

ith DGU output current ith DGU voltage Power line current ith DGU filter inductance ith DGU shunt capacitor ith DGU local load impedance Power line resistance Power line inductance ith DGU converter voltage ith DGU constant power load Incident matrix Communication graph Set of DGUs

Abbreviations, Nomenclature and Notations

E PI KpI KiI Iref ,i ψi Vdc,ref ΔVi,j xia vˆdc,i σi μi

Set of power lines Proportional Integral Gain of the proportional part of the PI controller Gain of the integral part of the PI controller Exchange current reference Information vector Local voltage reference Local voltage off-set FDIA value Average voltage estimate in the ith DGU Virtual state Auxiliary signal

xix

Chapter 1

Introduction

Considering the increasing energy demand, persistent scientific innovation, and environmental concerns, it is natural to observe that the energy sector is drastically expanding to a more efficient, resilient, responsive, and green architecture in all of its stages, from its generation to storage, transmission, distribution, and consumption. Indeed, it is clear that society is advancing to a fully electrical future. For example, electricity consumption may double in the transportation sector due to the rapid development of electrical vehicles. In parallel, the introduction of electrochemical devices in households and the electrical grid may provide new ways of storing and selling energy, from a business point of view and/or an individual one. The current perspective is that the sector is advancing to an architecture with larger renewable energy penetration, more distributed generation and storage, and higher degree of autonomy. In such context, control theory has to advance in parallel to the energy innovations in order to provide adequate solutions to the emerging problems related to identification, monitoring, and control. Otherwise, efficiency, resilience, lifetime, and stability may be endangered. Indeed, in this new framework, energy systems are frequently required to modify its current set-point and/or control inputs, in order to address a continuously changing energy demand/generation and/or reduce the effect of undesirable disturbances. Following this line, a clear limitation in the sector is that physical, technological and economical constraints limits the amount of sensors that can be placed in the system. Consequently, there will be significant fluctuations of multi-dimensional unmeasured internal variables during the system operation. The internal variables have to be properly monitored and controlled in order to guarantee a proper performance and lifespan of the elements. To solve this conflict, it comes natural the idea of designing an algorithm that generates an estimation of the internal variables based on the measurements from the available sensors. In the automatic control field, a computationally efficient solution to this problem is obtained by means of observers. Although the idea of observer can be traced back to more than 50 years ago, and the theory has been continuously improving since then, there is a significant © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Cecilia, Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems, Springer Theses, https://doi.org/10.1007/978-3-031-38924-5_1

1

2

1 Introduction

theory-practice gap in the energy sector. On the one hand, it is remarkable that the introduction of new ways of generating, distributing and storing energy is significantly increasing the complexity of the sector from a systems perspective. Indeed, strong nonlinearities are being introduced in the system, which makes standard linear control algorithms (that are more familiar for practitioners in the energy sector) unreliable and with limited theoretical guarantees. Furthermore, the system large scale demands for computationally-efficient and distributed solutions. From these points, it is clear that more recent and advanced nonlinear observer results have to be introduced in the energy sector. On the other hand, observer theory results are often based on assumptions that are hardly met in real energy systems. That is, the requirement of an exact model of the system and ideal sensors. For this reason, the effect of unmodelled factors have to be taken into account during observer design. Indeed, it is noticeable that most theoretical results in nonlinear observer design generally focus on proving the stability of the architecture. That is, the convergence to zero (or to a bounded value) of an estimation-error signal. Nonetheless, such results provide little to no result on how to tune the observer parameters in order to guarantee performance. For this reason, in practice, the commission of a nonlinear observer usually relies on a trial-and-error parameter tuning which ends in suboptimal and/or inadequate performance. It should be remarked that tuning the parameters of a nonlinear observer is a highly complex task which, in general, ends in a non-convex optimization. For this reason, there is a current tendency of guaranteeing the observer performance, not from an optimal parameter tuning, but from a redesign perspective. That is, to assume an already existing observer and modify its structure through a process that guarantees a performance improvement. The redesign framework may be a key-step in the process of deploying nonlinear observers in the energy sector and will be a central topic of this Ph.D thesis.

1.1 Thesis Objective In light of the discussion presented above, the main objective of this thesis can be defined as: “Develop a general observer design framework such that it: (a) (b) (c) (d) (e)

considers that the system presents strong nonlinearities, maintains adequate estimation accuracy in presence of significant sensor noise, presents adequate performance in the presence of model parametric uncertainty, can be implemented in a computationally efficient manner, can be used to solve relevant energy systems problems.”

In order to achieve this general objective the following steps will be undertaken • Study the current state-of-art of nonlinear observer design. • Develop a redesign framework that, under some assumptions on the observer, guarantees an improvement of the accuracy in the presence of sensor noise, without compromising other performance indices as robustness and transient behaviour.

1.2 Thesis Outline

3

• Develop a redesign framework (compatible with the first one), that guarantees an improvement of the observer accuracy in the presence of parametric uncertainty, without compromising other performance indices as robustness and transient behaviour. • Utilize the new theory to provide a solution for a relevant monitoring problem in a hydrogen fuel cell, which is a highly nonlinear and uncertain electrochemical system. • Implement the new theory to solve a cyber-security problem in a large scale DCmicrogrid with nonlinear unknown constant power loads. • Verify the proposed algorithms through numerical simulations and, whenever possible, in an experimental prototype.

1.2 Thesis Outline This thesis is separated in two parts. In the first on, the theoretical aspects of nonlinear observer design and redesign are studied for a relatively general nonlinear system. In the second one, the new developed theory will be utilized to solve relevant practical problems in the energy sector. It should be stated that the first part of the thesis does not focus on the target practical applications, thus, its results are not restricted to the cases considered in the second part of the thesis and, in future works, could be extended to other problems in energy systems. Specifically, the remainder of this thesis is organized as follows. In Chap. 2 the problem of estimating unmeasured internal variables will be formalized through what will be referred to as “the observation problem”. The sufficient conditions for the solvability of the observation problems will be presented. Finally, the reasoning behind using nonlinear observers to solve the observability problem will be studied. Chapter 3 focuses on the main theory related to nonlinear observer design. First, a formal definition of a nonlinear observer is provided. Moreover, the concept of observer stability (in the ideal case, that is, without unmodelled elements) is formalized through a Lyapunov analysis. Second, the chapter focuses on two types of observers. On the one hand, the high-gain observer and its more recent modifications. On the other, the parameter estimation-based observer. Following from this second observer, a novel implicitly-regularized observer will be proposed. Finally, the presented observer design framework is generalized to take into account unmodelled elements. That is, the presence of measurement noise and the presence of unmodelled disturbances/uncertainty. Chapter 4 proposes a filter-based redesign to address the effect of measurement noise. First, it theoretically analyzes the implementation of low-pass filters to reduce the effect of high-frequency sensor noise. Second, a novel “internal-model filter” will be proposed in order to address narrow-band measurement disturbances. Third, the low-pass filter and the internal model filter will be combined in a unified filter. Finally, the chapter presents some recent results on the implementation of nonlinear dead-zone filters in observers.

4

1 Introduction

Chapter 5 studies the adaptive redesign approach to address parametric uncertainty. At the beginning, the standard direct adaptive redesign is presented. Multiple theoretical and practical limitations of the direct adaptive redesign will be established and, later, relaxed. First, a strict Lyapunov function will be constructed to assess the stability of the direct redesign with robustness guarantees. Second, a highgain observer approach will be presented to circumvent the relative degree limitation of direct adaptive redesigns. Finally, the direct redesign will be extended to the case in which the parameter vector is sparse. In the end of the chapter, a new indirect adaptive redesign methodology based on the immersion and invariance approach will be presented and validated. Chapter 6 implements the developed theory in the problem of fuel cell cathode liquid water monitoring. Firstly, a low-complexity fuel cell model will be developed. After that, the model order is reduced to obtain a more tractable model, from an observer design viewpoint. Then, three observers are presented and validated through numerical simulations. First, an observer based on recent results on highgain observer and nonlinear dead-zone filters will be presented. Second, the observer performance is improved by including the voltage sensor signal. It will be established that the accuracy of this second observer is affected by periodic anode purges, and an internal model filter redesign will be proposed to address this disturbance. Third, a direct adaptive redesign will be presented to increase the robustness of the observer in front of parametric uncertainty in the liquid water dynamics equation. At the end of the chapter, this last observer is validated in an experimental setup. Chapter 7 utilizes the new observer theory to the problem of detection and mitigation of false data injection attacks in a DC microgrid with unknown constant power loads. The beginning of the chapter focuses on microgrid modelling and formulating the type of attack considered. Then, the attack detection and mitigation is transformed to an observation problem. A distributed nonlinear observer is proposed to solve the observation problem. The local observer (i.e. the observer that is implemented in each converter of the system in a distributed manner) is composed by two interconnected observers and a constant power load estimator based on the indirect adaptive approach proposed in Chap. 5. Finally, some conclusions are drawn in Chap. 8.

Chapter 2

The Observation Problem

Many control and monitoring problems require an estimation of unmeasured internal state variables from measurements of available sensors. This estimation process is usually referred as “The observation problem”. This chapter presents the observation problem for generic nonlinear systems and explains the sufficient conditions for this problem to be solvable. Finally, the reasons for the use of observers in the observation problem are given.

2.1 Problem Definition Before establishing what is being referred to as the problem of “estimating the system unmeasured internal variables”, it is important to clearly define which type of system is being considered in this work. Indeed, this Ph.D thesis main arguments will be developed under the state-space formalism. Specifically, consider a multi-input multioutput nonlinear system of the form x˙ = f(x, u) y = h(x),

(2.1)

where x ∈ Rn is the state vector, u ∈ Rq is the controlled input, y ∈ Rm is the measured output and the maps f : Rn → Rn , h : Rn → Rm are sufficiently smooth vector fields. Moreover, there exists some sets X0 ⊆ X ⊆ Rn and U ⊆ Rq , such that the trajectories of (2.1), with initial conditions x(0) in X0 and input u(t) belonging to U for all times, remain in X for all t ≥ 0. It is assumed that the system is initialized at time t = 0.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Cecilia, Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems, Springer Theses, https://doi.org/10.1007/978-3-031-38924-5_2

5

6

2 The Observation Problem

Moreover, define the following • X(x0 , u, t): Value of the states at time t of (2.1) with input u and initialized at the state x0 . • Y(x0 , u, t): Value of the output at time t of (2.1) with input u and initialized at the state x0 . Now that the considered system has been described it is possible to characterize the observation problem. Definition 2.1 (Observation Problem) Consider any input u ∈ U and any initial condition x 0 ∈ X0 . The objective is to generate a signal xˆ (t) based on the values of u and Y (x 0 , u, t) in the time range t ∈ [0, t], such that xˆ (t) converges to X(x 0 , u, t). Notice that a direct method to solve this problem would be to invert the output function h, i.e. xˆ = h−1 (y), where h−1 : Rm → Rn is an inverse function that satisfies h−1 (h(x)) = x. However, in most practical applications, the number of sensors is lower than the dimension of the states, m < n. Consequently, h is not an injective function and the inverse function h−1 does not exist. This fact means that from a single sample of the output is not possible to infer the state value. Nonetheless, instead of using a single sample, the observation problem can and has to be solved using the model in (2.1) and the whole trajectories of u and Y(x0 , u, t) in the time range t ∈ [0, t]. Naturally, before studying and developing an algorithm for the observation problem, it is crucial to establish if such problem can be solved for the considered system. Roughly speaking, it has to be studied if the trajectories of the output Y(x0 , u, t), together with the equations of the system model, contain enough information to estimate the true state trajectory. If such property is not satisfied, there cannot be any algorithm that retrieves the system states from the output. This analysis is commonly referred to as observability analysis.

2.2 Observability Analysis and Necessary Conditions for the Solvability of the Observation Problem As a point of departure, it can be interesting to establish a weak condition that makes the observation problem solvable. Similar to the linear case, the weakest condition is the well-known concept of detectability [1], which (in its asymptotic form) can be formalized as follows.

2.2 Observability Analysis and Necessary Conditions for the Solvability …

7

Fig. 2.1 a Non-detectable system, multiple state trajectories generate the same output trajectory. b Detectable system, multiple state trajectories generate the same output trajectory and condition (2.2) is satisfied

Definition 2.2 (Detectability) The system (2.1) is asymptotically detectable if any pair of output trajectories Y (x a , u, t) and Y (x b , u, t) generated from different initial conditions, i.e. x a = x b and x a , x b ∈ X0 , that satisfy Y(xa , u, t) = Y(xb , u, t),

∀t ≥ 0,

satisfies the following lim |X(xa , u, t) − X(xb , u, t)| = 0.

t→∞

(2.2)

A graphical representation of the detectability concept is depicted in Fig. 2.1. The concept of detectability explores a delicate situation, that is, the case in which multiple state trajectories generates the same output trajectory. In such scenario, an algorithm to solve the observation problem may retrieve a state trajectory that is not necessarily the true one. For example, the algorithm may retrieve the trajectory X(xb , u, t) while the system evolves with X(xa , u, t). Nonetheless, the detectability assumption ensures that all these indistinguishable solutions converge to the same one. Thus, even if, initially, the algorithm retrieves an erroneous state trajectory, this will converge asymptotically to the true one. There are multiple techniques that can be implemented in systems with a detectability-like condition, e.g. [2, 3]. However, there are some reasons that motivates imposing a stronger observability condition. First, there is no simple methodology to prove that a system is detectable and usually relies on finding a metric space in which the system is contractive [1, 2]. Although, there have been some recent attempts to learn this metric space from data [4, 5], this topic remains an open problem. Second, an algorithm designed from a detectability property do not have a tunable convergence, i.e. the type of convergence and rate in which xˆ approaches X(x0 , u, t) cannot be modified [6]. Finally, in general, observers designed from a detectability condition are not guaranteed to be robust to model uncertainty and sensor noise.

8

2 The Observation Problem

Fig. 2.2 a Non-detectable nor observable system, multiple state trajectories generate the same output trajectory and condition (2.2) is not satisfied. b Instantaneously observable system, different state trajectories generate different output trajectories that differentiate in an arbitrary small time

The issues of detectability-based algorithms arise from the fact that the algorithm, initially, will retrieve an erroneous state trajectory, that will eventually converge to the true one by means of (2.2). This convergence, cannot be modified by the algorithm and is fragile to model uncertainty and sensor noise. For this reason, it is usually convenient to demand a stronger condition, that is, different state trajectories cannot generate the same output trajectory. This property is referred to as instantaneous observability. Definition 2.3 (Local Instantaneous Observability) The system (2.1) is locally instantaneously observable if for any time td > 0, any pair of output trajectories Y (x a , u, t) and Y (x b , u, t), with x b ∈ X xa , where X xa is a neighborhood of x a , that satisfy ∀td > t ≥ 0, Y(xa , u, t) = Y(xb , u, t),

implies xa = xb . A graphical representation of the concept of instantaneous observability is depicted in Fig. 2.2. Remark 2.1 The term “instantaneous” comes from the fact that td can be arbitrarily small, i.e. different state trajectories generate trajectories that are different in an arbitrary small amount of time. Nonetheless, there are techniques that can be implemented when td does not satisfy this property, e.g. [7]. Remark 2.2 Most applications don’t require that all different initial conditions generate a different output trajectory. In practice, it is only required that all possible initial conditions, x a , generate an output trajectory that is different from the output trajectory generated by any possible initial condition in its neighbourhood. This fact facilitates the design of a test to check instantaneous observability. There is a well-known test that can be used to study if a system is locally instantaneously observable known as the observability rank condition [8, 9]. Notice that,

2.2 Observability Analysis and Necessary Conditions for the Solvability …

9

as the function f and h are sufficiently smooth, the state trajectory X(x0 , u, t) and output trajectory, Y(x0 , u, t), for an input u are uniquely determined by the initial condition, x0 . Moreover, by means of the Taylor theorem, any analytic function on a given interval t ∈ [0, td ] is equivalent to the knowledge of all of its derivatives at time 0, i.e.  ∞  d k y  t k . (2.3) Y(x0 , u, t) = dt k x0 k! k=0 Thus, the instantaneous observability analysis consists in finding the conditions in which the function in (2.3) can be inverted to retrieve a unique initial state x0 , i.e. any initial condition x0 and its corresponding trajectory X(x0 , u, t) generates a unique output trajectory. Indeed, the vector composed by the first k − 1 derivatives of the output ⎡ ⎤ y ⎢ y˙ ⎥ ⎢ ⎥ ˙ ..., u(k−2) ) = ⎢ . ⎥ (2.4) Ok (x, u, u, ⎣ .. ⎦ y(k−1)

is referred as the observability map of order k of the system [8]. Naturally, the is system instantaneously observable if the observability map, Ok , at some order k is injective. It should be remarked that there is no general method to prove that an observability map is globally injective. Nonetheless, by means of the inverse function theorem, it is possible to obtain a sufficient local condition. Definition 2.4 (Observability rank condition) A system is said to satisfy the observability rank condition at some point, x 0 with input u0 with derivatives u˙ 0 , ..., u(k−2) 0 if the observability map satisfies   ∂Ok  =n rank ∂x x0 ,u0 ,u˙ 0 ,...,u(k−2) 0

(2.5)

for some order k. From this it can be derived that [8]. Lemma 2.1 If the system satisfies the observability rank condition at x 0 for some , there is a neighbourhood around x 0 in which input u0 with derivatives u˙ 0 , ..., u(k−2) 0 ˙ ..., u(k−1) ) is injective in x. the function Ok (x, u, u, Remark 2.3 The fact that a point, x 0 , does not satisfy the observability rank condition does not imply that the observability map, Ok , is not injective for some order k. This is why the observability rank condition is a sufficient but not necessary condition for instantaneous observability.

10

2 The Observation Problem

For linear time-invariant systems of the form x˙ = Ax + Bu, y = Cx,

(2.6)

where A ∈ Rn×n , B ∈ Rn×q and C ∈ Rm×n , the observability rank condition becomes the well known Kalman rank condition [10], ⎡ ⎢ ⎢ rank ⎢ ⎣

C CA .. .

⎤ ⎥ ⎥ ⎥ = n. ⎦

(2.7)

CA(n−1) Comparing the Kalman rank condition in (2.7) and the observability rank condition (2.5) it is possible to summarize the main differences between an observability analysis in linear systems and the same analysis in nonlinear systems, which should clarify in which ways analysing the observability of a nonlinear system is a difficult task with no generic solution: • For general nonlinear systems, the observability rank condition depends on the state. Therefore, the observability in nonlinear systems is a local property that may vary depending on the system region of the operation. For linear systems, observability is a global property. • For nonlinear systems, the observability rank condition depends on the input and its derivatives. Consequently, there may be some input signals that drive a system unobservable. In the linear case, observability is an input-invariant property. If there are no inputs that makes the system unobservable, it is said that the system is uniformly observable on the inputs [11]. • In the nonlinear context, the observability rank condition may be satisfied for k > n and there is not an a priori method to know which is the value of k. For linear systems, it is well known that if the observability rank condition is not satisfied for an observability map of order k = n, it cannot be satisfied for an order k > n ([12], Lemma 2.6). This is why the Kalman rank condition is always derived from an observability map of order n. The observability analysis up to this point studies whether the observation problem is analytically solvable or not. However, it does not provide any information on how “easy” is to observe the system in practice, i.e. how complex is to distinguish between two different state trajectories from the measured output trajectory in the concerned practical application. For detectable systems, it is clear that such distinction between trajectories is not possible, however, if the presence of sensor noise is considered, even instantaneously observable systems may have trajectories that are close to be indistinguishable and produce significant numerical problems. To better understand this problem, it is convenient to introduce the concept of weak observability [13, 14].

2.2 Observability Analysis and Necessary Conditions for the Solvability …

11

Fig. 2.3 a Non-weak observable system, small variations in the initial conditions induces large variations in the output trajectories. b Weak observable system, large variations in the initial conditions induces small variations in the output trajectories

Definition 2.5 A system of the form (2.6) can be defined as weakly observable at x 0 with input u at a time range [0, t] if the value of the output L2 -norm of the output

t1

|Y(x0 , u, t)|2 dt,

0

presents a small singular value and/or large condition number.1 Remark 2.4 Not to confuse this idea with the concept of weak observability for nonlinear systems used in [8], which refers to the possibility of distinguish a state x from other states in a local neighbourhood from the measured output. The weak observability analysis studies how much does a variation in the initial conditions around x0 vary the output trajectory. Conversely, in a weak observability context a small variation in the output trajectory requires a large variation in the initial conditions. A graphical comparison between a non-weak observable system and a weak observable system is depicted in Fig. 2.3. The concept of weak observability is of interest if one considers sensor noise. In the presence of noise, the algorithm for the observation problem cannot distinguish between close measured output trajectories. This fact is mostly harmless in nonweak observable scenarios, as close output trajectories are produced by close state trajectories. Thus, even if the algorithm retrieves the wrong trajectory, it will be near the true one. However, in weak observable systems, the error induced by the noise will be significantly larger. To obtain more intuition on these ideas, it is convenient to study a more specific time-invariant linear case. Consider a linear time-invariant system of the form (2.6). The output trajectory in the time range [0, t] from an initial condition x0 can be computed as follows y(t) = Ce At x0 . 1

The condition number of a matrix A is computed as

maximum and minimum eigenvalue, respectively.

(2.8)

λmax (A) , where λmax (·) and λmin (·), are the λmin (A)

12

2 The Observation Problem

Pre-multiplying by (Ce At ) and integrating in the time range [0, t], one gets the following expression



t

 t (e ) C Ce dt x0 = (e At ) C y(t). At



At

0

(2.9)

0

A direct method to obtain the initial conditions of the system, and later the states by using the formula x(t) = e At x0 , is to isolate x0 from (2.9). Indeed, the recursive implementation of this approach results in the well known Kalman filter [10]. Notice that the isolation of x0 requires the inversion of the factor

t

(e At ) C Ce At dt.

(2.10)

0

Therefore, recovering the initial conditions from the output will become an ill-posed inverse problem if the factor in (2.10) presents a large condition number. The readers familiar with linear systems will identify the factor in (2.10) as the observability Grammian of the system. To connect this result with the concept of weak observability, consider the L2 norm of y, which can be directly computed from the solution (2.8) as x 0



t

At





(e ) C Ce dt x0 dt. At

0

If the factor (2.10) presents an low condition number, there will be directions of x0 in which the L2 -norm of y is low in relation to the other directions of the system, which shows that the system is weakly observable in that direction. The presented analysis may provide some intuition on the difficulties of solving the observation problem in weakly observable scenarios. The extension of this analysis to general nonlinear systems is still an open problem. Although, some local results can be obtained if the system is linearized around an operating point [13, 14]. The aim of this subsection was to briefly present the observability concepts that will be relevant for the rest of the thesis. An observability analysis provides the conditions in which the system state can be recovered from the measured output and input trajectories. Notice that observability is a structural property that is independent of the algorithm to be used to solve the observation problem. For this reason, the observability has to be studied a priori to analyze if the quantity, quality, and placement of the measured signals is enough to solve the observation problem. Once it has been established that the system is observable, the next step is to select and design the proper algorithm to solve the observation problem. The theoretical and practical interest of the observation problem have motivated multiple lines of research on the topic. From all the possible approaches, this work will focus on a technique referred to as observer. Next subsection will briefly introduce the idea behind an observer and why it has been selected for this thesis.

2.3 The Benefits of Observers

13

2.3 The Benefits of Observers Numerous theoretical methodologies to solve the observation problem have appeared in the past decades. When analysed from a conceptual point of view, these methodologies can be classified as computationally-oriented and analytically-oriented methods. The former ones do not require an analytical model of the system and are developed on the basis of numerical models collected from large amount of data from the states to be estimated and the outputs, e.g. [15]. Even though there is a growing interest on the topic, most results leverage off a continuous increase in the computation capacity and do not provide a theoretical insight on the conditions that make this approach work or not. For example, the observability analysis presented in the past section relies on properties of an analytical model of the system. Therefore, in the computationally-oriented context, there is no guarantee that the observation problem is solvable in the first place. Consequently, as long as it’s possible to analyze the system in such a way, analytically-oriented methods provide a more honest, consistent and interpretable solution to the observation problem. Analytically-oriented methods assumes the existence of an analytical model of the system, proceeds by studying the properties of said model, and provides a solution according to the properties. From this viewpoint, there are multiple strategies that can be developed. As the system model is continuously differentiable, the observation problem can be solved by estimating the initial condition of the system, x0 , and forward simulating the model to generate the state trajectory. A direct way to implement this method is to forward simulate the system (2.1) for a set of initial conditions and gradually remove the initial conditions that are not coherent with the measured output trajectory. If the system is instantaneously observable, there is a single initial condition that generates the measured output trajectory, and this process will eventually retrieve the true one. This strategy can be formulated in a stochastic framework [16] or a deterministic one [17, 18]. Nonetheless, this approach strongly relies on an adequate initialization of the algorithm in a region around the true initial state, and an accurate model in order to simulate forward in time. An alternative idea is based ˆ 0 , u, t), and on generating a set of output signals for multiple initial conditions, Y(x select the adequate initial condition with respect to some cost function [19, 20]

t

ˆ 0 , u, τ ) − y(τ )|2 dτ. |Y(x

t−T

Nonetheless, if the concerned system is nonlinear, the approach results in a complex nonlinear optimization problem, which presents the common issues of computational complexity, non-convexity and convergence to local-minima. In spite of these issues, some algorithms denoted as finite-horizon observers have been developed to solve this optimization problem online [20]. It should be remarked that most results focus on discrete-time systems. Notice that the issues in solving the observation problem are related to three factors: the unknown initial conditions of the system, the uncertainty in the analytical

14

2 The Observation Problem

model and the model nonlinearities. The main message of this section is that these issues can be addressed from a control theory perspective. First, the unknown initial condition problem is solved if the trajectories generated by the algorithm “forget” its initial condition. This behaviour can be depicted as a stability property of the error between the true states and the estimated states. Second, the presence of model uncertainty can be addressed if the algorithm is robust to uncertainty. Stability and robustness are the main topics in control theory and motivates the design of a state estimation algorithm that implements the concept of feedback. Such an algorithm is referred to as an observer. The last issue is related to model nonlinearities, which motivates the design of the algorithm from a nonlinear control theory perspective and develop nonlinear observers.

References 1. Andrieu V, Besançon G, Serres U (2013) Observability necessary conditions for the existence of observers. In: 52nd IEEE conference on decision and control, pp 4442–4447. https://doi. org/10.1109/CDC.2013.6760573 2. Sanfelice RG, Praly L (2016) Convergence of nonlinear observers on R n with a Riemannian metric (part ii). IEEE Trans Autom Control 61(10):2848–2860. https://doi.org/10.1109/TAC. 2015.2504483 3. Tsinias J (1989) Observer design for nonlinear systems. Syst & Control Lett 13(2):135–142. ISSN 0167-6911. https://doi.org/10.1016/0167-6911(89)90030-3 4. Manchester IR (2018) Contracting nonlinear observers: convex optimization and learning from data. In: 2018 annual American control conference (ACC), pp 1873–1880. https://doi.org/10. 23919/ACC.2018.8431837 5. Yi B, Wang R, Manchester IR (2021) Reduced-order nonlinear observers via contraction analysis and convex optimization. IEEE Trans Autom Control :1. https://doi.org/10.1109/TAC. 2021.3115887 6. Andrieu V, Jayawardhana B, Praly L (2016) Transverse exponential stability and applications. IEEE Trans Autom Control 61(11):3396–3411. https://doi.org/10.1109/TAC.2016.2528050 7. Besançon G, Ticlea A (2007) An immersion-based observer design for rank-observable nonlinear systems. IEEE Trans Autom Control 52(1):83–88. https://doi.org/10.1109/TAC.2006. 889867 8. Hermann R, Krener A (1977) Nonlinear controllability and observability. IEEE Trans Autom Control 22(5):728–740. https://doi.org/10.1109/TAC.1977.1101601 9. Martinelli A (2019) Nonlinear unknown input observability: extension of the observability rank condition. IEEE Trans Autom Control 64(1):222–237. https://doi.org/10.1109/TAC.2018. 2798806 10. Kalman RE, Bucy RS (1961) New results in linear filtering and prediction theory. J Basic Eng 82(1):34–45 11. Besançon G (2007) An overview on observer tools for nonlinear systems. Nonlinear observers and applications, pp 1–33 12. Korovin SK, Fomichev VV, Fomichev VV (2009) State observers for linear systems with uncertainty. de Gruyter 13. Krener AJ, Ide K (2009) Measures of unobservability. In: Proceedings of the 48h IEEE conference on decision and control (CDC) held jointly with 2009 28th Chinese control conference, pp 6401–6406. https://doi.org/10.1109/CDC.2009.5400067 14. Powel ND, Morgansen KA (2015) Empirical observability Gramian rank condition for weak observability of nonlinear systems with control. In: 2015 54th IEEE conference on decision and control (CDC), pp 6342–6348. https://doi.org/10.1109/CDC.2015.7403218

References

15

15. Masti D, Bernardini D, Bemporad A (2021) A machine-learning approach to synthesize virtual sensors for parameter-varying systems. Eur J Control 61:40–49. ISSN 0947-3580. https://doi. org/10.1016/j.ejcon.2021.06.005 16. Jazwinski AH (1970) Stochastic processes and filtering theory. Courier Corporation 17. Gouzé JL, Rapaport A, Hadj-Sadok MZ (2000) Interval observers for uncertain biological systems. Ecol Model. ISSN 03043800. https://doi.org/10.1016/S0304-3800(00)00279-9 18. Lin H, Zhai G, Antsaklis PJ (2003) Set-valued observer design for a class of uncertain linear systems with persistent disturbance. In: Proceedings of the 2003 American control conference, 2003. ISBN 0743-1619. https://doi.org/10.1109/ACC.2003.1243351 19. Zimmer G (1994) State observation by on-line minimization. Int J Control. ISSN 13665820. https://doi.org/10.1080/00207179408921482 20. Alamir M (2007) Nonlinear moving horizon observers: theory and real-time implementation. In: Nonlinear observers and applications. Lecture notes in control and information sciences, vol 363. Berlin, pp 139–179

Chapter 3

Nonlinear Observer Design

The observation problem can be efficiently solved by the use of observers. This chapter presents the idea of nonlinear observer for nonlinear systems. Then, two specific nonlinear observers are discussed. First, the high-gain observer and its more recent modifications. Second, the parameter estimation-based observer and a modification that implicitly regularizes the estimation. Finally, some comments on the performance of nonlinear observers in the presence of sensor noise and unmodelled disturbances are given.

3.1 Observer Definition and Design The objective is to design an algorithm to estimate the unknown states, x, of an observable system of the form (2.1). The observer approach is based on the design and implementation of the so-called “observer”, which generates in real-time such estimation, xˆ . Roughly speaking, an observer is a dynamical system that uses the mathematical model of the system to “mimic” the state trajectories of the true plant. In order to correct for model uncertainty, unmodelled disturbances and uncertain initial conditions, an observer includes a feedback term that corrects the estimation based on the difference between the measured outputs and the observer estimation of said outputs. A more rigorous mathematical formulation is given in the following definition (see Fig. 3.1 for a scheme of the observer). Definition 3.1 For a nonlinear system, consider the following set of dynamics ξ˙ˆ = ϕ(ξˆ , u) + κ(ξˆ , y − h(ˆx)), xˆ = ψ(ξˆ )

(3.1)

where ξˆ ⊆ Rn ξ , with n ξ ≥ n, is the observer state. The functions ϕ : Rn ξ × Rq → Rn ξ , κ : Rn ξ × Rm → Rn ξ and ψ : Rn ξ → Rn are locally Lipschitz. Function κ is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Cecilia, Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems, Springer Theses, https://doi.org/10.1007/978-3-031-38924-5_3

17

18

3 Nonlinear Observer Design

Fig. 3.1 General observer estimation scheme

the output feedback term which satisfies κ(ξˆ , 0) = 0 for all ξˆ ∈ Rn ξ . Finally, there exists a right inverse ψ −R of the function ψ, such that x = ψ(ψ −R (x)) for all x ∈ X. Such structure is defined as an observer if it satisfies the following properties: • xˆ (0) = x(0) ⇒ xˆ (t) = x(t), ∀t ≥ 0 • | xˆ (t) − x(t)| → 0 as t → ∞. From the presented definition it is important to remark the following possibilities in observer design for nonlinear systems. • The dimension of the observer state may be larger than the original state order. This possibility includes the extended Kalman filter [1], in which the observer gain is computed through some extra dynamics that depends on a linearization of the system. Moreover, it includes the possibility of immersing the state in a larger dimension in which the structure is more tractable [2–4]. • The observer feedback term, κ, may be a nonlinear function. For certain nonlinear systems, linear feedback terms cannot guarantee observer stability [5, 6]. • The observer coordinates, ξˆ , may be different from the original system coordinates, xˆ . Therefore, it is necessary to find a map, ψ, that relates the observer coordinates with the original coordinates [7]. Remark 3.1 As it will be shown in Sect. 3.2.2.3, in some cases, it may be convenient to have a map ψ that depends on the input, u, and its derivatives. Nonetheless, this case has been obviated here in order to simplify the notation. It can be seen that the observer design problem is reduced to designing the factors ϕ, κ and ψ. Although observer design for linear systems is a well-studied topic with general solutions even in the presence of uncertainty and noise [8–11], this generality is not preserved in the nonlinear context. The current state of nonlinear observer design is based on a three-step process. First, the original system is transformed to a canonical form with a more tractable structure. Second, a nonlinear observer is designed in the new coordinates. Third, the transformation is inverted to obtain an estimation of the states in the original coordinates. The existence of such an invertible transformation depends on structural and observability properties of the original system. A scheme of the three-step process is depicted in Fig. 3.2.

3.1 Observer Definition and Design

19

Fig. 3.2 Three-step process for observer design in nonlinear systems

In general, nonlinear observers are designed from a control theory perspective. Indeed, consider the state-estimation error signal x˜ = x − xˆ . A proper design of the factors ϕ, κ and ψ would be the ones that ensures the convergence of x˜ to zero (or to a bounded value in the presence of noise, disturbances and/or uncertainty). For the rest of the document, this convergence property will be referred to as the stability of the observer. Notice that the convergence of the estimation-error implies that any observer initial value, even if the initial condition of the true system is unknown, will eventually converge to the true system trajectory. Thus, an observer eventually forgets its “initial conditions” and the unknown initial conditions limitation of the observation problem is solved. Nonetheless, the stability study of the error signal, x˜ , is significantly complex due to the nonlinearities of the original system and the observer. For nonlinear systems, stability is commonly studied by means of Lyapunov functions [12–14]. Specifically, in the proposed observer context, it is assumed that there is a locally Lipschitz function (Lyapunov function) V : Rn × Rn ξ → R satisfying −R ¯ (x) − ξˆ |) α(|x − ψ(ξˆ )|) ≤ V (x, ξˆ ) ≤ α(|ψ

(3.2)

for all x, ξˆ ∈ X, Rn ξ , where α, α¯ ∈ K∞ , so that its derivative along solutions to the nonlinear system and (3.1) satisfies V˙ ≤ −α1 V (x, ξˆ ).

(3.3)

for all x, ξ ∈ X, Rn ξ and u ∈ U, where α1 is some positive constant. Conditions (3.2) and (3.3) ensures that the observer (3.1) is asymptotically convergent for system (2.1). Specifically, there exist a function β 1 ∈ KL such that the solutions (x(t), ξˆ (t)) belonging to X, Rn ξ for all t ≥ 0 satisfy

20

3 Nonlinear Observer Design

  |˜x(t)| ≤ β 1 |ψ −R (x(0)) − ξˆ (0)|, t .

(3.4)

It is out of the scope of this work to provide an in-depth review of all observer techniques that can be designed by using this framework. The interested readers are referred to [15–17] and references therein. Instead, the remainder of this chapter focuses on presenting in-depth the few techniques that have been explored in this thesis. On the one hand, the classic high-gain observer and its most modern modifications. On the other, the recent parameter estimation-based observer. Remark 3.2 Notice that the proposed framework only studies the idealized case in which the model is perfect, i.e. there is no uncertainties or disturbances, and there is no measurement noise. Considering the presence of uncertainties, disturbances, and measurement noise is crucial for the adequate implementation of the observer in a practical scenario. For this reason, these elements will be considered in the remainder of the chapter and will be presented in a more general framework in the end of the chapter.

3.2 High-Gain Observers Although the Lyapunov approach presented in the past subsection has theoretical interest, it is too general to be implementable in practice. First, there is no guide on how to design or which structure has the Lyapunov function V (x, ξˆ ). Second, even if the structure of V (x, ξˆ ) was known, the same problem appears in the observer factors ϕ, κ and ψ. For this reason, observer design is limited to specific nonlinear systems classes in which the structure of V (x, ξˆ ), ϕ, κ and ψ is fixed a priori. Linear systems are a prototypical example of this case. Specifically, consider s system of the form x˙ = Ax + Bu

(3.5)

y = cx where A ∈ Rn×n , B ∈ Rn×q and c ∈ R1×n are matrices assumed to be known. It is well-known that a linear Luenberger observer can be designed with ϕ(ξˆ , u) = Aξˆ + Bu, κ(ξˆ , y − cˆx) = K(y − cˆx), ψ(ξˆ ) = Iξˆ , with a quadratic (radially unbounded) Lyapunov function V (x) = x˜ P˜x.

(3.6)

where K ∈ Rn×1 and P ∈ Rn×n , with P = P , are matrices such that the following linear matrix inequality (LMI) is satisfied for some positive constant q

3.2 High-Gain Observers

21

P(A − Kc) + (A − Kc) P ≤ −qP,

(3.7)

Remark 3.3 The LMI in (3.7) is always solvable if the pair ( A, c) is Kalman rank observable. Then, an easy computation shows that (3.3) is satisfied. Specifically,  



˙ V = x˜ P(A − Kc) + (A − Kc) P x˜ ≤ −q V. The simplicity of such design arises from the fact the observability of the pair (A, c) can be easily analyzed through the Kalman rank condition, the Lyapunov function is quadratic and the observer feedback term is linear. For this reason, several authors tried to extend this type of design to semi-linear systems, i.e. nonlinear systems with a linear part. The idea was to utilize the linear part of the system to prove observability of the system and design the observer gain in such a way that the nonlinearity does not destabilize the observer. Specifically, consider a nonlinear system depicted by x˙ = Ax + Bu + φ(x, u) y = cx where φ : Rn × Rq → Rn is a nonlinear function and the pair is (A, c) is observable. Following the promising results for linear systems, the objective was to design the observer with linear gain κ assuming a quadratic Lyapunov function as in (3.6) and the following ϕ(ξˆ , u) = Aξˆ + Bu + φ(ξˆ , u), ψ(ξˆ ) = Iξˆ . Even in this restrictive structure, the problem was still not directly solvable, and some extra assumptions had to be imposed on the nonlinearities. For example, for systems with Lipschitz nonlinearities, it is possible to use the observer in [18]. Similar results have been obtained for nonlinearities that are bounded [19] or that satisfy a bounded Jacobian condition [20]. This line of research was promising, but was limited by two significant drawbacks. First, assuming that the system is semi-linear and the nonlinearities satisfy certain properties is very restrictive. Second, even if the original system is in the adequate semi-linear structure, it may be possible that the system is instantaneously observable, but the linear pair (A, c) is not observable (see [5, 21] for some examples). Following from the presented approach in Fig. 3.2, both issues could be solved if one finds an injective transformation ψ −R (x) that transforms the system in the adequate semi-linear form. Nonetheless, none of the mentioned references provides a methodology to design ψ −R . This coordinate limitation is practically solved if one restricts the observer coordinates to a triangular structure [22]. By restricting the target observer coordinates, there are a family of coordinate transformations that are relatively easy to compute, see Sect. 3.2.2. For this reason, most observer development for semi-linear systems

22

3 Nonlinear Observer Design

usually assume that the system is in a triangular form. Among all the possibilities, a remarkable technique is the high-gain observer. Due to its simplicity of design and analysis, it has been used in multiple control scenarios, some examples are examples are: output stabilization techniques [23], output regulation [24], fault detection and identification [25], consensus of multi-agent systems [26] and performance recovery of feedback-linearization methods [27], from others. The remainder of this section will focus on the high-gain observer technique. First, the classic high-gain observer will be presented. Second, some insights on how to design the map ψ −R and its inverse ψ to transform a nonlinear system to a triangular form will be presented. Finally, a recent modification of the high-gain observer, which reduces the major drawbacks of the technique will be presented.

3.2.1 Classic High-Gain Observer For simplicity, this section assume an autonomous nonlinear single-output system in the so-called phase-variable form [28] x˙i = xi+1 , i = 1, ..., n − 1 x˙n = φ(x, d),

(3.8)

y = x1 + v, where x = [x1 , .., xn ] ∈ Rn is the state, d ∈ Rnd depicts an unknown signal which may represent parametric uncertainties or unknown disturbances. Additionally, v is unknown bounded high frequency measurement noise. Both d and v are assumed to be bounded. Finally, the function φ(·, ·) is assumed to be Lipschitz with a Lipschitz constant L φ , (3.9) |φ(x, d) − φ(z, d)| ≤ L φ |x − z|. Remark 3.4 To simplify, this chapter focuses in phase-variable forms. It should be remarked that the high-gain observer can also be implemented in other triangular structures. Some remarkable examples are general triangular forms [17], non-strict feedback form systems [28] and strict-feedback form systems [29]. Naturally, motivated by the framework in Fig. 3.2, it can also be implemented in systems that can be transformed to a triangular form by means of a coordinate change [16]. Some comments on how to design such a transformation can be found in Sect. 3.2.2. Remark 3.5 Notice that the observer framework presented at the beginning of this chapter does not consider the presence of disturbances and noise. Nonetheless, in order to understand the drawbacks and benefits of the high-gain observer, it is interesting to consider these elements. To present a more complete observer design theory, the framework presented in the past section will be extended to consider the presence of noise, uncertainty, and disturbances in Sect. 3.4.

3.2 High-Gain Observers

23

A high-gain observer for the system in (3.8) is a copy of the triangular structure and a linear feedback term of the form αi x˙ˆi = xˆi+1 + i (y − xˆ1 ), i = 1, ..., n − 1 ε ˙xˆn = φ(ˆx, 0) + αn (y − xˆ1 ), εn

(3.10)

where ε is a design parameter and α1 , ..., αn are also design parameters that have to be tuned such that the polynomial s n + α1 s n−1 + · · · + αn−1 s + αn

(3.11)

has all the roots in the left half-plane. Then, the stability and performance properties of the high-gain observer can be summarized through the following theorem. Theorem 3.1 Consider a symmetric positive definite matrix P solution of 

PAcl + Acl P ≤ −In ⎡

where

−α1 −α2 .. .

⎢ ⎢ ⎢ Acl = ⎢ ⎢ ⎣−αn−1 −αn

(3.12)

⎤ 1 0 ··· 0 0 1 · · · 0⎥ ⎥ .. . . .. ⎥ . . . 0 .⎥ ⎥ ⎦ 0 ··· 1 0 0 ··· 0

1 , 1}, the estimation-error of the observer (3.10) sat2L φ | P| isfies the following bound for all t > 0 Then, for all ε ≤ min{

k2 k1 − t k4 |xi − xˆi | ≤ i−1 e ε |xi (0) − xˆi (0)| + εn+1−i k2 M + i−1 |v|∞ , i = 1, ..., n ε ε (3.13) where k1 , ..., k4 are some positive constants, M is a positive constant that bounds |φ(x, d) − φ(x, 0)| and | · |∞ is the maximum value of the norm. Proof It should be stated that this proof is not new and is based on the results in [30]. First, consider a scaled version of the errors. η1 =

x1 − xˆ1 x2 − xˆ2 , η2 = n−2 , . . . , ηn = xn − xˆn . n−1 ε ε

(3.14)

24

3 Nonlinear Observer Design



The dynamics in this scaled coordinates η  η1 , . . . , ηn can be computed as εη˙ = Acl η + εδ(x, xˆ , d) − where

1 Ev εn−1

δ = col(0, . . . , 0, δn ), E = [α1 , ..., αn ]

and δn is

δn = φ(x, d) − φ(ˆx, 0).

By means of the Lipschitz condition (3.9), it can be shown that for all ε ≤ 1 |δ| ≤ L φ |η| + |φ(x, d) + φ(x, 0)|  L φ |η| + M.

(3.15)

Consider the positive definite and radially unbounded Lyapunov function V = η Pη.

(3.16)

The derivative of (3.16) satisfies the following ε V˙ ≤ −η η + 2εη Pδ − + 2ε|P||η|M +

2  η PEv ≤ −|η|2 + 2εL φ |P||η|2 εn−1

2 |PE||η||v|∞ . εn−1

(3.17)

Now consider the first two elements in the right hand side of (3.17). See that for all 1 inequality (3.17) reduces to ε≤ 2L φ |P| 2 ε V˙ ≤ −c|η|2 + 2ε|P||η|M + n−1 |PE||η||v|∞ . ε Therefore, the following holds   c 2 1 ˙ ε V ≤ − |η| , ∀ |η| ≥ 4 ε|P||η|M + n−1 |PE||η||v|∞ , 2 ε where c is a positive constant dependent of ε. As a consequence, by applying Theorem 4.5 of [12] and inverting the coordinate change (3.14), the bound (3.13) can be proved.  Remark 3.6 The inequality in (3.12) is equivalent to the one in (3.7). By direct inspection of the estimation-error bound (3.13) it is possible to describe the main benefits and potential drawbacks of high-gain observers.

3.2 High-Gain Observers

25

On the one hand, the first factor k2 k1 − t e ε |xi (0) − xˆi (0)| εi−1

(3.18)

converges to zero with a rate proportional to ε−1 . Consequently, the estimation-error rate of convergence can be made arbitrary fast by means of reducing ε. 1 On the other hand, notice that the first term is multiplied by a factor i−1 , which ε does not alter the rate that the term converges to zero, but significantly increases the value of the term (3.18) at initial time instants. As a consequence, the observer states “peaks” during the transient and the amplitude of this peak is increased as the design parameter ε is decreased. This abrupt increase/decrease of the observer states is known as the peaking phenomena. The second term, εn+1−i k2 M, is related with the unknown disturbances and/or uncertainty of the model. In the absence of measurement noise (i.e. M = 0, |v|∞ = 0), the estimation-error converges to a bounded region proportional to the norm of the unknown disturbances and/or uncertainty. This error can be arbitrary reduced by decreasing the design parameter ε. Finally, the third element of (3.13), k4 |v|∞ εi−1 is related with the effect of the measurement noise on the estimation. In the absence of uncertainty and/or unknown disturbances (i.e. d = 0), the estimation-error converges to a bounded region proportional to the noise norm, |v|∞ . Notice that the bound of this 1 region is also proportional to i−1 . Consequently, the noise sensitivity is significantly ε deteriorated as the design parameter ε is decreased. All these benefits and drawbacks are presented though the following example. Lets consider a nonlinear system in triangular form depicted by x˙1 = x2 x˙2 = −2x1 + 2x2 + x1 x22 y = x1 Assume that the expression of x˙2 is incorrectly modelled as x˙2 = −2x1 − 2x2 . Then, a high-gain observer can be designed as follows

26

3 Nonlinear Observer Design

Fig. 3.3 Evolution of system true x2 (blue), high-gain observer estimation with ε = 0.5 (orange), estimation with ε = 0.1 (yellow) and estimation ε = 0.05 (purple)

2 x˙ˆ1 = xˆ2 + (y − xˆ1 ) ε 1 x˙ˆ2 = −2 xˆ1 − 2 xˆ2 + 2 (y − xˆ1 ). ε The only parameter that remains to be tuned is ε. In Fig. 3.3, it is depicted the evolution of the true value of x2 and the estimation of different high-gain observers with different values of ε. Notice that in all cases the estimation-error x˜2 = x2 − xˆ2 converges to a relatively low value despite the presence of uncertainty in the equation x˙2 . Moreover, the estimation converges to a lower error value as ε diminishes. Now, lets focus on the first two seconds. Notice that the observers with lower ε converge faster to the true value of x2 ; however, they present a peak during the first time instants, which is amplified as ε diminishes. This abrupt increase is what is referred as the peaking phenomena. Now, assume that the measured signal is corrupted with some high-frequency noise; i.e. y = x1 + v, where v is some measurement noise. Again, Fig. 3.4 depicts the evolution of the true value of x2 and the estimation of different high-gain observers with different value of ε in the presence of measurement noise. It can be seen that the effect of the noise in the accuracy is amplified as ε decreases, making the estimation with ε = 0.1 and ε = 0.05 practically useless. As a summary, the high-gain observer presents a clear trade-off. The appealing properties of fast and robust estimation, that is achieved by decreasing the parameter ε, are related with the peaking phenomena and an increase of the observer measurement noise sensitivity. Indeed, on the one hand, if the ε is too large, the observer accuracy will be deteriorated due to uncertainties. On the other, if ε is too small, the accuracy will be affected by the measurement noise. This trade-off can be depicted in a Pareto-style plot, see Fig. 3.5.

3.2 High-Gain Observers

27

Fig. 3.4 Evolution of system true x2 (blue), high-gain observer estimation with ε = 0.5 (orange), estimation with ε = 0.1 (yellow) and estimation ε = 0.05 (purple) Fig. 3.5 Observer estimation-error ultimate bound with respect to ε

Although it is accentuated in high-gain observers, this trade-off between disturbance rejection and noise sensitivity is common in observers in general [31]. More insights on the topic will be provided in Sect. 3.4. Remark 3.7 There are alternative techniques that can be implemented in systems with triangular structure, being the main difference, the assumption in the nonlinear function φ. Some remarkable examples are high-order differentiators [32–34], which assumes that the nonlinear function is bounded, homogenous observer [35] for nonlinearities that satisfy a Hölder inequality, or a combination of high-gain and highorder differentiators for more complex non-linearities [36]. This work will focus on the high-gain observer mainly due to two reasons. First, the systems considered in the practical applications satisfy the Lipschitz assumption. Second, high-gain observers are simpler to analyze through Lyapunov arguments.

28

3 Nonlinear Observer Design

3.2.2 Transformation to a Triangular Form The design of the high-gain observer is based on the premise that the system presents a triangular form. Naturally, many systems will not present such a structure. Nonetheless, an appealing property of triangular structures is that any system, under the proper observability assumption, can be transformed to a triangular structure. The aim of this section is to define an adequate transformation, ψ −R (x), that transforms the system to the correct form for high-gain observer design. Notice that the transformation has to satisfy the following properties • The resulting system presents the triangular structure in (3.8), or a similar triangular form, see Remark 3.4. • The nonlinear function in the last equation, φ, has to be Lipschitz. • The transformation has to be invertible, i.e. there is a map ψ such that x = ψ(ψ −R (x)).

3.2.2.1

Autonomous Case

To begin with, lets consider an autonomous single-output nonlinear system of the form x˙ = f(x); y = h(x) where f : Rn → Rn and h : Rn → R are sufficiently smooth functions. A potential candidate is a transformation formed by the output derivatives up to order n − 1 [37] ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ h(x) y ξ1 ⎢ y˙ ⎥ ⎢ L f h(x) ⎥ ⎢ξ2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ (3.19) ψ −R (x)  ⎢ . ⎥ = ⎢ ⎥  ⎢ .. ⎥ . .. ⎣ .. ⎦ ⎣ ⎦ ⎣.⎦ . y (n−1)

h(x) L n−1 f

ξn

It is relatively easy to see that this transformation puts the system in the triangular form (3.8) as the following relation is immediately satisfied ξ˙i = ξi+1 , ∀i = 1, . . . , n − 1. Nonetheless, as mentioned before, the transformation still has to ensure that the function in the last equation, φ, is Lipschitz and the map ψ −R (x) has to be invertible. First, invertibility of ψ −R (x) requires the map to be injective. In the literature, a system with an injective map made of the output derivatives up to order n − 1 is usually attributed as weakly differential observable of order n [28]. Second, the nonlinearities of the resulting system have to be Lipschitz. This condi−R tion will be fulfilled if the map ψ −R (x) is injective and the Jacobian ∂ψ∂x (x) has rank

3.2 High-Gain Observers

29

n. In the literature, a system that fulfills this condition is usually attributed as strong differential observable of order n [28]. Obviously, strong differential observability implies weak differential observability. Therefore, any autonomous system that is strong differential observable of order n can be transformed into a triangular form through the map ψ −R . It is noticeable that this property is immediately satisfied if the system is instantaneously observable with an observability map of order n, as presented in Sect. 2.2 of Chap. 2. Notice that this condition may not hold globally, but only on some compact set.

3.2.2.2

Non-autonomous Case

The next step consists in extending this result to non-autonomous systems, i.e. systems with controlled inputs defined as x˙ = f(x, u);

y = h(x).

The straightforward idea may be to directly construct the map of output derivatives, as in (3.19). ⎡ ⎤ y ⎢ y˙ ⎥ ⎢ ⎥ (3.20) ˙ . . . , u (n−2) )  ⎢ . ⎥ . ψ −R (x, u, u, ⎣ .. ⎦ y (n−1)

However, now, the input and some of its derivatives may appear in the mapping. In such case, the assumptions presented in the autonomous case have to be extended to include the presence of the input [28, 38]. In practice, the observability assumption remains the same, which means, the system is instantaneously observable with a map of order n. Notice that, there may be some inputs that make the system lose observability, and makes the transformation not injective. If there are no inputs that make the system unobservable, it is said that the system is uniformly observable on the inputs [17]. In some cases, the input derivatives are not accessible. For this reason, it is of interest to study the construction of an input independent transformation. This problem presents a simple solution in the context of control affine systems (nonetheless, it can be extended to more general systems [28]), i.e. x˙ = f0 (x) + g(x)u;

y = h(x)

(3.21)

where f0 : Rn → Rn and h : Rn → R are sufficiently smooth functions. The idea is to implement an input-independent mapping made of the Lie derivatives of h(x) along the vector field f0 (x),

30

3 Nonlinear Observer Design

⎡ ⎢ ⎢ ψ −R (x)  ⎢ ⎣

h(x) L f0 h(x) .. .

L n−1 f0 h(x)

⎡ ⎤ ξ1 ⎥ ⎢ ξ2 ⎥ ⎥ ⎢ ⎥ ⎥  ⎢ .. ⎥ , ⎦ ⎣.⎦ ⎤

(3.22)

ξn

where L f0 h(x) is the Lie derivative of the function h along the vector field f0 , which ∂h(x) f0 (x). is computed as ∂x Now, under the appropriate conditions, the map (3.22) will transform the system into the triangular form with a structure [28] ⎧ ˙ ⎪ ⎪ξ1 = ξ2 + φ1 (ξ1 , u) ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎨. ξ˙i = ξi+1 + φi (ξ1 , . . . , ξi , u), y = ξ1 ⎪ ⎪ . ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎩˙ ξn = φn (ξ, u),

(3.23)

where the functions φi are Lipschitz and a high-gain observer can still be implemented [30]. Again, it is necessary to invert the transformation in (3.22), in order to recover the states in the original coordinates. Invertibility requires the map to be injective, which is (locally) satisfied when the Jacobian of ψ −R (x) is full rank. It is crucial to remark that, in this scenario, the map in (3.22) may not be injective, but the original system be instantaneously observable with a map of order n. Precisely, if the original system was not uniformly observable in the inputs, even if the system is restricted to observable inputs and trajectories, the map in (3.23) will not be injective. This limitation is a consequence of three facts. First, the resulting system in (3.23) is uniformly observable in the inputs [39]. Second, if the Jacobian of the map in (3.22) is full rank, then, it is a diffeomorphism. Finally, if system (3.21) is diffeomorphic to the system in (3.23), both have the same observability properties. Therefore, both have to be uniformly observable in the inputs. Remark 3.8 Consider the multi-input scenario where the inputs are separated as 

u = u1 , u2 . Assume that the system is only observable uniformly in the inputs u2 . Then, it is possible to create a transformation that combines the map (3.19) and the map (3.22). Specifically, the system can be rewritten as x˙ = f0 (x, u1 ) + g(x)u2 ; y = h(x), and the transformation in (3.22) can still be implemented. Nonetheless, notice that the map ψ −R will depend on the inputs u1 and its derivatives in this case.

3.2 High-Gain Observers

3.2.2.3

31

Inverting the Transformation

The high-gain observer has exemplified the first two steps of observer design. First, the system is transformed in a favourable form (in this case, a triangular form) through a coordinate change, ψ −R . Second, an observer is designed in this new coordinates to generate the signal ξˆ . The last step consists in finding an inverse, ψ, to generate the estimation in the original coordinates, xˆ . Naturally, if the transformation is simple, it is possible to find an analytical expression of this inverse and the last step is solved, which results in the observer presented in (3.1). Nonetheless, in many cases, finding an analytical expression of the inverse may be difficult. In these cases, inversions relies on a minimization of the form xˆ = min |ψ −R (x) − ξˆ |. x∈X

However, this minimization approach presents two significant drawbacks. On the one hand, the high computational cost of the minimization. On the other, without the inverse of the transformation, it is not possible to find the analytical expression of the target system. To understand this second limitation, consider the transformation in (3.19). A quick computation shows that the target system satisfies ξ˙i = ξi+1 , i = 1, ..., n − 1 ξ˙n = φ(x), y = x1 . Notice that the last equation still depends on the original coordinates, and it is necessary to use the inverse to get the equation to the new coordinates φ(x) = φ(ψ(ξ )). These limitations motivate the implementation of the observer directly in the original coordinates, in order to avoid the computation of the inverse, ψ. The result can be summarized through the following theorem. Theorem 3.2 Consider a nonlinear observer of the form (3.1). Assume that the map −R ψ −R is a diffeomorphism, i.e ψ −R : Rn → Rn and the Jacobian ψ∂x (x) is full rank. Then, the observer can be directly implemented as x˙ˆ =



∂ψ −R (ˆx) ∂x

−1 

 ϕ(ψ −R (ˆx), u) + κ(ψ −R (ˆx), y − h(ˆx)) .

Proof From the equality ξˆ = ψ −R (ˆx) it can be deduced that ξ˙ˆ =

∂ψ −R (ˆx)x˙ˆ . ∂x

(3.24)

32

3 Nonlinear Observer Design

Therefore, the next equality holds x˙ˆ =



∂ψ −R (ˆx) ∂x

−1

ξ˙ˆ . 

Remark 3.9 The transformations to a triangular structure (3.19) and (3.22) satisfies the following property ξ˙ = ϕ(ξ , u).

Therefore, the following equality is satisfied 

∂ψ −R (x) ∂x

−1

ϕ(ψ −R (x), u) = f(x, u),

and the observer in the original coordinates (3.24) can be directly implemented as x˙ˆ = f(ˆx, u) +



∂ψ −R (ˆx) ∂x

−1   −R κ(ψ (ˆx), y − h(ˆx)) . −R

Remark 3.10 Systems that have some points/regions where the Jacobian ψ∂x (x) is not full rank present a subtle difficulty. Even if, somehow, the system trajectories are designed to avoid these regions, the observer states, specially during the transient, may cross these points. This problem can be solved by appropriately constraining the observer states in a convex set that avoids these singularities [40]. Remark 3.11 This section has focused on cases in which ψ −R is a diffeomorphism, as it will be sufficient for the applications considered in this work. Nonetheless, it should be remarked that the approach in (3.24) cannot be implemented in cases where ψ −R is an injective immersion, i.e. the map is injective, but the target system is of larger dimension than the original system. In such cases, it is still possible to implement the observer in the original coordinates, although, the process is more convoluted, see for example [7] and references therein.

3.2.3 Extension of the High-Gain Observer: Low-Power Peaking-Free Observer The last sections have presented the complete design process of a high-gain observer. As pointed out in Remark 3.7, the described process can be easily adapted to alternative observer techniques, e.g. the higher-order differentiator based on sliding modes [32]. Indeed, in practice, it is common to select these alternatives before the highgain observer, see for example the recent review on observer design in fuel cells [41].

3.2 High-Gain Observers

33

The main reason for this design choice is the presence of the peaking phenomena, see Fig. 3.3, which significantly aggravates the transient of the observer estimation and can have relevant practical problems. For example, it aggravates the problem in Remark 3.10. Nonetheless, multiple authors have developed techniques to reduce or eliminate the peaking phenomena. Initial results are based on modifying the observer dynamics in order to bound its states to a prescribed set. Remarkable examples are the use of projection algorithms [42], the implementation of hybrid instantaneous jumps [43] or modify the observer feedback term under some convexity assumptions [44]. Although the presented results reduced the peaking phenomena, the high-gain observer dynamics were significantly modified and its implementation in feedback loops was not trivial. Another strategy was based on interconnecting a cascade of reduced highgain observers of order 1 and adding saturations at various levels of the observer structure [45]. It was a promising result, as a similar strategy, without feedback interconnection, was shown to be applicable in output feedback control [46]. However, the estimation-error was only proven to converge to a bounded region. The extension of these results lead to the creation of the low-power peaking-free observer [47]. This new observer structure presented the outstanding results of eliminating the peaking phenomena in all its states, improving the observer sensitivity to noise and preserving the classic high-gain performance. The remainder of this subsection briefly summarizes the results in [47]. The low-power peaking-free observer scheme is based on interconnecting n − 1 blocks of dimension 2, which provides an estimation of (xi , xi+1 ) i = 1, ..., n − 1, with a one dimensional block which provides an estimation of xn . The observer includes n − 1 “virtual states”, η, that are going to be saturated in order to eliminate the peaking phenomena. Specifically, for phase-variable systems (3.8), the observer takes the following form (see Fig. 3.6) αi x˙ˆi = ηi + ei , i = 1, ..., n − 1 ε ˙xˆn = φ(ˆx, 0) + αn en ε βi η˙ i = satri+2 (ηi+1 ) + 2 ei , i = 1, ..., n − 2 ε βn−1 η˙ n−1 = φ(ˆx, 0) + 2 en−1 ε

(3.25)

(3.26)

with e1  y − xˆ1 , ei  satri (ηi−1 ) − xˆi , i = 2, ..., n where xˆ = [xˆ1 , ..., xˆn ] ∈ Rn is the estimation of x, η = [η1 , ..., ηn−1 ] ∈ Rn−1 is the virtual state, α = [α1 , ..., αn ] ∈ Rn and β = [β1 , ..., βn ] ∈ Rn−1 are positive

34

3 Nonlinear Observer Design

Fig. 3.6 Scheme of the low-power peaking-free observer

design parameters, ε is the design high-gain parameter and satk (·) is a saturation function to be designed which satisfies satk (s) = s ∀|s| ≤ k, satk (s) = k ∀|s| ≥ k

(3.27)

Some remarkable differences between the low-power peaking-free structure and the classic high-gain observer can be summarized as follows: • The output-estimation error, y − xˆ1 is only used in the first state-estimation dynamics, i.e. x˙ˆ1 and η˙ 1 . • The high-gain parameter, ε, is only raised to the power of 2, instead of the power n − 1. The parameter design in this new observer is more convoluted than in the classical high-gain observer and requires the definition of some extra matrices. Define the following  Bk 

   0i−k,1 −αi 0 ∈ Ri×1 ∀k ∈ N, Ei  ∈ R2i×21 , i = 1, ..., n − 1 1 −βi 0

Next, let Mn ∈ R(2n−1)×(2n−1) be a matrix recursively constructed as follows M1  E1 , ⎡ ⎤ B2(i−1) B2  M  i−1 ⎦ i = 2, ..., n − 1, Mi  ⎣ αi  B2(i−1) Ei βi   0 Mn−1 . Mn   αn B2(i−1) −αn Finally, let Λi (s) : [0, 1] → R2i×2i , i = 1, ..., n be a continuous matrices defined as

3.2 High-Gain Observers

35

Λ1 (s)  M1 , ⎡ ⎤ sB2(i−1) B2  M  i−1 ⎦ i = 2, ..., n − 1, Λi (s)  ⎣ αi  B Ei βi 2(i−1) Λn (s)  Mn Now, after defining these matrices, it is possible to summarize the low-power peakingfree observer design and performance properties in the following theorem [47]. Theorem 3.3 Design ri of the saturation functions as ri  max x∈X |xi | for i = 1, ..., n. Moreover design αi , i = 1, ..., n and βi , i = 1, .., n − 1 such that there  exists Pi = Pi > 0 and μi > 0 that satisfy the following Pi Λi (s) + Λi (s) Pi ≤ μi I f or i = 1, ...n, ∀s ∈ [0, 1].

(3.28)

Then, provided that |v|∞ is small enough, there exists a value ε∗ such that for all ε ≤ min{ε∗ , 1} the estimation-error of the observer (3.25) satisfies the following bound k2 k1 − t k4 |xi − xˆi | ≤ min{ i−1 e ε |xi (0) − xˆi (0)| + εn+1−i k2 M + i−1 |v|∞ , p¯ i } (3.29) ε ε where k1 , ..., k4 are some positive constants (different from the ones in (3.13)) and p¯ i is a positive constant independent from ε. Proof The reader is referred to [47] for the complete proof.



Remark 3.12 Condition (3.28) may seem very restrictive and convoluted to compute. However, it is always possible to find some parameters αi and βi that makes the matrix inequality feasible. Moreover, a methodology to design αi and βi is presented in [48]. Let compare the estimation-error convergence of the classical high-gain observer (3.13) to the one in the low-power peaking-free observer (3.29). First, notice that the new observer structure presents the same benefits of high-gain observation. In the absence of noise and uncertainty (i.e. d = |v|∞ = 0), the estimation-error converges to zero with a decay rate proportional to ε−1 . In the presence of uncertainty, the estimation-error converges to a bounded region that can be made arbitrary small by reducing ε. Therefore, the observer maintains the decay rate and robustness performance of the classic high-gain observer. Second, by inspection of (3.29), it is possible to see that |xi − xˆi | ≤ p¯ i , i = 1, ..., n where p¯ i is independent from ε.

36

3 Nonlinear Observer Design

Thus, decreasing ε does not induce any peak in the observer states and the peaking phenomena of the classic high-gain observer is practically solved. Recall the system presented in the past example. Now lets solve the same estimation problem with a low-power peaking-free observer. The structure of the observer would the following one αi x˙ˆ1 = η1 + (y − xˆ1 ), ε αn xˆ˙2 = −2 xˆ1 − 2 xˆ2 + (satr2 (η1 ) − xˆ2 ) ε β1 η˙ 1 = −2 xˆ1 − 2 xˆ2 + 2 (y − xˆ1 ). ε By simulation, it can be seen that the states of the system evolve in the following compact set |x1 | ≤ 0.75, |x2 | ≤ 1 Therefore, the saturation function parameter, r2 , can be designed as r2 = 1. Moreover, after some computations, it can be seen that the parameters α1 = 3, α2 = 3 and β1 = 2 satisfy the matrix inequality (3.28). Now, consider that it is required to have the same convergence rate as in the classic high-gain observer with ε = 0.05 presented in the past example. Consequently, the low-power peaking-free observer is designed with a gain ε = 0.05. In Fig. 3.7, it is compared the evolution of the state-estimation, xˆ2 , of the classic high-gain observer with ε = 0.05 presented in the past example and the low-power peaking-free estimation of the same state. Notice that the convergence rate of both observers is nearly identical. Moreover, the low-power peaking-free observer converges to a lower estimation-error. The crucial part in Fig. 3.7 is that the classic high-gain observer peaks to a value close to

Fig. 3.7 Evolution of system true x2 (blue), classic high-gain observer (HGO) estimation with ε = 0.05 (orange) and low-power peaking-free observer (LPPFO) estimation (yellow)

3.2 High-Gain Observers

37

Fig. 3.8 Evolution of system true x2 (blue), low-power peaking-free observer (LPPFO) estimation (orange) with ε = 0.05 (yellow) and low-power peaking-free observer (LPPFO) estimation with ε = 0.01 (orange)

4 and, as shown in the past example, the amplitude of this peak is incremented as ε is reduced. However, the low-power peaking-free observer converges to a value around 1 and after that converges to the true value. Notice that, the low-power peakingfree estimation xˆ2 is not being saturated at 1. The estimation is continuous and differentiable in all its evolution. Moreover, the fact that the low-power peaking-free estimation converges to 1 is independent of the gain ε. In Fig. 3.8 it is compared the estimation of the low-power peaking-free observer with ε = 0.05 presented in this example and another observer identical in all the design parameters except ε, which is 0.01. Notice that decreasing ε in the low-power peaking-free observer increases the observer convergence rate, but, does not increase the transient estimation-error or induce any peaking phenomena. A last remarkable property of the low-power peaking-free observer is that the the extra states, η, reduce the noise sensitivity of the observer for systems of order larger than 2 [47]. To understand this property, notice that the, except for the first state-estimation dynamics, xˆ1 and η1 , all the other states are estimated using the error satri (ηi−1 ) − xˆi instead of y − x1 . Consequently, the relative degree between the state-estimation and the measurement noise is increased, which reduces the sensitivity of the observer to high-frequency output disturbances [49]. For this reason, for systems of dimension larger than 2, it may be convenient to implement a low-power observer (with or without the saturations) in order to get better noise sensitivity.

38

3 Nonlinear Observer Design

3.3 Parameter Estimation-Based Observer Last section has presented the high-gain observer as an example of the three-step process in Fig. 3.2. The simplicity (relative to other nonlinear observer techniques) of the high-gain observer design derives from three convenient properties: • As the system is semi-linear, observer design for nonlinear triangular structures, as (3.8) and (3.23), is nearly as simple as in linear systems. • The design of a coordinate change that transforms a system to a triangular form is a relatively easy process, see Sect. 3.2.2. • Proving that the transformation is well-posed is based on well-understood observability assumptions. All these properties stem from the fact that in a triangular form, in general, the states depict the derivative of the output, i.e. ξ1 is the output and ξk is the k − 1-th derivative of the output. Therefore, in practice, the high-gain observer (or any other observer for triangular structures) is composed by two elements. First, a “differentiator” (3.10) that robustly estimates the derivatives of the output, and a map (3.19) that relates the output derivatives with the system states. Both can be implemented simultaneously by means of (3.24). The existence of a map between the output derivatives and the states is the well-known instantaneous observability definition, see Sect. 2.2 of Chap. 2. By looking at the high-gain observer in this way, a concerning issue appears. Indeed, it is well-known that estimating the derivatives of a signal is a process that is highly sensitive to measurement noise. Consequently, observers for triangular structures will have high sensitivity to measurement noise due to its inherent differentiator part. This limitation has motivated the development of alternative observer techniques. Specifically, the design of observer for non-triangular forms and the design of transformations to such alternative forms. A promising line of research is the design of observers for nonlinear systems with linearizable error dynamics. To understand this approach, consider a single-output system be depicted by: x˙ = f(x, u), y = h(x). (3.30) It is said that the observer in (3.1) linearizes the error dynamics if the resulting error dynamics are linear plus the observer term κ(ξˆ , y − h(ˆx)). To achieve such linearization, first, the system is transformed into an equivalent one, where the linearization is trivial. The simplest target system is a linear one with additive output-dependant nonlinearity: (3.31) ξ˙ = Aξ + fnl (y, u), y = Cξ where fnl : R × Rq → Rn is a known nonlienar vector field. Then, since the outputs are known, the observer design step consists on fixing ϕ as ϕ(ξˆ , y) = Aξˆ + fnl (y, u), resulting in the desired linear error dynamics:

3.3 Parameter Estimation-Based Observer

39

ξ˙˜ = Aξ + fnl (y, u) − Aξˆ − fnl (y, u) − κ(ξˆ , y − h(ˆx)) = Aξ˜ − κ(ξˆ , y − h(ˆx)), (3.32) where ξ˜ = ξ − ξˆ and κ(ξˆ , y − h(ˆx)) can be easily designed as a linear feedback term. Such an approach was first studied for single-output systems [50] (see also [51]), and was later extended to single-output controlled system [52] and to multi-ouptut controlled systems [53]. Moreover, see [16, 54] for more recent reviews on the topic. Remark 3.13 The approach can be extended to the case where the linear part depends on the input and the output [55, 56]: ξ˙ = A(u, y)ξ + fnl (y, u), y = C(u)ξ .

(3.33)

Remark 3.14 It is important to remark the difference between the idea of linearization via coordinate change that is being explored in this section, and the Taylor linearization approach that is typical in techniques as the extended Kalman filter [1, 57]. In the former approach, the system is transformed into an equivalent one, in the latter the system is locally approximated by a linear one. Consequently, observers based on linearization via coordinate change are global in the sense that any observer initial condition will converge to the true one. Alternatively, observers based on Taylor linearization are only local and require the observer initial states to be close to the true value in order to make the approximation valid. However, the approach presents a significant limitation: the existence of a transformation to the form in (3.31) requires very restrictive conditions and can only be implemented in a limited class of systems. Fortunately, the existence conditions of the transformation can be relaxed if the target system has a nonlinear output. Specifically, if the resulting system has the form ξ˙ = A(y, u)ξ + fnl (y, u), y = Ξ (ξ )

(3.34)

where A : Rm × Rq → Rn ξ ×n ξ , Ξ : Rn ξ → Rm are nonlinear vector fields. This approach has been recently being explored through the theory of the Koopman operator applied to observers [58, 59] and the Kazantzis-Kravaris/Luenberger Observer [2, 3, 60]. Some insights on how to transform a nonlinear system to this form will be commented in the Sect. 3.3.4. Following this line, the aim of this section is to explore an alternative technique that can be implemented to systems of the form (3.34), which is referred to as parameter estimation-based observer [61, 62]. For the interested readers, the relationship between the Koopman operator and the KazantzisKravaris/Luenberger Observer can be found in [63], and the relation between the Kazantzis-Kravaris/Luenberger and the parameter estimation-based observer can be found in [64].

40

3 Nonlinear Observer Design

3.3.1 Idea and Structure of the Observer The main idea of the parameter estimation-based observer is to transform the stateestimation problem into an on-line parameter estimation problem. Precisely, the observer takes the unknown initial conditions of the estimation-error as some constant parameters to estimate, and forward-simulates from these initial conditions to retrieve the current value of the states. Naturally, estimating the initial conditions of a generic nonlinear system is a complex task. Nonetheless, as presented in [62], it has a computational efficient solution for systems in the following state-affine form ξ˙ = A(y, u)ξ + fnl (y, u), y = C(u)ξ .

(3.35)

where C(u) : Rq → Rm×n ξ is a nonlinear vector field. Moreover, it is assumed that u is such that all the trajectories of ξ evolve in a compact set. The idea is to implement an open-loop copy of the system in (3.35) that generates an estimation of the states z z˙ = A(y, u)z + fnl (y, u). Then, consider the estimation-error eξ = ξ − z, the dynamics of which are depicted by e˙ ξ = A(u, y)eξ . It can be seen that the dynamics of the estimation-error are depicted by a linear time-varying system. Thus, all the solutions of the error dynamics in (3.3.1) can be computed as a linear combination of the columns of its fundamental matrix Φ A ∈ Rn ξ ×n ξ , which is the unique solution of ˙ A = A(u, y)Φ A . Φ Indeed, the solution of the estimation-error, eξ , is computed as eξ (t) = Φ A (t)[Φ A (0)]−1 eξ (0). Remark 3.15 To simplify the notation and computation, the fundamental matrix will be initialized as Φ A (0) = I n ξ . Now, by exploiting the fact that the output is affine in the states it is possible to derive the following relation. Define, y  y − C(u)ξˆ .

3.3 Parameter Estimation-Based Observer

41

Then, it is possible to see the following equality y = Ψ eξ (0),

(3.36)

where Ψ  C(u)Φ A . The main idea of the parameter estimation-based observer is to take the initial conditions of the estimation-error as some constant parameter to estimate, i.e. define θ  eξ (0). Then, the expression in (3.36) can be transformed in a linear regression equation of the form y = Ψ θ, (3.37) where y and Ψ are known signals and θ is a vector of parameters to estimate. Notice that, as the original system evolves in a compact set, the signal Ψ is also bounded. Then, by means of implementing any on-line parameter estimation algorithm for linear regression, see [65] for some recent review on the topic, it is possible to generate an estimation of the unknown estimation-error initial conditions, θˆ such that ˆ = 0. (3.38) lim |θ − θ| t→∞

The final step of the observer consists in using the parameter estimation, θˆ to retrieve the states. This can be achieved through the following expression ξˆ = z + Φ A θˆ . The only part of the algorithm that remains to be defined is the parameter estimation algorithm that is used to generate, θˆ , such that (3.38) is satisfied. Next section will provide some insights on the topic.

3.3.2 Standard Parameter Estimator The objective is to design an algorithm to retrieve the unknown parameters, θ, from the signals y and Ψ , knowing that the equality in (3.37) holds. In general, the number of parameters will be larger than the number of measured signals, and Ψ cannot be inverted to retrieve the parameters. Thus, a single sample of the measured signals is not sufficient to estimate the parameters. Consequently, similar to the observation problem case, the idea is to design an on-line algorithm that uses the whole trajectory instead of a single sample. There exists a solution for the parameter estimation problem that can be found in any adaptive control textbook [66, 67]. The algorithm is commonly referred to as

42

3 Nonlinear Observer Design

gradient-descent estimator and, for the concerned linear regression (3.37), is computed as follows ˆ θ˙ˆ = γ Ψ (y − Ψ θ) (3.39) where γ > 0 is the adaptation gain. The stability properties of the gradient-descent estimator (3.39) have been extensively studied in the literature. In order to present the stability results it is important to introduce the concept of persistence of excitation. Definition 3.2 A bounded vector signal Ψ ∈ Rn ξ is persistently excited if there exists some positive constants T > 0 and α > 0 such that  t

t+T

Ψ (τ ) Ψ (τ )dτ ≥ α In ξ

This definition allows to summarize the stability results through the following theorem. Theorem 3.4 Consider the linear regression (3.37) and the gradient-descent estimator (3.39). Then, the following condition is satisfied ˜ = 0, lim Ψ (t)θ(t)

t→∞

where θ˜ = θ − θˆ . Moreover, if the vector Ψ is persistently excited, then, θ˜ = 0 is a global exponential stable equilibrium. Proof The proof can be found in classic adaptive control textbooks as [66, 67].  A way of understanding this result is to see the dynamics in (3.39) as an algorithm that fits a linear model to the measured trajectory. If the system trajectories are not “rich” enough, there may be multiple parameters that may explain the measured signals. Thus, the algorithm will converge to some parameter combination that, in general, will not be the true one. For the parameter estimation-based observer, the persistence of excitation condition of the vector Ψ is satisfied if the state-affine system (3.35) is observable uniformly in the inputs. The focus of the next subsection will be to propose a solution to the case in which the system does not satisfy this persistence of excitation condition.

3.3.3 Implicitly-Regularized Observer One of the advantages of the parameter estimation-based observer is that the observability condition of the system is substituted by a persistence of excitation condition. This is a convenient fact because, recently, multiple authors have proposed parameter estimation algorithms with strictly weaker excitation conditions. Some relevant

3.3 Parameter Estimation-Based Observer

43

results are the concurrent learning applications in adaptive control [68], algorithms based on the concept of lack of persistence of excitation of order q [69], the dynamic regressor extension and mixing [70, 71] and its more recent modifications [72]. Motivated by recent results in adaptive control and machine learning [73], this section suggests using non-Euclidean adaptation algorithms that implicitly regularize the parameter estimation. To better understand what is being considered in this section, notice that Theorem 3.4 establishes that the estimation of the algorithm in (3.39) converges to the following set Ω = {θˆ | Ψ (t)θˆ = Ψ (t)θ ∀t}, i.e. the set of parameters that interpolates the dynamics Ψ (t)θ along the measured trajectory. If the system is persistently excited, then, this set only contains the true parameters. Otherwise, it may have multiple parameter combination that explains the observed trajectory. What is referred to as implicit regularization is an algorithm that, using the same measured signals, y and Ψ , converges to a set that is strictly smaller than Ω. Implicit regularization can be achieved by modifying the gradient-descent so it satisfies some local geometry imposed by the designer. The idea is to implement a natural gradientdescent like parameter estimation so, additionally to converge to the set Ω, the resulting set of parameters optimizes a potential function defined by the designer. The result can be summarized through the following theorem. Theorem 3.5 Consider a strongly convex function ϕ(·) and the natural gradient-like adaptation law:  −1   θ˙ˆ = ∇ 2 ϕ(θˆ ) Ψ  y − Ψ θˆ

(3.40)

Moreover, consider that θˆ (0) = minq ϕ(θ ). Then, the parameter-estimation, θˆ , converges to min ϕ(θ ).

θ ∈R

θ ∈Ω

Proof This proof is a sketched version of the results in [73]. The first step of the proof consists in showing that (3.40) converges to Ω. Define the Bregman divergence of a strictly convex function ϕ as [74]: dϕ (a, b)  ϕ(a) − ϕ(b) − (a − b) ∇ϕ(b). Consider the radially unbounded Lyapunov function: V = d(θ , θˆ ). Define eθ  θ − θˆ . The derivative of (3.41) satisfies the following:

(3.41)

44

3 Nonlinear Observer Design  V˙ = eθ ∇ 2 ϕ(θˆ )θ˙ˆ

=

 −eθ ∇ 2 ϕ(θˆ )

    −1    2  ˆ ˆ ˆ ∇ ϕ(θ ) Ψ y − Ψ θ ≤ −eθ Ψ y − Ψ θ

  ≤ − Ψ θ˜ Ψ θ˜ ,

which shows that the system converges to the set Ω. Second, it will be shown that the gradient-like dynamics (3.40) implicitly optimize the convex function ϕ(θˆ ). ˆ factor presents the following time derivative: The d(θ ||θ)     d d ˆ ˆ ˆ dϕ (θ, θ ) = − ∇ϕ(θ ) eθ = − y − Ψ θ Ψ eθ . dt dt

(3.42)

The integration of (3.42) results in: dϕ (θ , θˆ (0)) = dϕ (θ , θˆ (t)) +

  t y − Ψ θˆ Ψ eθ dτ. 0

Previously, it has been shown that the parameter-estimation converges to the set ˆ θˆ (t) ∈ Ω. Consequently, one can show that for θ ∈ Ω and θ(∞) ∈ Ω, the next condition is satisfied:     ∞ ˆ ˆ ˆ ˆ y − Ψθ y − Ψ θ dτ. (3.43) dϕ (θ , θ (0)) = dϕ (θ, θ (∞)) + 0

Only the first term in the right side of (3.43) depends on θ . Consequently, the ˆ minimum value over θ of the right hand side of (3.43) is found at θ(∞). Moreover, the minimum of the left-hand side factor is naturally obtained in arg min dϕ (θ , θˆ (0)). Therefore,

θ∈Ω

θˆ (∞) = min dϕ (θ , θˆ (0)). θ ∈Ω

(3.44)

If the observer states are initialized in θˆ (0) = minq ϕ(θ ), the equation in (3.44) θ ∈R

becomes:

θˆ (∞) = min ϕ(θ ). θ ∈Ω



Theorem 3.5 establishes that, in the case of persistently excited systems, the parameter-estimation dynamics (3.40) converges to the true value. In the nonpersistently excited case, the estimation converges to the value θˆ ∈ Ω that minimizes the convex function ϕ(·) to be defined. Intuitively, the function ϕ(·) is a potential function that can be implemented to exploit additional prior information to ensure

3.3 Parameter Estimation-Based Observer

45

that the state-estimation problem is well-posed. Some examples of prior information that can be exploited in this context may be physical consistency conditions of the system [75], physical bounds on the states [76], non-negativeness of the system [77] or an L p norm of the states to be minimized [73]. Relative to the latter, it is convenient to present the following remark. −1  in (3.40) may not be known for some p norms. Remark 3.16 The factor ∇ 2 ϕ(θˆ ) 1 ˆ2 Consequently, the potential function ϕ is implemented as, |θ| . The benefit of this 2 p modification is that the Jacobian of a squared p norm has an analytical inverse, which allows the adaptation (3.40) to be computed as follows [78],   w˙ = Ψ  y˜ − Ψ θˆ

(3.45)

d−1 θˆi = |w|2−d sign(wi ) f or i = 1, ..., q d |wi |

(3.46)

1 1 where w  ∇ϕ(θˆ ), | · |d is the d-norm and + = 1. d p As an example, consider a nonlinear system of the form x˙ = A(u)x, y = cx, with the state vector x = [x1 , x2 ] and the matrices: A(t) =

 

 0u ; c= 10 00

(3.47)

where the input is a time-varying signal computed as u = e−0.9t . In the proposed system, the vector Ψ of the parameter estimation-based observer will not be persistently excited. This can be deduced from the following fact. The fundamental matrix of system (3.47) in the interval [t0 , t] is: ⎡ φ(t, t0 ) =

⎣1 − 0

⎤  1 e−0.9t − e−0.9t0 ⎦ . 0.9 1

Consequently, the observability gramian of the system is:  m 12 (t)e−0.9t0 t − t0 . m 12 (t)e−0.9t0 m 22 (t)e−1.8t0

 M(t, t0 ) =

46

3 Nonlinear Observer Design

where    1 1 t − t0 − 1 − e−0.9(t−t0 ) 0.9 0.9  1 −1.8(t−t0 ) 1 1 −0.9(t−t0 ) − e e m 22 (t) = + 2 0.9 1.8 0.45  1 1 + − + t − t0 . 1.8 0.45

m 12 (t) =

The eigenvalues of the observability gramian are: λ1,2 =

   1 t + m 22 e−1.8t0 ± (t − t0 − m 22 e−1.8t0 ) + 4m 212 e−1.8t0 2

The eigenvalues are positive for all t0 and t > t0 . However, it can be seen that: lim λmin = 0,

t0 →∞

lim λmax = t − t0 .

t0 →∞

Therefore, the system loses observability with time and the vector Ψ cannot be persistently excited. Consequently, a parameter estimation-based observer with the standard gradient-descent (3.39) cannot estimate the second state. Indeed, this fact is validated in Fig. 3.9 where it can be seen that the estimation does not converge to the true value. Alternatively, one can exploit some properties of the system to design a potential function ϕ(·) that regularizes the estimation problem. In the considered system, an observer can be initialized as C xˆ (0) = C x(0). In such case, the following is satisfied for (3.37): cθ = 0. (3.48) Condition (3.48) means that, in the parameter-based observer approach, under the proper observer initialization, the parameter vector to be estimated presents multiple zeros. Following ideas of compressed sensing and sparse signal recovery [79, 80], it

Fig. 3.9 True evolution of the state x2 , estimation of the implicit regularized observer (IRO) with l1.1 norm and estimation of the ‘parameter estimation-based observer with standard gradient descent

3.3 Parameter Estimation-Based Observer

47

can be shown that an L 1 regularization factor improves the conditioning of an inverse problem if the parameter vector to be estimated is somewhat sparse. Consequently, ˆ 1, an implicitly regularized observer that optimizes the potential function ϕ(θˆ ) = |θ| where | · |1 is the 1-norm, should present better estimation accuracy. Remark 3.17 Notice that the 1-norm is not strictly convex, which prevents (3.41) 1 to be a Lyapunov function. For this reason, ϕ is implemented as the norm |θˆ |2p with 2 p = 1 + , where  is a small positive constant. This is the closest strictly convex function to the 1-norm. In Fig. 3.9 it is depicted the estimation of an implicitly regularized observer with 1 potential function |θˆ |1.1 computed through (3.45)–(3.46). It can be seen that the 2 estimation converges to a value closer to the true one. This validates the benefits of the proposed approach.

3.3.4 Transformation to a State-affine Form Most nonlinear systems do not present the structure of (3.35). Motivated by the framework in Fig. 3.2, the only option is to find a diffeomorphism (or injective immersion if the resulting coordinates are of larger dimension), ψ −R that transforms the system to the adequate form. The transformation has to satisfy the following properties • The transformation is the solution of the following partial differential equation ∂ψ −R (x)f(x, u) = A(u, h(x))ψ −R (x) + fnl (u, h(x)) ∂x

(3.49)

• The following algebraic constraint is satisfied C(u)ψ −R = L(u, h(x)),

(3.50)

where L : Rq × Rm → Rm is a nonlinear vector field. • The function ψ −R is injective. Unfortunately, contrary to the transformation to a triangular form, there is no methodology that can be used to circumvent the partial differential equation in (3.49). Some interesting results have been obtained for the existence and design of a transformation to the form in (3.34) [2, 3], but is not clear how to include the algebraic constraint (3.50) in these results. As a last step, the transformation has to be inverted to recover the states in the original coordinates. The reader is referred to the Sect. 3.2.2.3 for more details on the topic.

48

3 Nonlinear Observer Design

From this it should be clear the advantages and drawbacks of the parameter estimation-based observer in relation to the high-gain observer. On the one hand, the parameter estimation-based observer can be implemented under weaker observability assumptions. Moreover, as it does not have an inherent differentiator part, thus, the observer presents lower noise sensitivity. On the other hand, in a general nonlinear system, the parameter estimation-based observer requires finding a transformation through a partial differential Eq. (3.49). The high-gain observer transformation is much more simpler to find and compute.

3.4 Nonlinear Observers in the Presence of Unmodelled Elements Last sections have presented two examples of observer design using the framework presented in Fig. 3.2. First, the high-gain observer and its low-power peaking-free modification. Second, the parameter estimation-based observer and its implicitly regularized modification. However, a clear limitation of the proposed observer design framework is that it does not include the unavoidable presence of uncertainty, disturbances and measurement noise. As it has been shown in Sect. 3.2, these elements have a significant impact on the observer accuracy and, for practical applications, have to be considered during the observer design. Naturally, the observer has to be designed so the effect of measurement noise, uncertainty and disturbances on the estimation-error is minimized. Before proposing some methods to achieve such minimization, it is convenient to extend the observer design framework presented at the beginning of the chapter. Specifically, this section considers a multi-input multi-output nonlinear system of the form x˙ = f(x, u) + w y = h(x) + v.

(3.51)

where the factor v ∈ Dv ⊆ Rm depicts bounded measurement noise and w ∈ Dw ⊆ Rn models model uncertainty and/or unmodelled external disturbances. Moreover, there exists some sets X0 ⊆ X ⊆ Rn x and U ⊆ Rn u , such that the trajectories of (3.51), with initial conditions x(0) in X0 , input u(t) belonging to U and disturbance d(t) in Dw for all times, remain in X for all t ≥ 0. Remark 3.18 For the rest of this work, it will be assumed that, if w models uncertainty, it does not modify the structural properties of the system. For example, if f (x, u) is stable and observable, then, f (x, u) + w is also stable and observable. It is assumed that there exists a known observer providing an asymptotic estimate depicted by the expression in (3.1). Naturally, in the presence of unmodelled elements, the convergence to an exact estimation depicted in the bound (3.4)

3.4 Nonlinear Observers in the Presence of Unmodelled Elements

49

cannot be achieved. Consequently, the bound in (3.4) is relaxed to an input-tostate stability (ISS) property. Specifically, there exists a locally Lipschitz function V : Rn × Rn ξ → R satisfying −R α(|x − ψ(ξˆ )|) ≤ V (x, ξˆ ) ≤ α(|ψ ¯ (x) − ξˆ |)

(3.52)

for all x, ξˆ ∈ X, Rn ξ , where α, α¯ ∈ K∞ , so that its time derivative along solutions to (3.51) and (3.1) satisfies V˙ ≤ −α1 V (x, ξˆ ) + γ1 |w|2 + γ2 |v|2 .

(3.53)

for all x, ξˆ ∈ X, Rn ξ , u ∈ U and d ∈ D ⊆ R, where α1 and γ1 , γ2 are some positive constants. Conditions (3.52) and (3.53) state that the observer (3.1) is asymptotically convergent for system (3.51) and in particular the estimation-error x˜ is ISS with respect to the disturbances/uncertainty, w, v. Specifically, there exist a function β1 ∈ KL and a pair of functions ϑ1 , ϑ2 ∈ K∞ such that the solutions (x(t), ξˆ (t)) belonging to X, Rn ξ for all d ∈ D ⊆ R and t ≥ 0 satisfy   −R ˆ |˜x(t)| ≤ β1 |ψ (x(0)) − ξ (0)|, t + ϑ1 (|w|[0,t] ) + ϑ2 (|v|[0,t] ).

(3.54)

Remark 3.19 As studied in [81], the ISS property in (3.53) is coordinate dependent if the sets X and U are unbounded. This limitation is obviated in this work, as the considered practical applications evolve in bounded sets. Remark 3.20 In the case that ϑi (·) is a linear function of the type ϑi (a) = ka with k > 0, it is said that the system has an ISS gain k with respect to the input a. Roughly speaking, when t is small(during the transient), the  estimation-error will be mostly bounded by the factor, β1 |ψ −R (x(0)) − ξˆ (0)|, t , related with the initial

estimation-error of the observer. Nonetheless, this factor converges asymptotically to zero. For this reason, on the other hand, when t is large (during the steady-state), this factor will be near zero and the error will be bounded by the terms related to the unmodelled factors ϑ1 (|w|[0,t] ) and ϑ2 (|v|[0,t] ). See Fig. 3.10 for a scheme of this convergence. In this situation, it is said that the estimation-error, x˜ is ultimately bounded by the factors, ϑ1 (|w|[0,t] ) + ϑ2 (|v|[0,t] ). Naturally, from a practical perspective, it is interesting to minimize the ultimate bound in order to get an accurate estimation. Simply, the term related to the measurement noise, ϑ2 (|v|[0,t] ), can be reduced by implementing better sensors with lower noise, i.e. a reduction in the term |v|[0,t] . Similarly, the term related to uncertainty/disturbances, ϑ1 (|w|[0,t] ) can be reduced by means of improving the system model, i.e. a reduction of the term |w|[0,t] . Nonetheless, due to economical and technological limitations, it may not be possible to improve the

50

3 Nonlinear Observer Design

Fig. 3.10 Scheme of an ISS system convergence

system sensors or the mathematical model. In such cases, the minimization of the ultimate bound must come from a proper observer design. Following this line, a first solution may be to tune the observer design parameters in order to reduce the functions ϑ1 (·) and ϑ2 (·). For example, the parameters αi , for i = 1, . . . , n and ε of the high-gain observer (3.10) could be optimized to reduce the observer ultimate bound. However, such an approach leads to significant obstacles. • Observers present a trade-off between noise sensitivity and disturbance rejection [31]. Thus, in general, a reduction of the function ϑ1 (·) implies an increase of ϑ2 (·), and vice-versa. This fact has been exemplified through the standard highgain observer, see (3.13) and Fig. 3.5. • The relation between the observer parameters and the functions ϑ1 (·) and ϑ2 (·) is not trivial and mostly nonlinear. Thus, the optimization of the ultimate bound, in general, cannot be formulated as a convex minimization. In practice, the observer parameters are tuned by means of trial-and-error process, which mostly converges in a local minimum. • The disturbance/noise trade-off creates a bound in the minimum obtainable ultimate bound. In some practical scenarios, this bound may be too large to be useful. Taking into account these obstacles, the remainder of this work will focus on proposing solutions to the ultimate bound problem that do not rely on an improvement of the observer parameter tuning nor an improvement of the system sensors/model. Specifically, the following chapters will suggest a set of solutions to the next questions. • Considering an already existing observer of the form (3.1), it is possible to redesign the observer such that the factor related with the sensor noise ϑ2 (·) without a significant increase of ϑ1 (·)? • Considering an already existing observer of the form (3.1), it is possible to redesign the observer such that the factor related with the uncertainty ϑ1 (·) without a siginificant increase of ϑ2 (·)? Chapter 4 will provide a positive answer to the first question by means of using filters. Chapter 5 will provide a positive answer to the second question by transforming the original observer into an adaptive observer.

References

51

References 1. Bernard P, Mimmo N, Marconi L (2021) On the semi-global stability of an ek-like filter. IEEE Control Syst Lett 5(5):1771–1776. https://doi.org/10.1109/LCSYS.2020.3044030 2. Andrieu V, Praly L (2006) On the existence of a kazantzis-kravaris/luenberger observer. SIAM J Control Optim 45(2):432–456 3. Bernard P, Andrieu V (2019) Luenberger observers for nonautonomous nonlinear systems. IEEE Trans Autom Control 64(1):270–281. https://doi.org/10.1109/TAC.2018.2872202 4. Hammouri H, Ahmed FS, Othman S (2018) Observer design based on immersion technics and canonical form. Syst & Control Lett 114:19–26. ISSN 0167-6911. https://doi.org/10.1016/j. sysconle.2018.02.003 5. Arcak M, Kokotovic P (2001) Nonlinear observers: a circle criterion design and robustness analysis. Autom 37(12):1923–1930. ISSN 0005-1098. https://doi.org/10.1016/S00051098(01)00160-1 6. Fan X, Arcak M (2003) Observer design for systems with multivariable monotone nonlinearities. Syst & Control Lett 50(4):319–330. ISSN 0167-6911. https://doi.org/10.1016/S01676911(03)00170-1 7. Bernard P, Marconi L (2020) Hybrid implementation of observers in plant’s coordinates with a finite number of approximate inversions and global convergence. Autom 111:108654. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2019.108654 8. O’Reilly J (1983) Observers for linear systems, vol 170. Academic press 9. Korovin SK, Fomichev VV, Fomichev VV (2009) State observers for linear systems with uncertainty. de Gruyter 10. Kalman RE, Bucy RS (1961) New results in linear filtering and prediction theory. J Basic Eng 82(1):34–45 11. Luenberger D (1971) An introduction to observers. IEEE Trans Autom Control 16(6):596–602. https://doi.org/10.1109/TAC.1971.1099826 12. Khalil HK, Grizzle JW (2002) Nonlinear systems, vol 3. Prentice hall Upper Saddle River, NJ 13. Sepulchre R, Jankovic M, Kokotovic PV (2012) Constructive nonlinear control. Springer Science & Business Media 14. Isidori A, Sontag ED, Thoma M (1995) Nonlinear control systems, vol 3. Springer 15. Bernard P, Andrieu V, Astolfi D (2022) Observer design for continuous-time dynamical systems. Annu Rev Control. ISSN 1367-5788. https://doi.org/10.1016/j.arcontrol.2021.11.002 16. Bernard P (2019) Observer design for nonlinear systems. ISSN 978-3-030-11146-5 17. Besançon G (2007) An overview on observer tools for nonlinear systems. In: Nonlinear observers and applications, pp 1–33 18. Rajamani R (1998) Observers for Lipschitz nonlinear systems. IEEE Trans Autom Control 43(3):397–401. https://doi.org/10.1109/9.661604 19. Yan X-G, Edwards C (2007) Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Autom 43(9):1605–1614. ISSN 0005-1098. https://doi.org/10.1016/j. automatica.2007.02.008 20. Phanomchoeng G, Rajamani R, Piyabongkarn D (2011) Nonlinear observer for bounded jacobian systems, with applications to automotive slip angle estimation. IEEE Trans Autom Control 56(5):1163–1170. https://doi.org/10.1109/TAC.2011.2108552 21. Krener AJ, Xiao MQ (2002) Observers for linearly unobservable nonlinear systems. Syst & Control Lett 46(4):281–288. ISSN 0167-6911. https://doi.org/10.1016/S01676911(02)00153-6 22. Bernard P, Praly L, Andrieu V, Hammouri H (2017b) On the triangular canonical form for uniformly observable controlled systems. Autom 85:293–300. ISSN 0005-1098. https://doi. org/10.1016/j.automatica.2017.07.034 23. Teel A, Praly L (1994) Global stabilizability and observability imply semi-global stabilizability by output feedback. Syst & Control Lett 22(5):313–325. https://doi.org/10.1016/01676911(94)90029-9

52

3 Nonlinear Observer Design

24. Byrnes CI, Isidori A (2004) Nonlinear internal models for output regulation. IEEE Trans Autom Control 49(12):2244–2247. ISSN 2334-3303. https://doi.org/10.1109/TAC.2004.838492 25. Martinez-Guerra R, Mata-Machuca JL (2016) Fault detection and diagnosis in nonlinear systems. Springer. https://doi.org/10.1007/978-3-319-03047-04 26. Chowdhury D, Khalil HK (2020) Practical synchronization in networks of nonlinear heterogeneous agents with application to power systems. IEEE Trans Autom Control :1. https://doi. org/10.1109/TAC.2020.2981084 27. Wu Y, Isidori A, Lu R, Khalil HK (2020) Performance recovery of dynamic feedbacklinearization methods for multivariable nonlinear systems. IEEE Trans Autom Control 65(4):1365–1380. https://doi.org/10.1109/TAC.2019.2924176 28. Gauthier J-P, Kupka I (2001) Deterministic observation theory and applications. Cambridge University Press. https://doi.org/10.1017/cbo9780511546648 29. Gauthier JP, Hammouri H, Othman S (1992) A simple observer for nonlinear systems applications to bioreactors. IEEE Trans Autom Control 37(6):875–880. https://doi.org/10.1109/9. 256352 30. Khalil HK, Praly L (2014) High-gain observers in nonlinear feedback control. Int J Robust Nonlinear Control 24(6):993–1015. https://doi.org/10.1002/rnc.3051 31. Goodwin GC (2018) A critique of observers used in the context of feedback control. In: Chen Z, Mendes A, Yan Y, Chen S (eds) Intelligent robotics and applications. Springer International Publishing, Cham, pp 1–24 32. Levant A (2003) Higher-order sliding modes, differentiation and output-feedback control. Int J Control 76(9–10):924–941. https://doi.org/10.1080/0020717031000099029. Nov 33. Fridman L, Shtessel Y, Edwards C, Yan X-G (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. https://doi.org/10.1002/rnc.1198 34. Moreno JA (2021) Arbitrary order fixed-time differentiators. IEEE Trans Autom Control :1. https://doi.org/10.1109/TAC.2021.3071027 35. Bernard P, Praly L, Andrieu V (2017a) Observers for a non-lipschitz triangular form. Autom 82:301–313. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2017.04.054 36. Andrieu V, Astolfi D, Bernard P (2021) Observer design via interconnections of second-order mixed sliding-mode/linear differentiators. Int J Robust Nonlinear Control 31(9):3631–3657. https://doi.org/10.1002/rnc.5301 37. Zeitz M (1984) Observability canonical (phase-variable) form for non-linear time-variable systems. Int J Syst Sci. ISSN 14645319. https://doi.org/10.1080/00207728408926614 38. Jouan P, Gauthier JP (1996) Finite singularities of nonlinear systems. output stabilization, observability, and observers. J Dyn Control Syst. ISSN 10792724. https://doi.org/10.1007/ BF02259528 39. Gauthier JP, Bornard G (1981) Observability for any u(t) of a class of Nonlinear Systems. IEEE Trans Autom Control. ISSN 15582523. https://doi.org/10.1109/TAC.1981.1102743 40. Astolfi D, Bernard P, Postoyan R, Marconi L (2021b) Constrained state estimation for nonlinear systems: a redesign approach based on convexity. IEEE Trans Autom Control :1. https://doi. org/10.1109/TAC.2021.3064537 41. Yuan H, Dai H, Wei X, Ming P (2020) Model-based observers for internal states estimation and control of proton exchange membrane fuel cell system: a review. J Power Sources 468:228376. ISSN 0378-7753. https://doi.org/10.1016/j.jpowsour.2020.228376 42. Maggiore M, Passino KM (2003) A separation principle for a class of non-UCO systems. IEEE Trans Autom Control 48(7):1122–1133. https://doi.org/10.1109/TAC.2003.814110 43. Prieur C, Tarbouriech S, Zaccarian L (2012) Hybrid high-gain observers without peaking for planar nonlinear systems. In: 2012 IEEE 51st IEEE conference on decision and control (CDC), pp 6175–6180. https://doi.org/10.1109/CDC.2012.6426449 44. Astolfi D, Praly L (2017) Integral action in output feedback for multi-input multi-output nonlinear systems. IEEE Trans Autom Control 62(4):1559–1574. https://doi.org/10.1109/TAC. 2016.2599784

References

53

45. Teel AR (2016) Further variants of the Astolfi/Marconi high-gain observer. In: 2016 American control conference (ACC), pp 993–998. https://doi.org/10.1109/ACC.2016.7525044 46. Khalil HK (2017a) Cascade high-gain observers in output feedback control. Autom 80:110 – 118. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2017.02.031 47. Astolfi D, Marconi L, Praly L, Teel AR (2018) Low-power peaking-free high-gain observers. Autom 98:169–179. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2018.09.009 48. Astolfi D (2016) Observers and robust output regulation for nonlinear systems. PhD thesis. https://pastel.archives-ouvertes.fr/tel-01774046/document 49. Astolfi D, Zaccarian L, Jungers M (2021c) On the use of low-pass filters in high-gain observers. Syst & Control Lett 148:104856. ISSN 0167-6911. https://doi.org/10.1016/j.sysconle.2020. 104856 50. Krener AJ, Isidori A (1983) Linearization by output injection and nonlinear observers. Syst Control Lett. ISSN 01676911. https://doi.org/10.1016/0167-6911(83)90037-3 51. Isidori A (1995) Observers with linear error dynamics, 3rd edn. In: Nonlinear control systems. Springer-Verlag, London, pp 203–217 52. Marino R, Tomei P (1995) Nonlinear control design; geometric, adaptive and robust. PrenticeHall, London 53. Krener AJ, Respondek W (2005) Nonlinear observers with linearizable error dynamics. SIAM J Control Optim. ISSN 0363-0129. https://doi.org/10.1137/0323016 54. Boutat D, Zheng G (2021) Observer design for nonlinear dynamical systems 55. Hammouri H, Kinnaert M (1996) A new procedure for time-varying linearization up to output injection. Syst Control Lett. ISSN 01676911. https://doi.org/10.1016/0167-6911(96)00022-9 56. Besançon G, Bornard G (1997) On characterizing classes of observer forms for nonlinear systems. In: European control conference (ECC). https://doi.org/10.23919/ecc.1997.7082587 57. Krener AJ (2003) The convergence of the extended Kalman filter. In: Directions in mathematical systems theory and optimization. Springer, pp 173–182 58. Mauroy A, Mezi´c I, Susuki Y (2020) The Koopman operator in systems and control: concepts, methodologies, and applications, vol 484. Springer Nature 59. Amit Surana and Andrzej Banaszuk (2016) Linear observer synthesis for nonlinear systems using Koopman operator framework. IFAC-PapersOnLine 49(18):716–723. ISSN 2405-8963. https://doi.org/10.1016/j.ifacol.2016.10.250. 10th IFAC symposium on nonlinear control systems NOLCOS 2016 60. Kazantzis N, Kravaris C (1997) Nonlinear observer design using lyapunov’s auxiliary theorem. In: Proceedings of the 36th IEEE conference on decision and control, vol 5, pp 4802–4807. https://doi.org/10.1109/CDC.1997.649779 61. Romeo Ortega, Alexey Bobtsov, Anton Pyrkin, and Stanislav Aranovskiy (2015) A parameter estimation approach to state observation of nonlinear systems. Syst & Control Lett 85:84–94. ISSN 0167-6911. https://doi.org/10.1016/j.sysconle.2015.09.008 62. Ortega R, Bobtsov A, Nikolaev N, Schiffer J, Dochain D (2021b) Generalized parameter estimation-based observers: application to power systems and chemical-biological reactors. Autom 129:109635. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2021.109635 63. Yi B, Manchester IR (2021) On the equivalence of contraction and Koopman approaches for nonlinear stability and control. arXiv:2103.15033 64. Yi B, Ortega R, Zhang W (2019) On state observers for nonlinear systems: a new design and a unifying framework. IEEE Trans Autom Control 64(3):1193–1200. https://doi.org/10.1109/ TAC.2018.2839526 65. Ortega R, Nikiforov V, Gerasimov D (2020) On modified parameter estimators for identification and adaptive control. A unified framework and some new schemes. Annu Rev Control 50:278– 293. ISSN 1367-5788. https://doi.org/10.1016/j.arcontrol.2020.06.002 66. Sastry S, Bodson M (2011) Adaptive control: stability, convergence and robustness. Courier Corporation 67. Ioannou PA, Sun J (2012) Robust adaptive control. Courier Corporation 68. Chowdhary G, Yucelen T, Mühlegg M, Johnson EN (2013) Concurrent learning adaptive control of linear systems with exponentially convergent bounds. Int J Adapt Control Signal Process 27(4):280–301. https://doi.org/10.1002/acs.2297

54

3 Nonlinear Observer Design

69. Marino R, Tomei P (2022) On exponentially convergent parameter estimation with lack of persistency of excitation. Syst & Control Lett 159:105080. ISSN 0167-6911. https://doi.org/ 10.1016/j.sysconle.2021.105080 70. Aranovskiy S, Bobtsov A, Ortega R, Pyrkin A (2017) Performance enhancement of parameter estimators via dynamic regressor extension and mixing. IEEE Trans Autom Control 62(7):3546–3550. https://doi.org/10.1109/TAC.2016.2614889 71. Ortega R, Aranovskiy S, Pyrkin AA, Astolfi A, Bobtsov AA (2021) New results on parameter estimation via dynamic regressor extension and mixing: continuous and discrete-time cases. IEEE Trans Autom Control 66(5):2265–2272. https://doi.org/10.1109/TAC.2020.3003651 72. Wang L, Ortega R, Bobtsov A, Romero JG, Yi B (2021) Identifiability implies robust, globally exponentially convergent on-line parameter estimation: application to model reference adaptive control. arXiv:2108.08436 73. Boffi NM, Slotine J-JE (2021) Implicit regularization and momentum algorithms in nonlinearly parameterized adaptive control and prediction. Neural Comput 33(3):590–673. ISSN 08997667. https://doi.org/10.1162/necoa01360 74. Bregman LM (1967) The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput Math Math Phys 7(3):200–217. ISSN 0041-5553. https://doi.org/10.1016/0041-5553(67)90040-7 75. Lee T, Kwon J, Park FC (2018) A natural adaptive control law for robot manipulators. In: 2018 IEEE/RSJ international conference on intelligent robots and systems (IROS), pp 1–9 76. Wensing PM, Slotine J-J (2020) Beyond convexity—contraction and global convergence of gradient descent. PLOS ONE 15(8):1–29. https://doi.org/10.1371/journal.pone.0236661 77. Berman A, Neumann M, Plemmons RJ, Stern RJ (1989) Nonnegative matrices in dynamic systems, vol 3. Wiley-Interscience 78. Gentile C (2003) The robustness of the p-norm algorithms. Mach Learn 53:265–299. https:// doi.org/10.1109/TIT.2005.864420 79. Tropp JA (2006) Just relax: convex programming methods for identifying sparse signals in noise. IEEE Trans Inf Theory 52(3):1030–1051. https://doi.org/10.1109/TIT.2005.864420 80. Donoho DL, Elad M, Temlyakov VN (2006) Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans Inf Theory 52(1):6–18. https://doi.org/10.1109/ TIT.2005.860430 81. Shim H, Liberzon D (2016) Nonlinear observers robust to measurement disturbances in an ISS sense. IEEE Trans Autom Control 61(1):48–61. https://doi.org/10.1109/TAC.2015.2423911

Chapter 4

Adding Filters in Nonlinear Observers

One of the main performance limiting factors of an observer is the presence of sensor noise. This chapter proposes the use of filters to reduce the effect of noise on the accuracy of an observer. First, a framework to implement low-pass filters in nonlinear observers is presented. Second, a novel internal-model filter to exactly cancel noise from a known generating model is proposed. Finally, results on the use of dynamic dead-zones are recalled and linked with the low-pass filter design framework.

4.1 The Conflict of Measurement Noise in Nonlinear Observers In the end of the past chapter, it has been shown that the presence of measurement noise has a direct effect on the accuracy of the observer. The observer main source of information is the measured output. Thus, it is natural to expect a significant degradation of the observer performance if the measured signals are inaccurate or noisy. Indeed, it is well-known that one of the major factors that limits the potential performance in the pure observation context or in observer-based feedback control context is the presence of measurement noise [1]. For this reason, some authors are developing control algorithms that avoid the use of observers [2, 3]. Nonetheless, currently, some control applications cannot avoid the observer part, which motivates the development of some methodology to reduce the effect of measurement noise in the nonlinear observer estimation-error. Firstly, it is convenient to stablish the adequate framework to characterize the measurement noise and its effects. Some authors, model the measurement noise as a stochastic signal with a known statistical distribution, which propagates though the observer to induce a state-estimation with, also, a known statistical distribution. Then, the observer parameters are tuned to optimize the state-estimation distribution.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Cecilia, Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems, Springer Theses, https://doi.org/10.1007/978-3-031-38924-5_4

55

56

4 Adding Filters in Nonlinear Observers

This framework has been deeply studied in the discrete-time Kalman filter context and its modifications [4–6]. Nonetheless, it is not clear how to extend this approach to the continuous-time nonlinear observers explored in the past sections. Indeed, even if the measurement noise satisfies a simple Gaussian distribution, it is not clear which is the resulting distribution after the coordinate change Ψ −R , and how to optimize said distribution. Alternatively, the noise can be modelled by its frequencies components. In practice, the measurement noise has most of its spectral content at a different frequencies from the model dynamics. Then, by combining a minimization of the H∞ /L2 inputoutput gain and the bounded real lemma [7], it is possible to tune the observer gains to reduce the effect of disturbances at the noise frequencies, with minimal degradation on the observer performance at the other frequencies. Nonetheless, even if there is an original frequency separation between the noise and the system, the observer “mixes” these frequencies due to its nonlinear nature. Thus, the separation is no longer valid in the estimation-error signal. There are some remarkable attempts in including noise frequency information in the nonlinear observer performance analysis [8, 9], but, it is no clear how to develop a design methodology from these results. Finally, the most common noise modelling framework in the nonlinear observer context is to assume the noise as a bounded signal. Then, its effect on the estimationerror can be analysed by means of its ISS gain, see Sect. 3.4 of Chap. 3. Naturally, in general, an adequate observer parameter tuning will present a lower ISS gain than a bad parameter tuning. Therefore, it can be used to get an intuition of the noise effect on the estimation-error. However, the ISS gain is not a good metric to be minimized. Mainly, because it is a very conservative metric and strongly depends on the selected Lyapunov equation. Consequently, if the original observer stability and its redesign have been proved with different Lyapunov functions, the ISS gain cannot be compared. In summary, the stochastic and/or frequency framework have a clear theory for linear observers, but its main assumptions can hardly be extended to the nonlinear case. For this reason, most designers have focused on the very conservative ISS gain. This chapter explores an alternative route. The idea is to cascade the complex nonlinear observer with a simpler dynamical system which will be referred as “filter”, which will be specifically designed to reduce the amount of noise that is introduced in the observer. As the filter is conceptualized specifically to reject measurement noise, its parameter tuning is much simpler than the original nonlinear observer tuning. Although, some readers may find the use of filters as an “obvious” solution to the measurement noise conflict, next section will show that the theory of adding filters in nonlinear observers have not been properly developed in the literature.

4.2 On Adding Filters in Observers

57

4.2 On Adding Filters in Observers This section focuses on the idea of cascading a nonlinear observer with a filter. To ease the reading, this section will assume that the concerned system is single output and the implemented filter is linear and with relative degree one, i.e. its input appears in the first derivative of the output. Nonetheless, the conclusions of this section can be extended to any type of filter. Specifically, consider the set of filters that takes as an input the signal u ∈ R and generates as an output u f ∈ R. ν˙ = Fν + Hu z˙ = −qz + Bν + cu   ν =z uf = E z

(4.1)

where (ν, z) ∈ Rn ν × R is the state of the filter and F, H, B, E, resp. q, c, are matrices of suitable dimension, resp. constants, to be defined. Moreover, define the matrix G as   F 0 G= . B −q   It is assumed that the triplet (G, E, H c ) is controllable and observable. Remark 4.1 Although it may be convenient for most applications, notice that the considered filter is not required to be stable. A common misconception on the topic of implementing filters in observers is to take the noisy output as the input u = y and implement u f as the new measured output for the observer feedback term, i.e. ξ˙ = ϕ(ξ , u) + κ(ξ , u f − h(x)), ˆ xˆ = ψ(ξ ),

(4.2)

see for example [10]. Indeed, in the implementation of a filter as in (4.2), the filtered output u f is compared with the estimation of the unfiltered output h(ˆx). Consequently, in the pure observation context, such an approach necessarily introduces some delay in the state-estimation, which prevents the convergence of the estimation to the true value even in the case of no disturbances. To compensate the filter lag, it is therefore required to include the filter dynamics (4.1) in the observer. Specifically, consider the following copy of the filter ν˙ˆ = Fˆν + H yˆ + K1 (z − zˆ ) + K2 (ν − νˆ ) zˆ˙ = −q zˆ + Bˆν + c yˆ + k1 (z − zˆ ) + k2 (ν − νˆ )

(4.3)

58

4 Adding Filters in Nonlinear Observers

where νˆ and zˆ are an estimation of the filter states, ν and z, respectively, and the feedback gains K1 , K2 ∈ Rm×1 , k2 ∈∈ R1×n ν and k1 ∈ R are designed such that the matrix    K1 K2 G− k 1 k2 is Hurwitz. Then, the the observer is re-designed as ξ˙ = ϕ(ξ , u) + κ(ξ , z˜ ), xˆ = ψ(ξ ),

(4.4)

where z˜ = z − zˆ . The differences between both observers and filter architectures are depicted in the schemes (a) and (b) of Fig. 4.1. Note that the output filter (4.1) and the filter estimator (4.3) can be implemented in a single set of dynamics (so that to reduce the overall dimension) as ¯ ν + H y˜ − K1 z˜ ν˜˙ = F˜ ¯ ν + c y˜ , z˜˙ = −q¯ z˜ + B˜

(4.5)

where ν˜ = ν − νˆ and y˜ = h(x) − h(x). ˆ Moreover, F¯ = (F − K2 ) is Hurwitz, B¯ = B − k2 and q¯ = q − k2 are positive definite. Thus, the filter design and the feedback gain K1 , K2 , k1 , k2 tuning can be solved ¯ H, B¯ and the constants q¯ and c. simultaneously by directly tuning the matrices F, Then, the filtered observer is implemented as ξ˙ = ϕ(ξ , u) + κ(ξ , z˜ ), xˆ = ψ(ξ ).

(4.6)

See the scheme (c) in Fig. 4.1 for a comparison with the other filter architectures considered in this section. The ideas presented in this section have been summarized through the following example. Lets consider a nonlinear system in triangular form depicted by x˙1 = x2 x˙2 = −2x1 + 2x2 + x1 x22 y = x1 . Then, a high-gain observer can be designed as follows 2 (y − xˆ1 ) x˙ˆ1 = xˆ2 + 0.1 1 (y − xˆ1 ) x˙ˆ2 = −2 xˆ1 + 2 xˆ2 + xˆ1 xˆ22 + 0.12

4.2 On Adding Filters in Observers

59

Fig. 4.1 a Observer with filtered output. b Observer with filtered output and filter estimator. c Observer with filtered output error

Consider a low-pass filter of the form z˙ = 10(−z + u).

(4.7)

The aim of this example is to compare the performance of the proposed high-gain observer with two different implementations of the low-pass filter in (4.7). First, the filter is implemented as in (a) of Fig. 4.1, i.e. with u = y and the observer is implemented as in (4.2). Specifically, 2 (z − xˆ1 ) x˙ˆ1 = xˆ2 + 0.1 1 (z − xˆ1 ) x˙ˆ2 = −2 xˆ1 + 2 xˆ2 + xˆ1 xˆ22 + 0.12 Second, the filter is implemented as in (c) of Fig. 4.1, i.e. with u = y˜ and the observer is implemented as in (4.6). Specifically, 2 z x˙ˆ1 = xˆ2 + 0.1 1 x˙ˆ2 = −2 xˆ1 + 2 xˆ2 + xˆ1 xˆ22 + z 0.12 In Fig. 4.2, it is shown the estimation of both architectures. It can be seen that the first architecture cannot converge to the true value, and converges to a delayed version of the signal. On the other hand, the second architecture converges to the true value.

60

4 Adding Filters in Nonlinear Observers

Fig. 4.2 True value of the state x2 , estimation of the observer with filtered y and estimation of the observer with filtered y˜

The example validates the benefits of the proposed scheme. Specifically, the observer maintains convergence to the true value. Consequently, from the remainder of this work, it will always be assumed that the filter is implemented in the form (4.5), i.e. it filters the output-estimation error y˜ . Nonetheless, it should be stated that, if the output error, y˜ , is the filtered signal, the coupling between the filter and the observer not always creates a stable structure, even if the original observer was stable. For this reason, it is necessary to develop a stability theory of implementing filters in nonlinear observers. It should also be remarked that the idea of filtering the output error, y˜ , instead of the output signal, y, is not new. Indeed, an example of this methodology is the Proportional-Integral observer [11–13] which combines the output-estimation error of a high-gain observer with its integral. Moreover, some recent references have extended this approach to more general low-pass filters in the context of high-gain observers [14, 15]. The objective of this section is to extend these results in two directions. First, study more general types of filters. Second, provide the stability theory for a more general type of nonlinear observer. Precisely, the remainder of this chapter will explore three types of filters. First, it will present the stability conditions of a class of low-pass filter, i.e. a filter that reduces high-frequency measurement noise, implemented in a relatively general type of observer. Second, it will present a novel filter referred as “internal-model filter” that can exactly reject noise generated from a known exo-system. Finally, it will show that the proposed framework can be used to design some recently proposed non-linear filters [16].

4.3 Low-Pass Filters in Nonlinear Observer As explained at the beginning of the chapter, generally, there is a frequency separation between the noise and the system dynamics. Indeed, the noise spectral density is accumulated at higher frequencies in relation to the system. For this reason, it is common to reduce the effect of noise through a filter that has no effect on low frequencies while reduces the magnitude in the higher ones. Although it is standard to use low-pass filters as a noise reduction tool, the stability theory when it is imple-

4.3 Low-Pass Filters in Nonlinear Observer

61

mented in the y˜ signal of a nonlinear observer has not been developed. This section presents a result in this direction. Precisely, this section considers the observer framework presented in Sect. 3.4 of Chap. 3 and the filter (4.5). Intuitively, in order to guarantee the stability of the filtered observer, the filter dynamics (4.5) have to be “fast” enough with respect to the y˜ dynamics. To describe more precisely what is referred as “fast”, first, it is important to stablish an output growth condition, i.e. some bound on the possible increment of the output-estimation error. Precisely, it is assumed that the observer satisfies the following output growth condition  |h(x) − h(ˆx)| ≤ l0 V (x, ξˆ ),  d(h(x) − h(ˆx)) ≤ l1 V (x, ξˆ ) + ld |d|, dt

(4.8)

for some positive constants l0 , l1 , and ld , and d is the output disturbance. There are relevant classes of linear and nonlinear observers that satisfy this output growth condition. Some important examples are: the linear Luenberger observer [17], the Kalman filter [18] and its extended version [19], observers for Lipschitz systems [20], high-gain observers [21, 22] and its low-power formulation [9, 23] and observers based on the circle criterion [24, 25]. The reader may find a description on how to compute the constants α1 , γ1 , l0 , l1 and ld for some of these observers in [16]. For instance, the output growth condition (4.8) is easily verified when the output function, h, is linear and the Lyapunov function, V , is quadratic. By combining the observer ISS gain related to the measurement noise, See Sect. 3.4 of Chap. 3, and the output growth condition (4.8), it is possible to establish what is meant as a “fast” filter through a Lyapunov argument. Precisely, as the filter is assumed to be stable, it is possible to find a matrix P ∈ Rn ν ×n ν such that ¯ + PF¯  ≤ −In ν . FP

(4.9)

Then, the following stability theorem can be presented. Theorem 4.1 Consider an observer with a linear ISS gain, γ1 , with respect to output disturbances that satisfies the output growth assumption (4.8). Moreover, assume that the observer is filtered as in (4.4) with a filter of the form (4.5). Finally, consider that the filtered observer is disturbed by an additive output disturbance d¯ ∈ D ⊆ R, i.e. the filter-observer is implemented as ¯ − K1 z˜ ¯ ν + H(h(x) − h(ˆx) + d) ν˙˜ = F˜ ¯ ¯ ν + c(h(x) − h(ˆx) + d), z˙˜ = −q¯ z˜ + B˜ ξ˙ˆ = ϕ(ξˆ , u) + κ(ξˆ , z˜ ), xˆ = ψ(ξˆ ).

(4.10)

62

4 Adding Filters in Nonlinear Observers

Then, if

 K1 = H, 2PH = B¯

(4.11)

where P ∈ Rn ν ×n ν is defined in (4.9), there is a positive constant q¯ ∗ such that for all q¯ and c satisfying ¯ q¯ > q¯ ∗ , c = q, there exists a radially unbounded Lyapunov function V f (x, ξˆ , z˜ , ν˜ ) satisfying ¯ −R (x) − ξˆ |, z˜ , ν˜ ) β(|x − ψ(ξˆ )|, z˜ , ν˜ ) ≤ V f ≤ β(|ψ for all x, ξˆ ∈ X, Rn ξ and d¯ ∈ D ⊆ R, where β, β¯ ∈ K∞ , such that its derivative satisfies ¯2 (4.12) V˙ f ≤ −α2 V f + γ2 |d| for some positive constants α2 , γ2 . Moreover, the filter-observer satisfies the following output growth condition

|h(x) − h(ˆx)| ≤ l0,z V f , d(h(x) − h(ˆx))

≤ l1,z V f + ld,z |d|, ¯ dt

(4.13)

for some positive constants l0,z , l1,z and ld,z . Proof The first part of the proof consists in finding the conditions in which the inequality (4.12) is satisfied. Define the variable η ∈ R as: η = z˜ − y˜ . (4.14) As the original observer has a linear ISS gain, γ1 , with respect to output disturbances, it can be seen that the derivative of V (x, ξˆ ) of the filtered observer with output disturbance (4.10) satisfies V (x, ξˆ ) ≤ −α1 V (x, ξˆ ) + γ1 |η|2 for all x, ξˆ ∈ X, Rn ξ and u ∈ U.

(4.15)

1 Now, consider the Lyapunov function Vη = η2 . Furthermore, consider the output 2 growth assumption of the original observer (4.8). Then, the derivative of Vη satisfies:

4.3 Low-Pass Filters in Nonlinear Observer

63

¯ − η d(h(x) − h(ˆx)) ¯ 2 − ηq¯ y˜ + ηB˜ν + cη( y˜ + d) V˙η = −qη dt

2 ≤ −(q¯ − ld )|η| + (l0 |c − q| ¯ + l1 )|η| V (x, ξ ) ¯ + ηB˜ν + c|η||d|.

(4.16)

Define a second Lyapunov function Vν = ν˜  P˜ν , where P ∈ Rn ν ×n ν is defined in (4.9). By considering that K1 = H, then, the derivative of Vν satisfies:   ¯ ν + H( y˜ + d) ¯ − K1 z˜ ≤ −|˜ν |2 − 2ν˜  PHη + 2|˜ν ||PH||d|. ¯ (4.17) V˙ν = 2ν˜  P F˜ Now, consider the composite Lyapunov function V f = V (x, ξˆ ) + Vη + Vν . By means of (4.15), (4.16) and (4.17), it can be shown that the derivative of V f satisfies ¯ + 2|˜ν ||PH||d| ¯ − 2ν˜  PHη + ηB˜ν V˙ f ≤ −χ  Qχ + c|η||d| where

(4.18)

⎤ ⎡ ⎡

⎤ 1 V (x, ξ ) ¯ + l1 ) 0 α1 − (l0 |c − q| ⎥ ⎢ 2 ⎣ |η| ⎦ . Q=⎣∗ (q¯ − ld − γ1 ) 0⎦ , χ = |˜ν | ∗ ∗ 1

It can be seen that, by imposing the condition 2PH = B , the last two factors of the right hand-side in (4.18) exactly cancel. Consequently, a sufficient condition for the existence of a positive constants α2 and γ2 such that (4.12) holds is that the matrix Q is strictly positive definite. As the factor α1 is positive definite, by applying the Schur complement it can be shown that Q is positive definite if ¯ + l 1 )2 (l0 |c − q| > 0, q¯ − ld − γ1 − 4α1 which presents the solution q¯ > ld + γ1 + c = q. ¯ This ends the first part of the proof.

l12  q∗ 4α1

64

4 Adding Filters in Nonlinear Observers

The second part of the proof consists in proving that the inequalities in (4.13) hold true. To do so, consider the following inequality V (x, ξˆ ) ≤ V f .

(4.19)

Then, from the first inequality in (4.8) it is immediate to obtain that √

|h(x) − h(ˆx)| ≤ l0 V ≤ l0 V f .

(4.20)

Therefore, the first inequality in (4.13) is satisfied with l0,z = l0 . In order to prove the second inequality in (4.13), consider that from (4.12) it can be derived the following α2 2γ2 ¯ 2 |d| . V˙ f ≤ − V f , ∀ V f > 2 α2

(4.21)

From here, by using a comparison theorem, it can be determined that V f is ultimately bounded ∀t > ts where ts is a positive constant as Vf ≤

2γ2 ¯ 2 |d| α2

From this result the following is derived |η| =

 √

√

γ2 ¯ 2 Vη ≤ 2 V f ≤ 2 |d|, ∀t > ts . α2

From this result and the second inequality in (4.8) it can be shown the following 

d(h(x) − h(ˆx))

≤ l1 V (x, ξ ) + ld |η| ≤ l1 V f + 2ld γ2 |d| ¯ 2 , ∀t > ts dt α2 which proves the second inequality in (4.13). Roughly speaking, Theorem 4.1 establishes the conditions in which the filter and observer coupling creates a stable structure. In these conditions, if there is no disturbance in the measurement, d¯ = 0, the state-estimation error, x, ˜ and the filter states, ν˜ , z˜ , converge asymptotically to zero. Moreover, if there are some output disturbances, d¯ = 0, the observer re-design still satisfies an ISS condition and output growth condition similar to the original observer. This property will be used in the next subsection to design and implement more complex filters. Remark 4.2 This result can be generalized to the case in which 2PH − B  ≤ ,

4.3 Low-Pass Filters in Nonlinear Observer

65

where is a positive constant. Nonetheless, to simplify the notation of the proof, a more simpler case has been considered.

4.3.1 Cascaded Filters and Iterative Filter Design Notice that a limitation of the proposed low-pass filter framework is that the equations in (4.5) can only model filters with relative degree one between the measured output error, y˜ , and the filter output z˜ . Nonetheless, by exploiting the results in Theorem 4.1 it is possible to extend the results in order to design higher-order filters. Interestingly, the filtered observer (4.4) tuned as in Theorem 4.1 satisfies an output growth condition similar to the ones in (4.8). Therefore, this filter can be connected in cascade with a second filter with “faster” dynamics and still maintain the stability of the observer, see Fig. 4.3. Specifically, the filtered observer with cascaded filters can be implemented as follows: ν˙˜ 2 = F¯ 2 ν˜ 2 + H2 y˜ − H2 z˜ 2 z˜˙ 2 = −q¯2 z˜ 2 + B2 ν˜ 1 + c2 y˜ ν˜˙ 1 = F¯ 1 ν˜ 1 + H1 z˜ 2 − H1 z˜ 1

(4.22)

z˙˜ 1 = −q¯1 z˜ 1 + B1 ν˜ 1 + c1 z˜ 2 , ξ˙ˆ = ϕ(ξˆ , u) + κ(ξ , z˜ 1 ) xˆ = ψ(ξˆ ). where ν˜ i ∈ Rn ν,i . Furthermore, following similar arguments as in Theorem 4.1 proof, it is possible to show that the two filters in cascade still satisfy an output growth condition similar to (4.8). Thus, it is possible to cascade both filters with a third one with “faster dynamics”, and so on. The advantage of this process is that from iteratively concatenating relative degree 1 filters it is possible to obtain a filter with higher relative degree between the input y˜ and its output z˜ . It is well known that increasing the relative degree allows improving the filtering of high-frequency components, thus, this property may be desirable for some practical applications. See, e.g. [13].

Fig. 4.3 Scheme of two filters in cascade

66

4 Adding Filters in Nonlinear Observers

4.3.2 Filter Examples In the next paragraphs, the presented theory will be implemented in some common linear filters.

4.3.2.1

First-Order Low-Pass Filter

Consider that the implemented filter in (4.5) is a first order low-pass filter of the form z˙˜ = −q z˜ + c y˜

(4.23)

where q, c are some positive constants to be tuned. In the case of high-gain observers, this filter has been considered in [14] and in the work [13], where it was denoted as PI observer. However, to the best of author’s knowledge, the stability conditions have not been derived for a more general type of observer. For the low-pass filter, it can be seen that q¯ = q,

B¯ = 0,

F¯ = P = 0,

H = 0 c = c.

Therefore, the convergence of the filtered observer can be ensured if the following conditions q > l d + γ1 +

l12 4α1

c=q are verified.

4.3.2.2

Second-Order Filter

Consider a filter of the form ν˙˜ = − f 1 ν˜ + h 1 y˜ − h 1 z˜ z˜˙ = −q¯ z˜ + b1 ν˜ + c y˜ , ¯ c ∈ R>0 . where f 1 , h 1 , b1 , q, In this scenario, the matrix P from the inequality (4.9) can be derived as: P=

1 . 2 f1

4.4 On Internal-Model Filters

67

Now, we immediately get that the filtered observer satisfies the conditions in Theorem 4.1 if the following inequalities are satisfied: q¯ > ld + γ1 +

l12 4α1

c = q¯ h 1 = b1 f 1 , which leads into two degrees of freedom, q¯ and h 1 , in the filter design.

4.4 On Internal-Model Filters Notice that the low-pass filter framework can only achieve a reduction of the output disturbances effect. In some cases, it is desirable to have the stronger result of exactly cancelling the effect of the disturbance. Naturally, to achieve such cancellation, the observer cannot only rely on the feedback term, and requires to have the information of a known generating model of the output disturbance [26]. In such case, the generating model can be included in the observer in order to exactly cancel the disturbance. In the presence of a generating model, it is common to consider an extended system composed by the original system and the disturbance generating model and design an observer that jointly estimates the states of the original system and the internal states of the generating model [26]. However, this approach is not easily extendable to nonlinear systems, where the existence of an observer for the original system does not guarantee that the same observer technique can be used for the extended system. This section proposes using a filter route to implement the output disturbance generating model in the observer. First, it is assumed that there is an already existing observer for the system. Second, the observer is connected in cascade with a filter that includes the generating model of the disturbances. The design of the filter follows from an internal-model based philosophy [27] in order to exactly cancel the output disturbance. In the linear context, this section provides the sufficient conditions in which the filter observer coupling creates a stable structure and rejects the output disturbance. Moreover, through a numerical simulation, this section provides some insights on how the approach can be extended to the nonlinear case.

4.4.1 Problem Formulation This section considers a multi-input multi-output linear system of the form x˙ = Ax + Bu,

y = Cx + d

(4.24)

68

4 Adding Filters in Nonlinear Observers

where A ∈ Rn×n , B ∈ Rn×q and C ∈ Rm×n are known matrices and d ∈ Rm are additive output disturbances. The disturbance d is assumed to be generated through an exo-system of the form ˙ = Sw, w d = Γw (4.25) where w ∈ Rn w is an exogenous variable and the matrix S is assumed to be neutrally stable. Moreover, the pair (S, Γ ) is assumed to be observable, otherwise, some components of the exogenous variable vector, w, would have no effect on the signal d and could be discarded. Remark 4.3 Notice that the proposed generating model is not allowed to have stable nor unstable poles. If the matrix S has stable poles, it implies that some modes of the noise converge to zero and can be discarded. Alternatively, if there are unstable poles, the noise signal grow unbounded as time increases, which is not a reasonable setting. The only available information of the exogenous system in (4.25) is the minimal polynomial of the matrix S. The factor Γ and the initial conditions, w(0) are unknown. The identification of the minimal polynomial can be achieved by driving the system to a known constant state and recording the output of the measurement for a sufficiently long time. If the disturbance is assumed to be a periodic signal, a Fourier analysis can be used to estimate the main frequencies of the disturbance. If a more general linear model is assumed, sub-space identification methods can be utilised [28]. This section assumes that there is a previously designed linear observer of the form x˙ˆ = Aˆx + Bu + K˜y (4.26) where K ∈ Rn×m is the observer gain, see Sect. 3.2 Chap. 3 for more details. Supposing that K has been designed so that the matrix (A − KC) is Hurwitz, in the presence of the output disturbance, i.e. d = 0, the state of the observer asymptotically converges to a steady-state Πw which is fully characterized by w-dynamics (4.25), i.e. lim |˜x(t) − Πw(t)| = 0 t→∞

where Π is a matrix solution of the following Sylvester equation ΠS = (A − KC)Π + CΓ This can be shown, for instance, following [8]. The aim of this section is to redesign the observer in (4.26) in order to exactly cancel the output disturbance generated by the exo-system in (4.25) and obtain an exact estimation of the states, that is, lim t→∞ |˜x(t)| = 0. Naturally, under the proper assumptions, the problem can be solved by considering the extended system

4.4 On Internal-Model Filters

69

x˙ = Ax + Bu ˙ = Sw, w y = Cx + Γ w

(4.27)

and designing a linear observer to estimate the extended state composed by x and w. This is a perfectly valid approach for linear systems. However, it cannot be easily extended to the nonlinear system scenario. Contrary to the linear case, as shown in Chap. 3, designing an observer for a nonlinear system is a difficult task with no systematic solution. The design difficulty scales with the order of the system, thus, designing an observer for the extended system (as in (4.27)) in the nonlinear case can be an unfeasible task, even if it is possible to design an observer for the original system. Consequently, the aim is to propose an alternative method to reject the output disturbance generated by (4.25) that, can be extended to the nonlinear case. The following sections will show that the proposed problem can be solved by connecting the observer in cascade with what will be referred as internal-model filter.

4.4.2 Main Assumptions Following an internal-model based approach [27], it will be studied the idea of cascading the linear observer with an internal-model filter that presents a zero-blocking property in order to reject the disturbance d. Before presenting the structure of the filter and its properties, it is convenient to present a pair of assumptions that are necessary to solve the concerned problem. First, the concerned system (4.24) is assumed to satisfy a detectability condition. Assumption 4.1 The pair (A, C) is detectable. Second, in order to eliminate the disturbance, d, from the output signal without blocking relevant information of the states, it is necessary to assume that the extended system in (4.27) satisfies some observability condition. Based on the results in [29] and the fact that the pair (S, Γ ) is observable, it can be deduced that the extended system (4.27) is observable if ⎤ ⎡ S − sIn w 0 A − sIn ⎦ = n + n w rank ⎣ 0 Γ C

∀s ∈ σ (A).

This leads to a condition similar to the non-resonance condition of other internalmodel-based controller designs [27]. Assumption 4.2 The spectrum of the matrices A and S in (4.24) and (4.25) are disjoint, i.e. σ (A) ∩ σ (S) = ∅. (4.28)

70

4 Adding Filters in Nonlinear Observers

Assumption 4.2 states that the modes of the perturbations don’t coincide with those of the plant dynamics to be observed. In practice, this is a completely reasonable assumption, otherwise it couldn’t be possible to distinguish between the system trajectories and the measurement noise.

4.4.3 Internal-Model Filter Consider a filter that takes as an input the output-estimation error, y˜ , and generates a filtered output, z ∈ Rm , of the form η˙ = Fη + G˜y

(4.29)



z = y˜ − G η with F = [Im ⊗ (Φ − MM )], G = [Im ⊗ M], where η ∈ Rn w ×m is the filter state vector, the element Φ ∈ Rn w ×n w is a skewsymmetric matrix such that its minimal polynomial coincides with the one of S, and M ∈ Rn w ×m is a vector to be tuned such that the pair (Φ, M) is controllable. Note that, by construction, the matrix Φ − MM is Hurwitz. Then, the linear observer (4.26) is modified by using the filtered version z instead of y˜ in the output injection term as follows: x˙ˆ = Aˆx + Bu + Kz.

(4.30)

Notice that the filter in (4.29) can be understood as a bench of m identical filters applied at each output error equation, i.e. η˙ i = (Φ − MM )ηi + M y˜i ,

(4.31)



z i = y˜i − M ηi , for i = 1, . . . , m     where ηi ∈ Rm , y˜ = y˜1 . . . y˜m and z = z 1 , . . . z m . To better understand the motivation of the internal-model filter (4.29), notice that the filter is a relative degree zero linear system which has the zeros placed at the eigenvalues of S. To see this fact, consider the Rosenbrock matrix of the filter  P(s) =

 sIm×n w − F −G . −G Im

4.4 On Internal-Model Filters

71

Next, from the properties of the Kronecker product, the following equality is satisfied F = [Im ⊗ Φ] − [Im ⊗ M M ]. As the minimal polynomial of Φ and S coincide, the minimal polynomial of [Im ⊗ Φ] also coincides with S [30]. Therefore, as [Im ⊗ MM ] is not a full rank matrix, the Rosenbrock matrix, P(s), loses rank when s is in the spectrum of S. This shows that the zeros of the internal-model filter (4.29) coincide with the eigenvalues of the exo-system (4.25). The presence of a zero, μz , in the filter implies that the signal z does not have any spectral component at the frequency μz . As a consequence, the steady-state periodic solution of the filtered observer (4.30) cannot not have any spectral component at the frequency μz . Therefore, as the zeros of the filter are placed at the eigenvalues of S, the output disturbance, d, does not have any effect on the state-estimation, which solves the desired disturbance rejection problem. Notice that this property is independent of the type of observer that is connected to the filter. For this reason, the internal-model filter approach can be extended to the nonlinear case. Nonetheless, it is convenient to formalize the blocking properties of the filter for the linear case. To do so, first, define the following matrix  Acl =

 F GC . KG A − KC

(4.32)

Then, consider the following theorem. Theorem 4.2 Consider the filtered observer (4.29)–(4.30) and suppose that Assumptions 4.1 and 4.2 are satisfied. Then, if the matrix Acl is Hurwitz, the state-estimation error, x˜ , converges exponentially to zero. In other words, solutions to (4.24), (4.25), (4.29) and (4.30) satisfy lim |x(t) − xˆ (t)| = 0 t→∞

for any initial condition (x, w, z, xˆ ) ∈ Rn × Rn w × Rm × Rn . Proof Consider the filtered observer error dynamics in the presence of an output disturbance generated by (4.25)       η˙ η G = A + Γ w. cl x˜ −K x˙˜

(4.33)

If the matrix Acl is Hurwitz, the system (4.33) presents a stable eigenspace and a centre eigenspace which can be expressed as the solution, Σ, Π, of the following Sylvester equation [27]

72

4 Adding Filters in Nonlinear Observers

  ΣS = FΣ + G Γ + CΠ ,   ΠS = (A − KC)Π + K G Σ − Γ . Precisely, the system (4.33) converges asymptotically to a steady-state solution as follows lim |η(t) − Σw(t)| = 0. (4.34) lim |˜x(t) − Πw(t)| = 0, t→∞

t→∞

As the internal-model filter can be decomposed in a bank of filters applied at each output error Eq. (4.31), inspecting component wise the elements of Σ =   Σ 1 . . . Σ m , the following is obtained for i = 1, . . . , m Σ i S = ΦΣ i + M[−M Σ i + CΠ + Γ i ] where Γ i is the row of Γ associated with the output y˜i . Since the minimal polynomial of Φ and S coincide, and the pair (Φ, M) is controllable, the following equality has to be satisfied for i = 1, . . . , m (Sect. 1.3, [31]) − M Σ i + Γ i + CΠ = 0,

(4.35)

ΠS = AΠ.

(4.36)

which implies the following

Finally, the condition in (4.28) implies that the equality (4.36) has a unique solution Π = 0. Therefore, using (4.34) the proof concludes. Remark 4.4 Filters with a similar blocking property can be found in previous literature of control and signal processing. Some remarkable examples are the “notch filter” in the signal processing community [32] or the “washout filter” in the context of pre-processing output regulation [33]. It should be remarked that a crucial condition for the stability and the disturbance rejection property of the filter-observer structure is that the matrix Acl is Hurwitz. However, no insight has been provided on how to tune M and Φ, if there is any, so this condition is satisfied. To provide a sufficient stability condition, it is convenient to establish that the internal-model filter satisfies a certain passivity condition. This result is summarized through the following lemma. Lemma 4.1 The internal-model filter in (4.29) is passive from the output-estimation error, y˜ , to the filtered signal, z. Proof As the pair (Φ, M) is controllable and the pair (Φ, −M ) observable, according to the positive real lemma (Lemma 6.2, [34]), the system in (4.29) is passive from the output-estimation error, y˜ , to the filtered signal, z, if there exist a symmetric positive definite matrix Q ∈ Rn w m×n w m , matrices W ∈ Rn w m×n w m and L ∈ Rn w m×1 such

4.4 On Internal-Model Filters

73

that QF + F Q = −LL QG = −G − LW

(4.37)



W W = 2Im . As the matrix Φ√is skew-symmetric, the inequalities in (4.37) are satisfied with √  Q = In w , L = − 2G and W = 2Im . This passivity condition provides a clue related to the sufficient observer property to ensure that the matrix Acl is Hurwitz. In the following, we suppose that the observer satisfies a passivity condition from the measured output error, y˜ , to the state-estimation error, x˜ . Assumption 4.3 The triplet (A, K, C) is passive, i.e. there is a symmetric positive definite matrix P ∈ Rn×n such that PA + A P ≤ 0

(4.38)



PK = C . Remark 4.5 Notice that, since the pair (A, C) is detectable, the passivity condition in (4.38) is immediately satisfied when A is Hurwitz and the gain is designed as K = P−1 C . From the Lemma 4.1 and the Assumption 4.3, it can be seen that the filtered observer (4.30) can be depicted as the feedback connection of two passive systems. Therefore, following from standard results in passivity (Theorem 6.3 [34]) the filtered observer is asymptotically stable if the extended system (4.27) is zero-state observable. Remark 4.6 Notice that the non-resonance condition (4.28) implies zero-state observability of the extended system (4.27). This stability result can be formalized through the following theorem. Theorem 4.3 Consider the filtered observer (4.30) and the internal-model filter (4.29). Suppose that Assumptions 4.1–4.3 are satisfied. Then, the matrix Acl in (4.32) is Hurwitz. Proof Consider the radially unbounded Lyapunov function V = η η + x˜  P x˜ . where the matrix P is defined in (4.38).

74

4 Adding Filters in Nonlinear Observers

Taking into account (4.37), the derivative of the Lyapunov equation satisfies the following V˙ = −2η GG η + 2η GC˜x − 2˜x PKz = −2η G G η + 2η G(z + G η) − 2˜x C z = 2(η G − x˜  C )z = −2z z. Thus, the derivative of Lyapunov equation V is negative semi-definite. Now, consider the dynamics of x˜ and η, x˙˜ = A˜x − Kz η˙ = Fη + Gz

(4.39)



z = C˜x − G η. The non-resonance condition (4.28) implies that the set = {(˜x, η)| z = 0} does not contain any trajectory of the system (4.39) except the trivial solution x˜ = η = 0. Therefore, by means of the LaSalle’s invariance principle it can be concluded that the origin of (4.39) is asymptotically stable.  The remainder of the section validates the benefits of the approach in a set of numerical examples. Consider a linear system of the form (4.24) with the matrices ⎡

⎤ ⎡ ⎤ −1.13 0.49 −0.11 −0.38 0 0 ⎦ , B = ⎣ 0.59 ⎦ A = ⎣−1.0000 0 −1.0000 0 0.52     100 C= , x(0) = 1 1 1 , 011 where the measured output is corrupted with an additive disturbance generated by the neutrally stable exo-system (4.25) with ⎡

⎤ 0 0 0 0 0 ⎢0 0 31.416 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 ⎥ S = ⎢0 −31.416 0 ⎥, ⎣0 0 0 0 62.832⎦ 0 0 0 −62.832 0     11010 Γ = , w(0) = 0.5 0.2 0 0.2 0 . 11100

4.4 On Internal-Model Filters

75

Fig. 4.4 Evolution of the state-estimation error of the linear observer and estimation-error of the same observer with filtered output error

Assume that the states of the system are estimated through a linear observer of the form (4.26) with the gain ⎡

⎤ 1.18 0.48 K = ⎣ 0.7 1.56⎦ , −0.26 4.09   and initial conditions xˆ (0) = 0 0 0 . Naturally, the presence of the output disturbance prevents the convergence of the state-estimation error, x˜ , to zero. This fact can be seen in Fig. 4.4. The objective is to robustify the observer in front of the presented output disturbance. Note that the matrix A is Hurwitz. Furthermore, the pair (A, C) is detectable, and K satisfies Eq. (4.38). As a consequence, Assumptions 4.1–4.3 are satisfied. Moreover, the pair (A, S) satisfies the non-resonance condition of Assumption 4.2. Therefore, according to Theorem 4.2, if the output-estimation error, y˜ , is filtered through an internal-model filter (4.29) the state-estimation error converges asymptotically to zero. Precisely, Fig. 4.4 presents the state-estimation error of an observer filtered with an internal-model filter (4.29) with matrices Φ = S,

  M = 1 1 1 1 1 .

It can be seen, that the filter completely rejects the output disturbance and the stateestimation error converges to zero. As an example, consider a linear system of the form (4.24) with the matrices ⎡

⎤ ⎡ ⎤ 0 1 0 0 1 ⎦ , B = ⎣0 ⎦ A = ⎣0 0 1 −0.5 −2.5 0     C = 1 0 0 , x(0) = 1 1 1 ,

76

4 Adding Filters in Nonlinear Observers

Fig. 4.5 Evolution of the state-estimation error of the linear observer without the passivity assumption and estimation-error of the same observer with filtered output error

where the measured output is corrupted with a disturbance generated by the exosystem (4.25) with 

     0 10 S= , Γ = 1 0 , w(0) = 1 0 . −10 0 Assume that the states of the system are estimated through a linear observer of the form (4.26) with the gain ⎡ ⎤ 2 K = ⎣ 1 ⎦, 0.5   and initial conditions xˆ (0) = 0 0 0 . Again, the presence of the output disturbance prevents the convergence of the state-estimation error, x˜ , to zero. This fact can be seen in Fig. 4.5. Notice that, as the matrix A is not Hurwitz, the observer does not satisfy the passivity condition in Assumption 4.3. Nonetheless, consider an internal-model filter (4.29) with matrices Φ = S,

  M = 1 1 .

Even if the observer does not satisfy the passivity assumption, a simple computation can show that the matrix in (4.32) is Hurwitz. Therefore, according to Theorem 4.2, the inclusion of the internal-model filter ensures that the state-estimation error converges to zero. This result has been validated in a numerical simulation and the result is depicted in Fig. 4.5. This validates the fact that the passivity condition in Assumption 4.3 is just a sufficient but not necessary condition. Thus, the proposed technique is not restricted to passive observers. The aim of the following example is to explore the feasibility of the internalmodel filter in a case scenario consisting of a nonlinear system and an observer with a nonlinear feedback term. Specifically, consider the following system, which is the system in [35] with some slight modifications in the linear matrix A, x˙ = Ax + Gφ(Hx),

y = cx + d

(4.40)

4.4 On Internal-Model Filters

77

  where x = x1 , x2 , x3 , ⎡

−1 A=⎣ 0 0 ⎡ 0 G = ⎣−1 0

⎤ 1 0   1 0 ⎦, c = 1 0 0 , 1 −1 ⎤   0 010 0 ⎦, H = . 001 −1

and φ(·) : R2 → R2 is a multivariable nonlinearity φ(·) =

  1/3x23 + x2 x32 . 1/3x33 + x3 x22

where d ∈ D ⊆ R is a measurement disturbance to be defined. Interestingly, the multivariable term, φ, satisfies the following monotonic property ∂φ ∂φ  (ζ ) + (ζ ) ≤ Υ , ∀ζ ∈ R2 , ∂ζ ∂ζ

(4.41)

where Υ ∈ R2×2 is a symmetric positive definite matrix. This property, motivates the design of an observer for systems with multivariable monotonic nonlinearities [35, 36]. Precisely, following the methodology in [35], it can be shown that an observer of the form   ˙xˆ = Aˆx + L(y − cˆx) + Gφ Hˆx + K(cˆx − y) (4.42) where L ∈ R3×1 and K ∈ R2×1 are observer gains designed as ⎡

⎤   3.2 1.92 ⎦ ⎣ 11.72 L= , K= , 0.61 3.16

(4.43)

achieves an exact estimation of the states in the absence of the output disturbance, d = 0. The aim of this sub-section is to robustify the observer in front of a high-frequency output periodic disturbance of the form (4.25), with the matrices ⎡

⎤ 0 31.416 0 0 ⎢−31.416 0 0 0 ⎥ ⎥, S=⎢ ⎣ 0 0 0 62.832⎦ 0 0 −62.832 0     Γ = 1 0 1 0 , w(0) = 1 0 1 0 .

(4.44)

78

4 Adding Filters in Nonlinear Observers

Fig. 4.6 Evolution of the state-estimation error of the observer (4.42) and estimation-error of the same observer with filtered output error

In the presence of such a disturbance, the observer estimation not only converges to an oscillatory trajectory, but presents a significant bias with respect to the true value, see Fig. 4.6. This behaviour is a consequence of introducing the disturbance through the nonlinear feedback term (4.42). Moreover, due to the nonlinearity of the feedback term, it is specially difficult to optimize the gains of the observer in order to reduce the effect of the measurement disturbance. Alternatively, Fig. 4.6 presents the estimation of the same nonlinear observer (4.42) connected in cascade with the internal-model filter (4.29) with Φ = S,

  M = 1 0 1 0 .

It can be seen, that the internal-model filter rejects the output disturbance and the state-estimation error converges to zero. This result provides a strong case in favour of the approach viability in the nonlinear scenario. Nonetheless, the validity of the result has to be taken with care. This work has provided the stability and disturbance rejection conditions of the internalmodel-filtered observer in the linear context, and these conditions have not been extended to the nonlinear case. Indeed, it is still required to extend the passivity argument in Theorem 4.3 proof to the nonlinear observer case, and extend the nonresonance condition (4.28) to nonlinear systems. These points will be addressed in future works.

4.4.4 Combining Both Filters: Low-Pass Internal-Model Filter The past sections have proposed two frameworks of filter design in observers. In the first one, it has been assumed that the spectral density of the noise is placed at higher frequencies than the system ones. Consequently, the effect of the noise is reduced by adding a low-pass filter in the observer. In the second one, the measurement noise is assumed to be generated by a linear neutrally stable generating model with known minimal polynomial. Then, the measurement noise is exactly cancelled by means

4.4 On Internal-Model Filters

79

of an internal-model filter. In practical applications, it is more reasonable to assume a combination of both scenarios, i.e. that some part of the measurement noise is modelled and the other is high-frequency unmodelled noise. For example, this will be the case for periodic disturbances with known period. In such case, the periodic disturbance contains infinite harmonics in its Fourier expansion series, and a finite compensator can only cancel the finite number of harmonics contained in it, leaving the unmodelled harmonics as high-frequency noise. For this reason, it is convenient to derive a filter that combines both methodologies. To simplify the notation, this section will consider the single-output case (m = 1). Nonetheless, by means of the decomposition in (4.31), it is relatively easy to extend these results to the multi-output case. Notice that the internal-model filter (4.29) is relative degree zero with respect to the measurement noise. Consequently, any unmodelled disturbance will appear in the observer feedback term with minimal reduction by the filter. A way of implementing some low-pass property on the filter is to increase its relative degree. Specifically, consider the following filter η˙ = Fη + G y˜

(4.45) 

z˙ = −q(z + G η − y˜ ) where F and G are defined as in (4.29) and q is a positive constant to be tuned. Moreover, assume that the observer is implemented as ξ˙ˆ = ϕ(ξˆ , u) + κ(ξˆ , z), xˆ = ψ(ξˆ ). Then, the following theorem can be established. Theorem 4.4 Assume that the internal-model filter (4.29), if implemented in an observer, creates a stable structure that rejects the measurement noise harmonics modelled in Φ for some matrices G and F. Then, there exists a positive constant q ∗ such that for all q > q ∗ , the same observer filtered through a low-pass internalmodel filter (4.45) with the same matrices F and G also creates a stable structure that rejects the measurement noise harmonics modelled in Φ. Proof Define the following new coordinate ζ = z − y˜ + G η. Then, the observer filtered though (4.45) can be rewritten in the following singularly perturbed form

80

4 Adding Filters in Nonlinear Observers

η˙ = Fη + G y˜ 1 ζ˙ = −ζ, q ξ˙ˆ = ϕ(ξˆ , u) + κ(ξˆ , ζ + y˜ − G η),

(4.46)

xˆ = ψ(ξˆ ). As is usual in singular perturbation analysis [37], the system will be separated in the reduced (slow) subsystem and boundary-layer (fast) sub-system. Define the fast-time variable as t f = qt. Then the singular perturbed system in (4.46) can be rewritten in the t f time-scale as dη 1 = (Fη + G y˜ ) dt f q dζ = −ζ, dt f d ξˆ 1 = [ϕ(ξˆ , u) + κ(ξˆ , ζ + y˜ − G η)], dt f q xˆ = ψ(ξˆ ). By setting q = ∞, it can be seen that the boundary-layer system in the fast time scale, t f , is defined for all T f ≥ 0 by the following dynamics dη dζ dξ = 0, = −ζ, = 0, xˆ = ψ(ξ ). dt f dt f dt f which leads to the global asymptotic equilibrium point ζ = 0. Consequently the following reduced system arises in the slow time-scale η˙ = Fη + G y˜ ξ˙ˆ = ϕ(ξˆ , u) + κ(ξˆ , y˜ − G η),

(4.47)

xˆ = ψ(ξˆ ). Notice that the reduced system is the observer filtered through an internal-model filter (4.29). Thus, following from standard results in singular perturbation analysis [37]. There is a positive value q ∗ such that for all q > q ∗ , the stability of the reduced system (4.47) implies the stability of the original system. Finally, it can be shown (see for example [38]) that the zeros of the low-pass internal-model filter are placed at the eigenvalues of Φ. Therefore, the filter preserves the blocking properties of the internal-model filter (4.29). 

4.4 On Internal-Model Filters

81

Fig. 4.7 Evolution of the state x1 and the measured output disturbed by a constant bias, high-frequency noise and outliers

The benefits of the low-pass internal-model filter are validated through a numerical example. Consider an autonomous Van der Pol oscillator, the dynamics of which are depicted by the following expression: x˙ = Ax + Bφ(x) y = x1 where x = [x1 , x2 ] and       0 1 0 A= , B = , φ(x) = x1 , x2 , x12 x2 . 0 0 1 As the system presents a triangular structure and the states evolve in a compact set, the states can be estimated through a high-gain observer, see Sect. 3.2 of Chap. 3. Now it is considered the scenario in which the output function is disturbed by a constant bias, unmodeled high-frequency noise and outliers. Specifically, the bias takes a constant value of 2, the high-frequency noise is generated by a sinusoidal of the form sin(100t) and every 5 s there is a periodic outlier, see Fig. 4.7. The value of the bias, the frequency of the noise and the period of the outlier is assumed to be unknown by the designer of the observer. The presence of these disturbances prevents the convergence of the estimation to the true value. This is more significant in the second state where the estimation is practically useless, see Fig. 4.8. The aim of this example is to redesign the observer in order to reduce the effect of the disturbances on the state-estimation error. It will be assumed that the designer has only modelled the frequency of the bias, the rest is interpreted as high-frequency measurement noise. In the context of high-gain observers, multiple solutions have been proposed for the reduction of measurement noise [39–41], outliers [42], or both simultaneously [16]. Nonetheless, most of these solutions require a significant modification of the observer structure and/or cannot handle measurement biases. This subsection will show that all the components can be handled simultaneously through the filter framework presented in this work.

82

4 Adding Filters in Nonlinear Observers

Fig. 4.8 Estimation error of the second state, x˜2 , for the high-gain observer and error of the same observer with filter

Precisely, one can make two observations. First, the dynamics of the system persistently oscillates with frequencies sufficiently far from zero. Second, a constant bias can be modelled as in (4.25) with S = 0 and Γ = 1. Consequently, implementing a filter that blocks the zero-frequency does not eliminate any relevant information of the system dynamics and the low-pass internal-model filter in (4.45) with the following parameters Φ = 0, G = 1, exactly cancels the output bias and ensures the convergence of the estimation-error to zero for a sufficiently large q. An example of such parameter can be q = 1. Now, it is important to analyse if such filter actually reduces the effect of the high-frequency noise and measurement outliers. Comparing the performance of an observer with a redesigned version of itself, is usually a complex task. Nonetheless, as the filter is linear and connected in cascade with the observer, the noise reduction can be directly quantified by analyzing the magnitude of the filter at the frequencies of the noise. Precisely, the filter in (4.45) with q = 1, is a band-pass filter with magnitude −40 dB at 100 rad/s (the frequency of the sinusoidal component) and lower magnitude at higher frequencies (frequencies related to the outliers). Consequently, at least a 99% reduction of the error induced by the sinusoidal term and the outliers should be expected. This is coherent with the results obtained by simulation, see Fig. 4.8. This redesign did not require any modification of the original observer and only required the tuning of two parameters, q and G, from a linear filter.

4.5 Dynamic Dead-Zone Filter The past sections have provided a framework to implement several types of filters in nonlinear observers. The benefits of the filters have been validated through some numerical simulations. Although the improvement is significant, it is noticeable that only linear filters have been considered. It is well known that linear control systems and filters present fundamental performance limitations [43]. The idea of overcoming these limitations has motivated the implementation of many nonlinearities on

4.5 Dynamic Dead-Zone Filter

83

Fig. 4.9 Dead-zone function with σ = 4, σ = 9 and σ = 16

the control loop, some recent examples are the use of reset systems [44] or hybrid integrators [45, 46]. Following from these positive results, the reader may be considering if there is similar research in the use of nonlinearities for the design of filters. Fortunately, the answer is affirmative and has motivated the development of filters based on dynamic dead-zones [16, 47]. The objective of this section is to briefly present the results that have been obtained on the topic. The key observation to make is that the effect of the noise is minimal during the transient of the observer, i.e. while the state-estimation error is large. The significant effects appear during the steady-state, as the output-estimation error is low in relation to the values of the measurement noise. This fact has motivated observer tunings in which the parameters present different values during the transient and the steady-state. For example, in relation to the high-gain observer, see Sect. 3.2 of Chap. 3, some authors have proposed using a low ε during the transient to ensure fast convergence and robustness and a large ε during the steady-state to reduce the effect of measurement noise [40, 48]. This approach is simple in high-gain observers as there is a clear relationship between the parameter ε and the effect of the noise. The idea would be to extend this approach to more general types of observers, where this relation is not clear. The solution proposed in [16, 47] is to substitute the output error term y˜ of the observer by a “deadzonated” version, dz √σ ( y˜ ), where dz √σ (·) is the dead-zone function computed as (4.48) dz √σ (a) = a − sat√σ (a), √ where the factor σ is the dead-zone amplitude. See Fig. 4.9 for an example of different dead-zone functions with different amplitudes. The motivation behind this modification is that the dead-zone function is capable of “trimming” part of the persistent measurement noise. Specifically, when the output-estimation error is small, i.e. during the steady state, the dead-zone function

84

4 Adding Filters in Nonlinear Observers

eliminates the feedback term of the observer. Thus, the noise does not enter in the observer and has no effect in the state-estimation. Remark 4.7 Similar to the internal-model filter (4.31). If the considered system is multiple output, it is possible to use a different dead-zone function in each outputestimation error signal. Naturally, while the dead-zone function is zero, the observer runs in open-loop, which may have a destabilizing effect on the estimation-error dynamics. For this reason, the amplitude, σ , of the dead-zone has to be adequately tuned to the current trajectory of the observer and the amount of noise. Unfortunately, a single amplitude, σ , may be inadequate for the whole possible trajectory space. For this reason, what is proposed in [16, 47] is to dynamically adapt the amplitude to the output-estimation error. Specifically, the signal σ is computed as σ˙ = −qσ + p| y˜ |, where q and p are positive constants to be tuned. Notice that, for a constant outputp estimation error, the amplitude converges to σ = | y˜ |. q Simply, during the steady-state, if the output-estimation error, y˜ , is large, the deadzone amplitude, σ , will increase and the filter will eliminate more measurement noise. Otherwise, it will reduce the amplitude to avoid compromising the stability of the observer. The only remaining step is to properly tune the parameters q and p of the deadp increments the amplitude of the zone dynamics. On the one hand, a large ratio q dead-zone which increases the noise reduction capabilities of the filter. On the other, a large ratio may induce the observer to operate too much in open-loop, which may tamper the stability of the filtered observer. Fortunately, by exploiting the growth condition (4.8) presented in the low-pass filter design framework in Sect. 4.3, this stability result can be summarized through the following theorem. Theorem 4.5 Consider an observer with a linear ISS gain, γ1 , with respect to output disturbances that satisfies the output growth assumption (4.8). Moreover, assume that the observer is filtered with a dynamic dead-zone filter. Finally, consider that the filtered observer is disturbed by an additive output disturbance d¯ ∈ D ⊆ R, i.e. the filter-observer is implemented as ¯ σ˙ = −qσ + p| y˜ + d|, ¯ ξ˙ˆ = ϕ(ξˆ , u) + κ(ξˆ , dz √σ ( y˜ + d)), xˆ = ψ(ξˆ ).

(4.49)

4.5 Dynamic Dead-Zone Filter

85

Then, for any fixed p > 0, there exists a positive constant q ∗ such that for all q > q ∗ there is a radially unbounded Lyapunov function V f (x, ξˆ , σ ) satisfying ¯ −R (x) − ξˆ |, σ ) β(|x − ψ(ξˆ )|, σ ) ≤ V f ≤ β(|ψ for all x, ξ ∈ X, Ξ and d¯ ∈ D ⊆ R, where β, β¯ ∈ K∞ , such that its derivative satisfies ¯2 V˙ f ≤ −α2 V f + γ2 |d| for some positive constants α2 , γ2 . Moreover, the filter-observer satisfies the following output growth condition

|h(x) − h(ˆx)| ≤ l0,z V f , d(h(x) − h(ˆx))

≤ l1,z V f + ld,z |d|, ¯ dt for some positive constants l0,z , l1,z and ld,z . Proof The proof can be found in Theorem 2 of [16].



Similar to the low-pass filter case in Sect. 4.3, Theorem 4.5 establishes the conditions in which the filter and observer coupling creates a stable structure. In these conditions, if there is no disturbance in the measurement, d¯ = 0, the state-estimation error, x, ˜ and the dynamic dead-zone filter amplitude, σ , converge asymptotically to zero. Moreover, if there are some output disturbances, d¯ = 0, the observer re-design still satisfies an ISS condition and output growth condition similar to the original observer. Therefore, following the ideas presented in Sect. 4.3.1, it is possible to cascade the dynamic dead-zone filter with additional dead-zone filters or low-pass filters to achieve more noise reduction. This example explores the possibility of combining the dynamic dead-zone filter with the low-power peaking-free observer of [9] for systems in phase-variable form presented in Sect. 3.2.3 of Chap. 3. The idea is to filter each estimation-error signal, ei , of the observer with a dynamic dead-zone filter. Specifically, the observer takes the following form (see Fig. 4.10) αi x˙ˆi = ηi + dz √σi (ei ), i = 1, . . . , n − 1 ε αn xˆ˙n = φ(ˆx, 0) + dz √σn (en ) ε βi η˙ i = satri+2 (ηi+1 ) + 2 dz √σi (ei ), i = 1, . . . , n − 2 ε βn−1 η˙ n−1 = φ(ˆx, 0) + 2 dz √σn−1 (en−1 ) ε qi σ˙ i = − 2 σi + pi ε|ei |, i = 1, . . . , n ε

(4.50)

(4.51)

86

4 Adding Filters in Nonlinear Observers

Fig. 4.10 Scheme of the low-power peaking-free observer with dynamic dead-zone modification

with e1  y − xˆ1 , ei  satri (ηi−1 ) − xˆi , i = 2, . . . , n √ where dz √σi (·) is the dynamic dead-zone defined in (4.48), σ is the amplitude of the dead-zone which is adapted through (4.51), qi and pi are some positive design parameters. The rest of parameters are introduced in Sect. 3.2.3 of Chap. 3. The proposed observer is implemented in the numerical example studied in [9]. Specifically, the performance of the proposed filtered observer will be compared with a standard high-gain observer, see Sect. 3.2 of Chap. 3 for the design of the standard high-gain observer. Consider a nonlinear system be depicted by x˙i = xi+1 , i = 1, . . . , 4 x˙4 = 0.2(x12 − 1) − x2 − x3 − 4x4 − x5 ,

(4.52)

y = x1 + v where v is some zero-mean bounded unknown high-frequency noise with a small variance of 6 · 10−7 . Assume that a control application requires a high-gain observer with a settling time (98%) of approximately 3 s, which, for the concerned systems, is an unusually fast requirement. After some calculations and numerical simulations, it can be shown that the appropriate high-gain observer can be designed with α1 = 7.5, α2 = 22.2188, α3 = 32.3438, α4 = 23.188, α5 = 6.5625 and ε = 0.1429. Although the observer achieves the desired settling time, the peaking phenomena and the noise sensibility makes the observer practically unusable. This fact is magnified in the state x5 , see Fig. 4.11. On the one hand, the observer peaks to a relative error1 of 1.2251 · 108 % during the transient. On the other hand, the observer converges to a relative error of 4104.8% during the steady-state. Therefore, the observer’s estimation is hardly useful for any application. Now, the same problem can be solved by using a low-power peaking-free observer with dynamic dead-zone modification. After some computations, it can be shown that 1

The relative error [%] between x and xˆ is computed as

x − x ˆ · 100. x

4.5 Dynamic Dead-Zone Filter

87

Fig. 4.11 True system’s x5 evolution (blue) and high-gain observer (HGO) estimation (orange). The true system’s value evolves between −0.02 and 0.02

Fig. 4.12 True system’s x5 evolution (blue) and low-power peaking-free observer with dead-zone modification estimation (orange)

the design parameters αi = 3 for i = 1, . . . , 4, α5 = 2, β1 = 8.5714, β2 = 3.2122, β3 = 1.4267 and β4 = 0.5347, ε = 0.1429, pi = 3, qi = 3 and pi = 100 for i = 1, . . . , 5, achieve the desired convergence rate. By comparing the evolution of xˆ5 , it is clear that the new observer significantly reduces the peaking phenomena and the noise sensibility, compare Fig. 4.12 with Fig. 4.11 and its scales. First, the relative error reaches a maximum of 5.3251E05% which is a reduction of 99.565% with respect to the peaking phenomena of the classic high-gain observer. Second, the new observer converges to a relative error of 75.563% during the steady-state, which implies a reduction of 98.16%.

88

4 Adding Filters in Nonlinear Observers

References 1. Goodwin GC (2018) A critique of observers used in the context of feedback control. In: Chen Z, Mendes A, Yan Y, Chen S (eds) Intelligent robotics and applications. Springer International Publishing, Cham, pp 1–24 2. Antonio Loria A (2013) Observer-less output feedback global tracking control of lossless Lagrangian systems. arXiv:1307.4659 3. Loria A, Nuno E, Panteley E (2021) Observerless output-feedback consensus-based formation control of 2nd-order nonholonomic systems. IEEE Trans Autom Control, 1–1. https://doi.org/ 10.1109/TAC.2021.3136140 4. Reif K, Gunther S, Yaz E, Unbehauen R (1999) Stochastic stability of the discrete-time extended Kalman filter. IEEE Trans Autom Control 44(4):714–728. https://doi.org/10.1109/9.754809 5. Julier SJ, Uhlmann JK (1997) New extension of the Kalman filter to nonlinear systems. In: Signal processing, sensor fusion, and target recognition VI, vol 3068. International Society for Optics and Photonics, pp 182–193 6. Arasaratnam Ienkaran, Haykin Simon (2009) Cubature Kalman filters. IEEE Trans Autom Control 54(6):1254–1269. https://doi.org/10.1109/TAC.2009.2019800 7. Dufour F, Elliott RJ (2014) State observation and filtering. In: LMIs in control systems, pp 285–317 8. Astolfi D, Marconi L, Praly L, Teel A (2016) Sensitivity to high-frequency measurement noise of nonlinear high-gain observers. IFAC-PapersOnLine 49(18):862–866. ISSN 24058963. https://doi.org/10.1016/j.ifacol.2016.10.274. 10th IFAC symposium on nonlinear control systems NOLCOS 2016 9. Astolfi D, Marconi L, Praly L, Teel AR (2018) Low-power peaking-free high-gain observers. Automatica 98:69–179 (2018). ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2018. 09.009 10. Khalil HK, Priess S (2016) Analysis of the use of low-pass filters with high-gain observers. IFAC-PapersOnLine 49(18):488–492 (2016). ISSN 2405-8963. https://doi.org/10.1016/j. ifacol.2016.10.212. 10th IFAC symposium on nonlinear control systems NOLCOS 2016 11. Beale SR, Shafai B (1988) Robust control system design with proportional integral observer. In: Proceedings of the 27th IEEE conference on decision and control, vol 1, pp 554–557. https:// doi.org/10.1109/CDC.1988.194372 12. Shafai B, Beale S, Niemann HH, Stoustrup JL (1996) LTR design of discrete-time proportionalintegral observers. IEEE Trans Autom Control 41(7):1056–1062. https://doi.org/10.1109/9. 508916 13. Busawon Krishna K, Kabore Pousga (2001) Disturbance attenuation using proportional integral observers. Int J Control 74(6):618–627. https://doi.org/10.1080/00207170010025249 14. Astolfi D, Zaccarian L, Jungers M (2021) On the use of low-pass filters in high-gain observers. Syst Control Lett 148:104856. ISSN 0167-6911. https://doi.org/10.1016/j.sysconle. 2020.104856 15. Tréangle C, Farza M, M’Saad M (2019) Filtered high gain observer for a class of uncertain nonlinear systems with sampled outputs. Automatica 101:197–206. ISSN 0005-1098. https:// doi.org/10.1016/j.automatica.2018.12.002 16. Astolfi Daniele, Alessandri Angelo, Zaccarian Luca (2021) Stubborn and dead-zone redesign for nonlinear observers and filters. IEEE Trans Autom Control 66(2):667–682. https://doi.org/ 10.1109/TAC.2020.2989816 17. Luenberger D (1971) An introduction to observers. IEEE Trans Autom Control 16(6):596–602. https://doi.org/10.1109/TAC.1971.1099826 18. Kalman RE, Bucy RS (1961) New results in linear filtering and prediction theory. J Basic Eng 82(1):34–45 19. Krener AJ (2003) The convergence of the extended Kalman filter. In: Directions in mathematical systems theory and optimization. Springer, Berlin, pp 173–182 20. Rajamani R (1998) Observers for Lipschitz nonlinear systems. IEEE Trans Autom Control 43(3):397–401. https://doi.org/10.1109/9.661604

References

89

21. Khalil Hassan K, Praly Laurent (2014) High-gain observers in nonlinear feedback control. Int J Robust Nonlinear Control 24(6):993–1015. https://doi.org/10.1002/rnc.3051 22. Zemouche Ali, Zhang Fan, Mazenc Frédéric, Rajamani Rajesh (2019) High-gain nonlinear observer with lower tuning parameter. IEEE Trans Autom Control 64(8):3194–3209. https:// doi.org/10.1109/TAC.2018.2882417 23. Astolfi Daniele, Marconi Lorenzo (2015) A high-gain nonlinear observer with limited gain power. IEEE Trans Autom Control 60(11):3059–3064. https://doi.org/10.1109/TAC.2015. 2408554 24. Arcak M, Kokotovic P (2001) Nonlinear observers: a circle criterion design and robustness analysis. Automatica 37(12):1923–1930. ISSN 0005-1098. https://doi.org/10.1016/S00051098(01)00160-1 25. Zemouche A, Rajamani R, Phanomchoeng G, Boulkroune B, Rafaralahy H, Zasadzinski M (2017) Circle criterion-based H∞ observer design for lipschitz and monotonic nonlinear systems - enhanced LMI conditions and constructive discussions. Automatica 85:412–425. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2017.07.067 26. Goodwin GC, Seron MM (2020) Fundamental performance properties of a general class of observers for linear systems having predictable disturbances. Automatica 113:108717. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2019.108717 27. Francis BA, Wonham WM (1976) The internal model principle of control theory. Automatica 12(5):457–465 28. Oymak Samet, Ozay Necmiye (2022) Revisiting ho—Kalman-based system identification: robustness and finite-sample analysis. IEEE Trans Autom Control 67(4):1914–1928. https:// doi.org/10.1109/TAC.2021.3083651 29. Davison E, Wang S (1975) New results on the controllability and observability of general composite systems. IEEE Trans Autom Control 20(1):123–128. https://doi.org/10.1109/TAC. 1975.1100857 30. Isidori A (2017) Regulation and tracking in linear systems. In: Lectures in feedback design for multivariable systems. Springer, Berlin, pp 83–133 31. Byrnes CI, Delli Priscoli F, Byrnes CI, Isidori A (2012) Output regulation of uncertain nonlinear systems. Springer Science & Business Media 32. Marino Riccardo, Tomei Patrizio (2016) Adaptive notch filters are local adaptive observers. Int J Adapt Control Signal Process 30(1):128–146 33. Wang Lei, Marconi Lorenzo, Wen Changyun, Hongye Su (2020) Pre-processing nonlinear output regulation with non-vanishing measurements. Automatica 111:108616 34. Khalil HK, Grizzle JW (2002) Nonlinear systems, vol 3. Prentice hall Upper Saddle River, NJ 35. Fan X, Arcak M (2003) Observer design for systems with multivariable monotone nonlinearities. Syst Control Lett 50(4):319–330. ISSN 0167-6911. https://doi.org/10.1016/S01676911(03)00170-1 36. Andrieu V, Tarbouriech S (2019) LMI conditions for contraction and synchronization. IFACPapersOnLine 52(16):616–621 37. Kokotovi´c P, Khalil HK, O’reilly J (1999) Singular perturbation methods in control: analysis and design. SIAM 38. Isidori A (2017) Stabilization of minimum-phase linear systems. In: Lectures in feedback design for multivariable systems. Springer, Berlin, pp 21–42 39. Sanfelice RG, Praly L (2011) On the performance of high-gain observers with gain adaptation under measurement noise. Automatica 47(10):2165–2176. ISSN 0005-1098. https://doi.org/ 10.1016/j.automatica.2011.08.002 40. Ahrens JH, Khalil HK (2009) High-gain observers in the presence of measurement noise: a switched-gain approach. Automatica 45(4):936–943. ISSN 0005-1098. https://doi.org/10. 1016/j.automatica.2008.11.012 41. Boizot N, Busvelle E, Gauthier JP (2010) An adaptive high-gain observer for nonlinear systems. Automatica 46(9):1483–1488. ISSN 0005-1098. https://doi.org/10.1016/j.automatica. 2010.06.004

90

4 Adding Filters in Nonlinear Observers

42. Alessandri A, Zaccarian L (2018) Stubborn state observers for linear time-invariant systems. Automatica 88:1–9. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2017.10.022 43. Seron MM, Braslavsky JH, Goodwin GC (2012) Fundamental limitations in filtering and control. Springer Science & Business Media 44. Baños A, Barreiro A (2012) Reset control systems. Springer, Berlin 45. van den Eijnden SJAM, Heertjes MF, Heemels WPMH, Nijmeijer H (2020) Hybrid integratorgain systems: a remedy for overshoot limitations in linear control? IEEE Control Syst Lett 4(4):1042–1047. https://doi.org/10.1109/LCSYS.2020.2998946 46. Deenen DA, Sharif B, van den Eijnden S, Nijmeijer H, Heemels M, Heertjes M (2021) Projection-based integrators for improved motion control: formalization, well-posedness and stability of hybrid integrator-gain systems. Automatica 133:109830. ISSN 0005-1098. https:// doi.org/10.1016/j.automatica.2021.109830 47. Cocetti M, Tarbouriech S, Zaccarian L (2019) High-gain dead-zone observers for linear and nonlinear plants. IEEE Control Syst Lett 3(2):356–361. https://doi.org/10.1109/LCSYS.2018. 2880931 48. Prasov AA, Khalil HK (2013) A nonlinear high-gain observer for systems with measurement noise in a feedback control framework. IEEE Trans Autom Control 58(3):569–580. https://doi. org/10.1109/TAC.2012.2218063

Chapter 5

Adaptive Observers: Direct and Indirect Redesign

Observers are model-based estimation algorithms. Consequently, any model uncertainty affects directly to the accuracy of the estimation. This chapter explores the concept of adaptive redesign to reduce the effect of parametric uncertainty. Some limitations of classic adaptive observers are detected and novel solutions are proposed.

5.1 Adaptive Observers and the Conflict of Model Uncertainty The end of Chap. 3 has shown that the presence of model uncertainty has a direct effect on the accuracy of the observer. Indeed, observers can be classified as model-based estimation algorithms, which contrasts the information of the measured trajectories with a mathematical model of the system to generate an estimation. Therefore, any discrepancy between the reality and the model will have a direct effect on the accuracy of the estimation. Furthermore, as studied in Chap. 3, observer design for nonlinear systems is an intricate task which may be infeasible for highly complex systems. To make the process somewhat feasible, it is necessary to develop a low-order model of the target system, which will necessarily introduce discrepancies between the model and the reality. For this reason, it is crucial to account for model uncertainty during the observer design process. An advantage of observers in front of other estimation algorithms is its closedloop nature. Indeed, the existence of uncertainty, will prevent the convergence of the state-estimation to its true value, which, in a significant amount of cases, also prevents the convergence of the output-estimation error. Therefore, the output error can be used as an indicator of the model quality and can be exploited as a source to improve the observer performance. Following this line, some observer techniques exploit the model structure to “force” a reduction of the output-estimation error by © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Cecilia, Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems, Springer Theses, https://doi.org/10.1007/978-3-031-38924-5_5

91

92

5 Adaptive Observers: Direct and Indirect Redesign

means of a high-gain feedback term. Taking the high-gain observer as an example, see Sect. 3.2 of Chap. 3, the parameter ε can be arbitrary decreased to reduce the uncertainty effect. This framework is commonly referred to as robust observer design. Nonetheless, such an approach presents clear limitations: • Observers present a trade-off between measurement noise and disturbance rejection. Consequently, robust observer approaches commonly present high noise sensitivity. • High-gain feedback terms usually deteriorates the transient of the observer. For example, decreasing the factor ε increases the peaking phenomena of high-gain observers. • There may be some uncertainty terms that cannot be reduced by means of high-gain feedback terms. • Robust observer approaches provide minimum to no information about the uncertainty. These limitations motivate the development of alternative strategies. From a practical point of view, if the implemented observer presents a persistent large output-estimation error, it may be a symptom that the model implemented in the observer is not adequate. In such case, the designer may be interested in re-calibrating the model to the current operating point of the system. Following this line, it may be convenient to develop an algorithm that continuously and automatically calibrates the model implemented in the observer, taking the measured signals and the observer signals as inputs. This idea creates what is referred to as an adaptive observer. To better understand the ideas considered in this chapter. Consider the following nonlinear system: x˙ = f(x, u, θ ) y = h(x) where f : Rn × Rq × Rn θ → Rn , h : Rn → Rm are known nonlinear functions and θ ∈ Rn θ is a vector of constant parameters. Moreover, assume that there exists the following observer ξ˙ˆ = ϕ(ξˆ , u, θ ) + κ(ξˆ , y − h(ˆx)), xˆ = ψ(ξˆ ).

(5.1)

which achieves an asymptotic estimation of the states. The elements of the observer and the definition of asymptotic estimation are introduced in Chap. 3. Remark 5.1 For the rest of this work, it will be assumed that the constant parameters θ do not modify the structural properties, as observability and stability, of the system. For example, if f(x, u, 0) is stable and observable, then, f(x, u, θ ) is also stable and observable for any value θ . Now, consider the scenario in which the parameter vector θ is unknown. Then, the observer in (5.1) cannot be implemented. An approach to circumvent this limitation

5.1 Adaptive Observers and the Conflict of Model Uncertainty

93

is to implement the observer without the unknown parameters, i.e. the factor ϕ is implemented as ϕ(ξˆ , u, 0), at a cost of introducing an uncertainty term as follows ξ˙ˆ = ϕ(ξˆ , u, θ ) + κ(ξˆ , y − h(ˆx)) + ϕ(ξˆ , u, 0) − ϕ(ξˆ , u, θ ), xˆ = ψ(ξˆ ). The robust observer approach would seek to reduce the effect of the uncertainty term, ϕ(ξˆ , u, 0) − ϕ(ξˆ , u, θ ), by means of a proper tuning of the feedback term, κ, at a cost of increased noise sensitivity and worse transient. Alternatively, one can consider the idea of coupling the observer with an additional algorithm that generates a signal θˆ such that lim |ϕ(ξˆ , u, θˆ ) − ϕ(ξˆ , u, θ )| = 0.

t→∞

(5.2)

where the observer is implemented as ξ˙ˆ = ϕ(ξˆ , u, θˆ ) + κ(ξˆ , y − h(ˆx)), xˆ = ψ(ξˆ ).

(5.3)

Consequently, under some assumptions, if the original observer (5.1) achieves an asymptotic estimation of the states, then, the observer in (5.3) also achieves an asymptotic estimation of the states without relying on high-gain feedback terms. Before proceeding with the natural question about how to design an algorithm that achieves (5.2), it is interesting to remark the following points: • Even if the original observer was stable. Once is coupled with an algorithm that generates θˆ , the resulting system may become unstable [1]. • The property (5.2) does not imply nor does require the signal θˆ to converge to the true value, i.e. it does not imply that lim |θˆ − θ | = 0.

t→∞

(5.4)

The first point is crucial to understand the complexity of adaptive observer design. The state-estimation error, x˜ , of the observer in (5.3) depends on the function estimation-error, ϕ(ξˆ , u, θˆ ) − ϕ(ξˆ , u, θ ). In parallel, independently of the algorithm that is used to generate θˆ , the function estimation-error will depend on the current estimation of the observer. This interaction may generate an unstable structure, even if all the elements, independently, seemed to be stable. Consequently, it is necessary to have a clear stability theory of the resulting coupling, and is not sufficient to couple the observer with any algorithm that seems to generate an “accurate enough” estimation, θˆ . The second point establishes a relaxation on the type of algorithm that is required to achieve asymptotic estimation of the states. That is, it is not necessary to generate a value θˆ that correctly interpolates the function ϕ in the whole state space, but only in the current trajectory of the system. Consequently, with just the trajectory of the

94

5 Adaptive Observers: Direct and Indirect Redesign

measured signals, it is possible to significantly improve the state-estimation error. A drawback of such an approach is that some extra excitation assumptions on the system trajectory are required to achieve the parameter convergence in (5.4). Next chapters will explore two frameworks to design adaptive observers. The first one will study some adaptation mechanisms that achieve the function convergence property in (5.2). This approach will be referred to as direct adaptive observer. In the second, an adaptation mechanism will be designed assuming the parameter estimation convergence in (5.4). This approach will be referred as indirect adaptive observer. As an endpoint of this introduction, it should be remarked that this chapter studies adaptive observers from an adaptive redesign point of view. Specifically, it assumes that there is an already existing observer, and it is desired to redesign this observer with some parameter adaptation dynamics. Indeed, such framework does not include the common approach of adding the unknown parameters in the state vector with a null derivative, θ˙ = 0, and then, design an observer for the extended system, see [2] for an example of the approach. This case is obviated as it does not particularly exploit the previous existence of an observer.

5.2 Direct Adaptive Observer Redesign The direct adaptive observer redesign approach is based on designing a parameter adaptation dynamics, θ˙ˆ , such that the unknown parameters do not appear in the observer Lyapunov function derivative. Naturally, such cancellation can only be achieved under certain structural assumptions. To simplify this section, the type of observer and system considered is going to be restricted. Specifically, consider the following multi-input multi-output nonlinear system, x˙ = f(x, u) + φ(x, u)θ y = h(x)

(5.5)

where θ ∈ Rn θ is a vector of unknown constant parameters to be estimated. The output function h : Rn → Rm is assumed to be known. The functions f : Rn × Rq → Rn and φ : Rn × Rq → Rn×n θ are assumed to be known and Lipschitz, with L f and L φ as Lipschitz constant, respectively. Moreover, there exists some sets X0 ⊆ X ⊆ Rn and U ⊆ Rq , such that the trajectories of (5.5), with initial conditions x(0) in X0 and input u(t) belonging to U for all times, remain in X for all t ≥ 0. Finally, the linear regressor factor is also assumed to be upper bounded as |φ(·, ·)| ≤ φmax . It is assumed that there is an observer of the form x˙ˆ = f(ˆx, u) + κ(ˆx) y˜ + φ(ˆx, u)θˆ ,

(5.6)

5.2 Direct Adaptive Observer Redesign

95

where κ(·) is a Lipschitz and bounded function designed such that, there is a locally Lipschitz function (Lyapunov function) V : Rn → R satisfying ¯ x|), α(|˜x|) ≤ V (˜x) ≤ α(|˜

(5.7)

where α, α¯ ∈ K∞ , so that its derivative along solutions to (5.5) and (5.6) satisfies ∂V φ(ˆx, u)θ˜ . V˙ ≤ −α1 V (˜x) + ∂ x˜

(5.8)

for all x ∈ X and u ∈ U, where α1 is some positive constant and θ˜ = θ − θˆ . Moreover, it is assumed that there is a function g : Rm × Rm × Rq × Rn → Rn θ such that ∂V φ(ˆx, u). (5.9) g(˜y, y, u, xˆ ) = ∂ x˜ Before proceeding with the direct adaptive redesign, it is convenient to state the main differences between the presented problem and the one considered at the beginning of the chapter. First, the observer is in the same coordinates as the system. This will simplify the notation of the proofs presented in this chapter. Nonetheless, it should be relatively easy to extend the result to implement coordinate transformations. Second, the observer feedback term is linear in the output-estimation error. Third, the system is linear in the unknown parameters. There are some notable results on the topic of adaptive control and observer for nonlinearly parametrized systems [3–7]. Nonetheless, it is still not clear how to extend these results to a more general type of observer. The idea is to exploit the property (5.9) to design some parameter adaptation ∂V φ(ˆx, u)θ˜ is eliminated in the Lyapunov equation dynamics, such that the term ∂ x˜ derivative (5.8). This idea can be formalized through the following theorem. Theorem 5.1 Consider a system of the form (5.5) and the observer (5.6) that satisfies (5.8) and (5.9). Moreover, let θˆ be computed as θ˙ˆ = g(˜y, y, u, xˆ ) .

(5.10)

Then, x˜ and φ(ˆx, u)θ˜ converge asymptotically to zero. Proof This proof is based on the results in [8, 9]. Consider the following radially unbounded Lyapunov function 1  Vθ = V (˜x) + θ˜ θ˜ , 2

(5.11)

where V (˜x) is defined in (5.7)–(5.8). As θ˙ = 0 and the property (5.9) holds, it can be shown that the derivative of Vθ satisfies the following:

96

5 Adaptive Observers: Direct and Indirect Redesign

∂V ∂V   V˙θ ≤ −α1 V (˜x) + φ(ˆx, u)θ˜ − θ˜ θ˙ˆ = −α1 V (˜x) + φ(ˆx, u)θ˜ − θ˜ g(˜y, y, u, xˆ ) ∂ x˜ ∂ x˜ = −α1 V (˜˜x).

This result implies that x˜ , θ˜ and θˆ are bounded and, then, that x˜ ∈ L2 . Moreover, since f, φ and κ are Lipschitz it implies that x˙˜ is uniformly continuous and |x˙˜ | is bounded. Therefore, by the Barbalat’s lemma [10], it can be shown that limt→∞ x˜ = 0. Finally, notice the following 

∞ 0

x˙˜ dt = lim x˜ (t) − x˜ (0) = −˜x(0), t→∞

which is finite. Then, as x˙˜ is uniformly continuous, by the Barbalat’s lemma, it implies that limt→∞ x˙˜ = 0. As x˜ and x˙˜ converge asymptotically to zero, and the state-estimation error is computed as follows ˆ x˙˜ = f(x, u) − f(ˆx, u) − κ(ˆx) y˜ + φ(x, u)θ − φ(ˆx, u)θ.   ˆ It is deduced that limt→∞ φ(x, u)θ − φ(ˆx, u)θ = 0. Thus, as limt→∞ x˜ = 0 and φ is Lipschitz it can be shown that limt→∞ φ(ˆx, u)θ˜ = 0.



As explained in the introduction, in direct adaptive observers, the parameter adaptation dynamics only interpolates the function φ(x, u)θ to the measured trajectory. Therefore, in general, there will be multiple parameter vectors that can correctly explain the system trajectory. To guarantee that the algorithm retrieves the true parameters, it is sufficient to impose that the system trajectories are “rich enough” such that only a single parameter vector can interpolate the system trajectory. Indeed, this condition has been previously introduced in Definition 3.2 in Chap. 3 through the concept of persistence of excitation. This result is formulated through the following theorem. Theorem 5.2 Consider a system of the form (5.5) and the observer (5.6) that satisfies (5.8) and (5.9). Moreover, let the estimated parameters be computed through (5.10). Then, if the regressor vector, φ(x, u), is persistently exciting, the parameter estimation-error θ˜ converges asymptotically to zero. Proof The proof can be found in Lemma A.1 in [9]. As an example, consider a nonlinear system of the form x˙ = f(x) + Bφ(x)θ y = cx



5.2 Direct Adaptive Observer Redesign

97

where x = [x1 , x2 ] , θ = [−0.2, 1, −0.3] and f(x) =

        0 x2 , B = , c = 1 1 , φ(x) = x1 , x2 , x12 x2 . 0 1

The objective is to design an observer for the considered system, under the assumption that the vector fields f, φ and the vectors B, c are known, but, but the parameter vector, θ, is unknown. Following the insights presented in [11], it is possible to prove that the following observer (5.12) x˙ˆ = f(ˆx) + Bφ(ˆx)θˆ + κ(y − c xˆ ) with κ = [−0.314, 3.156] , satisfy the Lyapunov condition (5.8). Specifically, it is an asymptotic observer in the case that θˆ = θ, with a quadratic Lyapunov function of the form: 1 V = x˜  P˜x, 2 where P ∈ R2×2 is a positive symmetric definite matrix. The key step in the direct adaptive observer redesign is finding a function g(˜y, y, xˆ ) that satisfies (5.9). Notice that in the considered scenario, the term in the left-hand side of (5.9) is ∂V (5.13) φ(ˆx) = x˜  PBφ(ˆx)θ˜ . ∂ x˜ The difficulty is to design g such that it only depends on the measured signals. To find such function, consider the next constant M ∈ R B P = Mc.

(5.14)

Then, it can be seen that a function g that satisfies (5.9) is the following g(˜y, xˆ ) = y˜  Mφ(ˆx),

(5.15)

and the observer can be adaptive redesigned with (5.10). Indeed, in Fig. 5.1 it is depicted the state-estimation error of the adaptive observer. It can be seen that even if the parameters θˆ are unknown, the adaptive redesign allows to achieve asymptotic convergence of the estimation. Furthermore, since the considered system presents oscillatory dynamics, that makes the vector Bφ(x) persistently exciting. As a consequence, the parameter estimation, θˆ , also converges to the true value. This result can be seen in Fig. 5.2. The appeal of direct adaptive observer redesign is obvious. On the one hand, the unknown parameters effect in the state-estimation error is completely eliminated without relying on high-gain feedback terms. On the other, if the system

98

5 Adaptive Observers: Direct and Indirect Redesign

Fig. 5.1 Evolution of the state-estimation error

Fig. 5.2 Evolution of the parameter estimation-error

trajectories are persistently excited, then, the parameter estimation dynamics automatically identifies the system model. Nonetheless, such an approach is not free of significant drawbacks which limits its applicability in practice. First, the Lyapunov function (5.11), although it can be used to prove stability through a Barbalat’s lemma argument, is a weak Lyapunov function (weak in the sense that its derivative is just non-positive). This has a significant implication for practical applications. Specifically, a weak Lyapunov function cannot be used to prove that the observer is ISS with respect to measurement noise and model uncertainties (see Theorem 3.4 in [12]). Therefore, just from the presented results, there is no theoretical guarantees that the observer is not fragile in front of noise or uncertainty. Moreover, a weak Lyapunov function cannot be used to guarantee that the coupling of the adaptive observer with the filters presented in Chap. 4 (or any other system) creates a stable structure. Second, the necessary condition (5.9) requires the original system to be relative degree one or zero between the measured output, y and the unknown parameters θ , i.e. the unknown parameters have to appear in the first derivative of the output. This fact significantly restricts the type of uncertainty that can be dealt with direct adaptive observers. Finally, direct adaptive observers assume that the regressor vector φ is exactly known. Nonetheless, in practice, as the uncertainty is related with the unmodelled parts of the system, it is common that the functions that compose φ are unknown. The aim of the following sections is to propose a set of solutions in order to relax the mentioned limitations.

5.2 Direct Adaptive Observer Redesign

99

5.2.1 Constructing a Strict Lyapunov Function The combination of a weak Lyapunov function (5.11) and the Barbalat’s lemma argument is a common approach to prove convergence for direct adaptive observers [8, 9]. Nonetheless, as explained before, a weak Lyapunov function cannot be used to guarantee stronger stability conditions or robustness in front of unmodelled elements. For this reason, it is convenient to substitute the weak (weak in the sense that its derivative is only negative semidefinite) Lyapunov function (5.11) for a strict Lyapunov function that can be used to proceed with the analysis. In this section, it is proposed to use the Mazenc construction [13] to derive a strict Lyapunov function from (5.11), which allows to define the robustness and convergence rate of the adaptive observer through standard Lyapunov arguments. This approach has proved to be successful in multiple adaptive control problems [14, 15]. To simplify the developments, a variation of the problem presented in the past section will be considered. Specifically, consider a system of the form x˙ = f(x, u) + φ(x, u)θ + w y = Cx + v

(5.16)

where θ ∈ Rn θ is a vector of unknown constant parameters to be estimated. The functions f and φ are assumed to be known and Lipschitz, with L f and L φ as Lipschitz constant, respectively. Additionally, the factor v ∈ Dv ⊆ Rm depicts bounded measurement noise and w ∈ Dw ⊆ Rn models model uncertainty and/or unmodelled external disturbances. Moreover, there exists some sets X0 ⊆ X ⊆ Rn and U ⊆ Rq , such that the trajectories of (5.5), with initial conditions x(0) in X0 and input u(t) belonging to U for all times, remain in X for all t ≥ 0. It is assumed that the relative degree between the measured output and the unknown parameters is one. Finally, the linear regressor factor is also assumed to be upper bounded and Lipschitz as |φ(·, ·)| ≤ φmax . Notice that, different form the case considered in (5.5), now it is assumed that there are unmodelled factors and measurement noise. Moreover, it is assumed that there is an observer of the form (5.6) with a quadratic Lyapunov function (5.17) V = x˜  P˜x, P = P ≥ 0 which satisfies the conditions (5.8) and (5.9) and the following bounds λmin (P)|˜x|2 ≤ V ≤ λmax (P)|˜x|2

(5.18)

where λmax (·), λmin (·) represents the maximum and minim eigenvalue, respectively. Finally, the observer is adapted through the dynamics in (5.10). As the output is linear and the Lyapunov quadratic, the function g is assumed to be designed as in (5.15) with a matrix M of adequate dimension.

100

5 Adaptive Observers: Direct and Indirect Redesign

In order to derive a strict Lyapunov function, it is convenient to, first, define the following locally Lipschitz time-varying function Λ

λmin (φ(t) φ(t)).

(5.19)

As the function φ is Lipschitz and bounded, it can be shown that function (5.19) is also bounded with a bounded derivative. As consequence, there is a value Λ¯ such that ˙ ≤ Λ. ¯ max{|Λ|, |Λ|} Moreover, define the following signal: ¯ 0− ΥΛ = 1 + 2ΛT

2 T0



t+T0 t



m

Λ(s)2 ds dm.

(5.20)

t

If we assume that the system is persistently excited, as defined in Definition 3.2 in Chap. 3, function (5.20) is bounded as follows ¯ 0. 1 ≤ ΥΛ ≤ 1 + 2ΛT Furthermore, there exists an upper bound for the derivative of (5.20), Υ˙Λ ≤ −

2μ2 + 2Λ2 . T0

Then, the result can be summarized through the following theorem. Theorem 5.3 Consider system 5.16, the observer (5.6) with a quadratic Lyapunov equation (5.17) that, in the absence of unmodelled elements, i.e. w = v = 0, satisfies (5.18), (5.8) and (5.9). Assume that the system is persistently excited as defined in Definition 3.2 in Chap. 3, and assume that the parameters, θˆ , of the observer are adapted through the dynamics in (5.10) with the function g designed as in (5.15) with a matrix M of adequate dimension. Then, the system admits the following Lyapunov function 1  V1 = −˜x φ(ˆx, u)θ˜ + (ΥΛ + α)(˜x P˜x + θ˜ θ˜ ), (5.21) 2 where θ˜ = θˆ − θ , which satisfies the following μ2 μ2 ˜ 2 ˜ |θ | + k4 |θ˜ ||w| + k5 |˜x||w| + k6 |θ||v| + k7 |˜x||v| V˙1 ≤ − λmin (P)|˜x|2 − T0 2T0 (5.22) where ki f or i = 4, . . . , 7 are some positive constants to be defined, provided that

5.2 Direct Adaptive Observer Redesign

α ≥ max{α1−1

T0 μ2

2 (L f + L φ θ + |C|)2 φmax +

101

 T0 ˙ 2 |φ(ˆx, u)|2 , 2φmax λmin (P)−1 , 1}. μ2 (5.23)

Proof Let χ  [˜x, θ˜ ] . Then, in view of (5.23), the Lyapunov function (5.21) is positive definite and radially unbounded. Specifically, there exists some constants V1,min , V1,max > 0, such that

where

V1,min |χ |2 ≤ V1 ≤ V1,max |χ|2

(5.24)

1 1 1 2 1 , (1 + α) − } V1,min = min{ (1 + α)λmin (P) − φmax 2 2 2 2

(5.25)

1 1 2 1 α 1 α ¯ 0 + )λmax (P), + ( + ΛT ¯ 0 + )}. (5.26) V1,max = max{ φmax + ( + ΛT 2 2 2 2 2 2 Notice that, by considering the presence of measurement noise and uncertainty,  the derivative of the function x˜  P˜x + θ˜ θ˜ becomes:  ˜ d(˜x P˜x + θ˜ θ) ˜ max |M||v|. ≤ −α1 |˜x|2 + γ1 |˜x||w| + |˜x|λmax (P)κmax |v| + |θ|φ dt

for some positive constant γ1 . The derivative of (5.21) satisfies the following:

 V˙1 ≤ − f(x, u) − f(ˆx, u) − κ(ˆx)(˜y + v) + φ(x, u)θ − φ(ˆx, u)θˆ + w φ(ˆx, u)θ˜ ˙ x, u)θ˜ − x˜  φ(ˆx, u)φ(ˆx, u) P˜x − x˜  φ(ˆ 

1 d(˜x P˜x + θ˜ θ˜ ) ˜ ) + Υ˙Λ (˜x P˜x + θ˜ θ) dt 2 ˜ + (κmax |v| + w)φmax |θ| ˜ + |˜x||φ(ˆ ˜ − α1 α|˜x|2 ˙ x, u)||θ| ≤ |˜x|(L f + L φ θ + |C|)φmax |θ|

+ (ΥΛ + α)(

˜ max |M||v| ¯ 0 )|˜x|λmax (P)κmax |v| + (α + 1 + 2ΛT ¯ 0 )|θ|φ + (α + 1 + 2ΛT μ2 ˜ 2 μ2 ¯ 0 )|˜x|γ1 w − |θ | − λmin (P)|˜x|2 + (α + 1 + 2ΛT T0 T0

(5.27)

Then, if one applies Young’s inequality to (5.27) as follows: ˜2 ˜ ≤ (L f + L φ θ + |C|)2 φ 2 |˜x|2 + 1 |θ| |˜x|(L f + L φ θ + |C|)φmax |θ| max 2 2 ˜ ≤ |φ(ˆ ˜ 2, ˙ x, u)||θ| ˙ x, u)|2 |˜x|2 + 1 |θ| |˜x||φ(ˆ 2 2

102

5 Adaptive Observers: Direct and Indirect Redesign

2T0 and considers the relation (5.23), the bound depicted in (5.22) can μ2 be deduced, where defines =

k4 = φmax ¯ 0 )γ1 k5 = (α + 1 + 2ΛT ¯ 0 )φmax |M| k6 = φmax κmax + (α + 1 + 2ΛT ¯ 0 )λmax (P)κmax . k7 = (α + 1 + 2ΛT



Although Theorem 5.3 and its proof are relatively complex, its consequences are easier to understand. At least in the case considered in this section, it is possible to provide a direct adaptive redesign and prove its stability through a strict Lyapunov function (strict in the sense that its derivative is strictly negative). Therefore, there are theoretical guarantees that the resulting adaptive observer is ISS with respect to unmodelled factors in the state equation and measurement noise. Moreover, the adaptive observer can be presented in the framework proposed in Sect. 3.4 of Chap. 3. This is an interesting result as it allows further modifications of direct adaptive observer redesigns. For example, it is possible to derive theoretical guarantees that the coupling of the observer and some of the filters presented in Chap. 4 creates a stable structure. Nonetheless, notice that the results presented in this section are based on the premise that the system is persistently excited. Indeed, in the absence of excitation, the Lyapunov analysis provided Theorem 5.3 no longer holds true, and the observer cannot be proved to be robust in front of measurement noise or uncertainty. This is an unsurprising result, as it is well-known in the adaptive control community that small unmodelled disturbances can make adaptation dynamics “drift” to infinity in the absence of persistence of excitation [16–18]. However, in the author’s knowledge, this phenomenon was not properly introduced in the context of direct adaptive nonlinear observer redesign.

5.2.2 Addressing the Relative Degree Condition: A High-Gain Observer Approach Direct adaptive observer redesign is based on cancelling the factors that depend on the unknown parameters. Naturally, such cancellation requires stringent assumptions on the Lyapunov equation, observer and system considered. The main concern is that the elements that depends on the unknown parameters will, generally, also depend on the unknown state-estimation error, which makes this cancellation not trivial. For this reason, it is common to assume that the system satisfies a “passivity-like” condition between the output and the unknown parameters [19]. In the presented direct adaptive redesign, this condition is achieved through the equality in (5.9). The limitation is

5.2 Direct Adaptive Observer Redesign

103

that passivity conditions are restricted to systems with relative degree one or zero between the measured outputs and the unknown parameter vector, i.e. the unknown parameters appear in the first derivative of the output. There is a significant amount of systems that do not possess this relative degree property. This fact motivates the modification of direct adaptive redesigns to higher relative degree systems. The most common approach is to modify the structure of the original system with the so-called filtered transformation [9, 20] or the use of filters to compute auxiliary signals [21, 22]. However, such approaches presents some conflicts that should be considered. First, the dynamics of the filter depend on the initial conditions of said filter. Therefore, the stability analysis of the observer is “trajectory-dependent”, in the sense that it pertains only to the trajectory generated for the given initial conditions. This fact may have a great impact on the observer performance and the observability study, which is overlooked in some papers. A further discussion in the context of adaptive control can be found in [23]. Second, the stability and accuracy of the observers with filtered transformations in the presence of uncertainty and unknown states in the parameter regressor vector is far from trivial and require further study. Moreover, this analysis need to consider the initial conditions of the filter as mentioned in the first point. This fact motivates the design of alternative methodologies to circumvent the relative degree restriction without the use of filters. A solution was proposed in [24] for reduced order observers. However, such approach requires solving partial differential equations which is in general hard to achieve. Alternatively, a parameter estimation for higher relative degrees can be achieved by implementing an extended high-gain observer [25], and using the observer extra state to achieve a linear regression. Nonetheless, this approach is limited to systems where the relative degree is equal to the dimension of the state vector. In this section, an alternative method to address the relative degree conflict in adaptive observers is proposed. Instead of implementing a set of filters, the adaptive observer is coupled with a high-gain observer, which allows to overcome the relative degree restriction and have a probable convergence. A general scheme of the proposed approach is depicted in Fig. 5.3.

Fig. 5.3 Scheme of the proposed adaptive observer for higher relative degree systems

104

5 Adaptive Observers: Direct and Indirect Redesign

To simplify the developments, this section will consider a modification of the problem presented at the beginning of the chapter. Specifically, consider a multiinput multi-output nonlinear system of the form. x˙ = f(x, u) + Bφ(x, u)θ y = C(u, y)x

(5.28)

where θ ∈ Rn θ is a vector of unknown constant parameters to be estimated. The matrices B ∈ Rn×s and C(·, ·) ∈ Rm×n are assumed to be known and bounded. The functions f(·, ·) ∈ Rn×1 and φ(·, ·) ∈ Rs×n θ are assumed to be known and Lipschitz, with L f and L φ as Lipschitz constant, respectively. The linear regressor factor is assumed to be upper bounded as |φ(·, ·)| ≤ φmax . The idea is to find an adequate auxiliary signal, z  H(u, y)x with a bounded matrix |H(u, y)| ≤ Hmax , to be used as a new measured output in the observer (see Fig. 5.3) in order to make the condition (5.9) feasible. In this work, an auxiliary signal can be depicted as adequate if the following conditions are satisfied: 1. The concerned system is relative degree 1 from the auxiliary signal, z, to the unknown parameter vector θ. Therefore, the function H(u, y) is such that rank(H(u, y)B) = rank(B) ∀u, y 2. There exists an observer of the form: x˙ˆ = f(ˆx, u) + Ξ 2 (ˆx)˜z + Bφ(ˆx, u)θˆ ,

(5.29)

where z˜ = z − H(u, y)ˆx, and Ξ 2 (·) ∈ Rn×1 is a function bounded as |Ξ 2 (·)| ≤ Ξmax , designed such that there is a quadratic Lyapunov function of the form V = x˜  P˜x,

(5.30)

where P ∈ Rn×n is a (possibly time-varying) positive definite symmetric matrix, and satisfies (5.7) and (5.8). 3. There exists a vector function, T(·), with a Lipschitz constant L t independent of u, that allows to reconstruct the auxiliary signal as follows: z =T(u, y1 , y˙1 , . . . , y1(r1 −1) , . . . , ym , . . . , ym(rm −1) ),

(5.31)

where ri is the relative degree index between the i th output, yi , and the unknown parameter vector, θ . Definition 5.1 A system depicted by (5.28) has a relative degree ri from the i th output signal, ci (u, y)x, to the unknown parameters, θ , if:

5.2 Direct Adaptive Observer Redesign

105

L Bφ(x,u) L kf(x,u) ci (u, y)x = 0 ∀k < ri − 1 i −1 ci (u, y)x = 0. L φ(x,u) L rf(x,u)

where ci (u, y) is the i th row of the matrix C(u, y) and the factor L f(x,u) ci (u, y)x operation denotes the Lie derivative of the function ci (u, y)x along the vector field f(x, u). The first two points allow to proceed with the adaptive observer design independently of the original relative degree of the system. Specifically, the parameter adaptation can be designed as θ˙ˆ = φ(ˆx, u) M(t, x, u)H(u, y)˜x.

(5.32)

where M(t, x, u) satisfies: M(t, x, u)H(u, y) = B P.

(5.33)

It is noticeable that the auxiliary signal, z, is not directly computable as it depends on the unknown states. Nonetheless, by means of the point (3), there is a map that relates the auxiliary signal with the input, output and its derivative up to the relative degree. Therefore, it is possible to design a high-gain observer that can achieve an estimation of the auxiliary signal, zˆ , which is robust in an ISS sense with respect to the unknown parameters, θ and states, x. This property is crucial, as it allows the high-gain observer to be coupled with the adaptive observer and have a provable convergence. Following the methodology presented in Chap. 3, before designing the high-gain observer, the system will be transformed to an adequate triangular form. Precisely, consider a set of coordinates ξ = [ξ 1 , . . . , ξ m ] , where ξ i ∈ Rri is defined ∀i = 1, . . . , m as: ⎡ ⎤ yi ⎢ y˙i ⎥ ⎢ ⎥ ξ i = ⎢ .. ⎥ . ⎣ . ⎦ yi(ri −1)

The dynamics of the ξ i coordinates are depicted by i ¯ x, θ ), ∀i = 1, . . . , m ξ˙ = Aξ i + Ψ i (ξ i , u,

y i = Cξ i where Ai ∈ Rri ×ri , Bw,i ∈ Rri ×1 and Ci ∈ R1×ri are

(5.34)

106

5 Adaptive Observers: Direct and Indirect Redesign

⎡

0 ⎢0 ⎢ ⎢ Ai = ⎢ ... ⎢ ⎣0 0

⎤ ⎡ ⎤ ⎡ ⎤ 1 0 ··· 0 0 1 ⎥ 0 1 · · · 0⎥ ⎢0⎥ ⎢0⎥ ⎥ ⎢ ⎥ .. . .⎥ ; B = ⎢ ⎢ .. ⎥ ; Ci = ⎢ .. ⎥ w,i . 0 . . .. ⎥ ⎥ ⎣.⎦ ⎣.⎦ 0 ··· 1⎦ 1 0 0 0 ··· 0 ⎡

⎢ ⎢ ¯ x, θ ) = ⎢ Ψ i (ξ i , u, ⎣

0 0 .. .

⎤

⎥ ⎥ ⎥ ⎦ ψri (ξ , x, θ , u, . . . , u(ri −ru,i ) )

(5.35)

where ru,i is the relative degree between the output, yi , and the input u and Ψ i is Lipschitz. The expression (5.34) is a triangular structure in which a high-gain observer can be designed, see Sect. 3.2 of Chap. 3. Precisely, i i ¯ xˆ , θˆ ) + Ei li (y i − ξˆ1i ) ξ˙ˆ i = Ai ξˆ + Ψ i (ξˆ , u,

(5.36)

where Ei ∈ Rri ×ri and li ∈ Rri ×1 are: ⎡

⎤ 1 ⎡ ⎤ 0⎥ l1,i ⎢ε ⎢ . ⎥ ⎢ ⎥ ; li = ⎣ .. ⎥ Ei = ⎢ . ⎦. ⎢ .. ⎥ ⎣ ⎦ 1 lri ,i 0 ε ri Remark 5.2 The factor Ψ i of the observer (5.36) depends on the state-estimation, xˆ , and the parameter estimation, θˆ , of the adaptive observer. This is the feedback term between the adaptive observer and the high-gain observer depicted in Fig. 5.3. Coupling the observers in this way allows having provable convergence of the state and parameter estimation. If the feedback term is obviated, as in [26], only boundedness of the estimation-error can be guaranteed. Remark 5.3 The factor Ψ i depends on the derivative of the inputs, which, in some cases, may be unknown. This derivatives may be robustly estimated through a differentiator [27]. Alternatively, they may be considered as an unknown disturbance. Notice that the unknown parameters, θ, and unknown states states, x, all appear in the last equation of system (5.34). Therefore, as explained in Sect. 3.2 of Chap. 3, the high-gain observer (5.36), under the proper parameter tuning, will be ISS with respect to the estimation-errors x˜ and θ˜ . Specifically, define the vector χ  [˜x, θ˜ ] , then, the high-gain observer estimation-error is ultimately bounded as follows:

5.2 Direct Adaptive Observer Redesign

107

|ξ˜ | ≤ εk1 |χ|

(5.37)

where k1 is some positive constants. Form this result, it is possible to establish that the estimation-error of the auxiliary signal is also ISS with respect to the same inputs. Theorem 5.4 Consider the system 5.34 and the high-gain observer 5.36. Define the estimation of the auxiliary signal as zˆ = T(u, ξˆ ). Then, the auxiliary signal estimation-error is ultimately bounded as |z − zˆ| ≤ εk1 L T |χ|

(5.38)

Proof The function T(·) is assumed to be Lipschitz with a constant L T . Therefore, the following bound holds |z − zˆ | ≤ L T |ξ˜ | ≤ εk1 L T |χ |  Remark 5.4 The only property that is required for the observer that estimates the auxiliary signal is an ISS condition with respect to the adaptive observer estimationerror. Naturally, this property is not restricted to standard high-gain observer. Indeed, the high-gain observer could be substituted by a low-power peaking-free observer to achieve better performance. The idea is to exploit the ISS bound in (5.38) to derive the conditions in which the high-gain observer/adaptive observer coupling generates a stable structure. The next step consists in showing that the redesigned observer (5.29) with the adaptation dynamics (5.32) is ISS with respect to the auxiliary signal estimation-error z − zˆ . In general, this step would be significantly complex, as common stability proofs of direct adaptive redesign (see Theorem 5.1 proof) cannot guarantee an ISS bound. Nonetheless, the result obtained in Sect. 5.2.1 will simplify the step. First, it can be shown that the signal z − zˆ is an output disturbance of the observer. Precisely, the observer (5.29) and (5.32) implemented with zˆ instead of z satisfies the following x˙ˆ = f(ˆx, u) + Ξ 2 (ˆx)˜z + Bφ(ˆx, u)θˆ + Ξ 2 (ˆx)(z − zˆ ), θ˙ˆ = φ(ˆx, u) M(t, x, u)˜z + φ(ˆx, u) M(t, x, u)(z − zˆ ).

(5.39)

where the term z − zˆ can be taken as an output disturbance and H(u, y) satisfies the conditions specified before. Then, the ISS bound of the new adaptive redesign can be formalized through the following theorem. Theorem 5.5 Consider the adaptive observer in (5.39), assume that zˆ is generated through the high-gain observer (5.36) that satisfies (5.73) and the vector Bφ(x, u)

108

5 Adaptive Observers: Direct and Indirect Redesign

is persistently excited. Then, the adaptive observer estimation-error, χ = [˜x, θ˜ ] , presents the following ultimate bound |χ| ≤ k8 |z − zˆ|,

(5.40)

where k8 is a positive constant. Proof Considering w = 0 and the term z − zˆ as an output disturbance, the bound in (5.22) reduces to: μ2 μ2 ˜ − zˆ | + k7 |˜x||z − zˆ | |eθ |2 + k6 |θ||z V˙1 ≤ − λmin (P)|˜x|2 − T0 2T0 μ2 1 ≤ − min{λmin (P), }|χ|2 + max{k6 , k7 }|χ||z − zˆ |. T0 2

(5.41)

It can be seen that for the region |χ| ≥ 2

max{k6 , k7 } |z − zˆ |, μ2 1 min{λmin (P), } T0 2

the derivative (5.41) is bounded as: 1 V˙1 ≤ − |χ|2 . 2 Then, from the comparison lemma and input to state stability theory [28], it is possible to deduce the following ultimate bound for the adaptive observer state and parameter estimation:  |χ| ≤ 2

V1,max max{k6 , k7 } |z − zˆ | μ 1 V1,min 2 min{λmin (P), } T0 2



Finally, the definition of these ultimate bounds (5.38) and (5.40) makes the smallgain theorem [28] a convenient method to proof the stability of the observer coupling. Theorem 5.6 Consider the system 5.34, the high-gain observer depicted in 5.36 tuned to ensure the bound 5.38 and the adaptive observer (5.39), which satisfies the bound 5.40. Then, the auxiliary signal estimation-error, z − zˆ, and the adaptive observer estimation-error, χ, converges to zero provided that εk1 L T k8 < 1.

(5.42)

5.2 Direct Adaptive Observer Redesign

109

Proof If condition (5.42) is satisfied, the ultimate bounds (5.38) and (5.40) define a contraction. Therefore, by the small gain theorem [28] it can be shown that |z − zˆ |, and the adaptive observer estimation-error, |χ| converges to zero.  Remark 5.5 Notice that condition (5.42) can be rearranged as: ε
0 and K(t) is a time-varying matrix defined as follows (5.46) k(t) = P(t)−1 c, is an observer for the concerned system in the case θˆ = θ with a quadratic Lyapunov function of the form (5.30). See [29] for more details on the considered observer. The aim of this section is to generate θˆ through a direct adaptive redesign approach. The high-gain observer modification proposed in this section will be used to address the higher-than-one relative degree of the system. To do so, first, it is required to design an adequate auxiliary signal, z. It can be seen that the signal degree z = x1 + ux2 = h(t) x, where h(t) = [1, u], is relative   1 with respect to the unknown parameters, θ. Furthermore, the pair A(t), h(t) is observable with the considered input, thus, the states can be estimated through an observer of the form (5.29). Finally, this signal can be reconstructed as z = y˙ + 1.3y, thus, it satisfies condition (5.31) and the parameters can be adapted through (5.32) that satisfies (5.33). Specifically, the parameter adaptation takes the form (5.32) with M(t) = b(k(t) )†

(5.47)

where (k(t) )† is the left Moore-Penrose pseudo-inverse computed as (k(t) )† = (k(t)k(t) )−1 k(t).

(5.48)

To better understand this design of the matrix M, notice that from (5.46), from the fact that z is the new measured output and the fact that P is symmetric, it can be derived that P(t) = h(t)(K(t) )† . Then, the following can be deduced ∂V φ(ˆx, u) = x˜  P(t)bφ(ˆx) = (z − h(t)ˆx)M(t) φ(ˆx), ∂ x˜ and (5.9) is satisfied with g(z, xˆ , t) = (z − h(t)ˆx)M(t) φ(ˆx). The rest of the observer parameters are summarized in Table 5.1.

5.2 Direct Adaptive Observer Redesign Table 5.1 True model parameters and observer design parameters

Parameter True parameters θ1 θ2 Observer parameters σ α1 α2 ε

111 Value 0.3 1 1.1 3 2 0.81

It should be remarked that the considered system satisfies the persistent excitation condition, see Definition 3.2 in Chap. 3. Therefore, according to the theory presented in this chapter, both, state and parameter-estimation converges to a bounded error. One of the key-points of this section is to address the relative degree limitation of adaptive observers without relying on filters. For this reason, it is convenient to compare the performance of the proposed approach with an already existing technique that does use filters. Due to the structure of the proposed example, it would be reasonable to use the standard filter-based adaptive observer proposed in [30]. Therefore, the proposed technique will be compared with the technique in [30]. To ease the readability of the section, the details of the design of this adaptive observer will be obviated. The evolution of the state-estimation error of both observers can be seen in Fig. 5.4. Furthermore, the evolution of the parameter-estimation error can be observed in Fig. 5.5. As it can be observed, the estimation of both techniques converge to a relatively similar bounded error. Nonetheless, it is appreciable that the convergence rate of estimation-error in the proposed approach is significantly faster in the proposed approach. The slow convergence rate is a consequence of the introduction of filters, which reduces the signals excitation levels and induces a slow parameter-estimation convergence. It should be remarked, that faster convergence rate could be achieved by increasing the gain of the observer. Nonetheless, this would significantly increase the sensitivity to measurement noise. This result exemplifies the motivation of avoiding filters in order to solve the relative degree restriction in adaptive observers. Finally, in relation to the proposed approach, the estimation-error converges to a relative error below the 2%. This result validates the performance of the proposed scheme under measurement noise and unmodelled disturbances.

112

5 Adaptive Observers: Direct and Indirect Redesign

Fig. 5.4 Evolution of the state-estimation error. Blue and orange lines depict the state-estimation error of the proposed approach (PA). Yellow and purple lines depict the state-estimation error of the approach in [30] (OA)

Fig. 5.5 Evolution of the parameter-estimation error. Blue and orange lines depict the parameter-estimation error of the proposed approach (PA). Yellow and purple lines depict the parameter-estimation error of the approach in [30] (OA)

5.2.3 Library-Based Adaptive Observation: A Sparsity-Promoting Adaptive Observer In general, the idea of (linearly parametrized) adaptive control and observation is to locally interpolate part of the uncertainty through a linear combination of basis functions, φ. Therefore, in any practical application, it immediately arises the question of how to select the set of basis functions. In models derived from the system first-principles, it is common that the vector θ represents unknown parameters with physical interpretation. In such cases, the basis functions are a priori fixed and also have physical interpretation. Nonetheless, it may be the case that the system uncertainty does not have this a priori fixed structure nor this suitable physical interpretation. In such scenarios, it is common to select a set of basis functions such that it satisfies some type of universal approximation property [31]. Some remarkable examples are the use of neural networks [32], reproducing kernels [33] or wavelets [25]. However, the resulting adaptive redesign usually presents two drawbacks. First, as the approximation accuracy depends on the quantity of basis functions considered, the adaptive dynamics may be of significant order and complexity. Second, even if the observer correctly interpolates the uncertainty, it is not possible to derive any physical interpretation of said uncertainty. To address these limitations, this section explores a slightly different approach. Specifically, it is assumed that the regressor vector, φ is unknown, but there is some a priori information on the type of functions that compose, φ. More precisely, this information is enough to design a library

5.2 Direct Adaptive Observer Redesign

113

of non-linear function candidates, Θ, to model the uncertainty. Ideally, the adaptive redesign would just select the functions that appear in the true system. Thus, obtaining a low-complexity and interpretable uncertainty interpolation. To better understand this idea, consider the system in (5.5), being the only difference that, now, it is assumed that the regressor vector φ : Rn × Rq → Rn×n θ is unknown. Instead, it is considered that the designer constructs a matrix, Θ : Rn × Rq → Rn× p with p > n θ , of candidate linearly independent non-linear functions. The function Θ is assumed to be Lipschitz and satisfy the following assumption Assumption Define φ w (x, u) ∈ Rn×1 for w = 1, . . . , p, as the column vector functions that compose Θ. Each column vector of the regressor, φ(x, u), can be computed as a scaled vector cφ w (x, u) for some w, where c ∈ R. Assumption 5.2.3 allows to rewrite system (5.5) in the following equivalent form x˙ = f(x, u) + Θ(x, u)θ s y = h(x)

(5.49)

where θ s ∈ R p is a vector of unknown parameters, in which n θ elements are equal to the elements in the original parameter vector, θ , and p − n θ elements are zero. The matrix Θ is going to be referred as a library of candidate nonlinear functions. In this library, the column vectors, φ w (x, u), are defined as inactive if they do not compose the original regressor vector φ, otherwise, are defined as active. The elements of the parameter vector, θ s , associated with inactive functions will be zero, otherwise, will be non-zero. Similar to the standard direct adaptive redesign, it is assumed that there is an observer of the form (5.50) x˙ˆ = f(ˆx, u) + κ(ˆx) y˜ + Θ(ˆx, u)θˆ s , where κ(·) is a Lipschitz and bounded function designed such that, there is a locally Lipschitz function (Lyapunov function) V : x → R satisfying ¯ x|) α(|˜x|) ≤ V (˜x) ≤ α(|˜

(5.51)

where α, α¯ ∈ K∞ , so that its derivative along solutions to (5.49) and (5.51) satisfies ∂V Θ(ˆx, u)θ˜ s . V˙ ≤ −α1 V (˜x) + ∂ x˜

(5.52)

for all x ∈ X and u ∈ U, where α1 is some positive constant, where θ˜ s = θ s − θˆ s . Moreover, it is assumed that there is a function g : Rm × Rm × Rq such that g(˜y, y, u, xˆ ) =

∂V Θ(ˆx, u). ∂ x˜

(5.53)

114

5 Adaptive Observers: Direct and Indirect Redesign

In this formulation, some may be tempted to just use the direct adaptive redesign in (5.10), as the conditions in Theorem 5.1 still hold true. Nonetheless, there is a subtle difficulty that has to be analysed. Indeed, in the considered problem, as dim(θ s ) > dim(θ ), even if the original basis of functions, φ, was persistently excited, i.e. there exist a single linear combination of these functions that correctly interpolates the uncertainty, no conclusion can be drawn for the excitation of the library, Θ. Moreover, in a significant amount of cases, no trajectory of the original system can ensure that the library is persistently excited. This derives from the fact that the considered over-parametrized problem presents an infinite amount of solutions, and standard parameter-adaptation schemes will converge to some solution that cannot be ensured to be the true one. As discussed in Sect. 5.2.1, persistence of excitation is a necessary condition to guarantee robustness of the adaptive redesign. Therefore, even if Theorem 5.1 guarantees asymptotic state-estimation in the ideal case, the resulting adaptive observer may be fragile to noise and uncertainty. An alternative way of looking at the excitation problem is that, in general, most functions from the library will not be present in the real system. Therefore, the library is over-complete, which makes the parameter-estimation objective an ill-posed inverse problem. The key observation to make is that only a sparse selection of non-linear functions will be active in the designed library, Θ. The presence of an ill-posed inverse problem and these hints of sparsity motivates the implementation of a l1 regularization [34], as it is commonly used in the least absolute shrinkage and selection operator (LASSO) regression. However, even though this type of optimization has been deeply studied in the context of system identification [35], regularization in general cannot be directly implemented in adaptive control. Fortunately, similar to the case studied in Sect. 3.3 in Chap. 3, it is possible to design some non-euclidean parameter adaptation in order to implicitly regularize the problem.  The idea is to substitute the quadratic term θ˜ θ˜ of the composite Lyapunov function (5.11) with the Bregman divergence of a strictly convex function ϕ [36], dϕ (θ s , θˆ s )  ϕ(θ s ) − ϕ(θˆ s ) − (θ s − θˆ s ) ∇ϕ(θˆ s ). Consequently, the following Lyapunov function is obtained Vϕ = V + dϕ (θ s , θˆ s ).

(5.54)

where V satisfies (5.51)–(5.53). Furthermore, define the set of possible parameter solutions as  {θˆ s ∈ R p | s = Θ(x, u)θˆ s }, where s  φ(x, u)θ. Then, we can establish the following result. Theorem 5.7 Consider the system 5.49, the state observer 5.50, which satisfies the conditions (5.51)–(5.53) for a certain Lyapunov function. Moreover, consider the following natural gradient-like adaptation law

5.2 Direct Adaptive Observer Redesign

115

 −1 ˙θˆ = ∇ 2 ϕ(θˆ ) g(˜y, y, u, xˆ ) . s s

(5.55)

Then, the state-estimation error, x˜ , converges asymptotically to zero. Moreover, consider that θˆ s (ts ) = minq ϕ(θ s ). Then, for a sufficiently large ts , the θ s ∈R

parameter-estimation, θˆ s , converges to min ϕ(θ s ). θ s ∈

Proof First, the derivative of the Lyapunov function (5.54) satisfies the following ∂V Θ(ˆx, u)θ˜ s − θ˜ ∇ 2 ϕ(θˆ s )θ˙ˆ s V˙ϕ ≤ −α1 V (˜x) + ∂ x˜ ∂V  Θ(ˆx, u)θ˜ s − θ˜ s g(˜y, y, u, xˆ ) = −α1 (˜x) + ∂ x˜ = −α1 V (˜x). Then, following a similar development as in Theorem 5.1 proof (combination of a weak Lyapunov function and the Barbalat’s lemma) it is possible to show that x˜ converges asymptotically to zero and θˆ converges to the set . For the second part of the second part of the proof, it is assumed that t > ts for a sufficiently large ts , such that x˜ ≈ 0. Then, taking into account (5.55) and (5.53), the time derivative of the Bregman divergence is   d d ˆ ˆ dϕ (θ s , θ s ) = − ∇ϕ(θ ) θ˜ s dt dt  −1 ∂V Θ(x, u)θ˜ s . = − ∇ 2 ϕ(θˆ s ) g(˜y, y, u, xˆ )θ˜ s = − ∂ x˜

(5.56)

The integration of the expression (5.56) ∀t > ts can be approximated as dϕ (θ s , θˆ s (ts )) = dϕ (θ s , θˆ s (t)) −



t ts

∂V Θ(x, u)θ˜ s (τ )dτ. ∂ x˜

(5.57)

Following a similar reasoning as in Theorem 5.1 proof it can be shown that θˆ s and θ˜ s converge to a constant. Thus, one can compute the limit as t → ∞ of (5.57) and conclude that the following holds dϕ (θ s , θˆ s (ts )) = dϕ (θ s , θˆ s (∞)) +

 ts

∞

∂V (Θ(x, u)θˆ s (τ ) − s(τ ))dτ. ∂ x˜

(5.58)

The right hand-side of (5.58) has a minimum over θ s at θˆ s (∞). In parallel, the minimum of the left-hand side factor is obtained in arg min dϕ (θ s , θˆ s (ts )). Therefore, θ s ∈

116

5 Adaptive Observers: Direct and Indirect Redesign

θˆ s (∞) = min dϕ (θ s , θˆ s (ts )).

(5.59)

θ s ∈

By considering that the parameter-estimation satisfies θˆ s (ts ) = minq ϕ(θ s ), expresθ s ∈R

sion (5.59) reduces to θˆ s (∞) = min ϕ(θ s ). θ s ∈



The results of Theorem 5.7 can be interpreted as follows. First, similar to the direct adaptive redesign presented at the beginning of Sect. 5.2, the state-estimation error converges to zero and the parameter estimation converges to the set , i.e. it interpolates the model uncertainty. This is a stronger result than other regularization schemes, e.g. [37], where the presence of the regularization term introduced a bias in the state-estimation. Second, the parameter-estimation converges to a subset of that depends on the strictly convex function ϕ. Therefore, even if the regressor vector, Θ(x, u) is not persistently exciting, the problem is implicitly regularized and may still retrieve the true value of the parameters The only part that remains to be specified is the potential function ϕ. It has been established that a remarkable property of the vector parameter θ s is its sparsity, i.e. some of its entrances are zero. Consequently, this section suggests implementing the l1 norm as the function ϕ in order to promote sparsity in the parameter-estimation solution [34]. Specifically, ϕ = |θˆ s |1 . Remark 5.7 The l1 norm is not strictly convex, which prevents Vϕ to be a Lyapunov function. For this reason, ϕ is implemented as the norm |θˆ s | p with p = 1 + , where is small. This is the closest strictly convex norm to the l1 regularization. It should be remarked that the inverse of the Hessian in (5.55), may not be computable for all p norms. For this reason, the function ϕ is implemented as the squared 1 ˆ2 p norm |θ| , with p close to one. The motivation behind such decision is that the 2 p Jacobian of the squared p norm has an analytical inverse, which allows the adaptation (5.55) to be implemented through the following dynamics, ˙ = g(˜y, y, u, xˆ ) w θˆs,i =

d−1 |w|2−d sign(wi ) d |wi |

(5.60) for i = 1, . . . , q

(5.61)

1 1 where w  ∇ϕ(θˆ s ), | · |d is the d-norm and + = 1. d p In the following, we consider a variation of the example considered in page 97. Now, it is assumed that the vector φ(x) is unknown. Instead, the following library of non-linear function candidates is considered

5.2 Direct Adaptive Observer Redesign

117

Θ(x) = [1, x1 , x2 , x1 x2 , x12 x2 , x1 x22 , x12 x22 , x12 , x22 , sin(x1 ), sin(x2 ), cos(x1 ), cos(x2 ) sin(x1 ) cos(x2 ), cos(x1 ) sin(x2 )]. This library satisfies Assumption 5.2.3. Thus, system can be rewritten as follows x˙ = f(x) + BΘ(x)θ s where f and B are defined at the beginning of Example ?? and θ s = [θs,1 , . . . , θs,15 ] is a vector of zeros except for θs,2 = −0.2, θs,3 = 1, θs,5 = −0.3. Finally, it is assumed that the states are unknown, and only the signal y = c x = x1 + x2 is being measured. The objective is to design an adaptive observer that through the measurement of y can reconstruct the unknown state, x, and parameters, θ s . The parameter-estimation should be close enough to the true value so it is possible to identify the active nonlinear functions of the library, Θ. Following the insights presented in [11], it is possible to prove that the observer dynamics (5.62) x˙ˆ = f(ˆx) + Bφ(ˆx)θˆ + l(y − c xˆ ) with l = [−0.314, 3.156] , satisfy the conditions (5.51)–(5.53) for a Lyapunov function of the form V = x˜  P˜x where P ∈ R2×2 is a symmetric positive definite matrix. The key step in the direct adaptive observer redesign is finding a function g(˜y, y, xˆ ) that satisfies (5.53). Notice that in the considered example, the term in the left-hand side of (5.53) is ∂V φ(ˆx) = x˜  PBφ(ˆx)θ˜ . (5.63) ∂ x˜ The aim is to find a function g which only depends on the measured signals that is equivalent to the expression in (5.63). To find such function, consider a constant M ∈ R such that (5.64) B P = Mc. Then it can be seen that a function g that satisfies (5.53) is the following g(˜y, xˆ ) = y˜  Mφ(ˆx),

(5.65)

According to Theorem 5.1, the adaptation dynamics in (5.10) should provide a relatively accurate estimation of the states. However, this standard adaptation dynamics

118

5 Adaptive Observers: Direct and Indirect Redesign

Fig. 5.6 parameterestimation evolution through the standard adaptive observer. No measurement noise is considered v = 0

Fig. 5.7 Evolution of the system’s states and sparsity-promoting adaptive observer estimation

cannot identify the active non-linear functions of Θ and the estimation θˆ s does not converge to the true value. This fact can be seen in Fig. 5.6, where it is depicted the parameter-estimation evolution for the standard adaptive observer in the noiseless case (v = 0). It is noticeable that there are parameters, which are not θs,2 , θs,3 or θs,5 , that converge to a non-zero value. Moreover, the parameters θs,2 , θs,3 or θs,5 do not converge to its true value. It should be remarked that, the original regressor vector Bφ(x) is persistently excited in the considered simulation. This fact exemplifies the standard adaptive observer inability to recover the active functions of Θ. Nonetheless, the parameter vector, θ s , is somewhat sparse. Thus, it is reasonable to think that the proposed sparsity-promoting adaptive dynamics (5.55) should out-perform the standard adaptive observer.

5.3 Indirect Adaptive Observer Redesign

119

Fig. 5.8 parameter-estimation evolution through the sparsity-promoting adaptive observer

Indeed, a second simulation has been designed for the same system. In this simulation the measurement is corrupted with Gaussian noise with variance 0.01. Moreover, the parameter-adaptation (5.10) is substituted by the natural gradient-like adaptation (5.60)–(5.61) with p = 1.1. As it can be seen, the sparsity-promoting adaptive observer achieves an accurate estimation of the system states, even in the presence of significant measurement noise, Fig. 5.7. Furthermore, in Fig. 5.8 it is depicted the evolution of the parameter-estimation for the natural gradient-like adaptation (3.45)–(3.46). It can be seen that the algorithm recovers the active nonlinear functions of Θ, as the parameters related to the non-active functions converge to zero. Moreover, the relative error1 between the estimation of θs,2 , θs,3 , θs,5 and its true value converges below the 2%.

5.3 Indirect Adaptive Observer Redesign The direct adaptive redesign presented in the last section is based on cancelling in the Lyapunov equation derivative the factors that depend on the unknown parameters. Although this approach presented positive results in certain cases, its implementability strongly depends on the type of observer, the system and the considered Lyapunov function. This fact is concerning, because a slight modification on the model equations or the observer may require a large redesign of the adaptive observer. For this reason, in some applications, it may be interesting to have an adaptive redesign in which the parameter dynamics design is decoupled from the observer design, which leads to more flexible schemes. Again, this section assumes that there is an already existing observer with some part of the uncertainty parametrized by θ . The idea that follows this section is to, first, define a parameter estimation scheme that generates an accurate estimation of 1

The relative error [%] between x and xˆ is computed as

|x − x| ˆ · 100. |x|

120

5 Adaptive Observers: Direct and Indirect Redesign

the parameters in the case that the states, x, are exactly known. Second, develop the conditions in which such convergence is preserved if the only an estimation of the state, xˆ , is known. It should be remarked that if there is no strict limitations on the type of system, observer or Lyapunov function (as in the direct adaptive redesign case), the second step requires the parameter adaptation dynamics to be somewhat robust to the mismatch, x = xˆ . To achieve such property, this section proposes using the Immersion and Invariance (I&I) framework [38–40]. Specifically, to simplify the section, consider an autonomous system of the form x˙ = f(x) + φ(x)θ

(5.66)

y = h(x) where θ ∈ Rn θ is a vector of unknown constant parameters to be estimated. The functions f and φ are assumed to be known and Lipschitz. Moreover, there exists some sets X0 ⊆ X ⊆ Rn , such that the trajectories of (5.5), with initial conditions x(0) in X0 , remain in X for all t ≥ 0. The idea is to design some parameter adaptation dynamics, θ˙ˆ , such that the following manifold M  {(x, θˆ ) ∈ Rn × Rn θ | θˆ − θ + β(x) = 0}

(5.67)

where β : Rn → Rn θ is a Lipschitz function to be designed, is made attractive and invariant. Naturally, if the system evolves in the manifold M, the unknown parameters can be retrieved as θ = θˆ + β(x). This idea can be formalized through the following theorem. Theorem 5.8 Consider a system of the form (5.66). Moreover, let the parameters be adapted through the following dynamics   ˙θˆ = − ∂β f(x) + φ(x)(θˆ + β(x)) . ∂x Then, the manifold in (5.67) is attractive and invariant if the following dynamics are stable   ∂β φ(x) z. (5.68) z˙ = − ∂x Proof Define the off-the-manifold variable z ∈ Rn θ as z  θˆ − θ + β(x).

5.3 Indirect Adaptive Observer Redesign

121

The derivative of the off-the-manifold variable satisfies       ∂β ∂β ∂β ∂β f(x) + φ(x)(θˆ + β(x)) + f(x) + φ(x)θ = − z˙ = θ˙ˆ + x˙ = − φ(x) z. ∂x ∂x ∂x ∂x

Therefore, if the dynamics in (5.68) are stable, the off-the-manifold variable z converges to the origin and the manifold in (5.67) is attractive and invariant.  The next natural question is how to design the function β such that the dynamics in (5.68) are stable. A sufficient condition can be obtained if the function φ is a gradient function, i.e. there exists a function b :∈ Rn → R such that φ(x) =

∂b . ∂x

Remark 5.8 From Poincaré’s Lemma, a necessary and sufficient condition for φ to be gradient function is that   ∂φ ∂φ = . (5.69) ∂x ∂x Then, the function β can be designed as β(x) = b(x), which, by substituting in (5.68), leads to the following z dynamics z˙ = −φ(x) φ(x)z.

(5.70)

Finally, similar to Theorem 3.4 in Chap. 3, the stability of the z dynamics can be established if the vector φ is persistently excited (see Definition 3.2 in Chap. 3). Remark 5.9 A similar set of dynamics can be obtained by implementing a set of low-pass filters [41], at a cost of deteriorating the transient of the estimator due to the set of filters initial conditions. It is noticeable that the off-the-manifold variable convergence relies on the persistence of excitation condition, similar to the direct adaptive redesign case, see Theorem 5.2. This is an unsurprising result, as it is hardly possible to recover unknown parameters without some kind of excitation. The main difference between the direct and the indirect approach is that the dynamics in (5.70) are more easily tractable, and does not require constructing a complex strict Lyapunov function as in Sect. 5.2.1 to analyse it. Indeed, there are previous results [42, 43] that shows the robustness of the dynamics in (5.70) in front of disturbances. Precisely, assume that the dynamics in (5.70) are disturbed as z˙ = −φ(x) φ(x)z + w

122

5 Adaptive Observers: Direct and Indirect Redesign

where w ∈ Rn θ is some bounded disturbance. Then, if the vector φ is persistently exciting, it can be shown that the variable z is ultimately bounded for all t > ts , where ts is a sufficiently large positive constant, as |z| ≤ γ |w| where γ is a positive constant. This fact can be obtained from the Lyapunov development in [42]. This property is really useful. In practice, the estimator in (5.69) cannot be implemented as the states, x, are unknown. Thus, the parameter estimator has to be implemented with the state-estimation generated by the observer, xˆ , i.e.   ∂β(ˆx) f(ˆx) + φ(ˆx)(θˆ + β(ˆx)) . θ˙ˆ = − ∂x

(5.71)

which results in a z dynamics with a disturbance term that depends on the mismatch x = xˆ . Specifically, z˙ = −φ(x) φ(x)z + δ where       ∂β(x) ∂β(ˆx) ˆ ˆ f(x) + φ(x) θ + β(x) − f(ˆx) + φ(ˆx) θ + β(ˆx) . δ ∂x ∂x (5.72) The functions β, f, φ and

∂β(x) are (locally) Lipschitz. Moreover, the factors ∂x

∂β(x) , φ and θˆ are upper-bounded. Therefore, there exist some positive constant ∂x L δ such that |δ| ≤ L δ |˜x|.

Therefore, if the system is persistently excited, the z variable will be ultimately bounded as (5.73) |z| ≤ γ L δ |˜x| This allows to formalize the stability of the coupling between I&I parameter estimator in (5.71) and an observer for system (5.66) through the following theorem. Theorem 5.9 Consider an observer for system (5.66) with a linear ISS gain ϑ1 with respect to the off-the-manifold variable z. Moreover, assume that θˆ is generated

5.3 Indirect Adaptive Observer Redesign

123

through the I&I estimator (5.71) with a function β such that the dynamics in (5.68) are stable. Then, the off-the-manifold variable asymptotically converges to zero if ϑ1 γ L δ ≤ 1.

(5.74)

Proof As the state-estimation error presents a linear ISS gain ϑ1 with respect to the off-the-manifold variable, the observer estimation-error is ultimately bounded as |˜x| ≤ ϑ1 |z|. By substitution in (5.73), it can be established that z is ultimately bounded as |z| ≤ ϑ1 γ L δ |z|, which defines a contraction if condition (5.74) is satisfied. Then, by means of the small gain theorem [28], it can be shown that z asymptotically converges to zero.  Consider a second order nonlinear system of the form x˙ = f(x) + φ(x)θ y = cx where x = [x1 , x2 ] , θ = 0.4 and  f(x) =

     0 x2 , c = 1 0 , φ(x) = . −x1 + 1 − sin(x1 )

  The system is initialized at x = 1 0 , which induces a trajectory that satisfies x1 > 0. It is assumed that the parameter θ is unknown. Notice that the system presents a triangular structure, thus, it is relatively easy to design a high-gain observer, see Sect. 3.2 in Chap. 3. The objective of this example is to redesign said high-gain observer following the indirect redesign approach presented in this section. Firstly, it is necessary to establish a I&I estimator of the parameter θ in the fullinformation case, i.e. x is known. Precisely, notice that, from the structure of φ and for any vector β, the z dynamics satisfy the following z˙ =

∂β sin(x1 )z. ∂ x2

This motivates the design of β = −x2 sin(x1 ), which would result in the following z dynamics z˙ = − sin(x1 )2 z.

124

5 Adaptive Observers: Direct and Indirect Redesign

As the system trajectory satisfies x1 > 0, the function sin(x1 ) is persistently excited (as it is not L2 integrable) and the resulting z dynamics are stable. Moreover, from the insights presented in this section, if the estimator is implemented as in (5.71) the z dynamics are ISS with respect to the state-estimation error x1 − xˆ1 with some linear ISS gain γ . Second, it is necessary to establish that the high-gain observer with the estimation generated by the I&I estimator is ISS with respect to the off-the-manifold variable z. Precisely, the high-gain observer presents the following form α1 x˙ˆ1 = xˆ2 + (y − xˆ1 ) ε α2 ˆ sin(xˆ1 ) + 2 (y − xˆ1 ) xˆ˙2 = −xˆ1 + 1 − (θˆ + β(x)) ε where θˆ is generated through (5.71). Notice that the xˆ2 -dynamics can be rewritten as α2 x˙ˆ2 = −xˆ1 + 1 − θ sin(xˆ1 ) + 2 (y − xˆ1 ) − z sin(xˆ1 ), ε where the term z sin(x1 ) can be interpreted as a disturbance. Section 3.2 in Chap. 3 have already established that high-gain observers are ISS with respect to disturbances in the last equation and the ISS gain can be arbitrary reduced by decreasing the term ε. Indeed, as sin(xˆ1 ) is a bounded signal, it is possible to prove that the high-gain observer estimation is ultimately bounded as follows |˜x| ≤ εk|z| where k is some positive constant. Finally, as the z dynamics are ISS with respect to the high-gain observer estimation-error and the observer estimation-error is ISS with respect to the off-themanifold variable, according to Theorem 5.9, a sufficient condition for the stability of the coupling is that εkγ ≤ 1. Thus, for a sufficiently small high-gain parameter ε, the coupling is stable. Precisely, the state-estimation error and the parameter estimation-error is depicted in Figs. 5.9 and 5.10, respectively. It can be observed that all the estimation-errors converges asymptotically to zero, which validates the proposed approach.

References

125

Fig. 5.9 Evolution of the state-estimation error

Fig. 5.10 Evolution of the parameter estimation-error. In this case, θ˜ = θ − θˆ − β(x) ˆ

References 1. Narendra KS, Annaswamy AM (2012) Stable adaptive systems. Courier Corporation 2. Afri Chouaib, Andrieu Vincent, Bako Laurent, Dufour Pascal (2016) State and parameter estimation: a nonlinear Luenberger observer approach. IEEE Trans Autom Control 62(2):973– 980 3. Ortega R, Gromov V, Nuño E, Pyrkin A, Romero JG (2021) Parameter estimation of nonlinearly parameterized regressions without overparameterization: application to adaptive control. Automatica 127:109544. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2021.109544 4. Tyukin IY, Prokhorov DV, van Leeuwen C (2007) Adaptation and parameter estimation in systems with unstable target dynamics and nonlinear parametrization. IEEE Trans Autom Control 52(9):1543–1559. https://doi.org/10.1109/TAC.2007.904448 5. Wang L, Ortega R, Su H, Liu Z, Xu W (2014) On parameter convergence of nonlinearly parameterized adaptive systems: analysis via contraction and first Lyapunov’s methods. In: 53rd IEEE conference on decision and control, pp 539–544. https://doi.org/10.1109/CDC. 2014.7039437 6. Tyukin IY, Steur E, Nijmeijer H, Van Leeuwen C (2013) Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters. Automatica 49(8):2409–2423. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2013.05.008 7. Grip HF, Johansen TA, Imsland L, Kaasa GO (2010) Parameter estimation and compensation in systems with nonlinearly parameterized perturbations. Automatica 46(1):19–28. ISSN 00051098. https://doi.org/10.1016/j.automatica.2009.10.013 8. Cho YM, Rajamani R (1997) A systematic approach to adaptive observer synthesis for nonlinear systems. IEEE Trans Autom Control 42(4):534–537. https://doi.org/10.1109/9.566664 9. Besançon G (2000) Remarks on nonlinear adaptive observer design. Syst Control Lett 41(4):271–280. ISSN 0167-6911. https://doi.org/10.1016/S0167-6911(00)00065-7 10. Sastry S, Bodson M (2011) Adaptive control: stability, convergence and robustness. Courier Corporation

126

5 Adaptive Observers: Direct and Indirect Redesign

11. Rajamani R (1998) Observers for Lipschitz nonlinear systems. IEEE Trans Autom Control 43(3):397–401. https://doi.org/10.1109/9.661604 12. Sontag ED (2008) Input to state stability: basic concepts and results. In: Nonlinear and optimal control theory. Springer, Berlin, pp 163–220 13. Mazenc F (2003) Strict Lyapunov functions for time-varying systems. Automatica 39(2):349– 353. ISSN 0005-1098. https://doi.org/10.1016/S0005-1098(02)00233-9 14. Loría A, Panteley E, Maghenem M (2019) Strict Lyapunov functions for model reference adaptive control: application to Lagrangian systems. IEEE Trans Autom Control 64(7):3040– 3045. https://doi.org/10.1109/TAC.2018.2874723 15. Rueda-Escobedo JG, Moreno JA (2021) Strong Lyapunov functions for two classical problems in adaptive control. Automatica 124:109250. ISSN 0005-1098. https://doi.org/10.1016/ j.automatica.2020.109250 16. Anderson BD, Bitmead RR, Johnson Jr CR, Kokotovic PV, Kosut RL, Mareels IM, Praly L, Riedle BD (1986) Passivity and averaging analysis: stability of adaptive systems. MIT Press 17. Ioannou PA, Kokotovic PV (1984) Instability analysis and improvement of robustness of adaptive control. Automatica 20(5):583–594. ISSN 0005-1098. https://doi.org/10.1016/00051098(84)90009-8 18. Rohrs C, Valavani L, Athans M, Stein G (1985) Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics. IEEE Trans Autom Control 30(9):881–889. https://doi.org/10.1109/TAC.1985.1104070 19. Bastin G, Gevers MR (1988) Stable adaptive observers for nonlinear time-varying systems. IEEE Trans Autom Control 33(7):650–658. https://doi.org/10.1109/9.1273 20. Marino R, Tomei P (1992) Global adaptive observers for nonlinear systems via filtered transformations. IEEE Trans Autom Control 37(8):1239–1245. https://doi.org/10.1109/9.151117 21. Zhang Q (2005) Revisiting different adaptive observers through a unified formulation. In: Proceedings of the 44th IEEE conference on decision and control, pp 3067–3072. https://doi. org/10.1109/CDC.2005.1582632 22. Efimov D, Edwards C, Zolghadri A (2016) Enhancement of adaptive observer robustness applying sliding mode techniques. Automatica 72:53–56. ISSN 0005-1098. https://doi.org/10. 1016/j.automatica.2016.05.029 23. Ortega R, Panteley E (2016) Adaptation is unnecessary in l1-adaptive control: what makes an adaptive controller “adaptive”? IEEE Control Syst Mag 36(1):47–52. https://doi.org/10.1109/ MCS.2015.2495023 24. Stamnes O, Aamo OM, Kaasa G (2011) Adaptive redesign of nonlinear observers. IEEE Trans Autom Control 56(5):1152–1157. https://doi.org/10.1109/TAC.2011.2107090 25. Bin Michelangelo, Marconi Lorenzo (2021) Model identification and adaptive state observation for a class of nonlinear systems. IEEE Trans Autom Control 66(12):5621–5636. https://doi. org/10.1109/TAC.2020.3041238 26. Dimassi H, Hadj Said S, Loria A, M’Sahli F (2021) An adaptive observer for a class of nonlinear systems with a high-gain approach: application to the twin-rotor system. Int J Control 94(2):370–381. https://doi.org/10.1080/00207179.2019.1594387 27. Levant Arie (2003) Higher-order sliding modes, differentiation and output-feedback control. Int J Control 76(9–10):924–941. https://doi.org/10.1080/0020717031000099029 28. Jiang Z, Teel A, Praly L (1994) Small-gain theorem for ISS systems and applications. Math Control Signals Syst 7:95–120. https://doi.org/10.1007/BF01211469 29. Bernard Pauline, Mimmo Nicola, Marconi Lorenzo (2021) On the semi-global stability of an EK-like filter. IEEE Control Syst Lett 5(5):1771–1776. https://doi.org/10.1109/LCSYS.2020. 3044030 30. Besançon G, Zhang Q, Hammouri H (2004) High-gain observer based state and parameter estimation in nonlinear systems. IFAC Proc Vol 37(13):327–332. https://doi.org/10.1016/S14746670(17)31244-2 (6th IFAC symposium on nonlinear control systems 2004 (NOLCOS 2004), Stuttgart, Germany, 1–3 Sept 2004) 31. Cheney EW, Light WA (2009) A course in approximation theory, vol 101. American Mathematical Society

References

127

32. Farrell JA, Polycarpou MM (2006) Adaptive approximation based control: unifying neural, fuzzy and traditional adaptive approximation approaches, vol 48. Wiley 33. Bobade P, Majumdar S, Pereira S, Kurdila AJ, Ferris JB (2019) Adaptive estimation for nonlinear systems using reproducing Kernel Hilbert spaces. Adv Comput Math 45(2):869–896 34. Tropp JA (2006) Just relax: convex programming methods for identifying sparse signals in noise. IEEE Trans Inf Theory 52(3):1030–1051. https://doi.org/10.1109/TIT.2005.864420 35. Pillonetto G, Dinuzzo F, Chen T, De Nicolao G, Ljung L (2014) Kernel methods in system identification, machine learning and function estimation: a survey. Automatica 50(3):657–682. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2014.01.001 36. Bregman LM (1967) The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput Math Math Phys 7(3):200–217. ISSN 0041-5553. https://doi.org/10.1016/0041-5553(67)90040-7 37. Zhang Q, Giri F, Ahmed-Ali T (2019) Regularized adaptive observer to address deficient excitation. IFAC-PapersOnLine 52(29):251–256. ISSN 2405-8963. https://doi.org/10.1016/j. ifacol.2019.12.658 (13th IFAC workshop on adaptive and learning control systems ALCOS 2019) 38. Astolfi A, Karagiannis D, Ortega R (2008) Nonlinear and adaptive control with applications, vol 187. Springer, Berlin 39. Astolfi A, Ortega R (2003) Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems. IEEE Trans Autom Control 48(4):590–606. https://doi.org/10. 1109/TAC.2003.809820 40. Liu Xiangbin, Ortega Romeo, Hongye Su, Chu Jian (2010) Immersion and invariance adaptive control of nonlinearly parameterized nonlinear systems. IEEE Trans Autom Control 55(9):2209–2214. https://doi.org/10.1109/TAC.2010.2052389 41. Na Jing, Xing Yashan, Costa-Castelló Ramon (2021) Adaptive estimation of time-varying parameters with application to roto-magnet plant. IEEE Trans Syst, Man, Cybern: Syst 51(2):731–741. https://doi.org/10.1109/TSMC.2018.2882844 42. Jenkins BM, Annaswamy AM, Lavretsky E, Gibson TE (2018) Convergence properties of adaptive systems and the definition of exponential stability. SIAM J Control Optim 56(4):2463– 2484 43. Narendra K, Annaswamy A (1986) Robust adaptive control in the presence of bounded disturbances. IEEE Trans Autom Control 31(4):306–315. https://doi.org/10.1109/TAC.1986. 1104259

Chapter 6

PEM Fuel Cell Monitoring

Efficiency, reliability and lifetime of polymer electrolyte membrane fuel cells (PEMFCs) are significantly limited by inadequate water management. High-performance water active control algorithms cannot be implemented due to the absence of adequate online sensors that can measure the internal liquid water saturation. A promising technique that can be applied in this context is the state observer. However, fuel cell models present strong nonlinearities, model uncertainty, unmatched unknown parameters and sensor noise, which are major difficulties in observer design. This chapter presents multiple observer algorithms for the estimation of the liquid water saturation. The algorithms are shown to provide an accurate estimation of the liquid water saturation and the liquid water transport parameters even in the presence of sensor noise and model inaccuracies. The results are validated through numerical simulations and in a real experimental prototype.

6.1 Introduction and PEM Fuel Cell Principles Hydrogen has been established as a key element to tackle the present critical energy challenges. However, to make a significant contribution to a clean energy transition, it is crucial that hydrogen is incorporated in strategic sectors as transport, buildings and stationary power generation. A promising device to promote such introduction are fuel cells. A hydrogen fuel cell is an electrochemical device that converts the chemical energy of hydrogen in DC current without relying on moving parts and emission of pollutants, which makes it a promising alternative to traditional internal combustion engines. Among the different types of fuel cell, polymer electrolyte membrane fuel cells (PEMFC) are considered as the most promising fuel cell candidate in transport and stationary backup power applications, due to is low-operating temperature, highenergy density, quick start-up and zero to low emissions [1]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Cecilia, Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems, Springer Theses, https://doi.org/10.1007/978-3-031-38924-5_6

129

130

6 PEM Fuel Cell Monitoring

Fig. 6.1 General scheme of a single PEM fuel cell operation

Precisely, a PEMFC consists of a solid polymer that is used as an electrolyte between the anode and the cathode. The fuel cell anode is constantly delivered with pure hydrogen. This hydrogen is processed at a platinum based catalyst layer, which separates the H2 into protons and electrons. 2H2  4H + + 4e− . The protons travel to the cathode catalyst layer through the membrane. However, due to the membrane ionic properties, the electrons are forced to travel through an external circuit, which generates the electrical load of the device. In parallel, the cathode is feed with pure oxygen or air, which flows to the cathode catalyst layer. In this layer, the oxygen is combined with the protons to generate water and heat, O2 + 4H + + 4e−  2H2 O,

(6.1)

which closes the overall reaction. H2 +

1 O2 → H2 O + Electrical Ener g y + H eat. 2

(6.2)

Figure 6.1 depicts a schematic of the PEM fuel cell operation. In high power demand fuel cell systems, a compressor is used to supply pressurized oxygen to the cathode channel, and a compressed air cylinder is used to supply pure hydrogen to the anode. The advantage of this fuel cell operation is that the high pressure aides the reactant delivery through the cell channels. However, the cathode compressor presents a parasitic power draw of the order of 20% [2]. An alternative is the open-cathode architecture, in which the fuel cell is not pressurized, and the oxygen is obtained through the ambient air. The air flow is forced though a fan, which has drastically less parasitic power draw than compressor systems. However, the open-cathode architecture is significantly more complex to control, as there is a

6.2 PEM Fuel Cell Model

131

coupling between reactant delivery and fuel cell cooling, which motivates the design of advanced control algorithms. In practice, degradation problems limit the economical and technological viability of PEMFCs [3]. For example, transport applications require variable operating conditions, which induces significant fluctuations in the fuel cell internal states. Inadequate operating conditions and dynamics will eventually lead to performance and components degradation [4]. Some crucial internal variables cannot be measured due to physical, economical and technical limitations. In such situations, a viable approach is to develop an online estimation algorithm to monitor the internal variables. Following from the insights presented in this work, this chapter will focus on developing a fuel cell monitoring algorithm by means of nonlinear observers.

6.2 PEM Fuel Cell Model Naturally, before developing a monitoring algorithm for the fuel cell system, it is necessary to realize a mathematical model for the system. In the process of system modelling, a crucial step is the selection of the appropriate model type. Between others, models can be classified as physical-based (or white-models) and black-box models [5]. The former ones are built based on knowledge of the first-principles that describe behaviour of the system. The resulting model depends on unknown parameters which have direct physical meaning (for example, friction coefficients). For this reason, white models are highly preferred on the industry. The latter focuses on describing the input-output behaviour of the system and are constructed upon purely experimental data. The interest on these type of models arise when there is insufficient knowledge of the dynamics that govern the concerned system. Relative to PEM fuel cell systems, although there are still some unknowns, the underlying physics are well understood and are reported in the literature. For this reason, this chapter will develop a physics-based model of the fuel cell. Following this line, the objective is to define a low-order nonlinear model for the PEM fuel cell described through a set of ordinary differential equations (ODEs) derived from the system first principles. The motivation is twofold. First, nonlinear models commonly present richer dynamics than its linearized counterparts, which allows the development controllers and estimators with higher performance. Second, although the topic of observer design for PDE systems is rapidly growing, see for example the extension of the high-gain observer for a class of hyperbolic PDE [6], its extension to standard PEM fuel cell models is still an open problem. Third, high order nonlinear models are practically intractable from a control-theoretic perspective. Indeed, for the topic of observer design, nonlinear observability analysis does not scale adequately with the dimension of the system [7], nonlinear observer design also does not scale [8] and some common nonlinear observer approaches present low performance in high-dimensional systems, e.g. in high-gain observers the noise sensitivity scales with the system order, see Sect. 3.2 of Chap. 3. Finally, low-order

132

6 PEM Fuel Cell Monitoring

models are simpler to calibrate and validate, which simplifies the identification step in the observer design workflow. The idea is to start from the governing equations of the system and solve such equations in a small amount of control volumes, in order to get a tractable model without compromising relevant information of the fuel cell dynamics. Following this strategy, the fuel cell model presented in this work is a semi-empirical, dynamic, pseudo-1D cell-level model that depicts a non-isothermal, two-fluid model for a parallel channel configuration open-cathode fuel cell. The model consists of a submodel for the flow channels, a sandwich sub-model that describe the reactants and water transport across the porous media and the membrane, a voltage sub-model and a thermal sub-model. It should be remarked that this model is an extension of the one presented and validated [9].

6.2.1 Governing Equations The model solves a set of transport partial differential equations that describe: the distribution of vapour water, the distribution of oxygen, the membrane absorbed water content, the distribution of liquid water, the distribution of hydrogen and the cell temperature distribution. These variables are later used to compute the fuel cell stack voltage through a static equation. The motivation behind these equations will be explained through the section. Specifically, the governing equations are the following ∂ρv ∂t ∂ρ O2 ε ∂t ∂s ρl ε ∂t ρmem,dr y ∂λ E W ∂t ∂T (ρC p )e f f ∂t ε

ef f

(6.3)

= ∇ · (D O2 ,e ∇ · ρ O2 ) + M O2 SO2   ef f ρl K l ∂ Pl =∇· ∇ · s + Sl μl ∂s   ρmem,dr y j = ∇ · nd − Dm,H2 O ∇ · λ + Sλ F EW

ef f

(6.4)

= ∇ · (K T ∇ · T ) + ST

(6.7)

= ∇ · (D H2 O,e ∇ · ρv ) + M H2 O Sw

(6.5) (6.6)

where εi (−) is the porosity/volume fraction of the phase i, ρv (kg m3 ) is the density ef f of water vapour, D H2 O,e (m 2 s −1 ) is the effective water vapour diffusivity (computed with the molecular and Knudsen diffusivity), ρ O2 (kg m3 ) is the density of oxygen, ef f D O2 ,e (m 2 s −1 ) is the effective oxygen diffusivity, M H2 O is the water molar mass, M O2 is the oxygen molar mass, ρl (kg m−3 ) is the liquid water density, μl kg m−1 s −1 ) is the dynamic viscosity, s (−) is the liquid water volume fraction in the porous media, ef f K l (m2 ) is the effective permeability, Pl (Bar ) is the liquid pressure (computed through the capillary pressure), ρmem,dr y (kg m−3 ) is the dry membrane density,

6.2 PEM Fuel Cell Model

133

E W (kg mol−1 ) is the ionomer equivalent weight, λ is the membrane water content, n d (−) is the electro-osmotic drag coefficient, Dm,H2 O (m 2 s −1 ) is the membrane back diffusion coefficient (Fickian diffusion), C p (J K −1 kg−1 ) is the specific heat capacity and K T (W m−1 K−1 ) is the thermal conductivity. The factors Si represent different source/sink terms which will be presented throughout the section. In parallel, the reactant transport in the fuel cell channels is modelled by a general mass balance that depicts, for each reactant, the mass that enters and leaves the channel. Formally, as the gas composition is assumed to be ideal and homogeneously mixed, diffusive flux along the channel is neglected and the gas composition changes are driven by bulk convective flow. In each channel, the species concentration is obtained by solving a mass balance of the following form  ρg V

∂ (yi )d V = −ρg ∂t



 (n · u)d A + A

Sq d V

(6.8)

V

where ρg (Kg m−3 ) is the gas density, yi (−) is the mass fraction of the specie i, Sq are the source terms, V (m 3 ) is the volume of the channel, n is the vector normal to the cross-section of the channel and u is the flow velocity in the channel.

6.2.2 Control Volumes The objective is to define a low order nonlinear model for the PEM fuel cell. Consequently, the considered number of control volumes have been minimized, see Fig. 6.2. The membrane has been divided in two different control volumes. This allows to describe the spatial dependant transport mechanisms, e.g. back-diffusions. The gas diffusion layer (GDL), micro-porous layer (MPL) and the catalyst layer (CL) have been grouped in the same control volume. This allows to separate the membrane water content, the porous media water content and the channel water content, without solving the transport equations in the GDL, CL and MPL separately. Each channel will be solved as a different control volume. Naturally, not all the species are present in all the control volumes. Thus, each governing equation is solved in a set of control volumes. In Table 6.1 it is summarized the variables that are being solved in each control volume.

6.2.3 Membrane Sub-model PEM fuel cells commonly implement a perfluorosulfonic acid polymer membrane, being the most famous one the Nafion membrane, due to its high proton conductivity and relatively high durability [10]. The ionomer of the membrane contains hydrophilic clusters with H + S O3− . These clusters can absorb water to generate

134

6 PEM Fuel Cell Monitoring

Fig. 6.2 PEMFC one dimensional domain and considered control volumes used to discretize the PDEs Table 6.1 Model solution variables Solution variable

Anode chan.

Anode PM

ρv

 Eq. (6.40)

 Eq. (6.25)

PEM anode

PEM cathode

ρ O2 λ

 Eq. (6.17)

ρ H2

 Eq. (6.39)

T

 Eq. (6.43)

 Eq. (6.45)

Cathode chan.

 Eq. (6.26)

 Eq. (6.36)

 Eq. (6.29)

 Eq. (6.38)

 Eq. (6.16)

 Eq. (6.34)

s

Cathode PM

 Eq. (6.35)  Eq. (6.45)

 Eq. (6.45)

 Eq. (6.45)

 Eq. (6.41)

hydrated hydrophilic regions in which the hydrogen protons are weakly attracted to the S O3− and are easily transported in the form of hydronium ions, H3 O + . The basis of PEM fuel cell operation is the transport of hydrogen protons from the anode catalyst layer to the cathode catalyst layer through the Nafion membrane. Consequently, the membrane needs to be well hydrated to allow adequate proton transport and correct fuel cell operation. For this reason, a key variable to model in the membrane control volume is the water absorption level, which is often depicted as the number of water molecules per molecule of S O3− , referred to as the water content, λ.

6.2 PEM Fuel Cell Model

135

The membrane sub-model focuses in modelling the generation and transport of the water content, λ, in the membrane. The dynamics of this variable are described by the equation in (6.6). The aim of this sub-section is to explain the factors that appear in Eq. (6.6) and explain how is the equation solved in the proposed control volumes. It should be remarked that this model neglects the oxygen and hydrogen membrane crossover. Therefore, these variables are not solved in the membrane. The dynamics of the water content can be mainly explained by the factors related to the water transport across the membrane and the factors related to the sorption/desorption of water from the porous media.

6.2.3.1

Water Transport Across the Membrane

The major water transport mechanisms across the membrane are the back diffusion and the electro-osmotic drag [11]. In reality, there are additional phenomena that induce a water movement in the membrane. Some notorious examples are the thermal osmotic drag induced by the temperature gradient between the anode and cathode [12] or the hydraulic permeation that is generated by a pressure gradient between the anode and cathode [13]. Nonetheless, the value of these additional factors is order of magnitudes lower than the back diffusion or electro-osmotic drag value, and can be obviated. PEM fuel cells are commonly operated with dry hydrogen in the anode channel and humidified air in the cathode channel. Moreover, the oxygen reduction reaction, which generates water, takes place in the cathode catalyst layer. Consequently, a fuel cell system usually present a higher quantity of water in its cathode side. For this reason, a net water flux may be generated through the membrane due to the water content gradient. This process is usually referred to as back diffusion and is computed through Fick’s law ρmem,dr y Dm,H2 O ∇λ (6.9) − EW where ρm (kg m−3 ) is the density of the dry membrane, E W (kg kmol−1 ) is the equivalent weight and Dm,H2 O (m 2 s −1 ) is the diffusion coefficient in the membrane. The diffusion coefficient is computed through an Arrhenius equation derived from experimental data    1 1 , − Dm,H2 O = Dλ ex p 2416 303 Ts

(6.10)

where Dλ is the back diffusion coefficient at a reference temperature of 303 K and Ts is the MEA temperature. This model computes the back diffusion coefficient through the empirical relation developed in [14],

136

6 PEM Fuel Cell Monitoring

Dm,H2 O

⎧   2436 ⎪ −3 ⎪ 3.1 · 10 , 03 ⎩4.17 · 10 λ(161ex p(−λ) + 1)ex p − T

In parallel, the protons travel from the anode to the cathode in a hydronium ion form. During this process, some water molecules are dragged from the anode to the cathode. This phenomenon is referred as electro-osmotic drag [15] and is computed as a function of a drag coefficient, n d and the current density, j (A/m−2 ) nd

j . F

(6.11)

The drag coefficient, n d , is sensitive to the water content. Various correlations between membrane water content and membrane drag coefficient can be found in the literature. In this model, the following linear relationship is used [16] nd =

6.2.3.2

2.5 λ. 22

(6.12)

Sorption/desorption of Water from the Porous Media

Some models assume that water content inside the membrane is in equilibrium with the water in the porous media. Consequently, the water transport in the membrane is only defined by the back diffusion factor and the electro-osmotic drag [16]. In reality, there are some interficial resistances between the membrane and water vapour [17, 18]. Previous models have shown that the inclusion of such resistance drastically changes the dynamics of the water content [19]. Thus, it is of interest to include such interficial phenomena. To represent these interficial effects, some flux boundary conditions that model the membrane sorption/desorption of vapour water are included. These fluxes depend on the water activity in the boundary. The model includes an empirical fitting polynomial which relates the equilibrium water content with the water activity, ai [16] λeq,i

0.0043 + 17.81ai − 39.85ai2 + 36ai3 , 0 < ai ≤ 1 = 1 < ai 14 + 1.4(ai − 1),

(6.13)

where the water activity is computed as ai =

PH2 O,i . Psat,i

Then, the sorption/desorption boundary fluxes are computed as follows

(6.14)

6.2 PEM Fuel Cell Model

Si = kλ,i

137

ρm (λeq,i − λi ) i = cathode; anode, EW

(6.15)

where kλ,i (s −1 ) is the adsorption rate and λeq,i is the equilibrium membrane water content computed through (6.13). The rate of water absorption/desorption, kλ , is computed using the following empirical relation [17]

  2000 1 λVW α 1 ex p kλ = , − δC L Ve + λVW R 303 Tm where α is a fitting parameter, the factor δC L (m) is the catalyst layer thickness, the constant VW (m 3 ) is the partial molar volume of water, which is computed as VW =

M H2 O , ρl

and the constant Ve is the partial molar volume of Nafion Ve =

6.2.3.3

EW ρmem,dr y

.

Membrane Sub-model ODEs

The aim of this section was to introduce the factors that have been considered in Eq. (6.6). Notice that this equation is a PDE due to the gradient transport mechanisms of the back diffusion and electro-osmotic drag. For this reason, the membrane has been divided into 2 control volumes, which allows to model these mechanisms through a set of ODEs, see Fig. 6.2. Specifically, the first control volume is composed by the cathode-side half of the membrane and the second is composed by the anode-side half of the membrane. Cathode and anode control volumes may present different water content which are depicted by: λc and λa , respectively. Then, the Eq. (6.6) is solved as  EW 2kλ,ca j 2(λan − λca ) + n d + Dm,H2 O (λeq,ca − λca ) (6.16) δmem ρmem F δmem δmem   EW 2kλ,an j 2(λan − λca ) 2 + =− n d + Dm,H2 O (λeq,an − λan ). δmem ρmem F δmem δmem (6.17)

λ˙ ca = λ˙ an

2



where δmem (m) is the membrane thickness.

138

6 PEM Fuel Cell Monitoring

6.2.4 Porous Media Sub-model The porous media of a fuel cell is composed by three elements. First, the catalyst layer, where the main fuel cell reactions occur generating electrical power. Second, the gas diffusion layer, which provides a uniform pathway for the reactants from the channels to the catalyst layer. Third, a micro-porous layer that eases the evacuation of water from the cathode and facilitates the movement of water from the cathode side to the anode side. The porous nature of the catalyst layer, micro-porous layer and the catalyst layer induces a diffusive resistance between the fuel cell channel and the respective catalyst layer. Consequently, it is crucial to model the fuel cell porous media in order to describe the reactant dynamics between the channel and the catalyst layer. Following this line, it should come natural the idea to model each layer as a separated control volume. Nonetheless, there are some issues that discourage such a decision. First, separating each porous layer in three different control volume would drastically increase the dimension of the model. Second, certain interficial phenomena, for example cracks in the micro-porous layer surface, may be difficult to model [20]. For these reasons, this work has considered grouping the whole porous media in a single control volume and model the interficial phenomena through effective material properties of the media, see Fig. 6.2. As mentioned in Sect. 6.2.3, the oxygen and hydrogen crossover across the membrane has been neglected. Moreover, as the nitrogen does not react with any component of the system, the model does not solve for the nitrogen in the porous media. For this reason, on the one hand, only the water vapour, hydrogen and liquid water are being solved in the anode porous media. On the other hand, oxygen, water vapour and liquid water in the cathode porous media. It should be remarked that all gasses are considered to be ideal. Moreover, gas mixtures are assumed to be dilute, isobaric, and also ideal. Finally, the effect of gravitational forces are neglected for the computation of the reactant dynamics.

6.2.4.1

Water Vapour Density Dynamics

The water vapour dynamics in the anode porous media and cathode porous media are obtained by solving the Eq. (6.3). The aim of this subsection is to explain the factors that appear in this equation considering the porous media control volume. As the considered fuel cell has parallel channels, it is fairly reasonable to assume that only small inlet pressures are required for the adequate reactant flow. Consequently, water vapour transport will be diffusive dominated, where the diffusion factor is computed as follows ef f D H2 O,e ∇ · ρv . Specifically, there is a diffusion of vapour water between the porous media and the channel volume. The main mechanism of water vapour diffusion is the collision

6.2 PEM Fuel Cell Model

139

between the vapour molecules, which is referred to as binary diffusion. The binary diffusion coefficient, D B , is computed as follows [21]  D B = DrBe f

T Tr e f

1.75 

 Pr e f , P

(6.18)

where DrBe f (m 2 s −1 ) is the binary diffusion coefficient at a reference temperature, Tr e f and pressure, Pr e f . However, as the MPL and the CL present significantly small pores, there exist a second diffusive mechanism due to the collision between the vapour molecules and the walls, referred to as Knudsen diffusion. The Knudsen diffusion coefficient is computed as [22] 2r p (1.66ε1.65 + 0.289) D = 3



K

8RT π M H2 O

(6.19)

where r p (nm) is the primary-particle radius. In order to combine both types of diffusions, a combined binary and Knudsen diffusion coefficient has been computed as

D H2 O,i

⎧ B ⎪ i = G DL ⎨ Di ,   = 1 1 −1 ⎪ + K , i = C L, M PL ⎩ DiB Di

(6.20)

Then, the diffusion of water between in the porous media control volume and the channel is computed combining the diffusive coefficients of the CL, MPL and the GDL as resistances in series: 

ef f

D H2 O,e δtotal

=

δC L ef f

D H2 O,C L

+

δM P L ef f

D H2 O,M P L

+

δG DL ef f

D H2 O,G DL

−1 (6.21)

The diffusion coefficient of the channel is assumed to be relatively large compared to the porous media, thus, it is neglected in the computation of the combined diffusive coefficient. Finally, the diffusion coefficient has to be corrected taking into account the porous nature of the control volume and the potential liquid water blockage [23] ef f

ef f

D H2 O,e = D H2 O ε1.5 (1 − s)1.5 .

(6.22)

Remark 6.1 The CL, MPL and GDL will present different porosity values. In the concerned model, it will be assumed an average porosity for the whole porous media. Specifically, the averaged porus media is computed as

140

6 PEM Fuel Cell Monitoring

ε=

δtotal 3



 εC L εG DL εM P L . + + δC L δG DL δM P L

(6.23)

In parallel to the diffusion of vapour water across the control volume, there are a pair of source/sink terms, Sw , that accounts for the vapour water sorpted/desorpted at the membrane interface computed through the expression (6.15), and the water phase change, which is computed as follows [11], ⎧ P − Psat ⎪ ⎨ M H2 O εkcond (1 − s) g , if RT rv = − P P sat ⎪ ⎩εkevap ρl s g , if RT

Pg ≥ Psat Pg < Psat

(6.24)

where kcond (s −1 ) and kevap (s −1 ) are the volumetric phase change rates.

6.2.4.2

Porous Media Water Vapour ODEs

Solving the water vapour PDE (6.3) will result in two ODEs, one for the anode side porous media, ρv,an , and another for the cathode side, ρv,ca . 2 ∂ρv,an = ∂t εδtotal,an



ef f

D H2 O,e,an

 rv,an (ρan,v,ch − ρv,an ) − ε

δtotal,an ρmem,dr y (λeq,an − λan ) − M H2 O kλ,an EW

2 ∂ρv,ca = ∂t εδtotal,ca





ef f

D H2 O,e,ca

(6.25)

(ρca,v,ch − ρv,ca ) −

δtotal,ca ρmem,dr y (λeq,ca − λca ) − M H2 O kλ,ca EW

rv,ca ε (6.26)

where δtotal,i (m) for i = ca, an is the thickness of the porous media, and ρca,v,ch (kg m−3 ) and ρan,v,ch (kg m−3 ) are the water vapour density in the cathode channel and the anode channel, respectively.

6.2.4.3

Oxygen Density Dynamics

In relation to the oxygen density, similar to the water vapour case, it is reasonable to assume that the oxygen transport will also be diffusive dominated. Moreover, similarly, the oxygen diffusion can be described by a combination of binary and Knudsen diffusion, as in (6.18)–(6.20). The diffusion of oxygen between the porous media volume and the channel is computed using an effective diffusive coefficient that combines the catalyst layer, micro-porous layer and the gas diffusion layer

6.2 PEM Fuel Cell Model

141



ef f

D O2

δtotal

=

δC L ef f

D O2 ,C L

δM P L

+

ef f

D O2 ,M P L

+

δG DL

−1

ef f

D O2 ,G DL

,

(6.27)

which has to be corrected taking into account the porosity and liquid water blockage ef f

ef f

D O2 ,e = D O2 ε1.5 (1 − s)1.5 ,

(6.28)

where ε is the effective porosity of the porous media, computed by (6.23). In parallel to the diffusion, the governing equation includes a sink term that model the consumption of oxygen in the catalyst layer due to the reduction reaction. It should be remarked that this model assumes that the reduction reaction in the catalyst layer is uniformly distributed and the reaction kinetics are obviated. Consequently, the sink term of the oxygen Eq. (6.4) is computed as SO2 = −ε

6.2.4.4

1 j . δC L 4F

Porous Media Oxygen ODE

Solving the oxygen concentration PDE (6.4) will result in a single ODE for the cathode side porous media, c O2 ,ca . ∂ρ O2 2 = ∂t εδtotal,ca



ef f

D O2 ,e,ca δtotal,ca

 (ρca,O2 ,ch − ρ O2 ) −

M O2 j δC L 4F

(6.29)

where ρca,O2 ,ch (kg m−3 ) is the oxygen density in the cathode channel.

6.2.5 Liquid Water Dynamics in the Porous Media The considered model assumes that the water co-exists in two-phase form. The presence of liquid water is a consequence of two factors. First, the possibility of water vapour condensation in the porous media. Second, the generation of water in the catalyst layer due to the reduction reaction it is assumed to be in liquid form. The presence of liquid water in the porous media may block the transport of reactants from the channel to the catalyst layer, this fact is modelled through the Eqs. (6.22) and (6.28), which can be used to model flooding phenomena in the fuel cell. The aim of this section is to describe the factors of the liquid water governing Eq. (6.5) and present how this equation is solved in the concerned control volumes. It should be remarked that the effect of the gravity is obviated in the liquid water dynamics.

142

6 PEM Fuel Cell Monitoring

Moreover, the water transport is convective dominated and described through the Darcy’s law. Specifically, the term ef f

ρl K l μl

∂ Pl ∇ · s, ∂s

is the convective flux generated due to the difference in the hydraulic pressure across the porous media. The effective permeability is computed through the following expression [23] ef f K l = K 0,e s 2 where K 0,e [m 2 ] is the intrinsic permeability. The liquid pressure, Pl , can be obtained through the capillary pressure, Pc in the porous media, which, is defined as Pc = Pl − Pg ,

(6.30)

where Pg is the gas pressure, which is independent from the liquid water. In PEM fuel cells, it is common to approximate the capillary pressure through the Leverett J-function, which relates the pressure, Pc , to the liquid water saturation [24, 25]. It should be remarked that, the MPL and GDL are usually designed to be hydrophobic in order to facilitate the liquid water transport, while the CL is hydrophilic. As a consequence, the Leverett J-function approximation changes significantly from the MPL to the GDL and CL. Indeed, the capillary pressure is computed as follows   ⎧

εi ⎪ ⎪σ H2 O cos(θi ) 1.42s − 2.12s 2 + 1.26s 3 , i = M P L , G DL ⎨ K 

0,i  Pc,i = ε ⎪ i ⎪ 1.42(1 − s) − 2.12(1 − s)2 + 1.26(1 − s)3 , i = C L ⎩σ H2 O cos(θi ) K 0,i

(6.31) where θ [◦ ] is the contact angle, σ H2 O [N m −1 ] is the liquid water surface tension coefficient and K 0 [m 2 ] is the intrinsic permeability. ef f ρl K l ∂ Pl ∇ ·s Due to the heterogeneous nature of the porous media, the factor μl ∂s ef f ρl K l ∂ Pl may vary along the control volume. To solve this conflict, the factor μl ∂s for the CL,GDL and MPL can be taken as a set of convective resistances in series Rs,i f or i = C L , G DL , M P L. Consequently, it is possible to compute an effective convective resistance as:  Re f f =

δC L δG DL δM P L + + Rs,C L Rs,G DL Rs,M P L

−1

.

(6.32)

6.2 PEM Fuel Cell Model

143

Additionally, the governing equation includes some sink/source terms that model the generation of liquid water in the catalyst layer due to the reduction reaction and . Again, it is assumed that the reduction reaction is uniformly distributed and the reaction kinetics are obviated. Consequently, the source term of the liquid water Eq. (6.5) is computed through the Faraday law in the cathode catalyst layer as εC L

M H2 O j . ρl Vca 2F

(6.33)

Finally, the liquid water sub-model includes the factors related to the water phase change (6.24).

6.2.5.1

Porous Media Liquid Water ODEs

Solving the PDE (6.5) in the porous media control volumes leads to two ODE, one for the cathode side, sca , and another for the anode side, san .   1 2 ∂san = rv,an Re f f,an san,ch − san + ∂t ρl εδtotal,an ρl ε   1 ∂sca M H2 O j 2 Re f f,ca sca,ch − sca + = rv,ca + ∂t ρl εδtotal,an ρl ε ρl Vca 2F

(6.34) (6.35)

where sca,ch , san,ch (−) depict the liquid water in the cathode channel and the anode channel, respectively.

6.2.6 Channel Sub-model The fuel cell reactants are delivered to the porous media through the channels. A mass balance is implemented in the model to characterize the reactant flow through the channel. Formally, the dynamics of the reactants in the fuel cell channels are computed by solving the equation in (6.8). Precisely, the term in the left-hand side of (6.8) depicts the mass rate of change, while the terms in the right-hand side depict the convective flux along the channel and the different sources/sinks. In order to simplify the model, the channels are modelled as straight, square channels with a single inlet and outlet. Moreover, the gas density in both channels, ρca and ρan , is assumed to remain constant. Thus, the effect of the density variation is neglected. Finally, the mass of liquid water in the channel is assumed to be zero. This is a reasonable assumption, as fuel cells usually operate at a sufficiently high mass-flow rate such that liquid water can evacuate the channel. It should be remarked that the proposed model assumes the channel to be a single control volume. Thus, it cannot model the distribution of reactant concentrations

144

6 PEM Fuel Cell Monitoring

along the channel, and only depicts an “average” of the reactant concentration in the channel control volume.

6.2.6.1

Cathode Channel Sub-model

The fuel cell cathode is delivered with ambient air to obtain the oxygen required for the main reactions. For all the air components, from a fuel cell perspective, it is only interesting to study the dynamics of the oxygen and the water vapour. Consequently, in the cathode channel, a set of ODEs are computed to solve for the water vapour density, ρca,v,ch and the cathode oxygen density ρca,O2 ,ch . The cathode water density, ρca,v,ch , is computed through an ODE that describes the   ∂ρca,v,ch M H2 O I O n˙ ca,H2 O v − n˙ ca,H2 O v − Vca rv,ca,ch − n˙ trans,v . (6.36) = ∂t Vca I −1 The factor Vca (m 3 ) is the total volume of the cathode channels, n˙ ca,H ) v (kmol s 2O is the inlet water vapour mass flow per channel which is computed as follows I ˙ ca n˙ ca,H v = yca,H2 O n 2O

where yca,H2 O is the water molar fraction and n˙ ca (kmol s −1 ) is the total cathode inlet molar flow rate. O −1 depicts the cathode outlet water vapour molar flow The factor n˙ ca,H v kmol s 2O and is computed as O n˙ ca,H v = A ca cca,v,ch u ca 2O where Aca (m 2 ) is the cross-sectional area of the cathode channel and u ca (m s −1 ) is the average cathode outlet velocity, which is assumed to be driven by the the species pressure differential between the channel and the external environment. The fluid flow is characterized as a Poiseuille flow in a channel for laminar regime. As a consequence, the average velocity is computed as u ca =

2 ΔPca dca 32μca L ca

(6.37)

2 where dca (m) is the effective diameter of the channel, μca (kg m−1 s −1 ) is the viscosity of the gas, L ca (m) is the length of the channel and ΔPca (Pa) is the inlet/outlet pressure difference. The element rv,ca,ch depicts the phase change of water, which can be computed by solving (6.24) in the cathode channel. It should be remarked that, in the case of water vapour condensation, the resulting liquid water is immediately evacuated from the fuel channel due to the high inlet mass-flow.

6.2 PEM Fuel Cell Model

145

The factor n˙ trans,v kmol s −1 is the vapour molar flux that diffuses from the GDL to the channel, and is computed as n˙ trans,v =

2



ef f

D H2 O,e,ca

εM H2 O (δtotal,ca + δcha,ca )

δtotal,ca

 (ρca,v,ch − ρv,ca ) .

The cathode oxygen density, ρca,O2 ,ch is computed through the ODE:   M O2 I ∂ρca,O2 ,ch O = n˙ ca,O2 − n˙ ca,O − V n ˙ ca trans,O2 . 2 ∂t Vca

(6.38)

I The factor n˙ ca,O kmols −1 is the inlet oxygen molar flow which is computed as 2 follows I = yca,O2 n˙ ca , n˙ ca,O 2

where yca,O2 (−) is the oxygen molar fraction and n˙ ca (kmols −1 ) is the cathode molar flux. Finally, n˙ trans,O2 kmol s −1 depicts the oxygen flux from the channel to the cathode porous media, n˙ trans,O2 =

2 M O2 ε(δtotal,ca + δchan,ca )



ef f

D H2 O,e,ca δtotal,ca

 (ρca,v,ch − ρ O2 ) .

O The factor n˙ ca,O depicts the cathode outlet water vapour mass flow and is com2 puted as O = Aca cca,O2 ,ch u ca , n˙ ca,O 2

where u ca is the average anode outlet velocity computed as in (6.37).

6.2.6.2

Anode Channel Sub-model

The fuel cell anode is delivered with humidified hydrogen. Consequently, this model solves for the hydrogen and the water vapour in the anode side. Precisely, a set of ODE are going to be implemented to solve for the anode hydrogen density, ρan,H2 and anode vapour water density, ρan,v,ch . The anode hydrogen density, ρan,H2 , is computed through   M H2 I ∂ρan,H2 O n˙ an,H2 − n˙ an,H = − n ˙ H2 ,used . 2 ∂t Van

(6.39)

I The factor Van (m 3 ) is the anode channel volume, n˙ an,H (kmol s −1 ) is the inlet 2 hydrogen mass flow which is computed as follows

146

6 PEM Fuel Cell Monitoring I n˙ an,H = yan,H2 n˙ H2 , 2

where yan,H2 (−) is the hydrogen molar fraction and n˙ H2 (kmol s −1 ) is the anode molar flux. O (kmol s −1 ) depicts the anode outlet hydrogen mass flow and is The factor n˙ an,H 2 computed as O = Aan yan,H2 u an , n˙ an,H 2 where Aan (m 2 ) is the cross-sectional area of the anode channel and u an is the average anode outlet velocity. The factor n˙ H2 ,used (kmols −1 ) describes the hydrogen consumption in the catalyst layer due to the hydrogen oxidation reaction, and is computed as n˙ H2 ,used =

I , 2F

The anode water vapour density, ρan,v,ch , is computed trough the following ODE:   M H2 I ∂ρan,v,ch O n˙ an,H2 O v − n˙ an,H2 O v − Van rv,an,ch − Van n˙ trans . = ∂t Van

(6.40)

All the factors of the ODE (6.40) are solved as in the cathode channel (6.36).

6.2.7 Thermal Sub-model Fuel cell operation is altered by multiple sources and sinks of thermal energy, which modify its temperature distribution. Indeed, the main chemical reactions generate heat, which is dissipated through the fuel cell. As most physical properties depend on the temperature, it is convenient to include a sub-model that depicts the temperature dynamics. Specifically, the fuel cell temperature is computed through the equation in (6.7). Naturally, a direct approach would be to solve the governing equation in all the considered control volumes. However, this would result in an unnecessary (from an observer design perspective) increase of the model dimension. For this reason, the thermal model is only solved in three control volumes. First, a control volume that encapsulates the cathode channel. Second, another control volume for the anode channel. Finally, a control volume for the membrane electrode assembly, i.e. the composition of the porous media and the membrane. The temperature inside each control volume is assumed to be uniform.

6.2 PEM Fuel Cell Model

6.2.7.1

147

Cathode Channel Thermal Dynamics

Even if the inlet gases in the anode channel and the cathode one present the same temperature, the channels may present different temperatures due to the unique reactions occurring in each side of the fuel cell. For this reason, it is convenient to have separate temperature variables in the anode and the cathode side. Indeed, the temperature dynamics are computed using first laws of thermodynamics. Specifically, the change in thermal energy of the system is computed as the difference between the input/output energies. Precisely, in the cathode channel, the thermal sources and sinks are: the convective heat transfer between the membrane electrode assembly and the gas channel, the heat from the water phase change and the energy transported by the gases. Specifically, the cathode temperature dynamics are computed through the following ODE ∂Tca = ∂t



−

yca,i ρca Vca c p,i

i

 i

O m˙ ca,i c p,i Tca −

−1

Q˙ conv,ca + Q˙ ca, phase +

 ∂ yca,i i

∂t

 (ρca Vca c p,i Tca ) ,



I m˙ ca,i c p,i Tca

i

(6.41)

where Tca (K ) is the cathode temperature, yca,i (−) is the mass fraction for each I (kg s−1 ) is the inlet species, c p,i is the specific heat capacity of the species i, m˙ ca,i −1 O mass flow of the species i and m˙ ca,i (kg s ) is the outlet mass flow of the species i. The factor Q˙ conv,ca (W ) depicts the convective heat transfer from the cathode channel to the membrane electrode assembly, which is computed through the Newton’s law of cooling (6.42) Q˙ conv,ca = h ca As,ca (Tca − Ts ) where As,ca (m 2 ) is the contact surface and h ca (W m −2 K −1 ) is the convective heat transfer coefficient. The factor Q˙ ca, phase (W ) model the heat generated from the mass of water changing phase whether from evaporation or condensation. Precisely, the heat released/absorbed during phase changes is computed as ΔHvap rv,ca,ch , Q˙ ca, phase = M H2 O where Hvap (J/mol) is the water heat of vaporization.

6.2.7.2

Anode Channel Thermal Dynamics

The heat transfer in the anode channel can be modelled through the same factors as in the cathode one. Consequently, a similar equation is derived for the anode channel,

148

6 PEM Fuel Cell Monitoring

∂Tan = ∂t



−

yan,i ρan Van c p,i

i

 i

O m˙ an,i c p,i Tan −

−1

Q˙ conv,an + Q˙ an, phase +

 ∂ yan,i i

∂t



 (ρan Van c p,i Tan ) ,

I m˙ an,i c p,i Tan

i

(6.43)

where Tan (K ) is the anode channel temperature, Q˙ conv,an depicts the convective heat transfer between the channel and the membrane electrode assembly and is computed through the Newton’s law Q˙ conv,an = h an As,an (Tan − Ts ),

(6.44)

and Q˙ an, phase depicts the heat released/absorbed due to the water phase change, which is computed as ΔHvap Q˙ an, phase = rv,an,ch . M H2 O 6.2.7.3

Membrane Electrode Assembly Thermal Dynamics

The last part of the thermal sub-model aims to depict the temperature dynamics in the volume composed by the porous media and the membrane. It is noticeable that the main electrochemical reactions occur in this control volume, which generate useful electrical energy and irreversible losses that are converted to thermal heat. Moreover, the concerned control volume has to include the convective heat transports from the membrane electrode assembly to the channels and surrounding ambient. Finally, it is necessary to include the heat generated/absorbed due to the water phase change. The heat generated due the sorption/desorption processes and due to ohmic losses are obviated. Specifically, the membrane electrode assembly temperature is computed as m s c p,s

∂Ts = Q˙ r + Q˙ phase − W˙ elec − Q˙ amb − Q˙ conv,ca − Q˙ conv,an ∂t

(6.45)

where Ts (K ) is the electrode membrane assembly temperature, m s (kg) is the total membrane electrode assembly mass and c p,s (J kg−1 K−1 ) is the effective specific heat of the control volume. The factor Q˙ r depicts the heat produced from the formation of water due to the reduction reaction, which is computed as M H2 O j , Q˙ r = ΔH f ρl Vca 2F where ΔH f (J kg) is the heat formation of liquid water.

6.2 PEM Fuel Cell Model

149

The factor Q˙ phase depicts the heat related to the water phase change phenomena, which is computed as ΔHvap Q˙ phase = (rv,ca + rv,an ). M H2 O Naturally, part of the energy of the electrochemical reaction is used as electrical power and is not lost as thermal heat. For this reason, a factor W˙ elec that depicts the useful electrical work is included in the model. Specifically, this factor is computed as follows W˙ = I V f c , where V f c (V ) is the fuel cell voltage. The last terms of the equation are related with the convective thermal transports. The terms Q˙ conv,ca and Q˙ conv,an are the thermal convective flux from the membrane electrode assembly to the channels, which are computed through (6.42) and (6.44), respectively. The factor Q˙ amb depicts the convective flux from the control volume to the ambient, which is also computed through the Newton’s law of cooling Q˙ amb = h amb As,st (Ts − Tamb ), where Tamb (K ) is the ambient temperature. Remark 6.2 If the ambient air is fed to the fuel cell through a fan, the convective heat transfer coefficient has to be computed as in a forced convection case, i.e. h amb = ρair C p,air vair where ρair (kg m−3 ) is the air density, C p,air (J Kg−1 K−1 ) is the air heat capacity and vair (m s−1 ) is the cathode inlet air velocity.

6.2.8 Electrochemical Sub-model Finally, the model includes a static relation between the model variables and the cell voltage. At standard conditions, the fuel cell presents a theoretical potential, which is derived from the Gibb’s free energy of the reaction. In practice, this value has to be corrected taking into account non-standard conditions and unavoidable overpotentials that appear during fuel cell operation. Indeed, the fuel cell voltage is computed by subtracting the irreversible potential losses to the theoretical open circuit voltage [26] E cell = E ocv − ηact − ηconc − ηohm . where E ocv si the open-circuit voltage, ηact are the activation overpotentials, ηconc the concentration overpotentials and ηohm the ohmic overpotentials.

150

6 PEM Fuel Cell Monitoring

The open circuit voltage, E ocv , is computed as the theoretical fuel cell voltage corrected for non-standard temperature and species activity, E ocv = E 0 + E T + E N , where E 0 is the theoretical potential at standard conditions. Precisely, in general, the fuel cell will operate with a temperature different from the standard value. For this reason, a temperature correction factor E T is included in the computation of the open circuit voltage. Indeed, the correction factor is computed as ΔS (Ts − 273), ET = 2F where ΔS (J kg−1 K−1 ) is the entropy change of the reaction. Moreover, the theoretical potential is also corrected for the concentration of the species in the catalyst layer. Specifically, the correction term is computed as [10] EN =

  √ a H2 a O2 RTs , ln 2F a H2 O

where ai is the activity of the species, computed as the ratio between the partial Pi pressure of the gas to the standard pressure ai = . As the water in the catalyst P0 layer is generated in liquid form, the activity for water is one. The activation losses are computed as the sum of the activation overpotentials in the anode and the cathode [27] ηact =

RTP M,an F



j j0,an

 +

RTP M,ca ln αc F



j



j0,ca

(6.46)

where αc [−] is the cathodic transfer coefficient, j0,an [A cm−2 ] and j0,ca [A cm−2 ] are the reference exchange current densities. The reference exchange current in the cathode is corrected to take into account the potential voltage loss induced by the lack of oxygen in the cathode catalyst layer and temperature, j0,ca = j0,ca,r e f

 

1 −ΔG ∗ 1 √ a O2 ex p − R TP M,ca 303

(6.47)

where ΔG ∗ [K j mol−1 ] is the activation energy associated of the ORR in platinum and j0,ca,r e f [A cm−2 ] is the reference exchange current in the cathode at standard conditions. Remark 6.3 Some authors have pointed out that that the electrochemical active surface area depends on the cathode catalyst layer liquid water saturation, s [28]. If the liquid water saturation is too large, the protons cannot efficiently reach the cathode catalyst layer, effectively reducing the electrochemical active area. Taking

6.2 PEM Fuel Cell Model

151

into account these details, the model can be adapted to compute a modified reference exchange current density [29],     sopt − s 1/3 (−ΔG ∗ /(RT f c )(1−(T f c /Tr e f ))) √ e a O 2 ac 1 − , sopt (6.48) where ac [−] is the electrode rugosity and sopt [−] is the liquid water saturation in which the effective electrochemical active surface area is maximum. j0,ca = 0.21 j0,ca,r e f

The total concentration losses, ηconc , in the fuel cell reaction are computed as [27] ηconc =

    jL 1 RTmem ln 1+ nF αc jL − j

where jL [A cm−2 ] is the limiting current. The ohmic losses are computed with the ohm’s law ηohm = j Amem (Rcom + Rionic )

(6.49)

where Amem is the active area of the membrane, Rcom [Ω] is the ohmic resistance of the cell conductive components and Rionic [Ω] is the ionic resistance of the membrane which is computed as δmem (6.50) Rionic = σm where the membrane ionic conductivity σm , which is a function of the membrane water content [16] 

 1 1 . − σm = (0.005139λ − 0.00326)ex p 1268 303 Tmem Remark 6.4 If one wants to model a fuel cell stack, i.e. a system with multiple cells, then, it is possible to assume that the conditions in all the cells are similar, thus, the stack voltage is computed as V f c = n cell (E ocv − ηact − ηconc − ηohm ) where n cell (−) is the number of cells in the stack.

6.2.9 Partial Experimental Validation and Model Feasibility Although the proposed model has been developed from the first principles of the system, and a simplified version was experimentally validated in [9], it is interesting

152

6 PEM Fuel Cell Monitoring

Fig. 6.3 PEMFC stack load current

to validate the feasibility of the model in some additional experimental scenarios. Due to time limitations, an exhaustive validation of all the model components has not been possible. Nonetheless, a partial experimental validation has been conducted to provide some evidence in favour of the model. In the experimentation, the prediction capabilities of the model have been tested for three variables: The temperature of the cathode gases, the stack voltage and the relative humidity of the cathode channel. The model parameters have been selected by means of a fuel cell literature review. The parameters are summarized in Table 6.2. It should be remarked that the experimental data considered in this section is taken from the work in [9]. The precise description of the experimental set-up can be found in Chap. 4 of [9]. Specifically, in the experiment, the PEMFC starts from an equilibrium point and then provides a set of current steps, the exact current profile is depicted in Fig. 6.7. The remaining input signals are maintained at a constant value. The value of the signals are summarized in Table 6.3. The evolution of the measured stack voltage and the model prediction is depicted in Fig. 6.4. It can be seen that the model prediction follows accurately the experimental profile. A remarkable property to analyse is the asymmetry of the voltage values. Notice that the exchange current takes the same values in both halves of the simulation, see Fig. 6.3. Nonetheless, the stack voltage values do not take the same value. This is due to internal variables related to the water dynamics that cannot reach the steady-state and accumulate during the experiment. It can be observed that the model is capable to replicate this asymmetry. The evolution of the cathode channel temperature and the model prediction is presented in Fig. 6.5. Again, it can be seen that the model error is also relatively low. Naturally, the temperature of the system increases as the exchange current increases. Again, there is an asymmetry of the temperature due to the thermal inertia of the system. It can be observed that the proposed model also presents this asymmetry.

6.2 PEM Fuel Cell Model

153

Table 6.2 PEM fuel cell model parameters Symbol Description [units]

Value

Physical constants F M H2

96485 2

M O2 M H2 O R ρl μl σ H2 O cp O2 cp N2 cp H2 cpv cpl ΔH f ΔHvap ΔG ∗ Fuel cell parameters n cell Acell n ca n an L ca L an wca wan h ca h an

Faraday Constant [C mol−1 ] Hydrogen molar mass [g mol−1 ] Oxygen molar mass [g mol−1 ] Water molar mass [g mol−1 ] Universal gas constant [J mol−1 K−1 ] Liquid phase density [g cm−3 ] Liquid water viscosity [Pa s] Liquid water surface tension [N m−1 ] Oxygen specific heat [JKg−1 K −1 ] Nitrogen specific heat [JKg−1 K −1 ] Hydrogen specific heat [JKg−1 K −1 ] Vapour water specific heat [JKg−1 K −1 ] Liquid water specific heat [JKg−1 K −1 ] Heat formation of water [JKg−1 ] Water heat of vaporization [JKg−1 ] Activation energy for the ORR on Pt [KJmol−1 ] Number of cells in the stack [−] Cell active area [cm 2 ] Cathode number of channels [−] anode number of channels [−] Length cathode channel [m] Length anode channel [m] Width cathode channel [m] Width anode channel [m] High cathode channel [m] High cathode channel [m]

32 18 8.314 0.997 4.05 · 10−4 6.44 · 10−2 928 1041 14472 1881 4184 −285830 44000 66

13 [9] 13.1 [9] 34 [9] 5 [9] 0.02 [9] 0.05 [9] 0.008 [9] 0.012 [9] 0.0016 [9] 0.0001 [9] (continued)

154 Table 6.2 (continued) Symbol Electrochemical param. E0 j0,an j0,ca,r e f jL Rcom Membrane parameters εm δmem EW ρmem,dr y α

6 PEM Fuel Cell Monitoring

Description [units]

Value

Theoretical voltage at standard conditions [V] Anode reference exchange current [A cm−2 ] Cathode reference exchange current [A cm−2 ] Limiting current [A cm−2 ] Electrical components resistance [Ω]

1.2291

Ionomer-phase volume fraction [–] Thickness [µm] Ionomer Equivalent weight [g mol−1 ] Membrane dry density [g cm−1 ] Membrane sorption coefficient [–]

0.19 (an) 0.17 (ca) [31]

1 [30] 100 [9] 0.2 [9] 0.45 [10]

50 [31] 1100 [32] 1.9 [32] 4.59 · 10−2 (s) 41.14 · 10−2 (d) [33]

Catalyst Layer param εC L δC L rp K 0,C L θC L Gas diffusion layer param δG DL εG DL K 0,G DL θG DL Micro-porus Layer param εM P L δC L rp K 0,M P L θM P L

Porosity [–] Thickness [µm] Primary-particle radius [nm] Intrinsic permeability [m2 ] Contact angle [◦ ]

0.47 (an) 0.53 (ca) [33] 2.8 (an) 4.5 (ca) [33] 39.5 [31] 1 · 10−19 [34] 89 [35, 36]

Thickness [µm] Porosity [–] Intrinsic permeability [m2 ] Contact angle [◦ ]

109 [33] 0.81 [33] 2 · 10−11 [32] 122 [37, 38]

Porosity [–] Thickness [µm] Primary-particle radius [nm] Intrinsic permeability [m2 ] Contact angle [◦ ]

0.58 [33] 37 [39] 56 [33] 1.39 · 10−13 [40] 110 [41] (continued)

6.2 PEM Fuel Cell Model Table 6.2 (continued) Symbol Physical properties DrBe f,v

DrBe f,O2

kcond,evap αc hi

155

Description [units]

Value

Reference vap. binary diffusion coefficient [–] 298.2 K (ca), 307.1 K (an); 101.325 kPa Reference oxygen binary diffusion coefficient [–] 273.2 K , 101.325 kPa phase change rate [s −1 ] cathode transfer coefficient [–] Convective heat coefficient [W m −2 K −1 ]

2.93 · 10−5 (ca) 9.15 · 10−5 (an) [42]

Table 6.3 Experiment conditions Description [units] Anode Stoic [–] RH inlet cathode [%] RH inlet anode [%] Atm. pressure [kPa] Inlet pressure cath. [kPa] Inlet pressure an. [kPa] Inlet temperature cath. [K] Inlet temperature an. [K] Amb. temperature [K] Flow rate cathode [mol/s]

Value 2 18 1 101 101 101 297 297 297 0.1

Fig. 6.4 PEMFC stack voltage and model prediction

1.81 · 10−5 [43]

100 (c) 1000 (v) [44] 1.05 [9] 0.02 (an) 0.01 (ca) 0.7 (amb) [9]

156

6 PEM Fuel Cell Monitoring

Fig. 6.5 Cathode channel temperature and model prediction

Fig. 6.6 Cathode channel relative humidity and model prediction

Finally, the evolution of the cathode relative humidity and the model prediction is depicted in Fig. 6.6. It can be observed that the model follows the general trajectory of the experimental data, i.e. the relative humidity decreases as the current increases. Nonetheless, in this case, the error is significant. This discrepancy may be a consequence of an inadequate water diffusion coefficient tuning. Thus, it is expected that the error can be significantly reduced by a proper model parameter identification. This section has presented a brief experimental analysis in favour of the proposed model. It can be seen that by selecting reasonable model parameters, in the sense of selecting values similar to the ones used in previous PEM fuel cell works, the model is capable of replicating the major electric and thermal behaviour of the fuel cell. Naturally, higher estimation accuracy can be achieved if the model parameters are identified based on experimental data from the target fuel cell system.

6.2 PEM Fuel Cell Model

157

Nonetheless, motivated by reasons discussed in the next section, the observers applications proposed in this work will be developed on a reduced-order version of the model. Therefore, it is only necessary to identify a few model parameters in order to deploy (in a real experimental set-up) the observers that will be proposed in this chapter. For this reason, a complete model identification has not been carried in this work.

6.2.10 Estimation Objectives and Model Reduction The last section has proposed a mathematical model that is low-order in comparison to alternative models that are commonly used for fuel cell design. Nonetheless, the resulting state-space model presents 13 states to be solved, which is still too large for nonlinear observer design. Indeed, the instantaneous observability analysis studied in Chap. 2 is an infeasible task for systems of this size. Moreover, the observers studied in Chap. 3 (and the reference therein) are not applicable in the proposed model. First, the peaking phenomena and the noise sensitivity of high-gain observers scales with the system dimension, thus, an estimation generated by a high-gain observer of dimension 13 is practically useless. Second, the parameter estimation-based observer requires finding an adequate coordinate transformation by means of solving a PDE. This task is infeasible for systems of order 13 with no additional structural properties. Consequently, this section proposes the most amenable solution, that is, reducing the dimension of the model. To do a proper model reduction, first, it is convenient to establish which will be the objective of the observer to be designed. Otherwise, relevant information for the desired application may be lost. Multiple observer applications for fuel cell systems can be found in the literature. Some remarkable examples are: the use of observer to estimate the oxygen partial pressure in the cathode in order to avoid oxygen starvation [45–49], monitoring of the nitrogen that crosses the membrane [50], the estimation of the membrane water content in order to avoid fuel cell drying [51], the estimation of the anode relative humidity [50] and the estimation of hydrogen partial pressure to avoid hydrogen starvation [48, 52]. In order to develop an observer algorithm with value, it is convenient to select an objective that has not been previously analysed in the literature. Following this line, it is remarkable the importance of water for an adequate fuel cell operation. It can be seen that water enters the fuel cell system due to the humidity of the reactant gases, see Eqs. (6.36) and (6.40), and the generation of water due to the reduction reaction, see Eq. (6.33). Part of this water is sorpted by the membrane which maintains the humidity of the system (6.15). It is desirable to have a highly humidified membrane, as it reduces the ionic losses of the system depicted in (6.50). For these reasons, some authors have developed observers to monitor the membrane water content [51]. Nonetheless, too much water may flood the porous media and prevent the arrival of the reactants to the catalyst layer. This process is modelled by a reduction of the reactants diffusion coefficients due to the presence of liquid water, see Eqs. (6.22) and (6.28). The available liquid water measuring techniques as the

158

6 PEM Fuel Cell Monitoring

current distribution method [53], neutron radiography [54] or x-ray radiography [55], are far too expensive, slow and intrusive to be a viable option for on-line monitoring systems. Moreover, there exists few observer algorithms designed to monitor this fuel cell variable. To solve this gap, this chapter will focus on designing an observer algorithm to estimate the liquid water saturation in fuel cells. Now that the objective has been established, the proposed model can be reduced to the parts that are related to the fuel cell liquid water. As it will be shown in the next subsections, this process will lead to a system composed by an equation that depicts the membrane electrode assembly temperature, an equation that describes the liquid water saturation dynamics and an equation that relates these variables to the fuel cell stack voltage.

6.2.10.1

Thermal Sub-model

The membrane electrode assembly temperature has a direct impact in the liquid water as it drastically modifies the evaporation/condensation rates of the water (6.24). Therefore, Eq. (6.45) has to be included in the reduced model. In order to reduce the complexity of the equation, the following assumptions have been considered • The heat generated by the vaporization of water or by convection to the channels is order of magnitude lower than the heat that is exchanged with the ambient due to the open-cathode nature of the fuel cell. Therefore, Q˙ phase , Q˙ conv,ca , Q˙ conv,an ≈ 0. • The energy of the reaction that is lost as thermal heat can be approximated as the difference between the theoretical electrical work of the system and the actual electrical work obtained. • A fan forces a flux of air in the fuel cell cathode. Thus, the heat transfer coefficient, h amb , has to be computed as in the forced convection case. Taking into account this assumptions, the equation in (6.45) can be rewritten as m s c p,s T˙ f c = I (E ocv n cell − V f c ) − ρair Ainlet C p,air (T f c − Tamb )vair .

(6.51)

where Ainlet (m 2 ) is the inlet manifold cross-sectional area.

6.2.10.2

Liquid Water Saturation Sub-model

Naturally, in order to estimate the liquid water saturation, it is crucial to implement the liquid water saturation dynamics (6.35) in the observer. In practice, most of the fuel cell water accumulates in the cathode side. Consequently, the presence of liquid water in the anode side will be insignificant in relation to the cathode side. For this reason, the observer will focus on estimating the liquid water saturation in the cathode porous media and the anode will be obviated. It is noticeable that the liquid water dynamics are strongly nonlinear due to the effective convective resistance computed through (6.32). Nonetheless, as most liquid

6.2 PEM Fuel Cell Model

159

water accumulate in the catalyst layer, which is a hydrophilic surface, it is reasonable to assume that the capillarity pressure can be computed assuming that all the porous media is hydrophilic, i.e.

Pc = σ H2 O cos(θC L )

  εC L 2 3 1.42(1 − s) − 2.12(1 − s) + 1.26(1 − s) . K 0,C L

Moreover, it is noticeable that the liquid water dynamics present a hybrid behaviour due to the phase change factor (6.24). Nonetheless, in general, it is desirable to operate the fuel cell at relatively high temperatures in order to reduce the reaction kinetic losses. Therefore, it is reasonable to assume that water will always evaporate. Furthermore, the saturation pressure can be approximated through the following relation Psat = p 0 e−Ea/(kb T f c ) where p 0 (Pa) is a pre-exponential factor, k B (eV K −1 ) is the Boltzmann’s constant, E a (eV ) is the activation energy of the evaporation process. Finally, it will be assumed that the channels operate at sufficiently large flow rates, such that any liquid water that is transported from the porous media to the channel is immediately evacuated. Therefore, the amount of liquid water in the channel is zero sca,ch = 0. By taking these assumptions, the liquid water dynamics can be described through an equation similar to the one studied in [56, 57]. Specifically, M H 2O M H2 O I − K evap ( p 0 e−Ea/(kb T f c ) − Pg )s 2F Acell RT f c A por e  e f f ρl s(−0.96 + 3.32 s − 3.78 s 2 ), − σ H2 O cos(θC L ) εK l K s μl

K s s˙ =

(6.52)

where K s (Kg m−2 ) is a liquid water accumulation constant in the cathode catalyst layer, A por e (m2 ) is the effective pore area and K evap (m3 s −1 ) is the product of the evaporation constant, kevap , and the catalyst layer volume. Writing the liquid water dynamics in this form presents several benefits. First, the equation to be analysed is significantly simplified. Second, the equation has been experimentally validated and tested for a control application in the works [56, 57].

6.2.10.3

Electrochemical Sub-model

Finally, the reduced model includes the stack voltage equation. The reasons to include this equation are twofold. First, the stack voltage appears in the temperature Eq. (6.51). Second, it is a signal that is relatively easy to measure and provides a significant amount of information of the system. Nonetheless, to obtain a reducedorder model, it is important to include several simplifications on the voltage equations.

160

6 PEM Fuel Cell Monitoring

First, it will be assumed that the partial pressure of the gases remains constant. Therefore, modelling the transport of the gases in the porous media can be obviated, which drastically reduces the dimension of the system. Moreover, the factors that depend on the gas partial pressures can be obviated and will be compensated by a proper system parameter identification. In situations that this assumption is not reasonable, the partial pressures could be estimated through the observers in [52]. Second, as the anode activation overpotential is order of magnitudes lower than the cathode one [58], the anode activation overpotential is obviated. Thus, the activation losses are computed as   RT f c j . ln ηact = αc F j0,ca Furthermore, as it is desirable to have the maximum amount of information related to the liquid water, the modified equation of the reference exchange current (6.48). Specifically, j0,ca

    sopt − s 1/3 (−ΔG ∗ /(RT f c )(1−(T f c /Tr e f ))) e = 0.21 j0,ca,r e f ac 1 − . sopt

Third, it is assumed that the membrane is well hydrated. Thus, the ohmic losses are computed as ηohm = Rohm I, where Rohm [Ω] is a constant parameter accounts for the ionic conductivity of the membrane and the resistance of the fuel cell electric conductive components. To extend the results to cases where the membrane is not correctly hydrated, and consequently, the membrane water content significantly modifies the ohmic losses through (6.50), it is possible to estimate the water content through the observer in [51]. Finally, it is assumed that the fuel cell will not operate at too high current densities during long periods of time, as it significantly decreases the system performance and accelerates its degradation [3]. Consequently, concentration losses will, in general, be small and can be neglected, i.e. ηconc ≈ 0. Moreover, in the concerned fuel cell system, the effect of fuel crossover and internal currents is order of magnitudes lower than the other losses. Therefore, they are also neglected. It should be remarked that the resulting voltage equation is equivalent to the one that was proposed and experimentally validated in [29].

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers The aim of this section is to use the nonlinear observer theory to develop a high performance algorithm to monitor the cathode liquid water. The observer will be based on the reduced model developed in the past section. In order to present the

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

161

developments in a more coherent manner, the reduced-order fuel cell model will be rewritten in the following state-space form x˙ = fs (x, I ) + g(x)vair

(6.53)

y1 = T f c y2 = V f c (x, I ). The state vector, x, is defined as: x = [T f c , s] ; and the control inputs are the load current and the cathode air velocity, u = [I, vair ] . In this section, it is only considered two possible output signals, the membrane electrode assembly temperature, T f c , and the stack voltage V f c . Moreover, the vector functions fs , g are: ⎤ K 1 (E ocv n cell − V f c )I ⎦ K evap fs (x, I ) = ⎣ 1 (K 3 I − K 4 f p (T f c , s)) − K 5 f d (s) Ks A por e ⎡

g(x) =

 K 2 (Tamb − T f c ) 0

where the constants K 1 , .., K 5 are defined as: ρair Ainlet C p,air 1 M H2 O , K2 = , K3 = , m s c p,s m s c p,s 2F Acell  M H 2O e f f ρl , K 5 = σ H2 O cos(θC L ) εK l , K4 = R K s μl

K1 =

and the nonlinear functions f p and f d are computed as: f p (T f c , s) =

s ( p 0 e−Ea/(kb T f c ) − Pg ), Tfc

f d (s) = s(−0.96 + 3.32 s − 3.78 s 2 ).

Finally, the function V f c depicts the a static relation between the system states, x, and the fuel cell stack voltage. For the rest of the chapter, all the developments will be expanded using the model in (6.53).

6.3.1 Observability Analysis Naturally, as a first step in observer design, it is crucial to analyse the possibility of estimating the liquid water saturation from the available measurements. Following from the theory presented in Chap. 2, this accounts to proving that each possible

162

6 PEM Fuel Cell Monitoring

trajectory of the measured outputs y1 , y2 , for a given set of inputs, can only be generated by a single trajectory of the states. Specifically, that the observability map (2.4) of the system is injective, which proves that the system is instantaneously observable, see Definition 2.3 in Chap. 2. Precisely, the observability map of order 2 for the system (6.53) can be computed as 



Tfc y1 = O2 = y˙1 K 1 (E ocv n cell − V f c )I + K 2 (Tamb − T f c )vair A sufficient condition for the injectivity of the observability map, O2 , is that its Jacobian is full rank in all the considered operating region. Indeed, the Jacobian of the observability map can be computed as ⎡ ⎤ 1 0 ∂O2 ∂ L fs T f c ⎦ (x, I, vair ) = ⎣ ∂ L fs T f c (x) (x) − K 2 vair ∂x ∂T f c ∂s

(6.54)

where ∂ L fs T f c (x) = ∂s

−K 1 RT f c I      sopt − s 1/3 sopt − s 2/3 3αc F 1 − sopt sopt sopt

Due to the triangular nature of the Jacobian, it is relatively easy to show that (6.54) is full rank in the domain D = {T f c , s, I ∈ R : T f c = 0; s = 0, sopt ; I = 0},

(6.55)

∂ L fs T f c (x) is non-zero. ∂s Proving the observability of (6.53) consists in showing the system states evolves inside the domain (6.55). Relative to the temperature, the operating condition T f c = 0 is physical impossible. Moreover, complete dryness of the porous media layer, s = 0, is not a common nor desirable operating point, from an efficiency point of view and feasibility of the model [32]. Additionally, the liquid water saturation usually evolves below its optimal value, sopt . Finally, as the fuel cell is used to supply some electrical load, the current will be larger than zero. Therefore, the Jacobian (6.54) is full rank in all the valid operating conditions of the fuel cell and the system is instantaneously observable uniformly in the input vair . Thus, the unknown states can be uniquely expressed as a function of y1 , y˙1 , I and vair . i.e. in the domain that

Remark 6.5 Even though the values s = 0 and s = sopt may not be reached by the system, the observer states, during the transient, may reach these values, making the system unobservable. See Remark 3.10 in Chap. 3 for a discussion on the topic.

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

163

Remark 6.6 A similar analysis has been conducted taking the stack voltage as the measured output. In the analysis, the observability map was not injective, which shows that the system is not observable just using the stack voltage. Thus, it is necessary to use the temperature measurement in the liquid water estimation.

6.3.2 System Transformation Following the observer design framework presented in Chap. 3, the next step consists in transforming the system to an adequate from through an injective transformation. For the rest of the section, this injective map will be referred to as T. Last subsection has established that the system is instantaneously observable in the considered operating region, and the observability of the system is uniform in the input vair . Consequently, it is possible to design a vair -independent coordinate change that transforms the system into a triangular structure. See Sect. 3.2.2 in Chap. 3 for more details. Specifically, consider system (6.53) and define the map T : Rn × R → Rn as

T(x, I ) =

 Tfc  ξ, K 1 (E ocv n cell − V f c )I

(6.56)

which is a diffeomorphism, as the Jacobian of (6.56) is full rank in the fuel cell operating region (6.55). The map (6.56) is an invertible coordinate change that transforms system (6.53) to the following triangular form ξ˙ = Aξ + (ξ, u), y = cξ where A=





  ψ1 (ξ1 , u) 01 ; c = 1 0 , (ξ, u) = 00 ψ2 (ξ, u, I˙)

(6.57)

(6.58)

and the functions ψ1 and ψ2 are Lipschitz. Remark 6.7 Notice that the function ψ2 depends on the derivative of the current, I˙, which is not directly measured. Nevertheless, there are multiple ways to deal with this problem. In some types of control, one has access to the derivatives of the input, e.g. backstepping control or higher-order sliding mode control, between others. Alternatively, one can estimate its value through robust differentiators [59] and deal with the discrepancy by increasing the high-gain design parameter. If neither of these options is adequate for the considered scenario, it is still possible to follow the insights presented in Sect. 3.2.2.3 of Chap. 3 in order to formulate the observer

164

6 PEM Fuel Cell Monitoring

in the original coordinates, x, instead of the target coordinates, ξ. This will avoid the need of computing I˙.

6.3.3 Observer 1: Low-Power Peaking-Free Dead-Zone Observer Once the system is transformed to a triangular structure, it is relatively simple to implement a high-gain observer, as explained in Sect. 3.2 of Chap. 3. Nonetheless, high-gain observers presents two significant drawbacks. First, the peaking phenomena may drive the observer estimations outside the observability domain (6.55), which makes the map in (6.56) lose injectivity. Second, temperature sensors present significant noise and standard high-gain observers are significantly sensitive to sensor noise. In order to reduce the effect of these limitations, the modification proposed in the Example of Chap. 4 will be adapted for the considered case. Specifically, the peaking phenomena will be eliminated by changing the high-gain observer for the low-power peaking-free observer studied in Sect. 3.2.3 of Chap. 3. Second, the effect of sensor noise will be reduced by implementing the dynamic dead-zone filter studied in Sect. 4.5 of Chap. 4. Precisely, the proposed observer is implemented through the following dynamics, α1 √ ˙ dz σ1 (e1 ) ξˆ1 = η1 + ψ1 (ξˆ1 , u) + ε ˙ ˆ u, I˙) + α2 dz √σ2 (e2 ) ξˆ2 = ψ2 (ξ, ε β ˆ u, I˙) + 1 dz √σ1 (e1 ) η˙1 = ψ2 (ξ, ε2 qi σ˙ i = − 2 σi + pi |ei |2 , i = 1, 2. ε

(6.59)

where e1  y − ξˆ1 , e2  satr1 (η1 ) − ξˆ2 . and α1 , α2 , β1 , ε, r2 and qi , pi for i = 1, 2 are parameters to be tuned. See Theorem 3.3 in Chap. 3 and Theorem 4.5 in Chap. 4 for the details on the parameter tuning. The last step consists in inverting the transformation in (6.56) to recover the states in the original coordinates. Naturally, the temperature is directly obtained through ξ1 = T f c , thus, it is only required to invert the second equation of (6.56) to retrieve the liquid water saturation. As the equation is relatively simple, it is possible to find an analytical expression of the inverse function. Precisely,

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

165

Fig. 6.7 Current profile used to excite the simulated model



sˆ = sopt 1 − 1 −

 0.21Acell j0,ca,r e f ac ex p

I

3 

 ηact αc F ΔG ∗ ξˆ1 − (1 − ) Tr e f R ξˆ1 R ξˆ1 (6.60)

where the activation overpotential, ηact , is computed as ηact =

ξˆ2 − ηohm . K1 I

Alternatively, following the insights in Sect. 3.2.2.3 in Chap. 3, the observer could be implemented in the original coordinates. The proposed observer has been validated through a numerical simulation. In the simulation, the fuel cell model in (6.53) is going to be disturbed with some changes in the exchange current signal, I . The exact profile introduced in the model is depicted in Fig. 6.7. In order to make the simulation more realistic, two more elements have been included in the model. First, the temperature of the model will be controlled through a proportional integral anti-windup structure (PI+AW) that, through the cathode air velocity, vair , will maintain the temperature close to a reference point, Tr e f = 310 K, despite the changes in the current profile. The observer is not used in this feedback loop, so, this section does not focus on the design of the controller. The interested reader is referred to the paper [56] for more details on the controller. Second, the generated temperature profile is corrupted with random Gaussian noise with a realistic variance value of 0.011 (taken from the real experimental set-up). In Fig. 6.8, it is depicted the general scheme of the simulation. From the observer point of view, it is assumed that there is no prior information of the unknown states. Therefore, the state-estimation is initialized at an arbitrary feasible operating condition Tˆ f c = 300 and sˆ = 0.01. It should be remarked that the system model and the observer model coincides in this simulation.

166

6 PEM Fuel Cell Monitoring

Fig. 6.8 Simulation general scheme. The PI+AW box depicts the controller proposed in [56], the fuel cell model box depicts the model (6.53) with the parameters summarized in Table 6.7 and the low-power power-peaking-free dead-zone observer box depicts the proposed observer structure Table 6.4 PEM fuel cell LPPFDZO parameters

Parameter

Value

α1 α2 β1 r1 ε qi p1 p2

1.5 0.01 0.5 0.16 0.25 3 180 40

To show the benefits of the proposed observer structure. The estimation of the low-power peaking-free dead-zone observer will be compared with the estimation of a classic high-gain observer with similar convergence rate. The parameter of the classic high-gain observer are α1 = 0.102, α2 = 0.0002 and ε = 0.055. The low-power peaking-free dead-zone observer design parameters are summarized in the Table 6.4. In Fig. 6.9 it is depicted the evolution of the model true liquid water saturation, the classic high-gain observer estimation and the low-power peaking-free dead-zone observer estimation. It can be seen that both observers converge to a relative error below 5% within the first 150 s, which is an acceptable convergence rate as, in general, the water dynamics of PEMFCs requires around 1000 s to reach a steady-state [11]. Notice that the convergence rate of both observers is nearly identical. Nonetheless, the

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

167

Fig. 6.9 Model liquid water saturation, standard high-gain observer estimation (HGO) and lowpower peaking-free dead-zone observer (LPPFDZO) estimation in presence of measurement noise

persistent estimation-error induced by the noise is significantly reduced in the lowpower peaking-free dead-zone observer. In order to quantify this improvement, the mean square error (MSE) of the liquid water saturation estimation has been computed. MSE =

n 1 (s(i) − sˆ (i))2 . n 1

The implementation of the low-power peaking-free dead-zone observer has resulted in a reduction of 32.3% of the MSE. It should be remarked that further noise reduction could be achieved by adding a properly tuned low-pass filter before the dynamic dead-zone.

6.3.4 Adding the Voltage Sensor Past subsection has presented an observer to estimate the liquid water saturation. Although the results are positive and have been validated through numerical simulations, there is still an interesting detail to be considered. That is, the observer does not use the stack voltage sensor measurements y2 . Intuitively, including additional sensor information in the estimator should improve the accuracy of the observer. Thus, it may be convenient to include the voltage sensor in the observer designed in the past section. Notice that the proposed objective differs from designing an observer for multioutput systems. What is being desired here is to redesign an already existing observer in order to include extra sensor measurements. In the Kalman filter context and signal process community, this redesign is usually referred as sensor fusion [60]. The main concern is that there is no generic methodology to include additional sensors

168

6 PEM Fuel Cell Monitoring

in high-gain observers, without a major modification of the whole observer scheme. Fortunately, the considered stack voltage equation satisfies a certain monotonicity property that simplifies the process. This subsection will, first, describe how monotonicity can be exploited in high-gain observers. Second, introduce a modification of the high-gain observer presented in the past section, that includes the voltage sensor. Third, briefly discuss the benefits of such modification.

6.3.4.1

Adding Additional Outputs in High-Gain Observers: A Monotonic Approach

In order to show that the proposed approach is a quite general result, not restricted to the estimation of the liquid water in fuel cell, this subsection considers a more general system in the following triangular form ξ˙ = Aξ + Bϕ(ξ), y1 = cξ y2 = h(ξ)

(6.61)

where ϕ is a globally Lipschitz function and the triplet (A, B, c) is in a triangular form similar to the one considered in Sect. 3.2 of Chap. 3. The main assumption in this case is that the nonlinear function h(·) satisfies the following monotonic condition   ˆ Q(ξ) ˆ h(ξ) − h(ξ) ˆ ≥0 (ξ − ξ)

ˆ ∈ Rn , ∀(ξ, ξ)

(6.62)

where Q : Rn → Rn is a matrix to be designed. Remark 6.8 A sufficient condition for a function to be monotonic is that [61] ∇h + ∇h ≥ 0. From the developments in Sect. 3.2 of Chap. 3, it is relatively easy to design a highgain observer (and its low-power modification) taking y1 as the measured signal. The idea is to redesign the resulting high-gain observer in order to include the second output y2 . The result can be formalized through the following theorem. Theorem 6.1 Consider a system of the form (6.61) and assume a standard high-gain observer of the for ˙ ˆ ˆ + Dk l(y1 − cξ) (6.63) ξˆ = Aξˆ + Bϕ(ξ) with Dk = diag(k, . . . , k n ), k is related to the high-gain parameter as k = ε−1 , and l being so that the matrix (A − lc) is Hurwitz. Assume that ε is small enough so (6.63) is an asymptotic observer. Moreover, define the matrix Pk ∈ Rn×n as −1 Pk = D−1 k PDk

(6.64)

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

169

where P ∈ Rn×n is a symmetric positive definite matrix such that P(A − lc) + (A − lc) P ≤ −In .

(6.65)

Then, the following system ˙ ˆ + P−1 Q(ξ)(y ˆ 2 − h(ξ)), ˆ ˆ + Dk l(y1 − cξ) ξˆ = Aξˆ + Bϕ(ξ) k

(6.66)

is also an asymptotic observer for the system (6.61). Proof A Lyapunov function that can be used to show the convergence of (6.63) is given (in the ξ coordinates) by ˜ = ξ˜ Pk ξ. ˜ V (ξ)

(6.67)

Indeed, its time derivative is computed as  

ˆ . V˙ = 2ξ˜ Pk (A − Dk lc)ξ˜ + B(ϕ(ξ) − ϕ(ξ)) −1 −1 By using the following relations D−1 k A = kADk , c = kcDk , the next bound is obtained  

−1 ˜ −1 ˆ V˙ = 2ξ˜ D−1 k P k(A − lc)Dk ξ + Dk B(ϕ(ξ) − ϕ(ξ))  2 ≤ −(k − 2L ϕ p) ¯ D−1 ξ˜  k

with L ϕ being the Lipschitz constant of ϕ, and p¯ being the maximum eigenvalue of P. Taking k large enough (or the parameter ε low enough), the Lyapunov function derivative is negative definite. Now, consider the observer redesign in (6.66), it can be seen that computing the derivative of (6.67) along solutions to (6.61), (6.66) gives     ˆ ˜ 2 ˜ ˆ ¯ D−1 V˙ = −(k − 2L ϕ p) k ξ − ξ Q(ξ) h(ξ) − h(ξ)  2 ≤ −(k − 2L ϕ p) ¯ D−1 ξ˜  k

Therefore, the Lyapunov derivative is still negative definite for a large enough k.  Remark 6.9 The notation in the Lyapunov function in (6.67) is slightly different from the one considered in Theorem 3.1 proof of Chap. 3. The proof considered here avoids the coordinate change (3.14) which simplifies the inclusion of the extra output. Nonetheless, in practice, the Lyapunov function and the developments are exactly the same.

170

6.3.4.2

6 PEM Fuel Cell Monitoring

Monotonicity of the Voltage Equation

In order to exploit the presented result it is crucial to establish the monotonicity of the stack voltage equation in the coordinates ξ that transforms the system to the triangular form in (6.57). As the temperature is measured, we can consider a stack voltage function that depends on the measured temperature (as a time-varying signal) and the estimated variable, ξˆ2 . Thus, it is sufficient to show that the stack voltage equation is monotonic with respect to the variable ξ2 . On the one hand, the stack voltage equation is monotonic with respect to the liquid water saturation, s. This fact can be seen from the following. Consider the partial derivative of the stack voltage ∂V f c = ∂s

−RT f c .      sopt − s 2/3 sopt − s 1/3 3αc F 1 − sopt sopt sopt

(6.68)

An important property of the system is that 0 < s < sopt . Consequently, the following relation holds in the considered operating conditions ∂V f c

∂V f c + ≤ 0. ∂s ∂s

(6.69)

Therefore, according to Remark 6.8, the stack voltage equation is monotonic with respect to s. Following a similar process, it is easy to show, from the relation in (6.60), that the liquid water saturation is monotonic to the variable ξ2 . That is, ∂s ∂s

+ ≤ 0. ∂ξ2 ∂ξ2 As the composition of two monotonic functions is also monotonic, it can be concluded that V f c is a monotonic function with respect to the variable ξ2 . This result implies that   ˆ ≥ 0. ξ˜2 h(ξ) − h(ξ) Thus, the condition in (6.62) is satisfied by taking Q as any vector Q =

 0 where k

k is a positive constant.

6.3.4.3

Numerical Simulations and the Effect of Anode Periodic Purges

The natural question is whether adding the voltage sensor has any significant impact on the performance of the high-gain observer. To assess this question, the modified observer will be analysed through a numerical simulation. The simulation presents

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

171

a similar structure as the one in Fig. 6.8, but using the observer with the voltage sensor instead of the low-power peaking-free dead-zone observer. Indeed, it will be analysed the high-gain observer in (6.66) without the peaking-free redesign nor any filter modification. Moreover, the model will be initialized at T f c = 310 and s = 0.18. The exchange current of the model will be fixed at I = 3 A and a PI+AW controller is implemented to maintain the temperature at 304 K . The temperature sensor will be disturbed by zero-mean high-frequency noise of variance 1 · 10−3 and the voltage sensor by similar noise of variance 1 · 10−5 . Both values are similar to the ones considered in Sect. 6.3.3 numerical simulation. Precisely, the observer is implemented as in (6.66) with the following parameters

l=

 1.5 , k = 2. 0.001

(6.70)

Moreover, from the inequality in (6.65) it is possible to find the following matrix P P=

 0.33 −0.5 , ∗ 1083.67

(6.71)

and the matrix Pk can be computed through (6.64). The true evolution of the liquid water saturation and the observer estimation is presented in Fig. 6.10. It can be seen that the estimation converges to the true value, which validates the result in Theorem 6.1. The outstanding property of the observer is its high convergence rate to noise sensitivity ratio. Indeed, the low-power peaking-free dead-zone presented larger noise sensitivity and considerably lower convergence rate. In Fig. 6.9 it can be seen that the observer required around 200 s to converge, while the observer in this section required 2 s. This improvement in the convergence rate can be mainly explained due to the monotonic property of the voltage function, which allows to push the liquid water

Fig. 6.10 Model liquid water saturation and high-gain observer estimation with voltage measurement estimation

172

6 PEM Fuel Cell Monitoring

Fig. 6.11 Model liquid water saturation and high-gain observer estimation with voltage measurement estimation in the presence of voltage disturbances. The disturbance is modelled as a set of periodic pulses of amplitude −0.2, duration of 1 s and period of 100 s

saturation estimation in the “right direction” during the observer transient to reduce undesirable behaviours. Besides, the only option to increase the standard high-gain observer (or the low-power modification) convergence rate is the increase of its gain, which drastically increases the observer noise sensitivity. Therefore, as the inclusion of the voltage sensor directly increases the convergence rate, the observer can be implemented with lower gain which reduces the noise sensitivity. It should be remarked that, besides the liquid water saturation, the fuel cell stack voltage is also monotonic to additional system variables that are not being considered in the model (6.53) e.g. the oxygen partial pressure (see Eq. (6.47)). For this reason, the results presented in Theorem 6.1 are not limited to liquid water estimation observers and can be extended to other fuel cell observers that do not include the stack voltage signal, e.g. [49]. Nonetheless, the proposed redesign is not free of drawbacks that should be addressed in practical applications. Naturally, the inclusion of the voltage signal implies that voltage sensor disturbances are directly fed in the observer, which may deteriorate the estimation. This is a significant drawback because in some particular fuel cell operation modes, in which the voltage signal is affected by significant disturbances besides the sensor noise. To see the effect that such disturbances may have on the estimation consider Fig. 6.11. The aim of the following section is to study the origin of such disturbances and propose an observer redesign to reduce its effect.

6.3.5 Observer 2: High-Gain Observer with Voltage Sensor and Low-Pass Internal-Model Filter 6.3.5.1

Motivation: Dead-End Mode Operation

During fuel cell operation, hydrogen is delivered in the anode side by a compressed cylinder. To do so, one can implement two different operating strategies [27]. The first, usually referred to as flow-through operation, operates with an open anode

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

173

Fig. 6.12 Modes of anode operation. a Dead-end mode, b flow-through mode

outlet, which eases the evacuation of undesired components as water and/or nitrogen. However, in order to ensure that the reactant deliver is at an equal or higher rate than its consumption, the hydrogen has to be delivered in excess, which requires a flow controller design. The second, referred to as dead-end mode, operates with a normally closed anode outlet. As the hydrogen is delivered by a high-pressure cylinder, deadend operation does not require any flow control, that is, hydrogen is being supplied as it is being consumed. In Fig. 6.12, it is depicted a general scheme of both anode operation strategies. However, in dead-end mode operation, the anode has to be periodically purged due to the accumulation of inert gases as nitrogen that crosses the membrane or water. During these purges, the partial pressures of the gases drastically oscillates, which induce some perturbations on the output voltage. As an example, the stack voltage measured in a fuel cell experimental prototype operated in dead-end mode is depicted in Fig. 6.13. More details on the experimental set-up will be given in Sect. 6.3.8.1. It is noticeable that, every 20 s, there are sudden downside “peaks” in the signal, one for each anode purge. Unfortunately, this phenomenon is not modelled in the reduced model in (6.53). Therefore, anode purges can be understood as external disturbances that affect the observer accuracy. Naturally, a means to reduce the effect of this disturbance is to extend the considered mathematical model to include the presence of anode purges. Nonetheless, that would mostly imply the inclusion of extra states in (6.53) which induces a significant modification of the observers presented in this chapter. Specifically, it would require an additional observability analysis, and involves constructing an alternative coordinate change. For this reason, as it has been done throughout the whole work (see the beginning of Chap. 5 for a discussion), it will be assumed that the mathematical model cannot be longer improved, and the disturbance reduction has to come from an adequate observer redesign. An interesting property to consider is that the membrane electrode assembly temperature and the liquid water saturation evolve in a slow time-scale in relation to

174

6 PEM Fuel Cell Monitoring

Fig. 6.13 Output voltage sample under dead-end operation. There is a periodic anode purge every 20 s, which induces a downwards peak in the voltage

the gas transport dynamics [62]. Consequently, anode purges have a negligible effect on states considered in (6.53). Thus, the effect of anode purges can be modelled as a voltage sensor disturbance x˙ = fs (x, I ) + g(x)vair y1 = T f c y2 = V f c (x, I ) + d. where d ∈ R is the disturbance induce by the anode purges.

6.3.5.2

Modelling Periodic Purges

Following from the discussion on the last section, the objective is to redesign the observer in (6.66) in order to reduce the effect of the output disturbance, d, generated by the anode purges. Prior to any observer modification, it is crucial to study if the disturbance satisfies some property that can be exploited in the redesign. Indeed, as most fuel cell systems do not include the presence of anode purges in the channel monitoring system, the anode channel is not purged at the exact time that is required, but, is periodically purged with a constant period, T , to be designed [29, 63–66]. This periodicity is the key property that will be exploited. Indeed, under the assumption that the periodic disturbance, d, is continuous, the signal can be represented by its Fourier expansion d(t) =

    ∞

2πkt 2πkt a0  ak cos + + bk sin . 2 T T k=1

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

175

 2πkt  1 T 2 T 2 T and ak = where a0 = d(t) 0 d(t)dt, ak = 0 d(t) cos T T T T 0  2πkt  sin . T Therefore, the periodic disturbance can be approximated by a linear exo-system that includes the first l harmonics of the Fourier expansion. Precisely, define the following matrices ⎡ ⎢ 0 Sk  ⎣ 2πk − T

⎤ 2πk   T ⎥ ⎦, Γk  1 0 . 0

Then, the periodic output disturbance can be approximated as ˙ = Sw, d ≈ Γ w w

(6.72)

  where S = blckdiag(02×2 , S1 , . . . , Sl ) and Γ = 1 Γ 1 . . . Γ l and w(0) is properly initialized. Notice that this purge modelling is much more simpler than including the presence of purges in the model as it only requires knowing the period of the purges, T . This is not a restrictive assumption, as the period of the purges are defined by the designer.

6.3.5.3

Low-Pass Internal-Model Filter in the Voltage Error Signal

The topic of rejecting output disturbances generated from a known exo-system has been studied in Sect. 4.4.3 of Chap. 4. The idea is to cascade the observer with an internal-model filter that exactly reject the disturbance. This section proposes implementing an internal-model that includes the harmonics in the exo-system (6.72) in order to reject the effect of the periodic purges. Notice that as the exo-system in (6.72) only includes a finite number of harmonics. Consequently, the unmodelled harmonics will still be fed to the observer as highfrequency noise. For this reason, the internal-model filter is implemented in the lowpass variation presented in Sect. 4.4.4 Chap. 4. Precisely, the proposed observer is exactly the same as the one presented in Sect. 6.3.4, but the voltage estimation-error, y˜2 , is substituted by its filtered version, which is computed through the following dynamics η˙ = Sη + Γ y˜2 z˙ = z − Γ η + y˜2 where y˜2 = y2 − V f c (y1 , sˆ , I ). The proposed observer has been validated in a numerical simulation and compared with the observer presented in Sect. 6.3.4, the high-gain observer with unfiltered

176

6 PEM Fuel Cell Monitoring

Fig. 6.14 Model liquid water saturation, high-gain observer estimation with voltage sensor and high-gain observer estimation with voltage sensor and low-pass internal model filter

voltage signal. The simulation replicates the conditions in Sect. 6.3.4.3, both observers are implemented with the parameters (6.70) and (6.71), and the voltage signal is disturbed with a set of periodic pulses of amplitude −0.2, duration of 1 s and period of 100 s that mimics the effect of anode purges. In the simulation, the internal model filter only includes the first 3 harmonics of the disturbance, l = 3. The results are presented in Fig. 6.14. It can be seen that both observers present similar convergence rate, nonetheless, the benefits of including the low-pass internalmodel filter are notorious. First, the effect of unmodelled high-frequency noise is significantly reduced. Second, the effect of the periodic output disturbance is practically eliminated. Nonetheless, the obtained results should to be taken with care. While multiple numerical simulations have validated the viability of internal-model filters in some nonlinear systems. Due to time limitations, the complete theory regarding its stability has not been developed in the general nonlinear case. Consequently, its design still requires a trial-and-error process with no direct theoretical guarantees. Future works will focus in solving this limitation. It is expected that, under some assumptions, the internal-model filter approach can be used to robustify other fuel cell observers in front of anodic purges. This is an interesting result as, to the author’s knowledge, the design of observers in the presence of anode purges (and its induced voltage perturbations) has not been considered before.

6.3.6 Observer 3: Direct Adaptive Observer Past sections have proposed a set of observers for the estimation of the liquid water saturation. Moreover, it has provided a set of modifications to address sensor disturbances, including high-frequency sensor noise and the effect of periodic anode purges. The benefits of the proposed observers have been validated and discussed through a set of numerical simulations. Nonetheless, there have been a clearly unreasonable assumption during the simulations. That is, the mathematical model used to

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

177

simulate the fuel cell is equal to the model used in the observer. Naturally, due to the assumptions on the model considered in Sect. 6.2, and the further assumptions taken to reduce the model up to the equations in (6.53), there will be significant discrepancy between the model and the reality. Following this line, a proper observer design requires a model parameter identification that minimizes the uncertainty. This fact creates two significant problems. First, if some states cannot be directly measured, it may be difficult to identify the parameters related to its dynamics. Second, even if an adequate identification is achieved, the result will only be local, i.e. the parameters will only be adequate around the operating point in which the experiments and the identification have been conducted. This implies that, in general, the model parameters will not be adequately tuned, and the model will lose accuracy with time due to a change in the operating conditions and/or degradation. This fact has been previously detected by multiple authors, and has motivated the implementation of robust observers in the fuel cell field [67–70]. Nonetheless, such approach provides limited information of the model uncertainty, which could be useful for diagnostics purposes, and are sensitive to sensor noise. To overcome this limitations, this section proposes following the direct adaptive redesign explored in Sect. 5.2 of Chap. 5, in order to generate on-line an estimation of the unknown parameters that is consistent with the measured signals. As a first step, it is crucial to establish which parameters can be reliably estimated through an off-line identification process, and which ones have to be estimated online by the adaptive redesign. Indeed, liquid water saturation is a signal that is difficult to measure in practice. For this reason, it is reasonable to assume that the parameters related to the water dynamics cannot be identified and are unknown. Precisely, it will be assumed that the parameters K 5 (liquid water diffusion coefficient), K evap (evaporation time constant) and K s (liquid water accumulation coefficient) are unknown. The rest are related with physical properties of the fuel cell that are relatively easy to measure, thus, assumed to be adequately identified. Consequently, for the rest of the section, the unknown parameter vector will be referred as   θ = θ1 θ2 θ3 =



K evap 1 K5 K s K s A por e



.

With this choice, the model (6.53) can be rewritten as follows x˙ = A(u)x + f(V f c , u) + bφ(x, u)θ, y = cx, where

   f (x, u) φ(x, u) = φ1 (u), φ2 (x), φ3 (x) , f(x, u) = 1 , 0 A(u) =





  0 −K 2 vair 0 , b= c= 1 0 0 0 1

(6.73)

178

6 PEM Fuel Cell Monitoring

Fig. 6.15 Scheme of the proposed adaptive observer for PEMFC

and f 1 (y2 , u) = K 1 (E ocv n cell − V f c )I + K 2 Tamb vair φ1 (u) = K 3 I, φ2 (x) = −K 4 f p (T f c ), φ3 (x) = − f d (s). Following from the design presented in Sect. 5.2 of Chap. 5, the subsequent steps would consist on designing an observer for system (6.73) assuming that θ is known, ˙ and then, designing a dynamics θˆ such that the unknown parameters are eliminated from the observer Lyapunov function. Notice that the relative degree between the measured signal and the parameters is higher than one. Therefore, this elimination is not a trivial task. To circumvent this obstacle, it is proposed to implement the high-gain observer modification proposed in Sect. 5.2.2 of Chap. 5, i.e. a high-gain observer is used to estimate an auxiliary signal with appropriate relative degree, and use this signal as the measured output of a direct adaptive observer. A scheme of the proposed observer is presented in Fig. 6.15. For the proposed application, consider the following relative degree 1 auxiliary signal z = h x   where h = 1 1 . The fact that this signal is relative degree 1 from the measurd output to the unknown parameters can be deduced from the fact that h b = 0. Moreover, the pair (A(u), h ) is observable if the cathode air velocity is different from zero, vair = 0. As the cathode air is the main source of oxygen, the air velocity will always be different from zero. Thus, the system is observable. Notice that the structure of the system (6.73) and the fact that the pair (A(u), h ) is observable motivates the implementation of the observer presented in Example of Chap. 5. Precisely, the observer is implemented as x˙ˆ = A(u)ˆx + f(V f c , u) + bφ(ˆx, u)θˆ + k(t)(ˆz − h xˆ ) ˙ P(t) = −σP(t) − A(u) P(t) − P(t)A(u) + hh ˙ θˆ = φ(ˆx, u) b (k(t) )† (ˆz − h xˆ )

(6.74)

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

179

where zˆ is the estimation of the auxiliary signal generated from the high-gain observer, σ is a positive design parameter and k(t) = P−1 (t)h. The only part that remains to be defined is the high-gain observer that is used to estimate the auxiliary signal, zˆ . In theory, a standard high-gain observer could be implemented for this purpose. Nonetheless, this chapter has presented multiple high-gain observer modifications with significantly higher performance. This chapter proposes estimating the auxiliary signal through a low-power peaking-free observer filtered with dynamic dead-zones similar to the one presented in Sect. 6.3.3. Remark 6.10 The reader may be interested in the reasons behind choosing the observer in Sect. 6.3.3 instead of the high-gain observer with voltage measurements and internal-model filter presented in Sect. 6.3.5, as the latter one presents significantly better convergence rate and noise sensitivity. The main reason is that the former does not utilize the voltage sensor. Consequently, the voltage signal can be used to validate the observer estimation accuracy, which will be crucial during the experimental validation of the observer. Remark 6.11 The structure of the system in (6.73) is not exactly the same as the one considered in the Example in Chap. 5 due to the factor f(V f c , u). Nonetheless, notice that this is a factor that only depends on measured signals. Thus, similar to the approach considered at the beginning of Sect. 3.3 of Chap. 3, the factor f(V f c , u) can be included in the observer and is exactly cancelled in the state-estimation error, leading to a state-estimation error of the same structure as in the Example of Chap. 5. Specifically, the auxiliary signal is computed through the following dynamics, α1 √ ˙ dz σ1 (e1 ) ξˆ1 = η1 + ψ1 (ξˆ1 , u) + ε ˙ ˆ I˙) + α2 dz √σ2 (e2 ) ξˆ2 = ψ2 (ˆx, u, θ, ε β ˆ I˙) + 1 dz √σ1 (e1 ), η˙1 = ψ2 (ˆx, u, θ, ε2 qi σ˙ i = − 2 σi + pi |ei |2 , i = 1, 2. ε where e1  y − ξˆ1 , e2  satr1 (η1 ) − ξˆ2 . and xˆ , θˆ are the state-estimation and the parameter estimation generated by the adaptive observer in (6.74).

180

6 PEM Fuel Cell Monitoring

Finally, the estimated auxiliary signal, zˆ , used in the adaptive observer is computed as follows zˆ = ξˆ1 + sˆ , where sˆ is computed through (6.60).

6.3.7 Numerical Simulation The proposed adaptive observer has been validated in a set of numerical simulations. The simulations present a similar structure as the one considered in Fig. 6.8. Being the only differences that, now, the adaptive observer in (6.74) is considered instead of the low-power peaking-free dead-zone observer, and the reference temperature in the controller is 304 K . Moreover, the observer parameters have been tuned to provide a state-estimation settling time (98%) of 200 s and provide adequate noise rejection. Said parameters are summarized in the Table 6.5. It is assumed that there is no information of the current derivative, I˙, required ˆ u, θ, I˙). Therefore, the proposed observer has been for the computation of ψ2 (ξ, ˆ implemented with ψ2 (ˆx, u, θ, 0). This will induce some bias error in the estimation. It should be remarked that there are methods to reduce this bias, see Remark 6.7. However, as presented in Sect. 3.2.3 of Chap. 3, the low-power peaking-free observer is robust with respect to the unmodelled I˙, making any modification unnecessary. The evolution of the model true liquid water saturation and the observer estimation is depicted in Fig. 6.16. The estimation-error converges to a relative error below 3%, within the first 200 s. Therefore, the proposed scheme is capable of estimating the fuel cell catalyst layer liquid water saturation while satisfying the proposed observer performance objectives even in the presence of significant sensor noise. It is noticeable that there is a small bias between the estimation and the true value. This discrepancy is created by the absence of the factor I˙ in the observer equations. Nevertheless, the robustness of the low-power observer reduces the effect of the unknown I˙, and the bias induced by this discrepancy is negligible. Table 6.5 Adaptive observer parameters in numerical simulation

Parameter

Value

α1 α2 β1 r1 ε σ qi pi

1.5 0.01 0.5 0.16 0.11 0.0612 3 300

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

181

Fig. 6.16 Model liquid water saturation (orange) and adaptive observer estimation (blue) in presence of measurement noise

Fig. 6.17 Model unknown parameter θ true value (orange) and adaptive observer estimation (blue) in presence of measurement noise

Furthermore, the observer estimation of the unknown parameters and the true model values is depicted in Fig. 6.17. In all cases, before 200 s, the estimation converges to a relative error below 0.1% despite the presence of significant sensor noise and the unmodelled factor, I˙. This convergence time is of the same time-scale as the PEMFC water dynamics [11]. Thus, the estimation could be used for real-time fuel cell monitoring. Naturally, the model can present additional uncertainty that is not contained in the parameters, θ. For this reason, a second simulation has been executed to assess the robustness of the algorithm in front of extra parametric uncertainty. Indeed, consider the same simulation being the only difference, that a random 5% time-varying error has been introduced in the parameters K 2 , Rohm , αc and Pg of the observer model. These parameters are not going to be adapted, thus, some bias in the estimation will be induced. The objective is to study if this bias is significant or not. The true evolution of the liquid water saturation and the observer estimation are depicted in Fig. 6.18. It can be seen that even in the presence of significant parameter uncertainty and sensor noise, the estimation converges to a maximum relative error below the 3%. The true value of the unknown parameters, θ, and the observer estimation is depicted in Fig. 6.19. In the presence of parametric uncertainty, the estimation of θ1 has converged to a relative error of 0.634%, the estimation of θ2 to 1.22% and θ3 to 2.02%. Therefore, even in the presence of significant parametric uncertainty, the observer still estimates the liquid water saturation and the

182

6 PEM Fuel Cell Monitoring

Fig. 6.18 Model liquid water saturation (orange) and adaptive observer estimation (blue) in presence of measurement noise with 5% time-varying error in the model parameters K 2 , Rohm , αc and pv

Fig. 6.19 Model unknown parameter θ true value (orange) and adaptive observer estimation (blue) in presence of measurement noise with 5% time-varying error in the model parameters K 2 , Rohm , αc and pv

unknown parameters with the desired performance, which validates the robustness of the technique.

6.3.8 Experimental Validation To end this chapter, the proposed observers are validated in a true experimental prototype. The most complete observer has been provided in Sect. 6.3.6, as it combines the observer in Sect. 6.3.3 (which can be substituted by the observer in Sect. 6.3.5) with an adaptive observer that estimates unknown parameters. For this reason, this section will focus on experimentally validating the observer in Sect. 6.3.6.

6.3.8.1

Experimental Set-Up

This work considers a PEMFC stack H-100 of Horizon fuel cell technologies of 20 cells and a rated power of 100 W. Due to its small size, low weight and lack of peripherals (as a consequence of the open-cathode architecture) the PEMFC H-series

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

183

Fig. 6.20 a Environmental chamber and H-100 PEMFC. b H-100 experimental set-up scheme

are very attractive for small transport applications. The fuel cell cathode is selfhumidified and includes an attached fan that delivers air to the cathode and cools down the system. The cathode air velocity is measured through a hot film sensor model EE75 of E+E Elektronik. The cathode fan is controlled through a NI-9505 PWM module of National Instruments. Pure hydrogen is delivered in the anode side by a compressed hydrogen cylinder. In order to avoid the design of a flow controller, the H-100 fuel cell will be operated in dead-end mode [27]. A pressure regulator maintains the anode inlet pressure at 0.4 bar and 500 ms purges are executed every 20 s. Due to the open-cathode architecture, the PEMFC is fed with oxygen taken directly from the ambient air, which makes the system very sensible to the ambient conditions. In order to make the experiments reproducible, the fuel cell is enclosed in an environmental chamber that regulates the ambient temperature, relative humidity and oxygen concentration. The humidity and ambient temperature at the cathode are measured through a sensor model HMM211 from Vaisala. Moreover, the temperature of each cell is measured, individually, through a type K thermocouple. The average of the 20 temperature measurements is taken as the averaged membrane electrode assembly temperature T f c . The experimental set-up includes a programmable load that allows to control the demanded current and emulate a real application. The stack voltage, V f c , is measured through an isolation amplifier, model AD215 from Analog Devices and the exchange current, I , through a Hall effect sensor model LTS 6 NP of LEM. The experimental set-up and a photography of the environmental chamber and the H-100 PEMFC can be seen in Fig. 6.20. All the data acquisition devices are connected to a Compact Rio embedded controller cRIO-9047 of National Instruments, which includes a FPGA module and can be programmed in the LabView environment. All the data is acquired in a sampling time of two seconds, which is considered to be adequate, as the fuel cell thermal and water dynamics time scales are an order of magnitude larger and the observer computational cost is negligible at this frequency.

184

6 PEM Fuel Cell Monitoring

The use of the presented fuel cell system and peripherals has been motivated by the existence of previous works [56, 57] that implemented a similar PEMFC model in a similar experimental setup. This fact allows having a reasonable comparison between the proposed observer technique and the available literature.

6.3.8.2

Methodology

The main problem in validating the proposed observer strategy is the unavailability of liquid water saturation sensors. As a consequence, one can apply the proposed observer technique to generate an estimation of the water, sˆ , and the unknown paramˆ However, these estimations cannot be compared with the true values. The eters, θ. idea is to use extra measured signals, which are not implemented in the observer, to validate the estimation. Specifically, the estimation of the stack temperature, Tˆ f c , the estimation of the liquid water saturation, sˆ , and the stack voltage equation can be used to generate an estimation of the stack voltage, Vˆ f c . If we assume that there is sufficient air and hydrogen and the current is low enough, the fuel cell concentration losses are negligible. Thus, the accuracy of the stack voltage computation only depends on the stack temperature and the catalyst layer liquid water saturation. Consequently, in such conditions, one can verify the accuracy of the liquid water saturation estimation by computing the errors |T f c − Tˆ f c | and |V f c − Vˆ f c |. The H-100 PEMFC setup will be excited by a constant exchange current of 3.9 A and a step function from 0.21 m s −1 to 0.19 m s −1 in the cathode air velocity. As the current is maintained constant at a medium-low range value, it is not expected to be excessive water generation due to the cathode reduction reaction. However, due to the air velocity decrease and the absence of temperature control, the fuel cell stack temperature is expected to increase, which will boost the liquid water evaporation rates. Therefore, the catalyst layer liquid water saturation should reduce during the experiment, which should provide the necessary system excitation for the parameter estimation. The accuracy of the parameter estimation, as previously anticipated, will be validated by comparison to the results obtained in [56, 57]. As the liquid water transport parameters may vary depending on the operating conditions, it is crucial to operate the PEMFC in a similar range as in [56, 57] in order to have a reasonable comparison. Specifically, the experiment conditions are summarized in Table 6.6.

6.3.8.3

System Identification

The H-100 fuel cell parameters in relation to the reduced model in (6.53) have been identified in the operating conditions depicted in Table 6.6 and in the corresponding stack temperatures. The system identification experiment consisted in introducing a step function in the cathode air velocity and measuring the stack temperature,

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers Table 6.6 Operating conditions of the experiment Variable Value Anode reactant Cathode reactant Tamb R Hamb Anode pressure Anode RH Current

Dry H2 Ambient air 25 75 0.4 0 3.9

185

Units − − ◦C % Bar % A

T f c , and voltage, V f c . The temperature equation parameters, K 1 and K 2 , have been fitted through a linear least squares comparing T f c and Tˆ f c , and the voltage equation parameters Rohm and αc have been fitted through a linear least squares comparing V and Vˆ f c . The values of j0 and ac have been taken from [57]. Reference values of the unknown parameters K s , K evap and K 5 are taken from [57]. These last three parameters will be used to assess the validity of the parameter estimation accuracy, but are not considered in the observer design and computation. The rest of the parameters are constants that have been taken from the literature. All these parameter values are summarized in Table 6.7.

6.3.8.4

Results and Discussion

The observer design parameters have been tuned to provide a state-estimation settling time of 200 s and sufficient noise rejection, see Table 6.8. Moreover, again, it is assumed that there is no access to the derivative of the current. Thus, the ˆ I˙) of the low-power peaking-free observer is implemented as factor ψ2 (ˆx, u, θ, ˆ ψ2 (ˆx, u, θ, 0). The constant current, the air velocity step profile and the induced stack temperature have been introduced in the proposed adaptive observer and, consequently, an estimation of the liquid water saturation has been generated, Fig. 6.21a. As stated before, the true value of the liquid water cannot be measured online, so, this estimation cannot be compared with any signal. Nevertheless, some conclusions can be drawn from this result. In Fig. 6.21a, it can be noticed that the estimation converges to a value around 0.4, which is coherent with the optimal value sopt depicted in Table 6.7. Moreover, after the air velocity change in second 325, it can be seen that the liquid water saturation has slowly decreased. This tendency was predicted during the experiment design, as an increase of the temperature results in an increase of the water evaporation rates. Notice that the liquid water saturation estimation in Fig. 6.21a stabilizes in the first 200 s, and, after the air velocity change, it stabilizes again in 200 s. Moreover, the high-frequency oscillations induced by the temperature sensor noise presents

186

6 PEM Fuel Cell Monitoring

Table 6.7 Parameters of the fuel cell model Parameter Value K1 K2 K3 K4 E ocv Tamb Acell A por e p0 Ea Pg n cell αc ac j0,ca,r e f ΔG ∗ Rohm sopt kb R F Ks K evap K5

K J−1 m−1 Kg C−1 m−2 Kg K J−1 V K m2 m2 Pa eV Pa − − − A m−2 J mol−1 Ω − eV K−1 J K−1 mol−1 C mol−1 Kg m−2 m3 s−1 s−1

0.0255 4.1457 · 10−5 0.0022 1.23 298 0.00225 2.2E7 1.196E11 0.449 2380 20 0.311 238 4.7E − 3 70000 1.566 0.55 8.6170 · 10−5 8.314 96485 1.746 8.6E5 0.2972

Table 6.8 PEMFC observer parameters Parameter α1 α2 β1 r2 ε σ qi pi

Units

3.276 · 10−4

Value 1.5 0.01 0.5 0.15 0.35 0.219 3 50

6.3 Cathode Liquid Water Saturation Monitoring Through Nonlinear Observers

a

b

c vair

vair Change

d

e

187

Change

f

Fig. 6.21 a Adaptive observer liquid water saturation estimation. b Measured stack temperature profile (orange) and adaptive observer estimation (blue). c Measured stack voltage profile (orange) and adaptive observer estimation (blue). d–f Adaptive observer parameter estimation θ (blue) and parameter value estimated in [56, 57] (orange). The values reported by the adaptive observer (blue) have achieved a more accurate voltage and temperature prediction than the ones estimated in [56, 57]. Thus, are closer to the ideal parameter value in the studied experiment

a peak-to-peak amplitude of less than 5%. It should be remarked that if a classic high-gain observer was applied to estimate the auxiliary signal, this error would be significantly larger, and the estimation would be practically unusable. In order to validate the accuracy of the estimation, the estimated stack temperature profile and liquid water saturation profile have been used to generate an estimation of the output voltage, Vˆ f c . As stated before, if Vˆ f c and Tˆ f c are accurate, then, it can be induced that sˆ is also accurate. Specifically, the measured temperature profile and the observer estimation is depicted in Fig. 6.21b. It can be seen that, after the transient, the relative error is below the 0.1%. The measured voltage evolution and observer estimation is depicted in Fig. 6.21c, where it can be seen that the voltage estimation converges to a relative error below the 0.4%. Following the reasoning presented in the methodology sub-section, as the temperature and voltage estimation converges to a relative error below the 0.4%, it can be concluded that the liquid water saturation estimation is also accurate. In parallel to the liquid water saturation estimation, the adaptive observer has generated an estimation of the water dynamic unknown parameters, θ. The accuracy of the observer estimation has been assessed by direct comparison to the values reported in [56, 57] for an H-100 fuel cell in similar operating conditions. It should be remarked that the values reported in [56, 57] were not directly measured, but estimated through an off-line identification algorithm. Therefore, its validity should be treated carefully. Indeed, it will be shown that the values estimated through the observer proposed in Sect. 6.3.6 presented higher prediction capabilities. The evolution of the parameter estimation is depicted in Fig. 6.21d–f. It is noticeable that the estimation converges to a value of the same magnitude as the ones reported in [56, 57], within the first 1000 s, which is in the timescale of the liquid water saturation dynamics [11]. It is noticeable the bias between the estimated value and the ones reported in previous works. This discrepancy is mainly explained by the

188

6 PEM Fuel Cell Monitoring

algorithm that is used to estimate the parameters, K 5 , K s and K evap . In [56, 57], these parameters were estimated through an off-line identification algorithm. Alternatively, this work has estimated these parameters through the on-line adaptation dynamics in (6.74). It should be remarked that the values reported in [56, 57], achieved a voltage estimation-error of 0.5% and a temperature estimation-error of 2%, which is significantly larger than the one reported in this work. Therefore, the parameters estimated through the proposed algorithm significantly improves the prediction capabilities of the model. Consequently, should be closer to the parameter value that minimizes the effect of unmodelled uncertainty.

References 1. Lai J, Ellis MW (2017) Fuel cell power systems and applications. Proc IEEE 105(11):2166– 2190. https://doi.org/10.1109/JPROC.2017.2723561. Nov 2. Larminie J, Dicks A, McDonald MS (2003) Fuel cell systems explained, vol 2. J Wiley Chichester, UK. https://doi.org/10.1002/9781118878330 3. Jouin M, Gouriveau R, Hissel D, Péra M-C, Zerhouni N (2016) Degradations analysis and aging modeling for health assessment and prognostics of PEMFC. Reliab Eng Syst Saf 148:78–95. https://doi.org/10.1016/j.ress.2015.12.003. Apr. 4. Zhao D, Gao F, Massonnat P, Dou M, Miraoui A (2015) Parameter sensitivity analysis and local temperature distribution effect for a PEMFC system. IEEE Trans Energy Convers 30(3):1008– 1018. https://doi.org/10.1109/TEC.2015.2404793. Mar. 5. Schoukens J, Ljung L (2019) Nonlinear system identification: a user-oriented road map. IEEE Control Syst Mag 39(6):28–99. https://doi.org/10.1109/MCS.2019.2938121 6. Kitsos C, Besançon G, Prieur C (2022) Contributions to the problem of high-gain observer design for hyperbolic systems. Trends in nonlinear and adaptive control, pp 109–134 7. Kang W, Krener AJ, Xiao M, Xu L (2013) A survey of observers for nonlinear dynamical systems. In: Data assimilation for atmospheric, oceanic and hydrologic applications, vol II, pp 1–25. Springer 8. Bernard P (2019) Observer design for nonlinear systems. ISSN 978-3-030-11146-5 9. Miller S (2013) Analysis of an open-cathode fuel cell stack in an enclosure for varying operating conditions. PhD thesis 10. O’Hayre R, Fabian T, Litster S, Prinz FB, Santiago JG (2007) Engineering model of a passive planar air breathing fuel cell cathode. J Power Sour 167(1):118–129. ISSN 0378-7753. https:// doi.org/10.1016/j.jpowsour.2007.01.073 11. Jiao K, Li X (2011) Water transport in polymer electrolyte membrane fuel cells. Prog Energy Combust Sci 37(3):221–291. https://doi.org/10.1016/j.pecs.2010.06.002. Jun. 12. Kim S, Mench MM (2009) Investigation of temperature-driven water transport in polymer electrolyte fuel cell: thermo-osmosis in membranes. J Membr Sci 328(1):113 – 120. ISSN 0376-7388. https://doi.org/10.1016/j.memsci.2008.11.043 13. Husar A, Higier A, Liu H (2008) In situ measurements of water transfer due to different mechanisms in a proton exchange membrane fuel cell. J Power Sour 183:240–246. https://doi. org/10.1016/j.jpowsour.2008.04.042 14. Motupally S, Becker A, Weidner J (2000) Diffusion of water in nafion 115 membranes. J Electrochem Soc-J Electrochem Soc 147:09. https://doi.org/10.1149/1.1393879 15. Zawodzinski TA, Davey J, Valerio J, Gottesfeld S (1995) The water content dependence of electro-osmotic drag in proton-conducting polymer electrolytes. Electrochimica Acta 40(3):297–302. ISSN 0013-4686. https://doi.org/10.1016/0013-4686(94)00277-8. Polymer electrolyte fuel cells

References

189

16. Springer TE, Zawodzinski TA, Gottesfeld S (1991) Polymer electrolyte fuel cell model. J Electrochem Soc 138(8):2334 17. Ge S, Li X, Yi B, Hsing I-M (2005) Absorption, desorption, and transport of water in polymer electrolyte membranes for fuel cells. J Electrochem Soc 152:A1149. https://doi.org/10.1149/ 1.1899263 18. Zhao Q, Majsztrik P, Benziger J (2011) Diffusion and interfacial transport of water in Nafion. J Phys Chem B 115(12):2717–2727 19. Cheah MJ, Kevrekidis IG, Benziger J (2011) Effect of interfacial water transport resistance on coupled proton and water transport across Nafion. J Phys Chem B 115(34):10239–10250 20. Turhan A, Kim S, Hatzell M, Mench MM (2010) Impact of channel wall hydrophobicity on through-plane water distribution and flooding behavior in a polymer electrolyte fuel cell. Electrochimica Acta-Electrochim Acta 55:2734–2745. https://doi.org/10.1016/j.electacta.2009.11. 095 21. Wang C-Y (2004) Fundamental models for fuel cell engineering. Chem Rev 104(10):4727– 4766 22. Secanell M, Jarauta A, Kosakian A, Sabharwal M, Zhou J (2017) PEM fuel cells, modeling. Springer, New York, NY, pp 1–61. ISBN 978-1-4939-2493-6. https://doi.org/10.1007/978-14939-2493-61019-1 23. Jiao K, Li X (2009) Three-dimensional multiphase modeling of cold start processes in polymer electrolyte membrane fuel cells. Electrochimica Acta 54(27):6876–6891. ISSN 0013-4686. https://doi.org/10.1016/j.electacta.2009.06.072 24. Dullien FAL (2012) Porous media: fluid transport and pore structure. Academic Press 25. Udell KS (1985) Heat transfer in porous media considering phase change and capillarity—the heat pipe effect. Int J Heat Mass Trans 28:485–495 February 26. Yalcinoz T, Alam MS (2008) Dynamic modeling and simulation of air-breathing proton exchange membrane fuel cell. J Power Sour 182(1):168–174 27. Barbir F (2012) PEM fuel cells: theory and practice. Academic Press. http://repository.fue.edu. eg/xmlui/bitstream/handle/123456789/3989/8074.pdf?sequence=1 28. Wang Q, Eikerling M, Song D, Liu Z (2004) Structure and performance of different types of agglomerates in cathode catalyst layers of PEM fuel cells. J Electroanal Chem 573(1):61–69 29. Strahl S, Husar A, Riera J (2014) Experimental study of hydrogen purge effects on performance and efficiency of an open-cathode proton exchange membrane fuel cell system. J Power Sour 248:474–482 30. Bernardi DM, Verbrugge MW (1992) A mathematical model of the solid-polymer-electrolyte fuel cell. J Electrochem Soc 139(9):2477–2491. https://doi.org/10.1149/1.2221251 31. Lion Specialty Chemicals Co Ltd (2019) ketjenblack highly electro-conductive carbon black (product information). https://www.lion-specialty-chem.co.jp/en/product/carbon/carbon01. html. Accessed 10 Nov 2019 32. Goshtasbi A, Pence BL, Ersal T (2016) Computationally efficient pseudo-2d non-isothermal modeling of polymer electrolyte membrane fuel cells with two-phase phenomena. J Electrochem Soc 163(13):F1412 33. Kosakian A, Urbina LP, Heaman A, Secanell M (2020) Understanding single-phase watermanagement signatures in fuel-cell impedance spectra: a numerical study. Electrochimica Acta 350:136204. ISSN 0013-4686. https://doi.org/10.1016/j.electacta.2020.136204 34. Zhou J, Shukla S, Putz A, Secanell M (2018) Analysis of the role of the microporous layer in improving polymer electrolyte fuel cell performance. Electrochimica Acta 268:366–382. ISSN 0013-4686. https://doi.org/10.1016/j.electacta.2018.02.100 35. Kusoglu A, Kwong A, Clark KT, Gunterman HP, Weber AZ (2012) Water uptake of fuel-cell catalyst layers. J Electrochem Soc 159(9):F530–F535. https://doi.org/10.1149/2.031209jes 36. Yu HM, Ziegler C, Oszcipok M, Zobel M, Hebling C (2006) Hydrophilicity and hydrophobicity study of catalyst layers in proton exchange membrane fuel cells. Electrochimica Acta 51(7):1199–1207 37. Gostick JT, Fowler MW, Ioannidis MA, Pritzker MD, Volfkovich YM, Sakars A (2006) Capillary pressure and hydrophilic porosity in gas diffusion layers for polymer electrolyte fuel cells.

190

38. 39.

40.

41.

42. 43. 44.

45. 46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

6 PEM Fuel Cell Monitoring J Power Sour 156(2):375–387. ISSN 0378-7753. https://doi.org/10.1016/j.jpowsour.2005.05. 086 Zhou J, Putz A, Secanell M (2017) A mixed wettability pore size distribution based mathematical model for analyzing two-phase flow in porous electrodes. J Electrochem Soc 164(6):F530 Shukla S, Wei F, Mandal M, Zhou J, Saha MS, Stumper J, Secanell M (2019) Determination of PEFC gas diffusion layer and catalyst layer porosity utilizing archimedes principle. J Electrochem Soc 166(15):F1142–F1147. https://doi.org/10.1149/2.0251915jes Pant LM, Mitra SK, Secanell M (2012) Absolute permeability and knudsen diffusivity measurements in PEMFC gas diffusion layers and micro porous layers. J Power Sour 206:153—160. ISSN 0378-7753. https://doi.org/10.1016/j.jpowsour.2012.01.099 Weber AZ (2010) Improved modeling and understanding of diffusion-media wettability on polymer-electrolyte-fuel-cell performance. J Power Sour 195(16):5292–5304. ISSN 03787753. https://doi.org/10.1016/j.jpowsour.2010.03.011 Cussler EL, Cussler EL (2009) Diffusion: mass transfer in fluid systems. Cambridge university Press Bird RB (2002) Transport phenomena. Appl Mech Rev 55(1):R1–R4 Meng H (2007) A two-phase non-isothermal mixed-domain PEM fuel cell model and its application to two-dimensional simulations. J Power Sour 168(1):218–228. ISSN 0378-7753. https:// doi.org/10.1016/j.jpowsour.2007.03.012. 10th European Lead Battery Conference Pukrushpan JT, Stefanopoulou AG, Peng H (2004) Control of fuel cell breathing. IEEE Control Syst Mag 24(2):30–46. https://doi.org/10.1109/MCS.2004.1275430 Rakhtala SM, Noei AR, Ghaderi R, Usai E (2014) Design of finite-time high-order sliding mode state observer: a practical insight to PEM fuel cell system. J Process Control 24(1):203–224. ISSN 0959-1524. https://doi.org/10.1016/j.jprocont.2013.08.006 Xu L, Hu J, Cheng S, Fang C, Li J, Ouyang M, Lehnert W (2017) Nonlinear observation of internal states of fuel cell cathode utilizing a high-order sliding-mode algorithm. J Power Sour 356:56–71. ISSN 0378-7753. https://doi.org/10.1016/j.jpowsour.2017.04.068 Sankar K, Jana AK (2018) Nonlinear multivariable sliding mode control of a reversible PEM fuel cell integrated system. Energy Convers Manag 171:541–565. ISSN 0196-8904. https:// doi.org/10.1016/j.enconman.2018.05.079 Luna J, Usai E, Husar A, Serra M (2016) Nonlinear observation in fuel cell systems: a comparison between disturbance estimation and high-order sliding-mode techniques. Int J Hydrog Energy 41(43): 19737–19748. ISSN 0360–3199. https://doi.org/10.1016/j.ijhydene.2016.06. 041. The 5th Iberian Symposium on Hydrogen, Fuel Cells and Advanced Batteries (HYCELTEC (2015) 5–8 Jul 2015. Tenerife, Spain Piffard M, Gerard M, Bideaux E, Da Fonseca R, Massioni P (2015) Control by state observer of pemfc anodic purges in dead-end operating mode. IFAC-PapersOnLine 48(15):237–243. ISSN 2405-8963. https://doi.org/10.1016/j.ifacol.2015.10.034. 4th IFAC Workshop on Engine and Powertrain Control, Simulation and Modeling E-COSM 2015 Görgün H, Arcak M, Barbir F (2006) An algorithm for estimation of membrane water content in pem fuel cells. J Power Sour 157(1):389–394. ISSN 0378-7753. https://doi.org/10.1016/j. jpowsour.2005.07.053 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. https://doi.org/10. 1109/TCST.2003.821958 Stumper J, Campbell SA, Wilkinson DP, Johnson MC, Davis M (1998) In-situ methods for the determination of current distributions in PEM fuel cells. Electrochim Acta 43(24):3773–3783. https://doi.org/10.1016/S0013-4686(98)00137-6 Geiger AB, Tsukada A, Lehmann E, Vontobel P, Wokaun A, Scherer G (2002) In situ investigation of two? Phase flow patterns in flow fields of pefc?s using neutron radiography. Fuel Cells 2:92–98. https://doi.org/10.1002/fuce.200290007 Manke I, Hartnig C, Grunerbel M, Lehnert W, Kardjilov N, Haibel A, Hilger A, Banhart J, Riesemeier H (2007) Investigation of water evolution and transport in fuel cells with high resolution synchrotron x-ray radiograph. Appl Phys Lett 90. https://doi.org/10.1063/1.2731440

References

191

56. Strahl S, Costa-Castelló R (2016) Model-based analysis for the thermal management of opencathode proton exchange membrane fuel cell systems concerning efficiency and stability. J Process Control 47:201–212. https://doi.org/10.1016/j.jprocont.2016.09.004. Nov. 57. Strahl S, Husar A, Puleston P, Riera J (2014) Performance improvement by temperature control of an open-cathode PEM fuel cell system. In Fuel Cells. https://doi.org/10.1002/fuce. 201300211. Mar 58. Zhang Y, Mawardi A, Pitchumani R (2007) Numerical studies on an air-breathing proton exchange membrane (PEM) fuel cell stack. J Power Sour 173(1):264 – 276. ISSN 0378-7753. https://doi.org/10.1016/j.jpowsour.2007.05.008 59. Levant A (2003) Higher-order sliding modes, differentiation and output-feedback control. Int J Control 76(9–10):924–941. https://doi.org/10.1080/0020717031000099029. Nov. 60. Sun S-L, Deng Z-L (2004) Multi-sensor optimal information fusion kalman filter. Automatica 40(6):1017–1023. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2004.01.014 61. Pavlov A, Pogromsky A, van de Wouw N, Nijmeijer H (2004) Convergent dynamics, a tribute to Boris Pavlovich demidovich. Syst Control Lett 52(3):257–261. ISSN 0167-6911. https:// doi.org/10.1016/j.sysconle.2004.02.003 62. McCain BA, Stefanopoulou AG, Butts KR (2006) A study toward minimum spatial discretization of a fuel cell dynamics model. In: ASME international mechanical engineering congress and exposition, vol 47683, pp 1099–1107 63. Chen J, Siegel JB, Stefanopoulou AG, Waldecker JR (2013) Optimization of purge cycle for dead-ended anode fuel cell operation. Int J Hydrog Energy 38(12):5092–5105. ISSN 03603199. https://doi.org/10.1016/j.ijhydene.2013.02.022 64. Dumercy L, Péra M-C, Glises R, Hissel D, Hamandi S, Badin F, Kauffmann J-M (2004) PEFC stack operating in anodic dead end mode. Fuel Cells 4(4):352–357. https://doi.org/10.1002/ fuce.200400053 65. Himanen O, Hottinen T, Tuurala S (2007) Operation of a planar free-breathing PEMFC in a dead-end mode. Electrochem Commun 9(5):891–894. ISSN 1388-2481. https://doi.org/10. 1016/j.elecom.2006.12.002 66. Mokmeli A, Asghari S (2010) An investigation into the effect of anode purging on the fuel cell performance. Int J Hydrog Energy 35(17):9276–9282 67. Benallouch M, Outbib R, Boutayeb M, Laroche E (2018) Robust observers for a class of nonlinear systems using PEM fuel cells as a simulated case study. Trans Control Syst Technol 26(1):291–298. https://doi.org/10.1109/TCST.2017.2658181 68. Pilloni A, Pisano A, Usai E (2015) Observer-based air excess ratio control of a PEM fuel cell system via high-order sliding mode. IEEE Trans Ind Electron 62(8):5236–5246. https://doi. org/10.1109/TIE.2015.2412520 69. Luna J, Husar A, Serra M (2015) Nonlinear distributed parameter observer design for fuel cell systems. Int J Hydrog Energy 40(34):11322–11332. ISSN 0360-3199. https://doi.org/10.1016/ j.ijhydene.2015.05.132 70. Yuan H, Dai H, Wei X, Ming P (2020) Model-based observers for internal states estimation and control of proton exchange membrane fuel cell system: a review. J Power Sour 468:228376. ISSN 0378-7753. https://doi.org/10.1016/j.jpowsour.2020.228376

Chapter 7

Cybersecurity in DC-Microgrids

This chapter implements the nonlinear observer theory developed in this book to detect and mitigate false-date injection attacks in DC-microgrids. The problem is significantly complicated due to the presence of nonlinear unknown constant power loads. To solve the problem, a distributed nonlinear adaptive observer is proposed. Moreover, the sufficient conditions for the stability and viability of the proposed scheme are given.

7.1 Introduction In order to address current critical energy and environmental objectives, the electrical grid is shifting to an alternative architecture with higher penetration of distributed generation units (DGUs), renewable energy sources and energy storage systems. Nonetheless, the inclusion of these elements throughout the electrical grid creates additional problems that should be addressed. That is, degraded voltage profile, congestions in transmission lines, and reduction of frequency reserves [1]. These issues motivate the implementation of novel control strategies. In order to get a more traceable control, the electrical grid is divided in smaller flexible entities composed by variable renewable sources, energy storage systems and controllable loads. These smaller entities are usually referred to as microgrids [2]. Said microgrids can be implemented in an autonomous architecture or can be connected to the grid. Depending on the point of common coupling type, microgrids can be classified as DC or AC [3]. Although remarkable progress have been achieved in relation to integration and operation of AC-microgrids, DC-microgrids are usually preferred due to its higher efficiency, more natural interface with renewable sources and energy storage systems, better compliance with consumer electronics and simplicity to control due to a lack of reactive power flow [4].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Cecilia, Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems, Springer Theses, https://doi.org/10.1007/978-3-031-38924-5_7

193

194

7 Cybersecurity in DC-Microgrids

In this framework, distributed generation units are connected to the microgrid common bus through controllable power electronic interface converters. Consequently, regulation of the common DC bus voltage becomes the main control priority. To do so, a popular method is to follow the droop control philosophy. The idea is to deploy a cooperative control strategy to ensure equal current sharing among the agents (distributed generation units). Nonetheless, droop control operates with an error across the voltage reference at the same time. To compensate for this error, secondary controllers are employed to provide compensation terms to limit the offset [5]. In order to achieve consensus between the distributed generation units, it is necessary to implement a communication layer between the different agents. In this context, two different control frameworks can be found. On the one hand, there is the centralized control framework, where data from distributed generation units is sent to a common aggregator, processed and the feedback signals are retrieved to the agents. On the other hand, there is the decentralized control framework, where control strategies are computed locally and the information is transmitted through digital communication links between neighbouring agents. Due to its higher scalability properties, high bandwidth and resilience against the single point of failure [6], most modern microgrids control algorithms focus on the distributed cotrol framework. Consequently, the DC microgrid control relies on the integration of a communication cyber-layer to improve its performance [7]. The interaction between the physical-layer and the cyber-layer creates a risk of malicious cyber-attacks, which may endanger the performance of the DGUs and/or the whole DC microgrid. For this reason, the interest in developing attack detection and mitigation techniques, from a control viewpoint, has increased in the recent years [8]. The aim of this chapter is to use the nonlinear observer theory developed in this thesis to develop a cyber-attack detection and mitigation strategy.

7.2 System Model, Problem Formulation and Proposal 7.2.1 DC-Microgrid Model and Control This work considers a DC microgrid composed by a set of DGUs. The different DGUs are connected through a set of resistive power lines. Each DGU is comprised of a DC voltage source and a DC-DC converter, which is connected to the point of common coupling. Different from similar results on the field [9], this work considers that the load side converter is required to deliver a constant power load (CPL). Indeed, in some cases, local energy demands can be modeled by means of a constant power loads in the input terminal. This is the case, for example, when the DGU is connected to a DC/AC converter that drives an electric motor with tightly regulated speed and one-to-one speed-torque characteristics [10]. As it will be shown, the inclusion of

7.2 System Model, Problem Formulation and Proposal Table 7.1 Symbols used in Eq. (7.1)

States

195 Description

Iti Vi Ik Parameters L ti Ci Ri Rk Lk Inputs ui Pi

DGU output current Voltage Power line current Filter inductance Shunt capacitor Local load impedance Power line resistance Power line inductance Converter voltage CPL

this element significantly complicates the problem, as it converts a simple linear estimation problem to a complex nonlinear observation one. This work considers an averaged model of the DC-DC converter, which results in the following equations for the dynamics of the i th DGU: L ti I˙ti = −Vi + u i  1 1 Ci V˙i = Iti − Ik,i − Vi − Pi Ri Vi

(7.1)

k∈E i

L k I˙k = (Vi − V j ) − Rk Ik ∀k ∈ Ei , where u i depicts the output voltage of the converter and Ei is the set of incident power lines. A description of all the symbols used in (7.1) is depicted in Table 7.1. Among the different system signals, only the generated current, Iti , and the load voltage, Vi , can be measured in each DGU. It is assumed that the line current , Ik , is not measured since this is not a signals used for DGU control. Therefore, the output in the i th DGU is defined as yi = [y1,i , y2,i ] = [Iti , Vi ] . In the cyber-layer, the DC microgrid is modeled through an undirected and connected communication graph G = {V , E }, where V depicts the set of DGU and E depicts the power lines between the DGUs [11]. The graph is described through an incident matrix B ∈ Rn×m , where n is the number of DGUs and m the number of resistive power lines, defined as: ⎧ ⎪ ⎨+1, if i is the positive end of the line j Bi j = −1, if j is the negative end of the line j ⎪ ⎩ 0, otherwise.

196

7 Cybersecurity in DC-Microgrids

where Bi j are the components of the matrix B. As it will be shown, for the adequate operation of the DC microgrid, each DGU is only required to send and receive information from its neighbour DGUs. Consequently, the communication graph in the cyber-layer coincides with the physical power network of the microgrid. As explained in the introduction, DC-microgrids are commonly operated following the droop control philosophy. This chapter considers the most common case, that is, each DGU has two PI controllers connected in cascade that ensure the tracking of the voltage reference and current control. Specifically, the input voltage, u i , is generated through the following PI controller [9]     y1,i − Ir e f,i . u i = K p I y1,i − Ir e f,i + K i I

(7.2)

where K p I is the gain of the proportional part, K i I is the gain of the integral part and Ir e f,i is the current reference generated by the previous PI in the cascaded controller. For observer design purposes, it is convenient to mention that the PI controller ensures that the DGU load voltage and the DGU output current present bounded trajectories. Moreover, the PI controller includes saturation limits that prevents under/over-voltage. In the distributed control topology, each DGU local control is complemented by the information from cyber-layer neighbouring DGUs to establish a distributed coordination. Between DGUs, the information vector ψ i = [ψ1,i , ψ2,i ] = [vˆdc,i , Iti ] is transmitted, where vˆdc,i depicts the average voltage estimate in the i th DGU [12]. The information vector, ψ i , is used to create a voltage off-set to be compensated by the secondary controllers [6]. More precisely, the local voltage reference, Vdc,r e f , to be tracked by the i th DGU is disturbed as follows: Vdc,r e f,i = Vdc,r e f + ΔV1i + ΔV2i ,

(7.3)

where ΔV1i and ΔV2i are the two voltage off-set computed as: ΔV1i = H1 (s)(Vdc,r e f −



(vˆdc,k − vˆdc,i ))

k∈E i

ΔV2i = H2 (s)(Idc,r e f −



(Itk − Iti ))

(7.4)

k∈E i

where Idc,r e f is a global current reference quantities for the whole microgrid, and H1 (s) and H2 (s) are proportional integral controllers.

7.2 System Model, Problem Formulation and Proposal

197

7.2.2 Cyber-Attack Model and Problem Formulation From the last section, it is noticeable that the DC-microgrid control requires the transmission of the information vector, ψ i , through the digital link between neighbour agents. This communication opens the possibility of simple local malicious cyber-attacks, which can tamper the performance and/or stability of the whole microgrid. Amongst many different types of cyber-attacks, e.g. false data injection attacks (FDIAs) [13], denial of service [14] and replay attacks [15]; this chapter focuses on the FDIA as, currently, it is the most prominent cyber-attack [13]. This attack is conducted by injecting malicious data in hijacked measurements in order to modify the behaviour of the microgrid. Specifically, this chapter focuses in malicious data that can be injected in the DGU output current measurements. Explicitly, an attack on the i th DGU is depicted as: Sensor attack : Cyber-link attack :

yi = [Iti + xia , Vi ] ψ i = [vˆdc,i , Iti + xia ]

where xia represents the FDIA value. It is assumed that the FDIA can be classified as a deception attack [9], which means that it satisfies the instantaneous system objectives but may affect the performance of the microgrid later. It is also assumed that the voltage sensor is free of attacks, as previous works have proved that a stealth attack is not possible by manipulating the voltage due to the presence of a distributed observer [16]. The objective of this chapter is to develop a security strategy for the defined FDIA. Reliable security measures against FDIAs rely on detecting if an attack is present or not in the system and identifying the compromised attack. In this context, a successful strategy is based on implementing an observer that is independent of the attack signal and a detector that computes the presence of the attack by comparing the estimation with the measured signals. Some examples of this approach may be the use of a weighted least squares as the observer of and a sparse optimization as a detector [17], the use of Kullback-Leibler distance as the detector, a Kalman filter as the observer and a Euclidean distance as the detector [18], a Kalman filter with a cosine similarity approach as detector [19, 20], neural-network based detector [21] or a short-term state forecasting as the observer [22]. One of the main drawbacks of the mentioned approaches is that the observer and the detector were implemented in a centralized framework, which hinders the scalability of the solution to large systems. Consequently, recent detection algorithms are changing to the distributed framework [23]. Some notable examples are the use of constant gain distributed linear observers [24], distributed extended Kalman filters [25, 26], a bank of unknown input observers [27]. For a more in-depth review of the potential impacts, vulnerabilities, and detection strategies of FDIAs in power systems, the reader is referred to the surveys in [28, 29] and references within. Once a FDIA has been detected, the immediate objective is to mitigate the influence of the attack on the microgrid without a significant effect on the system perfor-

198

7 Cybersecurity in DC-Microgrids

mance. In relation to DC microgrids, an event-driven approach has been proposed to mitigate FDIA [30], man-in-the-middle attacks for a system of homogeneous agents [31] and for a system of heterogeneous agents [32]. A major limitation of available observer-based detection and mitigation methods is the assumption that the microgrid dynamics are linear. In many cases, the load side converter is required to deliver a constant power to the load. In such situations, the voltage dynamics behave non-linearly [33]. This fact can be seen in the model 1 introduces a strong nonlinearity in the system. Indeed, (7.1), where the term Pi Vi small deviations of the load voltage may produce large variations of the equilibrium point, thereby limiting the validity linear approximations. Moreover, it is reasonable to assume that the constant power load (CPL) is not known and may vary during the system operation, which makes the dynamics linearization process infeasible. Additionally, it should be mentioned that CPLs have a destabilizing effect on the DC microgrid. In such context, the interaction between FDIAs and CPLs may drive the power system to unstable equilibrium points, which may lead to significant oscillations or to network collapse. This instability behaviour cannot be replicated in linear DC microgrids with similar false data signal. To get a better theoretical understanding of this instability, consider the following. As the DGUs of the concerned microgrid are controlled through linear PI controllers, and the system is nonlinear, the stability can only be ensured in a region of attraction around the equilibrium point where the PI has been tuned [34]. Specifically, for a DGU modelled as in (7.1) and a cascaded PI as the DGU primary control, there is a region Di ⊂ R2 such that if Iti , Vi ∈ Di , then, the DGU voltage and output current converge to the desired references. Otherwise, the system becomes unstable [34]. The region of attraction of cooperative DC microgrids with linear controllers can be computed through a series of sum of square optimizations [34]. Suppose that at time t0 the i th DGU states are inside the region of attraction and the system is subjected to a cyber-attack. During the attack, the system response will certainly involve large variations of the state variables which may lead to an escape of the region of attraction and, consequently, may lead to an unstable system. This fact confirms that the interaction between FDIAs and CPLs can destabilize the plant. Now it should be clear the problematic that introduces the presence of local unknown CPLs. On the one hand, the system dynamics are strongly non-linear and uncertain. Consequently, more common detection and mitigation strategies that are based on linear approximations are inadequate for the considered problem, as the discrepancy between the system and the model may be detected as a cyber-attack. Second, the interaction between a simple FDIA and the CPL may destabilize the whole microgrid. Therefore, the presence of CPLs simplifies the attack design. This chapter will show that the problem of detecting and mitigating the considered FDIA can be jointly solved by an adequate nonlinear observer design. The main objective is to design an algorithm that can reconstruct the attack signal xia , i.e., the algorithm has to generate an estimation xˆia such that limt→∞ |xia − xˆia | = 0. It is worth noting that this objective is more restrictive than achieving an isolation of the attack, which only requires finding the compromised sensor or cyber-link, but does

7.2 System Model, Problem Formulation and Proposal

199

not necessarily acquire any information of the attack signal. The main advantage of reconstructing the attack signal is that the effect of the attack over the considered system can be mitigated. Specifically, assume that a sensor attack is reconstructed, i.e. xia = xˆia . Then, the attacked sensor (or cyber-link) can be cleaned using: yicleaned = [y1,i − xˆia , y2,i ] = [Iti , Vi ] ,

(7.5)

which completely eliminates the effect of the attack in the DC microgrid. Following this line, this chapter proposes the design of an observer that, irrespective of the presence of an attack, can estimate the actual value of the generated current, Iti , such that limt→∞ | Iˆti − Iti | =, where Iˆti depicts the load current estimation. If such an estimation is achieved, the attack can be reconstructed by comparing the estimation with the measured value of the current. Specifically, a sensor attack signal can be reconstructed by computing the following: xˆia = y1,i − Iˆti = Iti + xia − Iˆti

(7.6)

and a cyber-link attack can be reconstructed as: xˆia = ψ2,i − Iˆti = Iti + xia − Iˆti .

(7.7)

By direct inspection of (7.6) and (7.7), it can be seen that limt→∞ | Iˆti − Iti | = implies limt→∞ |xia − xˆia | = 0. Therefore, the problem of reconstructing the attack signal has been transformed to an observation problem. Remark 7.1 Naturally, the reconstruction of the attack signal (7.6) (7.7) can also be employed to detect the presence of an attack, which may be later used to activate secondary security protocols. Specifically, define the following residual for the detection of a sensor attack in the i th DGU: rs,i = y1,i − Iˆti ;

(7.8)

and a residual for a cyber link attack: rcl,i = ψ2,i − Iˆti .

(7.9)

The presence of an attack can be detected by evaluating the following inequalities: Sensor attack : C yber link attack :

rs,i > r¯i rcl,i > r¯i

(7.10) (7.11)

where r¯i is a positive constant parameter designed appropriately to avoid false alarms induced by the voltage sensor noise. The design of r¯i is related to the accuracy of the estimation scheme under measurement noise.

200

7 Cybersecurity in DC-Microgrids

As discussed previously, linear approximations of the proposed microgrid model are inadequate in the presence of CPLs. For this reason, it is required to design a non-linear observer through the direct study of the non-linear model in (7.1). A key property for the estimation algorithm design is the following: Lemma 7.1 The i th DGU output current, Iti , can be computed through the input u i , the voltage Vi , its derivative, V˙i , the CPL, Pi , and the line currents, Ik,i , by computing the following expression Iˆti = Ci V˙i +



Ik,i +

k∈E i

1 1 Vi + Pi . Ri Vi

(7.12)

Proof Expression (7.12) is obtained by isolating Iti from the second equation in (7.1).  Therefore, assuming that one generates an estimation, V˙ˆi , Iˆk,i and Pˆi such that limt→∞ |V˙ˆi − V˙i | = 0, limt→∞ | Iˆk,i − Ik,i | = 0 and limt→∞ | Pˆi − Pi | = 0. Then, limt→∞ | Iˆti − Iti | = 0, where Iˆti is computed through (7.12) using the estimations V˙ˆi , Iˆk,i and Pˆi . The next section aim is to present an observer for the estimation of V˙ˆ , Iˆ and Pˆ . i

k,i

i

The objective seems simple, but it involves significant difficulties, as the considered system includes sensor noise, uncertainty and locally unobservable states.

7.3 Proposed Nonlinear Observer The objective here is to design an observer algorithm that can accurately estimate, V˙ , Ik,i f or k ∈ Ei and Pi , of the i th DGU, even in the presence of false data, unmodelled disturbances, model uncertainty and sensor noise. In order to ease the scalability of the algorithm (and avoid the inclusion of an aggregator in the system), it is of prime interest to design a distributed observer algorithm. This means that the observer of the i th DGU has to generate an estimation of Iti based only on the signals measured in the i th DGU, yi , and the signals transmitted through the incident cyber-links, ψ i . In order to make the detection and mitigation algorithm invariant to the presence of FDIA, the observer will be designed just taking the voltage as the measured signal. The first two steps in the nonlinear observer design process consist in, first, establishing that the the system is observable and, second, transforming the system to an adequate form, see Chap. 3. This chapter proceeds with these two steps as in the standard high-gain observer methodology. That is, first, a coordinate transformation that puts the system in a triangular form is designed. Second, if the coordinate map is a diffeomorphism, it can be concluded that the system is instantaneously observable, see Chap. 2. Nonetheless, the coordinate transformation needs to be independent of the input, u i . As the DC-DC converter is controlled using the measured DGU output

7.3 Proposed Nonlinear Observer

201

current in (7.2), the signal u i is sensitive to the sensor FDIA. Therefore, an inputdependant coordinate transformation will be sensitive to the attack signal, which will introduce a bias in the state-estimation. Taking into account these details, the following result can be established. Lemma 7.2 Define m i as the number of incident edges in the i th vertex. Then, the following input-independent map ⎤ ⎤ ⎡ Vi ξ1,i ⎢ V˙i ⎥ ⎢ ξ2,i ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ η1,i ⎥ ⎥ = Ti (Vi , V˙i , I1,i , . . . , Im i ,i ) = ⎢ I1,i ⎥ ⎢ ⎢ .. ⎥ ⎢ .. ⎥ ⎣ . ⎦ ⎣ . ⎦ ηm,i Im i ,i ⎡

defines a diffeomorphism that transforms the system (7.1) into the following triangular form ξ˙1,i = ξ2,i ξ˙2,i = φi (ξi , u i , Pi , η i ) + w1 1 Rk (ξ1,i − ξ1, j ) − η j,i + w2, j η˙ j,i = Lk Lk

(7.13) f or j = 1, ..., m i

where ξi = [ξ1,i , ξ2,i ] are the local voltage and its derivative, η i = [I1,i , . . . Im i ,i ] are the incident power line currents, w1 and w2, j f or j = 1, ..., m i represent unknown disturbances or model uncertainty and   1  ξ2,i 1 φi (ξi , u i , Pi , η i )  − ξ1,i + u i + Pi 2 Ci L ti ξ1,i   1  Rk 1 − (ξ1,i − ξ1, j ) − ηk,i − ξ2,i . Lk Lk Ri

(7.14)

k∈E i

The measured signal in the new coordinates is yi = Vi = ξ1,i . Proof The Jacobian of Ti (Vi , V˙i , I1,i , . . . , Im i ,i ) is full rank in all DGU operating conditions, therefore, the function φi defines a diffeomorphism in the considered operating conditions.  By inspection of the structure in (7.13), it is possible to study the elements involved in the estimation problem that prevents the direct implementation of solutions available in the literature. First, for system in (7.13), if the local voltage, Vi = ξ1,i , is the measured signal, most observer techniques can only achieve an estimation of ξ1,i and ξ2,i , as the m i states η i are in the unobservable space of the system [35].

202

7 Cybersecurity in DC-Microgrids

Remark 7.2 Precisely, the observability map Jacobian [36] (see Eq. 2.4 of Chap. 2 for a definition of the observability map) of system in (7.13) with the load voltage, Vi , as the measured output, has a dimension of 3. As a consequence, m i − 1 power line currents will be unobservable. Therefore, it is possible to design an observer that can estimate ξi and at least one power line current. Nevertheless, this result has limited practical use and does not allow the computation of (7.12). Alternatively, this work exploits the extra dimension in the observation space in order to compute an observable function that can be used to estimate all the power line currents in a distributed manner. Second, standard robust observers give no information about the unknown CPL, Pi , and may present noise sensitivity problems, see beginning of Chap. 5 for a discussion on the topic. Finally, the inability of estimating the CPL, Pi , and the power line currents, Ik , prevents the computation of (7.12), which makes the proposed mitigation strategy difficult to implement. To solve these limitations, this work proposes designing the observer as the interconnection of three sub-systems. First, an extended low-power observer [37] will be designed to robustly estimate the states ξ1,i , ξ2,i and a virtual state, σi . Second, the virtual state, σi , will be used to compute an auxiliary signal, μi , that is going to be implemented as the measured signal in a Luenberger observer to robustly estimate the unobservable power line currents, η i . Finally, the unknown power load will be estimated through an indirect adaptive law based on the immersion and invariance (I&I) technique [38]. The reasoning behind such observer structure will be explained throughout this chapter. It should be remarked that the observer is implemented locally at each DGU. Moreover, the proper operation of the observer will require the communication of the observer with its neighbouring DGU observers. For this reason, the stability of the estimation scheme is separated in two steps. First, it will be proved that, under proper parameter tuning, the local observer is stable in a ISS sense. Second, the conditions in which the network of observers generate a stable structure will be established.

7.3.1 Estimation of ξ1,i , ξ2,i and Design of Extended Low-Power Observer Let us consider the first two equations of (7.13), and extend the system as follows: ξ˙1,i = ξ2,i ξ˙2,i = σi σ˙i = φ˙ i (ξi , u˙ i , Pi , η i , ξ 2,k ) + w˙ 1 where

(7.15)

7.3 Proposed Nonlinear Observer

203

∂φi (ξi , u i , Pi , η i ) ˙ ∂φi (ξi , u i , Pi , η i ) η˙ i φ˙ i (ξi , u˙ i , Pi , η i , ξ 2,k ) = ξi + ∂ξi ∂η i  ∂φi (ξi , u i , Pi , η i ) ∂φi (ξi , u i , Pi , η i ) u˙ i + + ξ2,k . ∂u ∂ξ1,k k∈E i

The objective is to design an algorithm to estimate ξ1,i , ξ2,i and σi in (7.15). System in (7.15) is uniformly observable in the inputs, u i , [39], which allows the implementation of a non-linear observer as the high-gain observer, see Sect. 3.2 in Chap. 3. In theory, the main advantages of a high-gain observer is its arbitrary convergence rate and robustness to uncertainty and disturbances in the expression σ˙i , which makes it an attractive option for the concerned estimation problem. However, the convergence rate and robustness relies on increasing the gain of the observer, which significantly increases the observer sensor noise sensitivity [40], i.e. small high-frequency noise in yi can significantly aggravate the state-estimation accuracy. See Sect. 3.2 of Chap. 3 for a discussion on the topic. For this reason, this work proposes implementing a low-power observer [37]. The low-power observer includes some extra dynamics that increases the relative degree between the state-estimation and the sensor noise, which attenuates the noise effect on the estimation accuracy without losing state and false data estimation convergence rate and robustness properties of the classic high-gain observer [37]. Indeed, the observer is the one in Sect. 3.2.3 of Chap. 3 without the saturation functions. By means of numerical simulation, it has been observed that the peaking phenomena is not relevant for the considered problem. For this reason, the saturation functions have been obviated to simplify the observer. Remark 7.3 The approach of extending the system with extra states, resembles the extended high-gain observer approach to the disturbance estimation problem [41]. Nonetheless, here, the proposed observer is not used to estimate some disturbance. As it will be shown, is used to estimate a signal that can be used by a secondary Luenberger observer to estimate the unknown line currents, Ik . For the concerned triangular structure (7.15), such observer takes the following form α1 ˙ (y − ξˆ1,i ) ξˆ1,i = λ1,i + ε α2 ˙ (λ1,i − ξˆ2,i ) ξˆ2,i = λ2,i + ε α3 (λ2,i − σˆi ) σ˙ˆi = φ˙ i (ξˆ i , u˙ i , Pˆi , ηˆ i , ξˆ 2,k ) + ε β1 λ˙ 1,i = λ2,i + 2 (y − ξˆ1,i ) ε β2 λ˙ 2,i = φ˙ i (ξˆ i , u˙ i , Pˆi , ηˆ i , ξˆ 2,k ) + 2 (λ1,i − ξˆ2,i ) ε

(7.16)

204

7 Cybersecurity in DC-Microgrids

where ξˆ i = [ξˆ1,i , ξˆ2,i ] is the estimation of ξi , σˆi is the estimation of σi , λ1,i and λ2,i are virtual states, α = [α1 , α2 , α3 ] and β = [β1 , β2 ] are positive design parameters, ε is the design high-gain parameter, Pˆi is the estimation of the CPL, ηˆ i is the estimation of η i and ξˆ 2,k is the estimation of ξ 2,k . The details related on the tuning of the observer parameters are discussed in Sect. 3.2.3 of Chap. 3. Relevant to the concerned problem is that, according to Theorem 3.3 in Chap. 3, if the parameters α and β, are properly tuned, then, there exists a positive value ε∗1 such that for all ε ≤ min{ε∗1 , 1} the estimation-error of the observer (7.16) satisfies the following ultimate bounds for q = 1, 2, |ξq,i − ξˆq,i | ≤ ε4−q k1 |Pi − Pˆi | + ε4−q k2 |η i − ηˆ i | + ε4−q k3 |ξ2,k − ξˆ 2,k | + ε4−q kw |w| ˙ |σi − σˆi | ≤ εk1 |Pi − Pˆi | + εk2 |η i − ηˆ i | + εk3 |ξ2,k

(7.17) − ξˆ 2,k | + εkw |w| ˙

(7.18)

where k1 , ..., k3 and kw are some positive constants independent from ε. By direct inspection of (7.17), it can be seen that if one achieves an accurate estimation of Pi , η i and ξ2,k f or all k ∈ Ei , then, the state-estimation errors |ξ j,i − ξˆ j,i | f or all j = 1, 2 and |σi − σˆi | of the low-power observer (7.16) will be ˙ In the noiseless case, this ultimate bound can be arbiultimately bounded by εkw |w|. trary reduced by decreasing the design parameter ε. Nonetheless, the designer should take into account that decreasing ε also increases the observer noise sensitivity and peaking phenomena. The implementation of the observer (7.16), requires the computation of ξˆ2,k . This factor is not locally estimated by the observer nor measured by any sensor, but is being computed by the observers allocated in the neighbour DGUs. Therefore, the computation of (7.16) requires the transmission of ξˆ2,i between observers in neighbour DGUs. Nonetheless, the communication of ξˆ2,i between observers is not a sufficient condition for the stability of the whole distributed scheme. The next subsections will stablish a sufficient stability condition. Finally, the computation of φ˙ i (ξˆ i , u˙ i , Pˆi , ηˆ i , ξˆ 2,k ) using the observer in (7.16), requires the calculation of u˙ i , which is not directly measured. Nevertheless, for systems of the form of (7.1), it is common to implement controllers that compute u˙ i , e.g. the authors in [9] track a reference voltage by implementing two PIs in cascade. In these cases, the factor u˙ i is a known expression of the states. Alternatively, it is common to implement controllers with small u˙ i , due to degradation and security concerns. In such cases, φ˙ i can be implemented as φ˙ i (ξˆ i , 0, Pˆi , ηˆ i , ξˆ 2,k ) and the induced persistent error can be attenuated (without significant performance degradation) by a reduction of ε.

7.3 Proposed Nonlinear Observer

205

7.3.2 Estimation of η i A crucial part for the computation of (7.12) and the observer implementation in (7.16) is to achieve an accurate estimation of the power line currents, η i . As stated before, the dynamics of η i remain in the unobservable space of (7.13), thus, it is not possible to implement an high-gain observer that jointly estimates ξ1,i , ξ2,i and η i . An interesting property of the η i dynamics is that, under an appropriate cooperative control of the DC microgrid, the power line currents are stable. Therefore, it is possible to achieve an estimation by integrating the power line equations of (7.13) in open loop, i.e. the estimator: 1 ˆ Rk (ξ1,i − ξˆ1, j ) − ηˆ j,i η˙ˆ j,i = Lk Lk

f or j = 1, ..., m i

(7.19)

converges to a bounded error, independent of ηˆ1,i (0). Indeed, this fact implies that the η i dynamics are not observable but are detectable, see Chap. 2 for an introduction to the concept of detectability. Nonetheless, similar to other observer designs that are based on the detectability property, this approach is simple but presents significant drawbacks. First, the estimation convergence rate (7.19) cannot be tuned. Second, the estimation accuracy is sensitive to the unknown disturbances/uncertainty w2, j of (7.13). Consequently, this work proposes a different estimation approach. The idea is to use the value of the virtual state σˆi to compute an auxiliary signal that allows the design of an observer for the η i dynamics. Assume that the extended low-power observer (7.16) is used to estimate ξ1,i , ξ2,i and the virtual state, σi . Moreover, notice that σi = φi (ξi , u i , Pi , η i ) + w1 . Then, by inspection of (7.14), the following holds: μi 

 Rk ηk,i + w1 − Δφ, i Lk

k∈E i

= Ci σˆi −

  1  ξˆ2,i 1  ˆ 1 − ξ1,i + u i + (ξˆ1,i − ξˆ1,k ) + ξˆ2,i − Pˆi (7.20) 2 L ti Lk Ri ξˆ1,i k∈E i

where Δφ, i  φi (ξi , u i , Pi , 0) − φi (ξˆ i , u i , Pˆi , 0) + σi − σˆi . Now, take the value μi (7.20) as the measured signal of the η i dynamics. Then, the following linear system is obtained: 1 Rk (ξ1,i − ξ1, j ) − η j,i + w2, j Lk Lk  Rk μi = ηk,i + w1 − Δφ, i. Lk

η˙ j,i =

k∈E i

f or j = 1, ..., m i (7.21)

206

7 Cybersecurity in DC-Microgrids

For convenience, (7.21) can be rewritten as η˙ i = Ai η i +

 1 (ξ1,i − ξ1,k ) + Im i w2 Lk

k∈E i

μi = Ci η i + w1 − Δφ, i

(7.22)

where Im i is a m i × m i identity matrix, ⎤ w2,1 ⎥ ⎢ w2  ⎣ ... ⎦ ⎡

w2,m i

and Ai ∈ Rm i ×m i and Ci ∈ R1×m i are matrices such that ⎡ R ⎤ 1 − 0 ⎢ L1 ⎥   ⎢ ⎥ R1 Rm i . ⎢ ⎥ .. Ai  ⎢ ⎥ , Ci = L 1 , . . . , L m . i ⎣ Rm i ⎦ 0 − L mi It can be seen that, the introduction of the auxiliary signal, μi , has transformed the η i dynamics into a LTI observable system with a non-linear disturbance in the sensor equation. Observer design for linear systems is a well-known topic, deeply studied in the literature [42]. However, not all observers are a valid option for system (7.22), as the implemented observer must ensure the stability of the whole estimation scheme once its coupled with the low-power observer (7.16). Moreover, the implemented observer should attenuate the effect of the uncertainty/disturbances w1 and w2 . Remark 7.4 System (7.22) is not strong observable [43], i.e. the disturbances of (7.22) cannot be geometrically decoupled from the system equations. Therefore, it is not possible to implement a robust observer that eliminates the effect of all the uncertainty/disturbances and/or the non-linear disturbance in the sensor. Remark 7.5 System (7.22) will present some unobservable dimension if there are identical incident power lines, i.e. power lines with the same resistance, Ri , and inductance, L i . This fact can be seen by computing the Kalman rank condition with the pair (Ai , Ci ), see Chap. 2 for a definition of the Kalman rank condition. Nonetheless, in such cases, the system is detectable, therefore, it is still possible to implement an observer following the approach presented at the beginning of the subsection. This work proposes the implementation of a Luenberger observer (similar to the one presented in the beginning of Sect. 3.2 in Chap. 3), due to its simplicity in design and tuning.

7.3 Proposed Nonlinear Observer

207

Lemma 7.3 Consider a linear Luenberger observer ˙ˆ = Ai ηˆ + η i i

 1 (ξˆ1,i − ξˆ1,k ) + Li (μi − Ci ηˆ i ) Lk

(7.23)

k∈E i

where μi is the auxiliary signal computed as (7.20), Li ∈ Rm i ×1 is a design matrix such that Ai − Li Ci has all the eigenvalues in the open left-half plane. Then, the estimation-error η i − ηˆ i is ultimately bounded as follows: |η i − ηˆ i | ≤ k4 |ξi − ξˆ i | + k5 |ξ1,k − ξˆ 1,k | + k6 |σi − σˆi | + k7 |Pi − Pˆi | + kw,2 |w1 | + kw,3 |w2 |

(7.24)

where k4 , ..., k6 and kw,2 , kw,3 are some positive constants. Proof As the matrix Ai − Li Ci is Hurwitz by design, there is a matrix P = P > 0 such that: (7.25) P(Ai − Li Ci ) + (Ai − Li Ci ) P = −Q, where Q is positive defined matrix. Consider the error dynamics between (7.23) and (7.22), eη  η i − ηˆ i , e˙ η = (Ai − Li Ci )eη + Im i (ξ1,i − ξˆ1,i ) + Im i (ξ 1,k − ξˆ 1,k ) − Li w1 − Li Δφ, i − Im i w2 .

Consider the radially unbounded Lyapunov candidate V = eη Peη . The derivative of the function is given by: V˙ = −eη Qeη + 2eη P(ξ1,i − ξˆ1,i ) + 2eη P(ξ 1,k − ξˆ 1,k ) − 2eη PLi w1 + 2eη PLi Δφ, i − 2eη Pw2 . As the function φi (ξi , u i , Pi , 0) is (locally) Lipschitz, there are some positive constants L 1 , L 2 , L 3 and L 4 such that |Δφ, i| ≤ L 1 |ξi − ξˆ i | + L 2 |ξ 1,k − ξˆ 1,k | + L 3 |σi − σˆi | + L 4 |Pi − Pˆi |. Therefore, the derivative of Lyapunov function candidate is upper bounded by: V˙ ≤ −λmin (Q)|eη | + 2|eη ||P|(Im i + Li L 1 )|ξi − ξˆ i | + 2|eη ||P|(Im i + Li L 2 )|ξ 1,k − ξˆ 1,k | + 2|eη ||P|(Im i + Li L 3 )|σi − σˆi | + 2|eη ||P|(Im i + Li L 4 )|Pi − Pˆi | + 2|eη ||PLi ||w1 | + 2|eη ||P||w2 | where λmin (·) depicts the minimum eigenvalue.

(7.26)

208

7 Cybersecurity in DC-Microgrids

The inequality in (7.26) shows that the system is ISS [44] taking ξi − ξˆ i , ξ 1,k − ˆξ 1,k , σi − σˆi , Pi − Pˆi , w1 and w2 as inputs. This fact proves the existence of the bound (7.24).  Notice that, similar to the low-power observer (7.16), the implementation of the linear observer (7.23), requires the transmission of ξˆ1,k between neighbour observers.

7.3.3 Estimation of Pi The computation of the DGU current estimation (7.12) and the auxiliary signal (7.20), requires an estimation of the CPL, Pi . A common approach in such situation is to include the unknown parameter as a new state of the system, i.e. include an state ξ3,i in system (7.13) as follows: ξ˙1,i = ξ2,i ξ˙2,i = φi (ξi , u i , ξ3,i , η i ) ξ˙3,i = 0 η˙ j,i =

1 Rk (ξ1,i − V j ) − η j,i Lk Lk

f or j = 1, ..., m i ,

and design an observer that can estimate ξ1,i , ξ2,i and ξ3,i . Nevertheless, in the concerned problem, this approach requires increasing the order of the low-power observer, which, drastically increases the noise sensitivity of the estimation algorithm. Moreover, this approach does not take advantage of the fact that Pi is constant and the resulting system may lose its uniform observability in the inputs. For this reason, it is interesting to implement an alternative parameter-estimation algorithm. It should be remarked that not all estimation algorithms that achieve limt→∞ |Pi − Pˆi | = 0 are adequate for the concerned problem. Indeed, there may be some optimization algorithms that achieve convergence of the parameter-estimation, but cannot ensure the stability once is coupled with the observer, see the beginning of Chap. 5 for a discussion on the topic. A common approach in the adaptive observer context is to design a direct adaptive redesign that cancels the unknown parameters in the Lyapunov function derivative [45], as explained in Sect. 5.2 of Chap. 5. However, this solution is fragile to uncertainty in the regressor vector. This limitation can be relaxed by implementing the indirect adaptive redesign algorithm based on the I&I technique [38] presented in Sect. 5.3 of Chap. 5, which offers some gain margin with respect uncertainties in the regressor vector and its stability can be proved through an L2 integrability condition that is generically satisfied in the considered system.

7.3 Proposed Nonlinear Observer

209

Lemma 7.4 Consider system (7.13), let ξˆ i be a state-estimation generated by a low-power observer (7.16) that satisfies (7.17) and let ηˆ i be a power line current estimation that satisfies (7.24). Define the following vector functions  f 0,i (ξˆ i , qˆi , ηˆ i , ξˆ 1,k ) =



ξˆ2,i , ˆ φi (ξi , qˆi , Pˆi , ηˆ i )

⎡

0

⎤

⎢ K pI ⎥ f 1,i (ξˆ i ) = ⎣ ξˆ2,i ⎦ + 2 Ci ξˆ1,i C L ti ξˆ1,i

where qˆi is the estimation of qi , which is defined as      1 ˙ qi = K p I Ci Vi + Ik,i + Vi − Ir e f,i + K i I y1,i − Ir e f,i . Ri k∈E i

and y1,i is the measured local current Iti . Define the following function   2 K p I ξˆ2,i γ ξˆ2,i ˆ + βi (ξi ) = 2 2 Ci ξˆ1,i C L ti ξˆ1,i

(7.27)

where γ is a positive constant to be tuned. Moreover, consider the following parameter-estimation dynamics:    ∂βi (ξˆ i ) ˙ˆ ˆ ˆ ˆ ˆ ˆ f 0,i (ξi , qˆi , ηˆ i , ξ 1,k ) + f 1,i (ξi ) θi + βi (ξi ) θi = − ∂ξi Pˆi = θˆ + βi (ξˆ i ).

(7.28)

Then, the following ultimate bound holds |Pi − Pˆi | ≤ k8 |ξi − ξˆ i | + k9 |ξ 1,k − ξˆ 1,k | + k10 |η i − ηˆ i | + kw,3 |w1 |

(7.29)

where k8 , . . . , k10 and kw,3 are some positive constants. Proof Notice that for (7.27), the following relation holds 2  K pI ∂βi (ξi ) ξ2,i f 1,i (ξi ) = γ + . 2 ∂ξi C L ti ξ1,i Ci ξ1,i Define the manifold

z  θˆ − Pi + βi (ξi ).

(7.30)

(7.31)

210

7 Cybersecurity in DC-Microgrids

Then, the dynamics of the off-the-manifold coordinates z are given by: z˙ = −

 2 K pI ξ2,i ∂βi (ξi ) f 1,i (ξi )z + δ = −γ + z+δ 2 ∂ξi C L ti ξ1,i Ci ξ1,i

where δ is defined as:     ∂βi (ξi ) ˆ f 0,i (ξi , qi , η i , ξ 1,k ) + f 1,i (ξi ) θi + βi (ξi ) + w1 δ ∂ξi    ∂βi (ξˆ i ) f 0,i (ξˆ i , qˆi , ηˆ i ) + f 1,i (ξˆ i ) θˆi + βi (ξˆ i ) . − ∂ξi The functions βi , f 0,i , f 1,i and

(7.32)

∂βi (ξi ) are (locally) Lipschitz. Moreover, the ∂ξi

∂βi (ξi ) is upper bounded. Therefore, there exist some positive constants ∂ξi L δ,1 , L δ,2 , L δ,3 and βmax such that

factor

|δ| ≤ L δ,1 |ξi − ξˆ i | + L δ,2 |η i − ηˆ i | + L δ,3 |ξ 1,k − ξˆ 1,k | + βmax |w1 |. Consider the Lyapunov function candidate: V =

1 2 (z) 2

(7.33)

The derivative of (7.33) satisfies the following V˙ = −zγ



2 2  K pI K pI ξ2,i + z + zδ ≤ −zγ + z + z L δ,1 |ξi − ξˆ i | 2 2 C L ti ξ1,i C L ti ξ1,i Ci ξ1,i Ci ξ1,i ξ2,i

+ z L δ,2 |η i − ηˆ i | + z L δ,3 |ξ1,k − ξˆ 1,k | + zβmax |w1 |

(7.34)

ξ2,i + Generically, the variable ξˆ1,i is upper and lower bounded. Thus, the factor 2 Ci ξ1,i K pI is not L2 integrable. Therefore, by inspection of (7.34), it is possible to show C L ti ξ1,i ˆ η i − ηˆ i that (7.33) is a ISS-Lyapunov function with linear ISS-gain [44] from ξ − ξ, and ξ 1,k − ξˆ 1,k to z. As a consequence, taking into account the relation (7.31) and the fact that βi is Lipschitz, the bound (7.29) can be deduced. 

7.3 Proposed Nonlinear Observer

211

7.3.4 Observer Stability The last subsections have presented the sub-systems that compose the observer that is locally implemented at each DGU. The observer can be seen as an interconnection of three estimation algorithms, a low-power observer for ξ1,i , ξ2,i , a Luenberger observer for the line currents, η i and an I&I adaptive law for the CPL, Pi . Moreover, the estimation algorithms require the communication of ξ1,k , ξ2,k from the observers in the neighbour DGUs. As it has been shown, each sub-observer can be represented in the framework presented in Sect. 3.4 of Chap. 3. Precisely, the error dynamics of each sub-observer is ISS stable taking as inputs the error dynamics of the other estimation algorithms and neighbour observers. Taking into account this detail, the small-gain theorem [46] is a convenient method that can be used to stablish the necessary conditions for the stability of the whole observer scheme. The stability proof is separated in two steps. First, it will be proved that, under proper parameter tuning, the local observer implemented in each DGU is ISS with respect to the neighbour observer estimation-error. Second, the conditions in which the network of observers generate a stable structure will be presented. In order to prove the stability of the observer, it is of interest to define the following vector ⎡ ⎤ ξ1,i − ξˆ1,i χi  ⎣ξ2,i − ξˆ2,i ⎦ . σi − σˆi Lemma 7.5 Consider the case without uncertainty/disturbances, i.e. w1 = 0 and w2, j = 0 f or j = 1, ..., m i . Moreover, consider an observer composed by (7.16), which satisfies the bound (7.17), the linear observer (7.22) that satisfies (7.24) and the parameter-estimation (7.28) that satisfies (7.29). Then, if the following condition holds, (7.35) k7 k10 < 1, there is a value ε∗2 such that, for all ε ≤ ε∗2 , the variable χi is ultimately bounded as follows, (7.36) |χi | ≤ εk11 |ξ 1,k − ξˆ 1,k | + εk12 |ξ2,k − ξˆ 2,k |. where k11 and k12 are some positive constants. Proof Taking into account (7.17) and the definition of χi , there are some positive constant n 1 and n 2 , such that the following bounds hold √ √ √ |χi | ≤ ε n 1 k1 |Pi − Pˆi | + ε n 1 k2 |η i − ηˆ i | + ε n 1 k3 |ξ 2,k − ξˆ 2,k |

(7.37)

212

7 Cybersecurity in DC-Microgrids

and √ √ √ |ξi − ξˆ i | ≤ ε2 n 2 k1 |Pi − Pˆi | + ε2 n 2 k2 |η i − ηˆ i | + ε2 n 2 k3 |ξ 2,k − ξˆ 2,k | ≤ |χi |, for some positive constants n 1 and n 2 . Consider (7.24), (7.29), the second equation of (7.37) and assume that k7 k10 < 1. Then, the following bound is obtained |Pi − Pˆi | ≤ (k8 + k10 max{k4 , k6 })|χi | + (k9 + k10 k5 )|ξ1,k − ξˆ 1,k | + k10 k7 |Pi − Pˆi | k9 + k10 k5 k8 + k10 max{k4 , k6 } |χi | + |ξ − ξˆ 1,k |. (7.38) ≤ 1 − k10 k7 1 − k10 k7 1,k Substituting (7.38) into (7.24), the following upper bound is obtained   k7 k8 + k7 k10 max{k4 , k6 } |η i − ηˆ i | ≤ max{k4 , k6 } + |χi | 1 − k10 k7   k7 k9 + k7 k10 k5 |ξ 1,k − ξˆ 1,k |. + k5 + 1 − k10 k7

(7.39)

Substituting (7.38) and (7.39) into the first equation of (7.37), the following is obtained,   √ k1 k8 + k1 k10 max{k4 , k6 } k2 k7 k8 + k2 k7 k10 max{k4 , k6 } |χ i | ≤ ε n 1 + k2 max{k4 , k6 } + |χ i | 1 − k10 k7 1 − k10 k7   √ k1 k9 + k1 k10 k5 k2 k7 k9 + k2 k7 k10 k5 |ξ 1,k − ξˆ 1,k | + ε n1 + k2 k5 1 − k10 k7 1 − k10 k7 √ + ε n 1 k3 |ξ 2,k − ξˆ 2,k |. (7.40)

Then, for ε ≤ ε∗2 , where ε∗2 

  √  k1 k8 + k1 k10 max{k4 , k6 } k2 k7 k8 + k2 k7 k10 max{k4 , k6 }  −1 + k2 max{k4 , k6 } + , n1 1 − k10 k7 1 − k10 k7

the bound (7.40) reduces to the ultimate bound (7.36).



Remark 7.6 The constant k7 depends on P and Q of (7.25). Therefore, the gain, L, of the Luenberger observer has to be designed in order to satisfy (7.35). The existence of the ultimate bound (7.36), shows that, the local observer at each DGU can be tuned so it is ISS taking the estimation-error of the neighbouring observers as inputs. This fact can be used to present a condition for the stability of an interconnection of observers for an arbitrary communication graph of the DC microgrid. Taking into account (7.36), the following ultimate bound for the i th DGU can be deduced

7.3 Proposed Nonlinear Observer

|χi | ≤ εk11



213

|ξ1,k − ξˆ1,k | + εk12

k∈E i

≤ ε max{k11 , k12 }





|ξ2,k − ξˆ2,k |

k∈E i

|χk |

(7.41)

k∈E i

Notice that the ISS gain, from the estimation-error of the neighbour DGUs, of the i th DGU is linear and multiplied by the design parameter ε. Taking into account this fact, one can stablish the following: Theorem 7.1 Define εi as the high-gain parameter of the low-power observer (7.16) in the i th DGU. Consider a DC microgrid of N DGUs modelled by (7.1) without uncertainty/disturbances (i.e. w1 and w2, j = 0 f or j = 1, ..., m i ), where an observer composed by (7.16), which satisfies the bound (7.17), a Luenberger observer (7.22), that satisfies (7.24) and a parameter-estimation (7.28), that satisfies (7.29), has been implemented. Moreover, the observer satisfies (7.35). Then, there are a set of positive constants ε∗3,i f or i = 1, ..., N , such that, for all εi ≤ ε∗3,i f or i = 1, ..., N , the estimations ξˆ1,i , ξˆ2,i , Pˆi and η i converges to its true value. Proof As the observer in the i th DGU satisfies the bound (7.41), the χi dynamics are ISS from χk f or k = 1, ..., m i . Define γi, j as the ISS gain from χ j to χi . As the communication graph of the microgrid is connected and without self-loops, and all the ISS gains are linear, the small-gain theorem reduces to a set of function compositions of γi, j and (1 − γi, j )−1 that have to be smaller than 1 (define a contraction) to ensure the estimation convergence [46]. The ISS gains, γi, j are proportional to the design parameter εi . Therefore, the functions γi, j and (1 − γi, j )−1 and its possible compositions can be made arbitrary small by reducing εi f or i = 1, ..., N , which proofs the theorem.  Remark 7.7 The observer parameter tuning that ensures the stability of the whole distributed scheme only relies on satisfying two inequalities. First, the gain L of the Luenberger observer (7.22) has to be tuned so condition (7.35) is satisfied. Notice that the power load estimation-error, Pi − Pˆi , matches the uncertainty w1 . Therefore, in practice, most gains L that are tuned in order to attenuate w1 , will make the observer satisfy (7.35). Second, the high-gain parameter, εi , has to satisfy εi ≤ min{1, ε∗1 , ε∗2 , ε∗3,i }. In general, a high-gain parameter that ensures an adequate convergence rate and robustness for the low-power observer (7.16) in the i th DGU will satisfy this inequality. Remark 7.8 The elements of the proposed observer have been designed to attenuate the effect of sensor and communication noise. The following numerical simulations and experimental validation will show that the effect of sensor noise is not significant for the considered application. Nonetheless, if required, further noise reduction could be achieved by implementing the low-pass filters or dynamic dead-zone filters presented in Chap. 4.

214

7 Cybersecurity in DC-Microgrids

7.4 Numerical Simulations and Experimental Validations Simulation and experimental validation of the proposed scheme can be found in the following publications: • Cecilia, A., Sahoo, S., Dragiˇcevi´c, T., Costa-Castelló, R., & Blaabjerg, F. (2021). On Addressing the Security and Stability Issues Due to False Data Injection Attacks in DC Microgrids-An Adaptive Observer Approach. IEEE Transactions on Power Electronics, 37(3), 2801–2814. • Cecilia, A., Sahoo, S., Dragiˇcevi´c, T., Costa-Castelló, R., & Blaabjerg, F. (2021). Detection and mitigation of false data in cooperative dc microgrids with unknown constant power loads. IEEE Transactions on Power Electronics, 36(8), 9565–9577.

References 1. Ipakchi A, Albuyeh F (2009) Grid of the future. IEEE Power Energy Mag 7(2):52–62 2. Lasseter RH (2002) Microgrids. In: 2002 IEEE power engineering society winter meeting. Conference proceedings (Cat No 02CH37309) 3. Guerrero JM, Vasquez JC, Matas J, de Vicuna LG, Castilla M (2011) Hierarchical control of droop-controlled ac and dc microgrids?a general approach toward standardization. IEEE Trans Ind Electron 58(1):158–172. https://doi.org/10.1109/TIE.2010.2066534 4. Dragiˇcevi´c T, Lu X, Vasquez JC, Guerrero JM (2016) Dc microgrids? part I: a review of control strategies and stabilization techniques. IEEE Trans Power Electron 31(7):4876–4891 5. Diaz NL, Dragiˇcevi´c T, Vasquez JC, Guerrero JM (2014) Intelligent distributed generation and storage units for dc microgrids? A new concept on cooperative control without communications beyond droop control. IEEE Trans Smart Grid 5(5):2476–2485 6. Sahoo S, Mishra S (2019) A distributed finite-time secondary average voltage regulation and current sharing controller for dc microgrids. IEEE Trans Smart Grid 10(1):282–292 7. Yazdanian M, Mehrizi-Sani A (2014) Distributed control techniques in microgrids. IEEE Trans Smart Grid 5(6):2901–2909 8. Sandberg H, Amin S, Johansson KH (2015) Cyberphysical security in networked control systems: an introduction to the issue. IEEE Control Syst Mag 35(1):20–23 9. Sahoo S, Peng JC, Devakumar A, Mishra S, Dragiˇcevi´c T (2020) On detection of false data in cooperative dc microgrids-a discordant element approach. IEEE Trans Ind Electron 67(8):6562–6571 10. Emadi A, Khaligh A, Rivetta CH, Williamson GA (2006) Constant power loads and negative impedance instability in automotive systems: definition, modeling, stability, and control of power electronic converters and motor drives. IEEE Trans Veh Technol 55(4):1112–1125. https://doi.org/10.1109/TVT.2006.877483 11. Trip Sebastian, Cucuzzella Michele, Cheng Xiaodong, Scherpen Jacquelien (2018) Distributed averaging control for voltage regulation and current sharing in dc microgrids. IEEE Contr Syst Lett 3(1):174–179 12. Nasirian V, Moayedi S, Davoudi A, Lewis FL (2015) Distributed cooperative control of dc microgrids. IEEE Trans Power Electron 30(4):2288–2303 13. Deng R, Zhuang P, Liang H (2019) False data injection attacks against state estimation in power distribution systems. IEEE Trans Smart Grid 10(3):2871–2881 14. Danzi P, Stefanovi? A, Dragiˇcevi´c T, Popovski P (2019) Software-defined microgrid control for resilience against denial-of-service attacks. IEEE Trans Smart Grid 10(5):5258–5268

References

215

15. Pasqualetti F, Dörfler F, Bullo F (2013) Attack detection and identification in cyber-physical systems. IEEE Trans Autom Control 58(11):2715–2729 16. Sahoo S, Mishra S, Peng JC, Dragiˇcevi´c T (2019) A stealth cyber-attack detection strategy for dc microgrids. IEEE Trans Power Electron 34(8):8162–8174. https://doi.org/10.1109/TPEL. 2018.2879886 17. Liu L, Esmalifalak M, Ding Q, Emesih VA, Han Z (2014) Detecting false data injection attacks on power grid by sparse optimization. IEEE Trans Smart Grid 5(2):612–621. https://doi.org/ 10.1109/TSG.2013.2284438 18. Manandhar K, Cao X, Hu F, Liu Y (2014) Detection of faults and attacks including false data injection attack in smart grid using kalman filter. IEEE Trans Control Netw Syst 1(4):370–379 19. Chen Po-Yu, Yang Shusen, McCann Julie A, Lin Jie, Yang Xinyu (2015) Detection of false data injection attacks in smart-grid systems. IEEE Commun Mag 53(2):206–213. https://doi. org/10.1109/MCOM.2015.7045410 20. Khalaf Mohsen, Youssef Amr, El-Saadany Ehab (2019) Joint detection and mitigation of false data injection attacks in AGC systems. IEEE Trans Smart Grid 10(5):4985–4995. https://doi. org/10.1109/TSG.2018.2872120 21. Sargolzaei A, Yazdani K, Abbaspour A, Crane III CD, Dixon WE (2020) Detection and mitigation of false data injection attacks in networked control systems. IEEE Trans Ind Inf 16(6):4281–4292. https://doi.org/10.1109/TII.2019.2952067 22. Zhao J, Zhang G, La Scala M, Dong ZY, Chen C, Wang J (2017) Short-term state forecastingaided method for detection of smart grid general false data injection attacks. IEEE Trans Smart Grid 8(4):1580–1590. https://doi.org/10.1109/TSG.2015.2492827 23. Li S, Yilmaz Y, Wang X (2015) Quickest detection of false data injection attack in widearea smart grids. IEEE Trans Smart Grid 6(6):2725–2735. https://doi.org/10.1109/TSG.2014. 2374577 24. Nishino H, Ishii H (2014) Distributed detection of cyber attacks and faults for power systems. IFAC Proc 47(3):11932–11937. ISSN 1474-6670. https://doi.org/10.3182/20140824-6-ZA1003.02425 25. Kurt MN, Yilmaz Y, Wang X (2018) Distributed quickest detection of cyber-attacks in smart grid. IEEE Trans Inf Foren Secur 13(8):2015–2030. https://doi.org/10.1109/TIFS.2018. 2800908 26. Khalid HM, Peng JC-H (2017) Immunity toward data-injection attacks using multisensor track fusion-based model prediction. IEEE Trans Smart Grid 8(2):697–707. https://doi.org/10.1109/ TSG.2015.2487280 27. Gallo AJ, Turan MS, Boem F, Parisini T, Ferrari-Trecate G (2020) A distributed cyber-attack detection scheme with application to dc microgrids. IEEE Trans Autom Control 65(9):3800– 3815 28. Liang G, Zhao J, Luo F, Weller SR, Dong ZY (2017) A review of false data injection attacks against modern power systems. IEEE Trans Smart Grid 8(4):1630–1638. https://doi.org/10. 1109/TSG.2015.2495133 29. Musleh AS, Chen G, Dong ZY (2020) A survey on the detection algorithms for false data injection attacks in smart grids. IEEE Trans Smart Grid 11(3):2218–2234. https://doi.org/10. 1109/TSG.2019.2949998 30. Sahoo S, Dragiˇcevi´c T, Blaabjerg F (2020) An event-driven resilient control strategy for dc microgrids. IEEE Trans Power Electron 35(12):13714–13724 May 31. Sahoo S, Dragiˇcevi´c T, Blaabjerg F (2021) Multilayer resilience paradigm against cyber attacks in dc microgrids. IEEE Trans Power Electron 36(3):2522–2532. https://doi.org/10.1109/TPEL. 2020.3014258. Mar 32. Sahoo S, Dragiˇcevi´c T, Blaabjerg F (2020) Resilient operation of heterogeneous sources in cooperative dc microgrids. IEEE Trans Power Electron 35(12):12601–12605 Apr 33. Soriano-Rangel CA, He W, Mancilla-David F, Ortega R (202) Voltage regulation in buck-boost converters feeding an unknown constant power load: an adaptive passivity-based control. IEEE Trans Control Syst Technol 1–8

216

7 Cybersecurity in DC-Microgrids

34. Severino B, Strunz K (2019) Enhancing transient stability of dc microgrid by enlarging the region of attraction through nonlinear polynomial droop control. IEEE Trans Circuits Syst I: Regul Pap 66(11):4388–4401. https://doi.org/10.1109/TCSI.2019.2924169. Nov 35. Isidori A, Sontag ED, Thoma M (1995) Nonlinear control systems, vol 3. Springer 36. Hermann R, Krener A (1977) Nonlinear controllability and observability. IEEE Trans Autom Control 22(5):728–740. https://doi.org/10.1109/TAC.1977.1101601 37. Astolfi Daniele, Marconi Lorenzo (2015) A high-gain nonlinear observer with limited gain power. IEEE Trans Autom Control 60(11):3059–3064. https://doi.org/10.1109/TAC.2015. 2408554 38. Astolfi A, Karagiannis D, Ortega R (2008) Nonlinear and adaptive control with applications, vol 187. Springer 39. Gauthier JP, Bornard G (1981) Observability for any u(t) of a gass of nonlinear systems. IEEE Trans Autom Control. ISSN 15582523. https://doi.org/10.1109/TAC.1981.1102743 40. Freidovich LB, Khalil HK (2008) Performance recovery of feedback-linearization-based designs. IEEE Trans Autom Control 53(10):2324–2334 Nov 41. Khalil HK (2017) Extended high-gain observers as disturbance estimators. SICE J Control Meas Syst Integ 10(3):125–134 42. Korovin SK, Fomichev VV, Fomichev VV (2009) State observers for linear systems with uncertainty. de Gruyter 43. Molinari B (1976) A strong controllability and observability in linear multivariable control. IEEE Trans Autom Control 21(5):761–764. https://doi.org/10.1109/TAC.1976.1101364. Oct 44. Sontag Eduardo D, Wang Yuan (1995) On characterizations of the input-to-state stability property. Syst Control Lett 24(5):351–359 Apr 45. Cho YM, Rajamani R (1997) A systematic approach to adaptive observer synthesis for nonlinear systems. IEEE Trans Autom Control 42(4):534–537. https://doi.org/10.1109/9.566664 46. Jiang Z, Teel A, Praly L (1994) Small-gain theorem for ISS systems and applications. Math Control Signals Syst 7:95–120. https://doi.org/10.1007/BF01211469. Jun

Chapter 8

Concluding Remarks

This chapter summarizes the main contents presented in the book and presents its main contributions. Moreover, some discussion on future perspectives of the lines of research opened by this book are given.

8.1 Summary The main research focus of this Ph.D. has been to develop an observer design framework that takes into account the presence of strong nonlinearities, significant measurement noise and parametric disturbances. The general idea throughout the work has been to, first, assume an already existing observer for the nonlinear system and, second, redesign its structure in order to improve its performance in front of measurement noise and unknown parameters. The starting point has been a type of observer that satisfies two properties: asymptotic stability which can be proved through a known Lyapunov function, and robustness in an ISS sense with respect to measurement disturbances and unknown parameters. Nonetheless, multiple limitations have been identified in the redesign process and the necessary improvements have been developed. All the theory has been utilized to solve relevant practical applications in the energy sector. In the following, a brief summary of the work achieved in this thesis is presented. The first part of the thesis in Chaps. 2 and 3 has focused on the observation problem and how it can be solved through continuous-time nonlinear observers. The sufficient observability conditions for the solvability of the observation problem have been introduced. Then, a standard three-step observer design process and its related Lyapunov stability analysis has been presented. This design framework has been exemplified with two different observer schemes. First, the high-gain observer and its more recent low-power peaking-free modification, which addresses the peaking phenomena issue. Second, the parameter estimation-based observer. In this second © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Cecilia, Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems, Springer Theses, https://doi.org/10.1007/978-3-031-38924-5_8

217

218

8 Concluding Remarks

scheme, an alternative observer approach has been proposed, which implicitly regularizes the estimation problem. Finally, the observer design framework has been extended to consider the presence of unmodelled elements as measurement noise and parametric uncertainty. The effect of these elements on the estimation error has been established by means of a Lyapunov analysis and the computation of an ISS function. The second part of the thesis in Chap. 4 studied the possibility of implementing filters to reduce the effect of measurement noise/perturbations. The beginning of the chapter has established that the correct way to implement a filter in an observer is to filter the output estimation error, not the output signal. This modification circumvents the phase-lag problem of the filter at a cost of tampering the stability of the observer. For this reason, a clear stability theory for adding filters in nonlinear observer has been developed. The first part of the chapter has focused on the topic of adding low-pass filters in nonlinear observers while preserving stability. It has been established that this filter can be implemented if the observer satisfies an output growth assumption. The second part has presented a novel internal-model filter that exactly rejects disturbances generated from a known exo-system. A complete filter stability analysis has been conducted for the linear observer case, and has validated in the nonlinear observer case through numerical simulations. After that, a third filter has been proposed which combines the low-pass filter and the internal-model filter. Finally, a discussion related to the similarities between the presented low-pass filter design framework and the recently proposed dynamic dead-zone filter has been included. The third part of the thesis in Chap. 5 has studied the adaptive redesign approach as a means to reduce the effect of parametric uncertainty. The beginning of the chapter has focused on the standard direct adaptive observer redesign. For this method, three theoretical and practical limitations have been detected and relaxed. First, it has been established that most direct adaptive observer redesigns proofs are based on a combination of a weak Lyapunov function and the Barbalat’s lemma, which has no robustness guarantees. To solve this issue, a strict Lyapunov function has been derived which shows that robustness (in an ISS sense) with respect to measurement noise and model uncertainty can be guaranteed if the system is persistently excited. Second, the very stringent relative degree condition that is required for the Lyapunov function cancellation is relaxed by means of implementing an auxiliary signal that is estimated through a high-gain observer. Finally, the direct redesign is extended to the case in which the regressor vector is modelled through a library of candidate functions. In this scenario, an observer that converges to a sparse parameter estimation solution is provided. In the final part of the chapter, an indirect adaptive observer redesign based on the immersion and invariance formalism has been proposed. In the fourth part of the thesis in Chap. 6, the developed theory has been implemented in the problem of liquid water monitoring in PEM fuel cells. First, a lowcomplexity model of the system has been derived from the fuel cell first-principles. Then, the order of the model has been reduced in order to get a tractable observation problem. Based on the resulting low-order model, three observers have been proposed. First, a low-power peaking-free observer that preserves the robustness and

8.2 Main Contributions

219

convergence rate of the classic high-gain observer while eliminating the peaking phenomena has been developed. The effect of sensor noise has been reduced by means of dynamic dead-zone filtering. Second, the theory and benefits related to including the voltage sensor signal has been discussed. It has been established that further convergence rate can be achieved by means of including the voltage sensor at a cost of accuracy loss in dead-end mode operation. To circumvent this limitation, the periodic anode purges have been modelled as a linear exo-system and eliminated by means of a low-pass internal-model filter. Third, an observer based on the direct adaptive redesign has been developed to reduce the effect of unknown parameters in the liquid water dynamics. This last observer has been validated in a real experimental prototype in the end of the chapter. In the last part of the thesis in Chap. 7 an algorithm for the detection and mitigation of false data injection attacks in DC microgrid with unknown constant power loads has been developed. The beginning of the chapter has focused on discussing in which ways the presence of unknown constant power loads increases the complexity of an attack detection algorithm, while eases the task of designing an attack. Moreover, it has been established that the cyber-attack can be simultaneously detected and mitigated if an observer is designed to estimate the voltage derivative, the line currents and the unknown constant power load in each DGU. To achieve such task, a distributed observer that combines an extended low-power observer, a Luenberger observer and an indirect adaptive redesign has been proposed. The stability of the network of observers has been established through a small-gain argument.

8.2 Main Contributions The specific technical contributions of this Ph.D. can be briefly summarized as follows: • Relative to adding filters in nonlinear observers, this thesis has proposed a stability theory for a quite general type of low-pass filter and nonlinear observer. Moreover, it has proposed a novel internal-model filter. The complete stability and disturbance rejection properties of the filter has been analysed in the linear scenario. A low-pass internal-model filter that combines both filters has also been proposed. • Relative to nonlinear observer adaptive redesign, this thesis has proposed a strict Lyapunov function for a relatively general type of direct adaptive redesign. Moreover, a high-gain observer method has been proposed to circumvent the relative degree condition of direct adaptive redesigns. Furthermore, a sparsity-promoting observer has been proposed to generalize the direct adaptive redesign to the case in which the unknown parameter vector is sparse. Finally, a indirect adaptive observer redesign based on the I&I technique has been proposed. • Relative to nonlinear observer design, a novel observer that implicitly optimizes a convex function to be designed has been presented. Moreover, a method

220

8 Concluding Remarks

to include secondary outputs in nonlinear observers based on a particular monotonicity property has been developed. • Relative to PEM fuel cell monitoring, a low-complexity model of the fuel cell has been developed. A set of observers to estimate the liquid water saturation in the catalyst porous media has been presented. An internal-model filter approach to reduce the effect of voltage perturbations has been developed. The effect of liquid water parametric uncertainty has been reduced by means of a direct adaptive observer redesign. The proposed techniques have been validated in a real experimental prototype. • Relative to DC microgrid cybersecurity, the detection and mitigation of false data injection attacks for DC microgrids with unknown constant power loads has been formulated as an observation problem. An algorithm that can estimate simultaneously and in a distributed manner the load current, derivative of the DGU voltage, the line currents and the unknown constant power loads has been presented. The proposed observer and mitigation strategy have been validated in a real experimental prototype.

8.3 Future Research Perspectives This thesis has presented some new results on the topic of nonlinear observer design in the presence of significant measurement noise and parametric uncertainty, which have been used to solve relevant problem in the field of energy systems. Nonetheless, there are still some details that require further investigation in order to get a more complete theory and to ease the implementation of the proposed techniques in other energy systems problems. Indeed, some of the lines that can be followed in future research can be summed up as follows: • The section related to low-pass filters focused on ensuring stability of the resulting scheme. An next step is to develop a methodology to tune the filters in order to guarantee adequate observer disturbance rejection and convergence rate. • The theory of internal model filters has to be extended to the nonlinear case. Specifically, develop the sufficient conditions in which an internal model filter and a nonlinear observer generates a stable structure. • The stability and robustness of the adaptive redesigns are based on the very restrictive assumption of persistence of excitation. Parameter estimators based on the Bregman divergence may serve as a way to relax this assumption. Nonetheless, a more in-depth analysis on the estimation convergence and the design of the potential function has to be carried on. • It may also be interesting to extend the proposed techniques to additional energy systems problem. Some promising fields are the state-of-charge and state-of-health estimation in Vanadium redox flow batteries and cyber-attack detection in ACmicrogrids. Moreover, the fuel cell model developed in Chap. 4 can be further studied to solve additional monitoring problems.

About the Author

Andreu Cecilia received the B.Eng. degree in industrial engineering, the double M.Sc. degree in automatic control/industrial engineering and the Ph.D. in automatic control, robotics and vision from the Universitat Politècnica de Catalunya, Barcelona, Spain, in 2017, 2020 and 2022, respectively. He is currently working as a postdoctoral researcher at Laboratoire d’Automatique, de Génie des Procédés et de Génie Pharmaceutique, Lyon, France. His research interests include observers and nonlinear system theory and its application to energy systems.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Cecilia, Advances in Nonlinear Observer Design for State and Parameter Estimation in Energy Systems, Springer Theses, https://doi.org/10.1007/978-3-031-38924-5

221