Boundary control and observation of coupled parabolic PDEs

Reaction-diffusion equations are parabolic Partial Differential Equations (PDEs) which often occur in practice, e.g., to model the concentration of one or more substances, distributed in space, under the in uence of different phenomena such as local chemical reactions, in which the substances are transformed into each other, and diffusion, which causes the substances to spread out over a surface in space. Certainly, reaction-diffusion PDEs are not confined to chemical applications but they also describe dynamical processes of non-chemical nature, with examples being found in thermodynamics, biology, geology, physics, ecology, etc. Problems such as parabolic Partial Differential Equations (PDEs) and many others require the user to have a considerable background in PDEs and functional analysis before one can study the control design methods for these systems, particularly boundary control design. Control and observation of coupled parabolic PDEs comes in roughly two settingsdepending on where the actuators and sensors are located \in domain" control, where the actuation penetrates inside the domain of the PDE system or is evenly distributed everywhere in the domain and \boundary" control, where the actuation and sensing are applied only through the boundary conditions. Boundary control is generally considered to be physically more realistic because actuation and sensing are nonintrusive but is also generally considered to be the harder problem, because the \input operator" and the "output operator" are unbounded operators. The method that this thesis develops for control of PDEs is the so-called backstepping control method. Backstepping is a particular approach to stabilization of dynamic systems and is particularly successful in the area of nonlinear control. The backstepping method achieves Lyapunov stabilization, which is often achieved by collectively shifting all the eigenvalues in a favorable direction in the complex plane, rather than by assigning individual eigenvalues. As the reader will soon learn, this task can be achieved in a rather elegant way, where the control gains are easy to compute symbolically, numerically, and in some cases even explicitly. In addition to presenting the methods for boundary control design, we present the dual methods for observer design using boundary sensing. Virtually every one of our control designs for full state stabilization has an observer counterpart. The observer gains are easy to compute symbolically or even explicitly in some cases. They are designed in such a way that the observer error system is exponentially stabilized. As in the case of finite-dimensional observer-based control, a separation principle holds in the sense that a closed-loop system remains stable after a full state stabilizing feedback is replaced by a feedback that employs the observer state instead of the plant state

Modelling and estimation of vanadium redox flow batteries: a review

Redox flow batteries are one of the most promising technologies for large-scale energy storage, especially in applications based on renewable energies. In this context, considerable efforts have been made in the last few years to overcome the limitations and optimise the performance of this technology, aiming to make it commercially competitive. From the monitoring point of view, one of the biggest challenges is the estimation of the system internal states, such as the state of charge and the state of health, given the complexity of obtaining such information directly from experimental measures. Therefore, many proposals have been recently developed to get rid of such inconvenient measurements and, instead, utilise an algorithm that makes use of a mathematical model in order to rely only on easily measurable variables such as the system’s voltage and current. This review provides a comprehensive study of the different types of dynamic models available in the literature, together with an analysis of the existing model-based estimation strategies. Finally, a discussion about the remaining challenges and possible future research lines on this field is presented.The research that gave rise to these results received support from “la Caixa” Foundation (ID 100010434. Fellowship code LCF/BQ/DI21/11860023) , the CSIC program for the Spanish Recovery, Transformation and Resilience Plan funded by the Recovery and Resilience Facility of the European Union, established by the Regulation (EU) 2020/2094, CSIC Interdisciplinary Thematic Platform (PTI+) Transición Energética Sostenible+ (PTI-TRANSENER+ project TRE2103000), the Spanish Ministry of Science and Innovation (project PID2021-126001OB-C31 funded by MCIN/AEI/10.13039/501100011033 / ERDF,EU) and the Spanish Ministry of Economy and Competitiveness under Project DOVELAR (ref. RTI2018-096001-B-C32).Peer ReviewedPostprint (published version

Observation and control of PDE with disturbances

In this Thesis, the problem of controlling and Observing some classes of distributed parameter systems is addressed. The particularity of this work is to consider partial differential equations (PDE) under the effect of external unknown disturbances. We consider generalized forms of two popular parabolic and hyperbolic infinite dimensional dynamics, the heat and wave equations. Sliding-mode control is used to achieve the control goals, exploiting the robustness properties of this robust control technique against persistent disturbances and parameter uncertainties

Observation and control of PDE with disturbances

In this Thesis, the problem of controlling and Observing some classes of distributed parameter systems is addressed. The particularity of this work is to consider partial differential equations (PDE) under the effect of external unknown disturbances. We consider generalized forms of two popular parabolic and hyperbolic infinite dimensional dynamics, the heat and wave equations. Sliding-mode control is used to achieve the control goals, exploiting the robustness properties of this robust control technique against persistent disturbances and parameter uncertainties

A Strict Control Lyapunov Function for a Diffusion Equation with Time-Varying Distributed Coefficients

International audienceIn this paper, a strict Lyapunov function is developed in order to show the exponential stability and input-to-state stability (ISS) properties of a diffusion equation for nonhomogeneous media. Such media can involve rapidly time-varying distributed diffusivity coefficients. Based on this Lyapunov function, a control law is derived to preserve the ISS properties of the system and improve its performance. A robustness analysis with respect to disturbances and estimation errors in the distributed parameters is performed on the system, precisely showing the impact of the controller on the rate of convergence and ISS gains. This is important in light of a possible implementation of the control since, in most cases, diffusion coefficient estimates involve a high degree of uncertainty. An application to the safety factor profile control for the Tore Supra tokamak illustrates and motivates the theoretical results. A constrained control law (incorporating nonlinear shape constraints in the actuation profiles) is designed to behave as closely as possible to the unconstrained version, albeit with the equivalent of a variable gain. Finally, the proposed control laws are tested under simulation, first in the nominal case and then using a model of Tore Supra dynamics, where they show adequate performance and robustness with respect to disturbances

Performance indicators for the dynamics modeling and control of PEMFC systems

Society is gradually becoming aware that the current energy industry, based on the use of fossil fuels, is inefficient, highly polluting and has a finite supply. Within the scientific community, there are indications that hydrogen (H2) as an energy vector, obtained from renewable energy sources, can represent a viable option to mitigate the problems associated with hydrocarbon combustion. In this context, the change from the current energy industry to a new structure with a significant involvement of H2 facilitates the introduction of fuel cells as elements of energy conversion. Polymer Electrolyte Membrane Fuel Cells (PEMFC) are gaining increased attention as viable energy conversion devices for a wide range of applications from automotive, stationary to portable. In order to optimize performance, these systems require active control and thus in-depth knowledge of the system dynamics which include fluid mechanics, thermal dynamics and reaction kinetics. One of the main issues, with respect to proper control of these systems, is the understanding of the water transport mechanisms through the membrane and the liquid water distribution. The thesis is based on the publication of nine international journal articles that are divided into 4 sub-topics: Dynamic fuel cell modeling, fuel cell system control-oriented analysis, identification of parameters and performance indicators and finally, fault and failure detection and system diagnosis. In the sub-topic of Dynamic Fuel cell modeling, experimentally validated Computational Fluid Dynamics (CFD) modeling is used to relate the effects of the physical phenomena associated with fluid mechanics and thermal dynamics, that occur inside the fuel cell [Alonso, 2009][Strahl, 2011], to water distribution. However, since these CFD models cannot be directly used for control, control-oriented models [Kunusch, 2008][Kunusch, 2011] have been developed in parallel. As well, another study is done in [Serra, 2006] which includes a controllability analysis of the system for future development and application of efficient controllers. The results of the above mentioned studies are limited because either they do not incorporate an electrochemical model or the model is not experimentally validated. Moreover, these models do not take into account the voltage losses due to liquid water inside the fuel cell. Therefore, there is a need to properly relate the relevant effects of fluid mechanics and thermal dynamics, including liquid water, to the fuel cell voltage. Primarily, methodologies are needed to determine the relevant indicators associated to the effect of water on the fuel cell performance. The works published in [Husar, 2008] and [Husar, 2011] treats experimental parameter identification, mainly focused on water transport through the membrane and fuel cell voltage loss indicators respectively. The implementation of the indicators indirect measurement methodology provides an experimental way for the isolation of three main types of voltage losses in the fuel cell: activation, mass transport and ohmic losses. Additionally since these voltage loss indicators relate the fuel cell operating conditions to the fuel cell voltage, they can be utilized to calibrate and validate CFD models as well as employed in novel control strategies. On the other hand, to develop reliable systems, the controller should not only take into account performance variables during standard operation but should also be able to detect failures and take the appropriate actions. A preliminary study on failure indicators is presented in [Husar 2007] and fault detection methodologies are described in [de Lira 2011]. As a whole, the compilation of articles represented in this thesis applies a comprehensive experimental approach which describes the implementation of novel methodologies and experimental procedures to characterize and model the PEMFC and their associated systems taking into consideration control oriented goals.La societat s'està adonant que la indústria energètica actual, basada en l'ús de combustibles fòssils, és ineficient, molt contaminant i té un subministrament limitat. Dins de la comunitat científica, hi ha indicis que el hidrogen (H2) com vector energètic, obtingut a partir de fonts d'energia renovables, pot representar una opció viable per a mitigar els problemes associats amb la combustió d'hidrocarburs. En aquest context, el canvi de la indústria energètica actual a una nova estructura amb una important participació de el hidrogen exigeix la introducció de les piles de combustible com elements de conversió d'energia. Les piles de combustible de membrana polimèrica (PEMFC) estan tenint cada vegada més atenció com a dispositius viables de conversió d'energia per a una àmplia gamma d'aplicacions com automoció, estacionàries o portàtils. Amb la finalitat d'optimitzar el seu rendiment, les piles PEM requereixen un control actiu i per tant un coneixement profund de la dinàmica del sistema, que inclou la mecànica de fluids, la dinàmica tèrmica i la cinètica de les reaccions. Un dels temes principals relacionat amb el control adequat d'aquests sistemes és la comprensió dels mecanismes de transport d'aigua a través de la membrana i la distribució d'aigua líquida. Aquesta tesi es basa en nou articles publicats en revistes internacionals que es divideixen en 4 subtemes: la modelització dinàmica de piles de combustible, l'anàlisi orientada al control del sistema, la identificació de paràmetres i d’indicadors de funcionament i, finalment, la detecció de fallades i la diagnosi dels sistemes. En el sub-tema de la modelització dinàmica de piles PEM, la modelització basada en la Dinàmica de Fluids Computacional (CFD) amb validació experimental s'ha utilitzat per a relacionar els efectes dels fenòmens físics de la mecànica de fluids i de la dinàmica tèrmica que es produeixen dintre de la pila [Alonso, 2009] [ Strahl, 2011] amb la distribució d'aigua. No obstant això, com aquests models CFD no poden ser utilitzats directament per al control, s'han desenvolupat models orientats a control [Kunusch, 2008] [Kunusch, 2011] en paral·lel. A més, en un altre estudi [Serra, 2006] s'inclou una anàlisi de control·labilitat del sistema per al desenvolupament i aplicació futurs de controladors eficaços. Però els resultats dels estudis esmentats anteriorment són limitats, ja sigui perquè no incorporen un model electroquímic o bé perquè no han estat validats experimentalment. A més, cap dels models té en compte les pèrdues de tensió degudes a l'aigua líquida dins de la pila de combustible. Per tant, hi ha una necessitat de relacionar adequadament els efectes rellevants de la mecànica de fluids i de la dinàmica tèrmica, incloent l'aigua líquida, amb el voltatge de la pila de combustible. Principalment, són necessàries metodologies per a determinar els indicadors rellevants associats a aquest efecte de l'aigua sobre el rendiment de la pila de combustible. Els treballs publicats en [Husar, 2008] i [Husar, 2011] tracten la identificació experimental de paràmetres, centrada en el transport d'aigua a través de la membrana i els indicadors de pèrdua de tensió, respectivament. L'aplicació d'una proposta de metodologia de mesura indirecte dels indicadors permet l'aïllament dels tres tipus principals de pèrdues de voltatge en la pila de combustible: l'activació, el transport de massa i les pèrdues ohmiques. Aquests indicadors de pèrdua de tensió relacionen les condicions d'operació amb el voltatge de la pila de combustible i per tant poden ser utilitzats per a calibrar i validar models CFD, així com per a definir noves estratègies de control. D'altra banda, per a aconseguir sistemes fiables, el controlador no només ha de considerar els indicadors de funcionament de l'operació normal, sinó que també ha de detectar possibles fallades per a poder prendre les accions adequades en cas de fallada. Un estudi preliminar sobre indicadors de fallades es presenta en [Husar 2007] i una metodologia de detecció de fallades completa es descriu en [Lira de 2011]. En el seu conjunt, el compendi d'articles que formen aquesta tesi segueix un enfocament experimental i descriu la implementació de noves metodologies i procediments experimentals per a la caracterització i el modelatge de piles PEM i els sistemes associats amb objectius orientats al control eficient d'aquests sistemes.La sociedad se ésta dando cuenta de que la industria energética actual, basada en el uso de combustibles fósiles, es ineficiente, muy contaminante y tiene un suministro limitado. Dentro de la comunidad científica, hay indicios de que el hidrógeno (H2) como vector energético, obtenido a partir de fuentes de energía renovables, puede representar una opción viable para mitigar los problemas asociados con la combustión de hidrocarburos. En este contexto, el cambio de la industria energética actual a una nueva estructura con una importante participación de H2 exige la introducción de pilas de combustible como elementos de conversión de energía. Las pilas de combustible de membrana polimérica (PEMFC) están ganando cada vez más atención como dispositivos viables de conversión de energía para una amplia gama de aplicaciones como automoción, estacionarias o portátiles. Con el fin de optimizar su rendimiento, las pilas PEM requieren un control activo y por lo tanto un conocimiento profundo de la dinámica del sistema, que incluye la mecánica de fluidos, la dinámica térmica y la cinética de las reacciones. Uno de los temas principales relacionado con el control adecuado de estos sistemas, es la comprensión de los mecanismos de transporte de agua a través de la membrana y la distribución de agua líquida. Esta tesis se basa en la publicación de nueve artículos en revistas internacionales que se dividen en 4 sub-temas: el modelado dinámico de pilas de combustible, el análisis orientado a control del sistema, la identificación de parámetros e indicadores de desempeño y, por último, la detección de fallos y la diagnosis. En el sub-tema de la modelización dinámica de pilas PEM, el modelado basado en Dinámica de Fluidos Computacional (CFD) con validación experimental se ha utilizado para relacionar los efectos de los fenómenos físicos de la mecánica de fluidos y la dinámica térmica que se producen dentro de la pila [Alonso, 2009] [ Strahl, 2011] con la distribución de agua. Sin embargo, como estos modelos CFD no pueden ser utilizados directamente para el control, modelos orientados a control [Kunusch, 2008] [Kunusch, 2011] se han desarrollado en paralelo. Además, en otro estudio [Serra, 2006] se incluye un análisis de controlabilidad del sistema para el futuro desarrollo y aplicación de controladores eficaces. Pero los resultados de los estudios mencionados anteriormente son limitados, ya sea porque no incorporan un modelo electroquímico o bien porque no son validados experimentalmente. Además, ninguno de los modelos tiene en cuenta las pérdidas de tensión debidas al agua líquida dentro de la pila de combustible. Por lo tanto, hay una necesidad de relacionar adecuadamente los efectos relevantes de la mecánica de fluidos y la dinámica térmica, incluyendo el agua líquida, con la tensión de la pila de combustible. Principalmente, son necesarias metodologías para determinar los indicadores relevantes asociados al efecto del agua sobre el rendimiento de la pila de combustible. Los trabajos publicados en [Husar, 2008] y [Husar, 2011] tratan la identificación experimental de parámetros, centrada en el transporte de agua a través de la membrana y los indicadores de pérdida de tensió, respectivamente. La aplicación de una metodología propuesta de medición indirecta de los indicadores permite el aislamiento de los tres tipos principales de pérdidas de tensión en la pila de combustible: la activación, el transporte de masa y las pérdidas óhmicas. Éstos indicadores de pérdida de tensión relacionan las condiciones de operación con la tensión de la pila de combustible y por lo tanto pueden ser utilizados para calibrar y validar modelos CFD, así como para definir nuevas estrategias de control. Por otro lado, para conseguir sistemas fiables, el controlador no sólo debe considerar los indicadores de desempeño de la operación regular, sino que también debe detectar posibles fallos para poder tomar las acciones adecuadas en caso de fallo. Un estudio preliminar sobre indicadores de fallos se presenta en [Husar 2007] y una metodología de detección de fallos completa se describe en [Lira de 2011]. En su conjunto, el compendio de artículos que forman esta tesis sigue un enfoque experimental y describe la implementación de nuevas metodologías y procedimientos experimentales para la caracterización y el modelado de pilas PEM y los sistemas asociados con objetivos orientados al control eficiente de estos sistemas

Lithium-Ion battery SOC estimation

Lithium-ion batteries are frequently used in Hybrid electric vehicles (HEVs), which are taking the place of gas-engine vehicles. An important but not measurable quantity in HEVs is the amount of charge remaining in the battery in a drive cycle. The remaining charge is normally identified by a variable called state of charge (SOC). A potential way of estimating the SOC is relating this variable with the state of a dynamical system. Afterwards, the SOC can be estimated through an observer design. As a precise model, electrochemical equations are chosen in this research to estimate the SOC. The first part of this thesis considers comparison studies of commonly-used finite-dimensional estimation methods for different distributed parameter systems (DPSs). In this part, the system is first approximated by a finite-dimensional representation; the observer dynamics is a copy of the finite-dimensional representation and a filtering gain obtained through observer design. The main outcome of these studies is comparing the performance of different observers in the state estimation of different types of DPSs after truncation. The studies are then expanded to investigate the effect of the truncated model by increasing the order of finite-dimensional approximation of the system numerically. The simulation results are also compared to the mathematical properties of the systems. A modified sliding mode observer is improved next to take care of the system's nonlinearity and compensate for the estimation error due to disturbances coming from an external input. It is proved that the modified SMO provides an exponential convergence of the estimation error in the existence of an external input. In most cases, the simulations results of the comparison studies indicate the improved performance of the modified SMO observer. Approximation and well-posedness of two general classes of nonlinear DPSs are studied next. The main concern of these studies is to produce a low-order model which converges to the original equation as the order of approximation increases. The available results in the literature are limited to specified classes of systems. These classes do not cover the lithium-ion cell model; however, the general forms presented here include the electrochemical equations as a specific version. In order to facilitate the electrochemical model for observer design, simplification of the model is considered in the next step. The original electrochemical equations are composed of both dynamical and constraint equations. They are simplified such that a fully dynamical representation can be derived. The fully dynamical representation is beneficial for real-time application since it does not require solving the constraint equation at every time iteration while solving the dynamical equations. Next, the electrochemical equations can be transformed into the general state space form studied in this thesis. Finally, an adaptive EKF observer is designed via the low-order model for SOC estimation. The electrochemical model employed here is a variable solid-state diffusivity model. Compared to other models, the variable solid-state diffusivity model is more accurate for cells with Lithium ion phosphate positive electrode, which are considered here, than others. The adaptive observer is constructed based on considering an adaptive model for the open circuit potential term in the electrochemical equations. The parameters of this model are identified simultaneously with the state estimation. Compared to the experimental data, simulation results show the efficiency of the designed observer in the existence of modeling inaccuracy

Modeling and control of fuel cell-battery hybrid energy sources

Environmental, political, and availability concerns regarding fossil fuels in recent decades have garnered substantial research and development in the area of alternative energy systems. Among various alternative energy systems, fuel cells and batteries have attracted significant attention both in academia and industry considering their superior performances and numerous advantages. In this dissertation, the modeling and control of these two electrochemical sources as the main constituents of fuel cell-battery hybrid energy sources are studied with ultimate goals of improving their performance, reducing their development and operational costs and consequently, easing their widespread commercialization. More specifically, Paper I provides a comprehensive background and literature review about Li-ion battery and its Battery Management System (BMS). Furthermore, the development of an experimental BMS design testbench is introduced in this paper. Paper II discusses the design of a novel observer for Li-ion battery State of Charge (SOC) estimation, as one of the most important functionalities of BMSs. Paper III addresses the control-oriented modeling and analysis of open-cathode fuel cells in order to provide a comprehensive system-level understanding of their real-time operation and to establish a basis for control design. Finally, in Paper IV a feedback controller, combined with a novel output-injection observer, is designed and implemented for open-cathode fuel cell temperature control. It is shown that temperature control not only ensures the fuel cell temperature reference is properly maintained, but, along with an uncertainty estimator, can also be used to adaptively stabilize the output voltage --Abstract, page iv

Vanadium redox flow batteries real-time State of Charge and State of Health estimation under electrolyte imbalance condition

This paper presents a novel observer architecture capable to estimate online the concentrations of the four vanadium species present in a vanadium redox flow battery (VRFB). The proposed architecture comprises three main stages: (1) a high-gain observer, to estimate the output voltage and its derivatives; (2) a dynamic inverter, to obtain a set of concentration candidate solutions; and (3) a static selector, to determine the actual concentrations. The methodology does not rely on the classic assumption of balanced electrolytes, thus significantly widening the application range in comparison with most of the literature previous studies. Furthermore, to perform the estimation, only a single voltage and current measurements are required, which eliminates the need of including complex and costly additional sensors. To validate the proposal, comprehensive simulation tests are conducted. These tests take into account typical side reactions that cause imbalance in VRFB systems, such as vanadium crossover and oxidation. The observer shows a remarkable performance when dealing with these realistic conditions, allowing to estimate with high accuracy and robustness the four vanadium concentrations, the State of Charge and the State of Health with a relative error below 2%.The project that gave rise to these results received the support of a fellowship from ”la Caixa” Foundation (ID 100010434). The fellowship code is LCF/BQ/DI21/11860023. This research was also supported by the Spanish Ministry of Science and Innovation, under the projects MAFALDA (PID2021-126001OB-C31) and MASHED (TED2021-129927B-I00). This work has been supported by the Spanish Ministry of Universities funded by the European Union - NextGenerationEU (2022UPC-MSC-93823).Peer ReviewedPostprint (published version