Computing an Inner and an Outer Approximation of the Viability Kernel

International audienceThe viability kernel corresponds to the set of all state vectors of a controlled dynamic system that are viable, i.e., such that there exists an input such that the system will not enter inside a forbidden zone. In this paper, we propose a method which computes an inner and an outer approximation of the viability kernel in a guaranteed way. Our method is based on interval analysis and uses the notions of V-viability and capture basin. We illustrate our approach on the car on the hill problem. A software package has been developed to solve any 2D-problem

Complete Model-Based Testing Applied to the Railway Domain

Testing is the most important verification technique to assert the correctness of an embedded system. Model-based testing (MBT) is a popular approach that generates test cases from models automatically. For the verification of safety-critical systems, complete MBT strategies are most promising. Complete testing strategies can guarantee that all errors of a certain kind are revealed by the generated test suite, given that the system-under-test fulfils several hypotheses. This work presents a complete testing strategy which is based on equivalence class abstraction. Using this approach, reactive systems, with a potentially infinite input domain but finitely many internal states, can be abstracted to finite-state machines. This allows for the generation of finite test suites providing completeness. However, for a system-under-test, it is hard to prove the validity of the hypotheses which justify the completeness of the applied testing strategy. Therefore, we experimentally evaluate the fault-detection capabilities of our equivalence class testing strategy in this work. We use a novel mutation-analysis strategy which introduces artificial errors to a SystemC model to mimic typical HW/SW integration errors. We provide experimental results that show the adequacy of our approach considering case studies from the railway domain (i.e., a speed-monitoring function and an interlocking-system controller) and from the automotive domain (i.e., an airbag controller). Furthermore, we present extensions to the equivalence class testing strategy. We show that a combination with randomisation and boundary-value selection is able to significantly increase the probability to detect HW/SW integration errors

Set-based state estimation and fault diagnosis using constrained zonotopes and applications

This doctoral thesis develops new methods for set-based state estimation and active fault diagnosis (AFD) of (i) nonlinear discrete-time systems, (ii) discrete-time nonlinear systems whose trajectories satisfy nonlinear equality constraints (called invariants), (iii) linear descriptor systems, and (iv) joint state and parameter estimation of nonlinear descriptor systems. Set-based estimation aims to compute tight enclosures of the possible system states in each time step subject to unknown-but-bounded uncertainties. To address this issue, the present doctoral thesis proposes new methods for efficiently propagating constrained zonotopes (CZs) through nonlinear mappings. Besides, this thesis improves the standard prediction-update framework for systems with invariants using new algorithms for refining CZs based on nonlinear constraints. In addition, this thesis introduces a new approach for set-based AFD of a class of nonlinear discrete-time systems. An affine parametrization of the reachable sets is obtained for the design of an optimal input for set-based AFD. In addition, this thesis presents new methods based on CZs for set-valued state estimation and AFD of linear descriptor systems. Linear static constraints on the state variables can be directly incorporated into CZs. Moreover, this thesis proposes a new representation for unbounded sets based on zonotopes, which allows to develop methods for state estimation and AFD also of unstable linear descriptor systems, without the knowledge of an enclosure of all the trajectories of the system. This thesis also develops a new method for set-based joint state and parameter estimation of nonlinear descriptor systems using CZs in a unified framework. Lastly, this manuscript applies the proposed set-based state estimation and AFD methods using CZs to unmanned aerial vehicles, water distribution networks, and a lithium-ion cell.Comment: My PhD Thesis from Federal University of Minas Gerais, Brazil. Most of the research work has already been published in DOIs 10.1109/CDC.2018.8618678, 10.23919/ECC.2018.8550353, 10.1016/j.automatica.2019.108614, 10.1016/j.ifacol.2020.12.2484, 10.1016/j.ifacol.2021.08.308, 10.1016/j.automatica.2021.109638, 10.1109/TCST.2021.3130534, 10.1016/j.automatica.2022.11042

Model Predictive Control of Complex Systems including Fault Tolerance Capabilities: Application to Sewer Networks

El control en temps real de xarxes de clavegueram (RTC) desenvolupa un paper fonamental dins de la gestió dels recursos hídrics relacionats amb el cicle urbà de l'aigua i, en general, amb el seu cicle natural. Un adequat disseny de control per a xarxes de clavegueram evita impactes mediambientals negatius originats per inundacions i/o alta pol·lució producte de condicions meteorològiques xtremes. No obstant, s'ha de tenir en compte que aquestes xarxes, a més de la seva grandària i quantitat de variables i instrumentació, són sistemes rics en dinàmiques complexes i altament no lineals. Aquest fet, unit a les condicions atmosfèriques extremes, fan necessari utilitzar una estratègia de control capaç¸ de suportar totes aquestes condicions. En aquest sentit, dins del camp del (RTC) de xarxes de clavegueram es destaquen les estratègies de control predictiu basat en model (MPC), les quals són alternatives adequades per al control de configuracions multivariable i de gran escala, aplicades com estratègies de control global del sistema. A m´es, permeten optimitzar la resposta del sistema tenint en compte diversos índexs de rendiment (control multiobjectiu). Aquesta tesi s'enfoca en el disseny de controladors MPC per a xarxes de clavegueram considerant diverses metodologies de modelat. Addicionalment, analitza les situacions en les quals es presenten fallades als actuadors de la xarxa, proposant estratègies per a mantenir la resposta del sistema amb la menor degradació possible dels objectius de control, malgrat la presència de la fallada. En la primera part s'introdueixen els conceptes principals dels temes a tractar en la tesi: xarxes de clavegueram, MPC i tolerància a fallades. Seguidament, es presenta la tècnica de modelat utilitzada per a definir el model d'una xarxa de clavegueram. Finalment, es presenta i descriu el cas d'aplicació en la tesi: la xarxa de clavegueram de Barcelona (Espanya). La segona part es centra en dissenyar controladors MPC per al cas d'estudi. S'han considerat dos tipus de model de xarxa: (i) un model lineal, el qual aproxima els comportaments no lineals de la xarxa, donant origen a estratègies MPC lineals amb les seves conegudes avantatges de l'optimització convexa i escalabilitat; i (ii) un model híbrid, el qual inclou les dinàmiques de commutació més representatives d'una xarxa de clavegueram com són els sobreeixidors. En aquest últim cas es proposa una nova etodologia de modelat híbrid per a xarxes de clavegueram i es dissenyen estratègies de control predictives basades en aquests models (HMPC), les quals calculen lleis de control globalment òptimes. Addicionalment, es proposa una estratègia de relaxació del problema d'optimització discreta per a evitar els grans temps de còmput requerits per a calcular la llei de control HMPC. Finalment, la tercera part de la tesi s'encarrega d'estudiar les capacitats de tolerància a fallades en actuadors de llaços de control MPC. En el cas de xarxes de clavegueram, la tesi considera fallades en les comportes de derivació i de retenció d'aigües residuals. A més, es proposa un modelat híbrid per a fallades que faci que el problema d'optimització associat no perdi la seva convexitat. Així, es proposen dos estratègies de HMPC tolerant a fallades (FTMPC): l'estratègia activa, la qual utilitza les avantatges d'una arquitectura de control tolerant a fallades (FTC), i l'estratègia passiva, la qual només depèn de la robustesa intrínseca de les tècniques de control MPC. Com a extensió a l'estudi de tolerància a fallades, es proposa una avaluació d'admissibilitat per a configuracions d'actuadors en fallada agafant com a referència la degradació dels objectius de control. El m-etode, basat en satisfacció de restriccions, permet avaluar l'admissibilitat d'una configuració d'actuadors en fallada i, en cas de no ser admesa, evitaria el procés de resoldre un problema d'optimització amb un alt cost computacional. Paraules clau: control predictiu basat en model, sistemes de clavegueram, sistemes híbrids, MLD, control tolerant a fallades, satisfacció de restriccions.El control en tiempo real de redes de alcantarillado (RTC) desempeña un papel fundamental dentro de la gestión de los recursos hídricos relacionados con el ciclo urbano del agua y, en general, con su ciclo natural. Un adecuado diseño de control para de redes de alcantarillado evita impactos medioambientales negativos originados por inundaciones y/o alta polución producto de condiciones meteorológicas extremas. Sin embargo, se debe tener en cuenta que estas redes, además de su gran tamaño y cantidad de variables e instrumentación, son sistemas ricos en dinámicas complejas y altamente no lineales. Este hecho, unido a unas condiciones atmosféricas extremas, hace necesario utilizar una estrategia de control capaz de soportar todas estas condiciones. En este sentido, dentro del campo del RTC de redes de alcantarillado se destacan las estrategias de control predictivo basadas en modelo (MPC), las cuales son alternativas adecuadas para el control de configuraciones multivariable y de gran escala, aplicadas como estrategias de control global del sistema. Además, permiten optimizar el desempeño del sistema teniendo en cuenta diversos índices de rendimiento (control multiobjetivo). Esta tesis se enfoca en el diseño de controladores MPC para redes de alcantarillado considerando diversas metodologías de modelado. Adicionalmente, analiza las situaciones en las cuales se presentan fallos en los actuadores de la red, proponiendo estrategias para mantener el desempeño del sistema y evitando la degradación de los objetivos de control a pesar de la presencia del fallo. En la primera parte se introducen los conceptos principales de los temas a tratar en la tesis: redes de alcantarillado, MPC y tolerancia a fallos. Además, se presenta la técnica de modelado utilizada para definir el modelo de una red de alcantarillado. Finalmente, se presenta y describe el caso de aplicación considerado en la tesis: la red de alcantarillado de Barcelona (España). La segunda parte se centra en diseñar controladores MPC para el caso de estudio. Dos tipos de modelo de la red son considerados: (i) un modelo lineal, el cual aproxima los comportamientos no lineales de la red, dando origen a estrategias MPC lineales con sus conocidas ventajas de optimización convexa y escalabilidad; y (ii) un modelo híbrido, el cual incluye las dinámicas de conmutación más representativas de una red de alcantarillado como lo son los rebosaderos. En este último caso se propone una nueva metodología de modelado híbrido para redes de alcantarillado y se diseñan estrategias de control predictivas basadas en estos modelos (HMPC), las cuales calculan leyes de control globalmente óptimas. Adicionalmente se propone una estrategia de relajación del problema de optimización discreto para evitar los grandes tiempos de cálculo que pudieran ser requeridos al obtener la ley de control HMPC. Finalmente, la tercera parte de la tesis se ocupa de estudiar las capacidades de tolerancia a fallos en actuadores de lazos de control MPC. En el caso de redes de alcantarillado, la tesis considera fallos en las compuertas de derivación y de retención de aguas residuales. De igual manera, se propone un modelado híbrido para los fallos que haga que el problema de optimización asociado no pierda su convexidad. Así, se proponen dos estrategias de HMPC tolerante a fallos (FTMPC): la estrategia activa, la cual utiliza las ventajas de una arquitectura de control tolerante a fallos (FTC), y la estrategia pasiva, la cual sólo depende de la robustez intrínseca de las técnicas de control MPC. Como extensión al estudio de tolerancia a fallos, se propone una evaluación de admisibilidad para configuraciones de actuadores en fallo tomando como referencia la degradación de los objetivos de control. El método, basado en satisfacción de restricciones, permite evaluar la admisibilidad de una configuración de actuadores en fallo y, en caso de no ser admitida, evitaría el proceso de resolver un problema de optimización con un alto coste computacional. Palabras clave: control predictivo basado en modelo, sistemas de alcantarillado, sistemas híbridos, MLD, control tolerante a fallos, satisfacción de restricciones.Real time control (RTC) of sewer networks plays a fundamental role in the management of hydrological systems, both in the urban water cycle, as well as in the natural water cycle. An adequate design of control systems for sewer networks can prevent the negative impact on the environment that Combined Sewer Overflow (CSO) as well as preventing flooding within city limits when extreme weather conditions occur. However, sewer networks are large scale systems with many variables, complex dynamics and strong nonlinear behaviour. Any control strategy applied should be capable of handling these challenging requirements. Within the field of RTC of sewer networks for global network control, the Model Predictive Control (MPC) strategy stands out due to its ability to handle large scale, nonlinear and multivariable systems. Furthermore, this strategy allows performance optimization, taking into account several control objectives simultaneously. This thesis is devoted to the design of MPC controllers for sewer networks, as well as the complementary modelling methodologies. Furthermore, scenarios where actuator faults occur are specially considered and strategies to maintain performance or at least minimizing its degradation in presence of faults are proposed. In the first part of this thesis, the basic concepts are introduced: sewer networks, MPC and fault tolerant control. In addition, the modelling methodologies used to describe such systems are presented. Finally the case study of this thesis is described: the sewer network of the city of Barcelona (Spain). The second part of this thesis is centered on the design of MPC controllers for the proposed case study. Two types of models are considered: (i) a linear model whose corresponding MPC strategy is known for its advantages such as convexity of the optimization problem and existing pro of sofstability, and (ii) a hybrid model which allows the inclusion of state dependent hybrid dynamics such as weirs. In the latter case, a new hybrid modelling methodology is introduced and hybrid model predictive control (HMPC) strategies based on these models are designed. Furthermore, strategies to relax the optimization problem are introduced to reduce calculation time required for the HMPC control law. Finally, the third part of this thesis is devoted to study the fault tolerance capabilities of MPC controllers. Actuator faults in retention and redirection gates are considered. Additionally, hybrid modelling techniques are presented for faults which, in the linear case, can not be treated without loosing convexity of the related optimization problem. Two fault tolerant HMPC strategies are compared: the active strategy, which uses the information from a diagnosis system to maintain control performance, and the passive strategy which only relies on the intrinsic robustness of the MPC control law. As an extension to the study of fault tolerance, the admissibility of faulty actuator configurations is analyzed with regard to the degradation of control objectives. The method, which is based on constraint satisfaction, allows the admissibility evaluation of actuator fault configurations, which avoids the process of solving the optimization problem with its related high computational cost. Keywords: MPC, sewer networks, hybrid systems, MLD, fault tolerant control, constraints satisfaction

Determination of Inner and Outer Bounds of Reachable Sets Through Subpavings

The computation of the reachable set of states of a given dynamic system is an important step to verify its safety during operation. There are different methods of computing reachable sets, namely interval integration, capture basin, methods involving the minimum time to reach function, and level set methods. This work deals with interval integration to compute subpavings to over or under approximate reachable sets of low dimensional systems. The main advantage of this method is that, compared to guaranteed integration, it allows to control the amount of over-estimation at the cost of increased computational effort. An algorithm to over and under estimate sets through subpavings, which potentially reduces the computational load when the test function or the contractor is computationally heavy, is implemented and tested. This algorithm is used to compute inner and outer approximations of reachable sets. The test function and the contractors used in this work to obtain the subpavings involve guaranteed integration, provided either by the Euler method or by another guaranteed integration method. The methods developed were applied to compute inner and outer approximations of reachable sets for the double integrator example. From the results it was observed that using contractors instead of test functions yields much tighter results. It was also confirmed that for a given minimum box size there is an optimum time step such that with a greater or smaller time step worse results are obtained