Array algorithms for H-infinity estimation

In this paper we develop array algorithms for H-infinity filtering. These algorithms can be regarded as the Krein space generalizations of H-2 array algorithms, which are currently the preferred method for implementing H-2 biters, The array algorithms considered include typo main families: square-root array algorithms, which are typically numerically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H-infinity filters, as these conditions are built into the algorithms themselves. However, since H-infinity square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H-2 case, further investigation is needed to determine the numerical behavior of such algorithms

Fault-tolerant Control Design to Enhance Damping of Inter-area Oscillations in Power Grids

SUMMARY In this paper passive and active approaches for the design of fault-tolerant controllers (FTCs) are presented. The FTCs are used to improve the damping of inter-area oscillations in a power grid. The effectiveness of using a combination of local and remote (wide-area) feedback signals is first demonstrated. The challenge is then to guarantee a minimum level of dynamic performance following a loss of remote signals. The designs are based on regional pole-placement using Linear Matrix Inequalities (LMIs). First, a passive FTC is proposed. It is shown that the computation of the controller reduces to the solution of bilinear matrix inequalities. An iterative procedure is then used to design the controller. Next, as an alternative to active, time varying controllers, one for each fault scenario, we propose an approach for the design of a 'minimal switching' FTC in which only one controller is designed, but where a simple switch is incorporated into the controller structure. A case study in a linear and nonlinear Nordic equivalent system is presented to show that the closed-loop response using a conventional control (CC) design could deteriorate the performance or even destabilize the system if the remote signals are lost and to demonstrate the effectiveness of the proposed FTC designs. KEY WORDS: Fault-tolerant control, regional pole-placement, simultaneous design, power oscillation damping, local and remote feedback

Fault-tolerant wide-area control of power systems

In this thesis, the stability and performance of closed-loop systems following the loss of sensors or feedback signals (sensor faults) are studied. The objective is to guarantee stability in the face of sensor faults while optimising performance under nominal (no sensor fault) condition. One of the main contributions of this work is to deal effectively with the combinatorial binary nature of the problem when the number of sensors is large. Several fault-tolerant controller and observer architectures that are suitable for different applications are proposed and their effectiveness demonstrated. The problems are formulated in terms of the existence of feasible solutions to linear matrix inequalities. The formulations presented in this work are described in a general form and can be applied to a large class of systems. In particular, the use of fault-tolerant architectures for damping inter-area oscillations in power systems using wide-area signals has been demonstrated. As an extension of the proposed formulations, regional pole placement to enhance the damping of inter-area modes has been incorporated. The objective is to achieve specified damping ratios for the inter-area modes and maximise the closed-loop performance under nominal condition while guaranteeing stability for all possible combinations of sensors faults. The performances of the proposed fault-tolerant architectures are validated through extensive nonlinear simulations using a simplified equivalent model of the Nordic power system.Open Acces

Fault tolerant control for nonlinear aircraft based on feedback linearization

The thesis concerns the fault tolerant flight control (FTFC) problem for nonlinear aircraft by making use of analytical redundancy. Considering initially fault-free flight, the feedback linearization theory plays an important role to provide a baseline control approach for de-coupling and stabilizing a non-linear statically unstable aircraft system. Then several reconfigurable control strategies are studied to provide further robust control performance:- A neural network (NN)-based adaption mechanism is used to develop reconfigurable FTFC performance through the combination of a concurrent updated learninglaw. - The combined feedback linearization and NN adaptor FTFC system is further improved through the use of a sliding mode control (SMC) strategy to enhance the convergence of the NN learning adaptor. - An approach to simultaneous estimation of both state and fault signals is incorporated within an active FTFC system.The faults acting independently on the three primary actuators of the nonlinear aircraft are compensated in the control system.The theoretical ideas developed in the thesis have been applied to the nonlinear Machan Unmanned Aerial Vehicle (UAV) system. The simulation results obtained from a tracking control system demonstrate the improved fault tolerant performance for all the presented control schemes, validated under various faults and disturbance scenarios.A Boeing 747 nonlinear benchmark model, developed within the framework of the GARTEUR FM-AG 16 project “fault tolerant flight control systems”,is used for the purpose of further simulation study and testing of the FTFC scheme developed by making the combined use of concurrent learning NN and SMC theory. The simulation results under the given fault scenario show a promising reconfiguration performance

Active fault-tolerant control of nonlinear systems with wind turbine application

The thesis concerns the theoretical development of Active Fault-Tolerant Control (AFTC) methods for nonlinear system via T-S multiple-modelling approach. The thesis adopted the estimation and compensation approach to AFTC within a tracking control framework. In this framework, the thesis considers several approaches to robust T-S fuzzy control and T-S fuzzy estimation: T-S fuzzy proportional multiple integral observer (PMIO); T-S fuzzy proportional-proportional integral observer (PPIO); T-S fuzzy virtual sensor (VS) based AFTC; T-S fuzzy Dynamic Output Feedback Control TSDOFC; T-S observer-based feedback control; Sliding Mode Control (SMC). The theoretical concepts have been applied to an offshore wind turbine (OWT) application study. The key developments that present in this thesis are:• The development of three active Fault Tolerant Tracking Control (FTTC) strategies for nonlinear systems described via T-S fuzzy inference modelling. The proposals combine the use of Linear Reference Model Fuzzy Control (LRMFC) with either the estimation and compensation concept or the control reconfiguration concept.• The development of T-S fuzzy observer-based state estimate fuzzy control strategy for nonlinear systems. The developed strategy has the capability to tolerate simultaneous actuator and sensor faults within tracking and regulating control framework. Additionally, a proposal to recover the Separation Principle has also been developed via the use of TSDOFC within the FTTC framework.• The proposals of two FTTC strategies based on the estimation and compensation concept for sustainable OWTs control. The proposals have introduced a significant attribute to the literature of sustainable OWTs control via (1) Obviating the need for Fault Detection and Diagnosis (FDD) unit, (2) Providing useful information to evaluate fault severity via the fault estimation signals.• The development of FTTC architecture for OWTs that combines the use of TSDOFC and a form of cascaded observers (cascaded analytical redundancy). This architecture is proposed in order to ensure the robustness of both the TSDOFC and the EWS estimator against the generator and rotor speed sensor faults.• A sliding mode baseline controller has been proposed within three FTTC strategies for sustainable OWTs control. The proposals utilise the inherent robustness of the SMC to tolerate some matched faults without the need for analytical redundancy. Following this, the combination of SMC and estimation and compensation framework proposed to ensure the close-loop system robustness to various faults.• Within the framework of the developed T-S fuzzy based FTTC strategies, a new perspective to reduce the T-S fuzzy control design conservatism problem has been proposed via the use of different control techniques that demand less design constraints. Moreover, within the SMC based FTTC, an investigation is given to demonstrate the SMC robustness against a wider than usual set of faults is enhanced via designing the sliding surface with minimum dimension of the feedback signals

Fault tolerant control system design for distillation processes

PhD ThesisThe complexity and sophistication of modern control systems deployed in the re nery operation, particularly the crude distillation unit as a result of increasing demand for higher performance and improved safety, are on the increase. This growing complexity comes with some level of vulnerabilities, part of which is the potential failure in some of the components that make up the control system, such as actuators and sensors. The interplay between these components and the control system needs to have some built-in robustness in the face of actuator and sensor faults, to guarantee higher reliability and improved safety of the control system and the plant respectively, which is fundamental to the economy and operation of the system. This thesis focuses on the application of frugally designed fault tolerant control systems (FTCS) with automatic actuator and sensor faults containment capabilities on distillation processes, particularly atmospheric crude distillation unit. A simple active actuator FTCS that used backup feedback signal, switchable references and restructurable PID controllers was designed and implemented on three distillation processes with varying complexities { methanol-water separation column, the benchmark Shell heavy oil fractionator, and an interactive dynamic crude distillation unit (CDU) to accommodate actuator faults. The fault detection and diagnosis (FDD) component of the actuator FTCS used dynamic principal component analysis (DPCA), a data-based fault diagnostic technique, because of its simplicity and ability to handle large amount of correlated process measurements. The recon gurable structure of the PID controllers was achieved using relative gain array (RGA) and dynamic RGA system interaction analysis tools for possible inputs { outputs pairing with and without the occurrence of actuator faults. The interactive dynamic simulation of CDU was developed in HYSYS and integrated with MATLAB application through which the FDD and the actuator FTCS were implemented. The proposed actuator FTCS is proved being very e ective in accommodating actuator faults in cases where there are suitable inputs { outputs pairing after occurrence of an actuator fault. Fault tolerant inferential controller (FTIC) was also designed and implemented on a binary distillation column and an interactive atmospheric CDU to accommodate sensor faults related to the controlled variables. The FTIC used dynamic partial least squared (DPLS) and dynamic principal component regression (DPCR) based soft sensor techniques to provide redundant controlled variable estimates, which are then used in place of faulty sensor outputs in the feedback loops to accommodate sensor faults and maintain the integrity of the entire control system. Implementation issues arising from the e ects of a sensor fault on the secondary variables used for soft sensor estimation were addressed and the approach was shown to be very e ective in accommodating all the sensor faults investigated in the distillation units. The actuator FTCS and the FTIC were then integrated with the DPCA FDD scheme to form a complete FTCS capable of accommodating successive actuator and sensor faults in the distillation processes investigated. The simulation results demonstrated the e ectiveness of the proposed approach. Lastly, fault tolerant model predictive control (FTMPC) with restructurable inputs { outputs pairing in the presence of actuator faults based on preassessed recon gurable control structures was proposed, and implemented on an interactive dynamic CDU. The FTMPC system used a rst order plus dead time (FOPDT) model of the plant for output prediction and RGA and DRGA tools to analyse possible control structure recon guration. The strategy helped improve the availability and performance of control systems in the presence of actuator faults, and can ultimately help prevent avoidable potential disasters in the re nery operation with improved bottom line { Pro t. Overall, the proposed approaches are shown to be e ective in handling actuator and sensor faults, when there are suitable manipulated variables and redundant analytical signals that could be used to contain the e ects of the faults on the system.University of Lagos, Nigeria & Petroleum Technology Development Fund (PTDF) for the Scholarship award at the later stage of my research programme

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

Advances in gain-scheduling and fault tolerant control techniques

This thesis presents some contributions to the state-of-the-art of the fields of gain-scheduling and fault tolerant control (FTC). In the area of gain-scheduling, the connections between the linear parameter varying (LPV) and Takagi-Sugeno (TS) paradigms are analyzed, showing that the methods for the automated generation of models by nonlinear embedding and by sector nonlinearity, developed for one class of systems, can be easily extended to deal with the other class. Then, two measures, based on the notions of overboundedness and region of attraction estimates, are proposed in order to compare different models and choose which one can be considered the best one. Later, the problem of designing state-feedback controllers for LPV systems has been considered, providing two main contributions. First, robust LPV controllers that can guarantee some desired performances when applied to uncertain LPV systems are designed, by using a double-layer polytopic description that takes into account both the variability due to the varying parameter vector and the uncertainty. Then, the idea of designing the controller in such a way that the required performances are scheduled by the varying parameters is explored, which provides an elegant way to vary online the behavior of the closed-loop system. In both cases, the problem reduces to finding a solution to a finite number of linear matrix inequalities (LMIs), which can be done efficiently using the available solvers. In the area of fault tolerant control, the thesis first shows that the aforementioned double-layer polytopic framework can be used for FTC, in such a way that different strategies (passive, active and hybrid) are obtained depending on the amount of available information. Later, an FTC strategy for LPV systems that involves a reconfigured reference model and virtual actuators is developed. It is shown that by including the saturations in the reference model equations, it is possible to design a model reference FTC system that automatically retunes the reference states whenever the system is affected by saturation nonlinearities. In this way, a graceful performance degradation in presence of actuator saturations is incorporated in an elegant way. Finally, the problem of FTC of unstable LPV systems subject to actuator saturations is considered. In this case, the design of the virtual actuator is performed in such a way that the convergence of the state trajectory to zero is assured despite the saturations and the appearance of faults. Also, it is shown that it is possible to obtain some guarantees about the tolerated delay between the fault occurrence and its isolation, and that the nominal controller can be designed so as to maximize the tolerated delay.Aquesta tesi presenta diverses contribucions a l'estat de l'art del control per planificació del guany i del control tolerant a fallades (FTC). Pel que fa al control per planificació del guany, s'analitzen les connexions entre els paradigmes dels sistemes lineals a paràmetres variants en el temps (LPV) i de Takagi-Sugeno (TS). Es demostra que els mètodes per a la generació automàtica de models mitjançant encastament no lineal i mitjançant no linealitat sectorial, desenvolupats per una classe de sistemes, es poden estendre fàcilment per fer-los servir amb l'altra classe. Es proposen dues mesures basades en les nocions de sobrefitació i d'estimació de la regió d'atracció, per tal de comparar diferents models i triar quin d'ells pot ser considerat el millor. Després, es considera el problema de dissenyar controladors per realimentació d'estat per a sistemes LPV, proporcionant dues contribucions principals. En primer lloc, fent servir una descripció amb doble capa politòpica que té en compte tant la variabilitat deguda al vector de paràmetres variants i la deguda a la incertesa, es dissenyen controladors LPV robustos que puguin garantir unes especificacions desitjades quan s'apliquen a sistemes LPV incerts. En segon lloc, s'explora la idea de dissenyar el controlador de tal manera que les especificacions requerides siguin programades pels paràmetres variants. Això proporciona una manera elegant de variar en línia el comportament del sistema en llaç tancat. En tots dos casos, el problema es redueix a trobar una solució d'un nombre finit de desigualtats matricials lineals (LMIs), que es poden resoldre fent servir algorismes numèrics disponibles i molt eficients. En l'àrea del control tolerant a fallades, primerament la tesi mostra que la descripció amb doble capa politòpica abans esmentada es pot utilitzar per fer FTC, de tal manera que, en funció de la quantitat d'informació disponible, s'obtenen diferents estratègies (passiva, activa i híbrida). Després, es desenvolupa una estratègia de FTC per a sistemes LPV que fa servir un model de referència reconfigurat combinat amb la tècnica d'actuadors virtuals. Es mostra que mitjançant la inclusió de les saturacions en les equacions del model de referència, és possible dissenyar un sistema de control tolerant a fallades que resintonitza automàticament els estats de referència cada vegada que el sistema es veu afectat per les no linealitats de la saturació en els actuadors. D'aquesta manera s'incorpora una degradació elegant de les especificacions en presència de saturacions d'actuadors. Finalment, es considera el problema de FTC per sistemes LPV inestables afectats per saturacions d'actuadors. En aquest cas, es porta a terme el disseny de l'actuador virtual de tal manera que la convergència a zero de la trajectòria d'estat està assegurada tot i les saturacions i l'aparició de fallades. A més, es mostra que és possible obtenir garanties sobre el retard tolerat entre l'aparició d'una fallada i el seu aïllament, i que el controlador nominal es pot dissenyar maximitzant el retard tolerat