Set-membership approach and Kalman observer based on zonotopes for discrete-time descriptor systems

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper proposes a set-membership state estimator and a zonotopic Kalman observer for discrete-time descriptor systems. Both approaches are developed in a set-based context considering system disturbances, measurement noise, and unknown inputs. This set-membership state estimation approach determines the set of consistent states with the model and measurements by constructing a parameterized intersection zonotope. Two methods to minimize the size of this intersection zonotope are provided: one inspired by Kalman filtering and the other based on solving an optimization problem involving a series of linear matrix inequalities. Additionally, we propose a zonotopic Kalman observer for discrete-time descriptor systems. Moreover, the relationship between both approaches is discussed. In particular, it is proved that the zonotopic Kalman observer in the current estimation type is equivalent to the set-membership approach. Finally, a numerical example is used to illustrate and compare the effectiveness of the proposed approaches.Peer ReviewedPostprint (author's final draft

Zonotopic fault detection observer design for Takagi–Sugeno fuzzy systems

This paper considers zonotopic fault detection observer design in the finite-frequency domain for discrete-time Takagi–Sugeno fuzzy systems with unknown but bounded disturbances and measurement noise. We present a novel fault detection observer structure, which is more general than the commonly used Luenberger form. To make the generated residual sensitive to faults and robust against disturbances, we develop a finite-frequency fault detection observer based on generalised Kalman–Yakubovich–Popov lemma and P-radius criterion. The design conditions are expressed in terms of linear matrix inequalities. The major merit of the proposed method is that residual evaluation can be easily implemented via zonotopic approach. Numerical examples are conducted to demonstrate the proposed methodPeer ReviewedPostprint (author's final draft

Advances in state estimation, diagnosis and control of complex systems

This dissertation intends to provide theoretical and practical contributions on estimation, diagnosis and control of complex systems, especially in the mathematical form of descriptor systems. The research is motivated by real applications, such as water networks and power systems, which require a control system to provide a proper management able to take into account their specific features and operating limits in presence of uncertainties related to their operation and failures from component malfunctions. Such a control system is expected to provide an optimal operation to obtain efficient and reliable performance. State estimation is an essential tool, which can be used not only for fault diagnosis but also for the controller design. To achieve a satisfactory robust performance, set theory is chosen to build a general framework for descriptor systems subject to uncertainties. Under certain assumptions, these uncertainties are propagated and bounded by deterministic sets that can be explicitly characterized at each iteration step. Moreover, set-invariance characterizations for descriptor systems are also of interest to describe the steady performance, which can also be used for active mode detection. For the controller design for complex systems, new developments of economic model predictive control (EMPC) are studied taking into account the case of underlying periodic behaviors. The EMPC controller is designed to be recursively feasible even with sudden changes in the economic cost function and the closed-loop convergence is guaranteed. Besides, a robust technique is plugged into the EMPC controller design to maintain these closed-loop properties in presence of uncertainties. Engineering applications modeled as descriptor systems are presented to illustrate these control strategies. From the real applications, some additional difficulties are solved, such as using a two-layer control strategy to avoid binary variables in real-time optimizations and using nonlinear constraint relaxation to deal with nonlinear algebraic equations in the descriptor model. Furthermore, the fault-tolerant capability is also included in the controller design for descriptor systems by means of the designed virtual actuator and virtual sensor together with an observer-based delayed controller.Esta tesis propone contribuciones de carácter teórico y aplicado para la estimación del estado, el diagnóstico y el control óptimo de sistemas dinámicos complejos en particular, para los sistemas descriptores, incluyendo la capacidad de tolerancia a fallos. La motivación de la tesis proviene de aplicaciones reales, como redes de agua y sistemas de energía, cuya naturaleza crítica requiere necesariamente un sistema de control para una gestión capaz de tener en cuenta sus características específicas y límites operativos en presencia de incertidumbres relacionadas con su funcionamiento, así como fallos de funcionamiento de los componentes. El objetivo es conseguir controladores que mejoren tanto la eficiencia como la fiabilidad de dichos sistemas. La estimación del estado es una herramienta esencial que puede usarse no solo para el diagnóstico de fallos sino también para el diseño del control. Con este fin, se ha decidido utilizar metodologías intervalares, o basadas en conjuntos, para construir un marco general para los sistemas de descriptores sujetos a incertidumbres desconocidas pero acotadas. Estas incertidumbres se propagan y delimitan mediante conjuntos que se pueden caracterizar explícitamente en cada instante. Por otra parte, también se proponen caracterizaciones basadas en conjuntos invariantes para sistemas de descriptores que permiten describir comportamientos estacionarios y resultan útiles para la detección de modos activos. Se estudian también nuevos desarrollos del control predictivo económico basado en modelos (EMPC) para tener en cuenta posibles comportamientos periódicos en la variación de parámetros o en las perturbaciones que afectan a estos sistemas. Además, se demuestra que el control EMPC propuesto garantiza la factibilidad recursiva, incluso frente a cambios repentinos en la función de coste económico y se garantiza la convergencia en lazo cerrado. Por otra parte, se utilizan técnicas de control robusto pata garantizar que las estrategias de control predictivo económico mantengan las prestaciones en lazo cerrado, incluso en presencia de incertidumbre. Los desarrollos de la tesis se ilustran con casos de estudio realistas. Para algunas de aplicaciones reales, se resuelven dificultades adicionales, como el uso de una estrategia de control de dos niveles para evitar incluir variables binarias en la optimización y el uso de la relajación de restricciones no lineales para tratar las ecuaciones algebraicas no lineales en el modelo descriptor en las redes de agua. Finalmente, se incluye también una contribución al diseño de estrategias de control con tolerancia a fallos para sistemas descriptores

Distributed Set-Based Observers Using Diffusion Strategy

Distributed estimation is more robust against single points of failure and requires less communication overhead compared to the centralized version. Among distributed estimation techniques, set-based estimation has gained much attention as it provides estimation guarantees for safety-critical applications and copes with unknown but bounded uncertainties. We propose two distributed set-based observers using interval-based and set-membership approaches for a linear discrete-time dynamical system with bounded modeling and measurement uncertainties. Both algorithms utilize a new over-approximating zonotopes intersection step named the set-based diffusion step. We use the term diffusion since our intersection of zonotopes formula resembles the traditional diffusion step in the stochastic Kalman filter. Our new zonotopes intersection takes linear time. Our set-based diffusion step decreases the estimation errors and the size of estimated sets and can be seen as a lightweight approach to achieve partial consensus between the distributed estimated sets. Every node shares its measurement with its neighbor in the measurement update step. The neighbors intersect their estimated sets constituting our proposed set-based diffusion step. We represent sets as zonotopes since they compactly represent high-dimensional sets, and they are closed under linear mapping and Minkowski addition. The applicability of our algorithms is demonstrated by a localization example. All used data and code to recreate our findings are publicly availabl

Robust fault estimation based on zonotopic Kalman filter for discrete-time descriptor systems

This paper proposes a set-based approach for robust fault estimation of discrete-time descriptor systems. The considered descriptor systems are subject to unknown-but-bounded uncertainties (state disturbances and measurement noise) in predefined zonotopes and additive actuator faults. The zonotopic fault estimation filter for descriptor systems is built based on fault detectability indices and matrix to estimate fault magnitude in a deterministic set. The zonotopic fault estimation filter gain is designed in a parameterized form. Within a set-based framework, following the zonotopic Kalman filter, the optimal filter gain is computed by minimizing the size of the corresponding zonotopes to achieve robustness against uncertainties and the identification of occurred actuator faults. Besides, boundedness of the proposed zonotopic fault estimation is analyzed, which proves that the size of obtained fault estimation bounds is not growing in time. Finally, the simulation results with two application examples are provided to show the effectiveness of the proposed approach.Peer ReviewedPostprint (author's final draft

Zonotopic unknown input observer of discrete-time descriptor systems for state estimation and robust fault detection

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper studies a set-based unknown input observer based on zonotopes for discrete-time descriptor systems affected by uncertainties with application to state estimation and robust fault detection. In this paper, two types of uncertainties are considered: (i) disturbances and noise both bounded by zonotopes; (ii) unknown inputs that can be decoupled. In terms of different applications, the observer gain for state estimation is designed to minimize the effects of unknown-but-bounded disturbances and noise as well as state estimation errors. On the other hand, for robust fault detection, in addition to attenuating uncertainties, the designed observer gain is also expected to be sensitive to faults. To achieve this goal, we propose an iterative algorithm to design the fault detection gain. Finally, some illustrative results in an application example show the effectiveness of the proposed algorithms.Peer ReviewedPostprint (author's final draft

Robust fault detection and isolation based on zonotopic unknown input observers for discrete-time descriptor systems

In this paper, we propose a robust fault detection and isolation (FDI) strategy based on zonotopic unknown input observers (UIOs) for discrete-time descriptor linear time-varying (LTV) systems subject to uncertainties and additive actuator faults. System uncertainties including state disturbances and measurement noise are unknown but bounded by predefined zonotopes. The uncertain state estimations and constructed residuals for robust FDI are propagated in a sequence of zonotopes. Based on a defined performance criterion, the fault detection (FD) observer gain is designed to be robust against uncertainties and meanwhile sensitive to faults. The explicit computational method for the FD observer gain is derived. In addition to include fault isolation, a bank of zonotopic UIOs are employed. Finally, we apply the proposed method into two case studies to show its effectiveness.Peer ReviewedPostprint (published version

Distributed estimation techniques forcyber-physical systems

Nowadays, with the increasing use of wireless networks, embedded devices and agents with processing and sensing capabilities, the development of distributed estimation techniques has become vital to monitor important variables of the system that are not directly available. Numerous distributed estimation techniques have been proposed in the literature according to the model of the system, noises and disturbances. One of the main objectives of this thesis is to search all those works that deal with distributed estimation techniques applied to cyber-physical systems, system of systems and heterogeneous systems, through using systematic review methodology. Even though systematic reviews are not the common way to survey a topic in the control community, they provide a rigorous, robust and objective formula that should not be ignored. The presented systematic review incorporates and adapts the guidelines recommended in other disciplines to the field of automation and control and presents a brief description of the different phases that constitute a systematic review. Undertaking the systematic review many gaps were discovered: it deserves to be remarked that some estimators are not applied to cyber-physical systems, such as sliding mode observers or set-membership observers. Subsequently, one of these particular techniques was chosen, set-membership estimator, to develop new applications for cyber-physical systems. This introduces the other objectives of the thesis, i.e. to present two novel formulations of distributed set-membership estimators. Both estimators use a multi-hop decomposition, so the dynamics of the system is rewritten to present a cascaded implementation of the distributed set-membership observer, decoupling the influence of the non-observable modes to the observable ones. So each agent must find a different set for each sub-space, instead of a unique set for all the states. Two different approaches have been used to address the same problem, that is, to design a guaranteed distributed estimation method for linear full-coupled systems affected by bounded disturbances, to be implemented in a set of distributed agents that need to communicate and collaborate to achieve this goal

Set-based state estimation and fault diagnosis of linear discrete-time descriptor systems using constrained zonotopes

This paper presents new methods for set-valued state estimation and active fault diagnosis of linear descriptor systems. The algorithms are based on constrained zonotopes, a generalization of zonotopes capable of describing strongly asymmetric convex sets, while retaining the computational advantages of zonotopes. Additionally, unlike other set representations like intervals, zonotopes, ellipsoids, paralletopes, among others, linear static constraints on the state variables, typical of descriptor systems, can be directly incorporated in the mathematical description of constrained zonotopes. Therefore, the proposed methods lead to more accurate results in state estimation in comparison to existing methods based on the previous sets without requiring rank assumptions on the structure of the descriptor system and with a fair trade-off between accuracy and efficiency. These advantages are highlighted in two numerical examples.Comment: This paper was accepted and presented in the 1st IFAC Virtual World Congress, 202

Set-membership estimation for linear time-varying descriptor systems

International audienceThis paper considers the problem of set-membership estimation for discrete-time linear time-varying descriptor systems subject to unknown but bounded disturbance and noise. We propose a set-membership estimation method based on a descriptor system observer and a zonotopic estimator of the observer error bounds. The observer parameters are optimized in order to minimize the sizes of the zonotopes enclosing all admissible state trajectories. Finally, two simulation results are provided to demonstrate the effectiveness of the proposed method