Set-valued Observer Research Articles

Secure virtual coupling control of connected train platoons under cyber attacks

Virtual coupling of automated trains has emerged as a promising technology to enhance the operational efficiency and line capacity of railway transportation systems. However, the vulnerability of such systems to malicious cyber attacks poses a significant threat to their stability, reliability and applicability. This article addresses the challenge of ensuring secure virtual coupling control of connected train platoons subject to various cyber attacks, including false data injection attacks, denial-of-service attacks, and replay attacks. First, based on the low-dimensional and noisy sensor measurements, novel set-valued observers are designed for each follower train to estimate their inaccessible full longitudinal motion states despite uncertainties and unknown attacks and noises. Then, distributed secure attack-compensation-based virtual coupling control laws are developed to ensure the platoon's stability and resilience against cyber attacks. Additionally, formal stability analysis of the resultant estimation error system and platoon tracking error system is provided. Furthermore, two algorithms are presented to elaborate the offline observer and controller design process as well as their online implementation. Finally, based on the data of a realistic urban rail transit line, simulation experiments are conducted to validate the efficacy of the proposed theoretical findings.

A Novel Set-Valued Sensor Fault Diagnosis Method for Lithium-Ion Battery Packs in Electric Vehicles

Sensor fault diagnosis is of great significance to ensure safe battery operation. This paper proposes a novel sensor fault diagnosis method that achieves the simultaneous fault detection, fault source and type identification, and fault estimation in a comprehensive way. Specifically, a set-valued observer, featuring a state predictor and a state estimator, is first constructed and designed to guarantee the inclusion of the unavailable actual battery state due to unknown modeling errors and noises at every instant of time. Compared with the traditional observers, a distinct feature of the proposed one lies in that the calculated state predictions and estimations of the battery system at each time step are ellipsoidal sets in state space rather than single vectors. The boundedness of state prediction and estimation errors is formally proved, and the tractable design criteria for determining the real-time optimal prediction and estimation ellipsoids are also derived. As for diagnosis algorithm, fault detection is implemented based on the intersection between the prediction and estimation ellipsoids. Then, a two-layer Pearson correlation coefficient analysis mechanism is developed to identify the source and type of sensor faults. Another set-valued observer based on an augmented battery model is further designed to estimate the fault level. Finally, experimental studies of a battery cell under different sensor fault sources, types and values are elaborated to verify the effectiveness of the proposed method.

Exclusion tendency-based observer design framework for active fault diagnosis

This paper proposes a new closed-loop observer-based active fault diagnosis (AFD) framework using a bank of set-valued observers (SVOs). Each SVO is designed for one healthy or faulty system mode to obtain the corresponding output estimation set (OES). The principle of fault diagnosis is to check whether the considered OES includes the system output. Compared with the existing works that mainly make use of SVOs to obtain accurate estimation results, our proposed method optimizes the observer gains to deform OESs such that the exclusion tendency of the system output in each OES is maximized. With this idea, inconsistent system modes can be removed rapidly, and faults can be diagnosed. The design logic is formulated as a bi-level max–min problem, which is transformed into a linear problem by Lagrangian duality and solved efficiently. At the end of this paper, a quadruple-tank example is used to compare the results of our proposed method with an existing fault diagnosis method to illustrate the effectiveness of the proposed method.

Adaptive Filtering Algorithms for Set-Valued Observations—Symmetric Measurement Approach to Unlabeled and Anonymized Data

Adaptive Filtering Algorithms for Set-Valued Observations—Symmetric Measurement Approach to Unlabeled and Anonymized Data

KKL set-valued observers for non-observable systems

KKL observer design consists in transforming the system dynamics into a filter of the output, which admits a trivial observer, and left-inverting the transformation to recover an estimate of the state in the system coordinates. This left-inversion is typically guaranteed under a backward-distinguishability condition. In this paper, instead, we demonstrate how this KKL approach may also be applied without any such observability assumption. We show that there exist appropriate choices of the filter such that any filter solution asymptotically contains the full information about the state indistinguishable class, namely the set of points generating the same output. Then, we investigate the existence of a set-valued left-inverse allowing to estimate asymptotically this indistinguishable class, in the Hausdorff sens. We prove that the estimate tends to be included asymptotically in the indistinguishable classes of the limit points of the system solution. Finally, we provide a numerical example illustrating this convergence.

A Novel Set-valued Observer Based State Estimation Algorithm for Nonlinear Systems

A Novel Set-valued Observer Based State Estimation Algorithm for Nonlinear Systems

Observer-based asymptotic active fault diagnosis: A two-layer optimization framework

This paper proposes a novel asymptotic online robust active fault diagnosis (AFD) framework for discrete linear time-invariant (LTI) systems using a bank of set-valued observers (SVO), each SVO designed to match a healthy/faulty actuator mode. Different from the existing set-based AFD methods, the proposed AFD method designs an input at each time instant by solving a two-layer optimization problem. At each time instant, an input is designed such that the output sets estimated by the SVOs at the next time instant keep away from each other as far as possible. With this idea, faults can be diagnosed after a sufficiently long period by designing and injecting inputs into the system online. The key point of the proposed AFD method is to establish a logic describing the process keeping a group of output estimation sets asymptotically away from each other as far as possible, which is done by formulating a two-layer optimization problem to simultaneously optimize the center distances and sizes of output estimation sets. At the end, a case study is used to illustrate the effectiveness of the proposed method.

Generalized set-theoretic interval observer using element-wise nonnegativity transformation

This paper proposes a generalized set-theoretic interval observer (GSIO) for robust state estimation of discrete linear time-invariant (LTI) systems with bounded uncertainties. Including the previous set-theoretic interval observer (SIO) as a special case, the proposed GSIO only requires to transform the system matrix of the first subsystem to be nonnegative. Based on this idea, a nonsmooth H∞ optimization method is used to design a nonnegative subsystem matrix for the first subsystem based on an element-wise nonnegativity transformation. The advantage of this method is that it can design a nonnegative subsystem matrix by transforming each of its elements to be nonnegative, which does not impact any structural constraints on matrix elements. This implies that more extra design degrees of freedom could be obtained from the coordinate transformation and observer gain matrices, which can optimize state estimation performance of GSIO (e.g., suppress the effects of disturbances and noises). In this way, we could not only achieve an expected balance between robust state estimation conservatism and computational complexity but also better robust state estimation performance. Besides, under the framework of the proposed GSIO, this paper further proves its existence, gives its stability condition and compares it with the set-valued observer (SVO) and the interval observer (IO) in state estimation conservatism, which together forms a complete theoretic system for the proposed GSIO. At the end of this paper, numerical examples are used to illustrate the effectiveness of the GSIO.

Interval set‐membership estimation for continuous linear systems

SummaryThis article proposes a mixed interval set‐membership estimation (ISME) method for continuous linear time‐invariant (LTI) systems by combining the positive system theory and the set theory. The proposed ISME method gives a new mixed interval‐set estimation framework for continuous LTI systems, whose benefit consists in that it has potential to achieve a balance of computational complexity and robust state estimation conservatism with respect to the interval observer (IO) and the set‐valued observer (SVO) for continuous LTI systems. Particularly, the proposed ISME method first uses a coordinate transformation such that the original system is transformed into an equivalent system. Second, the equivalent system is partitioned into two subsystems, where the first subsystem has a Meztler and Hurwitz subsystem matrix and then an IO is designed for the first subsystem based on the positive system theory. Because it is not guaranteed that the second subsystem also has a Meztler and Hurwitz subsystem matrix, a zonotopic SVO is further designed for the second subsystem based on the set theory. Consequently, an integration of the two steps above provides the whole SE results for the original system. At the end of this article, an example is used to illustrate the effectiveness of the proposed ISME method.

A Novel Set-Theoretic Interval Observer for Discrete Linear Time-Invariant Systems

This article proposes a novel robust state estimation method for discrete linear time-invariant systems by using the positive system theory and the set theory to overcome the disadvantages of interval observers and set-valued observers to finally achieve an expected balance between computational complexity and robust state estimation conservatism. The proposed method first transforms the system into an equivalent system with its system matrix in the canonical companion form. Then, based on this canonical companion form, we partition the equivalent system into two subsystems, where the first subsystem owns a nonnegative subsystem matrix and the second subsystem has a subsystem matrix including possible negative elements of the system matrix of the equivalent system. In this way, the design of interval observers is divided into two steps. The first step is to design an interval observer for the first subsystem based on the positive system theory. The second step is to design a zonotopic set-valued observer for the second subsystem based on the set theory. Consequently, a so-called set-theoretic interval observer (SIO) of the whole system can be synthesized by integrating the interval estimations and set-valued estimations of the two subsystems together. At the end, a numerical system is used to illustrate the effectiveness of the proposed SIO.

Fault Detection of WECSs With a Delayed Input and an Unknown Part Based on SVO

This paper studies an improved set-valued observer (SVO) for a complex model of a wind energy conversion system (WECS). To fit reality, an unknown function is used to describe the aerodynamic model, and a delay of the input signal is added to simulate the actual situation. The conservative SVO estimates the state error convergence within an adjustable interval. The extended Lipschitz condition, set-induced theory, and uniformly ultimately boundedness guarantees the convergence of errors. Based on the SVO, a fault detection system and corresponding fault detection strategies for common faults of WECSs are proposed. By adjusting the gain matrix L, the SVO shows excellent state tracking performance. The simulation results show that the proposed fault detection system can provide rapid and accurate fault detection.

FIR system identification with set-valued and precise observations from multiple sensors

This paper considers the system identification problem for FIR (finite impulse response) systems with set-valued and precise observations received from multiple sensors. A fusion estimation algorithm based on some suitable identification algorithms for different types of observations is proposed. In particular, least square method is chosen for FIR systems with precise observations, while empirical measure method and EM algorithm are chosen for FIR systems with set-valued observations in the cases of periodic and general system inputs, respectively. Then, the quasi-convex combination estimator (QCCE) fusing the two different estimators by a linear combination with appropriate weights is constructed. Furthermore, the convergence properties are theoretically analyzed in terms of strong consistency and asymptotic efficiency. The fused estimator QCCE is proved to achieve the Cramer-Rao (CR) lower bound asymptotically under periodic inputs. Extensive numerical simulations validate the superiority of the fusion estimation algorithm under both periodic and general inputs.

Design of set-valued observer for a particular situation of WECS: commensurate time delays

With the rapid development of industrial automation, the normal operation of large industrial systems under unattended condition has become the research target. Considering the time delays occurred in the long communication channels, a set-valued observer (SVO) is utilized for the simplified wind energy conversion system to track and estimate an interval of states, which covering all allowed operating conditions. For the commensuration of the communication time delays, a delay operator is introduced to transform the system into an unstable linear-like system. Based on the SVO, a state feedback control law is studied by using Sylvester pole placement. The simulation results demonstrate that the states estimation errors of SVO are generally bounded to the given C-set, and superior to the Kalman filter.

Self-Triggered and Event-Triggered Set-Valued Observers

This paper addresses the problem of reducing the required network load and computational power for the implementation of Set-Valued Observers (SVOs) in Networked Control System (NCS). Event- and self-triggered strategies for NCS, modeled as discrete-time Linear Parameter-Varying (LPV) systems, are studied by showing how the triggering condition can be selected. The methodology provided can be applied to determine when it is required to perform a full (“classical”) computation of the SVOs, while providing low-complexity state overbounds for the remaining time, at the expenses of temporarily reducing the estimation accuracy. As part of the procedure, an algorithm is provided to compute a suitable centrally symmetric polytope that allows to find hyper-parallelepiped and ellipsoidal overbounds to the exact set-valued state estimates calculated by the SVOs. By construction, the proposed triggering techniques do not influence the convergence of the SVOs, as at some subsequent time instants, set-valued estimates are computed using the conventional SVOs. Results are provided for the triggering frequency of the self-triggered strategy and two interesting cases: distributed systems when the dynamics of all nodes are equal up to a reordering of the matrix; and when the probability distribution of the parameters influencing the dynamics is known. The performance of the proposed algorithm is demonstrated in simulation by using a time-sensitive example.

Fault detection for LPV systems using Set-Valued Observers: A coprime factorization approach

This paper addresses the problem of fault detection for linear parameter-varying systems in the presence of measurement noise and exogenous disturbances using Set-Valued Observers (SVOs). The applicability of current methods is limited in the sense that, to increase accuracy, the detection requires a large number of past measurements and the boundedness of the set-valued estimates is only guaranteed for stable systems. In order to widen the class of systems to be modeled and also to reduce the associated computational cost, the aforementioned issues must be addressed. A solution involving left-coprime factorization and deadbeat observers is proposed that reduces the required number of past measurements without compromising accuracy and allowing the design of SVOs for fault detection of unstable systems by using the resulting coprime factorization stable subsystems. The algorithm is shown to produce bounded set-valued estimates and an example is provided. Performance is assessed through simulations, illustrating, in particular that small-magnitude faults (compared to exogenous disturbances) can be detected under mild assumptions.

Stochastic and deterministic fault detection for randomized gossip algorithms

This paper addresses the problem of detecting faults in linear randomized gossip algorithms, where the selection of the dynamics matrix is stochastic. A fault is a disturbance signal injected by an attacker to corrupt the states of the nodes. We propose the use of Set-Valued Observers (SVOs) to detect if the state observations are compatible with the system dynamics for the worst case in a deterministic setting. The concept of Stochastic Set-Valued Observers (SSVOs) is also introduced to construct a set that is guaranteed to contain all possible states with, at least, a pre-specified desired probability. The proposed algorithm is stable in the sense that it requires a finite number of vertices to represent polytopic sets and it allows for the computation of the largest magnitude of the disturbance that an attacker can inject in the network without being detected. Results are presented to reduce the computational cost of this approach and, in particular, by considering only local information and representing the remainder of the network as a disturbance. The case of a consensus algorithm is discussed leading to the conclusion that, by using the proposed SVOs, finite-time consensus is achieved in non-faulty environments. A novel algorithm is proposed that produces less conservative set-valued state estimates by having nodes exchanging local estimates. The algorithm inherits all the previous properties and also enables finite-time consensus computation regardless of the value of the horizon.

Closed-loop input design for guaranteed fault diagnosis using set-valued observers

Active fault diagnosis (AFD) can be used to improve the diagnosability of faults by injecting a suitably designed input into a process. When faults are described as discrete switches between linear systems with uncertainties bounded within zonotopes, an optimal open-loop input guaranteeing diagnosis within a specified time horizon can be computed efficiently by solving a Mixed Integer Quadratic Program (MIQP). In this article, the constrained zonotope (CZ) set representation recently developed by the authors is used to extend the MIQP approach to general polytopic uncertainties without sacrificing efficiency. Next, this approach is combined with a CZ-based set-valued observer in a moving horizon framework to achieve rigorous closed-loop AFD. This method can greatly accelerate diagnosis relative to the open-loop approach, but requires online optimization. To reduce the online cost, we propose a method for solving the open-loop problem explicitly with respect to past measurements and inputs, which requires only observability of the nominal and faulty models. The effectiveness of the proposed approaches is demonstrated through several numerical examples.

Localization from semantic observations via the matrix permanent

Most approaches to robot localization rely on low-level geometric features such as points, lines, and planes. In this paper, we use object recognition to obtain semantic information from the robot’s sensors and consider the task of localizing the robot within a prior map of landmarks, which are annotated with semantic labels. As object recognition algorithms miss detections and produce false alarms, correct data association between the detections and the landmarks on the map is central to the semantic localization problem. Instead of the traditional vector-based representation, we propose a sensor model, which encodes the semantic observations via random finite sets and enables a unified treatment of missed detections, false alarms, and data association. Our second contribution is to reduce the problem of computing the likelihood of a set-valued observation to the problem of computing a matrix permanent. It is this crucial transformation that allows us to solve the semantic localization problem with a polynomial-time approximation to the set-based Bayes filter. Finally, we address the active semantic localization problem, in which the observer’s trajectory is planned in order to improve the accuracy and efficiency of the localization process. The performance of our approach is demonstrated in simulation and in real environments using deformable-part-model-based object detectors. Robust global localization from semantic observations is demonstrated for a mobile robot, for the Project Tango phone, and on the KITTI visual odometry dataset. Comparisons are made with the traditional lidar-based geometric Monte Carlo localization.

Fault Detection and Isolation in Inertial Measurement Units Based on Bounding Sets

This work addresses the problem of Fault Detection and Isolation (FDI) for navigation systems equipped with sensors providing inertial measurements and vector observations. Assuming upper bounded sensor noise, two strategies are proposed: i) the first one takes advantage of existing hardware redundancy, requiring at least five sensor measurements to isolate faults; ii) the second approach exploits the analytical redundancy between the angular velocity measurements and the vector observations, by resorting to set-valued observers (SVOs). Necessary and sufficient conditions on the magnitude of the faults are provided, in order to guarantee successful detection and isolation, when hardware redundancy is available. Due to the set-based construction of the methods, none of the solutions generates false detections and no decision threshold is required. Using a simulation scenario, the proposed strategies are compared with two alternatives available in the literature.

Fault‐tolerant control of an air heating fan using set‐valued observers: An experimental evaluation

SummaryThis paper proposes a multiple‐model solution to the problem of controlling an air heating fan subject to faults. These faults are modeled by means of an abnormal unknown airflow input rate which the nominal controller is not designed for. Moreover, the average temperature of the air flowing through the system, which can be seen as an offset on the corresponding dynamics, is (slowly) time‐varying and highly dependent on the ambient temperature. The fault‐tolerant control (FTC) method adopted makes use of set‐valued observers (SVOs) to invalidate possible models of the system. Unlike classical fault detection, this approach does not rely on residuals to detect abnormal system operation. This fact allows to reduce the conservatism of the solution and enables a straightforward design from the faulty and nominal models of the plant. Moreover, the absolute distinguishability concept is used to derive input signals that bolster the detection of faults. Although SVOs require heavy real‐time calculations that hinder its implementability in systems with low computational power, it is shown that the architecture of the FTC strategy proposed is highly parallelizable and, thus, may take advantage of standard multi‐core processing units. Experimental results are presented. Copyright © 2015 John Wiley & Sons, Ltd.