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Defense against cyberattacks is an emerging topic related to fault-tolerant control. In order to avoid difficult mathematical modeling, model-free control (MFC) is suggested as an alternative to classical control. For illustration purpose a Load Frequency Control of multi-areas power network is considered. In the simulations, load altering attacks and Denial of Service (DoS) in the communication network are applied to the system. Our aim is to compare the impact of cyberattacks on control loops closed via respectively a classical controller in such situations and a model-free one. Computer experiments show impressive results with MFC., Comment: 5th International Conference on Control and Fault-Tolerant Systems, Saint-Rapha\"{e}l, France, September 29th - October 1st, 2021

This paper presents an active Fault-Tolerant Control (FTC) scheme for the temperature regulation of buildings subject to actuator faults. An optimal control strategy based on a regularized predictive economic criterion is proposed to determine the remedial actions to be taken when actuator faults are diagnosed. These corrective actions can be used either to generate a less “aggressive” control scenario by minimizing the number of on/off cycles of the faulty actuator or to reduce the number of actuators used at any time. One of the interests of this strategy is to prevent simple faults from developing into serious failures for actuators particularly vulnerable to repeated on/off cycles (in particular heat pump compressors). To account for the “aggressiveness” of a control scenario, the penalty term associated with the regularized economic criterion is defined using the parsimony property of $\ell_{\varepsilon}$ -norm ( $\varepsilon$ small).

Teams of networked autonomous agents have been used in a number of applications, such as mobile sensor networks and intelligent transportation systems. However, in such systems, the effect of faults and errors in one or more of the sub-systems can easily spread throughout the network, quickly degrading the performance of the entire system. In consensus-driven dynamics, the effects of faults are particularly relevant because of the presence of unconstrained rigid modes in the transfer function of the system. Here, we propose a two-stage technique for the identification and accommodation of a biased-measurements agent, in a network of mobile robots with time invariant interaction topology. We assume these interactions to only take place in the form of relative position measurements. A fault identification filter deployed on a single observer agent is used to estimate a single fault occurring anywhere in the network. Once the fault is detected, an optimal leader-based accommodation strategy is initiated. Results are presented by means of numerical simulations and robot experiments., 7 pages, 7 figures, conference

In computing science, resilience is the ability of system or network architecture to recover normal operation after a brutal crash. When malicious cyber act on control signals of a Networked Control System (NCS) is designed to remain undetectable from passive model-based Fault Detection and Isolation (FDI) schemes, we show that the unobservable consequence on the state variable of the plant becomes brutally observable after the disappearance of the damaging action. In order to quickly recover the nominal behavior of the Linear Quadratic Gaussian (LQG) controller, a resilient LQG controller is obtained from an active version of the Generalized Likelihood Ratio (GLR) test designed to detect the disappearance of the malicious act and to increase the tracking ability of the Kalman filter at detection time.

In this chapter, we shall present the solution to the problem of Fault-tolerant Control (FTC) by taking the behavioral system theoretic viewpoint.

This chapter reviews the current and past status of the progress made in the area of FTC research. Most of the FTC techniques depends on the type of fault occurred into the system, where the FD plays a significant role. So, the chapter starts with classifying the types of faults followed by the discussion on fault diagnosis and the analysis of various FTC schemes.

In this chapter, we introduce basic concepts of the behavioral approach that put the foundation to address active FTC approaches in this book. We shall see here how the behaviors are described for a dynamical system. The elementary properties (for example linearity, time/shift invariance) associated with a dynamical system are discussed taking this behavioral point of view. As we mentioned (very briefly) in the last chapter, the real essence of this approach lies in its representation-free description. However, for the brevity of explanation, we shall often utilize kernel representations. These representations consider the polynomial matrices for describing a system, which are considered more general way of representing dynamical systems since these representations can easily be translated into either external-type or internal-type of representations by Chen (Linear systsem theory and design. Oxford University Press, New York, 1999, [18, Sects. 6–7]), Polderman and Willems (Introduction to mathematical systems theory: a behavioral approach. Springer, Berlin, 1997, [84]). To motivate further use of behavioral framework, we have considered various follow-up examples.

The prime aim of a fault-tolerant control system is to maintain the system performance, defined in terms of the desired behavior, at anytime, i.e., even after an occurrence of a fault. In previous two chapters, we presented novel real-time controller reconfiguration mechanisms to deal with the aforementioned issue.

International audience; An investigation and analysis based on TRNSYS simulation are reported on the performance of a renewable energy system compound of two parts. In the first part (production) we consider a PV/wind hybrid system with battery storage. And in the second part (consumption) we consider a heat pump as HVAC system to insure the thermal comfort and the quality of the air for the occupants in a meeting room. Then based on Physical knowledge and data generated by TRNSYS, explicit models are obtained for all the equipments.

International audience; State estimation of stochastic discrete-time linear systems subject to unknown inputs or constant biases has been widely studied but no work has been dedicated to the case where a disturbance switches between unknown input and constant bias. We show that such disturbance can affect a networked control system subject to deception attacks and data losses on the control signals transmitted by the controller to the plant. This paper proposes to estimate the switching disturbance from an augmented state version of the intermittent unknown input Kalman filter recently developed by the authors. Sufficient stochastic stability conditions are established when the arrival binary sequence of data losses follows a Bernoulli randomprocess.

This paper deals with a multi-objective fault detection observer design problem in time and frequency domain for time invariant systems. Firstly, to improve the abilities of designed observer, different design criteria are proposed, which are evaluated by some suitable performance indices in time and frequency domain. Details about the relationships among different criteria are analyzed for two cases when the fault appears and disappears. With selected residual evaluation function and threshold, a formula with an envelope in time domain to realize fast fault detection is proposed for some typical faults. The designed observer considers the trade-off between fast transients of the residual for specified faults and the traditional criterion H – /H ∞ for general faults and disturbances. Compared with H – /H ∞ frequency design method, the effectiveness of the proposed method is demonstrated by the numerical simulation with a vehicle lateral dynamics system.

International audience; Modeling, analysis and control of networked control systems (NCS) have recently emerged as topics of significant interest to the control community. The defining feature of any NCS is that information is exchanged using digital band-limited serial communication channel among systems components and usually shared by other feedback control loops. Conventional control theory with many ideal assumptions, such as synchronized control and non-delayed sensing and actuation must be revisited so that the limitations on communication capabilities within the control design framework can be integrated. Recent achievements showed that it is possible to solve communication problems and control problems simultaneously, thus contributing to a more efficient NCS design. This paper aims at giving an overview of Fault Diagnosis methods dealing with enhancement of robustness against network induced effects and introducing co-design approaches making it possible to solve communication problems and control problems simultaneously, thus contributing to a more efficient design.

A mixed active and passive GLR test for a fault tolerant control system This paper presents an adaptive Generalized Likelihood Ratio (GLR) test for multiple Faults Detection and Isolation (FDI) in stochastic linear dynamic systems. Based on the work of Willsky and Jones (1976), we propose a modified generalized likelihood ratio test, allowing detection, isolation and estimation of multiple sequential faults. Our contribution aims to maximise the good decision rate of fault detection using another updating strategy. This is based on a reference model updated on-line after each detection and isolation of one fault. To reduce the computational requirement, the passive GLR test will be derived from a state estimator designed on a fixed reference model directly sensitive to system changes. We will show that active and passive GLR tests will be mixed and give interesting results compared with the GLR of Willsky and Jones (1976), and that they can be easily integrated in a reconfigurable Fault-Tolerant Control System (FTCS) to asymptotically recover the nominal system performances of the jump-free system.

Model-free reconfiguration mechanism for fault toleranceThe problem of fault tolerant control is studied from the behavioral point of view. In this mathematical framework, the concept of interconnection among the variables describing the system is a key point. The problem is that the behavior we intend to control is not known. Therefore, we are interested in designing a fault accommodation scheme for an unknown behavior through an appropriate behavioral interconnection. Here we deal simply with the trajectories that are generated by the system in real time. These trajectories determine the behavior of a system in various (faulty/healthy) modes. Based on the desired interconnected behavior, only the trajectories that obey certain laws are selected. These laws, representing the desired behavior, can indeed be achieved by a regular interconnection. Thus, when the trajectories do not belong to a certain desired behavior, it is considered to be due to the occurrence of a fault in the system. The vantage point is that the fault tolerant control problem now becomes completely a model-free scheme. Moreover, no explicit fault diagnosis module is required in our approach. The proposed fault tolerance mechanism is illustrated on an aircraft during the landing phase.

International audience; This paper presents a distributed state filtering strategy for large scale network controlled systems. The distributed state filtering scheme is obtained by considering the interconnection signals between subsystems as intermittent unknown inputs and by applying on each subsystem a modified version of the unknown input Kalman filter restricted, in its original version, to the treatment of permanent unknown inputs. The stochastic stability conditions of the filter are established from an upper bound of the mean prediction error covariance matrix and independent Bernoulli processes for the data exchange rates between subsystems.

In this paper, a restricted diagonal detection filter for multiple fault detection and isolation in linear discrete-time stochastic systems is presented. The proposed filter makes it possible to generate a minimum variance white innovation sequence having directional properties in response to each fault. For detection and isolation of multiple impulsive faults, a bank of generalized likelihood ratio (GLR) tests is applied to the innovation sequence generated by the detection filter. It is shown that this strategy extends the GLR test of Willsky and Jones (IEEE Trans. Automat. Control 1976; 21(1):108112) to the case of multiple impulsive faults appearing simultaneously. Copyright © 2010 John Wiley & Sons, Ltd.

International audience; This paper proposes a new method for actuator/sensor faults detection, isolation and magnitude estimation. By establishing new theoretical results relating to fault diagnosis and by defining a specific application domain, the research work described in this paper aims at developing a model-based fault detection filter combined with a conventional Luenberger observer. Based on an appropriate gain, the developed technique enables one to supervise a system through an accurate bank of residuals within a generalized observer scheme. The approach is evaluated by diagnosing a series of actuator/sensor faults in a speed control loop of a hot rolling mill used to carry out multi-pass forward–forward or forward–reverse rolling. The performance of the technique is illustrated via a wide range of simulations in the ‘faulty' case and a practical experiment in the ‘fault-free' case using a real rolling mill.

In this paper, the fault tolerant control problem is addressed in a networked framework. An isolation filter together with a fault compensation mechanism are proposed for FDI. Several design procedures are studied. First, in the centralized architecture, the inputs and outputs information used for FDI are collected and processed in a central node. Considering that sub-systems exchange information over the network, a decentralized architecture is then proposed for FDI. The autonomous capability of the decentralized design is also considered. But, whatever the design architecture is, fault detectability and isolability, are dependant on the available information. A method based on system structural analysis is then proposed,to provide fault detectability and fault isolability conditions, thus making possible fault tolerant control. Finally, we give an algorithm which allows to distribute a system into subsystems such that the problem of autonomous FDI is solvable for each of these subsystems.

Control reconfigurability, from the fault tolerance point of view, defines the capacity of the control system to allow restoration of performance in the presence of faults. For its evaluation it is required to calculate the controllability gramian. This calculation is usually carried on before control system be implemented, i.e., off-line. Instead, the on-line evaluation of the control reconfigurability from input/output data could be useful for the fault tolerance redesign. The contribution of the paper is to propose a way of calculating on-line the controllability gramian from input/output data in order to evaluate an index based on the control reconfigurability. The paper aims to estimate the reconfigurability for a linear system under partial loss of control effectiveness. The controllability gramian could be computed in direct form from the initial condition response. For practical aspects, an identification algorithm is proposed in order to compute the system controllability gramian, independently of initial conditions. An example is presented to illustrate the method.

International audience; This paper deals with static output feedback control of a class of discrete-time Networked Control Systems (NCSs) subject to random failures and random delays. The different random processes are modeled as Markovian chains, and the resulting closed-loop system belongs to the class of discrete-time Markovian Jump Linear Systems (MJLS). The main contribution is to formulate conditions for multi-performance design related to this class of stochastic hybrid systems. The specifications and objectives under consideration include stochastic stability, $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ performances. Results are formulated as matrix inequalities. A numerical algorithm is provided and its running is illustrated on a numerical example.

International audience; This paper addresses the problem of fault tolerant control over communication networks which induce time-varying delays in bounded intervals. The problem is formulated in a discrete-time setting by modeling the controlled plant which may be subject to failures as an uncertain discrete-time process connected to a discrete-time controller via a digital communication network. A procedure is presented for the design of a bank of fault-tolerant controllers capable to stabilize and guarantee an H2-performance bound for all faulty plant modes in the presence of network-induced time-varying delays.

Nous proposons dans cet article une nouvelle approche de synthèse d'un filtre de détection basée sur l'inversion du système. La méthode proposée présente une solution au problème de diagnostic multi-défaut dans le cas des systèmes linéaires discrets stochastiques. Basé sur une technique d'inversion du système, le filtre de détection est conçu sous une condition d'existence moins restrictive que celles rencontrées dans le cas des filtres classiques. Cette condition correspondra à la condition d'existence d'un inverse à gauche stable. Le calcul de l'inverse à gauche du système sera effectué par l'intermédiaire de la matrice d'interaction du système décrivant la structure des zéros infinis du système. La structure des zéros à l'infini ne sera plus contrainte de respecter une structure diagonale et l'estimation à temps minimal des amplitudes de défauts sera à minimum de variance par l'introduction d'un degré de liberté supplémentaire sur le gain de sortie du filtre.

International audience; In this paper, static output feedback stochastic stabilization and disturbance attenuation issues for a class of discrete-time Networked Control Systems (NCSs) subject to random failures and random delays are addressed. The different random processes are modeled as Markovian chains, and the resulting closed-loop system belongs to the class of discrete-time Markovian Jump Linear Systems (MJLS). Results are formulated as matrix inequalities. A numerical algorithm based on nonconvex optimization is provided and its running is illustrated on a classical example from literature.

The effect of network-induced delays introduced into the control loop is modelled as time-varying disturbance. A fault isolation filter (FIF) for fault detection of multiple input multiple output (MIMO) networked control systems (NCSs) with multiple faults is then parameterized. Directional residual generations decoupling from the disturbance ensure the treatment of multiple faults appearing simultaneously or sequentially. The remaining degrees of freedom in the design of the filter's gain are used to satisfy an H ∞ disturbance attenuation and poles assignment constraints in the frame of Markov jump linear systems (MJLS). The sufficient existence conditions of the free parameters are formulated as a convex optimization problem over a couple of linear matrix inequalities (LMIs). An illustrative example is given to show the efficiency of the proposed method for NCS.

International audience; In this paper, an Active Fault Tolerant Control System dedicated to the speed control of a hot roughing rolling mill is designed. Actuator and/or sensors faults are considered to occur on the hot roughing rolling mill which is used to carry out multi-pass forward-forward or forward-reverse rolling. The main contribution concerns the design of an accurate bank of filter-observers tandem able to isolate and estimate both the sensor and the actuator faults while preserving the system's integrity in terms of structure and performance. The effectiveness and performances of the proposed approach are illustrated via a computer simulation platform.

This paper provides a tutorial overview of a number of aspects and approaches to Control over the Network for Network Control Systems (NCS) that are likely to lead to good fault-tolerant control properties, subject to network faults. In order to analyze and derive the best strategies for fault tolerant NCS, it is initially assumed that the network communication bandwidth is infinite. This gives a simpler way to map the NCS structure in terms of computing nodes and control subsystems/components. Two Fault-tolerant NCS architectures have been described, analyzed and compared with view to demonstrating that the classical concepts of Fault-tolerant Control (FTC), namely of active and passive FTC can be related to equivalent (although more complex) concepts in NCS. The study confirms that the de-centralized approach to faulttolerant control of NCS suffers from a difficult challenge as to how to compensate for fault effects occurring throughout the NCS. On the other hand, the distributed hierarchical structure, requires a coordination function which is able to manage (a) the local control task, (b) the compensation of faults, and (c) the network reconfiguration, if required, subject to significant network subsystem faults.

In this paper, the problem of Fault diagnosis is addressed in a particular settings: Autonomous control systems (ACS) sharing a single communication channel (WIFI for example) are considered. The problem is to optimize the sampling rate for the exchange of information between the ACS and the FDI systems in order to have efficient fault diagnosis. In this setting, reducing the number of times that the FDI systems is activated implies a reduction in transmissions and thus a reduction in energy expenditures. The aim is to generate residual signals which, in the fault free case, are supposed to be identical to zero.

This paper proposes a method for actuator/sensors fault diagnosis in a speed control loop of a hot roughing rolling mill used to carry out multi-pass forward-forward or forward-reverse rolling. According to fault diagnosis theory and the specific application domain, the main objective of the work is to develop a model-based fault detection filter combined with a conventional Luenberger observer. Based on an appropriate gain, the developed technique enables to supervise the hot roughing rolling mill through an accurate bank of residuals within the Generalized Observer Scheme. The effectiveness and performances of the technique are illustrated via a computer simulation platform.

In this paper, we develop a Fault Tolerant Control (FTC) strategy designed to preserve closed-loop stability in spite of actuator failures. The main contribution is the design of a linear output feedback control function for a class of discrete-time switched linear systems with explicit modelling of multiple actuator failures. Assuming an on-line and real-time Fault Detection and Isolation (FDI) scheme such that, each time there is a change in status of one or more of the p many actuators, the FDI scheme will update a p-length vector of real parameters in the interval [0,1] recording the degree of loss of effectiveness (with 0 for normal operation and 1 for total failure), we can perform controller redesign on-line using LMI techniques. We conclude the paper with an example illustrating the performance of our proposed FTC strategy.

This paper deals with the description of a design procedure for hierarchical Fault Tolerant Control (FTC) of large, distributed system and explains the implementation of such procedure into ControlBuild software tool. Following a functional perspective, a procedure for the modular design of the diagnostic and reconfiguration algorithms which run at different levels of the hierarchy is presented. The whole procedure is applied to an hydraulic benchmark system and implemented into ControlBuild.

This paper proposes a method for actuator fault diagnosis in a speed control loop of a hot roughing rolling mill used to carry out multi-pass forward-forward or forward-reverse rolling. The main objectives of the work is to first improve an actuator fault detection and fault magnitude estimation problem for a particular system and second to estimate non measurable variables in order to provide suitable information to the monitoring system. According to fault diagnosis theory and the specific application domain, a model-based fault detection filter combined with a conventional Luenberger observer is proposed. Based on an appropriate gain, the developed technique enables to supervise the hot roughing rolling mill. The effectiveness and performances of the technique are illustrated on a computer simulation platform.

In this paper, a new Fault-Tolerant Control System strategy is presented for linear dynamic systems. The detection and isolation of each jump is achieved by the Kalman filter designed under a dead beat constraint allowing the maximum adaptability of the fault magnitude estimation in the case of abrupt change. By using the state estimation corrected by the estimation of jumps we will describe a integrated Fault Tolerant Control via a particular LQG approach.

This paper deals with the right-hand side observability of a linear function of the state and the unknown inputs for linear systems in descriptor form. For such systems, necessary and sufficient conditions ensuring the ability of reconstructing any given functional of the state and unknown inputs are determined by means of geometrical tools like subspace sequences.

The aim of Fault Tolerant Control (FTC) is to preserve the ability of the system to reach performances as close as possible to those which were initially assigned to it. The main goal of this paper consists in the development of a FTC strategy, based on both reliability and life cost of components. Once a fault has been detected and isolated and when it is not possible to reach the nominal performances of the system, the reconfiguration task proposed in this paper needs to find all possible structures of system that preserve pre-specified performances, calculate the system reliabilities and costs for all structures and finally search the optimal structure that has a highest reliability and/or a lowest cost. The proposed approach is illustrated through simulations considering a heating system benchmark used in the Intelligent Fault Tolerant Control in Integrated Systems European project (IFATIS EU-IST-2001-32122).

The problem of fault detection and isolation in a nonlinear framework is addressed. A systematic method that ensures the synthesis of asymptotic nonlinear filters is proposed for residual generation. In order to show the effectiveness of the proposed approach, each theoretical step is illustrated by means of simulations. The example of a three-tank system is then experimentally considered to verify the validity of our approach.

In this paper, we address the dynamic output feedback control problem of continuous time active fault tolerant control system with Markovian parameters (AFTCSMP). We will first derive a necessary and sufficient condition for the exponential stability in the mean square of the AFTCSMP under a dynamic output feedback control, in terms of coupled matrix inequalities, and then we will give an LMI (Linear Matrix Inequalities) characterization of dynamical compensators that stabilize the closed-loop system in the mean square sense.

An approach for Fault Tolerant Control dealing with critical fault is proposed in this paper. The aim of Fault Tolerant Control is to preserve after fault occurrence the ability for the degraded system to reach performances as close as possible to those which were initially assigned to the system. After fault detection and isolation, control reconfiguration is a usual way to recover the nominal performances. The reconfiguration strategy proposed here is based on the reliability analysis of the system components. This strategy relies on the existence of system reconfigurability properties related to different set of resources according to the selected structure. The effectiveness and performances of the method are validated in simulation considering a heating system benchmark used in the Intelligent Fault Tolerant Control in Integrated Systems European project (IFATIS EU-IST-2001-32122).

In this paper a method for fault tolerant control in dynamic systems is presented. The proposed approach is composed of two stages. The first stage is the detection and isolation of the failed component using a directionnal filter designed under a particular eigenstructure assignment. The second stage is represented by the reconfiguration mechanism which make possible to compensate the fault effects. This approach was applied to a benchmark proposed in the framework of the IFATIS project (IFATIS-IST 32122) and gave good results for actuator fault accommodation.

This article presents the synthesis and the real tests of three control law strategies to a winding process. The principal no linearity of this type of system is due to the variation, during operation, of the radius of the winding reels. A linear time varying model can thus describe the dynamic behavior of this system. A linear optimal control law, a gain scheduled controller and a non linear control law (input-output linearizing and decoupling approach) are selected for the control law synthesis. These control laws are then applied to the process with a DSP dSpace board interfaced with Matlab/Simulink. The experimental results illustrate the decoupling advantage of the nonlinear approach.

In this paper a method for fault-tolerant control in dynamic systems is presented. The proposed approach is composed of two stages. The first stage is the detection and isolation of the failed component using a directional filter designed under a particular eigenstructure assignment. The second stage is represented by the reconfiguration mechanism which makes possible the compensation of the fault effects. This approach was applied to the web tension control of winding machine and gave good results for actuator fault accommodation.

A new fault tolerant control (FTC) system devoted to non linear stochastic system described by multiple linear models is proposed in this paper. The proposed active FTC approach is composed of three stages: nominal controller, fault detection and isolation (FDI) scheme that is based on a bank of robust fault decoupling Kalman filters and a reconfiguration mechanism. Fault free residuals generated by a bank of Kalman Filter are used first to select the most appropriate model to characterize dynamic behavior of the system. Then, evaluation of the fault sensitive residuals provides information about the location and time occurrence of the faults. Once a fault has been detected and isolated, residuals are monitored to activate a fault compensation loop in order to cancel the fault effect on to the state vector. The effectiveness and performances of the method are validated in simulation considering a classical non linear benchmark.

In this paper, the main goal is to design an approach that performs fault detection and isolation in non linear system. Fault diagnosis is established by regarding system as an interpolation of multiple linear time invariant stochastic models and not as a single global model. In multiple model framework, the purpose of the paper consist of generate a robust model selection of the "best" representative linear model. Fault diagnosis method presented here is based on a bank of decoupled Kalman filters. The proposed method allows detection, isolation and estimation of multiple faults which appear simultaneously or in a sequential way. Robust model selection is obtained by regarding the residual vector insensitive to faults. Performances of the method are tested on a simulation example.

The paper deals with nonlinear filters for Fault Detection and Isolation. An alternative to the Fundamental Problem of Residual Generation is proposed according that faults do not occur simultaneously. The main result is the proposition of a systematic method for filters bank design in order to isolate all faults. The method application is achieved on an academic example.

In this paper, fault diagnosis for a large class of nonlinear systems is developed: nonlinear systems are supposed to be represented via a multiple model approach based on a convex combination of linear time invariant (LTI) models. Under fault isolation conditions, the main contribution is to design an adaptive filter which makes possible multiple faults detection, isolation and estimation over the whole operating range of nonlinear system. According to nonlinear representation via a gain scheduling interpolation method (successfully used in the control design even with fast varying parameters), the design is performed by extending robust fault isolation filter methods in LTI systems to nonlinear systems. The stability conditions of the adaptive filter are developed. These conditions result in convex linear matrix inequalities (LMIs) that can be solved efficiently with optimization techniques. The performances of the methods are demonstrated using a nonlinear system simulation example.

In this paper, we present a fault isolation filter (PIF) for fault detection in linear stochastic systems affected by disturbances. The fault isolation filter will generate a directional residual decoupled from unknown input allowing the treatment of multiple faults appearing simultaneously or sequentially. The output of the filter will be not only a directional residual but also a minimum-time fault magnitudes estimation. The effect of plant and state noises on the fault estimation will be minimized. An numerical example is given to illustrate the design of the proposed filter.