An LMI approach to design robust fault detection filter for uncertain LTI systems

In this paper, the robust fault detection filter design problem for uncertain linear time-invariant (LTI) systems with both unknown inputs and modelling errors is studied. The basic idea of our study is to use an optimal residual generator (assuming no modelling errors) as the reference residual model of the robust fault detection filter design for uncertain LTI systems with modelling errors and, based on it, to formulate the robust fault detection filter design as an H_∞ model-matching problem. By using some recent results of H_∞ optimization, a solution of the optimization problem is then presented via a linear matrix inequality (LMI) formulation. The main results include the development of an optimal reference residual model, the formulation of robust fault detection filter design problem, the derivation of a sufficient condition for the existence of a robust fault detection filter and a construction of it based on the LMI solution parameters, the determination of adaptive threshold for fault detection. An illustrative design example is employed to demonstrate the effectiveness of the proposed approach.     

Robust fault estimation of uncertain systems using an LMI‐based approach

General recent techniques in fault detection and isolation (FDI) are based on H_∞ optimization methods to address the issue of robustness in the presence of disturbances, uncertainties and modeling errors. Recently developed linear matrix inequality (LMI) optimization methods are currently used to design controllers and filters, which present several advantages over the Riccati equation‐based design methods. This article presents an LMI formulation to design full‐order and reduced‐order robust H_∞ FDI filters to estimate the faulty input signals in the presence of uncertainty and model errors. Several cases are examined for nominal and uncertain plants, which consider a weight function for the disturbance and a reference model for the faults. The FDI LMI synthesis conditions are obtained based on the bounded real lemma for the nominal case and on a sufficient extension for the uncertain case. The conditions for the existence of a feasible solution form a convex problem for the full‐order filter, which may be solved via recently developed LMI optimization techniques. For the reduced‐order FDI filter, the inequalities include a non‐convex constraint, and an alternating projections method is presented to address this case. The examples presented in this paper compare the simulated results of a structural model for the nominal and uncertain cases and show that a degree of conservatism exists in the robust fault estimation; however, more reliable solutions are achieved than the nominal design. Copyright © 2008 John Wiley & Sons, Ltd.     

A stable fault detection observer design in finite frequency domain

This article addresses the stable fault detection observer design problem for linear time-invariant continuous-time systems in finite-frequency domain. The fault detection filter design is a synthesised optimal Luenberger observer that guarantees two requested performance indexes of fault sensitivity and stability. With the aid of generalised Kalman–Yakubovich–Popov lemma and increasing dimensions of slack variable matrix, the stability and H _? performance analysis of the closed-loop system with a fault detection observer has been translated into a convex linear matrix inequality (LMI) optimisation problem to avoid the complexity of system associated with weight functions. An iterative LMI algorithm has been presented for the fault detection observer design. The effectiveness of proposed approaches is demonstrated by two numerical examples.     

一类不确定线性系统的鲁棒故障观测器设计

研究线性时不变系统的鲁棒故障观测器设计问题,给定故障检测率,降低误报率作为优化指标,对故障和未知扰动在残差信号中的输出进行滤波。再考虑不确定性影响,运用模型匹配方法,提出了具有不确定性线性系统的鲁棒故障观测器设计方法,并以线性矩阵不等式形式给出问题求解算法。通过求解具有线性矩阵不等式约束的凸优化问题获得观测器的全局最优解。通过仿真表明,改进的设计方法可有效保障残差信号对故障信号的灵敏度,并改善了诊断性能。     

Robust fault detection of uncertain linear systems via quasi-LMIs

A novel H_∞ deconvolution filter design method is discussed for fault detection of uncertain polytopic linear systems subject to unknown inputs. The enhancement of fault sensitivity is characterized in terms of an H_- index. By means of the Projection Lemma and Congruence Transformations, a quasi-convex LMI formulation of the design problem is obtained. The effectiveness of the filter is illustrated via a numerical example.     

Synthesis of stochastic fault tolerant control systems with random FDI delay

In this work, the synthesis of fault tolerant control (FTC) for stochastic stability and H _∞ performance is studied. Occurrence of faults in the system is governed by a Markov Chain, so the open-loop system is modelled as a linear system with Markovian jumping parameters. The fault detection and isolation (FDI) decision is modelled as another random process that will indicate the fault mode after an exponentially distributed random delay. This stochastic formulation of FTC concerns the random nature of faults and the effect of random fault detection delay on the overall system, and can be regarded as an extension to the traditional reconfigurable control problem. In this paper, output feedback controllers are designed using an iterative LMI algorithm for mean exponential stability (MES) and the H _∞ performance. Model uncertainties and external disturbance are also considered in the robust design.     

Optimal H∞ scaling for sensitivity optimization of detection filters

The paper deals with the sensitivity optimization of detection filters in linear time‐varying (LTV) systems which are subject to multiple simultaneous faults and disturbances. The robust fault detection filter design problem as a scaled H_∞ filtering problem is considered. The effect of two different input scaling approaches to the optimization process is investigated. The objective is to provide the smallest scaled L₂ gain of the unknown input of the system that is guaranteed to be less than a prespecified level, i.e., to produce a filter with optimal disturbance suppression capability in such a way that sufficient sensitivity to failure modes should still be maintained. It is shown how to obtain bounds on the scaled L₂ gain by transforming the standard H_∞ filtering problem into a convex feasibility problem, specifically, a structured, linear matrix inequality (LMI). Numerical examples demonstrating the effect of the scaled optimization with respect to conventional H_∞ filtering is presented. Copyright © 2002 John Wiley & Sons, Ltd.     

A Delay‐Dependent Approach to Robust Fast Adaptive Fault Estimation Design for Uncertain Neutral Systems with Time‐Varying Interval Delay

This paper studies the problem of robust fault estimation for neutral systems, which are subjected to uncertainties, actuator fault, time‐varying interval delay, and norm‐bounded external disturbance. Based on the fast adaptive fault estimation (FAFE) algorithm, we focus on the design of a fault estimation filter that guarantees stability in the filtering error system with a prescribed H_∞ performance. A novel Lyapunov‐Krasovskii functional is employed, which includes time delay information. A delay‐dependent criterion of robust fault estimation design is obtained by employing the free‐weighting matrices technique, and the proposed result has advantages over some existing results, in that it is less conservative and it enlarges the application scope. An improved sufficient condition for the existence of such a filter is proposed in terms of the linear matrix inequality (LMI) by the Schur complements and the cone complementary linearization algorithm. Finally, illustrative examples are provided to show the effectiveness of the proposed method.     

Robust fault detection filter design for uncertain LTI systems based on new bounded real lemma

This paper concerns the problem of robust fault detection filter design for uncertain linear time-invariant (LTI) systems with both model uncertainty and disturbances. Firstly, the fault detection filter design is formulated to H _∞ model-matching problem. Secondly, based on a new bounded real lemma, a sufficient condition for the existence of the robust fault detection filter is constructed in term of linear matrix inequalities (LMIs). Owing on the introduction of the tuning parameter and slack variables in obtained LMI condition, the proposed design method can provide higher fault detection sensitivity performance than the existing one. Finally, an illustrative example is employed to demonstrate the effectiveness of the proposed approach. Recommended by Editorial Board member Bin Jiang under the direction of Editor Jae Weon Choi. This work was supported by Postdoctoral Fundation of Jiangsu Province under grant 0901026c and Key Laboratory of Education Ministry for Image Processing and Intelligent Control under grant 200805. Tao Li received the Ph.D. degree in the Research Institute of Automation Southeast University, China. Now He is a postdoctoral researcher with the same university. His current research interests include time-delay systems, neural networks, robust control, fault detection and diagnosis. Lingyao Wu received the Ph.D. degree in the Research Institute of Automation Southeast University, China. Now He is an Assistant Professor in the Research Institute of Automation Southeast University. His current research interests include time-delay systems, neural networks, robust control, fault detection and diagnosis. Xinjiang Wei was born in Dongying, China, in 1977. He received the B.S. degrees from Yantai Normal University, China in 1999, M.S. degrees from Bohai University in 2002, and the Ph.D. degree in Department of Information from Northeastern University in 2005. From 2006 to Present, he was with Ludong University as an Associate Professor. From 2006 to 2009, he was a Postdoctoral Fellow at Southeast University. His research interests include robust control, nonlinear control, and fuzzy control.     

马尔可夫跳变系统的鲁棒故障检测与时域优化

研究马尔可夫跳变系统(MJS) 鲁棒故障检测滤波器(FDF) 的设计与优化问题. 基于观测器构建残差发生器, 将相应的FDF 设计问题转化为H_∞ 滤波问题, 以LMI 的形式得到并证明了FDF 存在的充分条件及求解方法. 为进一步改善故障检测系统的性能, 采用一种时域优化方法对其进行优化, 并以矩阵Moore-Penrose 逆的形式给出了该优化问题的最优解. 数值仿真表明该方法具有较好的检测效果.     

An ILMI Approach to RFDF for Uncertain Linear Systems with Nonlinear Perturbations

The robust fault detection filter design for uncertain linear systems with nonlinear perturbations is formulated as a two-objective optimization problem. Solvable conditions for the existence of such a robust fault detection filter are given in terms of matrix inequalities (MIs), which can be solved by applying iterative linear matrix inequality (ILMI) techniques. Particularly, compared with two existing LMI methods, the developed algorithm is more generalized and less conservative. An illustrative example is given to show the effectiveness of the proposed method.     

非线性摄动系统的鲁棒故障诊断滤波器设计ILMI算法

The robust fault detection filter design for uncertain linear systems with nonlinear perturbations is formulated as a two-objective optimization problem. Solvable conditions for the exis- tence of such a robust fault detection filter are given in terms of matrix inequalities (MIs), which can be solved by applying iterative linear matrix inequality (ILMI) techniques. Particularly, compared with two existing LMI methods, the developed algorithm is more generalized and less conservative. An illustrative example is given to show the effectiveness of the proposed method.     

Robust fault detection for networked systems with communication delay and data missing

In this paper, the robust fault detection problem is investigated for a class of discrete-time networked systems with unknown input and multiple state delays. A novel measurement model is utilized to represent both the random measurement delays and the stochastic data missing phenomenon, which typically result from the limited capacity of the communication networks. The network status is assumed to vary in a Markovian fashion and its transition probability matrix is uncertain but resides in a known convex set of a polytopic type. The main purpose of this paper is to design a robust fault detection filter such that, for all unknown inputs, possible parameter uncertainties and incomplete measurements, the error between the residual signal and the fault signal is made as small as possible. By casting the addressed robust fault detection problem into an auxiliary robust H_∞ filtering problem of a certain Markovian jumping system, a sufficient condition for the existence of the desired robust fault detection filter is established in terms of linear matrix inequalities. A numerical example is provided to illustrate the effectiveness and applicability of the proposed technique.     

A robust deconvolution scheme for fault detection and isolation of uncertain linear systems: an LMI approach

Optimal H_∞ deconvolution filter theory is exploited for the design of robust fault detection and isolation (FDI) units for uncertain polytopic linear systems. Such a filter is synthesized under frequency domain conditions which ensure guaranteed levels of disturbance attenuation, residual decoupling and deconvolution performance in prescribed frequency ranges. By means of the Projection Lemma, a quasi-convex formulation of the problem is obtained via LMIs. A FDI logic based on adaptive thresholds is also proposed for reducing the generation of false alarms. The effectiveness of the design technique is illustrated via a numerical example.     

A single dynamic observer-based module for design of simultaneous fault detection,isolation and tracking control scheme

The problem of simultaneous fault detection, isolation and tracking (SFDIT) control design for linear systems subject to both bounded energy and bounded peak disturbances is considered in this work. A dynamic observer is proposed and implemented by using the H_∞/H_?/L₁ formulation of the SFDIT problem. A single dynamic observer module is designed that generates the residuals as well as the control signals. The objective of the SFDIT module is to ensure that simultaneously the effects of disturbances and control signals on the residual signals are minimised (in order to accomplish the fault detection goal) subject to the constraint that the transfer matrix from the faults to the residuals is equal to a pre-assigned diagonal transfer matrix (in order to accomplish the fault isolation goal), while the effects of disturbances, reference inputs and faults on the specified control outputs are minimised (in order to accomplish the fault-tolerant and tracking control goals). A set of linear matrix inequality (LMI) feasibility conditions are derived to ensure solvability of the problem. In order to illustrate and demonstrate the effectiveness of our proposed design methodology, the developed and proposed schemes are applied to an autonomous unmanned underwater vehicle (AUV).     

Fault detection for a class of uncertain nonlinear Markovian jump stochastic systems with mode-dependent time delays and sensor saturation

This paper considers the problem of robust H_∞ fault detection for a class of uncertain nonlinear Markovian jump stochastic systems with mode-dependent time delays and sensor saturation. We aim to design a linear mode-dependent H_∞ fault detection filter that ensures, the fault detection system is not only stochastically asymptotically stable in the large, but also satisfies a prescribed H_∞-norm level for all admissible uncertainties. By using the Lyapunov stability theory and generalised Itô formula, some novel delay-dependent sufficient conditions in terms of linear matrix inequality are proposed to guarantee the existence of the desired fault detection filter. Explicit expression of the desired mode-dependent linear filter parameters is characterised by matrix decomposition, congruence transformation and convex optimisation technique. Sector condition method is utilised to deal with sensor saturation, a definite relation of sector condition parameters with fault detection system robustness against disturbances and sensitivity to faults is put forward, and weighting fault signal approach is employed to improve the performance of the fault detection system. A simulation example and an industrial nonisothermal continuous stirred tank reactor system are utilised to verify the effectiveness and usefulness of the proposed method.     

Robust Fault Detection for Discrete‐Time Nonlinear Impulsive Switched Systems

This paper deals with the problem of robust fault detection for discrete‐time nonlinear impulsive switched systems. The fault detection filter is used as the residual generator, in which the filter parameters are dependant on the system mode. Attention is focused on designing the robust fault detection filter. The problem of robust fault detection is converted into an H_∞‐filtering problem. A sufficient condition for the solvability of this problem is established by a convex optimization problem. A numerical example is provided to demonstrate the effectiveness of the proposed method.     

基于多目标约束的鲁棒故障检测滤波器设计

针对包含未知输入的线性时不变系统,研究了其鲁棒故障检测滤波器设计问题。由于故障信号往往分布在有限频域段内,设计的目标包括使特定有限频域段上的复合性能指标最小化以及满足区域极点配置的要求。一个基于LMI的方法被提出用于解决该设计问题。该方法的优点在于求解过程中可以获取给定有限频域段上的频域指标的真实值,并可求得满足目标的最优解。因此,设计的故障检测滤波器可以获得良好的故障检测性能。一个基于某国产歼击机的设计实例验证了该方法的有效性。     

Fault detection filter design for stochastic time-delay systems with sensor faults

This article considers the fault detection (FD) problem for a class of Itô-type stochastic time-delay systems subject to external disturbances and sensor faults. The main objective is to design a fault detection filter (FDF) such that it has prescribed levels of disturbance attenuation and fault sensitivity. Sufficient conditions for guaranteeing these levels are formulated in terms of linear matrix inequalities (LMIs), and the corresponding fault detection filter design is cast into a convex optimisation problem which can be efficiently handled by using standard numerical algorithms. In order to reduce the conservatism of filter design with mixed objectives, multi-Lyapunov functions approach is used via Projection Lemma. In addition, it is shown that our results not only include some previous conditions characterising H _∞ performance and H _? performance defined for linear time-invariant (LTI) systems as special cases but also improve these conditions. Finally, two examples are employed to illustrate the effectiveness of the proposed design scheme.