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.     

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.     

Unified Set Membership theory for identification, prediction and filtering of nonlinear systems

The problem of making inferences from data measured on nonlinear systems is investigated within a Set Membership (SM) framework and it is shown that identification, prediction and filtering can be treated as specific instances of the general presented theory. The SM framework presents an alternative view to the Parametric Statistical (PS) framework, more widely used for studying the above specific problems. In particular, in the SM framework, a bound only on the gradient of the model regression function is assumed, at difference from PS methods which assume the choice of a parametric functional form of the regression function. Moreover, the SM theory assumes only that the noise is bounded, in contrast with PS approaches, which rely on noise assumptions such as stationarity, uncorrelation, type of distribution, etc. The basic notions and results of the general inference making theory are presented. Moreover, some of the main results that can be obtained for the specific inferences of identification, prediction and filtering are reviewed. Concluding comments on the presented results are also reported, focused on the discussion of two basic questions: what may be gained in identification, prediction and filtering of nonlinear systems by using the presented SM framework instead of the widely diffused PS framework? why SM methods could provide stronger results than the PS methods, requiring weaker assumptions on system and on noise?     

Toeplitz matrices for LTI systems,an illustration of their application to Wiener filters and estimators

The Wiener–Kolmogorov theory of filtering has been with us since the first half of the twentieth century. A later matrix-based approach which was more general was derived with the steady-state Kalman filter. This approach uses a novel method of representing causal and uncausal systems in the form of convolution matrices and leads to a Wiener solution which is much easier to calculate than either the Kalman or Wiener approaches. For coloured additive noise, it avoids the use of Diophantine equations. The key idea missing in previous work is the close link between polynomials and Toeplitz matrices which are lower triangular in form. There is already a reasonably sized literature in the mathematics field on such matrices and so the area is ripe for exploration. Although the method does not offer a different or better solution, it shows a completely new way of defining linear time-invariant (LTI) systems which is neither transfer-function nor state-space-based. This is achieved by exploiting the connection between polynomials and Toeplitz matrices. The application here is the Wiener filter but there could well be many more as this is a generic approach.     

Robust Kalman filtering for uncertain discrete-time systems with probabilistic parameters bounded within a polytope

We consider the problem of finite horizon discrete-time Kalman filtering for systems with parametric uncertainties. Specifically, we consider unknown but deterministic uncertainties where the uncertain parameters are assumed to lie in a convex polyhedron with uniform probability density. The condition and a procedure for the construction of a suboptimal filter that minimizes an expected error covariance over-bound are derived.     

Design of state estimators for uncertain linear systems using quadratic boundedness

The notion of quadratic boundedness, which allows one to address the stability of a dynamic system in the presence of bounded disturbances, is applied to the design of state estimators for discrete-time linear systems with polytopic uncertainties. Necessary and sufficient stability conditions are stated and upper bounding sequences on the estimation error are derived. For the purpose of design, such conditions can be expressed in terms of linear matrix inequalities (LMIs), thus guaranteeing the numerical tractability. Simulation results are reported to show the effectiveness of the approach.     

Linear adaptive control for nonstationary uncertain systems under bounded noise

A discrete-time nonstationary linear control system is considered to be given by the algebraic difference equation in the state space. The control system is subject to a bounded additive noise. Uncertain parameters of the system take their values on the given polytopes which evolve in time. The objective is to generate a linear feedback, which provides the minimization of a given performance criterion in adaptive way. In general, the control problem is reduced to the convex programming one of an insignificant computational complexity. Therewith, the control problem can be solved analytically in the case of interval set-valued parameter estimates.     

不确定噪声下基于轴对称盒空间滤波的非线性系统状态估计

针对不确定噪声下的非线性系统状态估计问题, 本文提出了一种基于轴对称盒空间滤波的状态估计方法. 首先, 利用轴对称盒空间包裹线性化过程带来的误差项, 将状态函数线性化误差轴对称盒空间与噪声轴对称盒空间求取闵可夫斯基和, 得到干扰误差轴对称盒空间; 随后, 利用状态量、线性误差和测量噪声的轴对称盒空间的闵可夫斯基和, 得到系统状态预测集; 进而, 利用轴对称盒空间边界正交的性质, 将盒空间拆分为多组超平面, 构造测量更新的约束条件并得到集员包裹. 本文所提方法相比传统的椭球滤波方法而言, 降低了算法的复杂度, 减少了包裹状态可行集和线性化过程带来的余, 获得了更加紧致精确的系统状态集. 最后, 采用非线性弹簧–质量–阻尼器系统验证了本文所提算法的有效性.     

Robust adaptive observer design for uncertain systems with bounded disturbances.

This paper presents a robust adaptive observer design methodology for a class of uncertain nonlinear systems in the presence of time-varying unknown parameters with absolutely integrable derivatives, and nonvanishing disturbances. Using the universal approximation property of radial basis function (RBF) neural networks and the adaptive bounding technique, the developed observer achieves asymptotic convergence of state estimation error to zero, while ensuring boundedness of parameter errors. A comparative simulation study is presented by the end.     

Distributed estimation design for LTI systems: a linear quadratic approach

ABSTRACT

This paper deals with the problem of distributedly estimate the state of a plant through a network of interconnected agents. Each of these agents must perform a real-time monitoring of the plant state, counting on the measurements of local plant outputs and on the exchange of information with neighbouring agents. The paper introduces a distributed LQ-based design that is applied to a distributed observer structure based on a multi-hop subspace decomposition. Stability and optimality conditions are derived and tested in simulation. Finally, the design method presented allows the user, through the tune of two scalar parameters, to modify the observer gains according to their experience about the plant.     

A generalized autocovariance least-squares method for Kalman filter tuning

This paper discusses a method for estimating noise covariances from process data. In linear stochastic state-space representations the true noise covariances are generally unknown in practical applications. Using estimated covariances a Kalman filter can be tuned in order to increase the accuracy of the state estimates. There is a linear relationship between covariances and autocovariance. Therefore, the covariance estimation problem can be stated as a least-squares problem, which can be solved as a symmetric semidefinite least-squares problem. This problem is convex and can be solved efficiently by interior-point methods. A numerical algorithm for solving the symmetric is able to handle systems with mutually correlated process noise and measurement noise.     

Quasi-equality constrained risk-sensitive filtering for nonlinear discrete-time systems

More and more data fusion models contain state constraints with valuable information in the filtering process.In this study,an optimal filter of risk-sensitive with quasi-equality constraints is formul...     

A new design of robust filters for uncertain systems

In this paper, a structured polynomial parameter-dependent approach is proposed for robust H₂ filtering of linear uncertain systems. Given a stable system with parameter uncertainties residing in a polytope with s vertices, the focus is on designing a robust filter such that the filtering error system is robustly asymptotically stable and has a guaranteed estimation error variance for the entire uncertainty domain. A new polynomial parameter-dependent idea is introduced to solve the robust H₂ filtering problem, which is different from the quadratic framework that entails fixed matrices for the entire uncertainty domain, or the linearly parameter-dependent framework that uses linear convex combinations of s matrices. This idea is realized by carefully selecting the structure of the matrices involved in the products with system matrices. Linear matrix inequality (LMI) conditions are obtained for the existence of admissible filters and based on these, the filter design is cast into a convex optimization problem, which can be readily solved via standard numerical software. Both continuous and discrete-time cases are considered. The merit of the methods presented in this paper lies in their less conservatism than the existing robust filter design methods, as shown both theoretically and through extensive numerical examples.     

Suboptimal filter for continuous‐time linear systems with unknown parameters

The filtering problem for continuous‐time linear systems with unknown parameters is considered. A new suboptimal filter is herein proposed. It is based on the optimal mean‐square linear combination of the local Kalman filters. In contrast to the optimal weights, the suboptimal weights do not depend on current observations; thus, the proposed filter can easily be implemented in real‐time. Examples demonstrate high accuracy and efficiency of the suboptimal filter. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust tracking control design for uncertain robotic systems with persistent bounded disturbances

In this study, a robust nonlinear L_∞‐gain tracking control design for uncertain robotic systems is proposed under persistent bounded disturbances. The design objective is that the peak of the tracking error in time domain must be as small as possible under persistent bounded disturbances. Since the nonlinear L_∞ ‐gain optimal tracking control cannot be solved directly, the nonlinear L_∞ ‐gain optimal tracking problem is transformed into a nonlinear L_∞ ‐gain tracking problem by given a prescribed disturbance attenuation level for the L_∞ ‐gain tracking performance. To guarantee that the L_∞ ‐gain tracking performance can be achieved for the uncertain robotic systems, a sliding‐mode scheme is introduced to eliminate the effect of the parameter uncertainties. By virtue of the skew‐symmetric property of the robotic systems, sufficient conditions are developed for solving the robust L_∞ ‐gain tracking control problems in terms of an algebraic equation instead of a differential equation. The proposed method is simple and the algebraic equation can be solved analytically. Therefore, the proposed robust L_∞ ‐gain tracking control scheme is suitable for practical control design of uncertain robotic systems. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Comments on “State estimation for linear systems with state equality constraints” [Automatica 43 (2007) 1363-1368]

The state estimation problem for linear systems with linear state equality constraints was dealt with in Ko & Bitmead [Ko, S., & Bitmead, R. (2007). State estimation for linear systems with state equality constraints. Automatica, 43, 1363-1368]. In this correspondence, it is first shown that a necessary assumption on the covariance of the process noise is missing in the main result of the paper. It is then shown that the main result of the paper can be achieved in a convenient and more general way without any additional assumptions on the covariance of the process noise except positive definiteness.     

Adaptive variable structure state estimation for uncertain systems with persistently bounded disturbances

This paper develops an adaptive state estimator design methodology for nonlinear systems with unknown nonlinearities and persistently bounded disturbances. In the proposed estimation scheme, the boundary layer strategy in variable structure techniques is utilized to design a continuous state estimator such that the undesirable chattering phenomenon is avoided; and the adaptive bounding technique is used for online estimation of the unknown bounding parameter. The existence condition of the adaptive estimators is provided in terms of linear matrix inequality (LMI). Since the orthogonal projection of the state estimation error onto the null space of the linear measurement distribution matrix is used in the derivation process, the update law of bounding parameter estimate is represented in terms of the available measurement error. The proposed estimator can ensure that the state estimation error is uniformly ultimately bounded (UUB) with an ultimate bound. Furthermore, using the existing LMI optimization technique, a suboptimal adaptive state estimator can be obtained in the sense of minimizing an upper bound of the peak gains in the ultimate bound. Finally, a simulation example is given to illustrate the effectiveness of the proposed design method. Copyright © 2010 John Wiley & Sons, Ltd.     

An analysis and design method for uncertain variable structure systems with bounded controllers

We present a method for estimating the asymptotic stability region(ASR) of uncertain variable structure systems with bounded switching feedback controllers. Using linear matrix inequalities(LMIs) we estimate the ASR and we show the exponential stability of the closed-loop control system in the estimated ASR. We also give a simple LMI-based method for designing switching surfaces that will make the estimated ASR big. Finally, we give numerical examples in order to show the effectiveness of our method.