Brief paper filtering with randomly occurring sensor saturations and missing measurements☆
Introduction
The past few decades have witnessed an ever increasing research interest in the filtering or state estimation problems that are fundamental to control and signal processing areas. For example, the renowned Kalman filtering theory serves as an essential part of the development of space and military technology (Cattivelli & Sayed, 2010). A variety of performance requirements have been proposed in the literature for the filter design, such as the specification, the minimum variance requirement, the distributed collaborative behavior and the so-called admissible variance constraint. For example, the extended Kalman filters have been designed in Kallapur, Petersen, and Anavatti (2009) for nonlinear deterministic systems and in Spinello and Stilwell (2010) for nonlinear stochastic systems. The robust filtering problems have been extensively studied in Wang, Ho, and Liu (2003) and Xie, Soh, and de Souza (1994) for systems with norm-bounded uncertainties and in James and Petersen (1998), and Savkin and Petersen (1998) for uncertain systems with integral quadratic constraint. The filters with error variance constraints have been exploited in Wang et al. (2003) and Xie et al. (1994) for systems which are subject to noises with known statistics. The hybrid filtering problems have been investigated in Yin and Dey (2003) by using Markov chain approaches. The optimal filters have been designed in Basin (2003) and Basin and Garcia (2003) for polynomial systems. Moreover, the filtering problems have recently received much research attention by using the linear matrix inequality (LMI) approach, see e.g., Gao and Chen (2007), Shi, Mahmoud, Nguang, and Ismail (2006), Wu, Lam, Paszke, Galkowski, and Rogers (2008), and Li, Lam, and Shu (2010).
Most filter design approaches available rely on the ideal assumption that there is a continuous flow of measurement signals with unlimited amplitudes. However, perfect communication is not always possible in many engineering systems especially in a networked environment. For example, due to sensor temporal failure or network transmission delay/loss (Gao and Chen, 2007, Sun et al., 2008, Wang et al., 2003), at certain time points, the system measurement may contain noise only, which means the real signal is missing. The filtering problem with missing measurements has gained considerable research attention and many results have been reported in the literature, see Kluge, Reif, and Brokate (2010) and Wang et al. (2003). A common way for handling the missing measurement is to utilize the Bernoulli distributed (binary switching) white sequence specified by a conditional probability distribution in the output equation. Such kind of “binary” description has been employed in many papers such as Hounkpevi and Yaz (2007), Kluge et al. (2010), Wang et al. (2003), and Zhao, Lam, and Gao (2009) for filtering problems of linear/nonlinear systems with probabilistic measurement losses. It is worth mentioning that, comparing to a large amount of results for missing measurements, the corresponding filter design problem for signals with limited amplitudes or saturation has received much less focus of research despite the fact that sensor saturations occur very often in practical engineering.
In reality, the obstacles in delivering the high performance promises of traditional filter theories are often due to the physical limitations of system components, of which the most commonly encountered one stems from the saturation that occurs in any actuators, sensors, or certain system components. Saturation brings in nonlinear characteristics that can severely restrict the amount of deployable filter scheme. Such a characteristic not only limits the filtering performance that can otherwise be achieved without saturation, it may also lead to undesirable oscillatory behavior or, even worse, instability. Therefore, the control problem for systems under actuator/sensor saturations have attracted considerable research interest (see e.g., Cao et al., 2002, Hu et al., 2002, Zuo et al., 2010) and the related filtering problem has also gained some scattered research attention (Xiao et al., 2004, Yang and Li, 2009). It should be pointed out that, in almost all relevant literature, the saturation is implicitly assumed to occur already. However, in networked environments such as wireless sensor networks, the sensor saturation itself may be subject to random abrupt changes, for example, random sensor failures leading to intermittent saturation, sensor aging resulting in changeable saturation level, repairs of partial components, changes in the interconnections of subsystems, sudden environment changes, modification of the operating point of a linearized model of a nonlinear systems, etc. In other words, the sensor saturations may occur in a probabilistic way and are randomly changeable in terms of their types and/or intensity. Such a phenomenon of sensor saturation, namely, randomly occurring sensor saturation (ROSS), has been largely overlooked in this area. It is, therefore, the main purpose of this paper to bring the issue of ROSS to the readers’ attention in order to better reflect the random nature of sensor saturations in large-scale networked systems such as wireless sensor networks.
In this paper, we aim to deal with the filtering problem for a class of nonlinear systems with randomly occurring incomplete information. The considered incomplete information includes both the sensor saturations and the missing measurements. A regional filter with a certain ellipsoid constraint is designed such that the filtering error dynamics is locally mean-square asymptotically stable and the -norm requirement is satisfied. Here, the regional gain filtering feature is specifically developed for addressing the random saturation nonlinearity. The characterization of the desired filter gains is derived in terms of the solution to a convex optimization problem that can be easily solved by using the semi-definite program method. A simulation example is employed to show the effectiveness of the filtering scheme proposed. The main novelty lies in three aspects: (1) the phenomenon of ROSS that typically exists in networked environments is put forward for investigation; (2) a novel sensor model is established to take both the ROSS and missing measurement into account; and (3) a new notion of the domain of attraction in the mean square sense is introduced and a certain ellipsoid constraint is imposed on the desired filter in the presence of random saturation nonlinearity.
Notation. The notation used here is fairly standard except where otherwise stated. denotes the -dimensional Euclidean space. refers to the norm of a matrix defined by . The notation (respectively, ), where and are real symmetric matrices, means that is positive semi-definite (respectively, positive definite). represents the transpose of the matrix . denotes the identity matrix of compatible dimension. stands for a block-diagonal matrix and the notation is employed to stand for . stands for the expectation of the stochastic variable . means the occurrence probability of the event “ ⋅”. is the space of square summable -dimensional vector-valued functions. In symmetric block matrices, “ ∗” is used as an ellipsis for terms induced by symmetry. Matrices, if they are not explicitly specified, are assumed to have compatible dimensions.
Section snippets
Problem formulation and preliminaries
Consider a nonlinear discrete-time system and sensors with both saturation and missing measurements , where is the state vector, is the output vector to be estimated, is the measurement received by sensor , and represent, respectively, the process noise belonging to and the measurement noise for sensor belonging to . is a continuously vector-valued function. ,
Main results
Let us start with tackling the saturation function . According to the definition of the saturation function (3), it is easily known that the nonlinear function satisfies and where is a positive scalar satisfying .
Set and define Then, it can be verified that the diagonal matrix satisfies and the nonlinear function satisfies for each .
An illustrative example
Consider a nonlinear discrete-time system described by (1) with the matrix parameters and the nonlinear function It is not difficult to verify that the above nonlinear function satisfies (4) with The concerned sensors with both ROSSs and missing measurements are modeled by (2) with the following
Zidong Wang was born in Jiangsu, China, in 1966. He received the B.Sc. degree in mathematics in 1986 from Suzhou University, Suzhou, China, and the M.Sc. degree in applied mathematics in 1990 and the Ph.D. degree in electrical engineering in 1994, both from Nanjing University of Science and Technology, Nanjing, China.
He is currently a Professor of Dynamical Systems and Computing in the Department of Information Systems and Computing, Brunel University, UK. From 1990 to 2002, he held teaching
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Zidong Wang was born in Jiangsu, China, in 1966. He received the B.Sc. degree in mathematics in 1986 from Suzhou University, Suzhou, China, and the M.Sc. degree in applied mathematics in 1990 and the Ph.D. degree in electrical engineering in 1994, both from Nanjing University of Science and Technology, Nanjing, China.
He is currently a Professor of Dynamical Systems and Computing in the Department of Information Systems and Computing, Brunel University, UK. From 1990 to 2002, he held teaching and research appointments in universities in China, Germany and the UK. Prof. Wang’s research interests include dynamical systems, signal processing, bioinformatics, control theory and applications. He has published more than 100 papers in refereed international journals. He is a holder of the Alexander von Humboldt Research Fellowship of Germany, the JSPS Research Fellowship of Japan, and the William Mong Visiting Research Fellowship of Hong Kong.
Prof. Wang serves as an Associate Editor for 11 international journals, including IEEE Transactions on Automatic Control, IEEE Transactions on Control Systems Technology, IEEE Transactions on Neural Networks, IEEE Transactions on Signal Processing, and IEEE Transactions on Systems, Man, and Cybernetics—Part C. He is a Senior Member of the IEEE, a Fellow of the Royal Statistical Society and a member of program committees for many international conferences.
Bo Shen received his B.Sc. degree in Mathematics from Northwestern Polytechnical University, Xi’an, China, in 2003 and the Ph.D. degree in Control Theory and Control Engineering from Donghua University, Shanghai, China, in 2011.
He is currently with the School of Information Science and Technology, Donghua University, Shanghai, China. From 2009 to 2010, he was a Research Assistant in the Department of Electrical and Electronic Engineering, the University of Hong Kong, Hong Kong. From 2010 to 2011, he was a Visiting PhD Student in the Department of Information Systems and Computing, Brunel University, UK. His research interest is primarily in nonlinear stochastic control and filtering.
Dr. Shen is a very active reviewer for many international journals.
Xiaohui Liu received his B.E. degree in computing from Hohai University, Nanjing, China, in 1982 and the Ph.D. degree in computer science from Heriot-Watt University, Edinburg, UK, in 1988.
He is currently a Professor of Computing at Brunel University, West London, UK. He leads the Intelligent Data Analysis (IDA) Group, performing interdisciplinary research involving artificial intelligence, dynamic systems, image and signal processing, and statistics, particularly for applications in biology, engineering and medicine.
Prof. Liu serves on the editorial boards of four computing journals, founded the biennial international conference series on IDA in 1995, and has given numerous invited talks at bioinformatics, data mining, and statistics conferences.
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This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Royal Society of the UK, the National Natural Science Foundation of China under Grants 61028008, 60974030, 61134009 and 61104125, the National 973 Program of China under Grant 2009CB320600, and the Alexander von Humboldt Foundation of Germany. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Michael V. Basin under the direction of Editor Ian R. Petersen.
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