Recursive approach to networked fault estimation with packet dropouts and randomly occurring uncertainties
Introduction
During the past two decades, a large number of advance results have been reported to deal with the problems of the fault detection and isolation (FDI) for many practical engineering systems under high reliability requirements [1], [2], [3], [4], [5], [6]. An important issue surrounding the investigation of the FDI is to estimate the fault signal firstly and then provide a residual generation scheme for fault detection or compensate the unknown fault for synthesizing the fault-tolerant control systems. Therefore, the fault estimation problem has received increasing research attention and a variety of fault estimation approaches have been given in the literature, see e.g. [7], [8], [9], [10] and the references therein. To be more specific, a sliding mode observer scheme has been given in [7] to deal with the fault estimation problem for dynamical systems with time-varying coupling strength, and the faults have been reconstructed based on the equivalent output error injection signal. In [8], the fault estimation problem has been studied for a class of linear discrete periodic systems with the information scheduler, where the desired fault estimation algorithm has been presented in terms of the Riccati matrix difference equation approach. Very recently, the estimation problems of randomly occurring faults over finite-horizon have been tackled in [9] for nonlinear time-varying systems with different sources of disturbances via the recursive Riccati difference equation approach and in [10] for nonlinear time-varying systems with fading measurements by means of the recursive matrix inequality method.
As is well known, the nonlinearities and uncertainties contribute the complexities of the system modeling, and hence it is necessary to handle them carefully when analyzing the complex dynamical systems under increasing performance requirements [9], [11], [12], [13], [14], [15], [16], [17], [18]. For example, the fault signal has been estimated in [9] for a class of time-varying systems with stochastic nonlinearities characterized by the statistical characteristics, and it can be seen that a novel performance requirement against different sources of disturbances has been introduced. To cope with the uncertainties, the fault estimation scheme has been developed in [19] for a class of linear uncertain time-varying systems with known inputs in view of the Krein-space theory. It is worth pointing out that an auxiliary system has been introduced in [19] where the parameter uncertainties have been tackled as the disturbance terms, and a new performance index involving the known inputs has been provided in order to better reflect the effect of the known inputs onto the whole system performance. Recently, the so-called randomly occurring uncertainties have been modeled by the Bernoulli distribution in [20], [21] and some analysis methods have been given for discrete uncertain nonlinear systems by means of the linear matrix inequality technique. However, it should be noticed that the fault estimation problem suffers from more complexities if both the randomly occurring uncertainties and the stochastic nonlinearities are taken into account, and this is necessary to properly characterize the resulted effects onto the fault estimation performance which constitutes the focus of our present research.
On another research frontier, the study of the networked control systems (NCSs) has received increasing research attention due to its advantages such as enabling the execution of several tasks over long distance, increasing the flexibility of control system design, and reducing the complexity and the overall cost [22], [23], [24], [25], [26], [27]. Nevertheless, accompanying with the developments of the NCSs, the network-induced phenomena (e.g. packet dropouts, communication delays, signal quantization, uncertain sampling periods) are inherently inevitable which would degrade the whole system performance [28], [29], [30], [31], [32], [33], [34]. Hence, a great number of results have been reported for the analysis and design of NCSs with network-induced phenomena [35], [36]. For instance, based on the non-uniform sampled outputs, a novel fault estimation approach has been developed in [37] for NCSs subject to non-uniform uncertain sampling periods. Note that the packet dropout (often called missing measurement) frequently occurs and, therefore, the fault estimation problem with packet dropouts has gained persistent research attention [10], [38], [39], [40]. In [39], the authors have designed a sliding-mode observer to estimate the rotor speed and the unknown load torque for the networked dc motor system with packet dropouts. Recently, the problems of FDI and fault estimation have been studied in [40] for NCSs with data packet dropouts and communication delays by developing a novel residual matching scheme, where the simultaneous estimation of fault and system state has been given. It is worthwhile to note that, compared with the fruitful results on fault estimation problems for time-invariant networked systems subject to packet dropouts with identical missing probabilities [41], the corresponding results on time-varying fault estimation problems with individual missing probabilities are still in early stages probably due to the increasing complexities. In addition, the problem of the joint estimation of fault and system state for time-varying networked systems with randomly occurring uncertainties and packet dropouts has not been thoroughly studied, not to mention the case when both different sensors having individual missing probability and the algorithm performance evaluation are conducted simultaneously. Hence, the fundamental question that this paper will answer is how to design the fault estimation algorithm for addressed time-varying systems with network-induced phenomena and evaluate the algorithm performance.
Inspired by the above discussions, we aim to develop the fault estimation scheme for the addressed time-varying networked systems and conduct the algorithm evaluation issue in order to meet the increasing performance requirements correspondingly. For the addressed problem, the phenomena of the randomly occurring uncertainties and packet dropouts are characterized by several random variables obeying the Bernoulli probability distribution with known occurrence probabilities. In particular, the measurements may experience the packet dropouts during the signal transmissions and different sensors are allowed to have individual missing probability, which is more general. Moreover, the relationship between the upper bound of the estimation error covariance and the missing probability is revealed, i.e., the monotonicity of the trace of the estimation error covariance with respect to the missing probability is shown. The major advantages of this paper lie in: (1) the addressed time-varying networked system is comprehensive which could cover the randomly occurring uncertainties, stochastic nonlinearities as well as packet dropouts in a same framework; (2) a compensation scheme is provided by utilizing the statistical information of the packet dropouts when designing the fault estimation algorithm; (3) the proposed fault estimation approach is capable of estimating the unknown system state as a by-product; and (4) the developed fault estimation algorithm is of a recursive form applicable for online implementations. Finally, the simulations show that the presented fault estimation scheme is capable of attenuating the adverse effects from packet dropouts and maintaining the desired performance.
The reminder of this paper is organized as follows. In Section 2, the considered system model is given and the addressed problem is formulated. In Section 3, we provide a new fault estimation algorithm which can handle the stochastic nonlinearities, randomly occurring uncertainties and packet dropouts simultaneously. In the same section, the state estimation is also obtained as a by-product. In Section 4, the algorithm performance evaluation is provided. An illustrative example is used in Section 5 to demonstrate the usefulness of the main results. Finally, the conclusion is provided in Section 6.
Notations: The notations used throughout the paper are fairly standard. represents the n-dimensional Euclidean space. For the matrix X, XT is the transpose. represents a diagonal matrix with in the diagonal. denotes the mathematical expectation of the random variable x. I and 0 stand for an identity matrix and a zero matrix with appropriate dimensions, respectively. The Hadamard product is defined as . Moreover, the matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.
Section snippets
Problem formulation and preliminaries
In this paper, we consider the following class of discrete time-varying networked systems:where is the system state, is the initial state with mean , is the measurement output, ξk and ςk are zero-mean white noises, is the additive fault, characterizes the random link failure case, ϖk is the zero-mean process noise with covariance Qk, and υk is the zero-mean measurement
Design of estimation algorithm
In this section, both the one-step prediction error covariance and the estimation error covariance are firstly derived. Subsequently, an upper bound of the estimation error covariance is presented by using the matrix analysis technique and then the estimator gain is designed to minimize the obtained upper bound.
Let the one-step prediction error be and the estimation error be . It follows that
Performance analysis of algorithm
In this section, we aim to examine the influence from the phenomenon of packet dropouts onto the performance of the fault estimation algorithm. Accordingly, we will reveal the relationship between the trace of the upper bound of the estimation error covariance and the occurrence probability of the packet dropouts. To proceed, let , i.e., all have same statistical properties. Then, it is easy to obtain that and .
Next, the relationship (i.e., monotonicity)
An illustrative example
In this section, we provide some simulations to show the usefulness of the developed fault estimation algorithm.
Consider the time-varying networked system (1), (2) with the following parameters:and the state vector with being the i-th element of the system state, and are zero-mean Gaussian white noises with covariances 0.02 and 0.05, respectively. The
Conclusions
In this paper, we have addressed the fault estimation problem for time-varying networked systems subject to the randomly occurring uncertainties, stochastic nonlinearities and packet dropouts. The network-induced uncertainties have been modeled by employing a Bernoulli random variable with known occurrence probability. Moreover, the phenomena of the packet dropouts have been modeled by using a set of mutually independent random variables and all sensor measurements have been allowed to have
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grants 11301118 and 11271103, the Fok Ying Tung Education Foundation of China under Grant 151004, the Science Funds for the Young Innovative Talents of HUST, the Youth Science Foundation of Heilongjiang Province of China under Grant QC2015085, University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province, the China Postdoctoral Science Foundation under Grants 2015T80482 and
Yue Song received the B.Sc. degree in Mathematics from Harbin University, Harbin, China, in 2015. She is working toward the M.Sc. degree in Operations Science and Control Theory with Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, China. Her current research interests include fault estimation for networked control systems, neural networks and complex networks. She is an active reviewer for some international journals.
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2020, Information SciencesCitation Excerpt :The co-design of fault detection filter and controller was addressed for networked unmanned surface vehicles in [21]. The fault estimation problem was studied for NCSs with packet dropouts and uncertainties via recursive approach in [22]. The problem of fault estimation and accommodation was addressed for NCSs with nonuniform sampling periods in [23].
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2019, Journal of the Franklin InstituteCitation Excerpt :A fault detection method is developed in finite frequency domain for networked systems with access constraint in [20]. In [21], the fault estimation problem is addressed for a class of NCSs subject to uncertainties, stochastic nonlinearities and packet dropouts. As for FTC, a fault-tolerant controller is proposed for a class of nonlinear networked control systems [22].
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Yue Song received the B.Sc. degree in Mathematics from Harbin University, Harbin, China, in 2015. She is working toward the M.Sc. degree in Operations Science and Control Theory with Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, China. Her current research interests include fault estimation for networked control systems, neural networks and complex networks. She is an active reviewer for some international journals.
Jun Hu received the B.Sc. degree in Information and Computation Science and M.Sc. degree in Applied Mathematics from Harbin University of Science and Technology, Harbin, China, in 2006 and 2009, respectively, and the Ph.D. degree in Control Science and Engineering from Harbin Institute of Technology, Harbin, China, in 2013. From September 2010 to September 2012, he was a Visiting Ph.D. Student in the Department of Information Systems and Computing, Brunel University, U.K. From May 2014 to April 2016, he was an Alexander von Humboldt research fellow at the University of Kaiserslautern, Kaiserslautern, Germany. He is currently a Professor with the Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, China. His current research interests include nonlinear control, filtering and fault estimation, time-varying systems, multi-agent systems and complex networks. He has published more than 20 papers in refereed international journals. He serves as a reviewer for Mathematical Reviews, as an editor for Neurocomputing, Journal of Intelligent & Fuzzy Systems, Systems Science & Control Engineering, and as a guest editor for International Journal of General Systems and Information Fusion. He is an active reviewer for many international journals.
Dongyan Chen received the B.Sc. degree in Mathematics from Northeast Normal University, Changchun, China, in 1985, M.Sc. degree in Operational Research from Jilin University, Changchun, China, in 1988, and the Ph.D. degree in Aerocraft Design from Harbin Institute of Technology, Harbin, China, in 2000. She is now a Professor and Ph.D. Supervisor with the Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, China. Her current research interests include robust control, time-delay systems, optimization approach, system optimization and supply chain management.
Donghai Ji received the B.Sc. degree in Department of Mathematics from Mudanjiang Normal University, Mudanjiang, China, in 1983, and the Ph.D. degree in Mathematics from Wroclaw University of Science and Technology, Wroclaw, Poland in 1994. He is now a Professor and PhD Supervisor with the Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, China. His current research interests include convex geometric analysis, applied mathematics etc.
Fengqiu Liu received the B.S. degree in Mathematics from Northeast Normal University in 2000. She received the M.S. and Ph.D. degrees in Mathematics from Harbin Institute of Technology, Harbin, China, in 2004 and 2012, respectively. She is currently a Professor with the Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, China. Her current research interests include fuzzy control theory, neural networks, and machine learning with emphasis on kernel methods.