Elsevier

Automatica

Volume 129, July 2021, 109684
Automatica

Brief paper
Distributed set-membership filtering for time-varying systems: A coding–decoding-based approach

https://doi.org/10.1016/j.automatica.2021.109684Get rights and content

Abstract

In this paper, the distributed set-membership filtering problem is studied for a class of discrete time-varying systems over digital wireless sensor networks. In order to make full utilization of limited bandwidth of the communication channels connecting sensor nodes, a new coding–decoding communication strategy is implemented to regulate the data transmission between individual nodes. Under the scheduling of the proposed communication strategy, the transmitted signal will be converted to certain codewords belonging to a finite set before transmission according to the corresponding coding rules. Sufficient criteria are firstly established for the solvability of the addressed filtering problem in terms of a series of matrix inequalities. Subsequently, a sub-optimal problem is presented with the purpose of seeking the locally optimal distributed filter. Finally, two simulation examples are given to demonstrate the proposed filtering algorithm.

Introduction

The past decades have witnessed fast development and wide adoption of wireless sensor networks (WSNs) technology. As the extensive utilization of WSNs technology, the relevant WSN-based filtering issue has been garnering much research attention whose aim is to estimate the plant states using the information from sensor nodes in a distributed yet collaborative way, see, e.g. Battistelli and Chisci, 2016, Combastel and Zolghadri, 2019, Dong et al., 2014, Nedic et al., 2010, Notarstefano and Bullo, 2011, Olfati-Saber, 2017, Roecker and McGillem, 1988 and the references therein, among which the most frequently used techniques are based on either Kalman filtering framework (Battilotti et al., 2020, Chen et al., 2019, Gan and Harris, 2001) or H algorithm (Ma et al., 2019, Qu et al., 2019). The unknown-but-bounded (UBB) disturbances widely appear in practical engineering, for instance, electrical and electronics engineering (Wang and Poor, 1999, Wang, Puig et al., 2018), mechanical engineering (Mousavinejad, Yang, Han, & Vlacic, 2018) and control engineering (Kurzhanski & Vályi, 1997). Fortunately, the set-membership filtering scheme has been put forward to estimate the state of system subject to UBB noise (Witsenhausen, 1968). The corresponding set-membership filtering problem has also attracted some initial interest in the context of WSNs (Ma et al., 2017, Orihuela et al., 2017, Wei et al., 2016, Xia et al., 2018).

For the problem of distributed filtering over digital WSNs, the first issue we face is to select or design an appropriate digital communication scheme for propagation. Among several well studied techniques, the so-called coding–decoding communication scheme has been very popular in networked systems (Wang, Wang et al., 2018, Wang, Wang, Liu et al., 2018). Recently, with the hope to mitigate the effect from the coding–decoding-induced inaccuracy/error, the control and filtering problems under coding–decoding scheme have been stirring interests of researchers from a wide range of communities, see, e.g. Lam, 2010, Li and Xie, 2012, Savkin and Cheng, 2007, Wang, Wang et al., 2018, Xiao et al., 2013, Zhou and Lu, 2009 and the references therein. Moreover, some initial research attention has been paid to the distributed filtering problem under coding–decoding scheme (Movaghati and Ardakani, 2014, Song et al., 2013). However, the corresponding research is far from adequate and still open.

Note that most of the coding–decoding-based networked control systems use the periodic coding–decoding scheme with fixed coding cycle, mainly because of the simplicity and convenience for analysis and design. However, as operated at fixed steps, the periodic scheme may lead to missing of critical information and unnecessary data encoding. In particular, the unnecessary data encoding would cause considerable waste of energy and bring special challenges on WSNs since all the individual sensing nodes are usually equipped with very limited energy. At present, considering the limited energy/bandwidth, there have been some results on the distributed set-membership filtering realized over analog wireless sensor networks, see e.g. Ding et al., 2020, Ge et al., 2019 and Orihuela, Millán, Roshany-Yamchi, and García (2018). The event-triggered mechanism and the negotiation strategy have been utilized in Ding et al. (2020) and Orihuela et al. (2018), respectively. Thus, a dynamic coding–decoding scheme will be designed in this paper, which can execute coding–decoding-based communication adaptively.

In response to the above discussions, it is our purpose in this paper to design a distributed set-membership filter for WSNs-based systems where the communication among sensing nodes is under the regulation of the proposed communication strategy. The specific challenges of this study are listed below. (1) For the proposed distributed set-membership filtering problem, how to design a dynamic coding–decoding strategy to schedule the information transmission between sensor nodes? (2) In the local filter of each sensor node, how to utilize the coding information from the neighboring nodes? (3) For the constructed distributed filter, what method can be used to derive sufficient conditions to obtain the desired filter?

The contributions of this paper are summarized as below. (1) A new coding–decoding strategy is designed that is capable of lowering communication frequency and saving energy. (2) Based on the proposed communication strategy, a local filter is constructed for each sensor node to estimate system state by using both local measurement and neighboring information. (3) For the constructed distributed filter, by employing linear matrix inequality technique and mathematical induction, sufficient conditions are obtained to ensure pre-specified filtering objective. Moreover, the explicit filter parameters can be obtained by solving certain recursive linear matrix inequalities. (4) An optimization problem is put forward to examine the locally optimal filter. Consequently, by overcoming the challenges mentioned above, the new coding–decoding-based distributed set-membership filtering problem is solved for a class of time-varying systems, which gives rise to the new coding–decoding-based distributed set-membership filtering algorithm.

Notation: Rn denotes the n-dimensional Euclidean space. trace[A] means the trace of matrix A. For a scalar aR, a is the largest integer not greater than a.

Section snippets

Problem formulation

In this paper, we estimate the state of the time-varying plant with the following dynamics: xk+1=Akxk+Bkwk,where xkRnx represents the state of the plant; wkRnω is the disturbance; Ak and Bk are known real-valued time-varying matrices with appropriate dimensions.

Assumption 1

The disturbance wk is confined to an ellipsoid defined by Wk{wk:wkTWk1wk1}, where Wk>0 is a known positive definite matrix describing the size and shape of the ellipsoid.

The state of the plant (1) is estimated by utilizing the

Filter design

Define decoding error ξi,kx̃i,kxˆi,k. We first give some analysis on the boundedness of ξi,k.

Lemma 1

For the proposed coding–decoding strategy, the decoding error ξi,k(iV) satisfies ξi,kTΘi,k−1ξi,k1,kthi,k>0,ξi,kTΩξi,k1,k=thi,ξi,k=0,k=0,where Ωdiagnx4(nxδ2).

Proof

See Appendix A.

We now ready to give the design approach of the desired distributed filter. A scalar σi,k is introduced to indicate the occurrence of coding. If k=thi, σi,k=1. Otherwise, σi,k=0.

Theorem 1

Consider the plant (1), the measurement outputs

Illustrative example

Example 1

Set N=3. The matrices and parameters of the specifics system under study are given next: Ak=1.05000.20.900.101.1,Bk=0.30.50.5,C=011101110C1,k=00.50,D1,k=0.3,C2,k=001,D2,k=0.2,C3,k=0.800,D3,k=0.25,δ=0.05,Θi,k=I3,i=1,2,3.

Considering noises wk=0.5sin(k), v1,k=0.6sin(2k), v2,k=0.5cos(k) and v3,k=0.7cos(3k), set Wk=1, V1,k=1, V2,k=1 and V3,k=1 satisfying Assumption 1, Assumption 2.

The simulation results are depicted in Fig. 2, Fig. 3. Fig. 2 describes the Euclidean norm of estimation error ei,k (iV

Conclusion

In this paper, the dynamic coding–decoding strategy has been designed. The distributed set-membership filtering algorithm has been proposed for a class of time-varying systems under the scheduling of the proposed communication strategy.

Acknowledgments

The authors would like to thank the associate editor and the reviewers for their valuable comments and suggestions.

Lei Liu received the B.Sc. degree in automation from Nanjing Forestry University, Nanjing, China, in 2012. He is currently pursuing the Ph.D. degree in control science and engineering at the School of Automation, Nanjing University of Science and Technology, Nanjing, China. His current research interests include distributed filtering, multi-agent systems, and networked control systems.

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    Lei Liu received the B.Sc. degree in automation from Nanjing Forestry University, Nanjing, China, in 2012. He is currently pursuing the Ph.D. degree in control science and engineering at the School of Automation, Nanjing University of Science and Technology, Nanjing, China. His current research interests include distributed filtering, multi-agent systems, and networked control systems.

    Lifeng Ma received the B.Sc. degree in Automation from Jiangsu University, Zhenjiang, China, in 2004 and the Ph.D. degree in Control Science and Engineering from Nanjing University of Science and Technology, Nanjing, China, in 2010.

    He is currently a Professor in the School of Automation, Nanjing University of Science and Technology, Nanjing, China. His current research interests include nonlinear control and signal processing, variable structure control, distributed control and filtering, time-varying systems and multi-agent systems. He has published more than 40 papers in refereed international journals. He serves as an editor for Neurocomputing and International Journal of Systems Science. He is a very active reviewer for many international journals.

    Jian Guo received the B.Sc. degree in electrical technology and the Ph.D. degree in control theory and control engineering from Nanjing University of Science and Technology, Nanjing, China, in 1997 and 2002, respectively. Since 2002, he has been with the School of Automation, Nanjing University of Science and Technology, and was promoted as a Professor in 2013. His research interests include adaptive control, intelligent system and motion control.

    Jie Zhang received his B.Sc. degree in Automatic Control in 2002, his M.Sc. degree in Automatic Control in 2004 and his Ph.D. degree in Control Theory and Control Engineering in 2011, all from Nanjing University of Science and Technology, Nanjing, China. From April 2013 to March 2014, he was an Academic Visitor in the Department of Information Systems and Computing, Brunel University London, UK. He is currently a Professor in the School of Automation, Nanjing University of Science and Technology. His current research interests include stochastic systems, networked systems, stochastic control and neural networks.

    Yuming Bo received his B.Sc. degree in Automatic Control in 1984, his M.Sc. degree in Automatic Control in 1987 and Ph.D. degree in Control Theory and Control Engineering in 2005, all from Nanjing University of Science and Technology, Nanjing, China. He is now a Professor of Control Theory and Control Engineering in the School of Automation at Nanjing University of Science and Technology, Nanjing, China. His research interests include stochastic control and estimation, computer communication and programming. He has published more than 50 papers in referred journals and served as an associate editor for two journals.

    This work was supported in part by the National Natural Science Foundation of China under Grant 61773209, the six talent peaks project in Jiangsu Province, China under Grant XYDXX-033, the Fundamental Research Funds for the Central Universities, China under Grant 30916011337 and the Postdoctoral Science Foundation of China under Grant 2014M551598. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Andrea Garulli under the direction of Editor Torsten Söderström.

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