Consensus based overlapping decentralized estimation with missing observations and communication faults☆
Introduction
A great deal of attention has been paid to the problem of decentralized state estimation of complex large scale systems. The key requirement in all the approaches is that a large scale system be modelled as an interconnection of subsystems, and that each subsystem have a decision maker (intelligent agent) associated with it. Depending on the available resources, an agent might have access to different local measurements, subsystem models, local estimators, and communication channels between the agents. Basic principles and structures for decentralized estimation have been discussed in, e.g., Sanders et al., 1974, Sanders et al., 1978, Šiljak (1991), Speyer (2004) and Tacker and Sanders (1980). One of the general design methodologies has been derived from the inclusion principle, using the expansion/contraction paradigm, where a complex system is expanded, decomposed into subsystems, and contracted back into the original system space after designing local estimators for the extracted subsystems, e.g., Iftar (1993a), Ikeda and Šiljak, 1981, Ikeda and Šiljak, 1986, Šiljak (1991) and Stanković, Chen, Mataušek, and Šiljak (1999). However, none of the existing methodologies is able to provide a systematic and general way of designing a communication strategy between the agents without recurring to a strong fusion center (Bar-Shalom and Li, 1995, Hashimpour et al., 1988, Zhu et al., 2001). Also, the important problems of intermittent observations and lossy networks have not yet been treated in this context.
On the other hand, important results have been obtained in the area of distributed iterations in parallel computation and distributed optimization, e.g., Bertsekas and Tsitsiklis, 1989, Bertsekas and Tsitsiklis, 2007, Tsitsiklis (1984), Tsitsiklis, Bertsekasand Athans (1986), Baran, Kaszkurewicz, and Bhaya (1996) and Blondel, Hendrickx, Olshevsky, and Tsitsiklis (2005), as well as in the fields of multi-agent systems and sensor networks, including numerous applications (see, e.g., Cassandras and Li (2005), Fax and Murray (2004), Gharavi and Kumar (2003), Jadbabaie, Lin, and Morse (2003), Lin, Francis, and Maggiore (2005), Moreau (2005), Olfati-Saber and Murray (2004), Olfati-Saber, Fax, and Murray (2007), Ren and Beard (2005) and Ren, Beard and Kingston (2005)). The noted references treat different problems, but they share a common general methodology: they all use some kind of agreement or consensus strategy between the agents. The decentralized state estimation problem itself is deeply embedded in this line of thought either implicitly, through the very definition of the consensus algorithms (e.g., see Ren et al. (2005)), or explicitly, where a dynamic consensus averaging strategy between multiple agents is used to obtain the required estimates (e.g., see Olfati-Saber (2005), Xiao and Boyd (2004) and Xiao, Boyd, and Lall (2005)). However, none of the mentioned schemes is aimed at establishing any type of real-time collaboration between local estimators in the overlapping decentralized estimation problem.
In this paper a novel state estimation algorithm for linear discrete-time systems is proposed based on: (1) overlapping system decomposition and design of local state estimators by intelligent agents according to their sensing and computing resources; (2) application of a consensus strategy enabling generation of the estimates of the whole state vector to all the agents in the network; (3) taking into account the influence of intermittent local observations and communication faults. The proposed algorithm represents a discrete-time counterpart of the continuous-time state estimation algorithm proposed in Stanković et al., 2007a, Stanković et al., 2009 and an extension to the state estimation problem of the parameter estimation algorithm proposed in Stanković, Stanković, and Stipanović (2007b), structurally resembling the algorithms considered in Tsitsiklis (1984) and Tsitsiklis et al. (1986).
The organization of the paper is as follows. The problem of overlapping decentralized state estimation is presented in Section 2. In Section 3 the proposed estimation algorithm is formulated. In Section 4, stability of the proposed estimation scheme is analyzed taking into account missing observation and communication faults, starting from the methodology of Nilsson (1996), Nilsson, Benhardsson, and Wittenmark (1998) and Sinopoli et al. (2004). Using a specially constructed matrix norm adapted to the structure of the network model, sufficient conditions for convergence to zero of the estimation error mean and boundedness of the mean-square estimation error are derived. The next section presents a strategy aimed at obtaining the consensus gains on the basis of minimization of the overall mean-square error. Section 6 is devoted to the interesting and important problem of denoising, with an emphasis on the connection between the suppression of the measurement noise influence and complexity of the multi-agent network. A number of characteristic examples is given within all the sections in order to illustrate the theoretically derived conclusions. They show applicability of the derived stability conditions (Section 4), characteristic average performance of the algorithm in the time domain (Section 4), practical aspects of the consensus scheme optimization and local adaptation (Section 5), and denoising effects for different network topologies (Section 6).
Section snippets
Overlapping decentralized estimation
Let a finite-dimensional discrete-time stochastic system be represented by where is the discrete-time instant, , , and are its state, output, input and measurement noise vectors, respectively, while , and are constant , and matrices, respectively. It is assumed that and are white zero-mean sequences of independent vector random variables with covariance matrices and ,
Consensus based estimator
Our task is to formulate a scheme which would provide to all the agents in the network reliable estimates of the whole state vector starting from the local estimation performed within each node and a decentralized communication strategy qualitatively uniform for all the nodes, in spite of missing measurements and communication faults. We propose the following algorithm based on the introduction of a consensus scheme:
Stability
In the stability analysis of the proposed estimator, we shall use the following results from the matrix analysis.
Lemma 1 Letbe a matrix norm having the propertyfor twomatricesandsatisfying(means that all the elements ofare nonnegative). Letbe any matrix norm and letbe partitioned into square blocks. Then,is a matrix norm, where ■
Lemma 2 Letbe anmatrix and. Then, there exists a matrix normsuch thatwhereis the spectral radius of a matrix
Optimization
In the previous section general conditions for stability of the proposed algorithm have been given: there has been, however, no indication of how to choose the parameters of the consensus scheme in practice. If the network topology (elements with nonzero probabilities) is chosen according to some a priori physical constraints, is completely defined by specifying diagonal matrices with nonnegative elements, giving relative weights to the communicated estimates. According to
Denoising
In this section we shall pay our attention to an important aspect of the analyzed scheme related to its capability to reduce the measurement noise influence by increasing the complexity of the multi-agent network. The basic problem of consensus averaging has been studied for different network topologies in, e.g., Xiao and Boyd (2004). It has been shown in Stanković et al. (2007a, 2007b, 2009) that, in the case of consensus based continuous-time estimation algorithms and stochastic approximation
Conclusion
In this paper a new algorithm for overlapping decentralized state estimation of discrete-time systems is proposed on the basis of a combination of local steady-state Kalman filters with intermittent observations and a consensus strategy connecting the local estimators with possible communication faults. Sufficient conditions for the convergence of the estimation error mean to zero and for preserving boundedness of the mean-square estimation error are derived using a new, specially defined
Srdjan S. Stanković got his Dipl. Ing. Degree from the Faculty of Electrical Engineering, University of Belgrade, Yugoslavia, in 1968. He got his M.Sc. and Ph.D. degrees from the same Faculty in 1972 and 1975, respectively. He was with the Institute for Nuclear Sciences, Vinca, Belgrade, Yugoslavia, from 1968 to 1972. Since 1973 he has been with the Faculty of Electrical Engineering, University of Belgrade, where he is currently Professor of Automatic Control and Head of the Department for
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Srdjan S. Stanković got his Dipl. Ing. Degree from the Faculty of Electrical Engineering, University of Belgrade, Yugoslavia, in 1968. He got his M.Sc. and Ph.D. degrees from the same Faculty in 1972 and 1975, respectively. He was with the Institute for Nuclear Sciences, Vinca, Belgrade, Yugoslavia, from 1968 to 1972. Since 1973 he has been with the Faculty of Electrical Engineering, University of Belgrade, where he is currently Professor of Automatic Control and Head of the Department for Signals and Systems. He held visiting positions at the Eindhoven University of Technology, Eindhoven, the Netherlands and at the Santa Clara University, Santa Clara, California. He is currently President of the National Council of Higher Education of Serbia. He has published numerous scientific papers in diverse fields, including estimation and identification, adaptive systems, digital signal processing, processing of medical images, large scale systems and neural networks. He has also been the leader of numerous scientific projects for government and industry. His research interests currently include large scale systems, networked control systems, vehicle formation control and fault detection and isolation.
Milos˘ S. Stanković received his Bachelor and Master degrees from the School of Electrical Engineering at the University of Belgrade in 2002 and 2004. He is currently pursuing his Ph.D. degree at the Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign. He is also affiliated with the Control and Decision Group of the Coordinated Science Laboratory at the UIUC. His research interests include decentralized estimation, coordination and control, multi-agent systems, mobile sensor networks, extremum seeking control, stochastic optimisation and machine learning.
Dus˘an M. Stipanović received his B.S. degree in Electrical Engineering from the University of Belgrade, Belgrade, Serbia, in 1994, and his M.S.E.E. and Ph.D. degrees (under the supervision of Professor Dragoslav S˘iljak) in Electrical Engineering from Santa Clara University, Santa Clara, California, in 1996 and 2000, respectively. Dr. Stipanović was an Adjunct Lecturer and Research Associate with the Department of Electrical Engineering at Santa Clara University in the period between 1998 and 2001, and a Research Associate in Professor Claire Tomlin’s Hybrid Systems Laboratory of the Department of Aeronautics and Astronautics at Stanford University, Stanford, California, in the period between 2001 and 2004. In 2004 he joined the University of Illinois at Urbana-Champaign as an Assistant Professor in the Department of Industrial and Enterprise Systems Engineering and a Research Assistant Professor in the Control and Decision Group of the Coordinated Science Laboratory. His research interests include decentralized control of interconnected systems with application to control of formations of vehicles and sensor networks, stability of discontinuous dynamic systems, differential game theory, and optimization with application to multiple vehicle coordination and systems safety verification. Dr. Stipanović is a member of the IEEE, the AIAA, and the International Society of Dynamic Games. In 2008, he received the Alexander von Humboldt research fellowship award. Currently, Dr. Stipanović is serving as an Associate Editor for the IEEE Transactions on Circuits and Systems I and he served as an Associate Editor for the IEEE Transactions on Circuits and Systems II from 2006 till 2008.
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The material in this paper was partially presented at 17th IFAC World Congress, Seoul, Korea, 2008 and American Control Conference, Seattle, 2008. This paper was recommended for publication in revised form by Associate Editor Giuseppe De Nicolao under the direction of Editor Ian R. Petersen.