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## Distributing the Kalman filters for large-scale systems

Venue: | IEEE Trans. on Signal Processing, http://arxiv.org/pdf/0708.0242 |

Citations: | 53 - 11 self |

### Citations

3707 |
A new approach to linear filtering and prediction problems
- Kalman
- 1960
(Show Context)
Citation Context ...stributed estimation, information filters, iterative methods, Kalman filtering, large-scale systems, matrix inversion, sparse matrices. I. INTRODUCTION CENTRALIZED implementation of the Kalman filter =-=[1]-=-, [2], although possibly optimal, is neither robust nor scalable to complex large-scale dynamical systems with their measurements distributed on a large geographical region. The reasons are twofold: i... |

1035 |
Optimal Filtering
- Anderson, Moore
- 1979
(Show Context)
Citation Context ...e elements outside the band defined by the Lth upper and Lth lower diagonal are 0. 4) The DICI Algorithm and Information Filters: We implement the Kalman filter in the Information filter format [11], =-=[20]-=-, which propagates in time the information matrices (inverse of the error covariances). The information matrices are inverted by the DICI algorithm in a distributed manner. Iterative matrix inversion ... |

580 | and R.Bucy, “New results in linear filtering and prediction theory
- Kalman
- 1961
(Show Context)
Citation Context ...uted estimation, information filters, iterative methods, Kalman filtering, large-scale systems, matrix inversion, sparse matrices. I. INTRODUCTION CENTRALIZED implementation of the Kalman filter [1], =-=[2]-=-, although possibly optimal, is neither robust nor scalable to complex large-scale dynamical systems with their measurements distributed on a large geographical region. The reasons are twofold: i) the... |

418 | Fast linear iterations for distributed averaging
- Xiao, Boyd
- 2004
(Show Context)
Citation Context ...are observed by several subsystems. To fuse this shared information, we implement a fusion algorithm using bipartite fusion graphs, which we introduced in [12], and local average consensus algorithms =-=[13]-=-. The interactions required by the fusion procedure are constrained to a small neighborhood and with a particular choice of the communication topology the observation fusion procedure remains single h... |

353 |
Kronecker Products and Matrix Calculus with Applications
- Graham
- 1981
(Show Context)
Citation Context ...cretization of (1) as where is the discrete-time index and the matrix is given by . .. . .. (2) (3) (4) . .. (5) where and the constants , , and are in terms of , in (2), and is the Kronecker product =-=[27]-=-. Putting , the discrete-time dynamical system takes the form In the above model, are the state initial conditions, is the model matrix, is the state noise vector and is the state noise matrix. Remark... |

310 |
Reducing the bandwidth of sparse symmetric matrices
- Cuthill, McKee
- 1969
(Show Context)
Citation Context ...assumptions, exhibit banded structures [31]–[33]. As a final comment, systems that are sparse but not localized can be converted to sparse and localized by using matrix bandwidth reduction algorithms =-=[34]-=-. (6) 2) Observation Model: Let the system described in (6) be monitored by a network of sensors. Observations at sensor and time are where is the local observation matrix for sensor , is the number o... |

277 | Kalman filtering with intermittent observations
- Sinopoli, Schenato, et al.
- 2004
(Show Context)
Citation Context ...o achieve convergence. It is worth mentioning that, with a finite number of iterations (true for any practical implementation), the resulting Kalman filter does not remain optimal. References [4] and =-=[9]-=- incorporate packet losses, intermittent observations, and communication delays in the data fusion process. Because they replicate an -dimensional Kalman filter at each sensor, they communicate and in... |

135 |
Distributed Kalman filters with embedded consensus filters
- Olfati-Saber
- 2005
(Show Context)
Citation Context ...orporate the distributed observations, which is also referred to in the literature as “data fusion”; see also [6]. Data fusion for Kalman filters over arbitrary communication networks is discussed in =-=[7]-=-, using iterative consensus protocols provided in [8]. The consensus protocols in [8] are assumed to converge asymptotically; thus, between any two time steps of the Kalman filter, the consensus proto... |

127 | Positive definite completions of partial Hermitian matrices. Linear Algebra and its Applications
- Grone, Johnson, et al.
- 1984
(Show Context)
Citation Context ... introduced in [19]. 1 In the error covariance domain, this approximation corresponds to the determinant/entropy maximizing completion of a partially specified (L-band, in our case) covariance matrix =-=[15]-=-–[18]. Such a completion results into a covariance matrix whose inverse is L-banded. We refer to a matrix as an L-banded matrix (L 0), if the elements outside the band defined by the Lth upper and Lth... |

121 | Consensus filters for sensor networks and distributed sensor fusion - Olfati-Saber, Shamma - 2005 |

91 |
Decentralized estimation and control for multisensor systems
- Mutambara
- 1998
(Show Context)
Citation Context ...In such problems, replication of the global dynamics in the local Kalman filters is either not practical or not possible. Kalman filters with reduced-order models have been studied in, e.g., [10] and =-=[11]-=- to address the computation burden posed by implementing th-order models. In these works, the reduced models are decoupled, which is suboptimal, as important coupling among the system variables is ign... |

63 |
Decentralized structures for parallel Kalman filtering
- Hashemipour, Roy, et al.
- 1988
(Show Context)
Citation Context ...r target tracking [3]–[5]. The problem in such scenarios reduces to how to efficiently incorporate the distributed observations, which is also referred to in the literature as “data fusion”; see also =-=[6]-=-. Data fusion for Kalman filters over arbitrary communication networks is discussed in [7], using iterative consensus protocols provided in [8]. The consensus protocols in [8] are assumed to converge ... |

57 | On a decentralized active sensing strategy using mobile sensor platforms in a network
- Chung, Gupta, et al.
- 1914
(Show Context)
Citation Context ...ter at each sensor, which is only practical, when the dimension of the state is small, for example, when multiple sensors mounted on a small number of robot platforms are used for target tracking [3]–=-=[5]-=-. The problem in such scenarios reduces to how to efficiently incorporate the distributed observations, which is also referred to in the literature as “data fusion”; see also [6]. Data fusion for Kalm... |

54 | Distributed Kalman filtering based on consensus strategies
- Carli, Chiuso, et al.
- 2008
(Show Context)
Citation Context ... ;E , is such that there exists a path from any vertex in S to any other vertex in S . 7In this case, we assume that the communication is fast enough so that the consensus algorithm can converge, see =-=[40]-=- for a discussion on distributed Kalman filtering based on consensus strategies. The convergence of the consensus algorithm is shown to be geometric and the convergence rate can be increased by optimi... |

50 |
A structure preserving model for power system stability analysis
- Bergen, Hill
- 1981
(Show Context)
Citation Context ...d localized structure) also occur. In image processing, the dynamics at a pixel depends on neighboring pixel values [29], [30]; power grid models, under certain assumptions, exhibit banded structures =-=[31]-=-–[33]. As a final comment, systems that are sparse but not localized can be converted to sparse and localized by using matrix bandwidth reduction algorithms [34]. (6) 2) Observation Model: Let the sys... |

40 |
Šiljak, Decentralized Control of Complex Systems
- D
- 1991
(Show Context)
Citation Context ... in a meaningful way. If the local error covariances evolve independently at each subsystem they may lose any coherence with the centralized error covariance. For example, in the estimation scheme in =-=[14]-=-, the coupled states are applied as inputs to the local observers, but, the error covariances remain decoupled and no structure of the centralized error covariance is retained by the local filters. To... |

37 | Optimal control of spatially distributed systems
- Motee, Jadbabaie
(Show Context)
Citation Context ... PDEs. We can relax this to sparse and localized matrices as when the coupling among the states decays with distance (in an appropriate measure); for example, see the spatially distributed systems in =-=[28]-=-. We mention briefly two other examples where such discrete-space-time models (with sparse and localized structure) also occur. In image processing, the dynamics at a pixel depends on neighboring pixe... |

37 | Sensor networks with random links: Topology design for distributed consensus
- Kar, Moura
- 2008
(Show Context)
Citation Context ...x for the consensus iterations using semidefinite programming [41]. The communication topology of the sensor network can also be improved to increase the convergence speed of the consensus algorithms =-=[42]-=-. Authorized licensed use limited to: Carnegie Mellon Libraries. Downloaded on January 16, 2010 at 17:55 from IEEE Xplore. Restrictions apply.KHAN AND MOURA: DISTRIBUTING THE KALMAN FILTER FOR LARGE-... |

35 |
Noncausal Gauss-Markov random fields: parameter structure and estimation
- Balram, Moura
- 1993
(Show Context)
Citation Context ...; see, for instance, [15], [17], and [18], and the references within. Furthermore, such covariance matrices result from approximating the Gaussian error processes (25) to Gauss–Markov of the th order =-=[16]-=- (for , this has also been studied in [36]). Reference [17] presents an algorithm to derive the approximation that is optimal in Kullback–Leibler or maximum entropy sense in the class of all -banded m... |

30 | Tests of sequential data assimilation for retrieving profile soil moisture and temperature from observed Lband radiobrightness - Galantowicz, Entekhabi, et al. - 1999 |

27 | Simultaneous recursive displacement estimation and restoration of noisy-blurred image sequences
- Brailean, Katsaggelos
- 1995
(Show Context)
Citation Context ...ion briefly two other examples where such discrete-space-time models (with sparse and localized structure) also occur. In image processing, the dynamics at a pixel depends on neighboring pixel values =-=[29]-=-, [30]; power grid models, under certain assumptions, exhibit banded structures [31]–[33]. As a final comment, systems that are sparse but not localized can be converted to sparse and localized by usi... |

26 |
Modeling future cyber-physical energy systems
- Ilic, Khan
- 2008
(Show Context)
Citation Context ...alized structure) also occur. In image processing, the dynamics at a pixel depends on neighboring pixel values [29], [30]; power grid models, under certain assumptions, exhibit banded structures [31]–=-=[33]-=-. As a final comment, systems that are sparse but not localized can be converted to sparse and localized by using matrix bandwidth reduction algorithms [34]. (6) 2) Observation Model: Let the system d... |

24 | Sequential filtering for multi-frame visual reconstruction
- Chin, Karl, et al.
- 1992
(Show Context)
Citation Context ...y approximate the information matrices and ,tobe -banded matrices, and . We refer to the CIF with this approximation as the centralized -banded Information filter (CLBIF). This approach is studied in =-=[35]-=-, where the information loss between and , is given by the divergence Define the -dimensional global observation variables as (16) (17) and the -dimensional local observation variables at sensor as (1... |

23 | Preventing future blackouts by means of enhanced electric power systems control: From complexity to order - Ilic, Allen, et al. - 2005 |

21 | Matrices with Banded Inverses: Inversion Algorithm and Factorization of Gauss-Markov Processes
- Kavcic, Moura
- 2000
(Show Context)
Citation Context ...ediction step of the local Information filters is in Section VII. We conclude the paper with results in Section VIII and conclusions in Section IX. Appendix A discusses the -banded inversion theorem, =-=[17]-=-. II. BACKGROUND In this section, we motivate the type of applications and largescale dynamical systems of interest to us. The context is that of a time-varying random field governed by partial differ... |

21 |
Block Matrices with L-Block-Banded Inverse: Inversion Algorithms
- Asif, Moura
- 2005
(Show Context)
Citation Context ... is given by the filter step in (22a)–(22b) and the prediction step in (23a)–(23b), where the optimal information matrices and are replaced by their -banded approximations. The algorithms in [17] and =-=[37]-=- reduce the computational complexity of the CLBIF to , but the resulting algorithm is still centralized and deals with the -dimensional state. To distribute the CLBIF, we start by distributing the glo... |

16 |
Fully decentralized algorithm for multisensor Kalman filtering
- Rao, Durrant-Whyte
- 1991
(Show Context)
Citation Context ... filter at each sensor, which is only practical, when the dimension of the state is small, for example, when multiple sensors mounted on a small number of robot platforms are used for target tracking =-=[3]-=-–[5]. The problem in such scenarios reduces to how to efficiently incorporate the distributed observations, which is also referred to in the literature as “data fusion”; see also [6]. Data fusion for ... |

16 | A bibliography on semiseparable matrices
- Vandebril, Barel, et al.
(Show Context)
Citation Context ... ) can be obtained from the DICI-OR algorithm by setting . 1) Convergence of the DICI-OR Algorithm: We introduce the following definition. Definition 1: Let be the set of semiseparable matrices [48], =-=[49]-=- defined by is an -banded symmetric positive definite matrix (59) The iterate and the collapse steps of the DICI algorithm can be combined in matrix form as follows: Iterate Step: (60) Collapse Step: ... |

15 | Estimation in sensor networks: A graph approach - Zhang, Moura, et al. - 2005 |

13 |
Model distribution in decentralized multi-sensor data fusion. American Control Conference
- Berg, Durrant-Whyte
- 1991
(Show Context)
Citation Context ... of to . In such problems, replication of the global dynamics in the local Kalman filters is either not practical or not possible. Kalman filters with reduced-order models have been studied in, e.g., =-=[10]-=- and [11] to address the computation burden posed by implementing th-order models. In these works, the reduced models are decoupled, which is suboptimal, as important coupling among the system variabl... |

13 |
Sur les matrices complètement non negatives et oscillatoires
- GANTMACHER, KREĬN
- 1937
(Show Context)
Citation Context ...meter, ) can be obtained from the DICI-OR algorithm by setting . 1) Convergence of the DICI-OR Algorithm: We introduce the following definition. Definition 1: Let be the set of semiseparable matrices =-=[48]-=-, [49] defined by is an -banded symmetric positive definite matrix (59) The iterate and the collapse steps of the DICI algorithm can be combined in matrix form as follows: Iterate Step: (60) Collapse ... |

12 |
Reliable distributed estimation with intermittent communications
- Saligrama, Castanon
- 2006
(Show Context)
Citation Context ...ations to achieve convergence. It is worth mentioning that, with a finite number of iterations (true for any practical implementation), the resulting Kalman filter does not remain optimal. References =-=[4]-=- and [9] incorporate packet losses, intermittent observations, and communication delays in the data fusion process. Because they replicate an -dimensional Kalman filter at each sensor, they communicat... |

10 | Distributed Kalman filters in sensor networks: Bipartite fusion graphs
- Khan, Moura
(Show Context)
Citation Context ...overlap. In particular, some state variables are observed by several subsystems. To fuse this shared information, we implement a fusion algorithm using bipartite fusion graphs, which we introduced in =-=[12]-=-, and local average consensus algorithms [13]. The interactions required by the fusion procedure are constrained to a small neighborhood and with a particular choice of the communication topology the ... |

7 | Reduced-rank Kalman filters applied to an idealized model of the wind-driven ocean circulation - Buehner, Malanotte-Rizzoli - 2003 |

5 | An Extended Kalman-Bucy Filter for Atmospheric Temperature Profile Retrieval with a Passive Microwave Sounder - Ledsham, Staelin - 1978 |

5 | A block-parallel Newton method via overlapping epsilon decomposition - Zecevic, Siljak - 1994 |

4 | A generalized Levinson algorithm for covariance extension with application to multiscale autoregressive modeling - Frakt, Lev-Ari, et al. - 2003 |

4 |
Statistical processing of large image sequences
- Khellah, Fieguth, et al.
- 2005
(Show Context)
Citation Context ...iefly two other examples where such discrete-space-time models (with sparse and localized structure) also occur. In image processing, the dynamics at a pixel depends on neighboring pixel values [29], =-=[30]-=-; power grid models, under certain assumptions, exhibit banded structures [31]–[33]. As a final comment, systems that are sparse but not localized can be converted to sparse and localized by using mat... |

4 |
Gaussian families and a theorem of patterned matrices
- Barrett, Feinsilver
- 1978
(Show Context)
Citation Context ... and the references within. Furthermore, such covariance matrices result from approximating the Gaussian error processes (25) to Gauss–Markov of the th order [16] (for , this has also been studied in =-=[36]-=-). Reference [17] presents an algorithm to derive the approximation that is optimal in Kullback–Leibler or maximum entropy sense in the class of all -banded matrices approximating the inverse of the e... |

4 |
Modern graph theory
- Bela
- 1998
(Show Context)
Citation Context ... is a graph whose vertices can be divided into two disjoint sets X and S , such that every edge connects a vertex in X to a vertex in S , and there is no edge between any two vertices of the same set =-=[39]-=-. 4Mathematically, this can be captured as follows. Let the number of nonzero elements in the jth column, h , of the global observation matrix H be given by N and let be the set of the locations of th... |

4 |
Block methods for the solution of linear interval equations
- Garloff
- 1990
(Show Context)
Citation Context ...identity matrix I] is neither a direct extension nor a generalization of (block) Jacobi or Gauss–Seidel type iterative algorithms (that solve a vector version, Zs = b with s; b 2 ,ofZS = I; see [21], =-=[43]-=-–[45]). Using the Jacobi or Gauss-Seidel type iterative schemes for solving ZS = I is equivalent to solving n linear systems of equations, Zs = b; hence, the complexity scales linearly with n. Instead... |

4 |
Parallel Asynchronous Team Algorithms: Convergence and Performance Analysis
- Barfin, Kaszkurewicz, et al.
- 1996
(Show Context)
Citation Context ...te matrices, , for sufficiently small , see [21], an alternate convergence proof is provided in [46] via convex -matrices, whereas convergence for parallel asynchronous team algorithms is provided in =-=[47]-=-. Since the information matrix is the inverse of an error covariance matrix; is symmetric positive definite by definition, and the JOR algorithm always converges. Plugging in (46) gives us the central... |

3 | Distributed Iterate-Collapse inversion (DICI) algorithm for L−banded matrices
- Khan, Moura
(Show Context)
Citation Context ...y a low dimensional Gauss–Markov error process. 1 The assimilation procedure is carried out with a distributed iterate-collapse inversion [(DICI), pronounced die-see] algorithm, briefly introduced in =-=[19]-=-. 1 In the error covariance domain, this approximation corresponds to the determinant/entropy maximizing completion of a partially specified (L-band, in our case) covariance matrix [15]–[18]. Such a c... |

3 | Estimation of the Ocean Geoid near the Blake Escarpment Using GEOS-3 Satellite Altimetry Data - Brammer |

3 |
Estimation of the tropical Atlantic circulation from altimetry data using a reduced-rank stationary Kalman filter,” Interhemispheric water exchanges
- Buehner, Malanotte-Rizzoli, et al.
- 2003
(Show Context)
Citation Context ...ur goal here is to motivate how discrete linear models occur that exhibit a sparse and localized structure that we use to distribute the model in Section III. Examples include physical phenomena [22]–=-=[26]-=-, e.g., ocean/wind circulation and heat/propagation equations, that can be broadly characterized by a PDE of the Navier–Stokes type. These are highly nonlinear and different regimens arise from differ... |

3 | Designing fast distributed iterations via semi-definite programming’, presented at
- Xiao, Boyd
- 2004
(Show Context)
Citation Context ...e convergence of the consensus algorithm is shown to be geometric and the convergence rate can be increased by optimizing the weight matrix for the consensus iterations using semidefinite programming =-=[41]-=-. The communication topology of the sensor network can also be improved to increase the convergence speed of the consensus algorithms [42]. Authorized licensed use limited to: Carnegie Mellon Librarie... |

2 |
Zečević, “Overlapping block-iterative methods for solving algebraic equations
- Šiljak, I
- 1995
(Show Context)
Citation Context ...ity matrix I] is neither a direct extension nor a generalization of (block) Jacobi or Gauss–Seidel type iterative algorithms (that solve a vector version, Zs = b with s; b 2 ,ofZS = I; see [21], [43]–=-=[45]-=-). Using the Jacobi or Gauss-Seidel type iterative schemes for solving ZS = I is equivalent to solving n linear systems of equations, Zs = b; hence, the complexity scales linearly with n. Instead the ... |

2 |
Šiljak, “Jacobi and gauss-seidel iterations for polytopic systems: convergence via convex m-matrices
- Stipanovic, D
- 2000
(Show Context)
Citation Context ...enotes the spectral norm of a matrix. The JOR algorithm (46) converges for all symmetric positive definite matrices, , for sufficiently small , see [21], an alternate convergence proof is provided in =-=[46]-=- via convex -matrices, whereas convergence for parallel asynchronous team algorithms is provided in [47]. Since the information matrix is the inverse of an error covariance matrix; is symmetric positi... |