Tianju Sui

Stability of MMSE state estimators over lossy networks using linear coding

This paper studies the state estimation problem for a stochastic discrete-time system over a lossy channel where the packet loss is modeled as an independent and identically distributed binary process. To counter the effect of random packet loss, we propose a linear coding method to preprocess the measured output, and prove that the coded output is information preserving when packet loss is void and is information enhancing when packet loss is present. An optimal state estimator under the minimum mean square error (MMSE) criterion is derived for the coded output when subject to packet loss. The maximum packet loss rate for ensuring a stable estimator is then derived and shown to be very close to a well-known lower bound. Also considered is a compressed linear coding method where the measured output is first compressed onto a lower dimensional space before encoding, and it is shown that the similar packet rate condition for stability holds.

Stability conditions for multi-sensor state estimation over a lossy network

This paper studies a networked state estimation problem for a spatially large linear system with a distributed array of sensors, each of which offers partial state measurements. A lossy communication network is used to transmit the sensor measurements to a central estimator where the minimum mean square error (MMSE) state estimate is computed. Under a Markovian packet loss model, we provide necessary and sufficient conditions for the stability of the estimator for any diagonalizable system in the sense that the mean of the state estimation error covariance matrix is uniformly bounded. In particular, the stability conditions for the second-order systems with an i.i.d. packet loss model are explicitly expressed as simple inequalities in terms of the largest open-loop pole and the packet loss rate.

Convergence analysis of Gaussian belief propagation for distributed state estimation

Belief propagation (BP) is a well-celebrated iterative optimization algorithm in statistical learning over network graphs with vast applications in many scientific and engineering fields. This paper studies a fundamental property of this algorithm, namely, its convergence behaviour. Our study is conducted through the problem of distributed state estimation for a networked linear system with additive Gaussian noises, using the weighted least-squares criterion. The corresponding BP algorithm is known as Gaussian BP. Our main contribution is to show that Gaussian BP is guaranteed to converge, under a mild regularity condition. Our result significantly generalizes previous known results on BPs convergence properties, as our study allows general network graphs with cycles and network nodes with random vectors. This result is expected to inspire further investigation of BP and wider applications of BP in distributed estimation and control.

Stability of the Kalman filtering with two periodically switching sensors over lossy networks

This paper considers the stability of Kalman filtering of a discrete-time stochastic system using two periodically switching sensors over a network subject to random packet losses, which is modeled by an independent and identically distributed Bernoulli process. It is proved that this problem can be converted into the stability of Kalman filtering using two sensors at each time instant, where the measurements of each sensor are transmitted via an independent lossy channel. Some necessary and sufficient conditions for stability of the estimation error covariance matrices are respectively established, and the effect of the periodic switching on the stability is revealed. Their implications and relationships with related results in the literature are discussed.

Stability of Kalman Filtering with Multiple Sensors Involving Lossy Communications

Abstract We study a networked state estimation problem for a linear system with multiple sensors, each of which transmits its measurements to a central estimator via a lossy communication network for computing the minimum mean-square-error (MMSE) state estimate. Under a general Markov packet loss process, we establish necessary and sufficient conditions for the stability of the estimator for any diagonalizable system in the sense that the mean of the state estimation error covariance matrix is uniformly bounded. For the second-order systems under an i.i.d. packet loss model, the stability condition is expressed as a simple inequality in terms of open-loop poles and the packet loss rate.

Kalman filtering with intermittent observations using measurements coding

This paper studies the state estimation problem of a stochastic discrete-time system over a lossy channel. The packet loss is modeled as an independent and identically distributed (i.i.d.) binary process. To reduce the effect of the random packet losses on the stability of the minimum mean square error estimator, we propose a linear coding method on the measurement of the system. In particular, the linear combination of the current and finite previous measurements is to be transmitted to the estimator over the lossy channel. Some necessary and sufficient conditions for the stability of the estimator are established, and the advantage of the linear coding method is exploited.

Accuracy analysis for distributed weighted least-squares estimation in finite steps and loopy networks

Distributed parameter estimation for large-scale systems is an active research problem. The goal is to derive a distributed algorithm in which each agent obtains a local estimate of its own subset of the global parameter vector, based on local measurements as well as information received from its neighbors. A recent algorithm has been proposed, which yields the optimal solution (i.e., the one that would be obtained using a centralized method) in finite time, provided the communication network forms an acyclic graph. If instead, the graph is cyclic, the only available alternative algorithm, which is based on iterative matrix inversion, achieving the optimal solution, does so asymptotically. However, it is also known that, in the cyclic case, the algorithm designed for acyclic graphs produces a solution which, although non optimal, is highly accurate. In this paper we do a theoretical study of the accuracy of this algorithm, in communication networks forming cyclic graphs. To this end, we provide bounds for the sub-optimality of the estimation error and the estimation error covariance, for a class of systems whose topological sparsity and signal-to-noise ratio satisfy certain condition. Our results show that, at each node, the accuracy improves exponentially with the so-called loop-free depth. Also, although the algorithm no longer converges in finite time in the case of cyclic graphs, simulation results show that the convergence is significantly faster than that of methods based on iterative matrix inversion. Our results suggest that, depending on the loop-free depth, the studied algorithm may be the preferred option even in applications with cyclic communication graphs.

Stability analysis for Kalman filters with random measurement matrices

This paper addresses the stability of a Kalman filter when measurements are intermittently available due to an unreliable communication channel between sensors and the estimator. This intermittent behaviour can be modelled as a random and time-varying measurement system. We consider a general discrete-time system with a random measurement matrix and study the stability condition for the associate Kalman filter. By deep dissecting the system structure, the necessary and sufficient stability condition for the Kalman filter is given. We also give methods to checking the stability condition for specific linear systems. Our results generalize previously known stability conditions where the system matrix structure is restricted.

Convergence analysis for Guassian belief propagation: Dynamic behaviour of marginal covariances

Despite of its wide success in many distributed statistical learning applications, the well-known Gaussian belief propagation (BP) algorithm still lacks sufficient understanding at the theoretical level. This paper studies the convergence of Gaussian BP by analyzing the dynamic behaviour of the marginal covariances. We show, under a mild technical assumption, that the information matrices (i.e., the inverses of marginal covariances) are guaranteed to converge exponentially to positive-definite matrices. The convergence rate is explicitly characterized. This result is a key step to the understanding of the dynamic behaviour of the BP iterations.