19 research outputs found

    Gossip and Distributed Kalman Filtering: Weak Consensus under Weak Detectability

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    The paper presents the gossip interactive Kalman filter (GIKF) for distributed Kalman filtering for networked systems and sensor networks, where inter-sensor communication and observations occur at the same time-scale. The communication among sensors is random; each sensor occasionally exchanges its filtering state information with a neighbor depending on the availability of the appropriate network link. We show that under a weak distributed detectability condition: 1. the GIKF error process remains stochastically bounded, irrespective of the instability properties of the random process dynamics; and 2. the network achieves \emph{weak consensus}, i.e., the conditional estimation error covariance at a (uniformly) randomly selected sensor converges in distribution to a unique invariant measure on the space of positive semi-definite matrices (independent of the initial state.) To prove these results, we interpret the filtered states (estimates and error covariances) at each node in the GIKF as stochastic particles with local interactions. We analyze the asymptotic properties of the error process by studying as a random dynamical system the associated switched (random) Riccati equation, the switching being dictated by a non-stationary Markov chain on the network graph.Comment: Submitted to the IEEE Transactions, 30 pages

    Distributed Signal Processing over Large-Scale Complex Systems

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    Large-scale and complex dynamical networks with high-dimension states have been emerging in the era of big data, which potentially generate massive data sets. To deal with the massive data sets, one promising method is the distributed collaboration strategy over the network. This dissertation proposes the schemes of distributed estimation and distributed quickest detection and also studies the performance of the distributed schemes with the large deviation analysis, which answers a fundamental question on how to quantify the rate at which the distributed scheme approaches the centralized performance. First, the distributed Kalman filtering scheme with the Gossip interaction among sensors is proposed to estimate the high-dimension states at each node, where sensors exchange their filtered states (estimates and error covariance) and propagate their observations via inter-sensor communications. The conditional estimation error covariance sequence at each sensor under this scheme is proven to evolve as a random Riccati equation (RRE) with Markov modulated switching. By formulating the RRE as a random dynamical system, it is shown that the network consensus over the estimation at each node is achieved. The large deviation analysis further shows that the distributed scheme converges to the optimal centralized one at an exponentially fast rate. By considering the energy and bandwidth constrains, a Quantized Gossip-based Interactive Kalman Filtering algorithm for scalar dynamic systems is also proposed, where the sensors exchange their quantized states with neighbors via inter-sensor communications. It is shown that, in the countable infinite quantization alphabet case, the network can still achieve weak consensus with the additional information loss caused by quantization. It is also proved that, under certain conditions, the network can also achieve weak consensus with the finite quantization alphabet, which is more restricted and practical. Then, the distributed quickest detection scheme is proposed with multiple rounds of inter-sensor communications to propagate observations during the sampling interval. By modeling the information propagation dynamics in the network as a Markov process, the two-layer large deviation analysis is used to analyze the performance of the distributed scheme. The first layer analysis proves that the probability of false alarm decays to zero exponentially fast with the increasing of the averaged detection delay, where the Kullback-Leibler (KL) information number is established as a crucial factor. The second-layer analysis shows that the probability of the rare event that not all observations are available at a sensor decays to zero at an exponentially fast rate when the number of communications increases, where the large deviation upper and lower bounds for this rate are also derived, based on which it is shown that the performance of the distributed algorithm converges exponentially fast to that of the centralized one, by proving that the defined distributed KL information number converges to the centralized KL information number

    Stochastics of Environmental and Financial Economics

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    Systems Theory, Contro

    Fundamental limits in Gaussian channels with feedback: confluence of communication, estimation, and control

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    The emerging study of integrating information theory and control systems theory has attracted tremendous attention, mainly motivated by the problems of control under communication constraints, feedback information theory, and networked systems. An often overlooked element is the estimation aspect; however, estimation cannot be studied isolatedly in those problems. Therefore, it is natural to investigate systems from the perspective of unifying communication, estimation, and control;This thesis is the first work to advocate such a perspective. To make Matters concrete, we focus on communication systems over Gaussian channels with feedback. For some of these channels, their fundamental limits for communication have been studied using information theoretic methods and control-oriented methods but remain open. In this thesis, we address the problems of characterizing and achieving the fundamental limits for these Gaussian channels with feedback by applying the unifying perspective;We establish a general equivalence among feedback communication, estimation, and feedback stabilization over the same Gaussian channels. As a consequence, we see that the information transmission (communication), information processing (estimation), and information utilization (control), seemingly different and usually separately treated, are in fact three sides of the same entity. We then reveal that the fundamental limitations in feedback communication, estimation, and control coincide: The achievable communication rates in the feedback communication problems can be alternatively given by the decay rates of the Cramer-Rao bounds (CRB) in the associated estimation problems or by the Bode sensitivity integrals in the associated control problems. Utilizing the general equivalence, we design optimal feedback communication schemes based on the celebrated Kalman filtering algorithm; these are the first deterministic, optimal communication schemes for these channels with feedback (except for the degenerated AWGN case). These schemes also extend the Schalkwijk-Kailath (SK) coding scheme and inherit its useful features, such as reduced coding complexity and improved performance. Hence, this thesis demonstrates that the new perspective plays a significant role in gaining new insights and new results in studying Gaussian feedback communication systems. We anticipate that the perspective could be extended to more general problems and helpful in building a theoretically and practically sound paradigm that unifies information, estimation, and control

    Generalized Sampling-Based Feedback Motion Planners

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    The motion planning problem can be formulated as a Markov decision process (MDP), if the uncertainties in the robot motion and environments can be modeled probabilistically. The complexity of solving these MDPs grow exponentially as the dimension of the problem increases and hence, it is nearly impossible to solve the problem even without constraints. Using hierarchical methods, these MDPs can be transformed into a semi-Markov decision process (SMDP) which only needs to be solved at certain landmark states. In the deterministic robotics motion planning community, sampling based algorithms like probabilistic roadmaps (PRM) and rapidly exploring random trees (RRTs) have been successful in solving very high dimensional deterministic problem. However they are not robust to system with uncertainties in the system dynamics and hence, one of the primary objective of this work is to generalize PRM/RRT to solve motion planning with uncertainty. We first present generalizations of randomized sampling based algorithms PRM and RRT, to incorporate the process uncertainty, and obstacle location uncertainty, termed as "generalized PRM" (GPRM) and "generalized RRT" (GRRT). The controllers used at the lower level of these planners are feedback controllers which ensure convergence of trajectories while mitigating the effects of process uncertainty. The results indicate that the algorithms solve the motion planning problem for a single agent in continuous state/control spaces in the presence of process uncertainty, and constraints such as obstacles and other state/input constraints. Secondly, a novel adaptive sampling technique, termed as "adaptive GPRM" (AGPRM), is proposed for these generalized planners to increase the efficiency and overall success probability of these planners. It was implemented on high-dimensional robot n-link manipulators, with up to 8 links, i.e. in a 16-dimensional state-space. The results demonstrate the ability of the proposed algorithm to handle the motion planning problem for highly non-linear systems in very high-dimensional state space. Finally, a solution methodology, termed the "multi-agent AGPRM" (MAGPRM), is proposed to solve the multi-agent motion planning problem under uncertainty. The technique uses a existing solution technique to the multiple traveling salesman problem (MTSP) in conjunction with GPRM. For real-time implementation, an ?inter-agent collision detection and avoidance? module was designed which ensures that no two agents collide at any time-step. Algorithm was tested on teams of homogeneous and heterogeneous agents in cluttered obstacle space and the algorithm demonstrate the ability to handle such problems in continuous state/control spaces in presence of process uncertainty
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