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We study distributed estimation of dynamic random fields observed by a sparsely connected network of agents/sensors. The sensors are inexpensive, low power, and they communicate locally and perform computation tasks. In the era of large-scale systems and big data, distributed estimators, yielding robust and reliable field estimates, are capable of significantly reducing the large computation and communication load required by centralized estimators, by running local parallel inference algorithms. The distributed estimators have applications in estimation, for example, of temperature, rainfall or wind-speed over a large geographical area; dynamic states of a power grid; location of a group of cooperating vehicles; or beliefs in social networks. The thesis develops distributed estimators where each sensor reconstructs the estimate of the entire field. Since the local estimators have direct access to only local innovations, local observations or a local state, the agents need a consensus-type step to construct locally an estimate of their global versions. This is akin to what we refer to as distributed dynamic averaging. Dynamic averaged quantities, which we call pseudo-quantities, are then used by the distributed local estimators to yield at each sensor an estimate of the whole field. Using terminology from the literature, we refer to the distributed estimators presented in this thesis as Consensus+Innovations-type Kalman filters. We propose three distinct types of distributed estimators according to the quantity that is dynamically averaged: (1) Pseudo-Innovations Kalman Filter (PIKF), (2) Distributed Information Kalman Filter (DIKF), and (3) Consensus+Innovations Kalman Filter (CIKF). The thesis proves that under minimal assumptions the distributed estimators, PIKF, DIKF and CIKF converge to unbiased and bounded mean-squared error (MSE) distributed estimates of the field. These distributed algorithms exhibit a Network Tracking Capacity (NTC) behavior – the MSE is bounded if the degree of instability of the field dynamics is below a threshold. We derive the threshold for each of the filters. The thesis establishes trade-offs between these three distributed estimators. The NTC of the PIKF depends on the network connectivity only, while the NTC of the DIKF and of the CIKF depend also on the observation models. On the other hand, when all the three estimators converge, numerical simulations show that the DIKF improves 2dB over the PIKF. Since the DIKF uses scalar gains, it is simpler to implement than the CIKF. Of the three estimators, the CIKF provides the best MSE performance using optimized gain matrices, yielding an improvement of 3dB over the DIKF. Keywords: Kalman filter, distributed state estimation, multi-agent networks, sensor networks, distributed algorithms, consensus, innovation, asymptotic convergence, mean-squared error, dynamic averaging, Riccati equation, Lyapunov iterations, distributed signal processing, random dynamical systems.

Distributed networks, meaning a network in which several agents work together unanimously to perform some task in order to reach goals has become a field with a wide range of applications. One such applications may exist in the form of drones with a purpose of observing and detecting forest fires. In such applications it can be of paramount importance to be able to agree over some opinions or values between the agents. This value could be something such as event detection or a general direction to fly in. However in such a network there might not exist a central hub and it would not be possible for all drones to communicate directly with each other. In order for such a network to be able to reach consensus or agreement, values have to be exchanged between the agents. This thesis focuses on a subset of this problem known as distributed averaging. In the thesis it is investigated how a networks ability to detect forest fires and communicate both efficiently and quickly can change when the number of agents are adjusted in the network. The results showed that, when operating in a fixed area, for a small network of drones the increasing effective energy cost per drone were higher, than that of a larger network. It was also discovered that the speed at which a network could reach an agreement was not necessarily affected by the size of the network. But as the field area being observed was increased, adverse effects were observed in terms of communication and event detection. / Distribuerade nätverk bestående av flera agenter som har som uppgift att tillsammans nå gemensamma resultat har blivit allt mer populärt. Ett sådant användningsområde är hur drönare kan användas för att observera och upptäcka skogsbränder över en given yta. I en sådan tillämpning är det av stor vikt att drönarnätverket kan kommunicera och kongruera över värden nätverket delar med varandra. Dessa värden kan representera händelser som nätverket har som uppgift att upptäcka eller en riktning för drönarna att flyga i. Det är inte alltid garanterat att det finns en central kommunikationscentral för sådana nätverk, utan blir beroende på att kommunicera med varandra för att utbyta och kongruera över värden. Den här rapporten fokuserar på en avgränsad del av det ovanstående problemet som kallas för distribuerat konsensusvärde (eng. distributed averaging). Rapporten undersöker hur ett sådant nätverks konvergeringsförmåga, totala energikostnad samt täckning påverkas när fler drönare tillförs till nätverket. När arbetsytan var satt till statisk storlek visade resultaten att den tillförda energikostnaden per drönare var högre för små nätverk än för större nätverk. Det visades också att hastigheten som nätverket når ett kongruerande värde inte nödvändigtvis påverkas av storleken av nätverket. När arbetsytan ökade i takt med storleken på nätverket observerades däremot motsatt effekt för energikostnad och hastigheten för att nå ett konsensusvärde.