A time-invariant, linear, distributed observer is described for estimating the state of an m > 0 channel, ndimensional continuous-time linear system of the formẋ = Ax, yi = Cix, i ∈ {1, 2, . . . , m}. The state x is simultaneously estimated by m agents assuming each agent i senses yi and receives the state zj of each of its neighbors' estimators. Neighbor relations are characterized by a constant directed graph N whose vertices correspond to agents and whose arcs depict neighbor relations. For the case when the neighbor graph is strongly connected, the overall distributed observer consists of m linear estimators, one for each agent; m − 1 of the estimators are of dimension n and one estimator is of dimension n + m − 1. Using results from classical decentralized control theory, it is shown that subject to the assumptions that (i) none of the Ci are zero, (ii) the neighbor graph N is strongly connected, (iii) the system whose state is to be estimated is jointly observable, and nothing more, it is possible to freely assign the spectrum of the overall distributed observer. For the more general case when N has q > 1 strongly connected components, it is explained how to construct a family of q distributed observers, one for each component, which can estimate x at a preassigned convergence rate.

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A hybrid observer is described for estimating the state of an m > 0 channel, n-dimensional, continuous-time, distributed linear system of the forṁ x = Ax, yi = Cix, i ∈ {1, 2, . . . , m}. The system's state x is simultaneously estimated by m agents assuming each agent i senses yi and receives appropriately defined data from each of its current neighbors. Neighbor relations are characterized by a time-varying directed graph N(t) whose vertices correspond to agents and whose arcs depict neighbor relations. Agent i updates its estimate xi of x at "event times" t1, t2, . . . using a local observer and a local parameter estimator. The local observer is a continuous time linear system whose input is yi and whose output wi is an asymptotically correct estimate of Lix where Li a matrix with kernel equaling the unobservable space of (Ci, A). The local parameter estimator is a recursive algorithm designed to estimate, prior to each event time tj, a constant parameter pj which satisfies the linear equations w k (tj −τ ) = L k pj +µ k (tj −τ ), k ∈ {1, 2, . . . , m}, where τ is a small positive constant and µ k is the state estimation error of local observer k. Agent i accomplishes this by iterating its parameter estimator state zi, q times within the interval [tj − τ, tj), and by making use of the state of each of its neighbors' parameter estimators at each iteration. The updated value of xi at event time tj is then xi(tj) = e Aτ zi(q). Subject to the assumptions that (i) none of the Ci are zero, (ii) the neighbor graph N(t) is strongly connected for all time, (iii) the system whose state is to be estimated is jointly observable, (iv) q is sufficiently large and nothing more, it is shown that each estimate xi converges to x exponentially fast as t → ∞ at a rate which can be controlled.

A distributed algorithm is proposed for solving a linear algebraic equation Ax = b over a multi-agent network, where A ∈ Rn ×n and the equation has a unique solution x * ∈ R n . Each agent knows only a subset of the rows of A b , controls a state vector x i (t) of size smaller than n and is able to receive information from its nearby neighbors. Neighbor relations are characterized by time-dependent directed graphs. It is shown that for a large class of time-varying networks, the proposed algorithm enables each agent to recursively update its own state by only using its neighbors' states such that all x i (t) converge exponentially fast to a specific part of x * of interest to agent i. Applications of the proposed algorithm include solving the least square solution problem and the network localization problem.

Recently, an opinion dynamics model has been proposed to describe a network of individuals discussing a set of logically interdependent topics. For each individual, the set of topics and the logical interdependencies between the topics (captured by a logic matrix) form a belief system. We investigate the role the logic matrix and its structure plays in determining the final opinions, including existence of the limiting opinions, of a strongly connected network of individuals. We provide a set of results that, given a set of individuals' belief systems, allow a systematic determination of which topics will reach a consensus, and which topics will disagreement in arise. For irreducible logic matrices, each topic reaches a consensus. For reducible logic matrices, which indicates a cascade interdependence relationship, conditions are given on whether a topic will reach a consensus or not. It turns out that heterogeneity among the individuals' logic matrices, including especially differences in the signs of the off-diagonal entries, can be a key determining factor. This paper thus attributes, for the first time, a strong diversity of limiting opinions to heterogeneity of belief systems in influence networks, in addition to the more typical explanation that strong diversity arises from individual stubbornness.