A hybrid observer for a distributed linear system with a changing neighbor graph

Abstract

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.

Finite-Time Distributed State Estimation over Time-Varying Graphs: Exploiting the Age-of-Information

“…(i) For a given sub-state j, there may not exist a common spanning tree rooted at node j in each graph G[k], k ∈ N. (ii) Assuming that a specific spanning tree rooted at node j is guaranteed to repeat at various points in time (not necessarily periodically), is restrictive, and qualifies as only a special case of conditions (C1)-(C3). (iii) Suppose for simplicity that G[k] is strongly-connected at each time-step (as in [11]), and hence, there exists a spanning tree T j [k] rooted at node j in each such graph. For estimating sub-state j, suppose consensus at time-step k is performed along the spanning tree T j [k].…”

“…Node 1 is the only node with non-zero measurements, and thus acts as the source node for this network. Suppose for simplicity that it perfectly measures the state at all timesteps, i.e., its state estimate isx 1 [k] = x[k], ∀k ∈ N. Given this setup, a standard consensus based state estimate update rule would take the form (see for example [5], [6], [11]): (7) where the weights w ij [k] are non-negative, and satisfy…”

“…To this end, we show that under a suitable selection of the observer gains, one can achieve finite-time convergence based on our approach. To put our results into context, it is instructive to compare them with the work closest to ours, namely [11]. In [11], the authors study a continuoustime analog of the problem under consideration, and develop a solution that leverages an elegant connection to the problem of distributed linear-equation solving [16].…”

“…For dynamic networks, there is much less work. Two recent papers [11] and [15] provide theoretical guarantees for certain specific classes of time-varying graphs; while the former proposes a two-time-scale approach, the latter develops a single-timescale algorithm. However, the extent to which such results can be generalized has remained an open question.…”

“…To put our results into context, it is instructive to compare them with the work closest to ours, namely [11]. In [11], the authors study a continuoustime analog of the problem under consideration, and develop a solution that leverages an elegant connection to the problem of distributed linear-equation solving [16]. In contrast to our technique, the one in [11] is inherently a two-time-scale approach, requires each node to maintain and update auxiliary state estimates, and works under the assumption that the communication graph is strongly-connected at every instant.…”