A Hybrid Observer for a Distributed Linear System with a Changing Neighbor Graph

TL;DR

This paper introduces a hybrid observer for distributed linear systems with time-varying neighbor graphs, enabling each agent to accurately estimate the system state exponentially fast despite changing communication links.

Contribution

A novel hybrid observer design that combines local continuous-time observers with recursive parameter estimation for distributed systems with dynamic neighbor relations.

Findings

Estimates converge exponentially fast to the true state.

Convergence rate can be controlled.

Effective under strong connectivity and joint observability assumptions.

Abstract

A hybrid observer is described for estimating the state of an $m>0$ channel, $n$-dimensional, continuous-time, distributed linear system of the form $\dot{x} = Ax,\;y_i = C_ix,\;i\in\{1,2,\ldots, m\}$. The system's state $x$ is simultaneously estimated by $m$ agents assuming each agent $i$ senses $y_i$ and receives appropriately defined data from each of its current neighbors. Neighbor relations are characterized by a time-varying directed graph $\mathbb{N}(t)$ whose vertices correspond to agents and whose arcs depict neighbor relations. Agent $i$ updates its estimate $x_i$ of $x$ at "event times" $t_1,t_2,\ldots $ using a local observer and a local parameter estimator. The local observer is a continuous time linear system whose input is $y_i$ and whose output $w_i$ is an asymptotically correct estimate of $L_ix$ where $L_i$ a matrix with kernel equaling the unobservable space of $(C_i,A)$. The local parameter estimator is a recursive algorithm designed to estimate, prior to each event time $t_j$, a constant parameter $p_j$ which satisfies the linear equations $w_k(t_j-\tau) = L_kp_j+\mu_k(t_j-\tau),\;k\in\{1,2,\ldots,m\}$, where $\tau$ is a small positive constant and $\mu_k$ is the state estimation error of local observer $k$. Agent $i$ accomplishes this by iterating its parameter estimator state $z_i$, $q$ times within the interval $[t_j-\tau, t_j)$, and by making use of the state of each of its neighbors' parameter estimators at each iteration. The updated value of $x_i$ at event time $t_j$ is then $x_i(t_j) = e^{A\tau}z_i(q)$. Subject to the assumptions that (i) the neighbor graph $\mathbb{N}(t)$ is strongly connected for all time, (ii) the system whose state is to be estimated is jointly observable, (iii) $q$ is sufficiently large, it is shown that each estimate $x_i$ converges to $x$ exponentially fast as $t\rightarrow \infty$ at a rate which can be controlled.