[Submitted on 13 Jun 2017]

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

$dot{x} = Ax,;y_i = C_ix,;iin{1,2,ldots, m}$. 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),;kin{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^{Atau}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

$trightarrow infty$ at a rate which can be controlled.

## Submission history

From: Lili Wang [view email]

**[v1]**

Tue, 13 Jun 2017 19:38:31 UTC (96 KB)

## More Stories

## Shell sees 2024 aviation fuel demand recovering to pre-pandemic levels

## Nam Joo-hyuk returns to big screen in crime action ‘Remember’

## Don Spike apprehended over drug use charges