Scheduling Kalman Filters in Continuous Time

TL;DR

This paper develops optimal scheduling policies for Kalman filters observing multiple systems with resource constraints, providing bounds and practical policies that nearly achieve optimal performance.

Contribution

It introduces a tractable relaxation and decomposition approach for scheduling Kalman filters, deriving analytical policies in scalar cases and near-optimal open-loop policies in general.

Findings

Derived a convex relaxation providing performance bounds.

Decomposed the scheduling problem into smaller subproblems.

Developed policies that closely match the performance bounds.

Abstract

A set of N independent Gaussian linear time invariant systems is observed by M sensors whose task is to provide the best possible steady-state causal minimum mean square estimate of the state of the systems, in addition to minimizing a steady-state measurement cost. The sensors can switch between systems instantaneously, and there are additional resource constraints, for example on the number of sensors which can observe a given system simultaneously. We first derive a tractable relaxation of the problem, which provides a bound on the achievable performance. This bound can be computed by solving a convex program involving linear matrix inequalities. Exploiting the additional structure of the sites evolving independently, we can decompose this program into coupled smaller dimensional problems. In the scalar case with identical sensors, we give an analytical expression of an index policy proposed in a more general context by Whittle. In the general case, we develop open-loop periodic switching policies whose performance matches the bound arbitrarily closely.