Geometric methods for optimal sensor design

TL;DR

This paper introduces a geometric approach to design optimal sensors for the Kalman filter, minimizing estimation error by characterizing and computing sensors that commute with a specific operator.

Contribution

It provides a novel geometric characterization of optimal sensors, a gradient flow method for their computation, and convergence proofs, applicable to sensor and actuator design.

Findings

Optimal sensors commute with a specific positive definite operator.

Gradient flow converges to the unique optimal sensor.

Optimal sensors minimize estimation error for fixed SNR.

Abstract

An observer is an estimator of the state of a dynamical system from noisy sensor measurements. The need for observers is ubiquitous, with applications in fields ranging from engineering to biology to economics. The most widely used observer is the Kalman filter, which is known to be the optimal estimator of the state when the noise is additive and Gaussian. Because its performance is limited by the sensors to which it is paired, it is natural to seek an optimal sensor for the Kalman filter. The problem is however not convex and, as a consequence, many ad hoc methods have been used over the years to design sensors. We show in this paper how to characterize and obtain the optimal sensor for the Kalman filter. Precisely, we exhibit a positive definite operator which optimal sensors have to commute with. We furthermore provide a gradient flow to find optimal sensors, and prove the convergence of this gradient flow to the unique minimum in a broad range of applications. This optimal sensor yields the lowest possible estimation error for measurements with a fixed signal to noise ratio. The results presented here also apply to the dual problem of optimal actuator design.