Geometric methods for optimal sensor design

The Kalman-Bucy filter is the optimal estimator of the state of a linear dynamical system from sensor measurements. Because its performance is limited by the sensors to which it is paired, it is natural to seek optimal sensors. The resulting optimization problem is however non-convex. Therefore, many ad hoc methods have been used over the years to design sensors in fields ranging from engineering to biology to economics. We show in this paper how to obtain optimal sensors for the Kalman filter. Precisely, we provide a structural equation that characterizes optimal sensors. We furthermore provide a gradient algorithm and prove its convergence to the optimal sensor. This optimal sensor yields the lowest possible estimation error for measurements with a fixed signal-to-noise ratio. The results of the paper are proved by reducing the optimal sensor problem to an optimization problem on a Grassmannian manifold and proving that the function to be minimized is a Morse function with a unique minimum. The results presented here also apply to the dual problem of optimal actuator design.