Sensor Placement for Optimal Kalman Filtering: Fundamental Limits, Submodularity, and Algorithms
Abstract
In this paper, we focus on sensor placement in linear dynamic estimation, where the objective is to place a small number of sensors in a system of interdependent states so to design an estimator with a desired estimation performance. In particular, we consider a linear timevariant system that is corrupted with process and measurement noise, and study how the selection of its sensors affects the estimation error of the corresponding Kalman filter over a finite observation interval. Our contributions are threefold: First, we prove that the minimum mean square error of the Kalman filter decreases only linearly as the number of sensors increases. That is, adding extra sensors so to reduce this estimation error is ineffective, a fundamental design limit. Similarly, we prove that the number of sensors grows linearly with the system's size for fixed minimum mean square error and number of output measurements over an observation interval; this is another fundamental limit, especially for systems where the system's size is large. Second, we prove that the logdet of the error covariance of the Kalman filter, which captures the volume of the corresponding confidence ellipsoid, with respect to the system's initial condition and process noise is a supermodular and nonincreasing set function in the choice of the sensor set. Therefore, it exhibits the diminishing returns property. Third, we provide efficient approximation algorithms that select a small number sensors so to optimize the Kalman filter with respect to this estimation error the worstcase performance guarantees of these algorithms are provided as well. Finally, we illustrate the efficiency of our algorithms using the problem of surfacebased monitoring of CO2 sequestration sites studied in Weimer et al. (2008).
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 arXiv:
 arXiv:1509.08146
 Bibcode:
 2015arXiv150908146T
 Keywords:

 Mathematics  Optimization and Control;
 Electrical Engineering and Systems Science  Systems and Control
 EPrint:
 Article appeared in ACC 2016