Matrix Concentration Inequalities for Sensor Selection
arxiv(2024)
摘要
In this work, we address the problem of sensor selection for state estimation
via Kalman filtering. We consider a linear time-invariant (LTI) dynamical
system subject to process and measurement noise, where the sensors we use to
perform state estimation are randomly drawn according to a sampling with
replacement policy. Since our selection of sensors is randomly chosen, the
estimation error covariance of the Kalman filter is also a stochastic quantity.
Fortunately, concentration inequalities (CIs) for the estimation error
covariance allow us to gauge the estimation performance we intend to achieve
when our sensors are randomly drawn with replacement. To obtain a non-trivial
improvement on existing CIs for the estimation error covariance, we first
present novel matrix CIs for a sum of independent and identically-distributed
(i.i.d.) and positive semi-definite (p.s.d.) random matrices, whose support is
finite. Next, we show that our guarantees generalize an existing matrix CI.
Also, we show that our generalized guarantees require significantly fewer
number of sampled sensors to be applicable. Lastly, we show through a numerical
study that our guarantees significantly outperform existing ones in terms of
their ability to bound (in the semi-definite sense) the steady-state estimation
error covariance of the Kalman filter.
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