On the optimal solutions of the infinite-horizon linear sensor scheduling problem

This paper studies the infinite-horizon sensor scheduling problem for linear Gaussian processes with linear measurement functions. Several important properties of the optimal infinite-horizon schedules are derived. In particular, it is proved that under some mild conditions, both the optimal infinite-horizon average-per-stage cost and the corresponding optimal sensor schedules are independent of the covariance matrix of the initial state. It is also proved that the optimal estimation cost can be approximated arbitrarily closely by a periodic schedule with a finite period, and moreover, the trajectory of the error covariance matrix under this periodic schedule converges exponentially to a unique limit cycle. These theoretical results provide valuable insights about the problem and can be used as general guidelines in the design and analysis of various infinite-horizon sensor scheduling algorithms.