# 1An Error Bound for the Sensor Scheduling Problem

## Abstract

Notation: Let A be the semi-definite cone, namely, the set of all the positive semidefinite matrices. Denote by λmin(·) and λmax(·) the smallest and the largest eigenvalues, respectively, of a given matrix in A. Let R+ and Z+ be the set of nonnegative real numbers and integers, respectively. Let ‖ · ‖ be the standard Euclidean norm. Denote by | · | the cardinality of a give set. I. PROBLEM FORMULATION Consider the following linear time-invariant stochastic system defined over a finite time horizon TN = {0, 1,..., N − 1}: x(t+ 1) = Ax(t) + w(t), ∀t ∈ TN, (1) where x(t) ∈ Rn is the state of the system and w(t) is the process noise. The initial state, x(0), is assumed to be Gaussian with zero mean and covariance matrix Σ0, i.e., x(0) ∼ N (0,Σ0). At each time step, M sensors are available to take measurement. The dynamics of the ith sensor is given by: yi(t) = Cix(t) + vi(t), ∀t = 0,..., N, (2) where yi(t) ∈ Rp and vi(t) ∈ Rp are the measurement output and measurement noise of the ith sensor at time t, respectively. We assume that the process noise and all the measurement noises are mutually independent Gaussian white noises given by: w(t) ∼ N (0,Σw), vi(t) ∼ N (0,Σ v i). Define λ−w = λmin(Σw) and λ−v = mini∈M{λmin(Σvi)}. Assume that λ−w> 0 and λ−v> 0. Denote by Mt the set of all the sequences of sensor indices of length t ≤ N. An element σ ∈ Mt is called a t-horizon sensor schedule. Under a given N-horizon sensor schedule σ ∈ MN, the measurement sequence is determined by: y(t) = yσ(t)(t) = Cσ(t)x(t) + vσ(t)(t), ∀t = 0,..., N