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Given a linear dynamical system, we consider the problem of selecting (atdesign-time) an optimal set of sensors (subject to certain budget constraints)to minimize the trace of the steady state error covariance matrix of the Kalmanfilter. Previous work has shown that this problem is NP-hard for certainclasses of systems and sensor costs; in this paper, we show that the problemremains NP-hard even for the special case where the system is stable and allsensor costs are identical. Furthermore, we show the stronger result that thereis no constant-factor (polynomial-time) approximation algorithm for thisproblem. This contrasts with other classes of sensor selection problems studiedin the literature, which typically pursue constant-factor approximations byleveraging greedy algorithms and submodularity of the cost function. Here, weprovide a specific example showing that greedy algorithms can performarbitrarily poorly for the problem of design-time sensor selection for Kalmanfiltering.

On the Complexity and Approximability of Optimal Sensor Selection for Kalman Filtering