Nonlinear POMDPs for Active State Tracking with Sensing Costs

Abstract: Active state tracking is needed in object classification, target tracking, medical diagnosis and estimation of sparse signals among other various applications. Herein, active state tracking of a discrete-time, finite-state Markov chain is considered. Noisy Gaussian observations are dynamically collected by exerting appropriate control over their information content, while incurring a related sensing cost. The objective is to devise sensing strategies to optimize the trade-off between tracking performance and sensing cost. A recently proposed Kalman-like estimator \cite{ZoisTSP14} is employed for state tracking. The associated mean-squared error and a generic sensing cost metric are then used in a partially observable Markov decision process formulation, and the optimal sensing strategy is derived via a dynamic programming recursion. The resulting recursion proves to be non-linear, challenging control policy design. Properties of the related cost functions are derived and sufficient conditions are provided regarding the structure of the optimal control policy enabling characterization of when passive state tracking is optimal. To overcome the associated computational burden of the optimal sensing strategy, two lower complexity strategies are proposed, which exploit the aforementioned properties. The performance of the proposed strategies is illustrated in a wireless body sensing application, where cost savings as high as $60\%$ are demonstrated for a $4\%$ detection error with respect to a static equal allocation sensing strategy.