Optimal estimation of multidimensional data with limited measurements

Recent results indicate how to optimally schedule transmissions of a measurement to a remote estimator when there are limited uses of the communication channel available. The resulting optimal encoder and estimation policies solve an important problem in networked control systems when bandwidth is limited. Previous results were obtained only for scalar processes, and the previous work was unable to address questions regarding informational relevance. We extend the state of-the art by treating the case where the source process and measurements are multi-dimensional. To this end, we develop a non-trivial re-working of the underlying proofs. Specifically, we develop optimal encoder policies for Gaussian and Gauss Markov measurement processes by utilizing a measure of the informational value of the source data. Explicit expressions for optimal hyper-ellipsoidal regions are derived and utilized in these encoder policies. Interestingly, it is shown in this paper that analytical expressions for the hyper-ellipsoids exist only when the state´s dimension is even; in odd dimensions (as in the scalar case) the solution requires a numerical look up (e.g., use of the erf function). We also have extended the previous analyses by introducing a weighting matrix in the quadratic cumulative cost function, whose purpose is to allow the system designer to designate which states are more important or relevant to total system performance.