Estimation and control performance for space-time point-process observations

Estimation and control problems are examined for a class of models involving a linear system, a quadratic cost, and observations that include a space-time point process as well as the familiar "signal in additive Wiener process" measurements. Motivation for this class of models is given in terms of position sensing and tracking for quantum-limited optical communication problems. These models include as special eases several simpler ones considered previously. As in the simpler cases, the optimum estimator is finite-dimensional and nonlinear, and the optimum controller separates into the optimum estimator followed by the certainty-equivalent control law. Although the optimum estimator and the optimum controller are finite-dimensional, the corresponding expected error covariance and optimum cost require infinite-dimensional calculations. This motivates the derivation of easily-computed upper and lower bounds on estimator and controller performance. The upper bounds are derived by evaluating exactly the performance of a parametrized family of suboptimum designs; one of these is identified as having smaller performance than any other, thus providing a minimal upper bound within this family. The lower bounds are obtained directly by calculations involving inequalities.