A new and improved MILP formulation to optimize observabitity, redundancy and precision for sensor network problems

New mathematical formulations of observability, redundancy, and an improved formulation for precision are provided which can be explicitly and analytically solved using mixed integer linear programming (MILP). By using the Schur complement found at the heart of both Gaussian elimination and Cholesky factorization for direct block matrix reduction and the variable classification and covariance calculations found in the reconciliation, regression, and regularization approach of Kelly, it is possible to efficiently optimize the overall instrumentation cost considering both estimability and variability as constraints during the branch-and-bound search of the MILP. Two illustrative examples are highlighted which minimize the cost of sensor placement subject to software and hardware redundancy of the measured variables, observability of the unmeasured variables, and their precision (i.e., inverse of their variance). This formulation is well suited to the problems of designing as well as retrofitting sensor locations in arbitrary networks.