Optimal Sensor Placement Design for Profile Estimation of Distributed Parameter Systems

This work addresses the problem of sensor placement design (SPD) in distributed parameter systems (DPS), where states evolve both in space and time. Existing approaches, often focused on linear systems and/or steady-state conditions, may not be appropriate for SPD of nonlinear DPS undergoing significant dynamic variations in the operating conditions. Additionally, many existing approaches aim to minimize estimation error properties only at discrete locations, thus leading to inferior profile performance as well as restricting sensor placement choices. In the current work, we propose an SPD approach for nonlinear DPS, considering dynamic variations in operating conditions. Toward this end, utilizing Lagrange polynomials and the orthogonal collocation (OC) method, we propose an objective function for SPD that accounts for estimation error properties across the entire spatial domain over the time period of interest. The approach leads to well-behaved estimates throughout the spatial domain and also enables sensor locations to be continuous variables, thereby eliminating the need for pre-specifying candidate locations. This results in a nonlinear programming optimization problem which does not involve integer variables. Extensive simulations of transport reaction processes described by partial differential equations (PDEs) validate the effectiveness of our SPD approach for state estimation in DPS. Furthermore, we also establish analogies between the optimal sensor placement problem in DPS and the optimal design of experiments problem, providing a conceptual framework for posing various other types of sensor placement objectives.