Optimal location of measurements for parameter estimation of distributed parameter systems

Systematic methodologies for the optimal location of spatial measurements, for efficient estimation of parameters of distributed systems, are investigated. A review of relevant methods in the literature is presented, and a comparison between the results obtained with three distinctive existing techniques is given. In addition, a new approach based on the Proper Orthogonal Decomposition (POD), to address this important problem is introduced and discussed with the aid of illustrative benchmark case studies from the literature. Based on the results obtained here, it was observed that the method based on the Gram determinant evolution (Vande Wouwer et al., 2000), does not always produce accurate results. It is strongly dependent on the behaviour of sensitivity coefficients and requires extensive calculations. The method based on max–min optimisation (Alonso, Kevrekidis, Banga, & Frouzakis, 2004) assigns optimal sensor locations to the positions where system outputs reach their extrema values; however, in some cases it produces more than one optimal solution. The D-optimal design method, Uciński (2003, June 18–20), produces as results the optimal number and spatial positions of measurements based on the behaviour (rather than the magnitude) of the sensitivity functions. Here we show that the extrema values of POD modes can be used directly to compute optimal sensor locations (as opposed e.g. to Alonso, Kevrekidis, et al., 2004, where PODs are merely used to reduce the system and further calculations are needed to compute sensor locations). Furthermore, we demonstrate the equivalence between the extrema of POD modes and of sensitivity functions. The added value of directly using PODs for the computation of optimal sensor locations is the computational efficiency of the method, side-stepping the tedious computation of sensitivity coefficient Jacobian matrices and using only system responses and/or experimental results directly. Furthermore, the inherent combination of model reduction and sensor location estimation in this method becomes more important as the complexity of the original distributed parameter system increases.