208 C. Papadimitriou 23.3 Optimal Sensor Placement Formulation Information theory and utility functions are used to formulate the optimal sensor placement problem so that the most reliable response reconstruction is achieved that is robust to modeling and input uncertainties. The objective is to select the sensor locations (and DOF) to maximize the information contained in the data for predicting the output response QoI. Using Lindley’s work [10] and extending the expected utility to include the uncertainty in the model parameters, one maximizes the expected utility function U δ = ϒ ! u δ;z,ϕ,y p z,y,ϕ|δ dz dy dϕ (23.9) that quantifies the usefulness of learning from the data in predicting output QoI, where u δ;z,ϕ,y is the utility function given a particular value of the model parameter set ϕ and the data y, p z,y,ϕ|δ = p z|y,ϕ,δ p y|ϕ,δ p ϕ , p z|y,ϕ,δ is the posterior uncertainty in the response prediction given the datay and the model parameter set ϕ, p y|ϕ,δ is the uncertainty in the outcome y given the model parameters, and p ϕ is the uncertainty in the model parameters. A rational choice of the utility function is the information gained by the data, quantified by the Kullback-Leibler divergence [11] between the prior and posterior probability distribution given an outcome y, obtained from an experimental design δ and the model parameters ϕ. The expected utility function is an average of the utility function over all possible values of the response predictions as they are inferred from the data, and all the possible data outcomes. In Eq. (23.9), the expected utility function has been extended to a robust measure that takes into account the uncertainty in the model parameters ϕ that are usually uncertain. It can be readily shown that the expected utility function can be formulated in terms of the change in the information entropy before and after the data are collected, given by U δ = Hz ϕ p ϕ dϕ− ϒ Hz|D y,ϕ,δ p y|ϕ,δ dy p ϕ dϕ (23.10) whereHz ϕ is the prior information entropy given the model parameter set, andHz|D y,ϕ,δ is the posterior information entropy given the data set and the model parameter set. For Gaussian probability distribution of the response vector z(t), the posterior information entropy given the values of the data set and the model parameter set ϕ is given by Hz|D y,ϕ,δ = 1 2 nz ln(2π) − 1 2 lndet Pz δ,ϕ (23.11) and thus it depends on the covariance P δ,ϕ of the error of the state and input estimation, as well as the sensor locations, while it is independent of the data. Taking into account that the prior information entropy Hz ϕ in Eq. (23.10) is constant, independent of the sensor configuration vector δ, and that the posterior information entropy Hz|D y,ϕ,δ does not depend on the data, the expected utility function finally takes the form U δ =c + 1 2 ! lndet 5P δ,ϕ 6p ϕ dϕ (23.12) which is a probability integral over the space of uncertain parameters ϕ. The integral Eq. (23.12) represents the robust information entropy over all possible values of the model parameters quantified by the PDF p ϕ . The multidimensional integral can be evaluated using Monte Carlo techniques or sparse grid methods [12].
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