Polynomial Chaos Based Method for State and Parameter Estimation
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Two new recursive approaches have been developed to provide accurate estimates for posterior moments of both parameters and system states while making use of the generalized Polynomial Chaos (gPC) framework for uncertainty propagation. The main idea of the generalized polynomial chaos method is to expand random state and input parameter variables involved in a stochastic differential/difference equation in a polynomial expansion. These polynomials are associated with the prior pdf for the input parameters. Later, Galerkin projection is used to obtain deterministic system of equations for the expansion coefficients. The first proposed approach (gPC-Bayes) provides means to update prior expansion coefficients by constraining the polynomial chaos expansion to satisfy the desired number of posterior moment constraints derived from the Bayes' rule. The second proposed approach makes use of the minimum variance formulation to update polynomial chaos expansion coefficients. The main advantage of proposed methods is that they not only provide point estimate for the state and parameters but they also provide statistical confidence bounds associated with these estimates. Numerical experiments involving four benchmark problems are considered to illustrate the effectiveness of the proposed ideas.