Sampling time design with misspecified Cramer-Rao bounds under input uncertainty

In the context of parameter estimation, under input uncertainty, the probability distribution function (pdf) of the measurement data mismatches the true pdf of measurement with accurate input. In this scenario, the Cramer-Rao bound (CRB), which is widely used in optimal experimental design, may become an overoptimistic lower bound on parameter estimation error covariance. To tackle this issue of mismatched measurement distribution subject to input uncertainty, in this work, a novel optimal sampling time design is proposed that employs the misspecified Cramer-Rao bound (MCRB), with the aim to collect informative data for high-quality parameter estimation. The MCRB is formed following the Cauchy-Schwarz inequality using the true pdf of the measurement, approximated by the statistics of measurement samples. In the numerical study, large samples from the input uncertainty space are generated and applied to the underlying system model; the outputs are calculated and used to approximate the true measurement pdf. The proposed MCRB-based sampling time design is formulated as a non-convex integer programming optimisation problem solved by a conjugate direction method. Three sampling time designs, the uniform sampling, the CRB-based design and the MCRB-based design, are tested on a benchmark enzyme reaction system model. The results show the necessity and superiority of using MCRB for experimental design under input uncertainty.