The aim of this work is estimating model parameters in simplified fundamental models intended for on-line process monitoring and state estimation. Stochastic error terms are often included in these differential equation models to account for process disturbances , time-varying parameters and model mismatch. Three maximum-likelihood-based parameter estimation techniques have been developed to estimate the model parameters and the intensity of the stochastic disturbances. These methods , which are designed to be less computationally intensive than Monte Carlo methods , rely on B-spline approximations for the state trajectories. Three different objective functions for parameter estimation have been developed using different approximations for likelihood functions. The first objective function is developed by approximating the expected value of the likelihood for the states and measurements , given the model parameters and disturbance intensities , using the mode of the corresponding probability distribution , assuming that measurement noise variances are known. The second method uses a Laplace approximation for the likelihood function of the measurements given the model parameters , disturbance intensity and noise variances. The third uses a more accurate fully Laplace approximation. These latter methods are more powerful than the first because they can be used when measurement noise variances are unknown. The three techniques , which result in relatively simple objective functions for parameter estimation , are compared with existing approximate maximum likelihood methods using a simple stochastic CSTR example.
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