Approximate Maximum Likelihood Parameter Estimation for Nonlinear Dynamic Models: Application to a Laboratory-Scale Nylon Reactor Model

Abstract

Parameter estimation in dynamic models that are described by a combination of nonlinear algebraic and differential equations is a challenging problem. The complexity of the problem increases significantly if it is acknowledged that there are two different types of random errors that influence the measurements obtained from dynamic processes: measurement errors and process disturbances. Measurement errors are problematic because they can make it difficult for modelers to obtain reliable parameter estimates, but random process disturbances can be even more problematic because they influence the future behavior of the process and therefore future measurements of process outputs. For example, consider an unknown disturbance that influences the temperature in a chemical reactor. The change in temperature can alter the rates of chemical reactions and can influence several different types of process measurements and how they change over time. Modelers often have knowledge about the quality of the measurements that are available for parameter estimation (e.g., good estimates of measurement variance from repeated measurements or from sensor suppliers), but they do not have a priori knowledge about the quality of their model equations, which are only approximate representations of the true physical process because of disturbances that are not included in the model equations and simplifying assumptions that are made during model development. Approximate maximum likelihood parameter estimation (AMLE) is a novel parameter estimation algorithm that we recently developed to address the problem of parameter estimation in continuous-time nonlinear dynamic models, in which model discrepancies are significant. 4-7 A convenient way to account for modeling errors and process disturbances is to include Gaussian noise terms on the right-hand side of the state equations, thereby converting ordinary differential equation models into stochastic differential equation (SDE) models (as