Development of grey box state estimators for systems subjected to time correlated unmeasured disturbances

Unmeasured disturbances, which arise from uncertainties in the physical input sources, are commonly encountered in a process operation. For the purpose of developing Bayesian state estimators, such disturbances have been traditionally treated as Gaussian white noise processes. In practice, however, such disturbances are often correlated in time and the simplistic white noise assumption may not hold. Thus, to generate accurate estimates of the states, it is essential to obtain a reasonably accurate characterisation of the dynamics associated with the unmeasured disturbances. In this work, a systematic approach has been developed for identifying discrete time stochastic disturbance models, which captures the dynamics associated with such unmeasured disturbances. Under certain simplifying assumptions, the discrete time unmeasured disturbance models are combined with a continuous time mechanistic model to derive a discrete nonlinear grey box model. The grey box model is further used to formulate a nonlinear Bayesian state estimator. A constrained optimisation problem, that maximizes the log likelihood function of the innovation sequence generated by the state estimator, is formulated and solved for estimation of the parameters of the unmeasured disturbance model and the measurement noise covariance from the input–output data. The efficacy of this approach is demonstrated by simulating a benchmark continuous fermenter system and using experimental data obtained from a heater-mixer setup. The simulation studies demonstrate that the proposed approach is able to identify correlated disturbance models that closely match the characteristics of the true unmeasured disturbance models.