Moving horizon estimator for nonlinear and non-Gaussian stochastic disturbances

Abstract

Over the last two decades, moving horizon estimation (MHE) has increasingly been used by researchers and industrial practitioners of nonlinear model predictive control as an alternative to recursive Bayesian estimation schemes. The MHE formulations available in the literature assume that uncertainties in the state dynamics can be modelled as additive and Gaussian stochastic processes. In practice, the unmeasured disturbances affect the system dynamics in a much more complex way and Gaussian assumption may prove inadequate to represent their behaviour. However, unlike the recursive Bayesian estimation area, handling of non-additive and non-Gaussian state disturbances has not received much attention in the MHE literature. In this work, a novel formulation of MHE is proposed in the Bayesian framework to solve state estimation problems associated with systems subjected to nonlinear non-Gaussian stochastic disturbances. To begin with, a Bayesian formulation that maximizes the joint posterior density function of the initial state and stochastic inputs is developed for a batch of data, which is further extended to the moving horizon framework. The proposed formulation associates the random fluctuations affecting the system dynamics with the unmeasured inputs entering the systems. The probabilistic formulation of MHE together with modelling of the state disturbances arising from physical sources enables us to work with any (non-Gaussian or Gaussian) probability density function (PDF) and systematically handle the stochastic inputs affecting the dynamics in a nonlinear manner without making any simplifying assumptions. We proceed to show that the disturbances/states with constraints can also be incorporated in our proposed formulation and can be interpreted as following truncated probability density functions. Efficacy of the proposed MHE approach is demonstrated using three benchmark simulation case studies and an experimental case study. Simulation studies show that the proposed MHE framework is able to perform similar to EnKF, a sampling based sequential Bayesian estimation approach which is capable of handling nonlinear non-Gaussian disturbances. It is further shown that the proposed MHE performs much better than the conventional MHE approach which assumes additive Gaussian noise in the state dynamics. Thus, the proposed MHE scheme provides an optimization based alternative to sampling based estimation schemes, such as Ensemble Kalman filtering, which can handle state estimation problems when state and measurement densities are non-Gaussian.