On the Nonlinear Dynamic State Reconstruction Problem for Chemical/Biochemical Reaction Systems in the Presence of Model Uncertainty

Abstract

A new approach to the unmeasurable state reconstruction problem for nonlinear chemical reaction systems in the presence of model uncertainty is proposed. In particular, a new robust nonlinear state estimation method is developed that explicitly uses all the available useful information associated with: (i) a dynamic model inevitably characterized by uncertainty, and (ii) a set of sensor measurements in order to accurately reconstruct other key quantities/variables that cannot be measured on-line due to physical and/or technical limitations. The problem of interest is conveniently formulated and addressed within the context of singular partial differential equations (PDE) theory, leading to a nonlinear state estimator that possesses a state-dependent gain computed through the solution of a system of first-order singular PDEs. A set of necessary and sufficient conditions is presented that ensure the existence and uniqueness of a locally analytic solution to the aforementioned system of singular PDEs, and a series solution method that can be easily implemented via a MAPLE code is developed. Under these conditions, the convergence of the estimation error or the mismatch between the actual unmeasurable states and their estimates is analyzed and characterized in the presence of model uncertainty. Finally, the performance of the proposed nonlinear etimator and its convergence properties are evaluated in an illustrative biochemical reaction system that exhibits nonlinear behavior coupled with parametric uncertainty, and the estimation objective is to accurately reconstruct the unmeasurable substrate concentration using the available cell mass concentration measurements and the model of the system under consideration.