Luenberger observer design for a dynamic system with embedded linear program, applied to bioprocesses

Microbial dynamics are fundamental to many processes in medicine and biotechnology. To model, estimate, and control such growth dynamics, methods of systems theory and control engineering are applied. In this paper, we use a modelling framework of dynamic constraint-based models, which appears as a system of ordinary differential equations of which the right hand side depends linearly on the optimal solution of a linear program (LP). This model describes the changes in the concentrations of extracellular metabolites and the amounts of all considered biomass components. The trajectories of the models are characterized by state-dependent switches among different optimal bases of the LP problem. The dynamics corresponding to each of these optimal bases are denoted as modes of the system. Based on such models, we study an online estimation problem in which the state variables are to be estimated from measurements according to a linear output equation. Due to the switching nature of the trajectories, we propose to use a bank of linear Luenberger observers for the different optimal bases of the LP. The system mode is estimated by a moving average of the error norm. An observer gain for each mode is determined by solving a set of Riccati equations with a common Lyapunov matrix. Simulation studies for a toy model with two bacterial species show the feasibility of this approach; from measurements of substrate and total biomass only, the observer is capable of correctly predicting the individual biomasses of the two species during exponential growth.