Stabilization of chemostats using feedback linearization

Stabilization via feedback linearization of two-species competition models based in the chemostat is considered. Competition is for a single growth-limiting resource in the first model, and for two perfectly substitutable resources in the second. The dilution rate and the input concentration of one nutrient are taken as input controls in each model. The main issues we investigate are limitations on stabilizable states and stabilizer design due to the structure of the linearized system. It is shown that determination of feasible stabilizable states requires computation of an equilibrium submanifold in the state space. Once this is accomplished, the potential exists for failure of a proportional-derivative stabilizer strategy due to singular structure in the resulting closed-loop dynamics caused by failure of the linearizing coordinate transformation to be a global diffeomorphism. This is exhibited explicitly in the three-dimensional model, and it is shown numerically that similar phenomena persist in four dimensions.