Title :
Further Results on Stabilization of Periodic Trajectories for a Chemostat With Two Species
Author :
Mazenc, Frédéric ; Malisoff, Michael ; Harmand, Jérôme
Author_Institution :
Inst. Nat. de la Recherche Agronomique (INRA 2), Montpellier
Abstract :
We discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model. We provide new tracking results for chemostats with two species and one limiting substrate, based on Lyapunov function methods. In particular, we use a linear feedback control of the dilution rate and an appropriate time-varying substrate input concentration to produce a locally exponentially stable oscillatory behavior. This means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast. We also obtain a globally stable oscillatory reference trajectory for the species concentrations, using a nonlinear feedback control depending on the dilution rate and the substrate input concentration. This guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory. Finally, we construct an explicit Lyapunov function for the corresponding global error dynamics. We demonstrate the efficacy of our method in a simulation.
Keywords :
Lyapunov methods; biocontrol; closed loop systems; feedback; microorganisms; nonlinear control systems; substrates; time-varying systems; Lyapunov function methods; chemostat; dilution rate; global error dynamics; linear feedback control; nonlinear feedback control; oscillatory reference trajectory; periodic trajectories stabilization; substrate input concentration; time-varying substrate input concentration; Biological system modeling; Control theory; Equations; Feedback control; Inductors; Lyapunov method; Mathematical model; Robust stability; Systems biology; Trajectory; Chemostats; Lyapunov functions; stabilization; systems biology; tracking;
Journal_Title :
Automatic Control, IEEE Transactions on
DOI :
10.1109/TAC.2007.911315