A Vector Lyapunov Function Characterization of Input-to-State Stability with Application to Robust Global Stabilization of the Chemostat

This paper develops a novel characterization of uniform input-to-state stability (UISS) in terms of vector Lyapunov functions. The motivation is coming from the problem of robust global stabilization of the chemostat and in particular from anaerobic digestion processes, which are characterized by narrow stability regions and extreme sensitivity to external perturbations, if operated under optimal steady state conditions. The theoretical results enable the derivation of precise robustness margins for a proportional output feedback law that is globally stabilizing the chemostat under nominal conditions. The derivations and results are extended for the case where a feedforward measurement is incorporated in the control law for the chemostat, and it is shown that this leads to improved robustness margins.