(266b) Relaxed Lyapunov Criteria for Robust Global Stabilization of Nonlinear Systems with Application to Chemostat Control

The motivation of the present work arises from control problems in continuous stirred microbial bioreactors, often called chemostats. In many applications, the design conditions for the chemostat represent a locally asymptotically stable steady state but the stability region could be too small to allow proper operation in the presence of disturbances. Thus, the need for control arises in the sense of enlargement of the stability region of the design steady state in the presence of disturbances ([1]). Under these circumstances, achieving global stability of the bioreactor in the presence of disturbances will be best possible outcome from the design of a control system.

The problem of existence and design of a feedback law that achieves robust global stabilization of a nonlinear system is closely related to the existence of a Robust Control Lyapunov Function. However, it may be a big challenge to derive a Control Lyapunov Function satisfying global properties for a given nonlinear system. In the present work, under appropriate hypotheses, we will derive relaxed Lyapunov-like sufficient conditions for Uniform Robust Global Asymptotic Stability.The Lyapunov-like conditions will be ?relaxed? in the sense that the Lyapunov differential inequality will not be required to hold over the entire state space, but only over an appropriate absorbing set, having the property that every trajectory of the system enters the set in finite time.

The theoretical results will be applied to a chemostat stabilization problem, where the dynamics is adequately represented by a two-state model involving the microbial biomass and the limiting organic substrate, with manipulated input the dilution rate. The growth rate of the microorganisms will be assumed follow Haldane kinetics, whereas the death rate of the microorganisms as well as the substrate consumption for cell maintenance will be accounted for assuming first-order kinetics. Moreover, the biomass balance will involve a time-varying uncertainty, accounting for the adjustment of the biomass to changes in the substrate levels. Applying the theoretical results on relaxed Lyapunov crireria, a robust globally stabilizing state feedback control law will be derived. Simulation results will also be used to illustrate the robustness properties of the derived control law.

References:

[1] I. Karafyllis, C. Kravaris, L. Syrou and G. Lyberatos, ?A Vector Lyapunov Function Characterization of Input-to-State Stability with Application to Robust Global Stabilization of the Chemostat?, European Journal of Control, Vol. 14, 2008, 47-61