Feedback control for chemostat models

Abstract

It is shown that a chemostat with two organisms can be made coexistent by means of feedback control of the dilution rate. Remaining freedom in the feedback law can be used to guarantee robustness or improve particular performance indices. Unfortunately a topological property prevents coexistence by feedback control for chemostats with more than two organisms. We apply our results to control bioreactors aimed at producing commercial products through genetically altered organisms. In all our results the coexistence takes its simplest form: a global asymptotically stable equilibrium point in the interior of the non-negative orthant.

Global Output Feedback Stabilization of a Chemostat With an Arbitrary Number of Species

“…This is at odds with the observation that in real ecological systems, it is common for many species to coexist in equilibrium on one nutrient source. This paradox has motivated a great deal of research: many contributions are devoted to modifying the model (1) to ensure coexistence of the species. Generally speaking, the past literature devoted to this problem can be classified into two categories.…”

“…The other category of papers is devoted to the control problem consisting in determining feedback laws which ensure persistence, when D and s in are regarded as inputs which can be chosen by an operator. In [1] and in [3], stabilizing feedbacks laws depending only on a sum of the species concentrations are used to stabilize a chemostat with two species. The paper [9] is concerned with the problem of stabilizing a periodic trajectory of the system (1) when there are two species.…”

Stabilization in a Two-Species Chemostat With Monod Growth Functions

“…See [2], [4], [12] for the fundamental role of chemostats in bioengineering. The basic model for a (well mixed) chemostat with two competing species iṡ…”

“…This is at odds with the observation that in real ecological systems, it is common for many species to coexist in equilibrium on one limiting nutrient. This paradox has motivated a great deal of research [4], [6]. Also, the species concentrations may not be available for measurement, and there may be actuator errors caused e.g.…”

Global stability for a model of competition in the chemostat with microbial inputs

“…b) Inputs as function of the state variables: models where D becomes a function of the state variables (called a feedback in the framework of control theory) as in [5], [9], [27] (all of them in a two-dimensional framework) and [6], [20] in a threedimensional framework. c) Heterogeneity of the liquid medium, which was described by using either PDE (see [11], [19], [32], [34]) or gradostat equations (see [15], [33] and references therein).…”