Robust control for an uncertain chemostat model

Abstract

Abstract. In this paper we consider a control problem for an uncertain chemostat model with a general growth function and cell mortality. This uncertainty affects the model (growth function) as well as the outputs (measurements of substrate). Despite this lack of information, an upper bound and a lower bound for those uncertainties are assumed to be known a priori. We build a family of feedback control laws on the dilution rate, giving a guaranteed estimation on the unmeasured variable (biomass), and stabilizing the two variables in a rectangular set, around a reference value of the substrate. We give two realistic applications of this control law to a depollution process and to phytoplankton culture.

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

“…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.…”

Remarks on output feedback stabilization of two-species chemostat models

“…The chemostat model plays a central role in bioengineering and population biology [2,3,[6][7][8][9]13]. For the case of two species growing on a single limiting substrate in a biological reactor, the model has the form…”

“…In classical cases where D is constant, the well known competitive exclusion principle implies that generically, only one species can survive in an equilibrium [11]. We are interested in globally asymptotically stabilizing a componentwise positive equilibrium for (1) on (0, ∞) 3 , thereby guaranteeing persistence of both species, so we take D to be a nonconstant controller.…”

“…For example, [4] assumes that µ i (s) < 0 for all s ≥ 0 and i = 1, 2, which is the case for Michaelis-Menten growth functions, but can be difficult to check in practice. See also [3] for feedbacks that decrease in the output and stabilize a suitable rectangle in the state space. By contrast, our result stabilizes a componentwise positive set point, without any assumptions on the higher derivatives, and so allows more general uptake functions that violate the usual concavity conditions; see, e.g., [12] for the importance of nonconcave uptake functions, but this earlier work does not involve feedback design.…”

Stability and stabilization for models of chemostats with multiple limiting substrates

“…The basic mathematical model for the chemostat was first derived in [40,41]. See [9,10,[18][19][20]28,39,46] for more recent discussions on the chemostat and its role in microbial ecology. For well-mixed chemostats, the competitive exclusion principle [2] specified conditions on the growth rates under which only one species can persist generically.…”