Robust Control of a Competitive Environment in the Chemostat using Discontinuous Control Laws

Abstract

This work addresses the problem of robust stabilization of the concentration of two different species competing for a single limiting substrate. This stabilization is performed by means of discontinuous feedback control laws that ensure coexistence of all species. The control laws are designed considering bounded uncertainties on the kinetic rates.

Robust adaptive estimation in the competitive chemostat

“…However, by stopping the estimation after a fixed-time (which is upper estimated, see section 3.2), the monitoring is no longer depreciated by noise in the unobservable regions (compare the green and red curves in Figure 4), confirming the intended features of this observer. In a possible control problem in which each species have to be stabilized at a certain level (which can only be attained by setting S = S i , see [23]), the feature discussed in Remark 6 is of great interest.…”

Robust stabilization of competing species in the chemostat

“…and hence, selecting D as given by (16) and stressing that x 1 (0) can be either lower or greater than…”

“…Remark 5. A question that might arise concerns the switching nature of control laws (16), (18) and (21). Although the presented results are theoretical, the slow nature of the studied systems can be advantageous for their implementation (indeed, D(t) is actuated by an electro-mechanical pump and S in (t) can also be actuated by pumping media with a higher concentration into a lower one).…”

“…This work is an extension of a previous conference paper [16], but this extended version presents substantial improvements.…”

“…Let S(0) ≤ S m . Then controls(16),(18) with χ ∈ (0, µ 1 (S(0))) and > 0 ensure that x 1 (t) = x d 1 and S(t) = S m for all t ≥ T , where T > 0 is a finite time dependent on the initial conditions.…”