Nonlinear control for algae growth models in the chemostat

Abstract

This paper deals with output feedback control of phytoplanktonic algae growth models in the chemostat. The considered class of model is of variable yield type, meaning that the ratio between the environmental nutrient absorption rate and the cells’ growth rate varies, which is different from classical bioprocesses assumptions. On the basis of weak qualitative hypotheses on the analytical expressions of the involved biological phenomena (which guarantee robustness of the procedure toward modeling uncertainties) we propose a nonlinear controller and prove its ability to globally stabilize such processes. Finally, we illustrate our approach with numerical simulations and show its benefits for biological laboratory experiments, especially for ensuring persistence of the culture facing classical experimental problems.

Appendices

A Nomenclature

Parameter values and units are according to [1] (for all the simulations):

Table 1 Nomenclature

B Barbalat’s lemma

Lemma 2 (Barbalat, [6])

Let $\phi : {\mathbb R}\rightarrow {\mathbb R}$ be a uniformly continuous function on [0, ∞). Suppose that $\lim_{t\rightarrow +\infty}\int_0^t \phi(t)\hbox{d}t$ exists and is finite. Then:

$$\lim_{t\rightarrow +\infty}\phi(t)=0$$

C Corollary of the Perron-Frobenius theorem

Definition 1, (Metzler matrix, [8])

A is a Metzler matrix iff all its off-diagonal elements are non-negative.

Corollary 1, (Perron-Frobenius, [17])

Let A be an irreducible Metzler matrix. Then, λ _M , the eigenvalue of A of largest real part is real, and the elements of its associated eigenvector v _M are positive. Moreover, any eigenvector of A with non-negative elements belongs to span {v _M }.

Remark 4

Actually, Smith proves more in his corollary (see [17]), but the remaining results are of no use for our purpose.

D Asymptotically autonomous systems

Definition 2 [13, 19]

Consider the systems:

$$\dot x= f(t,x)$$

(13)

$$\dot y=g(y)$$

(14)

with f(x,t) and g(x) continuous in x and t and locally Lipschitz in x on an open set $\theta\subset{\mathbb R}^n.$ System Eq. 13 is asymptotically autonomous with limit system Eq. 14 if for all compact $K\subset\theta:$

$$\lim_{t\rightarrow+\infty}f(t,x)=g(x),\quad \forall x \in K $$

Theorem 1 [13, 19]

Consider the asymptotically autonomous system Eq. 13 with limit system Eq. 14 . Let e be a locally asymptotically stable equilibrium of Eq. 14 and ω the ω -limit set of a bounded solution x(t,x ₀) of Eq. 13 . If ω contains a point y ₀ such that the forward trajectory y (t,y ₀) of Eq. 14 converges to e, then:

$$\lim_{t\rightarrow+\infty}x(t)=e$$

Remark 5

Observe that in our case, each forward trajectory of the limit system Eq. 11 initiated in $\mathcal{E}.$converges toward $s^\star,$ and each trajectory of the asymptotically autonomous system Eq. 12 converges to $\mathcal{E}.$ Then, each trajectory of the asymptotically autonomous system Eq. 12 converges to $s^\star.$