We consider now the coupling of observer (ll) with the estimation (a, в) provided by observer

(8) . This amounts to study the robustness of the second observer with respect to uncertainties of parameters a and в.

Proposition 5. Consider the observer (11) with (a, f) replaced by (a(-), в(:)) such that (a(t),f(t)) e [a-,a+] x [в-,в+], vt > 0,

then there exists positive numbers b>2, cT2, d2 such that for any є > 0 there exists 02 large enough to guarantee the inequalities

m(t) — m< є + b2e-c2t||£(0) — Z(0)|| (l2)

|x(t) — x(t) < є + d-2 ‘в(Є) — в + b2e-c21 ||t(0) — Z(0)|| (l3)

for any t > 0.

Proof. As for the proof of Proposition 3, we fix an initial condition of system (3) and consider the bounded set Oi = {yl (t) }t>Q. The dynamics of e = Z — Z is

e = (A + Ke2C)e + (ф(yl, Z, a, f,) — f (yl, Z, a, в))v

For 62 large enough, one has — 02/2 + /bL/ij < 0 and then, using again (18), obtains

from which we deduce the exponential convergence of the error vector e toward any arbitrary small neighbourhood of 0 provided that -2 is large enough.

The Lipschitz continuity of the map lm( ) w. r.t. Z uniformly iny Є O provides the inequality

(12) .

For the estimation of x(-), one has the inequality

x — x = m — eZ2 <e — в\Ы + в+ h — Z2

provided the estimation (13), the variable £2 being bounded. |

Corollary 2. At any time t > 0, the coupled observer

f (yi, Z, a (s2), в&))/ 9l)

integrated for si Є [0, min(t, r(t))j and s2 Є [0, t],with

T(t)= yi(0) — yi (t)+ У2(0) — У2(t),

а (s2) = sat (a-, a+, la (yi (min(s2, r(t)))), jf(min(s2, r(t)))), e(s2) = sat (в-, , lp (yi (min(s2, r(t)))), £(min(s2, r(t)))),

provides the estimations

m(t) = l m(yi (t), £(t)) ^

(x(t), xd (t)) = (-в(t)^Z2(t), y2(t) + в (t)Z2 (t)) .

The convergence of the estimator is exponentially practical, provided 9i and 92 to be sufficiently large.

2. Numerical simulations

We have considered a Monod’s growth function (2) with the parameters pmax = 1 and Ks = 100 and the initial conditions s(0) = 50, x(0) = 1, xd(0) = 0. The parameters to be reconstructed have been chosen, along with a priory bounds, as follows:

parameter		к	m
value	0.2	0.2	0.1
bounds	[0.1,0.3]	[0.1,0.3]	[0.05,0.2]

Those values provide an effective growth that is reasonably fast (s(0) is about Ks/2), and a value T (see (6)) we find by numerical simulations is not too small. For the time interval 0 < t < tmax = 80, we found numerically the interval 0 < t < Tmax = T(tmax) ~ 37.22 (see Figure 1). For the first observer, we have chosen a gain parameter 9i = 3 that provides

a small error on the estimation of the parameters a and в at time Tmax (see Figures 2 and 3). These estimations have been used on-line by the second observer, with 82 = 2 as a choice

for the gain parameter. On Figures 4 and 5, one can see that the estimation error get small when the estimations provided by the first observer are already small. Simulations have been also conducted with additive noise on measurements y1 and y2 with a signal-to-noise

ratio of 10 and a frequency of 0.1Hz (see Figures 6 and 7). In presence of a low frequency

noise (as it can be usually assumed in biological applications), one finds a good robustness of the estimations of parameters а, в and variables x and xd. Estimation of parameter m is more affected by noise. This can be explained by the structure of the equations (5): the estimation of m is related to the second derivative of both observations yi and y2, and consequently is more sensitive to noise on the observations.

3.

Conclusion

The extension of the Monod’s model with an additional compartment of dead cells and substrate recycling terms is no longer identifiable, considering the observations of the substrate concentration and the total biomass. Nevertheless, we have shown that the model can be written in a particular cascade form, considering two time scales. This decomposition allows to design separately two observers, and then to interconnect them in cascade. The first one works on a bounded time scale, explaining why the system is not identifiable at steady state, while the second one works on unbounded time scale. Finally, this construction provides a practical convergence of the coupled observers. Each observer has been built considering the variable high-gain technique proposed in [10] with an explicit construction of Lipschitz extensions of the dynamics, similarly to the work presented in [19]. Other choices of observers techniques could have been made and applied to this particular structure. We believe that such a decomposition might be applied to other systems of interest, that are not identifiable or observable at steady state.