Playing with the structure of the dynamics, we are able to write the model as a particular cascade of two sub-models. We first present a practical observer for the reconstruction of the parameters a and k using the observation y1 only, but with a change of time that depends on yi and y2. We then present a second observer for the reconstruction of the parameter m
and the state variables ж and, using both observations yi and y2 and the knowledge of the parameters a and k. Finally, we consider the coupling of the two observers, the second one using the estimations of a and k provided by the first one. More precisely, our model is of the form

Z = F(Z, P) , y = H(Z)

where F is our vector field with the state, parameters and observation vectors Z, P and y of dimension respectively 3, 3 and 2. We found a partition

Zi Pi dimZi = 1, dimPi = 2

Z2 P2 dimZ2 = 2, dimP2 = 1

y = ( yi ) = ( Hi (Z1 ’ )

W H2 (Z2)!

and the dynamics is decoupled as follows

1

Щу)

dt

Z2 = F2 (z2, Уl, P1, P2 )

with d<p(y) /dt > 0. Moreover, the following characteristics are fulfilled:

i. (Zi, Pi) is observable for the dynamics (Fi, Hi) i. e. without the term d<p(y)/dt,

ii. (Z2, P2) is observable for the dynamics (F2, H2) when Pi is known. Then, the consideration of two observers Fi(-) and F2(Pi, ■) for the pairs (Zi, Pi) and (Z2, P2) respectively, leads to the construction of a cascade observer

ir(Zp)=^ZbKin),

Ш(%)=Р2ІРЬЇ2,Р2,У2)

with r(t) = <p(y(t)) — ф(у(0)), that we make explicit below. Notice that the coupling of two observers is made by Pi, and that the term d^i(y)/dt prevents to have an asymptotic convergence when lim r(t) < +ro.

Definition 1. An estimator Z7 (■) of a vector Zf), where 7 Є Г is a parameter, is said to have a practical exponential convergence if there exists positive constants Ki, K2 such that for any є > 0 and в > 0, the inequality

\Z7(t) — Z(t)\< є + Kie—Kel, Vt > 0

is fulfilled for some 7 Є Г.

In the following we shall denote by sat(l, u, 1) the saturation operator max(l, min(u, 1)).