Let us consider the new variable

T(t) = yi (0) — yi (0 + У2 (о) — У2 (0

that is measured on-line. From Proposition 1, one deduces that r(-) is bounded. One can also easily check the property

dT

— = (1 — k) a x(t) > 0 , Vf > 0 . Consequently, t(-) is an increasing function up to

t = lim T(t) < +ro

and t(-) defines a diffeomorphism from [0, +ro) to [0,T). Then, one can check that the dynamics of the variable s in time t is decoupled from the dynamics of the other state variables:

where a and в are parameters defined as combinations of the unknown parameters a and k: к

1 — it’

1

я(1 — к)

and from (4) one has (a, в) Є [a-, a+] x [в-, в+]. For the identification of the parameters a, в, we propose below to build an observer. Other techniques, such as least squares methods, could have been chosen. An observer presents the advantage of exhibiting a innovation vector that gives a real-time information on the convergence of the estimation.

^,,S) = § + M3A|i

and the pair (A, C) in the Brunovsky’s canonical form:

/0 1 0

A = 0 0 1 and C = (10 0) . (7)

0 0j

The unknown parameters a and в can then be made explicit as functions of the observation yi and the state vector g:

One can notice that functions <p(y1, •), la(y, •) and їв(y1, •) are not well defined on R3, but along the trajectories of (3) one has g3/g2 = —ви'(У1) and g2 = a — вц(У1), that are bounded. Moreover Assumption A2 guarantees that ц’ (y1) is always strictly positive. We can consider (globally) Lipschitz extensions of these functions away from the trajectories of the system, as follows:

g2 — h1 (У1, g)

Ігі(УіЛ)

У'(Уі) with

ІЧ (Уі/ g) = sat ^-^+y'(yi),-^“y'(yi),^ , h2 (У1, g) = sat (a— — в+ Ц(У1), a+ — ГЦ(У1), g2) .

Then one obtains a construction of a practical observer.

Proposition 3. There exist numbers > 0 and C1 > 0 such that the observer

(8)

guarantees the convergence

max (|a(t) — a|,в(т) — в) < he-^1 т||g(0) — g(0)|| for any 61 large enough and т Є [0, T).

Proof. Consider a trajectory of dynamics (3) and let O1 = {y 1 (t) }t>0. From Proposition 1, one knows that the set O1 is bounded.

Define Kg1 = — (301 362 б3 )T. One can check that Kg1 = —P—1CT, where Pg1 is solution of the algebraic equation

61 P6l + ATPei + P6l A = CTC.

Consider then the error vector e = £ — £. One has

0 0

ф (уь £) — Ф (У1/£)/

where ф(yi, • ) is (globally) Lipschitz on R3 uniformly in yi Є Oi. We then use the result in [10] that provides the existence of numbers C1 > 0 and q1 > 0 such that

for 01 large enough. Finally, functions la (y, •), Ip (y1, •) being also (globally) Lipschitz on R3 uniformly in yi Є 0, one obtains the inequality (9). |

Corollary 1. Estimation of a and k with the same convergence properties than (9) are given by

k(r),d(r)=sat(k-,k+, &{T) ),sat(a-,a+,1+/X{T))

y y 1 +4?)) f(T) )

Remark. The observer (8) provides only a practical convergence since r(t) does not tend toward +ro when the time t get arbitrary large. For large values of initial x, it may happens that y(t) > t for some times t > 0. Because the present observer requires the observation y1 until time t, it has to be integrated up to time min(r(t), t) when the current time is t.

5.1. A second observer for m and x

We come back in time t and consider the measured variable z = y 1 + y2. When the parameters

T

a and в are known, the dynamics of the vector Z = [z z ‘i can be written as follows:

/

with z = CZ

and i/>(i/i, Z,rt,^) = ^ +Z2?»(yi)(Myi) -«)

Parameter m and variable x(-) can then be made explicit as functions of y1 and Z:

m = Zm(yi, Z) = ?'(Уі) — 7^ , x = — fZl Z2

Functions f(y1, •, a, в) and lm(y 1, •) are not well defined in R3 but along the trajectories of the dynamics (3), one has Z3/Z2 = y(y1) — m and Z2 = —x/в that are bounded. These functions can be extended as (globally) Lipschitz functions w. r.t. Z:

f(yh Z, a, в) = h3 (У1, Z )Z3 + min(Z2, z(0)2/e2 )ц'(У1 )(eP(y1) — a)

lm (УЪ Z) = У(У1) — h3 (У1, Z)

with

Proposition 4. When a and в are known, there exists numbers b2 > 0 and C2 > 0 such that the observer

/0 302

for any 02 large enough and t > 0.

Proof. As for the proof of Proposition 3, it is a straightforward application of the result given in [10]. 1