Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
such that
d ¯ S
dt =
F(θ, I)
¯θ 2
� � �
NS ¯ + SN ¯ − A ′ �
S ¯ + SA ¯
3.6 Boundedness of the Riccati Matrix
− 1
¯θ
+ C ′
R −1 C − ¯ SQ¯ S (3.13)
�
.
� ¯S ˜ b ∗ (¯z,ū)+ ˜ b ′ ∗ (¯z,ū) ¯ S
We complete the description of the observer in the τ time scale with the equation
d¯ θ
dτ = F(¯ θ, Ī)
¯θ
.
This last set of equations is established in order to investigate the properties of the Riccati
matrix, particularly the fact that it is bounded (Cf. Section 3.6).
Remark 37
The Lipstchitz constant of the vector field b(., u) is the same in the x(t), ˜x(t) and ¯x(τ)
coordinates. It occurs quite naturally in the single output case but implies a special definition
of the observer in the multiple output case. Consequently this fact is proven in Chapter 5,
Lemma 51, for systems having multiple outputs.
3.6 Boundedness of the Riccati Matrix
The Riccati matrix S(t) has some very important properties:
− S(t) can be upper and lower bounded by matrices of the form c.Id, and
− if S(0) is symmetric definite positive, then S(t) is also symmetric definite positive for
all times.
Those � properties are established in the book [57], with the difference being that the term
�
�
� F(θ,I)
�
�
�
� doesn’t appear in the Riccati equation (Cf. Section 2.4 of the 6th chapter of the
θ 2
book), and that the result there is also only valid for times t ≥ T ∗ , for some T ∗ > 0. As we
will see in the present section, when θ(0) is set to 1 we can say a little bit more.
Lemma 38 ([57])
�
�
Let us consider the Riccati equation (3.13). If the functions ai (u (t)), ��b ∗ i,j (z,u)
� �
� �
�, �
� F(θ,I)
θ 2
�
�
�
�are smaller than aM > 0, and if ai (u (t)) >am > 0 then for all τ0 > 0 there exist two constants
0 < �α < � β (depending on τ0, aM, am) such that, for all τ ≥ τ0, the solution of the Riccati
equation satisfies the inequality
�α Id ≤ S (τ) ≤ � β Id.
This first relation is extended to all times t>0 via a second lemma.
Lemma 39
We still consider the Riccati equation (3.13) with S (0) = S0 being a symmetric definite
positive matrix taken in a compact subset of the form aId ≤ S0 ≤ bId, 0 < a < b and
θ (0) = 1. Then there exist two constants 0 < α 0 (and therefore α Id ≤ � S (t) ≤ β Id for all t ≥ 0).
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