Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
B.1 Bounds on the Riccati Equation
observation matrix is fixed (i.e. not u-dependent), the derivative of such a m matrix is
bounded.
If we want to consider C matrices defined with a function α1 as in the continuous case,
the derivative of this function must have a bounded time derivative. The consequence is that
bang-bang like control inputs become inadmissible.
We now have gathered all the elements necessary to prove the second half of this subsection’s
result.
Lemma 95
Consider equation (B.1) and the assumptions of Lemma 77. Let T∗ > 0 be fixed. There
exist α2 > 0 and µ>0 such that for all subdivision {τi}i∈N with τi − τi−1 β |Q|:
S1(+) = e−λτ1φu(τ1, 0)S0φ ′
� u(τ1, 0)
τ1
+λ e −λ(τ1−v)
φu(τ1,v)
In the same manner we obtain at time τ2:
τ1
0
S2(+) = e−λ(τ2−τ1) φu(τ2, τ1)S1(+)φ ′
� u(τ2, τ1)
τ2
+λ e −λ(τ2−v)
�
φu(τ2,v) S(v) − S(v)QS(v)
λ
�
S(v) − S(v)QS(v)
�
φ
λ
′
u(τ1,v)dv + C ′
R −1 Cτ1.
�
φ ′
u(τ2,v)dv + C ′
R −1 C(τ2 − τ1).
A few computations (see Appendix A.1 for the resolvent’s properties) lead to:
S2(+) = e−λτ2φu(τ2, 0)S0φ ′
� u(τ2, 0)
τ2
+λ e
0
−λ(τ2−v)
�
φu(τ2,v) S(v) − S(v)QS(v)
�
φ
λ
′
u(τ2,v)dv
+e−λ(τ2−τ1) φu(τ2, τ1)C ′
R−1C(τ2 − τ1).
R −1 Cφ ′
u(τ2, τ1)τ1 + C ′
4 The matrix inequality we want to achieve implies, in particular, the invertibility of the matrices Sk(+),
k ∈ N. It implies the invertibility of the sum that appears in (B.13). This matrix cannot be inverted if there
are less than n summation terms.
As is explained at the end of the proof, the result is achieved provided that the step size of the subdivision
used in the summation is small enough. Since we have the freedom to make µ as small as we want, the
summation above mentioned can be performed with sufficiently many points whatever the value of T ∗ . It
makes the requirement T∗ ≥ nµ redundant.
Recall now that µ represents the maximum sampling that allows us to use the observer. Lemma 89 provides
us with a first value for µ, and it is pretty much system dependent. By removing the assumption, we probably
have to shorten µ for the sake of mathematical elegance only.
We think that the proof is better that way.
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