Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
Proof.
The first bound of (B.5) gives:
and the second is rewritten:
x2(−) ≤ 2b
a
x1(+) ≤ 2b
a
+ 2b
a
+ 2b
a
B.1 Bounds on the Riccati Equation
1
e2bτ1 + rτ1,
− 1
x1(+) − 2b
a
x1(+) � e 2b(τ2−τ1) − 1 � + 2b
a
We want to replace x1(+) by the upper bound found above.
Let us define the function
2bxe
h(x) =
2bτ
ax (e2bτ − 1) + 2b .
Its derivative w.r.t. x is
h ′
(x) = e2bτ 2b � ax(e 2bτ − 1) + 2b � − a(e 2bτ − 1)xe 2bτ 2b
[ax(e 2bτ − 1) + 2b] 2
=
e 2bτ (2b) 2
[ax(e 2bτ − 1) + 2b]
It is positive for all τ > 0, and we can replace x1(+) by its upper bound:
x2(−) ≤ 2b 2b
+
a a
�
2b 2b 1
a + a e2bτ �
1−1
+ rτ1 − 2b
a
� �e �
1 + rτ1
2b(τ2−τ1) 2b − 1 +
≤ 2b
a
+ 2b
a
� 2b
a
2b
a
+ 2b
a
1
e2bτ1 − 1
e 2bτ 1−1
� 2b
a
+ 2b
a
+ 2b
a
� 2b
a
2 .
1
e2bτ1−1 + 2b
a
1
� �e �
+ rτ1
2b(τ2−τ1) 2b − 1 +
1
e2bτ1−1 We upper bounds the denominator of the last term with:
�
2b 2b 1
+
a a e2bτ1 � �
+ rτ1 e
− 1 2b(τ2−τ1)
�
− 1
and the denominator of the second term with:
�
2b 2b 1
+
a a e2bτ1 � �
+ rτ1 e
− 1 2b(τ2−τ1)
�
− 1
We also simplify (2b/a) in those two terms:
x2(−) ≤ 2b
a
≤ 2b
a
+ 2b
a
+ 2b
a
+ 2b
a ≥
� 2b
a
+ 2b
a
.
a
rτ1
� �e � .
+ rτ1
2b(τ2−τ1) 2b − 1 +
+ 2b
a
≥ 2b
a ,
1
e2bτ1 � �
e
− 1
2b(τ2−τ1)
�
− 1 + 2b
a .
1
e2bτ1 1
��
− 1 1+ 1
e2bτ � �e � � + rτ1,
2b(τ2−τ1)
1−1
− 1 +1
1
e2bτ2 − e2bτ1 + e2bτ1 + rτ1.
− 1
138
a
a