Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
B.1 Bounds on the Riccati Equation
− Sn is the set of (n × n) symmetric matrices having their values in R,
− Sn(+) is the set of positive definite matrices of Sn,
− Tr(S) denotes the trace of the matrix S,
− for any matrix S, |S| = � Tr(S ′ S) is the Frobenius norm, when S ∈ Sn, |S| = � Tr(S 2 ),
− for any matrix S, �S�2 = sup
�x�2=1
�Sx�2, is the norm induced by the second euclidean
norm, we also write it �S� by omission,
− we keep the τ time scale notation, but we use S instead of ¯ S to ease the reading of
equations,
� kδt
− τk = θ(v)dv. Since θ is fixed during prediction periods, the time elapsed between
0
two correction steps is (τk − τk−1) =θk−1δt.
�
− A stands for the matrix
θ.
A(ū)+ ¯ b(¯z,ū)
θ
�
, we omit to write the dependencies to u, z and
Matrix facts
Notice that according to equation (B.1), if S(0) is symmetric then S(t) is symmetric.
1. On the Frobenius norm (Cf. [67], Section 5.6):
(a) If U and V are orthogonal matrices then |UAV | = |A|,
(b) if A is symmetric semi positive then |A| = |DA|, where DA is the diagonal form of
A,
(c) if A is as in (b), |A| = �� λ 2 i
(d) �A�2 ≤ |A| ≤ � (n)�A�2,
(e) A is symmetric, A ≤�A�2Id ≤ |A|Id.
2. On the trace of square matrices (Cf. [67]):
� 1
2 , where λi ≥ 0 denotes the eigenvalues of A,
(a) If A, B are (n × n) matrices, then |Tr(AB)| ≤ � Tr(A ′ A) � Tr(B ′ B),
(b) if S is (n × n), and symmetric semi positive, then Tr(S 2 ) ≤ Tr(S) 2 ,
(c) if S is as in (b) then Tr(S 2 ) ≥ 1
nTr(S) 2 ,
(d) if S is as in (b) then, Tr(SQS) ≥ q
ntr(S)2 , with q = min
�x�=1 x′ Qx, and Q symmetric
definite positive,
(e) as a consequence of (b) and (c), if S is as in (b), and �.� denotes any norm on
(n × n) matrices, then there exist l, n > 0 such that
l�S� ≤Tr(S) ≤ m�S�.
3. On inequalities between semi positive matrices (Cf. [67], Sections 7.7 and 7.8), A and
B are symmetric in this paragraph:
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