Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
(of matrices). The corresponding matrix (denoted S or P ) is called the Riccati matrix. It is
a symmetric and positive definite matrix 12 . Since S is a (n × n) symmetric square matrix,
we only need to compute the upper or the lower part of the matrix (i.e. there are n(n+1)
2
equations to solve).
Definition 15
The high-gain extended Kalman filter is defined by the two equations below:
�
dz
dt = A(u)z + b(z, u) − S−1C ′
R−1 (Cz − y)
dS
dt = −(A(u)+b∗ (z, u)) ′
S − S(A (u)+b∗ (z, u)) + C ′
R−1C − SQθS.
(2.9)
The matrices Q and R are originally the covariance matrices of the state and output
noise respectively, and therefore are expected to be symmetric and positive definite. Since
this observer is developed within the frame of the deterministic observation theory, those two
matrices will be used as tuning parameters. Qθ is defined as Qθ = θ 2 ∆ −1 Q∆ −1 where θ > 1
is a fixed parameter and
⎛
1 0 · · · 0
⎜
∆ = ⎜
0
⎜
⎝ .
1
θ
. ..
. ..
. ..
.
0
0 · · · 0 1
θn−1 ⎞
⎟
⎠ .
In most cases, normal form representations are not used when implementing an extended
Kalman filter (case θ = 1). The Jacobian matrices of f and h (computed with respect to the
variable x) are used in the Riccati equation:
dS
dt
� �′ � � � �′
∂f
∂f ∂h
= −
S − S
+
R
∂x |x=z ∂x |x=z ∂x |x=z
−1
� �
∂h
− SQθS.
∂x |x=z
Refer to Chapter 1 for a more detailed explanation.
Theorem 16 ([47, 57])
For θ large enough and for all T>0, the high-gain extended Kalman filter (2.9) satisfies
the inequality below for all t> T
θ :
�z(t) − x(t)� 2 ≤ θ n−1 k(T )�z( T
) − x(T
θ θ )�2 T
−(θω(t)−µ(T ))(t−
e θ )
for some positive continuous functions k(T ), ω(T ) and µ(T ).
2.6 On Adaptive High-gain Observers
In this section, we review a few adaptation strategies for the high-gain parameter that
can be found in the literature. Strategies that are based on Luenberger like 13 observers are
12 The fact that the solution of the Riccati equation of the observer (2.9) remains definite positive is not
obvious and must be proven. Such a proof can be found in [57], Lemma 6.2.15 and Appendix B in the
continuous discrete case.
13 Here Luenberger like has to be understood in a broad sense. That is: observers with a correction gain
matrix that is not computed online and, most of the time, computed following a pole placement-like scheme.
21