Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2. the use of a variant of extended observers.
1.2 Motivations
A story is best told when started from the beginning. Thus we begin by introducing the
Kalman filter in the linear case. A linear system is given by:
� dx(t)
dt = Ax(t)+Bu(t),
y(t) = Cy(t),
where A, B and C are matrices (having the appropriate dimensions) that may or may not
depend on time. The two archetypal observers for such a system were proposed in the 1960’s
by D. G. Luenberger [86], R. E. Kalman. and B. S. Bucy [75, 76]. They are known as the
Luenberger observer and the Kalman-Bucy filter respectively.
The leading mechanism in those two algorithms is the prediction correction scheme. The
new estimated state is obtained by means of:
− a prediction based on the model and the old estimated state, and
− the correction of the obtained values by the measurement error weighted by a gain
matrix.
We denote the estimated state by z(t). The corresponding equation for the estimated state
is:
dz(t)
= Az(t)+Bu(t) − K (Cz(t) − y(t)) .
dt
In the Luenberger observer, the matrix K is computed once and for all. The real part of
all the eigenvalues of (A − KC) have to be strictly negative.
In the Kalman filter, the gain matrix is defined as K(t) =S−1 (t)C ′
R−1 where S is the
solution of the differential equation:
d
S(t) =−A′ S(t) − S(t)A − S(t)QS(t)+C
dt ′
R −1 C.
This equation is a Riccati equation of matrices and is referred to as the Riccati equation.
The matrices A, B and C are expected to be time dependent (i.e. A(t), B(t) and C(t)).
Otherwise, the Kalman filter is equivalent to the Luenberger observer.
The Kalman filter is the solution to an optimization problem where the matrices Q and
R play the role of weighting coefficients (details can be found in [75]). These matrices must
be symmetric definite positive. According to this definition, the Kalman filter is an optimal
solution to the observation problem. We will see below that the Kalman filter has a stochastic
interpretation that gives sense to those Q and R matrices.
Those two observers are well known algorithms, i.e. convergence can be proven. However,
when the matrices A, B and C are time dependent, the Kalman filter has to be used since
the gain matrix is constantly being updated.
Observability in the linear case is characterized by a simple criterion. The loop closure
problem has an elegant solution called the separation principle: controller and observer can
be designed independently and the overall loop remains stable. Interesting and detailed exposes
on the Kalman filter can be found in [43, 45, 58].
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