noise cannot be fulfilled, and the classical stability results are not applicable. The present paper
aims at completing this missing case.
The optimal state estimation realized by the Kalman filter can be viewed as a trade-off
between the uncertainties in the state equation and in the output equation. In an OE system,
the state equation is assumed noise-free. This point of view suggests that the state estimation
should solely rely on the state equation, provided that the initial state of the OE system is exactly
known. In practice the Kalman filter remains useful when the initial state is not exactly known
or when the OE system is unstable. Of course, if the state of an unstable system diverges, so does
its state estimate by the Kalman filter. Typically in practice, unstable systems are stabilized
by feedback controllers so that the system state remains bounded. The Kalman filter can be
applied either to the controlled system itself or to the entire closed loop system. In the latter
case, the controller must be linear and completely known, excluding the saturation protection
and any other nonlinearities.
The classical optimality results of the Kalman filter are also valid in the case of OE systems
(Jazwinski, 1970, chapter 7). However, it is necessary to complete the stability analysis, as the
classical results are not applicable in this case.
The main results presented in this paper are as follows. Under the uniform complete observ-
ability condition, it is first shown that the dynamics of the Kalman filter applied to a continuous
time OE LTV system is asymptotically stable, regardless of the stability of the system itself. The
exponential or polynomial convergence of the Kalman filter is further established, depending on
the stability or the instability property of the considered system. The boundedness of the solu-
tion of the Riccati equation, and thus also of the Kalman gain, is also proved under the same
condition. These results complete the classical results (Kalman, 1963; Jazwinski, 1970), which
exclude the case of OE systems.
For linear time invariant (LTI) systems, it is a common practice to design the Kalman filter
by solving an algebraic Riccati equation (in contrast to dynamic differential Riccati equation for
general LTV systems as considered in the present paper). In this case, the controllability and
observability conditions can be replaced by the weaker stabilizability and detectability conditions
(Laub, 1979; Arnold and Laub, 1984).
Some preliminary results about the Riccati equation of time invariant systems have been
presented in the conference paper (Ni and Zhang, 2013), and some further results on the asymp-
totic stability of the Kalman filter have been presented in the conference paper (Ni and Zhang,
2015), without the exponential or polynomial convergence rate analysis and part of the numerical
examples reported in the present paper.
The rest of the paper is organized as follows. Some preliminary elements are introduced in
Section 2. The problem considered in this paper is formulated in Section 3. The asymptotic
stability of the Kalman filter for OE systems is established in Section 4. Exponential and
polynomial convergence rates of the Kalman filter are analyzed in Section 5. Some analytical
and numerical examples are presented in Sections 6 and 7. Finally, concluding remarks are
drawn in Section 8.
2. Definitions and preliminary elements
Let us shortly recall some definitions and basic facts about LTV systems, which are necessary
for the following sections.
Let m and n be any two positive integers. For a vector x ∈ R
n
, kxk denotes its Euclidean
2