Finite time convergent observers for linear
time-varying systems
Patrick H. Menold, Rolf Findeisen, Frank Allg¨ower
Institute for Systems Theory in Engineering
University of Stuttgart
70550 Stuttgart
Pfaffenwaldring 9
Germany
Email:
menold,findeise,allgower
@ist.uni-stuttgart.de
Abstract— This paper focuses on the design of observers
with finite convergence time for continuous time linear time–
varying systems. For this purpose we outline how recent results
for finite time convergent observers for time-invariant systems
can be expanded to the time–varying case. Besides deriving
general conditions that guarantee finite time convergence we
furthermore show how for systems that can be transformed
to observer canonical form suitable observer parameters can
be obtained. The obtained results are exemplified considering
a simple example system.
I. INTRODUCTION
Observing the state of a continuous time system is an
important problem and many different observer designs exist
by now (see for example[1]). However, most of the existing
observer designs guarantee only that the estimation error tends
to zero asymptotically. In principle it is desirable to have
observers that allow to reconstruct (in the nominal case) the
system state in finite time. Such observers are for example of
interest for the application of state feedback controllers.
So far there exist only a couple of observers with a
finite convergence time. Examples are sliding mode based
observers [2], [3], [4], moving horizon based observers [5],
[6], [7], [8] and the observer design presented in [9]. The
Sliding mode based observers can be applied to linear and
nonlinear systems and guarantee finite time convergence of the
estimation error. However, the differential equations describing
the estimation error can only be rendered semi–global stable
in the sense that a bounded region of system states must
be considered. Moving horizon observers are based on the
online solution of a dynamic optimization problem involving
the output measurements and the system model. They can
be applied to linear as well as nonlinear systems. Under
the assumption that the global solution to the dynamic op-
timization problem can be found and that a certain finite time
observability assumption holds, moving horizon observers can
in principle guarantee finite time convergence of the estimation
error. The main drawback of moving horizon observers is
that a dynamic optimization problem must be solved online.
The observer design presented in [9] is, in comparison to
the moving horizon and sliding mode observer designs rather
simple to implement. However, it can only be applied to linear
time–invariant systems. Basically two identity observers with
different speeds of convergence are used and the state estimate
is constructed based on delayed and current state estimates
of the identity observers. One of the key advantages of this
approach is that no bounded region of system states is required.
In this note we propose an extension of the design presented
in [9] to linear time-varying systems. For this purpose we
derive conditions for the (time–varying) observer feedback
matrices and the delay used that lead to guaranteed finite time
convergence of the observer error. In general, however, it is
rather difficult to obtain suitable observer feedback matrices
and a suitable delay that satisfy the derived conditions. How-
ever, as outlined, for systems that can be brought to observer
canonical form it is trivial to pick suitable parameters, thus
allowing the application of the derived observer design to a
wide class of systems.
The overall paper is structured as follows: In Section II
we present the proposed observer for time-varying continuous
time MIMO systems and state conditions that guarantee the
finite time convergence of the observer error. Since it is in
general difficult to obtain suitable observer parameters, we
show in Section III how for MISO systems that can be
transformed to observer canonical form suitable parameters
can be obtained. Note that while the results are only derived for
MISO systems for notational simplicity, they can be trivially
expanded to the MIMO case. Section IV contains a small
example, showing the application of the derived observer to
a fourth order system with two outputs. The paper finally
concludes with a short discussion of the results in Section V.
II. GENERAL LINEAR TIME-VARYING SYSTEMS
Consider a continuous time linear time–varying MIMO
system of the form
˙x
t
A
t
x
t
B
t
u
t
x
t
0
x
0
(1a)
y
t
C
t
x
t
(1b)
with the state x
t
n
, the output y
t
p
and the input
u
t
m
. We assume that u
, A
, B
and C
are
piecewise continuous and bounded over
t
0
∞
and are known
exactly. This implies that the solution of (1) exists, is unique
and bounded for all times [10], [11].
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