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observers and Kalman filters, an observer consists of a copy of the original system
plus a correction term which is a function of the measured data.
Asymptotic observers are widely used in control systems to achieve stable
estimates of state variables. The design emphasizes the stability and simplicity of
the estimation process. In general it does not optimize any performance measure.
The Luenberger observer for linear systems is a simplest example that illustrates the
fundamental idea of asymptotic observers.
1.3.1
Luenberger Observer
Given a dynamical system with an output
x D Ax
y D Cx
(1.10)
x 2 R n is the state variable,
y 2 R p is the output which can be measured,
where
A 2 R n n and
C 2 R p n are matrices. We assume that
A
,
C
and the output
y.t/
are known information. The goal is to find an estimate, estimate, denoted by
x.t/
,
of the state variable so that
x.t/
asymptotically approaches
x.t/
. The observer has
the following form
x D Ax C G.y C x/
(1.11)
G 2 R n p is called the observer gain, which is used to stabilize the
estimation error. Define
The matrix
e D x x
then the error dynamics has the following form
e D Ae G.y C x/
D .A GC/e
(1.12)
It is obvious that
e.t/
asymptotically approaches zero if the eigenvalues of
A GC
are all located in the left half plane. To estimate
x.t/
, one can use any initial guess
x.0/
.Then x.t/
from ( 1.11 ) satisfies
t !1 e.t/ D 0
lim
When applying the observer,
is measured online and the ( 1.11 ) is numerically
propagated in real-time to provide an estimate of
y.t/
x.t/
.
It can be proved that, for any set of
n
complex numbers, there always exists an
observer gain,
, so that the eigenvalues of the error dynamics ( 1.12 ) are placed
at these locations, if the pair
G
.A;C/
is observable, i.e. the following observability
matrix has full rank
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