A Survey of Observers for Nonlinear Dynamical Systems - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 6

were introduced in Kang and Barbot ( 2007 ). Meanwhile, quantitatively measure
partial observability has been rapidly developed in a sequence of papers ( Kang 2011 ;
Kang and Xu 2009a , b , 2011 ). For PDEs, the observability is defined and computed
for the finite dimensional approximations of the original model. In Kang and Xu
( 2009a , b ), dynamic optimization is used as a tool for the definition.

W R n be a subspace. Let

Definition 1.1. Given a trajectory x.t/ , t 2 Œt 0 ;t 1 .Let

>0 be a constant. Define as follows

x.t/ jj h.x.t// h.x.t// jj
subject to
x D f.x/;
jj x.0/ x 0 jj D
x.0/ x 0 2 W

D min

Then the ratio = is a measure of observability for the W -component of x.0/ .

can be considered as an extension of the observability
Gramian. Consider a linear system ( 1.1 ). Suppose

If ! 0 , the ratio =

W D R n . Then the observability

Gramian, G , satisfies ( Kailath 1980 ; Krener and Ide 2009 )

jj y jj L 2 D x 0 Px 0

(1.8)

Given jj x 0 jj D ,wehave

2 D min 2

(1.9)

2 = 2 equals the
reciprocal of the smallest eigenvalue of the observability Gramian. In Kang and Xu
( 2009a , b ), the concept of partial observability was applied to more general problems
using various types of norms and knowledge of the system. An example of optimal
sensor location by maximizing the observability for data assimilations was given in
Kang and Xu ( 2011 ).

where min is the smallest eigenvalue of G . Therefore, the ratio

1.3 Asymptotic Observers

Following control theory, asymptotic observers are systems defined by differential
or difference equations and associated computational algorithms which accepts the
measured data from another system as input and returns an estimate of the state of
the other system. In the case of a perfect model without noise and uncertainties,
the estimated state should converge to the true state of the system being observed.
Also if the initial state of the observer equals the true state, then the estimation error
is zero along the entire trajectory. In most observer designs, such as Luenberger