Nudging Methods: A Critical Overview - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 45

2 1
22

and

M D H D Œ1;0

1 D .2 C p

It can be verified that the eigenvalues of M are 2/ > 1 and 2 D

.2 p

2/ < 1 . Hence, the true system x.k C 1/ D M x.k/ is unstable with one

G D Œg 1 ;g 2 T 2 R 2 1 and consider growing mode and one decaying mode. Let

.2 g 1 / 1
.2 C g 2 /2

A D M GH D

D . 1 ; 2 /

the eigenvalues are given by the roots of

.A D I/ D 2 .4 g 1 / C .2 2g 1 g 2 / 0 D p./ D det

g 2 D 1 4

Setting g 1 D 3 and , it follows that

2

0 2 C 1
4

1
2

D

1 D 2 D 2

Hence, are the eigenvalues which in turn implies that .M GH/ is

a Hurwitz matrix.

2.4.3 Observer Based Nudging: Nonlinear Dynamics

There is a vast corpus of books and papers in control literature relating to the
design of observers for nonlinear dynamical systems, simply known as nonlinear
observers ( Isidori ( 1995 ), Marquez ( 2003 ), Bonnabel et al. ( 2009 ), Auroux ( 2011 )).
While it is tempting to provide a comprehensive survey of results from this area,
it turns out that nonlinear observer design theory is deeply rooted in some of the
fundamental results from differential geometry. Even an elementary introduction to
these beautiful results will take us too far from our stated goals. So, quite reluctantly,
we content ourselves with a very simple approach based on the classical Lyapunov
theory of stability.
Let the given nonlinear dynamical system be given by

x.k C 1/ D M x.k/ C F.x.k// (2.88)

with as the initial condition where the right hand side of ( 2.88 )isthesumof

the linear part

x.0/

F W R n ! R n is Mx and the nonlinear part F.x.k// where the map

assumed to satisfy the (global) Lipschitz condition
ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ F.x 1 / F.x 2 / ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 2 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ x 1 x 2 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 2

L F (2.89)

x 1 x 2 2 R n where

for all L F >0 is called the Lipschitz constant.