A Survey of Observers for Nonlinear Dynamical Systems - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 11

Using the following notation

/ D z 2 z 3 z n .

T

f.

z /

z

2
4

3
5

010 00
001 00
: : : : : : : : : : : : : : :

A D

C D 10 0

the observer has the form

z D f. z

/ S 1 C.C z y/

where S is the solution of the equation

A T S S A C C T C D 0

S

where is a constant. It is proved in Gauthier et al. ( 1992 ) that the error of the

observer

e.t/ D z z

approaches zero if is large enough (thus the name “high gain observer”). While

the proof is carried out in the z -space, the observer can be constructed in the original
state space

@ 1

z
@x

x D f.x/

S 1 C.h.x/ y/

The simplicity in the construction of a high gain observer makes it a convenient
tool for nonlinear systems ( Gauthier and Kupka 1994 ). However, in the presence
of noise, a high gain observer should be used with caution. It may significantly
enlarge the impact of the noise and result in large estimation errors. In addition,
the “homomorphism” requirement for ( 1.17 ) limits the region in the state space in
which the observer is applicable. In Krener and Kang ( 2003 ), a nonlinear observer is
constructed without a global homomorphism requirement. In addition, the observer
gain depends on the state of the system so that it is not constantly high. Global or
semi-global observers can also be derived based on Lyapunove functions for systems
with a triangular structure or bounded nonlinear terms in its differential equations
( Lei et al. 2007 ; Krener and Kang 2003 ; Tsinias 1989 ). Deriving Lyapunov
functions for nonlinear systems is always difficult. An alternative is to directly apply
convergent numerical algorithms in nonlinear observers. For instance, an Euler-
Newton observer is introduced in Kang ( 2006 ). Moving horizon observers are also
computational based methodologies ( Findeisen et al. 2002 ; Michalska and Mayne
1995 ).
The sliding mode observer is another Lyapunov function based approach. It has
the capability of handling unknown inputs or unknown parameters ( Floquet and
Barbot 2007 ). A survey of various types of sliding mode observers can be found in