A Survey of Observers for Nonlinear Dynamical Systems - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 22

where the constant and kk 1 is the Sobolev norm

of the first order. Therefore, the observer design for the PDE system ( 1.33 ) boils
down to the observer design for ODE ( 1.35 ), and the methods in previous sections
are applicable.

>0 does not depend on u .t/

1.6 Conclusions

The Kalman filter, invented initially for control systems, has been widely used in
science and engineering including data assimilation. For the last several decades,
the estimation theory for dynamical systems have been actively developed in control
theory. We have surveyed some but not all of the ways of observers for a nonlinear
system. Some approaches have been applied to engineering problems for many
years, and some others are relatively new. It is not clear which of them is scalable
for systems with extremely high dimensions, like atmospheric models or ocean
dynamics. However, we certainly hope that some of these ideas will benefit the data
assimilation community. The high gain observer is a theoretical finite dimensional
solution to a broad class of systems with small noise. The minimum energy and
H 1 observers are theoretical infinite dimensional solutions to broad classes of
noisy problems. However, it is not trivial to implement them for nonlinear systems.
The linearization techniques give local and sometimes only approximate solutions
for narrower classes of problems. The extended Kalman filter is probably still
the most robust and practical approach for most problems. If there are substantial
nonlinearities, e.g., multiple stable equilibria and/or stable limit cycles then the use
of multiple extended Kalman filters is probably the preferred approach. However, a
disadvantage for the extended Kalman Filter is the requirement of linearization in
real-time. This is why the unscented Kalman filter is getting increasingly popular
in engineering applications, although it suffers the requirement of doubling the
dimension of the system. To summarize, there is no best estimation method for
general nonlinear systems. Observers should be designed to fit the specific behavior
and form of a system and its model.
All these methods rely on the observability of the system to insure convergence.
The concept of observability has the potential to benefit data assimilations in several
ways, including optimal sensor network design, data thinning, targeted sensing, etc.
For these applications, numerically computing the observability for large systems is
a challenge that needs further research.

References

Anderson JL (2003) A local least squares framework for ensemble filtering. Mon Weather Rev
131:634-642
Balas M (1980) Towards a (more) practical control theory for distributed parameter systems,
control and dynamic systems. In: Leondes CT (ed) Advances in theory and applications, vol 18.
Academic, New York