A Survey of Observers for Nonlinear Dynamical Systems - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 2

returns an estimate of the state of the other system. Observers play a critical role in
control systems because many feedback controllers depend on the accurate estimate
of state variables of the system to be controlled. An accurate estimation of the state
in the presence of noise and uncertainties is essential for a controller to achieve high
quality performance.
Estimation from data with random noise can be traced to Gauss about 200 years
ago who invented the technique of deterministic least-squares for orbit measure-
ments. In the early twentieth century, Fisher introduced maximum likelihood
estimation. Then in the middle of the twentieth century Wiener invented his
well known optimal filter for stationary processes. Around 1960s, Kalman and
Bucy introduced an optimal recursive filter for dynamical systems. This filter,
now known as the Kalman filter, is “the very foundation for data mixing in
modern multisensor systems ( Gelb 1974 ).” The estimation for systems governed by
differential equations has been an active research field in control theory for more
than 50 years. In addition to the Kalman filter, which is essentially a recursive
solution to the least square problem, estimation processes have been developed
for various performance requirements, such as asymptotically stable estimation,
H 1 estimation, and minimum energy estimation. Fundamental theory has been
developed to analyze observability, an intrinsic property of systems with sensors that
largely determines the invertibility from past measurement to the state of the system.
Data assimilation is an area of estimation theory and an application to systems
with extremely high dimensions. Both filtering and smoothing methods are critical
to date assimilation. Although we focus on nonlinear filtering methods in this paper,
smoothing algorithms can be developed using similar ideas. Approaches such as
ensemble Kalman filters and 4D-Var are based on the theory of optimal estimation,
especially the Kalman filter and minimum energy estimation. The data assimilation
community has done extensive research on these topics for over 30 years. While this
topic is focused on problems in data assimilation, this article is to provide a survey
on some ideas and results that have been actively developed in control theory, but
not widely used in data assimilation. The goal is to lay out some related but different
concepts and methods. We hope that some of them may inspire different approaches
that benefit the area of data assimilation.

1.2 Observability

In this paper, we consider systems defined by differential equations. The sensor
measurement is defined by an output function. For example,

x D Ax
y D Cx
x.0/ D x 0

(1.1)