A Survey of Observers for Nonlinear Dynamical Systems - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 8

2
4

3
5

C
CA
CA 2
: : :
CA n 1

This result guarantees that one can always find linear observers with stable error
dynamics for observable systems. For systems in which is not observable,

it is still possible to achieve asymptotic stability of ( 1.12 ). This depends on the
spectrum of

.A;C/

, which can be divided into observable modes and unobservable
modes. Details are referred to Kailath ( 1980 ). If all the unobservable modes are
on the left half plane, then there always exists a

A

that stabilizes ( 1.12 ).
The error dynamics does not include measurement error. If the output is corrupted
by noise, the asymptotic stability of the observer guarantees that

G

is stabilized
around the true value. There are infinitely many observer gains to stabilize the
observer. A high gain observer has fast convergence to the true value of the system,
however it is very sensitive to sensor noise. Although asymptotic observers do not
guarantee optimal performance in any sense, their advantage lies in the simplicity.
For real time applications, each estimate at a given time is simply computed by
one step integration of the observer equation, which can be implemented using any
numerical algorithm for solving ordinary differential equations (ODEs). Luenberger
observers can be found as a standard topic in almost all textbooks on control theory,
for instance ( Kailath 1980 ; Khalil 2002 ).

x.t/

1.3.2 Observers with Linear Error Dynamics

For nonlinear systems, observer design with a guaranteed asymptotically stable error
dynamics is a difficult task ( Hermann and Krener 1977 ). The Luenberger observer
works for linear systems because its error dynamics is decoupled from the unknown
trajectory being observed. For nonlinear systems, however, this is not true in general.
There is a large volume of literature on the construction of nonlinear observers
that admit a linear error dynamics. In the pioneering work ( Krener and Isidori
1983 ) a technique called output injection was introduced. In addition, necessary
and sufficient conditions are found under which the error dynamics of the nonlinear
observer is equivalent to a linear ODE. Consider a nonlinear dynamical system with
an output

x D f.x/
y D h.x/

(1.13)

x 2 R n is the state variable, y 2 R p is the output which can be measured, in which

f.x/ are vector valued functions with adequate smoothness. In Krener and

Isidori ( 1983 ), it is propose to find a change of coordinates around a fixed point

and h.x/

x 0