A Survey of Observers for Nonlinear Dynamical Systems - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 9

z D z .x/

(1.14)

z .x 0 / D 0

so that ( 1.13 ) is transformed into a linear system with a nonlinear output injection

z D A z C .y/

y D C

(1.15)

z

C 2 R p n . If this is the case, then we can easily
construct a Luenberger type of observer as follows.

A 2 R n n and

for some matrices

z D A z C .y/ C G.y C z

/ (1.16)

Let

e D z z

then the error dynamics is a linear system decoupled from z .t/

e D .A GC/e

If G , the observer gain, is chosen so that the eigenvalues of .A GC/ are all in the

left half plane, then

t !1 e.t/ D 0
Not all nonlinear systems can be transformed into a linear system with output
injection. The existence of the change of coordinates ( 1.14 ) can be determined
using Lie differentiation. Given a function

lim

h.x/ ,let dh represents the 1-form, or

the gradient,

@h

@x 1 .x/ @h @x 2 .x/ @h

dh.x/ D

@x n .x/

The Lie derivative is defined as follows

L f .h/ D dh f
L f .dh/ D f T @ 2 h
@x

C dh @f
@x

The following theorem was proved in Krener and Isidori ( 1983 ).

Theorem 1.1. There exists a local change of coordinates ( 1.14 ) that trans-
forms ( 1.13 ) into a linear system with output inject ( 1.15 ) if and only if

f.x 0 / D 0
h.x 0 / D 0

L f .dh/ L f .dh/

and is a linear combination of for k D 0;1; ;n 1 .