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298 CHAPTER 11. NONLINEAR OBSERVERS
which is of the form.
with
If u = 0, we have
and thus
x = f (x) + g(x)u
y = h(x)
X1 0
rank({Oh,VLfh}) = rank({[1 0], [0 1]}) = 2
J i =f(x)
y = h(x)
is observable according to Definition 11.4. Now consider the same system but assume that
u = 1. Substituting this input function, we obtain the following dynamical equations:
{
A glimpse at the new linear time-invariant state space realization shows that observability
has been lost.
11.2.1 Nonlinear Observers
There are several ways to approach the nonlinear state reconstruction problem, depending
on the characteristics of the plant. A complete coverage of the subject is outside the scope
of this textbook. In the next two sections we discuss two rather different approaches to
nonlinear observer design, each applicable to a particular class of systems.
11.3 Observers with Linear Error Dynamics
Motivated by the work on feedback linearization, it is tempting to approach nonlinear state
reconstruction using the following three-step procedure:
(i) Find an invertible coordinate transformation that linearizes the state space realization.
(ii) Design an observer for the resulting linear system.
(iii) Recover the original state using the inverse coordinate transformation defined in (i).