Nonlinear Control Sy.. - Free

296 CHAPTER 11. NONLINEAR OBSERVERS
11.2 Nonlinear Observability
Now consider the system
x=f(x)+g(x)u f :R"->lR',g:]R'->1R
y = h(x) h : PJ - 1R
(11.13)
For simplicity we restrict attention to single-output systems. We also assume that f (),
are sufficiently smooth and that h(O) = 0. Throughout this section we will need the following
notation:
Clearly
xu(t,xo): represents the solution of (11.13) at time t originated by the input u and
the initial state xo.
y(xu(t, xo): represents the output y when the state x is xu(t, xo).
y(xu(t, xo)) = h(xu(t, xo))
Definition 11.3 A pair of states (xo, xo) is said to be distinguishable if there exists an
input function u such that
y(xu(t, xo)) = y(xu(t, xo))
Definition 11.4 The state space realization ni is said to be (locally) observable at xo E R"
if there exists a neighborhood U0 of xo such that every state x # xo E fZ is distinguishable
from xo. It is said to be locally observable if it is locally observable at each xo E R".
This means that 0,,, is locally observable in a neighborhood Uo C Rn if there exists an input
u E li such that
y(xu(t, xo)) y(xu(t, xo)) Vt E [0, t] xo = x2
There is no requirement in Definition 11.4 that distinguishability must hold for all
functions. There are several subtleties in the observability of nonlinear systems, and, in general,
checking observability is much more involved than in the linear case. In the following
theorem, we consider an unforced nonlinear system of the form
'Yn1
I x=f(x) f :1 -*
y = h(x) h:ii. [F
and look for observability conditions in a neighborhood of the origin x = 0.
(11.14)