Nonlinear Control Sy.. - Free

11.2. NONLINEAR OBSERVABILITY 297
Theorem 11.1 The state space realization (11.14) is locally observable in a neighborhood
Uo C D containing the origin, if
Vh
rank = n dx E Uo
VLf 'h
Proof: The proof is omitted. See Reference [52] or [36].
The following example shows that, for linear time-invariant systems, condition (11.15)
is equivalent to the observability condition (11.7).
Example 11.1 Let
= Ax
y = Cx.
Then h(x) = Cx and f (x) = Ax, and we have
Vh(x) = C
VLfh = V(- ) = V(CAx) = CA
VLf-1h =
CAn-1
and therefore '51 is observable if and only if S = {C, CA, CA2, . . , CAn-1 } is linearly independent
or, equivalently, if rank(O) = n.
Roughly speaking, definition 11.4 and Theorem 11.1 state that if the linearization
of the state equation (11.13) is observable, then (11.13) is locally observable around the
origin. Of course, for nonlinear systems local observability does not, in general, imply
global observability.
We saw earlier that for linear time-invariant systems, observability is independent
of the input function and the B matrix in the state space realization. This property is a
consequence of the fact that the mapping xo --1 y is linear, as discussed in Section 11.2.
Nonlinear systems often exhibit singular inputs that can render the state space realization
unobservable. The following example clarifies this point.
Example 11.2 Consider the following state space realization:
-k1 = x2(1 - u)
0.1 x2 = x1
y = xl