Nonlinear Control Sy.. - Free

11.1. OBSERVERS FOR LINEAR TIME-INVARIANT SYSTEMS 293
or, equivalently
N(CeAt) = 0.
Thus, L is one-to-one [and so (11.1) is observable] if and only if the null space of CeAt is
empty (see Definition 2.9).
Notice that, according to this discussion, the observability properties of the state
space realization (11.1) are independent of the input u and/or the matrix B. Therefore we
can assume, without loss of generality, that u - 0 in (11.1). Assuming for simplicity that
this is the case, (11.1) reduces to
x= Ax
(11.6)
l y=Cx.
Observability conditions can now be easily derived as follows. From the discussion above,
setting u = 0, we have that the state space realization (11.6) [or (11.1)] is observable if and
only if
y(t) = CeAtxo - 0 = x - 0.
We now show that this is the case if and only if
To see this, note that
rank(O)'Irank
y(t) = CeAtxo
C(I+tA+
C
CA
2
CAn-1
= n (11.7)
By the Sylvester theorem, only the first n - 1 powers of AZ are linearly independent. Thus
or,
y = 0 b Cxo = CAxo = CA2xo = ... = CAn-1xo
y=0
C
CA
CAn-1
xp = 0, b Ox0 = 0
and the condition Oxo = 0 xo = 0 is satisfied if and only if rank(O) = n.
The matrix 0 is called the observability matrix. Given these conditions, we can
redefine observability of linear time-invariant state space realizations as follows.