Section 9–7 / Observability 685
Note that
Hence, the rank of the matrix is less than 3. Therefore, the system is not
completely observable.
In fact, in this system, cancellation occurs in the transfer function of the system. The transfer
function between X 1 (s)andU(s)is
and the transfer function between Y(s)andX 1 (s)is
Therefore, the transfer function between the output Y(s)and the input U(s)is
Clearly, the two factors (s+1)cancel each other. This means that there are nonzero initial states
x(0),which cannot be determined from the measurement of y(t).
Comments. The transfer function has no cancellation if and only if the system is com-
pletely state controllable and completely observable. This means that the canceled transfer
function does not carry along all the information characterizing the dynamic system.
Alternative Form of the Condition for Complete Observability. Consider the
system described by Equations (9–63) and (9–64), rewritten
(9–66)
(9–67)
Suppose that the transformation matrix PtransformsAinto a diagonal matrix, or
whereDis a diagonal matrix. Let us define
Then Equations (9–66) and (9–67) can be written
Hence,
y(t)=CPeDt z(0)
y=CPz
z
=P-^1 APz=Dz
x=Pz
P-^1 AP=D
y=Cx
x
=Ax
Y(s)
U(s)=
(s+1)(s+4)
(s+1)(s+2)(s+3)Y(s)
X 1 (s)=(s+1)(s+4)X 1 (s)
U(s)=
1
(s+1)(s+2)(s+3)CCAC(A)^2 C*D
3
4
5
1
- 6
- 7
- 1
6
5
- 1
3 = 0