or
The system is completely observable if none of the columns of the m*nmatrixCP
consists of all zero elements. This is because, if the ith column of CPconsists of all zero
elements, then the state variable zi(0)will not appear in the output equation and there-
fore cannot be determined from observation of y(t).Thus,x(0),which is related to z(0)
by the nonsingular matrix P, cannot be determined. (Remember that this test applies only
if the matrix is in diagonal form.)
If the matrix Acannot be transformed into a diagonal matrix, then by use of a suitable
transformation matrix S, we can transform Ainto a Jordan canonical form, or
whereJis in the Jordan canonical form.
Let us define
Then Equations (9–66) and (9–67) can be written
Hence,
The system is completely observable if (1) no two Jordan blocks in Jare associated with
the same eigenvalues, (2) no columns of CSthat correspond to the first row of each
Jordan block consist of zero elements, and (3) no columns of CSthat correspond to
distinct eigenvalues consist of zero elements.
To clarify condition (2), in Example 9–16 we have encircled by dashed lines the
columns of CSthat correspond to the first row of each Jordan block.
EXAMPLE 9–16 The following systems are completely observable.
E
x# 1
x# 2
x# 3
x# 4
x# 5U = E
2
0
0
0
1
2
0
0
1
2
- 3
0
0
1
- 3
UE
x 1
x 2
x 3
x 4
x 5U, c
y 1
y 2 d= B1
0
1
1
1
1
0
1
0
0
RE
x 1
x 2
x 3
x 4
x 5U
C
x# 1
x# 2
x# 3S = C
2
0
0
1
2
0
0
1
2
SC
x 1
x 2
x 3S, c
y 1
y 2 d= B3
4
0
0
0
0
RC
x 1
x 2
x 3S
B
x# 1
x# 2R = B
- 1
0
0
- 2
RB
x 1
x 2R, y=[1 3]B
x 1
x 2R
y(t)=CSeJt z(0)
y=CSz
z
=S-^1 ASz=Jz
x=Sz
S-^1 AS=J
P-^1 AP
y(t)=CPF
el^1 t
0
el^2 t
0
eln^ t
Vz(0)=CPF
el^1 tz 1 (0)
el^2 tz 2 (0)
eln^ tzn(0)
V
686 Chapter 9 / Control Systems Analysis in State SpaceOpenmirrors.com