The following systems are not completely observable.
Principle of Duality. We shall now discuss the relationship between controllability
and observability. We shall introduce the principle of duality, due to Kalman, to clarify
apparent analogies between controllability and observability.
Consider the system S 1 described by
where
and the dual system S 2 defined by
where
The principle of duality states that the system S 1 is completely state controllable
(observable) if and only if system S 2 is completely observable (state controllable).
To verify this principle, let us write down the necessary and sufficient conditions for
complete state controllability and complete observability for systems S 1 andS 2.
C*=conjugate transpose of C
B*=conjugate transpose of B
A*=conjugate transpose of A
n=output vector (r-vector)
v=control vector (m-vector)
z=state vector (n-vector)
n=B*z
z
=Az+Cv
C=m*n matrix
B=n*r matrix
A=n*n matrix
y=output vector (m-vector)
u=control vector (r-vector)
x=state vector (n-vector)
y=Cx
x# =Ax+Bu
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
0
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= B0
0
1
2
3
4
RC
x 1
x 2
x 3S
B
x# 1
x# 2R = B
- 1
0
0
- 2
RB
x 1
x 2R, y=[0 1]B
x 1
x 2R
Section 9–7 / Observability 687