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0066_Frame_C24 Page 37 Thursday, January 10, 2002 3:45 PM

Assuming the same component values as in Example 24.14 for R 1 , R 2 , R 3 , C 1 , and C 3 we have

v ˙ C3 t() = – 1 0 v C3 t() + 1 v i t()
20
i R1 t() 10 – 3 – ----- iR1 t() 0 (24.217)
˙
9
v C3 t()
3
v o t() = [ 1 – 10 ] (24.218)
i R1 t()
If we look at the relative magnitude of the components of C matrix, we can foretell a priori that the
output v o (t) will be mainly determined by state i R1 (t). To verify this we compute the observability gramian
defined in (24.172), solving

T
0 = A Q + QA + C C (24.219)
T
– 1 10 – 3 – 1 0 1
0 = q 11 q 12 + q 11 q 12 20 + [ 110 ] (24.220)
3
–
20
–
0 ----- q 21 q 22 q 21 q 22 10 – 3 – ----- – 10 3
9 9
We have
0.57 69.83
Q = (24.221)
69.83 225000
From there we can compute the contribution of each state to the total energy in the output. Doing
this, we verify that the state variable i R1 (t) has an effect over the output greater than the effect of v C3 (t),
as defined in Eq. (24.212):

()
x 0 = [ 1, 0] ⇒ E x 0 = 0.57 (24.222)
T
x 0 = [ 0, 1] ⇒ E x 0 = 225000 (24.223)
T
()
The transfer function is

1
V o s() s ++ 1 -- 1
9
⋅
------------ = -------------------- ----------- (24.224)
V i s() s + 20 s + 1
-----
9
We observe that there is a pole-zero quasi-cancellation.
Duality Principle
We observe a remarkable similarity between the results in Theorem 24.3 and in Theorem 24.4, and also
for the definitions of the gramians (24.172) and (24.213). This is known as the duality principle, and it
can be formalized as follows:
Theorem 24.5 (Duality) Consider a state space model described by the 4-tuple (A, B, C, D). Then the
T T T T
system is completely controllable if and only if the dual system (A , C , B , D ) is completely observable.
Canonical Decomposition and Detectability
The above theorem can often be used to go from a result on controllability to one on observability and
vice versa. The dual of Lemma 24.1 is:

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