Page 791 - The Mechatronics Handbook

0066_Frame_C24 Page 39 Thursday, January 10, 2002 3:45 PM

Canonical Decomposition
Further insight into the structure of linear dynamical systems is obtained by considering those systems
which are only partially observable or controllable. These systems can be separated into completely
observable and completely controllable systems.
The two results of Lemmas 24.1 and 24.4 can be combined for those systems, which are neither completely
observable nor completely controllable. We can see it as follows.

Theorem 24.6 (Canonical Decomposition Theorem) Consider a system described in state space form.
Then, there always exists a similarity transformation T such that the transformed model for x = T x takes
–
1
the form

A co 0 A 13 0
B 1
A = A 21 A 22 A 23 A 24 , B = B 2 , C = [ C 1 0C 2 0] (24.231)
0 0 A 33 0 0
0 0 A 34 A 44 0

where
i) The subsystem [A co, B 1, C 1 ] is both completely controllable and completely observable and has the
same transfer function as the original system (see Lemma 24.6).
ii) The subsystem

A co 0 B 1 [ C 1 0]
, , (24.232)
A 21 A 22 B 2

is completely controllable.
iii) The subsystem

A co A 13 B 1 [ C 1 C 2]
, , (24.233)
0 A 33 0
is completely observable.

The canonical decomposition described in Theorem 24.6 leads to an important consequence for the
transfer function of the model, which will take only the completely observable and completely controllable
subspace.
Lemma 24.6 Consider the transfer function matrix H(s) given by

Y s() = H s()U s() (24.234)

Then

–
(
H = C sIA) B + D = C 1 sIA co) B 1 + D (24.235)
(
1
1
–
–
–
B 1
where C 1, A co , and are as in Eq. (24.231). This state description is a minimal realization of the transfer
function.
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