Canonical Decomposition of Linear Time-Invariant System
Consider the linear time-invariant dynamical equation
 :

x  Ax  Bu
y  Cx  Du
(*)
where A, B, C and D are n × n, n × p, q × n, and q × p
constant matrices.
What can we say if the system is uncontrollable and/or
unobservable?
Equivalence Transformation
Let
x  Px ,
where P is any n × n nonsingular matrix.
The substitution of
x  Px into (*) yields
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
x  Ax  Bu

(**)
y  C x  Du
where
A  PAP 1
B  PB
C  CP -1
DD
Controllability of the Transformed System
U  B
 PB
 PB
 PB
A B ... A n 1 B 
PAP -1 PB ... PA n 1 P 1 PB 
PAB ... PA n 1 B 
AB ... An 1 B   PU
 (U )   ( PU )   (U )
Σ is controllable iff
 is controllable
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Theorem
The controllability and observability of a LTI dynamical
equation are invariant under any equivalence transformation.
Representation of Noncontrollable Realizations
Let {A, B, C, D} be such that
Rank U(A,B) = r < n
Than we shall show that a transformation matrix P can always
be found such that the realization
{ A  PAP 1
B  PB
C  CP -1
D  D}
has the form
 Ac
A
0
Acc  r
Ac  n  r
C  Cc
Cc 
 Bc 
B 
0
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Any realization in this form the following properties:
1.
The r × r subsystem { Ac , Bc , Cc ,
D } is
controllable.
2.
C ( sI  A ) 1 B  Cc ( sI  Ac ) 1 Bc
The controllable subsystem has the same T.F. of the
original system.
To show the second statement
xx
( sI  Ac ) 1
  Bc 
C ( sI  A ) B  C 

1  
0
0
(
sI

A
)




c
( sI  Ac ) 1 Bc 
 Cc Cc 

0


 Cc ( sI  Ac ) 1 Bc
1
xx  ( sI  Ac ) 1 Acc ( sI  Ac ) 1
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Note that
 Bc
U ( A, B )  
0
Ac Bc
0
... Acn 1 Bc 

...
0 
How do we find P?
Let P-1 = Q
QU ( A , B )  U ( A, B)
or
Q1
U ( Ac , Bc ) xx 
Q2 
 B

0
0

AB ... A r 1 B
xx
From which
Q1  B
AB
...
Ar 1 BU 1 ( Ac , Bc )
If
U 1 ( Ac , Bc )  I
Then
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Q1  B
AB
...
Ar 1 B 
and
Q2 = [
]
Any linearly independent columns which makes Q nonsingular
If the state variables
x
are correspondingly partitioned as
 xc 
x 
 xc 
then
xc
xc
is said to be controllable
is said to be uncontrollable
The realization { A, B , C } can be decomposed as
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Representation of Unobervable Realizations
It should be clear that similar statements could be made about
unobservable realization
Thus if
Rank O(A,C) = r < n
Than we can find a nonsingular transformation matrix P such
that the realization
{ A  PAP 1
B  PB
C  CP -1
D  D}
has the form
A
A o
 Aoo
0 r

Ao  n  r
B 
B   o
 Bo 
C  Co 0
and {Co , Ao } is observable
and
T ( s)  C ( sI  A) 1 B  Co ( sI  Ao ) 1 Bo
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The separation into observable and unobservable parts can be
represented by
General Decomposition Theorem
We can always find a nonsingular transformation that will
allow us to rewrite the state equation

x  Ax  Bu
y  Cx
in the form

x  Ax  Bu
y  Cx
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where

xcTo
 Aco

A
A   21
 0

 0
0
A13
Aco
A23
0
Ac o
0
A43
0 

A24 
0 

Ac o 
0
Cc o
0
C  Cco
xcTo
xcTo

T
x T  xco
 Bco 


B
c
o

B 
 0 


 0 
where
1. The system
Aco , Bco , Cco 
is controllable and observable and
T ( s)  C ( sI  A ) 1 B  Cco ( sI  Aco ) 1 Bco
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2.
 Aco

 A21
0 
,
Aco 
 Bco 

,
 Bco 
Cco

0

Cco

Cc o 

is controllable.
3.
 Aco

 0
A13 
,
Ac o 
 Bco 
 0 ,


is observable
4.
The system
Aco ,
0, 0
is neither controllable nor observable.
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