§2-3 Observability of Linear Dynamical
Equations
1. Definition of observability
Observability studies the possibility of estimating the
state from the output.
Definition 2-6 The dynamical equation
x  A (t )x  B(t )u
y  C(t )x  D(t )u, t  [t0 ,  ) (2  1)
is said to be observable at t0, if there exists a finite t1>t0
such that for any state x(t0) at time t0, the knowledge of the
y
input
and the output ut0 ,tover
the
time
interval

t0 ,t1 
1
t0 ,tthe
 state x(t ). Otherwise, the
suffices to determine
1
0
dynamical equation is said to be unobservable at t0.
Remark: The system is said to be unobservable if there
exists a state x(t0) such that x(t0) can not be determined
yt0 ,t1 
ut0 ,t1 
by
and
uniquely.
Question How to determine the state x(t0) with
output
? yt0 ,t1 
Example 2-11 Consider the following system
 x1  0 1   x1  0 
x   0 0  x   1  u
 2  
 2 
y  cx  1 0x
whose state transition matrix is
1 t  t 0 
F (t  t 0 )  

0
1


ut0 ,t1 
and
The solution is
t
  (t  t )u (t )d t
 x1  1 (t  t 0 )   x10  t 0



x   0

t
x
1
  20  
 2 
u (t )d t


 t0







Note that
t
1 (t  t 0 )   x10 
y =cx  [1 0] 
  (t  t )u (t )d t



1  x 20  t
0
0
c (t , t0 )
Pre-multiplying
0  1 
 1
(t  t ) 1  0 
0

 
*( t ,t0 ) c*
0  1   1
0  1 
 1
1 (t  t 0 )   x10 
(t  t ) 1  0  y = (t  t ) 1  0  [1 0]  0
 x 
1

  20 
0
0

  
 
*(t ,t 0 )c*
*(t ,t 0 )c*
0  1  t
 1

(t  t )u (t )d t




(t  t 0 ) 1  0  t 0
*(t ,t 0 )c*
 (t ,t 0 )c
x10 
  *(t ,t 0 )c * y (t )dt    *(t ,t 0 )c *c *(t ,t 0 )dt x 20 
t
t
t1
t1
0
0
t1
t
t0
t0
  { *(t ,t 0 )c *  (t  t )u (t )d t }dt

 (t1  t 0 )
h (t 0 ,t1, y )  
 1 (t  t ) 2
1
0

2
known
1
2
(t1  t 0 ) 
 x10 
2
    g (t 0 ,t1, u )
1
3  x 20 
(t1  t 0 )

3
known
we have

 (t1  t 0 )

 1 (t  t ) 2
 2 1 0
1
2
(t1  t 0 ) 
 x10 
2
    h (t 0 ,t1, y )  g (t 0 ,t1, u )
1
3  x 20 
(t1  t 0 )

3
known
It is easy to verify that

 (t1  t 0 )
det 
 1 (t  t ) 2
 2 1 0
for t1  t.0
1
2
(t1  t 0 ) 
1
2
4

(
t

t
)
0

1
0
1
12
3
(t1  t 0 )

3
Hence,

(t1  t 0 )

x
 10 
x    1
 20   (t  t ) 2
 2 1 0
1
1
2
(t1  t 0 ) 
2
 [h (t 0 ,t1, y )  g (t 0 ,t1, u )]
1
(t1  t 0 )3 

3
2. Criteria for observability
Theorem 2-8 Dynamical equation
x  A (t )x  B(t )u
y  C(t )x  D(t )u, t  [t0 ,  ) (2  1)
is observable at time t0 if and only if there exists a
finitet1>t0, such that the n columns of matrix
C(t )Φ(t ,t 0 )
is linearly independent over [t0, t1].
Proof
Sufficiency:
1). Consider
t
y (t )  C(t )Φ(t ,t 0 )x (t 0 )   C(t )Φ(t , t )B(t )u (t )d t (＊)
t0
2). Pre-multiplying both sides of the equation (＊) with
[C(t )Φ(t ,t 0 )]  Φ (t ,t 0 )C (t )
we have
Φ (t ,t 0 )C (t )C(t )Φ(t ,t 0 )x (t 0 )  Φ (t ,t 0 )C (t )y1(t )
t
y1 : y (t )   C(t )Φ(t , t )B(t )u (t )d t
t0
3). Integrating both sides from t0 to t1, we have
V (t 0 ,t1 )x (t 0 ) 
t1
t1

Φ (t ,t 0 )C (t )y1 (t )d t
t0
V (t 0 ,t1 ) :  Φ (t ,t 0 )C (t )C(t )Φ(t , t 0 )d t
t0
Form Theorem 2-1, it follows that V(t0, t1) is
nonsingular if and only if the columns of C(t) (t, t0) is
linearly independent over [t0, t1].
Necessity: the proof is by contradiction.
Assume that the system is observable but the columns
C(tlinearly
)Φ(t , t0 )dependent for any
of
are
. Then
 t0 a
a 0
theret1exists
, such that
C(t )Φ(t ,t 0 )a  0，t  [t 0 ,t1 ].
a we have
If we choose x (t 0 ) , then
y (t )  C(t )Φ(t ,t 0 )a  0t  t 0
which means that x(t0) can not be determined by y.
Corollay 2-8 The dynamical equation (2-1) is
observable at time t0 if and only if there exists a finite
time t1>t0 such that the matrix V(t0, t1) is nonsingular,
where
t1
V (t 0 ,t1 )   Φ (t ,t 0 )C (t )C(t )Φ(t ,t 0 )d t .
t0
Theorem 2-10 Suppose that A(t) and C(t) of the state
equation (A(t), B(t), C(t)) are n-1 times continuously
differentiable. Then the dynamical equation is observable
at t0 if there exists a finite t1>t0 such that
 N 0 (t1 ) 
 N (t ) 
rank  1 1   n


 N (t ) 
 n 1 1 
where
N k (t ) = N k -
N 0 (t ) = C(t )
dN k - 1(t )
1 (t )A(t ) +
dt
k = 1, 2, L , n - 1
5. The observability criteria for LTI systems
1. Observability criteria
Theorem 2-11 For the n-dimensional linear time invariant
dynamical equation
x  Ax  Bu
（2-21）
y  Cx  Du
the following statements are equivalent:
(1)All columns of CeAt are linearly independent on
+).
[t0,
[0, )
(2)All columns of C(sIA)1are linearly independent over
C.
(3) The matrix
t
V (t 0 ,t ) 
A* ( t t 0 )
A ( t t 0 )
e
C*C
e
dt

t0
is nonsingular for any t0 ≥0 and t > t0.
(4) The n q n observability matrix
 C 
 CA 
 n
rank 


 n 1 
CA 
i A,
(6) For every eigenvalue of
A  l i I
rank 
n

 C 
(2  15)
§2- 4 Controllability and observability of
Jordan canonical form
1. Equivalence transformation
Consider
x  Ax  Bu
x
Let x  Pand
y  Cx  Du
det( P.)Then
 0 we have
x  Ax  B u
y  Cx  Du
where
A  PAP 1, B  PB, C  CP 1, D  D
Theorem 2-13: The controllability and observability of
a linear time-invariant dynamical equation are invariant
under any equivalence transformation.
Proof
From the Theorem 2-6,
x  Ax  Bu
y  Cx  Du
is controllable if and only if
rank [B AB A(n 1)B]  n
It is easy to verify that
[B AB A( n1)B]  P[B AB A( n1)B]
2. Criteria for controllability and observability
of the jordan-form dynamical equations
Typical Jordan-canonical form matrix is as follows
2 1



2


A1
2 1




2



3 1 
A2 

3 

Example Determine the controllability and observability of
the following system
 250 1

1
 052

1
A1





2
520 1
A
B

520


0

4
-5
03 1 
A 2-5 


03

1
1 1 0 7 3 0 
C

0
4

1
3
3
1


Using PBH rank test: rank [ A  l i I B] or
l 1  2, l 2  3
2

0

3

1
5

0
b L11
b L12
b L 21
A  l i I
rank 

C


 250 1
 205


A


c111

1 1
C
0 4
520 1
250
c112
0
7
1 3






-5
03 1 
c121-5 
03
3 0
3 1 
1
1

2
B
0
4

1
2
0

3

1
5

0
A  l i I
Using PBH rank test: rank [ A  l i I B] or rank 

C


l 1  2, l 2  3
li
当系统矩阵有重特征值时，常常可以化为若当
ri
A
ij
形，这时A、 B、C的形式如下：
Ai
 A1

 B1 
 b1ij 

 l i 1A


B
 b  2 
2


li 1
A=  
B
ij 
 2

 B 




Aij  
li
ij
 




A
B
1 m
 

 m 

C  [C
1
C2
 Ai 1

Ai 2

Ai 



Ci  [Ci1 Ci 2

l iC
l l
j
m]
j
 bLij 



 Bi1 

B 
 B   i2 
i

 



Airi 
 Biri 
Ciri ], i  1,2, , m
l i


Aij  



Cij  [c1ij
1
li
c 2ij
1
li





1
l i l l
j
j
 b1ij 
b 
 2ij 

Bij  




 bLij 


cLij ], i  1,2, , m ; j  1,2, , ri
Theorem 2-14
 System in Jordan canonical form is controllable if
and only if the rows of the following matrix
 bLi 1 
b

 Li 2 


b

 Liri 
are linearly independent.
i  1, 2,
,m
 System in Jordan canonical form is observable if and
only if the columns of
C1i  [c1i1 c1i 2 c1iri ], i  1, 2, m
are linearly independent.
proof
 A1  i I

[ A  i I B]  



Ai  i I
B1 


Bi 


Let Ai be an nith order block, we only need to check
rank [Ai  l i I Bi ]  n i
Because other sub blocks are full row rank, and
 Ai 1

 Bi1 


B 
Ai 2
i2 



Ai 
Bi 


 


B 
A
iri 

 iri 
By using PBH test
 Ai 1  l i I

 Bi1 


B 
Ai 2  l i I
i2 



Ai  l i I 
Bi 


 


B 
A

l
I
iri
i 

 iri 
the last row and the first column are zero because Aij is
of Jordan canonical form. Therefore, if the matrix
 bLi 1 
b

 Li 2 


b

 Liri 
formed by the last rows of
independent, then
Bi1、Bi 2、
are Blinearly
iri
rank [A i  i I B i ]  ni
Similarly, we can prove the observability for system in a
Jordan canonical form.
0


Aij  l i I  



Cij  [c1ij
1
0
1
0
c 2ij





1
0 l l
j
j
 b1ij 
b 
 2ij 

Bij  




 bLij 


cLij ], i  1,2, , m ; j  1,2, , ri
Example
Determine the controllability and observability of the
following system
1



A






1
1
A1
1
1
2
1
A22









2
 77
1 1 2 0 0 2 0 
C  1 0 1 2 0 1 1 


1 0 2 3 0 2 2 
0
1

0
B  0

1

0

0
0
0
1
0
1
3
0
0
0

0
1

2

1
2

Substituting
0




A





Bhave
 A  1I , we
in1
1
0
0
0







1 1

1

1
 b L11  1 0 0 
 b   0 1 0 
 L12  

 b L13  0 0 1 
which is of full row rank .
0
1

0

B  0
1

0
0

0
0
1
0
1
3
0
0
0 
0

1
2

1
2 
b L11
bL12
bL13
Substituting
in2
 A  2I, weBhave

 1 1




1




1


A
1


0 1 


0 


0 

 b L 21  0 3 1 
 b   0 0 2 

 L 22  
which is of full row rank .
0
1

0

B  0
1

0
0

0
0
1
0
1
3
0
0
0 
0

1
2

1
2 
bL 21
b L 22
 b L11  1 0 0
 b   0 1 0
 L12  

 b L13  0 0 1 
,
 b L 21  0 3 1 
 b   0 0 2 

 L 22  
are linearly independent. Therefore, the system is
controllable.
Determine the observability of the following system:
1



A






1
C  1


1
1
1
1
1
2
1
2
1
2
0
0
2
0
1
2
0
1
0
2
3
0
2









2
 77
0
1

2

Substituting
in1
 A  1I 
rank  , we have

C


0



A






1
1
C  1


1
1
2
0
0
2
0
1
2
0
1
0
2
3
0
2
c111
0
0
0
1
1
1
c112 c113









1
 77
0
1

2

The sub-block
1
1


1
2
1
2
0
2

3

is of full column
rank.
Substituting
 A  2I 
rank 
, wehave
 C 
in
2
 1



A






1
C  1


1
1
1
1
1
0
1
0
1
2
0
0
2
0
1
2
0
1
0
2
3
0
2
c121









0
 77
0
1

2

Because the
column
C121 is zero,
the system is
unobservable.
Example Consider the single input system
2 1 0 0
0 
 2 0 0
1 
, b   
A
3 1

0




3

1 
It is easy to check that the system is controllable by
using PBH test.