Outline
• Definitions.
• Tests
Controllability
and Observability
– Diagonal form.
– Eigenvector test.
– Rank test.
M. Sami Fadali
Professor EE
University of Nevada, Reno
• Examples
1
2
Diagonal Form
Observability Definition
A LTI system in diagonal form
୬
A LTI system is observable if we can uniquely
from the input
determine any initial state
and output history
୬
ଵ
ଶ
௡
is observable if and only if its output matrix has no
zero columns.
ஃ௧
௓ூ
over a finite time interval
୬
୬ଵ
ఒభ ௧
ఒమ ௧
ఒ೙ ௧
ஃ௧
ఒభ ௧
୬ଶ
୬௡
ఒమ ௧
ఒ೙ ௧
3
୬௜
ଵ
ଶ
௡
Unobservable mode
ఒ೔ ௧
4
Rank Test
Eigenvector Test
A LTI system is completely observable if and only if
the
observability matrix has rank .
A LTI system is observable if and only if the
product of the output matrix and the matrix of
right eigenvectors has no zero columns.
௜
௜ unobservable state
•
•
Unobservable mode
For any pair
୬ଵ
୬
୬௜
௡ିଵ
ఒ೔ ௧
௜
୬ଶ
௜ ௜
Rank deficit = number of unobservable modes.
୬௡
௜
ఒ೔ ௧
௜
unobservable (mode, state)
௜
5
6
Diagonal Form
Controllability Definition
୬
A LTI system is controllable if for any initial
there exists an input
that can
state
in a
drive the system to any final state
finite time interval T.
௜
୬
ିଵ
்
௡
்
௜
௜ ௜
்
ଵ
்
ଶ
A LTI system in diagonal form is controllable if and only
if its input matrix has no zero rows.
௧
‫ݖ‬௜ ‫ ݐ‬ൌ න ఒ೔ఛ
• Originally defined to origin (problematic).
• Reachability from the origin to any state.
଴
if and only if
்
௜
7
்
௜
்
௜
is zero
்
ఒ೔ ௧
uncontrollable mode
8
Eigenvector Test
Rank Test
A LTI system is controllable if and only if
the product of the output matrix and the
matrix of left eigenvectors has no zero rows.
୬
A LTI system is controllable if and only if
the controllability matrix has full rank n.
௡ିଵ
C
C
• Rank deficit = number of uncontrollable
modes.
்
ଵ
்
ଶ
ିଵ
்
௡
்
௜
்
ఒ೔ ௧
்
௜ C
uncontrollable mode
்
ఒ೔ ௧
uncontrollable mode
9
Example: Diagonal Form
Example: Eigenvector Test
0
0   x1 (t )  1 
 x1 (t )    1 0
 x (t )   0  4 0
0   x 2 (t )   1 
 2 
    r (t )

0  5 0   x3 (t )  0
 x3 (t )   0
 x (t )   0
0
0  11  x4 (t )  1 
 4  
• Check controllability and observability
5
0 
 1 0
1 
 0 4 4
1 
0 
, B   
A
0 5 2
0
0 
0
1 
0
0  11

 
10 0 2 0
C

 0 0 0 1
 x1 (t ) 
 x (t ) 
y (t )  100 4 2 0  2 
 x3 (t ) 
 x (t ) 
 4 
e 5t uncontroll able, e 11t unobservab le
10
11
12
Example Continued: Observability
0  0.2901  0.1537 
1  0.9284  0.1757 

0 0.2321
0.3075 
0
0
0.9225 
5
0 
1
 1 0
0
 0 4 4

0


A
,V  
0 5 2
0
0
0
0

0
0  11


10 0  2.4371  0.9225 
10 0 2 0
C
CV

0 0

0
0.9225 

 0 0 0 1
e  4 t unobservab le
Example Continued: Controllability
5
0 
 1 0
1
 0 4 4
0

0
 ,W  
A
0 5 2
0
0
0
0

0
0  11


1 
1 
B 
0 
1 
 
0 1.25
1
4
0 4.3084
0
0
 0.25 
 1.1429 

- 1.4361 
1.0841 
 0.75 
  0.1429 

WB  
 1.4361 
 1.0841 


controllab le
13
Example: Controllability Rank Test
C  B
AB
2
AB
14
Example: Observability Rank Test
2
0 
 10 0
 0 0
0
1 


40
4 
 10 0
 0 0
0
 11 

O
 10 0  250  36 


0 0
0
121 
10 0 2 0

C

 10 0 1300
896 
 0 0 0 1


0
 1331
 0 0
rank O   3  4  1one unobservable mode
5
0 
 1 0
 0 4 4
0 

A
0 5 2
0
0
0
0  11

A B
3
1  1  9 169 
1  4 8
96 



0  2 32  402 
1  11 121  1331


rank C   4
controllable
MATLAB: >> rank(ctrb(A,B))
15
MATLAB: >> rank(obsv(A,C))
16