Controllability and Observability

1. Week 16
Controllability
and
Observability
Prof Charlton S. Inao
Defence University
College of Engineering
16/3/2016

2. Instructional Objectives
6/3/2016 2

3. CONTROLLABILITY
6/3/2016 3

4. 6/3/2016 4

5. 6/3/2016 5

6. 6/3/2016 6

7. 76/3/2016

8. State Controllability
• Controllability Matrix CM
• System is said to be state controllable if
 BABAABBCM n 12 
 
)( nCMrank 

9. State Controllability (Example)
• Consider the system given below
• State diagram of the system is
 xy
uxx
21
0
1
30
01
















1
1
)(sU
)(sY
1
-1
s
3
-1
s
2
1x
2x

10. State Controllability (Example)
 ABBCM 





 

00
11
CM







0
1
B 






0
1
AB
System order(state
variable) is 2 but rank is
1, therefore not
controllable

11. 116/3/2016

12. 6/3/2016 12

13. 136/3/2016

14. Workout Exercise
146/3/2016

15. OBSERVABILITY
156/3/2016

16. 166/3/2016

17. 176/3/2016

18. 186/3/2016

19. State Observability
• Observable Matrix (OM)
• The system is said to be completely state observable if

















1
2
MMatrixityObservabil
n
CA
CA
CA
C
O

nOMrank )(
n= system order ,based on
the number of state
variable

20. State Observability (Example)
• Consider the system given below
• OM is obtained as
• Where
 xy
uxx
40
1
0
20
10






















CA
C
OM
 40C
   120
20
10
40 






CA

21. State Observability (Example)








120
40
MO
1)(sU -1
s
-1
s 1x2x
2
4
)(sY
Rank =1
n=system order =2

23. Output Controllability
• Output controllability describes the ability of an external
input to move the output from any initial condition to any
final condition in a finite time interval.
• Output controllability matrix (OCM) is given as
 BCABCACABCBCM n 12
O 
 

24. Work Out Exercise
• Check the state controllability, state observability
and output controllability of the following system
 10,
1
0
,
01
10












 CBA

25. 256/3/2016

26. 6/3/2016 26

27. Reference/Basis
6/3/2016 27

28. 6/3/2016 28
Reference/Basis

29. 296/3/2016
If determinant is zero , i.e singular… the system is non
observable
N=rank=3
System order=full
rank, there fore it
is observable

30. Finding the determinant
6/3/2016 30
Down (+)
UP (-)

31. Unobservability via Observability Mtrix
316/3/2016
If determinant of the observability
matrix is zero , the system is
unobservable

32. Calculation of Determinant
6/3/2016 32
If determinant of the observability
matrix is zero , the system is
unobservable

33. 6/3/2016 33

34. 346/3/2016

35. 356/3/2016

36. 366/3/2016

37. 376/3/2016

38. 386/3/2016

39. 396/3/2016

40. 406/3/2016

41. 416/3/2016
All zero
column
System order =2
Rank=1
Not equal , therfore
UNOBSERVABLE

42. 426/3/2016

43. 436/3/2016
Identical but
negated(opposite
sign)
Identical but
negated(opposite
sign)

44. Controllability and Observability
Using Matlab
Prof Charlton S. Inao

49. • % State Space Representation % x' = Ax + Bu % y = Cx + Du % %
Problem 1 --------------------------------------------------------------- %
• Check Controllability and Observability of a 2nd order System %
• Given ------------------------------------------------------------------- MatrixA =
[0 1;-2 -3]; MatrixB = [0;1]; MatrixC = [1 -1]; MatrixD = 0; %
• Objective --------------------------------------------------------------- %
• 1) To Find Controllable Matrix Qc, its rank and check controllability
• % 2) To Find Observable Matrix Qb, its rank and check observability
%------
• --- % Controllable Matrix -----------------------------------------------------
Qc = ctrb(MatrixA,MatrixB); rankQc = rank(Qc); disp('Controllable
Matrix is Qc = '); disp(Qc); if(rankQc == rank(MatrixA)) disp('Given
System is Controllable.'); else disp('Given System is Uncontrollable');
end % Observable Matrix -------------------------------------------------------
Qb = obsv(MatrixA, MatrixC); rankQb = rank(Qb); disp('Observable
Matrix is Qb = '); disp(Qb); if(rankQb == rank(MatrixA)) disp('Given
System is Observable.'); else disp('Given System is Unobservable');
end % End of Program ----------------