week 16 controllability and observability june 1 final
TRANSCRIPT
05/01/2023 1
Week 16 Controllability
and ObservabilityProf Charlton S. InaoDefence University
College of Engineering
05/01/2023 2
Instructional Objectives
05/01/2023 3
CONTROLLABILITY
05/01/2023 4
05/01/2023 5
05/01/2023 6
05/01/2023 7
State Controllability• Controllability Matrix CM
• System is said to be state controllable if
BABAABBCM n 12
)( nCMrank
State Controllability (Example)• Consider the system given below
• State diagram of the system is
xy
uxx
2101
3001
1
1
)(sU
)(sY1
-1s
3-1s
2
1x
2x
State Controllability (Example)•
ABBCM
0011
CM
01
B
01
A B
System order(state variable) is 2 but rank is
1, therefore not controllable
05/01/2023 11
05/01/2023 12
05/01/2023 13
05/01/2023 14
Workout Exercise
05/01/2023 15
OBSERVABILITY
05/01/2023 16
05/01/2023 17
05/01/2023 18
State Observability• Observable Matrix (OM)
• The system is said to be completely state observable if
1
2MMatrix ity Observabil
nCA
CACAC
O
nOMrank )(
n= system order ,based on the number of state variable
State Observability (Example)• Consider the system given below
• OM is obtained as
• Where
xy
uxx
4010
2010
CAC
OM
40C
12020
1040
CA
State Observability (Example)•
12040
MO
1)(sU -1s -1s 1x2x
2
4
)(sY
Rank =1n=system order =2
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 12O
Work Out Exercise
• Check the state controllability, state observability and output controllability of the following system
10,10
,0110
CBA
05/01/2023 25
05/01/2023 26
05/01/2023 27
Reference/Basis
05/01/2023 28
Reference/Basis
05/01/2023 29
If determinant is zero , i.e singular… the system is non observable
N=rank=3
System order=full rank, there fore it is observable
05/01/2023 30
Finding the determinant
Down (+)
UP (-)
05/01/2023 31
Unobservability via Observability Mtrix
If determinant of the observability matrix is zero , the system is unobservable
05/01/2023 32
Calculation of Determinant
If determinant of the observability matrix is zero , the system is unobservable
05/01/2023 33
05/01/2023 34
05/01/2023 35
05/01/2023 36
05/01/2023 37
05/01/2023 38
05/01/2023 39
05/01/2023 40
05/01/2023 41
All zero column
System order =2Rank=1Not equal , therfore UNOBSERVABLE
05/01/2023 42
05/01/2023 43
Identical but negated(opposite sign)
Identical but negated(opposite sign)
Controllability and Observability Using Matlab
Prof Charlton S. Inao
• % 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 ----------------