written 6.2 years ago by | • modified 6.2 years ago |
x=[06−5102324] x+[012] u
Topic : State Variable Models
Difficulty : High
Marks : 5M or 10M
written 6.2 years ago by | • modified 6.2 years ago |
x=[06−5102324] x+[012] u
Topic : State Variable Models
Difficulty : High
Marks : 5M or 10M
written 6.2 years ago by | • modified 6.1 years ago |
Controllability
˙x(k) = A x(k) + B u(k) ---(1)
y(k) = C x(k) + D u(k)
The necessary and sufficient condition for controllability is that rank of the composite matrix Qc is n,
Qo = [B: AB : A2B : ---------An−1B]
Defination : The equation (1) is said to be completely state controllable if for initial state X(0) and any final state X(N), there exists an input sequence, K= 0,1,2,----,N, which transfers X(0) to X(N) for some finite N otherwise the equation (1) is uncontrollable
Observability
Defination :- The equation (1) is said be observative if any initial state X(0) can be uniquely determined from the knowledge of output y(k) and input sequence u(k), for k=0,1,2______,N where N is some finite time, otherewise the state moded/equation (1) is unobservable.
A system is completely observable if and only if the rank of the composite matrix Qo is n.
where => Qo = [cT:ATcT:−−−−−−(AT)n−1cT]
OR
Qo = [C CA CA2 CAn−1]
Given =>
˙x=[06−5 102 324] x+[0 1 2] u
y=[130] x
we first check for controllability,
Here, A= [06−5 102 324]
A = 3*3 matrix
n = 3
B = [0 1 2] , C= [130]
The necessary and sufficient condition for controllability is,
Qo = [B : AB : ------An−1B]
since, n=3
Qo = [B:AB:A2B] -----(1)
B = [0 1 2] -------(2)
AB= [06−5 102 324] [0 1 2]
AB = [−4 4 10] -----------(3)
A2B = A.[A B] = [06−5 102 324][−4 4 10]
A2B = [−26 16 36] -----------(4)
Put (2),(3),(4) in (1)
Qo = [0−4−26 1416 21036]
Now find determinant of |Qc|
|Qo| = -36
|Qo| ≠ 0
Since the determinant of Qc is non zero, therefore the rank of Qc = n = 3
Hence the system is completely controllable.
Observability =>
Given,
B = [0 1 2] , B' = [012]
C = [130] , C' = [1 3 0]
A = [06−5 102 324] , A' = [013 602 −524]
The necessary and sufficient condition for observability.
Qo=[CTATCT−−−−(AT)n−1CT]
For n=3,
Qo=[CTATCT(AT)2CT]−−−−−(5)
CT=[1 3 0]-----(6)
ATCT=[013 602 −524][1 3 0]
ATCT=[3 6 1]-----(7)
(AT)2CT=AT⋅(ATCT)=[013 602 −524][3 6 1]
(AT)2CT=[9201]-----(8)
Put equations (6), (7), (8) in equation (5),
Qo=[139 3620 011]
Now, find the determinant of Qo
|Qo|=|139 3620 011|
|Qo|=4
Since, |Qo|≠0, the rank of Qo is n=3.
∴ The system is completely observable.