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Explain controllability and observability both necessary condition for stability. Check controllability and observability for the system

x=[065102324] x+[012] u

y=[130] x


Topic : State Variable Models

Difficulty : High

Marks : 5M or 10M

1 Answer
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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 : ---------An1B]

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)n1cT]

OR

Qo = [C CA CA2 CAn1]

Given =>

˙x=[065 102 324] x+[0 1 2] u

y=[130] x

we first check for controllability,

Here, A= [065 102 324]

A = 3*3 matrix

n = 3

B = [0 1 2] , C= [130]

The necessary and sufficient condition for controllability is,

Qo = [B : AB : ------An1B]

since, n=3

Qo = [B:AB:A2B] -----(1)

B = [0 1 2] -------(2)

AB= [065 102 324] [0 1 2]

AB = [4 4 10] -----------(3)

A2B = A.[A B] = [065 102 324][4 4 10]

A2B = [26 16 36] -----------(4)

Put (2),(3),(4) in (1)

Qo = [0426 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 = [065 102 324] , A' = [013 602 524]

The necessary and sufficient condition for observability.

Qo=[CTATCT(AT)n1CT]

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.

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