Correct Answer - Option 3 : Both 1 and 2

__Concept:__

State space representation:

ẋ(t) = A(t)x(t) + B(t)u(t)

y(t) = C(t)x(t) + D(t)u(t)

y(t) is output

u(t) is input

x(t) is a state vector

A is a system matrix

This representation is continuous time-variant.

Controllability:

A system is said to be controllable if it is possible to transfer the system state from any initial state x(t0) to any desired state x(t) in a specified finite time interval by a control vector u(t)

Kalman’s test for controllability:

ẋ = Ax + Bu

Qc = {B AB A2B … An-1 B]

Qc = controllability matrix

If |Qc| = 0, system is not controllable

If |Qc|≠ 0, system is controllable

Observability:

A system is said to be observable if every state x(t0) can be completely identified by measurement of output y(t) over a finite time interval.

Kalman’s test for observability:

Q0 = [CT ATCT (AT)2CT …. (AT)n-1 CT]

Q0 = observability testing matrix

If |Q0| = 0, system is not observable

If |Q0| ≠ 0, system is observable.

__Calculation:__

\(A = \left[ {\begin{array}{*{20}{c}} 0&1\\ 2&-3 \end{array}} \right],\;B = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right],\;C = \left[ {\begin{array}{*{20}{c}} 1&1 \end{array}} \right]\)

Controllability:

\(AB = \left[ {\begin{array}{*{20}{c}} 0&1\\ 2&-3 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ -3 \end{array}} \right]\)

Controllability matrix,

\(M = \left[ {\begin{array}{*{20}{c}} 0&1\\ 1&-3 \end{array}} \right]\)

|M| = -1

Therefore, the system is controllable.

Observability:

\(CA = \left[ {\begin{array}{*{20}{c}} 1&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&1\\ 2&-3 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2&-2 \end{array}} \right]\)

Observability matrix,

\(N = \left[ {\begin{array}{*{20}{c}} 1&1\\ 2&-2 \end{array}} \right]\)

|N| = -4

Therefore, the system is observable