Correct Answer - Option 3 : both controllable and observable
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:
\(A = \left[ {\begin{array}{*{20}{c}} 2&1\\ -1&2 \end{array}} \right],\;B = \left[ {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right],\;C = \left[ {\begin{array}{*{20}{c}} 1&1 \end{array}} \right]\)
Controllability:
\(AB = \left[ {\begin{array}{*{20}{c}} 2&1\\ -1&2 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3\\ 1 \end{array}} \right]\)
Controllability matrix,
\(M = \left[ {\begin{array}{*{20}{c}} 1&3\\ 1&1 \end{array}} \right]\)
|M| = -2
Therefore, the system is controllable.
Observability:
\(CA = \left[ {\begin{array}{*{20}{c}} 1&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 2&1\\ -1&2 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&3 \end{array}} \right]\)
Observability matrix,
\(N = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&3 \end{array}} \right]\)
|N| = 2
Therefore, the system is observable
