Correct Answer - Option 3 : Controllable but not observable
Concept:
Controllability:
It is the internal states of the system are changed from one value to another value in a finite time by a finite input, then we can say the system is controllable otherwise it is not controllable.
To check Controllability, consider the controllability matrix (ϕc)
\(\left[ {{\phi _c}} \right] = {\left[ {{A^0}B\;\;\;{A^1}B} \right]_{2 \times 2}}\;\;\;;\left[ {{\phi _c}} \right] = {\left[ {{A^0}B\;\;\;\;\;{A^1}B\;\;\;\;{A^2}B} \right]_{3 \times 3}}\)
If |ϕc| = 0; then the system is uncontrollable
If |ϕc| ≠ 0, then the system is controllable.
Observability:
If the internal states of the system can be evaluated from the output of the system of any time, then we can say, the system is observable otherwise it is not observable.
To check observability, consider the observability matrix
\(\left[ {{\phi _0}} \right] = {\left[ {{C^T}\;\;\;\;A{C^T}} \right]_{2 \times 2}}\; \to {2^{nd}}\;order\:\:\:;\:\:\:\left[ {{\phi _0}} \right] = {\left[ {{C^T}\;\;\;\;\;A{C^T}\;\;\;\;\;{A^2}{C^T}} \right]_{3 \times 3}}\)
If |ϕo| = 0; Then system is not-observable
If |ϕo| ≠ 0; then the system is observable.
Calculation:
Controllability matrix, C = [Q PQ]
\(PQ = \left[ {\begin{array}{*{20}{c}} { - 1}&1\\ 0&{ - 3} \end{array}} \right]\;\left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ { - 3} \end{array}} \right]\)
\(C = \left[ {\begin{array}{*{20}{c}} 0&1\\ 1&{ - 3} \end{array}} \right]\)
|C| = -1 ≠ 0
System is controllable.
Observability matrix, \(O = \left[ {\begin{array}{*{20}{c}} R\\ {RP} \end{array}} \right]\)
\(RP = \left[ {0\;1} \right]\left[ {\begin{array}{*{20}{c}} { - 1}&1\\ 0&{3 - } \end{array}} \right] = \left[ {0\; - 3} \right]\)
\(O = \left[ {\begin{array}{*{20}{c}} 0&1\\ 0&{ - 3} \end{array}} \right]\)
|O| = 0
System is not observable.