Controllability and Observability — MCQs – EE

1. A system is said to be completely controllable if:
(A) All inputs are zero
(B) Every state can be reached from the initial state by suitable input
(C) Output depends only on input
(D) The system matrix A is diagonal

2. A system is said to be completely observable if:
(A) Every state affects the output
(B) Output is independent of states
(C) System has no poles
(D) The input matrix B is zero

3. The controllability matrix of a system (A, B) is given by:
(A) [A B]
(B) [B AB A²B … Aⁿ⁻¹B]
(C) [C CA CA² … CAⁿ⁻¹]
(D) [A C B D]

4. The observability matrix of a system (A, C) is given by:
(A) [C CA CA² … CAⁿ⁻¹]
(B) [A B]
(C) [B AB … Aⁿ⁻¹B]
(D) [C B]

5. A system is controllable if the rank of the controllability matrix equals:
(A) Number of inputs
(B) Number of outputs
(C) Number of states
(D) Number of poles

6. A system is observable if the rank of the observability matrix equals:
(A) Number of inputs
(B) Number of outputs
(C) Number of states
(D) Number of zeros

7. If a system is not completely controllable, it means:
(A) The system cannot be stabilized
(B) Some states cannot be influenced by any input
(C) Output does not depend on input
(D) System poles cannot be placed

8. If a system is not completely observable, it means:
(A) Some states cannot be measured from the output
(B) Input does not affect the state
(C) System has no poles
(D) System gain is infinite

9. The concepts of controllability and observability are duals of each other. This means:
(A) Both are independent
(B) Controllability of (A, B) implies observability of (Aᵀ, Cᵀ)
(C) Controllability implies stability
(D) Observability implies damping

10. For a controllable and observable system, all poles:
(A) Can be assigned arbitrarily by feedback
(B) Are fixed
(C) Must be real
(D) Depend only on C matrix

11. The Kalman rank condition is used to test:
(A) Stability
(B) Controllability and observability
(C) Frequency response
(D) Steady-state error

12. Which of the following pairs are dual concepts?
(A) Stability and controllability
(B) Controllability and observability
(C) Damping and gain
(D) Feedback and feedforward

13. A system that is uncontrollable but observable means:
(A) Some states cannot be controlled but can be measured
(B) Some states can be controlled but not measured
(C) All states are inaccessible
(D) The system is stable

14. The state feedback controller requires:
(A) Controllability
(B) Observability
(C) Both controllability and observability
(D) None

15. The state observer requires:
(A) Controllability
(B) Observability
(C) Both controllability and observability
(D) None

16. If the controllability matrix is singular, the system is:
(A) Completely controllable
(B) Uncontrollable
(C) Observable
(D) Stable

17. If the observability matrix is singular, the system is:
(A) Observable
(B) Unobservable
(C) Controllable
(D) Stable

18. The pole placement method of control design requires the system to be:
(A) Observable
(B) Controllable
(C) Both
(D) None

19. The Luenberger observer is used to:
(A) Improve controllability
(B) Estimate unmeasured states
(C) Reduce system order
(D) Increase damping

20. The Controllability Gramian is used to check:
(A) Observability
(B) Stability
(C) Controllability in continuous systems
(D) Phase margin

21. The Observability Gramian is used to check:
(A) Controllability
(B) Observability in continuous systems
(C) Frequency response
(D) Transfer gain

22. The rank condition for controllability and observability applies to:
(A) Linear time-invariant systems
(B) Nonlinear systems only
(C) Discrete systems only
(D) Frequency-domain systems

23. A system is both controllable and observable if:
(A) Both controllability and observability matrices have full rank
(B) A matrix is diagonal
(C) Output matrix C is invertible
(D) System poles are complex

24. If a system is controllable but not observable, one cannot:
(A) Stabilize the system
(B) Design a state observer
(C) Apply state feedback
(D) Measure input

25. The controllability and observability concepts were introduced by:
(A) Bode
(B) Routh
(C) Kalman
(D) Nyquist

26. The duality between controllability and observability implies:
(A) Interchange of A and B matrices
(B) Interchange of B and C matrices
(C) Interchange of A and C matrices
(D) Interchange of C and D matrices

27. If a system has unobservable states, those states:
(A) Do not affect the output
(B) Affect the input only
(C) Affect the poles
(D) Are always unstable

28. A reduced-order observer is designed when:
(A) All states are measurable
(B) Some states are unmeasured
(C) System is uncontrollable
(D) Input is constant

29. Controllability ensures that:
(A) System poles can be moved to desired locations
(B) System zeros can be modified
(C) System gain can be increased
(D) Output is measurable

30. Observability ensures that:
(A) All states can be reconstructed from output
(B) All inputs can be determined from output
(C) System is stable
(D) System is controllable