State Transition Matrix — MCQs – EE

1. The state transition matrix is denoted by:
(A) Φ(t)
(B) A(t)
(C) C(t)
(D) D(t)

2. The state transition matrix describes:
(A) The relationship between input and output
(B) The evolution of the state vector over time
(C) The steady-state behavior of the system
(D) The transfer function

3. The state transition matrix Φ(t) satisfies the condition:
(A) Φ(0) = 0
(B) Φ(0) = I
(C) Φ(0) = A
(D) Φ(0) = B

4. The state transition matrix is used to find:
(A) The transfer function
(B) The output response
(C) The state vector at any time t
(D) The system gain

5. The state transition matrix Φ(t) is a function of:
(A) The input only
(B) The output only
(C) The system matrix A
(D) The control matrix B

6. The state equation of a linear time-invariant system is given by
(A) The transfer function
(B) The state transition matrix
(C) The Nyquist plot
(D) The Bode plot

7. The state transition matrix Φ(t) represents:
(A) The input effect
(B) The system’s natural response
(C) The external disturbance
(D) The steady-state error

8. For a time-invariant system, the state transition matrix depends only on:
(A) Initial conditions
(B) Time difference (t – t₀)
(C) System input
(D) System output

9. The state transition matrix is used in the solution:
(A) y(t) = Φ(t) y(0) + ∫Φ(t−τ)u(τ)dτ
(B) x(t) = Φ(t)x(0) + ∫Φ(t−τ)Bu(τ)dτ
(C) x(t) = Ax + Bu
(D) y(t) = Cx + Du

10. The derivative of the state transition matrix satisfies:
(A) dΦ/dt = AΦ(t)
(B) dΦ/dt = Φ(t)A
(C) dΦ/dt = BΦ(t)
(D) dΦ/dt = Φ(t)B

11. The state transition matrix Φ(t) is also called the:
(A) System response matrix
(B) Fundamental matrix solution
(C) Transfer function
(D) Controllability matrix

12. The state transition matrix provides information about:
(A) How inputs affect the system
(B) How the system evolves without input
(C) Only steady-state response
(D) Only damping ratio

13. The inverse of the state transition matrix satisfies:
(A) Φ⁻¹(t) = Φ(t)
(B) Φ⁻¹(t) = Φ(−t)
(C) Φ⁻¹(t) = AΦ(t)
(D) Φ⁻¹(t) = Φ(t)A

14. For a time-invariant system, Φ(t) can be expressed as:
(A) Laplace transform of A
(B) Inverse Laplace of transfer function
(C) Matrix exponential of A
(D) Product of A and B

15. The state transition matrix is n × n if the system has:
(A) n inputs
(B) n outputs
(C) n states
(D) n poles

16. The state transition matrix helps to determine:
(A) Stability and transient response
(B) Phase margin
(C) Frequency response
(D) Root locus shape

17. The state transition matrix is always:
(A) Symmetric
(B) Orthogonal
(C) Non-singular
(D) Diagonal

18. The state transition matrix for t = 0 is:
(A) A
(B) B
(C) I
(D) 0

19. The superposition property holds for:
(A) Linear time-invariant systems only
(B) Nonlinear systems
(C) Random systems
(D) All systems

20. The stability of the system can be determined from:
(A) The determinant of Φ(t)
(B) The eigenvalues of A
(C) The rank of Φ(t)
(D) The trace of B

21. The state transition matrix relates:
(A) Input and output
(B) Initial state and future state
(C) System parameters and poles
(D) Input gain and damping

22. The state transition matrix Φ(t₁, t₂) satisfies:
(A) Φ(t₁, t₂) = Φ(t₂ − t₁)
(B) Φ(t₁, t₂) = Φ(t₂)Φ(t₁)
(C) Φ(t₁, t₂) = Φ(t₁) − Φ(t₂)
(D) Φ(t₁, t₂) = A(t₂ − t₁)

23. The homogeneous solution of a state equation involves:
(A) Only Φ(t)
(B) Only u(t)
(C) Both Φ(t) and u(t)
(D) Only A and B

24. The transition property of Φ(t) is:
(A) Φ(t₁ + t₂) = Φ(t₁)Φ(t₂)
(B) Φ(t₁ + t₂) = Φ(t₁) + Φ(t₂)
(C) Φ(t₁ + t₂) = Φ(t₂) − Φ(t₁)
(D) Φ(t₁ + t₂) = AΦ(t₁)Φ(t₂)

25. The state transition matrix Φ(t) can be obtained using:
(A) Laplace transform
(B) Fourier transform
(C) Nyquist method
(D) Root locus

26. The Laplace transform of the state transition matrix is:
(A) (sI + A)⁻¹
(B) (sI − A)⁻¹
(C) (A − sI)⁻¹
(D) (sA − I)⁻¹

27. The state transition matrix is also known as the:
(A) Propagation matrix
(B) Feedback matrix
(C) Control matrix
(D) Output matrix

28. The determinant of Φ(t) is always:
(A) Zero
(B) Nonzero
(C) Negative
(D) Positive

29. The state transition matrix for a stable system tends to:
(A) Zero as t → ∞
(B) Infinity as t → ∞
(C) A constant value
(D) Oscillate indefinitely

30. The state transition matrix helps in computing:
(A) Input signal
(B) State response for any input
(C) Frequency response
(D) Gain margin