GATE Control Systems 3 — Questions and Answers
Question 1: The characteristic equation of a closed-loop system is 1 + G(s)H(s) = 0. The roots of this equation are:
- Open-loop poles
- Open-loop zeros
- Closed-loop poles (Correct answer)
- Closed-loop zeros
Correct answer: Closed-loop poles
The roots of the characteristic equation 1 + G(s)H(s) = 0 are the closed-loop poles of the system.
Question 2: Which criterion is used to determine the ABSOLUTE stability of a linear time-invariant system without finding the roots of the characteristic polynomial?
- Bode criterion
- Routh-Hurwitz criterion (Correct answer)
- Nyquist criterion
- Root locus criterion
Correct answer: Routh-Hurwitz criterion
The Routh-Hurwitz criterion uses an array of coefficients to determine the number of RHP roots without solving the polynomial.
Question 3: For a unity negative feedback system, if the open-loop transfer function is G(s) = K/[s(s+2)], the system is critically damped when K equals:
- 0.5
- 1 (Correct answer)
- 2
- 4
Correct answer: 1
Closed-loop characteristic equation is s² + 2s + K = 0; critical damping requires ζ = 1, so (2/(2√K)) = 1, giving K = 1.
Question 4: The phase margin of a system is defined as 180° plus the phase angle of the open-loop transfer function at:
- Phase crossover frequency
- Gain crossover frequency (Correct answer)
- Natural frequency
- Bandwidth frequency
Correct answer: Gain crossover frequency
Phase margin = 180° + ∠G(jωgc)H(jωgc), where ωgc is the gain crossover frequency where |GH| = 1 (0 dB).
Question 5: State transition matrix Φ(t) = e^(At) satisfies which property?
- Φ(t1 + t2) = Φ(t1) - Φ(t2)
- Φ(t1 + t2) = Φ(t1) · Φ(t2) (Correct answer)
- Φ(-t) = Φ(t)
- Φ(0) = 0
Correct answer: Φ(t1 + t2) = Φ(t1) · Φ(t2)
The state transition matrix satisfies the semigroup property: Φ(t1 + t2) = Φ(t1)·Φ(t2), analogous to the scalar exponential.
Question 6: In the Nyquist stability criterion, the number of clockwise encirclements N of the (-1, j0) point is related to Z (RHP closed-loop poles) and P (RHP open-loop poles) by:
- N = Z + P
- N = Z - P (Correct answer)
- N = P - Z
- N = Z × P
Correct answer: N = Z - P
Nyquist criterion states N = Z - P; for a stable system with no open-loop RHP poles, Z = 0 requires N = 0.
Question 7: A system is said to be observable if:
- All states can be driven to zero by a suitable input
- The initial state can be determined from the output over a finite time interval (Correct answer)
- The transfer function has no RHP zeros
- All eigenvalues are in the left half-plane
Correct answer: The initial state can be determined from the output over a finite time interval
Observability means the initial state x(0) can be uniquely determined from the output y(t) over a finite time interval [0, T].
GATE Control Systems 3
The characteristic equation of a closed-loop system is 1 + G(s)H(s) = 0.
The roots of this equation are: