List of top Control Systems Questions asked in GATE EC

Consider the unity-negative-feedback system shown in Figure (i) below, where gain \( K \geq 0 \). The root locus of this system is shown in Figure (ii) below. 
For what value(s) of \( K \) will the system in Figure (i) have a pole at \( -1 + j1 \)? 


 

Which of the following best describes the phase and sinusoidal components in the given control system plot?
Let \( x[n] \) be a discrete-time signal whose \( z \)-transform is \( X(z) \). Which of the following statements is/are TRUE?

Consider a message signal \( m(t) \) which is bandlimited to \( [-W, W] \), where \( W \) is in Hz. Consider the following two modulation schemes for the message signal:
• Double sideband-suppressed carrier (DSB-SC): \[ f_{DSB}(t) = A_c m(t) \cos(2\pi f_c t) \] • Amplitude modulation (AM): \[ f_{AM}(t) = A_c \left( 1 + \mu m(t) \right) \cos(2\pi f_c t) \] Here, \( A_c \) and \( f_c \) are the amplitude and frequency (in Hz) of the carrier, respectively. In the case of AM, \( \mu \) denotes the modulation index. Consider the following statements: 
(i) An envelope detector can be used for demodulation in the DSB-SC scheme if \( m(t)>0 \) for all \( t \). 
(ii) An envelope detector can be used for demodulation in the AM scheme only if \( m(t)>0 \) for all \( t \).
Which of the following options is/are correct?

For the closed-loop control system shown, with $r(t)=0$, the steady-state error $e(\infty)$ due to a unit step disturbance $d(t)$ is ____________ (rounded off to two decimal places).
Two linear time-invariant systems with transfer functions \[ G_1(s) = \frac{10}{s^2 + s + 1}, \qquad G_2(s) = \frac{10}{s^2 + s\sqrt{10} + 10} \] have unit step responses $y_1(t)$ and $y_2(t)$, respectively. Which of the following statements is/are true?
Consider an even polynomial $p(s)$ given by \[ p(s) = s^4 + 5s^2 + 4 + K, \] where $K$ is an unknown real parameter. The complete range of $K$ for which $p(s)$ has all its roots on the imaginary axis is ________________.
The root-locus plot of a closed-loop system with unity negative feedback and transfer function $KG(s)$ in the forward path is shown in the figure. Note that $K$ is varied from 0 to $\infty$. Select the transfer function $G(s)$ that results in the shown root-locus.
Consider a closed-loop control system with unity negative feedback and $K G(s)$ in the forward path, where the gain $K = 2$. The complete Nyquist plot of the transfer function $G(s)$ is shown in the figure. Note that the Nyquist contour has been chosen to have the clockwise sense. Assume $G(s)$ has no poles on the closed right-half of the complex plane. The number of poles of the closed-loop transfer function in the closed right-half of the complex plane is ____________.