# GATE Questions & Answers of Control Systems Electronics and Communication Engg

#### Control Systems 84 Question(s) | Weightage 12 (Marks)

A closed-loop control system is stable if the Nyquist plot of the corresponding open-loop transfer function

The open-loop transfer function of a unity-feedback control system is

The value of K at the breakaway point of the feedback control system’s root-locus plot is ________

The open-loop transfer function of a unity-feedback control system is given by

For the peak overshoot of the closed-loop system to a unit step input to be 10%, the value of K is ____________

The response of the system $G\left(s\right)=\frac{s-2}{\left(s+1\right)\left(s+3\right)}$ to the unit step input $u(t)$ is $y(t).$ The value at =0+ is ________

The number and direction of encirclements around the point -1+0 in the complex plane by the Nyquist plot of $G\left(s\right)=\frac{1-s}{4+2s}$ is

In the feedback system shown below $G\left(S\right)=\frac{1}{\left({s}^{2}+2s\right)}$ .
The step response of the closed-loop system should have minimum settling time and have no overshoot.

The required value of gain k to achieve this is ________

In the feedback system shown below $G\left(S\right)=\frac{1}{\left(s+1\right)\left(s+2\right)\left(s+3\right)}$ .

The positive value of k for which the gain margin of the loop is exactly 0 dB and the phase margin of the loop is exactly zero degree is ________

The asymptotic Bode phase plot of $G\left(S\right)=\frac{k}{\left(s+0.1\right)\left(s+10\right)\left(s+{p}_{1}\right)}$ , with k and p1 both positive, is shown below.

The value of p1 is ________

The block diagram of a feedback control system is shown in the figure. The overall closed-loop gain G of the system is

For the unity feedback control system shown in the figure, the open-loop transfer function G(s) is given as

$G\left(s\right)=\frac{2}{s\left(s+1\right)}$

The steady state error ess due to a unit step input is

A second-order linear time-invariant system is described by the following state equations

$\frac{d}{dt}{x}_{1}\left(t\right)+2{x}_{1}\left(t\right)=3u\left(t\right)$

$\frac{d}{dt}{x}_{2}\left(t\right)+{x}_{2}\left(t\right)=u\left(t\right)$

where x1(t) and x2(t) are the two state variables and u(t) denotes the input. If the output c(t) = x1(t), then the system is

The forward-path transfer function and the feedback-path transfer function of a single loop negative feedback control system are given as

respectively. If the variable parameter K is real positive, then the location of the breakaway point on the root locus diagram of the system is __________

In the circuit shown, the switch SW is thrown from position A to position B at time t= 0. The energy (in $\mathrm{μJ}$) taken from the 3 V source to charge the 0.1 $\mathrm{\mu }$F capacitor from 0 V to 3 V is

Negative feedback in a closed-loop control system DOES NOT

A unity negative feedback system has the open-loop transfer function $G\left(s\right)=\frac{K}{s\left(s+1\right)\left(s+3\right)}$. The value of the gain K (>0) at which the root locus crosses the imaginary axis is ______.

The polar plot of the transfer will be in the

In the circuit shown, switch SW is closed at t = 0. Assuming zero initial conditions, the value of vc(t) (in Volts) at t = 1 sec is _____.

The open-loop transfer function of a plant in a unity feedback configuration is given as . The value of the gain K (>0) for which the point –1 + 2 lies on the root locus is _____.

By performing cascading and/or summing/differencing operations using transfer function blocks G1(s) and G2(s), one CANNOT realize a transfer function of the form

For the signal flow graph shown in the figure, the value of $\frac{C\left(s\right)}{R\left(s\right)}$ is

A unity negative feedback system has an open-loop transfer function $G\left(s\right)=\frac{K}{s\left(s+10\right)}$ . The gain $K$ for the system to have a damping ratio of 0.25 is ________.

In the circuit shown, the initial voltages across the capacitors C1 and C2 are 1 V and 3 V, respectively. The switch is closed at time t =0. The total energy dissipated (in Joules) in the resistor R until steady state is reached, is _______.

The output of a standard second-order system for a unit step input is given as

$y\left(t\right)=1-\frac{2}{\sqrt{3}}{e}^{-t}\mathrm{cos}\left(\sqrt{3}t-\frac{\mathrm{\pi }}{6}\right)$; The transfer function of the system is

The transfer function of a mass-spring damper system is given by

$G\left(s\right)=\frac{1}{M{s}^{2}+Bs+k}$

The frequency response data for the system are given in the following table.

 ω in rad/s |G(jω)| in dB arg (G(jω)) in deg 0.01 -18.5 -0.2 0.1 -18.5 -1.3 0.2 -18.4 -2.6 1 -16 -16.9 2 -11.4 -89.4 3 -21.5 -151 5 -32.8 -167 10 -45.3 -174.5

The unit step response of the system approaches a steady state value of____.

Consider the Bode plot shown in the figure. Assume that all the poles and zeros are real-valued.

The value of fH – fL (in Hz) is ______.

The phase margin (in degrees) of the system $G\left(s\right)=\frac{10}{s\left(s+10\right)}$ is _____.

network is described by the state model as

 $x_2=-4x_2-\mathrm u$

$y=3{x}_{1}-2{x}_{2}$

The transfer function $H\left(s\right)\left(=\frac{Y\left(s\right)}{U\left(s\right)}\right)$ is

The position control of a DC servo-motor is given in the figure.The values of parameters are $K_T=1\;\mathrm N\operatorname{- }\mathrm m/\mathrm A,\;R_a=1\mathrm\Omega,\;L_a=0.1\mathrm H,\;J=5\mathrm{kg}-\mathrm m^2,\;B=1\;\mathrm N\operatorname{- }\mathrm m/$ (rad/sec) and $K_b=1\mathrm V/$ (rad/sec). The steady-state position response (in radians) due to unit impulse disturbance torque ${T}_{d}$ is ________.

For the system shown in the figure, s=-2.75 lies on the root locus if K is__________

The forward path transfer function of a unity negative feedback system is given by

$G\left(s\right)=\frac{K}{\left(s+2\right)\left(s-1\right)}$

The value of K which will place both the poles of the closed-loop system at the same location, is _______.

Consider the feedback system shown in the figure. The Nyquist plot of G(s) is also shown. Which one of the following conclusions is correct?

Consider the state space model of a system, as given below

The system is

The phase margin in degrees of $G\left(s\right)=\frac{10}{\left(s+0.1\right)\left(s+1\right)\left(s+10\right)}$ calculated using the asymptotic Bode plot is ________.

For the following feedback system $G\left(s\right)=\frac{1}{\left(s+1\right)\left(s+2\right)}$The 2%-settling time of the step response is required to be less than 2 seconds.

Which one of the following compensators C(S) achieves this?

The natural frequency of an undamped second-order system is 40 rad/s. If the system is damped with a damping ratio 0.3, the damped natural frequency in rad/s is ________.

For the following system,

when X1(s)=0, the transfer function $\frac{Y\left(s\right)}{{X}_{2}\left(s\right)}$ is

An unforced linear time invariant (LTI) system is represented by

$\left[\begin{array}{c}\stackrel{.}{{x}_{1}}\\ \stackrel{.}{{x}_{1}}\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& -2\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]$

If the initial conditions are x1(0)=1 and x2(0)=−1, the solution of the state equation is

The Bode asymptotic magnitude plot of a minimum phase system is shown in the figure.

If the system is connected in a unity negative feedback configuration, the steady state error of the closed loop system, to a unit ramp input, is_________.

Consider the state space system expressed by the signal flow diagram shown in the figure.

The corresponding system is

Consider the following block diagram in the figure.

The transfer function $\frac{C\left(s\right)}{R\left(s\right)}$ is

The steady state error of the system shown in the figure for a unit step input is _______.

The state equation of a second-order linear system is given by

$\stackrel{.}{x}\left(t\right)=Ax\left(t\right),x\left(0\right)={x}_{0}$

For and for

when ${x}_{0}=\left[\begin{array}{c}3\\ 5\end{array}\right],x\left(t\right)$ is

In the root locus plot shown in the figure, the pole/zero marks and the arrows have been removed. Which one of the following transfer functions has this root locus?

In a Bode magnitude plot, which one of the following slopes would be exhibited at high frequencies by a 4th order all-pole system?

For the second order closed-loop system shown in the figure, the natural frequency (in rad/s) is

The state transition matrix $\phi$(t) of a system $\left[\begin{array}{c}\stackrel{.}{{x}_{1}}\\ \stackrel{.}{{x}_{2}}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]$ is

Consider a transfer function ${G}_{p}\left(s\right)=\frac{p{s}^{2}+3ps-2}{{s}^{2}+\left(3+p\right)s+\left(2-p\right)}$ with p a positive real parameter. The maximum value of p until which Gp remains stable is ________.

The characteristic equation of a unity negative feedback system is $1+KG\left(s\right)=0$. The open loop transfer function G(s) has one pole at 0 and two poles at -1. The root locus of the system for varying K is shown in the figure.

The constant damping ratio line, for ξ=0.5, intersects the root locus at point A. The distance from the origin to point A is given as 0.5. The value of K at point A is ________ .

The Bode plot of a transfer function G(s) is shown in the figure below.

The gain is 32 dB and -8 dB at 1 rad/s and 10 rad/s respectively. The phase is negative for all ω. Then G(s) is

Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?

The signal flow graph for a system is given below. The transfer function $\frac{Y\left(s\right)}{U\left(s\right)}$ for this system is

The state diagram of a system is shown below. A system is described by the state-variable equations



The State-variable equations of the system shown in the figure above are

The state diagram of a system is shown below. A system is described by the state-variable equations



The state transition matrix ${e}^{{A}_{t}}$ of the system shown in the figure above is

A system with transfer function

G(s)=$\frac{\left({s}^{2}+9\right)\left(s+2\right)}{\left(s+1\right)\left(s+3\right)\left(s+4\right)}$

is excited by $\mathrm{sin}\left(\omega t\right)$. The steady-state output of the system is zero at

The state variable description of an LTI system is given by

$y=\left(\begin{array}{ccc}1& 0& 0\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)$

where y is the output and u is the input. The system is controllable for

The differential equation $100\frac{{d}^{2}y}{d{t}^{2}}-20\frac{dy}{dt}+y=x\left(t\right)$  describes a system with an input x(t) and an output y(t). The system, which is initially relaxed, is excited by a unit step input. The output y(t) can be represented by the waveform

For the transfer function $G\left(\mathrm{j\omega }\right)=5+\mathrm{j\omega },$ the corresponding Nyquist plot for positive frequency has the form

The root locus plot for a system is given below. The open loop transfer function corresponding to this plot is given by

The block diagram of a system with one input u and two outputs y1 and y2 is given below.

A state space model of the above system in terms of the state vector $\overline{)x}$ and the output vector $\overline{)y}={\left[\begin{array}{cc}{y}_{1}& {y}_{2}\end{array}\right]}^{T}$ is

The input-output transfer function of a plant $H\left(s\right)=\frac{100}{s{\left(s+10\right)}^{2}}$  The plant is placed in a unity negative feedback configuration as shown in the figure below

The signal flow graph that DOES NOT model the plant transfer function H(s) is

The input-output transfer function of a plant $H\left(s\right)=\frac{100}{s{\left(s+10\right)}^{2}}$  The plant is placed in a unity negative feedback configuration as shown in the figure below

The gain margin of the system under closed loop unity negative feedback is

The transfer function Y(s)/R(s) of the system shown is

A system with transfer function $\frac{Y\left(s\right)}{X\left(s\right)}=\frac{s}{s+p}$ has an output $y\left(t\right)=\mathrm{cos}\left(2t-\frac{\mathrm{\pi }}{3}\right)$ for the input signal Then, the system parameter ‘p’ is

For the asymptotic Bode magnitude plot shown below, the system transfer function can be

The signal flow graph of a system is shown below.

The state variable representation of the system can be

The signal flow graph of a system is shown below.

The transfer function of the system is

Consider the system $\frac{dx}{dt}=Ax+Bu$ with $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $B=\left[\begin{array}{c}p\\ q\end{array}\right]$ where p and q are arbitrary real numbers. Which of the following statements about the controllability of the system is true?

The feedback configuration and the pole-zero locations of $G\left(s\right)=\frac{{s}^{2}-2s+2}{{s}^{2}+2s+2}$ are shown below. The root locus for negative values of k, i.e. for −$\infty$ < k < 0, has breakaway/break in points and angle of departure at pole P (with respect to the positive real axis) equal to

The unit step response of an under-damped second order system has steady state value of -2. Which one of the following transfer functions has these properties?

The Nyquist plot of a stable transfer function G(s) is shown in the figure. We are interested in the  stability of the closed loop system in the feedback configuration shown.

Which of the following statements is true?

The Nyquist plot of a stable transfer function G(s) is shown in the figure. We are interested in the  stability of the closed loop system in the feedback configuration shown.

The gain and phase margins of G(s) for closed loop stability are

The pole-zero plot given below corresponds to a

Step responses of a set of three second-order underdamped systems all have the same percentage overshoot. Which of the following diagrams represents the poles of the three systems?

A linear, time-invariant, causal continuous time system has a rational transfer function with simple poles at s=-2 and s=-4, and one simple zero at s=-1. A unit step u(t) is applied at the input of the system. At steady state, the output has constant value of 1. The impulse response of this system is

 Group I $P=\frac{25}{{s}^{2}+25}$ $Q=\frac{36}{{s}^{2}+20s+36}$ $R=\frac{36}{{s}^{2}+12s+36}$ $S=\frac{49}{{s}^{2}+7s+49}$