# GATE Questions & Answers of Control Systems Electrical Engineering

#### Control Systems 84 Question(s) | Weightage 11 (Marks)

Match the transfer functions of the second-order systems with the nature of the systems given below.

 Transfer functions Nature of system $\style{font-family:'Times New Roman'}{P:\;\frac{15}{s^2+5s+15}}$ I:  Overdamped $\style{font-family:'Times New Roman'}{Q:\;\frac{25}{s^2+10s+25}}$ II: Critically damped $\style{font-family:'Times New Roman'}{R:\;\frac{35}{s^2+18s+35}}$ III: Underdamped

Consider a unity feedback system with forward transfer function given by

$G(s)=\frac1{(s+1)(s+2)}$

The steady-state error in the output of the system for a unit-step input is _________(up to 2 decimal places).

Consider a system governed by the following equations

$\frac{dx_1(t)}{dt}=x_2(t)-x_1(t)$

$\frac{dx_2(t)}{dt}=x_1(t)-x_2(t)$

The initial conditions are such that $x_1(0)<x_2(0)<\infty.$ Let $x_{1f}=\lim\limits_{t\rightarrow\infty}x_{1(t)}\;$ and $x_{2f}=\lim\limits_{t\rightarrow\infty}x_{2(t)}\;$ . Which one of the following is true?

The number of roots of the polynomial, $\style{font-family:'Times New Roman'}{s^7+s^6+7s^5+14s^4+31s^3+73s^2+25s+200}$ , in the open left half of the complex plane is

The unit step response $y(t)$ of a unity feedback system with open loop transfer function $G(s)H(s)\;\frac K{(s+1)^2(s+2)}$ is shown in the figure. The value of $K$ is _______ (up to 2 decimal places).

The phase cross-over frequency of the transfer function $G\left(S\right)=\frac{100}{{\left(S+1\right)}^{3}}$ in rad/s is

Consider the following asymptotic Bode magnitude plot (ω is in rad/s).

Which one of the following transfer functions is best represented by the above Bode magnitude plot?

Loop transfer function of a feedback system is $G\left(s\right)H\left(s\right)=\frac{s+3}{{s}^{2}\left(s-3\right)}$ Take the Nyquist contour in the clockwise direction. Then, the Nyquist plot of $G\left(s\right)H\left(s\right)$ encircles $-1+j0$

Given the following polynomial equation

the number of roots of the polynomial, which have real parts strictly less than −1, is ________.

For the network shown in the figure below, the frequency (in rad/s) at which the maximum phase lag occurs is, ___________.

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

The closed loop system will be stable if,

Consider a linear time invariant system $\stackrel{.}{x}=Ax$, with initial condition $x\left(0\right)$ at $t=0$. Suppose $\alpha$ and $\beta$ are eigenvectors of (2 x 2) matrix A corresponding to distinct eigenvalues ${\lambda }_{1}$ and ${\lambda }_{2}$ respectively. Then the response $x\left(t\right)$ of the system due to initial condition $x\left(0\right)=\alpha$ is

A second-order real system has the following properties:

a) the damping ratio $\zeta =0.5$ and undamped natural frequency ${\omega }_{n}=10$ rad/s,
b) the steady state value of the output, to a unit step input, is 1.02.

The transfer function of the system is

The gain at the breakaway point of the root locus of a unity feedback system with open loop transfer function $G\left(s\right)=\frac{Ks}{\left(s-1\right)\left(s-4\right)}$ is

A Bode magnitude plot for the transfer function G(s) of a plant is shown in the figure. Which one of the following transfer functions best describes the plant?

In the signal flow diagram given in the figure, ${u}_{1}$ and ${u}_{2}$ are possible inputs whereas ${y}_{1}$ and ${y}_{2}$ are possible outputs. When would the SISO system derived from this diagram be controllable and observable?

The transfer function of a second order real system with a perfectly flat magnitude response of unity has a pole at $\left(2-j3\right)$. List all the poles and zeroes.

The open loop poles of a third order unity feedback system are at 0,-1,-2. Let the frequency corresponding to the point where the root locus of the system transits to unstable region be K. Now suppose we introduce a zero in the open loop transfer function at -3, while keeping all the earlier open loop poles intact. Which one of the following is TRUE about the point where the root locus of the modified system transits to unstable region?

An open loop control system results in a response of for a unit impulse input. The DC gain of the control system is _________.

Nyquist plot of two functions  ${G}_{1}$(s) and ${G}_{2}$(s) are shown in figure.

Nyquist plot of the product of ${G}_{1}$ (s) and ${G}_{2}$(s) is

The following discrete-time equations result from the numerical integration of the differential equations of an un-damped simple harmonic oscillator with state variables x and y. The integrationtime step is h.
$\frac{{x}_{k+1}-{x}_{k}}{h}={y}_{k}$
$\frac{{y}_{k+1}-{y}_{k}}{h}=-{x}_{k}$
For this discrete-time system, which one of the following statements is TRUE?

The unit step response of a system with the transfer function $G\left(s\right)=\frac{1-2s}{1+s}$ is given by which one of the following waveforms?

An open loop transfer function G(s) of a system is

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

For a unity feedback system, the breakaway point of the root loci on the real axis occurs at,

For the system governed by the set of equations:

$\mathrm{y}=3{x}_{1}$

${}_{}$the transfer function Y(s)/U(s) is given by

In the formation of Routh-Hurwitz array for a polynomial, all the elements of a row have zero values. This premature termination of the array indicates the presence of

The root locus of a unity feedback system is shown in the figure

The closed loop transfer function of the system is

For the given system, it is desired that the system be stable. The minimum value of $\alpha$ for this condition is _______________.

The Bode magnitude plot of the transfer function $G\left(S\right)=\frac{K\left(1+0.5s\right)\left(1+as\right)}{s\left(1+\frac{s}{8}\right)\left(1+bs\right)\left(1+\frac{s}{36}\right)}$ is shown below: Note that -6 dB/octave = -20 dB/decade. The value of $\frac{a}{bk}$ is_____________.

The closed-loop transfer function of a system is $\mathrm{T}\left(\mathrm{S}\right)=\frac{4}{\left({\mathrm{s}}^{2}+0.4\mathrm{s}+4\right)}$. The steady state error due to unit step input is __________.

The state transition matrix for the system

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

is

A system with the open loop transfer function

$G\left(S\right)=\frac{K}{s\left(s+2\right)\left({s}^{2}+2s+2\right)}$

is connected in a negative feedback configuration with a feedback gain of unity. For the closed loop system to be marginally stable, the value of K is ______

For the transfer function

$G\left(S\right)=\frac{5\left(s+4\right)}{s\left(s+0.25\right)\left({s}^{2}+4s+25\right)}$

The values of the constant gain term and the highest corner frequency of the Bode plot respectively are

The second order dynamic system

$\frac{dx}{dt}=PX+Qu$

$y=RX$

has the matrices P, Q and R as follows:

The system has the following controllability and observability properties:

The signal flow graph of a system is shown below. U(s) is the input and C(s) is the output.

Assuming, h1=b1 and h0=b0-b1a1, the input-output transfer function, $G\left(S\right)=\frac{C\left(s\right)}{U\left(s\right)}$ of the system is given by

A single-input single-output feedback system has forward transfer function $G\left(s\right)$ and feedback transfer function $H\left(s\right)$. It is given that $\left|G\left(s\right)H\left(s\right)\right|<1.$ Which of the following is true about the stability of the system?

The block diagram of a system is shown in the figure

If the desired transfer function of the system is

$\frac{C\left(s\right)}{R\left(s\right)}=\frac{s}{{s}^{2}+s+1}$

then G(s) is

Consider the system described by following state space equations

If u is unit step input, then the steady state error of the system is

The magnitude Bode plot of a network is shown in the figure

The maximum phase angle $\phi_m$ and the corresponding gain Gm respectively, are

Assuming zero initial condition, the response y(t) of the system given below to a unit step input u(t) is

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

The gain $\left(20log\left|G\left(s\right)\right|\right)$ 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

The open-loop transfer function of a dc motor is given as $\frac{\omega \left(s\right)}{{V}_{a}\left(s\right)}=\frac{10}{1+10s}$ When connected in feedback as shown below, the approximate value of Ka that will reduce the time constant of the closed loop system by one hundred times as compared to that of the open-loop system is

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 variable formulation of a system is given as

The system is

The state variable formulation of a system is given as

The response y(t) to a unit step input 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 feedback system shown below oscillates at 2 rad/s when

The transfer function of a compensator is given as

${G}_{c}\left(s\right)=\frac{s+a}{s+b}$.

${G}_{c}\left(s\right)$ is a lead compensator if

The transfer function of a compensator is given as

${G}_{c}\left(s\right)=\frac{s+a}{s+b}$.

The phase of the above lead compensator is maximum at

The frequency response of a linear system G(jω) is provided in the tubular form below

 $\left|G\left(j\omega \right)\right|$ 1.3 1.2 1.0 0.8 0.5 0.3 $\angle G\left(j\omega \right)$ −130oc −140oc −150oc −160oc −180oc −200oc

The gain margin and phase margin of the system are

The steady state error of a unity feedback linear system for a unit step input is 0.1. The steady state error of the same system, for a pulse input r(t) having a magnitude of 10 and a duration of one second, as shown in the figure is

An open loop system represented by the transfer function $G\left(s\right)=\frac{\left(s-1\right)}{\left(s+2\right)\left(s+3\right)}$ is

The open loop transfer function $G\left(s\right)$ of a unity feedback control system is given as

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

From the root locus, it can be inferred that when k tends to positive infinity,

The response $h\left(t\right)$ of a linear time invariant system to an impulse $\delta \left(t\right)$, under initially relaxed condition is $h\left(t\right)={e}^{-t}+{e}^{-2t}$. The response of this system for a unit step input $u\left(t\right)$ is

A two loop position control system is shown below

The gain K of the Tacho-generator influences mainly the

As shown in the figure, a negative feedback system has an amplifier of gain 100 with ±10% tolerance in the forward path, and an attenuator of value 9/100 in the feedback path. The overall system gain is approximately

For the system $2}{\left(s+1\right)}\phantom{\rule{0ex}{0ex}}$ the approximate time taken for a step response to reach 98% of its final value is

The frequency response of G(s) = 1 / [s (s + 1) (s + 2) ] plotted in the complex G(jω) plane (for o < ω < ∞) is

The system x = Ax + Bu with $A=\left[\begin{array}{cc}-1& 2\\ 0& 2\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right]$ is

The characteristic equation of a closed-loop system is s(s+1)(s+3)+k(s+2)=0, k>0. Which of the following statements is true?

The measurement system shown in the figure uses three sub-systems in cascade whose gains are specified as G1,G2 and $\frac{1}{{G}_{3}}$. The relative small errors associated with each respective subsystem G1,G2 and G3 are ε1,ε2 and ε3. The error associated with the output is:

The polar plot of an open loop stable system is shown below. The closed loop system is

The first two rows of Routh's tabulation of a third order equation are as follows.

$\begin{array}{ccc}{\mathrm{s}}^{3}& 2& 2\\ {\mathrm{s}}^{2}& 4& 4\end{array}$

This means there are

The asymptotic approximation of the log-magnitude vs frequency plot of a system containing only real poles and zeros is shown. Its transfer function is

The unit-step response of a unity feedback system with open loop transfer function G(s) = K/((s + 1)(s + 2)) is shown in the figure. The value of K is

The open loop transfer function of a unity feedback system is given by G(s) = (e-0.1s)/s. The gain margin of this system is

A system is described by the following state and output equations

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

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

$y\left(t\right)={x}_{1}\left(t\right)$

where u(t) is the input and y(t) is the output

The system transfer function is

A system is described by the following state and output equations

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

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

$y\left(t\right)={x}_{1}\left(t\right)$

where u(t) is the input and y(t) is the output

The state transition matrix of the above system is

A function y(t) satisfies the following differential equation :

$\frac{dy\left(t\right)}{dt}+y\left(t\right)=\delta \left(t\right)$

where δ(t) is the delta function. Assuming zero initial condition, and denoting the unit step function by u(t), y(t) can be of the form

The transfer function of a linear time invariant system is given as

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

The steady state value of the output of the system for a unit impulse input applied at time instant t = 1 will be

The transfer functions of two compensators are given below :

${C}_{1}=\frac{10\left(s+1\right)}{\left(s+10\right)},{C}_{2}=\frac{s+10}{10\left(s+1\right)}$

Which one of the following statements is correct ?

The asymptotic Bode magnitude plot of a minimum phase transfer function is shown in the figure :

This transfer function has

Figure shows a feedback system where K > 0

The range of K for which the system is stable will be given by

The transfer function of a system is given as

$\frac{100}{{s}^{2}+20s+100}$

The system is

$\mathrm{x}=\mathrm{Ax}+\mathrm{Bu}\phantom{\rule{0ex}{0ex}}\mathrm{y}=\mathrm{Cx}$