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

#### Control Systems 103 Question(s) | Weightage 12 (Marks)

The Nyquist stability criterion and the Routh criterion both are powerful analysis tools for determining the stability of feedback controllers. Identify which of the following statements is FALSE:

##### Show Answer

Consider $\begin{array}{l}p\left(s\right)=s^3+a_2s^2+a_1s+a_0\\\end{array}$ with all real coefficients. It is known that its derivative $\begin{array}{l}p\left(s\right)\\\end{array}$ has no real roots. The number of real roots of $p(s)$ is

##### Show Answer

The state equation and the output equation of a control system are given below:

X$=\begin{bmatrix}-4&-1.5\\4&0\end{bmatrix}$X$+\begin{bmatrix}2\\0\end{bmatrix}u$,

$y=\begin{bmatrix}1.5&0.625\end{bmatrix}$X.

The transfer function representation of the system is

##### Show Answer

For a unity feedback control system with the forward path transfer function

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

The peak resonant magnitude $M_r$ of the closed-loop frequency response is 2. The corresponding value of the gain $K$ (correct to two decimal places) is _________.

##### Show Answer

The figure below shows the Bode magnitude and phase plots of a stable transfer function $G\left(s\right)=\frac{n_0}{s^3+d_2s^2+d_1s+d_0}$ .

Consider the negative unity feedback configuration with gain $k$ in the feedforward path. The closed loop is stable for $k<k_0$ . The maximum value of $k_0$ is ______.

##### Show Answer

The open loop transfer function $G\left(s\right)=\frac{\left(s+1\right)}{{s}^{p}\left(s+2\right)\left(s+3\right)}$ where p is an integer, is connected in unity feedback configuration as shown in the figure.

Given that the steady state error is zero for unit and 6 for unit ramp input, the value of the parameter p is_________

##### Show Answer

Consider a stable system with transfer function $G\left(s\right)=\frac{{s}^{p}+{b}_{1}{s}^{p-1}+···+{b}_{p}}{{s}^{q}+{a}_{1}{s}^{q-1}+···+{a}_{q}}$ where b1, …, bp and a1, … , aq are real valued constant. The slop of the Bode log magnitude curve of G(s) converges to -60 dB /decade as ω→∞. A possible pair of values for p and q is

##### Show Answer

Which of the following can be the pole-zero configuration of a phase-lag controller (lag compensator)?

##### Show Answer

In the circuit shown, the voltage VIN(t) is described by: ${\mathrm V}_\mathrm{IN}\left(t\right)=\left\{\begin{array}{lc}0,&\mathrm{for}\;\mathrm t<0\\15\;\mathrm{Volts},&\mathrm{for}\;\mathrm t\geq0\end{array}\right.$ where t is in second. The time (in seconds) at which the current I in the circuit will reach the value 2 Amperes is_________.

##### Show Answer

A linear time invariant (LTI) system with the transfer function ${G}_{\left(s\right)}=\frac{K\left({s}^{2}+2s+2\right)}{\left({s}^{2}-3s+2\right)}$ is  connected in unity feedback configuration as shown in the figure.

For the closed loop system shown, the root locus for 0 < K < ∞ intersect the imaginary axis for K = 1.5. The closed loop system is suitable for

##### Show Answer

Which one of the following options correctly describes the location of the roots of the equation ${s}^{4}+{s}^{2}+1=0$ on the plane?

##### Show Answer

The Nyquist plot of the transfer function $G\left(S\right)=\frac K{\left(s^2+2s+2\right)\left(s+2\right)}$ does not encircle the point $\left(-1+j0\right)$ for K=10 but does encircle the point $\left(-1+j0\right)$ for = 100. Then the closed loop system (having unity gain feedback) is

##### Show Answer

For the system shown in the figure, Y(s)/ X(s)=____________

##### Show Answer

Consider the state space realization $\left[\begin{array}{c}{x}_{1}\left(t\right)\\ {x}_{2}\left(t\right)\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& -9\end{array}\right]\left[\begin{array}{c}{x}_{1}\left(t\right)\\ {x}_{2}\left(t\right)\end{array}\right]+\left[\begin{array}{c}0\\ 45\end{array}\right]u\left(t\right)$ , with the initial condition $\left[\begin{array}{c}{x}_{1}\left(0\right)\\ {x}_{2}\left(0\right)\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$, where u(t) denotes the unit step function. The value of $\begin{array}{c}\mathrm{lim}\\ t\to \infty \end{array}\left|\sqrt{{{x}_{1}}^{2}\left(t\right)+{{x}_{2}}^{2}\left(t\right)}\right|$ is_____________.

##### Show Answer

Which of the following is incorrect?

##### Show Answer

The switch in the circuit, shown in the figure, was for a long time and is closed at t = 0.

The current i(t) (in ampere) at t = 0.5 seconds is________

##### Show Answer

A unity feedback control system is characterized by the open-loop transfer function

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

The value of k for which the system oscillates at 2 rad/s is___________

##### Show Answer

A second-order LTI system is described by the following state equation,

$\frac{d}{dt}{x}_{1}\left(t\right)-{x}_{2}\left(t\right)=0\phantom{\rule{0ex}{0ex}}\frac{d}{dt}{x}_{2}\left(t\right)+2{x}_{1}\left(t\right)+3{x}_{2}\left(t\right)=r\left(t\right)$

where x1() and x2() are the two state variables and r() denotes the input. The output () = x().

The system is.

##### Show Answer

A unity feedback control system is characterized by the open-loop transfer function

$G\left(s\right)=\frac{10K\left(s+2\right)}{{s}^{3}+3{s}^{2}+10}$

The Nyquist path and the corresponding Nyquist plot of () are shown in the figure below.

If 0 < K < 1, then the number of poles of the closed-loop transfer function that lie in the right-half of the s-plane is

##### Show Answer

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

##### Show Answer

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 ________

##### Show Answer

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 ____________

##### Show Answer

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 ________

##### Show Answer

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

##### Show Answer

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 ________

##### Show Answer

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 ________

##### Show Answer

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 ________

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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 __________

##### Show Answer

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

##### Show Answer

Negative feedback in a closed-loop control system DOES NOT

##### Show Answer

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 ______.

##### Show Answer

The polar plot of the transfer will be in the

##### Show Answer

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 _____.

##### Show Answer

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 _____.

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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 ________.

##### Show Answer

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 _______.

##### Show Answer

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

##### Show Answer

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____.

##### Show Answer

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 ______.

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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 ________.

##### Show Answer

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

##### Show Answer

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 _______.

##### Show Answer

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?

##### Show Answer

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

The system is

##### Show Answer

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 ________.

##### Show Answer

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?

##### Show Answer

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 ________.

##### Show Answer

For the following system,

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

##### Show Answer

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

##### Show Answer

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_________.

##### Show Answer

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

The corresponding system is

##### Show Answer

Consider the following block diagram in the figure.

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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?

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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 ________.

##### Show Answer

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 ________ .

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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