# 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:

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

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

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 _________.

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 ______.

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_________

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

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

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_________. 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

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?

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

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

Which of the following is incorrect?

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________

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___________

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.

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

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