GATE Questions & Answers of Control Systems Electronics and Communication Engg

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(s)=(s+1)sp(s+2)(s+3) 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(s)=sp+b1sp-1+···+bpsq+a1sq-1+···+aq 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(s)=K(s2+2s+2)(s2-3s+2) 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 s4+s2+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 x1(t)x2(t)=000-9x1(t)x2(t)+045u(t) , with the initial condition x1(0)x2(0)=00, where u(t) denotes the unit step function. The value of limtx12(t)+x22(t) 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(s)=2(s+1)s3+ks2+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,

ddtx1(t)-x2(t)=0ddtx2(t)+2x1(t)+3x2(t)=r(t)

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(s)=10K(s+2)s3+3s2+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

       Gs=Ks2+5s+5

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

    Gs=Kss+2

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 Gs=s-2s+1s+3 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 Gs=1-s4+2s is

In the feedback system shown below GS=1s2+2s .
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 GS=1s+1s+2s+3 .

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 GS=ks+0.1s+10s+p1 , 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

Gs=2ss+1

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

ddtx1t+2x1t=3ut

ddtx2t+x2t=ut

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

                           Gs=Ks+2s2+2s+2 and Hs=1

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 μF 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 Gs=Kss+1s+3. The value of the gain K (>0) at which the root locus crosses the imaginary axis is ______.

The polar plot of the transfer Gs=10s+1s+10 for 0ω< 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 Gs =Ks+4s+8s2-9 . 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 CsRs is

A unity negative feedback system has an open-loop transfer function Gs=Kss+10 . 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

yt=1-23e-tcos3t-π6; The transfer function of the system is

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

Gs=1Ms2+Bs+k

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

ω in rad/s |G()| in dB arg (G()) 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 Gs=10ss+10 is _____.

network is described by the state model as

        x·1=2x1-x2+3u

 $ x_2=-4x_2-\mathrm u $

y=3x1-2x2

The transfer function Hs=YsUs 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 Td 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

Gs=Ks+2s-1

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

x1.x2.x3.=-1100-1000-2x1x2x3+040u; y=111x1x2x3

The system is

The phase margin in degrees of Gs=10s+0.1s+1s+10 calculated using the asymptotic Bode plot is ________.

For the following feedback system Gs=1s+1s+2The 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 YsX2s is

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

x1.x1.=-100-2x1x2

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 CsRs 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

x.t=Axt,x0=x0

For x0=1-1,xt=e-t-e-t  and for x0=01,xt=e-t-e-2te-t+e-2t 

when x0=35,xt 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 x1.x2.=0100x1x2 is

Consider a transfer function Gps=ps2+3ps-2s2+3+ps+2-p 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+KGs=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 (20 logG(s)) 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 Y(s)U(s) for this system is

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

X=AX+Bu;     y=CX+Du   

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

X=AX+Bu;     y=CX+Du   

The state transition matrix e A t of the system shown in the figure above is

A system with transfer function

G(s)=s2+9s+2s+1s+3s+4

is excited by sinωt. The steady-state output of the system is zero at

The state variable description of an LTI system is given by

x1.x2.x3.= 0a1000a2a300 x1x2x3+001u

y=100x1x2x3

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

The differential equation 100d2ydt2-20dydt+y=xt  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=5+, the corresponding Nyquist plot for positive frequency has the form