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The system shown in figure below.







can be reduced to the form with

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 ____ when a = 1 and b = 2

For a tachometer, if $\theta \left(t\right)$ is the rotor displacement in radians, $e\left(t\right)$ is the output voltage and $K_t$ is the tachometer constant in $V/rad/sec$, then the transfer function,

$\frac{E\left(s\right)}{Q\left(s\right)}$ will be

The polar plot of

$G\left(s\right)=\frac{10}{s(s+1{)}^2}$

intercepts real axis at $\omega ={\omega }_0$. Then, the real part and ${\omega }_0$ are respectively given by:

The matrix of any state space equations for the transfer function $C\left(s\right)/R\left(s\right)$ of the system, shown below in figure is

The transfer function of a compensator isgiven as

$G_c\left(s\right)=\frac{s+a}{s+b}$

$G_c\left(s\right)$ is a lead compensator if

For the system governed by the set of equation:

$\frac{d{x}_1}{dt}=2x_1+x_2+u$

$\frac{d{x}_2}{dt}=-2x_1+u$

$y=3x_1$

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

The transfer function of a plant is

$T\left(s\right)=\frac{5}{(s+5)(s^2+s+1)}.$ The second-order approximation of $T\left(s\right)$ using dominant pole concept is

A state space representation for the transfer function $\frac{y\left(s\right)}{u\left(s\right)}=\frac{s+6}{s^2+5s+6}$ is $\dot{x}=Ax=Bu,$ where

$A=\left[\begin{array}{cc}0& 1\\ -6& -5\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right]$, and the value of C is

A unity feedback system has an open loop transfer function of the form



$KG\left(s\right)=\frac{K(s+a)}{s^2(s+b)};b>a$



Which of the loci shown in figure can be valid rootloci for the system?

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

$\dot{x}\left(t\right)=Ax\left(t\right),x\left(0\right)=x_0$



$For{x}_0=\left[\begin{array}{c}1\\ -1\end{array}\right],x\left(t\right)=\left[\begin{array}{c}{e}^{-t}\\ -e^{-t}\end{array}\right]\text{and for}{x}_0=\left[\begin{array}{c}0\\ 1\end{array}\right]$



$x\left(t\right)=\left[\begin{array}{cc}{e}^{-t}& e^{-2t}\\ -e^{-t}& +2e^{-2t}\end{array}\right]$



$When{x}_0=\left[\begin{array}{c}3\\ 5\end{array}\right],x\left(t\right)is$

1. 5

The output y(t) of a system is related to its input x(t) as

$y\left(t\right)={\int }_0^{t}x(\tau -2)d\tau$

where, x(t) = 0 and y(t) = 0 for t $\le$ 0. The transfer function of the system is

The loop transfer function of a negative feedback system is

$G\left(s\right)H\left(s\right)=\frac{K(s+11)}{s(s+2)(s+8)}$

The value of $K$ , for which the system is marginally stable is

1. 160

A state variable system $x\left(t\right)=\left[\begin{array}{cc}0& 1\\ 0& -3\end{array}\right]X\left(t\right)+\left[\begin{array}{c}1\\ 0\end{array}\right]U\left(t\right),$ with the initial condition $X\left(0\right)=[-13{]}^T$ and the unit step input $u\left(t\right)$ has



The state transition matrix

The state model of a system is given as,



$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]\left[\begin{array}{c}{x}_2\\ x_1\end{array}\right]$



$x^T\left(0\right)=\left[10\right]$

The solution of homogenous equation is given by,

A conttrol system with a PD controller is shown in the figure. If the velocity error constant $K_v=1000$ and the damping ratio $\zeta =0.5$, then the values of $K_P$ and $K_D$ are

Let a causal LTI system be characterized by the following differential equation, with initial rest condition.

$\frac{d^{2}y}{d{t}^2}+7\frac{dy}{dt}+10y\left(t\right)=4x\left(t\right)+5\frac{dx\left(t\right)}{dt}$



Where, x(t) and y(t) are the input and output respectively. The impulse response of the system is [u(t) is the unit step function]

A system described by the transfer function $H\left(s\right)=\frac{1}{s^3+\alpha s^2+Ks+3}$ is stable.

The constraints on $\alpha$ and $K$ are

The magnitude of frequency response of an underdamped second order system is $5$ at $0rad/sec$ and peaks to $\frac{10}{\sqrt{3}}$ at $5\sqrt{2}rad/sec.$ The transfer function of the system is

1. 4

Consider a closed-loop system shown in Figure(A) below. the root locus for its is shown in Figure (B). the closed loop transfer function for the system is

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

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

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

1. 2.8

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

$\dot{x}=AX+Bu;y=CX+Du$



The state -variable equations of the system in the figure above are

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 $G\left(s\right)$ is

A state variable system $x\left(t\right)=\left[\begin{array}{cc}0& 1\\ 0& -3\end{array}\right]X\left(t\right)+\left[\begin{array}{c}1\\ 0\end{array}\right]U\left(t\right),$

with the initial condition $X\left(0\right)=[-13{]}^T$ and the unit step input $u\left(t\right)$ has



The state transition equation:

For the system shown in figure, $S=-2.75$ lies on the root locus if $K$ is

1. 0.3

The root locus of the feedback control system having the characteristic equation,

$s^2+6Ks+2s+5=0$



Where K>0, enters into the real axis at

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



The gain $\left(20log|G\right(s\left)|\right)$ is 32 dB and -8 dB at 1 rad/s and 10 rad/s respectively. The phase is negative for all $\omega$. The G(s) is.

For the system described by the state equation

$\dot{x}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0.5& 1& 2\end{array}\right]x+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u$

If the control signal $u$ is given by $u=[-0.5-3-5]x+v$, then the eigen values of the closed-loop system will be

The open loop transfer function

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

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 step input and is $6$ for unit ramp input, the value of the parameter $p$ is .

1. 1

Which of the following points is NOT on the root locus of a system with the open-loop transfer function

$G\left(s\right)H\left(s\right)=\frac{K}{s(s+1)(s+3)}?$

1. 1.913

Match the unit-step responses(1),(2) and (3) with transfer functions P(s), Q(s) and R(s), given below.



$P\left(s\right)=-\frac{1}{(s+1)};Q\left(s\right)=\frac{2(s-1)}{(s+10)(s+2)};R\left(s\right)=\frac{1}{(s+1{)}^2}$

A lead compensator network includes a parallel combination of $R$ and $C$ in the feed-forward path. If the transfer function of the compensator is$G_c\left(s\right)=\frac{s+2}{s+4},$ the value of $RC$ is

1. 0.5

The asymptotic Bode plot for the gain magnitude of a minimum phase system G(s) is shown in the figure. The transfer function of the system G(s) is

The transfer function of a system is given by, $\frac{V_0\left(s\right)}{V_i\left(s\right)}=\frac{1-s}{1+s}.$ Let the output of the system be $V_0\left(t\right)=V_{m}sin(\omega t+j)$ for the input, $V_i\left(t\right)=V_{m}sin\left(\omega t\right)$ then the minimum and maximum values of $j$(in radians) are respectively

The first two rows in the Routh table for the characteristic equation of a certain closed-loop control system are given as



$\begin{array}{ccc}{s}^3& 1& (2k+3)\\ s^2& 2K& 4\end{array}$



The range of $K$ for which the system is stable is

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







The closed-loop transfer function of the system is.

Match the inferences $X,Y$ and $Z$ about a system, to the corresponding properties of the elements of fiirst column in Routh's Table of the system characteristic equation.

List- I

X.The system is stable..

Y. The system is unstable..

Z. The test breaks down..

List II

P. when all elements are positive

Q. when any one elements is zero

R. when there is a change in sign of coefficients

For the Bode plot shown below the transfer function is,

A linear system is described by the following state equation

$\dot{X}\left(t\right)=AX\left(t\right)+BU\left(t\right),A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

The state- transition matrix of the system is

Nyquist plots of two fucntions $G_1\left(s\right)$ and $G_2\left(s\right)$ are shown in figure.



Nyquist plot of the product of $G_1\left(s\right)$ and $G_2\left(s\right)$ is

The output of a standard second-order system for a unit step input is given as $y\left(t\right)=1-\frac{2}{\sqrt{3}}{e}^{-t}\cos (\sqrt{3}t-\frac{\pi }{6}).$The transfer function of the system is

The open-loop transfer function of a unity feedback configuration is given as $G\left(s\right)=\frac{K(s+4)}{(s+8)(s^2-9)}$

The value of a gain $K(>0)$ for which $-1+j2$ lies on the root locus is

1. 25.54

A casual system having the transfer function $H\left(s\right)=\frac{1}{s+2}$ is excited with $10u\left(t\right)$. The time at which the output reaches 99% of its steady state value is

For a second-order system with the closed-loop transfer function $T\left(s\right)=\frac{9}{s^2+4s+9}.$ The settling time for 2 % band, in seconds, is

1. 0.316

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

Unit step response of second order system is shown in figure.





Transfer function of the system is