Impulse response of Second Order System

Q. A control system with certain excitatioin is governed by the following mathematical equation

$\frac{d^{2}x}{d{t}^2}+\frac{1}{2}\frac{dx}{dt}+\frac{1}{18}x=10+5e^{-4t}+2e^{-5t}$

The natural time constants of the response of the system are

1. 2s and 5s
2. 3s and 6s
3. 4s and 5s
4. 1/3s and 1/6s

Q. A second-order real system has the following properties:

(a) The damping ratio $\xi =0.5$ and undamped natural frequency ${\omega }_n=10rad/s.$

(b) The steady state value of the output to a unit step input is 1.02.

The transfer function of the system is

1. $\frac{102}{s^2+5s+100}$
2. $\frac{102}{s^2+10s+100}$
3. $\frac{102}{s^2+s+100}$
4. $\frac{102}{s^2+5s+100}$

Q. A linear, time - invarient, casual continuous time system has a rational transfer function with simple poles at $s=-2$ and $s=-4$, and one simple zero at $s=-1$.A unit step $u\left(t\right)$ is applied at the input of the system. At steady state, the output has constant value of $1$. The impulse response of this system is

1. $[e^{-2t}+e^{-4t}]u\left(t\right)$
2. $[-4e^{-2t}+12e^{-4t}-e^{-t}]u\left(t\right)$
3. $[-4e^{-2t}+12e^{-4t}]u\left(t\right)$
4. $[-0.5e^{-2t}+1.5e^{-4t}]u\left(t\right)$

Q. 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 $x\left(t\right)$ and an output $y\left(t\right)$.The system, which is initially relaxed is excited by a unit step input. The output $y\left(t\right)$ can be represented by the waveform

1. 
2. 
3. 
4. 

Q. Given figure shows a closed loop unity feedback system. The controller block has transfer function denoted by $G_c\left(s\right)$. The controller is cacaded to plant, which is denoted by $G_p\left(s\right)$



The loop transfer function $G_c\left(s\right)$ is

1. $\frac{1+0.1s}{s}$
2. $-\frac{1+0.1s}{s}$
3. $-\frac{s}{s+1}$
4. $\frac{s}{s+1}$

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

1. $16$
2. $4$
3. $2$
4. $1$

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

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

1. $P\left(s\right)-\left(3\right),Q\left(s\right)-\left(2\right),R\left(s\right)-\left(1\right)$
2. $P\left(s\right)-\left(1\right),Q\left(s\right)-\left(2\right),R\left(s\right)-\left(3\right)$
3. $P\left(s\right)-\left(2\right),Q\left(s\right)-\left(1\right),R\left(s\right)-\left(3\right)$
4. $P\left(s\right)-\left(1\right),Q\left(s\right)-\left(3\right),R\left(s\right)-\left(2\right)$