Time Response Analysis · Control Systems · GATE ECE

In the feedback control system shown in the figure below $G(s) = \dfrac{6}{s(s+1)(s+2)}$.

$R(s), Y(s),$ and $E(s)$ are the Laplace transforms of $r(t), y(t),$ and $e(t)$, respectively. If the input $r(t)$ is a unit step function, then __________

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

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 of $$\frac{\mathrm{dy}}{\mathrm{dt}}\;\mathrm{at}\;\mathrm t\;=\;0^+\;\mathrm{is}$$ ___________.

For the unity feedback control system shown in the figure, the open-loop transfer function G(s) is given as $$G\left(s\right)\;=\;\frac2{s\left(s\;+\;1\right)}$$ .The steady state error e_ss due to a unit step input is

A unity negative feedback system has an open–loop transfer function $$G\left(s\right)=\frac K{s\left(s+1\right)}$$.The gain K for the system to have a damping ratio of 0.25 is _____________.

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

The input $$-3\mathrm e^{2\mathrm t}\;\mathrm u\left(\mathrm t\right)$$, where u(t) is the unit step function, is applied to a system with transfer function $$\frac{s-2}{s+3}$$. If the initial value of the output is -2, then the value of the output at steady state is __________.

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 ________.

The differential equation $$$100\frac{\mathrm d^2\mathrm y}{\mathrm{dt}^2}-20\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm y=\mathrm x\left(\mathrm t\right)$$$ 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

Step responses of a set of three second-order underdamped systems all have the same percentage overshoot. Which of the following diagrams represents the poles of the three systems?

If the Laplace transform of a signal y(t) is $$Y\left(s\right)\;=\;\frac1{s\left(s\;-\;1\right)}$$ , then its final value is:

Consider a system with the transfer function $$$G\left(s\right)=\frac{s+6}{Ks^2+s+6}$$$ Its damping ratio will be 0.5 when the value of K is

If the characteristic equation of a closed-loop system is s² + 2s + 2 =0, then the system is

For a second-order system with the closed-loop transfer function $$$T\left(s\right)=\frac9{s^2+4s+9}$$$ the settling time for 2% band in seconds is

If $$F\left(s\right)\;=\;\frac\omega{s^2\;+\;\omega^2}$$, then the value of $$\underset{t\rightarrow\infty}{\lim\;}f\left(t\right),\;\left\{where\;F\left(s\right)\;is\;the\;L\left[f\left(t\right)\right]\right\}$$

Consider a feedback control system with loop transfer function $$$G\left(s\right)H\left(s\right)=\frac{K\left(1+0.5s\right)}{s\left(1+s\right)\left(1+2s\right)}$$$ The type of the closed loop system is

Consider a unity feedback control system with open-loop transfer function $$G\left(s\right)\;=\;\frac K{s\left(s\;+\;1\right)}$$ . The steady state error of the system due to a unit step input is

The step error coefficient of a system G(s)=$$\frac1{\left(s+6\right)\left(s+1\right)}$$ with unity feedback is

If $$L\left(f\left(t\right)\right)=\frac{2\left(s+1\right)}{s^2+2s+5}$$ then f(0⁺) and f($$\infty$$) are given by [Note: 'L' stands for 'Laplace Transform of']

For a second order system, damping ratio $$\left(\xi\right)$$ , is 0 < $$\xi$$ < 1 ,then the roots of the characteristic polynomial are

The final value theorem is used to find the

The poles of a continuous time oscillator are ___________.

Consider a unity negative feedback control system with forward path gain $G(s) = \frac{K}{(s + 1)(s + 2)(s + 3)}$ as shown.

The impulse response of the closed-loop system decays faster than $e^{-t}$ if ________.

A closed loop system is shown in the figure where $$k>0$$ and $$\alpha > 0$$. The steady state error due to a ramp input $$(R(s) = \alpha /{s^2})$$ is given by

Two linear time-invariant systems with transfer functions

$${G_1}(s) = {{10} \over {{s^2} + s + 1}}$$ and $${G_2}(s) = {{10} \over {{s^2} + s\sqrt {10} + 10}}$$

have unit step responses y₁(t) and y₂(t), respectively. Which of the following statements is/are true?

The block diagram of a closed-loop control system is shown in the figure. R(s), Y(s), and D(s) are the Laplace transforms of the time-domain signals r(t), y(t), and d(t), respectively. Let the error signal be defined as e(t) = r(t) $$-$$ y(t). Assuming the reference input r(t) = 0 for all t, the steady-state error e($$\infty$$), due to a unit step disturbance d(t), is __________ (rounded off to two decimal places).

In the feedback system shown below $$$G\left(S\right)\;=\frac1{(s^2\;+\;2s)}$$$ 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 __________.

The open-loop transfer function of a unity-feedback control system is $$$G\left(S\right)\;=\frac K{s(s\;+\;2)}$$$ For the peak overshoot of the closed-loop system to a unit step input to be 10%, the value of K is __________.

The output of a standard second–order system for a unit step input is given as $$$y\left(t\right)=1-\frac2{\sqrt3}e^{-t}\cos\left(\sqrt3t\;-\;\frac{\mathrm\pi}6\right)$$$ The transfer function of the system is

The damping ratio of a series RLC circuit can be expressed as

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

For the following feedback system $$G\left(s\right)=\frac1{\left(s+1\right)\left(s+2\right)}$$.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 open-loop transfer function of a dc motor is given as $$\;\frac{\omega\left(s\right)}{V_a\left(s\right)}=\frac{10}{1+10s}$$, when connected in feedback as shown below, the approximate value of K_a that will reduce the time constant of the closed-loop system by one hundred times as compared to that of the open-loop system is

The unit step response of an under-damped second order system has steady state value of -2. Which one of the following transfer function has these properties?

Group I lists a set of four transfer functions. Group II gives a list of possible step responses y(t). Match the step responses with the corresponding transfer functions

Group I $$$P=\frac{25}{s^2+25}\;\;\;\;\;Q=\frac{36}{s^2+20s+36}$$$ $$$R=\frac{36}{s^2+12s+36}\;\;\;\;\;S=\frac{49}{s^2+7s+49}$$$

Group II

A linear, time-invariant, causal 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(t) 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

The frequency response of a linear, time-invariant system is given by $$H\left(f\right)\;=\;\frac5{1\;+\;j10\mathrm{πf}}$$ .The step response of the system is:

The transfer function of a plant is $$$T\left(s\right)=\frac5{\left(s+5\right)\left(s^2+s+1\right)}$$$ The second-order approximation of T (s) using dominant pole concept is:

The unit-step response of a system starting from rest is given by $$$\mathrm c\left(\mathrm t\right)=1-\mathrm e^{-2\mathrm t}\;\mathrm{for}\;\mathrm t\geq0$$$The transfer function of the system is:

The unit impulse response of a system is: $$$h\left(t\right)\;=\;e^{-t},\;t\geq0$$$ For this system, the steady-state value of the output for unit step input is equal to

A ramp input applied to an unity feedback system results in 5% steady state error. The type number and zero frequency gain of the system are respectively.

In the derivation of expression for peak percent overshoot,$$$M_p=exp\left(\frac{-\mathrm{πξ}}{\sqrt{1-\xi^2}}\right)\times100\%$$$ .Which one of the following conditions is NOT required?

A system described by the following differential equation $$$\frac{d^2y}{dt^2}+3\frac{dy}{dt}+2y=x\left(t\right)$$$ is initially at rest. For input x(t) = 2u(t), the output y(t) is

A causal system having the transfer function $$G\left(s\right)\;=\;\frac1{s\;+\;2}$$ is excited with 10u(t). The time at which the output reaches 99% of its steady state value is

A second-order system has the transfer function $$\frac{C\left(s\right)}{R\left(s\right)}=\frac4{s^2+4s+4}$$. With r(t) as the unit-step function, the response c(t) of the system is represented by

The transfer function of a system is $$G\left(s\right)\;=\;\frac{100}{\left(s\;+\;1\right)\left(s\;+\;100\right)}$$.For a unit step input to the system the approximate settling time for 2% criterion is

If the closed-loop transfer function T(s) of a unity negative feedback system is given by $$$T\left(s\right)=\frac{a_{n-1}s+a_n}{s^n+a_1s^{n-1}+.....+a_{n-1}s+a_n}$$$ then the steady state error for a unit ramp input is

The unit impulse response of a linear time invariant system is the unit step function u(t). For t>0, the response of the system ot an excitation e^-at u(t), a > 0 will be

Match the following with List-1 with List-2

List-1
(a) Very low response at very high frequencies.
(b) Overshoot
(c) Synchro-control transformer output

List-2
(1) Low pass systems
(2) Velocity damping
(3) Natural frequency
(4) Phase-sensitive modulation
(5) Damping ration

The response of an LCR circuit to a step input is
(a) Over damped
(b) Critically damped
(c) Oscillatory

If the transfer function has

(1) poles on the negative real axis
(2) poles on the imaginary axis
(3) multiple poles on the positive real axis
(4) poles on the positive real axis
(5) multiple poles on the negative real axis

A second order system has a transfer function given by $$\mathrm G\left(\mathrm s\right)\;=\;\frac{25}{\mathrm s^2\;+\;8\mathrm s\;+\;25}$$ .If the system, initially at rest is subjected to a unit step input at t = 0, the second peak in response will occur at

A unity feedback control system has the open loop transfer function $$G\left(s\right)\;=\;\frac{4\left(1\;+\;2s\right)}{s^2\left(s\;+\;2\right)}$$ .If the input to the system is a unit ramp, the steady-state error will be

The steady state error of a stable 'type 0' unity feedback system for a unit step function is

A critically damped, continuous-time, second order system, when sampled, will have ( in Z domain)

The block diagram of a feedback system is shown in Figure.
(a) Find the closed loop transfer function.
(b) Find the minimum value of G for which the step response of the system would exhibit an overshoot, as shown in Figure.
(c) For G equal to twice this minimum value, find the time period T indicated in Figure.

Following fig. shows the block diagram representation of control system. The system in block A has an impulse response h(t ) = e^−t u(t ). The system in block B has an impulse response h(t ) = e^−2t u(t ). The block 'K' amplifies its inputs by a factor k. For the overall system with input x(t) and output y(t). (a) Find the transfer function $$\frac{y\left(s\right)}{x\left(s\right)}$$ when k =1.
(b) Find the impulse response when k = 0.
(c) Find the values of k for which the system becomes unstable.

For the linear, time-invariant system whose block diagram is shown in Fig. with input x(t) and output y(t),
(a) Find the transfer function.
(b) For the step response of the system [i.e. find y(t) when x(t) is a unit step function and the initial conditions are zero]
(c) Find y(t), if x(t) is as shown in Fig. and the initial conditions are zero.

Block diagram model of a position control system is shown in figure.
(a) In absence of derivative feedback (K_t=0) , determine damping ratio of the system for amplifier gain K_A=5. Also find the steady state error to unit ramp input.

(b) Find suitable values of the parameters K_A and K_t so that damping ratio of the system is increased to 0.7 without affecting the steady state error as obtained in part (a).

Consider the system shown in Fig. Determine the value of 'a' such that the damping ratio is 0.5. Also obtain the values of the rise time 't_r' and maximum overshoot 'M_p' in its step response.