GATE-CH-1988-8-c-i-ctrl-1mark

A rectangular tank is fitted with a valve at the bottom and is used for storing a liquid. The area of cross-section of the tank is 10 m${}^2$ and the flow resistance of the valve (assumed constant) is 0.1 s/m${}^2$. The time constant of the tank will be:

• 1

• 100

• 10.1

• 9.9

Solution

GATE-CH-1988-8-c-ii-ctrl-1mark

The transfer function of a system is given by, $Y/X=1/(s^2+5s+6)$. The roots of the characteristic equation are located,

• to the left of the imaginary axis and on the real axis

• on the imaginary axis

• right of the imaginary axis and on the real axis

• at the origin

Solution

GATE-CH-1991-9-i-ctrl-2mark

A certain thermocouple has a specific time constant of 2 s. If the process temperature changes abruptly from 800 to 900${}^{\circ }$C, the temperature reading in an indicator attached to the thermocouple after 6 s will be approximately,

• 860${}^{\circ }$C

• 900${}^{\circ }$C

• 890${}^{\circ }$C

• 895${}^{\circ }$C

Solution

GATE-CH-1992-9-a-ctrl-2mark

When a bare thermocouple is covered by a protective sheath, the response becomes:

• Faster and oscillatory

• Faster and non-oscillatory

• Slower and oscillatory

• Slower and non-oscillatory

Solution

GATE-CH-1995-1-u-ctrl-1mark

The response of two tanks of same size and resistance in series is

• under damped

• critically damped

• over damped

• none of the above

Solution

GATE-CH-1995-1-v-ctrl-1mark

The transfer function of a pure dead time system with dead time ${\tau }_d$ is

• ${\displaystyle \frac{1}{{\tau }_{d}s+1}}$

• ${\tau }_{d}s+1$

• $e^{-{\tau }_{d}s}$

• $e^{{\tau }_{d}s}$

Solution

GATE-CH-1995-2-s-ctrl-2mark

Identify an unbounded input from four inputs whose functions are given below:

• 1

• ${\displaystyle \frac{1}{s}}$

• ${\displaystyle \frac{1}{s^2}}$

• ${\displaystyle \frac{1}{s^2+1}}$

Solution

GATE-CH-1997-1-24-ctrl-1mark

The transfer function for a first-order process with time delay is

• ${\displaystyle \frac{e^{{\tau }_{d}s}}{(\tau s+1)}}$

• ${\displaystyle \frac{e^{-{\tau }_{d}s}}{(\tau s+1)}}$

• ${\displaystyle \frac{1}{(\tau s+1)({\tau }_{d}s+1)}}$

• ${\displaystyle \frac{{\tau }_{d}s}{(\tau s+1)}}$

Solution

GATE-CH-1997-1-26-ctrl-1mark

For an input forcing function, $X(t)=2t^2$, the Laplace transform of this function is

• ${\displaystyle \frac{2}{s^2}}$

• ${\displaystyle \frac{4}{s^2}}$

• ${\displaystyle \frac{2}{s^3}}$

• ${\displaystyle \frac{4}{s^3}}$

Solution

GATE-CH-1997-2-19-ctrl-2mark

The transfer function of a process is ${\displaystyle \frac{1}{16s^2+8s+4}}$. If a step change is introduced into the system, then the response will be

• underdamped

• critically damped

• overdamped

• none of the above

Solution

GATE-CH-1999-1-26-ctrl-1mark

A system with a double pole at the origin is unstable since the corresponding form in the time domain

• is a constant

• grows exponentially with time

• grows linearly with time

• decays linearly with time

Solution

GATE-CH-1999-1-28-ctrl-1mark

A typical example of a physical system with under-damped characteristics is a

• U-tube manometer

• spring-loaded diaphragm valve

• CSTR with first-order reaction

• thermocouple kept immersed in a liquid-filled thermowell

GATE-CH-1999-2-16-ctrl-2mark

A control system has the following transfer function, $$F(s)=\frac{(s-1)(s+1)}{s(s-2)(s+4)}$$ The initial value of the corresponding time function is

• 1

• 1/8

• 7/8

• $-1$

GATE-CH-2000-1-25-ctrl-1mark

The unit step response of the transfer function ${\displaystyle \frac{2s-1}{(3s+1)(4s+1)}}$ reaches its final steady state asymptotically after

• a monotonic increase
• a monotonic decrease
• initially increasing and then decreasing
• initially decreasing and then increasing

GATE-CH-2000-1-26-ctrl-1mark

The unit step response of the transfer function ${\displaystyle \frac{1}{s^2+2s+3}}$ 

• has a non-zero slope at the origin 
  

• has a damped oscillatory characteristic
  

• is overdamped 
  

• is unstable 
  

GATE-CH-2000-2-22-ctrl-2mark

The initial value ($t=0^{+}$) of the unit step response of the transfer function ${\displaystyle \frac{s+1}{2s+1}}$ is 

• 0

• 1/2

• 1

• 2

GATE-CH-2001-1-18-ctrl-1mark

A process is initially at steady state with its output $y=1$ for an input $u=1$. The input is suddenly changed to 2 at $t=0$. The output response is $y(t)=1+2t$. The transfer function of the process is

• ${\displaystyle \frac{2}{s}}$

• $1+{\displaystyle \frac{2}{s^2}}$

• $1+{\displaystyle \frac{2}{s}}$

• ${\displaystyle \frac{1}{s}(1+{\displaystyle \frac{2}{s}})}$

GATE-CH-2001-2-20-ctrl-2mark

A second order system can be obtained by connecting two first order systems $1/({\tau }_{1}s+1)$ and $1/({\tau }_{2}s+1)$ in series. The damping ratio of the resultant second order system for the case ${\tau }_1\ne {\tau }_2$ will be

• > 1

• = 1

• < 1

• $={\tau }_2/{\tau }_1$

GATE-CH-2004-27-ctrl-1mark

For the time domain function $f(t)=t$, the Laplace transform of ${\displaystyle {\int }_0^{t}f(t)dt}$ is given by

• ${\displaystyle \frac{1}{2s^3}}$

• ${\displaystyle \frac{2}{s^3}}$

• ${\displaystyle \frac{1}{s^3}}$

• ${\displaystyle \frac{2}{s^2}}$

GATE-CH-2004-78-ctrl-2mark

Match first order system given in Group I with the appropriate time constant in Group II.

Group I	Group II
(P) Thermometer	(1) $(m{C}_P)/(hA)$
(Q) Mixing	(2) $q/V$
	(3) $V/q$
	(4) $(hA)/(m{C}_P)$

• P-4, Q-2

• P-4, Q-3

• P-1, Q-2

• P-1, Q-3

GATE-CH-2004-79-ctrl-2mark

The experimental response of a controlled variable $y(t)$ for a step change of magnitude $P$ in the manipulated variable $x(t)$ is shown below.

The appropriate transfer function of the process is

• ${\displaystyle \frac{(Q/P)e^{-(Q/R)s}}{{\tau }_{d}s+1}}$

• ${\displaystyle \frac{(Q/R)e^{-{\tau }_{d}s}}{(Q/P)s+1}}$

• ${\displaystyle \frac{(Q/P)e^{-{\tau }_{d}s}}{(Q/R)s+1}}$

• ${\displaystyle \frac{(Q/R)e^{-(P/Q)s}}{{\tau }_{d}s+1}}$

GATE-CH-2005-8-ctrl-1mark

The unit step response of a first order system with time constant $\tau$ and steady state gain $K_p$ is given by

• $K_p(1-e^{-t/\tau })$

• $K_p(1+e^{-t/\tau })$

• $K_p(1-e^{-2t/\tau })$

• $K_p{e}^{-t/\tau }/\tau$

GATE-CH-2005-9-ctrl-1mark

An example of an open-loop underdamped system is

• liquid level in a tank

• U-tube manometer

• thermocouple in a thermo-well

• two non-interacting first order systems in series

GATE-CH-2009-17-ctrl-1mark

The roots of the characteristic equation of an underdamped second order system are

• real, negative and equal
• real, negative and unequal
• real, positive and unequal
• complex conjugates

GATE-CH-2014-10-ctrl-1mark

A unit impulse response of a first order system with time constant $\tau$ and steady state gain $K_p$ is given by

• ${\displaystyle \frac{1}{K_p\tau }{e}^{t/\tau }}$
• ${\displaystyle K_p{e}^{-t/\tau }}$
• ${\displaystyle \tau K_p{e}^{-t/\tau }}$
• ${\displaystyle \frac{K_p}{\tau }{e}^{-t/\tau }}$

GATE-CH-2016-18-ctrl-1mark

What is the order of response exhibited by a U-tube manometer?

• Zero order
• First order
• Second order
• Third order

GATE-CH-1992-19-a-ctrl-6mark

A thermometer follows first-order dynamics with a time constant of 0.2 min. It is placed in a temperature bath at 100${}^{\circ }$C and is allowed to reach steady state. It is suddenly transferred to another bath at 150${}^{\circ }$C at time $t$ = 0 and is left there for 0.2 min. It is immediately returned to the original bath at 100${}^{\circ }$C.
Calculate the readings (in ${}^{\circ }$C) at:
(i) $t$ = 0.1 min
{#1}

(ii) $t$ = 0.4 min
{#2}

GATE-CH-1995-25-ctrl-5mark

When a thermometer at 30${}^{\circ }$C is placed in water bath at 90${}^{\circ }$C, the initial rate of rise in thermometer temperature is found to be 2${}^{\circ }$C/s.
(i) What is the time constant (in s) of the thermometer, assuming it is a first order device with unity steady state gain?
{#1}

(ii) What will thermometer (in ${}^{\circ }$C) read after one minute?
{#2}

GATE-CH-1991-9-iii-ctrl-2mark

A system has the transfer function $Y/X=10/(s^2+1.6s+4)$. A step change of 4 units magnitude is introduced in this system. The percent overshoot is:

• 20

• 30

• 25

• 35

GATE-CH-2003-76-ctrl-2mark

Water is entering a storage tank at a temperature $T_0$ and flow rate $Q_0$ and leaving at a flow rate $Q$ and temperature $T$. There are negligible heat losses in the tank. The area of cross section of the tank is $A_c$. The model that describes the dynamic variation of water temperature in the tank with time is given as

• $Q_0(T_0-T)=A_{c}h{\displaystyle \frac{dT}{dt}}$

• $Q_0{T}_0-QT=A_{c}h{\displaystyle \frac{dT}{dt}}$

• $Q(T_0-T)=A_{c}h{\displaystyle \frac{dT}{dt}}$

• $Q(T_0-T)=A_c{\displaystyle \frac{d(Th)}{dt}}$

GATE-CH-2006-57-ctrl-2mark

A 2-input, 2-output process can be described in the Laplace transform domain as given below $$\begin{array}{rl}({\tau }_{1}s+1)Y_1(s)& =K_1{U}_1(s)+K_2{U}_2(s)\\ ({\tau }_{2}s+1)Y_2(s)& =K_3{U}_2(s)+K_4{Y}_1(s)\end{array}$$ where $U_1$ and $U_2$ are the inputs and $Y_1$ and $Y_2$ are the outputs. The gains of the transfer functions $Y_1(s)/U_2(s)$ and $Y_2(s)/U_2(s)$, respectively, are

• $K_2$ and $K_3$
• $K_1$ and $K_3+K_2{K}_4$
• $K_2$ and $K_3+K_1{K}_4$
• $K_2$ and $K_3+K_2{K}_4$

GATE-CH-2007-60-ctrl-2mark

The dynamic model for a mixing tank open to atmosphere at its top as shown below is to be written. The objective of mixing is to cool the hot water stream entering the tank at a flow rate $q_2$ and feed temperature of $T_s$, with a cold water feed stream entering the tank at a flow rate $q_1$ and feed temperature of $T_0$. A water stream is drawn from the tank bottom at a flow rate of $q_4$ by a pump and the level in the tank is proposed to be controlled by drawing another water stream at a flow rate $q_3$. Neglect evaporation and other losses from the tank.

The dynamic model for the tank is given as

• ${\displaystyle \frac{dV}{dt}}=q_1+q_2-q_3,V{\displaystyle \frac{dT}{dt}}=q_1{T}_o+q_2{T}_s-q_{3}T$
• ${\displaystyle \frac{dV}{dt}}=q_1-q_4,{\displaystyle \frac{d(VT)}{dt}}=q_1{T}_s+q_{4}T$
• ${\displaystyle \frac{dV}{dt}}=q_1+q_2-q_4,{\displaystyle \frac{d(VT)}{dt}}=q_1{T}_o+q_2{T}_s-q_{4}T$
• ${\displaystyle \frac{dV}{dt}}=q_1+q_2-q_3-q_4,V{\displaystyle \frac{dT}{dt}}=q_1(T_o-T)+q_2(T_s-T)$

GATE-CH-2007-61-ctrl-2mark

Match the transfer functions with the response to a unit step input shown in the figure. 

• i-E, ii-C, iii-A, iv-D, v-B
• i-A, ii-B, iii-C, iv-D, v-E
• i-B, ii-A, iii-C, iv-E, v-D
• i-E, ii-A, iii-C, iv-B, v-D

GATE-CH-2008-60-ctrl-2mark

The unit impulse response of a first order process is given by $2e^{-0.5t}$. The gain and time constant of the process are, respectively,

• 4 and 2

• 2 and 2

• 2 and 0.5

• 1 and 0.5

GATE-CH-2008-62-ctrl-2mark

A tank of volume 0.25 m³ and height 1 m has water flowing in at 0.05 m³/min. The outlet flow rate is governed by the relation $F_{\text{out}}=0.1h$ where $h$ is the height of the water in the tank in m and $F_{\text{out}}$ is the outlet flow rate in m³/min. The inlet flow rate changes suddenly from its nominal value of 0.05 m³/min to 0.15 m³/min and remains there. The time (in minutes) at which the tank will begin to overflow is given by

• 0.28
• 1.01
• 1.73
• $\mathrm{\infty }$

GATE-CH-2008-63-ctrl-2mark

Which ONE of the following transfer functions corresponds to an inverse response process with a positive gain?

• ${\displaystyle \frac{1}{2s+1}}-{\displaystyle \frac{2}{3s+1}}$
• ${\displaystyle \frac{2}{s+1}}-{\displaystyle \frac{5}{s+10}}$
• ${\displaystyle \frac{3(0.5s-1)}{(2s+1)(s+1)}}$
• ${\displaystyle \frac{5}{s+1}}-{\displaystyle \frac{3}{2s+1}}$

GATE-CH-2009-44-ctrl-2mark

For a tank of cross-sectional area 100 cm² and inlet flow rate ($F_i$ in cm³/s), the outlet flow rate ($F_o$ in cm³/s) is related to the liquid height ($H$ in cm) as $F_o=3\sqrt{H}$ (see the figure below).

Then the transfer function ${\displaystyle \frac{\overline{H}(s)}{\overline{F_i}(s)}}$ (over-bar indicates deviation variables) of the process around the steady-state point, $F_{i0}=18$ cm³/s and $H_0=36$ cm, is

• ${\displaystyle \frac{1}{100s+1}}$
• ${\displaystyle \frac{2}{200s+1}}$
• ${\displaystyle \frac{3}{300s+1}}$
• ${\displaystyle \frac{4}{400s+1}}$

GATE-CH-2013-44-ctrl-2mark

A unit gain second order underdamped process has a period of oscillation 1 second and decay ratio 0.25. The transfer function of the process is

• ${\displaystyle \frac{1}{0.024s^2+0.067s+1}}$
• ${\displaystyle \frac{1}{0.067s^2+0.024s+1}}$
• ${\displaystyle \frac{1}{0.021s^2+0.1176s+1}}$
• ${\displaystyle \frac{1}{0.1176s^2+0.021s+1}}$

GATE-CH-2015-50-ctrl-2mark

Which one of the following transfer functions, upon a unit step change in disturbance at $t=0$, will show a stable time domain response with a negative initial slope (i.e., slope at $t=0$):

• ${\displaystyle G(s)=\frac{1}{s+1}-\frac{2}{s+4}}$
• ${\displaystyle G(s)=\frac{1}{s+1}+\frac{2}{s+4}}$
• ${\displaystyle G(s)=\frac{1}{s+1}+\frac{2}{s-4}}$
• ${\displaystyle G(s)=\frac{1}{s-1}+\frac{2}{s-4}}$

GATE-CH-2016-53-ctrl-2mark

For a unit step input, the response of a second order system is

$$y(t)=K_p[1-\frac{1}{\sqrt{1-{\zeta }^2}}{e}^{-\zeta t/\tau }\sin (\frac{\sqrt{1-{\zeta }^2}}{\tau }t+\varphi )]$$

where, $K_p$ is the steady state gain, $\zeta$ is the damping coefficient, $\tau$ is the natural period of oscillation and $\varphi$ is the phase lag. The overshoot of the system is ${\displaystyle \exp (-\frac{\pi \zeta }{1-{\zeta }^2})}$. For a unit step  input, the response of the system from an initial steady state condition at $t=0$ is shown in the figure below.

What is the natural period of oscillation (in seconds) of the system?

• 15.9

• 50

• 63.2

• 100

GATE-CH-2018-31-ctrl-2mark

The decay ratio for a system having complex conjugate poles as ${\displaystyle (-\frac{1}{10}+j\frac{2}{15})}$ and ${\displaystyle (-\frac{1}{10}-j\frac{2}{15})}$ is

• $7\times {10}^{-1}$
• $8\times {10}^{-2}$
• $9\times {10}^{-3}$
• $10\times {10}^{-4}$

GATE-CH-2019-30-ctrl-2mark

Consider two non-interacting tanks-in-series as shown in figure. Water enters Tank 1 at $q$ cm³/s and drains down to Tank 2 by gravity at a rate $k\sqrt{h_1}$ (cm³/s). Similarly, water drains from Tank 2 by gravity at a rate of $k\sqrt{h_2}$ (cm³/s) where $h_1$ and $h_2$ represent levels of Tank 1 and Tank 2 respectively (see figure). Drain valve constant $k=4$ cm^2.5/s and cross-sectional areas of the two tanks are $A_1=A_2=28$ cm².

At steady state operation, the water inlet flow rate is $q_s=16$ cm³/s. The transfer function relating the deviation variables $H_2$ (cm) to flow rate $Q$ (cm³/s) is,

• ${\displaystyle \frac{2}{(56s+1{)}^2}}$

• ${\displaystyle \frac{2}{(62s+1{)}^2}}$

• ${\displaystyle \frac{2}{(36s+1{)}^2}}$

• ${\displaystyle \frac{2}{(49s+1{)}^2}}$

GATE-CH-1989-18-ii-ctrl-3mark

A temperature alarm unit, a unity gain first order system with a time constant of two minutes, is subjected to a sudden 100 K rise because of fire. If an increase of 50 K is required to activate the alarm, what will be the delay (in minutes) in signaling the temperature change?

• Answer

GATE-CH-1990-18-i-ctrl-6mark

A thermometer of time constant 10 seconds, initially at 30${}^{\circ }$C, is suddenly immersed into water at 100${}^{\circ }$C. How long (in s) will it take for the thermometer reading to reach 90${}^{\circ }$C?

• Answer

GATE-CH-2000-13-ctrl-5mark

The response of a thermocouple can be modeled as a first order process to changes in the temperature of the environment. If such a thermocouple at 25^oC is immersed suddenly in a fluid at 80^oC and held there, it is found that the thermocouple reading (in ^oC) reaches 63.2% of the final steady value in 40 seconds. Find the time constant of the thermocouple (in seconds).

• Answer

GATE-IN-2015-32-ctrl-2mark

A system with transfer function $G(s)={\displaystyle \frac{1}{s^2+1}}$ has zero initial conditions. The percentage overshoot in its step response is ______%

• Answer

GATE-CH-1991-19-ii-ctrl-6mark

An aqueous solution (density = 1000 kg/m${}^3$, specific heat = 4 kJ/kg.${}^{\circ }$C) at 300 K is continuously fed at a flow rate of 1 m${}^3$/min to a continuous flow stirred tank of volume 1 m${}^3$ containing a heater having a heating capacity of 1000 kW. If the liquid in the tank is also at 300 K to start with, find the equation which predicts the exit temperature of the solution as a function of time after the heater is switched on.

• Answer

GATE-CH-1993-26-a-ctrl-5mark

A first order reaction $A\to B$ with the rate constant $k$ is taking place in CSTR fed with $A$ at concentration $C_{AF}$ which remains unchanged. There are likely to be some deviations in feed rate ($F$) of $A$. Derive linearized transfer function between concentration of $A$ in the outlet and feed rate of $A$ assuming that volume $V$ of reacting mixture remains unchanged.

• Answer

GATE-CH-1994-26-ctrl-5mark

Derive an analytical expression for a unit impulse response of a system whose transfer function is given by $$\frac{Y(s)}{X(s)}=\frac{1.5}{s^2+3s+2}$$

• Answer

GATE-CH-1995-26-ctrl-5mark

A first order system with transfer function ${\displaystyle G_p=\frac{Y(s)}{X(s)}=\frac{1}{s+1}}$ is subjected to input $x(t)=t$. Derive the expression for change in output $y(t)$ as a function of time.
(i) What is the maximum and minimum difference between input and output?
(ii) At what time does these difference occur?

• Answer