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

• \(\displaystyle \frac {1}{\tau _ds+1}\)

• \(\tau _ds+1\)

• \(e^{-\tau _ds}\)

• \(e^{\tau _ds}\)

Solution

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

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

• 1

• \(\dfrac {1}{s}\)

• \(\dfrac {1}{s^2}\)

• \(\dfrac {1}{s^2+1}\)

Solution

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

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

• \(\displaystyle \frac {e^{\tau _ds}}{(\tau s +1)}\)

• \(\displaystyle \frac {e^{-\tau _ds}}{(\tau s +1)}\)

• \(\displaystyle \frac {1}{(\tau s+1)(\tau _ds +1)}\)

• \(\displaystyle \frac {\tau _ds}{(\tau s +1)}\)

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 \(\dfrac{2s-1}{(3s+1)(4s+1)}\) 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 \( \dfrac{1}{s^2 + 2s + 3}\) 

• 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 \(\dfrac{s + 1}{2s + 1}\) 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

• \(\dfrac{2}{s}\)

• \(1+\dfrac{2}{s^2}\)

• \(1+\dfrac{2}{s}\)

• \( \displaystyle \frac{1}{s}\left(1+\dfrac{2}{s}\right) \)

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

A second order system can be obtained by connecting two first order systems \(1/(\tau_1s+1)\) and \(1/(\tau_2s+1)\) 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

• \(\dfrac{1}{2s^3}\)

• \(\dfrac{2}{s^3}\)

• \(\dfrac{1}{s^3}\)

• \(\dfrac{2}{s^2}\)

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) \((mC_P)/(hA)\)
(Q) Mixing	(2) \(q/V\)
	(3) \(V/q\)
	(4) \((hA)/(mC_P)\)

• 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

• \(\dfrac{(Q/P)e^{-(Q/R)s}}{\tau_d s+1}\)

• \(\dfrac{(Q/R)e^{-\tau_ds}}{(Q/P) s+1}\)

• \(\dfrac{(Q/P)e^{-\tau_ds}}{(Q/R) s+1}\)

• \(\dfrac{(Q/R)e^{-(P/Q)s}}{\tau_d s+1}\)

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

• \(\displaystyle \frac{1}{K_p\tau}e^{t/\tau}\)
• \(\displaystyle K_pe^{-t/\tau} \)
• \(\displaystyle \tau K_pe^{-t/\tau} \)
• \(\displaystyle \frac{K_p}{\tau}e^{-t/\tau} \)

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

• \(Q_0(T_0-T) = A_ch\dfrac{dT}{dt}\)

• \(Q_0T_0-QT = A_ch\dfrac{dT}{dt}\)

• \(Q(T_0-T) = A_ch\dfrac{dT}{dt}\)

• \(Q(T_0-T) = A_c\dfrac{d(Th)}{dt}\)

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

A 2-input, 2-output process can be described in the Laplace transform domain as given below \[ \begin{align*} (\tau_1s+1)Y_1(s) &= K_1U_1(s) + K_2U_2(s) \\
 (\tau_2s+1)Y_2(s) &= K_3U_2(s) + K_4Y_1(s) \end{align*} \] 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 \(K_3+K_2K_4\)
• \(K_2\) and \(K_3+K_1K_4\)
• \(K_2\) and \(K_3+K_2K_4\)

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

• \(\dfrac{dV}{dt} = q_1+q_2-q_3, \qquad V\dfrac{dT}{dt} = q_1T_o+q_2T_s-q_3T\)
• \(\dfrac{dV}{dt} = q_1-q_4,\qquad \dfrac{d(VT)}{dt} = q_1T_s+q_4T\)
• \(\dfrac{dV}{dt} = q_1+q_2-q_4,\qquad \dfrac{d(VT)}{dt} = q_1T_o+q_2T_s-q_4T\)
• \(\dfrac{dV}{dt} = q_1+q_2-q_3-q_4,\qquad V\dfrac{dT}{dt} = q_1(T_o-T)+q_2(T_s-T)\)

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 \(F_{\text{out}} = 0.1\,h\) 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
• \(\infty\)

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

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

• \(\dfrac{1}{2s+1} - \dfrac{2}{3s+1}\)
• \(\dfrac{2}{s+1} - \dfrac{5}{s+10}\)
• \(\dfrac{3(0.5s-1)}{(2s+1)(s+1)}\)
• \(\dfrac{5}{s+1} - \dfrac{3}{2s+1}\)

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

• \(\dfrac{1}{100s+1}\)
• \(\dfrac{2}{200s+1}\)
• \(\dfrac{3}{300s+1}\)
• \(\dfrac{4}{400s+1}\)

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

• \(\dfrac{1}{0.024 s^2 + 0.067 s + 1}\)
• \(\dfrac{1}{0.067 s^2 + 0.024 s + 1}\)
• \(\dfrac{1}{0.021 s^2 + 0.1176 s +1}\)
• \(\dfrac{1}{0.1176 s^2 + 0.021 s +1}\)

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

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

where, \(K_p\) is the steady state gain, \(\zeta\) is the damping coefficient, \(\tau\) is the natural period of oscillation and \(\phi\) is the phase lag. The overshoot of the system is \(\displaystyle \exp\left(-\frac{\pi \zeta}{1-\zeta^2} \right)\). 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 \(\displaystyle \left(-\frac{1}{10}+j\frac{2}{15}\right)\) and \(\displaystyle \left(-\frac{1}{10}-j\frac{2}{15}\right)\) is

• \(7\times10^{-1}\)
• \(8\times10^{-2}\)
• \(9\times10^{-3}\)
• \(10\times10^{-4}\)

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,

• \(\dfrac{2}{(56s+1)^2}\)

• \(\dfrac{2}{(62s+1)^2}\)

• \(\dfrac{2}{(36s+1)^2}\)

• \(\dfrac{2}{(49s+1)^2}\)

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 \(G(s) = \dfrac{1}{s^2+1}\) 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 \(A \rightarrow B\) 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