## Open-Loop Response

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

1988-8-c-i-ctrl

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

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

1988-8-c-ii-ctrl

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

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

1991-9-i-ctrl

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

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

1992-9-a-ctrl

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

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

1995-1-u-ctrl

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

• under damped

• critically damped

• over damped

• none of the above

[Index]

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

1995-1-v-ctrl

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

• $$\displaystyle \frac {1}{\tau _ds+1}$$

• $$\tau _ds+1$$

• $$e^{-\tau _ds}$$

• $$e^{\tau _ds}$$

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

1995-2-s-ctrl

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

• 1

• $$\dfrac {1}{s}$$

• $$\dfrac {1}{s^2}$$

• $$\dfrac {1}{s^2+1}$$

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

1997-1-24-ctrl

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

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

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

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

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

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

1997-1-26-ctrl

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

• $$\displaystyle \frac {2}{s^2}$$

• $$\displaystyle \frac {4}{s^2}$$

• $$\displaystyle \frac {2}{s^3}$$

• $$\displaystyle \frac {4}{s^3}$$

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

1997-2-19-ctrl

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

[Index]

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

1999-1-26-ctrl

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

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

1999-1-28-ctrl

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

• U-tube manometer

• CSTR with first-order reaction

• thermocouple kept immersed in a liquid-filled thermowell

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

1999-2-16-ctrl

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

2000-1-25-ctrl

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

2000-1-26-ctrl

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

[Index]

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

2000-2-22-ctrl

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

2001-1-18-ctrl

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

2001-2-20-ctrl

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

2004-27-ctrl

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

2004-78-ctrl

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

 Group I Group II (1) $$(mC_P)/(hA)$$ (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

[Index]

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

2004-79-ctrl

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

2005-8-ctrl

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

2005-9-ctrl

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

2009-17-ctrl

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

2014-10-ctrl

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

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

[Index]

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

2016-18-ctrl

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

1992-19-a-ctrl

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

1995-25-ctrl

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

1991-9-iii-ctrl

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

2003-76-ctrl

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

• $$Q_0(T_0-T) = A_ch\dfrac{dT}{dt}$$

• $$Q_0T_0-QT = A_ch\dfrac{dT}{dt}$$

• $$Q(T_0-T) = A_ch\dfrac{dT}{dt}$$

• $$Q(T_0-T) = A_c\dfrac{d(Th)}{dt}$$

[Index]

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

2006-57-ctrl

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

2007-60-ctrl

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

2007-61-ctrl

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

2008-60-ctrl

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

2008-62-ctrl

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
• $$\infty$$

[Index]

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

2008-63-ctrl

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

2009-44-ctrl

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

2013-44-ctrl

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

2015-50-ctrl

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

2016-53-ctrl

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

[Index]

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

2018-31-ctrl

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

2019-30-ctrl

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 $$k=4$$ cm2.5/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

1989-18-ii-ctrl

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?

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

1990-18-i-ctrl

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?

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

2000-13-ctrl

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 25oC is immersed suddenly in a fluid at 80oC 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).

[Index]

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

IN-2015-32-ctrl

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

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

1991-19-ii-ctrl

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.

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

1993-26-a-ctrl

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.

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

1994-26-ctrl

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

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

1995-26-ctrl

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?

[Index]