Tutorial problems
1) Find the overall transfer function for the given system
1) A control system is shown in the following figure. The transfer functions G2 and H2
are fixed. Determine the transfer function G1 and H1 so that the closed loop transfer
function Y/R is exactly equal to 1.
2) Determine the transfer function for the given system
5) A control system is shown in the following figure. The process Gp is a first order with
unity steady state gain and a 10 s time constant. The controller Gc is a gain with value 10.
- Compute the closed loop time constant and comment on the results.
6) A mercury thermometer is a first-order measurement system with a time constant of 5
seconds. Describe how the reading given by the thermometer will change with time when it is
suddenly placed in a hot liquid. About how long will it take for a steady reading to be
obtained?
7) Explain how the responses of the given systems vary with time when subject to a unit step
input:
a) Static gain = 12
b) Static gain =12 and time constant = 4 sec.
c) Static gain = 12, ζ= 0.5, ωn= 0.78
8) The open loop transfer function of a system is given with ζ= 0.4, ωn= 0.5. Using XCOS,
determine the approximate values of peak overshoot, time to peak, and settling time. Plot the
approximate unit step response of the system.
9) A thermometer requires 1 min to indicate 98% of the response to a step input. Assuming
the thermometer is a first order system, find the time constant. If the thermometer is placed in
a bath, the temperature of which is changing linearly at a rate 10oC/min, how much error does
the thermometer show?
10) A photocell with a 35 ms time constant is used to measure light flashes. How long after a
sudden dark to light flash before the cell output is 80 % of the final value?
11) A pressure sensor has a resistance that changes with pressure according to
R = (0.15kΩ/psi) P + 2.5 kΩ
This resistance is then converted to a voltage with the transfer function
V = 10 R/(R+10 kΩ) volts
12) The sensor time constant is 350 ms. At t=0, the pressure P changes suddenly from 40 psi
to 150 psi. What is the voltage output at 0.5 seconds? What is the indicated pressure at this
time?
13) The time constant of the first order system means that the output is reached to ______
from its steady state value.
A) 98 %
B) 63 %
C) 50 %
14) A ______________ can oscillate for a step change of input.
A) First order system
B) Second order system
C) Dynamic system
15) The settling time for the first order system is equal to _______ of its time constant to
reach 95 % of steady state output.
A) Two
B) Three
C) Four
16) The 2nd order system with _______ roots has a response without overshoot.
A) Real
B) Complex
C) Real and negative
17) The second order system is highly under-damped when the damping factor is close to
______.
A) One
B) Zero
C) Negative value
18) As the damping factor increases, the overshoot ___________.
A) Decreases
B) Increases
C) Is stable
19) Increasing the negative real part of complex roots, __________ the system damping.
A) Decreasing
B) Increasing
C) Not change
20) As the imaginary part of complex roots increases, the maximum overshoot __________.
A) Decreases
B) Increases
C) Is stable
21) A stable system has all roots with __________ real parts.
A) Positive
B) Negative
C) Zero
22) What are the orders of the systems described by the following transfer functions:
1
G(s)=𝑚𝑠2 +𝑏𝑠+𝑘
1
G(s)=𝑅𝐶𝑠+1
1
G(s)=𝐿𝐶𝑠2 +𝑅𝐶𝑠+1
23) Identify the gain and time constant of the following first-order transfer function:
24) A RC circuit has the following transfer function:
For a step input u(t) = 2V for t 0:
a) What is the steady-state response of the circuit?
b) What is the time taken for the output of the RC circuit to reach 95% of its steady-state
response?
c) Check your result with XCOS
25) The general form of a first-order system is described by:
where T is the time constant, k is the gain, u(t) and y(t) are the input and output of the system
respectively. The unit-step response of a first-order system is shown in the following figure.
Determine the parameters k and T from this figure.
26) Consider the first-order system,
Obtain the unit-step response curves for T = 0.1, 0.5, 1.0, 5.0 and 10.0 respectively, with
XCOS.
27) Consider the first-order system,
Obtain the unit-step response curves for k = 0.1, 0.5, 1.0, 5.0, and 10.0 respectively, with
XCOS.
28) A general second-order system has the form:
What are the values of k, , and n for the following system:
29) A second-order system is described by the differential equation:
a) Write down the transfer function Y(s)/U(s) of the system, where U(s) and Y(s) are the
Laplace transforms of u(t) and y(t), respectively.
b) Obtain the damping ratio and the natural frequency n of the system.
c) Calculate the rise time and percent overshoot of the system.
d) Evaluate y(t) for a unit-step input u(t).
e) Check your answers of the above with XCOS.
30) This question is very difficult and recommended only for students who feel they require
an extra challenge
For the control system shown by the block diagram, the numerical value of J = 1 kg-m2 and
B=1 N-m/(rad/sec).
a) Find the transfer function Y(s)/R(s).
b) Determine the values of the gain K1 and velocity feedback constant K2 so that the
maximum
overshoot in the unit-step response is 0.2 and peak time is 1 sec.
c) With these values of K1 and K2, obtain the rise time and settling time.
d) Obtain the response y(t) for the unit-step input r(t).
e) Check the above calculations with XCOS.
31) When the second-order system
is subjected to a unit-step input, the system output responds as shown in the following figure.
Determine K and T from the response curve.
32) The figure below shows a block diagram of a space vehicle attitude control system where
R and Y are the Laplace transforms of the reference (or desired) and actual attitude angles
respectively. Determine the values of KP and KD to yield a settling time of 0.5 second and
20% overshoot in the close-loop system for a unit-step input.
Check your results using XCOS
33) Consider the second-order system
Obtain the unit-impulse response curves for = 0.1, 0.3, 0.5, 0.7, 1.0, and 4.0 respectively,
with XCOS.
34) Consider the second-order system
Assuming that n = 2, k = 2, obtain the unit-impulse response curves for = 0.1, 0.3, 0.5, 0.7,
1.0, and 4.0 respectively, with XCOS.