THAI NGUYEN UNIVERSITY OF TECHNOLOGY
FACULTY OF INTERNATIONAL TRAINING
System 1
(EE0010)
Assignments
Nguyen Tien Hung
TNUT
2020
Copyright © 2020 by Nguyen Tien Hung
Published by FIT
fit.tnut.edu.vn
First printing, December 2020
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1
Dynamic systems
2
Mechanical and electrical systems . . . . . . . . . . . . . 3
3
Input-output mathematical models . . . . . . . . . . . . . 5
3.1
Analytical solutions of system input-output equations
5
3.2
Numerical solutions of ordinary differential equations
6
4
System transfer functions . . . . . . . . . . . . . . . . . . . . . . . 7
4.1
Single-input single-output transfer functions
5
Frequency analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.1
Frequency-response transfer functions
5.2
Bode diagrams
10
5.3
Polar diagrams
10
1
7
9
6
Closed-loop systems and system stability . . . . . . 11
6.1
Algebraic stability criteria
11
6.2
Nyquist stability criterion
13
6.3
Root-locus method
13
7
Control systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.1
Steady-state control error
15
7.2
Industrial controllers
16
Preface
This is a collection of exercises for the EE0010 course. Please refer to [1] for
some of them.
Chapter 1. Introduction
1.1
Dynamic systems
Exercise 1.1.1 A motorized wheelchair uses a battery pack to supply two
dc motors that, in turn, drive the left and right wheels through a belt transmission. The wheelchair is controlled through a joystick that allows the
user to select forward and backward rotation of the wheels. The user accomplishes turning by running the two motors in opposite directions, thus
rotating the chair about a vertical axis. The battery voltage is Eb , the internal moving parts of the motors have rotational inertia (Ja ), and the electrical
coils of the motors have both resistance (Ra ) and inductance (La ). The belt
transmissions drive the wheels, which have rotational inertia (Jw ) through
a drive ratio given by the parameter N . The mass of the chair and rider
is given by mt and the grade is given by the angle γ. Is the energy stored
in the motor armatures, wheels, and chair (because of kinetic energy of the
inertia) independent? Why or why not?
Chapter 2. Mechanical and
electrical systems
Exercise 2.0.1 An automobile weighing 3000 lb is put in motion on a level
highway and then allowed to coast to rest. Its speed is measured at successive
increments of time, as recorded in the following table:
Time (s) Speed (ft/s, approx.)
0
15.2
10
10.1
20
6.6
30
4.5
40
2.9
50
2.1
60
1.25
a. Using a mass-damper model for this system, draw the system diagram
and set up the differential equation for the velocity v1 of the vehicle.
b. Evaluate m and then estimate b by using a graph of the data given in
the preceding table. (Hint: Use the slope dv1 /dt and v1 itself at any
time t).
Exercise 2.0.2 An electric motor has been disconnected from its electrical
driving circuit and set up to be driven mechanically by a variable-speed
electric hand drill mounted in a simple dynamometer arrangement. The
Chapter 2. Mechanical and electrical systems
4
torque versus speed data given in the following table were obtained. Then,
starting at a high speed, the motor was allowed to coast so that the speed
versus time data given in the table could be taken
Driven
Shaft speed
Torque
(rpm)
(Nm)
100
0.85
200
1.35
300
2.10
400
2.70
500
3.70
600
4.50
-
Time
(s)
0
10
20
30
40
50
60
Coasting
Shaft speed
(rpm)
600
395
270
180
110
70
40
a. Evaluate the Coulomb friction torque Tc and the linear rotational
damping coefficient B for the electric motor.
b. Draw a system model for the electric motor and set up the differential
equation for the shaft speed Ω1 during the coasting interval.
c. Estimate the rotational inertia J for the rotating parts of the electric
motor.
Chapter 3. Input-output mathematical models
3.1
Analytical solutions of system input-output equations
Exercise 3.1.1 A first-order model of a dynamic system is
2ẏ + 5y = 5f (t)
a. Find and sketch the response of this system to the unit step input
signal f (t) = us (t), for y(0) = 2.
b. Repeat part (3.1.1.a) for zero initial condition y(0) = 0.
c. Repeat part (3.1.1.a) for a unit impulse input f (t) = δ(t).
d. Repeat part (3.1.1.b) for a unit impulse input f (t) = δ(t).
Exercise 3.1.2 The roots of a second-order model are p1 = −1 + j and
p2 = −1 − j.
a. Find and sketch the system unit step response assuming zero initial
conditions, ẏ(0) = 0 and y(0) = 0.
b. Repeat part (3.1.2.a) for the roots of the characteristic equation p1 =
1 + j and p2 = 1 − j. Explain the major difference between the step
responses found in parts (3.1.2.a) and (3.1.2.b).
Exercise 3.1.3 A rotational mechanical system has been modeled by the
Chapter 3. Input-output mathematical models
6
equation
j Ω̇ + BΩ = T (t)
Determine the values of J and B for which the following conditions are
met:
ˆ Steady-state rotational velocity for a constant torque, T (t) = 10 Nm,
is 50 rpm (revolutions per minute).
ˆ The speed drops below 5%of its steady-state value within 160 ms after
the input torque is removed, T (t) = 0.
Exercise 3.1.4 Output voltage signals y( t) of a linear first-order electrical
circuits was measured, as shown in Figs. P4.4(a). Write analytical expressions describing the signal.
12
4.416
0
0.2
0.4
0.6
0.8
Figure 3.1: Output voltage signal
Exercise 3.1.5 A mass m = 1.5 lbs2 /ft sliding on a fixed guideway is
subjected to a suddenly applied constant force F (t) = 100lb at time t =
0. The coefficient of linear friction between the mass and the guideway is
b = 300 lb s/ft. Find the system time constant. Write the system model
equation and solve it for the response of mass velocity v as a function of
time, assuming v(0) = 0. Sketch and label the system response versus time.
3.2
Numerical solutions of ordinary differential equations
Exercise 3.2.1 Another well-known nonlinear equation is the Van der Poll
equation, a second-order nonlinear equation that results in an oscillatory
response for some values of the parameters:
d2 y
dy
+x=0
− µ(1 − y 2 )
dt2
dt
Write a script to solve this equation by using the Runge-Kutta method.
Use µ = 3 and experiment with a wide variety of initial conditions for both
y and dy/dt.
Chapter 4. System transfer
functions
4.1
Single-input single-output transfer functions
Exercise 4.1.1 Consider a tracking control system shown in the below
figure
+
Controller
Process
Figure 4.1: Tracking control system
a. Find the closed-loop transfer function relating the output Y (s) to the
input R(s).
b. Compare the poles of the closed-loop transfer function with those of
the system for which U (s) is the input and Y (s) is the output. What
do you conclude?
Exercise 4.1.2 The transfer function block diagram of a system designed
to control the liquid level in a chemical process is shown in the below figure.
In this diagram, Kp is an adjustable gain that can be set by the machine
Chapter 4. System transfer functions
8
operator.
+
Controller
Process
Figure 4.2: Liquid level control system
a. Derive the closed-loop transfer function relating R(s) to Y (s).
b. How does the value of Kp affect the locations of the closed-loop poles
relative to the poles of the open-loop system?
Chapter 5. Frequency analysis
5.1
Frequency-response transfer functions
Exercise 5.1.1 A given system input-output equation is
a2
d2 y
dy
d2 u
du
+ a1
+ a0 y = b2 2 + b1
+ b0 u
2
dt
dt
dt
dt
Find the expression for the output y(t) when the input is a sinusoidal
function of time, u(t) = U sin Ωf t with ωf = (a0 /a2 )0.5 .
Exercise 5.1.2 A closed-loop system consisting of a process of transfer
function Tp (s) and a controller of transfer function Tc (s) has been modeled
as shown by the transfer function block diagram in the below figure, where
+
Controller
Plant
Figure 5.1: A closed-loop system
Chapter 5. Frequency analysis
10
Tp (s) =
5
,
5s + 1
Tc (s) =
2(s + 1)
2s + 1
a. Find the system closed-loop transfer function Tcl (s).
b. Find the amplitude and the phase angle of the output signal of the
closed-loop system, y(t), when u(t) = 0.2 sin 3t.
Exercise 5.1.3 A hot-water storage tank has been modeled by the equation
8000
dTw
= 3Ta − 3Tw
dt
where Tw is the temperature of the water and Ta is the temperature of the
ambient air. For several days, the ambient air temperature has been varying
in a sinusoidal fashion from a maximum of 10o C at noon to a minimum
of –10o C at midnight of each day. Determine the maximum and minimum
temperatures of the water in the storage tank during those days and find
at what times of the day the maximum and minimum temperatures have
occurred.
5.2
Bode diagrams
Exercise 5.2.1 Sketch asymptotic Bode diagrams for the following transfer
functions:
5s + 1
a. T (s) =
.
(s + 1)(s + 2)
5s + 1
b. T (s) =
.
s(s + 1)(s + 2)
5.3
Polar diagrams
Exercise 5.3.1 Use Matlab’s nyquist commands to plot polar diagrams for
the following transfer functions:
1
a. T (s) =
s(5s + 1)
10s
b. T (s) =
2s + 1
6
c. T (s) =
(5s + 1)2
Chapter 6. Closed-loop systems
and system stability
6.1
Algebraic stability criteria
Exercise 6.1.1 A closed-loop transfer function of a dynamic system is
s + 10
+ 10s3 + 20s2 + s + 1
Use the Hurwitz criterion to determine the stability of this system.
Tcl (s) =
10s4
Exercise 6.1.2 The transfer functions of the system represented by the
block diagram shown in the below figure are
+
-
Figure 6.1: Closed-loop control system
G(s) =
2s + 1
3s3 + 2s2 + s + 1
Chapter 6. Closed-loop systems and system stability
12
H(s) = 10
a. Determine the stability of the open-loop system.
b. Determine the stability of the closed-loop system.
Exercise 6.1.3 Figure 6.2 shows a block diagram of a control system. The
transfer functions of the controller Tc (s) and of the controlled process Tp (s)
are
+
Controller
Process
Figure 6.2: Block diagram of a control system
1
Tc (s) = K 1 +
4s
Tp (s) =
5
100s2 + 20s + 1
Using the Hurwitz criterion, determine the stability conditions for the
open-loop and closed-loop systems in terms of the controller gain K.
Exercise 6.1.4 The transfer functions of the system shown in Figure 6.2
are
Tc (s) = K
Tp (s) =
2
s(τ s + 1)2
Determine the stability condition for the closed-loop system in terms of
the controller gain K and the process time constant. Show the area of the
system stability in the (τ, K) coordinate system.
Exercise 6.1.5 The transfer function block diagram of a system designed
to control the liquid level in a chemical process is shown in figure 6.3 (see
Exercise 4.1.2). In this diagram, Kp is an adjustable gain that can be set by
the machine operator.
+
Controller
Process
Figure 6.3: Liquid level control system
6.2 Nyquist stability criterion
13
Use the Routh criterion to find the limits for Kp to ensure the stability
of the closed-loop system.
Exercise 6.1.6 The block diagram for a control system has been developed
as shown in Figure 6.4. The system parameters are
+
-
Figure 6.4: Block diagram of a feedback control system
Kc
=
3.0v/v,
Kp = 4.6m/V
τi
=
3.5s,
τp = 1.4s
τc
=
0.1s,
Kf = 1.0V/m
a. Determine whether the system is stable or unstable.
b. If the system is stable, find the stability gain and phase margins.
6.2
Nyquist stability criterion
Exercise 6.2.1 The transfer functions of the system shown in Figure 6.2
are
Tc (s) = K
Tp (s) =
1
s(0.2s + 1)(0.008s + 1)
Sketch the polar plot of the open-loop system Tc Tp , and determine the
stability condition for the closed-loop system in terms of the controller gain
K by using the Nyquist criterion.
6.3
Root-locus method
Exercise 6.3.1 Consider the feedback system represented by the block diagram shown in Figure 6.2 with the following transfer functions:
Tc (s) = K
14
Chapter 6. Closed-loop systems and system stability
Tp (s) =
10
(s + 5)(s + 0.2)
a. Construct the root locus for this system.
b. Determine the locations of the roots of the system characteristic equation required for 20 percent overshoot in the system step response. Find
the value of K necessary for the roots to be at the desired locations.
What will be the period of damped oscillations Td in the system step
response?
Exercise 6.3.2 The transfer functions for the system represented by the
block diagram shown in Figure 6.2 are
Tc (s) = K
Tp (s) =
1
(s + 1)(s + 2)(s + 5)
a. Construct the root locus for this system.
b. Use the constructed root locus to determine the value of K for which
the closed-loop system is marginally stable.
Chapter 7. Control systems
7.1
Steady-state control error
Exercise 7.1.1 The open-loop transfer function of a system was found to
be
K
Tol (s) =
(s + 5)(s + 2)2
Determine the range of K for which the closed-loop system meets the
following performance requirements:
ˆ the steady-state error for a unit step input is less than 10% of the input
signal, and
ˆ the system is stable.
Exercise 7.1.2 A simplified block diagram of the engine speed control system known as a flyball governor, invented by JamesWatt in the 18th century,
is shown in Figure 7.1
a. Develop the closed-loop transfer function Tcl (s) = Ω0 (s)/Ωd (s) and
write the differential equation relating the actual speed of the engine
Ω0 (t) to the desired speed Ωd (t) in the time domain.
b. Find the gain of the hydraulic servo necessary for the steady-state
value of the error in the system, e(t) = Ω0 (t) − Ωd (t), to be less than
1% of the magnitude of the step input.
Chapter 7. Control systems
16
Flyball
+
Servo
Steam engine
-
Figure 7.1: Block diagram of steam engine speed control system
Exercise 7.1.3 A system open-loop transfer function is
Tol (s) =
K
s2 (τ s + 1)
Find the steady-state control error in the closed-loop system subjected
to input u(t) = t2 . Express the steady-state error in terms of the staticacceleration error coefficient Ka , defined as
Ka = lim s2 Tol (s)
s→0
7.2
Industrial controllers
Exercise 7.2.1 The block diagram of the control system developed for a
thermal process is shown in Figure 7.2.
+
-
Figure 7.2: Block diagram of temperature control system
a. Determine the gain of the proportional controller Kp necessary for the
stability gain margin Kg = 1.2.
b. Find the steady-state control error in the system when the input temperature changes suddenly by 10o C, r(t) = 10us (t) by using the value
of the proportional gain obtained from the stability requirement in part
7.2.1.a.
Exercise 7.2.2 The block diagram of a control system is shown in Figure
7.3. The process transfer function, Tp (s), is
Tp (s) =
1
10s + 1
Exercises
17
+
-
Figure 7.3: Block diagram of a control system
Compare the performance of the control system with proportional and
PI controllers. The controller transfer functions are
Tc (s) = 9
for the proportional controller and
1
Tc (s) = 9 1 +
1.8s
for the PI controller. In particular, compare the percentage of overshoot of
the step responses and the steady-state errors for a unit step input obtained
with the two controllers.
Exercise 7.2.3 The process transfer function of the control system shown
in Figure 7.3 has been found to be
Tp (s) =
K
(τ1 s + 1)(τ2 s + 1)
Compare the damping ratios and the steady-state errors for a step input
obtained in this system with proportional and PD controllers. The controller
transfer functions are
Tc (s) = Kp
for the proportional controller and
Tc (s) = Kp (1 + Td s)
for the ideal PD controller.
Bibliography
[1] B. T. Kulakowski, J. F. Gardner, and J. L. Shearer. Dynamic Modeling
and Control of Engineering Systems. Cambridge University Press, 2007.