Department of Electronics Telecommunications and
Informatics
University of Aveiro
Subject: Sistemas de Controlo II
(Control Systems II)
2014/2015
Lecture 1 (12/Feb 2015)
Petia Georgieva (petia@ua.pt)
SC2 2014/2015
1
SYLLABUS
• 0. Introduction and historical perspectives
Continuous (analog) control systems - Petia Georgieva
Chapter 1. Review of the main topics of control system theory (SC1)
(chapters I-VII of the book of Prof. Melo)
Chapter 2. Basic compensators (chapter VIII)
Chapter 3. PID compensator (chapter VIII)
Chapter 4. State-feedback control (chapters X, XI)
Digital control systems - Telmo Cunha
• Chapter 5. Design of discrete-time controllers through emulation
• Chapter 6. Controller design in the discrete time
• Chapter 7. Direct digital control
• Chapter 8. Identification of discrete-time models through the least
squares technique
• Chapter 9. RST controllers – polynomial approach
2
0. Introduction
• System – An interconnection of elements and devices
for a desired purpose.
• Control System – An interconnection of elements and
devices that will provide a desired response.
• The interaction is defined in terms of
i. System input
ii. System output
iii. Disturbances
3
Historical Perspectives of Control Systems
1. Ancient Greece (1 to 300 BC)
- water float regulation, water clock, automatic oil lamp
2. Cornellis Drebbel (17th century)
- Temperature control
3. Industrial revolution (18th century) – development of
steam engine. How to control the speed of rotation of the
engine without human intervention ?
4. 18th Century (1769 ?) James Watt’s flyball governor
for the speed control of a steam engine (the first system
for automatic control of a machine).
Watt’s Flyball Governor (speed limiter)
Historical Perspectives of Control Systems
Laplace (1749-1827) and Fourier (1758-1830) – developed essential
mathematical framework for theoretical analysis.
Maxwell (1868) in his paper “On governors” developed the differential eq.
model of the governor , linearized around an equilibrium point and proved
that the stability of the system depends on the roots of a characteristics
equation having negative real parts.
Hurwitz (1875) and Routh (1905) - developed stability criteria for linear
systems.
Lyapunov (1983), russian mathematician, developed stability criteria for
non-linear systems.
Bell Telephone Labs (1930) - work on feedback amplifier design based on
the concept of frequency response and backed by the maths of complex
variables. Nyquist (1932) developed stability criteria using frequency
domain methods.
Bode (1945) and Nickols extended it to the most comonly used control
system design in the frequncy domain.
Evans (1948) –based on the work of Maxwell and Routh, Evans developed
the Root Locus method to display grafically the roots of the characteristic eq.
SC1 (Routh- Hurwitz table & Root Locus method)
Routh- Hurwitz table
Root Locus method ( Matlab rlocus)
Open-Loop Control System
Single Input-Single Output (SISO)
Example of Open Loop Control System: Missile Launcher
Example of Open Loop Control System :
control of the speed of a turntable
Closed-Loop Control System
Single Input-Single Output (SISO)
Has a negative feedback to compare the actual output to the desired
output response.
Disturbance
Set-point or
Reference
input
Controlled
Signal
Error
+
+
Controller
Manipulated
Variable
Actuator
-
Feedback Signal
Sensor
+
+
Actual
Output
+
Process
Example of a SISO Closed Loop Control
System (negative feedback)
Missile Launcher System
Example of a Closed Loop Control System
model of the national income
Example of a Closed Loop Control
System of the speed of a turntable
Multi Input Multi Output (MIMO)
Closed Loop Control System
Desired
Output
Response
Controller
Process
Measurement
Output
Variables
Example of MIMO Closed Loop Control System Boiler Generator
Example of a Modern MIMO Control System
The Future of Control Systems
CAMBADA IEETA ROBOTICS – part of the future control systems
2008: World champion of middle size robocup
2009-2015: Leading position in all national&intern. competitions
18
Chapter 1:Review of the main topics of control
system theory (SC1)
1.
2.
3.
4.
5.
6.
7.
Linear system modeling
Time (transient) response of 1st and 2nd order systems
Stability (Routh-Hurwitz)
Control architectures (open-loop; closed-loop)
Negative feedback systems
Root-locus
Steady-state response (steady-state error)
20
What is a Mathematical Model?
A set of mathematical equations (e.g., differential equations)
that describes the input-output behavior of a system.
What is a model used for?
•System Simulation
•Prediction/Forecasting
•Control System Design
•Design/Performance Evaluation
Dynamic Systems
A system is said to be dynamic if its current output may depend on the past
history as well as the present values of the input variables. Mathematically,
y ( t )   [ u (  ), 0    t ]
u : Input, t : Time
Example: A moving mass
y
Model: Force=Mass x Acceleration
u
M
  u
My
Example of a Dynamic System
Velocity-Force:
v ( t )  y ( t )  y ( 0 ) 
1
M
t
 u ( ) d 
0
Position-Force:
y ( t )  y ( 0 )  t y ( 0 ) 
1
M
t t1
  u ( ) d  dt 1
0 0
Therefore, this is a dynamic system.
If the torque force (bdy/dt) is included, then
M y  b y  u
Homework –find the models (SC1 examples)
24
Mathematical Modeling Basics
• Mathematical model of a real world system is
derived using a combination of physical laws (1st
principles) and/or experimental tools.
• Physical laws are used to determine the model
structure (linear or nonlinear) and order.
• The parameters of the model are often estimated
and/or validated experimentally.
• Mathematical model of a dynamic system is often
expressed as a system of differential (difference in
the case of discrete-time systems) equations.
Mathematical Modeling Basics
•
A nonlinear model is often linearized about a
certain operating point.
•
Model reduction (or approximation) may be needed.
•
Numerical values of the model parameters are
often approximated from experimental data by
curve fitting.
Example: Accelerometer
Consider the mass-spring-damper (may be used as
accelerometer or seismograph) system:
u
x
M
fs(y): the spring reaction force; nonlinear function of y=x-u
fd(y): the torque reaction force; nonlinear function of y=x-u
Newton’s 2nd law
Linearized model:
Mx  M
y  u 
 fd ( y)  fs ( y)
M y  b y  ky  M u
Transfer Function (TF)
Transfer Function is the algebraic input-output relationship of a linear
time-invariant system in s domain.
U
G(s)
Y
Example: Accelerometer System
M y  b y  ky  u  G ( s ) 
Y (s)
U (s)

Ms
2
M s  bs  k
2
,s 
d
dt
Comments on TF
• Transfer Function is a property of the system
independent from the input-output signals.
• It is an algebraic representation of differential
equations applying the Laplace Transformation.
• Systems from different disciplines (e.g., mechanical,
chemical, electrical) may have the same transfer
function
Mixed Systems
• Most real world systems (processes, plants) are of the
mixed type, e.g., electromechanical,
hydromechanical, etc
• Each subsystem within a mixed system can be
modeled as single discipline system first.
• The subsystems are integrated into the entire system
applying for example the block diagram rools.
• Overall mathematical model may be assembled into a
system of equations, or a transfer function.
Electro-Mechanical Example
Input: voltage u
Output: Angular velocity 
Ra
u
La
ia
Elecrical Subsystem (loop method):
u  R a i a  La
dia
dt
 eb ,
Mechanical Subsystem
e b - e m f v o lta g e
T motor  J ω  B ω
B
dc

J
Electro-Mechanical Example
Power Transformation:
Torque-Current:
Voltage-Speed:
Ra
B
T motor  K t i a
e b  K bω
La
u
ia
dc
where Kt: torque constant, Kb: velocity constant
For an ideal motor K t  K b
Combing previous equations results in the
following mathematical model:
di a

L
 R a i a  K b  u
 a
dt

 J ω  B ω - K i  0
t a


Transfer Function
of Electromechanical Example
Taking Laplace transform of the system’s differential
equations with zero initial conditions gives:
Ra
 L a s  R a I a ( s )  K b W ( s )  U ( s )

 Js  B W (s) - K t I a ( s )  0
La
B
ia
u
Kt

Eliminating Ia yields the input-output transfer function
of 2nd order system.
W(s)
U(s)

L a Js
2
 JR
Kt
a
 BL a   BR
a
 KtKb
Reduced Order Model
Assuming small inductance, La 0 we get a
transfer function of 1st order system.
Ω (s )
U (s )
K 
T 

K t
Ra 
Js  B  K t K b Ra 
K t
Ra 
B  Kt Kb Ra
J
B  Kt Kb Ra

K
Ts  1
 system gain
 tim e constant
,
What is the Control System Engineer
trying to achieve?
• First, to understand well the application in order to
apply a suitable control system.
• A good control system has to
– generate a response quickly and without strong
oscillations (good transient response),
– have low error once settled (good steady-state
response),
– and will not oscillate wildly or damage that
system (stability).
35
Summary
The central problem in control is to find
a technically feasible way to act on a
given process so that the process
behaves, as closely as possible, to
some desired behavior. Furthermore,
this approximate behavior should be
achieved in the face of uncertainty of
the process and in the presence of
uncontrollable external disturbances
acting on the process.
36
BIBLIOGRAPHY
(in Portugues)
- 37 -
BIBLIOGRAPHY
- 38 -