‫ق الوا عن ثورة ‪ 25‬يناير‬
‫وزير الخارجية األلمانى [فستر فيله]‪:‬‬
‫“أتطلع إلى زيارة مصر والحديث مع الذين ق اموا بالثورة”‪.‬‬
‫سيلفيو برلسكوني رئيس وزاراء إيطاليا‪:‬‬
‫“ال جديد في مصر فقد صنع المصريون التاريخ كالعادة”‬
‫ستولتنبرج رئيس وزراء النرويج ‪:‬‬
‫“اليوم كلنا مصريين”‬
‫ق الوا عن ثورة ‪ 25‬يناير‬
‫هاينز فيشر رئيس النمسا ‪:‬‬
‫“شعب مصر أعظم شعوب األرض و يستحق جائزة نوبل للسالم”‬
‫الرئيس األمريكي باراك أوباما ‪:‬‬
‫“ي جب أن نربي أب ناءنا ليصبحوا كشباب مصر”‬
‫شبكة ‪ CNN‬اإلخبارية‪:‬‬
‫“ألول مرة نرى شعبا يقوم بثورة ثم ينظف الشوارع بعده”‪.‬‬
Course Name :-
Lecture 1:
Course organization and
introduction to feedback control
Feb. 2012
Instructor: Mohamed Sayed Bayoumi
A. Professor
Aerospace Engineering
Cairo University
Course Description
• This course is concerned with both analysis and
design of feedback linear control systems
– Analysis:
System Modeling,
(mech.-Translational-Rotational), Elect.,
Fluid, Thermal system.
Response (Partial function,
Sensitivity, and Stability
-Design: PID controller
- Implementation: Op-Amp. & Pneumatic.
Reading material
• Text book:
– Charles L. Phillips and Royce D. Harbor, Feedback
Control Systems,
– Van de vegte, Feedback Control Systems,
– Nice, Feedback Control Systems,
– Ogata, Feedback Control Systems,
– Harison, Introduction to Control Systems,
– Raven, Introduction to Control Systems
– R.C. Dorf and R.H. Bishop, Modern Control Systems,
11th Edition, Prentice Hall, 2008,
Course Evaluation
• Homework: 5% (late homework will not be accepted)
• Quiz I: 3%
• Mid-Term: 20%
• Class participation: 2%
• Final exam: 70%
Disturbance
(Noise)
Input R(t)
Reference
desired output
uk
+
uact
Controller
(+)
Actuator
Control
signal
Process
Actuating
signal
Feedback signal b(t)
measurement
Output c(t)
(actual outpu)
A modern Feedback Control
System
Figure 2.1: WATT’S SPEED GOVERNOR
Figure 2.3: MISSILE LAUNCHING AND GUIDANCE SYSTEM
A design example : Open loop
A design example
Closed loop
What is a control system?
• Generally speaking, a control system is a system that is
used to realize a desired output or objective
• Open-loop control systems
Chapter 1 Introduction
1.2.3 Fundamental structure of control systems
1) Open loop control systems
Disturbance
(Noise)
Input r(t)
Reference
desired output
uk
Controller
uact
Actuator
Control
signal
Process
Output c(t)
(actual output)
Actuating
signal
Fig1.10 .
Features: Only there is a forward action from the input to the
output.
Chapter 1 Introduction
Notes: 1) Positive feedback; 2) Negative feedback—Feedback.
1.3 types of control systems
1) linear systems versus Nonlinear systems.
2) Time-invariant systems vs. Time-varying systems.
3) Continuous systems vs. Discrete (data) systems.
4) Constant input modulation vs. Servo control systems.
1.4 Basic performance requirements of control systems
1) Stability.
2) Accuracy (steady state performance).
3) Rapidness (instantaneous characteristic).
– Closed-loop control systems
(this is what we are most interested in for this course)
• Definition of a closed-loop (or feedback) control system
– Plant: part of the system to be controlled
– Sensor: used for the measurement of a variable
– Controller (or compensator): used to obtain satisfactory
characteristics for the total system
Chapter 1 Introduction
2) Closed loop (feedback) control
systems
Input r)t(
Reference
desired output
uk
+
uact
Controller
(+)
Disturbance
)Noise(
Actuator
Control
signal
Process
Output c)t(
)actual output(
Actuating
signal
Feedback signal b)t(
measurement
Fig1.11 .
Features:
not only there is a forward action , also a backward action
between the output and the input (measuring the output and
comparing it with the input).
1) measuring the output (controlled variable) . 2) Feedback.
Advantages/Disadvantages
Open-Loop Systems
Simple
Inexpensive
Cannot correct for
disturbances or plant
variations
Closed-Loop Systems
Complex &
expensive
Less sensitive to noise,
disturbances, plant
variations
Better control of transient
steady-state response
Better accuracy
Self-sustained oscillations
possible
Chapter 1 Introduction
1. Establish control
goals
6. Describe a controller and select
key parameters to be adjusted
2. Identify the variables to control
3. Write the specifications
for the variables
7. Optimize the parameters and
analyze the performance
Performance
meet the
specifications
Performance does not
4. Establish the system configuration Meet the specifications
Identify the actuator
Finalize the design
5. Obtain a model of the process,
the actuator and the sensor
Fig.1.12
• Advantages of feedback
– Feedback allows high performance in the presence
of uncertainty
– Feedback allows the dynamics of a system to be
modified
• One major disadvantage of feedback
– It may create instability
Lecture 2:
Mathematical foundation and system modeling
Outline of this lecture
• Mathematical foundation
– Complex variables
– Differential equations
– Laplace transform
• System modeling
– Definition of mathematical model
– Definition of linear system
– Transfer functions
System modeling
• Definition of mathematical model:
– Mathematical relationships that relate the output of a
system to its input
– It should be understood that no mathematical model of a
physical system is exact
– We generally strive to develop a model that is adequate for
the problem at hand without making the model overly
complex
• Definition of linear system:
– A system is linear if superposition
applies
Definition
• Transforms -- a mathematical
conversion from one way of thinking to
another to make a problem easier to
solve
problem
in original
way of
thinking
transform
solution
in transform
way of
thinking
2. Transforms
solution
in original
way of
thinking
inverse
transform
problem
in time
domain
Laplace
transform
solution
in
s domain
• Other transforms
• Fourier
• z-transform
• wavelets
2. Transforms
inverse
Laplace
transform
solution
in time
domain
A correction
• About the differential theorem of Laplace transform
– An example: to calculate L[du(t)/dt]
• The inverse Laplace transform is given by
– Mechanical translational systems
X(t)
K
kx (t )
m
Mass Spring System
X(t)
m
f(t)
f(t)
Free body
diagram
mx&&(t ) = -kx(t )
+ f(t)
mx&&(t ) +kx(t ) = 0
+ f(t)
ms2 X ( s ) + kX ( s ) = 0 + F(s)
( ms + k ) X ( s ) = 0 + F(s)
2
X ( s)
F ( s)
=
m x& &( t )
1
= Transfer function
2
ms +k
Static balance
K
kd
ky (t )
d
m
f(t)
f (t)
my&&(t )
m&y&(t ) = mg - k{d + y (t )} + f (t )
m&y&(t ) = -ky(t ) + f (t )
y(t)
mg
mg = kd
One degree of freedom
Forced Vibration
K
C
m
f(t)
Cx& (t )
K X(t)
m
X(t)
m&x&(t ) = -Cx& (t ) - kx(t ) + f (t )
X(t)
f(t)
Free body
diagram
Forced vibration
m&x&(t ) + Cx& (t ) + kx(t ) = f (t )
ms X ( s) + CsX ( s) + kX ( s) = F ( s)
2
X ( s)
1
=
= T .F .
2
F ( s ) ms + Cs = k
Two degree of
freedom
c1
k1
m
m11
f1
k2
C2
x1
m22
x1  x2
f2
x2  x1
C1 x&1
k1 x1
k1 x1
m1
m1
x1
f1
k2 ( x1 - x2 )
C2 ( x&1 - x&2 )
f1
x1
C2 ( x&2 - x&1 )
k2 ( x2 - x1 )
m2
f2
C1 x&1
m2
x2
f2
x2
x2
x1  x2
m1&x&1 = -k1 x1 - C1x&1 - k2 ( x1 - x2 ) - C2 ( x&1 - x&2 ) + f1
m1&x&1 + (C1 + C2 ) x&1 + (k1 + k2 ) x1 = f1 + C2 x&2 + k2 x2
m2 &x&2 = k2 ( x1 - x2 ) + C2 ( x&1 - x&2 ) + f 2
m2 &x&2 + C2 x&2 + k2 x2 = f 2 + C2 x&1 + k2 x1
k1 x1
C1 x&1
m1
x1
f1
k2 ( x1 - x2 ) C2 ( x&1 - x&2 )
m2
f2
x2
x2  x1
m1&x&1 = -C1 x&1 - k1x1 + C2 ( x&2 - x&1 ) + k2 ( x2 - x1 ) + f1
k1 x1
m1&x&1 + (C1 + C2 ) x&1 + (k1 + k2 ) x1 = f1 + C2 x&2 + k2 x2
C1 x&1
m1
f1
m2 &x&2 = -C2 ( x&1 - x&2 ) - k2 ( x1 - x2 ) + f 2
k2 ( x2 - x1 )
m2 &x&2 + C2 x&2 + k2 x2 = f 2 + C2 x&1 + k2 x1
x1
C2 ( x&2 - x&1 )
m2
f2
x2
x1  x2
k1 x1
x2  x1
C1 x&1
k1 x1
x1
f1
k2 ( x1 - x2 )
C2 ( x&1 - x&2 )
f1
k2 ( x2 - x1 )
m1
x2
f2
x1
f1
x1
k2 ( x1 - x2 )
C2 ( x&1 - x&2 )
k2 ( x2 - x1 )
C2 ( x&2 - x&1 )
C2 ( x&2 - x&1 )
m2
m2
f2
C1 x&1
m1
m1
C1 x&1
k1 x1
x2
m2
f2
x2
k1 x1
m1&x&1 = -k1 x1 - C1x&1 - k2 ( x1 - x2 ) - C2 ( x&1 - x&2 ) + f1
m1&x&1 + (C1 + C2 ) x&1 + (k1 + k2 ) x1 = f1 + C2 x&2 + k2 x2
m2 &x&2 = -C2 ( x&2 - x&1 ) - k2 ( x2 - x1 ) + f 2
C1 x&1
m1
x1
f1
k2 ( x1 - x2 )
k2 ( x2 - x1 )
m2 &x&2 + C2 x&2 + k2 x2 = f 2 + C2 x&1 + k2 x1
C2 ( x&1 - x&2 )
C2 ( x&2 - x&1 )
m2
f2
x2
Ta( t) - Ts( t)
Ta( t)
( t )
Ta( t)
0
Ts ( t )
s( t ) - a( t )
= through - variable
angular rate difference = across-variable
Gear Ratio = n = N1/N2
N2  L
N1  m
L
n  m
L
n  m
x
r 
converts radial motion to linear motion
System with Gears
Power = constant
T11 = T22
1
2 = n
b
x1
x1*a =x2*b
x2
a
e
y
x1
a
x2
b
Figure 2.27
A gear system
Figure 2.31
Gear train
Figure 2.30
a. Rotational mechanical system with
gears;
b. system after reflection of torques
and impedances to the output shaft;
c. block diagram
Motor shaft
Output shaft
Table 2.3
Voltage-current, voltage-charge, and
impedance relationships for capacitors,
resistors, and inductors
Mathematical models of electrical systems
R
RC network
v1(t)
dv2
i (t ) = C
dt
v1 (t ) - v2 (t )
i (t ) =
R
dv2 v1 (t ) - v2 (t )
C
=
dt
R
dv2
RC
+ v2 (t ) = v1 (t )
dt
i(t)
C
v2(t)
ei
eo
ei
eo
V2( s )
R2
R2
V1( s )
R
R1 + R2
R2

R
 max

V2( s )
ks  1( s ) -  2( s )
V2( s )
ks  error( s )
ks
Vbattery
 max

Figure 2.9
Three-loop
electrical network