AMSTERDAM
UNI ERSIT OF
APPLIED SCIENCES
ACA C A 2019-2020
A ED
A
FLIGH
A ED
FLIGH
A
A
, AMSTERDAM UNIVERSITY OF APPLIED SCIENCES
Course Overview
Week Subject
1 General introduction in automated flight, basics in complex numbers.
2 Historical developments; Laplace transform.
3 Control Theory; Laplace transform and Partial fraction expansion.
4 Control Theory; Partial Fraction expansion, mass spring system.
5 Control Theory; Block Diagrams and Stability
6 Control Theory; Stability, Poles, System Response and Block Diagrams
7 Control Theory: Proportional Control and Root Locus
8 Control Theory: PID Controllers
AUTOMATED FLIGHT SUMMARY 2
, AMSTERDAM UNIVERSITY OF APPLIED SCIENCES
Week 1: General Introduction and basics in Complex Numbers
Control Theory
System A set of connected things or devices that operate together.
input C System output
Summing Point
feedback Branch Point
= Closed-Loop System (+feedback)
= Open-Loop System
Closed-Loop A system that utilizes a measurement of the output and makes a comparison with
the desired output.
Open-Loop A system that uses a device to control the process without using feedback. The
output of the system does not affect the signal necessary to control the system.
Feedback A signal from the output of the system that is applied as feedback to control the
system.
Feedback control can be:
- Positive Feedback; Regenerative Feedback
Output signal is fed back, so it adds to the input signal.
- In phase with input;
- Increase oscillatory motion.
- Negative Feedback; Degenerative Feedback
Output signal is fed back, so it subtracts to the input signal.
- Out of phase with input;
- Improves stability.
When feedback is used in a system, a controller is needed to rectify the input [ C ].
Summing Point where signals come together.
Point
Branch Point Point where signals divert.
SISO Single Input – Single Output
SIMO Single Input – Multiple Output
MISO Multiple Input – Single Output
MIMO Multiple Input – Multiple Output
AUTOMATED FLIGHT SUMMARY 3
, AMSTERDAM UNIVERSITY OF APPLIED SCIENCES
Complex Numbers
A complex number is a number that can be expressed in the form a + bj, where a and b are real
numbers. For the complex number a + bi, a is called the real part, and b is called the imaginary part.
Cartesian notation
Complex numbers expressed in their Cartesian form are written as:
𝑧 = a + bj
Where a is the real part and b the imaginary part.
Complex Conjugate
The conjugate of a complex number is the expressed by changing the sign + or – of the
imaginary from + to –, or vice versa.
If z = a + bj, its complex conjugate, denoted by 𝑧, is
𝑧 = a − bj
Operations
If 𝑧1 = 𝑎1 + 𝑏1 j and 𝑧2 = 𝑎2 + 𝑏2 j
Sum Adding two complex numbers together;
𝑧1 + 𝑧2 = 𝑎1 + 𝑎2 + (𝑏1 + 𝑏2 )j
Multiply Multiplying two complex numbers;
𝑧1 ∙ 𝑧2 = 𝑎1 ∙ 𝑎2 − 𝑏1 ∙ 𝑏2 + (𝑎1 ∙ 𝑏2 + 𝑎2 ∙ 𝑏1 )j
𝑧1 ∙ 𝑧1 = 𝑎2 + 𝑏2
Divide Dividing two complex numbers;
𝑧1
𝑧2
= 𝑎(𝑎1 ∙2𝑎)22 ++ 𝑏(𝑏1 ∙2𝑏)22 + (−𝑎(𝑎12∙ )𝑏22 + 𝑎2 ∙ 𝑏1 )
+ (𝑏2 )2
j
To multiply and divide complex numbers it is easier to re-write the cartesian complex number
to its Polar Form.
Argand Diagram
The complex number z = a+bj can be plotted as a
point with coordinates (a, b). Because the real part of
z is plotted on the horizontal axis, we often refer to
this as the real axis. The imaginary part of z is plotted
on the vertical axis and so we refer to this as the
imaginary axis. Such a diagram is called an Argand
diagram.
AUTOMATED FLIGHT SUMMARY 4
UNI ERSIT OF
APPLIED SCIENCES
ACA C A 2019-2020
A ED
A
FLIGH
A ED
FLIGH
A
A
, AMSTERDAM UNIVERSITY OF APPLIED SCIENCES
Course Overview
Week Subject
1 General introduction in automated flight, basics in complex numbers.
2 Historical developments; Laplace transform.
3 Control Theory; Laplace transform and Partial fraction expansion.
4 Control Theory; Partial Fraction expansion, mass spring system.
5 Control Theory; Block Diagrams and Stability
6 Control Theory; Stability, Poles, System Response and Block Diagrams
7 Control Theory: Proportional Control and Root Locus
8 Control Theory: PID Controllers
AUTOMATED FLIGHT SUMMARY 2
, AMSTERDAM UNIVERSITY OF APPLIED SCIENCES
Week 1: General Introduction and basics in Complex Numbers
Control Theory
System A set of connected things or devices that operate together.
input C System output
Summing Point
feedback Branch Point
= Closed-Loop System (+feedback)
= Open-Loop System
Closed-Loop A system that utilizes a measurement of the output and makes a comparison with
the desired output.
Open-Loop A system that uses a device to control the process without using feedback. The
output of the system does not affect the signal necessary to control the system.
Feedback A signal from the output of the system that is applied as feedback to control the
system.
Feedback control can be:
- Positive Feedback; Regenerative Feedback
Output signal is fed back, so it adds to the input signal.
- In phase with input;
- Increase oscillatory motion.
- Negative Feedback; Degenerative Feedback
Output signal is fed back, so it subtracts to the input signal.
- Out of phase with input;
- Improves stability.
When feedback is used in a system, a controller is needed to rectify the input [ C ].
Summing Point where signals come together.
Point
Branch Point Point where signals divert.
SISO Single Input – Single Output
SIMO Single Input – Multiple Output
MISO Multiple Input – Single Output
MIMO Multiple Input – Multiple Output
AUTOMATED FLIGHT SUMMARY 3
, AMSTERDAM UNIVERSITY OF APPLIED SCIENCES
Complex Numbers
A complex number is a number that can be expressed in the form a + bj, where a and b are real
numbers. For the complex number a + bi, a is called the real part, and b is called the imaginary part.
Cartesian notation
Complex numbers expressed in their Cartesian form are written as:
𝑧 = a + bj
Where a is the real part and b the imaginary part.
Complex Conjugate
The conjugate of a complex number is the expressed by changing the sign + or – of the
imaginary from + to –, or vice versa.
If z = a + bj, its complex conjugate, denoted by 𝑧, is
𝑧 = a − bj
Operations
If 𝑧1 = 𝑎1 + 𝑏1 j and 𝑧2 = 𝑎2 + 𝑏2 j
Sum Adding two complex numbers together;
𝑧1 + 𝑧2 = 𝑎1 + 𝑎2 + (𝑏1 + 𝑏2 )j
Multiply Multiplying two complex numbers;
𝑧1 ∙ 𝑧2 = 𝑎1 ∙ 𝑎2 − 𝑏1 ∙ 𝑏2 + (𝑎1 ∙ 𝑏2 + 𝑎2 ∙ 𝑏1 )j
𝑧1 ∙ 𝑧1 = 𝑎2 + 𝑏2
Divide Dividing two complex numbers;
𝑧1
𝑧2
= 𝑎(𝑎1 ∙2𝑎)22 ++ 𝑏(𝑏1 ∙2𝑏)22 + (−𝑎(𝑎12∙ )𝑏22 + 𝑎2 ∙ 𝑏1 )
+ (𝑏2 )2
j
To multiply and divide complex numbers it is easier to re-write the cartesian complex number
to its Polar Form.
Argand Diagram
The complex number z = a+bj can be plotted as a
point with coordinates (a, b). Because the real part of
z is plotted on the horizontal axis, we often refer to
this as the real axis. The imaginary part of z is plotted
on the vertical axis and so we refer to this as the
imaginary axis. Such a diagram is called an Argand
diagram.
AUTOMATED FLIGHT SUMMARY 4
