This document provides an introduction to digital control systems. It discusses how control systems consisting of interconnected components can be designed to achieve a desired purpose. Modern control engineering uses digital control strategies to improve various processes. The design gap exists between physical systems and their models. The iterative nature of design allows engineers to effectively handle this gap. Digital control offers advantages over analog control like accuracy, flexibility and lower costs. However, digital control can introduce delays. Examples of digitally controlled systems include automotive engine control and aircraft autopilots. Controller design involves modeling systems, transforming between differential and difference equations, and mapping between the s-plane and z-plane.

1. DIGITAL CONTROL SYSTEMS
LECTURE NO.1
PROF. DR. KHALAF S GAEID
ELECTRICAL ENGINEERING DEPARTMENT/ TIKRIT
UNIVERSITY
GAEIDKHALAF@GMAIL.COM

2. 1. A control system consisting of interconnected components is designed to achieve a desired
purpose.
2. Modern control engineering practice includes the use of control design strategies for
improving manufacturing processes, the efficiency of energy use, advanced automobile
control, including rapid transit, among others.
3. We also discuss the notion of a design gap. The gap exists between the complex physical
system under investigation and the model used in the control system synthesis.
4. The iterative nature of design allows us to handle the design gap effectively while
accomplishing necessary tradeoffs in complexity, performance, and cost in order to meet
the design specifications.
Introduction to Control Systems Objectives

3. Introduction
System – An interconnection of elements and devices for a desired purpose.
Control System – An interconnection of components forming a system configuration that
will provide a desired response.
Process – The device, plant, or system under control. The input and output relationship
represents the cause-and-effect relationship of the process.

4. What is Digital Control?
Automatic control is the science that develops techniques to steer, guide, control dynamic
systems.
These systems are built by humans and must perform a specific task. Examples of such
dynamic systems are found in biology, physics, robotics, finance, etc.
Digital Control means that the control laws are implemented in a digital device, such as a
microcontroller or a microprocessor. Such devices are light, fast and economical.
Digital Control Systems z-Plane Analysis of Discrete Time Control Systems

5. INTRODUCTION
Digital control offers distinct advantages over analog control that explain its
popularity.
Accuracy: Digital signals are more accurate than their analogue counterparts.
Implementation Errors: Implementation errors are negligible.
Flexibility: Modification of a digital controller is possible without complete
replacement.
Speed: Digital computers may yield superior performance at very fast speeds
Cost: Digital controllers are more economical than analogue controllers.
5

6. DISADVANTAGES OF DIGITAL COMPUTERS
FROM THE TRACKING PERFORMANCE SIDE, THE ANALOG CONTROL SYSTEM
EXHIBITS GOOD PERFORMANCES THAN DIGITAL CONTROL SYSTEM.
DIGITAL CONTROL SYSTEM WILL INTRODUCE A DELAY IN THE LOOP.
6
https://www.mathworks.com/academia/books/search.html?q=digital%20control&
page=1

7. STRUCTURE OF A DIGITAL CONTROL SYSTEM
7

8. Examples of Digitally Controlled Systems
Nowadays, digitally controlled systems are everywhere,
• Automotive industry: speed regulators in cars,
• Aeronautic/space industry: autopilots, automatic take off/landing, cruise control
• Chemistry: pharmaceutical industries, oil transformation, liquid level in tanks
• Robotics: robot-arm trajectory control, manipulation,
• Housing: in-house temperature regulation

9. INTRODUCTION
Modeling of dynamic systems
Model: A representation of a system.
Types of Models:
1. Physical models (prototypes)
2. Mathematical models (e.g., input-output relationships)
3. Analytical models (using physical laws)
4. Computer (numerical) models
5. Experimental models (using input/output experimental data)
Models for physical dynamic systems: Lumped-parameter models
Continuous-parameter models. Example: Spring element (flexibility, inertia, damping)

10. INTRODUCTION
Signal categories for identifying control system types
Continuous-time signal & quantized signal
Continuous-time signal is defined continuously in the time domain. Figure on the left
shows a continuous-time signal, represented by x(t).
Quantized signal is a signal whose amplitudes are discrete and limited. Figure on the
right shows a quantized signal.
Analog signal or continuous signal is continuous in time and in amplitude. The real
word consists of analog signals.
The Simulink quantizers as

11. INTRODUCTION
Discrete-time signal & sampled-data signal
Discrete-time signal is defined only at certain time instants. For a discrete-time signal, the
amplitude between two consecutive time instants is just not defined. Figure on the left shows a
discrete-time signal, represented by y(kh), or simply y(k), where k is an integer and h is the time
interval.
Sampled-data signal is a discrete-time signal resulting by sampling a continuous-time signal.
Figure on the right shows a sampled-data signal deriving from the continuous-time signal, shown
in the figure at the center, by a sampling process. It is represented by x
∗
(t).
Discrete time signal sampled data signal

12. INTRODUCTION
Digital signal or binary coded data signal
Digital signal is a sequence of binary numbers. In or out from a microprocessor, a
semiconductor memory, or a shift register.
In practice, a digital signal, as shown in the figures at the bottom, is derived by two
processes: sampling and then quantizing.

13. INTRODUCTION
Control System Types

14. INTRODUCTION
Controller Design in Digital Control Systems

15. INTRODUCTION
Controller design in digital control systems - Design in S-domain
Digitization (DIG) or discrete control design
The above design works very well if sampling period T is sufficiently small.

16. INTRODUCTION
How to design a Controller:
For the approximation methods, used to convert the continuous controller to
digital controller, are: Euler method, Trapezoidal method …etc and we can use
forward or backward approximation.
Afterwards, the differential equation of the controller can be transformed to
difference equation where this difference equation can be easily programmed as a
control algorithm.
Note that the sampling time T should be close to zero.
Hw. Design digital controller using simulink

17. backward approximation

18. INTRODUCTION
Controller design in digital control systems -Design in Z domain
Direct (DIR) control design

19. Figure. Piecewise-constant signal xZOH(t).

20. EXAMPLES OF DIGITAL CONTROL SYSTEMS
20
Closed-Loop Drug Delivery System
HW(optional ): Implement this system with Matlab/simulink

21. FIGURE 1.CONVERSION OF ANTENNA AZIMUTH POSITION CONTROL SYSTEM FROM:
A. ANALOG CONTROL TO B. DIGITAL CONTROL

22. EXAMPLES OF DIGITAL CONTROL SYSTEMS
22
Aircraft Turbojet Engine

23. INTRODUCTION
Mathematical comparison between analog and digital control systems

24. DIFFERENCE EQUATION VS DIFFERENTIAL EQUATION
A difference equation expresses the change in some variable as a result of a
finite change in another variable.
A differential equation expresses the change in some variable as a result of an
infinitesimal change in another variable.
24

25. DIFFERENTIAL EQUATION
Rearranging above equation in following form
25
𝑦 𝑡 =
1
𝑚
𝐹 𝑡 −
𝐷
𝑚
𝑦 𝑡
𝐾
𝑚
𝑦(𝑡)
𝑑𝑡 𝑑𝑡
1
𝑚
−
𝐷
𝑚
−
𝐾
𝑚
𝑦 𝑦 𝑦
𝐹(𝑡)


26. DIFFERENCE EQUATION
26
𝑦(𝑘 + 2) =
1
𝑚
𝐹(𝑘) −
𝐷
𝑚
𝑦(𝑘 + 1)
𝐾
𝑚
𝑦(𝑘)
1
𝑧
1
𝑧
1
𝑚
−
𝐷
𝑚
−
𝐾
𝑚
𝑦(𝑘 + 2) 𝑦(𝑘)
𝐹(𝑘)

𝑦(𝑘 + 1)

27. DIFFERENCE EQUATIONS
Example-1: For each of the following difference equations, determine the (a) order
of the equation. Is the equation (b) linear, (c) time invariant, or (d) homogeneous?
1. 𝑦 𝑘 + 2 + 0.8𝑦 𝑘 + 1 + 0.07𝑦 𝑘 = 𝑢 𝑘
2. 𝑦 𝑘 + 4 + sin(0.4𝑘)𝑦 𝑘 + 1 + 0.3𝑦 𝑘 = 0
3. 𝑦 𝑘 + 1 = −0.1𝑦2
𝑘
Try to implement all the above difference equations with simulink as HW
27

28. DIFFERENCE EQUATIONS
Example-1: For each of the following difference equations, determine the
(a) order of the equation. Is the equation (b) linear, (c) time invariant, or (d)
homogeneous?
1. 𝑦 𝑘 + 2 + 0.8𝑦 𝑘 + 1 + 0.07𝑦 𝑘 = 𝑢 𝑘
Solution:
a) The equation is second order.
b) All terms enter the equation linearly
c) All the terms if the equation have constant coefficients. Therefore the
equation is therefore LTI.
d) A forcing function appears in the equation, so it is nonhomogeneous.
28

29. DIFFERENCE EQUATIONS
Example-1: For each of the following difference equations, determine the
(a) order of the equation. Is the equation (b) linear, (c) time invariant, or (d)
homogeneous?
2. 𝑦 𝑘 + 4 + sin(0.4𝑘)𝑦 𝑘 + 1 + 0.3𝑦 𝑘 = 0
Solution:
a) The equation is 4th order.
b) All terms are linear
c) The second coefficient is time dependent
d) There is no forcing function therefore the equation is homogeneous.
29

30. DIFFERENCE EQUATIONS
Example-1: For each of the following difference equations, determine the
(a) order of the equation. Is the equation (b) linear, (c) time invariant, or
(d) homogeneous?
3. 𝑦 𝑘 + 1 = −0.1𝑦2
𝑘
Solution:
a) The equation is 1st order.
b) Nonlinear
c) Time invariant
d) Homogeneous
30

31. FIGURE 2A. PLACEMENT OF THE DIGITAL COMPUTER WITHIN THE LOOP; B. DETAILED BLOCK
DIAGRAM SHOWING PLACEMENT OF A/D AND D/A CONVERTERS

32. FIG.3.DIGITAL-TO-ANALOG CONVERTER
HTTP://WWW.UOBABYLON.EDU.IQ/EPRINTS/PUBLICATION_7_11855_163.PDF
HW. What are the main D/A and A/D methods, try to implement it in Simulink or any other software?

33. FIG.4 STEPS IN ANALOG-TO-DIGITAL CONVERSION:
A. ANALOG SIGNAL; B. ANALOG SIGNAL AFTER SAMPLE-AND-HOLD;
C. CONVERSION OF SAMPLES TO DIGITAL NUMBERS

34. FIG.5. TWO VIEWS OF UNIFORM-RATE SAMPLING:
a. switch opening and closing;
b. product of time waveform and sampling waveform

35. FIG.6. MODEL OF SAMPLING WITH A UNIFORM RECTANGULAR PULSE TRAIN

36. FIG.7.IDEAL SAMPLING AND THE ZERO-ORDER HOLD
http://ctms.engin.umich.edu/CTMS/index.php?example=MotorPosition&section=SimulinkControl

37. TABLE1. PARTIAL TABLE OF Z- AND S-TRANSFORMS

38. TABLE2.Z-TRANSFORM THEOREMS

39. FIG.8.SAMPLED-DATA SYSTEMS:
A. CONTINUOUS; B. SAMPLED INPUT; C. SAMPLED INPUT AND OUTPUT

40. FIG.9. SAMPLED-DATA SYSTEMS AND THEIR Z-TRANSFORMS

41. FIG.10.STEPS IN BLOCK DIAGRAM REDUCTION OF A SAMPLED-DATA SYSTEM

42. FIG.11DIGITAL SYSTEM FOR SKILL-ASSESSMENT EXERCISE TRY TO SOLVE IT

43. CONFORMAL MAPPING BETWEEN S-PLANE TO Z-PLANE
Where 𝑠 = 𝜎 + 𝑗𝜔.
Then 𝑧 in polar coordinates is given by
43
𝑧 = 𝑒(𝜎+𝑗𝜔)𝑇
𝑧 = 𝑒𝜎𝑇
𝑒𝑗𝜔𝑇
∠𝑧 = 𝜔𝑇
𝑧 = 𝑒𝜎𝑇
𝑧 = 𝑒𝑠𝑇

44. CONFORMAL MAPPING BETWEEN S-PLANE TO Z-PLANE
We will discuss following cases to map given points on s-plane to z-plane.
 Case-1: Real pole in s-plane (𝑠 = 𝜎)
 Case-2: Imaginary Pole in s-plane (𝑠 = 𝑗𝜔)
 Case-3: Complex Poles (𝑠 = 𝜎 + 𝑗𝜔)
44
𝑠 − 𝑝𝑙𝑎𝑛𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒

45. CONFORMAL MAPPING BETWEEN S-PLANE TO Z-PLANE
Case-1: Real pole in s-plane (𝑠 = 𝜎)
We know
Therefore
45
∠𝑧 = 𝜔𝑇
𝑧 = 𝑒𝜎𝑇
𝑧 = 𝑒𝜎𝑇 ∠𝑧 = 0

46. CONFORMAL MAPPING BETWEEN S-PLANE TO Z-PLANE
46
When 𝑠 = 0
∠𝑧 = 𝜔𝑇
𝑧 = 𝑒𝜎𝑇
𝑧 = 𝑒0𝑇 = 1
∠𝑧 = 0𝑇 = 0
𝑠 = 0
𝑠 − 𝑝𝑙𝑎𝑛𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒
1
Case-1: Real pole in s-plane (𝑠 = 𝜎)

47. CONFORMAL MAPPING BETWEEN S-PLANE TO Z-PLANE
47
When 𝑠 = −∞
∠𝑧 = 𝜔𝑇
𝑧 = 𝑒𝜎𝑇
𝑧 = 𝑒−∞𝑇
= 0
∠𝑧 = 0
−∞
𝑠 − 𝑝𝑙𝑎𝑛𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒
0
Case-1: Real pole in s-plane (𝑠 = 𝜎)

48. CONFORMAL MAPPING BETWEEN S-PLANE TO Z-PLANE
Case-2: Imaginary pole in s-plane (𝑠 = ±𝑗𝜔)
We know
Therefore
48
∠𝑧 = 𝜔𝑇
𝑧 = 𝑒𝜎𝑇
𝑧 = 1 ∠𝑧 = ±𝜔𝑇

49. CONFORMAL MAPPING BETWEEN S-PLANE TO Z-PLANE
49
Consider 𝑠 = 𝑗𝜔
∠𝑧 = 𝜔𝑇
𝑧 = 𝑒𝜎𝑇
𝑧 = 𝑒0𝑇 = 1
∠𝑧 = 𝜔𝑇
𝑠 = 𝑗𝜔
𝑠 − 𝑝𝑙𝑎𝑛𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒
1
−1
−1
1
𝜔𝑇
Case-2: Imaginary pole in s-plane (𝑠 = ±𝑗𝜔)

50. FIG.13.MAPPING REGIONS OF THE S-PLANE ONTO THE Z-PLANE

51. MAPPING REGIONS OF THE S-PLANE ONTO THE Z-PLANE
51

52. FIG.14.FINDING STABILITY OF A MISSILE CONTROL SYSTEM:
A. MISSILE; B. CONCEPTUAL BLOCK DIAGRAM; C. BLOCK DIAGRAM;
D. BLOCK DIAGRAM WITH EQUIVALENT SINGLE SAMPLER

53. FIG.25 A. DIGITAL CONTROL SYSTEM SHOWING THE DIGITAL COMPUTER PERFORMING COMPENSATION;
B. CONTINUOUS SYSTEM USED FOR DESIGN;
C. TRANSFORMED DIGITAL SYSTEM

54. FIG.27.BLOCK DIAGRAM SHOWING COMPUTER EMULATION OF A DIGITAL COMPENSATOR

55. 55
DISCRETE-TIME SIGNALS: SEQUENCES
Discrete-Time signals are represented as
In sampling of an analog signal xa(t):
1/T (reciprocal of T) : sampling frequency
 
  integer
:
,
, n
n
n
x
x 





    period
sampling
T
nT
x
n
x a :
,

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Cumbersome, so just use
 
x n
difficult to do or manage and taking a lot
of time and effort

56. 56
FIGURE. GRAPHICAL REPRESENTATION OF A DISCRETE-TIME SIGNAL
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56
Abscissa: continuous line
: is defined only at discrete instants
 
x n

57. 57
Figure 2
EXAMPLE Sampling the analog waveform
)
(
|
)
(
]
[ nT
x
t
x
n
x a
nT
t
a 
 

58. 58 10/5/2023
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Sum of two sequences
Product of two sequences
Multiplication of a sequence by a number α
Delay (shift) of a sequence
BASIC SEQUENCE OPERATIONS
]
[
]
[ n
y
n
x 
integer
:
]
[
]
[ 0
0 n
n
n
x
n
y 

]
[
]
[ n
y
n
x 
]
[n
x



59. 59
BASIC SEQUENCES
Unit sample sequence (discrete-time impulse, impulse, Unit
impulse)
 






0
1
0
0
n
n
n

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59
continuous-time unit impulse function δ(t) )

60. 60
BASIC SEQUENCES
[ ] [ ] [ ]
k
x n x k n k



 

         
7
2
1
3 7
2
1
3







 
n
a
n
a
n
a
n
a
n
p 



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60
arbitrary sequence
A sum of scaled, delayed impulses

61. 1. The acronym DCS stands for?
2. Many digital control systems utilize Ethernet as a communications network, because . .
3.Resolution refers to in the analog-to-digital conversion portion of a digital control system.
4,Parity bits are used for the purpose of in digital
systems.
5.A typical use for an integer variable in a digital control
system is:
6.A watchdog timer is a device or a programmed routine used for what purpose in a digital
control system?
Projects
Design Efficient DC-to-DC Power Converters
Electric vehicles and charging stations
Renewable energy
Convert SPICE models into Simscape components
Convert SPICE models into Simscape components
Implement Power Electronics Control on an Embedded Platform
Replace Hand Coding with Code Generation
Optimize for C2000 to Improve Execution Performance
Strategies Hardware-in-the-Loop (HIL) Testing
FPGA-based HIL Relevant to Power Electronics
Adaptive control for stability optimization for Induction Mo
Intelligent control for stability optimization for Induction M
Intelligent robotic control for stability optimization

62. 62
THANKS
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