Digital Control Faculty of Engineering Helwan University

1. ‫ر‬َ‫ـد‬ْ‫ق‬‫ِـ‬‫ن‬،،،‫لما‬‫اننا‬ ‫نصدق‬ْْ‫ق‬ِ‫ن‬‫ر‬َ‫د‬
LECTURE (3)
Z-Analysis of Discrete Time
Control Systems
ASCO. Prof. Amr E. Mohamed

2. Agenda
 Mathematical representation of impulse sampling.
 The convolution integral method for obtaining the z-transform.
 Examples
 Properties
 Inverse Z-Transform
 Long Division
 Partial Fraction
 Solution of Difference Equation
 Mapping between s-plane to z-plane
2

3. Impulse Sampling
 Fictitious Sampler.
 The output of the sampler is a train of impulses.
 Let’s define the train of impulses
𝛿 𝑇 𝑡 =
𝑘=0
∞
𝑥 𝑘𝑇 𝛿 𝑡 − 𝑘𝑇
3

4. Impulse Sampling
 The sampler may be considered a modulator
𝑥∗
𝑡 = 𝑥 𝑡 𝛿 𝑇 𝑡 =
𝑘=0
∞
𝑥 𝑘𝑇 𝛿 𝑡 − 𝑘𝑇
𝑥∗
𝑡 = 𝑥 0 𝛿 𝑡 + 𝑥 𝑇 𝛿 𝑡 − 𝑇 + ⋯ + 𝑥 𝑘𝑇 𝛿(𝑡 − 𝑘𝑇) + ⋯
4

5. Impulse Sampling
 The Laplace transform of 𝑥∗
𝑡
𝑋∗ 𝑠 = 𝑥 0 𝛿 𝑡 + 𝑥 𝑇 𝑒−𝑠𝑇 + ⋯ + 𝑥 𝑘𝑇 𝑒−𝑠𝑘𝑇 + ⋯
𝑋∗ 𝑠 =
𝑘=0
∞
𝑥 𝑘𝑇 𝑒−𝑠𝑘𝑇
 If we define 𝑧 = 𝑒 𝑠𝑇
⟹ 𝑠 =
1
𝑇
ln 𝑧
𝑋∗ 𝑠
𝑠=
1
𝑇
ln 𝑧
= 𝑋 𝑧 =
𝑘=0
∞
𝑥 𝑘𝑇 𝑧−𝑘
 The Laplace transform of sampled signal 𝑥∗ 𝑡 has been shown to be the
same as z-transform of the signal 𝑥 𝑡 if 𝑒 𝑠𝑇
is defined as z.
5

6. Data Hold Circuits
 Data hold is a process of generating a continuous-time signal ℎ(𝑡) from
a discrete time sequence 𝑥∗ 𝑡 .
 A hold circuit approximately reproduces the signal applied to the
sampler.
ℎ 𝑘𝑇 + 𝑡 = 𝑎 𝑛 𝑡 𝑛 + 𝑎 𝑛−1 𝑡 𝑛−1 + ⋯ + 𝑎1 𝑡 + 𝑎0
 Note that the signal ℎ 𝑘𝑇 must equal 𝑥 𝑘𝑇 , hence
ℎ 𝑘𝑇 + 𝑡 = 𝑎 𝑛 𝑡 𝑛 + 𝑎 𝑛−1 𝑡 𝑛−1 + ⋯ + 𝑎1 𝑡 + 𝑥 𝑘𝑇
 The simplest data-hold is obtained when 𝑛 = 0 [Zero-Order Hold (ZOH)]
ℎ 𝑘𝑇 + 𝑡 = 𝑥 𝑘𝑇
 When 𝑛 = 1 [First-Order Hold (FOH)]
ℎ 𝑘𝑇 + 𝑡 = 𝑎1 𝑡 + 𝑥 𝑘𝑇
6

7. Zero-Order Hold (ZOH)
ℎ 𝑘𝑇 + 𝑡 = 𝑥 𝑘𝑇 𝑓𝑜𝑟 0 ≤ 𝑡 ≤ 𝑇
7

8. Zero-Order Hold (ZOH)
 The output ℎ(𝑡)
ℎ1 𝑡 = 𝑥 0 [1 𝑡 − 1(𝑡 − 𝑇)] + 𝑥 𝑇 [1 𝑡 − 𝑇 − 1(𝑡 − 2𝑇)] + ⋯
ℎ1 𝑡 =
𝑘=0
∞
𝑥 𝑘𝑇 [1 𝑡 − 𝑘𝑇 − 1(𝑡 − (𝑘 + 1)𝑇)]
 Since, ℒ 1 𝑡 − 𝑘𝑇 =
𝑒−𝑘𝑠𝑇
𝑠
 Then the Laplace transform of the output ℎ1 𝑡 is 𝐻1 𝑠
𝐻1 𝑠 =
𝑘=0
∞
𝑥 𝑘𝑇
𝑒−𝑘𝑠𝑇 − 𝑒−(𝑘+1)𝑠𝑇
𝑠
=
1 − 𝑒−𝑠𝑇
𝑠
𝑘=0
∞
𝑥 𝑘𝑇 𝑒−𝑘𝑠𝑇
8

9. Zero-Order Hold (ZOH)
 Since, ℒ ℎ1 𝑡 = 𝐻1 𝑠 = ℒ ℎ2 𝑡 = 𝐻2 𝑠
𝐻2 𝑠 =
1 − 𝑒−𝑠𝑇
𝑠
𝑘=0
∞
𝑥 𝑘𝑇 𝑒−𝑘𝑠𝑇
=
1 − 𝑒−𝑠𝑇
𝑠
𝑋∗
(𝑠) = 𝐺ℎ0 𝑠 𝑋∗
(𝑠)
 Then the transfer function of the ZOH is
𝐺ℎ0 𝑠 =
1 − 𝑒−𝑠𝑇
𝑠
 Thus , the real sampler and zero-order hold can be replaced by a
mathematically equivalent continuous time system that consists of an
impulse sampler and a transfer function
1−𝑒−𝑠𝑇
𝑠
.
9

10. First Order Hold (FOH)
 The equation of the first order hold is ℎ 𝑘𝑇 + 𝑡 = 𝑎1 𝑡 + 𝑥 𝑘𝑇 𝑓𝑜𝑟 0 ≤ 𝑡 ≤ 𝑇
 By applying the condition ℎ (𝑘 − 1)𝑇 = 𝑥 (𝑘 − 1)𝑇
 The constant 𝑎1 can be determined as follows:
ℎ (𝑘 − 1)𝑇 = −𝑎1 𝑇 + 𝑥 𝑘𝑇 = 𝑥 (𝑘 − 1)𝑇
𝑎1 =
𝑥 𝑘𝑇 − 𝑥 (𝑘 − 1)𝑇
𝑇
10

11. First Order Hold (FOH)
 Hence,
ℎ (𝑘 − 1)𝑇 = 𝑥 𝑘𝑇 +
𝑥 𝑘𝑇 − 𝑥 (𝑘 − 1)𝑇
𝑇
 Suppose that the input 𝑥 𝑡 is unit- step function
ℎ 𝑡 = 1 +
𝑡
𝑇
1 𝑡 −
𝑡 − 𝑇
𝑇
1 𝑡 − 𝑇 − 1(𝑡 − 𝑇)
11

12. First Order Hold (FOH)
 The Laplace transform of the last equation
𝐻 𝑠 =
1
𝑠
+
1
𝑇𝑠2
−
1
𝑇𝑠2
𝑒−𝑠𝑇 −
1
𝑠
𝑒−𝑠𝑇 =
1 − 𝑒−𝑠𝑇
𝑠
+
1 − 𝑒−𝑠𝑇
𝑇𝑠2
𝐻 𝑠 = 1 − 𝑒−𝑠𝑇
𝑇𝑠 + 1
𝑇𝑠2
 The Laplace transform of the input 𝑥∗
𝑡 is
𝑋∗
𝑠 =
𝑘=0
∞
1 𝑘𝑇 𝑒−𝑘𝑠𝑇
=
1
1 − 𝑒−𝑠𝑇
12

13. First Order Hold (FOH)
 Since, 𝐻 𝑠 = 1 − 𝑒−𝑠𝑇 𝑇𝑠+1
𝑇𝑠2 = 𝐺ℎ1 𝑠 𝑋∗ 𝑠
 Hence, the transfer function of the FOH is
𝐺ℎ1 𝑠 =
𝐻 𝑠
𝑋∗ 𝑠
= 1 − 𝑒−𝑠𝑇 2
𝑇𝑠 + 1
𝑇𝑠2
𝑮 𝒉𝟏 𝒔 =
𝟏 − 𝒆−𝒔𝑻
𝒔
𝟐
𝑻𝒔 + 𝟏
𝑻
13

14. Obtaining the z-Transform by the Convolution Integral Method
 Calculating 𝑋∗
𝑠 from the original
signal 𝑋 𝑠
 By substituting 𝒛 for 𝒆 𝒔𝑻
to obtain 𝑋(𝑧) from the sampled signal 𝑋∗
𝑠
 For simple pole
 For multiple pole of order n
14

15. Example
 Solution:
15

16. Example
 Solution:
16

17. Reconstructing Original Signals from Sampled Signals
 Sampling Theorem
 If the sampling frequency is sufficiently high compared with the highest-
frequency component involved in the continuous-time signal, the amplitude
characteristics of the continuous-time signal may be preserved in the
envelope of the sampled signal.
 To reconstruct the original signal from a sampled signal, there is a certain
minimum frequency that the sampling operation must satisfy.
 We assume that 𝑥(𝑡) does not contain any frequency components above 𝜔1
rad/sec.
17

18. Reconstructing Original Signals from Sampled Signals
 Sampling Theorem
 If 𝜔𝑠 , defied as 2𝜋/𝑇 is greater than 2 𝜔1 , where 𝜔1 is the highest-
frequency component present in the continuous-time signal 𝑥(𝑡), then the
signal 𝑥(𝑡) can be reconstructed completely from the sampled signal 𝑥(𝑡).
 The frequency spectrum:
18

19. Reconstructing Original Signals from Sampled Signals
19

20. Reconstructing Original Signals from Sampled Signals
 Ideal Low-pass filter
 The ideal filter attenuates all complementary components to zero and will
pass only the primary component.
 If the sampling frequency is less than twice the highest-frequency
component of the original continuous-time signal, even the ideal filter
cannot reconstruct the original continuous-time signal.
20

21. Reconstructing Original Signals from Sampled Signals
 Ideal Low-pass filter is NOT physically realizable
 For the ideal filter an output is required prior to the application of the input
to the filter
21

22. Reconstructing Original Signals from Sampled Signals
 Frequency response of the Zero-Order Hold.
 The transfer function of the ZOH
22

23. Reconstructing Original Signals from Sampled Signals
 Frequency response characteristics of the Zero-Order Hold.
23

24. Reconstructing Original Signals from Sampled Signals
 Frequency response characteristics of the Zero-Order Hold.
 The comparison of the ideal filter and the ZOH.
 ZOH is a Low-pass filter, although its function is not quite good.
 The accuracy of the ZOH as an extrapolator depends on the sampling
frequency.
24

25. Reconstructing Original Signals from Sampled Signals
 Folding
 The phenomenon of the overlap in the frequency spectra.
 The folding frequency (Nyquist frequency): 𝜔 𝑁
𝜔 𝑁 =
1
2
𝜔𝑠 =
𝜋
𝑇
 In practice, signals in control systems have high-frequency components, and
some folding effect will almost always exist.
25

26. Reconstructing Original Signals from Sampled Signals
 Aliasing
 The phenomenon that frequency component n 𝜔𝑠 ± 𝜔2 shows up at
frequency 𝜔2 when the signal 𝑥(𝑡) is sampled.
 To avoid aliasing, we must either choose the sampling frequency high
enough or use a prefilter ahead of the sampler to reshape the frequency
spectrum of the signal before the signal is sampled.
26

27. The Pulse Transfer Function - Convolution Summation
27

28. The Pulse Transfer Function – Starred Laplace transform
28

29. The Pulse Transfer Function – Starred Laplace transform
29

30. The Pulse Transfer Function of Cascade Elements
30

31. The Pulse Transfer Function of Cascade Elements
31

32. The Pulse Transfer Function of The Closed Loop Systems
32

33. The Pulse Transfer Function of The Closed Loop Systems
33

34. Closed-Loop Pulse Transfer Function of a Digital Control System
34

35. Closed-Loop Pulse Transfer Function of a Digital Control System
35

36. Closed-Loop Pulse Transfer Function of a Digital PID Controller
36

37. Closed-Loop Pulse
Transfer Function
of a Digital PID
Controller
37

38. Closed-Loop Pulse Transfer Function of a Digital PID Controller
38

39. PID Controller
39

40. Obtaining Response Between Consecutive Sampling Instants
 The z-transform analysis will not give information on the reponse
between two consecutive sampling instants.
 Three methods for providing a response between consecutive sampling
instants are commonly available:
1) Laplace transform method
2) Modified z-Transform method
3) State-Space method.
40

41. Laplace Transform Method
41

42. Realization of Digital Controllers and Digital Filters
 A digital filter is a computational algorithm that converts an input
sequence of numbers into an output sequence in such a way that the
characteristics of the signal are changed in some prescribed fashion.
 A digital filter processes a digital signal by passing desirable frequency
components of the digital input signal and rejecting undesirable ones.
 In general, a digital controller is a form of digital filter.
 In general, “Realization” means determining the physical layout for the
appropriate combination of arithmetic and storage operation.
 Realization techniques are
 Direct realization.
 Standard realization.
 Series realization.
 Parallel realization.
 Lader realization. 42

43. Direct realization.
 The general form of the digital filter is
 Then
43

44. Standard Programming
 In direct programming, the numerator uses a set of 𝑚 delay elements
and the denominator uses a different set of 𝑛 delay elements. Thus the
total number of delay elements used in direct programming is (𝑛 + 𝑚).
 The standard programming uses a minimum number of delay elements
(𝑛).
44

45. Standard Programming
45

46. Standard Programming
46

47. Note:
 In realizing digital controllers or digital filters, it is important to have a
good level of accuracy . Basically, three sources of errors affect the
accuracy:
1) The quantization error due to the quantization of the input signal into a
finite number of discrete levels. The quantization noise may be considered
white noise; the variance of the noise is 𝜎2
= 𝑄2
/12.
2) The error due to the accumulation of round-off errors in the arithmetic
operations in the digital system.
3) The error due to quantization of the coefficients 𝑎𝑖 and 𝑏𝑖 of the pulse
transfer function. This error may become large as the order of the pulse
transfer function is increased.
• That is, in a higher–order digital filter in direct structure, small errors in the
coefficients cause large errors in the locations of the poles and zeros of the
filter.
47

48. Decomposition Techniques
 Note that the third type of error listed may be reduced by
mathematically decomposing a higher-order pulse transfer function into
a combination of lower-order pulse transfer functions.
 For decomposing higher-order pulse transfer functions in order to avoid
the coefficient sensitivity problem, the following three approaches are
commonly used.
1) Series Programming.
2) Parallel Programming
3) Lader Programming
48

49. Series Programming
 The 𝐺(𝑧) may be decomposed as follows:
 Then the block diagram for the digital filter 𝐺(𝑧) is a series connection
of p component digital filters as shown.
49

50. Parallel Programming
 The 𝐺(𝑧) is expanded using the partial
fractions as follows:
 Then the block diagram for the digital
filter 𝐺(𝑧) is a parallel connection of
p component digital filters as shown.
50

51. Lader Programming
 The 𝐺(𝑧) is decomposed into a continued-fractions form as follows:
 The G(z) may be written as follows:
51

52. Lader Programming (Cont.)
 Then the block diagram for the digital filter 𝐺(𝑧) is a Lader connection
of p component digital filters as shown.
52

53. 53