1. ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ (22Δ802) Β΄ ΕΞΑΜΗΝΟ 2016-17  2610 996449
Καθηγητής Πέτρος Π. Γρουμπός 
Ώρες Γραφείου: Τετάρτη Πέμπτη Παρασκευή 11:00-12:00
Γραφείο: 1ος όροφος
Τομέας Συστημάτων & Αυτομάτου Ελέγχου
Τμήμα ΗΜ&ΤΥ
Μάθημα 3

2. 1. Introduction Control system Gc(s) GP(s) R(s) E(s) M(s) Y(s)
Controller
Plant
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

3. Introduction (συνέχεια)
Computer controlled system
Gc(z)
ZOH
GP(s)
R(z)
E(z)
M(z)
GHP(z)
Computer system
Y(z)
Plant
A/D
D/A

4. 2.Digital control loop: Components
GHP(z) is the transfer function of control object + ZOH, where z indicates the discrete time domain
GC(z) is a controller implemented in computer languages.
A/D is the Analog-to-Digital converter (Voltage Binary number).
D/A is the Digital-to-Analog converter (Binary number  Voltage).
The little switch indicate a sampling operation.

5. Digital control loop: Signals
Discrete time domain
R(z) is the desired output
E(z) is the error signal
M(z) is the controller output/control action
C(z) is the actual output
Continuous time domain
In continuous time domain, R(z), E(z), M(z) and C(z) are corresponding to r(t), e(t) m(t) and c(t).

6. Digital control loop: Sequence of events
Get desired output r(t) at this instant in time
Measure actual output c(t)
Calculate error e(t)=r(t)-c(t)
Derive control signal m(t) based on proper control algorithm
Output this control signal m(t) to controlled object
Save previous history of error and output for later use
Repeat step 1 to 6 (go to 1)

7. Digital control loop: Forms of signals
Computer cannot sample while calculating, so there is a sample frequency 1/T for data acquisition through a A/D, where T is sampling interval.
The data of a signal are recorded and represented as a sequence of number in memory.
Based on these numbers, a control signal is derived and then conveyed to controlled object through a D/A

8. Digital control loop: Forms of signals
In between sample instants, the input is supposed as constant and the output is held as a constant by a device termed as zero-order-hold (ZOH).
The reconstruction of a signal will be a ‘stair-step, and a low-pass filter is employed to smooth out the rough edges

9. 3. ADC and DAC
time
f(t)
Time kT
f(kT)
Sampling
A/D

10. 3 ADC and DAC(συν)
D/A is used as a ZOH.
Time kT
f(kT)
D/A

11. 3. ADC and DAC (συν) Output
Have a discrete number of quantization levels
Number of levels L=2N, where N is the number of bits
eg N=3 bits, L=23=8 levels
Output
Input 
ADC
Analog
Digital

12. 3. ADC and DAC ADC Bits Level Signal Error 1 2 5 5/2=2.5 4 5/4=1.25 38
5/8=0.625 16
5/16=0.3125

13. 3 ADC and DAC ADC More bits more accuracy. The commonly used ADC has
8-bits: L=28=256 (coarse)
10-bits: L=210=1024 (adequate)
12-bits: L=212=4096 (works well)
16-bits: L=216=65536 (almost overkill)

14. 3 ADC and DAC ADC
Distances between sequential levels are the same. eg 5v/28=0.0195v
The weight of each bit is different. The most significant bit is the most left bit and the least significant bit is the most right bit.
Bit 0
Bit N-1 20
2N-1


15. 3 ADC and DAC ADC
Example: For N=8, find the number range of the ADC in binary, decimal and hexadecimal numbers. If the input signal is from 0 to 5 voltage for the above number range, what will be the number for a 2 voltage signal in decimal and binary numbers?
Solution: In binary: B B
In decimal: 0  = 255
In hexadecimal: 0  F=15; 00H FFH

16. 3 ADC and DAC DAC
Output
Input 
DAC
Digital
Analog
Example: For N=8 and the signal is from 0 to 5, find the output value for the number 145.
Solution:5/255=x/145, x=5*145/255=2.8431=2.84

17. Multi-channel A/D converter
3 ADC and DAC
Multi-channel A/D converter 
MUX AD
Digital
signal
Status
Control …
Analog signal

18. Multi-channel D/A converter
3 ADC and DAC
Multi-channel D/A converter  DA
MUX
Analog
signals
Control …
Digital signals

19. 4 Errors 
Input
ADC 
Output
Errors

20. 4 Errors
The quantization error or resolution error is the difference between the analog input value and the equivalent digital value. On average it is one half of the LSB.
Linearity error: the maximum deviation in step size from ideal step size, expressed as a percentage of full scale.
Settling time: the time it takes for the output to reach within +/- half of the step size of the final output.

21. 4 Errors
Gain error 
ADC
Input
Output
Output
Digital
Analog
Input

22. 5 Sampling theorem f(kT) f(t) Sampling time Time kT A/D
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

23. 5 Sampling theorem
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

24. 5 Sampling theorem
If we need the sampled data to keep all the features of the original signal, what is the minimum sampling frequency?
Or what conditions should we meet if we wish that the sampled data can represent the original data exactly?
The answer to the above question forms the Sampling theorem/Shannon’s sampling theorem/Shannon’s theorem.
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

25. 5 Sampling theorem
A continuous-time signal f(t) with a finite bandwidth 0 (the highest frequency component in the signal, or the Nyquist frequency) can be uniquely described by the sampled signal f(kT){k=…,-1,0,1….}, when the sampling frequency s is greater than 20.
In other words, if a signal is sampled twice faster than its highest frequency component, the sampled date can represent all the features of this signal.
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

26. 6 The proven of sampling theorem
The proven is based on Fourier Transform
Fourier transform: A transformation from time domain to frequency domain f(t)  F(), where t is time and  is frequency.
For a continuous time function f(t), we can uniquely find F(). If given F(), we can also unique determine f(t).
It means that f(t) and F() are equivalent.
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

27. 6 The proven of sampling theorem
Fourier
Transform 
F() 0
-0
f(t)
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

28. 6 The proven of sampling theorem
2. For a sampled signal fs(t), we have
Fourier
Transform 
Fs() 0
-0
-s s
2s
-2s
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

29. 6 The proven of sampling theorem
3. The relationship between f(t) and fs(t), and F() and Fs().
Fourier
Transform 
F() 0
-0
Fs()
-s s
2s
-2s
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

30. 6 The proven of sampling theorem
4. If we change the sampling frequency, what will happen with fs(t) and Fs(). 
Fs() 0
-0
-s s
2s
-2s
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

31. 6 The proven of sampling theorem
Fs() 0
-0
-s s
2s
-2s
For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplification, or power gain.
A remote controlled robot arm can be used to pick up material in a radioactive environment.
In a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.
If you are going to drive a carat at a fix speed, can you simply hold the accelerator pedal as it is? No, you have to adjust the position of accelerator pedal according to the road condition to keep a constant speed. The change of the road condition can be consider as sort of disturbance.

32. 6 The proven of sampling theorem
5. Conclusions
If our sampling frequency s is faster enough, that is s>20, there will be gaps between the shifting F() in Fs(). We can always put a filter to figure out F() from Fs(). Otherwise if the repeating F() figures overlap in Fs(), we cannot put a filter to figure out F() from Fs(). The turning point from possible to impossible is s =20, where 0 is the highest frequency component or Nyquist Frequency of the signal.

33. 7 Aliasing
1. Aliasing problem

34. 7 Aliasing
Ambiguity: alias

35. 7 Aliasing 2. Finding aliases
The fundamental alias frequency is given by
=| (0+ n)mod(s) - n|
where mod() means the remainder of an division operation, 0 is signal bandwidth, n Nyquist frequency, and s sampling frequency
Example: For f0=90Hz & fs=100, find alias.
Solution: =2f, fn=fs/2=50Hz,
f=| (f0+ fn)mod(fs) - fn|=|(90+50)mod(100)-50|
=|40-50|=10Hz

36. 7 Aliasing 3. Preventing aliases
Make sure your sampling frequency is greater than twice of the highest frequency component of the signal
Pre-filtering
Set your sampling frequency to the maximum if possible

37. 7 Aliasing
Suppose that the Nyquist frequency of a signal is 100Hz. If we use an 8-bit ADC to sample this signal at the frequency of 200Hz, can the sample data represents this signal exactly? Why?

38. 7 Aliasing
Theoretically, as long as the sampling frequency is greater than or equal to twice the Nyquist frequency, aliases will not happen. However, because of the conversion/quantisation error, the practical sampling frequency is much higher than that (5 to 10 times of the Nyquist frequency). Fortunately, most of the time the speeds of ADC and computer are also much greater than signal’s Nyquist frequency.

39. Exercise
Exercise 1: The frequency spectrum of a continuous-time signal is shown below.
What is the minimum sampling frequency for this signal to be sampled without aliasing.
If the above process were to be sampled at 10 Krad/s, sketch the resulting spectrum from –20 Krad/s to 20 Krad/s. -8 -4 4 8
 Krad/s
F()

40. Hints   The relationship between f(t) and fs(t), and F() and Fs().
Fourier
Transform 
-0 0
Fs() 
-0-s
-0-2s
-0 0
0+s
0+2s
-0+s
0-2s
0-s
-0+2s

41. Answers 4 8
 Krad/s 12 2 6 14 18 16 10
F()