Prof. Wahied Gharieb Ali Abdelaal CSE 502: Control Systems(1) Topic#2 Mathematical Tools for Analysis Faculty of Engineering Computer and Systems Engineering Department Master and Diploma Students

2 Outline Ordinary Differential Equations (ODE) Laplace Transform and Its Inverse Laplace Transform Properties Sampling and Digital Systems Selection of Sampling Frequency Z-Transform and Its Properties Summary

3 Ordinary Differential Equations (ODE) An ordinary differential equation of order n is given by: Example: Second order linear ordinary differential equation. It is convenient to define the differential operators: and

4 Characteristic Polynomial & Characteristic Equation The polynomial: is called the Characteristic Polynomial and the equation : is called the characteristic equation The characteristic polynomial is D 2 +3D +2. The characteristic equation is D 2 +3D+2=0 (D+2)(D+1)=0. The roots are D 1 =-2 & D 2 =-1 Example: Ordinary Differential Equations (ODE)

5 Solution of Linear Ordinary Differential Equation The solution of a differential equation contains two parts: Free response Forced response The free response; is the solution of the differential equation when the input is zero. The forced response ; is the solution of the differential equation when all initial conditions are zero. The total response is the sum of the free response and the forced response. Ordinary Differential Equations (ODE)

6 Example The free response (y 1 ) D 2 + 3D +2 =0  (D+2)(D+1)=0  D=-1 ; D=-2 The forced response (y 2 ) depends on the forcing function f(t). If f(t)= cos  t  y 2 (t) = A 1 cos  t + A 2 sin  t f(t)= t 2  y 2 (t) = A 1 + A 2 t + A 3 t 2 f(t)= te -t  y 2 (t) = A 1 e -t + A 2 te -t f(t)= e  t  y 2 (t) = Ae  t The above forms will usually work if the forcing function is not a part of the free response ! Ordinary Differential Equations (ODE)

7 In this example, f(t) = -4e -3t  y 2 (t)= Ae -3t because The total response y(t)= y 1 (t) + y 2 (t) We have to find A, A 1 and A 2 satisfying the differential equation and the initial conditions 9Ae -3t –9Ae -3t +2Ae -3t = -4e -3t  A=-2 y 2 (t)=-2e -3t Ordinary Differential Equations (ODE)

8 Using initial conditions to find A 1 and A 2 y(0)=0  A 1 +A 2 -2=0  A 1 =2-A 2 ý (0)= 1  -A 1 -2A 2 +6=1  A 1 =-1 and A 2 =3 y(t)= -e -t + 3e -2t - 2e -3t The total response y(t)= -2e -3t + A 1 e -t + A 2 e -2t Ordinary Differential Equations (ODE)

Example The free response (y 1 ) D 2 + D +1/2 =0  Complex poles  D 1,2 = -1/2  j1/2 9 Ordinary Differential Equations (ODE)

Forced response Since f(t)=1/2t  y 2 (t)=B 1 +B 2 t 0+B 2 +1/2B 1 +1/2B 2 t =1/2t  B 1 =-2 and B 2 =1 y 2 (t)=-2+t 10 Ordinary Differential Equations (ODE)

11 Laplace Transform Given a real function f(t) that satisfies Laplace Transform of f(t) is defined as : Where F(s) = L[f(t)] or f(t) = L -1 [F(s)] Definition

12 Laplace Transform 1.f(t) =  (t) = impulse signal defined by: where for t  0,  otherwise Examples

13 2. f(t) = u(t) = Unit Step A shift in the time domain is equivalent to an exponential term in the s-plane. Laplace Transform

14 3. f(t) = t u(t) = Ramp Function Laplace Transform 4. Exponential Function f(t)=e -  t u(t)

15 5. Sinusoidal Functions f(t) = e jωt u(t) Laplace Transform

16 Laplace Transform

17 Laplace Transform

18 Laplace Transform

19 Laplace Transform

20 If the limit exist Initial Value Theorem 1. F(s) has a pole of order 2 at zero and theorems can not be applied. Provided that sF(s) does not have any poles on the j  axis and in the right half s-plane Final Value Theorem Examples 2. ; No poles in the right half s-plane (Stable). Initial and Final Value Theorems

21 Inverse Laplace Transform using Partial Fraction Expansion The inverse Laplace Transform does not relay on the use of the inversion Integral. Rather the inverse Laplace transform operation involving rational functions can be carried out using a Laplace Transform table and Partial fraction expansion. f(t)=L -1 [F(s)] Suppose that all poles of the transfer function are simple From the Laplace Transform Table, Inverse Laplace Transform

22 Example Consider : The first term is the steady-state solution; the last two terms represent the transient solution. Unlike the classical method, which requires separate steps to give the transient and the steady state solutions, the Laplace transform method gives the entire response. By tacking the inverse Laplace transform of this equation, we get the complete solution as: Inverse Laplace Transform

23 Example Inverse Laplace Transform

24 Example Inverse Laplace Transform

25 Inverse Laplace Transform

26 f(t)F(s) 1.δ(t) 1 2.u(t) 3.t u(t) 4.t n u(t) 5.e -at u(t) 6. sin  t u(t) 7. cos  t u(t) Laplace Transform of Basic Functions

27 Laplace Transform Properties

28 Laplace Transform Properties

29 Laplace Transform Properties

30 Sampling and Digital Systems

31 Advantages of digital control  Hardware is replaced by software, which is costly-effective  Complex function can be implemented in software so easily rather than hardware  Reliability in implementation, that means, you can simply modify the control function in software without extra cost.  Computers can be used in data logging (monitoring), supervisory control and can control multiple loop simultaneously as the computers are well fast. Sampling and Digital Systems

32 Sampling and Digital Systems

33 Sampling and Digital Systems

34 Sampling and Digital Systems

35 Sampling and Digital Systems

36 Digital controllers could take one of the forms: A computer or simply microprocessor board. Once they have developed and started to be manufactured commercially, digital controllers are developed. Microcontroller is a microprocessor system on chip as a single integrated circuit. It is used in embedded control applications such as TV, mobile phones, Air conditioner, Video Camera, Hard disk controllers, Robots, Smart car manufacturing,...etc. It is used for a limited number of I/O signals in real time applications. Programmable logic controller (PLCs). PLC can handle a very large number of I/O signals (as hundreds or thousands) in industrial control applications. It has a standard interfaces with the field measurements. The PLC technology replaces the old hardwired control (relay logic control) cabinets in the industry. Sampling and Digital Systems

37 The analog signal is a continuous representation of a signal, that it takes different values with time. Digital signals have two values only or two level corresponding to logic 1 and logic zero Sampling and Digital Systems

38 The ADC requires three operations in sequence: 1- Sampling, we need to sample the analog signal at a constant rate. The sampler could be an electronic switch. The critical question is how to select the sampling frequency. 2- Holding, that holds the sample in during the sampling period until a new sample is captured. This is necessary to convert a constant value into digital word. 3- Conversion, it is often sequential circuit that takes a considerable time to convert the holding sample into digital word. Sampling and Digital Systems

39 Sampling and holding process A/D Converters Digitized Value  Sampling and Digital Systems

40 Sampling and Digital Systems

41 DAC requires two operations in sequence: 1- DAC, in general is faster than ADC ones and easier in implementation. 2- Holding, it is very difficult to apply the discrete signal that outputs from DAC directly to an analog process. It will excite the system and fatigue the actuator. Therefore, holding these samples makes them in a continuous form (stepping levels). Sampling and Digital Systems

42 Sampling and Digital Systems

43 It is imperative that an ADC's sample time is fast enough to capture essential changes in the analog waveform. In data acquisition terminology, the highest-frequency waveform that an ADC can theoretically capture is called Nyquist frequency, which equals to one-half of the ADC's sample frequency. Therefore, if an ADC circuit has a sample frequency of 5000 Hz, the highest frequency waveform will be the Nyquist frequency of 2500 Hz. If an ADC is subjected to an analog input signal whose frequency exceeds the Nyquist frequency for that ADC, the converter will output a digitized signal of falsely low frequency.This phenomenon is known as aliasing effect. Selection of Sampling Frequency

44 Selection of Sampling Frequency

45 Selection of Sampling Frequency

46 Aliasing Phenomenon In practice, the sampling frequency = 10 *frequency bandwidth of the analog signal. Selection of Sampling Frequency

47 The system bandwidth frequency is not the only limit to select the sampling frequency, there is also other constraints due to time considerations in ADC, DAC, and microprocessor to execute the control program. In general, the sampling period Ts to control a single loop can be selected using the following relationship: 1/(2 f B ) > Ts > (T ADC + T μp +T DAC ) Where f B = frequency bandwidth of the analog signal T ADC = conversion time of ADC T DAC = conversion time of DAC(can be ignored) T μp = Execution time of the control program in microprocessor, it depends the speed of microprocessor Selection of Sampling Frequency

48 Fourier Transform  F(  ) 00 -0-0  Fs()Fs() 00 -0-0 -s-s ss 2s2s -2  s Fs()Fs() 00 -0-0 -s-s ss 2s2s Aliasing effect Selection of Sampling Frequency

49 Preventing aliases  Make sure your sampling frequency is greater than twice of the highest frequency component of the signal. In practice, take it ten times the highest frequency component.  Pre-filtering of the analog signal  Set your sampling frequency to the maximum if possible Selection of Sampling Frequency

50 Selection of Sampling Frequency without pre- filtering With pre- filtering

51 Z-Transform and Its Properties

52 Z-Transform and Its Properties

53 Z-Transform and Its Properties

54 Z-Transform and Its Properties

55 Z-Transform and Its Properties

56 Z-Transform and Its Properties

57 Z-Transform and Its Properties

58 Z-Transform and Its Properties

59 Z-Transform and Its Properties

60 Z-Transform and Its Properties

61 Inverse of Z-Transform Z-Transform and Its Properties

62 Z-Transform and Its Properties

63 Z-Transform and Its Properties

64 Z-Transform and Its Properties

65 Inverse of Z-Transform For simple poles of X(z) z k-1 at z=z i the corresponding residue K is given by If X(z) z k-1 contains multiple pole of order q at z=z j the corresponding residue K is given by Z-Transform and Its Properties

66 Examples Z-Transform and Its Properties

67 Z-Transform and Its Properties

68 Z-Transform and Its Properties

69  Laplace transform is a necessary mathematical tool for analysis and design of continuous control systems  Z-Transform is a necessary mathematical tool for analysis and design of sampled and digital control systems  MATLAB/SIMULINK software is a powerful tool for analysis, simulation, and design for both continuous and digital control systems Summary

70 In real time digital control applications, the sampling period is bounded by lower and upper limits  The lower sampling limit depends on the computational and conversion delays  The upper sampling limit depends on the frequency bandwidth of the sampled signals  Noise rejection in the measured signals by using low pass filter is necessary to avoid aliasing effect in the frequency spectrum Summary