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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

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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

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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

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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

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

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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

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

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

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16 Laplace Transform

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17 Laplace Transform

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18 Laplace Transform

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19 Laplace Transform

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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

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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

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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

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23 Example Inverse Laplace Transform

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24 Example Inverse Laplace Transform

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25 Inverse Laplace Transform

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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

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27 Laplace Transform Properties

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28 Laplace Transform Properties

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29 Laplace Transform Properties

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30 Sampling and Digital Systems

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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

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32 Sampling and Digital Systems

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33 Sampling and Digital Systems

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34 Sampling and Digital Systems

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35 Sampling and Digital Systems

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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

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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

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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

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

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40 Sampling and Digital Systems

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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

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42 Sampling and Digital Systems

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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

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44 Selection of Sampling Frequency

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45 Selection of Sampling Frequency

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

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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

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48 Fourier Transform F( ) 00 -0-0 Fs()Fs() 00 -0-0 -s-s ss 2s2s -2 s Fs()Fs() 00 -0-0 -s-s ss 2s2s Aliasing effect Selection of Sampling Frequency

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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

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

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51 Z-Transform and Its Properties

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52 Z-Transform and Its Properties

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53 Z-Transform and Its Properties

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54 Z-Transform and Its Properties

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55 Z-Transform and Its Properties

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56 Z-Transform and Its Properties

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57 Z-Transform and Its Properties

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58 Z-Transform and Its Properties

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59 Z-Transform and Its Properties

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60 Z-Transform and Its Properties

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

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62 Z-Transform and Its Properties

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63 Z-Transform and Its Properties

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64 Z-Transform and Its Properties

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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

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66 Examples Z-Transform and Its Properties

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67 Z-Transform and Its Properties

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68 Z-Transform and Its Properties

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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

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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

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