1. 1 MATLAB for Engineering Applications Introduced by دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل Department of Cintrol Engineering دانشگاه صنعتی بابل دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

2. 2 Introduction مقدمه Matlab: Matrix Laboratory لابراتوار ماتریسها Matlab is both a computer programming language and a software environment for using language effectively. هم یک زبان برنامه نویسی است هم یک نرم افزار Matlab has a number of add-on software modules, called toolboxes, that perform more specialized computations, dealing with applications such as: بسته های نرم افزاری به نام جعبه ابزار برای انجام امور کاملا تخصصی دارد. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

3. 3 Tool Boxes جعبه ابزار دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل پردازش سیگنال و تصویرImage and signal processing سیستمهای اقتصادی طراحی سیستمهای کنترل Financial analysis Control systems design سیستمهای مخابراتی منطق فازی Communication systems Fuzzy logic شبکه های عصبی طراحی فیلتر Neural networks Filter design بهینه سازی معادلات دیفرانسیلی با مشتقات جرئی Optimization Partial differential equations آمار شناسایی سیستمها Statistics System identification Virtual reality Wavelet 5 اسفند ماه 1389

4. 4 Contents Matlab Windows and Menus Basic Operations and Arrays Files, Functions Plotting with Matlab Programming with Matlab Numerical Methods for Differential Equations دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

5. 5 1. Matlab Windows and Menus دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

6. 6 1. Matlab Windows and Menus Enter commands and expressions at the prompt position (>>) in command window. Ex: r = 8/10 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

7. 7 1. Matlab Windows and Menus u = [0:.1:10]  u = 0,0.1,0.2,0.3,…,9.8,9.9,10 z = 5*sin(u)  z = 5sin(0), 5sin(0.1), …5sin(9.9), 5sin(10). دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

8. 8 1. Matlab Windows and Menus ایجاد فایل بر حسب احتیاج با استفاده ار ادیتور MATLAB به نام.m file دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

9. 9 1. Matlab Windows and Menus دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

10. 10 2. Basic Operations and Arrays Row vector; بردارهای سطری >> r = [2,4,10]; >> n = [2,4,10] n = 2 4 10 >> n = [2 4 10] n = 2 4 10 Column vector; بردارهای ستونی >> r = [2; 4;10] r = 2 4 10 >> y = [ 2 4 10]’ y =2 4 10 Ex: r =[ 2 4 10] >> v = 5*r v = [10 20 50] >>w = r + v w = [12 24 60] >> u = [r, w] u = [2 4 10 12 24 60] Ex: Creat a row vector from 5 to 30, incremented by 0.1. >> t = [5:0.1:30] t = 5, 5.1,5.2,…, 29.8,29.9,30. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

11. 11 2. Basic Operations and Arrays Matrix; ماتریس >> a = [2,4,10;16,3,7] a = 2 4 10 16 3 7 Ex: A = [6, -2;10,3;4,7], B = [9,8;-5,12] >> A*B ans 6424 75116 1 116 Array operations; عملیات بردارها Ex: a =[1:5]; b = [3:7]; >> a = [1:5] a = 12345 >> b = [3:7] b = 34567 >> c = a.* b c = 38152435 >> c = a./ b c = 0.333 0.5 0.6 0.6670.7143 >> c = a.^ b c = 116 243409678125 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

12. 12 2. Basic Operations and Arrays Ex: >> x = [0:0.01:1]; >> y = exp(-x).* sin(x)./sqrt(x.^2 +1); >>plot(x,y) دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

13. 13 3. Files and Functions Matlab uses two types of M-files: script files and function files. فایل نوشتاری و فایل توابع A script file contains a sequence of Matlab commands and is useful when you need to use many commands or arrays with many elements. A function file is useful when you need to repeat a set of commands several times. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

14. 14 3. Files and Functions Example of a simple script file computes the matrix C=AB and displays it on the screen. % Program ‘prod_abc.m’ %This program computes the %matrix product C = A * B,and %displays the result. A=[1,4;2,5]; B=[3,6;1,2]; C=A*B In the Matlab command window type the script file’s name prod_abc to execute the program. >>prod_abc C= 7 14 11 22 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

15. 15 3. Files and Functions Function file: All the variables in a function file are local, which means their values are available only within the function. Function files are useful when you need to repeat a set of commands several times. Function files are like functions in C, subroutines in BASIC, and procedures in Pascal. Its syntax is as follows: function [output variables] = function_name(input variables); Note: (i) output variables are enclosed in square brackets, the square brackets are optional when there is only one output.[ ] نقش (ii) input variables are enclosed with parentheses نقش پرانتز (iii) function_name must be the same as the filename in which it is saved (with the.m extension) یکسانی نام فایل و تابع (iv) function is called by its name. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

16. 16 3. Files and Functions Ex:, find the value of x that gives a minimum of y for. function y=f2(x) y=1-x.* exp (-x); Save it as f2.m In command window, type x=fmin(‘f2’, 0, 5). The answer is x = 1. To find the minimum value of y, type y = f2(x). دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

17. 17 3. Files and Functions دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

18. 18 4. Plotting with Matlab XY plotting functions Ex:>>x = [0:0.1:52]; >>y = 0.4*sqrt(1.8*x); >>plot(x,y) >>xlabel(‘Distance (miles)’) >>ylabel(‘Height (miles)’) >>title(‘Tocket height as a function of downrange distance’) دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

19. 19 4. Plotting with Matlab The function plot command fplot Syntax fplot(‘string’,[xmin xmax]) or fplot(‘string’,[xmin xmax ymin ymax]) Ex:>> f = ‘cos(tan(x))-tan(sin(x))’; >>fplot(f, [1 2]) دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

20. 20 4. Plotting with Matlab Ex: plot the hyperbolic sine and tangent x = [0:0.01:2]; y = sinh(x); z = tanh(x); plot(x,y,x,z,’- -‘), xlabel(‘x’), … ylabel(‘Hyperbolic sine and tangent’),… legend(‘sinh(x)’, ‘tanh(x)’) دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

21. 21 4. Plotting with Matlab Data markers, line types and colors Data markers:. * x o + Line types: - -- -. : Colors: black(k) Blue(b) Green(g) Red(r) Yellow(y) دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

22. 22 4. Plotting with Matlab Surface mesh plots (z=f(x,y)) [X,Y]=meshgrid(x,y), if x=[xmin:xspacing:xmax], y=[ymin:yspacing:ymax] The meshgrid function generates the grid point in the xy plane, and then evaluate the function f(x,y) at these points. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

23. 23 4. Plotting with Matlab Ex: with a spacing of 0.1. >>[X,Y]=meshgrid(-2:0.1:2); >>Z=X.*exp(-((X-Y.^2).^2+Y.^2)); >>mesh(X,Y,Z), xlabel(‘x’), ylabel(‘y’),zlabel(‘z’) دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

24. 24 4. Plotting with Matlab Contour plots Ex: >>[X,Y]=meshgrid(-2:0.1:2); >>Z=X.*exp(-((X-Y.^2).^2+Y.^2)); >>contour(X,Y,Z), xlabel(‘x’), ylabel(‘y’) دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

25. 25 5. Programming with Matlab Relational operators < less than <=less than or equal to > greater than >=greater than or equal to = =equal to ~=not equal to Logical operations ~ not & and |or xor(a,b)exclusive or دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

26. 26 5. Programming with Matlab Conditional statements (1) if logical expression statements end Ex: if x >= 0 y = sqrt(x) end or it can be written on a single line if x >= 0, y = sqrt(x), end دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

27. 27 5. Programming with Matlab (2) Nest if statements if logical expression 1 statement group 1 if logical expression 2 statement group 2 end دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

28. 28 5. Programming with Matlab (3) The else statement if logical expression statement group 1 else styatement group 2 end Ex: and that if x >= 0 y= sqrt(x) else y = exp(x) – 1 end دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

29. 29 5. Programming with Matlab (4) The elseif statement if logical expression 1 statement group 1 elseif logical expression 2 statement group end Ex: y = lnx if x >= 5, y = sqrt(x) if 0<= x < 5 if x >= 5 y = log(x) elseif x >= 0 y = sqrt(x) end دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

30. 30 5. Programming with Matlab Loops for loop variable = m:s:n Statements end m:s:n=>initial value m, incremented by the value s and terminated by the value n. Note: s may be negative, k=10:-2:4 produce k = 10,8,6,4. if s is omitted, the step value defaults to one. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

31. 31 5. Programming with Matlab Ex: implied loops x= [0:5:100]; y = cos(x); To achieve the same result using a for loop, we must type for k = 1: 21 x = (k-1) *5; y(k) = cos(x); end دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

32. 32 5. Programming with Matlab While loop is used when the looping process terminates because a specified condition is satisfied, and thus the number of passes is not known in advance. while logical expression statements End Ex: x = 5; while x <25 disp (x) x = 2*x-1; end The results displayed by the disp statement are 5, 9, and 17. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

33. 33 5. Programming with Matlab For the while loop to function properly, the following two conditions must occur; The loop variable must have a value before the while statement is executed. The loop variable must be changed somehow by the statements. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

34. 34 5. Programming with Matlab Switch structure: Anything programmed using switch can also be programmed using if structures. However, for some applications the switch structure is more readable than code using the if structures. switch input expression (scalar or string) case value1 statement group 1 case value2 statement group 2 … otherwise statement group n end دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

35. 35 5. Programming with Matlab Ex: variable angle switch angle case 45 disp(‘Northeast’) case 135 disp(‘Sortheast’) case 225 disp(‘Southwest’) case 315 disp(‘Northwest’) otherwise disp(‘Direction Unknown’) end دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

36. 36 6. Numerical Methods for Differential Equations Euler method: first order differential equation, high order differential equation In general, as the accuracy of the approximation is increased, so is the complexity of the programming involved. Understanding the concept of step size and its effects on solution accuracy is important. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

37. 37 6. Numerical Methods for Differential Equations Consider the equation r(t) is a known function We have, where is the step size. For convenience, دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

38. 38 6. Numerical Methods for Differential Equations Example:, r = -10, y(0) = 2, plots the y trajectory over the range, step size, which is 20% of the time constant. True solution is, with time constant دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

39. 39 6. Numerical Methods for Differential Equations eulermethod.m r = -10; delta = 0.02; y(1) = 2; k = 0; for time = [delta:delta:0.5] k = k+1; y(k+1) = y(k)+r*y(k)*delta; end t = [0:delta:0.5]; y_true = 2 * exp(-10*t); plot(t,y,’o’,t,y_true); دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

40. 40 6. Numerical Methods for Differential Equations دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

41. 41 6. Numerical Methods for Differential Equations There is some noticeable error from the result of above figure. If we use a step size equal to 5% of the time constant, the error would not be noticeable on the following plot. However, the smaller step size requires longer run times. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

42. 42 6. Numerical Methods for Differential Equations Matlab ODE Solvers ode23 and ode45 can be used to solve differential equations. Basic syntax for solvers: [t,y]=ode23(‘ydot’,tspan,y0) where ydot is the name of the function file whose inputs must be t and y and whose output must be a column vector representing dy/dt;that is f(t,y). tspan contains the starting and ending values of the independent variable t. y0 is the value دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

43. 43 6. Numerical Methods for Differential Equations Ex: Response of an RC circuit suppose the value of RC = 0.1 second. Using the numerical method to find the free response for the case where the applied voltage v is 0 and the initial capacitor voltage is y(0)=2 volts دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

44. 44 6. Numerical Methods for Differential Equations rccirc.m function ydot=rccirc(t,y) ydot = -10*y; test2.m clear; [t,y] =ode45(‘rccirc’, [0,0.4],2); y_true = 2*exp(-10*t); plot(t,y,’o’,t,y_true) xlabel(‘Time(sec)’) ylabel(‘Capacitor Voltage’) دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

45. 45 6. Numerical Methods for Differential Equations Ex: The following equation describes the motion of a mass connected to a spring with viscous friction acting between the mass and the surface. Another force f(t) also acts on the mass. Let We have These two equations can be written as one matrix equation as follows: دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

46. 46 6. Numerical Methods for Differential Equations msd.m function xdot = msd(t,x) % function file for mass with spring and damping % position is first variable, velocity is second variable. global c f k m A=[0,1;-k/m,-c/m]; b=[0;1/m]; xdot=A*x+b*f; test.m global c f k m m=1;c=2;k=5;f=10; [t,x]=ode23(‘msd’,[0,5],[0,0]); plot(t,x), xlabel(‘Time(sec)’),… ylabel(‘Displacement (m) and Velocity (m/sec)’),… gtext(‘Displacement’), gtext(‘Velocity’) دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

47. 47 6. Numerical Methods for Differential Equations دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

48. 48 6. Numerical Methods for Differential Equations دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389

49. 49 Summary 1. Matlab Windows and Menus 2. Basic Operations and Arrays 3. Files and Functions 4. Plotting with Matlab 5. Programming with Matlab 6. Numerical Methods for Differential Equations دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 5 اسفند ماه 1389