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Introduction to Scilab application to feedback control Yassine Ariba Brno University of Technology - April 2015 Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 1 / 142 Sommaire 1 What is Scilab ? 4 Application to

  1. What is Scilab ? Matrices Some useful functions : return the dimensions of a matrix size compute the determinant of a matrix det compute the inverse matrix inv rank return the rank of a matrix diag extract the diagonal of a matrix triu extract the upper triangular part of a matrix tril extract the lower triangular part of a matrix spec return the eigenvalues of a matrix --> B = [1 0 ; 2 2]; --> det(B) ans = 2. --> inv(B) ans = 1. 0. - 1. 0.5 --> triu(A) ans = 1. 2. 3. 0. 5. 6. 0. 0. 9. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 22 / 142

  2. What is Scilab ? Matrices Matrix operations Basic operations +, -, *, /, ˆ can be directly performed Watch out for dimension compatibility ! transpose operator : “ .’ ” , transpose and conjugate operator : “ ’ ” --> C = [ 1 0 ; 3 1 ; 0 2]; --> D = [1 1 ; 4 0]; --> B + D ans = 2. 1. 6. 2. --> B * inv(B) ans = 1. 0. 0. 1. --> A * C ans = 7. 8. 19. 17. 31. 26. --> A + B !--error 8 Inconsistent addition. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 23 / 142

  3. What is Scilab ? Matrices Elementary functions are applied element-wise --> M = [0 %pi /2 ; -%pi /2 %pi ]; --> sin(M) ans = 0. 1. - 1. 1.225D -16 --> t = [0:0.2:1]; --> exp(t) ans = 1. 1.2214 1.4918 1.8221 2.2255 2.7182 There are specific versions of those functions for matrix operations expm logm sqrtm sinm cosm ˆ Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 24 / 142

  4. What is Scilab ? Matrices Element-wise operations .* ./ . ˆ --> A = [0 4 ; 1 2]; --> B = [1 2 ; 5 -3]; --> A * B ans = 20. - 12. 11. - 4. --> A .* B ans = 0. 8. 5. - 6. --> A.^2 ans = 0. 16. 1. 4. --> exp(t)./(t+1) ans = 1. 1.0178 1.0655 1.1388 1.2364 1.3591 Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 25 / 142

  5. What is Scilab ? Plotting Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 26 / 142

  6. What is Scilab ? Plotting 2D graphics To plot a curve in the x-y plan use function plot --> x = [0:0.1:2* %pi ]; --> y = cos(x); --> plot(x,y,’*’) Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 27 / 142

  7. What is Scilab ? Plotting 2D graphics To plot a curve in the x-y plan use function plot --> x = [0:0.1:2* %pi ]; --> y = cos(x); --> plot(x,y,’*’) plot traces a point for each couple x(i) - y(i) . x and y must have the same size. By default, a line is drawn between points. The third argument defines the style of the plot. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 27 / 142

  8. What is Scilab ? Plotting --> x = [0:0.1:2* %pi ]; --> y2 = cos (2*x); --> y3 = cos (4*x); --> y4 = cos (6*x); --> plot(x,y1); --> plot(x,y2 ,’r’); --> plot(x,y3 ,’k:’); --> plot(x,y4 ,’g--’); Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 28 / 142

  9. What is Scilab ? Plotting --> x = [0:0.1:2* %pi ]; --> y2 = cos (2*x); --> y3 = cos (4*x); --> y4 = cos (6*x); --> plot(x,y1); --> plot(x,y2 ,’r’); --> plot(x,y3 ,’k:’); --> plot(x,y4 ,’g--’); Several graphics can be displayed. clf : clear the current graphic figure. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 28 / 142

  10. What is Scilab ? Plotting 3D graphics To plot a parametric curve in 3D space use function : param3d --> t = 0:0.01:10* %pi; --> x = sin(t); --> y = cos(t); --> z = t; --> param3d(x,y,z); Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 29 / 142

  11. What is Scilab ? Plotting 3D graphics To plot a parametric curve in 3D space use function : param3d --> t = 0:0.01:10* %pi; --> x = sin(t); --> y = cos(t); --> z = t; --> param3d(x,y,z); Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 29 / 142

  12. What is Scilab ? Plotting To plot a surface in 3D space use function : surf --> x = [-%pi :0.2: %pi ]; --> y = [-%pi :0.2: %pi ]; --> [X,Y] = meshgrid(x,y); --> Z = cos(X).* sin(Y); --> surf(X,Y,Z) --> f=gcf (); --> f.color_map = jetcolormap (32); Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 30 / 142

  13. What is Scilab ? Plotting To plot a surface in 3D space use function : surf --> x = [-%pi :0.2: %pi ]; --> y = [-%pi :0.2: %pi ]; --> [X,Y] = meshgrid(x,y); --> Z = cos(X).* sin(Y); --> surf(X,Y,Z) --> f=gcf (); --> f.color_map = jetcolormap (32); Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 30 / 142

  14. What is Scilab ? Plotting Overview Scilab provides several graphical functions : 2D graphic plot level curves in x-y plan contour surf 3D surface pie “pie” plot histplot histogram plot hist3d 3D histogram plot bar bar plot polar coordinate plot polarplot Some instructions allow to add features to the figure : add a title title xtitle add a title and labels on axis legend add a legend Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 31 / 142

  15. What is Scilab ? Plotting --> x = linspace ( -20 ,20 ,1000); --> y1 = x.* sin(x); --> y2 = -x; --> plot(x,y1 ,’b’,x,y2 ,’r’) --> xtitle(’mon�graphique ’,’label�axe�x’,’label�axe�y’); --> legend(’y1=x*sin(x)’,’y2=-x’); Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 32 / 142

  16. What is Scilab ? Plotting --> x = linspace ( -20 ,20 ,1000); --> y1 = x.* sin(x); --> y2 = -x; --> plot(x,y1 ,’b’,x,y2 ,’r’) --> xtitle(’mon�graphique ’,’label�axe�x’,’label�axe�y’); --> legend(’y1=x*sin(x)’,’y2=-x’); Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 32 / 142

  17. What is Scilab ? Programming Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 33 / 142

  18. What is Scilab ? Programming Scripts A script is a set of instructions gathered in a file. Scilab provides a programming language (interpreted). Scilab includes an editor, but any text editor may be used. File extension should be “ .sce ” (but this is not mandatory). Editor launched from “ Applications > SciNotes ” or by typing editor on the console. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 34 / 142

  19. What is Scilab ? Programming Scripts A script is a set of instructions gathered in a file. Scilab provides a programming language (interpreted). Scilab includes an editor, but any text editor may be used. File extension should be “ .sce ” (but this is not mandatory). Editor launched from “ Applications > SciNotes ” or by typing editor on the console. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 34 / 142

  20. What is Scilab ? Programming Example of a script : myscript.sce // radius of a sphere r = 2; // calculation of the area A = 4* %pi*r^2; // calculation of the volume V = 4* %pi*r^3/3; disp(A,’Area:’); disp(V,’Volume:’); On the console : -->exec(’myscript.sce ’, -1) Area: 50.265482 Volume: 33.510322 The file must be located in the current directory Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 35 / 142

  21. What is Scilab ? Programming Comments : words following // are not interpreted. The current directory can be modified in menu File of the console. The path may be specified instead exec(’C:\Users\yassine\scilab\myscript.sce’, -1) Scripts may also be run from the shortcut in the toolbar. Variables defined in the workspace (from the console) are visible and can be modified in the script. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 36 / 142

  22. What is Scilab ? Programming Another example : myscript2.sce x1 = -1; x2 = 1; x = linspace(x1 ,x2 ,n); y = exp (-2*x).* sin (3*x); plot(x,y); disp(’see�plot�on�the�figure ’); On the console : --> n = 50; -->exec(’myscript2.sce ’, -1) see plot on the figure Here the variable n must be defined beforehand. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 37 / 142

  23. What is Scilab ? Programming Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 38 / 142

  24. What is Scilab ? Programming Looping and branching Scilab language includes classical control structures Conditional statements if boolean expression if then instructions 1 else instructions 2 end if (x >=0) then disp("x�is�positive"); else disp("x�is�negative"); end Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 39 / 142

  25. What is Scilab ? Programming Branching with respect to the value of a variable select variable select case value 1 instructions 1 case value 2 instructions 2 else instructions 3 end select i case 1 disp("One"); case 2 disp("Two"); case 3 disp("Three"); else disp("Other"); end Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 40 / 142

  26. What is Scilab ? Programming Loop control statements for variable = start : step : end for instructions end n = 10; for k = 1:n y(k) = exp(k); end Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 41 / 142

  27. What is Scilab ? Programming Loop control based on a boolean expression while ( boolean expression ) while instructions end x = 16; while ( x > 1 ) x = x/2; end Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 42 / 142

  28. What is Scilab ? Programming And also : instruction break interrupt and exit a loop. instruction continue skip to the next iteration of a loop. Note that as much as possible, use vector / matrix operations instead of loops. The code may run 10 to 100 times faster. This feature of Scilab is known as the vectorization . tic S = 0; for k = 1:1000 S = S + k; end t = toc (); disp(t); tic N = [1:1000]; S = sum(N); t = toc (); disp(t); -->exec(’myscript.sce ’, -1) 0.029 0.002 Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 43 / 142

  29. What is Scilab ? Programming Functions A function is a command that makes computations from variables and returns a result outvar = afunction( invar ) afunction is the name of the function invar is the input argument outvar is the output argument, returned by the function Examples : --> y = sin (1.8) y = 0.9738476 --> x =[0:0.1:1]; --> N = length(x) N = 11. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 44 / 142

  30. What is Scilab ? Programming User can define its own functions function [ out1 , out2 ,...] = myfunction( in1 , in2 ,...) body of the function endfunction once the environment function ... endfunction is executed myfunction is defined and loaded in Scilab after any change in the function, it must be reloaded to be taken into account files including functions generally have the extension “ .sci ” Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 45 / 142

  31. What is Scilab ? Programming Example 1 : calculation of the roots of a quadratic equation. Define and load the function function [x1 ,x2] = roots_equ2d (a,b,c) // roots of ax^2 + bx + c = 0 delta = b^2 - 4*a*c x1 = (-b - sqrt(delta ))/(2*a) x2 = (-b + sqrt(delta ))/(2*a) endfunction Then, you can use it as any other Scilab function --> [r1 ,r2] = roots_equ2d (1,3,2) r2 = - 1. r1 = - 2. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 46 / 142

  32. What is Scilab ? Programming Example 2 : functions are appropriate to define mathematical functions. f ( x ) = ( x + 1) e − 2 x function y = f(x) y = (x+1).* exp (-2*x); endfunction --> y = f(4) y = 0.0016773 --> y = f(2.5) y = 0.0235828 --> t = [0:0.1:5]; --> plot(t,f) Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 47 / 142

  33. What is Scilab ? Programming Variables from workspace are known inside the function but any change inside the function remain local. function z=mytest(x) z = x + a; a = a +1; endfunction --> a = 2; --> mytest (3) ans = 5. --> a a = 2. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 48 / 142

  34. For MATLAB users Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 49 / 142

  35. For MATLAB users MATLAB vs Scilab Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 50 / 142

  36. For MATLAB users MATLAB vs Scilab For MATLAB users Many instructions have the same syntax, but some others not... A dictionary gives a list of the main MATLAB functions with their Scilab equivalents http://help.scilab.org/docs/5.4.1/en_US/section_36184e52ee88ad558380be4e92d3de21.html Some tools are provided to convert MATLAB files to Scilab (e.g. mfile2sci ) http://help.scilab.org/docs/5.4.1/en_US/About_M2SCI_tools.html A good note on Scilab for MATLAB users Eike Rietsch, An Introduction to Scilab from a Matlab User’s Point of View , May 2010 http://www.scilab.org/en/resources/documentation/community Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 51 / 142

  37. For MATLAB users MATLAB vs Scilab Somme differences about the syntax In MATLAB In Scilab search with keywords lookfor search with keywords apropos comments % comments // predefined constants i , pi , inf , true predefined constants %i , %pi , %inf , %t special characters in name of variables special characters in name of variables , # , ! , ? , $ continuation of a statement ... continuation of a statement .. flow control switch case otherwise flow control select case else last element of a vector x(end) last element of a vector x($) Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 52 / 142

  38. For MATLAB users MATLAB vs Scilab Different responses for a same command In MATLAB In Scilab length , the larger of the number of length , the product of the number of rows and columns rows and columns after a first plot , a second one clears after a first plot , a second one holds the current figure the previous division by a vector division by a vector >> x = 1/[1 2 3] --> x = 1/[1 2 3] Error using / Matrix dimensions must x = agree. 0.0714286 0.1428571 operators == and ∼ = compare elements 0.2142857 >> [1 2 3] == 1 x is solution of [1 2 3]*x = 1 ans = 1 0 0 operators == and ∼ = compare objects >> [1 2 3] == [1 2] --> [1 2 3] == 1 Error using == ans = Matrix dimensions must agree. T F F >> [1 2] == [’1’,’2’] --> [1 2 3] == [1 2] ans = ans = 0 0 F --> [1 2] == [’1’,’2’] ans = F Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 53 / 142

  39. For MATLAB users MATLAB vs Scilab Different responses for a same command In MATLAB In Scilab for a matrix A=[1 2 4;4 8 2;6 0 9] for a matrix A=[1 2 4;4 8 2;6 0 9] >> max(A) --> max(A) ans = ans = 7 8 9 9. >> sum(A) --> sum(A) ans = ans = 12 10 18 36. disp must have a single argument disp may have several arguments >> a=3; --> a = 3; >> disp([’the result is --> disp(a,’the result is ’ + ’,int2str(a),’ ...bye!’]) string(a),’hello!’) the result is 3 ...bye! hello! the result is 3 3. note that : prettyprint generates the Latex code to represent a Scilab object Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 54 / 142

  40. For MATLAB users MATLAB vs Scilab Difference when running a script In MATLAB script is invoked by typing its name myscript the m-file must be in a directory of the search path (or specify the path in the call) use a semi-colon to print or not the result of an instruction In Scilab script is invoked with the exec command --> exec(’myscript.sce’) the file must be the working directory (or specify the path in the call) a second argument may be appended (mode) to specify what to print it does not seem to do what the documentation says... not clear for me Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 55 / 142

  41. For MATLAB users MATLAB vs Scilab a simple example, myscript.sce : // a simple script: myscript a = 1 b = a+3; disp(’result�is�’+string(b)) the second argument mode Value Meaning 0 the default value -1 print nothing 1 echo each command line 2 print prompt −− > 3 echo + prompt 4 stop before each prompt 7 stop + prompt + echo Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 56 / 142

  42. For MATLAB users MATLAB vs Scilab --> exec(’myscript.sce ’ ,0) a = 1. result is 4 (as Matlab works) --> exec(’myscript.sce ’ ,-1) result is 4 (only output of disp is printed) --> exec(’myscript.sce ’ ,1) -->// a simple script: myscript -->a = 1 a = 1. -->b = a+3; -->disp(’result�is�’+string(b)) result is 4 (everything is printed (instructions and outputs) Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 57 / 142

  43. For MATLAB users MATLAB vs Scilab Difference when using user defined functions In MATLAB a function is a file, they must have the same name variables in the function are local variables any other functions defined in the file are local functions In Scilab a function is a variable variables in the function are local variables and variables from the calling workspace are known when defined ( function ... endfunction ), functions are not executed but loaded any change in the function requires to reload it (executing the environment) Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 58 / 142

  44. For MATLAB users MATLAB vs Scilab A function may be defined directly in the console : --> -->function [y] = addition(u1 ,u2) --> y = u1 + u2; -->endfunction --> --> addition (2 ,3) ans = 5. or in a file anyscript.sce , and the file must be executed : --> addition (2 ,3) !--error 4 Variable non d´ efinie : addition --> exec(’anyscript.sce ’, -1) --> addition (2 ,3) ans = 5. Any change needs to redefine (reload) the function to be taken into account. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 59 / 142

  45. For MATLAB users Exercices Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 60 / 142

  46. For MATLAB users Exercices Exercices Exercice 1 Calculate the sum of the first 100 integers squared, 1 + 2 2 + 3 2 + 4 2 + . . . + 100 2 in 3 different ways : with instruction for , with instruction while , with instruction sum . Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 61 / 142

  47. For MATLAB users Exercices Exercice 2 Let us consider a monotonic increasing function f over an interval [ a, b ] such that f ( a ) < 0 and f ( b ) > 0 Write an algorithm, based on a dichotomic search, to find the root x 0 such that f ( x 0 ) = 0. f 1 ( x ) = 2 x 4 + 2 . 3 x 3 − 16 x 2 − 8 x − 17 . 5 with x ∈ [0 , 100], f 2 ( x ) = tan( x 2 ) − x with x ∈ [0 . 5 , π/ 3]. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 62 / 142

  48. For MATLAB users Exercices Exercice 3 The Fourier expansion of a square wave f with a period T and an amplitude A is of the form : + ∞ a n = 2 A nπ ω = 2 π � � � f ( t ) = a n cos( nωt ) with nπ sin and T . 2 n =1 Plot the Fourier expansion for different numbers of harmonics. numerical values : A = 2, T = 0 . 5 s , t ∈ [0 , 2]. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 63 / 142

  49. For MATLAB users Exercices Exercice 4 Numerical integration for estimating π π = surface of a unit disc x 2 + y 2 = 1 Compute the area of a quarter of the disc with the rectangle method. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 64 / 142

  50. Xcos Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 65 / 142

  51. Xcos Basics Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 66 / 142

  52. Xcos Basics Xcos Xcos is a graphical environment to simulate dynamic systems. � counterpart of Scilab. R It is the Simulink It is launched in Application/Xcos or by typing xcos Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 67 / 142

  53. Xcos Basics Xcos Xcos is a graphical environment to simulate dynamic systems. � counterpart of Scilab. R It is the Simulink It is launched in Application/Xcos or by typing xcos Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 67 / 142

  54. Xcos Basics A simple example Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 68 / 142

  55. Xcos Basics A simple example block sub-palette sinus Sources/GENSIN f gain Math. Operations/GAINBLK f scope Sinks/CSCOPE clock Sources/CLOCK c drag and drop blocks from the palette browser to the editing window k is variable from the workspace (or from Simulation/Set context) black lines are data flows and red lines are event flows Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 68 / 142

  56. Xcos Basics Settings : frequency = 2 π/ 3, k = 2, final integral time = 12, Ymin= − 3, Ymax= 3, Refresh period = 12 Run simulation from Simulation/Start Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 69 / 142

  57. Xcos Basics Let simulate a mass-spring-damper system The system can be described by the equation of motion m ¨ x ( t ) + f ˙ x ( t ) + kx ( t ) = 0 with the initial conditions : x (0) = 5 and ˙ x (0) = 0 Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 70 / 142

  58. Xcos Basics Let simulate a mass-spring-damper system The system can be described by the equation of motion m ¨ x ( t ) + f ˙ x ( t ) + kx ( t ) = 0 with the initial conditions : x (0) = 5 and ˙ x (0) = 0 The acceleration of the mass is then given by x ( t ) = − 1 � � ¨ kx ( t ) + f ˙ x ( t ) m Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 70 / 142

  59. Xcos Basics modeling and simulation with Xcos block sub-palette sum Math. Operations/BIGSOM f gain Math. Operations/GAINBLK f integral Cont. time systems/INTEGRAL m scope Sinks/CSCOPE x-y scope Sinks/CSCOPXY clock Sources/CLOCK c Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 71 / 142

  60. Xcos Basics parameters : m = 1, k = 2 and f = 0 . 2 Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 72 / 142

  61. Xcos Basics Let add an external force Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 73 / 142

  62. Xcos Basics Let add an external force Define a superblock : Edit/Region to superblock Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 73 / 142

  63. Xcos Basics Example 3 : simulation of a PWM signal Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 74 / 142

  64. Xcos Basics Example 3 : simulation of a PWM signal Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 74 / 142

  65. Xcos Physical modeling Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 75 / 142

  66. Xcos Physical modeling Physical modeling Xcos includes, as a module extension, Modelica blocks. It provides a physical approach for modeling Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 76 / 142

  67. Xcos Physical modeling Physical modeling Xcos includes, as a module extension, Modelica blocks. It provides a physical approach for modeling causal modeling physical modeling with or � = input/output relations acausal modeling Modelica blocks in Xcos require a C compiler Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 76 / 142

  68. Xcos Physical modeling Examples Electrical system Mechanical system Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 77 / 142

  69. Xcos Physical modeling Installation requirements : install a gcc compiler (in the OS), install MinGW toolbox (in Scilab), install Coselica Toolbox for more Modelica blocks (in Scilab, optional) Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 78 / 142

  70. Xcos Physical modeling Installation requirements : install a gcc compiler (in the OS), install MinGW toolbox (in Scilab), install Coselica Toolbox for more Modelica blocks (in Scilab, optional) Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 78 / 142

  71. Xcos Exercices Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 79 / 142

  72. Xcos Exercices Exercices Exercice 1 Let us consider a RC circuit The system can be described by the linear differential equation RC ˙ v ( t ) + v ( t ) = e ( t ) with the initial condition : v (0) = 0. Simulate the response v ( t ) to a step input e ( t ) = 5 V . Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 80 / 142

  73. Xcos Exercices Exercice 2 Let us consider the predator-prey model based on Lotka-Volterra equations  � � x ( t ) ˙ = x ( t ) a − by ( t )    � � y ( t ) ˙ = y ( t ) − c + dx ( t )    where x ( t ) is the number of prey y ( t ) is the number of predator a and d are parameters describing the growth of the prey population and the predator population b and c are parameters describing the death rate of the prey population and the predator population Simulate 2 the evolution of populations with initial conditions x (0) = 4 and y (0) = 2. 2. numerical values : a = 3, b = 1, c = 2, d = 1 Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 81 / 142

  74. Application to feedback control Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 82 / 142

  75. Application to feedback control A brief review Sommaire 1 What is Scilab ? 4 Application to feedback control Introduction A brief review Basics System analysis in Scilab Matrices Bode plot Plotting Simulation with Xcos Programming Exercices 2 For MATLAB users 5 Classical control design MATLAB vs Scilab Loopshaping Exercices Phase lag controller 3 Xcos Phase lead controller Basics PID controller Physical modeling Exercices Exercices Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 83 / 142

  76. Application to feedback control A brief review A brief review Objective : Design a controller to control a dynamical system. The output to be controlled is measured and taken into account by the controller. ⇒ feedback control Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 84 / 142

  77. Application to feedback control A brief review Example : angular position control of a robotic arm. Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 85 / 142

  78. Application to feedback control A brief review Example : angular position control of a robotic arm. u ( t ) is the control voltage of the DC motor (actuator) θ ( t ) is the angular position of the arm (measured with a sensor) The input-output relationship is given by : θ ( t ) + ˙ ¨ θ ( t ) = u ( t ) Y. Ariba - Icam, Toulouse. Brno University of Technology - April 2015 85 / 142

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