1. Introduction to MATLAB and Simulink

2. Outline Section I (Jan 30 th, 2016) ◦ Background ◦ Basic Syntax and Commands ◦ Linear Algebra ◦ Loops Section II (Feb 6 st, 2016) ◦ Graphing & Plots ◦ Scripts and Functions Section III (Feb 20 th, 2016) ◦ Linear & System of Equations Solving ◦ ODE Solving Section IV (Mar 5 th, 2016) ◦ Simulink

3. Section I January30 th, 2015 ◦ Background ◦ Basic Syntax and Commands ◦ Linear Algebra ◦ Loops

4. Background MATLAB = Matrix Laboratory (developed by The MathWorks). Opening MATLAB Working Memory Command History Command Window Working Path

5. Variables Have not to be previously declared Variable names can contain up to 63 characters Variable names must start with a letter followed by letters, digits, and underscores( Valid: x1 = 5, Invalid: 1x = 5) Variable names are case sensitive (a1 does NOT equal A1)  EX: Using command window >> x = 5; % this is used as a comment >> y = 3; >> z = x + y z = 8  Semicolon (;) : Suppresses output  Percentage (%) : Commenting. Only good for that line Operators

6. MATLAB Special Variables ansDefault variable name for results piValue of pi number (3.14) epsSmallest incremental number infInfinity NaNNot a number e.g. 0/0 realminThe smallest usable positive real number realmaxThe largest usable positive real number i,jImaginary unit

7. MATLAB Assignment & Operators Assignment=a = b (assign b to a) Addition+a + b Subtraction-a - b Multiplication*or.* a*b or a.*b Division/or./ a/b or a./b Power^.^ a^b or a.^b

8. MATLAB Matrices MATLAB treats all variables as matrices. For our purposes a matrix can be thought of as an array, in fact, that is how it is stored. Vectors are special forms of matrices and contain only one row OR one column. Scalars are matrices with only one row AND one column

9. MATLAB Matrices  A matrix with only one row is called a row vector. A row vector can be created in MATLAB as follows (note the commas): » rowvec = [12, 14, 63] rowvec = 12 14 63  A matrix with only one column is called a column vector. A column vector can be created in MATLAB as follows (note the semicolons): » colvec = [13 ; 45; -2] colvec = 13 45 -2

10. MATLAB Matrices  A matrix can be created in MATLAB as follows (note the commas AND semicolons): » matrix = [1, 2, 3 ; 4, 5,6 ; 7, 8, 9] matrix = 1 2 3 4 5 6 7 8 9

11. Index MATLAB index starts with 1, NOT 0! Vector Index: ◦ a = [22 17 7 4 42]  a(1) = 22  a(3) = 7 Matrix Index: ◦ a = [7 12 42; 5 1 23; 4 9 10];  a(1, 3) = 42  a(3, 2) = 9

12. Some matrix functions in MATLAB X = ones(r,c)Creates matrix full with ones X = zeros(r,c)Creates matrix full with zeros A = diag(x)Creates squared matrix with vector x in diagonal [r,c] = size(A)Return dimensions of matrix A X = A’Transposed matrix X = inv(A)Inverse matrix squared matrix X = pinv(A)Pseudo inverse d = det(A)Determinant [X,D] = eig(A)Eigenvalues and eigenvectors [U,D,V] = svd(A)singular value decomposition

13. Dot Operator For scalar operations, nothing new is needed. Example: a = 5; b = 3; ==> c = a*b %c = 15 For element operations, a dot must be used before the operator. Note: Dot operator not the same as dot product! Example: ◦ a = [1 2 3 4]; ◦ b = [5 6 7 8]; ◦ c = a*b ◦ Result: ??? Error using ==> mtimes inner matrix dimensions must agree Now, try: ◦ c = a.*b %notice the dot! ◦ Result: c=[5 12 21 32] Notice what it is doing: a(1)*b(1), a(2)*b(2), etc.

14. Dot Product Check Dimensions ◦ A*B ◦ A*B’ ◦ A’*B Use MATLAB command >> C=dot(A,B) Cross Product By definition Use MATLAB command >> C=cross(A,B) Vector Products Consider & generate two random 3*1 vectors, A & B A=rand(3,1), B=rand(3,1) A(2)*B(3)-A(3)*B(2) A(3)*B(1)-A(1)*B(3) A(1)*B(2)-A(2)*B(1)

15. Loops Loop statements do not need parenthesis. The statements are recognized via tabs. 3 general types of loops: ◦ If/else/elseif loops ◦ For loops ◦ While loops There is a 4 th type, called nested loop, that can be any of the above 3 (and any combinations of them).

16. If/else/elseif Loops The condition must be previously defined!

17. For Loops Counter variable does not have to be i; it can be any variable Iterations can be tightly controlled with min:stepsize:max. No need to pre-define the counter because you are declaring in the for loop itself! for i=1:10 statement end

18. While Loops As long as output of condition is a logic true, it will continue looping until the condition becomes false. BE CAREFUL OF INFINITE LOOPS. while condition statement end

19. Nested Loops Can use a mix of the different types of loops. Very useful for performing algorithm/operations on vectors and matrices.

20. Examples Write a code to print out integer numbers from 5 to 36 Write a code to print out integers between 45 to 109 which are divisible to 3 Write a code to print out integers between 45 to 109 which are divisible to 5 Write a code : ◦ If n is divisible to 3 print “divisible to 3” ◦ elseIf n is divisible to 5 print “divisible to 5” ◦ Elseif n is divisible by 5 and 3 print “divisible by 15”

21. Section II Feb 6 st, 2016 ◦ Graphing & Plots ◦ Scripts and Functions ◦ Linear & System of Equations Solving

22. Plots plot –plot(x,y,linespec) ◦ x and y vectors must be same length! subplot –subplot(m,n,p) where mxn matrix, p = current graph figure –creates new window for graph title –creates text label at top of graph xlabel/ylabel–horizontal and vertical labels for graph

23. Example x=0:pi/100:2*pi; y=sin(x); figure; plot(x,y) ◦ xlabel ◦ ylabel ◦ title ◦ figure

24. example x= 0:pi/100:2*pi; y1=sin(x); y2=sin(x-0.25); y3=sin(x-0.5); figure; plot(x,y1,x,y2,’--’,x,y3,’:’); legend

25. subplot subplot –subplot(m,n,p) where mxn matrix, p = current graph

26. Axis-specific Helps focus on what is important on the graph. Change only the x or y axis limits: ◦ xlim([xminxmax]) or ylim([yminymax]) ◦ min and max can be positive or negative. example

27. Some useful commands Grid on Hold on Legend Plot3 : example ◦ t=0:pi/50:10*pi; ◦ st=sin(t); ◦ ct=cos(t) ◦ figure; ◦ plot3(st,ct,t)

28. plotyy 2D line plots with y-axis on both left and right side ◦ [AX,H1,H2]=plotyy(x1,y1,x2,y2)

29. Scripts & functions Two kind of M-files: ◦ Scripts ◦ Functions:  With parameters and returning values  Only visible variables defined inside the function or parameters  Usually one file for each function defined  Structure:

30. Function example In function: ◦ function y=average(x) ◦ y=sum(x)/length(x); ◦ end In script: ◦ z=1:99; ◦ average(z)

31. Polynomials We can use an array to represent a polynomial. To do so we use list the coefficients in decreasing order of the powers. For example x3+4x+15 will look like [1 0 4 15] To find roots of this polynomial we use roots command. roots ([1 0 4 15]) To create a polynomial from its roots poly command is used. poly([1 2 3]) where r1=1, r2=2, r3=3 To evaluate the new polynomial at x =5 we can use polyval command. polyval([ 1 -6 11 -6], 5)

32. Systems of Equations Consider the following system of equations ◦ x+5y+15z=7 ◦ x-3y+13z=3 ◦ 3x-4y-15z=11 One way to solve this system of equations is to use matrices. First, define matrix A: ◦ A=[1 5 15; 1 -3 13; 3 -4 15]; Second, matrix b: ◦ b=[7;3;11]; Third, we solve the equation Ax=b for x, taking the inverse of A and multiply it by b: ◦ x=inv(A)*b Note that we cannot solve equation Ax=b by dividing b by A because vectors A and b have different dimensions!

33. Section 3 MATLAB February 20 th, 2016 ◦ Symbolic Solving ◦ Systems of Equations ◦ ODE Solving Farhad Goodarzi

34. Systems of Equations Consider the following system of equations ◦ x+5y+15z=7 ◦ x-3y+13z=3 ◦ 3x-4y-15z=11 One way to solve this system of equations is to use matrices. First, define matrix A: ◦ A=[1 5 15; 1 -3 13; 3 -4 15]; Second, matrix b: ◦ b=[7;3;11]; Third, we solve the equation Ax=b for x, taking the inverse of A and multiply it by b: ◦ x=inv(A)*b Note that we cannot solve equation Ax=b by dividing b by A because vectors A and b have different dimensions!

35. Symbolic toolbox Use syms command ◦ syms F3 x a b ◦ F3=sqrt(x) ◦ Int(F3,a,b)

36. Symbolic toolbox example Define functions F1=6x3-4x2+bx-5, F2= Sin (y), and F3= Sqrt(x). Use int() function to determine:

37. Solving ODE with dsolve, 1 st order First order equations ◦ y’=xy  y=dsolve(‘Dy=y*x’,’x’)  y=dsolve(‘Dy=y*x’,’y(1)=1’,’x’) ◦ Or  eq1=‘Dy=y*x’;  y=dsolve(eq1,’x’); ◦ Issues  our expression for y(x) isn’t suited for array operations: vectorize()  y, as MATLAB returns it, is actually a symbol : eval() ◦ Plotting  x = linspace(0,1,20);  z = eval(vectorize(y));  plot(x,z)

38. Solving ODE with dsolve, 2 nd order Second order equations ◦ EQ:  eqn2 = ’D2y + 8*Dy + 2*y =cos(x)’;  inits2 = ’y(0)=0, Dy(0)=1’;  y=dsolve(eqn2,inits2,’x’) ◦ Plotting  x = linspace(0,1,20);  z = eval(vectorize(y));  plot(x,z)

39. Solving ODE with dsolve, systems Systems of equations ◦ EQs:  [x,y,z]=dsolve(’Dx=x+2*y-z’,’Dy=x+z’,’Dz=4*x-4*y+5*z’)  inits=’x(0)=1,y(0)=2,z(0)=3’;  [x,y,z]=dsolve(’Dx=x+2*y-z’,’Dy=x+z’,’Dz=4*x-4*y+5*z’,inits)  Notice that since no independent variable was specified, MATLAB used its default, t. ◦ Plotting  t=linspace(0,.5,25);  xx=eval(vectorize(x));  yy=eval(vectorize(y));  zz=eval(vectorize(z));  plot(t, xx, t, yy, t, zz)

40. Solving ODE with dsolve, systems solve the given differential equations symbolically.

41. ODE Solving Numerically Defining an ODE function in an M-file Solving First-Order ODEs Solving Systems of First-Order ODEs Solving higher order ODEs

42. Numerical methods are used to solve initial value problems where it is difficult to obtain exact solutions MATLAB has several different built-ins functions for numerical solution of ODEs

43. Solvers

44. Solving first-order ODEs Define the function M-file  function y_dot=EOM(y,t)  global alpha gamma  y_dot=alpha * y - gamma * y^2;  end Make a script M-file  global alpha gamma  alpha=0.15;  gamma=3.5;   y0=1;  [y t]=ode45(@EOM,[0 10],y0);  plot(t,y);

45. Solving systems of ODEs Example Create function containing the equations Change the error tolerance using odeset Plotting the columns of returned vector

46. Solving a Stiff system Equations Create the function Ode and plotting

47. Example Use “ode45” to solve the following differential equation and plot y(x) in the interval of [0,6 π ]. Put your name in the plot title.

48. Solving higher-order ODE Project: Simple pendulum Equation of motion is given by: 2 nd order Nonlinear ODE Convert the 2 nd order ODE to standard form

49. Simple Pendulum Initial conditions and constants Coding ◦ Make a function M-file for equation of motion  function z_dot=EOM_pendulum(t,z)  global G L  theta=z(1);  theta_dot=z(2);  theta_dot2=-(G/L)*sin(theta);  z_dot=[theta_dot;theta_dot2];  end ◦ Make a script M-file to run the code  global G L  G=9.8;  L=2;  tspan=[0 2*pi];  inits=[pi/3 0];  [t, y]=ode45(@EOM_pendulum,tspan,inits);

50. Lorenz Equations Initials and constants, T=[0 20] Plot x vs. z, check if you get same results as

51. Section 4 March 5 th, 2016 ◦ Simulink

52. Simulink >> simulink ◦ Continuous and discrete dynamics blocks, such as Integration, Transfer functions, Transport Delay, etc. ◦ Math blocks, such as Sum, Product, Add, etc ◦ Sources, such as Ramp, Random Generator, Step, etc

53. Useful blocks Math and Control inputs and outputs

54. Example Open a new window and drag the following blocks into that file Run the simulation and double click on the scope to see the results

55. Example Open a new window and drag the following blocks into that file Run the simulation and double click on the scope to see the results

56. Example Open a new window and drag the following blocks into that file Run the simulation and double click on the scope to see the results

57. simple model Build a Simulink model that solves the differential equation Initial condition First, sketch a simulation diagram of this mathematical model (equation) Input is the forcing function 3sin(2t) Output is the solution of the differential equation x(t) Now build this model in Simulink

58. simple model Build a Simulink model that solves the differential equation Double-click on the Sine Wave block to set amplitude = 3 and freq = 2. This produces the desired input of 3sin(2t)

59. simple model It should look like this, change the initial condition in integrator to -1

60. Example – Toy Train consisting of an engine and a car. Assuming that the train only travels in one direction, we want to apply control to the train so that it has a smooth start-up and stop, along with a constant-speed ride The mass of the engine and the car will be represented by M1 and M2, respectively. The two are held together by a spring, which has the stiffness coefficient of k. F represents the force applied by the engine, and the Greek letter, mu, represents the coefficient of rolling friction.

61. Example – Toy Train System

62. Example – Toy Train System

63. Example – Toy Train Build this

64. Example – Toy Train Before running the model, we need to assign numerical values to each of the variables used in the model. Create an new m-file and enter the following commands. M1=1; M2=0.5; k=1; F=1; mu=0.002; g=9.8; Execute your m-file to define these values. Simulink will recognize MATLAB variables for use in the model.

65. Pendulum Example Try the following blocks