MATLAB Premier. Middle East Technical University Department of Mechanical Engineering ME 304 1/50

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1 MATLAB Premier Middle East Technical University Department of Mechanical Engineering ME 304 1/50

2 Outline Introduction Basic Features of MATLAB Prompt Level and Basic Arithmetic Operations Scalars, Vectors, and Matrices Matrix Operations and MATLAB Functions Graphics and Data Visualization Programming in MATLAB MATLAB in Control Engineering Operations on Transfer Functions Simulation of LTI Systems Simulink ME 304 /50

3 Introduction MATrix LABoratory MATLAB is an interactive software for numerical / symbolic computations and graphics. Originally designed for matrix computations, solving linear equation sets, data visualization,, etc. The capabilities of MATLAB can be extended through programs written in its own programming language called M-script script. Over the years, the capabilities of MATLAB in every aspect have widened rapidly. It has now become an industry-standard standard computing environment and an outstanding tool for engineering education. ME 304 3/50

4 MATLAB Prompt To get started, type one of these: helpwin, helpdesk, or demo. For product information, type tour or visit (45 + 7)/3 ans = Arithmetic operations: +, -, *, /, ^ ans (ANSwer) is a temporary variable storing the result of an instant calculation.» 100^ 3.5*ans Previous answer can be recalled. ans = e+004 Scientific notation: » sqrt(ans) MATLAB offers a large number of predefined ans = functions like sqrt, sin, cos, tan, exp, log, atan, abs, tanh, and more... ME 304 4/50

5 MATLAB Objects» a =.1 a =.100» b = -9.75;» x = [1; ; 3] x = 1 3» y = [1 3] y = 1 3 Result of various operations can be assigned to the variables you want to define. Variable names are case sensitive. Note that semicolon (;) suppresses the display. Column vector (3 1) is defined: x = 1 3 y is a row vector (1 3): y = [1 3] ME 304 5/50

6 MATLAB Objects (Cont d)» A = [1 3; 4 5 6; 7 8 9] A =» whos Name Size Bytes Class A 3x3 7 double array a 1x1 8 double array ans 1x1 8 double array b 1x1 8 double array x 3x1 4 double array y 1x3 4 double array A is a matrix (3 3): A = whos displays the names of all variables defined in the MATLAB workspace (memory). Grand total is 18 elements using 144 bytes ME 304 6/50

7 Special Variables» i ans =» -5 + a*i i ans = i» cos(pi) ans = -1» a = inf;» 1/a ans = 0 By default, i and j are defined as imaginary unit ( -1). MATLAB allows storage of complex numbers and relevant operations on them. cos(π) a = ME 304 7/50

8 Important Notes» sin(x) ans = » help ans Most arithmetic operators and certain functions work for matrices and vectors as well. An extensive online help is available. ANS Most recent answer. ANS is the variable created automatically when expressions are not assigned to anything else. ANSwer. ME 304 8/50

9 Array Manipulations» A(,3) ans = 6» A(:,) ans = 5 8» A(3,:) ans = 7 8 9» A(1:,:3) ans = Returns the element at nd row, 3rd column. Returns all the elements at nd column. Returns all the elements at 3rd row. Returns submatrix. Column 1 3 A = Row 1 3 ME 304 9/50

10 Array Manipulations (Cont d)» B = [A; zeros(1,3); y] B = Forms a new matrix B using the variables previously defined. Here, the command zeros(m,n) creates an (m by n) matrix consisting of 0 elements. 1 3 B = [A] 3x3 [0] 1x3 [y] 1x3 B = [ ] [ 1 3 ] Matrix A Zero vector Vector y ME /50

11 Array Manipulations (Cont d)» B = [x ones(3,1) A] B = Redefines matrix B using the variables previously defined. Here, the command ones(m,n) creates an (m by n) matrix consisting of 1s B = [x] 3x1 [1] 3x1 [A] 3x3 B = Vector x Unity vector Matrix A ME /50

12 Matrix Operations» A = [1 3; 4 5 6; 7 8 9];» B = [1 0; 4 0 6; 0 8 9];» C = [1 ; 3 4; 5 6];» C' ans = » det(a) ans = 0» inv(b) ans = Let us define these matrices. Matrix transpose using (') operator. The command transpose(c) serves for the same purpose. DETerminant of matrix A. INVerse of matrix B. ME 304 1/50

13 Matrix Operations (Cont d)» A+B ans = MATLAB enables you to perform arithmetic operations (+,-,*) on matrices.» A+C??? Error using ==> + Matrix dimensions must agree.» A*C ans = » C*A??? Error using ==> * Inner matrix dimensions must agree. Matrices must be conformable to carry out a certain operation. Matrix multiplication. Invalid operation due to a mismatch in matrix inner dimensions. ME /50

14 Matrix Operations (Cont d)» A-3 ans = » 3*A ans = » A.*B ans = Scalars (1 1) are exceptions in matrix operations. Multiplies every element of the matrix A by 3. Multiplies the corresponding elements of matrix A and B. The operator (.*) is called array multiplication. ME /50

15 Solving Equation Sets Consider this equation set: x x 5x x 3x 8x 4x 3 + x + 10x = = 1 = Alternatively, we have x = x x 10 x = 1 A x = Therefore, the solution becomes (A A) x = A b x = A b b To implement this in MATLAB, one defines the following:» A = [1 1-4; -3 ; ];» b = [8; 1; ];» x = A\b The operator (\) is called left matrix divide and it essentially solves the linear system using Gauss-Seidel method. Note that x = inv(a)*b yields the same result. ME /50

16 Advanced Matrix Functions» eig(a) Returns all the eigenvalues of matrix A. ans = » [V,D] = eig(a) V = Eigenvectors D = Eigenvalues lu (command) computes LU factorization of a matrix. chol computes the cholesky factorization of a symmetric positive definite matrix. qr computes the QR factorization of a matrix. svd obtains singular value decomposition of a matrix. norm computes various matrix or vector norms. cond, condest, rcond estimates various condition numbers. ME /50

17 Vectors» t = 1:5 t = » t = 0:.1:1 Creates a row vector whose components increase arithmetically. Components can change by non-unit steps. t = Columns 1 through Columns 8 through » t = linspace(0,1,11) Generates a vector with 11 equally spaced elements that lie between 0 and 1. t = Columns 1 through Columns 8 through ME /50

18 D Graphics MATLAB offers a wide variety of built-in in data visualization functions. Function plot is used to generate D graphs. To obtain a graph of y=f(x f(x), one needs to create the domain x and then form the range y with the corresponding function values.» x = linspace(0,*pi,100);» y = sin(x);» plot(x,y);» xlabel( x [rad] );» ylabel( y );» grid on; ME /50

19 D Graphics (Cont d) As the second example, let us plot the following function: x y = f (x) = 1+ x The corresponding MATLAB commands are as follows:» x = linspace(-5,5,100);» y = x./(1+x.*x);» plot(x,y);» xlabel( x );» ylabel( y );» grid on; ME /50

20 Programming Entering all the commands at the MATLAB prompt sequentially is a tedious and slow process. One can type in all the commands in a text file having an extension of m. The file containing these commands is called M-script. When the name of this m file is keyed in just like a command at the prompt, MATLAB executes all the commands sequentially. Similar to a high-level language, MATLAB provides a number of standard programming constructs such as loops and conditionals. ME 304 0/50

21 Conditionals IF Statement if <boolean expression 1> MATLAB statements elseif <boolean expression > MATLAB statements else MATLAB statements end Example if x == 0 y = 0; elseif x == 1 y = 1; else y = 1 *(x.5); end The use of conditionals in MATLAB programs are very similar to those of common high-level languages. The logical operators in MATLAB are <, >, <=, >=, == (logical equal), ~= (not equal). Boolean expressions take the values 1 (true) or 0 (false). ME 304 1/50

22 For Loop FOR Statement for variable = expression MATLAB statements end Example 1 for k = 1:4 disp(k); end Example for k = [ ] disp(k); end for loop repeats the statements as the loop index takes on the values in a given vector. Like an if construct, the loop must be terminated by end statement. disp command simply displays its argument without showing the variable name. ME 304 /50

23 While Loop WHILE Statement while <boolean expression> MATLAB statements end Example x = 1; while 1+x > 1 x = x/; end disp(*x) Output.04e-016 while loop repeats the statements as long as the given boolean expression is true. Another important command is break which terminates the current for or while loop. The sample program is used to test the floating point accuracy ( machine( epsilon ) ) of a particular computer system. ME 304 3/50

24 MATLAB Functions While programming, it is often times necessary to define one s own subroutines (procedures/functions). Unlike high-level languages, these functions cannot be included in the main program. Each function has to be an individual m file. This m file that begins with a line of the following form: function [out1,out,...] = cmd_name(in1,in,...) When a function is invoked, MATLAB creates a temporary workspace. The statements inside the function have no access to the variables used in the main workspace unless they are passed as inputs. When execution ends, all local variables are erased. ME 304 4/50

25 MATLAB Functions (Cont d) function [y,z] = fcn(x) z = 1 + x.*x; y = x./z; plot(x,y) M-file named fcn.m.» x = linspace(-5,5,100);» fcn(x) Calls M-file named fcn.m and passes the vector x.» y1 = fcn(x);» [y,z] = fcn(x); Different output arguments were passed from the MATLAB function fcn. ME 304 5/50

26 Useful MATLAB Functions Poles of a System D(s) = s 5 + 3s 4 + 7s 3 + 5s» den = [ ];» roots(den) ans = i i i i + s + 1 Characteristic Polynomial Let the poles of a system be p 1 = -3; p = -; p 3 = -1; p 3,4 = -5 ± 5i;» poles = [ *i 5-5*i];» poly(poles) ans = The resulting characteristic polynomial is D(s) = s s s s + 610s ME 304 6/50

27 Useful Commands (Cont d) Partial Fractions Expansion» num = [ 5 3 6];» den = [ ];» [r,p,k] = residue(num,den) Consider the following TF: G(s) s s s + 6s + 3s s + 6 r = = Residues The resulting fractions are G(s) = N i= 1 ri s p i + k G (s) = s + 3 s + s + 1 p = k = Poles Remainder ME 304 7/50

28 Useful Commands (Cont d) Polynomial Multiplication 3 P(s) = s + s + 3s + 1 D(s) = P(s)Q(s) =? Q(s) = s + s + 5 Zero/Pole Locations» num = [ 5 3 6];» den = [ ];» pzmap(num,den)» P = [1 3 1];» Q = [1 1 5];» D = conv(p,q) D = Pole-Zero Map The resulting polynomial becomes Imag Axis D(s) = s 5 + 3s s s + 16s Real Axis ME 304 8/50

29 Block Diagram Reduction Transfer Functions in Series X(s) G 1 (s) Y(s) X(s) = G(s) = G As an illustration, let G (s) 1 (s)g (s) Y(s)» num1 = [1 5]; den1 = [1 3 5];» num = [1]; den = [1 1];» [num,den] = series(num1,den1,num,den) num = den = The resulting TF becomes s + 5 G1(s) = ; G(s) = s + 3s s + 1 G(s) = s 3 + 4s s s + 5 NUMerator DENominator ME 304 9/50

30 BD Reduction (Cont d) Transfer Functions in Parallel G 1 (s)» num1 = [1 5]; den1 = [1 3 5];» num = [1]; den = [1 1];» [num,den]=parallel(num1,den1,num,den) X(s) + Y(s) num = G (s) Y(s) X(s) = G(s) = G1 (s) + G As an illustration, let (s) den = The resulting TF becomes s + 5 G1(s) = ; G(s) = s + 3s s + 1 G(s) = s 3 s + 4s + 9s s + 5 ME /50

31 BD Reduction (Cont d) Closed Loop Transfer Functions X(s) + Y(s) X(s) _ G 1 (s) G (s) G (s) = G(s) = 1 1+ G (s)g As an illustration, let 1 (s) Y(s)» num1 = [1 5]; den1 = [1 3 5];» num = [1]; den = [1 1];» [num,den]=feedback(num1,den1,num,den) num = den = The resulting TF becomes s + 5 G1(s) = ; G(s) = s + 3s s + 1 G(s) = s 3 s + 4s + 6s s + 10 ME /50

32 Unit Impulse Response As an illustration, let G(s) = s s s + 4 MATLAB commands are» num = [ 4]; den = [1 4];» impulse(num,den,10) Duration of simulation (10 sec.) ME 304 3/50

33 Unit Step Response As an illustration, let G(s) = s s s + 4 MATLAB commands are» num = [ 4]; den = [1 4];» step(num,den,10) Duration of simulation (10 sec.) ME /50

34 Simulink Run MATLAB (Latest version 6.5 Rev13) Welcome screen of Matlab Select Simulink Simulink Library Browser ME /50

35 Select NEW or OPEN (a previous model file) ME /50

36 An Untitled model is opened and using Simulink Library you can create your model. ME /50

37 Under Sources Select Signal Generator and drag into model ME /50

38 Under Sinks Select Scope and drag into model ME /50

39 Under Signals and Systems Select Mux and drag into model ME /50

40 Select PID Controller Under Simulink Extras Additional Linear and drag into model ME /50

41 Under Discontinuities Select Saturation and drag into model ME /50

42 Under Math Operations Select Sum and drag into model ME 304 4/50

43 Complete Model ME /50

44 Select Simulation Parameters under Simulation Menu ME /50

45 Set Solver Parameters ME /50

46 Set Scope Parameters ME /50

47 Set Format to Array, remove the tick on Limit data points to last, tick Save data to workspace,, and give a variable name. ME /50

48 You may plot the response by using plot command on the command window by using the data saved in the workspace. ME /50

49 Examples from System Responses (PD-control) ME /50

50 END ME /50