Lab #1 Revision to MATLAB

Transcription

1 Lab #1 Revision to MATLAB Objectives In this lab we would have a revision to MATLAB, especially the basic commands you have dealt with in analog control. 1. What Is MATLAB? MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include: Math and computation. Algorithm development. Data acquisition. Modeling, simulation, and prototyping. Data analysis, exploration, and visualization. Scientific and engineering graphics. Application development, including graphical user interface building. 2. Basic Operations: 2.1 Computation % This is an example of arithmetic >> 1+2 Ans = Suppressing Display of Results >> 1+2; 2.3 The workspace Current values are stored in the MATLAB workspace. The 'who' command tells you what variable are currently defined. The 'whos' command tells you more about the variables. Use the 'clear' command to delete variables from the workspace. >> x=2; >> y=3; >> z=x+y z = 5

2 >> who Your variables are: x y z >> whos Name Size Bytes Class x 1x1 8 double array y 1x1 8 double array z 1x1 8 double array Grand total is 3 elements using 24 bytes >> 2.4 Command Line Editing and Recall: MATLAB does not work like a spreadsheet. It will not go back and repeat calculations unless you tell it to. A comma (, ) separates statements on a line. Three periods ( ) continues on the next line. The up arrow ( ) goes to the previous command. The down arrow ( ) goes to the next command. The sideways arrows move within the command. MATLAB is case sensitive. 2.5 MATLAB Operations and Conventions: Expression follow the standard of precedence Exponentiation. Multiplication and Division. Addition and Subtraction. Expressions are evaluated from left to right. Parentheses work from inner to outer. Complex numbers are entered using the characters ' i ' and ' j ', unless other definitions have been assigned to them. 2.6 To ask for online help with a command, type 'help ' followed by the command name. For instance >> help cos COS Cosine. COS(X) is the cosine of the elements of X. 3. Matrices: MATLAB works essentially with only one kind of object a rectangular numerical matrix with possibly complex entries. All variables represent matrices. A scalar is a 1x1 matrix and a vector is a matrix with only one row or column. There are several ways to enter matrices into MATLAB including: Entering an explicit list of elements

3 Using built-in statement or function. Created in a disk file with your local editor. Loaded from external data files or applications 3.1 Generating Vectors and the Colon Notation We can create a vector (matrix with only one row) by directly inputting the elements >> W=[ ] W = We could also from a vector by using the colon ( : ) operator. " Colon Notation " is used to generate vectors and to reference sub matrices. The colon notation and subscripting by integral vectors and keys to efficient manipulation in MATLAB. Creative use of these features to factorize operations lets you minimize the use of loops (which slows MATLAB down) and makes code simple and readable. >> Q=[1:5] Q = We could also use any increment to construct matrices >> E=[2:2:10] E = Generating Matrices: Column matrices can be created with the semicolon ( ; ) >> A=[1,2,3;4,5,6;7,8,9] A = Referencing Individual Entries: Round parentheses are used to reference individual elements of a matrix

4 >> A(1,2) ans = 2

5 3.4 The colon notation can be used to access sub matrices of a matrix. A(1:4,3) is the column vector consisting of the first 4 entries of the 3rd column of matrix A. 3.5 Deleting Rows and Columns You can delete rows and columns from a matrix using just a pair of square brackets. Start with >> X=[ ; ; ; ] X = >> %To delete the second column of X >> X(:,2)=[] X = Matrix Operations: The following matrix operation are available + addition - subtraction * multiplication ^ power transpose ( real ) or conjugate transpose ( ' complex ).' transpose ( real or complex ) \ left division / right division 4.1 Matrix Division If A is an invertible square matrix and b is a compatible column vector, or respectively a compatible row vector, then

6 x=a\b is the solution of A*x = b x= b/a is the solution of x*a=b 4.2 Entry-Wise Operations The matrix operations addition and subtraction are already entry-wise but the other operations are not; they are matrix operations. The other operation,*,^,\,/ can be made to operate entry-wise by preceding them with a period. >> B=[1 2;3 4] B = >> B*B ans = >> B.*B ans = Matrix Building Functions Eye Zeros Ones Triu Tril Rand Randn For example, >> zeros(2,3) ans = Identity matrix Matrix of zeros Matrix of ones Upper triangular part of matrix Lower triangular part of matrix Uniformly distributed random elements Normally distributed random elements

7 >> A=rand(3) A = >> triu(a) ans = A particular element of a matrix can be assigned: >> M(1,2) = 5; % places the number 5 in the first row, second column Here are some examples: >> Z = zeros(2,4) Z = >> F = 5*ones(3,3) F = >> N = fix(10*rand(1,10)) N = >> R = randn(4,4) R =

8 6. The Colon Operator The colon, :, is one of the most important MATLAB operators. It occurs in several different forms: The expression 1:10 is a row vector containing the integers from 1 to 10, i.e To obtain non-unit spacing, specify an increment. For example, >> 100:-7:50 % a row vector containing the integers from Subscript expressions involving colons refer to portions of a matrix: >> A(1:k,j) % is the first k elements of the jth column of A. So >> sum(a(1:4,4)) % computes the sum of the fourth column. But there is a better way. The colon by itself refers to all the elements in a row or column of a matrix and the keyword end refers to the last row or column. So sum(a(:,end)) % computes the sum of the elements in the last column of A. Exercise 1 1- Generate a matrix of size 4 4 of normally distributed random numbers. 2- Change the value of the 3rd column to [ ]. 3- Delete the 2nd row. Exercise 2 Define the following vector x [n]= n n n Representing Polynomials MATLAB represents polynomials as row vectors containing coefficients ordered by descending powers. For example, consider the equation 3 p(x) = x 2x 5 To enter this polynomial into MATLAB, use >> p = [ ];

9 7.1 Polynomial Roots The roots function calculates the roots of a polynomial. >> r = roots(p) r = i i By convention, MATLAB stores roots in column vectors. The function poly returns to the polynomial coefficients. >> p2 = poly(r) p2 = e poly and roots are inverse functions. 7.2 Characteristic Polynomials The poly function also computes the coefficients of the characteristic polynomial of a matrix. >> A = [ ; ; 9 0 1]; >> poly(a) ans = The roots of this polynomial, computed with roots, are the characteristic roots, or eigenvalues, of the matrix A. (Use eig to compute the eigenvalues of a matrix directly.) 7.3 Polynomial Evaluation The polyval function evaluates a polynomial at a specified value. To evaluate p at s = 5, use >> polyval(p,5) ans = 110 It is also possible to evaluate a polynomial in a matrix sense. In this case p(x) = x3 2x 5 becomes p(x ) = X 3 2X 5I, where X is a square matrix and I is the identity matrix. For example, create a square matrix X and evaluate the polynomial p at X. >> X = [2 4 5; ; 7 1 5]; >> Y = polyvalm(p,x) Y =

10 Exercise 3 Consider the polynomial p(x) = x4 5x3 + 7x Find the roots of this polynomial. 2- From these roots, reconstruct p(x). 3- Find the value of p(x) at x=3. 4- Evaluate the polynomial at X ( matrix sense).

11 8. Graphics in MATALB: 8.1 Creating a Plot The plot function has different forms, depending on the input arguments. If y is a vector, plot(y) produces a piecewise linear graph of the elements of y versus the index of the elements of y. If you specify two vectors as arguments, plot(x,y) produces a graph of y versus x. For example, these statements use the colon operator to create a vector of x values ranging from zero to 2π, compute the sine of these values, and plot the result. >> x = 0:pi/100:2*pi; >> y = sin(x); >> plot(x,y) %Now label the axes and add a title. The characters \pi create the symbol π. >> xlabel('x = 0:2\pi') >> ylabel('sine of x') >> title('plot of the Sine Function','FontSize',12) 8.2 Multiple Data Sets in One Graph Figure 1: Output of the above script Multiple x-y pair arguments create multiple graphs with a single call to plot. MATLAB automatically cycles through a predefined list of colors to allow discrimination among sets of data. For example, these statements plot three related functions of x, each curve in a separate distinguishing color. >> X=0:pi/40:2*pi; >> Y=sin(x); >> y2 = sin(x-.25); >> y3 = sin(x-.5); >> plot(x,y,x,y2,x,y3) % The legend command provides an easy way to identify the individual plots. >> legend('sin(x)','sin(x-.25)','sin(x-.5)')

12 8.3 Specifying Line Styles and Colors It is possible to specify color, line styles, and markers (such as plus signs or circles) when you plot your data using the plot command:. plot(x,y,'color_style_marker') color_style_marker is a string containing from one to four characters (enclosed in single quotation marks) constructed from a color, a line style, and a marker type: Color strings are 'c', 'm', 'y', 'r', 'g', 'b', 'w', and 'k'. These correspond to cyan, magenta, yellow, red, green, blue, white, and black. Line style strings are '-' for solid, '--' for dashed, ':' for dotted, '-.' for dash-dot. Omit the line style for no line. The marker types are '+', 'o', '*', and 'x' and the filled marker types are 's' for square, 'd' for diamond, '^' for up triangle, 'v' for down triangle, '>' for right triangle, '<' for left triangle, 'p' for pentagram, 'h' for hexagram, and none for no marker. 8.4 Adding Plots to an Existing Graph The hold command enables you to add plots to an existing graph. When you type >> hold on MATLAB does not replace the existing graph when you issue another plotting command; it adds the new data to the current graph, rescaling the axes if necessary. For example, these statements first create a contour plot of the peaks function, then superimpose a pseudo color plot of the same function. [x,y,z] = peaks; contour(x,y,z,20,'k') hold on pcolor(x,y,z) shading interp hold off 8.5 Multiple Plots in One Figure The subplot command enables you to display multiple plots in the same window or print them on the same piece of paper. Typing >> subplot(m,n,p) partitions the figure window into an m-by-n matrix of small subplots and selects the pth subplot for the current plot. The plots are numbered along first the top row of the figure window, then the second row, and so on. For example, these statements plot data in four different sub regions of the figure window. >> t = 0:pi/10:2*pi; >> [X,Y,Z] = cylinder(4*cos(t)); >> subplot(2,2,1); mesh(x) >> subplot(2,2,2); mesh(y) >> subplot(2,2,3); mesh(z) >> subplot(2,2,4); mesh(x,y,z)

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14 8.6 Figure Windows Graphing functions automatically open a new figure window if there are no figure windows already on the screen. If a figure window exists, MATLAB uses that window for graphics output. If there are multiple figure windows open, MATLAB targets the one that is designated the current figure (the last figure used or clicked in). To make an existing figure window the current figure, you can click the mouse while the pointer is in that window or you can type >> figure(n) where n is the number in the figure title bar. The results of subsequent graphics commands are displayed in this window. To open a new figure window and make it the current figure, type figure stem Plot discrete sequence data. stem(y) plots the data sequence Y as stems that extend from equally spaced and automatically generated values along the x-axis. When Y is a matrix, stem plots all elements in a row against the same x value. stem(x,y) plots X versus the columns of Y. X and Y are vectors or matrices of the same size. Additionally, X can be a row or a column vector and Y a matrix with length(x) rows. stem(...,'fill') specifies whether to color the circle at the end of the stem. The following example creates a stem plot of a circular function. >> y = linspace(0,2*pi,10); >> h = stem(cos(y),'fill','-.'); >> set(h(3),'color','r','linewidth',2) % Set base line properties >> axis ([ ]) Figure 2: Stem demonstration

15 Exercise 4 Using MATLAB, plot the function f (t) = e 2t cos(3πt)u(t) where t ranges from 2 to 10. Give the x-axis the title (t sec) and the y-axis the title (f(t) volts ). Repeat using stem function. Then use subplot to combine the two figures in one figure and give it the name: difference between plot and stem. 9. Dimension Functions Size returns the array dimensions d = size(x) returns the sizes of each dimension of array X in a vector d with ndims(x) elements. [m,n] = size(x) returns the size of matrix X in separate variables m and n. m = size(x,dim) returns the size of the dimension of X specified by scalar dim. [d1,d2,d3,...,dn] = size(x) returns the sizes of the first n dimensions of array X in separate variables. length returns the length of vector The statement length(x) is equivalent to max(size(x)) for nonempty arrays and 0 for empty arrays. n = length(x) returns the size of the longest dimension of X. If X is a vector, this is the same as its length. >> x = ones(1,8); >> n = length(x) n = Linear Model Representations The Control System Toolbox supports the following model representations: State-space models (SS) of the form dx dt Ax Bu y= Cx+ Du where A, B, C, and D are matrices of appropriate dimensions, x is the state vector, and u and y are the input and output vectors. Transfer functions (TF), for example,

16 H( s) 2 s s 2 s 10 Zero-pole-gain (ZPK) models, for example, 3( z 1 j)( z 1 j) H( z) ( z 0.2)( z 0.1) Frequency response data (FRD) models, which consist of sampled measurements of a system s frequency response. For example, you can store experimentally collected frequency response data in an FRD model Converting Between Model Representations Now that you have a specific representation of a system, you can convert to other model representations. Transfer Function Representation. You can use tf or ss2tf to convert from the state-space representation to the transfer function. Sys_ tf= ss2 tf (sys_ss) Or Sys_ tf= tf( sys_ss) Zero/ Pole/ Gain Representation. Similarly, the zpk function converts from state-space or transfer function representations to the zero/pole/gain format. sys _ zpk = zpk (sys _ ss) Steady-state Representation. You can use tf2ss to convert from the transfer function representation to state-space model. sys _ ss = tf 2ss(sys _ tf ) 10.2 Constructing Transfer Function and Zero/Pole/Gain Models You can construct a control system model using tf, zpk, ss, or frd. Sys= tf(num,den) %Transfer function Sys= ss (a, b, c, d) %State- space sys= zpk (z,p, k ) %Zero/ pole/ gain sys = frd(response,frequencies)

17 10.3 Data Retrieval The functions tf, zpk, ss, and frd pack the model data and sample time in a single LTI object. Conversely, the following commands provide convenient data retrieval for any type of a TF, SS, or ZPK model sys, or FRD model sysfr. [num, den, Ts] = tfdata (sys) % Ts sample time [z, p, k, Ts] = zpkdata (sys) [a, b, c, d, Ts] = ssdata(sys) [a, b, c, d, e, Ts] = dssdata (ss) [response, frequency, Ts] = frdata (sysfr) Furthermore, you can do the following operations on transfer functions: Zero: Find zeros of the transfer function Pole: Find poles of the transfer function Dc gain: Find the dc gain of the transfer function Exercise 5 Consider the transfer function ( s 3 20( s 2 3s 2 4) 2s 5) 1- Find the poles, zeros, dc gain of the transfer function. 2- Find the equivalent state space model of the system. 3- Find the zero/pole/gain representation of the system. Compare with the results in part Partial Fraction Expansion: You can find the Partial Fraction Expansion (PFE) using the MATLAB command residue. [r,p,k]=residue(num,den); Where r is the vector of PFE coefficients, p the vector of roots of den1, and k=0 if the degree of num1 < the degree of den1.