Chapter 1 MATLAB Preliminaries

Chapter 1 MATLAB Preliminaries 1.1 INTRODUCTION MATLAB (Matrix Laboratory) is a high-level technical computing environment developed by The Mathworks, Inc. for mathematical, scientific, and engineering applications. It is design to perform complex, high-performance numerical analysis and computation. This software integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notations. Powerful applications such as modeling, data analysis, simulation, exploration, and visualization can be done with this tool. For these reasons, MATLAB has been commonly used in the analysis and design of feedback control systems. 1.2 MATLAB FAMILIARIZATION When you start MATLAB, the MATLAB desktop appears, containing tools (graphical user interfaces) for managing files, variables, and applications associated with MATLAB. The first time MATLAB starts, the desktop appears as shown in Fig. 1.1. The following important entities in the MATLAB user interface are explained in detail. Command Window This is where you input commands like entering values in variables, or running scripts (m-files). M-files are scripts that simply execute a series of MATLAB statements, or they can be functions that also accept arguments and produce outputs. The prompt >> is an indicator where you will input values, basic expressions and scripts. Command History Lines you enter in the Command Window are logged in the Command History window. In the Command History, you can view previously used functions, and copy then execute those selected lines. The text %-- 11:18 AM 4/04/04 --% indicates the Timestamp in which the command was executed, while the succeeding texts indicate the commands performed on that particular time. 4

Fig. 1.1 The MATLAB 7.0 environment. Fig. 1.2 The Workspace Browser. 5

Workspace Browser The MATLAB Workspace Browser (see Fig. 1.2) consists of the set of variables (named arrays) built up during a MATLAB session and stored in memory. You add variables to the workspace by using these functions, running m-files, and loading saved workspaces. To view the workspace and information about each variable, use the Workspace Browser, or use the functions who and whos. Start Menu The Start Menu consists of various MATLAB Toolboxes and Blocksets. These toolboxes and blocksets are used depending on a specific application, and contain various predefined functions and demos. The Control System Toolbox is used in this text. 1.3 MATLAB FUNDAMENTALS: FUNCTIONS, EXPRESSIONS, AND COMMANDS Once you are familiar with the MATLAB environment, it is time to enter basic expressions. In the command window, all commands are straightforward; just type the expressions such as entering values in a variable or running an m-file. Expressions and Variables Expressions typed without a variable name are evaluated by MATLAB and the result is stored and displayed by a variable called ans. The result of an expression can be assigned to a variable name. Variable names consist of a letter, followed by any number of letters, digits, or underscores. MATLAB uses only the first 31 characters of a variable name and MATLAB is case sensitive. MATLAB is so straightforward that you may enter also mathematical expressions and formulas as you enter the variables. Notice that MATLAB display the result with 5 significant digits (default format: format short). The commands format short e, format long, and format long e display 5 digits floating point, 15 digits fixed point, and 15 digits floating point, respectively. The first example shows a no typed variable expressions, answer is automatically stored in ans. The first expression involves direct value in a variable, while the second one shows a direct mathematical expression using the function cos. Further explanations on Math functions are explained in Elementary Mathematics Functions. Listing 1.1 >> 13 13 6

>> cos(3.1416/3) 0.49999787927255 The second example shows an expression with values stored in variables a andb. Listing 1.2 >> a= 234.56778 a = 2.3457e+002 >> b=3.1416*(cos(3.1416/6))+2 b = 4.7207e+000 In displaying answers and expressions, the % indicates that any typed expression is converted into a comment while typing a semicolon, ;, after a mathematical expression, omits the MATLAB response or does not display the result. An example is shown in basic display manipulation. The first expression does not display the value of a1 but it is still in the workspace while the second expression is converted into a comment. Listing 1.3 >> a1=3.5445/64; >> % a1=3.5445/64 Using the command fprintf you can directly manipulate the format of the output of an expression or function. This command displays the result with a desired format on the screen or to specified filename. The %8.4f, indicates that the output value is a float number that has 4 decimal values and has 8 characters in length, if length of characters is less than 8 (7 in the example) the 8th would be a space. All expressions inside the single quote sign are the ones to be displayed. The \n indicates that the next output to be displayed (if there s any) would be on the next line. The expression to be evaluated is typed after the comma, sign. Listing 1.4 >> fprintf('area of a circle is %8.4f Square meters\n', 3.1416*3^2) Area of a circle is 28.2744 Square meters 7

MATLAB has several predefined variables and are listed in Table 1.1. You may try the following variables in the Command Window to check their functionality. There are also several commands that perform specific functions. These commands are listed in Table 1.2. It is recommended to try them for you to see how it works. Table 1.1 Special variables and constants. ans Most recent answer computer Computer type eps Floating point relative accuracy i orj Imaginary unit inf Infinity NaN Not-a-number pi 3.14159265358 or ( ) realmax Largest floating point number realmin Smallest floating point number Table 1.2 MATLAB special commands. clc clear variable diary exit help command home load save who whos variable Clears all input and output from the Command Window display, giving you a "clean screen." Clear the contents and the variable itself in the workspace. Without the variable, it clears the whole workspace Causes a copy of all subsequent command window input and most of the resulting command window output to be appended to the named file. If no file is specified, the file 'diary' is used. Exits MATLAB Ask help on a particular command Same as clc. Read the contents of the MAT-file saved. Saves the workspace to a binary file called a MAT-file, which has a.mat extension. Displays variables in workspace Displays the number of elements in a variable, if no variable is indicated, it displays the values of each variable. 8

A sequence of characters in single quotes is called a character string or text variable. Characters can be augmented or combined by using a vector [ first character, second character ] An example is shown below. Listing 1.5 >> c='manila' c = Manila >> cs=[c,',philippines'] cs = Manila,Philippines 1.4 MATLAB COMMANDS AND MATRIX FUNCTIONS MATLAB treats the single value element, like in the first and second example earlier on Expressions and Variables as a single element matrix. From the word MATLAB which means Matrix Laboratory, it explains why values are treated as matrices. Matrices are entered into MATLAB by listing the elements of the matrix and enclosing them within a pair of brackets, [ ]. Elements of a single row are separated by commas or blanks, and rows are separated by semicolons or carriage return. Entering Vectors and Matrices A single row matrix A 4 32 31 5 is entered in MATLAB in two ways, either using spaces or using commas. Listing 1.6 >> A=[4 32 31 5] A = 4 32 31 5 >> A=[4,32,31,5] A = 4 32 31 5 9

3 A single column matrix B 4 is entered in MATLAB by using semicolons or carriage 5 returns (in this example, using semicolons). Listing 1.7 >> B=[3;4;5] B = 3 4 5 Combining the single column and single row matrix instructions, you can create an m x n 1 2 matrix. If you want to create matrixc 3 4, it is entered in MATLAB by using spaces or commas with semicolons or carriage returns, as shown below. Listing 1.8 >> C=[1 2;3 4] C = 1 2 3 4 The entire row or column can be addressed using colon ;. For example, if you want to get the first row of matrix C, follow the example below. The number 1 denotes first and its location tells whether it is a column or row. If the number is placed on the first position, then it denotes that the output is an entire row and if it is placed in the second position, it denotes an entire column. In our example, the number 1 is placed on the first position; therefore its output is the first row. Try to interchange the colons and numbers to see the corresponding change on the output. Listing 1.9 >> frow=c(1,:) frow = 1 2 >> 10

You can also address a single element in a given matrix, we know that a matrix element can be addressed in an expression (i,j), where i is the row and j is the column of the element. Same thing is done here in MATLAB. For example, to get the 1 st row, 2 nd column element of matrix C, which is 2, see the example below. Listing 1.10 >> f12=c(1,2) f12 = 2 Table 1.3 Element-by-element math operations. + Addition - Subtraction.* Element-by-element multiplication../ Element-by-element division..\ Element-by-element left division..^ Element-by-element power. Unconjugated array transpose Matrix / Vector Basic Operations Arithmetic operations on arrays are done element-by-element. This means that addition and subtraction are the same for arrays and matrices, but that multiplicative and division operations are different. Basic operations such as addition and subtraction on matrices with the same dimensions can be done. If they are conformable, two matrices can be multiplied and divided. For element-by-element multiplication and division or array operations, MATLAB uses a dot, or decimal point, as part of the notation for multiplication and division. Element by element operations are listed in Table 1.3. Given two matrices C and D, multiplication can be done by typing >>C*D, element by element multiplication can be done by typing >>C.*D, >>C\D is equivalent to C -1 D, and >>C/D is equivalent to CD -1. The inverse of a matrix, denoted by C -1 can be done by using the command >>inv(c). An example is shown below. 11

Listing 1.11 >> C=[1 2;3 4] C = 1 2 3 4 >> D=[5 6;7 8] D = >> C*D >> C\D 5 6 7 8 19 22 43 50-3.0000-4.0000 4.0000 5.0000 >> inv(c)*d -3.0000-4.0000 4.0000 5.0000 >> C.*D 5 12 21 32 12

Transpose and Conjugate Transpose An n vector is a row vector or a column array of n numbers. In MATLAB, elements enclosed by brackets and separated by semicolon generate a column vector. The conjugate transpose of a row vector is a column vector, and vice versa. This can be done in MATLAB using the symbol (apostrophe). An example is shown below, the transpose of matrix D is E. If the matrix is real, the conjugate transpose is simply a transpose. Listing 1.12 >> E=D' E = 5 7 6 8 Special Matrices and Matrix Manipulations Special matrices can be generated by using the functions given in Table 1.4. Table 1.4 Elementary matrices. eye Identity matrix meshgrid x and y arrays for 3-D plots ones Ones matrix zeros Zeros matrix rand Uniformly distributed random numbers randn Normally distributed random numbers Information on a particular matrix can be extracted. This can aid in certain calculations and operations. Try these commands and see how it works. See Table 1.5 for details. Table 1.5 Basic information of a matrix. disp Display array isempty True for empty matrix isequal True if arrays are identical isnumeric True for numeric arrays issparse True for sparse matrix length Length of vector ndims Number of dimensions numel Number of elements size Size of matrix 13

Table 1.6 Commands for manipulating matrices. diag Create or extract diagonal fliplr Flip matrix in the left/right direction flipud Flip matrix in the up/down direction reshape change size rot90 Rotate matrix 90 degrees tril Extract lower triangular part triu Extract upper triangular part : Index into matrix, rearrange matrix Table 1.7 Some operations on matrices. det Determinant norm Matrix vector norm trace Sum of diagonal elements inv Matrix inverse eig Eigenvalues and eigenvectors poly Characteristic polynomial exp Matrix exponential logm Matrix logarithm sqrtm Matrix square root Manipulation and creation of matrices can be done using the following MATLAB functions. See Tables 6 and 7. Try the following commands and see how it works. Generating Sequences Vectors can be generated by just specifying the first, last and the increment desired for each element. For example, if you want to create a row vector with a first element of 1 and a last element of 9 and an increment of 1, just use the syntax in the example below. The default increment is 1 so even if you omit the second parameter 1, >>F=(1:9), you still can get the same result. You may try to experiment and change the increment, and see what happens. Listing 1.13 >> F=(1:1:9) F = 1 2 3 4 5 6 7 8 9 >> F=(1:9) 14

F = 1 2 3 4 5 6 7 8 9 Complex Numbers We encounter some problems that include operations and manipulation of complex numbers. Cases like getting the roots of a polynomial or solutions on partial fraction expansion that involves complex roots are common. In MATLAB, imaginary numbers are represented using i or j. Mathematical operations are straightforward, similar to operations in real numbers. As an example, let us perform basic operations of two complex numbers. Given: (25 j65) and (30 j80) (addition and division) Listing 1.14 >> (25+65j)+(30+80j) %Addition 5.5000e+001 +1.4500e+002i >> (25+65j)/(30+80j) %Division 0.8151-0.0068i We can also extract some parameters from complex numbers like phase angle and magnitude, real part and imaginary part. As an example, we need to get the magnitude and phase angle of the complex number 25 j65. We have to convert the angle in degrees so we multiplied /180 to the answer of angle (always in radians). Listing 1.15 >> abs(25-65i) 69.6419 >> angle(25-65i)*(180/pi) 15

-68.9625 >> real(25-65i) ans 25 >> imag(25-65i) ans -65 Matrix Exponential expm(a) is the matrix exponential of an n n matrix A. That is, 2 3 A A expm( A) I A 2! 3! Note that a transcendental function is interpreted as a matrix function if an m is appended to the function name, as in expm(a) or sqrtm(a). Eigenvalues The eigenvalues of a matrix A is found by using the function >>eig(a), which returns the eigenvalues in a column vector. If A is real and symmetric, the eigenvalues will be real. But if A is not symmetric, the eigenvalues are frequently complex in nature. Elementary Mathematics Functions and Operators in MATLAB Elementary math functions and operators in MATLAB are very easy to use since they are basic and straightforward in nature. Basic operations like addition, subtraction, multiplication, division can be represented by +, -, *, / respectively. If you want to raise a number to a certain exponent, just simply insert ^ after the number and before the exponent. For example, if you want to evaluate 2 6, just type >>2^6. For matrices or arrays, operations are different (for exponents, multiplication and division) as stated earlier. Some MATLAB basic math functions automatically operate element by element on an array. Functions that operate element by element are given in Table 1.8. 16

Table 1.8 Elementary math functions. abs Absolute value acos Inverse cosine acosh Inverse hyperbolic cosine angle Phase angle asin Inverse sine asinh Inverse hyperbolic sine atan Inverse tangent atanh Inverse hyperbolic tangent conj complex conjugate cos Cosine exp Exponential cosh Hyperbolic cosine floor Round towards infinity fix Round towards zero imag Complex Imaginary part log Natural logarithm log10 Common logarithm real Complex real part rem remainder after division round Round towards nearest integer sign Signum function sinh Hyperbolic sine sqrt Square root tan Tangent tanh Hyperbolic tangent Logical Operators MATLAB relational operators and logical operators also work on an element-byelement basis. Relational operators compare two scalars and produce a 1 if the operation is true and a 0 if it is false. For example, if you enter >>t=17>55, MATLAB will respond with t =0. When used with two matrices, relational operators compare corresponding matrix elements. For example, >>L = D <= X will check every element of D against the corresponding element of X. If the element of D is less than or equal to the corresponding element of X, the corresponding element of L will be 1. Otherwise, the corresponding element of L will be zero. The logical operators & for logical AND, for logical OR, and ~ for logical NOT all return 1 for logical TRUE and 0 for logical FALSE. An example is shown below. Listing 1.16 >> H=[1 3 5 7 9]; >> I=[0 2 3 4 5]; >> J=[2 2 6 3 4]; 17

>> lgcal1=h<i lgcal1 = 0 0 0 0 0 >> lgcal2=h>i lgcal2 = 1 1 1 1 1 >> lgcal3=(h>i)&(h>=j) lgcal3 = 0 1 0 1 1 1.5 CREATING SCRIPTS AND FUNCTIONS (m-files) Files that contain code in the MATLAB language are called m-files. You create m-files using a text editor, then use them as you would any other MATLAB function or command. There are two kinds of m-files: 1. Scripts, which do not accept input arguments or return output arguments. They operate on data in the workspace. 2. Functions, which can accept input arguments and return output arguments. Internal variables are local to the function. Creating Script Files If you're a new MATLAB programmer, just create the m-files that you want to try out in the current directory. As you develop more of your own m-files, you will want to organize them into other directories and personal toolboxes that you can add to MATLAB search path. If you duplicate function names, MATLAB executes the one that occurs first in the search path. Each script file should have a name that ends in.m The commands in the script file are executed in the MATLAB environment by simply entering the name of the script file without the extension.m. An m-file can be created by clicking >FILE>NEW>M-FILE and then a window will appear. You may type your script or code in the m-file window. See Fig. 1.3. 18

Fig. 1.3 MATLAB m-file editor. Creating Functions Function files are m-files that are very similar to script files. The first major difference is that the first line of a function file begins with the word function, followed by a statement identifying the name of the function and the input and output argument in the form: function [output arguments] = function_name(input arguments) Control Flow In your functions or scripts, it is common that you include control flow commands. MATLAB control flow is given in Table 1.9. The control flow commands for, while, and if are statements similar to those used in many computer languages to produce conditional statements or loops. Every for,while, and if statements must be matched with an end statement. The break command can be used to terminate the execution of a loop permanently. The if statement can be used in conjunction with the nargin, 19

nargout, and error functions to perform error checking of a function. Inside a function file, nargin and nargout are equal to the number of input and output arguments, respectively, that were used in function call. The function error( message ) returns control to the keyboard and displays message. For more information regarding the commands, please type >> help <command>. Table 1.9 Control flow commands in MATLAB. break Terminate execution of loop else used with if elseif used with if and Terminate the loop for, while, and if statement error Display message and abort function for Repeat statement a specified number of times if conditionally executed statements return return to invoking function while Repeat statements an indefinite number of times 1.6 GENERATING PLOTS MATLAB can create high-resolution, publication-quality 2-D, 3-D, linear, log, semilog, polar, bar chart and contour plots on plotters, dot-matrix printers, and laser printers. Some of the 2-D graph types are plot, loglog, semilogx, semilogy, polar, and bar. The command grid adds a grid to the graph, and the command title( text ), xlabel( text ), ylabel( text ), and text( text ) can be used for labeling and placing text on the graph. MATLAB provides automatic scaling. The function axis([xmin, xmax, ymin, ymax])enforces manual scaling. As an example, let us plot a sinusoidal function, let y 2sin x, where x is the abscissa (an angle) and y is the ordinate. Take note that angles are in radians. The listing is shown below. Listing 1.17 >> x=(0:0.002:2*pi); >> y=2*sin(x); >> plot(x,y); title('sinusoidal waveform'); 20

The output waveform is shown in Fig. 1.4. Fig. 1.4 Plot of the function y = 2sinx. 1.7 OTHER MATLAB FUNCTIONS AND COMMANDS FOR CONTROL SYSTEMS ENGINEERING APPLICATIONS The functions to be presented are basic and are commonly used in control systems applications. These functions are not used directly for solving control system problems but rather application of these commands will greatly help in solving them. Polynomials Polynomials can be represented as row vectors containing the coefficients. As an 3 2 example, the polynomial s 32s 32 can be represented as Listing 1.18 >>x=[1 32 0 32] x = >> 1 32 0 32 The first element of the vector is the coefficient of the highest exponent while the last is the coefficient of the lowest exponent. A 0 is included because there is no s on the polynomial. Polynomials can also be represented in a factor form of its roots such as, 21

2 4 5 P s s s s, which can be written using the function poly(), as shown below. Listing 1.19 >> x=poly([0 2 4 5]) x = 1 3-18 -40 0 For a given characteristic polynomial, its roots can be obtained by using the function roots(). Listing 1.20 >> roots([1 3-18 -40 0]) 0 4.0000-5.0000-2.0000 Two polynomials can also be multiplied and divided using the conv() and deconv() functions. >> x = [1 2 1]; >> y = [2 5]; >> m = conv(x,y) m = 2 9 12 5 >> n = deconv(x,y) n = 0.5000-0.2500 Other important functions used in control engineering analysis and design are shown in Table 1.10. 22

Table 1.10 Other basic functions for control engineering applications. ilaplace Generates the inverse-laplace transform of a polynomial in the s domain. (Requires Symbolic Math Toolbox) laplace Generates the Laplace transform of a polynomial in the time domain. (Requires Symbolic Math Toolbox) polyval Polynomial evaluation. residue Partial fraction expansion. 1.8 MATLAB EXERCISES 1. Generate a 1 3 matrix whose entries are the coefficients of the polynomial 2 N s 2s 3s 1. 2. Generate a 1 4 matrix whose entries are the coefficients of the polynomial 3 2 D s s 6s 11s 6. 3. Determine the product N(s)D(s). 4. Assume a rational function H s H(s). N s D s, determine the partial fraction expansion of 5. Determine the poles and zeros of H(s). (Determine the roots of the numerator and denominator polynomials.) 6. What is the value of H(s) if s = 2? 7. Generate the following matrices: 1 0 2 A 2 1 3 1 2 0 0 3 0 0 0 2 2 1 B 1 0 D I C 3 If possible, determine the following: T T a. AB b. BA c. A A d. A A C e. Show that I3C CI3 C. f. Eigenvalues of C 23

2 8. Generate a plot of the function f t 1 (0.9) cos t 9. Solve the system of linear equations for 5 cycles. x 2y 3z 2w 2 2y 6w 1 2x z w 0 y 6z 1 10. Determine the rank of the coefficient matrix developed in No. 9 using the function rank. 11. Augment the constant matrix in No. 9 to the coefficient matrix also in No. 9. Determine the Reduced Row Echelon Form of this augmented matrix by using the function rref. What is the significance of this result? 12. Generate the vector x 0 0 0 1 1 2 2 0 0 0 without typing the entries manually, and by using the functions ones and zeros only. 13. Determine the size of the matrix generated in No. 9 using the function size. 14. Determine the length of the matrix generated in No. 9 using the function length. 15. What is the difference between size and length? 24