A Mini course on MATLAB

Transcription

1 A Mini course on MATLAB MATLAB Basics, Matrix Algebra and Scripting Arun K. Tangirala 1 1 Department of Chemical Engineering Indian Institute of Technology Madras Mini Course on MATLAB A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

2 About MATLAB About 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. - from Mathworks MATLAB is available on all three major operating systems SIMULINK, a powerful simulator for modelling and simulating dynamic systems, usually accompanies MATLAB (requires separate orde) One of the biggest advantages of using MATLAB is one does not need to declare ahead of time, the type of variable being used The base version of MATLAB/SIMULINK is aptly supported by a variety of toolboxes A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

3 Outline 1 Matrices 2 Matrix manipulations 3 Graphics 4 General purpose commands 5 Programming basics 6 Scripts and Functions A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

4 Matrices Matrices and Arrays 1 % C r e a t e a column v e c t o r 2 xvec = [ 1 ; 2 ; 4 ; 5 ] ; % OR 3 xvec = [ ] ; % N o t i c e the t r a n s p o s e 4 5 % One c o u l d a l s o c r e a t e a u n i f o r m l y spaced v e c t o r 6 uvec = ( 0 : : ) ; % This v e c t o r has 1000 samples 7 8 % Now how to c r e a t e a m a t r i x 9 A = [ 1 2 ; 4 5 ] ; 10 % OR a s s i g n each element 11 A( 1, 1 ) = 1 ; A( 2, 1 ) = 2 ; A( 1, 2 ) = 4 ; A( 2, 2 ) = 5 ; 12 % OR 13 A= zeros ( 3, 3 ) ; 14 A( 1 ) = 1 ; A( 2 ) = 2 ; A( 3 ) = 4 ; A( 4 ) = 5 ; 15 % N o t i c e above how the m a t r i x is d e f i n e d l i k e an array Removing the semicolon at the end allows you to echo the output A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

5 Matrices Creating matrices/arrays Both matrices and vectors are enclosed in square brackets Elements are accessed using the parentheses One can always append and/or insert elements to these objects A wide variety of manipulations are possible. A vector is a special case of a matrix 1 % Access the f i r s t two and l a s t two e l e m e n t s of uvec 2 uvec ( [ 1 2 end 1 end ]) 3 % Take e v e r y t h i r d element of uvec and s t o r e it in yvec 4 yvec = uvec ( 1 : 3 : end ); 5 % Size of a m a t r i x / v e c t o r & the t o t a l number of e l e m e n t s 6 [ nrow, n c o l ] = s i z e (A ) ; n e l A = numel (A ) ; 7 % Convert any m a t r i x or a v e c t o r i n t o a column v e c t o r 8 Acol = A ( : ) ; A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

6 Matrices Special matrices/vectors MATLAB allows one to create a variety of special matrices that appear in several problems One can even produce a large matrix containing replicates of a smaller matrix 1 % C r e a t e an i d e n t i t y m a t r i x 2 Imat = eye ( 3, 3 ) ; % Try eye ( 3, 2 ) 3 % C r e a t e a m a t r i x of ones or z e r o s 4 Amat = ones ( 3, 3 ) ; % OR 5 Amat = ones ( s i z e ( Imat ) ) ; 6 % S p e c i a l m a t r i c e s 7 hadamard ( 2 ) % Hadamard m a t r i x 8 magic ( 3 ) % Magic m a t r i x 9 t o e p l i t z ( [ 3 4 ], [ 3 2 ] ) % T o e p l i t z m a t r i x 10 % R e p l i c a t e a s m a l l m a t r i x 11 A=randn ( 2, 2 ) ; B = repmat (A, 2, 3 ) ; A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

7 Matrices Simple checks on matrices/arrays In some situations, it becomes necessary to make some important checks on the matrices of interest 1 % Check if a m a t r i x is empty 2 A = [ ] ; isempty (A) 3 % Check if two m a t r i c e s are e q u a l 4 A=exp ( [ 1 2 ; 3 4 ] ) ; B = expm ( [ 1 2 ; 3 4 ] ) ; 5 i s e q u a l (A, B) 6 % Check if a m a t r i x c o n t a i n s numeric e l e m e n t s 7 A = [ red ; blue ]; % N o t i c e the t r a i l i n g space a f t e r 8 i s n u m e r i c (A) 9 % Check if a m a t r i x c o n t a i n s r e a l e l e m e n t s 10 A = ones ( 2, 2 ) ; B = zeros ( 2, 2 ) ; C = A + j B; 11 i s r e a l (C) 12 i s r e a l ( complex (A) ) 13 % Check if m a t r i x e l e m e n t s are NaN 14 A = [ 1 2 ; i n f 4 ] ; 15 isnan (A) A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

8 Simple matrix operations Matrices Several operations can be carried out using simple commands 1 % Transpose a m a t r i x 2 A = [ ; ] ; A 3 A = [ i ; i ] ; A 4 % F l i p a m a t r i x 5 A = [ ; ] ; f l i p u d (A) 6 f l i p l r (A) 7 % E x t r a c t t r i a n g u l a r part 8 t r i l (A) 9 % Find i n d i c e s of z e r o e l e m e n t s 10 f i n d (A) 11 f i n d (A >= 4) 12 % C r e a t e a 3D m a t r i x s t a r t i n g from a 2D m a t r i x 13 A = [ 1 2 ; 3 4 ] ; A ( :, :, 2 ) = [ 5 6 ; 7 8 ] 14 % Remove s i n g l e t o n d i mension 15 s q u e e z e ( rand ( 3, 2, 1 ) ) A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

9 Matrix manipulations Mathematical operations on matrices We now learn how to perform some important mathematical operations 1 % Matrix m u l t i p l i c a t i o n s 2 A = [ ; ] ; B = randn ( 2, 2 ) ; A B % R e g u l a r product 3 A = [ 1 2 ; 3 4 ] ; B = [ 5 6 ; 7 8 ] ; A. B % Element w i s e m u l 4 % F i n d the maximum and minimum 5 A = [ ; 2 1]; max(a) 6 min( abs(a) ) 7 % Find the i n v e r s e and d e t e r m i n a n t of a m a t r i x 8 A = [ 1 2 ; 4 6 ] ; inv (A) 9 A = [ ; ] ; det(a A) 10 pinv (A) % Pseudo i n v e r s e of A 11 % Norm and t r a c e of a m a t r i x 12 A = [ 1 2 ; 3 4 ] ; norm(a) 13 norm(a, i n f ) 14 trace (A) % Trace of a m a t r i x 15 sum( diag(a) ) % This s h o u l d e q u a l t r a c e A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

10 Matrix manipulations Mathematical operations on matrices 1 % Matrix d i v i s i o n ( r e g u l a r and element w i s e ) 2 A = [ 1 2 ; 3 4 ] ; B = [ 5 6 ; 7 8 ] ; B/A 3 B inv (A) 4 B. /A 5 % E i g e n v a l u e c a l c u l a t i o n s 6 A = [ 1 2 ; 3 4 ] ; [VA, lama ] = eig (A) 7 inv (VA) expm( lama ) VA 8 expm(a) 9 % Rank of a m a t r i x 10 rank ( [ 4 5 ; ] ) 11 % C h a r a c t e r i s t i c e q u a t i o n 12 cpa = poly ( [ 1 2 ; 3 4 ] ) ; 13 roots( cpa ) % Compare with e i g e n v a l u e s of A 14 % LU f a c t o r i z a t i o n 15 [ L,U] = lu ( [ 1 2 ; 3 4 ] ) 16 % O r t h o g o n a l i z a t i o n 17 Q= orth ( [ 1 2 ; 3 4 ] ) continued A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

11 Matrix manipulations Special numbers and variables pi : The number π eps : Floating point relative accuracy inf : The number (too large a number) i : The imaginary number i = 1 (could use j) NaN : Not-a-Number (e.g., addition or subtraction of Inf) ans : The most recent answer end : The last element of a vector; OR to indicate the end of a loop or a conditional statement all : A function that returns 1 if none of the elements of a vector are zero; used with clear command to clear all variables A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

12 Elementary functions Matrix manipulations Trigonometric: sin, sinh, atan, sec, etc. Exponential: exp, log2, pow2, nextpow2, sqrt, etc. Complex: abs, angle, conj, imag, unwrap, etc. Rounding: fix, floor, ceil, mod, rem, sign, etc. Specialized: bessel, beta, erf, dot, gamma, etc. Number theoretic: factor, primes, factorial, gcd, nchoosek, etc. A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

13 Matrix manipulations Examples 1 % C r e a t e 1000 samples of a s i n e and c o s i n e 2 x = cos (2 pi 0. 2 ( 0 : ) ) ; y = s i n (2 pi 0. 2 ( 0 : ) ) ; 3 f i n d ( ( x. ˆ 2 + y. ˆ 2 ) = 1) % Sum s q u a r e s s h o u l d e q u a l 1 4 % C r e a t e a complex number 5 z = x + i y; i s r e a l ( z + conj ( z ) ) 6 a r g z = unwrap ( angle ( z ) ) ; 7 % Rounding 8 v = [ ]; round ( v ) 9 c e i l ( v ) 10 sign ( v ) 11 % Generate p r i mes 12 p r i m e s ( 10) 13 % Find dot p r oduct 14 v1 = ( 1 : 4 ) ; v2 = s q r t ( v1 ) ; 15 dot( v1, v2 ) 16 sum( v1. v2 ) A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

14 Graphics Graphics We now turn to learning how to plot data and modify the resulting plots 1 % C r e a t e the time v e c t o r and s i n e v e c t o r 2 t v e c = ( 0 : : 9 9 ) ; 3 x t = s i n (2 pi 0.2 t v e c ) ; 4 % P l o t the first 20 samples of s i n e 5 p l o t ( t v e c ( 1 : 2 0 ), x t ( 1 : 2 0 ) ) ; 6 % Give a t i t l e and l a b e l s, and t u r n the g r i d on 7 t i t l e ( Plot of f i r s t 20 samples of s i n u s o i d ); 8 x l a b e l ( Samples ); y l a b e l ( Amplitude ); 9 g r i d on 10 % Also p l o t a c o s i n e of the same f r e q u e n c y with a r ed lin 11 y t = cos (2 pi 0.2 t v e c ) ; 12 hold on % Hold the c u r r e n t f i g u r e 13 p l o t ( t v e c ( 1 : 2 0 ), y t ( 1 : 2 0 ), r ); A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

15 Graphics Subplots One can draw different figures in different plots but on the same figure! 1 t v e c = l i n s p a c e ( , 4, ) ; % C r e a t e a l i n e a r l y spaced t 2 % C r e a t e a new f i g u r e 3 f i g u r e 4 % P l o t the first c u r v e at the bottom 5 subplot ( ) ; % D i v i d e s the f i g u r e i n t o 2 rows and 1 colu 6 p l o t ( tvec, log10 ( t v e c ) ) ; 7 t i t l e ( Parabola ); x l a b e l ( Time ); y l a b e l ( Amplitude ); 8 g r i d on 9 % P l o t the second c u r v e at the top 10 subplot ( ) ; 11 semilogx ( tvec, log10 ( t v e c ) ) ; % Plot with l o g 1 0 ( t ) on x a x i 12 t i t l e ( Parabola ); x l a b e l ( Time ); y l a b e l ( Amplitude ); 13 g r i d on The general subplot(m,n,p) refers to the p th plot in the m n subplots A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

16 Graphics Modifying plots Each figure has a handle which provides access to all of its properties. 1 % C r e a t e x and y v e c t o r s 2 t = 2 pi ( 0 : ) ; x = s i n ( t ) ; y = cos ( t ) ; 3 % P l o t y vs. x 4 h f i g = f i g u r e ; % F i g u r e command r e t u r n s a h a n d l e 5 p l o t ( x, y,, l i n e w i d t h, 2 ) ; % S o l i d and thick l i n e 6 % Modify p l o t 7 g r i d on ; a x i s s q u a r e % Use s q u a r e axis to see the c i r c l e 8 set (h f i g, Color, [ ], Name, Famous I d e n t i t y ); 9 h ax = gca ; % Get h a n d l e to c u r r e n t axis 10 set (h ax, f o n t s i z e,12, fontweight, bold ); 11 % Set l a b e l s and t i t l e with d e s i r e d f o n t 12 x l a b e l ( x : s i n (\ theta ), f o n t s i z e,14) ; 13 y l a b e l ( y : cos (\ theta ), f o n t s i z e,14) ; 14 t i t l e ( s i n ˆ2(\ theta ) + cos ˆ2(\ theta ) = 1 ); 15 set ( get ( gca, T i t l e ), fontweight, bold ); A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

17 Graphics Some tips on graphics The get and set commands are used to retrieve the current properties and set the desired properties of that object The figure is the parent object, the axis its child and then the curves are the children of the axes There are a variety of plots possible - plot3d, contour, mesh, surf, image, etc. One can use LaTeX commands (see texlabel) to insert Greek symbols One can further change the tick labels and the tick numbers as desired. A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

18 Graphics Saving and printing Every object in the figure can be assigned a tag. A specific object in the figure can be easily accessed by using the findobj command. All figures can be edited directly by means of the GUI under Edit Figure Properties (or type plotedit). Figures can be saved as a MATLAB figure that can be loaded at any later time Every figure can exported to a variety of format including EPS, PDF and JPG formats. Use commands clf to clear and close the current figure. A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

19 General purpose commands General purpose commands General: help, demo, ver, doc Debugging: who, keyboard, disp, sprintf, return, error, debug Workspace: whos, save, load, clear, diary, format Editing: edit, open, pcode, which, pwd Miscellaneous: path, pathtool, cd, copyfile, mkdir, computer, clc Reading up the documentation and help on any function beforehand is very useful. MATLAB comes with a built-in editor which is very powerful. There is also an editor which allows you to edit variables directly. Use the clear command with caution The keyboard and return commands are two extremely useful commands for debugging. A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

20 Examples General purpose commands 1 % Open diary to r e c o r d commands and output 2 d i a r y r e c o r d. t x t 3 % C r e a t e v a r i a b l e s in the workspace 4 A=randn ( 2, 2 ) ; B = { red ; log10 ( 2 ) ; rand ( 2, 1 ) } ; 5 C = s t r u c t ( Name, Rama, Age,24, Place, I n d i a ); 6 % Check if v a r i a b l e s have been c r e a t e d and t h e i r s i z e 7 whos 8 % D i s p l a y the s t r u c t u r e v a r i a b l e 9 disp (C) 10 % Save them i n t o a f i l e 11 save t r i a l. mat A B C 12 % Clear t h e s e v a r i a b l e s and then l o a d them 13 c l e a r a l l ; 14 load t r i a l. mat 15 % E d i t m a t r i x A 16 open A 17 d i a r y o f f % Switch o f f the d i a r y A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

21 Programming basics Controlled flow 1 xvec = ( 1 0 : : 1 0 ) ; y f = [ ] ; 2 % Begin the f o r l o o p 3 f o r k = 1 : length ( xvec ), 4 if ( x ( k ) < 0 ), 5 y f ( k ) = 2 x ( k )ˆ2 + 3 ; 6 e l s e 7 y f ( k ) = x ( k ) + 3 ; 8 end 9 end 10 % Try u s i n g w h i l e l o o p 11 count = 1 ; yw = [ ] ; 12 while ( count <= length ( xvec ) ) 13 if ( x ( count ) < 0 ), 14 yw = [ y 2 x ( count )ˆ2+3]; 15 e l s e 16 yw = [ y x +3]; 17 end 18 count = count + 1 ; 19 end 20 % Check if both r e s u l t s a r e e q u a l 21 i s e q u a l ( yf, yw ) ; A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

22 Programming basics Controlled flow using switch Here, we do the same task using switch command. 1 x = ( 1 0 : : 1 0 ) ; ys = [ ] ; 2 f o r k = 1 : length ( x ) 3 s w i t c h ( x ( k ) < 0) 4 c a s e 1 5 ys ( k ) = 2 x ( k )ˆ2 + 3 ; 6 o t h e r w i s e 7 ys ( k ) = x ( k ) + 3 ; 8 end 9 end 10 % Try u s i n g a d i f f e r e n t method 11 % Use the f i n d f u n c t i o n 12 indneg = f i n d (x < 0 ) ; indnonneg = f i n d (x >= 0 ) ; 13 yn = [ ] ; 14 yn ( i n dneg ) = 2 x ( indneg ). ˆ ; 15 yn ( indnonneg ) = x ( indnonneg ) + 3 ; 16 % Check if both are e q u a l 17 i s e q u a l ( ys ( : ), yn ( : ) ) A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

23 Programming basics Useful language commands Controlled flow: break, continue, try, catch Evaluation: eval, feval, run, assignin, Scripting: function, global, mfilename, nargin, varargin, nargchk Messaging: warning, display, fprintf, Interactive: input, pause, uimenu A function in MATLAB is similar to a subroutine in C. A function takes in input arguments, processes them and produces output arguments. The variables used within the function have a limited scope. They exist as long as only the function is being executed. Global variables are useful to access variables in the function workspace from the regular workspace and, therefore, should be used with caution. Commands such as input and uimenu could be used for interactively seeking inputs from the user. A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

24 Polynomial functons Programming basics Polynomials: poly, roots, polyval, polyfit, polyder, conv Interpolation: interpft, interp1, spline, interp1q 1 % C r e a t e a p o l y n o m i a l 2 px = poly ( [ 1 2 ] ) % Takes in the r o o t s of the p o l y n o m i a l 3 % E v a l u a t e the p o l y n o m i a l at x = 1, 2, 3 4 p o l y v a l ( px, [ ] ) 5 % Find r o o t s of a p o l y n o m i a l : x ˆ3 + 6x ˆ2 + 11x x r = roots ( [ ] ) % Takes in the c o e f f i c i e n t s 7 px = poly ( x r ) ; % Check if you get back the same answer 8 % Convolve p o l y n o m i a l s ( x+1) and ( x ˆ2 + 5x + 6) 9 gx = conv ( [ 1 1 ], [ ] ) % P r o v i d e the c o e f f i c i e n t s 10 % F i t t i n g p o l y n o m i a l to data 11 xk = ( 0 : : 9 9 ) ; 12 yk = xk. ˆ xk randn( length ( xk ), 1 ) ; 13 [c e s t, s e s t ] = p o l y f i t ( xk, yk, 2 ) A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

25 Scripts and Functions Script file A script file in MATLAB is any set of MATLAB commands which are executed in that sequence. Script files have a.m extension. Use a meaningful name for the filename A script file can be run by simply typing the filename without its extension. The scope of the variables used in a script file is the general workspace. They can be accessed even after the execution. A script file can contain functions which can be used within that script. MATLAB executes the script file by interpreting every line, which makes it somewhat slow at times. An m-file script can be converted into a psuedo code using the pcode command. The resulting file runs faster than the m-file. A script file can be put on path so that it can be called from any directory. A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

26 Script file: Example Scripts and Functions 1 % SCRIPT FILE TO SOLVE A SET OF LINEAR EQUATIONS 2 % S o l v e A p = b f o r p 3 A=magic ( 3 ) ; b = ( 1 : 3 ) ; 4 p = inv (A) b 5 % A l t e r n a t i v e l y 6 p2 = A \ b 7 % L i n e a r e q u a t i o n s a r i s e in system i d e n t i f i c a t i o n f o r example 8 % Generate input output data mixed with n o i s e 9 Phi = randn ( , 3 ) ; 10 Y = Phi p randn ( , 1 ) ; 11 % E s t i m a t e the p a r a m e t e r s u s i n g LS method 12 p l s = Phi \ Y % OR 13 p l s = pinv ( Phi ) Y 14 % Use l e a s t s q u a r e s f u n c t i o n from o p t i m i z a t i o n 15 p l s = l s q l i n ( Phi, Y ) ; 16 % Compare the p r e d i c t e d vs. a c t u a l output 17 Yhat = Phi p l s ; 18 f i g u r e 19 p l ot (Y, Yhat, x,y, Y, r ); 20 g r i d on ; a x i s t i g h t ; t i t l e ( Predicted vs. Actual ); 21 x l a b e l ( Y ); y l a b e l ( Yhat ); legend ({ Predicted ; I d e a l } ); A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

27 Scripts and Functions Function files Every function file should begin as: function yout = myfun(xin) where yout and xin are generic output and input arguments of that function In principle, a function can be written without the output argument or input arguments Any comments immediately below the function declaration line and until the next blank or declaration line will be used by the help command to give help on that function. The input and output variable names are dummy names. The variables only exist as long as the function is being executed. A function is available for execution as soon as it is saved. Functions can be accessed by handles, which are passed on to several other functions such as ode45, feval, etc. A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

28 Scripts and Functions Function file: Simple Example 1 function r s o l = q u a d s o l ( cvec ) ; 2 % F u n c t i o n to compute the r o o t s of a q u a d r a t i c e q u a t i o n 3 % 4 % Usage : r s o l = quad sol( cvec ); 5 6 % Read the c o e f f i c i e n t s 7 a = cvec ( 1 ) ; b = cvec ( 2 ) ; c = cvec ( 3 ) ; 8 9 % Compute the s o l u t i o n 10 r s o l = [ ] ; 11 r s o l ( 1 ) = ( b + s q r t ( bˆ2 4 a c ) ) / ( 2 a ) ; 12 r s o l ( 2 ) = ( b s q r t ( bˆ2 4 a c ) ) / ( 2 a ) ; Ideally one should also describe the input and output arguments The output could be of any type and any in number Variable number of input arguments could be supplied A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

29 Scripts and Functions Function file: Advanced Example 1 function pspec = p s p e c f u n ( x, p l o t o p t ) 2 % F u n c t i o n to compute and p l o t the power spectrum 3 % of a s i g n a l u s i n g s t a n d a r d and smoothed t e c h n i q u e s 4 % Usage : pspec = p s p e c f u n ( x, p l o t o p t ); 5 6 % Set o p t i o n a l argument if not s u p p l i e d 7 if ( nargin == 1 ), p l o t o p t = 1 ; end 8 % Read the time s e r i e s i n f o r m a t i o n 9 x = x ( : ) ; nsamp = length ( x ) ; 10 % Compute the raw power spectrum 11 x f = fft( x ) ; xps raw = abs( x f ( 1 : end / 2 ) ). ˆ 2 ; 12 f = ( 0 : 1 / nsamp :0.5 1/ nsamp ) ; 13 % Compute the smoothed power spectrum 14 [ xps welch, F ] = pwelch ( x, ) ; 15 % Return the output in pspec 16 pspec = s t r u c t ( Xpsraw, [ f xps raw ], Xpswelch [ F x p s w e l c h ] ) ; 17 % P l o t if opted 18 if ( p l o t o p t == 1) 19 f i g u r e 20 subplot ( ) ; 21 p l o t ( f, xps raw ) ; g r i d on ; a x i s t i g h t 22 subplot (212) 23 p l o t (F/(2 max(f ) ), x p s w e l c h ) ; g r i d on ; a x i s t i g h t 24 end A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30

30 Scripts and Functions Function related functions Optimization: fzero, fsolve, fminbnd, lsqnonlin, lsqcurvefit, linprog, etc. Calculus: quad, ode45, pdepe, dde23, etc. Plotting: fplot, odeplot, ezplot, ezmesh, etc. Miscellaneous: inline, eval, argnames, formula, etc. Most of the functions require handles of functions to be passed to them. The functions related to numerical integration require the functions to return derivatives. A.K. Tangirala (IIT Madras) Introduction to MATLAB/SIMULINK October / 30