Introduction to MATLAB Fall Bruno Abreu Calfa Department of Chemical Engineering Carnegie Mellon University

Transcription

1 Introduction to MATLAB Fall 2014 Bruno Abreu Calfa Department of Chemical Engineering Carnegie Mellon University 1

2 What is MATLAB? Main MATLAB windows Simple calcula9ons Easy plo<ng with ezplot MATLAB BASICS 2

3 Matrix Laboratory What is MATLAB? Everything in MATLAB is a matrix Enhanced by toolboxes (functions for specimic applications) Optimization Statistics Bioinformatics Signal Processing Finance Widely used in industry and academia. Getting help: MATLAB Help, Google, books 3

4 Main Window 4

5 MATLAB as a Calculator Basic mathematical operators >> >> 2*3 + 4 >> 4/3-3/4 + 2^3 >> 4.2/ Use parentheses to enforce precedence >> 2*(3 + 4) >> 4.2/( ) >> 15/((2 + 3)*(4-1)) >> 2ˆ(3/2) 5

6 Easy Plo<ng with ezplot Plot simple func9ons with the command ezplot Example >> ezplot('x^2') Set limits on the horizontal axis >> ezplot('exp(-0.5*t)*sin(3*t)',[0 10]) Plo<ng implicit func9ons >> ezplot('x^2 + y^2-1',[-2 2]) >> axis equal 6

7 Graphical Solu9on to Linear Equa9ons Solve the following system graphically Rewrite it as follows Plot two implicit func9ons >> hold all >> ezplot('2*x1 + 3*x2-6') >> ezplot('4*x1 + 9*x2-15') >> hold off 7

8 Scalars, vectors, matrices DEFINING VARIABLES 8

9 DeMining Variables (I/II) Use the = sign for assignment >> a = 1 % Scalar variable 'a' stores the value 1 >> % This is a comment and is ignored by MATLAB >> b = -5.2; % ; at the end of a command avoids displaying the result on Command Window >> v1 = [1 2 3] % Row vector or 1-by-3 matrix >> v2 = [1,2,3] % Same as v1 >> v3 = [4;5;6] % Column vector or 3-by-1 matrix >> v1 + v3 % Error! Matrix dimensions must match! >> v1 + v3.' % OK. Transpose v3 to become 1-by-3 >> 2*v1 % Element-wise multiplication 9

10 DeMining Variables (II/II) >> m1 = [1 2 3; 4 5 6] % 2-by-3 matrix >> m2 = [4 5 6; 1 2 3] % Another 2-by-3 matrix >> m3 = m1*m2 % Error! Cannot multiply m1 and m2 >> m3 = m1.'*m2 % OK. Transpose m1 to 3-by-2, then multiply by m2. Store result in variable m3. >> 0.5*m1 % Element-wise multiplication Some useful func9ons >> m = ; >> fprintf('mass flowrate = %.2f kg/s', m) >> v = zeros(10,1) % 10-by-1 matrix full of zeros >> m = zeros(5) % 5-by-5 matrix full of zeros 10

11 MATLAB Quiz How do you use ezplot to graph? (a) ezplot('2x^2y + 4xy - 5xy^2-6') (b) (c) ezplot('2*x^2*y + 4*x*y - 5*x*y^2-6') ezplot('2*x^2*y + 4*x*y - 5*x*y^2') How do you define the matrix? (a) [2 1 0; 0-3 4] (b) [2 1; 0-3; 0 4] (c) [2 0; 1-3; 0 4] What is the solu9on to [3 5 2; 1 0 1] * [1 0; 2 0]? (a) Error (b) [13 0; 1 0] (c) [7 0; 2 0] 11

12 Everyone knows A\b SOLVING LINEAR EQUATIONS 12

13 Basics Given matrix A and vector b, solve for vector of unknowns x General idea In prac9ce, NEVER invert A and then mul9ply by b. More efficient and numerically stable methods (06-262). Applica9on: Linear material balances 13

14 Solving Ax = b in MATLAB Define matrix A and vector b Use the backslash operator \ to solve for x x = A\b Example >> A = [2 3; 4 9] >> b = [6; 15] >> x = A\b % Solution = [1.5; 1] 14

15 Sample Exam II: Problem 3 Flowsheet of the process (pure W) 100 kg/hr 15% S 85% W Boiler 80% S 20% W 60% S 40% W 15

16 Sample Exam II: Problem 3 (pure W) 100 kg/hr 15% S 85% W Boiler 80% S 20% W 60% S 40% W Calculate total amount of water driven off in the boiler. I. Overall solids balance II. Overall balance III. Mixer Overall: Solids: In MATLAB: >> A = [1 1; ] >> b = [25; 15] >> x = A\b 16

17 Sample Exam II: Problem 3' Unknown composi9on of stream leaving boiler. Addi9onal specifica9on: bypass stream flowrate = 10 kg/hr. Solve for all flows. Flowsheet of the process (pure W) 100 kg/hr 15% S 85% W Boiler 60% S 40% W 17

18 Sample Exam II: Problem 3' Hint: write all balances in terms of component flowrates. 100 kg/hr 15 kg/hr S 85 kg/hr W Boiler 60% S 40% W 10 kg/hr Degree of freedom analysis: 9 variables (component flowrates) Need to write 9 independent equa9ons 18

19 Sample Exam II: Problem 3' 100 kg/hr 15 kg/hr S 85 kg/hr W Boiler 60% S 40% W 10 kg/hr I. Outlet specification II. Splitter III. Boiler inlet composition (2) (1) (3) IV. Bypass stream composition V. Boiler (5) VI. Mixer (7) (4) (6) (8) VII. Bypass specification (9) 19

20 Sample Exam II: Problem 3' System of equa9ons in matrix form (1) (2) (3) (4) (5) (6) (7) (8) (9) In MATLAB Enter A and b (use the spreadsheet tool to fill in the nonzero elements of A and b) Solve for x >> x = A\b 20

21 Ethanol Production: Block Diagram Produc9on of ethanol from ethylene and water. µ 51 Feed (given) C 2 H 4 H 2 O C 2 H 6 O Biegler, L. T., Grossmann, I. E., Westerberg, A. W. (1997). Systematic Methods of Chemical Process Design. Prentice Hall. 21

22 Linear Material Balances Component material balances for ethylene (inlet of ethylene μ 01 and split frac9ons for separators are given). 11 equations 11 unknowns 22

23 Solving System of Linear Equations System in matrix form In MATLAB Define A and b Execute A\b 23

24 Gaussian Elimination: Basics The opera9on A\b may perform Gaussian Elimina9on (GE). There are a few varia9ons of GE (with complete or par9al pivo9ng). We will cover the pure GE algorithm. Objec9ve: perform elementary row opera9ons that make the A matrix upper triangular. Apply same opera9ons to vector b. Elementary row opera9ons Interchanging two rows or columns Adding a mul9ple of one row or column to another Mul9plying any row or column by a nonzero element Easier to explain with an example. Addi9onal reading: hdp:// of- linear- equa9ons.html zeros Upper Triangular Matrix 24

25 Gaussian Elimination: Example Convert to augmented matrix equation Elementary row operations: 25

26 Gaussian Elimination: Example Restoring original system: Back-substitution step: Row 3: Row 2: Row 1: 26

27 Reduced Row Echelon Form (RREF) If GE con9nues further to make the original matrix A the iden9ty matrix, then the augmented matrix is said to be in the RREF. For computa9onal reasons, when solving systems of linear equa9ons, it is some9mes preferable to stop row opera9ons before the matrix is completely reduced. RREF of the previous example: In MATLAB, rref func9on. >> A = [1 1 1; 4 3 4; 9 3 4] >> b = [3; 8; 7] >> Ab = [A b] >> rref(ab) 27

28 Using Editor Window CREATING MATLAB SCRIPTS 28

29 Editor Window

30 Creating a Script Type your code Save file (M- file) Click on Run See results on Command Window Publishing features (generate HTML, PDF etc.) Look it up for more details (other tutorials, Google etc.) 30

31 Scripting Example Using the colon (:) operator. start:stride:end t = -3:3; y = sin(t); figure plot(t,y) Decrease the stride from 1 (default) to 0.1. t = -3:0.1:3; y = sin(t); figure plot(t,y) title('plot of sine') xlabel('t') ylabel('y') See file Demo_Script.m for more examples. 31

32 Sample Exam II: Problem 3' Unknown composi9on of stream leaving boiler. Addi9onal specifica9on: bypass stream flowrate. Plot water flowrate afer boiler for different values of bypass flowrate (e.g, from 1 kg/hr to 10 kg/hr). Flowsheet of the process (pure W) 100 kg/hr 15% S 85% W Boiler 60% S 40% W 32

33 Sample Exam II: Problem 3' Hint: write all balances in terms of component flowrates. 100 kg/hr 15 kg/hr S 85 kg/hr W Boiler 60% S 40% W 1, 2,, 10 kg/hr Degree of freedom analysis: 9 variables (component flowrates) Need to write 9 independent equa9ons 33

34 Sample Exam II: Problem 3' 100 kg/hr 15 kg/hr S 85 kg/hr W Boiler 60% S 40% W p kg/hr I. Outlet specification II. Splitter III. Boiler inlet composition (2) (1) (3) IV. Bypass stream composition V. Boiler (5) VI. Mixer (7) (4) (6) (8) VII. Bypass specification (9) 34

35 Sample Exam II: Problem 3' System of equa9ons in matrix form (1) (2) (3) (4) (5) (6) (7) (8) (9) In MATLAB Enter A and b (use the spreadsheet tool to fill in the nonzero elements of A and b). Enter any number for p (e.g., 0). Write script to solve system for different values of p and then create a plot. 35

36 Sample Exam II: Problem 3' MATLAB script bprates = 1:10; waterrate = zeros(size(bprates)); for i = 1:length(bprates) b(end) = bprates(i); x = A\b; waterrate(i) = x(5); end figure plot(bprates,waterrate) xlabel('bypass Rate (kg/hr)','fontsize',16) ylabel('water Rate after Boiler (kg/hr)','fontsize',16) 36

37 Sample Exam II: Problem 3' Solu9on plot 37

38 plot, scatter, and bar commands MORE PLOTTING 38

39 General Usage Many functions for plotting ezplot, plot, scatter, loglog, semilogx, semilogy, surf, surfc, mesh, meshc Typical syntax: command(data1,data2,...,['prop1name',prop1value,...]) Formatting properties and values are described in the Help: 2D plots: lineseries properties 3D plots: surface properties 39

40 plot Command x = linspace(0,2*pi); y = sin(x); figure plot(x,y); x = linspace(0,2*pi); y = sin(x); figure plot(x,y,'color,'red'); title('plot of sin(x)'); xlabel('x'); ylabel('y'); 40

41 plot and scatter Commands Multiple plots on same Migure using hold all and hold off x = [0.5; 1.0; 1.5; 2.0; 2.5]; y = [0.135; 0.165; 0.195; 0.207; 0.223]; yp = *x ; figure hold all scatter(x,y,300,'fill') plot(x,yp,'linewidth',5) hold off xlabel('x','fontsize',24) ylabel('y','fontsize',24) leg = legend('data','linear Model','location','northwest'); set(leg,'fontsize',24) 41

42 bar Command x = [ ]; y = [ ]; figure bar(x,y) 42

43 Additional Help Two- part tutorial given at CMU (with exercises and solu9ons) hdp:// MATLABTutorials.html Prof. John Kitchin s MATLAB blog hdp://matlab.cheme.cmu.edu/ 43