Project 3: Least-Squares Curve Fitting

Transcription

1 ECE 309 Project #3 Page i Project 3: Least-Squares Curve Fitting Read the entire instructions for the project before you begin. Pay special attention to what you are asked to turn in (see pp. 6-7), including the format that is requested. Preliminary Preparation and Calculations 1. Part of this project concerns a physics experiment in which students drop an object from a roof, simulating a free-fall. Do a google search to review the assumptions for free-fall. Required equations: From basic physics, we know that the displacement (d, meters) of a moving object is given by: d = vit +.5 a t 2, where t is time (seconds), vi is the initial velocity (m/s), and a is the acceleration due to gravity (= -9.8 m/s/s). 2. Part of this project concerns an RC circuit. For the circuit below, assume that the capacitor is initially charged to V0 volts before the switch is thrown at time t = 0: + v(t) - (Note that the battery that initially charged the capacitor is not shown.) Figure 1: Source-free RC Circuit a. Use Kirchoff s current law (KCL) at the top node to write the differential equation describing the voltage, v(t), across the capacitor after the switch is thrown. Justify each step of your derivation. b. Solve your differential equation to obtain the equation for the voltage across the capacitor as it discharges. c. Recall that the time constant, for this circuit is: =. Procedure, Part 1. Least-Squares Curve Fitting for Polynomials Writing the m-file In this section we will be writing a function m-file, called pnfit, that you will be able to use to generate plots for your lab reports. This will be useful whenever the experimental data is modeled by a polynomial, say of the form: y = anx n + an-1 x n a1x + a0; Your function m-file will have inputs:

2 ECE 309 Project #3 Page ii n, the degree of the polynomial that we expect to fit the data; x, a vector of the x-components of the observed data; and y, a vector of the y-components of the observed data. and outputs ŷ, the least-squares estimate of y, as a vector of polynomial coefficients; and e 2 2 2, the sum of the squared errors e (y ŷ). Your m-file will also generate a single plot containing the experimental data points (x, y), with o s for data markers, and no line connecting the points; and a smooth plot of ŷ, the least-squares fit to the data, over the range of x-values represented in the experiment. Your m-file should also prompt the user to supply the x-axis label, the y-axis label, and the plot title. We will create a flow chart for pnfit, together in class. Testing the m-file, Part 1 Write a second m-file, called test_drive_pnfit, to test the m-file called pnfit. In this file, use polyval to generate data vectors y (for a given test vector x) for 2 known polynomials (of your choice): a 2 nd degree polynomial, say P2; call the data vectors x2_test and y2_test; and a 4 th degree polynomial, say P4; call the data vectors x4_test and y4_test. Use trial and error to choose a reasonable range of x values for P2 and P4, to ensure that you ve covered the interesting regions (where the polynomial may change directions). The purpose of test_drive_pnfit is to test the previous program, pnfit, and see if it fits the correct polynomial for each of your 2 test vectors above. (Of course it should obtain the exact coefficients doing a least-squares fit, because our test vectors are noise-free.) In the program test_drive_pnfit, call your pnfit program to find the appropriate least-squares fit polynomial for both pairs of test vectors. Keep debugging your code until you can obtain the correct least-squares fit for both test cases. Testing the m-file, Part 2 Suppose that some beginning physics students are conducting a free-fall lab experiment that requires them to drop a small steel ball from the top of a 10-storey building of height 50 meters, and measure the distance that it falls as a function of time. The students place observers from the various lab teams on different floors of the building with stop-watches, and obtain the following measured data: Table 1. Free-fall Experiment Data t, sec. d, m

3 ECE 309 Project #3 Page iii 1) Use your function m-file pnfit to generate the least-squares fit equation for the data given in Table 1, the sum of the squared errors for your best-fit, and the plot. 2) Write a very short paragraph explaining the results, from the perspective of the physics students writing their lab report. 1 Include a table as shown below to compare your calculated (experimental) values for vi and a to the theoretical values for vi and a in a free-fall, from the equation : d = vit +.5 a t 2. Parameters Theoretical Value Experimental Value % Error a vi Procedure, Part 2. Constrained Least-Squares Curve Fitting 1) Complete the separate tutorial/worksheet: Least-squares Fit, with Constraints. When you are done, ask your instructor or the TA to check your answers. 2) Note that the 2nd-degree polynomial found as the least-squares fit for the physics free-fall problem (in Procedure, Part 1) was an unconstrained quadratic of the form: d = Et 2 + Dt + C. The physical model for the physics problem was a constrained quadratic (d = vit +.5 a t 2 ) of the form: d = Et 2 + Dt. Using the techniques in the tutorial/worksheet, find the vector/matrix equation for the constrained quadratic least-squares fit of the form: d = Et 2 + Dt for the data given in Table 1. (Plug in the appropriate numerical values for the left-hand side of the equation.) = ) Make an m-file called const_pnfit to contain the MATLAB commands you use for Procedure, Part 2. This m-file should solve the vector/matrix equation (obtained in question 2 above) for the vector of E D unknowns,. According to this model, what is your constrained quadratic least-squares fit for d, and what are your experimental estimates (based on the given data) for the parameters a and vi? 1 For the physics students, the purpose of the experiment is to verify the distance equation and to validate its parameters in the case of a free-fall.

4 ECE 309 Project #3 Page iv Procedure, Part 3. Least-Squares Curve Fitting for an Exponential Function Planning the m-file First use the lecture notes provided in class to create a flow chart showing the steps required to generate a least-squares fit exponential, ŷ = ce ax, fitting a set of points {x, y}. (It will be helpful to look ahead to the next section to see what your m-file needs to do.) Writing the m-file Write a function m-file called myexpfit, to find the least-squares fit exponential, of the form ŷ = ce ax, given input vectors x and y. Your function m-file will have inputs: x, a vector of the x-components of the observed exponential data; and y, a vector of the y-components of the observed exponential data. and outputs parameters c and a. Your function should transform x to X and y to Y as described in the lecture notes; call the function pnfit 2 (passing in X, Y and n = 1, since we want a linear fit) to obtain A and B of Y = AX + B; use the appropriate equations to obtain parameters c and a. Your m-file should also generate a single plot containing the experimental data points (x, y), with o s for data markers, and no line connecting the points; and a smooth plot of ŷ, the least-squares fit to the data, over the range of x-values represented in the experiment. (You will probably want to create a vector xdense, similar to the one we did in pnfit.) Your m-file should label the x and y axis (call them x and y ), and include a grid for the plots. Testing the m-file, Part 1 Write a program called test_drive_myexpfit, designed to generate a vector x and a vector y for some known exponential function, c e ax (your choice of c, a). Then input the x and y vectors to your function myexpfit, to see if it correctly determines the c and a parameters. Keep debugging your code until your test case is successful. 2 You should first modify pnfit (call your new function pnfit2) so that it only returns the A and B values (i.e., the vector of coefficients for Y). You no longer need to compute the sum of the squared errors or do the graphs that were drawn for pnfit.

5 ECE 309 Project #3 Page v Testing the m-file, Part 2 For the RC Circuit shown in Figure 1, the capacitor is initially completely charged at time t = 0, and students took measurements of the voltage across the capacitor during its discharge phase, after the switch is thrown, for a range of time values given in milliseconds. (Note the units!) This experimental data will be available from your instructor. From your answer to preliminary calculation question 2b, you know that the discharge function is exponential. Use your m-file myexpfit to generate a plot showing experimental data (not connected by a line) and the smooth exponential least-squares fit to the data, and determine the estimated parameters of the exponential decay: V0 = ; =

6 ECE 309 Project #3 Page 1 of 7 Student name: Joshua Rothe Cover-sheet: MATLAB Project 3 Curve Fitting Preliminary Calculation Answers (handwritten for 2b; show derivation, justifying each line): 2b. KCL Diff. Eq v(t): Ic = C (dv/dt) and Ir = v/r When switch closes, Ic = Ir, thus C (dv/td) = v/r Thus, C*v(t) =(1/r) (C*r) v (t) + v(t) = V 2c. = R*C + v(t) - Figure 1: Source-free RC Circuit Procedure, Part 1: Least-Squares Fit for Polynomials Testing the m-file, Part 2: Least-squares fit equation obtained for the physics experiment (Question 1, p. 3): d = t^ t Sum of squared errors for the physics experiment: Calculated values for parameter estimates, based on experimental data: a = 9.8 vi = 0 (from Question 2, p.3) Procedure, Part 2: Curve Fitting with Constrained Polynomials Least-squares fit equation obtained for the physics experiment (Question 3, p. 4): d = x^ x Calculated values for parameter estimates, based on experimental data: a = vi = (from Question 3, p.4) Procedure, Part 3: Curve Fitting for Exponential Functions v(t) = *e^( *t) (least-squares fit for voltage in the RC Circuit) estimate for : ; estimate for V0:

7 ECE 309 Project #3 Page 2 of 7 function [y_hat,e_squared] = pnfit(n,x,y) %pnfit solves Project 3 y_hat = polyfit(x,y,n); new_y = polyval(y_hat,x); errors = (y-new_y).^2; e_squared = sum(errors); x1 = input('enter the x label: ','s'); y1 = input('enter the y label: ','s'); ptitle = input('enter the plot title: ','s'); plot(x,y,'o'); hold on plot(x,new_y); xlabel(x1),ylabel(y1),title(ptitle); grid disp('y_hat: '); disp(y_hat); disp('e_squared: '); disp(e_squared); >> test_drive_pnfit_p2 Type the name of the assignment here: Project 3, Part 2 Joshua Rothe ECE 309 Project 3, Part 2 29-Nov-2014 Enter the x label : X Enter the y label : Y Grid enabled? y/n (case sensitive!) : y Enter the plot title : Part 2 y_hat: e_squared: nd degree Y values : Part Y X

8 ECE 309 Project #3 Page 3 of 7 Procedure, Part 1 Parameters Theoretical Experimental % Error Value Value a % vi N/A Although we cannot obtain a % error value for vi (diving by zero), we can assume since it is relatively close to zero that the percentage error is similar to a, which is 8.2%. We can expect this deviation from the theoretical value of gravity's acceleration due to things such as wind resistance etc. Overall, the data followed what we expected it to follow before we conducted the experiment.

9 ECE 309 Project #3 Page 4 of 7 Worksheet: Least-squares Fit, with Constraints Column 1 consists of xi^2 values. I would delete column 2. The vector of unknowns would be x = [x^2', 0', 1'] Vector form: [1 1; 3 1; 5 1; 6 1] * [xi]' = [ ]' Quadratic least-squares fit for the data is x^ [x21 x11 0; x22 x12 0; x23 x13 0; x2m x1m 0] * [x1 x2 xm] = [y1 y2 ym] Solving for constrained least-squares fit: >> A = [1 1; 4 2; 16 4; 25 5] A = >> b = [ ]' b = >> x = ((A'*A)^-1)*(A'*b) x = Y = x^ x

10 ECE 309 Project #3 Page 5 of 7 Procedure Part 2. Constrained Least-Squares Curve Fitting % Program const_pnfit % Program solves project 3, procedure part 2, constrained least-squares % curve fitting header_rothe; A = [0 0; ; ; ; ; ]; b = [ ]'; x = ((A'*A)^-1)*(A'*b) >> const_pnfit Type the name of the assignment here: Procedure 2 Joshua Rothe ECE 309 Procedure 2 01-Dec-2014 x = Least-squares fit is y = x^ x. Experimental estimates are a = (bringing the 9.8 m/s^2 up to , close to the theoretical) and vi = (not too far off from 0).

11 ECE 309 Project #3 Page 6 of 7 Procedure Part 3 Initial data v Logarithmic scale for y input v Polyfit to obtain least-squares equation, first degree v Exponential of the b value becomes c v Plot: x axis is x, y axis is c*e^(a*x)

12 ECE 309 Project #3 Page 7 of 7 function [ c,a ] = myexpfit( x,y ) %myexpfit Function solves project 3, procedure 3 % Function finds least-squares fit exponential X = x; Y = log(y); % sets up x and y y_hat = polyfit(x,y,1); a = y_hat(1,1); b = y_hat(1,2); c = exp(b); % sets up exponential values labelx = input('enter x label: ','s'); labely = input('enter y label: ','s'); ptitle = input('enter plot title: ','s'); plot(x,y,'o') hold on plot(x,c.*exp(a.*x)) xlabel(labelx),ylabel(labely),title(ptitle) grid disp('a ='); disp(a); disp('c = '); disp(c); end % After importing the data: >> time = timems.*.001; >> myexpfit( time, vvolts ); Enter x label: time Enter y label: volts Enter plot title: Part 3 a = c = Part volts time