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ENGR 132 Final Project

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ENGR 132 Final Project

  1. 1. ENGR 132 - FOS Project Spring 2016 Technical Brief Teammate FNs: Hannah Snow Jessica Murray Senzeyu Zhang Mia Sheppard Purdue Logins: snowh murra119 zhan2196 shepparm Section Number: 15 Team Number: 04 To: President Frank O. Simpson From: Section : 15, Team 4 RE: FOS Project Date: Class 31 The problem involves a quality analysis of thermocouples designed by First Order Systems so that they may provide an ethical statement about their thermocouples to customers. FOS needs accurate information on the parameters and functionality of their thermocouples, which is provided by our team’s executive function, algorithm, and regression function, and they need to be able to utilize these functions in the future to examine future products. FOS needs to be able to trust the documents we present to them; needs to know that they will work generally and accurately for all of their products; need to know they have a set of functions that run efficiently and accounts for possible errors. Our constraints are: the time frame that we have been provided; the fact that we are creating a functions for the analysis of clean and noisy data; the need for the functions to be generalizable; the need to determine the parameters of the first order system via our own methods. The main purpose of our algorithm is to use for and while loops to find the parameters of ts, ys, yss, and �. The algorithm then takes the values of the parameters, solves the piecewise equation for yt using a for loop, generates a calibration plot, and finds SSE in order to assess how well our parameters fit the data obtained from the thermocouples. The process our team has followed so far is: determining parameter identification methods; generalizing them so they can fit any data set; generating algorithms based off our chosen methods; working on improving one algorithm; creating an executive and regression function. For method generation, we had a group thinking session to consider the multiple possibilities. This helped us determine several methods and select ones we thought were the most appropriate for this project. We made this determination by considering the errors that could occur, whether or not it was possible to code in MATLAB, and whether or not it fit our skill sets as a team so everyone could be involved. We settled on a method that set limitations on the data to allow us to pinpoint ts, ys, and yss. This then allowed us to find �, and since this method was applicable to both clean and noisy data, it was a clear choice for the development of our two algorithms. Throughout the development of our two separate algorithms, we communicated on what was working and what wasn't to better develop our chosen method. Once the algorithms were finished, we discussed limitations and improvements for each. We also compared the parameter results using the piecewise function to find yt, applying those values to the plot of the data, and then determining SSE to assess how accurate the parameters were. As both sets of code were similar in their effectiveness, it was difficult to choose one to improve upon for our final
  2. 2. ENGR 132 - FOS Project Spring 2016 Technical Brief algorithm. Since we were going to work together to improve and develop whichever code was chosen, we decided to simply choose one and commit to it for the rest of the FOS project. Since the determination of which algorithm to use and develop, we have communicated as a group to determine faults in the code, how to improve efficiency, and how to improve accuracy in determining parameters. We’ve refined the code as we’ve progressed, making sure each line worked in the best possible way. We’ve made sure the algorithm, regression function, and executive function fit as an effective triad for quality analysis to complete all goals. These goals were: identify parameters; complete a statistical analysis of �; execute regression to complete a price analysis; determine SSE for each model. We’ve made sure our variables lined up across functions, appropriate function calls have been made, we’ve all contributed to the structuring and execution of the functions, and have checked that each part is working before progressing. We’ve made sure to delegate so that everyone has had something to do the entire process. The first step in our algorithm is to load the data and separate it into temperature and time variables. Then, to find ys, limits are set on the data to better evaluate it. To determine the bottom limit, we find the minimum of the total data. To determine the top limit, we double the minimum. To determine ys, a while loop is executed that checks for when the temperature exceeds the limits. For this to occur, we initialize the index at 1, then after running through the loop, we check to see if the result is accurate. We then execute another while loop that checks subsequent data points to see if they are still between the limits. If they are, the algorithm returns to the initial loop, and if they aren’t, then the algorithm stops running the loops. Once the loops terminate, we set the index to index = index -1, as the function result is one value over the desired result for ys. Then ys is produced with ys = Temp(index) and ts is produced with ts = Time(index). To determine yss, we set new limits and initialize the index to execute the same loop. However, instead of starting with the first data point and progressing forward in index, the loops begins at the end of the data and progresses backwards in index. We then produce yss with yss = Temp(index). To determine the value of �, we execute the following calculation to find y�: y� = ys + (.632 * (yss - ys)). From there, we set limits so that we can determine which time value corresponds with y�. To set the upper bound we execute the following calculation: topBound = y� + y� * .008. To set the lower bound we execute the following calculation: bottomBound = y� - y� * .008. We then use conditional statements to determine x� so that we can then determine � by subtracting ts from x� Finally, ys, ts, yss, and � are printed to the command window. Table 1 displays the mean and standard deviation for � throughout the five FOS thermocouple models. After refining the code, the � values became less varied, and the regression plot is now exponential instead of linear. In Table 2, the r2 value is 0.9519, so it accounts for 95% variance. This means the line represents the data well. The SSE value and SST value from Table 2 are related to Figure 1. SSE is the sum of squared error and SST is the total sum of squares. Figure 1 is a representation of the time constants (�) from the 100 time histories plotted against the price of each model. As stated above, it is seen that the regression line now accurately represents the time constants. In a perfect experiment, the data would be clean, but the data we received is noisy, so this is a place error occurs. This accounts for the quality of the experiments themselves. The error in this process also occurs due to the range of the data, which means that there were instances
  3. 3. ENGR 132 - FOS Project Spring 2016 Technical Brief where the range between two data points was quite large and exceeded the limits that we had to set in order to determine ts, ys, and yss. At first, our parameter identification range was extremely inaccurate and large because we had to account for such instances. It caused large SSE and SST values, along with � values that had too much variation. After refining ranges, as well as the regression plot, the SSE and SST values became much smaller and proved that the products were consistent. In our first trial, the r2 value came out to be around 0.4, but with our refinements we were able to get the r value to 0.952. FOS can say that their products are consistent in performance, pricing and manufacturing. They can say this and support their statement with the regression model and r2 valued. The r2 value is 0.952, so this shows that the data fits well and that it is consistent. Niemann, H., & Miklos, R. (2014). A Simple Method for Estimation of Parameters in First Order Systems [Abstract]. J. Phys.: Conf. Ser. Journal of Physics: Conference Series, 570(1), 012001. Retrieved March 28, 2016. Table 1 Model Number τ Characteristics SSEmod,ave Mean Standard Deviation FOS-1 0.189660 0.028182 2.4022 FOS-2 0.474560 0.031535 2.6576 FOS-3 0.735350 0.054435 3.3350 FOS-4 1.166220 0.067187 4.2280 FOS-5 1.688610 0.069588 4.5325 Figure 1
  4. 4. ENGR 132 - FOS Project Spring 2016 Technical Brief Table 2 Parameter τ Value SSE 1.3537 SST 28.1362 r2 0.9519