Curve Fitting Matlab 2013 A 14

vth and R are constants already known, so I only want to achieve I01, I02, n1, n2. The problem is: as you can see, I is dependent on itself. I was trying to use the curve fitting toolbox, but it doesn't seem to work on recursive equations.

curve fitting matlab 2013 a 14

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Several solutions [4-9] have been proposed to perform curve fitting. One of the widely used nonlinear curve fitting algorithms was introduced by Levenberg-Marquardt (LM) [5,6]. This method belongs to the gradient-descent family. However, due to the sensitivity of the method, the data quality and the initial guess, the curve fitting algorithm is unfortunately susceptible to becoming trapped in local minimums. To obtain proper curve fitting results, there is a need for automatic outlier handling, usage of predefined initial curves, and adaptive weighting for data points [10,11]. All of these demands can, without doubt, be applicable to large sets of data. Nonlinear and non-iterative least square regression analysis was presented in [7] for robust logistic curve fitting with detection of possible outliers. This non-iterative algorithm was implemented in a microcomputer and assessed using different biological and medical data. A review of popular fitting models using linear and nonlinear regression is given in [4]. The study serves as a practical guide to help researchers who are not statisticians understand statistical tools more clearly, especially dose-response curve fitting using nonlinear regression. This book also describes the robust fitting algorithm and the outlier detection mechanism used in the commercial software GraphPad Prism;. In [8], an automatic best-of-fit estimation procedure is introduced based on the Akaike information criterion. This best-of-fit model guarantees not only the estimation of the parameters with smallest sum-of-squares errors but also good prediction of the model.

Simulated data for DRC estimation was used in [9] to examine the performance of the proposed Grid algorithm. The peculiarity of the Grid algorithm is that it visits all the points in a grid of four curve parameters and searches for the point with the optimum sum-of-squares error. A coarse-to-fine grid model and a threshold-based outlier detection mechanism were used to make the algorithm more efficient. This paper also provided Java-based software together with a sample dataset for academic use. However, the software is applicable only when the measurements are taken without replicates (one data point at each concentration). Furthermore, various popular computer software and code packages have been presented for DRC fitting such as the DRC package in R [12], the nlinfit function in MATLAB; [13], and the XLfit add-in for Microsoft; Excel; [14].

To overcome the situation in which data spreads differently at concentrations, various weighting techniques are considered, including relative weighting, Poisson weighting, and observed variability-based weighting [4]. Relative weighting extends the idea of standard weighting by dividing the squared distance by the square of the corresponding response value Y; hence, the relative variability is consistent. Similarly, Poisson weighting and weighting by observed variability use different forms of dividing the response value Y. Indeed, minimizing the sum-of-squares might yield the best possible curve when all variations obey a Gaussian distribution (without considering how different the standard deviations at concentrations are). However, it is common for one data point to be far from the rest (caused by experimental mistakes); then, this point does not belong to the same Gaussian distribution as the remaining points and it contributes erroneous impact to the fitting. The Tukey biweight function [10] was introduced to reduce the effect of outliers. This weighting function considers large residuals and treats them with low weights, or even zero weights, so that they do not sway the fitting much. In this section, we present a modification of the Tukey function and apply it to our fitting.

QIBA test data corresponding to the Tofts model. Left: RCE values of 30 combinations of Ktrans and v e values of simulated QIBA test data without added noise. Upper-right: curve-fitting with the Tofts model over two random points of the QIBA test data without noise. Lower-right: curve-fitting of the Tofts model adding Gaussian noise of zero mean and σ=20% of the signal baseline.

The problem is that Linest will only fit a straight line to the data, so if we want a non-linear fit we have to convert the data into a form that is a straight line, get the coefficients for that line, then convert it back to the non-linear form we want.In the case of the exponential curve we fit a straight line to ln(y) = ax + b, so y = Exp(ax + b) = Exp(b) * Exp(ax)For the example in the spreadsheet, Exp(b) is in cell Q51, so to return the y value for x = 2 (in cell A37) you need:= Q51 * EXP(O51 * A37)Note that this returns 10.411, rather than 9.82, because you are fitting a smooth curve to scattered data. Looking at the Exponential chart you can see that in this case the exponential curve is not a good fit to the data. The power curve gives a much better fit, but even in that case extrapolating past x = 11 would probably rapidly diverge.

The current research of state of charge (SoC) online estimation of lithium-ion battery (LiB) in electric vehicles (EVs) mainly focuses on adopting or improving of battery models and estimation filters. However, little attention has been paid to the accuracy of various open circuit voltage (OCV) models for correcting the SoC with aid of the ampere-hour counting method. This paper presents a comprehensive comparison study on eighteen OCV models which cover the majority of models used in literature. The low-current OCV tests are conducted on the typical commercial LiFePO4/graphite (LFP) and LiNiMnCoO2/graphite (NMC) cells to obtain the experimental OCV-SoC curves at different ambient temperature and aging stages. With selected OCV and SoC points from experimental OCV-SoC curves, the parameters of each OCV model are determined by curve fitting toolbox of MATLAB 2013. Then the fitting OCV-SoC curves based on diversified OCV models are also obtained. The indicator of root-mean-square error (RMSE) between the experimental data and fitted data is selected to evaluate the adaptabilities of these OCV models for their main features, advantages, and limitations. The sensitivities of OCV models to ambient temperatures, aging stages, numbers of data points, and SoC regions are studied for both NMC and LFP cells. Furthermore, the influences of these models on SoC estimation are discussed. Through a comprehensive comparison and analysis on OCV models, some recommendations in selecting OCV models for both NMC and LFP cells are given.

Due to the logarithmic function exists in some OCV models, the entire or whole SoC region is selected from 0.001% to 99.999%. The fitted OCV-SoC curves of different OCV models at 25 C for both NMC and LFP fresh cells are shown in Figure 3. Note that the SoCs change dramatically when drop to 0% or rise to 100%, the RMSEs are collected the SoC regions between 2.5% and 97.5%. The RMSEs of different SoC regions are given in Figure 4. Although these models perform well in fitting the SoC regions between 10% and 90%, but some models perform poorly in fitting the entire SoC regions for both NMC and LFP fresh cells, especially in the low and high SoC regions.

Theoretically, the fitting performance is better with higher order polynomial function, however, the 2nd to 4th order polynomial functions perform well in middle SoC regions, but perform poorly in low and high SoC regions. But for 5th and 6th order polynomial functions, the OCV-SoC curves pass through data points 0% and 100% and fluctuate sharply in the middle SoC regions. The fitting performances are become better for the 7th to 12th order polynomial functions. Therefore, we may conclude from Figure 4 that the models 16, 17 and 18 are suitable for NMC cell and models 7, 17 and 18 are suitable for LFP cell, which perform well in both the entire and middle SoC regions. In the following sections, all the OCV models are determined from the experimental data of entire SoC regions.

As mentioned above, the OCV models are determined by fitting certain numbers of experimental data points with MATLAB curve fitting toolbox. Some studies used 21 data points (i.e., in every 5% SoC interval) to determine the OCV model [13, 14, 42], but some other studies used 11 data points (i.e., in every 10% SoC interval) [43,44,45] or 51 data points (i.e., in every 2% SoC interval) [30] to determine the OCV models. The comparison results depicted in Figure 5 indicate that the OCV models are more sensitive to the data points for LFP cell than NMC cell. These may be caused by their differences in low and high SoC regions of OCV-SoC curves. In addition, the RMSEs of NMC cell change slightly with increasing the numbers of data points except the models 5, 7, 12 and 13, but for the LFP cell, the RMSEs are generally decrease greatly with 21 data points than with 11 data points, but the RMSEs virtually unchanged if change the number of data points from 21 to 51 except models 8 and 14. Note that the models 8 and 18 require more than 12 and 13 data points, respectively, to fit by MATLAB curve fitting toolbox.

Several solutions [4]-[9] have been proposed to perform curve fitting. One of the widely used nonlinear curve fitting algorithms was introduced by Levenberg-Marquardt (LM) [5],[6]. This method belongs to the gradient-descent family. However, due to the sensitivity of the method, the data quality and the initial guess, the curve fitting algorithm is unfortunately susceptible to becoming trapped in local minimums. To obtain proper curve fitting results, there is a need for automatic outlier handling, usage of predefined initial curves, and adaptive weighting for data points [10],[11]. All of these demands can, without doubt, be applicable to large sets of data. Nonlinear and non-iterative least square regression analysis was presented in [7] for robust logistic curve fitting with detection of possible outliers. This non-iterative algorithm was implemented in a microcomputer and assessed using different biological and medical data. A review of popular fitting models using linear and nonlinear regression is given in [4]. The study serves as a practical guide to help researchers who are not statisticians understand statistical tools more clearly, especially dose-response curve fitting using nonlinear regression. This book also describes the robust fitting algorithm and the outlier detection mechanism used in the commercial software GraphPad Prism;. In [8], an automatic best-of-fit estimation procedure is introduced based on the Akaike information criterion. This best-of-fit model guarantees not only the estimation of the parameters with smallest sum-of-squares errors but also good prediction of the model. 2ff7e9595c