Nonlinear Regression Using MATLAB

Step 1 Construct a function which calculates the (sum of the squares)/A, using these commands. 

TABLE E.8. Correlation of Rate Expression Using Solver in Excel 

1 A pa B pb C Measured Rate D Calculated Rate E Residual F Residual Squared 

Step 2 Test this function by removing the semi-colons. Then issue the following command 

Step 3 The minimum of value is found using the fminsearch function  

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Nonlinear regression using Matlab was used to fit Equation (12.3) to Km values obtained from the literature,15 experimentally relevant concentrations and scaled [EP]/[E]0 data calculated as the area under EP divided by the sum of the EP and E areas. 

To use nonlinear regression, you minimize Eq. (E.3) with respect to the unknown parameters. Polynomial and multiple regression do this too (behind the scenes), but for nonlinear curve fits it is necessary to use functions such as Solver in Excel and fminsearch in MATLAB. This is demonstrated using the same example given above for multiple regression. 

Based on these constraints, the following MATLAB function was written. It should be saved in the same location as the script that will be used to run the nonlinear regression. 

The differential and integral methods are used primarily in analyzing batch reactor data. Because a number of software packages (e.g.. Polymath, MATLAB) are now available to analyze data, a rather extensive discussion of nonlinear regression is included. 

St data have been obtained for the off-gas CO2 concentration response obtained from changing the feed rate to a bioreactor. Use Smith s method as well as nonlinear regression based on Excel and MATLAB to estimate parameters in a second-order model from experimental... 

Numerical Methods for Chemical Engineers Using Excel , VBA, and MATLAB Exercise 9.2 Nonlinear regression. 

As with the coverage of VBA in this text, this chapter has only touched the tip of the iceberg with respect to MATLAB. It is intended that with the background of the introductory material present here, a student can explore the vastness of available MATLAB features and functions. For example, the nonlinear regression example (Example 10.7) used the function minsearch, which is not a highly robust minimization algorithm. Another MATLAB function that is particularly suited to nonlinear regression is nlf it and its companion nlparci, which provides confidence... 

The Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

The built-in function minsearch is based on a rather unsophisticated algorithm. There are more robust unconstrained minimization functions available in some of the MATLAB Toolboxes, but unfortunately, these are not standard. For simple problems, minsearch often works well enough. It is used here to minimize the sum of squares between fictitious data (program generated data) and a function in which the regression coefficients appear nonlinearly. 

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