Matlab optimisation toolbox

If basic assumptions concerning the error structure are incorrect (e.g., non-Gaussian distribution) or cannot be specified, more robust estimation techniques may be necessary, e.g., Maria and Heinzle (1998). In addition to the above considerations, it is often important to introduce constraints on the estimated parameters (e.g., the parameters can only be positive). Such constraints are included in the simulation and parameter estimation package ACSL-OPTIMIZE and in the MATLAB Optimisation Toolbox. Because of numerical inaccuracy, scaling of parameters and data may be necessary if the numerical values are of greatly differing order. Plots of the residuals, difference between model and measurement value, are very useful in identifying systematic or model errors. 

All the four PSO algorithms can find the global optimal solutions whereas the gradient based optimisation algorithm from the MATLAB Optimisation Toolbox, fminunc, fails to find the global optimal solutions when the initial values are not close to the global optimal solutions. 

For the purpose of comparison, optimisation using a single neural network was first carried out. Table 2 shows the obtained results. As can be seen from the table, the values for the difference between the final amounts of product and by-product using the PSO codes were similar to the ones obtained using the MATLAB Optimisation Toolbox function,yJwzwcow, in this fed-batch reactor. However, PSO can cope with multiple local minima in general as shown in Section 2. 

As the MATLAB software packages with Optimisation Toolbox provides both effective ordinary differential equation (ODE) solvers as well as powerful optimization algorithms, the dynamic simulations reported in this paper are carried out by using the MATLAB Optimisation Toolbox (8). 

Matlab does not include a routine of the kind of fzero for more than one variable. Only the function fsolve, which is part of the Optimisation Toolbox, can deal with systems of equations with several variables. Here we demonstrate the application of fsolve to the system of equations (3.70). 

The MATLAB software consists of a basic package of mathematical routines as well as optional toolboxes that cover specific engineering application areas such as process control, optimisation, signal processing and symbolic mathematics. It also has a toolbox that makes available many of the Numerical Algorithms Group (NAG) routines. These routines are some of the best implementations of numerical methods that are currently available. MATLAB offers a convenient computing environment for the solution of process simulation problems, and is used here. 

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