MATLAB®

This is essentially the program eqbrm (Anderson and Crerar, 1993, Appendix E) written in matlab . It uses input prepared by a spreadsheet and is easily modified to handle multiple inputs. 

The spreadsheet input for the CO2 speciation in Chapter 16 looks like this  

Only the number matrix is copied to Matlab the labels are for clarity. 

The program assumes that H+ is the species in column 1. If it is not, edit the last line of the program so that the reported pH uses a different column number. 

Rows 11 and 12 are optional. If they are included, the program uses the extended form of the Debye-Hiickel equation. If only one number is entered (in column 1) in row 10, it is intepreted as the A term in the Davies equation, and that is used, as in the original eqbrm. 

There are many numerical software products available, both commerical and open-source. It is good practice to use the available algorithms for solving ODFs and PDFs rather than to develop new ones from scratch. In this section we Ust some of most popular algorithms in MATLAB, since it is used widely in both academia and industry. Another popular software product is GNU Octave (freeware). Besides these software products, there is a series of useful books on numerical analysis, Numerical Recipes.  

Methods for solving BVPs and PDFs are summarized in Table 6.6. The standard Matlab installation has one PDF solver. But there is also a toolbox for solving PDFs 

Solver Explicit/ implicit Single step/ multi-step Non- stiff/stiff Accuracy Method 

For the convenience of MATLAB beginners, we list some other usefiil algorithms in the Tables 6.7 and 6.8. Some of these algorithms and commands might assist in completing the practice problems. 

Linear equation systems can be solved with the linsolve algorithm. Various postprocessing and visualization commands are available, as shown in Table 6.8. 

Menu bar Contains the commands you can use to perform certmn tasks, for example, to save your workspace or to change the View settings. 

Current Directory Shows the active directory, but you can also use it to change the directory. 

Current Directory wind.ow Shows all files, their types, sizes, and descriptions in Current Directory. 

Moving window outside the desktop Clicking on this icon will move the window outside the desktop. 

To get started, select WATLftB Help or Oetaos from t 

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Box-Jenkins model is chosen for building up a TF for identification of the material. MATLAB 5.0 was used for calculation of characteristics of the model. In the Figure 5 the main results of modeling are presented. 

After the assembling of the stochastic matrix Pd we have to solve the associated non-selfadjoint eigenvalue problem. Our present numerical results have been computed using the code speig by Radke AND S0RENSEN in Matlab,... 

Using a program such as MATLAB, these are easily calculated. 

The holistic thermodynamic approach based on material (charge, concentration and electron) balances is a firm and valuable tool for a choice of the best a priori conditions of chemical analyses performed in electrolytic systems. Such an approach has been already presented in a series of papers issued in recent years, see [1-4] and references cited therein. In this communication, the approach will be exemplified with electrolytic systems, with special emphasis put on the complex systems where all particular types (acid-base, redox, complexation and precipitation) of chemical equilibria occur in parallel and/or sequentially. All attainable physicochemical knowledge can be involved in calculations and none simplifying assumptions are needed. All analytical prescriptions can be followed. The approach enables all possible (from thermodynamic viewpoint) reactions to be included and all effects resulting from activation barrier(s) and incomplete set of equilibrium data presumed can be tested. The problems involved are presented on some examples of analytical systems considered lately, concerning potentiometric titrations in complex titrand + titrant systems. All calculations were done with use of iterative computer programs MATLAB and DELPHI. 

It was indicated that the original method can be extended on systems where two or three analytes can be determined from a single titration curve. The shifts DpH affected by j-th PT addition should be sufficiently high it depends on pH value, a kind and concentration of the buffer chosen and its properties. The criterion of choice of the related conditions of analysis has been proposed. A computer program (written in MATLAB and DELPHI languages), that enables the pH-static titration to be done automatically, has also been prepared. 

In compensator design, hand calculation is cumbersome, and a suitable computer package, such as MATLAB is generally used. 

Construet, on log-linear graph paper, using asymptotes, and validate using MATLAB or a similar tool, the Bode diagrams for... 

The results in this example were obtained using the MATLAB Robust Control Toolbox. 

The 11 and 22 set rulebase simulations were undertaken using SIMULINK, together with the fuzzy logie toolbox for use with MATLAB. More details on the... 

MATLAB Fuzzy Inference System (FIS) editor can be found in Appendix 1. Figure 10.16 shows the control surface for the 11 set rulebase fuzzy logic controller. 

Appendix 1 Control system design using MATLAB... 

MATLAB, its Toolboxes and SIMULINK have beeome, over a number of years, the industry standard software paekage for eontrol system design. The purpose of this Appendix is to introduee the reader to some of the more useful aspeets of MATLAB, and to illustrate how the software may be used to solve examples given in the main text of the book. 

The examples given in this Appendix were generated using MATLAB Version 5.3. Onee the software has been installed, MATLAB is most easily entered by elieking the MATLAB ieon. Alternatively, in a Windows environment, MATLAB ean be entered by elieking the following sequenee... 

The user should now be in the MATLAB eommand window, whieh eontains some helpful eomments together with the MATLAB prompt . MATLAB eommands are typed after the prompt, and entered using Return (or Enter ). Terminating the eommand with will suppress the result of the eommand being printed in the eommand window. Comments are preeeded by the % symbol. 

This tutorial introduees the reader to matrix operations using MATLAB. All text in eourier font is either typed into, or printed into the eommand window. 

MATLAB may be elosed by elieking File —> Exit MATLAB... 

This tutorial introduees the reader to time domain analysis using MATLAB. It uses eommands from the Control System Toolbox. A list of the eommands ean be found using... 

Script files A seript file is an ASCII text file of MATLAB eommands, that ean be ereated using... 

A seript file should have a name that ends in .m , and is run by typing the name of the file (without .m ) after the MATLAB prompt, or by typing the sequenee... 

The advantage of a seript file is that it only needs to be ereated onee and saves the labour of eontinually typing lists of eommands at the MATLAB prompt. 

With MATLAB Version 5.3, typing simulink at the MATLAB prompt brings up the SIMULINK Library Browser. Clieking on the Create new model ieon in the top left-hand eorner ereates a new window ealled untitled . 

This tutorial shows how MATLAB ean be used to build up and test elosed-loop eontrol systems. The examples given relate to those solved in Chapter 4. The eom-mands used in MATLAB to ereate a single model from the elements in the eontrol loop are... 

The tutorial demonstrates how MATLAB is used to generate root loeus diagrams, and lienee how to design eontrol systems in the. v-plane. Examples given in Chapter 5 are used to illustrate the MATLAB eommands. The roots of the eharaeteristie equation (or any polynomial) ean be found using the roots eommand. 

Using MATLAB to design a system, it is possible to superimpose lines of constant ( and ajn on the root locus diagram. It is also possible, using a cursor in the graphics window, to select a point on the locus, and return values for open-loop gain K and closed-loop poles using the command... 

Here a line of ( = 0.5(/3 = 60°) is drawn together with four circles (+ = 2, 4, 6 and 8rad/s.) At the MATLAB prompt, the user is asked to select a point in the graphics window. If the intersection of the complex locus with the ( = 0.5 line is selected (see Figure 5.14), the following response is obtained... 

When run, the program invites the user to seleet a point in the graphies window, whieh may be used to find the value of A" when ( = 0.25. If the last line of examp59.m is typed at the MATLAB prompt, the eursor re-appears, and a further seleetion ean be made, in this ease to seleet the value of K for marginal stability. This is demonstrated below... 

This tutorial shows how MATLAB can be used to construct all the classical frequency domain plots, i.e. Bode gain and phase diagrams, Nyquist diagrams and Nichols charts. Control system design problems from Chapter 6 are used as examples. 

Script file fig627.m produces the Nichols chart for Example 6.4 when K = 4, as illustrated in Figure 6.27. The command ngrid produces the closed-loop magnitude and phase contours and axis provides user-defined axes. Some versions of MATLAB appear to have problems with the nichols command. 

This tutorial looks at the application of MATLAB to digital control system design, using the problems in Chapter 7 as design examples. 

Note that r locus and r locf ind works for both continuous and discrete systems. The statement squar e provides square axes and so provides a round unit circle. The command zgr id creates a unit circle together with contours of constant natural frequency and damping, within the unit circle. When examp76.m has been run, using r locf ind at the MATLAB prompt allows points on the loci to be selected and values of K identified (see Figure 7.20)... 

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