MATLAB function

We can use the MATLAB function tf 2 ss () to convert the transfer function in (E4-2) to state space form ... 

We can check with MATLAB that the model matrix A has eigenvalues -0.29, -0.69, and -10.02. They are identical with the closed-loop poles. Given a block diagram, MATLAB can put the state space model together for us easily. To do that, we need to learn some closed-loop MATLAB functions, and we will defer this illustration to MATLAB Session 5. 

We should find that for values of Xj > 0.5, the system stays stable. For = 0.5, the system may become unstable, but only at infinitely large Kc. The system may become unstable for x < 0.5 if Kc is too large. Finally, for the choice of x = 0.1, we should find with the MATLAB function riocf ind that the ultimate gain is roughly 0.25, the same answer from Example 7.3. How close you get depends on how accurate you can click the axis crossover point. 

The MATLAB function bode () returns the actual magnitude even though the documentation says dB. This is a detail that we can check with the function f reqresp () as explained on our Web Support, especially with future upgrades of the software. 

Similarly, we can use the MATLAB function obsv () for the observability matrix ... 

Check 1. The same result can be obtained with the MATLAB function acker () which uses the Ackermann s formula. The statements are ... 

To evaluate the matrix polynomial in Eq. (9-23), we use the MATLAB function polyvalm () which applies the coefficients in p2 to the matrix A. 

Many other matlab functions, for example, impulse (), isim (), etc., take both transfer function and state space arguments (what you can call polymorphic). There is very little reason to do the conversion back to transfer function once you can live in state space with peace. 

It can be synthesized with the MATLAB function feedback (). As an illustration, we will use a simple first order function for Gp and Gm, and a PI controller for Gc. When all is done, we test the dynamic response with a unit step change in the reference. To make the reading easier, we break the task up into steps. Generally, we would put the transfer function statements inside an M-file and define the values of the gains and time constants outside in the workspace. 

We may want to find the ultimate gain when the loci cross the imaginary axis. Again there are many ways to do it. The easiest method is to estimate with the MATLAB function rlocf ind (), which we will introduce next. 

Initially, we develop Matlab code and Excel spreadsheets for relatively simple systems that have explicit analytical solutions. The main thrust of this chapter is the development of a toolbox of methods for modelling equilibrium and kinetic systems of any complexity. The computations are all iterative processes where, starting from initial guesses, the algorithms converge toward the correct solutions. Computations of this nature are beyond the limits of straightforward Excel calculations. Matlab, on the other hand, is ideally suited for these tasks, as most of them can be formulated as matrix operations. Many readers will be surprised at the simplicity and compactness of well-written Matlab functions that resolve equilibrium systems of any complexity. 

Instead of adapting the NewtonRaphson.m function we just use the Matlab function fzero which is a general routine for that kind of one-dimensional problem. ... 

In contrast to our preferred standard mode in this book, we do not develop a Matlab function for the task of numerical integration of the differential equations pertinent to chemical kinetics. While it would be fairly easy to develop basic functions that work reliably and efficiently with most mechanisms, it was decided not to include such functions since Matlab, in its basic edition, supplies a good suite of fully fledged ODE solvers. ODE solvers play a very important role in many applications outside chemistry and thus high level routines are readily available. An important aspect for fast computation is the automatic adjustment of the step-size, depending on the required accuracy. Also, it is important to differentiate between stiff and non-stiff problems. Proper discussion of the difference between the two is clearly outside the scope of this book, however, we indicate the stiffness of problems in a series of examples discussed later. So, instead of developing our own ODE solver in Matlab, we will learn how to use the routines supplied by Matlab. This will be done in a quite extensive series of examples. 

If required, the initial value for the Marquardt parameter, mp, in case of divergence, has to be chosen sensibly as well the original suggestion was to use the value of the largest diagonal element of J J. This is, however, not crucial and in the Matlab function nglm. m further below, we set this initial value, if required, to one. 

Let us start with an example the Matlab function Data AB. m models the absorption spectra of a reacting solution as a function of time. They are stored as rows of the matrix Y. The reaction is a simple first order reaction A - B as introduced in Chapter 3.4.2, Rate Laws with Explicit Solutions. 

EFA. m is a short Matlab function that computes forward and backward EFA matrices for a given number, ne, of singular values. Its structure is essentially identical to the one discussed for Main EFA2, m. 

Figure 5-53. ALS using the Matlab function Isqnonneg. m for nonnegative linear least-squares fitting.

In Constraint Positive Component Spectra (p.168), we introduced an improved, much faster matrix based function nonneg. m (provided by C. Andersson) that is more efficient than the Matlab function Isqnonneg. m. The result of implementing the function constra ints nonneg. m is identical but achieved much faster. 

Figure 5-54. ALS using the Matlab function constra ints nonneg un imod. m. performing positive linear least-squares and removing secondary maxima in the concentration profiles.

The Newton-Gauss algorithm (ng Jm3. m), is called from Main RFA.m, and requires a Matlab function that computes the residuals as a function of the parameters T, as defined in equation (5.54). This calculation is performed in the Matlab function Rcalc RFA. m. 

The Matlab function PCR ca librat ion. m performs the PCR calibration according to equation (5.62). Note that we use ne=12 eigenvectors in the above calculations. This is the optimal number for prediction, as we show in Cross Validation (p.303j. The reader is invited to play with this number and observe the effect. 

MATLAB has a built-in root finder for scalar equations f(x) = 0 in one real variable x that are in standard form. The built-in MATLAB function is fzero. The use of fzero hinges on a user-defined function, such as the function f inside the following fzero tester, called fzerotryl, that we apply to our previously studied third degree polynomial. 

We note that fzerotryl is a MATLAB function m file that is stored in its folder with the extension. m as fzerotryl.m. Our program code is annotated with comments following the % symbol. Anything that follows after a % symbol on a line of code is not executed in MATLAB. More on MATLAB files, their storage, creation, etc. is given in Section 1.2.5. Specific built-in MATLAB functions and their use, such as fzero in the code below, should always be scrutinized by our students for their input/output syntax etc. using the built-in help MATLAB command help fzero for example. 

The first call of fzero inside fzerotryl takes 24 iterations to arrive at the real root x = -1.1304 of our trial polynomial p(x) x — 2x2 + 4 when starting at x0 = —2, while the second call converges after 9 iterations when looking for real roots of p inside the interval [—2, —1]. Please look up help fzero to learn more about this MATLAB function and how it was used. 

The code above is a MATLAB function m file with two inputs xstaxt and k and one output X. Its first code line... 

Modify the MATLAB function m file newtonpoly9.m on p. 30 to print out three columns of iterates with their iteration indices for all values of k. 

MATLAB m and Other Files and Built-in MATLAB Functions... 

In the previous sections we have used MATLAB commands that were typed in on the desktop command line, as well as MATLAB commands that invoke special stored MATLAB files, called m files. When starting to work with MATLAB, we advise our users to create a special folder for their personal MATLAB m files once and to continue to use this folder to store personal m files. To access the stored m files from the MATLAB desktop, the user has to point the small desktop window called Current Directory to the personal m file folder. Then all stored m files, as well as all built in MATLAB functions can be accessed, called upon, and used from the command line. All stored personal... 

The last part of this section lists a few standard MATLAB operations, functions, and commands, collected into groups, together with short descriptions. This may help our readers to more easily find and use built-in MATLAB functions in their own MATLAB program codes. Please note that our MATLAB function descriptions below are very few and very short by necessity. The user should use the help. . . command to find the full length MATLAB reference guide entry for each MATLAB function when the need arises. This will help our readers use the full power and functionality of MATLAB commands and will enable them to browse for and find related built-in MATLAB functions. 

The printed MATLAB Function Reference Guide book consists of three volumes amounting to about 2,000 pages that list every one of about 1,000 built-in MATLAB functions. 

Creates a function handle, so that functions can be passed to MATLAB function m files as variables are. 

The MATLAB codes printed in this book give our readers a varied and solid introduction into the above and many other built-in MATLAB functions, as well as into their functionality. 

Note that almost all calling sequences of MATLAB function m files start with the function s name, such as runsolveadiabxy above, followed by a list of parameters in parentheses (. .. ). Our particular call runsolveadiabxy(285,305,1,8.5) uses the interval limits 285 and 305 for a as its first two parameters, followed by the values of / and 7 for a specific chemical reaction. In our m files the list of possible parameters is always explained in the first comment lines of code. Often one or several of the parameters are optional. If they are not specified in the calling sequence, they are internally set to default values inside the program, such as n and anno are here. 

The MATLAB function adiabNisoplot.m finds the zero crossings of the graph of / in (3.4) graphically. Graphs are generally very helpful when trying to solve equations since they help us visualize the points of intersection of the function / in question with the... 

Next we draw a grayscale contour map of this surface, as well as the level curve F(a,y) = 0 drawn in black using the MATLAB function adiabNisocolorcontour. m. On a computer screen the same coloring scheme as in Figure 3.6 is used. 

Compared to solveadiabxy. m for the adiabatic CSTR case in Section 3.1, the above MATLAB function solveNadiabxy. m depends on the two extra parameters Kc and yc that were defined following equation (3.9). It uses MATLAB s built-in root finder fzero.m. As explained in Section 3.1, such root-finding algorithms are not very reliable for finding multiple steady states near the borders of the multiplicity region. The reason - as pointed out earlier in Section 1.2 - is geometric the points of intersection of the linear and exponential parts of equations such as (3.16) are very shallow, and their values are very hard to pin down via either a Newton or a bisection method, especially near the bifurcation points. 

Our CD also contains the MATLAB function m file runNadiabNisokccurve. m. A call of runNadiabNisokccurve (180000,1,15,1,100,. 001, .7,1.2), for example, plots only the bifurcation curve with respect to Kc in Figure 3.16, i.e., it repeats the bottom plot in Figure 3.14. 

Since we are interested in generating the maximal amount of the intermediate component B in the consecutive reaction A —> B —> C, we look at the concentration of B defined by xb v) in equation (4.82). This we do using the MATLAB function xBversusy.m. 

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