MATLAB command

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

The script file kalfild.m solves, in forward-time, the discrete solution of the Kalman filter equations, using equations (9.74), (9.75) and (9.76) in a recursive process. The MATLAB command Iqed gives the same result. 

The very first step is to find the ultimate gain. With the given third order process transfer function, we use the following MATLAB commands,... 

MATLAB is most at home dealing with arrays, which we will refer to as matrices and vectors. They are all created by enclosing a set of numbers in brackets, [ ]. First, we define a row vector by entering in the MATLAB Command Window ... 

To launch the LTI Viewer, enter in the MATLAB command window ltiview... 

The LTI Viewer runs in its own workspace, which is separate from the MATLAB workspace. The Viewer also works with only LTI objects generated by functions such as tf () and zpk (), and after Chapter 4, state space objects, ss (). So let s generate a couple of objects in the MATLAB command window first ... 

Simulink shares the main MATLAB workspace. When we enter information into, say, the transfer function block, we can use a variable symbol instead of a number. We then define the variable and assign values to it in the MATLAB command window. This allows for a much quicker route to do parametric studies than changing the numbers within the Simulink icons and dialog boxes. 

To launch the root focus design tool, enter within the MATLAB command window ... 

Example Consider the following example from Noble and Daniel [Applied Linear Algebra, Prentice-Hall (1987)] with the MATLAB commands to do the analysis. Define the following real matrix with m = 3 and n = 2 (whose rank k = 1). 

Now using the MATLAB command line software, we can easily demonstrate this solution (for the multivariate problem we have identified) using a series of simple matrix operations as shown in Table 21-1 below ... 

The first graphical representation using MATLAB software is that of a two-dimensional contour surface plot of the data from Table 75-1 [2], This Figure 75-3 plot can represent multiple levels of j-axis data (absorbance) by the use of contours and color schemes. The MATLAB commands for generating this image are given in Table 75-2 where A represents the raster data matrix shown in Table 75-1. 

Table 75-2 MATLAB commands for generating a contour plot of data matrix A found in Table 75-1...

Table 75-4 MATLAB commands for generating a 2-D contour plot over a 3-D surface plot...

Note that Matlab directly adds/subtracts a single scalar element-wise to/from matrices. Suppose you want to subtract element foi,2 (a scalar) from all elements of matrix A. A valid Matlab command would be ... 

In Matlab the plus (+) and minus (-) operators cannot be directly applied to equations that involve vectors or matrices of different dimensions. In order to perform the same operation as in the former Excel example, column vector b ,3 must be replicated three times to match the dimensions of A. For this, the Matlab command repmat can be used. 

Note that the Matlab command sum(B.A2,2) performs a row-wise addition of all squared elements of B. For a column-wise summation sum (B. A2,1) or just sum(B. A2) could be used. 

The main program Main chrom, m loads the measurement, generates initial guesses for the parameters, calls nglm. m and reports the fitted parameters, fprintf is a C based Matlab command that allows compact output. The reader is invited to read up on this command in the Matlab manuals. 

Note that entry-wise defined matrices and vectors are delimited by square brackets [ and ] in MATLAB commands. 

The first command generates the matrix A. A is stored in MATLAB s workspace, but it is not displayed on screen due to the following the command. With the second command, the matrix B is generated, stored, and displayed on screen because there is a blank and no following the second command. Note that a comma, after a MATLAB command has the same effect as a blank as regards not suppressing screen output. In MATLAB, one can put as many comma or semicolon delimited commands on one line as one wishes and will fit. 

The commas after our MATLAB commands above are necessary as delimiters to be able to place several commands onto one line of code. They create screen output, whereas semicolons would have suppressed it. The final entry result = [size(A ), length(A )] on the command line creates the vector (4, 3, 4), called result by us. It contains the number of rows (4) of AT, followed by AT,s number of columns (3) and its length or maximal dimension (4). 

Note that we have to place zeros in the positions of A that correspond to unused variables in any equation in (1.1). Below are the corresponding MATLAB commands that let us find the solution vector... 

In this book we shall use MATLAB codes and explain more involved features of MAT-LAB as we encounter them. MATLAB has a built-in help menu typing help format at the prompt, or help, for example, will show the syntax and variations of these two commands format and backslash . Whenever a student encounters a MATLAB command that is not self explanatory, we suggest using this built-in help function of MATLAB. 

To illustrate we first verify the identical behavior of the MATLAB QR based polynomial-root finder roots and MATLAB s QR based matrix eigenvalue finder eig for p s companion matrix P = C(p) First we define p by its coefficient vector in MATLAB s workspace, then we invoke the MATLAB polynomial-root finder roots, followed by its matrix eigenvalue finder eig on the companion matrix ofp. Finally we display the companion matrix P of p. As an example we use p(x) = x3 — 2.x2 + 4 here and represent p by its coefficient vector [1 -2 0 4] in the following line of MATLAB commands. 

We advise our students to follow our text and to always copy our MATLAB command lines onto their MATLAB desktop as they read through this chapter. Our students and readers should observe the execution of these commands, and we encourage them to alter these commands slightly to learn more about MATLAB.]... 

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. 

A script m file in MATLAB does not start with a function. . . declaration as the function m file fzerotryl. m does. A script m file rather consists only of MATLAB commands. See Section 1.2.5 for more details on these two types of m files in MATLAB. 

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... 

MATLAB m files can contain any number of MATLAB commands. They are handy for long sets of MATLAB instructions or data and afford easy repeated use and easy modification. There are two kinds of MATLAB m files function and script m files. Script m files such as poly9. m on p. 28 use the data that is defined inside of them and any undefined data is taken from the data (with the proper name) that is currently active on the workspace look at the top left window commonly displayed on the MATLAB desktop for a list of defined variables and their type. Script files overwrite the workspace data as instructed in their code. 

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. 

Three dots. .. indicate a line break inside a MATLAB command. Useful to break long lines of contiguous code, such as with title text. 

Exercise your command of MATLAB by trying to replicate the three graphs in Figure 3.15. Learn about the subplot, xlabel, ylabel, and title MATLAB commands and how to use them. 

It is advisable to graph and inspect the reaction more closely at the entrance of the reactor. After a few experiments with Vend = 0.1, 0.04, and 0.01, we have set the desired final volume to Vend = 0.01 m3 in Figure 4.4, which is drawn by the MATLAB command fixedbedreactf.003,160,1200,1030,.01,15,0,1) . ... 

This code is typically called by a MATLAB command such as f luidbed(4000,. 4,. 9,1,1, 45/99,10 s8,10 sll,.4,.6,18,27,1,0,10), which uses a large time limit of 4000 dimensionless time units so that the individual solution curves of the IVPs have time to run to the global steady state at (0.17619, 0.72225, 0.86446). We do this with the original parameter data of p. 183, a controller gain of K = 10, and several different initial values. Since the value of K is greater than 4, the middle steady state with maximal x i> yield is both unique and (statically) stable. 

MATLAB allows the user to choose the IVP integrator, such as ode23 or ode45 etc, and to select a stiff or nonstiff integrator, each as warranted by the specific problem. Moreover, each of the MATLAB s ODE solvers ode... allows us to specify certain options , as done in fluidbed.m in the fourth MATLAB command line options = odeset( RelTol ,10 -6, AbsTol ,10 -8, Vectorized , on ) for example. 

A detailed reading of the fluidbed.m program reveals that a hold on MATLAB command is put onto the plot that is never released. This helps us draw multiple trajectories onto one 3D plot such as in Figure 4.34. Note that Figure 4.34 depicts the actual MATLAB output after a suitable rotation as explained before on p. 186. If the user wants... 

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