Matlab loops

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... 

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... 

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 transfer function has closed-loop poles at -0.29, -0.69, and -10.02. (Of course, we computed them using MATLAB.)... 

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 will not write out the entire closed-loop function C/R, or in this case, T/Tsp. The main reason is that our design and analysis will be based on only the characteristic equation. The closed-loop function is only handy to do time domain simulation, which can be computed easily using MATLAB. Saying that, we do need to analysis the closed-loop transfer function for several simple cases so we have abetter theoretical understanding. 

Example 5.6 Provide illustrative closed-loop time response simulations. Most texts have schematic plots to illustrate the general properties of a feedback system. This is something that we can do ourselves using MATLAB. Simulate the observations that we have made in previous examples. Use a unity feedback system. 

Example 7.6 Construct the root locus plots of some of the more common closed-loop equations with numerical values. Make sure you try them yourself with MATLAB. 

There are two important steps that we must follow. First, make sure you go through the MATLAB tutorial (Session 6) carefully to acquire a feel on the probable shapes of root locus plots. Secondly, test guidelines 3 and 4 listed above for every plot that you make in the tutorial. These guidelines can become your most handy tool to deduce, without doing any algebra, whether a system will exhibit underdamped behavior. Or in other words, whether a system will have complex closed-loop poles. 

However, MATLAB allows us to get the answer with very little work—something that is very useful when we deal with more complex systems. Consider a numerical problem with values of the process gain Kp = 1, and process time constants X = 2 and x2 = 4 such that the closed-loop equation is... 

We should find a gain margin of 1.47 (3.34 dB) and a phase margin of 12.3°. Both margins are a bit small. If we do a root locus plot on each case and with the help of riocf ind () in MATLAB, we should find that the corresponding closed-loop poles of these results are indeed quite close to the imaginary axis. 

Where do we go from here We may stay with the design or we may increase the margins We also can use MATLAB to simulate the closed-loop time domain response and from the... 

To achieve a damping ratio of 0.8, we can find that the closed-loop poles must be at -4.5 3.38j (using a combination of what we learned in Example 7.5 and Fig. 2.5), but we can cheat with MATLAB and use root locus plots ... 

Hence, our first step is to use root locus to find the closed-loop poles of a PI control system with a damping ratio of 0.8. The MATLAB statements to continue with Example 4.7B are ... 

One could generate a closed-loop function with, for example, Gc Gp/ (i + Gc Gp), but this is not recommended. In this case, MATLAB simply multiplies everything together with no reduction and the resulting function is very unclean. 

MATLAB automatically selects a reasonable vector for k, calculates the roots, and plots them. The function rlocus () also adds the open-loop zeros and poles of G(s) to the plot. 

Optional reading In the initial learning stage, it can be a bad habit to rely on MATLAB too much. Hence the following tutorial goes the slow way in making root locus plots, which hopefully may make us more aware of how the loci relate to pole and zero positions. The first thing, of course, is to identify the open-loop poles. 

MATLAB will wait for us to click on a point (the chosen closed-loop pole) in the root locus plot and then returns the closed-loop gain (ck) and the corresponding closed-loop poles (cpole). MATLAB does the calculation with the root locus magnitude rule, which is explained on our Web Support. 

What if we click a point not exactly on a root locus When we select a point s, MATLAB calculates the value k = -p(s )/q(s ), which will only be a real positive number if s satisfies the closed-loop equation. Otherwise, k is either complex, or negative if the pole is a real number. In this case, MATLAB calculates the magnitude of k, uses it as the gain and computes the corresponding closed-loop poles. Thus we find the chosen points are always right on the root loci no matter where we click. 

The few short programs presented in this chapter may also serve as a very rudimentary introduction into Matlab. Readers not familiar with Matlab but otherwise proficient in an alternative language will be surprised at the almost complete lack of for. .. end loops. We also introduce matrix operations in Excel, assuming that the other, more common aspects of Excel are known to the reader. While there is a reasonable collection of matrix operations available in Excel, their usage is rather cumbersome. We believe that many readers will appreciate the short introduction into this aspect of Excel. 

It is a matter of opinion whether the loop or the repmat option is preferable. One is shorter and the other is easier to comprehend one is Matlab specific and the other is more general. In this book, we tend to use the shorter, Matlab-style version, at least as long as the readability is not severely compromised. 

Note that this kind of normalisation, via the norm function, can only be performed column- (or row-) wise via a loop as seen in the Matlab box above. Calling norm with one matrix argument determines a different kind of normalisation coefficients. We refer to the Matlab help and function references for more detail. 

Defying Matlab elegance, one could write equation (4.46) as a loop, but it is certainly faster to vectorise the equation. The vectorised Matlab code (note that the polynomial degree equals the number of parameters minus one, nd=np-1) ... 

The Matlab program Main NG2. m has implemented the additions for a termination criterion and numerical derivatives. Refer to the Matlab Help Desk for information on the while end loop and also the break command. 

Matlab supplies the function Isqnonneg that performs a non-negative least-squares fit of the kind y=Ca+r, where y and a are column vectors. The function computes the best vector a with only positive entries. This equation corresponds to data acquired at only one wavelength. In our application, the columns of A have to be computed individually in a loop over all wavelengths, in each instance using the appropriate column of Y. C is the complete matrix of concentrations. It is, of course, the same for all wavelengths. 

Preprocessing is slightly awkward in Matlab. One way is to write a small program with loops as described in Section A.5.6. If you think in terms of vectors and matrices, however, it is fairly easy to come up with a simple approach. If W is our original 2x3 matrix and we want to mean centre file columns, we can easily obtain a 1 x 3 vector w... 

This paper presents the application of a model based predictive control strategy for the primary stage of the freeze drying process, which has not been tackled until now. A model predictive control framework is provided to minimize the sublimation time. The problem is directly addressed for the non linear distributed parameters system that describes the dynamic of the process. The mathematical model takes in account the main phenomena, including the heat and mass transfer in both the dried and frozen layers, and the moving sublimation front. The obtained results show the efficiency of the control software developed (MPC CB) under Matlab. The MPC( CB based on a modified levenberg-marquardt algorithm allows to control a continuous process in the open or closed loop and to find the optimal constrained control. 

MATLAB and SIMULINK are invaluable tools for the finequency- and time-domain calculations required for C R analysis. In this section, several examples are carried out using MATLAB, it being assumed that the reader is familiar with the MATLAB syntax. The reader is referred to Bequette (1998) for details of MATLAB usage in dynamical analysis and control, and to the multimedia CD-ROM that accompanies this text for sources of these and other useful MATLAB functions and scripts for C R analysis. In particular, the interactive C R Tutorial CRGUI can be used to test three example linear processes for controllability and resiliency and simulate their closed-loop response under single-loop PI control. 

Consider the component parts in the LSF configuration represented by Eqs. (21.56) and (21.57). In this example, the elements of the transfer function matrices are entered into MATLAB and used to compute the DC contour maps for this configuration. P ju> and ju>] are computed for each frequency, and used to compute DC for all of the disturbancedirections. By looping over all frequencies, the entire DC map is calculated and repeated for each manipulated variable separately. Note that, as mentioned in Example 21.8, the inputs are nominally at 50% of the full range. Here, the nominal inputs are taken as Lf, = Li= 11 kmol/min, = 0.222 X 10 kcal/min, and the maximum disturbance magnitudes are taken as F = 18 kmol/min and xp = 0.2 ( 20% of the full range). 

MatLab is used to obtain graphs of the step response of this closed loop system. The transfer function of BF... 

The use of MatLab to check the robustness of the closed loop system gives The transfer function ... 

Figure 2 The nested loop program in pseudo-MATLAB code for the Floyd-Steinberg algorithm.

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