Plotting with MATLAB

MATLAB offers many choices when it comes to creatii charts. For example, you can create x-y diarts, column charts (or histograms), contour, or surface plots. As we mentioned in Chapter 14, as an ei ineering student, and later as a practicing engineer, most of the charts that you will create will be x-y type diarts. Therefore, we will explain in detail how to create an x—y chart. 

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Let us now discuss the MATLAB commands that commonly are used when plotdi data. The plot (Xf y) command plots y values versus xvalues. You can use various line types, plot symbols, or colors with the command plot (x,y, s), where s is a charaaer string that defines a particular line type, plot symbol, or line color. The 8 can take on one of the properties shown in Table 15.12. 

For example, ifyou issue the coounand plot (x,y, k - ), MATLAB will plot the curve using a black solid line with an marker shown at each data point. If you do not specify a line color, MATLAB automatically assigns a color to the plot. 

With MATLAB, you can generate other types of plots, including contour and surfttce plots. You can also control the x- and y-axis scales. For example, the MATLAB s loglog (X, y) uses the base-10 logarithmic scales forx- andy-axes. Notexandy are the variables that you want to plot. The command loglog(X/y) is identical to the plot (x,y),  

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What is the phase angle of the minimum phase function (s + 3)/(s + 6) versus the simplest nonminimum phase function (s - 3)/(s + 6) Also try plot with MATLAB. The magnitude plots are identical. The phase angle of the nonminimum phase example will go from 0° to -180°, while you d see a minimum of the phase angle in the minimum phase function. Thus for a transfer function that is minimum phase, one may identify the function from simply the magnitude plot. But we cannot do the same if the function is nonminimum phase. 

Note 2 The iterative solution in solving the ultimate frequency is tricky. The equation has poor numerical properties—arising from the fact that tan9 "jumps" from infinity at 9 = (ir/2) to negative infinity at 9 = (ir/2)+. To better see why, use MATLAB to make a plot of the function (LHS of the equation) with 9 < co < 1. With MATLAB, we can solve the equation with the f zero () function. Create an M-file named f. m, and enter these two statements in it ... 

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. 

To determine the shape of a root locus plot, we need other rules to determine the locations of the so-called breakaway and break-in points, the corresponding angles of departure and arrival, and the angle of the asymptotes if the loci approach infinity. They all arise from the analysis of the characteristic equation. These features, including item 4 above, are explained in our Web Support pages. With MATLAB, our need for them is minimal. 

Even with MATLAB, we should still know the expected shape of the curves and its telltale features. This understanding is crucial in developing our problem solving skills. Thus doing a few simple hand constructions is very instructive. When we sketch the Bode plot, we must identify the comer (break) frequencies, slopes of the magnitude asymptotes and the contributions of phase lags at small and large frequencies. We ll pick up the details in the examples. 

From here on, we will provide only the important analytical equations or plots of asymptotes in the examples. You should generate plots with sample numerical values using MATLAB as you read them. 

This is how we may do the plots with time delay (details in MATLAB Session 7). Half of the work is taken up by the plotting statements. 

We ll skip the proportional controller, which is just Gc = Kc. Again, do the plots using sample numbers with MATLAB as you read the examples. 

How do I know the answer is correct Just "plug" Kc back into G0l and repeat the Bode plot using G0l It does not take that much time to check with MATLAB. Now, we are finally ready for some examples. Again, run MATLAB to confirm the results while you read them. 

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 ... 

With MATLAB, we can easily prepare the root locus plots of this equation for the cases of Xj = 0.05, 0.5, and 5 s. (You should do it yourself. We ll show only a rough sketch in Fig E10.1. Help can be found in the Review Problems.)... 

What a piece of cake Not only does MATLAB perform the calculation, it automatically makes the plot with a properly chosen time axis. Nice 2 As a habit, find out more about a function with help as in... 

Let us solve the Lorenz equations below with MATLAB between t = 0 and 20 and prepare plots of v, versus t, and a state-space representation ofty versus yu andy3 versus y2 by using two different sets of initial conditions Ti(0) =y2(0) =y3(0) = 5.0 andyffO) =y2(0) = y3(0) = 5.0. 

Let us solve the Lorenz equations below with MATLAB between t = 0 and 20 and prepare plots ofy, versus t,... 

From the total sample set (48 samples), 45 samples were used as calibration samples. The three samples excluded from the calibration set were selected on the basis of a representative variation of their active ingredient concentrations, and finally used as unknown test samples to predict the content of their active ingredients. Partial least squares (PLS) models for each active ingredient were developed with the Unscrambler Software (version 9.6 CAMO Software AS, Oslo, Norway) from the MSC-pretreated median spectra of all pixels of each of the 45 calibration sample images. Based on these calibration models, the predictions of the active ingredient content for each pixel of the imaging data of the three test samples and their evaluation as histograms, contour plots and RGB plots was performed with Matlab v. 7.0.4 software (see below). 

Fig. 9.10. The signal at left was obtained with a voltammetric sensor and was processed with Matlab to compute the wavelet coefficients plotted at right. Such time-scale plotting of the coefficients is known as scalogram.

A surface response fitting of second-order polynomial fimction with = 0.71 was plotted in MATLAB to fiuther explore the coupling effects of T. Thus, the thermal conductivity for the GSA-SDS/FMWNT composites can be expressed as cn =f T(, Tc, Aabuik( i , Tm), Agd) and can be evaluated as shown in Eq. 6.14. The fuU polynomial equation for (Tf Tc) is given in Appendix 6B. 

The data were processed with Matlab 7.1 software (The MathWorks, Inc., Natick, MA, USA) to visualize the friction coefficient in colour plots and to determine the friction coefficients in dependence on the applied load (average of 2000 values/load). 

The PSO algorithm was used in MATLAB platform. The experiments were carried out with presently taking one supplier and one customer, i.e. i = 1 and j = 1. The input of variable cost per unit (C,y), transportation cost per unit (t,y), inventory holding cost per unit (hy) and penalty cost (Sg) was taken from a case study (Teimoury et al. 2010). The number of iterations made was 30. Once the values are entered for different products, the minimum cost was calculated by PSO algorithm in MATLAB and the graph for each product was plotted in MATLAB. 

It is simple to model this equation using a spreadsheet (such as EXCEL), or programs such as MATLAB, or graphics calculators. However, it must be borne in mind that the logarithmic model does not work at very low currents, especially at zero. It is best to start the plots with a current of l.OmAcm. As an example, we have given below the MATLAB script file that was used to produce the graph in Figure 3.1. 

Alternatively, online help is available through the menu bar under Help. The question mark button on the tool bar will also invoke the Help Browser. Starting with 2008 releases, you can invoke a context-sensitive function browser, which lies on the left of the command prompt, and has the symbol f. You may type, for example, the command plot, and MATLAB will attempt to find all related MATLAB functions that include the word plot as part of their function names (Fig. 1.7). 

Convert this problem to dimensionless form to reduce the munber of independent parameters. Then, use the finite difference method to convert this boimdary value problem into a set of nonlinear algebraic equations and solve with MATLAB. Plot the dimensionless concentration as a function of the remaining adjustable dimensionless parameteifs). To speed up your calculations, have your function routine return the Jacobian matrix. 

We have given up the pretense that we can cover controller design and still have time to do all the plots manually. We rely on MATLAB to construct the plots. For example, we take a unique approach to root locus plots. We do not ignore it like some texts do, but we also do not go into the hand sketching details. The same can be said with frequency response analysis. On the whole, we use root locus and Bode plots as computational and pedagogical tools in ways that can help to understand the choice of different controller designs. Exercises that may help such thinking are in the MATLAB tutorials and homework problems. 

Plot the unit step response using just the first and second order Pade approximation in Eqs. (3.30) and (3-31). Try also the step response of a first order function with dead time as in Example 3.2. Note that while the approximation to the exponential function itself is not that good, the approximation to the entire transfer function is not as bad, as long as td x. How do you plot the exact solution in MATLAB ... 

MATLAB calculation details and plots can be found on our Web Support. You should observe that Cohen-Coon and Ziegler-Nichols tuning relations lead to roughly 74% and 64% overshoot, respectively, which are more significant than what we expect with a quarter decay ratio criterion. 

For now, we ll take a look at the construction of Bode and Nyquist plots of transfer functions that we have discussed in Chapters 2 and 3. Keep in mind that these plots contain the same information G(jco). It is important that you run MATLAB with sample numerical values while reading the following examples. Yes, you need to go through MATLAB Session 7 first. 

Fig. E8.13 is a rough hand sketch with the high and low frequency asymptotes. It is meant to help interpret the MATLAB plots that we will generate next.

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. 

Derive Eqs. (8-19) and (8-20). Use MATLAB to plot the resonant frequency and maximum magnitude as a function of damping ratio with K = 1. 

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. 

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