Root locus plots

This produces the (pole-zero cancellation) root locus plot shown in Figure 5.18. When run, exampSlO.m allows the user to select the value of K that corresponds to ( = 0.7, and then uses this selected value to plot the step response. The text that appears in the command window is... 

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. 

By and large, a quarter decay ratio response is acceptable for disturbances but not desirable for set point changes. Theoretically, we can pick any decay ratio of our liking. Recall Section 2.7 (p. 2-17) that the position of the closed-loop pole lies on a line governed by 0 = cos C In the next chapter, we will locate the pole position on a root locus plot based on a given damping ratio. 

Root locus plots of the closed-loop poles... 

The entire range of stability for x = 0.1 is 0 < Kc < 0.25. We will revisit this problem when we cover root locus plots we can make much better sense without doing any algebraic work ... 

Note that with this very specific case by choosing x = 1, the open-loop zero introduced by the PI controller cancels one of the open-loop poles of the process function at -1. If we do a root locus plot later, we d see how the root loci change to that of a purely second order system. With respect to this example, the value is not important as long as x > 1/2. 

One may question whether direct substitution is a better method. There is no clear-cut winner here. By and large, we are less prone to making algebraic errors when we apply the Routh-Hurwitz recipe, and the interpretation of the results is more straightforward. With direct substitution, we do not have to remember ary formulas, and we can find the ultimate frequency, which however, can be obtained with a root locus plot or frequency response analysis—techniques that we will cover later. 

The idea of a root locus plot is simple—if we have a computer. We pick one design parameter, say, the proportional gain Kc, and write a small program to calculate the roots of the characteristic polynomial for each chosen value of as in 0, 1, 2, 3,., 100,..., etc. The results (the values of the roots) can be tabulated or better yet, plotted on the complex plane. Even though the idea of plotting a root locus sounds so simple, it is one of the most powerful techniques in controller design and analysis when there is no time delay. 

Example 7.5 Construct the root locus plots of a first and second order system with a proportional controller. See how the loci approach infinity. 

The root locus plot (Fig. E7.5) is simply a line on the real axis starting at s = -1/tp when Kc = 0, and extends to negative infinity as Kc approaches infinity. As we increase the proportional gain, the system response becomes faster. Would there be an upper limit in reality (Yes, saturation.)... 

Figure E7.5. Root locus plots of (a) first order, and (b) second order systems.

This is the idea behind the plotting of the closed-loop poles—in other words, construction of root locus plots. Of course, we need mathematical or computational tools when we have more complex systems. An important observation from Example 7.5 is that with simple first and second order systems with no open-loop zeros in the RHP, the closed-loop system is always stable. 

In other words, on a root locus plot, we expect the "trace" of the root loci to begin at the open-loop poles and terminate at the open-loop zeros (if there is one). For real systems, m > n, and n>... 

Example 7.2B Do the root locus plot and find the ultimate gain of Example 7.2 (p. 7-5). The closed-loop equation from that example is ... 

After entering the riocf ind () command, MATLAB will prompt us to click a point on the root locus plot. In this problem, we select the intersection between the root locus and the imaginary axis for the ultimate gain. 

Select various values of X and use MATLAB to construct the root locus plots. Sample statements are ... 

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. 

In most classic control texts, we find plotting guidelines to help hand sketching of root locus plots. After going over Example 7.6, you should find that some of them are quite intuitive. These simple guidelines are ... 

The root locus plot is symmetric about the real axis. 

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. 

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. 

In terms of controller design, the closed-loop poles (or now the root loci) also tell us about the system dynamics. We can extract much more information from a root locus plot than from a Routh criterion analysis or a s = jco substitution. In fact, it is common to impose, say, a time constant or a damping ratio specification on the system when we use root locus plots as a design tool. 

To begin with, this is a second order system with no positive zeros and so stability is not an issue. Theoretically speaking, we could have derived and proved all results with the simple second order characteristic equation, but we take the easy way out with root locus plots. 

Note 1 Theoretically speaking, the point C on the root locus plot is ideal—the fastest possible response without any oscillation. We rarely can do that in practice the proportional gain would have been so large that the controller would be saturated. 

For as instructive as root locus plots appear to be, this technique does have its limitations. The most important one is that it cannot handle dead time easily. When we have a system with dead time, we must make an approximation with the Pade formulas. This is the same limitation that applies to the Routh-Hurwitz criterion. 

From the perspective of a root locus plot, a phase-lag compensator adds a large open-loop zero... 

From the characteristic polynomial, it is probable that we ll get either overdamped or underdamped system response, depending on how we design the controller. The choice is not clear from the algebra, and this is where the root locus plot comes in handy. From the perspective of a root-locus plot, we can immediately make the decision that no matter what, both Zq and p0 should be larger than the value of l/xp in Gp. That s how we may "steer" the closed-loop poles away from the imaginary axis for better system response. (If we know our root locus, we should know that this system is always stable.)... 

The root locus plot resembles that of a real PD controller. The system remains overdamped with... 

On the Bode plot, the comer frequencies are, in increasing order, l/xp, Zq, and p0. The frequency asymptotes meeting at co = l/xp and p0 are those of a first-order lag. The frequency asymptotes meeting at co = z0 are those of a first-order lead. The largest phase lag of the system is -90° at very high frequencies. The system is always stable as displayed by the root locus plot. 

The shape of the root locus plot resembles that of a PI controller, except of course we do not have an open-loop pole at the origin anymore. The root loci approach one another from -xp and -p0, then break away from the real axis to form a circle which breaks in to the left of the open-loop zero at -z0. One locus approaches negative infinity and the other toward -z0. One may design the controller with an approach similar to that in Example 7.7 (p. 7-16). 

The concept of gain and phase margins derived from the Nyquist criterion provides a general relative stability criterion. Frequency response graphical tools such as Bode, Nyquist and Nichols plots can all be used in ensuring that a control system is stable. As in root locus plots, we can only vary one parameter at a time, and the common practice is to vary the proportional gain. 

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. 

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 ... 

To find the new state feedback gain is a matter of applying Eq. (9-29) and the Ackermann s formula. The hard part is to make an intelligent decision on the choice of closed-loop poles. Following the lead of Example 4.7B, we use root locus plots to help us. With the understanding that we have two open-loop poles at -4 and -5, a reasonable choice of the integral time constant is 1/3 min. With the open-loop zero at -3, the reactor system is always stable, and the dominant closed-loop pole is real and the reactor system will not suffer from excessive oscillation. 

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.)... 

From the root locus plots, it is clear that the system may become unstable when x = 0.05 s. The system is always stable when = 5 s, but the speed of the system response is limited by the dominant pole between the origin and -0.2. The proper choice is xt = 0.5 s in which case the system is always stable but the closed-loop poles can move farther, loosely speaking, away from the origin. 

Do the root locus plots in Example 10-l(d). Confirm the stability analysis in Example 10-... 

In the simplest scenario, we can think of the equation as a unity feedback system with only a proportional controller (i.e., k = Kc) and G(s) as the process function. We are interested in finding the roots for different values of the parameter k. We can either tabulate the results or we can plot the solutions s in the complex plane—the result is the root-locus 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. 

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