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To determine what part of the locus is acceptable, we can use the command to plot lines of constant damping ratio and natural frequency.In our problem, we need an overshoot less than 5% (which means a damping ratio Zeta of greater than 0.7) and a rise time of 1 second (which means a natural frequency Wn greater than 1.8).Figure 4: Root Locus Plot Back to Top The plot in Figure 4 above shows all possible closed-loop pole locations for a pure proportional controller.Obviously not all of those closed-loop poles will satisfy our design criteria.You do not need to learn many of these procedures because they can be done approximately and more easily using the Matlab root locus GUI rltool( ). Any parts of the real axis with an odd number of open-loop poles and zeros to the right of them are part of the root locus. The pole/zero plot for The pole/zero furthest to the right on the real axis is the zero at 0. For the portion of the real axis between -4 and -1, there is one pole and one zero to the right, two poles/zeros should be able to draw an approximate root locus by hand, so that you can get a general idea of how a system's behavior changes with K without going to a computer. Since this is an even number, the -4 to -1 section of the real axis is not on the root locus.

Figure 6: Root Locus Plot with Grid Lines On the plot above, the diagonal lines indicate constant damping ratios (Zeta), and the semicircles indicate lines of constant natural frequency (Wn).Since the root locus is actually the locations of all possible closed loop poles, from the root locus we can select a gain such that our closed-loop system will perform the way we want.If any of the selected poles are on the right half plane, the closed-loop system will be unstable.Going back to our problem, to make the overshoot less than 5%, the poles have to be in between the two red lines, and to make the rise time shorter than 1 second, the poles have to be outside of the green semicircle.So now we know only the part of the locus outside of the semicircle and in between the two lines are acceptable.

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If H(s) has more poles than zeros (as is often the case), m infinity is zero.