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Essentials of Control Techniques and Theory_8

Tham khảo tài liệu 'essentials of control techniques and theory_8', kinh doanh - tiếp thị, quản trị kinh doanh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The Root Locus 175 the sum of the roots of P while this new coefficient will represent the sum of the spare poles. So we can say that as k becomes large the sum of the roots of the spare poles will be the difference between the sum of the open loop poles and the sum of the open loop zeros. Another way to say this is Give each open loop pole a weight of 1. Give each zero a weight of 1. Then the asymptotes will meet at the center of gravity. Q 12.5.1 Show that the root locus of the system 1 s s 1 2 has three asymptotes which intersect at s 2 3. Make a very rough sketch. Q 12.5.2 Add a zero at s 2 so that the system becomes s 2 s s 1 2. What and where are the asymptotes now Q 12.5.3 A zoomed out version of the root-locus plotter is to be found at www.esscont. com 12 rootzoom.htm. Edit the values of the poles and the zeros to test the assertions of this last section. Figure 12.5 shows the plot for a zero at 2 and poles at 0 and 1. Figure 12.5 Screen grab of www.esscont.com 12 rootzoom.htm for G s 2 s s 1 . 176 Essentials of Control Techniques and Theory There are more rules that can be derived for plotting the locus by hand. It can be shown that at a breakaway point where the poles join and split away in different directions then the derivative G s 0. It can be shown that those parts of the real axis that have an odd number of poles or zeroes on the axis to the right of them will form part of the plot. But it is probably easier to make use of the root-locus plotting software on the website. One warning is that some operating systems will put up an error message if the JavaScript is kept busy for more than five seconds. The plot can be made much neater by reducing ds to 0.2 but reducing it to 0.01 might provoke the message. 12.6 Compensators and Other Examples We have so far described the root locus as though it were only applicable to unity feedback. Suppose that we use some controller dynamics either at the input to the system or in the feedback loop see Figure 12.6 .