Question A.5: Plot the root locus for the variable K in the following open...
Plot the root locus for the variable K in the following open-loop gain transfer function:
F(s) = \frac{K . s .(s + 4) . (s + 5)}{(s + 1)(s + 6)(s + 7)(s + 1 + j)(s + 1 − j)}
The root locus for this system begins with the open-loop poles and zeros. These can be found by inspection from this conveniently factored expression.
The poles are located at −1, −6, −7, and −1 \pm j. These are plotted in Figure A.17 using Xs. Also, there are three finite open-loop zeros at 0, −4, and −5. These are plotted in the figure using Os. This also means that the system has two zeros at infinity, which in turn means that there will be two root locus lines that head towards infinity. The next thing to determine is where the root locus exists on the real axis. Remembering that it exists to the right of an odd number of real axis poles and zeros, we can determine that it will exists on the real axis between 0 and −1, between −4 and −5, and between −7 and −6. This makes the plot of the root-locus line that begins at −1 fairly obvious. It will run from the pole at −1 to the zero at 0. Next, we need to find the real axis intercept of the asymptotes of the two lines that will go to infinity. From (A.77),
it will be at ,
\sigma = \frac{\left[(−7) + (−6) + (−1) + (−1 + j) + (−1 − j)] − [(−5) + (−4) + 0\right] }{5-3}=-3.5From (A.76),}
\theta = \frac{(2k + 1)\pi }{\# \text{finite poles} −\# \text{finite zeros}}it is easy to see that these will go in the direction of 90° and 270°. These will be the root locus lines for the poles starting at −1 \pmj. Now there are only two poles left for which to draw root-locus lines. Those are the ones at −6 and −7. These lines will both exist on the real axis between −6 and −7 until they meet somewhere in the middle. Once they meet, they will break away from the real axis and travel in the complex plane until they rejoin between −4 and −5.
Note that the root locus has to be symmetric about the real axis, the two poles have to end up at −4 and −5, and they have to be on the real axis between −4 and −5, so this is the only choice. Once they break into the real axis again, they both split, and one heads to −4 and one heads to −5. The complete root locus plot is shown in Figure A.17.
