Given the transfer function from Example 13.6, find the value of gain K that results in a pair of complex poles with equal real and imaginary parts (corresponding to a damping ratio of 0.707).
Question 13.9: Given the transfer function from Example 13.6, find the valu...
Again, start the process by entering the transfer function. In this case, enter the transfer function by using unity gain, knowing that the root-locus method will compute a gain factor that, when multiplied by the original open-loop gain, will result in a closed-loop system of desired properties:
>> tf139=tf([1],[6 11 6 1])
Transfer function:
1
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6 s∧3 + 11 s∧2 + 6 s + 1
>> rlocus(tf139)
The resulting plot shows the three different loci (one for each of the system poles).
However, this particular plot is interactive. If any of the loci is selected with a mouse click, a small black block appears indicating a pole location along with a callout showing the properties of that particular pole and the gain associated with that pole location.
By dragging the block, any pole location along any of the loci can be found (and hence, the corresponding gain factor). Figure 13.13 shows a screen shot of the root-locus plot, with the pole indicator at the location desired by the problem statement. Therefore a closed-loop gain of 0.561 results in a pair of complex poles whose damping ratio is near 0.7071.
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