Design a Lead controller using root locus techniques for the plant,

Question

such that the following performance specifications are met:



System is stable.



What will be an ideal response?

Answer 1

Step 1:

Given the specifications on the desired closed loop performance in terms of identify the specification region or point, s*, in the s-plane. The spec region is the cone shown in the following figure.



We can design to any point in the total spec cone. For demonstration purposes we will select our design point well within the cone at



Step 2:

Select the lead zero to move the rightmost pole(s) to s*. The rightmost poles are the dominant poles, that is, they are the poles that govern the fundamental parts of the response. In this example the plant only has two poles located at If a zero were placed at about s = –10, the plant poles would tend to travel leftward in a circular path with the center at the zero. Along their trajectory the poles would be dragged through the design point, so this zero seems to be satisfactory at this point in the design.



Step 3:

Select the lead pole The pole becomes s = –100



Step 4:

The lead compensator transfer function becomes,

To find the gain value, K, we simply solve the root locus magnitude condition at s*. This calculation is performed below.



This compensator is implemented in the cascade configuration shown in the following figure.





The unit step response of the control system and the spec system, is shown below,





The difference in the transient response is due to interaction of the lead pole with the dominant closed loop poles. At K=32.25, the closed loop poles are actually located at, To reduce this effect the lead zero could be moved rightward or the lead pole moved leftward.