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 line shown in the following figure.





We can design to any point n the total spec line. For demonstration purposes we will select our design point at the start of the line,



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 has three poles located at Let’s get rid of the two complex poles using a cancellation compensator of the form,

Applying this compensator in series with the plant will shift the poles at leaving a plant with one dominat pole If a zero were placed at about s = –10, the dominant plant pole would tend to travel leftward in a linear path 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 with the cancellation compensator as shown in the following figure.





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