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 two bold lines shown in the following figure.





We can design to any point on the spec. For demonstration purposes we will select our design point at the start point of one of the spec lines,



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 four 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 two dominant poles at

We could continue with the lead compensator design, however, simple gain feedback will force the two dominant poles to pass through the spec point. We’ll try this approach instead of using a lead compensator.



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.