Design a PI controller using root locus techniques for the plant
such that the following performance specifications are met.
System is stable.
What will be an ideal response?
Step 1:
The steady state step error spec is satisfied by the pole of the PI compensator which makes the control system a type 1 system.
Step 2:
In this step we will design the PI zero and the K for desired closed loop pole locations. A root locus sketch is necessary. The following figure presents the PZ plot for the control system 
and the performance specification which is just a time constant. This time constant value, 0.5, identifies a pole location at
The root locus
sketch is shown below,
To find the gain value, 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 slight difference between the two responses is due to the interaction of the plant pole located at -6.
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