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

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 at the corner of the cone,



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 three poles located at Let’s

get rid of the real pole using a cancellation compensator, Applying this compensator in series with the plant will shift the pole at –1 to –100 leaving a plant with two dominant poles at



Since s* is known, the zero for the lead compensator is computed by equating the two lengths (radii) from the root locus sketch shown below.





The zero location is determined by equating two radii, and of the same circle. is the distance from the zero, z, to the open loop pole point, and is the distance from the zero to the spec point. and are defined as,





The zero location is solved by equating and ,





Step 3:

Select the lead pole The pole becomes s = –17.5. The lead compensator transfer function becomes,





Step 4:

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.



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:





The difference in the transient response is again due to interaction of the lead pole with the dominant closed loop poles. To reduce this effect the lead zero could be moved rightward or the lead pole moved leftward.