Design a Lead controller using Bode techniques for the plant

Question

such that the following performance specifications are met:



System is stable.



What will be an ideal response?

Answer 1

The plant spec bode plots are shown below.



A lead compensator is used to add phase at specific frequencies and gain compensation moves the LM plot vertically without affecting the phase. Adding gain is easy, from the LM plot and the time constant spec we need to add 1 LM (or a factor of 10) gain to meet the time constant spec Also, we need to add about 50–55 degrees of phase at this frequency. The following table shows the relationship between the lead ratio, and the phase produced by a first order lead compensator of the form,



55° of phase corresponds to a lead ratio of and the lead zero location becomes:



The lead pole is then computed as:



The lead compensator becomes (including the scaling gain of 10):





Note: The gain of 100 includes the scaling gain of 10 and the gain required for unity dc gain, 10.

The Bode plot of the lead compensator plus plant is shown below.



Notice the gain is a little high, our LM crossover frequency is about 23 rad/sec instead of the desired 10 rad/sec. This is because the lead compensator does add some magnitude as well as phase. However, we’ll leave it even though a slight gain reduction should be made.



The final control system is shown in the following figure.



The unit step response is presented below.