Code LV

Here's a problem I had to do for homework. Some of the equation formatting didn't copy over, sorry... Given an system G(s) = 10/(s(s+4)) and it’s open loop bode plots, design an appropriate compensator so that the closed loop phase margin is at least 45 o and the velocity constant is 50. The bode plot shows that the PM for the open loop system is 64.2 o at ω = 1.931rad/s, at the same frequency, we see that the P.M. of the closed loop system is 129 o . The velocity constant indicates the steady state error to a ramp input must be e ss =1/K v = 1/50 = 0.02.  First, the velocity constant of the uncompensated system is: Kv=limit as s->0 of sG(s) =limit as s->0 of s10/s(s+4)=10/4=2.5 since we want K v  50, the compensator must have a gain of at least 50/2.5 = 20.   If our compensator G c (s) = K c = 20, we get the following new responses by simply shifting the open loop bode up by about 26dB. Thus the new open loop 0dB crossing is at   ω = 13.83 rad/s

In this post we can see how to make root locus plots in python. This requires the setup from part 1. The problem is from Dorf's modern control systems AP 10.1. A three-axis pick-and-place application requires the precise movement of a robotic arm in three-dimensional space. The overshoot for a step input should be less than 13%. a) Let Gc(s) = K, and determine the gain K that satisfies the requirement. Determine the resulting settling time (with a 2% criterion). First we compute the Routh Hurwitz table to determine the valid range of K. We see 0<K<20 so If we let K = 2, then the step response becomes: which shows the settling time is 8.68 seconds and overshoot is less than 13%. b) Use a lead network and reduce the settling time to less than 3 seconds. Since we are dealing with time response parameters the root-locus method will be used. The new controller is: the overshoot and settling time criteria lead to a damping ratio of 0.545 or more and the real part of the dom