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Showing posts from March, 2013

Control Systems in Python - Phase Lag Compensator Design

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 45o and the velocity constant is 50. The bode plot shows that the PM for the open loop system is 64.2o at ω = 1.931rad/s, at the same frequency, we see that the P.M. of the closed loop system is 129o. The velocity constant indicates the steady state error to a ramp input must be ess =1/Kv = 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 Kv  50, the compensator must have a gain of at least 50/2.5 = 20.  
If our compensator Gc(s) = Kc = 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 and the open and closed loop phase mar…

Control Systems in Python - Part 3 - Root Locus Plots

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 dominant pole…