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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 poles must be to the left of -1.33 respectively.  

If we choose our dominant poles to be at s1 = -2 +- 2j they fall in the correct region on the root locus and should get the desired response. The angle of the system L(s) at s1 is Ls 63.4o.  So letting m=0,
∠Gc(s1) =180 - 63.4 = 116.6o
the controller must add this amount. If we pick our zero to be at ½ , then using the relationship:
∠( s1+ z ) -∠(s1+p)  = 106.26o
we find p = 6.34 and K = 34.19.  This leads to the step response of:
from which we can see the settling time has been reduced but not enough to meet the specifications. Let's move it closer to the system pole at -1. 

Reiterating through the root locus design procedure, if we move our zero out to z=¾ and recalculate p=5.86 and K=32.9, the response now is:
which shows another good improvement in the settling time but the zero still needs moved closer to the pole at -1.

Lastly let’s set our z=0.9, then p=5.63 and K=32.5. The root locus is computed using:def rootLocus(Ts,*args,**kwargs):    num = Poly(Ts.as_numer_denom()[0],s).all_coeffs()    den = Poly(Ts.as_numer_denom()[1],s).all_coeffs()    tf = matlab.tf(map(float,num),map(float,den))    r,k = matlab.rlocus(tf,*args,**kwargs)    plt.title("Root Locus")    #plt.plot(k,r)    plt.grid()    plt.show()and the output is:resulting in the final controller of:who's step response is:which shows the controller has reduced the overshoot and settling time to within the specifications (the settling time is less than 3 seconds).

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