W9_ControlRootLocus

English translation

!pip install control

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Installing collected packages: control
Successfully installed control-0.10.0

import numpy as np
import control as ctl
import matplotlib.pyplot as plt

Plotting the root locus

$L(s) = \frac{s + 2}{s^2+2s+3}$

L = ctl.tf([1,2],[1,2,3])
roots, gains = ctl.root_locus(L, xlim=[-5, 2], ylim=[-2, 2], plot=True)

<Figure size 640x480 with 1 Axes>

Choosing gain (K) using the root locus (P controller)

$L(s) = \frac{s + 7}{s(s + 5)(s + 15)(s + 20)}$

Objectives

%overshoot < 5% $\rightarrow \zeta<0.7$
$T_r < 1s \rightarrow \omega_n > 1.8$

s = ctl.tf([1,0],[1])
L = (s + 7)/(s*(s + 5)*(s + 15)*(s + 20))
roots, gains = ctl.root_locus(L, xlim=[-22, 3], ylim=[-15, 15], plot=True)

<Figure size 640x480 with 1 Axes>

roots, gains

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        [-1.43764200e+01+0.00000000e+00j, -2.24239643e+00+0.00000000e+00j],
        [-1.44553034e+01+0.00000000e+00j, -2.24086125e+00+0.00000000e+00j],
        [-1.45341671e+01+0.00000000e+00j, -2.23934578e+00+0.00000000e+00j],
        [-1.46130115e+01+0.00000000e+00j, -2.23784962e+00+0.00000000e+00j],
        [-1.46918369e+01+0.00000000e+00j, -2.23637240e+00+0.00000000e+00j],
        [-1.47706438e+01+0.00000000e+00j, -2.23491376e+00+0.00000000e+00j],
        [-1.48494325e+01+0.00000000e+00j, -2.23347335e+00+0.00000000e+00j],
        [-1.49282032e+01+0.00000000e+00j, -2.23205081e+00+0.00000000e+00j],
        [-1.51582485e+02+0.00000000e+00j, -2.02005582e+00+0.00000000e+00j],
        [-1.51602342e+03+0.00000000e+00j, -2.00198148e+00+0.00000000e+00j],
        [-1.51602538e+04+0.00000000e+00j, -2.00019791e+00+0.00000000e+00j]]),
 array([0.00000000e+00, 7.73482349e-02, 1.54696470e-01, 2.32044705e-01,
        3.09392940e-01, 6.18785879e-01, 9.28178819e-01, 1.23757176e+00,
        1.54696470e+00, 1.85635764e+00, 2.16575058e+00, 2.47514352e+00,
        2.78453646e+00, 3.09392940e+00, 3.40332234e+00, 3.48067057e+00,
        3.55801880e+00, 3.63536704e+00, 3.71271527e+00, 3.79006351e+00,
        3.86741174e+00, 3.94475998e+00, 4.02210821e+00, 4.09945645e+00,
        4.17680468e+00, 4.25415292e+00, 4.33150115e+00, 4.40884939e+00,
        4.48619762e+00, 4.56354586e+00, 4.64089409e+00, 4.71824233e+00,
        4.79559056e+00, 4.87293880e+00, 4.95028703e+00, 5.02763527e+00,
        5.10498350e+00, 5.18233174e+00, 5.25967997e+00, 5.31078538e+00,
        5.36189079e+00, 5.41299620e+00, 5.42577256e+00, 5.43854891e+00,
        5.45132526e+00, 5.46410162e+00, 5.47066232e+00, 5.47722303e+00,
        5.48378373e+00, 5.49034444e+00, 5.51658726e+00, 5.54283009e+00,
        5.56907291e+00, 5.58840997e+00, 5.60774703e+00, 5.62708409e+00,
        5.64642115e+00, 5.72376938e+00, 5.80111762e+00, 5.87846585e+00,
        5.95581409e+00, 6.03316232e+00, 6.11051056e+00, 6.18785879e+00,
        6.26520703e+00, 6.34255526e+00, 6.41990350e+00, 6.49725173e+00,
        6.57459997e+00, 6.65194820e+00, 6.72929644e+00, 6.80664467e+00,
        6.88399290e+00, 6.96134114e+00, 7.03868937e+00, 7.11603761e+00,
        7.19338584e+00, 7.27073408e+00, 7.34808231e+00, 7.42543055e+00,
        7.50277878e+00, 7.58012702e+00, 7.65747525e+00, 7.73482349e+00,
        7.81217172e+00, 7.88951996e+00, 7.96686819e+00, 8.04421643e+00,
        8.12156466e+00, 8.19891290e+00, 8.27626113e+00, 8.35360937e+00,
        8.43095760e+00, 8.50830584e+00, 8.58565407e+00, 8.66300231e+00,
        8.74035054e+00, 8.81769878e+00, 8.89504701e+00, 8.97239525e+00,
        9.04974348e+00, 9.12709172e+00, 9.20443995e+00, 9.28178819e+00,
        9.35913642e+00, 9.43648466e+00, 9.51383289e+00, 9.59118113e+00,
        9.66852936e+00, 9.74587760e+00, 9.82322583e+00, 9.90057407e+00,
        9.97792230e+00, 1.00552705e+01, 1.01326188e+01, 1.02099670e+01,
        1.02873152e+01, 1.03646635e+01, 1.04420117e+01, 1.05193599e+01,
        1.05967082e+01, 1.06740564e+01, 1.07514046e+01, 1.08287529e+01,
        1.09061011e+01, 1.09834494e+01, 1.10607976e+01, 1.11381458e+01,
        1.12154941e+01, 1.12928423e+01, 1.13701905e+01, 1.14475388e+01,
        1.15248870e+01, 1.16022352e+01, 1.16795835e+01, 1.17569317e+01,
        1.18342799e+01, 1.19116282e+01, 1.19889764e+01, 1.20663246e+01,
        1.21436729e+01, 1.22210211e+01, 1.22983693e+01, 1.23757176e+01,
        1.24530658e+01, 1.25304141e+01, 1.26077623e+01, 1.26851105e+01,
        1.27624588e+01, 1.28398070e+01, 1.29171552e+01, 1.29945035e+01,
        1.30718517e+01, 1.31491999e+01, 1.32265482e+01, 1.33038964e+01,
        1.33812446e+01, 1.34585929e+01, 1.35359411e+01, 1.36132893e+01,
        1.36906376e+01, 1.37679858e+01, 1.38453340e+01, 1.39226823e+01,
        1.40000305e+01, 1.40773787e+01, 1.41547270e+01, 1.42320752e+01,
        1.43094235e+01, 1.43867717e+01, 1.44641199e+01, 1.45414682e+01,
        1.46188164e+01, 1.46961646e+01, 1.47735129e+01, 1.48508611e+01,
        1.49282093e+01, 1.50055576e+01, 1.50829058e+01, 1.51602540e+01,
        1.51602540e+02, 1.51602540e+03, 1.51602540e+04]))

K = 500
roots, gains = ctl.root_locus(L, initial_gain=K, xlim=[-22, 3], ylim=[-15, 15], plot=True)
plt.show()
T = ctl.feedback(K*L,1)
t,y = ctl.step_response(T, np.arange(0,2.5,0.01))
plt.plot(t,y)
plt.xlabel('Time')
plt.title('Response of the feedback sytem')
plt.axis([0, 2.5, 0, 1.1])

<Figure size 640x480 with 1 Axes>

(0.0, 2.5, 0.0, 1.1)

<Figure size 640x480 with 1 Axes>