Root Locus · Lesson

Execution setup

The Python option uses python-control. If the module is missing, run %pip install control==0.10.2 in a Colab or Jupyter code cell.

Topics covered

Examine how choosing a gain relates to pole locations and the time response.

Overall workflow

Open-loop model → candidate gains → closed-loop poles → time response

Paths of poles as gain changes

The closed-loop poles with unity feedback are the roots of 1+K L(s)=0. The root locus shows how these roots move as K changes.

When choosing a K, examine all poles corresponding to that K together.

Checking pole choices through responses

If any pole has a positive real part, stable step-response metrics cannot be calculated. Even for stable candidates, zeros, residues, and inputs affect the response. Compare damping and speed predicted from poles with the actual curves before selecting a gain.

Closed-loop poles that change with gain

Apply $K\in\{1,4,9\}$ to $G(s)=\frac1{s(s+2)}$. Examine the pole map and step response together.

import numpy as np
import matplotlib.pyplot as plt
import control as ctl
s=ctl.tf('s')
G=1/(s*(s+2));t=np.linspace(0,10,1001)
fig,axes=plt.subplots(1,2,figsize=(9,3))
for gain in [1,4,9]:
    closed=ctl.feedback(gain*G,1);poles=ctl.poles(closed)
    axes[0].scatter(poles.real,poles.imag,marker='x',label=f'K={gain}')
    tm,y=ctl.step_response(closed,t);axes[1].plot(tm,y,label=f'K={gain}')
axes[0].set(xlabel='Real',ylabel='Imaginary');axes[1].set(xlabel='Time [s]',ylabel='Output')
for ax in axes:ax.grid();ax.legend()
fig.tight_layout()

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