Root Locus

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.

Learning objectives

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

Gain and closed-loop poles

Starting conditions

$L(s)=\frac{s+7}{s(s+5)(s+15)(s+20)}$. Use $K\in\{100,500,1000\}$.

Procedure

1. Plot the root locus and mark the poles of feedback(K*L,1) for each K using distinct symbols.
2. Tabulate the real parts of the poles and stability for each case.
3. Predict pole movements and response changes relative to the baseline condition $K=500$.

Results to retain

• 1 root-locus plot and a pole table for each K.

import numpy as np
import matplotlib.pyplot as plt
import control as ctl
s = ctl.tf('s')
L=(s+7)/(s*(s+5)*(s+15)*(s+20))
ctl.root_locus_plot(L)
for k in [100,500,1000]: print(k,ctl.poles(ctl.feedback(k*L,1)))

Record results and self-check

K:

Closed-loop Pole:

Maximum real part:

Stability:

Points to check

• Do the root-locus points match the directly calculated closed-loop poles?
• Are complex roots still conjugate pairs?

Testing predictions with time responses

Starting conditions

Apply a unit step over $0\le t\le 8$ s to the same model and gain.

Procedure

1. Calculate overshoot and settling time only for stable cases.
2. Explain the relationship between the slowest pole pair and oscillation and decay in the time response.
3. $K=500$: add one value above or below this and test your prediction.

Results to retain

• A response plot and performance table including all 4 conditions.

import numpy as np
import matplotlib.pyplot as plt
import control as ctl
s = ctl.tf('s')
L=(s+7)/(s*(s+5)*(s+15)*(s+20))
t=np.linspace(0,8,1601)
T=ctl.feedback(500*L,1)
t,y=ctl.step_response(T,t)
plt.plot(t,y);plt.xlabel('Time [s]');plt.grid()
# TODO: Add the remaining gains.

Record results and self-check

K:

Overshoot [%]:

Settling Time [s]:

Agreement with predictions:

Points to check

• Did you avoid forcing convergence metrics for unstable cases?
• Did you distinguish predictions based on poles from actual observations?

Applying the same procedure to other models

Starting conditions

Use $L_2(s)=\frac{s+2}{s^2+2s+3}$ and $K\in\{.5,2,10\}$.

Procedure

1. Create a root-locus plot, pole table, and step plot for L2 in the same format.
2. Determine whether gain changes produce the same pattern in the higher-order model L and in L2.
3. Explain whether conclusions from one model can be applied directly to another.

Results to retain

• A comparison summary table for both models and a brief interpretation.

import numpy as np
import matplotlib.pyplot as plt
import control as ctl
s = ctl.tf('s')
L2=(s+2)/(s*s+2*s+3)
for k in [.5,2,10]:print(k,ctl.poles(ctl.feedback(k*L2,1)))

Record results and self-check

Model:

K:

Pole:

Overshoot [%]:

Settling Time [s]:

Points to check

• Did you reconstruct the closed-loop expression for each model?
• Did you avoid treating equal gain values as evidence of equal performance in the two systems?