NiiL
Programming language

Root Locus

English translation

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

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.

MATLAB · Control System Toolbox

Before practice

01 What does a point on the root locus represent?

Description

These are candidate closed-loop poles at the selected gain. Examine all poles corresponding to a single gain together.

02 Can pole locations alone reveal every aspect of the actual response?

Description

Zeros, residues, and inputs also affect the response. Check the actual time response for the selected gain.

Practice

Preview practice notebookDownload practice notebookPreview practice mlx scriptDownload practice mlx script

1. Gain and closed-loop poles

Starting conditions

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

Results to retain

Items to record and points to 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?

2. Testing predictions with time responses

Starting conditions

Apply a unit step over 0≤t≤80\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=500K=500: add one value above or below this and test your prediction.

Results to retain

Items to record and points to 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?

3. Applying the same procedure to other models

Starting conditions

Use L2(s)=s+2s2+2s+3L_2(s)=\frac{s+2}{s^2+2s+3} and K∈{.5,2,10}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

Items to record and points to 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?