Root locus for L(s)=1/(s(s+1)(s+2))

Python Code

"""Control Systems — Root locus for L(s)=1/(s(s+1)(s+2))."""
from pathlib import Path

# ── Palette + theme (matching the Plotly Studio gallery these charts ship in) ──
VIOLET, TEAL, GREEN, PINK, ORANGE = "#845EEE", "#52B3D0", "#55B685", "#DA5597", "#E9A23B"
PRIMARY, SECONDARY = VIOLET, TEAL
COLORWAY = [VIOLET, TEAL, GREEN, PINK, ORANGE]
BG, TEXT, GRID, MUTED = "#ffffff", "#1c2024", "#d9d9e0", "#60646c"
FONT = "Inter, -apple-system, BlinkMacSystemFont, sans-serif"
COLORSCALE = [[0, "rgba(132, 94, 238, 0.05)"], [1, "rgba(132, 94, 238, 0.9)"]]


def apply_theme(fig):
    """Light gallery theme: white background, Inter font, soft gridlines."""
    fig.update_layout(
        paper_bgcolor=BG, plot_bgcolor=BG, colorway=COLORWAY,
        font=dict(family=FONT, color=TEXT, size=12),
        legend=dict(font=dict(color=TEXT)),
        hoverlabel=dict(bgcolor="#f0f0f3", font=dict(color=TEXT, family=FONT), bordercolor=GRID),
    )
    fig.update_xaxes(gridcolor=GRID, linecolor=GRID, zerolinecolor=GRID)
    fig.update_yaxes(gridcolor=GRID, linecolor=GRID, zerolinecolor=GRID)


def fetch_csv(url, **kwargs):
    import io
    import pandas as pd
    import requests
    r = requests.get(url, timeout=60)
    r.raise_for_status()
    return pd.read_csv(io.StringIO(r.text), **kwargs)


def fetch_json(url):
    import requests
    r = requests.get(url, timeout=60)
    r.raise_for_status()
    return r.json()

import numpy as np
import plotly.graph_objects as go


def generate():
    K_values = np.linspace(0, 30, 3000)

    roots_real = []
    roots_imag = []
    k_colors = []

    for K in K_values:
        # Characteristic polynomial: s^3 + 3s^2 + 2s + K
        coeffs = [1, 3, 2, K]
        roots = np.roots(coeffs)
        for r in roots:
            roots_real.append(r.real)
            roots_imag.append(r.imag)
            k_colors.append(K)

    fig = go.Figure()

    # Root locus colored by K
    fig.add_trace(go.Scatter(
        x=roots_real, y=roots_imag,
        mode="markers",
        name="Root locus",
        marker=dict(
            color=k_colors,
            colorscale="Viridis",
            size=2,
            opacity=0.6,
            colorbar=dict(title="K", thickness=14, len=0.7),
        ),
        hovertemplate="Re=%{x:.3f}, Im=%{y:.3f}<extra></extra>",
    ))

    # Open-loop poles at s=0, -1, -2
    fig.add_trace(go.Scatter(
        x=[0, -1, -2], y=[0, 0, 0],
        mode="markers",
        name="Open-loop poles",
        marker=dict(color="red", size=12, symbol="x", line=dict(width=2.5)),
        hovertemplate="%{x:.0f}+j%{y:.3f}<extra>OL pole</extra>",
    ))

    # Imaginary axis crossing at K≈6, ω≈±√2
    omega_c = np.sqrt(2)
    fig.add_trace(go.Scatter(
        x=[0, 0], y=[omega_c, -omega_c],
        mode="markers",
        name="Crossing (K≈6, ω≈±√2)",
        marker=dict(color=TEAL, size=11, symbol="circle-open", line=dict(width=2.5)),
        hovertemplate="±j√2 (K≈6)<extra></extra>",
    ))

    fig.update_layout(
        xaxis=dict(
            title="Real",
            range=[-4, 1.5],
            zeroline=True,
            zerolinecolor=GRID,
            zerolinewidth=1,
        ),
        yaxis=dict(
            title="Imaginary",
            range=[-5, 5],
            zeroline=True,
            zerolinecolor=GRID,
            zerolinewidth=1,
        ),
        legend=dict(orientation="h", y=1.08),
        margin=dict(t=40, b=60, l=70, r=40),
        height=600,
    )
    apply_theme(fig)
    return fig


fig = generate()
fig.show()

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