A root locus plot traces how the closed-loop poles of a transfer function migrate through the complex plane as a system parameter (typically gain K) varies from 0 to infinity. It is a fundamental tool in classical control theory for analyzing system stability and designing controllers. The plot reveals critical information about pole trajectories, stability boundaries, and gain margins.
""" anyplot.ai
root-locus-basic: Root Locus Plot for Control Systems
Library: plotly 6.8.0 | Python 3.13.14
Quality: 86/100 | Updated: 2026-06-18
"""
import os
import numpy as np
import plotly.graph_objects as go
# Theme tokens — Imprint palette, theme-adaptive chrome
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
ELEVATED_BG = "#FFFDF6" if THEME == "light" else "#242420"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"
GRID = "rgba(26,26,23,0.15)" if THEME == "light" else "rgba(240,239,232,0.15)"
REF_LINE = "rgba(107,106,99,0.28)" if THEME == "light" else "rgba(168,167,159,0.28)"
# Imprint categorical palette — positions 1–3 for the three root locus branches
BRANCH_COLORS = ["#009E73", "#C475FD", "#4467A3"]
# Data: G(s) = 1 / (s(s+1)(s+3))
# Open-loop poles at s = 0, -1, -3; no finite zeros
# Characteristic equation: s^3 + 4s^2 + 3s + K = 0
open_loop_poles = np.array([0.0, -1.0, -3.0])
gains = np.concatenate(
[
np.linspace(0, 0.5, 200),
np.linspace(0.5, 4, 400),
np.linspace(4, 12, 400),
np.linspace(12, 50, 300),
np.linspace(50, 200, 200),
]
)
branches = {i: {"real": [], "imag": [], "gain": []} for i in range(3)}
prev_roots = open_loop_poles.copy().astype(complex)
for K in gains:
roots = np.roots([1, 4, 3, K])
roots = np.sort_complex(roots)
used = [False] * 3
assignment = [0] * 3
for i in range(3):
best_j, best_dist = -1, np.inf
for j in range(3):
if not used[j]:
d = abs(prev_roots[i] - roots[j])
if d < best_dist:
best_dist = d
best_j = j
used[best_j] = True
assignment[i] = best_j
for i in range(3):
r = roots[assignment[i]]
branches[i]["real"].append(r.real)
branches[i]["imag"].append(r.imag)
branches[i]["gain"].append(K)
prev_roots = np.array([roots[assignment[i]] for i in range(3)])
branch_names = ["Branch 1 (from s=0)", "Branch 2 (from s=−1)", "Branch 3 (from s=−3)"]
# Plot
fig = go.Figure()
# Real axis root locus segments: [−1, 0] and (−∞, −3]
for seg in [[-1, 0], [-5.5, -3]]:
fig.add_trace(
go.Scatter(
x=seg,
y=[0, 0],
mode="lines",
line={"width": 7, "color": "rgba(0,158,115,0.18)"},
showlegend=False,
hoverinfo="skip",
)
)
# Constant damping ratio lines (ζ = 0.2, 0.4, 0.6, 0.8)
r_max = 5.3
for zeta in [0.2, 0.4, 0.6, 0.8]:
r_line = np.linspace(0, r_max, 2)
x_vals = -r_line * zeta
y_vals = r_line * np.sqrt(1 - zeta**2)
for sign in [1, -1]:
fig.add_trace(
go.Scatter(
x=x_vals,
y=sign * y_vals,
mode="lines",
line={"width": 1, "color": REF_LINE, "dash": "dash"},
showlegend=False,
hoverinfo="skip",
)
)
fig.add_annotation(
x=x_vals[-1], y=y_vals[-1] + 0.12, text=f"ζ={zeta}", showarrow=False, font={"size": 10, "color": INK_MUTED}
)
# Constant natural frequency arcs (ωn = 1, 2, 3, 4, 5)
theta = np.linspace(np.pi / 2, np.pi, 100)
for wn in [1, 2, 3, 4, 5]:
for sign in [1, -1]:
fig.add_trace(
go.Scatter(
x=wn * np.cos(theta),
y=sign * wn * np.sin(theta),
mode="lines",
line={"width": 1, "color": REF_LINE, "dash": "dot"},
showlegend=False,
hoverinfo="skip",
)
)
fig.add_annotation(
x=wn * np.cos(np.pi * 0.55),
y=wn * np.sin(np.pi * 0.55) + 0.12,
text=f"ωn={wn}",
showarrow=False,
font={"size": 10, "color": INK_MUTED},
)
# Stability boundary — imaginary axis shaded band
fig.add_shape(
type="line", x0=0, x1=0, y0=-5.5, y1=5.5, line={"color": "rgba(174,48,48,0.15)", "width": 20}, layer="below"
)
fig.add_annotation(
x=0.4,
y=4.7,
text="Stability<br>Boundary",
showarrow=False,
font={"size": 10, "color": "rgba(174,48,48,0.6)", "family": "Arial, sans-serif"},
)
# Root locus branches
for i in range(3):
fig.add_trace(
go.Scatter(
x=branches[i]["real"],
y=branches[i]["imag"],
mode="lines",
line={"width": 2.5, "color": BRANCH_COLORS[i]},
name=branch_names[i],
legendgroup=f"branch{i}",
hovertemplate=(
f"<b>Branch {i + 1}</b><br>σ = %{{x:.3f}}<br>jω = %{{y:.3f}}<br>K = %{{customdata:.2f}}<extra></extra>"
),
customdata=branches[i]["gain"],
)
)
# Direction arrows indicating increasing gain
for i in range(3):
n = len(branches[i]["real"])
for frac in [0.3, 0.65]:
idx = int(n * frac)
if idx < n - 5:
dx = branches[i]["real"][idx + 5] - branches[i]["real"][idx]
dy = branches[i]["imag"][idx + 5] - branches[i]["imag"][idx]
norm = np.sqrt(dx**2 + dy**2)
if norm > 1e-6:
fig.add_annotation(
x=branches[i]["real"][idx],
y=branches[i]["imag"][idx],
ax=-dx / norm * 25,
ay=dy / norm * 25,
xref="x",
yref="y",
axref="pixel",
ayref="pixel",
showarrow=True,
arrowhead=3,
arrowsize=1.8,
arrowwidth=2,
arrowcolor=BRANCH_COLORS[i],
text="",
)
# Open-loop poles (× markers)
fig.add_trace(
go.Scatter(
x=open_loop_poles,
y=np.zeros(3),
mode="markers+text",
marker={"symbol": "x-thin", "size": 16, "color": INK, "line": {"width": 3}},
text=["s=0", "s=−1", "s=−3"],
textposition="top center",
textfont={"size": 10, "color": INK},
name="Open-loop poles",
hovertemplate="Pole at s = %{x:.1f}<extra></extra>",
)
)
# jω-axis crossing (Routh–Hurwitz: K_crit = 12, roots at ±j√3)
K_crit = 12.0
jw_cross = np.sqrt(3)
fig.add_trace(
go.Scatter(
x=[0, 0],
y=[jw_cross, -jw_cross],
mode="markers",
marker={"symbol": "diamond", "size": 14, "color": "#AE3030", "line": {"width": 2, "color": PAGE_BG}},
name=f"jω crossing (K={K_crit:.0f})",
hovertemplate="s = %{y:+.3f}j<br>K = 12 (critical gain)<extra></extra>",
)
)
fig.add_annotation(
x=0,
y=jw_cross,
text=f" K={K_crit:.0f}",
showarrow=True,
arrowhead=0,
arrowwidth=1,
arrowcolor="#AE3030",
ax=50,
ay=-20,
font={"size": 10, "color": "#AE3030"},
)
# Breakaway point (σ ≈ −0.451)
s_break = -0.451
K_break = -(s_break**3 + 4 * s_break**2 + 3 * s_break)
fig.add_trace(
go.Scatter(
x=[s_break],
y=[0],
mode="markers",
marker={"symbol": "star", "size": 18, "color": "#BD8233", "line": {"width": 2, "color": PAGE_BG}},
name=f"Breakaway (K≈{K_break:.2f})",
hovertemplate="Breakaway point<br>s ≈ −0.451<br>K ≈ %{customdata:.2f}<extra></extra>",
customdata=[K_break],
)
)
fig.add_annotation(
x=s_break,
y=0,
text=f" K≈{K_break:.2f}",
showarrow=True,
arrowhead=0,
arrowwidth=1,
arrowcolor="#BD8233",
ax=-55,
ay=30,
font={"size": 10, "color": "#BD8233"},
)
# Layout — square canvas preserves equal axis scaling for the complex plane
axis_range = 5.5
fig.update_layout(
autosize=False,
title={
"text": "root-locus-basic · python · plotly · anyplot.ai",
"font": {"size": 16, "color": INK, "family": "Arial, sans-serif"},
"x": 0.5,
"xanchor": "center",
"y": 0.975,
},
xaxis={
"title": {
"text": "Real Axis (σ)",
"font": {"size": 12, "color": INK, "family": "Arial, sans-serif"},
"standoff": 12,
},
"tickfont": {"size": 10, "color": INK_SOFT},
"zeroline": True,
"zerolinewidth": 1.5,
"zerolinecolor": INK_SOFT,
"showgrid": True,
"gridwidth": 1,
"gridcolor": GRID,
"range": [-axis_range, axis_range],
"constrain": "domain",
"dtick": 1,
"linecolor": INK_SOFT,
},
yaxis={
"title": {
"text": "Imaginary Axis (jω)",
"font": {"size": 12, "color": INK, "family": "Arial, sans-serif"},
"standoff": 12,
},
"tickfont": {"size": 10, "color": INK_SOFT},
"zeroline": True,
"zerolinewidth": 1.5,
"zerolinecolor": INK_SOFT,
"showgrid": True,
"gridwidth": 1,
"gridcolor": GRID,
"range": [-axis_range, axis_range],
"scaleanchor": "x",
"scaleratio": 1,
"dtick": 1,
"linecolor": INK_SOFT,
},
paper_bgcolor=PAGE_BG,
plot_bgcolor=PAGE_BG,
font={"color": INK},
legend={
"font": {"size": 10, "color": INK_SOFT, "family": "Arial, sans-serif"},
"bgcolor": ELEVATED_BG,
"bordercolor": INK_SOFT,
"borderwidth": 1,
"x": 0.99,
"y": 0.01,
"xanchor": "right",
"yanchor": "bottom",
"itemsizing": "constant",
},
margin={"l": 80, "r": 40, "t": 80, "b": 60},
hoverlabel={"bgcolor": ELEVATED_BG, "font_size": 12, "bordercolor": INK_SOFT},
)
# Transfer function subtitle — lower-left, clear of legend (lower-right) and ζ labels (upper-left)
fig.add_annotation(
text="G(s) = 1 / s(s+1)(s+3)",
xref="paper",
yref="paper",
x=0.03,
y=0.03,
xanchor="left",
yanchor="bottom",
showarrow=False,
font={"size": 10, "color": INK_MUTED, "family": "Courier New, monospace"},
)
# Save — square canvas (2400×2400)
fig.write_image(f"plot-{THEME}.png", width=600, height=600, scale=4)
fig.write_html(f"plot-{THEME}.html", include_plotlyjs="cdn")
Part of Root Locus Plot for Control Systems on anyplot.ai.