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: letsplot 4.10.1 | Python 3.13.13
Quality: 86/100 | Updated: 2026-06-18
"""
import os
import numpy as np
import pandas as pd
from lets_plot import *
from lets_plot import aes # explicit import to satisfy F405 for multi-line calls
from lets_plot.export import ggsave
LetsPlot.setup_html()
# Theme-adaptive chrome — Imprint palette
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_RULE = "rgba(26,26,23,0.18)" if THEME == "light" else "rgba(240,239,232,0.18)"
# Imprint categorical palette — hybrid-v3 canonical order
IMPRINT_PALETTE = ["#009E73", "#C475FD", "#4467A3", "#BD8233", "#AE3030", "#2ABCCD", "#954477", "#99B314"]
# Data — G(s) = (s + 3) / [s(s+1)(s+2)(s+4)]
# Open-loop poles: 0, −1, −2, −4 | Open-loop zero: −3
den_coeffs = np.polymul(np.polymul([1, 0], [1, 1]), np.polymul([1, 2], [1, 4]))
open_loop_poles = np.array([0.0, -1.0, -2.0, -4.0])
open_loop_zeros = np.array([-3.0])
# Gain sweep: dense near origin to capture breakaway, sparser at high gain
k_values = np.concatenate(
[np.linspace(0, 0.5, 200), np.linspace(0.5, 5, 400), np.linspace(5, 30, 400), np.linspace(30, 120, 400)]
)
all_real, all_imag, all_gain, all_branch = [], [], [], []
for k in k_values:
char_poly = np.polyadd(den_coeffs, k * np.array([0, 0, 0, 1, 3]))
roots = np.sort_complex(np.roots(char_poly))
for b, root in enumerate(roots):
all_real.append(root.real)
all_imag.append(root.imag)
all_gain.append(k)
all_branch.append(f"Branch {b + 1}")
df = pd.DataFrame({"real": all_real, "imaginary": all_imag, "gain": all_gain, "branch": all_branch})
# Pole and zero marker data
poles_df = pd.DataFrame({"real": open_loop_poles, "imaginary": 0.0})
zeros_df = pd.DataFrame({"real": open_loop_zeros, "imaginary": 0.0})
# Imaginary-axis stability crossings (real ≈ 0, nonzero imaginary part)
crossing_mask = (np.abs(df["real"]) < 0.08) & (np.abs(df["imaginary"]) > 0.3)
crossings = df[crossing_mask].copy()
if len(crossings) > 0:
crossings = crossings.sort_values("imaginary")
crossing_pts = pd.concat(
[crossings[crossings["imaginary"] > 0].head(1), crossings[crossings["imaginary"] < 0].head(1)]
)
else:
crossing_pts = pd.DataFrame(columns=df.columns)
# Direction arrows at selected gain values
arrow_gains = [5, 15, 50]
arrow_rows = []
for ag in arrow_gains:
idx = np.argmin(np.abs(k_values - ag))
subset = df[(df["gain"] >= k_values[max(0, idx - 1)]) & (df["gain"] <= k_values[min(len(k_values) - 1, idx + 1)])]
for _, row in subset.drop_duplicates(subset="branch").iterrows():
k_next = k_values[min(len(k_values) - 1, idx + 5)]
next_pts = df[(np.abs(df["gain"] - k_next) < 1.0) & (df["branch"] == row["branch"])]
if len(next_pts) > 0:
npt = next_pts.iloc[0]
dx, dy = npt["real"] - row["real"], npt["imaginary"] - row["imaginary"]
mag = np.hypot(dx, dy)
if mag > 0.01:
s = 0.25 / mag
arrow_rows.append(
{
"x": row["real"],
"y": row["imaginary"],
"xend": row["real"] + dx * s,
"yend": row["imaginary"] + dy * s,
}
)
arrows_df = pd.DataFrame(arrow_rows) if arrow_rows else pd.DataFrame(columns=["x", "y", "xend", "yend"])
# Damping ratio guide lines (ζ = constant), radial from origin in LHP
r_max = 5.8
zeta_values = [0.2, 0.4, 0.6, 0.8]
zeta_segs = []
for z in zeta_values:
theta = np.arccos(z)
zeta_segs += [
{"x": 0, "y": 0, "xend": -r_max * np.cos(theta), "yend": r_max * np.sin(theta)},
{"x": 0, "y": 0, "xend": -r_max * np.cos(theta), "yend": -r_max * np.sin(theta)},
]
zeta_df = pd.DataFrame(zeta_segs)
# Zeta labels — staggered radial distances to avoid crowding in upper LHP
zeta_labels = pd.DataFrame(
[
{
"x": -r_max * (0.36 + i * 0.14) * np.cos(np.arccos(z)),
"y": r_max * (0.36 + i * 0.14) * np.sin(np.arccos(z)),
"label": f"ζ={z}",
}
for i, z in enumerate(zeta_values)
]
)
# Natural frequency arcs (ωn = constant semicircles, LHP only)
wn_values = [1, 2, 3, 4]
wn_rows = []
for wn in wn_values:
theta = np.linspace(np.pi / 2, 3 * np.pi / 2, 120)
wn_rows += [{"real": wn * np.cos(t), "imaginary": wn * np.sin(t), "wn": str(wn)} for t in theta]
wn_df = pd.DataFrame(wn_rows)
# Critical gain annotation at stability boundary
crit_k = crossing_pts["gain"].iloc[0] if len(crossing_pts) > 0 else None
crit_y = float(crossing_pts["imaginary"].max()) if len(crossing_pts) > 0 else 2.0
annot_df = pd.DataFrame(
{"x": [0.2], "y": [crit_y + 0.45], "label": [f"K ≈ {crit_k:.1f}" if crit_k is not None else ""]}
)
plot = (
ggplot()
# ── Reference overlays (behind data) ───────────────────────────────
+ geom_path(
aes(x="real", y="imaginary", group="wn"),
data=wn_df,
color=GRID_RULE,
size=0.5,
linetype="dashed",
tooltips="none",
)
+ geom_segment(
aes(x="x", y="y", xend="xend", yend="yend"),
data=zeta_df,
color=GRID_RULE,
size=0.5,
linetype="dashed",
tooltips="none",
)
# Stable LHP shading with Imprint brand green tint
+ geom_rect(
aes(xmin="xmin", ymin="ymin", xmax="xmax", ymax="ymax"),
data=pd.DataFrame({"xmin": [-6.5], "xmax": [0.0], "ymin": [-5.5], "ymax": [5.5]}),
fill=IMPRINT_PALETTE[0],
alpha=0.07,
inherit_aes=False,
tooltips="none",
)
# ── Axis reference lines ────────────────────────────────────────────
+ geom_vline(xintercept=0, color=INK_SOFT, size=0.7)
+ geom_hline(yintercept=0, color=INK_SOFT, size=0.4)
# ── Locus branches (Imprint palette, interactive tooltips) ──────────
+ geom_path(
aes(x="real", y="imaginary", color="branch"),
data=df,
size=2.0,
alpha=0.9,
tooltips=layer_tooltips()
.format("gain", ".2f")
.format("real", ".3f")
.format("imaginary", ".3f")
.line("Branch @branch")
.line("K = @gain")
.line("Re = @real")
.line("Im = @imaginary"),
)
+ scale_color_manual(values=IMPRINT_PALETTE[:4], name="Locus branch")
# ── Direction arrows (increasing gain) ─────────────────────────────
+ geom_segment(
aes(x="x", y="y", xend="xend", yend="yend"),
data=arrows_df,
color=INK,
size=1.0,
inherit_aes=False,
tooltips="none",
arrow=arrow(length=10, ends="last", type="open"),
)
# ── Open-loop poles (×) and zero (○) ───────────────────────────────
+ geom_point(
aes(x="real", y="imaginary"),
data=poles_df,
shape=4,
size=6,
color=INK,
stroke=2.0,
inherit_aes=False,
tooltips=layer_tooltips().line("Pole: s = @real"),
)
+ geom_point(
aes(x="real", y="imaginary"),
data=zeros_df,
shape=1,
size=6,
color=INK,
stroke=2.0,
inherit_aes=False,
tooltips=layer_tooltips().line("Zero: s = @real"),
)
# ── Stability-boundary crossings ────────────────────────────────────
+ geom_point(
aes(x="real", y="imaginary"),
data=crossing_pts,
shape=18,
size=6,
color=IMPRINT_PALETTE[4],
inherit_aes=False,
tooltips=layer_tooltips().line("Stability boundary").line("K ≈ @gain"),
)
# ── Text annotations ────────────────────────────────────────────────
# Zeta labels — staggered for legibility
+ geom_text(
aes(x="x", y="y", label="label"),
data=zeta_labels,
size=3.5,
color=INK_MUTED,
inherit_aes=False,
family="monospace",
)
# Complex-plane axis symbols
+ geom_text(
aes(x="x", y="y", label="label"),
data=pd.DataFrame({"x": [0.15], "y": [4.55], "label": ["jω"]}),
size=4.5,
color=INK_MUTED,
hjust=0,
family="serif",
inherit_aes=False,
)
+ geom_text(
aes(x="x", y="y", label="label"),
data=pd.DataFrame({"x": [3.2], "y": [-0.22], "label": ["σ"]}),
size=4.5,
color=INK_MUTED,
hjust=1,
family="serif",
inherit_aes=False,
)
# Stability region labels
+ geom_text(
aes(x="x", y="y", label="label"),
data=pd.DataFrame({"x": [-4.8], "y": [4.3], "label": ["STABLE"]}),
size=5.0,
color=IMPRINT_PALETTE[0],
alpha=0.55,
fontface="bold",
family="monospace",
inherit_aes=False,
)
+ geom_text(
aes(x="x", y="y", label="label"),
data=pd.DataFrame({"x": [0.6], "y": [4.3], "label": ["UNSTABLE"]}),
size=4.0,
color=IMPRINT_PALETTE[4],
alpha=0.5,
fontface="bold",
family="monospace",
inherit_aes=False,
)
# Critical gain annotation
+ geom_text(
aes(x="x", y="y", label="label"),
data=annot_df,
size=4.0,
color=IMPRINT_PALETTE[4],
hjust=0,
family="monospace",
fontface="bold",
inherit_aes=False,
)
# ── Title, axis labels, layout ──────────────────────────────────────
+ labs(
x="Real axis (σ)",
y="Imaginary axis (jω)",
title="root-locus-basic · python · letsplot · anyplot.ai",
caption="G(s) = (s+3)/[s(s+1)(s+2)(s+4)] · × = poles · ○ = zero · ◆ = stability crossing",
)
# Equal-axis scaling preserves geometry (circles stay circular)
+ coord_fixed(ratio=1, xlim=[-5.5, 3.5], ylim=[-4.5, 4.5])
+ ggsize(600, 600)
+ theme_minimal()
+ theme(
axis_text=element_text(size=10, color=INK_SOFT),
axis_title=element_text(size=12, color=INK, face="bold"),
plot_title=element_text(size=16, color=INK, face="bold"),
plot_caption=element_text(size=9, color=INK_MUTED, face="italic"),
legend_text=element_text(size=10, color=INK_SOFT),
legend_title=element_text(size=11, color=INK, face="bold"),
legend_position="right",
panel_grid_major=element_blank(),
panel_grid_minor=element_blank(),
plot_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
panel_background=element_rect(fill=PAGE_BG),
axis_line=element_line(color=INK_SOFT, size=0.5),
legend_background=element_rect(fill=ELEVATED_BG, color=INK_SOFT),
plot_margin=[20, 25, 15, 15],
)
)
# Save PNG (scale=4 → 2400 × 2400) and interactive HTML
ggsave(plot, f"plot-{THEME}.png", path=".", scale=4)
ggsave(plot, f"plot-{THEME}.html", path=".")
Part of Root Locus Plot for Control Systems on anyplot.ai.