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: makie 0.22.10 | Julia 1.11.9
# Quality: 88/100 | Created: 2026-06-18
using CairoMakie
using Colors
using LinearAlgebra
# Theme tokens — Imprint palette
const THEME = get(ENV, "ANYPLOT_THEME", "light")
const PAGE_BG = THEME == "light" ? colorant"#FAF8F1" : colorant"#1A1A17"
const ELEVATED_BG = THEME == "light" ? colorant"#FFFDF6" : colorant"#242420"
const INK = THEME == "light" ? colorant"#1A1A17" : colorant"#F0EFE8"
const INK_SOFT = THEME == "light" ? colorant"#4A4A44" : colorant"#B8B7B0"
const IMPRINT = [
colorant"#009E73", # 1 — brand green (first categorical series)
colorant"#C475FD", # 2 — lavender
colorant"#4467A3", # 3 — blue
colorant"#BD8233", # 4 — ochre
colorant"#AE3030", # 5 — matte red
colorant"#2ABCCD", # 6 — cyan
colorant"#954477", # 7 — rose
colorant"#99B314", # 8 — lime
]
# Data
# Open-loop: G(s) = K / (s(s+2)(s+4))
# Characteristic equation: s³ + 6s² + 8s + K = 0
# Open-loop poles: s = 0, −2, −4 | No finite zeros
# Breakaway point: s ≈ −0.845 at K ≈ 3.08 (two branches merge → complex pair)
# jω-axis crossings: s = ±j√8 ≈ ±j2.83 at K = 48 (stability boundary)
const OL_POLES = [0.0, -2.0, -4.0]
const N_BR = 3
function companion_roots(K::Float64)
A = Float64[0 1 0
0 0 1
-K -8 -6]
return complex.(eigvals(A)) # always ComplexF64, even when roots are real
end
K_range = range(0.0002, 200.0; length = 1200)
# Sort descending by real part: branch 1 (green) ← pole at s=0 (closest to jω axis)
init = sort(companion_roots(Float64(K_range[1])), by = real, rev = true)
branches = [[c] for c in init]
for k in K_range[2:end]
curr = companion_roots(Float64(k))
prev_tips = [b[end] for b in branches]
remaining = collect(1:N_BR)
assign = zeros(Int, N_BR)
for b in 1:N_BR
best_pos, best_d = 1, Inf
for (pos, j) in enumerate(remaining)
d = abs(curr[j] - prev_tips[b])
if d < best_d
best_d = d
best_pos = pos
end
end
assign[b] = remaining[best_pos]
deleteat!(remaining, best_pos)
end
for b in 1:N_BR
push!(branches[b], curr[assign[b]])
end
end
bx = [real.(branches[b]) for b in 1:N_BR]
by = [imag.(branches[b]) for b in 1:N_BR]
# Grid color (INK at 12% opacity)
gc = RGBAf(INK.r, INK.g, INK.b, 0.12f0)
# Figure — square canvas → 2400×2400 output; DataAspect preserves s-plane geometry
fig = Figure(
size = (1200, 1200),
fontsize = 14,
backgroundcolor = PAGE_BG,
)
ax = Axis(
fig[1, 1];
title = "root-locus-basic · julia · makie · anyplot.ai",
titlesize = 20,
titlecolor = INK,
xlabel = "Real Axis",
ylabel = "Imaginary Axis",
xlabelsize = 14,
ylabelsize = 14,
xlabelcolor = INK,
ylabelcolor = INK,
xticklabelsize = 12,
yticklabelsize = 12,
xticklabelcolor = INK_SOFT,
yticklabelcolor = INK_SOFT,
xtickcolor = INK_SOFT,
ytickcolor = INK_SOFT,
backgroundcolor = PAGE_BG,
topspinevisible = false,
rightspinevisible = false,
leftspinecolor = INK_SOFT,
bottomspinecolor = INK_SOFT,
xgridcolor = gc,
ygridcolor = gc,
xminorgridvisible = false,
yminorgridvisible = false,
aspect = DataAspect(),
)
# Imaginary axis (stability boundary): thin vertical reference at x=0
vlines!(ax, [0.0]; color = RGBAf(INK.r, INK.g, INK.b, 0.15f0), linewidth = 1.0)
# Constant natural-frequency arcs (left half-plane semicircles, dotted)
for wn in (2.0, 4.0, 6.0)
th = range(π / 2, 3π / 2; length = 120)
lines!(ax, wn .* cos.(th), wn .* sin.(th);
color = RGBAf(INK.r, INK.g, INK.b, 0.15f0),
linestyle = :dot,
linewidth = 1.0,
)
end
# Constant damping-ratio lines (dashed radial lines from origin)
for (zeta, lbl) in ((0.3, "ζ=0.3"), (0.5, "ζ=0.5"), (0.7, "ζ=0.7"))
phi = acos(zeta) # angle from negative real axis
r_ext = 9.0
sx, sy = -cos(phi), sin(phi) # unit direction into upper half-plane
col = RGBAf(INK.r, INK.g, INK.b, 0.22f0)
lines!(ax, [0.0, r_ext * sx], [0.0, r_ext * sy]; color = col, linestyle = :dash, linewidth = 1.2)
lines!(ax, [0.0, r_ext * sx], [0.0, -r_ext * sy]; color = col, linestyle = :dash, linewidth = 1.2)
text!(ax, 0.52 * r_ext * sx + 0.15, 0.52 * r_ext * sy;
text = lbl,
color = INK_SOFT,
fontsize = 13,
align = (:left, :center),
)
end
# Locus branches with direction arrows
branch_colors = IMPRINT[1:3]
branch_labels = ["Branch from s=0", "Branch from s=−2", "Branch from s=−4"]
for b in 1:N_BR
xs, ys = bx[b], by[b]
lines!(ax, xs, ys; color = branch_colors[b], linewidth = 2.5, label = branch_labels[b])
# Direction-of-increasing-K arrow (rotated triangle at 40% along branch)
n = length(xs)
ai = clamp(round(Int, n * 0.40), 2, n - 1)
dx = xs[ai + 1] - xs[ai - 1]
dy = ys[ai + 1] - ys[ai - 1]
scatter!(ax, [xs[ai]], [ys[ai]];
marker = :utriangle,
markersize = 15,
color = branch_colors[b],
strokewidth = 0.0,
rotation = atan(dy, dx) - π / 2,
)
end
# Open-loop poles (× markers)
scatter!(ax, OL_POLES, zeros(N_BR);
marker = :xcross,
markersize = 24,
color = INK,
label = "Open-loop poles (×)",
)
# jω-axis crossings at K=48 (stability boundary)
jw_y = sqrt(8.0) # ≈ 2.83
scatter!(ax, [0.0, 0.0], [jw_y, -jw_y];
marker = :diamond,
markersize = 16,
color = IMPRINT[5],
strokewidth = 1.5,
strokecolor = PAGE_BG,
label = "jω crossings (K=48)",
)
text!(ax, 0.2, jw_y + 0.38;
text = "±j$(round(jw_y; digits = 2))\n(K = 48)",
color = INK_SOFT,
fontsize = 13,
align = (:left, :center),
)
xlims!(ax, -9.0, 3.0)
ylims!(ax, -6.0, 6.0)
axislegend(ax;
position = :rt,
framecolor = INK_SOFT,
framewidth = 0.5,
backgroundcolor = ELEVATED_BG,
labelcolor = INK,
labelsize = 12,
rowgap = 3,
)
save(joinpath(@__DIR__, "plot-$(THEME).png"), fig; px_per_unit = 2)
Part of Root Locus Plot for Control Systems on anyplot.ai.